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\begin{document}
\begin{abstract}
We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a tangent space on a 4-dimensional differentiable manifold. The manifold is equip\-ped with a positive definite metric and an additional tensor structure of type $(1, 1)$. The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold.
\end{abstract}
\maketitle
\section{Introduction}
There are various applications of the correspondences between circles and ellipses (circles and hyperbolas, circles and parabolas), as well as between spheres and other quadratic surfaces, for example in geometry, mechanics, astrophysics. Circles and spheres could be determined with respect to an indefinite metric and then their images could be obtained in Euclidean space. In this vein, we consider a circle determined with respect to an associated indefinite metric on a Riemannian manifold and the corresponding quadratic curve in Euclidean space. Also, we study a sphere (a hypersphere) determined with respect to an associated indefinite metric on a Riemannian manifold and the corresponding quadratic surface (hyper-surface) in Euclidean space.
We will mention some papers which concern models of hyper-spheres, spheres and circles with respect to some indefinite metrics and their relations with the corresponding quadratic geometrical objects (\cite{Abe, dok-dzhe, ikawa, konderak, lopez}).
The Hermitian manifolds form a class of manifolds with an integrable almost complex structure $J$ (\cite{GrHer}). One subclass consists of the so-called locally conformal K\"{a}hler manifolds, determined by a special property of the covariant derivative of $J$. Some of the recent investigations of locally conformal K\"{a}hler manifolds are made in \cite{angella, cherevko, huang, moroianu-ornea, prvanovic, vilcu}.
We consider a 4-dimensional Riemannian manifold $M$, endowed with a positive definite metric $g$ and an endomorphism $S$ in a tangent space $T_{p}M$ at an arbitrary point $p$ on $M$. The fourth power of $S$ is minus identity and the components of $S$ form a skew-circulant matrix with respect to some basis of $T_{p}M$. It is supposed that $S$ is compatible with $g$. Such a manifold $(M, g, S)$ is defined in \cite{dok-raz}. In \cite{raz-dok} it is proved that $(M, g, J)$, where $J=S^{2}$, is a locally conformal K\"{a}hler manifold.
We consider the associated metric $\tilde{g}$ on $(M, g, S)$ defined by both structures $g$ and $S$. The metric $\tilde{g}$ is necessarily indefinite and it determines space-like vectors, isotropic vectors and time-like vectors in every $T_{p}M$. We study hyper-spheres in $T_{p}M$, spheres and circles in some special subspaces of $T_{p}M$ with respect to $\tilde{g}$.
The paper is organized as follows. In Sect.~\ref{sec:2}, we recall some necessary facts, definitions and statements about the manifold $(M, g, S)$ obtained in \cite{dok-raz} and \cite{raz-dok}. In Sect.~\ref{sec:3}, we find the equation of a central hyper-sphere in $T_{p}M$ with respect to the associated metric $\tilde{g}$. In Sect.~\ref{sec:4}, we consider spheres with respect to $\tilde{g}$ in special 3-dimensional subspaces of $T_{p}M$ and obtain their equations. In Sect.~\ref{sec:5}, we consider some special 2-planes in $T_{p}M$ and we get the equations of circles with respect to $\tilde{g}$ in these 2-planes. We interpret all equations of the curves and surfaces, studied in Sect.~\ref{sec:3}, Sect.~\ref{sec:4} and Sect.~\ref{sec:5}, in terms of $g$.
\section{Preliminaries}
\label{sec:2}
The skew-circulant matrices are Toeplitz matrices, which are well-studied in \cite{davis} and \cite{grayR}. In our work we consider a tensor structure on a 4-dimensional differentiable manifold, whose component matrix is skew-circulant. Therefore we recall the following definition.
{\em A real skew-circulant matrix\/} with the first row $(a_{1}, a_{2}, a_{3}, a_{4})\in R^{4}$ is a square matrix of the form
\[ \left(
\begin{array}{cccc}
a_{1} & a_{2} & a_{3} & a_{4}\\
-a_{4} & a_{1} & a_{2} & a_{3} \\
-a_{3} & -a_{4} & a_{1} & a_{2}\\
-a_{2} & -a_{3} & -a_{4} & a_{1}\\
\end{array}
\right).\]
We now introduce a manifold $(M, g, S)$ in detail. Let $M$ be a $4$-dimensional Riemannian manifold equipped with a tensor $S$ of type $(1,1)$. Let the components of $S$ form the following skew-circulant matrix in a local coordinate system:
\[ S=\left(
\begin{array}{cccc}
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1\\
-1 & 0 & 0 & 0\\
\end{array}
\right).\]
Then $S$ has the property
\begin{equation}\label{q4}
S^{4}=-\mathrm{id}.
\end{equation}
We assume that $g$ is a positive definite metric on $M$, which satisfies the equality
\begin{equation}\label{2.12}
g(Su, Sv)=g(u,v), \quad u, v\in \mathfrak{X}M.
\end{equation}
Such a manifold $(M, g, S)$ is introduced in \cite{dok-raz}.
The manifold $(M, g, J)$, where $J=S^{2}$, is a locally conformal K\"{a}hler manifold (Theorem 5.3 in \cite{raz-dok}).
The associated metric $\tilde{g}$ on $(M, g, S)$, defined by
\begin{equation}\label{defF}
\tilde{g}(u, v)=g(u, Sv)+g(Su, v),
\end{equation}
is necessarily indefinite.
Consequently, having in mind (\ref{defF}), for an arbitrary vector $v$ it is valid:
\begin{equation}\label{s}
\tilde{g}(v, v)=2g(v, Sv)=a,\qquad a\in \mathbb{R}.
\end{equation}
According to the physical terminology we give the following
\begin{definition}\label{D1}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$. If a vector $u$ satisfies $\tilde{g}(u, u)>0$ (resp. $\tilde{g}(u, u)<0$), then $u$ is a space-like (resp. a time-like) vector.
If $u$ is nonzero and satisfies $\tilde{g}(u, u)=0$, then $u$ is an isotropic vector.
\end{definition}
It is well-known that the norm of every vector $u$ of the tangent space $T_{p}M$ and the cosine of the angle between two nonzero vectors $u$ and $v$ of $T_{p}M$ are given by
\begin{equation}\label{size}
\|u\|=\sqrt{g(u, u)},
\end{equation}
\begin{equation}\label{cos}
\cos\angle(u, v)=\frac{g(u, v)}{\|u\| \|v\|}.
\end{equation}
A basis of type $\{u, Su, S^{2}u, S^{3}u\}$ of $T_{p}M$ is called an $S$-\textit{basis}. In this case we say that {\em the vector $u$ induces an $S$-basis of \/} $T_{p}M$. In \cite{dok-raz} the following assertions are proved.
If a vector $u$ induces an $S$-basis, then
(i) the angles between the basis vectors are
\begin{eqnarray}\label{ugli}\nonumber
& \angle(u,Su)=\angle(Su,S^{2}u)=\angle(S^{2}u, S^{3}u)=\pi-\angle(S^{3}u,u),\\
& \angle(u,S^{2}u)=\angle(Su,S^{3}u)=\frac{\pi}{2}.
\end{eqnarray}
(ii) the angle $\varphi$, determined by
\begin{equation}\label{oznachenia}
\varphi=\angle(u,Su),
\end{equation}
satisfies inequalities
\begin{equation}\label{inequalities}
\frac{\pi}{4}<\varphi<\frac{3\pi}{4}.
\end{equation}
Next we have
\begin{theorem}\label{thmR}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$ and let the vector $u$ induce an $S$-basis. The following
propositions hold true.
\begin{itemize}
\item[(i)] Vector $u$ is space-like if and only if $\varphi\in \big(\frac{\pi}{4},\frac{\pi}{2}\big)$.
\item[(ii)] Vector $u$ is isotropic if and only if $\varphi=\frac{\pi}{2}$.
\item[(iii)] Vector $u$ is time-like if and only if $\varphi\in(\frac{\pi}{2},\frac{3\pi}{4})$.
\end{itemize}
\end{theorem}
\begin{proof} Using (\ref{s}), (\ref{size}), (\ref{cos}) and (\ref{oznachenia}) we get
$\tilde{g}(u, u)=2\|u\|^{2}\cos\varphi$. Having in mind Definition~\ref{D1} and inequalities (\ref{inequalities}) the proof follows. \end{proof}
Evidently, due to (\ref{q4}), (\ref{2.12}) and (\ref{defF}), we state
\begin{corollary}
If $u$ is a space-like (isotropic or time-like) vector, then $Su$, $S^{2}u$ and $S^{3}u$ are space-like (isotropic or time-like) vectors, respectively.
\end{corollary}
In the next sections we get equations of hyper-spheres, spheres and circles with respect to $\tilde{g}$ in some subspaces of $T_{p}M$ on $(M, g, S)$. Obviously, the obtained curves and surfaces do not depend on the choice of the basis. We use orthonormal bases with respect to the metric $g$ on $(M, g, S)$ to find their equations easier. In Section~\ref{sec:3}, we use an orthonormal $S$-\textit{basis} of $T_{p}M$. The existence of such bases is proved in \cite{dok-raz}. In Section~\ref{sec:4} and Section~\ref{sec:5}, we construct orthonormal bases of 3-dimensional subspaces of $T_{p}M$ and of 2-dimensional subspaces of $T_{p}M$ with the help of an arbitrary $S$-\textit{basis}.
\section{Hyper-spheres with respect to the associated metric}\label{sec:3}
Let $\{u, Su, S^{2}u, S^{3}u\}$ be an orthonormal $S$-\textit{basis} of $T_{p}M$ with respect to the metric $g$ on $(M, g, S)$.
If $p_{xyzt}$ is a coordinate system such that the vectors $u$, $Su$, $S^{2}u$ and $S^{3}u$ are on the axes $p_{x}$, $p_{y}$, $p_{z}$ and $p_{t}$, respectively, then $p_{xyzt}$ is orthonormal.
The radius vector $v$ of an arbitrary point $(x, y, z, t)$ of $T_{p}M$ is expressed by the equality
\begin{equation}\label{V1}
v=xu+ySu+zS^{2}u+tS^{3}u.
\end{equation}
A hyper-sphere $s$ centered at the origin $p$, with respect to $\tilde{g}$ on $(M, g, S)$, is defined by (\ref{s}).
We apply (\ref{V1}) into (\ref{s}), and bearing in mind that $p_{xyzt}$ is an orthonormal coordinate system and also equalities (\ref{q4}) and (\ref{2.12}), we obtain the equation of $s$ as follows:
\begin{equation}\label{hyp1}
2(xy-xt+yz+zt)=a.
\end{equation}
Now, we transform the coordinate system $p_{xyzt}$ into $p_{x'y'z't'}$ by
\begin{equation}\label{transl}
\begin{array}{ll}
x =\frac{1}{2}(x'-y'+z'-t'),& y = \frac{\sqrt{2}}{2}(-y'+t')\\
z = -\frac{1}{2}(x'+y'+z'+t'),& t= \frac{\sqrt{2}}{2}(-x'+z').
\end{array}
\end{equation}
We substitute (\ref{transl}) into (\ref{hyp1}) and it takes the form
\begin{equation}\label{hyprot}
x'^{2}+y'^{2}-z'^{2}-t'^{2}=\frac{a}{\sqrt{2}}.
\end{equation}
Evidently, in terms of $g$, we have that (\ref{hyprot}) is an equation of a 3-dimensional hyperboloid.
Therefore, we state the following
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$ and let the vector $u$ induce an orthonormal $S$-basis of $T_{p}M$. If $p_{xyzt}$ is a coordinate system such that $u\in p_{x}$, $Su\in p_{y}$, $S^{2}u\in p_{z}$, $S^{3}u\in p_{t}$, then the hyper-sphere (\ref{s}) has the equation (\ref{hyprot}) with respect to the coordinate system $p_{x'y'z't'}$, obtained by the transformation (\ref{transl}) of $p_{xyzt}$.
\end{theorem}
\begin{corollary} Let $s$ be the 3-dimensional hyperboloid (\ref{hyprot}). The following
propositions are valid.
\begin{itemize}
\item [i)] Every point on $s$, where $a<0$, has a time-like radius vector;
\item [ii)] Every point on $s$, where $a=0$, has an isotropic radius vector;
\item [iii)] Every point on $s$, where $a>0$, has a space-like radius vector.
\end{itemize}
\end{corollary}
\begin{proof} According to Definition~\ref{D1} and due to (\ref{s}) the statement holds. \end{proof}
\begin{corollary}
Let $s$ be the 3-dimensional hyperboloid (\ref{hyprot}). Then the intersections $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ and $\sigma_{4}$ between $s$ and the coordinate planes of $p_{x'y'z't'}$, respectively, are the following surfaces:
\begin{itemize}
\item [i)] $\sigma_{1}$, $\sigma_{2}$ are hyperboloids of two sheets and $\sigma_{3}$, $\sigma_{4}$ are hyperboloids of one sheet, in case $a>0$;
\item [ii)] $\sigma_{1}$, $\sigma_{2}$ are hyperboloids of one sheet and $\sigma_{3}$, $\sigma_{4}$ are hyperboloids of two sheets, in case $a<0$;
\item [iii)] $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ and $\sigma_{4}$ are circular cones, in case $a=0$.
\end{itemize}
\end{corollary}
\begin{proof} Using (\ref{hyprot}) and the equation of the coordinate plane $x'=0$ we get the surface $\sigma_{1}:\ \sqrt{2}(y'^{2}-z'^{2}-t'^{2})=a,\ x'=0$. Consequently, if $a>0$, then $\sigma_{1}$ is a hyperboloid of two sheet, if $a<0$, then $\sigma_{1}$ is a hyperboloid of one sheet and if $a=0$, then $\sigma_{1}$ is a circular cone. Similarly, we consider the other cases of intersections $\sigma_{2}$, $\sigma_{3}$ and $\sigma_{4}$.
\end{proof}
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$ and let the vector $u$ induce an orthonormal $S$-basis of $T_{p}M$. If $p_{xyzt}$ is a coordinate system such that $u\in p_{x}$, $Su\in p_{y}$, $S^{2}u\in p_{z}$ and $S^{3}u\in p_{t}$, then $u$, $Su$, $S^{2}u$ and $S^{3}u$ are isotropic vectors and their heads lie at the surface with equations
\begin{equation}\label{circles}
x'^{2}+y'^{2}=\frac{1}{2},\quad
z'^{2}+t'^{2}=\frac{1}{2},
\end{equation}
where $p_{x'y'z't'}$ is the coordinate system obtained by the transformation (\ref{transl}) of $p_{xyzt}$.
\end{theorem}
\begin{proof}
Bearing in mind (\ref{defF}) and Definition~\ref{D1} we get that $u$, $Su$, $S^{2}u$ and $S^{3}u$ are isotropic vectors with respect to $\tilde{g}$. Therefore, their heads are on the hyper-cone (\ref{hyprot}) in case $a=0$. On the other hand, these heads lie at the unit hyper-sphere with respect to $g$.
This hyper-sphere with respect to $p_{x'y'z't'}$ has the equation \begin{equation}\label{sphere}
x'^{2} + y'^{2} + z'^{2}+t'^{2}=1.
\end{equation}
The system of (\ref{hyprot}), where $a=0$, and (\ref{sphere}) gives the intersection of a hyper-cone with a hyper-sphere. This intersection, with respect to the coordinate system $p_{x'y'z't'}$, is represented by the equivalent system (\ref{circles}).
\end{proof}
\section{Spheres in a 3-dimensional subspace of $T_{p}M$}\label{sec:4}
Let the unit vector $u$ induce an $S$-\textit{basis} of $T_{p}M$.
Hence $u$ induces four different pyramids spanned by the following triples $\{u, Su, S^{2}u\}$, $\{Su, S^{2}u, S^{3}u\}$, $\{u, Su, S^{3}u\}$ and $\{u, S^{2}u, S^{3}u\}$. According to (\ref{2.12}) and (\ref{ugli}), the first and the second pyramid constructed on these basis vectors are equal, as well as the third and the fourth pyramid are also equal. Thus we will investigate only the subspaces with bases defined by the first and the third pyramid.
\subsection{A sphere in the $3$-dimensional subspace of $T_{p}M$, spanned by vectors $u$, $Su$ and $S^{2}u$}
\begin{lemma} Let $\alpha_{1}$ be a subspace of $T_{p}M$ with a basis $\{u, Su, S^{2}u\}$. The system of vectors $\{e_{1}, e_{2}, e_{3}\}$, determined by the equalities
\begin{eqnarray}\label{orth-base}
e_{1}=u,\quad
e_{2}=\frac{(-\cos\varphi)u+Su-(\cos\varphi)S^{2}u}{\sqrt{1-2\cos^{2}\varphi}}, \quad e_{3}=S^{2}u
\end{eqnarray}
form an orthonormal basis of $\alpha_{1}$.
\end{lemma}
\begin{proof}
Using (\ref{2.12}), (\ref{ugli}) and (\ref{orth-base}) we obtain
$g(e_{1},e_{1})=g(e_{2},e_{2})=g(e_{3},e_{3})=1$ and $g(e_{1},e_{2})=g(e_{2},e_{3})=g(e_{1},e_{3})=0.$
\end{proof}
The coordinate system $p_{xyz}$ such that $e_{1}\in p_{x}$, $e_{2}\in p_{y}$ and $e_{3}\in p_{z}$ is orthonormal.
A sphere $s_{1}$ in $\alpha_{1}$ centered at the origin $p$, with respect to $\tilde{g}$ on $(M, g, S)$, is defined by (\ref{s}). In the next statement we get the equation of $s_{1}$ with respect to the orthonormal coordinate system $p_{xyz}$.
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M,g,S)$ and let $\alpha_{1}$ be a 3-dimensional subspace of $T_{p}M$ with a basis $\{u, Su, S^{2}u\}$. If $e_{1}$, $e_{2}$ and $e_{3}$ are determined by (\ref{orth-base}) and $p_{xyz}$ is a coordinate system such that $e_{1}\in p_{x}$, $e_{2}\in p_{y}$, $e_{3}\in p_{z}$, then the equation of the sphere $s_{1}$ in $\alpha_{1}$ is given by
\begin{equation}\label{s2}
2(\cos\varphi)(x^{2}- y^{2}+ z^{2})+2\sqrt{1-2\cos^{2}\varphi}(xy+yz)=a.
\end{equation}
\end{theorem}
\begin{proof}
The radius vector $v$ of an arbitrary point $(x, y, z)$ on $\alpha_{1}$ is expressed by
$ v=xe_{1}+ye_{2}+ze_{3}.$
We apply the latter equality into \eqref{s} and we find
\begin{eqnarray}\label{s-pom}\nonumber
\tilde{g}(e_{1},e_{1})x^{2}+\tilde{g}(e_{2},e_{2})y^{2}+ \tilde{g}(e_{3},e_{3})z^{2}+2\tilde{g}(e_{1},e_{2})xy\\+2\tilde{g}(e_{1},e_{3})xz+2\tilde{g}(e_{2},e_{3})yz=a.
\end{eqnarray}
Using \eqref{2.12}, \eqref{defF}, \eqref{ugli} and \eqref{orth-base}, we obtain
\begin{eqnarray*}
\tilde{g}(e_{1},e_{1})=2\cos\varphi,\quad \tilde{g}(e_{3},e_{3})=2\cos\varphi,\quad \tilde{g}(e_{2},e_{2})=-2\cos\varphi,\\ \tilde{g}(e_{1},e_{2})=\tilde{g}(e_{2},e_{3})=\sqrt{1-2\cos^{2}\varphi}, \quad \tilde{g}(e_{1},e_{3})=0.
\end{eqnarray*}
Substituting the latter equalities into \eqref{s-pom} we get \eqref{s2}.
\end{proof}
Now, we transform the coordinate system $p_{xyz}$ into $p_{x'y'z'}$ by
\[
x=\frac{1}{\sqrt{2}}x'+\lambda_{1}y'+\mu_{1}z',\quad y=\lambda_{2}y'+\mu_{2}z',\quad z=-\frac{1}{\sqrt{2}}x'+\lambda_{1}y'+\mu_{1}z',
\]
where
\begin{equation}\label{lambda-mu}
\begin{array}{ll}
\lambda_{1}=\frac{1}{2}\sqrt{1+\sqrt{2}\cos\varphi},& \lambda_{2}=\frac{\sqrt{2}}{2}\sqrt{1-\sqrt{2}\cos\varphi},\\
\mu_{1}=\frac{1}{2}\sqrt{1-\sqrt{2}\cos\varphi},& \mu_{2}=-\frac{\sqrt{2}}{2}\sqrt{1+\sqrt{2}\cos\varphi}.
\end{array}
\end{equation}
Therefore the equation \eqref{s2} takes the form
\begin{equation}\label{tr-s}
2\cos\varphi x'^{2}+\sqrt{2}y'^{2}-\sqrt{2}z'^{2}=a.
\end{equation}
\begin{corollary} Let $s_{1}$ be the surface, determined by \eqref{tr-s} in case $a=0$. The following statements hold true.
\begin{itemize}
\item [i)] If $\varphi\neq \frac{\pi}{2}$, then $s_{1}$ is a cone;
\item [ii)] If $\varphi=\frac{\pi}{2}$, then $s_{1}$ separates into two planes $z'=\pm y'$.
\end{itemize}
\end{corollary}
\begin{corollary} Let $s_{1}$ be the surface, determined by \eqref{tr-s} in case $a>0$. The following statements hold true.
\begin{itemize}
\item [i)] If $\varphi\in(\frac{\pi}{4}, \frac{\pi}{2})$, then $s_{1}$ is a hyperboloid of one sheets;
\item [ii)] If $\varphi\in(\frac{\pi}{2}, \frac{3\pi}{4})$, then $s_{1}$ is a hyperboloid of two sheet;
\item [iii)] If $\varphi=\frac{\pi}{2}$, then $s_{1}$ is a hyperbolic cylinder.
\end{itemize}
\end{corollary}
\begin{corollary}\label{cor4.5} Let $s_{1}$ be the surface, determined by \eqref{tr-s} in case $a<0$. The following statements hold true.
\begin{itemize}
\item [i)] If $\varphi\in(\frac{\pi}{4}, \frac{\pi}{2})$, then $s_{1}$ is a hyperboloid of two sheets;
\item [ii)] If $\varphi\in(\frac{\pi}{2}, \frac{3\pi}{4})$, then $s_{1}$ is a hyperboloid of one sheet;
\item [iii)] If $\varphi=\frac{\pi}{2}$, then $s_{1}$ is a hyperbolic cylinder.
\end{itemize}
\end{corollary}
\subsection{A sphere in the $3$-dimensional subspace of $T_{p}M$, spanned by vectors $u$, $Su$ and $S^{3}u$}
\begin{lemma} Let $\alpha_{2}$ be a subspace of $T_{p}M$ with a basis $\{u, Su, S^{3}u\}$. The system of vectors $\{e_{1}, e_{2}, e_{3}\}$, determined by the equalities
\begin{equation}\label{orth-base2}
e_{1}=Su,\quad
e_{2}=\frac{u-(\cos\varphi)Su+(\cos\varphi)S^{3}u}{\sqrt{1-2\cos^{2}\varphi}}, \quad e_{3}=S^{3}u,
\end{equation}
is an orthonormal basis of $\alpha_{2}$.
\end{lemma}
\begin{proof}
Using \eqref{q4}, \eqref{2.12}, \eqref{ugli} and \eqref{orth-base2} we obtain
$g(e_{1}, e_{2})=g(e_{2}, e_{3})=g(e_{1}, e_{3})=0$ and $g(e_{1}, e_{1})=g(e_{2}, e_{2})=g(e_{3}, e_{3})=1$.
\end{proof}
The coordinate system $p_{xyz}$ such that the vectors $e_{1}$, $e_{2}$ and $e_{3}$ are on the axes $p_{x}$, $p_{y}$ and $p_{z}$, respectively, is orthonormal.
A sphere $s_{2}$ in $\alpha_{2}$ centered at the origin $p$, with respect to $\tilde{g}$ on $(M, g, S)$, is defined by (\ref{s}). In the next statement we get the equation of $s_{2}$ with respect to the orthonormal coordinate system $p_{xyz}$.
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M,g,S)$ and let $\alpha_{2}$ be a 3-dimensional subspace of $T_{p}M$ with a basis $\{u, Su, S^{2}u\}$. If $e_{1}$, $e_{2}$ and $e_{3}$ are determined by (\ref{orth-base2}) and $p_{xyz}$ is a coordinate system such that $e_{1}\in p_{x}$, $e_{2}\in p_{y}$, $e_{3}\in p_{z}$, then the equation of the sphere $s_{2}$ in $\alpha_{2}$ is given by
\begin{equation}\label{surf2}
2(\cos\varphi)(x^{2}- y^{2}+ z^{2})+2\sqrt{1-2\cos^{2}\varphi}(xy-yz)=a.
\end{equation}
\end{theorem}
\begin{proof}
The radius vector $v$ of an arbitrary point $(x, y, z)$ on $\alpha_{2}$ is expressed by
$v=xe_{1}+ye_{2}+ze_{3}.$
Then (\ref{s}) takes the form
\begin{eqnarray}\label{s-pom2}\nonumber
\tilde{g}(e_{1},e_{1})x^{2}+\tilde{g}(e_{2},e_{2})y^{2}+ \tilde{g}(e_{3},e_{3})z^{2}+2\tilde{g}(e_{1},e_{2})xy\\+2\tilde{g}(e_{1},e_{3})xz+2\tilde{g}(e_{2},e_{3})yz=a.
\end{eqnarray}
By (\ref{2.12}), (\ref{defF}), (\ref{ugli}) and (\ref{orth-base2}) we obtain
\begin{eqnarray*}
\tilde{g}(e_{1},e_{1})=2\cos\varphi,\quad \tilde{g}(e_{3},e_{3})=2\cos\varphi,\quad \tilde{g}(e_{2},e_{2})=-2\cos\varphi,\\ \tilde{g}(e_{1},e_{2})=\sqrt{1-2\cos^{2}\varphi}, \quad \tilde{g}(e_{1},e_{3})=0, \quad \tilde{g}(e_{2},e_{3})=-\sqrt{1-2\cos^{2}\varphi}.
\end{eqnarray*}
Substituting the latter equalities into \eqref{s-pom2} we get \eqref{surf2}.
\end{proof}
After transformation of the coordinate system $p_{xyz}$ into $p_{x'y'z'}$ by
\[
x=\frac{1}{\sqrt{2}}x'+\lambda_{1}y'+\mu_{1}z',\quad y=\lambda_{2}y'+\mu_{2}z',\quad z=\frac{1}{\sqrt{2}}x'-\lambda_{1}y'-\mu_{1}z'
\]
with \eqref{lambda-mu}, the equation \eqref{surf2} takes the form
\[
2\cos\varphi x'^{2}+\sqrt{2}y'^{2}-\sqrt{2}z'^{2}=a.
\]
The above equation is the same as (\ref{tr-s}).
\section{Circles in a special 2-planes of $T_{p}M$}\label{sec:5}
Let the unit vector $u$ induce an $S$-\textit{basis} of $T_{p}M$. Now we study circles in three different subspaces $\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ spanned by 2-planes $\{u, S^{2}u\}$, $\{u, Su\}$ and $\{u, S^{3}u\}$, respectively.
\subsection{Circles in the 2-plane $\beta_{1}$}
Due to (\ref{ugli}) it is true that both vectors $u$ and $S^{2}u$ form an orthonormal basis of $\beta_{1}$.
We construct a coordinate system $p_{xy}$ on $\beta_{1}$, such that $u$ is on the axis $p_{x}$ and $S^{2}u$ is on the axis $p_{y}$.
Therefore $p_{xy}$ is an orthonormal coordinate system of $\beta_{1}$.
\begin{lemma} The system $\{u, S^{2}u\}$ satisfies the following equalities:
\begin{equation}\label{circle1}
\tilde{g}(u,u)=\tilde{g}(S^{2}u,S^{2}u)=2\cos\varphi ,\quad \tilde{g}(u,S^{2}u)=0.
\end{equation}
\end{lemma}
\begin{proof}
From (\ref{2.12}), (\ref{defF}), (\ref{ugli}) we get (\ref{circle1}) by direct calculations.
\end{proof}
A circle $k_{1}$ in $\beta_{1}$ centered at the origin $p$, with respect to $\tilde{g}$ on $(M,g,S)$, is defined by (\ref{s}). Now we obtain the equation of $k_{1}$ with respect to $p_{xy}$.
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$ and let $\beta_{1}$ be a $2$-plane in $T_{p}M$ with a basis $\{u, S^{2}u\}$. If $p_{xy}$ is a coordinate system such that $u\in p_{x}$, $Su\in p_{y}$, then the equation of the circle $k_{1}$ in $\beta_{1}$ is given by
\begin{equation}\label{k-P}
2\cos\varphi x^{2} +2\cos\varphi y^{2}=a,\quad \varphi\neq \frac{\pi}{2}.
\end{equation}
\end{theorem}
\begin{proof}
The radius vector $v$ of an arbitrary point on $\beta_{1}$ is expressed by
\begin{equation}\label{v-za-p}
v=xu+yS^{2}u,
\end{equation}
which implies $S^{2}v=-yu+xS^{2}u$. Then from (\ref{s}), (\ref{circle1}) and (\ref{v-za-p}) it follows (\ref{k-P}).
\end{proof}
\begin{corollary} Let $k_{1}$ be the curve determined by (\ref{k-P}). Then $k_{1}$ is a circle in case when $a> 0$ and $\varphi\in \big(\frac{\pi}{4},\frac{\pi}{2}\big)$, or in case when $a< 0$ and $\varphi\in \big(\frac{\pi}{2},\frac{3\pi}{4}\big)$. The curve $k_{1}$ degenerates into the point $p$ in case $a=0$.
\end{corollary}
We note that a 2-plane $\delta=\{u, Ju\}$, where $\delta=J\delta$, is known as $J$-invariant section of $T_{p}M$ on an almost Hermitian manifold $(M, g, J)$.
Therefore, the 2-plane $\beta_{1}=\{u, S^{2}u\}$ is a $J$-invariant section of $T_{p}M$ on the manifold $(M, g, J)$, $J=S^{2}$.
\subsection{Circles in the 2-plane $\beta_{2}$}
\begin{lemma} Let $\beta_{2}$ be the $2$-plane spanned by unit vectors $u$ and $Su$. The system of vectors $\{e_{1}, e_{2}\}$, determined by the equalities
\begin{equation}\label{i-j}
e_{1}=\frac{1}{\sqrt{2(1+\cos\varphi)}}(u+Su),\quad e_{2}=\frac{1}{\sqrt{2(1-\cos\varphi)}}(-u+Su),
\end{equation}
is an orthonormal basis of $\beta_{2}$.
\end{lemma}
\begin{proof}
Using (\ref{ugli}) and (\ref{i-j}), we calculate $g(e_{1}, e_{2})=0$, $g(e_{1}, e_{1})=g(e_{2}, e_{2})=1$.
\end{proof}
We construct a coordinate system $p_{xy}$ on $\beta_{2}$, such that $e_{1}$ is on the axis $p_{x}$ and $e_{2}$ is on the axis $p_{y}$, i.e. $p_{xy}$ is orthonormal.
\begin{lemma} The system $\{e_{1}, e_{2}\}$ satisfies the following equalities:
\begin{equation}\label{circle2}
\tilde{g}(e_{1},e_{1})=\frac{2\cos\varphi+1}{1+\cos\varphi},\quad\tilde{g}(e_{2},e_{2})=\frac{2\cos\varphi-1}{1-\cos\varphi} ,\quad \tilde{g}(e_{1},e_{2})=0.
\end{equation}
\end{lemma}
\begin{proof}
Using (\ref{2.12}), (\ref{defF}) and (\ref{ugli}) we get (\ref{circle2}) by direct calculations.
\end{proof}
A circle $k_{2}$ in $\beta_{2}$ centered at the origin $p$, with respect to $\tilde{g}$ on $(M, g, S)$, is defined by (\ref{s}).
Further we obtain the equation of $k_{2}$ with respect to $p_{xy}$.
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$ and let $\beta_{2}=\{u, Su\}$ be a $2$-plane in $T_{p}M$ with an orthonormal basis (\ref{i-j}). If $p_{xy}$ is a coordinate system such that $e_{1}\in p_{x}$, $e_{2}\in p_{y}$, then the equation of the circle $k_{2}$ in $\beta_{2}$ is given by
\begin{equation}\label{k-q}
\frac{2\cos\varphi+1}{1+\cos\varphi} x^{2}+\frac{2\cos\varphi-1}{1-\cos\varphi}y^{2}=a.
\end{equation}
\end{theorem}
\begin{proof}
The radius vector $v$ of an arbitrary point on $\beta_{2}$ is
$ v=xe_{1}+ye_{2}.$
Using the latter equality, from (\ref{s}) we get
\[ \tilde{g}(v,v)=\tilde{g}(e_{1},e_{1})x^{2}+2\tilde{g}(e_{1},e_{2})xy+\tilde{g}(e_{2},e_{2})y^{2}=a.\]
Applying (\ref{circle2}) into the above equation we find (\ref{k-q}).
\end{proof}
According to the parameters $a$ and $\varphi$ the equation (\ref{k-q}) describes different quadratic curves. All possible values of these parameters and the corresponding types of the curve (\ref{k-q}) are studied in the Table~\ref{tab:1}.
\subsection{Circles in the 2-plane $\beta_{3}$}
\begin{lemma} Let $\beta_{3}$ be the $2$-plane spanned by unit vectors $u$ and $S^{3}u$. The system of vectors $\{e_{1}, e_{2}\}$, determined by the equalities
\begin{equation}\label{e1-e2}
e_{1}=\frac{1}{\sqrt{2(1-\cos\varphi)}}(u+S^{3}u),\quad e_{2}=\frac{1}{\sqrt{2(1+\cos\varphi)}}(-u+S^{3}u),
\end{equation}
is an orthonormal basis of $\beta_{3}$.
\end{lemma}
\begin{proof}
Using (\ref{ugli}) and (\ref{e1-e2}), we calculate $g(e_{1}, e_{2})=0$, $g(e_{1}, e_{1})=g(e_{2}, e_{2})=1$.
\end{proof}
We construct a coordinate system $p_{xy}$ on $\beta_{3}$, such that $e_{1}\in p_{x}$ and $e_{2}\in p_{y}$, i.e. $p_{xy}$ is orthonormal.
\begin{lemma} The system $\{e_{1}, e_{2}\}$ satisfies the following equalities:
\begin{equation}\label{circle3}
\tilde{g}(e_{1},e_{1})=\frac{2\cos\varphi-1}{1-\cos\varphi},\quad\tilde{g}(e_{2},e_{2})=\frac{2\cos\varphi+1}{1+\cos\varphi} ,\quad \tilde{g}(e_{1},e_{2})=0.
\end{equation}
\end{lemma}
\begin{proof}
Using (\ref{q4}), (\ref{2.12}), (\ref{defF}) and (\ref{ugli}) we get (\ref{circle3}) by direct calculations.
\end{proof}
A circle $k_{3}$ in $\beta_{3}$ centered at the origin $p$, with respect to $\tilde{g}$ on $(M, g, S)$, is defined by (\ref{s}).
In the next statement we obtain the equation of $k_{3}$ with respect to $p_{xy}$.
\begin{theorem}
Let $\tilde{g}$ be the associated metric on $(M, g, S)$ and let $\beta_{3}=\{u, S^{3}u\}$ be a $2$-plane in $T_{p}M$ with an orthonormal basis (\ref{e1-e2}). If $p_{xy}$ is a coordinate system such that $e_{1}\in p_{x}$, $e_{2}\in p_{y}$, then the equation of a circle $k_{3}$ in $\beta_{3}$ is given by
\begin{equation}\label{k-q2}
\frac{2\cos\varphi-1}{1-\cos\varphi} x^{2}+\frac{2\cos\varphi+1}{1+\cos\varphi}y^{2}=a.
\end{equation}
\end{theorem}
\begin{proof}
The radius vector $v$ of an arbitrary point on $\beta_{3}$ is
$v=xe_{1}+ye_{2}.$
Then (\ref{s}) imply
\begin{equation}\label{k-pom3}
\tilde{g}(v,v)=\tilde{g}(e_{1},e_{1})x^{2}+2\tilde{g}(e_{1},e_{2})xy+\tilde{g}(e_{2},e_{2})y^{2}=a.
\end{equation}
We apply (\ref{circle3}) into (\ref{k-pom3}) and we find (\ref{k-q2}).
\end{proof}
The equation (\ref{k-q2}) determines curves which are the same as the obtained ones by (\ref{k-q}). They are described in the Table~\ref{tab:1}.
\begin{table}
\caption{Curves $k_{2}$ and $k_{3}$}
\label{tab:1}
\begin{tabular}{llll}
\hline\noalign{
}
$\varphi$ & $a$ & & $k_{2}$, $k_{3}$ \\
\noalign{
}\hline
$(\frac{\pi}{4},\frac{\pi}{3})$ & $a>0$ & & an ellipse\\
& $a=0$ & & the point $p$ \\
& $a<0$ & & the empty set \\
\noalign{
}\hline
$\frac{\pi}{3}$ & $a>0$ & & two lines $x=\pm\frac{\sqrt{3a}}{2}$ \\
& $a=0$ & & the line x=0 \\
& $a<0$ & & the empty set \\
\noalign{
}\hline\noalign{
}
$(\frac{\pi}{3}, \frac{2\pi}{3})$ & $a>0$ & & a hyperbola \\
& $a=0$ & & two lines $y=\pm cx$, $c$ is a constant \\
& $a<0$ & & a hyperbola \\
\noalign{
}\hline
$\frac{2\pi}{3}$ & $a>0$ & &the empty set \\
& $a=0$ & & the line y=0 \\
& $a<0$ & &two lines $y=\pm\frac{\sqrt{-3a}}{2}$\\
\noalign{
}\hline
$(\frac{2\pi}{3},\frac{3\pi}{4})$ & $a>0$ & & the empty set\\
& $a=0$ & & the point $p$ \\
& $a<0$ & & an ellipse\\
\noalign{
}\hline
\end{tabular}
\end{table}
{\small\rm\baselineskip=10pt
\baselineskip=10pt
\qquad Georgi Dzhelepov\par
\qquad Faculty of Economics\par
\qquad Department of Mathematics and Informatics\par
\qquad Agricultural University of Plovdiv\par
\qquad 4000 Plovdiv, Bulgaria\par
\qquad {\tt [email protected]}
\qquad Iva Dokuzova\par
\qquad Faculty of Mathematics and Informatics\par
\qquad Department of Algebra and Geometry\par
\qquad University of Plovdiv Paisii Hilendarski\par
\qquad 24 Tzar Asen, 4000 Plovdiv, Bulgaria\par
\qquad {\tt [email protected]}
\qquad Dimitar Razpopov\par
\qquad Faculty of Economics\par
\qquad Department of Mathematics and Informatics\par
\qquad Agricultural University of Plovdiv\par
\qquad 4000 Plovdiv, Bulgaria\par
\qquad {\tt [email protected]}
}
\end{document}
|
\begin{document}
\title{Proof terms for infinitary rewriting, progress report}
\section{Preliminaries}
\label{sec:prelim}
\subsection{Ordinal arithmetics}
\label{sec:ordinal-arithmetics}
One of the main foundations for this work is the theory of countable ordinals; (citation needed) and (citation needed) are good references on this subject.
We want to point out some definitions and results which are critical in order to prove some of the basic properties of infinitary proof terms.
\newcommand{\onlyLongVersion}[1]{}
\newcommand{\onlyShortVersion}[1]{#1}
\input{ordinals-main-results}
\subsection{Positions and terms}
\begin{definition}[Position, depth of a position]
\label{dfn:pos}
A \emph{position} is a finite sequence of $\Nat_{>0}$.
The empty sequence is denoted by the symbol $\epsilon$.
The \emph{depth} of a position $p$, notation $\posln{p}$, is defined as its length as a sequence; observe that $\posln{\epsilon} = 0$.
\end{definition}
\begin{definition}[Concatenation of positions]
\label{dfn:pos-concatenation}
Let $p,q$ be positions.
Then we define $p \cdot q$, the concatenation of $p$ and $q$, as follows: $\epsilon \cdot q \eqdef q$ and $(i p) \cdot q \eqdef i (p \cdot q)$.
Moreover, given $P,Q$ sets of positions, then we define also $P \cdot q \eqdef \set{p \cdot q \setsthat p \in P}$ and $p \cdot Q \eqdef \set{p \cdot q \setsthat q \in Q}$.
We will omit the dot to denote concatenation, \ie\ we will write $pq, pQ, Pq$ instead of $p \cdot q, p \cdot Q, P \cdot q$ wherever no confusion arises.
\end{definition}
\begin{definition}[Signature, function symbol, constant]
\label{dfn:signature}
A \emph{signature} is a finite set of symbols along with a function from this set to $\Nat_{\geq 0}$, called \emph{arity} and noted $\arityfn$.
The usual notation is $\Sigma \eqdef \set{f_i/n_i}_{i \in I}$, where each $f_i$ is a symbol and $n_i = \arity{f_i}$.
We will follow the custom of writing $f \in \Sigma$ as a shorthand notation for $\exists n . n \in \Nat_{\geq 0} \,\land\, f/n \in \Sigma$.
A \emph{constant} is a function symbol $c$ such that $\arity{c} = 0$.
\end{definition}
\begin{definition}[Tree domain]
\label{dfn:tree-domain}
A \emph{tree domain} is any set of positions $P$ satisfying the following conditions ($p, q$ positions; $i,j \in \Nat_{>0})$:
$P \neq \emptyset$;
$P$ is prefix closed, \ie\ $pq \in P$ implies $p \in P$ (particularly, $\epsilon \in P$);
if $pj \in P$ and $1 \leq i < j$, then $pi \in P$.
\end{definition}
\begin{definition}[Term, positions of a term, symbol at a position, sets of finitary and infinitary terms]
\label{dfn:term}
A \emph{term} over a signature $\Sigma$ and a countable set of variables $\thevar$ is any pair $\pair{P}{F}$, such that $P$ is a tree domain, $F : P \to \Sigma \cup \thevar$, and the following condition holds:
if $p \in P$ and $F(p) = h$, then $pi \in P$ iff $i \leq ar(h)$, where we consider $ar(x) = 0$ if $x \in \thevar$
\footnote{in some texts, \eg\ \cite{Courcelle83} and \cite{Gallier86}, a term is defined just as a function from positions to symbols; the set of positions is implicitly determined by being the domain of the function. We prefer to explicitly include the set of positions in the definition, I guess that such a decision leads to a clearer definition of terms by describing the tree domain first, and the function afterwards. I guess that we are following the idea expressed in \cite{terese} page 670, ``a (\ldots) term can be described as the set of its positions, together with a function (\ldots)''}
.
If $t = \pair{P}{F}$ is a term, we will denote $P$ by $\Pos{t}$, and $F$ just by $t$; therefore, we will write $t(p)$ to denote $F(p)$.
A term is \emph{finite} iff its tree domain is, otherwise it is \emph{infinite.}
Given a signature $\Sigma$ and a countable set of variables $\thevar$, the set of \emph{finitary terms} over $\Sigma$, notation $Ter(\Sigma,\thevar)$, is the set of finite terms over $\Sigma$; and the set of \emph{infinitary terms} over $\Sigma$, notation $Ter^\infty(\Sigma,\thevar)$, is the set of finite or infinite terms over $\Sigma$.
We will often drop the set of variables, writing just $Ter(\Sigma)$ or $Ter^\infty(\Sigma)$.
\end{definition}
We will name \emph{head symbol} of a term $t$ the symbol $t(\epsilon)$. The name \emph{root symbol} will be used as well.
\begin{notation}[Intuitive notation for terms]
\label{dfn:term-intuitive-notation}
An alternative notation will be often used for terms in $Ter^\infty(\Sigma,\thevar)$: if $x \in \thevar$ and $f/n \in \Sigma$, then we will write \\
\begin{tabular}{@{$\quad\bullet\quad$}p{.9\textwidth}}
$x$ for $\pair{\set{\epsilon}}{F}$ where $F(\epsilon) = x$, and \\
$f(t_1, \ldots, t_n)$ for $\pair{P}{F}$, where
$P = \set{\epsilon} \cup \bigcup_{1 \leq i \leq n} \set{ip \setsthat p \in \Pos{t_i}}$,
$F(\epsilon) = f$, and
$F(ip) = t_i(p)$.
\end{tabular}
\noindent
We will use $t \in \thevar$ as shorthand notation for $t = \pair{\set{\epsilon}}{F}$, $F(\epsilon) = x$, and $x \in \thevar$.
\noindent
If $f/1 \in \Sigma$, then we will write $f^\omega$ for the term $t = f(f(f( \ldots )))$, \ie\ $\Pos{t} = \set{1^n \setsthat n \in \Nat}$ and $t(p) = f$ for all $p \in \Pos{t}$
\footnote{This convention could generalise to any $f/n \in \Sigma$, by defining $f^\omega = \pair{P}{F}$ where $P$ is the set of all the sequences that can be built using the numbers $\set{1,2,\ldots,n}$, and $F(p) \eqdef f$ for all $p \in P$. Roughly speaking, $f^\omega$ would be defined as the infinite tree all filled with $f$.}.
\end{notation}
We observe that all terms can be described using \refnotation{term-intuitive-notation}.
\begin{proposition}
\label{rsl:term-then-intuitive-notation}
Let $t \in Ter^\infty(\Sigma,\thevar)$. Then either $t = x$ or $t = f(t_1, \ldots, t_n)$ where $f/n \in \Sigma$ and $t_i \in Ter^\infty(\Sigma,\thevar)$ for all $i \leq n$; \confer\ \refnotation{term-intuitive-notation}.
\end{proposition}
\begin{proof}
\refDfn{tree-domain} implies that $\epsilon \in \Pos{t}$.
Assume $t(\epsilon) = x \in \thevar$. Moreover, assume for contradiction the existence of some $p \in \Pos{t} \sthat p \neq \epsilon$. In that case there should be some $n \in \Nat$ being the minimum of the depths of such positions, \ie\ $n = min(\posln{p} \setsthat p \in \Pos{t} \land p \neq \epsilon)$.
Observe that $n = 1$ would imply the existence of some $i \in \Nat$ verifying $i \in \Pos{t}$, contradicting \refdfn{term} since we consider $ar(x) = 0$. In turn, $n > 1$ would entail $p = p'i \in \Pos{t}$ for some $p$ verifying $\posln{p} = n$ and $\posln{p'} > 0$, implying $p' \in \Pos{t}$ by \refdfn{tree-domain}, thus contradicting minimality of $n$.
Consequently, $\Pos{t} = \set{\epsilon}$, hence $t = x$.
Assume $t(\epsilon) = f \in \Sigma$.
For each $i \in \Nat$ we define $P_i \eqdef \set{p \ \setsthat ip \in \Pos{t}}$, and $F_i : P_i \to \Sigma \cup \thevar$ such that $F_i(p) \eqdef t(ip)$.
If $i \leq ar(f)$, then $P_i \neq \emptyset$ since $\epsilon \in P_i$. Moreover, $\Pos{t}$ being a tree domain implies immediately that $P_i$ enjoys the remaining conditions in \refdfn{tree-domain}; and also the condition on $F_i$ described in \refdfn{term} stems immediately from the fact that $t$ is a term.
Therefore, $t_i \eqdef \pair{P_i}{F_i}$ is a term.
On the other hand, $i > ar(f)$ implies that $P_i = \emptyset$, thus $\Pos{t} = \set{\epsilon} \cup \bigcup_{1 \leq i \leq ar(f)}\set{ip \ \setsthat p \in P_i}$.
We conclude by observing that $t = f(t_1, \ldots, t_n)$.
\end{proof}
\begin{definition}[Occurrence]
\label{dfn:occurrence}
Let $t$ be a (either finite or infinite) term over $\Sigma$ and $a \in \Sigma \cup \thevar$. An \emph{occurrence} of $a$ in $t$ is a position $p \in \Pos{t}$ such that $t(p) = a$.
We define $\Occs{a}{t}$ as the set of occurrences of $a$ in $t$.
A symbol $a \in \Sigma \cup \thevar$ \emph{occurs in} a term $t$ iff $\Occs{a}{t} \neq \emptyset$, \ie\ iff there is at least one occurrence of $a$ in $t$; $a$ occurs exactly $n \in \Nat$ times in $t$ iff
$|\ \Occs{a}{t} | = n$,
where $|\ S \ |$ denotes the cardinal of any set $S$.
\end{definition}
\begin{definition}[Closed term, linear term]
\label{dfn:closed-linear}
A term $t$ is said to be \emph{closed} iff it includes no occurrences of variables; it is said to be \emph{linear} iff no variable occurs in it more than once.
\end{definition}
\begin{definition}[Subterm at a position]
\label{dfn:subtat}
Let $t = \pair{P}{F}$ be a term, and $p \in P$. We define the \emph{subterm} of $t$ at position $p$, notation $\subtat{t}{p}$, as $\pair{\subtat{P}{p}}{\subtat{F}{p}}$, where $\subtat{P}{p}$ and $\subtat{F}{p}$ are the \emph{projections} of $P$ and $F$ over $p$ respectively; \ie,
$\subtat{P}{p} \, \eqdef \set {q \setsthat pq \in P}$ and
$\subtat{F}{p} \, : \subtat{P}{p} \ \to \Sigma \cup \thevar$ such that $\subtat{F}{p}(q) \eqdef F(pq)$.
\end{definition}
\refDfn{subtat} allow a straightforward and direct (i.e. non-inductive) proof of a basic result about subterms. Namely
\begin{lemma}
\label{rsl:subtat-composition}
$\subtat{t}{pq} = \subtat{(\subtat{t}{p})}{q}$.
\end{lemma}
\begin{proof}
If we call $\pair{P}{F} \eqdef \subtat{t}{pq}$ and $\pair{P'}{F'} \eqdef \subtat{(\subtat{t}{p})}{q}$, then \refdfn{subtat} yields \\
\minicenter{
$\begin{array}{ccl@{\qquad}ccl}
P & = & \set{r \setsthat pqr \in \Pos{t}} & P' & = & \set{r \setsthat qr \in \Pos{\subtat{t}{p}}} \\
F(r) & = & t(pqr) & F'(r) & = & \subtat{t}{p} (qr) = t(pqr)
\end{array}$
} \\[2pt]
We conclude by observing that $pqr \in \Pos{t} \textiff qr \in \Pos{\subtat{t}{p}}$.
\end{proof}
\noindent
Particularly, if $t = f(t_1, \ldots, t_n)$, then $\subtat{t}{ip} = \subtat{t_i}{p}$; \confer\ \refnotation{term-intuitive-notation}.
\begin{definition}[Replacement at a position]
\label{dfn:repl}
Let $\,t$ and $u$ be terms, and $p \in \Pos{t}$. We define the \emph{replacement} of $\,t$ under position $p$ with $u$, notation $\repl{t}{u}{p}$, as $\pair{P'}{F'}$ such that
$P' \eqdef \set{q \in \Pos{t} \setsthat p \not\leq q} \cup \set{pq \setsthat q \in \Pos{u}}$ and \\
$F'(q) \eqdef \left\{
\begin{array}{r@{ \ \textiff \ }l}
t(q) & p \not\leq q \\
u(q') & q = pq'
\end{array}
\right.$.
\end{definition}
We state and prove some basic properties about replacement.
It is worth mentioning that the definition of term we use (\confer\ \refdfn{term}) is different from the definition in \cite{terese} for finitary terms (Dfn 2.1.2, page 26) or \cite{trat} (Dfn. 3.1.2, page 35), so that it is necessary to verify these properties.
\begin{lemma}
\label{rsl:repl-homo}
Let $t = f(t_1, \ldots, t_n)$ and $u$ be terms, and $p \in \Pos{t_i}$.
Then $\repl{t}{u}{ip} = f(t_1, \ldots, \repl{t_i}{u}{p}, \ldots, t_n)$.
\end{lemma}
\begin{proof}
Let us call $t' = \pair{P'}{F'} \eqdef \repl{f(t_1, \ldots, t_n)}{u}{ip} \ $ and \\ $t'' = \pair{P''}{F''} \eqdef f(t_1, \ldots, \repl{t_i}{u}{p}, \ldots, t_n)$.
By joining \refnotation{term-intuitive-notation} and \refdfn{repl} we obtain
$P' = \set{\epsilon} \cup \set{jq \setsthat q \in \Pos{t_j} \land j \neq i} \cup \set{iq' \setsthat q' \in \Pos{t_i} \land p \not\leq q'} \cup \set{ipq \setsthat q \in \Pos{u}}$. It is straightforward to verify that $P' = P''$; particularly, notice that $\Pos{\repl{t_i}{u}{p}} = \set{q' \setsthat q' \in \Pos{t_i} \land p \not\leq q'} \cup \set{pq \setsthat q \in \Pos{u}}$.
Let us compare $F'(p)$ and $F''(p)$, for any $p \in P' = P''$.
$F'(\epsilon) = F''(\epsilon) = f$.
If $j \neq i$ then $ip \not\leq jq$, then $F'(jq) = F''(jq) = t_j(q)$.
If $p \not\leq q'$, then $F'(iq') = F(iq') = t_i(q')$, and $F''(iq') = \repl{t_i}{u}{p}(q') = t_i(q')$.
Finally, if $q = pq'$, then $F'(iq) = u(q')$ and $F''(iq) = \repl{t_i}{u}{p}(pq') = u(q')$.
Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:repl-ctx}
Let $t$ and $u$ be terms and $pq \in \Pos{t}$.
Then $\repl{t}{u}{pq} = \repl{t}{\repl{\subtat{t}{p}}{u}{q}}{p}$.
\end{lemma}
\begin{proof}
By induction on $p$.
If $p = \epsilon$, then both $\repl{t}{u}{pq}$ and $\repl{t}{\repl{\subtat{t}{p}}{u}{q}}{p}$ are equal to $\repl{t}{u}{q}$.
Assume that $p = i p'$, in this case $t = g(t_1, \ldots, t_n)$.
\refLem{repl-homo} implies that $\repl{t}{u}{pq} = \repl{t}{u}{ip'q} = g(t_1, \ldots, \repl{t_i}{u}{p'q}, \ldots t_n)$
and also
$\repl{t}{\repl{\subtat{t}{p}}{u}{q}}{p} = \repl{t}{\repl{\subtat{t}{ip'}}{u}{q}}{ip'} =
g(t_1, \ldots, \repl{t_i}{\repl{\subtat{t_i}{p'}}{u}{q}}{p'}, \ldots, t_n)$. We conclude by \ih\ on $p'$, $t_i$ and $u$.
\end{proof}
\begin{lemma}
\label{rsl:repl-disj}
Let $t,s$ be terms and $p,q \in \Pos{t}$ such that $p \disj q$.
Then $\subtat{(\repl{t}{s}{q})}{p} = \subtat{t}{p}$.
\end{lemma}
\begin{proof}
Say $t = \pair{P}{F}$, $\repl{t}{s}{q} = \pair{P'}{F'}$, $\subtat{t}{p} = \pair{P_p}{F_p}$, and $\subtat{(\repl{t}{s}{q})}{p} = \pair{P'_p}{F'_p}$.
We prove $P_p = P'_p$ by double inclusion.
\begin{tabular}{lp{.86\textwidth}}
$\subseteq )$ &
Let $p' \in P_p$, so that $p p' \in P$. Observe that $p \disj q$ implies $p p' \disj q$, so that $q \not\leq p p'$, implying $p p' \in P'$, and therefore $p' \in P'_p$. \\
$\supseteq )$ &
Let $p' \in P'_p$, so that $p p' \in P'$. We have already verified $q \not\leq p p'$, so that the only valid option \wrt\ Dfn.~\ref{dfn:repl} is $p p' \in P$, implying $p' \in P_p$.
\end{tabular}
Let $p' \in P'_p = P_p$, so that $p p' \in P \cap P'$ and $q \not\leq p p'$.
Dfn.~\ref{dfn:subtat} implies $F'_p(p') = F'(p p')$ and $F_p(p') = F(p p')$.
In turn, Dfn.~\ref{dfn:repl} yields $F'(p p') = F(p p')$, since $q \not\leq p p'$.
Consequently $F_p = F'_p$. Thus we conclude.
\end{proof}
\subsection{Contexts}
\begin{definition}[Context, one-hole context]
\label{dfn:ctx}
A \emph{context} over $\Sigma$ is a term (either finite or infinite) over $\Sigma \cup \set{\Box/0}$.
A \emph{one-hole context} is a context in which the symbol $\Box$ occurs exactly once.
\end{definition}
\begin{definition}[Position of a variable/hole in a linear term/context]
\label{dfn:vpos}
Let $t$ be a term. Then we define ${\tt VOccs}(t) \eqdef \set{p \setsthat t(p) \in \thevar}$.
Given a term $t$, if $| {\tt VOccs}(t) | = n \in \Nat$, then for any $i$ such that $1 \leq i \leq n$ we define $\VPos{t}{i}$, the $i$-th variable occurrence in $t$, as the $i$-th element of the set ${\tt VOccs}(t)$, considering the order given by $p < q$ iff $\posln{p} < \posln{q}$ or $\posln{p} = \posln{q}$, $p = rip'$, $q = rjq'$, $i < j \ $
\footnote{orderings among positions will be studied in the analysis of different standard concepts}.
Analogously, if $C$ is a context including a finite number of occurrences of the box, then we define $\BPos{C}{i}$ as the $i$-th element of $\Occs{\Box}{C}$, considering the order just described.
\end{definition}
\begin{definition}[Context replacement]
\label{dfn:ctx-repl}
Let $C$ be a context including exactly $n$ occurrences of the box, and $t_1, \ldots, t_n$ terms.
We define the replacement of $C$ using $t_1, \ldots, t_n$
as $C[t_1, \ldots, t_n] \eqdef \pair{P}{F}$, where \\
$P \eqdef \set{p \in \Pos{C} \setsthat C(p) \neq \Box} \cup \bigcup_{i} \set{\BPos{C}{i} \cdot p \setsthat p \in \Pos{t_i}}$, \\ and
$F'(p) \eqdef
\left\{
\begin{array}{rcl}
C(p) & \textiff & C(p) \neq \Box \\
t_i(q) & \textiff & p = \BPos{C}{i} \cdot q
\end{array}
\right.
$
\end{definition}
We remark that, given $\BPos{C}{i} \disj \BPos{C}{j}$ if $i \neq j$, and that $\repl{\repl{t}{u_1}{p}}{u_2}{q} = \repl{\repl{t}{u_2}{q}}{u_1}{p}$ if $p \disj q$, it should be possible to prove that \\
$C[t_1, \ldots, t_n] =
\repl{
\repl{
\repl{C \,}{t_1}{\BPos{C}{1}\,}
}{t_2}{\BPos{C}{2}} \ldots
}{t_n}{\BPos{C}{n}}$. We leave the verification of this conjecture as future work.
It is easy to verify an expected result about context replacement, namely:
\begin{lemma}
\label{rsl:ctx-repl-composition}
$\subtat{C[t_1, \ldots, t_n]}{\BPos{C}{i} \cdot p} = \subtat{t_i}{p}$
\end{lemma}
\begin{proof}
Immediate from Dfn.~\ref{dfn:ctx-repl}.
\end{proof}
\subsection{Distance between terms}
In this section, the notion of distance between terms to be used in this work, and the corresponding definition of limit of an infinite sequence of (possibly infinite) terms, are introduced.
\begin{definition}[Distance between terms, \confer\ \cite{terese} p. 670]
\label{dfn:distance}
Let $t,u$ be terms. We define the \emph{distance} between $t$ and $u$, notation $\tdist{t}{u}$, as follows: \\
\begin{tabular}{@{$\quad\bullet\quad$}p{.9\textwidth}}
$0$ iff $t = u$, and \\
$2^{-k}$ otherwise, where $k$ is the length of the shortest position at which the two terms differ; \ie\
$k = \posln{p} \sthat p$ is minimal for $p \in \Pos{t} \cup \Pos{u}$ and $t(p) \neq u(p)$.
\end{tabular}
\end{definition}
This definition of distance implies that, for any $t$, $u$ terms, obtaining $\tdist{t}{u} < 2^{-k}$ for all $k < \omega$ is a sufficient condition to conclude $t = u$.
In turn, to check $\tdist{t}{u} < 2^{-k}$ it is enough to verify, for any position $p$, that $\posln{p} \leq k$ and $p \in \Pos{t} \cup \Pos{u}$ entails $p \in \Pos{t} \cap \Pos{u}$ and $t(p) = t(u)$.
\begin{definition}[Limit of a sequence of terms]
\label{dfn:limit-terms}
Let $<t_i>_{i < \alpha}$ a sequence of terms where $\alpha$ is a countable limit ordinal.
We say that the sequence $<t_i>$ has the term $t$ as its limit (notation $\lim_{i \to \alpha} t_i = t$) iff the following limit condition holds: for any $p \in \Nat$ there exists $k_p < \alpha$ such that for all $j$ satisfying $k_p < j < \alpha$, $\tdist{t_j}{t} < 2^{-p}$.
\end{definition}
Since the set of infinitary terms is proven to be equal to the metric completion of $Ter(\Sigma)$ w.r.t. the metric given by \refdfn{distance} (so it is trivially metric-complete \wrt\ that metric), given this definition of limit, if a sequence has limit then it is Cauchy-convergent \wrt\ distance.
We observe that the set $\iSigmaTerms$ for a given signature $\Sigma$, along with the distance given in Dfn.~\ref{dfn:distance}, form an \emph{ultrametric} space
\footnote{references here?}.
Formally:
\begin{lemma}
\label{rsl:tdist-is-ultrametric}
Let $t, u, w$ be terms. Then $\tdist{t}{w} \leq max(\tdist{t}{u}, \tdist{u}{w})$.
\end{lemma}
\begin{proof}
If $t = u = w$, then all distances are $0$.
Oteherwise, we analyse $k$ where $max(\tdist{t}{u}, \tdist{u}{w}) = 2^{-k}$.
If $k = 0$ we conclude immediately isince the distance between any pair of terms cannot be more than one.
Assume $k = k' + 1$. Then $\tdist{t}{u} < 2^{-k'}$, implying that for any position $p$ such that $\posln{p} \leq k'$, it is easy to verify that $p \in \Pos{t}$ iff $p \in \Pos{u}$, and moreover, $p \in \Pos{t}$ implies $t(p) = u(p)$.
On the other hand, the same properties hold for $u$ \wrt\ $w$, since $\tdist{u}{w} < 2^{-k'}$.
Hence $\tdist{t}{w} \leq 2^{-k}$, thus we conclude.
\end{proof}
The distance between a term and the result of a replacement on that term is limited by the depth of the position corresponding to the replacement. Namely:
\begin{lemma}
\label{rsl:repl-dist}
Let $t,s$ be terms and $p \in \Pos{t}$. Then $\tdist{t}{\repl{t}{s}{p}} \leq 2^{-\posln{p}}$.
\end{lemma}
\begin{proof}
We proceed by induction on $p$.
If $p = \epsilon$ then we conclude immediately since $\tdist{t}{u} \leq 2^0 = 1$ for any term $u$.
Otherwise, \ie\ if $p = i p'$, observe that $i p' \in \Pos{t}$ implies $t = f(t_1, \ldots, t_i, \ldots, t_m)$.
Then $\repl{t}{s}{p} = f(t_1, \ldots, \repl{t_i}{s}{p'}, \ldots, t_m)$, \confer\ Lem.~\ref{rsl:repl-homo}, implying $\tdist{t}{\repl{t}{s}{p}} = \frac{1}{2} * \tdist{t_i}{\repl{t_i}{s}{p'}}$.
In turn, \ih\ yields $\tdist{t_i}{\repl{t_i}{s}{p'}} \leq 2^{-\posln{p'}}$.
Therefore, easy exponent arithmetics recalling $\posln{p} = \posln{p'} + 1$ suffices to conclude.
\end{proof}
\subsection{Substitutions}
\begin{definition}[Substitution]
\label{dfn:substitution}
Given a set of variables $\thevar$ and a signature $\Sigma$, a \emph{substitution} is a function $\sigma : \thevar \to \sinfterms$ where $\sigma(x) = x$ except for a finite subset of $\thevar$
\footnote{Even when removing the finite support condition is not needed so far, I wonder whether something is broken if we consider arbitrary substitutions, allowing those with infinite support as well. -- Carlos May 25th, 2013.}
.
Any substitution is extended into a function, bearing the same name $\sigma$, where $\sigma: \sinfterms \to \sinfterms$, defined as follows:
$\sigma t \eqdef \pair{P}{F}$ where \\
$P = \set{p \in \Pos{t} \setsthat t(p) \notin \thevar} \cup \set{pq \setsthat t(p) = x \in \thevar \land q \in \Pos{\sigma x}}$ and \\
$F(p) =
\left\{
\begin{array}{rcl}
t(p) & \textiff & p \in \Pos{t} \land t(p) \notin \thevar \\
\sigma x (q') & \textiff & p = q q' \land t(q) = x \in \thevar
\end{array}
\right.$
\end{definition}
\subsubsection{Uniqueness of the extension to terms}
For finitary terms, the extension of the domain of a substitution from variables to terms can be defined by relying to the concept of \emph{$\Sigma$-algebra}; \confer\ \cite{trat} Chapter 3.
Given a signature $\Sigma$, we can define a $\Sigma$-algebra whose carrier set is $Ter(\Sigma,\thevar)$, which we will denote by $Ter(\Sigma,\thevar)$ as well. For any $f/n \in \Sigma$, the corresponding function is defined simply as follows: \\
\minicenter{$f^{Ter(\Sigma,\thevar)}(t_1, \ldots, t_n) \eqdef f(t_1, \ldots, t_n)$} \\
(\confer\ \refprop{term-then-intuitive-notation}).
Moreover, this $\Sigma$-algebra is \emph{generated} by $\thevar$, \confer \cite{trat} dfn. 3.2.2.
A similar $\Sigma$-algebra can be defined having \sinfterms\ as carrier set. On the other hand, \sinfterms\ considered as a $\Sigma$-algebra is not generated by $\thevar$; notice that the $\Sigma$-subalgebra generated by $\thevar$ for \sinfterms\ is exactly $Ter(\Sigma,\thevar)$.
The following result relates substitutions with the $\Sigma$-algebra \sinfterms\ in an expected way.
In the sequel, we will distinguish between the two functions introduced in \refdfn{substitution}.
We will use $\sigma$ for the function whose domain is the set of variables, and $\wid{\sigma}$ for the function whose domain is the set of terms.
\begin{lemma}
Let $\wid{\sigma}$ be a substitution on terms.
Then $\wid{\sigma}$ is an endomorphism on \sinfterms\ which extends the corresponding $\sigma$ defined on variables.
\end{lemma}
\begin{proof}
It is enough to show that $\wid{\sigma}(f(t_1, \ldots, t_n)) = f(\wid{\sigma}(t_1), \ldots, \wid{\sigma}(t_n)))$; \confer\ \refprop{term-then-intuitive-notation}; let us call these terms $t' = \pair{P'}{F'}$ and $t'' = \pair{P''}{F''}$ respectively.
By applying notation~\ref{dfn:term-intuitive-notation} and \refdfn{substitution}, we obtain \\[2pt]
$\begin{array}{rcl@{}l}
P' & = &
\set{\epsilon} \cup
\ \bigcup_i \ (&
\set{ip \setsthat p \in \Pos{t_i} \land t_i(p) \notin \thevar} \ \cup
\\ & & &
\set{ipq \setsthat t_i(p) = x \in \thevar \land q \in \Pos{\sigma x}})
\\
F'(\epsilon) & = & f \\
F'(ip) & = & t_i(p) & \textif p \in \Pos{t_i} \land t_i(p) \notin \thevar \\
F'(ipq) & = & \sigma x(q) & \textif t_i(p) = x \in \thevar \land q \in \Pos{\sigma x}
\end{array}
$ \\[2pt]
An analogous analysis for $P''$ and $F''$ is enough to conclude.
\end{proof}
Nonetheless, we cannot use the result on uniqueness of homomorphisms on generated $\Sigma$-algebras given the values for the generator set (\confer\ \cite{trat} lemma 3.3.1) to assert that $\wid{\sigma}$ is the only endomorphism on \sinfterms\ which extends $\sigma$.
The reason is that \sinfterms\ is not generated by $\thevar$.
Fortunately, an analogous uniqueness result can be proved for endomorphisms on \sinfterms.
\begin{proposition}
\label{rsl:endomorphism-uniqueness-for-infinitary-terms}
Let $\Sigma$ be a signature, and $\phi, \psi$ two endomorphisms on the $\Sigma$-algebra \sinfterms\ which coincide on $\thevar$.
Then $\phi = \psi$.
\end{proposition}
\begin{proof}
We will prove the following statement, which entails the desired result (\ie\ that for any term $t$, $\psi(t) = \phi(t)$):
for any $k < \omega$, given a term $t$ and a position $p$ such that $\posln{p} \leq k$ and $p \in \Pos{\psi(t)} \cup \Pos{\phi(t)}$, then $\psi(t)(p) = \phi(t)(p)$.
\Confer\ comment following Dfn.~\ref{dfn:distance}.
We proceed by induction on $k$. There is one case which does not need to resort to the inductive argument:
if $t \in \thevar$, then $\psi(t) = \phi(t)$ since hypotheses assert that these functions coincide on $\thevar$.
Thus assume $t = f(t_1, \ldots, t_m)$; \confer\ Prop.~\ref{rsl:term-then-intuitive-notation}.
In this case hypotheses entail $\psi(t) = f(\psi(t_1), \ldots, \psi(t_m))$ and $\phi(t) = f(\phi(t_1), \ldots, \phi(t_m))$.
If $k = 0$, then $\posln{p} \leq k$ implies $p = \epsilon$, hence it is enough to observe that $\psi(t)(\epsilon) = \phi(t)(\epsilon) = f$.
Assume $k = k' + 1$. If $\posln{p} \leq k'$ then applying \ih\ on $k'$ \wrt\ $t$ and $q$ suffices to conclude. If $\posln{p} = k$, then $p = iq$ (recall $k > 0$) where $\posln{q} = k'$ and $q \in \Pos{\psi(t_i)} \cup \Pos{\phi(t_i)}$.
Therefore we can apply \ih\ on $k'$ \wrt\ $t_i$ and $q$, obtaining $\psi(t_i)(q) = \phi(t_i)(q)$. Thus we conclude by observing $\psi(t)(p) = \psi(t_i)(q)$ and analogously for $\phi$.
\end{proof}
Consequently, we can assert that $\wid{\sigma}$ is the only endomorphism on \sinfterms\ which extends $\sigma$, as desired.
\subsection{Term rewriting systems}
\label{sec:trs}
\begin{definition}[Reduction rule, term rewriting system]
\label{dfn:trs}
Assuming a set of variables $\thevar$ and given a signature $\Sigma$, a \emph{reduction rule} (just \emph{rule} if no confusion arises) over $\Sigma$ is a pair of terms $\pair{l}{r}$ satisfying the following conditions: $l$ is a finite term, $l \notin \thevar$, and each variable occurring in $r$ occurs also in $l$. Notation for a reduction rule: $l \to r$, also $\mu: l \to r$ if assigning explicit names to rules is desirable.
The terms $l$ and $r$, respectively, are the \emph{left-hand side} and \emph{right-hand} side, \emph{lhs} and \emph{rhs} for short, of the rule $l \to r$.
A \emph{term rewriting system} (shorthand TRS) is a pair $T = \pair{\Sigma}{R}$, where $\Sigma$ is a signature and $R$ is a set of rules over $\Sigma$.
If the right-hand sides of all the rules are finite terms, then $T$ can be considered as a TRS over either $Ter(\Sigma)$ or $Ter^\infty(\Sigma)$; otherwise, only the infinitary interpretation is valid.
In either case, a TRS over $Ter^\infty(\Sigma)$ is known as a \emph{infinitary TRS}, or iTRS for short.
\end{definition}
We define that a \TRS\ is \emph{left-linear} iff for any $l$ left-hand side of a rule, and for any $x$ variable, $x$ occurs in $l$ at most once.
This work will study reductions in left-linear \iTRSs\ only.
Additionaly, we will say that a reduction rule $\mu : l \to r$ is \emph{collapsing} iff $r \in \thevar$.
\subsection{Reduction, redex occurrence}
\label{sec:reduction}
\begin{definition}[Reduction step, source, target, active position, depth]
\label{dfn:step}
Let $T = \pair{\Sigma}{R}$ be a TRS, $t \in Ter^\infty(\Sigma)$, $p \in \Pos{t}$, $\mu : l \to r \in R$ and $\sigma$ a substitution, such that $\subtat{t}{p} = \sigma l$.
Then the 4-tuple $a = \langle t, p, \mu, \sigma \rangle$ is a \emph{reduction step}.
We define $src(a) \eqdef t$, $tgt(a) \eqdef \repl{t}{\sigma r}{p}$, $\RPos{a} \eqdef p$, and $\sdepth{a} \eqdef \posln{p}$. They are, respectively, the \emph{source}, \emph{target}, \emph{redex position} and \emph{depth} of $a$.
\end{definition}
\noindent
If the source term of a reduction step is clear from the context, it can be omitted when describing the step. On the other hand, if the substitution is unimportant \wrt\ the subject being discussed, it can be omitted as well.
Therefore, we will sometimes refer to a reduction step $\langle t, p, \mu, \sigma \rangle$ as $\langle p, \mu, \sigma \rangle$, or even just $\pair{p}{\mu}$.
Notice that, given a term $t$, the reduction steps having $t$ as source term are in an obvious bijection with the occurrences of redexes (\ie\ of subterms having the form $\sigma l$ for some rule $\mu: l \to r$) inside $t$.
Namely, the reduction step $\langle t, p, \mu, \sigma \rangle$ correspond to the occurrence, at position $p$, of a redex with rule $\mu$ and substitution $\sigma$.
Therefore, we will take the (maybe rather unusual) convention of considering reduction steps from $t$ and \textbf{redex occurrences in $t$} as synonyms.
We also want to remark that the definition of a reduction step is given in terms of the \emph{position} of the corresponding redex occurrence, opposed to the \emph{context} which surrounds it (\confer\ \cite {terese} dfn. 2.2.4).
The choice of position is motivated by the fact that in infinitary rewriting reasonings, induction on terms (and therefore in contexts which are terms for an extended signature) is not valid, whereas induction on positions is allowed.
Finally, notice that if $t$, $p$ and $\mu$ are known in advance, then the specification of $\sigma$ is redundant. Nonetheless, I prefer to include the substitution in the definition because it will permit to describe with precision a redex occurrence whose existence is asserted.
Notice also that the inclusion of the rule is redundant for orthogonal TRSs; it is included in the characterisation of reduction steps because proof terms are intended to describe reductions in any, maybe non-orthogonal, left-linear TRS.
A \textbf{normal form} is a term having no redex occurrences, or equivalently, a term being the source of no reduction step.
Some examples of reduction steps follow: consider the TRS whose rules are $\mu: f(x) \to g(x)$ and $\nu: h(i(x),y) \to j(y,x)$, and the term
$t = g \,\big( h(i(f(a)),f(i(b))) \,\big)$.
Then there are three reductions steps from $t$, namely: \\
$\langle t, 1, \nu, \set{x \eqdef f(a), y \eqdef f(i(b))} \rangle$,
$\langle t, 111, \mu, \set{x \eqdef a} \rangle$, and
$\langle t, 12, \mu, \set{x \eqdef i(b)} \rangle$.
Next we will give a precise formal definition for the concept of \emph{\redseq}.
Producing a precise definition is needed, particularly since proof terms are meant as a tool to study precisely \redseqs.
Formal definitions of infinitary \redseqs\ are given and discussed throughout the literature on the subject, \confer\ \eg\ \cite{orthogonal-itrs-90}, \cite{orthogonal-itrs-95}, \cite{terese}, \cite{inf-normalization}.
A \emph{\redseq} will be defined as a sequence of reduction steps, having any (finite or infinite) ordinal as length.
This approach, and also the idea of concatenating reduction sequences, is in line with the description given in \cite{terese}, Sec. 2. We quote from page 38
\begin{quote}
Concatenating reduction steps we have (possibly infinite) \emph{reduction sequences} $t_0 \to t_1 \to t_2 \ldots$, or \textit{reductions} for short.
\end{quote}
Notice that in the definition which follows, focus is set on \emph{steps} rather than \emph{terms}.
Not all sequences of steps are \redseqs; some conditions must hold.
Obviously, if $\stepa$ and $\stepb$ are consecutive steps in a sequence, then $tgt(\stepa)$ must coincide with $src(\stepb)$.
This coherence condition must hold also for steps having \emph{limit} positions in the sequence. \Eg\ in a sequence $\stepa_0 ; \stepa_1 ; \ldots ; \stepa_n; \ldots, \stepa_\omega \ldots$, there must be some relation between the step $\stepa_\omega$ and the sequence of the steps previous to it.
This relation is commonly formalised in the literature by asking the sequence of targets of the previous steps, \ie\ the sequence $tgt(\stepa_0); tgt(\stepa_1); \ldots; tgt(\stepa_n); \ldots$ to have a limit, and that limit to coincide with $src(\stepa_\omega)$. This requirement is related with the characterisation of \emph{weakly convergent} infinitary rewriting.
In order to obtain a notion of \redseq\ enjoying some desired properties, a further condition is imposed. Namely, the \emph{depth} of successive steps is required to tend to $\omega$ at each limit in the sequence, \ie\ up to the $\omega$-th step, up to the $\omega * 2$-th step, and so on. \Redseqs\ for which this requirement, and also the coherence requirements described before, hold, are known as \emph{strongly convergent} in the literature.
These considerations motivate the following definitions.
\begin{definition}[Reduction sequence, convergence]
\label{dfn:sred}
A (well-formed) \emph{\redseq} is: either $\redid{t}$, the \emph{empty \redseq\ for} the term $t$, or else a non-empty sequence of reduction steps $\reda \eqdef \langle \redel{\reda}{\alpha} \rangle_{\alpha < \beta}$, where $\beta > 0$ and $\reda$ verifies all the following conditions:
\begin{enumerate}
\item \label{it:dfn-sred-successor-coherence}
For all $\alpha$ such that $\alpha + 1 < \beta$, $src(\redel{\reda}{\alpha+1}) = tgt(\redel{\reda}{\alpha})$.
\item \label{it:dfn-sred-limit}
For all limit ordinals $\beta_0 < \beta$:
\begin{enumerate}
\item \label{it:dfn-sred-limit-existence}
The sequence $\langle tgt(\redel{\reda}{\alpha}) \rangle_{\alpha < \beta_0}$ has a limit.
\item \label{it:dfn-sred-limit-coherence}
$\lim_{\alpha \to \beta_0} tgt(\redel{\reda}{\alpha}) = src(\redel{\reda}{\beta_0})$.
\item \label{it:dfn-sred-depth}
For all $n < \omega$, there exists $\beta' < \beta_0$ such that $\sdepth{\redel{\reda}{\alpha}} > n$ if $\beta' < \alpha < \beta_0$.
\end{enumerate}
\end{enumerate}
We say that a reduction sequence $\reda$ is \emph{convergent} iff either $\reda = \redid{t}$ for some term $t$, or else $\reda = \langle \redel{\reda}{\alpha} \rangle_{\alpha < \beta}$, and either $\beta$ is a successor ordinal, or else $\beta$ is a limit ordinal and conditions (\ref{it:dfn-sred-limit-existence}) and (\ref{it:dfn-sred-depth}) hold for $\beta$ as well.
\end{definition}
\begin{definition}[Source of a \redseq]
\label{dfn:redseq-src}
Let $\reda$ be a \redseq. We define the \emph{source} term of $\reda$, notation $src(\reda)$, as follows:
if $\reda = \redid{t}$, then $src(\reda) \eqdef t$,
if $\reda = \langle \redel{\reda}{\alpha} \rangle_{\alpha < \beta}$, then $src(\reda) \eqdef src(\redel{\reda}{0})$.
\end{definition}
\begin{definition}[Target of a \redseq]
\label{dfn:redseq-tgt}
Let $\reda$ be a convergent \redseq. We define the the \emph{target} term of $\reda$, notation $tgt(\reda)$, as follows:
if $\reda = \redid{t}$, then $tgt(\reda) \eqdef t$;
if $\reda = \langle \redel{\reda}{\alpha} \rangle_{\alpha < \beta}$, then $\beta = \beta' + 1$ implies $tgt(\reda) \eqdef tgt(\redel{\reda}{\beta'})$, and $\beta$ being a limit ordinal implies $tgt(\reda) \eqdef \lim_{\alpha \to \beta} tgt(\redel{\reda}{\alpha})$.
\end{definition}
\begin{definition}[Length of a \redseq]
\label{dfn:redseq-length}
Let $\reda$ be a \redseq. We define the \emph{length} of $\reda$, notation $\redln{\reda}$, as follows:
if $\reda = \redid{t}$, then $\redln{\reda} \eqdef 0$,
if $\reda = \langle \redel{\reda}{\alpha} \rangle_{\alpha < \beta}$, then $\redln{\reda} \eqdef \beta$.
\end{definition}
\begin{definition}[Minimum activity depth of a \redseq]
\label{dfn:redseq-mind}
Let $\reda$ be a \redseq. We define the \emph{minimum activity depth} of $\reda$, notation $\mind{\reda}$, as follows:
if $\reda = \redid{t}$, then $\mind{\reda} \eqdef \omega$,
if $\reda = \langle \redel{\reda}{\alpha} \rangle_{\alpha < \beta}$, then $\mind{\reda} \eqdef min \set{\sdepth{\redel{\reda}{\alpha}} \setsthat \alpha < \beta}$.
\end{definition}
\begin{definition}[Section of a \redseq]
\label{dfn:redseq-section}
Let $\reda$ be a \redseq\ and $\alpha, \beta$ ordinals verifying $\alpha < \redln{\reda}$, $\beta \leq \redln{\reda}$ and $\alpha \leq \beta$.
We define the \emph{section} of $\reda$ from $\alpha$ to $\beta$, notation $\redsublt{\reda}{\alpha}{\beta}$, as follows:
if $\alpha = \beta < \redln{\reda}$, then $\redsublt{\reda}{\alpha}{\beta} \eqdef \redid{src(\redel{\reda}{\alpha})}$, otherwise, \ie\ if $\alpha < \beta$, then $\redsublt{\reda}{\alpha}{\beta} \eqdef \langle \redel{\reda}{\alpha+\gamma} \rangle_{\alpha+\gamma < \beta}$.
\end{definition}
\begin{remark}
\label{rmk:tgt-mentioned-then-defined}
Any mention of $tgt(\reda)$ implies that the target of the \redseq\ $\reda$ is defined, \ie\ that $\reda$ is a \emph{convergent} \redseq.
\end{remark}
It is worth remarking that the requirement about depths of successive steps, \ie\ condition~(\ref{it:dfn-sred-depth}) in Dfn.~\ref{dfn:sred}, is not enough to guarantee the well-formedness of \redseqs.
Let us discuss briefly this issue. Some examples will be given using the rules $f(x) \to g(x)$, $h(x) \to j(x)$, and $g(x) \to f(x)$, and denoting concatenation of sequences by semicolons.
Depth requirement alone does not guarantee coherence at limit positions, as discussed before defining \redseqs. \Eg, the sequence of steps $f\om \infred g\om ; h\om \infred j\om$, which length is $\omega * 2$, does not produce a well-formed reduction sequence, even when depths tend to infinity at each limit ordinal in the sequence of steps; a target (namely $g\om$) can be determined for the prefix of the first $\omega$ steps, but it does not coincide with the source of the $\omega$-th step, \ie\ $h\om$.
Moreover, the depth condition alone does not even guarantee the existence of a limit for each limit ordinal prefix. \Eg\ consider the sequence of steps, having length $\omega^2$, informally described as follows:
$f\om \infred g\om ; g\om \infred f\om ; g(f\om) \infred g\om ; f(g\om) \infred f\om ; g^2(f\om) \infred g\om ; f^2(g\om) \infred f\om ; \ldots g^n(f\om) \infred g\om ; f^n(g\om) \infred f\om ; \ldots \ $.
This sequence of steps obeys the depth condition at each limit ordinal, including $\omega^2$ itself, but even though, a limit cannot be determined for it.
Therefore, the requirement about the existence of a limit, \ie\ condition~(\ref{it:dfn-sred-limit-existence}), cannot be removed by the mere fact of including the depth requirement.
It could possibly be proved, by means of a careful transfinite induction on limit ordinals, that for any sequence of steps, and each limit ordinal $\beta$ up to the length of that sequence, the depth requirement on each limit ordinal $\leq \beta$, plus coherence (\ie\ condition~(\ref{it:dfn-sred-limit-coherence})) at all limit ordinals $< \beta$, imply the existence of a limit in the sequence of targets at ordinal $\beta$.
Since this issue is not in the focus of the present work, we leave it as subject of further investigation.
Notice that the way in which the concept of \redseq\ is formalised here differs from the approach taken in \cite{terese}, Sec. 8.2,
which cannot be adapted for infinitary rewriting (perhaps unless coinduction is involved, \confer\ \cite{endrullis-wir-2013}) since the construction of \redseqs\ is based there on simple induction, and therefore can only describe finite sequences.
As the correspondence between proof terms and \redseqs\ given in \cite{terese} is based on the mentioned characterisation, this observation suggests that the adequacy between proof terms and \redseqs\ for the infinitary case should be treated in a way different than what is described in \cite{terese}.
Given a term $t$, we will refer to the reduction sequences having $t$ as source term as the reduction sequences \textbf{from} $t$.
Moreover, if $s$ is the only normal form verifying $t \infred s$, then we will say that $s$ is the (infinitary) \textbf{normal form of $t$}.
We can define reduction steps and sequences which model applications of rules to contexts rather than terms.
\begin{remark}
\label{rmk:trs-ctx}
For any TRS $T = \pair{\Sigma}{R}$ we can think of an associated TRS $T^\Box \eqdef \pair{\Sigma \cup \set{\Box/0}}{R}$, which makes it possible to describe reductions on contexts.
In the sequel we will include references to reduction steps and reduction sequences whose source and target are contexts; they must be understood as defined in $T^\Box$.
\end{remark}
\subsection{Patterns, pattern depth}
\label{sec:patterns}
Given a reduction rule $\mu: l \to r$ and a reduction step $\stepa = \langle t,p,\mu,\sigma \rangle$, the role of the \emph{function symbol} occurrences in $l$ differs from that of the \emph{variable} occurrences:
the former must be present explicitly in $src(a)$ having the same structure as in $l$;
while the latter are included in the domain of $\sigma$.
We will sometimes need to refer to the positions of all the occurrences of function symbols in (the lhs of) a rule, and also in (the source term of) a reduction step.
\Eg\ if $\mu = f(g(x,h(y)) \to f(y)$, then the occurrences of function symbols in (the lhs of) $\mu$ are at positions $\epsilon$, $1$ and $12$.
The corresponding formal definitions follow.
\begin{definition}[Pattern, pattern positions, pattern depth]
\label{dfn:patt}
Let $t$ be a term. The \emph{pattern} of $t$, notation $\patt{t}$, is the context which results of changing all the variable occurrences in $t$ with boxes; \confer\ \cite{terese} dfn. 2.7.3, pg. 49.
The set of \emph{pattern positions} of $t$, notation $\PPos{t}$ is defined as $\set{p \setsthat p \in \Pos{t} \textand t(p) \notin \thevar}$.
The \emph{pattern depth} of $t$, notation $\Pdepth{t}$, is defined as $max(\set{|p| \setsthat p \in \PPos{t}})$; if $x \in \thevar$ then $\Pdepth{x}$ is undefined.
Let $\mu:l \to r$ be a reduction rule. The set of pattern positions and the pattern depth of $\mu$ are defined as follows:
$\PPos{\mu} \eqdef \PPos{l}$, $\Pdepth{\mu} \eqdef \Pdepth{l}$.
Let $a = \langle t,p,\mu,\sigma \rangle$ be a reduction step. The set of pattern positions of $a$ is defined as follows:
$\PPos{a} \eqdef p \cdot \PPos{\mu}$.
\end{definition}
For example, if $\mu : h(i(x),g(i(y)),c) \to h(x,x,y)$, $t = g(h(i(g(a)),g(i(b)),c))$ and $a = \langle t,1,\mu,\set{x \eqdef g(a), y \eqdef b}\rangle$, then
$\PPos{\mu} = \set{\epsilon,1,2,21,3}$, $\Pdepth{\mu} = 2$, and $\PPos{a} = 1 \cdot \PPos{\mu} = \set{1,11,12,121,13}$.
\includeStandardisation{
\noindent
Sometines we will need to consider the maximum rule pattern depth of a TRS.
\begin{definition}[Maximum rule pattern depth]
\label{dfn:maximum-rule-pd}
Let $T = \langle \Sigma, R \rangle$ be a (finitary or infinitary) TRS.
We define the maximum rule pattern depth of $T$ as follows:
$\maxPdepth{T} \eqdef max(\set{\Pdepth{\mu} \setsthat \mu \in R})$.
\end{definition}
}
\subsection{Some properties about infinitary rewriting}
We include in this Section the statement and proof of some properties on infinitary rewriting which are needed in following Sections.
In turn, these properties require some definitions to be given.
We say that a term $t$ is \emph{infinitary weakly normalising}, shorthand notation $WN^\infty$, iff there exists at least one \redseq\ $\reda$ such that $t \infredx{\reda} u$ and $u$ is a normal form. We say that a term $t$ is \emph{strongly normalising}, shorthand notation $SN^\infty$, iff there is no divergent \redseq\ whose source term is $t$.
A term $t$ has the \emph{unique normal-form property}, shorthand notation $UN^\infty$, iff whenever $t \infred u_1$, $t \infred u_2$ and both $u_1$ and $u_2$ are normal forms, then $u_1 = u_2$.
A \TRS\ is $WN^\infty$ ($SN^\infty$, $UN^\infty$) iff all its terms are.
\Confer\ \cite{inf-normalization} for a study of normalisation for infinitary rewriting.
A \TRS\ is \emph{left-linear} iff for any rule $\mu$ and for any $x \in \thevar$, $x$ occurs in the left-hand side of $\mu$ at most once.
A \TRS\ \trst\ is \emph{orthogonal} iff it is left-linear and there is no term $t$ such that $t = \sigma_1 l_1$ and $\subtat{t}{p} = \sigma_2 l_2$, where $l_1$ and $l_2$ are left-hand sides of rules in \trst, and $p \in \PPos{l_1}$.
Some examples of left-hand sides of rules leading to non-orthogonal \TRSs\ follow.
No \TRS\ including a rule whose left-hand side is $f(g(x))$ and another having as left-hand side either $g(x)$ or $g(h(x))$, is orthogonal: $t = f(g(h(a)))$ is a counterexample for the corresponding condition.
Also, no \TRS\ including rules whose left-hand sides are $h(f(x),y)$ and $h(x,g(y))$ is orthogonal, a counterexample is $t = h(f(a),g(b))$. In this case the position $p$ mentioned in the definition is $\epsilon$ for the given counterexample.
Finally, no \TRS\ including a rule whose left-hand side is $f(f(x))$ is orthogonal, a counterexample is $t = f(f(f(a)))$. In this case the same rule corresponds to $l_1$ and $l_2$.
Properties of first-order infinitary orthogonal \TRSs\ are studied \eg\ in \cite{orthogonal-itrs-95}.
A \TRS\ $\trst$ is \emph{disjoint} iff
the set of all the function symbols occurring in the left-hand sides of the rules of $\trst$
is disjoint from
the set of all the function symbols occurring in the right-hand sides of the rules of $\trst$.
The results to be given in this Section are particularly needed for the study of the class of proof terms corresponding to coinitial sets of redexes, which involves the definition of TRSs which are `companions' to the TRS under study. \Confer\ the concept of \emph{2-rewriting system}, notation 8.2.12 in \cite{terese}.
The `companion' TRSs enjoy some desirable properties.
First of all, they are all orthogonal, and therefore they enjoy the property $UN^\infty$; \confer\ \cite{inf-normalization} Section 5.
Some of them are Recursive Program Schemes (\confer\ \cite{terese} dfn. 3.4.7), \ie, they are orthogonal and all their rules have the form $f(\ldots, x_i, \ldots) \to t$, so that we can distinguish the subset $\mathcal{F} \eqdef \set{f \setsthat f(\ldots, x_i, \ldots) \to t \in R}$ within their signature.
Furthermore, the following additional restriction is imposed.
Notice that for Recursive Program Schemes, the disjointness condition amounts to assert that no symbol in $\mathcal{F}$ appears in the right-hand side of any rule.
Sections of \redseqs, \confer\ Dfn.~\ref{dfn:redseq-section}, enjoy some basic properties.
\begin{lemma}
\label{rsl:redseq-proper-section-convergent}
Let $\reda$ be a \redseq, and $\alpha < \redln{\reda}$. Then $\redupto{\reda}{\alpha}$ is convergent.
\end{lemma}
\begin{proof}
It is immediate to verify that $\redupto{\reda}{\alpha}$ is a well-formed \redseq.
If $\alpha = 0$, \ie\ $\redupto{\reda}{\alpha} = \redid{src(\reda)}$, or if $\alpha$ is a successor ordinal, then it is immediately convergent.
If $\alpha$ is a limit ordinal, the fact that $\reda$ is well-formed implies that conditions~(\ref{it:dfn-sred-limit-existence}) and (\ref{it:dfn-sred-depth}) hold for $\alpha < \redln{\reda}$, hence $\redupto{\reda}{\alpha}$ is convergent.
\end{proof}
\begin{lemma}
\label{rsl:redseq-section-src-tgt-coherence}
Let $\reda$ be a \redseq\ and $\alpha < \redln{\reda}$. Then $src(\redel{\reda}{\alpha}) = tgt(\redupto{\reda}{\alpha})$.
\end{lemma}
\begin{proof}
Notice that Lem.~\ref{rsl:redseq-proper-section-convergent} implies that $\redupto{\reda}{\alpha}$ is convergent, so that its limit is defined.
If $\alpha = 0$, \ie\ $\redupto{\reda}{\alpha} = \redid{src(\redel{\reda}{0})}$, then we conclude immediately.
Otherwise, $\alpha = \alpha' + 1$ implies $src(\redel{\reda}{\alpha}) = tgt(\redel{\reda}{\alpha'})$, and $\alpha$ limit implies $src(\redel{\reda}{\alpha}) = \lim_{\alpha' \to \alpha} tgt(\redel{\reda}{\alpha'})$, \confer\ conditions~(\ref{it:dfn-sred-successor-coherence}) and (\ref{it:dfn-sred-limit-coherence}) resp. in Dfn.~\ref{dfn:sred}. In either case, this coincides with $tgt(\redupto{\reda}{\alpha})$, \confer\ Dfn.~\ref{dfn:redseq-tgt}. Thus we conclude.
\end{proof}
We prove some expected properties of targets of convergent \redseqs.
\begin{lemma}
\label{rsl:redseq-mind-big-src-tgt}
Let $\reda$ be a convergent \redseq\ and $n < \omega$ such that $\mind{\reda} > n$.
Then $\tdist{src(\reda)}{tgt(\reda)} < 2^{-n}$.
\end{lemma}
\begin{proof}
We proceed by induction on $\redln{\reda}$.
If $\redln{\reda} = 0$, \ie\ $\reda = \redid{t}$ for some term $t$, then $tgt(\reda) = src(\reda) = t$, so that we conclude immediately.
Assume that $\redln{\reda}$ is a successor ordinal, so that $\reda = \reda'; \stepa \ $ where $\redln{\reda'} < \redln{\reda}$.
Then \ih\ can be applied to obtain
$\tdist{src(\reda')}{tgt(\reda')} = \tdist{src(\reda)}{src(\stepa)} < 2^{-n}$.
In turn, $tgt(\reda) = tgt(\stepa) = \repl{src(\stepa)}{s}{p}$ for some term $s$, where $p = \RPos{\stepa}$, so that hypotheses imply $\mind{\stepa} > n$.
Then Lem.~\ref{rsl:repl-dist} implies $\tdist{src(\stepa)}{tgt(\reda)} \leq 2^{-\posln{p}} < 2^{-n}$.
Hence Lem.~\ref{rsl:tdist-is-ultrametric} allows to conclude.
Assume that $\alpha \eqdef \redln{\reda}$ is a limit ordinal.
In this case $tgt(\reda) = \lim_{\alpha' \to \alpha} tgt(\redel{\reda}{\alpha'})$.
Let $\alpha_n < \alpha$ such that $\tdist{tgt(\redel{\reda}{\alpha'})}{tgt(\reda)} < 2^{-n}$ if $\alpha_n < \alpha' < \alpha$.
Then particularly
$\tdist{tgt(\redel{\reda}{\alpha_n + 1})}{tgt(\reda)}
= \tdist{tgt(\redupto{\reda}{\alpha_n + 2})}{tgt(\reda)}
< 2^{-n}$; recall $\alpha_n < \alpha$ limit implies $\alpha_n + 2 < \alpha$.
In turn, \ih\ can be applied on $\redupto{\reda}{\alpha_n + 2}$ to obtain $\tdist{src(\redupto{\reda}{\alpha_n + 2})}{tgt(\redupto{\reda}{\alpha_n + 2})} < 2^{-n}$.
Hence we conclude by Lem.~\ref{rsl:tdist-is-ultrametric}.
\end{proof}
\begin{lemma}
\label{rsl:redseq-disj}
Let $t \infredx{\reda} u$ and $p \in \Pos{t}$ such that $\RPos{\redel{\reda}{\alpha}} \disj p$ for all $\alpha < \redln{\reda}$.
Then $\subtat{t}{p} = \subtat{u}{p}$.
\end{lemma}
\begin{proof}
We proceed by induction on $\redln{\reda}$.
If $\redln{\reda} = 0$, \ie\ $\reda = \redid{t}$, then we conclude immediately since $u = t$.
Assume that $\redln{\reda}$ is a successor ordinal, so that $t \infredx{\reda'} u' \sstepx{\stepa} u$. In this case, \ih\ applies to $\reda'$, yielding $\subtat{t}{p} = \subtat{u'}{p}$.
In turn, $u = \repl{u'}{s}{q}$ for some term $s$, where $q = \RPos{\stepa} \disj p$. Then Lem.~\ref{rsl:repl-disj} implies $\subtat{u'}{p} = \subtat{u}{p}$. Thus we conclude.
Assume that $\alpha \eqdef \redln{\reda}$ is a limit ordinal.
In this case $u = tgt(\reda) = \lim_{\alpha' \to \alpha} tgt(\redel{\reda}{\alpha'})$.
Let $n < \omega$, and $\alpha_n < \alpha$ such that
$\tdist{tgt(\redel{\reda}{\alpha'})}{u} < 2^{-(n + \posln{p})}$, implying
$\tdist{\subtat{tgt(\redel{\reda}{\alpha'})}{p}}{\subtat{u}{p}} < 2^{-n}$, if $\alpha_n < \alpha' < \alpha$. Particularly, $\tdist{\subtat{tgt(\redel{\reda}{\alpha_n+1})}{p}}{\subtat{u}{p}} < 2^{-n}$.
Recall that $\alpha_n < \alpha$ limit implies $\alpha_n + i < \alpha$ if $i < \omega$. Then \ih\ can be applied to $\redupto{\reda}{\alpha_n + 2}$, yielding
$\subtat{src(\redupto{\reda}{\alpha_n + 2})}{p} = tgt(\subtat{\redupto{\reda}{\alpha_n + 2})}{p}$, so that $\subtat{t}{p} = \subtat{tgt(\redel{\reda}{\alpha_n + 1})}{p}$. Hence $\tdist{\subtat{t}{p}}{\subtat{u}{p}} < 2^{-n}$ for all $n < \omega$.
Consequently, we conclude.
\end{proof}
The just introduced properties allow to define the \emph{projection} of a \redseq\ not including head steps over an index.
We verify that the definition yields a well-formed \redseq; in the infinitary setting, this verification involves a fair amount of work.
The following definition involves the use of a sequence of non-contiguous ordinals which we will call $A$. We use $\seqln{A}$ and $\seqel{A}{\alpha}$ to denote the order type of $A$ and its $\alpha$-th element respectively, where $\alpha < \seqln{A}$.
In turn, this sequence is built from a set of ordinals $S$ as follows. If $S = \emptyset$, then $A$ is the empty sequence, so that $\seqln{A} = 0$. Otherwise, we define $\seqel{A}{0}$ as the minimal element of $S$.
Let $\alpha > 0$ such that $\seqel{A}{\alpha'}$ is defined for all $\alpha' < \alpha$. If $\alpha = \alpha' + 1$ then we consider the set $\set{\beta \in S \setsthat \beta > \seqel{A}{\alpha'}}$, and if $\alpha$ is a limit ordinal then we consider $\set{\beta \in S \setsthat \beta \geq sup(\set{\seqel{A}{\alpha'} \setsthat \alpha' < \alpha})}$.
In either case, if the considered set is empty then we state that $\seqel{A}{\alpha_1}$ as undefined for all $\alpha_1 \geq \alpha$, so that $\seqln{A} = \alpha$. Otherwise, we define $\seqel{A}{\alpha}$ as the minimum of the considered set.
\newcommand{\redai}{\proj{\reda}{i}}
\begin{definition}
\label{dfn:proj-redseq}
Let $\reda$ a \redseq\ such that $\mind{\reda} > 0$, and $i$ such that $1 \leq i \leq m$ where $src(\reda) = f(t_1, \ldots, t_m)$.
We define the \emph{projection of $\reda$ over $i$}, notation $\redai$, as the \redseq\ whose specification follows.
Let $A$ be sequence built from the set $\set{ \alpha \setsthat \alpha < \redln{\reda} \,\land\, i \leq \RPos{\redel{\reda}{\alpha}} }$, \wrt\ the usual order of ordinals.
If $A$ is empty, then $\redai \eqdef \redid{t_i}$.
Otherwise $\redln{\redai} \eqdef \seqln{A}$, and $\redel{(\redai)}{\alpha} \eqdef \langle s_i, p, \mu \rangle$ where $\redel{\reda}{\seqel{A}{\alpha}} = \langle f(s_1, \ldots, s_i, \ldots, s_m), ip, \mu \rangle$.
Observe that Lem.~\ref{rsl:redseq-proper-section-convergent} implies $\redupto{\reda}{\seqel{A}{\alpha}}$ to be convergent, and in turn Lem.~\ref{rsl:redseq-mind-big-src-tgt} implies $tgt(\redupto{\reda}{\seqel{A}{\alpha}})(\epsilon) = src(\reda)(\epsilon) = f$; therefore, $tgt(\redupto{\reda}{\seqel{A}{\alpha}}) = src(\redel{\reda}{\seqel{A}{\alpha}}) = f(s_1, \ldots, s_i, \ldots, s_m)$. \Confer\ also Lem.~\ref{rsl:redseq-section-src-tgt-coherence}.
\end{definition}
\begin{lemma}
\label{rsl:proj-redseq-well-defined}
Let $\reda$ be a \redseq\ such that $\mind{\reda} > 0$, and $i$ such that $1 \leq i \leq m$ where $src(\reda)(\epsilon) = f/m$.
Then $\redai$ is a well-formed \redseq\ and $src(\redai) = \subtat{src(\reda)}{i}$.
Moreover, if $\reda$ is convergent, then $\redai$ is convergent as well, and $tgt(\redai) = \subtat{tgt(\reda)}{i}$.
\end{lemma}
\begin{proof}
Let $A$ be the sequence of positions of steps in $\reda$ at or below position $i$. We proceed by induction on $\redln{\redai} = \seqln{A}$.
Assume $A$ is empty, so that $\redai = \redid{\subtat{src(\reda)}{i}}$.
Then just Dfn.~\ref{dfn:sred} implies immediately that $\redai$ is a well-formed and convergent \redseq, and Dfn.~\ref{dfn:redseq-src} that $src(\redai) = \subtat{src(\reda)}{i}$.
If $\reda$ is convergent, then observe that $A$ being empty implies $\RPos{\redel{\reda}{\alpha}} \disj i$ for all $\alpha < \redln{\reda}$; recall $\mind{\reda} > 0$. Then Lem~\ref{rsl:redseq-disj} implies $\subtat{tgt(\reda)}{i} = \subtat{src(\reda)}{i} = tgt(\redai)$. Thus we conclude.
Assume that $\seqln{A} = \alpha + 1$, \ie, $\seqln{A}$ is a successor ordinal.
Observe that $\redupto{(\redai)}{\alpha} = \proj{\redupto{\reda}{\seqel{A}{\alpha}}}{i}$, and that Lem.~\ref{rsl:redseq-proper-section-convergent} implies that $\redupto{\reda}{\seqel{A}{\alpha}}$ is convergent.
Then \ih\ on $\redupto{\reda}{\seqel{A}{\alpha}}$ yields that $\redupto{(\redai)}{\alpha}$ is a well-formed and convergent \redseq, that $src(\redai) = src(\redupto{(\redai)}{\alpha}) = \subtat{src(\reda)}{i}$, and that $tgt(\redupto{(\redai)}{\alpha}) = \subtat{tgt(\redupto{\reda}{\seqel{A}{\alpha}})}{i} = \subtat{src(\redel{\reda}{\seqel{A}{\alpha}})}{i}$, \confer\ Lem.~\ref{rsl:redseq-section-src-tgt-coherence}.
On the other hand, $src(\redel{(\redai)}{\alpha}) = \subtat{src(\redel{\reda}{\seqel{A}{\alpha}})}{i}$.
We verify that the conditions in Dfn.~\ref{dfn:sred} hold for $\redai$. The analysis depends on $\alpha$. \\[5pt]
\begin{tabular}{@{$\ \ \bullet\quad$}p{.9\textwidth}}
If $\alpha = 0$, then $\redupto{\redai}{\alpha} = \redid{\subtat{src(\reda)}{i}}$. In this case, conditions~(\ref{it:dfn-sred-successor-coherence}) and (\ref{it:dfn-sred-limit}) hold immediately.
\\[2pt]
If $\alpha = \alpha' + 1$, then $\redupto{(\redai)}{\alpha}$ being a well-formed \redseq\ implies that condition~(\ref{it:dfn-sred-successor-coherence}) holds for all $\alpha_0$ such that $\alpha_0 + 1 < \alpha$; \ie\ for all needed indexes but $\alpha'$. In turn,
$tgt(\redel{(\redai)}{\alpha'}) = tgt(\redupto{(\redai)}{\alpha}) = \subtat{src(\redel{\reda}{\seqel{A}{\alpha}})}{i} = src(\redel{(\redai)}{\alpha}) = src(\redel{(\redai)}{\alpha'+ 1})$. On the other hand, $\redupto{(\redai)}{\alpha}$ being well-formed implies also that condition~(\ref{it:dfn-sred-limit}) holds for $\redai$; indeed, $\alpha_0 < (\alpha' + 1) + 1$ and $\alpha_0$ limit implies $\alpha_0 < \alpha' + 1$.
\\[2pt]
If $\alpha$ is a limit ordinal, then $\redupto{(\redai)}{\alpha}$ being a well-formed \redseq\ implies that condition~(\ref{it:dfn-sred-successor-coherence}) holds for $\redai$; notice $\alpha_0 + 1 < \alpha + 1$ implies $\alpha_0 < \alpha$, so that $\alpha$ limit implies in turn $\alpha_0 + 1 < \alpha$.
Furthermore, $\redupto{(\redai)}{\alpha}$ being convergent implies that conditions~(\ref{it:dfn-sred-limit-existence}) and (\ref{it:dfn-sred-depth}) hold for all $\alpha_0$ limit ordinals verifying $\alpha_0 < \alpha + 1$, particularly for $\alpha$; and also that condition~(\ref{it:dfn-sred-limit-coherence}) holds for all limit $\alpha_0 < \alpha$.
In turn, $\lim_{\alpha' \to \alpha} tgt(\redel{(\redai)}{\alpha'}) = tgt(\redupto{(\redai)}{\alpha}) = \subtat{src(\redel{\reda}{\seqel{A}{\alpha}})}{i} = src(\redel{(\redai)}{\alpha})$, so that condition~(\ref{it:dfn-sred-limit-coherence}) to hold also for $\alpha$
\end{tabular} \\[5pt]
Hence, in either case, we have verified that $\redai$ is a well-formed \redseq.
In turn, $\redln{\redai} = \seqln{A}$ being a successor ordinal implies immediately that $\redai$ is convergent.
If $\reda$ is convergent, then we must verify $tgt(\redai) = \subtat{tgt(\reda)}{i}$.
Let $\redel{(\redai)}{\alpha} = \langle t_i, p, \mu \rangle$ where $\redel{\reda}{\seqel{A}{\alpha}} = \langle f(t_1, \ldots, t_i, \ldots, t_m), ip, \mu \rangle$.
Then $tgt(\redai) = tgt(\redel{(\redai)}{\alpha}) = \repl{t_i}{s}{p}$ for some term $s$, and $tgt(\redel{\reda}{\seqel{A}{\alpha}}) = \repl{f(t_1, \ldots, t_i, \ldots, t_m)}{s}{ip} = $ \\ $f(t_1, \ldots, \repl{t_i}{s}{p}, \ldots, t_m)$, \confer\ Lem.~\ref{rsl:repl-homo}, therefore $tgt(\redai) = \subtat{tgt(\redel{\reda}{\seqel{A}{\alpha}})}{i}$.
If $\redln{\reda} = \seqel{A}{\alpha} + 1$, then $tgt(\reda) = tgt(\redel{\reda}{\seqel{A}{\alpha}})$.
Otherwise, for all $\alpha'$ verifying $\seqel{A}{\alpha} < \alpha' < \redln{\reda}$, it is immediate that $\RPos{\redel{\reda}{\alpha'}} \disj i$. Then Lem.~\ref{rsl:redseq-disj} implies $\subtat{tgt(\redsublt{\reda}{\seqel{A}{\alpha} + 1}{\redln{\reda}})}{i} = \subtat{src(\redsublt{\reda}{\seqel{A}{\alpha} + 1}{\redln{\reda}})}{i}$.
In either case, $\subtat{tgt(\reda)}{i} = \subtat{tgt(\redel{\reda}{\seqel{A}{\alpha}})}{i} = tgt(\redai)$.
Thus we conclude.
Assume that $\alpha \eqdef \seqln{A}$ is a limit ordinal.
Let $\alpha'$ such that $\alpha' + 1 < \alpha$, then $\alpha$ limit implies $\alpha' + 2 < \alpha$.
Therefore \ih\ can be applied to obtain that $\redupto{(\redai)}{\alpha' + 2}$ is a well-formed \redseq, implying that $src(\redel{(\redai)}{\alpha' + 1}) = tgt(\redel{(\redai)}{\alpha'})$.
Consequently, $\redai$ verifies condition~(\ref{it:dfn-sred-successor-coherence}) in Dfn.~\ref{dfn:sred}.
Let $\alpha_0$ be a limit ordinal verifying $\alpha_0 < \alpha$.
Observe that $\seqel{A}{\alpha_0} < \redln{\reda}$, then Lem.~\ref{rsl:redseq-proper-section-convergent} implies that $\redupto{\reda}{\seqel{A}{\alpha_0}}$ is convergent.
We apply \ih\ to obtain that $\redupto{(\redai)}{\alpha_0}$ is a well-formed and convergent \redseq. Therefore conditions~(\ref{it:dfn-sred-limit-existence}) and (\ref{it:dfn-sred-depth}) hold for $\redai$ \wrt\ $\alpha_0$.
Moreover $\lim_{\alpha' \to \alpha_0} tgt(\redel{(\redai)}{\alpha'}) = tgt(\redupto{(\redai)}{\alpha_0}) = src(\redel{(\redai)}{\alpha_0}$, \confer\ Dfn.~\ref{dfn:redseq-tgt} and Lem.~\ref{rsl:redseq-section-src-tgt-coherence} resp.. Hence $\redai$ enjoys condition~(\ref{it:dfn-sred-limit-coherence}) \wrt\ $\alpha_0$ as well.
Consequently, $\redai$ is a well-formed \redseq.
Observe that $src(\redai) = src(\redel{(\redai)}{0}) = src(\proj{\redupto{\reda}{\seqel{A}{1}}}{i})$. Since obviously $1 < \alpha$, we can use \ih\ to obtain $src(\redai) = \subtat{src(\redupto{\reda}{\seqel{A}{1}})}{i} = \subtat{src(\reda)}{i}$.
Assume that $\reda$ is convergent.
Let $B \eqdef \set{\beta' \setsthat \beta' < \redln{\reda} \,\land\, \seqel{A}{\alpha'} < \beta' \textforall \alpha' < \alpha}$.
We define $\beta$ as follows: $\beta \eqdef \redln{\reda}$ if $B$ is empty, and $\beta \eqdef min(B)$ otherwise.
Assume for contradiction that $\beta = \beta' + 1$ for some $\beta'$.
If $B$ is empty, so that $\redln{\reda} = \beta' + 1$, then $\beta' \notin B$ implies the existence of some $\alpha' < \alpha$ such that $\beta' \leq \seqel{A}{\alpha'}$ and then $\beta' < \seqel{A}{\alpha' + 1}$, contradicting $\seqel{A}{\alpha'+1} < \redln{\reda}$.
Otherwise $\beta = min(B)$, implying that $\beta' \leq \seqel{A}{\alpha'}$ for some $\alpha' < \alpha$. But this would imply $\beta \leq \seqel{A}{\alpha' + 1}$, contradicting $\beta \in B$.
Consequently, $\beta$ is a limit ordinal.
We verify conditions~(\ref{it:dfn-sred-limit-existence}) and (\ref{it:dfn-sred-depth}) for $\redai$ \wrt\ $\alpha$. \\[5pt]
\begin{tabular}{@{$\ \ \bullet\quad$}p{.9\textwidth}}
To verify condition~(\ref{it:dfn-sred-limit-existence}), it is enough to show that $\lim_{\alpha' \to \alpha} tgt(\redel{(\redai)}{\alpha'}) = \subtat{u}{i}$, where $u = \lim_{\beta' \to \beta} tgt(\redel{\reda}{\beta'}) = tgt(\redupto{\reda}{\beta})$.
Let $n < \omega$, and $\beta_n < \beta$ such that $\tdist{tgt(\redel{\reda}{\beta'})}{u} < 2^{-(n+1)}$, implying $\tdist{\subtat{tgt(\redel{\reda}{\beta'})}{i}}{\subtat{u}{i}} < 2^{-n}$, if $\beta_n < \beta' < \beta$.
Then $\beta_n < \beta$ implies that $\beta_n \leq \seqel{A}{\alpha_n}$ for some $\alpha_n < \alpha$, then $\alpha_n < \alpha' < \alpha$ implies $
\tdist{tgt(\redel{(\redai)}{\alpha'})}{\subtat{u}{i}} < 2^{-n}$, recalling that $tgt(\redel{(\redai)}{\alpha'}) = \subtat{tgt(\redel{\reda}{\seqel{A}{\alpha}})}{i}$.
Consequently, $\lim_{\alpha' \to \alpha} tgt(\redel{(\redai)}{\alpha'}) = \subtat{tgt(\redupto{\reda}{\beta})}{i}$, and then $\redai$ verifies condition~(\ref{it:dfn-sred-limit-existence}) \wrt\ $\alpha$.
\\[2pt]
Let $n < \omega$, let $\beta_n < \beta$ such that $\sdepth{\redel{\reda}{\beta'}} > n + 1$ if $\beta_n < \beta' < \beta$. By an argument similar to that used for condition~(\ref{it:dfn-sred-limit-existence}), we obtain the existence of some $\alpha_n < \alpha$ such that $\sdepth{\redel{\reda}{\seqel{A}{\alpha'}}} > n + 1$, implying $\sdepth{\redel{(\redai)}{\alpha'}} > n$, if $\alpha_n < \alpha' < \alpha$.
Consequently, $\redai$ verifies condition~(\ref{it:dfn-sred-depth}) for $\alpha$.
\end{tabular} \\[5pt]
Hence, $\redai$ is a convergent \redseq.
In turn, Dfn.~\ref{dfn:redseq-tgt} yields $tgt(\redai) = \lim_{\alpha' \to \alpha} tgt(\redel{(\redai)}{\alpha'})$, then we have already verified that $tgt(\redai) = \subtat{tgt(\redupto{\reda}{\beta})}{i}$.
If $\beta = \redln{\reda}$, then immediately $tgt(\redai) = \subtat{tgt(\reda)}{i}$.
Otherwise, it is immediate to observe that $\RPos{\redel{\reda}{\beta'}} \disj i$ if $\beta \leq \beta' < \redln{\reda}$.
Hence $tgt(\redai)
= \subtat{tgt(\redupto{\reda}{\beta})}{i}
= \subtat{src(\redsublt{\reda}{\beta}{\redln{\reda}})}{i}
= \subtat{tgt(\redsublt{\reda}{\beta}{\redln{\reda}})}{i}
= \subtat{tgt(\reda)}{i}$;
by already obtained result, Lem.~\ref{rsl:redseq-section-src-tgt-coherence} (recall $src(\redsublt{\reda}{\beta}{\redln{\reda}}) = src(\redel{\reda}{\beta})$, Lem.~\ref{rsl:redseq-disj}, and simple analysis of Dfn.~\ref{dfn:redseq-tgt} resp..
Thus we conclude.
\end{proof}
The following result extends the idea of a projection of a \redseq\ from arguments of function symbols to arguments of contexts.
\begin{lemma}
\label{rsl:redseq-respects-src-tgt}
Let $C$ a context having exactly $m$ holes, and $C[t_1, \ldots, t_m] \infredx{\reda} u$, such that for all $\alpha < \redln{\reda}$, there exists some $i$ verifying $1 \leq i \leq m$ and $\BPos{C}{i} \leq \RPos{\redel{\reda}{\alpha}}$.
Then $u = C[u_1, \ldots, u_m]$ and for all $i$ such that $1 \leq i \leq m$, there is a \redseq\ $\reda_i$ verifying $t_i \infredx{\reda_i} u_i$.
\end{lemma}
\begin{proof}
Straightforward induction on $max \set{\posln{\BPos{C}{i}} \setsthat 1 \leq i \leq m}$, resorting on Lem.~\ref{rsl:proj-redseq-well-defined} for the inductive case.
\end{proof}
Two properties about normalisation follow.
\begin{lemma}
\label{rsl:orthogonal-leading-head-steps-to-nf}
Let $\trst$ an orthogonal \TRS, and $t,s,u$ terms such that $t \infredx{\redc} u$, $t \sredx{\reda} s$, $u$ is a normal form, and $\sdepth{\redel{\reda}{i}} = 0$ for all $i < \redln{\reda}$.
Then $s \infredx{\redc'} u$ for some \redseq\ $\redc'$.
\end{lemma}
\begin{proof}
We proceed by induction on $\redln{\reda}$, observe that $\reda$ is finite, and then only numeral induction is needed.
If $\redln{\reda} = 0$, \ie\ $\reda$ is the empty reduction for $t$, then $s = t$ so that we conclude by taking $\redc' \eqdef \redc$.
Assume $\redln{\reda} = n + 1$, so that $t \sstepx{\stepa} s_0 \sredx{\reda'} s$ where $\stepa = \langle t, \epsilon, \mu \rangle$ for some rule $\mu: l[x_1, \ldots, x_m] \to h$, and $\redln{\reda'} = n$.
We will resort to a result presented and proved in \eg\ \cite{orthogonal-itrs-95} and \cite{terese}, where it is called \emph{Strip Lemma}
\footnote{in \cite{orthogonal-itrs-90}, a preliminary version of \cite{orthogonal-itrs-95}, the same property is called \emph{Parallel Moves Lemma}}
. This result implies that whenever $t \infredx{\redb} t'$ and $t \sstepx{\stepb} s_0$, then $t' \infredx{\stepb_r} s'$ and $s_0 \infredx{\redb_r} s'$,
where $\stepb_r$ is the \emph{residual} of $\stepb$ \emph{after} $\redb$
\footnote{the statement in \cite{terese}, and also in \cite{orthogonal-itrs-90}, describe also the nature of $\redb_r$. We will not give the details here since they are not needed for this proof.}
. The result of the lemma can be described graphically as follows: \\[5pt]
\newcommand{\arInfR}{
\ar@{*{}*@{-}*{\hspace{-3mm}\scriptscriptstyle >\hspace{-1mm}>\hspace{-1mm}>}}[r]
}
\newcommand{\arInfD}{
\ar@{*{}*@{-}*{\textnormal{
\rotatebox{270}{$\hspace{-4mm}\scriptscriptstyle >\hspace{-1mm}>\hspace{-1mm}>$}
}}}[d]
}
$\xymatrix@C=12mm@R=12mm{
t \arInfR^{\redb} \ar[d]_{\stepb} & t' \arInfD^{\stepb_r} \\
s_0 \arInfR_{\redb_r} & s'
}$
While we will not include here the formal definition of residual, we mention a feature valid for orthogonal \TRSs\ which is crucial for this proof.
Assume $\stepb = \langle t, \epsilon, \mu \rangle$ such that $\mu: l \to h$, and $\stepc = \langle t, p, \nu \rangle$ where $p \neq \epsilon$ and $t \sstepx{\stepc} v$. Then $t = l[t_1, \ldots, t_m]$, $q \leq p$ for some $q$ such that $l(q) \in \thevar$, and therefore $v = l[t'_1, \ldots, t'_m]$.
In this case, there is exactly one residual of $\stepb$ after $\stepc$, namely $\langle v, \epsilon, \mu \rangle$.
This property carries on for the residual of $\stepb$ after a reduction $\redd$ where $\mind{\redd} > 0$, even if $\redln{\redd}$ is a limit ordinal.
Graphically: \\[5pt]
$
\xymatrix@C=18mm@R=15mm{
t \ar[r]^{\stepc \,=\, \langle t, p, \, \nu \rangle} \ar[d]_{\stepb \,=\, \langle t, \epsilon, \, \mu \rangle} &
v \ar[d]^{\langle v, \epsilon, \, \mu \rangle}
\\
t' & w'
}
\qquad \qquad
\xymatrix@C=18mm@R=15mm{
t \arInfR^{\redd} \ar[d]_{\stepb \,=\, \langle t, \epsilon, \, \mu \rangle} &
v \ar[d]^{\langle v, \epsilon, \, \mu \rangle}
\\
t' & w'
}
$
We return to the proof. Observe that $t = l[v_1, \ldots, v_m]$ since $\langle t, \epsilon, \mu \rangle$ is a redex.
Then a simple transfinite induction yields that $\redc$ not including any root step would imply $u = l[v'_1, \ldots, v'_m]$, contradicting that $u$ is a normal form.
Let $\alpha$ be the minimum index corresponding to a root step in $\redc$.
Then the described property of residuals implies that $\stepa$ has exacly one residual after $\redupto{\redc}{\alpha}$, which is $\stepa' \eqdef \langle t_\alpha, \epsilon, \mu \rangle$ where $t_\alpha$ is the target term of $\redupto{\redc}{\alpha}$.
Moreover, $\redel{\redc}{\alpha}$ being a root step implies that the rule used in that step is also $\mu$, \ie\ $\redel{\redc}{\alpha} = \langle t_\alpha, \epsilon, \mu \rangle = \stepa'$. Therefore we can build the following graphic: \\[5pt]
$
\xymatrix@C=28mm@R=15mm{
t \arInfR^{\redupto{\redc}{\alpha}} \ar[d]_{\stepa \,=\, \langle t, \, \epsilon, \, \mu \rangle} &
t_{\alpha} \ar[r]^{\redel{\redc}{\alpha} \,=\, \langle t_\alpha, \, \epsilon, \, \mu \rangle}\ar[d]_{\stepa' \,=\, \langle t_\alpha, \, \epsilon, \, \mu \rangle} &
t_{\alpha + 1} \arInfR^{\redsublt{\redc}{\alpha+1}{\redln{\redc}}} \ar@{=}[d] &
u
\\
s_0 \arInfR_{\redc_1} &
t_{\alpha+1} \ar@{=}[r] &
t_{\alpha+1} \arInfR_{\redsublt{\redc}{\alpha+1}{\redln{\redc}}} &
u
}
$
Hence \ih\ on $s_0 \sredx{\reda'} s$ suffices to conclude.
\end{proof}
\begin{proposition}
\label{rsl:disjoint-then-isn}
Let \trst\ be a disjoint \TRS\ which does not include collapsing rules. Then \trst\ has the property $SN^\infty$.
\footnote{I guess that this property can be generalised to any \TRS\ in which the sets of \textbf{head} symbols of lhss and rhss are disjoint, with exactly the same proof. I don't know whether change the statement of the proposition, which is used through this text only for disjoint \TRSs.}
\end{proposition}
\begin{proof}
First we prove the following auxiliary result: for any \redseq\ \reda, limit ordinal $\beta$ such that $\beta \leq \redln{\reda}$, and $n < \omega$,
\begin{equation}
\begin{array}{rl}
\textif & \exists \beta_1 < \beta \sthat \forall i \ (\beta_1 < i < \beta \textnormal{ implies } \sdepth{\redel{\reda}{i}} \geq n) \\
\textthen & \exists \beta' < \beta \sthat \forall i \ (\beta' < i' < \beta \textnormal{ implies } \sdepth{\redel{\reda}{i'}} > n)
\end{array}
\label{eq:disjoint-sn-01}
\end{equation}
Assume for any $\reda$, $\beta$ and $n$ that the premise holds.
The term $src(\redel{\reda}{\beta}) = tgt(\redsublt{\reda}{\beta_1}{\beta})$ can include only a finite number of redexes at depth $n$.
Additionally, the hypothesis yields that any reduction step included in $\redsublt{\reda}{\beta_1}{\beta}$, say $\redel{\reda}{j}$, satisfies $\sdepth{\redel{\reda}{j}} \geq n$, and moreover leaves at its redex position (\confer\ \refdfn{step})
a symbol not being the head symbol of a left-hand side, since $T$ is disjoint and it does not include collapsing rules.
Therefore,
no redex occurrence can be created at depth $n$, implying that any reduction step at depth exactly $n$ included in $\redsublt{\reda}{\beta_1}{\beta}$ must correspond to a redex occurrence already included in $src(\redel{\reda}{\beta_1})$ and being at the same position.
Consequently, if we call $k$ the number of steps at depth exactly $n$ included in $\redsublt{\reda}{\beta_1}{\beta}$, we obtain $k < \omega$.
Thus we conclude the proof of the auxiliary result by taking $\beta'$ to be the ordinal such that $\redel{\reda}{\beta'}$ is the last of such steps if $k > 0$, and $\beta' \eqdef \beta_1$ if $k = 0$.
Now we prove, for any \redseq\ \reda\ in $T$, that \reda\ is convergent; \ie\ that for any $n < \omega$ and $\beta$ limit ordinal such that $\beta \leq \redln{\reda}$,
\begin{equation}
\exists \beta' < \beta \sthat \forall i \ (\beta' < i < \beta \textnormal{ implies } \sdepth{\redel{\reda}{i}} > n)
\label{eq:disjoint-sn-02}
\end{equation}
We conclude the proof of the proposition by proving \refeqn{disjoint-sn-02} by induction on $n$.
If $n = 0$, then the premise of \refeqn{disjoint-sn-01} holds taking $\beta_1 = 0$, then we conclude by \refeqn{disjoint-sn-01}.
If $n > 0$, then the premise of \refeqn{disjoint-sn-01} holds for some $\beta_1$ by \ih\ of \refeqn{disjoint-sn-02} considering $n - 1$ instead of $n$, then we conclude again by \refeqn{disjoint-sn-01}.
\end{proof}
\section{Proof terms}
\label{sec:pterm}
The intent of the definition of proof terms is to provide a tool to formally denote, or witness, \redseqs\ in infinitary rewriting.
Proof terms are, indeed, terms, in a signature extending that of the \iTRS\ whose \redseqs\ are to be described.
This \TRS\ will be referred to as the \textbf{object} \TRS\; we will also use the terminology `object terms' and `object \redseqs' analogously.
As already noticed, the scope of this work is limited to \emph{left-linear} \iTRSs.
The proof terms for infinitary rewriting we introduce in this Section generalise the definition given in \cite{terese} for finitary first-order rewriting, \confer\ their Dfn. 8.2.18.
The idea of using terms to denote reduction sequences has been proposed also for simply-typed lambda-calculus in \cite{Hilken96}, and for higher-order rewriting in \cite{bruggink2008}.
For each proof term we define: its \emph{source} and \emph{target} which are object terms, if it is \emph{convergent}, and its \emph{minimum depth}. All these concepts refer to the \redseqs\ which are denoted by the proof term.
In this section, a formal definition of the set of infinitary proof terms for a given \iTRS\ will be given.
Then a simplified transfinite induction principle on the set of valid proof terms is given. The form of induction we introduce allows for simpler proofs for many properties to be verified in the rest of this work. Also, we will verify that proof terms enjoy some basic properties.
The definition of the set of proof terms is extensive, because it is given in two different stages, and also some auxiliary notions need to be defined simultaneously.
Therefore, we give firstly an informal introduction to the idea of proof term, and how it is used to describe the reduction space of a \TRS.
For each reduction rule in the object TRS, a \emph{rule symbol} is introduced in the signature for proof terms. The arity of a rule symbol coincides with the number of different variables occurring in the left-hand side of the rule it represents.
\Eg, the signature of proof terms for a first-order \TRS\ \trst\ including the rules $f(x) \to g(x)$, $h(j(x),j(y)) \to f(x)$ and $g(x) \to k(x)$ adds the rule symbols $\mu/1$, $\nu/2$ and $\rho/1$, corresponding respectively to each of the described rules.
We describe some valid proof terms along with the \trst-reductions they denote $\mu(a) : f(a) \to g(a)$, $g(\nu(a,b)) : g(h(j(a),j(b))) \to g(f(a))$, $h(\mu(a), \mu(b)) : h(f(a),f(b)) \sred h(g(a),g(b))$, $\nu(\mu(a),b) : h(j(f(a)),j(b)) \sred f(g(a))$.
In the infinitary setting, infinite proof terms denote \redseqs\ involving infinite terms, and/or having infinite length.
We give some examples of infinite proof terms corresponding to the the \TRS\ \trst\ introduced in the previous paragraph:
$\mu(j\om) : f(j\om) \to g(j\om)$, $\mu\om : f\om \infred g\om$
\footnote{In the following, a formal way to compute the source and target corresponding to any proof term will be developed.}
.
Proof terms, as described up to this point, can be used to denote arbitrarily complex \emph{developments}, \ie\ \redseqs\ in which all the contracted redexes are present in its source term.
On the other hand, dealing with the contraction of redexes which are \emph{created} by previous steps in a \redseq\ require the idea of \emph{concatenation}, or \emph{composition}, to be taken into account in the definition of proof terms.
This proposal takes from \cite{terese} the idea of describing concatenation by means of a binary symbol which is added to the signature of proof terms. This symbol is called ``the dot'', because of its graphical representation, \ie\ $\comp$.
Some examples of proof terms including occurrences of the dot follow:
$\mu(a) \comp \rho(a) : f(a) \to g(a) \to k(a)$,
$\nu(\mu(b),c) \comp \mu(\rho(b)) \comp \rho(k(b)) :
h(j(f(b)), j(c)) \sred f(g(b)) \sred g(k(b)) \sred k(k(b))$,
$\mu\om \comp \rho\om : f\om \infred g\om \infred k\om$.
As the concatenation symbol have no special ``status'' in the signature, it can be freely combined with rule as well as object symbols, \eg\
$j(\mu(a) \comp \rho(a)) : j(f(a)) \sred j(k(a))$ denotes a two-step contraction being ``local'' to the argument of the $j$ symbol, while $\nu(\mu(a) \comp \rho(a), b) : h(j(f(a)),j(b)) \sred f(k(a))$ denote a parallelism between an outer step and the concatenation of two inner steps.
We observe that not any term in the extended signature correspond to a valid proof term.
Each occurrence of the dot imposes a coherence condition: (the \redseqs\ corresponding to) its operands must be composable.
\Eg\ neither $\mu(a) \comp \nu(a,a)$ nor $\mu(a) \comp \rho(b)$ are valid proof terms, because the step $f(a) \to g(a)$ is not left-composable, neither with $h(j(a),j(a)) \to f(a)$, nor with $g(b) \to h(b)$.
Therefore, some rules must be provided in order to specify the subset of valid proof terms out of the set of all terms corresponding to the extended signature. As suggested by the just given example, these rules will be related with the occurrences of the dot.
We want to stress that the denotational capabilities of proof terms allow for a great variety in the description of reductions.
Particularly, parallel/nested steps can be explicitly described, and thus differentiated from its sequential counterparts.
\Eg, the proof terms $\mu(f(a)) \comp g(\mu(a))$ and $\mu(\mu(a))$ are different, so that the model of reductions given by proof terms allow to recognise $f(f(a)) \to g(f(a)) \to g(g(a))$ and $f(f(a)) \mulstep g(g(a))$ as different objects in the reduction space of the same \TRS.
Furthermore, as we have already observed, proof terms allow to combine in different ways the concatenation symbol with the other symbols in the extended signature.
This capability brings new ways to differentiate subtly different reductions, by describing them using different proof terms.
These considerations motivate the following assertion: proof terms denote different forms of \emph{contraction activity}, a concept broader than that of \emph{\redseq}.
We claim that proof terms as a way to describe contraction activity allow for a very detailed study of the reduction/derivation space of a calculus.
\subsection{Multisteps}
\label{sec:mstep}
\label{sec:mstep-defs}
Since the restrictions on the set of valid proof terms pertain to the dot occurrences, ``dotless'' proof terms form the foundation from which the definition of proof terms is built.
We will give the name \emph{multistep} to any proof terms without dot occurrences.
As we have discussed in the informal introduction, multisteps correspond to sets of coinitial redexes.
We have also seen that sequencing is explicitly denoted in proof terms by means of the concatenation symbol, \ie\ the dot.
Therefore, multisteps are intended to denote the contraction activity consisting in the \textbf{simultaneous} contraction of a set of redexes, \ie\ a multistep, \confer\ \cite{terese}, Dfn. 4.5.11.. Hence the name we have given to the proof terms to be defined next.
In the sequel, we define the set of \emph{\imsteps}, along with some basic features of a multistep, namely: how to determine its \emph{source} and \emph{target} terms, whether it is \emph{convergent} or not, and its \emph{minimum activity depth}.
These concepts are needed to properly define the restrictions to be imposed to occurrences of the dot in the general definition of the set of proof terms.
\begin{definition}[Signature for multisteps]
\label{dfn:sigma-r}
Let $T = \pair{\Sigma}{R}$ be a (either finitary or infinitary) TRS. We define the signature for the \imsteps\ over $T$ as follows:
$\Sigma^R :=
\Sigma \cup
\set{ \mu/n \setsthat \mu : l \to r \in R \land |FV(l)| = n }$
.
\end{definition}
\begin{definition}[\Imsteps]
\label{dfn:imstep}
The set of \imsteps\ for an iTRS $T \pair{\Sigma}{R}$ is exactly the set of the closed (\confer\ Dfn.\ref{dfn:closed-linear}) terms
\footnote{By restricting \imsteps, and later proof terms (\confer\ Sec.~\ref{sec:pterm}) to be closed terms, we follow the idea expressed in \cite{terese}, Remark~8.2.21 (pg. 324): ``Since here we are interested in \peqence, we may simply assume that reductions/proof terms are closed.''.
Moreover, this decision simplifies our treatment of \peqence\ given in Sec.~\ref{sec:peqence}.
}
in $Ter^\infty(\Sigma^R)$.
\end{definition}
To define the source and target terms of a multistep, we define `companion' ad-hoc iTRSs; \confer\ \refsec{trs}.
\begin{definition}[$src_T$, $tgt_T$]
\label{dfn:srct-tgtt}
Let $T = \pair{\Sigma}{R}$ be a (either finitary or infinitary) TRS. We define the TRSs $src_T$ and $tgt_T$ as follows.
The signature of both $src_T$ and $tgt_T$ is $\Sigma^R$.
The rules of $src_T$ are $\set{\mu(x_1, \ldots, x_n) \to l[x_1, \ldots, x_n] \setsthat \mu : l \to r \in R}$.
The rules of $tgt_T$ are $\set{\mu(x_1, \ldots, x_n) \to r[x_1, \ldots, x_n] \setsthat \mu : l \to r \in R}$.
\end{definition}
We remark that for any object TRS $T$, both $src_T$ and $tgt_T$ are orthogonal and disjoint; moreover, $src_T$ does not include collapsing rules, since the lhs of a reduction rule cannot be a variable (\confer\ \refdfn{trs}).
Therefore, both $src_T$ and $tgt_T$ enjoy the property $UN^\infty$ (\confer\ the comment about $UN^\infty$ in \refsec{trs}) and $src_T$ enjoys also $SN^\infty$ (\confer\ \refprop{disjoint-then-isn}).
Consequently, any \imstep\ has exactly one $src_T$-normal form, and at most one $tgt_T$-normal form.
This observations entail the soundness of the following definition.
\begin{definition}[Source and target of an \imstep]
\label{dfn:src-tgt-imstep}
Let $\psi$ be an \imstep.
Then we define $src(\psi)$ to be the $src_T$-normal form of $\psi$.
Moreover, if $\psi$ is weakly normalising in $tgt_T$, we define $tgt(\psi)$ to be the corresponding normal form; otherwise, $tgt(\psi)$ is undefined.
\end{definition}
For the kind of
\multistepsAfterDevelopments{object \redseqs}
\multistepsIndependent{contraction activity}
we intend to denote with \imsteps, it is correct to identify convergence with existence of target. Formally:
\begin{definition}[Convergent \imsteps]
\label{dfn:imstep-convergence}
An \imstep\ $\psi$ is convergent iff $\tgtt(\psi)$ is defined.
\end{definition}
\begin{definition}[Minimum activity depth of an \imstep]
\label{dfn:dmin-imstep}
Let $\psi$ be an \imstep.
We define the minimum activity depth of $\psi$, notation $\mind{\psi}$, as follows. \\
If $\psi$ does not include occurrences of rule symbols, \ie\ if it is a term in $Ter^\infty(\Sigma)$, then $\mind{\psi} \eqdef \omega$. \\
Otherwise $\mind{\psi}$ is the minimum $n$ such that exists at least one position $p$ verifying $\psi(p) = \mu$ where $\mu$ is a rule symbol, and $n = \posln{p}$. This case admits an equivalent inductive definition based on \refnotation{term-intuitive-notation}: \\
\hspace*{1cm}
$
\begin{array}{rcl}
\mind{f(\psi_1 \ldots \psi_n)} & := & 1 + min(\mind{\psi_1} \ldots \mind{\psi_n}) \\
\mind{\mu(\psi_1 \ldots \psi_n)} & := & 0
\end{array}
$
\end{definition}
\multistepsAfterDevelopments{As examples of the definitions we have just given, we show \imsteps\ corresponding to each \orthoredexset\ in the examples of \refsec{dev-max-dev}, along with the computation of their source and target terms.
We recall the rules of the object iTRS considered: }
\multistepsIndependent{In the following, we will give some examples of \imsteps. We will consider the following object rules: }
$\mu: f(i(x),y) \to h(y)$, $\nu: g(x) \to x$, $\rho: a \to b$, $\pi: m(x) \to n(x)$, $\tau: n(x) \to f(x,x)$. Then the rules of the companion iTRSs are \\
$src_T$:
$\mu(x,y) \to f(i(x),y)$ \quad $\nu(x) \to g(x)$ \quad $\rho \to a$ \quad
$\pi(x) \to m(x)$ \quad $\tau(x) \to n(x)$ \\
$tgt_T$:
$\mu(x,y) \to h(y)$ \quad $\nu(x) \to x$ \quad $\rho \to b$ \quad
$\pi(x) \to n(x)$ \quad $\tau(x) \to f(x,x)$
\multistepsAfterDevelopments{Each set of redexes $A_i$ is represented by the underlined term $usrc(A_i)$.}
\multistepsIndependent{For each of the examples, we show the source term, underlining the head symbols of some of its redexes, and the \imstep\ denoting contraction of underlined redexes. Then we develop source and target computation.
To keep notation compact, we will omit parenthesis for unary symbols.}
\begingroup
\multistepsIndependent{\renewcommand{\ulnrule}[2]{\uln{#1}}}
\begin{itemize}
\item
The \imstep\ corresponding to
$h(\ulnrule{f}{\mu}(i\ulnrule{a}{\rho},n\ulnrule{m}{\pi} b))$ is
$\psi_1 \eqdef h(\mu(\rho,n \pi b))$. Computation of $src(\psi_1)$ and $tgt(\psi_1)$ follow: \\
$\psi_1 = h(\mu(\rho,n \pi b)) \ssteptrs{src_T} h(f(i \rho, n \pi b)) \ssteptrs{src_T} h(f(i a, n \pi b)) \ssteptrs{src_T} h(f(i a, n m b))$ \\
$\psi_1 = h(\mu(\rho,n \pi b)) \ssteptrs{tgt_T} hhn \pi b \ssteptrs{tgt_T} hhn n b$.
\item
$\psi_2 \eqdef \pi^\omega$ corresponds to
${\ulnrule{m}{\pi}}^\omega$. Let us compute source and target: \\
$\psi_2 = \pi^\omega \ssteptrs{src_T} m(\pi^\omega) \ssteptrs{src_T} mm(\pi^\omega) \infredtrs{src_T} m^\omega$ \\
$\psi_2 = \pi^\omega \ssteptrs{tgt_T} n(\pi^\omega) \ssteptrs{tgt_T} nn(\pi^\omega) \infredtrs{tgt_T} n^\omega$.
\item
$\psi_3 \eqdef \nu^\omega$ corresponds to ${\ulnrule{g}{\nu}}^\omega$. \\
The computation of source runs as in the previous case:
$\psi_3 = \nu^\omega \infredtrs{src_T} g^\omega$.
On the other hand, the target of all $tgt_T$ redex occurrences in $\nu^\omega$ (namely, $\langle 1^i, \nu(x) \to x, \set{x \to \nu^\omega} \rangle$) is again $\nu^\omega$. Therefore $tgt(\psi_3)$ is undefined.
\item
Finally, $\psi_4 = h(\mu(\nu^\omega, \rho))$ corresponds to
$h(\ulnrule{f}{\mu}(i {\ulnrule{g}{\nu}}^\omega, \ulnrule{a}{\rho}))$. \\
Computation of source follows: \\
$\psi_4 = h(\mu(\nu^\omega, \rho)) \ssteptrs{src_T} h(f(i \nu^\omega, \rho)) \ssteptrs{src_T} h(f(i \nu^\omega, a)) \infredtrs{src_T} h(f(i g^\omega, a))$. \\
Many $tgt_T$ \redseqs\ from $\psi_4$ are possible, \eg: \\
$\psi_4 = h(\mu(\nu^\omega, \rho))
\ssteptrs{tgt_T} hh \rho \ssteptrs{tgt_T} hhb$ \\
$\psi_4 = h(\mu(\nu^\omega, \rho))
\ssteptrs{tgt_T} h(\mu(\nu^\omega, b))
\infredtrs{tgt_T} h(\mu(\nu^\omega, b))
\ssteptrs{tgt_T} hhb$ where the $i$-th step for $1 \leq i < \omega$ is $\langle h(\mu(\nu^\omega, b)), 11 \cdot 1^i, \nu(x) \to x, \set{x \eqdef \nu^\omega} \rangle$ \\
$\psi_4 = h(\mu(\nu^\omega, \rho))
\infredtrs{tgt_T} h(\mu(\nu^\omega, \rho))$
where all steps are $\langle \psi_4, 11, \nu(x) \to x, \set{x \eqdef \nu^\omega} \rangle$, a divergent $tgt_T$ \redseqs. \\
Then $\psi_4$ admit both convergent and divergent \redseqs\ in $tgt_T$. As $\psi_4$ is $tgt_T$-weakly normalising, we get $tgt(\psi_4) = hhb$.
\end{itemize}
\endgroup
\subsection{Adding dots properly}
\usingContractionActivity{
In this section we will give the definition of the set of all legal proof terms, by taking \imsteps\ as the foundation, and giving precise rules for the addition of occurrences of the dot.
As we have discussed in the informal introduction, for a term like $\psi \comp \phi$ to be a well-defined proof term, the concatenation of the contraction activities denoted by $\psi$ and $\phi$ must make sense.
}
\usingRedseqOnly{
For a term like $\psi \comp \phi$ to be a well-defined proof term, it must coherently denote any \redseq\ consisting of the concatenation of a \redseq\ denoted by $\psi$ with one denoted by $\phi$.
}
Two conditions, related with this coherence requirement, are imposed.
\usingContractionActivity{
Firstly, the activity denoted by $\psi$ must be \emph{convergent}, \ie, it should exist at least one way to render such activity as a convergent \redseq; this condition implies particularly that the target term of $\psi$ can be uniquely determined.
Secondly, the target term of (the activity denoted by) $\psi$ must coincide with the source term of (that corresponding to) $\phi$.
}
\usingRedseqOnly{
Firstly, $\psi$ must denote at least one convergent \redseq. Secondly, the target of any \redseq\ denoted by $\psi$ must coincide with the source of any \redseq\ denoted by $\phi$.
We remark that these conditions are similar to those required for a pre\redseq\ in order to be a \redseq, \confer\ \refdfn{sred}
\footnote{Another remark: while the first condition is unique to infinitary rewriting, the second one must be imposed to proof terms denoting finite \redseqs\ as well; \confer\ \eg\ the transitivity rule in \cite{terese}, dfn. 8.2.18.}.
}
The need of imposing such conditions on the occurrences of the dot implies that the set of proof terms must be defined along with the source, target and convergence condition for each proof term, in a joint definition.
Convergence depends in turn of the depth of the \redseqs\ being denoted; therefore, \emph{minimal activity depth} of proof terms must be merged within the same, huge definition.
An additional goal is to define the set of proof terms by an \emph{inductive} construction, taking the set of \imsteps\ as the base case.
By doing so, we will be able to reason about proof terms in an inductive, opposed to coinductive, fashion, taking properties about \imsteps\ as the foundation for the inductive reasonings.
Since the occurrences of the dot are defined inductively, a special treatment is needed to allow a proof term to include an infinite number of them. Such a proof term should denote the concatenation of an infinite series of \redseqs
\usingContractionActivity{ or, more generally, of contraction activities}
.
Therefore, special care is taken to guarantee that no component is lost in the construction of the infinite concatenation; \ie, that any component is at a finite distance from the root in the corresponding proof term.
In turn, the separate treatment of binary and infinite concatenation gives rise to potential ambiguities in the construction of a proof term
\footnote{it is a good idea to cite \cite{Gallier86}, and/or other work, here?}.
To avoid the possibility of such ambiguities, the definition of the set of proof terms is \emph{layered}, such that the proof terms included in a layer can be built taking as components proof terms in previous layers only.
Countable ordinals are used as layers for proof terms, and each proof term belongs to exactly one layer.
Therefore, layers give a transfinite induction principle to reason about the set of valid proof terms. An alternative, simpler induction principle for proof terms is given later in this section.
The aforementioned restrictions and considerations try to justify the intricacies of the following definitions.
\begin{definition}[Signature for proof terms]
\label{dfn:sigma-pt}
Let $T = \pair{\Sigma}{R}$ be a (either finitary or infinitary) TRS. We define the signature for the proof terms over $T$ as follows:
$\Sigma^{PT} := \Sigma^R \cup \set{\comp / 2}$
. \Confer\ \refdfn{sigma-r} of $\,\Sigma^R$.
\end{definition}
\begin{definition}[\layerpterm{\alpha}, layer $\alpha$ in the definition of proof terms]
\label{dfn:layer-pterm}
Let \trst\ be a \TRS, and $\alpha$ a countable ordinal. We define \layerpterm{\alpha}, the $\alpha$-th layer in the construction of the set of proof terms for \trst, along with the source, target, convergence condition, and minimal activity depth of any proof term in \layerpterm{\alpha}.
If $\psi \in \layerpterm{\alpha}$, we will write $src(\psi)$, $tgt(\psi)$ and $\mind{\psi}$ for the source, target and minimal activity depth of $\psi$ respectively.
If $\alpha = 0$, then $\layerpterm{\alpha} \eqdef \emptyset$.
Otherwise, we proceed inductively on $\alpha$, defining \layerpterm{\alpha} to be the smallest set in $Ter^\infty(\Sigma^{PT})$ verifying the following conditions.
\begin{enumerate}
\item
\label{rule:ptmstep}
If $\alpha = 1$ and $\psi$ is an \imstep\ for \trst, then $\psi \in \layerpterm{\alpha}$. The source, target, convergence condition and minimal activity depth of $\psi$ coincide with the definitions given for \imsteps\ in \refsec{mstep-defs}.
\item
\label{rule:ptinfC}
Assume that for any $i < \omega$, $\psi_i \in \layerpterm{\alpha_i}$, such that $\alpha = \underset{i < \omega}{\Sigma} \alpha_i$; \confer\ \refdfn{ordinal-infAdd}. Moreover, assume that for all $n$, $\psi_n$ is convergent, and $tgt(\psi_n) = src(\psi_{n+1})$. \\[1mm]
\begin{tabular}{@{}p{72mm}c}
\begin{minipage}{72mm}
Then $\psi \eqdef \pair{P}{F} \in \layerpterm{\alpha}$, where \\
$P \eqdef \set{2^n \setsthat n < w} \cup (\underset{n < \omega}{\bigcup} 2^n 1 \cdot \Pos{\psi_n})$, \\
$F(2^n) \eqdef \comp$, and $F(2^n 1 p) \eqdef \psi_n(p)$. \\
A concise term notation for $\psi$ is $\icomp \psi_i$; \\
being in fact an abbreviation for
$\psi_1 \comp (\psi_2 \comp (\psi_3 \comp \ldots))$. \\
\end{minipage}
&
\begin{minipage}{55mm}
A graphical representation is \\
\hspace*{\stretch{1}}
$\xymatrix@C=3mm@R=3mm{
& \ar[dl] \ar[dr] \cdot \\
\psi_1 & & \ar[dl] \ar[dr] \cdot \\
& \psi_2 & & \ar[dl] \ar[dr] \cdot \\
& & \psi_3 & & \ddots \\
}$
\hspace*{\stretch{1}}
\end{minipage}
\end{tabular}
We define $src(\psi) \eqdef src(\psi_0)$, $tgt(\psi) \eqdef \lim_{i \to \omega} tgt(\psi_i)$ and $\mind{\psi} \eqdef min(\mind{\psi_i}_{i < \omega})$; notice that $tgt(\psi)$ can be undefined.
We define that $\psi$ is convergent iff for all $k < \omega$, there is some $n < \omega$ such that $\mind{\psi_j} > k$ if $j > n$.
\item
\label{rule:ptbinC}
Assume that $\psi_1 \in \layerpterm{\alpha_1}$, $\psi_2 \in \layerpterm{\alpha_2}$, $\alpha_2$ is a successor ordinal, $\psi_1$ is convergent, $tgt(\psi_1) = src(\psi_2)$, and $\alpha = \alpha_1 + \alpha_2 + 1$.
Then $\psi \eqdef \pair{P}{F} \in \layerpterm{\alpha}$, where
$P \eqdef \set{\epsilon} \cup (1 \cdot \Pos{\psi_1}) \cup (2 \cdot \Pos{\psi_2})$,
$F(\epsilon) \eqdef \comp$, and $F(ip) \eqdef \psi_i(p)$ for $i = 1,2$.
A concise term notation for $\psi$ is $\psi_1 \comp \psi_2$. A graphical notation is
$\xymatrix@C=3mm@R=4mm{
& \ar[dl] \ar[dr] \cdot \\
\psi_1 & & \psi_2
}$
We define $src(\psi) \eqdef src(\psi_1)$, $tgt(\psi) \eqdef tgt(\psi_2)$ and $\mind{\psi} = min(\mind{\psi_1}, \mind{\psi_2})$; $\psi$ is convergent iff $\psi_2$ is.
\item
\label{rule:ptsymbol}
Assume that $\psi_i \in \layerpterm{\alpha_i}$ for $i = 1, 2, \ldots, n$, that $\alpha_i > 1$ for at least one $i$, $f/n \in \Sigma$ (resp. $\mu/n$ is a rule symbol), and $\alpha = \alpha_1 + \ldots + \alpha_n + 1$.
Then $\psi \eqdef \pair{P}{F} \in \layerpterm{\alpha}$, where
$P \eqdef \set{\epsilon} \cup (\underset{1 \leq i \leq n}{\bigcup}i \cdot \Pos{\psi_i})$, $F(\epsilon) \eqdef f$ (resp. $F(\epsilon) \eqdef \mu$), and $F(ip) \eqdef \psi_i(p)$ for $i = 1,2,\ldots,n$.
A concise term notation for $\psi$ is $f(\psi_1, \ldots, \psi_n)$ (resp. $\mu(\psi_1, \ldots, \psi_n)$).
For $f \in \Sigma$, we define $src(\psi) = f(src(\psi_1), \ldots, src(\psi_n))$, $tgt(\psi) = f(tgt(\psi_1), \ldots, tgt(\psi_n))$, and $\mind{\psi} \eqdef 1+min(\mind{\psi_1}, \ldots, \mind{\psi_n})$. $\psi$ is convergent iff all $\psi_i$ are. We observe that $tgt(\psi)$ is undefined if at least one $tgt(\psi_i)$ is.
For $\mu$ being a rule symbol such that $\mu : l \to r$, we define $src(\psi) = l[src(\psi_1), \ldots, src(\psi_n)]$, $tgt(\psi) = r[tgt(\psi_1), \ldots, tgt(\psi_n)]$, and $\mind{\psi} \eqdef 0$. $\psi$ is convergent iff all $\psi_i$ \textbf{corresponding to some $x_i$ occurring in $r$} are. We observe that $tgt(\psi)$ is undefined if at least one $tgt(\psi_i)$ is for the $\psi_i$ already mentioned.
\end{enumerate}
\end{definition}
\begin{definition}[\setpterm, the set of proof terms]
\label{dfn:pterm}
We define the set of proof terms as follows:
$\setpterm \eqdef \underset{\alpha < \omega_1}{\bigcup} \layerpterm{\alpha}$.
\end{definition}
We notice that all proof terms are \emph{closed} terms in $Ter^\infty(\Sigma^{PT})$. This fact is a consequence of the definition of the set of \imsteps, which are the base layer in the definition of \setpterm. \Confer\ the footnote on Dfn.~\ref{dfn:imstep}.
We will say that a proof term $\psi$ is an \emph{infinite concatenation} iff $\psi(2^n) = \comp$ for all $n < \omega$.
Observe that all infinite concatenations admit the concise term notation $\psi = \icomp \psi_i$, where $\psi_n = \subtat{\psi}{2^n 1}$.
Furthermore, $\psi$ not being an infinite concatenation implies the existence of some $n < \omega$ such that $2^n \in \Pos{\psi}$ and $\psi(2^n) \neq \comp$.
\subsection{Soundness of the definitions}
In this section we will study the definition of the set of valid proof terms in some detail, stating and proving properties related to its soundness.
\begin{lemma}
\label{rsl:ptinfC-iff-limit}
Let $\psi$, $\alpha$ such that $\psi \in \layerpterm{\alpha}$.
Then $\psi$ is an infinite concatenation iff $\alpha$ is a limit ordinal iff $\psi$ is generated by case \ref{rule:ptinfC} in Dfn.~\ref{dfn:layer-pterm}.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$, analysing the rules in Dfn.~\ref{dfn:layer-pterm}.
\noindent
Case \ref{rule:ptmstep}: in this case $\psi$ is an \imstep, so that $\psi(2^0) = \psi(\epsilon) \neq \comp$.
\noindent
Case \ref{rule:ptinfC}:
in this case $\psi = \icomp \psi_i$, that is, an infinite concatenation.
It is enough to observe that $\layerpterm{0} = \emptyset$, and that $\alpha_i > 0$ for all $i$ implies that $\sum_{i < \omega} \alpha_i$ is a limit ordinal.
\noindent
Case \ref{rule:ptbinC}:
in this case $\psi = \psi_1 \comp \psi_2$ where $\psi_i \in \layerpterm{\alpha_i}$, $\alpha_2$ is a successor ordinal, and $\alpha = \alpha_1 + \alpha_2 + 1$, \ie\ a successor ordinal.
\ih\ on $\psi_2$ implies that $\psi_2(2^n) \neq \comp$ for some $n < \omega$.
We conclude by observing that $\psi(2^{n+1}) = \psi_2(2^n)$.
\noindent
Case \ref{rule:ptsymbol}:
in this case it is immediate that $\psi(2^0) = \psi(\epsilon) \neq \comp$, and that $\alpha$ is a successor ordinal.
\end{proof}
\begin{lemma}
\label{rsl:ptmstep-iff-one}
Let $\psi$, $\alpha$ such that $\psi \in \layerpterm{\alpha}$.
Then $\psi$ is an \imstep\ iff $\alpha = 1$ iff $\psi$ is generated by case 1 in Dfn.~\ref{dfn:layer-pterm}.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$, analysing the rules in Dfn.~\ref{dfn:layer-pterm}.
\noindent
Case \ref{rule:ptmstep}: we conclude immediately.
\noindent
Case \ref{rule:ptinfC}:
in this case $\psi$ is not an \imstep, observe \eg\ that $\psi(\epsilon) = \comp$, and $\alpha$ is a limit ordinal, \confer\ Lem.~\ref{rsl:ptinfC-iff-limit}. Thus we conclude.
\noindent
Case \ref{rule:ptbinC}:
in this case $\psi$ is not an \imstep, observe \eg\ that $\psi(\epsilon) = \comp$, and $\alpha > \alpha_1 + 1 > 1$, recall $\layerpterm{0} = \emptyset$.
Thus we conclude.
\noindent
Case \ref{rule:ptsymbol}:
in this case $\psi = f(\psi_1, \ldots, \psi_n)$ where $\psi_i \in \layerpterm{\alpha_i}$ for all $i$, and exists some $k$ such that $\alpha_k > 1$.
Observe that $\alpha > \alpha_k > 1$, then we can apply \ih\ to obtain that $\psi_k$ is not an \imstep, hence $\psi$ is neither. Thus we conclude.
\end{proof}
The set \setpterm\ is closed by operations, formally:
\begin{proposition}[Completeness of \setpterm]
\label{prop:compl-pt}
\hspace*{1mm} \\
\begin{enumerate}
\item \label{it:compl-pt:mul}
If $\psi$ is an infinite multistep, then $\psi \in \setpterm$.
\item \label{it:compl-pt:binC}
If $\psi_1, \psi_2 \in \setpterm$, $\psi_1$ is convergent, and $src(\psi_2) = tgt(\psi_1)$, then $\psi_1 \comp \psi_2 \in \setpterm$.
\item \label{it:compl-pt:infC}
Given a sequence $\iomegaseq{\psi_i}$ such that for all $i$, $\psi_i \in \setpterm$, $\psi_i$ are convergent, and $tgt(\psi_i) = src(\psi_{i+1})$,
then $\icomp \psi_i \in \setpterm$.
\item \label{it:compl-pt:symbol}
If $\psi_1, \ldots, \psi_n \in \setpterm$ and $f \in \Sigma$, then $f(\psi_1, \ldots, \psi_n) \in \setpterm$.
\item \label{it:compl-pt:rule}
If $\psi_1, \ldots, \psi_n \in \setpterm$ and $\mu$ is a rule symbol, then $\mu(\psi_1, \ldots, \psi_n) \in \setpterm$.
\end{enumerate}
\end{proposition}
\begin{proof}
We prove each item separately, referring to cases in Dfn.~\ref{dfn:layer-pterm}.
\noindent
Item \ref{it:compl-pt:mul}:
in this case $\psi \in \layerpterm{1}$, this is immediate from case \ref{rule:ptmstep}.
\noindent
Item \ref{it:compl-pt:binC}:
Let $\alpha_1$, $\alpha_2$ such that $\psi_i \in \layerpterm{\alpha_i}$ for $i = 1,2$.
If $\alpha_2$ is a successor ordinal, then $\psi_1 \comp \psi_2 \in \layerpterm{\alpha_1 + \alpha_2 + 1} \subseteq \setpterm$.
If $\alpha_2$ is a limit ordinal, then Lem.~\ref{rsl:ptinfC-iff-limit} implies that $\psi_2 = \icomp \phi_i$, where for all $i$, $\phi_i$ is convergent and $tgt(\phi_i) = src(\phi_{i+1})$; \confer\ case~\ref{rule:ptinfC}.
On the other hand, hypotheses imply that $\psi_1$ is convergent and $tgt(\psi_1) = src(\psi_2) = src(\phi_0)$.
Then $\psi_1 \comp \psi_2 \in \layerpterm{\alpha_1 + \alpha_2}$, again by case~\ref{rule:ptinfC}. Observe that $\psi_1 \comp \psi_2 = \psi_1 \comp (\icomp \phi_i) = \icomp \phi'_i$ where $\phi'_0 \eqdef \psi_1$ and $\phi'_{i+1} \eqdef \phi_i$ for all $i < \omega$.
\noindent
Item \ref{it:compl-pt:infC}:
we conclude just by observing that case~\ref{rule:ptinfC} implies that $\icomp \psi_i \in \layerpterm{\beta}$, where $\psi_i \in \layerpterm{\alpha_i}$ for all $i < \omega$ and $\beta \eqdef \sum_{i < \omega} \alpha_i$.
\noindent
Item \ref{it:compl-pt:symbol} and Item \ref{it:compl-pt:rule}:
it is enough to observe that case~\ref{rule:ptsymbol} applies.
\end{proof}
Now we prove uniqueness of formation, \wrt\ the layered definition, for any valid proof term.
\begin{lemma}
\label{rsl:pterm-layer-uniqueness}
Let $\psi \in \setpterm$. Then there exists a unique $\alpha$ such that $\psi \in \layerpterm{\alpha}$, and moreover there is exactly one case in Dfn.~\ref{dfn:layer-pterm} justifying $\psi \in \layerpterm{\alpha}$.
\end{lemma}
\begin{proof}
We will prove the following statement, which is equivalent to the desired result.
\begin{quote}
Let $\psi \in \setpterm$, $\alpha$ minimal for $\psi \in \layerpterm{\alpha}$, and $\beta$ such that $\psi \in \layerpterm{\beta}$.
Then $\beta = \alpha$, and there is exactly one case in Dfn.~\ref{dfn:layer-pterm} justifying $\psi \in \layerpterm{\alpha}$.
\end{quote}
We proceed by induction on $\alpha$, analysing which case in Dfn.~\ref{dfn:layer-pterm} could justify $\psi \in \layerpterm{\alpha}$.
\noindent
Case \ref{rule:ptmstep}.
In this case $\alpha = 1$ and $\psi$ is an \imstep. We conclude by Lem.~\ref{rsl:ptmstep-iff-one}.
\noindent
Case \ref{rule:ptinfC}.
In this case $\psi = \icomp \psi_i$ such that $\psi_i \in \layerpterm{\alpha_i}$ and $\alpha = \sum_{i < \omega} \alpha_i$. Observe that $\alpha > \alpha_i$ for all $i$, recall $\layerpterm{0} = \emptyset$.
Assume $\psi \in \layerpterm{\beta}$.
Lem.~\ref{rsl:ptinfC-iff-limit} implies that this assertion is generated by case \ref{rule:ptinfC}, implying that $\beta = \sum_{i < \omega} \beta_i$ and $\psi_i \in \layerpterm{\beta_i}$.
Let $i < \omega$ and $\gamma_i$ minimal for $\psi_i \in \layerpterm{\gamma_i}$.
Then $\gamma_i \leq \alpha_i < \alpha$, and therefore \ih\ can be applied twice on each $\psi_i$ obtaining $\beta_i = \alpha_i = \gamma_i$.
Thus we conclude.
\noindent
Case \ref{rule:ptbinC}.
In this case $\psi = \psi_1 \comp \psi_2$, $\alpha = \alpha_1 + \alpha_2 + 1$, $ \alpha_2$ is a successor ordinal, and $\psi_i \in \layerpterm{\alpha_i}$ for $i = 1,2$.
Then Lem.~\ref{rsl:ptinfC-iff-limit} applied to $\psi_2$ implies that it is not an infinite concatenation, thus neither is $\psi$.
On the other hand, observe that $\alpha$ is a successor ordinal verifying $\alpha > \alpha_i$ for $i = 1,2$.
Assume $\psi \in \layerpterm{\beta}$.
Then applying again Lem.~\ref{rsl:ptinfC-iff-limit} yields that this assertion is not justified by case 2 (since $\psi$ is not an infinite concatenation); therefore, the shape of $\psi$ (recall $\psi(\epsilon) = \comp$) leaves case 3 as the only valid option.
Hence $\beta = \beta_1 + \beta_2 + 1$ where $\psi_i \in \layerpterm{\beta_i}$ for $i = 1,2$.
An argument analogous to that used in the previous case, \ie\ resorting to the \ih\ on each $\psi_i$, yields $\beta_i = \alpha_i$. Thus we conclude.
\noindent
Case \ref{rule:ptsymbol}.
In this case $\psi = f(\psi_1, \ldots, \psi_m)$ and $\alpha = \alpha_1 + \ldots + \alpha_m + 1$, where $\psi_i \in \layerpterm{\alpha_i}$ for all $i$, and exists some $k$ veriyfing $\alpha_k > 1$.
Then Lem.~\ref{rsl:ptmstep-iff-one} implies that $\psi_k$ is not an \imstep, so that neither is $\psi$.
Therefore, the shape of $\psi$ (recall $\psi(\epsilon) \neq \comp$) leaves case 4 as the only valid option, implying that $\beta = \beta_1 + \ldots + \beta_m + 1$ where $\psi_i \in \layerpterm{\beta_i}$ for all $i$.
We conclude by obtaining $\beta_i = \alpha_i$ through an argument resorting to the \ih, like in the previous cases.
\end{proof}
\subsection{A simplified induction principle}
\label{sec:pterm-induction-principle}
The layered definition of \setpterm\ allows to perform inductive reasonings over proof terms, based in their concise notation. This makes an induction principle easy to work with. Formally:
\begin{proposition}[Simple induction principle for \setpterm]
\label{rsl:pterm-induction-principle}
Let $P$ an unary predicate satisfying all the following conditions: \\
\begin{tabular}{lp{125mm}}
1. & If $\psi$ is an infinitary multistep, then $P(\psi)$ holds. \\
2. & For all $\psi_1, \psi_2$ such that $\psi_1 \comp \psi_2 \in \setpterm$, $P(\psi_1)$ and $P(\psi_2)$ imply $P(\psi_1 \comp \psi_2)$. \\
3. & Given $\iomegaseq{\psi_i}$ such that $\icomp \psi_i \in \setpterm$, $P(\psi_i)$ for all $i$ imply $P(\icomp \psi_i)$. \\
4. & For all $\psi_1, \ldots, \psi_n \in \setpterm$ and for all $f \in \Sigma$, $P(\psi_1), \ldots, P(\psi_n)$ imply $P(f(\psi_1, \ldots, \psi_n))$. \\
5. & For all $\psi_1, \ldots, \psi_n \in \setpterm$ and for any rule symbol $\mu$, $P(\psi_1), \ldots, P(\psi_n)$ imply $P(\mu(\psi_1, \ldots, \psi_n))$. \\
\end{tabular}
\noindent
Then $P(\psi)$ holds for all $\psi \in \setpterm$.
\end{proposition}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \in \layerpterm{\alpha}$, referring to the conditions in the lemma statement.
\noindent
If $\alpha = 1$, then Lem.~\ref{rsl:ptmstep-iff-one} implies $\psi$ to be an \imstep, so that we conclude by condition 1.
\noindent
Assume that $\alpha$ is a successor ordinal.
If $\psi(\epsilon) = \comp$, then Lem~\ref{rsl:ptinfC-iff-limit} implies that $\psi = \psi_1 \comp \psi_2$, such that for $i = 1,2$, $\psi_i \in \layerpterm{\alpha_i}$ for some $\alpha_i$ satisfying $\alpha > \alpha_i$.
Then \ih\ can be applied on each $\psi_i$ yielding $P(\psi_1)$ and $P(\psi_2)$ to hold. We conclude by condition 2.
Otherwise, \ie\ if $\psi = f(\psi_1, \ldots, \psi_m)$ or $\psi = \mu(\psi_1, \ldots, \psi_m)$, then Lem.~\ref{rsl:ptmstep-iff-one} implies that $\psi$ is not an \imstep, therefore for all $i$, $\psi_i \in \layerpterm{\alpha_i}$ where $\alpha > \alpha_i$. Then \ih\ on each $i$ yield $P(\psi_i)$ to hold for all $i$. We conclude by condition 4.
\noindent
Assume that $\alpha$ is a limit ordinal.
In this case, Lem~\ref{rsl:ptinfC-iff-limit} implies that $\psi = \icomp \psi_i$, such that for all $i < \omega$, $\psi_i \in \layerpterm{\alpha_i}$ where $\alpha_i < \alpha$.
Then we can apply \ih\ on each $\psi_i$ obtaining that $P(\psi_i)$ holds for all $i < \omega$. We conclude by condition 3.
\end{proof}
We will resort to the induction principle given by Prop.~\ref{rsl:pterm-induction-principle} in forthcoming proofs, where we will indicate as \emph{induction hypotheses} the hypotheses of each case in the Proposition.
\Eg\ when proving a property for proof terms having the form $\psi_1 \comp \psi_2$, we will refer to the hypohteses of case 2 in Prop.~\ref{rsl:pterm-induction-principle}, namely that the property holds for $\psi_1$ and $\psi_2$, as induction hypothesis in the proof.
The intent is to produce intuitively simple yet rigorously valid proofs of properties on the set of proof terms.
\subsection{Basic properties of proof terms}
The following lemma shows that the target of a convergent proof term is always defined, and also a correspondence between $mind(\psi)$ and the existence of a fixed prefix for the activity denoted by $\psi$.
These two results are merged in the same lemma because they need to be proved simultaneously.
\begin{lemma}
\label{rsl:mind-big-then-tdist-little}
Let $\psi$ be a convergent proof.
Then
\begin{enumerate}[(a)]
\item \label{it:convergent-then-has-tgt}
$tgt(\psi)$ is defined.
\item \label{it:mind-big-then-tdist-little}
For all $n < \omega$, $mind(\psi) > n$ implies $\tdist{src(\psi)}{tgt(\psi)} < 2^{-n}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \in \layerpterm{\alpha}$, analysing the case in Dfn.~\ref{dfn:layer-pterm} corresponding to $\psi$.
If $\psi$ is an \imstep, then item~(\ref{it:convergent-then-has-tgt}) is immediate from Dfn.~\ref{dfn:imstep-convergence}, and for item~(\ref{it:mind-big-then-tdist-little}) an easy induction on $n$ suffices.
Assume $\psi = \psi_1 \comp \psi_2$.
Item~(\ref{it:convergent-then-has-tgt}) can be proved by just applying \ih\ on $\psi_2$.
To obtain item~(\ref{it:mind-big-then-tdist-little}), observe that \ih\ applies to $\psi_i$ for $i = 1,2$, since $mind(\psi_i) \geq mind(\psi) > n$, yielding $\tdist{src(\psi_i)}{tgt(\psi_i)} < 2^{-n}$.
Moreover Lemma~\ref{rsl:tdist-is-ultrametric} implies $\tdist{src(\psi)}{tgt(\psi)} \leq
max(\tdist{src(\psi)}{src(\psi_2)}, \tdist{src(\psi_2)}{tgt(\psi)}$. Thus we conclude by observing $src(\psi) = src(\psi_1)$, $src(\psi_2) = tgt(\psi_1)$, and $tgt(\psi) = tgt(\psi_2)$.
Assume $\psi = \icomp \psi_i$.
We prove item~(\ref{it:convergent-then-has-tgt}).
For any $i < \omega$, $\psi_i$ being convergent implies that \ih\ applies to obtain that $tgt(\psi_i)$ is defined.
Let $n < \omega$, and $k_n$ such that $\mind{\psi_i} > n$ if $k_n < i < \omega$.
Let $j$ such that $k_n < j$.
Then \ih:(\ref{it:mind-big-then-tdist-little}) applies on $\psi_{k_n+1} \comp \ldots \comp \psi_j$, implying $\tdist{tgt(\psi_{k_n+1})}{tgt(\psi_j)} < 2^{-n}$
\footnote{A possible shortcut from here is observing that the sequence $\langle tgt(\psi_i) \rangle_{i < \omega}$ is Cauchy-convergent, and therefore has a limit. We can refer to Thm.~12.2.1 in \cite{terese}, or its proof.}
.
Therefore, for any position $p$ and $j \geq k_{\posln{p}}+1$, $p \in \Pos{tgt(\psi_j)}$ iff $p \in \Pos{tgt(\psi_{k_{\posln{p}}+1})}$, and in such case, $tgt(\psi_j)(p) = tgt(\psi_{k_{\posln{p}}+1})(p)$.
We define $t = \langle P, F \rangle$ as follows:
$p \in P$ iff $p \in \Pos{tgt(\psi_{k_{\posln{p}}+1})}$, and
$F(p) \eqdef tgt(\psi_{k_{\posln{p}}+1})(p)$ for all $p \in P$.
To conclude this part of the proof, it is enough to verify that $tgt(\psi) = \lim_{i \to \omega} tgt(\psi_i) = t$.
\begin{itemize}
\item
We verify that $P$ is a tree domain, \confer\ Dfn.~\ref{dfn:tree-domain}.
Let $pq \in P$, then \linebreak $pq \in \Pos{tgt(\psi_{k_{\posln{pq}}+1})}$, implying that $p \in \Pos{tgt(\psi_{k_{\posln{pq}}+1 })}$. Then $p \in \Pos{tgt(\psi_{k_{\posln{p}+1}})}$, hence $p \in P$.
Let $pj \in P$ and $i$ such that $1 \leq i \leq j$. Observing $\posln{pj} = \posln{pi}$, a straightforward argument based on $\psi_{k_{\posln{pj}}+1}$ yields $pi \in P$.
\item
We verify that $t$ is a well-defined term, \confer\ Dfn.~\ref{dfn:term}.
Let $p \in P$, $f/m \eqdef F(p)$, and $i < \omega$.
Observe $f = \psi_{k_{\posln{p}}+1}(p) = \psi_{k_{\posln{p}+1}+1}(p)$.
Then $pi \in P$ iff $pi \in \Pos{\psi_{k_{\posln{pi}}+1}}$ iff $i \leq m$.
\item
We verify that $t = \lim_{i \to \omega} tgt(\psi_i)$.
Let $n < \omega$, $j > k_n$, and $p$ a position verifying $\posln{p} \leq n$, so that $k_{\posln{p}} \leq k_n$, implying in turn $k_{\posln{p}} + 1 \leq j$.
Then $p \in \Pos{t}$ iff $p \in \Pos{tgt(\psi_{k_{\posln{p}}+1})}$ iff $p \in \Pos{tgt(\psi_j)}$, and in such case, $t(p) = tgt(\psi_{k_{\posln{p}}+1})(p) = tgt(\psi_j)(p)$.
Hence $\tdist{tgt(\psi_j)}{t} < 2^{-n}$.
Consequently, $t = \lim_{i \to \omega} tgt(\psi_i)$.
\end{itemize}
We prove item~(\ref{it:mind-big-then-tdist-little}).
For all $i < \omega$, $mind(\psi_i) \geq mind(\psi) > n$, and then an easy induction on $i$ using an argument similar to the one just described for binary composition yields $\tdist{src(\psi)}{tgt(\psi_i)} < 2^{-n}$.
Recall that $tgt(\psi) = \lim_{i \to \omega} tgt(\psi_i)$, then there exists some $k$ such that $\tdist{tgt(\psi_j)}{tgt(\psi)} < 2^{-n}$ if $j > k$.
Then $\tdist{src(\psi)}{tgt(\psi_{k+1})} < 2^{-n}$ and $\tdist{tgt(\psi_{k+1})}{tgt(\psi)} < 2^{-n}$.
We conclude by Lemma~\ref{rsl:tdist-is-ultrametric}.
Assume $\psi = f(\psi_1, \ldots, \psi_m)$ and that it is not an \imstep.
Then $\psi$ being convergent implies that all $\psi_i$ are.
Therefore a straightforward argument based on \ih\ implies item~(\ref{it:convergent-then-has-tgt}) to hold.
Moreover, the way in which $src$, $tgt$ and $\mindfn$ for this case, implies that a natural inductive argument yields also item~(\ref{it:mind-big-then-tdist-little}).
Assume $\psi = \mu(\psi_1, \ldots, \psi_m)$, and that it is not an \imstep.
Then $\psi$ being convergent implies that $\psi_i$ is if $x_i$ occurs in the right-hand side of $\mu$, thus \ih:(\ref{it:convergent-then-has-tgt}) implies that $tgt(\psi_i)$ is defined for those $\psi_i$. Hence, definition of $tgt$ for this case yields item~(\ref{it:convergent-then-has-tgt}).
On the other hand, $\mind{\psi} = 0$ contradicting the hypotheses of item~(\ref{it:mind-big-then-tdist-little}).
Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:mind-ctx}
Let $C$ be a context in $\SigmaTerms$ having $k$ holes, and $\psi_1, \ldots, \psi_k$ proof terms.
Then $\mind{C[\psi_1, \ldots, \psi_k]} = min \set{\mind{\psi_i} + \posln{\BPos{C}{i}} \setsthat 1 \leq i \leq k}$.
\end{lemma}
\begin{proof}
An easy, although somewhat cumbersome, induction on $max \set{\posln{\BPos{C}{i}}}$ suffices.
If $C = \Box$, then both sides of the equation in the lemma conclusion equates to $\psi$, thus we conclude.
Assume $C = f(C_1, \ldots, C_m)$. \\
Observe that $C[\psi_1, \ldots, \psi_k] = f(C_1[\psi_{1_1}, \ldots, \psi_{1_{q1}}], \ldots, C_m[\psi_{m_1}, \ldots, \psi_{m_{qm}}])$, where $\set{\psi_{j_i}} = \set{\psi_1, \ldots, \psi_k}$.
Consequently, for any $i$ such that $1 \leq i \leq k$, $\BPos{C}{i} = e \, p$ for some $e$ verifying $1 \leq e \leq m$, and therefore $p = \BPos{C_e}{l}$ for some $l$. In turn, this implies $\posln{\BPos{C}{i}} = 1 + \posln{\BPos{C_e}{l}}$.
Conversely, for any $e$ such that $1 \leq e \leq m$, and for any $\BPos{C_e}{i}$, there is an index $j$ such that $\BPos{C}{j} = e \cdot \BPos{C_e}{i}$.
Furthermore, $\mind{C[\psi_1, \ldots, \psi_k]} = 1 + min \{\mind{C_j[\psi_{j_1}, \ldots, \psi_{j_{qj}}]} $ $\setsthat 1 \leq j \leq m\}$.
Let $j$ minimal for $\mind{\psi_j} + \posln{\BPos{C}{j}}$, so that showing $\mind{C[\psi_1, \ldots, \psi_k]} = \mind{\psi_j} + \posln{\BPos{C}{j}}$ is enough to conclude.
Let $e, i$ such that $\BPos{C}{j} = e \cdot \BPos{C_e}{i}$.
The existence of some $j', i'$ such $\BPos{C}{j'} = e \cdot \BPos{C_e}{i'}$ and $\mind{\psi_j'} + \posln{\BPos{C_e}{i'}} < \mind{\psi_j} + \posln{\BPos{C_e}{i}}$ would contradict minimality of $\psi_j$ \wrt\ $C$, so that $j, i$ are minimal for $\mind{\psi_j} + \posln{\BPos{C_e}{i}}$.
Therefore, applying \ih\ on $C_j$, yields that $\mind{C_e[\psi_{e_1}, \ldots, \psi_{e_{qe}}]} = \mind{\psi_j} + \posln{\BPos{C_e}{i}}$.
Assume for contractiction the existence of some $m,h$ such that \\
$\mind{C_h[\psi_{h_1}, \ldots, \psi_{h_{qh}}]} < \mind{C_e[\psi_{e_1}, \ldots, \psi_{h_{eh}}]}$.
Applying \ih\ on $C_h$ we obtain $\mind{C_h[\psi_{h_1}, \ldots, \psi_{h_{qh}}]} = \mind{\psi_g} + \posln{\BPos{C_h}{f}}$ for some $f$ and $g$ such that $\BPos{C}{g} =h \comp \BPos{C_h}{f}$.
But then our assumption would imply $\mind{\psi_g} + \posln{\BPos{C}{g}} = \mind{\psi_g} + \posln{\BPos{C_h}{f}} + 1 < \mind{\psi_j} + \posln{\BPos{C_e}{i}} + 1 = \mind{\psi_j} + \posln{\BPos{C}{j}}$, contradicting minimality of $j$ \wrt\ $C$.
Hence, $\mind{C[\psi_1, \ldots, \psi_k]} = 1 + \mind{C_e[\psi_{e_1}, \ldots, \psi_{e_{qe}}]} = \mind{\psi_j} + \posln{\BPos{C}{j}}$.
Thus we conclude.
\end{proof}
Some properties related with convergence follow.
\begin{lemma}
\label{rsl:imstep-convergent-args}
Let $\psi = f(\psi_1, \ldots, \psi_m)$ be a convergent \imstep, and $i$ such that $1 \leq i \leq m$.
Then $\psi_i$ is a convergent \imstep.
\end{lemma}
\begin{proof}
Dfn.~\ref{dfn:imstep} yields immediately that $\psi_i$ is an \imstep.
Moreover, $f(\psi_1, \ldots, \psi_m)$ being convergent means the existence of a convergent $\tgtt$-\redseq\ $\reda$ such that $f(\psi_1, \ldots, \psi_m) \infredxtrs{\reda}{\tgtt} t$ and $t$ is a $\tgtt$-normal form, \ie\ $t \in \iSigmaTerms$. Observe that $\mind{\reda} > 0$, since $f$ does not occur in any left-hand side of a rule in $\tgtt$.
Then Lem.~\ref{rsl:redseq-mind-big-src-tgt} implies $t = f(t_1, \ldots, t_m)$. In turn, Lem.~\ref{rsl:proj-redseq-well-defined} implies $\psi_i \infredxtrs{\proj{\reda}{i}}{\tgtt} t_i$. Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:fnsymbol-convergence}
Let $\psi = f(\psi_1, \ldots, \psi_m)$ be a convergent proof term. Then $\psi$ is convergent iff $\psi_i$ is convergent for all suitable $i$.
\end{lemma}
\begin{proof}
If $\psi$ is an \imstep, then the $\Rightarrow )$ direction is an immediate corollary of Lem.~\ref{rsl:imstep-convergent-args}.
For the $\Leftarrow )$ direction, recall that for any $i$, $\psi_i$ being convergent means the existence of a $\tgtt$-\redseq\ $\reda_i$ verifying $\psi_i \infredxtrs{\reda_i}{\tgtt} t_i$ where $t_i \in \iSigmaTerms$.
Then $f(\psi_1, \ldots, \psi_m) \infredxtrs{\reda}{\tgtt} f(t_1, \ldots, t_m)$, where $\reda \eqdef (1 \cdot \reda_1 ) ; \ldots ; (m \cdot \reda_m)$, and $i \cdot \reda_i$ is defined as follows: $\redln{i \cdot \reda_i} \eqdef \redln{\reda_i}$ and $\redel{i \cdot \reda_i}{\alpha} \eqdef \langle f(t_1, \ldots, \phi, \ldots \psi_m), ip, \mu \rangle$ where $\redel{\reda_i}{\alpha} = \langle \phi, p, \mu \rangle$. A simple transfinite induction yields $f(t_1, \ldots, t_{i-1}, \psi_i, \psi_{i+1}, \ldots, \psi_m) \infredxtrs{i \cdot \reda_i}{\tgtt} f(t_1, \ldots, t_{i-1}, t_i, \psi_{i+1}, \ldots, \psi_m)$.
If $\psi$ is not an \imstep, then the result is an immediate consequence of Dfn.~\ref{dfn:layer-pterm}, case~(\ref{rule:ptsymbol}).
Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:ctx-convergence}
Let $C$ be a context in $\iSigmaTerms$ having exactly $m$ holes, and $\psi_1, \ldots, \psi_m$ proof terms.
Then $C[\psi_1, \ldots, \psi_m]$ is convergent iff $\psi_i$ is convergent for all suitable $i$.
\end{lemma}
\begin{proof}
A straightforward induction on $max \set{\posln{\BPos{C}{i}} \setsthat 1 \leq i \leq m}$, resorting to Lem.~\ref{rsl:fnsymbol-convergence} in the inductive case, suffices to conclude.
\end{proof}
\begin{lemma}
\label{rsl:rulesymbol-convergence}
Let $\mu : l[x_1, \ldots, x_m] \to h[x_1, \ldots, x_m]$ be a rule included in a certain \TRS; and $\psi_1, \ldots, \psi_m$ proof terms.
Then $\psi = \mu(\psi_1, \ldots, \psi_m)$ is convergent iff $\psi_i$ is convergent for all $i$ such that $x_i$ occurs in $h[x_1, \ldots, x_m]$.
\end{lemma}
\begin{proof}
Assume that $\psi$ is an \imstep.
We verify $\Rightarrow )$.
Convergence of $\psi$ implies $\psi \infredxtrs{\reda}{\tgtt} t$ for some \redseq\ $\reda$, where $t \in \iSigmaTerms$.
Notice that $\mind{\reda} > 0$ would imply $t(\epsilon) = \mu$ (\confer\ Lem.~\ref{rsl:redseq-mind-big-src-tgt}), contradicting $t \in \iSigmaTerms$.
Therefore $\mind{\reda} = 0$, implying $\reda = \reda_1; \langle \chi, \epsilon, \uln{\nu} \rangle, \reda_2$ where $\mind{\reda_1} > 0$.
In turn, $\mind{\reda_1} > 0$ implies that $tgt(\reda_1) = \chi = \mu(\chi_1, \ldots, \chi_m)$ where $\psi_i \infredxtrs{\proj{\reda}{i}}{\tgtt} \chi_i$, \confer\ Lem.~\ref{rsl:redseq-mind-big-src-tgt} and Lem~\ref{rsl:proj-redseq-well-defined}.
Hence $\uln{\nu} = \uln{\mu} : \mu(x_1, \ldots, x_m) \to h[x_1, \ldots, x_m]$, implying $src(\reda_2) = h[\chi_1, \ldots, \chi_m]$.
Observe that $\chi_i$ occurs in $src(\reda_2)$ iff $x_i$ occurs in $h$.
We analyse two cases: \\[5pt]
\begin{tabular}{@{$\ \ \bullet\ \ $}p{.9\textwidth}}
$h[x_1, \ldots, x_m] = x_j$, so that $src(\reda_2) = \chi_j$.
In this case $\psi_j \infredx{\proj{\reda}{j}} \chi_j \infredx{\reda_2} t$. We conclude by observing that only convergence of $\psi_j$ is required in this case. \\
$h \notin \thevar$.
In this case $h[\chi_1, \ldots, \chi_m] \infredx{\reda_2} t$.
Observe that all the steps in $\reda_2$ lies ``below'' (an argument of) $h$. Then Lem.~\ref{rsl:redseq-respects-src-tgt} implies $t = h[t_1, \ldots, t_m]$ and, moreover, that a \redseq\ $\reda'_i$ exists which verifies $\chi_i \infredx{\reda'_i} t_i$ for all $i$ such that $x_i$ occurrs in $h[x_1, \ldots x_m]$.
Therefore, for any of those indices, say $i$, $\psi_i \infredx{\proj{\reda_1}{i}} \chi_i \infredx{\reda'_i} t_i$.
Thus we conclude.
\end{tabular}
To verify the $\Leftarrow )$ direction, observe that all the $\psi_i$ corresponding to variables occurring in $h$ being convergent implies
$\psi \to h[\psi_1, \ldots \psi_m] \infredx{\reda_1} h[t_1, \ldots, \psi_m] \ldots \infredx{\reda_m} h[t_1, \ldots, t_m]$, where eventually some $\reda_i$ are performed more than once, if the corresponding $x_i$ occurs more than once in $h[x_1, \ldots, x_m]$.
Hence $\psi$ is $\tgtt$-$WN^\infty$, \ie\ it is a convergent \imstep.
Finally, if $\psi$ is not an \imstep, then Dfn.~\ref{dfn:layer-pterm}, case~(\ref{rule:ptsymbol}), allows to conclude immediately.
\end{proof}
\subsection{Trivial proof terms}
This section deals with the proof terms denoting no activity, which will be termed \emph{trivial proof terms}.
The structure of trivial proof terms can be arbitrarily complex, \ie\ $\compomega{j}(\icomp a)$ is a trivial proof term.
We prove that some expected properties hold for these proof terms. These properties will be used later in this work.
\begin{definition}
\label{dfn:trivial-pterms}
Let $\psi$ be a proof term. We will say that $\psi$ is a \emph{trivial proof term} iff it does not include any rule symbol occurrences.
\end{definition}
\begin{lemma}
\label{rsl:trivial-pterm-mind-omega}
Let $\psi$ be a proof term. Then $\psi$ is trivial iff $\mind{\psi} = \omega$.
\end{lemma}
\begin{proof}
For the $\Rightarrow )$ direction, a straightforward induction on $\psi$ (\ie\ on $\alpha$ such that $\psi \in \layerpterm{\alpha}$) suffices.
For the base case, \ie\ when $\psi$ is an \imstep, we just refer to Dfn.~\ref{dfn:dmin-imstep}.
For the $\Leftarrow )$ direction, a similar induction on $\psi$ yields the counterpositive, \ie\ that if $\psi$ includes at least one rule symbol occurrence, then $\mind{\psi} < \omega$.
If $\psi$ is an \imstep, then we define $n$ to be the least depth of a rule symbol occurrence in $\psi$. An easy induction on $n$ yields $\mind{\psi} = n$. If $\psi = \mu(\psi_1, \ldots, \psi_m)$, then $\mind{\psi} = 0$. For the other cases, \ih\ suffices to conclude.
\end{proof}
\section{\Peqence}
\label{sec:peqence}
\usingRedseqOnly{
Two proof terms denoting reduction processes which are essentially the same, or more generally consist of the same steps performed in different order, should be recognised as being \emph{\peqent}.}
\usingContractionActivity{
Two proof terms can be the result of arranging the same contraction activity in different ways, regarding parallelism/nesting degree, sequential order, and/or localisation of contractions.
Such proof terms should be recognised as being \emph{\peqent}.}
In this section we give a criterion to decide equivalence between proof terms.
The approach is to extend the \emph{\peqence} criterion, as it is defined in \cite{terese} Sec. 8.3, to the infinitary setting.
\Peqence, for which the notation \peq\ will be used henceforth in this document, is defined there for finitary proof terms as the congruence generated by the following equation schemes
\[
\begin{array}{lrcl}
\peqidleft & 1 \comp \psi & \eqnpeq & \psi \\
\peqidright & \psi \comp 1 & \eqnpeq & \psi \\
\peqassoc & \psi \comp (\phi \comp \chi)
& \eqnpeq &
(\psi \comp \phi) \comp \chi \\
\peqstruct & f(\psi_1, \ldots, \psi_m) \comp f(\phi_1, \ldots, \phi_m)
& \eqnpeq &
f(\psi_1 \comp \phi_1, \ldots, \psi_m \comp \phi_m) \\
\peqoutin & \mu(\psi_1, \ldots, \psi_m)
& \eqnpeq &
\mu(s_1, \ldots, s_m) \comp r[\psi_1, \ldots, \psi_m] \\
\peqinout & \mu(\psi_1, \ldots, \psi_m)
& \eqnpeq &
l[\psi_1, \ldots, \psi_m] \comp \mu(t_1, \ldots, t_m)
\end{array}
\]
where $\mu: l \to r$, $s_i = src(\psi_i)$ and $t_i = tgt(\psi_i)$.
\usingStrEqIdEq{
Additionally, two subrelations of \peq\ are remarked in \cite{terese}, namely the congruence generated by the first three and first four equations.
These relations are known as \emph{identity equivalence} and \emph{structural equivalence} respectively.
In this document, the notation $\ideq$ and $\streq$ resp. will be used for these relations
\footnote{The notations used for the three introduced relations differs from those used in \cite{terese}, which I find somewhat confusing.}.
}
Some challenges must be addressed in order to extend the \peqence\ definition to the infinitary setting.
Consider \eg\ the rules $\mu: f(x) \to g(x)$, $\nu: g(x) \to h(x)$ and $\rho: j(x) \to k(x)$, and the \redseqs\ \\[2pt]
\minicenter{
$j(f\om) \infred j(g\om) \sstep k(g\om)$ \qquad\qquad
$j(f\om) \to k(f\om) \infred k(g\om)$} \\[2pt]
which can be denoted by the proof terms \\[2pt]
\minicenter{
$\icomp j(g^i(\mu(f\om))) \comp \rho(g\om)$ \qquad\qquad
$\rho(f\om) \comp \icomp k(g^i(\mu(f\om)))$} \\[2pt]
respectively.
\usingRedseqOnly{
These reduction sequences are \peqent.
Both consist of a $\rho$ head step and an infinite number of $\mu$ steps forming a convergent reduction sequence of length $\omega$.
}
\usingContractionActivity{
These proof terms denote the same contraction activity, namely a $\rho$ step transforming the head $j$ into a $k$, and an infinite number of $\mu$ steps transforming each occurrence of $f$ into one of $g$.
Therefore, they should be stated as \peqent.
Observe that both proof terms are sequential, denoting precisely each of the described \redseqs.
}
The difference lies in the order in which the two operations are performed: first the $\mu$ steps and then the $\rho$ step in the sequence to the left, and viceversa in the sequence of the right. The difference is apparent in the proof terms who describe the sequences.
Both considered \redseqs\ are convergent.
In order to equate
\usingRedseqOnly{these \redseqs}
\usingContractionActivity{the given \redseqs}
, an \emph{infinite} number of step permutations must be performed: the $\rho$ step must be permuted in turn with each of the infinite $\mu$ steps.
It is even impossible to determine which should be the \emph{first} $\mu$ step to be permuted with the $\rho$ step in order to transform the sequence to the left into that to the right.
If we proceed the other way around, we can by finite means permute the initial $\rho$ step with a finite prefix of the infinite $\mu$ reduction, obtaining
$j(f\om) \to j(g(f\om)) \to \ldots \to j(g^n(f\om)) \to k(g^n(f\om)) \infred k(g\om)$,
but there will always be an infinite $\mu$ sequence ``still to be permuted'' with the $\rho$ step.
This situation is reflected in the
\usingRedseqOnly{proof terms}
\usingContractionActivity{sequential proof terms}
. There is no way to extract a ``last'' component in the infinite composition $\icomp j(g^i(\mu(f\om)))$, in order to permute it with $\rho(g\om)$. On the other hand, by applying the congruence on \peqence\ equations to infinitary terms, we can permute the leading $\rho(f\om)$ with a finite number of component of the following infinite composition in $\rho(f\om) \comp \icomp k(g^i(\mu(f\om)))$, \ie \\
$
\begin{array}{@{\hspace*{1cm}}cl}
\multicolumn{2}{l}{\rho(f\om) \comp \icomp k(g^i(\mu(f\om)))} \\
\peq &
\rho(f\om) \comp k(\mu(f\om)) \comp \icomp k(g^{i+1}(\mu(f\om))) \\
\peq &
j(\mu(f\om)) \comp \rho(g(f\om)) \comp \icomp k(g^{i+1}(\mu(f\om))) \\
\peq &
j(\mu(f\om)) \comp \rho(g(f\om)) \comp k(g(\mu(f\om))) \comp \icomp k(g^{i+2}(\mu(f\om))) \\
\peq &
j(\mu(f\om)) \comp j(g(\mu(f\om))) \comp \rho(g^2(f\om)) \comp \icomp k(g^{i+2}(\mu(f\om))) \\
\peq & \ldots \\
\peq &
j(\mu(f\om)) \comp \ldots \comp \rho(g^n(f\om)) \comp \icomp k(g^{i+n}(\mu(f\om))) \\
\end{array}$ \\
therefore having still an infinite composition to the right of the $\rho$ step.
An adequate characterisation of \peqence\ for the infinitary setting should sanction the equivalence of these sequential proof terms.
\usingContractionActivity{
Moreover, notice that all the redexes contracted in (the activity included in) either considered \redseq\ are present in the source term $j(f\om)$, so that the same activity can be denoted also by an \imstep\ (\ie\ a fully nested proof term), which is $\nu(\mu\om)$. Combinations of sequential and nested descriptions are possible as well, \eg\ $\rho(\icomp g^i(\mu(f\om)))$ and $\rho(f\om) \comp k(\mu\om)$. A sound \peqence\ characterisation should allow to state the equivalence of either of these proof terms \wrt\ any of the sequential versions introduced before.
To conclude the \peqence\ of either sequential proof term and (say) the multistep counterpart, an infinite number of step (de)nesting, using the \peqoutin\ or \peqinout\ equations, should be performed.
Using congruence on equations, a finite (though arbitrary) number of (de)nestings can be performed.
\Eg\ the equivalence between
$\rho(f\om) \comp \icomp k(g^i(\mu(f\om)))$ and
$\rho(\mu^3(f\om)) \comp \icomp k(g^{i+3}(\mu(f\om)))$ can be proved by nesting the three outer $\mu$ steps inside the $\rho$-step, as follows: \\
$
\begin{array}{@{\hspace*{1cm}}cl}
\multicolumn{2}{l}{\rho(f\om) \comp \icomp k(g^i(\mu(f\om)))} \\
\peq &
\rho(f\om) \comp k(\mu(f\om)) \comp k(g(\mu(f\om))) \comp k(g(g(\mu(f\om)))) \comp \icomp k(g^{i+3}(\mu(f\om))) \\
\peq &
\rho(f\om) \comp k(\mu(f\om)) \comp k(g(\mu(f\om) \comp g(\mu(f\om)))) \comp \icomp k(g^{i+3}(\mu(f\om))) \\
\peq &
\rho(f\om) \comp k(\mu(f\om)) \comp k(g(\mu(\mu(f\om)))) \comp \icomp k(g^{i+3}(\mu(f\om))) \\
\peq &
\rho(f\om) \comp k(\mu(f\om) \comp g(\mu(\mu(f\om)))) \comp \icomp k(g^{i+3}(\mu(f\om))) \\
\peq &
\rho(f\om) \comp k(\mu^3(f\om)) \comp \icomp k(g^{i+3}(\mu(f\om))) \\
\peq &
\rho(\mu^3(f\om)) \comp \icomp k(g^{i+3}(\mu(f\om))) \\
\end{array}$ \\
We describe briefly this schematic description of the \peqence\ derivation.
Firstly, \peqassoc\ is used to separate the first components of the infinite composition; notice that in this abrigded description, other uses of \peqassoc\ are left implicit.
Then \peqstruct\ is used twice (albeit described as one ``step'' in this description) from $k(g(\mu(f\om))) \comp k(g(g(\mu(f\om))))$, \wrt\ the symbols $k$ and $g$ respectively, thus obtaining $k(g(\mu(f\om) \comp g(\mu(f\om))))$.
This allows to subsequently apply \peqoutin\ on $\mu(f\om) \comp g(\mu(f\om)$, yielding $\mu(\mu(f\om))$.
The fourth and fifth lines describe a similar process, applied in order to obtain a concise description of the first three $\mu$ steps. This description is furthermore condensed with the leading $\rho(f\om)$ step, by applying \peqoutin\ once more.
An analogous process can be performed with any finite number of $\mu$ steps, yielding $\rho(f\om) \comp \icomp k(g^i(\mu(f\om))) \peq \rho(\mu^n(f\om)) \comp \icomp k(g^{i+n}(\mu(f\om)))$.
In any case, there will always remain an infinite quantity of $\mu$ steps separated from the nested part.
}
Let us analyse an additional example using the same rules. Consider the \redseqs \\[2pt]
\minicenter{
$f\om \infred g\om \infred h\om$ \quad and \quad
$f\om \sstep g(f\om) \sstep h(f\om) \sstep h(g(f\om)) \sstep h^2(f\om) \infred h\om$} \\[2pt]
which can be denoted by the sequential proof terms \\[2pt]
\minicenter{
$\icomp g^i(\mu(f\om)) \comp \icomp h^i(\nu(g\om))$ \qquad and \qquad
$\icomp (h^i(\mu(f\om)) \comp h^i(\nu(f\om)))$} \\[2pt]
respectively.
Again, the reduction sequences are equivalent: they consist of an infinite number of $\mu$ steps and an infinite number of $\nu$ steps. In the left-hand sequence, first all the $\mu$ steps are performed, followed by the $\nu$ steps. In the right-hand sequence, $\mu$ and $\nu$ steps are interleaved.
Therefore, the proof terms describing these reductions should be sanction as \peqent.
We remark that in this case, \emph{each of the infinite number} of $\nu$ steps must be permuted with an infinite number of $\mu$ steps.
We will see that this added complexity of the needed permutations on \redseqs\ is reflected in the \peqent\ characterisation for proof terms, by means of an additional device needed to cope with this case.
\usingContractionActivity{
The contraction activity included in either \redseq\ can also be described by non-sequential proof terms, remarkably $\mu\om \comp \nu\om$, but also \eg\
$\mu(f\om) \comp \icomp k^i(\nu(\mu(f\om)))$.
In this case, as the contraction of \emph{created} redexes is involved (since each $\nu$ step is created by the corresponding $\mu$ step), there is no way of describing this contraction activity by an \imstep.
}
\includeStandardisation{
Notice that in both examples given so far, the \redseq\ shown to the right is standard, so that the permutation which equate the \redseq\ to the left to that to the right corresponds to a \emph{standardisation} of the former.
}
\usingRedseqOnly{
Proof terms denoting sequential vs. nested version of the same reduction must be considered as well.
\Eg, the characterisation of \peqence\ for the finitary case allows to obtain
$\mu(\mu(a)) \peq \mu(f(a)) \comp g(\mu(a))$.
In the infinitary realm, an infinite number of steps could be nested in an infinite multistep.
The corresponding characterisation of \peqence\ should take such cases into account, allowing to obtain \eg\
$\mu\om \peq \icomp g^i(\mu(f\om))$.
}
We remark that even when the characterisation of \peqence\ to be introduced can be applied to any well-formed proof term, the study of infinitary rewriting based on this characterisation we develop afterwards, mostly applies only to \emph{convergent} proof terms. Therefore, most of the additional definitions and results to come assume that the proof terms under consideration are convergent.
A study of \peqence\ considering also \emph{divergent} proof terms is left as future work.
\subsection{The formal infinitary \peqence\ relation}
In the following, we formally state the \peqence\ criterion we propose for infinitary proof terms.
As we have indicated in the introduction to this Section, the definition will be based on equational logic, so that a set of basic equations and another of equational rules will be introduced.
The basic equations model the basic operations needed to perform a permutation of steps using the description of contraction activity given by proof terms, while the rules model the equivalence closure and the closure by the operations corresponding to the symbols in the signature of proof terms.
The need to reason about (proof terms including) infinite concatenations implies the inclusion of one equation schema and one rule which specifically account for their infinite nature.
Therefore, the relations which formalise the notion of \peqence\ use an explicit form of \emph{infinitary equational logic}.
In order to obtain a formal \peqence\ relation that is intuitively adequate, \ie\ which models adequately the concept of \peqence\ behaving as expected in a variety of examples, a very special rule must be added to the rules corresponding to equivalence and operations closure.
This rule allows to incorporate the idea of \emph{limit} into infinitary equational logic judgements.
In turn, to obtain an intuitively reasonable ``limit rule'', some particular requirements must be put in its premises, to limit the way in which this rule can be applied in a judgement. These requirements force to define a separate, previous ``base'' relation, which is used to define the ``limit rule'' for the \peqence\ relation. We will use $\peqe$ to denote the ``base'' relation, and $\peq$ for \peqence.
In the rest of this work, we will need to reason about the base \peqence\ relation. As we want to be able to proceed by some sort of transfinite induction on the complexity of the \peqence\ judgement, we will give a \emph{layered} definition of \peqence, like we did for the definition of proof terms in Sec.~\ref{sec:pterm}. Therefore, we will define, for each countable ordinal $\alpha$, the relations $\layerpeqe{\alpha}$ and $\layerpeq{\alpha}$.
Induction on \peqent\ terms can be performed by induction on the (say, minimal) layer to which the pair of terms belongs. The same holds for terms related by the ``base'' \peqence\ relation.
Formal definitions of the $\peqe$ and $\peq$ relations follow:
\begin{definition}[Layer of base \peqence]
\label{dfn:layer-peqe}
Let $\alpha$ be a countable ordinal.
We define the $\alpha$-th \emph{level of base \peqence}, notation $\layerpeqe{\alpha}$, as follows: given $\psi$ and $\phi$ proof terms, $\psi \layerpeqe{\alpha} \phi$ iff the equation $\psi \layerpeqx{\alpha} \phi$ can be obtained by means of the equational logic system whose basic equations are the instances of the following schemata for which both lhs and rhs are proof terms
\footnote{hence they are particularly \emph{closed} terms, \confer\ Dfn.~\ref{dfn:imstep} and Dfn.~\ref{dfn:layer-pterm}.}
\[
\begin{array}{lrcl}
\peqidleft & 1 \comp \psi & \eqnpeq & \psi \\
\peqidright & \psi \comp 1 & \eqnpeq & \psi \\
\peqassoc & \psi \comp (\phi \comp \chi)
& \eqnpeq &
(\psi \comp \phi) \comp \chi \\
\peqstruct & f(\psi_1, \ldots, \psi_m) \comp f(\phi_1, \ldots, \phi_m)
& \eqnpeq &
f(\psi_1 \comp \phi_1, \ldots, \psi_m \comp \phi_m) \\
\peqinfstruct &
\icomp f(\psi^1_i, \ldots, \psi^m_i)
& \eqnpeq &
f(\icomp \psi^1_i, \ldots, \icomp \psi^m_i) \\
\peqoutin & \mu(\psi_1, \ldots, \psi_m)
& \eqnpeq &
\mu(s_1, \ldots, s_m) \comp r[\psi_1, \ldots, \psi_m] \\
\peqinout & \mu(\psi_1, \ldots, \psi_m)
& \eqnpeq &
l[\psi_1, \ldots, \psi_m] \comp \mu(t_1, \ldots, t_m)
\end{array}
\]
verifying also the following conditions: $1 = src(\psi)$ for $\peqidleft$;
$\psi$ convergent and $1 = tgt(\psi)$ for $\peqidright$;
$\mu: l \to r$ for both \peqoutin\ and \peqinout;
$s_i = src(\psi_i)$ for \peqoutin;
$\psi_i$ convergent and $t_i = tgt(\psi_i)$ for all $i$ for \peqinout.
Equational logic rules are defined by transfinite recursion on $\alpha$ as follows
\footnote{
An alternative could be to consider \emph{open} instances of the equations, \ie\ one instance of \peqstruct\ and \peqinfstruct\ for each object function symbol plus one instance of \peqinout\ and \peqoutin\ for each rule symbol, where all the $\psi_i$, $\phi_i$, $\chi_i$, $s_i$ and $t_i$ would be considered as variables.
In order to equate instances of the such generated equations, a \emph{substitution} rule should be added at the equational logic level.
In this way, considering the rules $\nu(x) : g(x) \to h(x)$, $\rho(x) : j(x) \to k(x)$ and $\pi : a \to b$, the equivalence $\nu(\rho(\pi)) \peqe \nu(j(a)) \comp h(\rho(\pi))$ would be justified by a two-step reasoning: a step using \eqleqn\ to obtain $\nu(\psi) \peqe \nu(s) \comp k(\psi)$ by the $\nu$ instance of the \peqoutin\ equtation, followed by the replacement of the $\psi$ and $s$ variables by the proof term $\rho(\pi)$ and its source, namely $j(a)$, by resorting to the substitution rule.
Unfortunately, this would be a rather inadequate approach because of the characteristics of proof terms in general, and of some of the equations in particular.
On one hand, an eventual extension of the set of proof terms in order to encompass open terms \emph{would not be closed by substitutions}.
A simple example considering the rule $\pi : a \to b$ follows: while $x \comp x$ would be a legal proof term, $\pi \comp \pi$ is not.
I guess this fact lies behind the difficulties for handling concatenation in the proposal of proof terms for HRS described in \cite{bruggink2008}; \confer\ particularly page 33.
On the other hand, not any instance of the equations correspond to their intent.
Firstly, the equation instance should correspond to valid proof terms at both lhs and rhs. Additionally, for the \peqinout equation, the $t_i$s are intended to be precisely $tgt(\psi_i)$, and not an arbitrary proof term verifying $tgt(\psi_i) = src(t_i)$. A similar condition holds for \peqoutin.
Observe that all these restrictions are considered when defining the set of legal instances of equations which can be used when applying the \eqleqn\ rule.
}
\\[5pt]
$\begin{array}{c}
\begin{array}{c} \\ \hline \psi \layerpeqx{1} \psi \end{array}
\ \ \eqlrefl
\qquad
\begin{array}{c} \psi \eqnpeq \phi \textnormal{ is a basic equation}\\ \hline \psi \layerpeqx{1} \phi \end{array}
\ \ \eqleqn
\\ \\
\begin{array}{c}
\psi \layerpeqx{\alpha_1} \phi \\
\hline
\phi \layerpeqxb{\alpha_1 + 1} \psi
\end{array}
\ \ \eqlsymm
\qquad
\begin{array}{c}
\psi \layerpeqx{\alpha_1} \phi \quad \phi \layerpeqx{\alpha_2} \xi \\
\hline
\psi \layerpeqxb{\alpha_1 + \alpha_2 + 1} \xi
\end{array}
\ \ \eqltrans
\\ \\
\begin{array}{c}
\psi_1 \layerpeqx{\alpha_1} \phi_1 \quad \ldots \quad \psi_n \layerpeqx{\alpha_n} \phi_n \quad
f/n \in \Sigma \\
\hline
f(\psi_1, \ldots, \psi_n) \layerpeqxb{\alpha_1 + \ldots + \alpha_n + 1} f(\phi_1, \ldots, \phi_n)
\end{array}
\ \ \eqlfun
\\ \\
\begin{array}{c}
\psi_1 \layerpeqx{\alpha_1} \phi_1 \quad \ldots \quad \psi_n \layerpeqx{\alpha_n} \phi_n \quad
\mu/n \textrm{ is a rule symbol} \\
\hline
\mu(\psi_1, \ldots, \psi_n) \layerpeqxb{\alpha_1 + \ldots + \alpha_n + 1} \mu(\phi_1, \ldots, \phi_n)
\end{array}
\ \ \eqlrule
\\ \\
\begin{array}{c}
\psi_1 \layerpeqx{\alpha_1} \phi_1 \quad \psi_2 \layerpeqx{\alpha_2} \phi_2 \\
\hline
\psi_1 \comp \psi_2 \layerpeqxb{\alpha_1 + \alpha_2 + 1} \phi_1 \comp \phi_2
\end{array}
\ \ \eqlcomp
\qquad
\begin{array}{c}
\psi_i \layerpeqx{\alpha_i} \phi_i \quad \textforall i < \omega \\
\hline
\icomp \psi_i \ \layerpeqxb{\Sigma_{i < \omega} \alpha_i}\ \icomp \phi_i
\end{array}
\ \ \eqlinfcomp
\end{array}$
\end{definition}
\begin{definition}[Base \peqence]
\label{dfn:peqe}
Let $\psi$, $\phi$ be proof terms.
We say that $\psi$ and $\phi$ are \emph{base-\peqent}, notation $\psi \peqe \phi$, iff $\psi \layerpeqe{\alpha} \phi$ for some $\alpha < \omega_1$.
\end{definition}
\includeStandardisation{
The subrelation of $\peqe$ whose definition follows will be used in some proofs. We say that $\psi \peqefs \phi$ (where $EFS$ stands for ``equational, finitary and structural'') iff $\psi \peqe \phi$ can be obtained by using only the equations \peqidleft, \peqidright, \peqassoc\ and \peqstruct, and not resorting to the \eqlinfcomp\ equational rule.
}
\begin{definition}[Layer of \peqence]
\label{dfn:layer-peq}
Let $\alpha$ be a countable ordinal.
We define the $\alpha$-th \emph{level of \peqence}, notation $\layerpeq{\alpha}$, as follows: given $\psi$ and $\phi$ proof terms, $\psi \layerpeq{\alpha} \phi$ iff the equation $\psi \layerpeqx{\alpha} \phi$ can be obtained by means of the equational logic system whose basic equations are those described in Dfn.~\ref{dfn:layer-peqe}, and the set of equational logic rules is the result of adding the rule \eqllim\ defined as follows
\\[5pt]
$\begin{array}{c}
\begin{array}{c}
\begin{array}{ll}
\left.
\begin{array}{l}
\psi \layerpeqe{\alpha_k} \chi_k \comp \psi'_k \ \ \mind{\psi'_k} > k
\\
\phi \layerpeqe{\beta_k} \chi_k \comp \phi'_k \ \ \mind{\phi'_k} > k
\end{array}
\right\}
&
\textforall k < \omega
\end{array}
\\
\hline
\psi \ \layerpeqx{\alpha}\ \phi
\qquad
\textnormal{where } \alpha = \sum_{i < \omega} \alpha_i + \sum_{i < \omega} \beta_i
\end{array}
\ \ \eqllim
\end{array}$
\\[5pt]
to the rules introduced in Dfn.~\ref{dfn:layer-peqe}.
\end{definition}
Notice that the explicit reference to the relations $\layerpeqe{\alpha_k}$ and $\layerpeqe{\beta_k}$ prevents the ``stacking'' of uses of the rule \eqllim\ in a \peqence\ judgement, \ie, that judgements leading to the premises of an application of the \eqllim\ rule cannot include other applications of the same rule.
This condition does not imply that a valid \peqence\ judgement can include at most one occurrence of \eqllim. \Eg\ a \peqence\ derivation having the following shape
$
\prooftree
\[
\ldots
\ \
\begin{array}{l}
\psi_1 \peqe \xi_k \comp \psi'_1 \\
\phi_1 \peqe \xi_k \comp \phi'_1
\end{array}
\ \
\ldots
\justifies
\psi_1 \peq \phi_1
\using
\eqllim
\]
\quad
\[
\begin{array}{l}
\psi_2 \peqe \chi_k \comp \psi'_2 \\
\phi_2 \peqe \chi_k \comp \phi'_2
\end{array}
\ \
\ldots
\justifies
\psi_2 \peq \phi_2
\using
\eqllim
\]
\justifies
\psi_1 \comp \psi_2 \peq \phi_1 \comp \phi_2
\using
\eqlcomp
\endprooftree
$
\noindent
is valid according to Dfn.~\ref{dfn:layer-peq}.
\begin{definition}[\Peqence]
\label{dfn:peq}
Let $\psi$, $\phi$ be proof terms.
We say that $\psi$ and $\phi$ are \emph{\peqent}, notation $\psi \peq \phi$, iff $\psi \layerpeq{\alpha} \phi$ for some $\alpha < \omega_1$.
\end{definition}
Observe that for any countable ordinal $\alpha$, $\layerpeqe{\alpha} \,\subseteq\, \layerpeq{\alpha}$, and therefore $\peqe \,\subseteq\, \peq$.
As discussed prior to the formal definitions, this characterisation of \peqence\ for infinitary proof terms adds, to the rules corresponding to the closure of the description of step permutation, a rule which allows to resort to the concept of \emph{limit} inside judgements. We found this necessary to obtain a complete characterisation, \ie, one which covers all the examples we have studied.
If the difference between the activity denoted by two proof terms can be proven to tend to zero, then we can resort to limits to assert that such difference is equal to zero, and therefore, that the proof terms must be considered equivalent.
The measure used to compute the difference between two proof terms \wrt\ their denoted activity is the \emph{minimal activity depth}.
The equational logic used to reason about infinitary derivations adds three features to its finitary counterpart, besides operating on infinitary proof terms instead of just finite ones.
These additions are: the \peqinfstruct\ \emph{equation schema}, and the \eqlinfcomp\ and \eqllim\ \emph{equational rules}.
The first addition is the generalisation of \peqstruct\ to the infinite composition.
It allows \eg\ the following \peqence\ reasoning \\
$\icomp j(g^i(\mu(f\om))) \comp \rho(g\om)
\peq
j(\icomp g^i(\mu(f\om))) \comp \rho(g\om)
\peq
\rho(\icomp g^i(\mu(f\om)))
\peq
\rho(f\om) \comp k(\icomp g^i(\mu(f\om)))$ \\
thus addressing the first example given in the introduction of this Section.
We observe that \peqinfstruct\ includes occurrences of an infinite number of variables: for each $j$ from 1 to the arity of $f$, $\psi^j_i$ is a distinct variable for each $i$ verifying $0 \leq i < \omega$.
On the other hand, the restriction to convergent proof terms imposes a convergence condition to the substitutes for these variables when applying this equation
\footnote{more precisely, an equation corresponding to this equation scheme}.
The use, in equational logic, of a convergence condition as a restriction for the application of an equation having occurrences of an infinite number of different variables, could be the object of further analysis.
The equational rule \eqlinfcomp\ allows transformations to be performed in each term of an infinite composition.
Consider the proof terms
$\psi_1 \eqdef \icomp (j(h^i(\mu(f\om))) \comp j(h^i(\nu(f\om)))) \comp \rho(h\om)$
and \\
$\psi_2 \eqdef \rho(f\om) \comp \icomp (k(h^i(\mu(f\om))) \comp k(h^i(\nu(f\om))))$, which represent equivalent \redseqs.
In order to transform $\psi_1$ into $\psi_2$, the $\rho$ step must be permuted with the preceding infinite composition, which in turn must be transformed into a proof term having the form $j(\psi'_1)$ in order to enable the permutation to be applied using the equations \peqinout\ and then \peqoutin.
To perform the desired transformation to
$\icomp (j(h^i(\mu(f\om))) \comp j(h^i(\nu(f\om))))$,
the equation \peqstruct\ must be applied \emph{on each of the infinite number of components}, so obtaining
$\icomp j(h^i(\mu(f\om)) \comp h^i(\nu(f\om)))$,
and then the equation \peqinfstruct\ transforms the latter into
$j(\icomp h^i(\mu(f\om)) \comp h^i(\nu(f\om)))$.
The rule \eqlinfcomp\ allows to obtain
$\icomp (j(h^i(\mu(f\om))) \comp j(h^i(\nu(f\om))))
\peqe
\icomp j(h^i(\mu(f\om)) \comp h^i(\nu(f\om)))$, taking as premises
$j(h^i(\mu(f\om))) \comp j(h^i(\nu(f\om))) \peqe
j(h^i(\mu(f\om)) \comp h^i(\nu(f\om)))$
for each $i < \omega$.
Therefore, the assertion $\psi_1 \peqe \psi_2$ can be justified by the following schematic equational judgement \\[5pt]
$\begin{array}{@{\hspace*{2cm}}cll}
\multicolumn{2}{l}{\icomp (j(h^i(\mu(f\om))) \comp j(h^i(\nu(f\om)))) \comp \rho(h\om)} \\
\peqe &
\icomp j(h^i(\mu(f\om)) \comp h^i(\nu(f\om))) \comp \rho(h\om) \\
\peqe &
j(\icomp h^i(\mu(f\om)) \comp h^i(\nu(f\om))) \comp \rho(h\om) & \textnormal{by } \peqinfstruct \\
\peqe &
\rho(\icomp h^i(\mu(f\om)) \comp h^i(\nu(f\om))) \\
\peqe &
\rho(f\om) \comp k(\icomp h^i(\mu(f\om)) \comp h^i(\nu(f\om))) \\
\peqe &
\rho(f\om) \comp \icomp k(h^i(\mu(f\om)) \comp h^i(\nu(f\om))) & \textnormal{by } \peqinfstruct \\
\peqe &
\rho(f\om) \comp \icomp (k(h^i(\mu(f\om))) \comp k(h^i(\nu(f\om)))) \\
\end{array}$ \\
where the first and last ``steps'' involve, in fact, an infinite number of equation occurrences.
When reasoning about convergent proof terms, the convergence conditions on the sequence $\langle \psi_i \rangle_{i < \omega}$ (resp. $\langle \phi_i \rangle_{i < \omega}$) for $\icomp \psi_i$ (resp. $\icomp \phi_i$) are implicit conditions to apply \eqlinfcomp.
Particularly, the minimal activity depth of the components must tend to $\omega$ for both $\psi$ and $\phi$, thus entailing a convergence condition on the infinite number of premises.
As we have remarked for the \peqinfstruct\ equation, the implications of such convergence conditions on equational reasoning could be object of future work.
To motivate the inclusion of the \eqllim\ equational rule, and consequently the need to define a separated \emph{base} relation, we recall the proof terms
$\psi_3 \eqdef \icomp g^i(\mu(f\om)) \comp \icomp h^i(\nu(g\om))$
and
$\psi_4 \eqdef \icomp (h^i(\mu(f\om)) \comp h^i(\nu(f\om)))$
from the introduction to this Section.
By using the base \peqence\ relation given in Dfn.~\ref{dfn:peqe}, we can permute the first $\nu$ step with all the $\mu$ steps but the first, obtaining
$\psi_3 \peqe \mu(f\om) \comp \nu(f\om) \comp h(\icomp g^i(\mu(f\om)) \comp \icomp h^i(\nu(g\om)))$; so that the first component in $\psi_4$ can be ``extracted'' from $\psi_3$.
Such a process can be repeated in order to ``extract'' more components, arriving to
$\psi_3 \peqe \mu(f\om) \comp \nu(f\om) \comp \ldots \comp h^n(\mu(f\om)) \comp h^n(\nu(f\om)) \comp h^{n+1}(\icomp g^i(\mu(f\om)) \comp \icomp h^i(\nu(g\om)))$
for each $n < \omega$.
On the other hand, it is straightforward to observe that $\psi_4 \peqe \mu(f\om) \comp \nu(f\om) \comp \ldots \comp h^n(\mu(f\om)) \comp h^n(\nu(f\om)) \comp h^{n+1}(\icomp h^{i}(\mu(f\om)) \comp h^{i}(\nu(f\om)))$.
In order to conclude $\psi_3 \peq \psi_4$, it is needed to resort to the \eqllim\ rule added in Dfn.~\ref{dfn:peq}.
We observe that the minimal activity depth of the successive ``differences''
$h^{n+1}(\icomp g^i(\mu(f\om)) \comp \icomp h^i(\nu(g\om)))$ and
$h^{n+1}(\icomp h^{i}(\mu(f\om)) \comp h^{i}(\nu(f\om)))$ tend to infinity, as required in the premises of the \eqllim\ rule.
\begin{remark}
\label{rmk:peqinout-restriction}
We notice that the requirement of lhs and rhs convergence put on the instances of the equation schemata does not imply that every variable in a scheme must necessarily be replaced by a convergent proof term.
\Eg, considering $\mu: f(x) \to g(x)$, $\nu: g(x) \to k(x)$, $\rho: h(x,y) \to j(y)$, and $\tau : i(x) \to x$, the following instance of \peqoutin:
$\rho(\tau\om, \mu(a) \cdot \nu(a)) \peqe \rho(i\om, f(a)) \comp j(\mu(a) \comp \nu(a))$,
is legal even when $\psi_1$ is replaced by the divergent proof term $\tau\om$.
Observe particularly that $\rho(\tau\om, \mu(a) \cdot \nu(a))$ is a convergent proof term: convergence of $\tau\om$ is not asked since the corresponding variable in the lhs of the $\rho$ rule does not occur in the rhs; \confer\ Dfn~\ref{dfn:layer-pterm}, case \ref{rule:ptsymbol}.
On the other hand, let us try to decompose the proof term $\rho(\tau\om, \mu(a) \cdot \nu(a))$ ``the other way around'', namely by using \peqinout\ instead of \peqoutin.
The form of \peqinout\ for proof terms having $\rho$ as root symbol is \\
\hspace*{1cm}
$\rho(\psi_1, \psi_2) \peq h(\psi_1, \psi_2) \comp \rho(tgt(\psi_1), tgt(\psi_2))$ \\
In turn, replacing $\psi_1$ with $\tau\om$ and $\psi_2$ with $\mu(a) \comp \nu(a)$ yields \\
\hspace*{1cm}
$\rho(\tau\om ,\, \mu(a) \comp \nu(a)) \peq h(\tau\om ,\, \mu(a) \comp \nu(a)) \comp \rho(tgt(\tau\om) ,\, tgt(\mu(a) \comp \nu(a)))$ \\
Therefore, applying the equation having the given proof term as left-hand side would require $tgt(\tau\om)$ to be defined, which is not the case.
A similar situation occurs with the (intuitively very simple) equation \peqidright. In this case, the target of the proof term at the right-hand side of an instance must be defined in order for the corresponding left-hand side to make sense.
To avoid this kind of situations, an additional requirement will be put to the uses of the \eqleqn\ equational rule, when the equation involved is either \peqinout\ or \peqidright.
For \peqidright, we ask $\psi$ to be convergent.
For \peqinout\, convergence must be asked, not only of the proof term $\mu(\psi_1, \ldots, \psi_m)$ at the left-hand side of the equation, but convergence must be required to all the (proof terms taking the place of each variable) $\psi_i$ as well.
Therefore, \wrt\ the motivating example, $\rho(\tau\om ,\, \mu(a) \comp \nu(a))$ is not a valid left-hand side to apply \peqinout, even if it is a convergent proof term.
When using either \peqinout\ or \peqidright\ in proofs involving the relation $\peqe$, it should be checked those uses to correspond to valid instances
\footnote{We notice that in the development of the
\includeStandardisation{standardisation proof}
\doNotIncludeStandardisation{compression proof in Section~\ref{sec:compression}}
, the uses of \peqinout\ correspond to situations in which the convergence of the proof term to be put at the right-hand side is known in advance. In fact, the intent of the uses of this equation is to ``obtain'' a condensed form of some contraction activity, corresponding to the left-hand side, in order to subsequently decomposing the obtained condensed form in a top-to-bottom fashion, through the \peqoutin\ equation. \Confer\ the proof of Lem.~\ref{rsl:jump-one-step}.}
.
The equations \peqidright\ and \peqinout\ are the only elements in Dfn.~\ref{dfn:peqe} for which a well-formed proof term being the element for one side in a possible instance does not have a convergent proof term as the correspondent element for the other side
\footnote{This claim will be proved shortly.}.
\end{remark}
\includeStandardisation{\subsubsection{Basic properties of \peqence}}
\doNotIncludeStandardisation{\subsection{Basic properties of \peqence}}
\label{sec:peqence-basic-properties}
\begin{lemma}
\label{rsl:peq-then-same-src-mind-tgt}
Let $\psi$, $\phi$ be convergent proof terms such that $\psi \peq \phi$. Then $src(\psi) = src(\phi)$, $tgt(\psi) = tgt(\phi)$ and $mind(\psi) = mind(\phi)$.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \layerpeq{\alpha} \phi$, analysing the equational logic rule used in the final step of that judgement.
Observe particularly that Lem~\ref{rsl:mind-big-then-tdist-little}:(\ref{it:convergent-then-has-tgt}) implies both $tgt(\psi)$ and $tgt(\phi)$ to be defined.
If the rule is \eqleqn, then we analyse the equation of which the pair $\pair{\psi}{\phi}$ is an instance.
It turns out that the only non-trivial cases are those corresponding to the \peqinfstruct\ equation and the \eqlinfcomp\ and \eqllim\ rules. We prove the result for each of these cases.
Assume that $\pair{\psi}{\phi}$ is an instance of the \peqinfstruct\ equation, \ie, that \\
$\psi = \icomp f(\psi^1_i, \ldots, \psi^m_i)$ and
$\phi = f(\icomp \psi^1_i, \ldots, \icomp \psi^m_i)$.
\begin{itemize}
\item
We verify $\mind{\psi} = \mind{\phi}$. \\
Observe that
$\mind{\psi}
= min_{i < \omega}(\mind{f(\psi^1_i, \ldots, \psi^m_i)})
= \mind{f(\psi^1_a, \ldots, \psi^m_a)}
= 1 + min(\mind{\psi^1_a}, \ldots, \mind{\psi^m_a})
= 1 + \mind{\psi^b_a}$
where
\begin{eqnarray}
\mind{f(\psi^1_a, \ldots, \psi^m_a)} & \leq & \mind{f(\psi^1_i, \ldots, \psi^m_i)}
\quad \textforall i < \omega
\label{eq:infstruct-mind-1} \\
\mind{\psi^b_a} & \leq & \mind{\psi^j_a}
\quad \textif 1 \leq j \leq m
\label{eq:infstruct-mind-2}
\end{eqnarray}
On the other hand,
$\mind{\phi}
= 1 + min(\mind{\icomp \psi^1_i}, \ldots, \mind{\icomp \psi^m_i})
= 1 + \mind{\icomp \psi^{b'}_i}
= 1 + \mind{\psi^{b'}_{a'}}$
where
\begin{eqnarray}
\mind{\icomp \psi^{b'}_i} & \leq & \mind{\icomp \psi^{j}_i}
\quad \textif 1 \leq j \leq m
\label{eq:infstruct-mind-3} \\
\mind{\psi^{b'}_{a'}} & \leq & \mind{\psi^{b'}_{i}}
\quad \textforall i < \omega
\label{eq:infstruct-mind-4}
\end{eqnarray}
Assume for contradiction $\mind{\psi^{b}_{a}} < \mind{\psi^{b'}_{a'}}$.
Then $b \neq b'$ would imply
$\mind{\icomp \psi^{b}_i} \leq \mind{\psi^{b}_{a}}
< \mind{\psi^{b'}_{a'}} = \mind{\icomp \psi^{b'}_i}$, contradicting (\ref{eq:infstruct-mind-3}), and $b = b'$ would immediately contradict (\ref{eq:infstruct-mind-4}).
Analogously, if we assume $\mind{\psi^{b'}_{a'}} < \mind{\psi^{b}_{a}}$, then $a \neq a'$ would imply
$\mind{f(\psi^1_{a'}, \ldots, \psi^m_{a'})} \leq 1 + \mind{\psi^{b'}_{a'}}
< 1 + \mind{\psi^{b}_{a}} = \mind{f(\psi^1_a, \ldots, \psi^m_a)}$, contradicting (\ref{eq:infstruct-mind-1}), and $a = a'$ would immediately contradict (\ref{eq:infstruct-mind-2}).
Hence we conclude.
\item
To verify the condition about source terms, it is enough to observe that
$src(\psi) = src(\phi) = f(src(\psi^1_0), \ldots, src(\psi^m_0))$.
\item
We verify $tgt(\psi) = tgt(\phi)$.
Observe that
$tgt(\psi) = \lim_{i \to \omega} f(tgt(\psi^1_i), \ldots, tgt(\psi^m_i))$ and
$tgt(\phi) = f(\lim_{i \to \omega} tgt(\psi^1_i), \ldots, \lim_{i \to \omega} tgt(\psi^m_i))$. \\
Let $t_j \eqdef \lim_{i \to \omega} tgt(\psi^j_i)$, so that $tgt(\phi) = f(t_1, \ldots, t_m)$.
Then it is enough to prove that $\tdist{tgt(\psi)}{f(t_1, \ldots, t_m)} = 0$. \\
Let $n < \omega$. Let $k$ such that for all $j$, $i > k$ implies $\tdist{tgt(\psi^j_i)}{t_j} < 2^{-(n-1)}$ and also
$\tdist{f(tgt(\psi^1_i), \ldots, tgt(\psi^m_i))}{tgt(\psi)} < 2^{-n}$. \\
Let $i \eqdef k+1$. Then
$\tdist{f(tgt(\psi^1_i), \ldots, tgt(\psi^m_i))}{f(t_1, \ldots, t_m)} = $\\
$\frac{1}{2} * max(\tdist{tgt(\psi^1_i)}{t_1}, \ldots, \tdist{tgt(\psi^1_m)}{t_m})
< 2^{-n}$.
Hence Lem.~\ref{rsl:tdist-is-ultrametric} yields
$\tdist{tgt(\psi)}{f(t_1,\ldots,t_m)} < 2^{-n}$.
Thus we conclude.
\end{itemize}
Assume that the rule justifying $\psi \layerpeq{\alpha} \phi$ is \eqlinfcomp, so that $\psi = \icomp \psi_i$, $\phi = \icomp \phi_i$, and for all $i < \omega$, $\psi_i \layerpeq{\alpha_i} \phi_i$ where $\alpha_i < \alpha$. \\
Source terms: it is enough to apply \ih\ on $\psi_0 \layerpeq{\alpha_0} \phi_0$ obtaining $src(\psi) = src(\psi_0) = src(\phi_0) = src(\phi)$. \\
Target terms and $\mindfn$:
Observe that \ih\ can be applied on each $\psi_i \layerpeq{\alpha_i} \phi_i$, yielding $tgt(\psi_i) = tgt(\phi_i)$ and $\mind{\psi_i} = \mind{\phi_i}$.
Then recalling the definitions of target and $\mindfn$ on $\psi$ and $\phi$ suffices to conclude.
Assume that the rule used in the last step of the judgement $\psi \layerpeq{\alpha} \phi$ is \eqllim, so that for all $n < \omega$,
$\psi \layerpeqe{\alpha_n} \chi_n \comp \psi'_n$ and $\phi \layerpeqe{\alpha_n} \chi_n \comp \phi'_n$, where $\mind{\psi'_n} > n$, $\mind{\phi'_n} > n$, $\alpha_n < \alpha$ and $\beta_n < \alpha$.
Observe that $\layerpeqe{\alpha} \,\subseteq\, \layerpeq{\alpha}$ for any ordinal $\alpha$, so that \ih\ can be applied to any premise of the \eqllim\ rule. \\[2pt]
Source terms: applying \ih\ on $\psi \layerpeq{\alpha_0} \chi_0 \comp \psi'_0$ and $\phi \layerpeq{\alpha_0} \chi_0 \comp \phi'_0$, we obtain $src(\psi) = src(\phi) = src(\chi_0)$. \\[2pt]
Target terms: we prove $\tdist{tgt(\psi)}{tgt(\phi)} = 0$.
Let $n < \omega$.
Then \ih\ on $\psi \layerpeq{\alpha_n} \chi_n \comp \psi'_n$ and $\phi \layerpeq{\alpha_n} \chi_n \comp \phi'_n$ yields
$tgt(\psi) = tgt(\psi'_n)$ and $tgt(\phi) = tgt(\phi'_n)$. Moreover, it is immediate to obtain $src(\psi'_n) = src(\phi'_n) = tgt(\chi_n)$. \\
Recalling that $\mind{\psi'_n} > n$ and $\mind{\phi'_n} > n$, Lem.~\ref{rsl:mind-big-then-tdist-little} can be applied to obtain
$\tdist{tgt(\chi_n)}{tgt(\psi)} = \tdist{src(\psi'_n)}{tgt(\psi'_n)} < 2^{-n}$ and analogously
$\tdist{tgt(\chi_n)}{tgt(\phi)} = \tdist{src(\phi'_n)}{tgt(\phi'_n)} < 2^{-n}$.
Therefore Lem.~\ref{rsl:tdist-is-ultrametric} yields $\tdist{tgt(\psi)}{tgt(\phi)} < 2^{-n}$. Thus we conclude. \\[2pt]
Minimal activity depth:
Assume for contradiction $n \eqdef \mind{\psi} < \mind{\phi}$.
Observe $\psi \peq \chi_n \comp \psi'_n$ and $\phi \peq \chi_n \comp \phi'_n$, where $\mind{\psi'_n} > n$ and $\mind{\phi'_n} > n$.
Then $\mind{\psi} = n$ implies $\mind{\chi_n} = n$, and therefore $\mind{\phi} = n$, contradicting the assumption.
The assertion $\mind{\phi} < \mind{\psi}$ can be contradicted analogously.
Thus we conclude.
\end{proof}
The result about $\mindfn$ and $src$ allows to prove that $\peqe$ is closed \wrt\ the set of convergent proof terms.
\begin{lemma}
\label{rsl:peqe-soundness-convergent}
Let $\psi$ and $\phi$ proof terms such that $\psi \peqe \phi$.
Then $\psi$ is a well-formed and convergent proof term iff $\phi$ is.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \layerpeqe{\alpha} \phi$, analysing the equational rule used in the last step in the corresponding $\peqe$ derivation.
If the rule is \eqleqn, then we analyse the basic equation used.
\begin{itemize}
\item
\peqidleft, \ie\ $\psi = src(\phi) \comp \phi$. It is immediate to verify the desired result.
\item
\peqidright, \ie\ $\psi = \phi \comp tgt(\phi)$. Observe that Remark~\ref{rmk:peqinout-restriction} implies that $\phi$ must be a convergent proof term. Thus we conclude immediately.
\item
\peqassoc, \ie\ $\psi = \chi \comp (\xi \comp \gamma)$ and $\phi = (\chi \comp \xi) \comp \gamma$. In this case, $\psi$ is well-formed iff $\phi$ is well-formed iff $\chi$, $\xi$ and $\gamma$ are well formed, and moreover $\chi$ and $\xi$ are convergent. Moreover, $\psi$ is convergent iff $\phi$ is convergent iff $\gamma$ is convergent. Thus we conclude.
\item
\peqstruct, \ie\ $\psi = f(\chi_1, \ldots, \chi_m) \comp f(\xi_1, \ldots, \xi_m)$ and $\phi = f(\chi_1 \comp \xi_1, \ldots, \chi_m \comp \xi_m)$.
In this case, $\psi$ is well formed iff $\phi$ is well-formed iff all $\chi_i$ and $\xi_i$ are well-formed, all the $\chi_i$ are also convergent (\confer\ Lem.~\ref{rsl:fnsymbol-convergence} for $\psi$), and $tgt(\chi_i) = src(\xi_i)$ for all $i$.
Moreover,
$\psi$ is convergent
iff all the $\xi_i$ are convergent (\confer\ again Lem.~\ref{rsl:fnsymbol-convergence})
iff all the $\chi_i \comp \xi_i$ are convergent
iff $\phi$ is convergent.
Thus we conclude.
\item
\peqinfstruct, \ie\ $\psi = \icomp f(\chi^1_i, \ldots, \chi^m_i)$ and $\phi = f(\icomp \chi^1_i, \ldots, \icomp \chi^m_i)$.
$\Rightarrow )$
Assume that $\psi$ is well-formed and convergent. Given $n < \omega$, let $k_n < \omega$ be an index verifying $\mind{f(\chi^1_i, \ldots \chi^m_i)} > n$ if $k_n < i$.
Let $j$ such that $1 \leq j \leq m$.
Then for all $i < \omega$, $f(\chi^1_i, \ldots \chi^m_i)$ convergent implies $\chi^j_i$ convergent, \confer\ Lem~\ref{rsl:fnsymbol-convergence}.
In turn $src(f(\chi^1_{i+1}, \ldots \chi^m_{i+1})) = tgt(f(\chi^1_i, \ldots \chi^m_i))$ implies immediately $src(\chi^j_{i+1}) = tgt(\chi^j_i)$.
Finally, if $i > k_{n+1}$, then $\mind{f(\chi^1_i, \ldots \chi^m_i)} > n + 1$ implies $\mind{\chi^j_i} > n$. Hence $\icomp \chi^j_i$ is well-formed and convergent. Consequently, so is $\phi$.
$\Leftarrow )$
Assume that $\phi$ is well-formed and convergent. Given $j$ such that $1 \leq j \leq m$ and $n < \omega$, let $k_{(n,j)}$ be an index verifying $\mind{\psi^j_i} > n$ if $k_{(n,j)} < i$.
Let $i < \omega$. Then $\chi^j_i$ convergent and $src(\psi^j_{i+1}) = tgt(\psi^j_i)$ for all $j$ implies $f(\chi^1_i, \ldots, \chi^m_i)$ convergent and $src(f(\chi^1_{i+1}, \ldots \chi^m_{i+1})) = tgt(f(\chi^1_i, \ldots \chi^m_i))$. Then $\psi$ is a well-formed proof term.
Moreover, for all $n < \omega$, if $i > max \set{k_{(n,j)} \setsthat 1 \leq j \leq m}$, then $\mind{f(\chi^1_i, \ldots, \chi^m_i)} > n$. Consequently, $\psi$ is convergent.
\item
\peqinout, \ie\ $\psi = \mu(\chi_1, \ldots, \chi_m)$ and $\phi = l[\chi_1, \ldots, \chi_m] \comp \mu(t_1, \ldots, t_m)$.
In this case, Remark~\ref{rmk:peqinout-restriction} implies that all $\chi_i$ are convergent proof terms.
Then both $\psi$ and $\phi$ are well-formed and convergent.
\item
\peqoutin, \ie\ $\psi = \mu(\chi_1, \ldots, \chi_m)$ and $\phi = \mu(s_1, \ldots, s_m) \comp r[\chi_1, \ldots, \chi_m]$.
In this case $\psi$ is well-formed iff $\phi$ is well-formed iff $\chi_i$ are well-formed.
Moreover, $\psi$ is convergent iff $\phi$ is convergent iff all $\chi_i$ corresponding to variables occurring in the right-hand side $r$, which are exactly those occurring in $r[\chi_1, \ldots, \chi_m]$, are convergent; \confer\ Lem.~\ref{rsl:rulesymbol-convergence} and Lem.~\ref{rsl:ctx-convergence} respectively.
\end{itemize}
If the equational rule used in the last step of the derivation ending in $\psi \layerpeqe{\alpha} \phi$ is \eqlrefl, \eqlsymm\ or \eqltrans, then a straightforward argument suffices to conclude.
If the rule is \eqlfun, \eqlrule\ or \eqlcomp, then a simple argument based on Lem.~\ref{rsl:fnsymbol-convergence}, Lem~\ref{rsl:rulesymbol-convergence} or just Dfn.~\ref{dfn:layer-pterm} case~(\ref{rule:ptbinC}) respectively, and \ih, suffices to conclude.
Assume that the rule used in the last step of the derivation is \eqlinfcomp. As the rule is symmetric, then it suffices to prove one side of the biconditional in the lemma statement.
Then assume that $\psi = \icomp \psi_i$ is a well-formed and convergent proof term.
Let $i < \omega$. Then $\psi_i$ is convergent and $src(\psi_{i+1}) = tgt(\psi_i)$.
Therefore \ih\ implies convergence of $\phi_i$, and Lem.~\ref{rsl:peq-then-same-src-mind-tgt} yields $src(\phi_{i+1}) = tgt(\phi_i)$. Hence $\phi$ is well-formed.
Let $n < \omega$. Then convergence of $\psi$ implies the existence of some $k_n < \omega$ verifying $\mind{\psi_i} > n$ if $k_n < i$. In turn, Lem.~\ref{rsl:peq-then-same-src-mind-tgt} implies $\mind{\phi_i} > n$ if $k_n < i$. Consequently, $\psi$ is convergent.
\end{proof}
The following lemma shows that \peqence\ is compatible with infinitary contexts.
\begin{lemma}
\label{rsl:peqe-compatible-ctx}
Let $C$ be a context having $k < \omega$ holes, and $\langle \psi_i \rangle_{i \leq k}$ and $\langle \phi \rangle_{i \leq k}$ two sequences of proof terms verifying $\psi_i \peqe \phi_i$ for all $i$.
Then $C[\psi_1, \ldots, \psi_k] \peqe C[\phi_1, \ldots, \phi_k]$.
\end{lemma}
\begin{proof}
An easy induction on $max \set{\posln{\BPos{C}{i}}}$ suffices. Resort to the \eqlfun\ equational rule for the inductive case.
\end{proof}
\doNotIncludeStandardisation{\renewcommand{\peqefs}{\peqe}}
The following lemma shows that the \peqstruct\ equation can be extended to contexts having a finite number of holes.
\begin{lemma}
\label{rsl:struct-ctx}
Let $C$ be a context in $\Sigma$ (\ie\ built from function symbols only) having exactly $n < \omega$ occurrences of the box; and $\psi_1, \ldots, \psi_n$, $\phi_1, \ldots, \phi_n$ proof terms.
Then
$C[\psi_1, \ldots, \psi_n] \comp C[\phi_1, \ldots, \phi_n]
\peqefs
C[\psi_1 \comp \phi_1, \ldots, \psi_n \comp \phi_n]$.
\end{lemma}
\begin{proof}
We proceed by induction on
$max(\set{\posln{\BPos{C}{i}}})$.
If $C = \Box$, then we conclude immediately, notice that in this case $n = 1$.
Otherwise $C = f(C_1, \ldots, C_m)$. In this case \\
$C[\psi_1, \ldots, \psi_n] \comp C[\phi_1, \ldots, \phi_n] \ = $ \\
\hspace*{1cm} $f(C_1[\psi_1, \ldots, \psi_{k1}], \ldots, C_m[\psi_{k(m-1)+1}, \ldots, \psi_{n}]) \ \comp $ \\
\hspace*{1cm} $f(C_1[\phi_1, \ldots, \phi_{k1}], \ldots, C_m[\phi_{k(m-1)+1}, \ldots, \phi_{n}])$, and \\
$C[\psi_1 \comp \phi_1, \ldots, \psi_n \comp \phi_n] \ = $ \\
\hspace*{1cm} $f(C_1[\psi_1 \comp \phi_1, \ldots, \psi_{k1} \comp \phi_{k1}], \ldots, C_m[\psi_{k(m-1)+1} \comp \phi_{k(m-1)+1}, \ldots, \psi_{n} \comp \phi_{n}])$.
We conclude by \ih\ on each $C_i$, and then by the \eqlfun\ equational rule.
\end{proof}
\begin{lemma}
\label{rsl:trivial-pterm-peq-src}
Let $\psi$ be a trivial proof term. Then $\psi \peq src(\psi)$.
\end{lemma}
\begin{proof}
Observe $\psi \peqe src(\psi) \comp \psi$ by \peqidleft.
On the other hand, $src(\psi) \peqe src(src(\psi)) \comp src(\psi) = src(\psi) \comp src(\psi)$, by \peqidleft\ and Dfn.~\ref{dfn:src-tgt-imstep} respectively; recall that $src(\psi)$ is a trivial \imstep.
Moreover, for any $n < \omega$, $\mind{\psi} = \mind{src(\psi)} = \omega > n$, \confer\ Lem.~\ref{rsl:trivial-pterm-mind-omega}.
Therefore the rule \eqllim\ can be applied to obtain $\psi \peq src(\psi)$.
\end{proof}
\section{Denotation of \redseqs}
\label{sec:pterm-denotation}
As stated in Sec.~\ref{sec:pterm}, the aim of the introduction of proof terms is to denote and study \redseqs\ in infinitary rewriting.
A basic question arises: can \emph{any} \redseq\ be denoted by a proof term?
In order to answer this question, we will resort to proof terms which denote a \redseq\ in a close, stepwise way, without condensing parallel or embedded steps.
Formally, we will define a proper subset of the set of valid proof terms, which we will call \emph{\ppterms}, which include only (denotation of) single steps and dots.
Then we will prove that any \redseq\ whose length is a countable ordinal can be denoted by means of a \ppterm. Observe that particularly this result applies to all convergent \redseqs, \confer\ Thm.~2 in \cite{inf-normalization}.
Once denotation of all countable-length \redseqs\ is stated, the issue of \emph{uniqueness of stepwise denotation} arises.
It is easy to realize that stepwise denotation of a \redseq\ is not unique, because of different ``bracketings'', \ie\ different ways to associate dots.
A simple example follows, using the rules
$\mu(x) : f(x) \to g(x)$, $\nu(x) : g(x) \to k(x)$, $\rho(x,y) : h(x,y) \to x$.
The proof terms $(\rho(f(a),b) \comp \mu(a)) \comp \nu(a)$ and $\rho(f(a),b) \comp (\mu(a) \comp \nu(a))$ are different stepwise denotations of the same \redseq, namely $h(f(a),b) \to f(a) \to g(a) \to k(a)$.
On the other hand, observe that these proof terms are \peqent, and moreover, its equivalence can be stated by using only the equation \peqassoc.
In the finitary setting, it is fairly intuitive that \ppterms\ being \emph{denotationally equivalent}, \ie\ such that they denote the same \redseq, can be proven to be \peqent\ by ``rebracketing'', \ie\ by applying equational logic using only the \peqassoc\ equation.
The reciprocal property also holds: if two \ppterms\ are \emph{rebracketing equivalent} (or, phrased differently, ``equal up to rebracketing'') then they denote the same \redseq.
The concepts we have just introduced allow to state the question about denotation uniqueness in a more precise way:
\textbf{do denotational and rebracketing equivalences coincide}?
For the finitary case, it is fairly simple to prove that the answer to this question is positive.
Indeed, by orienting the \peqassoc\ equation in either direction, \emph{standard} denotations of \redseqs\ can be obtained.
These standard \ppterms\ can also be seen as the result of coherently associating dots to the left or to the right.
For \ppterms\ denoting infinite \redseqs, the question seems less obvious.
\Eg\ consider the sequence
$f\om \to g(f\om) \to g(g(f\om)) \infred g\om$
which can be denoted \eg\ by the \ppterms\
$\icomp g^i(\mu(f\om))$ and $\icomp g^{2*i}(\mu(f\om)) \comp g^{2*i+1}(\mu(f\om))$.
For any $n < \omega$, it is easy to obtain, using only the equation \peqassoc, that
$\psi \peqe
( \mu(f\om) \comp \ldots \comp g^{2*n+1}(\mu(f\om)) ) \comp g^{2*(n+1)}(\psi)$
and
$\phi \peqe
( \mu(f\om) \comp \ldots \comp g^{2*n+1}(\mu(f\om)) ) \comp g^{2*(n+1)}(\phi)$.
Then we can obtain $\psi \peq \phi$ \textbf{by resorting to a limit argument}, \ie\ by applying the \eqllim\ rule.
On the other hand, we did not find a way to justify \peqence\ between these \ppterms\ which avoids the use of \eqllim.
In this Section we will prove that, provided the characterisation of \peqence\ given in Sec.~\ref{sec:peqence}, denotational and rebracketing equivalences do coincide for infinitary term rewriting.
The corresponding proofs make evident the role of the limit \peqence\ argument in order to verify this coincidence.
\subsection{\Ppterms}
\label{sec:ppterm}
\denotationDistributed{
In this section we will introduce a particular subclass of the set of proof terms, namely the \emph{\ppterms}.
Just like \imsteps\ are in correspondence with maximal developments of \orthoredexsets, so that any maximal development can be \emph{denoted} by an \imstep (\confer\ \refsec{mstep-orthoredexset}); \ppterms\ are related with the set of all \redseqs\ for a given \TRS, so that \emph{any \redseq} can be denoted by a \ppterm.
A formal characterisation of the idea of denoting a \redseq\ by a (stepwise) proof term will be given in the following Section~\ref{sec:pterm-denotation}.
}
\denotationInOwnChapter{
In the following, we introduce the set of \ppterms, give some additional related definitions and state some basic properties of this subset of the set of valid proof terms.
}
\begin{definition}[One-step]
\label{dfn:one-step}
A \emph{one-step} is an \imstep\ including exactly one occurrence of a rule symbol.
If $\psi$ is a one-step, then we define the redex position of $\psi$, notation $\RPos{\psi}$, as the position of the unique rule symbol occurrence in $\psi$, and the depth of $\psi$, notation $\sdepth{\psi}$, as $\posln{\RPos{\psi}}$; \confer\ \refdfn{step} for the analogy with the corresponding notions as defined for a reduction step.
\end{definition}
\begin{definition}[\Ppterm, \Pnpterm]
\label{dfn:ppterm}
A \emph{\ppterm} is any proof term $\psi$ whose formation satisfies any of the following conditions, where we refer to cases in \refdfn{layer-pterm}:
\begin{itemize}
\item $\psi$ is a one-step, so it is built by case \ref{rule:ptmstep},
\item $\psi$ is built by case \ref{rule:ptinfC}, so that $\psi = \icomp \psi_i$, and all of the $\psi_i$ are \ppterms, or
\item $\psi$ is built by case \ref{rule:ptbinC}, so that $\psi = \psi_1 \comp \psi_2$, and both $\psi_1$ and $\psi_2$ are \ppterms.
\end{itemize}
A \emph{\pnpterm} is any proof term $\psi$ such that either $\psi$ is a \ppterm\ or $\psi \in \iSigmaTerms$.
\end{definition}
\begin{definition}[Steps of a \pnpterm]
\label{dfn:steps}
For any $\psi$ \pnpterm, we define the number of \emph{steps} of $\psi$, notation $\ppsteps{\psi}$, as the countable ordinal defined as follows: \\
\begin{tabular}{l}
if $\psi \in \iSigmaTerms$, then $\ppsteps{\psi} \eqdef 0$. \\
if $\psi$ is a one-step, then $\ppsteps{\psi} \eqdef 1$. \\
if $\psi = \icomp \psi_i$ then $\ppsteps{\psi} \eqdef \sum_{i < \omega} \ppsteps{\psi_i}$; \confer \refdfn{ordinal-infAdd}. \\
if $\psi = \psi_1 \comp \psi_2$ then $\ppsteps{\psi} \eqdef \ppsteps{\psi_1} + \ppsteps{\psi_2}$.
\end{tabular}
\end{definition}
\begin{lemma}
\label{rsl:steps-ordinal-coherence}
Let $\psi$ be a \ppterm, and let $\alpha$ the ordinal such that $\psi \in \layerpterm{\alpha}$. Then $\ppsteps{\psi}$ is a limit ordinal iff $\alpha$ is.
\end{lemma}
\begin{proof}
Easy induction on $\alpha$ where $\psi \in \layerpterm{\alpha}$.
\end{proof}
\begin{definition}[$\alpha$-th component of a \ppterm]
\label{dfn:ppterm-component}
Let $\psi$ be a \ppterm\ and $\alpha$ an ordinal such that $\alpha < \ppsteps{\psi}$. We define the \emph{$\alpha$-th component} of $\psi$, notation $\psi[\alpha]$, as the one-step defined as follows: \\
\begin{tabular}{p{.95\textwidth}}
if $\psi$ is a one-step, then $\psi[0] \eqdef \psi$. \\
if $\psi = \icomp \psi_i$, then there are unique $k$ and $\gamma$ such that $\alpha = \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_{k-1}} + \gamma$ and $\gamma < \ppsteps{\psi_k}$; \confer\ \reflem{ordinal-lt-infAdd-then-unique-representation}. We define $\psi[\alpha] \eqdef \psi_k[\gamma]$. \\
if $\psi = \psi_1 \comp \psi_2$ and $\alpha < \ppsteps{\psi_1}$ then $\psi[\alpha] \eqdef \psi_1[\alpha]$. \\
if $\psi = \psi_1 \comp \psi_2$ and $\ppsteps{\psi_1} \leq \alpha$, then $\psi[\alpha] \eqdef \psi_2[\beta]$ such that $\ppsteps{\psi_1} + \beta = \alpha$.
\end{tabular}
\end{definition}
\begin{definition}
\label{dfn:maxd}
Let $\psi$ be a \ppterm\ such that $\ppsteps{\psi} < \omega$.
Then we define the \emph{maximal depth activity} of $\psi$ as $\maxd{\psi} \eqdef max(\sdepth{\redel{\psi}{n}} \setsthat n < \ppsteps{\psi})$.
We also define the \emph{maximal step depth} of $\psi$ as $\maxsd{\psi} \eqdef max(\Pdepth{\mu} \setsthat \mu \in R)$ where $R$ is the set of all the rule symbols occurring in $\psi$.
\end{definition}
We show some expected properties of the components of a \ppterm.
These properties particularly entail that a \ppterm\ can be seen as the concatenation of its components, so that the particular way in which they are associated is irrelevant.
\denotationDistributed{
More on this in Section~\ref{sec:peqence}, specifically in Section~\ref{sec:peqence-basic-properties} and Section~\ref{sec:frso}.}
\begin{lemma}
\label{rsl:ppterm-mind-big-then-tdist-little}
Let $\psi$ be a \ppterm, $\alpha$ an ordinal and $n < \omega$, such that $\mind{\psi} > n$ and $\alpha < \ppsteps{\psi}$. Then
\begin{enumerate}
\item \label{it:ppterm-mind-big-then-step-mind-big}
$\sdepth{\redel{\psi}{\alpha}} > n$.
\item \label{it:ppterm-mind-big-then-tdist-little-step}
$\tdist{src(\redel{\psi}{\alpha})}{tgt(\redel{\psi}{\alpha})} < 2^{-n}$.
\item \label{it:ppterm-mind-big-then-tdist-little-src}
$\tdist{src(\psi)}{tgt(\redel{\psi}{\alpha})} < 2^{-n}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$, \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
If $\psi$ is a one-step then $\alpha = 0$ and $\redel{\psi}{\alpha} = \psi$.
Then we conclude immediately; \confer\ Lemma~\ref{rsl:mind-big-then-tdist-little} for (\ref{it:ppterm-mind-big-then-tdist-little-step}) and (\ref{it:ppterm-mind-big-then-tdist-little-src}).
Assume $\psi = \psi_1 \comp \psi_2$.
If $\alpha < \ppsteps{\psi_1}$, so that $\redel{\psi}{\alpha} = \redel{\psi_1}{\alpha}$, then we conclude by \ih\ on $\psi_1$.
Otherwise $\alpha = \ppsteps{\psi_1} + \beta$, so that $\redel{\psi}{\alpha} = \redel{\psi_2}{\beta}$. Then by applying \ih\ on $\psi_2$ we obtain (\ref{it:ppterm-mind-big-then-step-mind-big}) and (\ref{it:ppterm-mind-big-then-tdist-little-step}) immediately, and also $\tdist{src(\psi_2)}{tgt(\redel{\psi}{\alpha})} < 2^{-n}$.
On the other hand we can apply Lemma~\ref{rsl:mind-big-then-tdist-little} to $\psi_1$, obtaining $\tdist{src(\psi)}{tgt(\psi_1)} < 2^{-n}$. Thus we conclude by Lemma~\ref{rsl:tdist-is-ultrametric} since $tgt(\psi_1) = src(\psi_2)$.
Assume $\psi = \icomp \psi_i$. Let $k$, $\beta$ such that $\redel{\psi}{\alpha} = \redel{\psi_k}{\beta}$, so that $\beta < \ppsteps{\psi_k}$.
Then \ih\ on $\psi_k$ yields immediately (\ref{it:ppterm-mind-big-then-step-mind-big}) and (\ref{it:ppterm-mind-big-then-tdist-little-step}), and also $\tdist{src(\psi_k)}{tgt(\redel{\psi}{\alpha})} < 2^{-n}$.
On the other hand, for each $i < k$ it is immediate that $\mind{\psi_i} \geq \mind{\psi} > n$, then an easy induction on $k$ using Lemma~\ref{rsl:mind-big-then-tdist-little} and Lemma~\ref{rsl:tdist-is-ultrametric} yields $\tdist{src(\psi)}{src(\psi_{k})} < 2^{-n}$. Thus we conclude by Lemma~\ref{rsl:tdist-is-ultrametric}.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-mind-big-then-tdist-little-tgt}
Let $\psi$ be a convergent \ppterm\ such that $\mind{\psi} > p$, and $\alpha < \ppsteps{\psi}$.
Then $\tdist{tgt(\redel{\psi}{\alpha})}{tgt(\psi)} < 2^{-p}$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$.
If $\psi$ is a one-step then $\alpha = 0$ and it suffices to observe that $\redel{\psi}{0} = \psi$.
Assume $\psi = \psi_1 \comp \psi_2$.
If $\alpha < \ppsteps{\psi_1}$, then \ih\ on $\psi_1$ yields $\tdist{tgt(\redel{\psi}{\alpha})}{tgt(\psi_1)} < 2^{-p}$.
On the other hand, Lemma~\ref{rsl:mind-big-then-tdist-little} implies $\tdist{src(\psi_2)}{tgt(\psi)} < 2^{-p}$.
We conclude by Lemma~\ref{rsl:tdist-is-ultrametric} since $tgt(\psi_1) = src(\psi_2)$.
Otherwise, $\alpha = \ppsteps{\psi_1} + \beta$, then $\redel{\psi}{\alpha} = \redel{\psi_2}{\beta}$.
In this case we can apply \ih\ on $\psi_2$ obtaining $\tdist{tgt(\redel{\psi_2}{\beta})}{tgt(\psi_2)} < 2^{-p}$, thus we conclude.
Assume $\psi = \icomp \psi_i$ and let $k$, $\gamma$ such that $\redel{\psi}{\alpha} = \redel{\psi_k}{\gamma}$.
Then \ih\ on $\psi_k$ yields $\tdist{tgt(\redel{\psi}{\alpha})}{tgt(\psi_k)} < 2^{-p}$.
Moreover, Lemma~\ref{rsl:mind-big-then-tdist-little} on $\icomp \psi_{k+1+i}$ implies $\tdist{src(\psi_{k+1})}{tgt(\psi)} < 2^{-p}$. Ths we conclude by Lemma~\ref{rsl:tdist-is-ultrametric}.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-src}
Let $\psi$ be a \ppterm. Then $src(\redel{\psi}{0}) = src(\psi)$.
\end{lemma}
\begin{proof}
Easy induction on $\psi$.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-tgt-successor}
Let $\psi$ be a \ppterm\ such that $\ppsteps{\psi} = \alpha + 1$. Then $tgt(\psi) = tgt(\redel{\psi}{\alpha})$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$.
If $\psi$ is a one-step then $\alpha = 0$ and we conclude immediately.
Assume $\psi = \psi_1 \comp \psi_2$.
Then $\alpha < \ppsteps{\psi_1}$ would imply $\alpha + 1 = \ppsteps{\psi} \leq \ppsteps{\psi_1}$, which is not possible since $\ppsteps{\psi_2} > 0$.
Then let $\beta$ be the ordinal verifying $\ppsteps{\psi_1} + \beta = \alpha$, so that $\redel{\psi}{\alpha} = \redel{\psi_2}{\beta}$.
We observe that $\ppsteps{\psi_1} + \beta + 1 = \alpha + 1 = \ppsteps{\psi}$, then $\ppsteps{\psi_2} = \beta + 1$.
We conclude by \ih\ on $\psi_2$.
Finally, $\psi = \icomp \psi_i$ contradicts $\ppsteps{\psi}$ to be a successor ordinal. Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-tgt-limit}
Let $\psi$ be a convergent \ppterm\ such that $\ppsteps{\psi}$ is a limit ordinal.
Then $tgt(\psi) = \lim_{\alpha \to \ppsteps{\psi}} tgt(\redel{\psi}{\alpha})$.
\end{lemma}
\begin{proof}
Observe $\ppsteps{\psi}$ being a limit ordinal implies $\psi = \icomp \psi_i$
\denotationInOwnChapter{(\confer\ Lem.~\ref{rsl:steps-ordinal-coherence} and Lem.~\ref{rsl:ptinfC-iff-limit})}
, so that $tgt(\psi)$ is defined to be equal to $\lim_{i \to \omega} tgt(\psi_i)$. Observe that Lem~\ref{rsl:mind-big-then-tdist-little}:(\ref{it:convergent-then-has-tgt}) implies this limit to be defined.
Let $p \in \Nat$, let $k'$ such that $k' < j < \omega$ implies $\tdist{tgt(\psi_j)}{tgt(\psi)} < 2^{-p}$, $k''$ such that $\mind{\psi_j} > p$ if $j > k''$, and $k \eqdef max(k', k'')$.
Let $\beta = \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_k}$ and $\gamma > \beta$.
Then $\gamma = \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_j} + \gamma'$ where $\gamma' < \ppsteps{\psi_{j+1}}$ and $j \geq k$, so that $\redel{\psi}{\gamma} = \redel{\psi_{j+1}}{\gamma'}$.
Then $j + 1 > k \geq k''$, so that Lemma~\ref{rsl:ppterm-mind-big-then-tdist-little-tgt} implies $\tdist{tgt(\redel{\psi}{\gamma})}{tgt(\psi_{j+1})} < 2^{-p}$.
On the other hand, $j + 1 > k \geq k'$ implies $\tdist{tgt(\psi_{j+1})}{tgt(\psi)} < 2^{-p}$.
Hence Lemma~\ref{rsl:tdist-is-ultrametric} yields $\tdist{tgt(\redel{\psi}{\gamma})}{tgt(\psi)} < 2^{-p}$.
Consequently, we conclude.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-tgt-src-coherence}
Let $\psi$ be a \ppterm\ and $\alpha < \ppsteps{\psi}$ such that $\alpha = \alpha' + 1$. Then $src(\redel{\psi}{\alpha}) = tgt(\redel{\psi}{\alpha'})$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$. Observe $\psi$ is a one-step would imply $\alpha = 0$, contradicting $\alpha = \alpha' + 1$.
Assume $\psi = \psi_1 \comp \psi_2$. We consider three cases
\begin{itemize}
\item
If $\alpha < \ppsteps{\psi_1}$ then we conclude just by \ih\ on $\psi_1$.
\item
If $\alpha = \ppsteps{\psi_1}$, then $\redel{\psi}{\alpha} = \redel{\psi_2}{0}$ and $\redel{\psi}{\alpha'} = \redel{\psi_1}{\alpha'}$ where $\alpha' + 1 = \alpha = \ppsteps{\psi_1}$.
Then $tgt(\redel{\psi}{\alpha'}) = tgt(\psi_1)$ and $src(\redel{\psi}{\alpha}) = src(\psi_2)$, by Lemma~\ref{rsl:ppterm-tgt-successor} and Lemma~\ref{rsl:ppterm-src} respectively. Thus we conclude.
\item
If $\alpha > \ppsteps{\psi_1}$, then $\alpha' = \ppsteps{\psi_1} + \beta'$ and $\alpha = \ppsteps{\psi_1} + (\beta' + 1)$, therefore $\redel{\psi}{\alpha} = \redel{\psi_2}{\beta' + 1}$ and $\redel{\psi}{\alpha'} = \redel{\psi_2}{\beta'}$. Observe that $\alpha < \ppsteps{\psi}$ implies $\beta' + 1 < \ppsteps{\psi_2}$. Hence we conclude by \ih\ on $\psi_2$.
\end{itemize}
Assume $\psi = \icomp \psi_i$. Let $k$, $\gamma$ such that $\alpha = \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_{k-1}} + \gamma$ and $\gamma < \ppsteps{\psi_k}$, so that $\redel{\psi}{\alpha} = \redel{\psi_k}{\gamma}$.
If $\gamma = 0$, then $\ppsteps{\psi_{k-1}} = \beta + 1$ for some $\beta$, and $\alpha' = \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_{k-2}} + \beta$, so that $\redel{\psi}{\alpha'} = \redel{\psi_{k-1}}{\beta}$.
Therefore $src(\redel{\psi}{\alpha}) = src(\psi_k)$ and $tgt(\redel{\psi}{\alpha'}) = tgt(\psi_{k-1})$, by Lemma~\ref{rsl:ppterm-src} and Lemma~\ref{rsl:ppterm-tgt-successor} respectively. Thus we conclude.
Otherwise $\gamma = \gamma' + 1$; notice that $\gamma$ being a limit ordinal would contradict $\alpha$ being a successor one.
In this case $\redel{\psi}{\alpha'} = \redel{\psi_k}{\gamma'}$, thus we conclude by \ih\ on $\psi_k$.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-seq-mind}
Let $\psi$ be a \ppterm.
Then \\
$\begin{array}{rcl}
\mind{\psi} & = &
min(\sdepth{\redel{\psi}{\alpha}} \setsthat \alpha < \ppsteps{\psi})
\denotationDistributed{
\\
& = &
\posln{\,min_{\leq_{DL}} (\RPos{\redel{\psi}{\alpha}} \setsthat \alpha < \ppsteps{\psi})}
}
\\
& = &
min(\mind{\redel{\psi}{\alpha}} \setsthat \alpha < \ppsteps{\psi})
\end{array}$
\denotationDistributed{
, \\
where if $P$ is a set of positions, then $p \eqdef min_{\leq_{DL}} (P)$ is the element of $P$ verifying $p \leq_{DL} q$ if $q \in P$.}
\end{lemma}
\begin{proof}
We prove that $\mind{\psi} = min(\mind{\redel{\psi}{\alpha}} \setsthat \alpha < \ppsteps{\psi})$.
The rest of the statement follows immediately since it is trivial to verify $\sdepth{\redel{\psi}{\alpha}} = \mind{\redel{\psi}{\alpha}}$ for any $\alpha$; \confer\ Dfn.~\ref{dfn:dmin-imstep}.
We proceed by induction on $\psi$; \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
We define $\mindp{\psi} \eqdef min(\mind{\redel{\psi}{\alpha}} \setsthat \alpha < \ppsteps{\psi})$, so we must verify $\mind{\psi} = \mindp{\psi}$.
If $\psi$ is a one-step then the result holds immediately.
Assume $\psi = \psi_1 \comp \psi_2$.
In this case, \ih\ on $\psi_i$ yields $\mind{\psi_i} = \mindp{\psi_i}$ for each $i = 1,2$, and Dfn.~\ref{dfn:layer-pterm} implies $\mind{\psi} = min(\mind{\psi_1}, \mind{\psi_2})$.
Then it suffices to verify $\mindp{\psi} = min(\mindp{\psi_1}, \mindp{\psi_2})$.
From the definition of $\mindpfn$, it is immediate that $\mindp{\psi} \leq \mindp{\psi_i}$ for $i = 1,2$.
Assume $\mindp{\psi_1} \leq \mindp{\psi_2}$. Notice $\mindp{\psi} < \mindp{\psi_1}$ would imply the existence of some $\gamma$ verifying $\mindp{\redel{\psi}{\gamma}} < \mindp{\psi_1}$, contradicting either the definition of $\mindp{\psi_1}$ (if $\gamma < \ppsteps{\psi_1}$) or the assertion $\mindp{\psi_1} \leq \mindp{\psi_2}$ (otherwise). Hence $\mindp{\psi} = \mindp{\psi_1}$.
A similar argument for the case $\mindp{\psi_2} < \mindp{\psi_1}$ is enough to conclude.
If $\psi = \icomp \psi_i$, then an argument similar to that used for binary composition applies.
To verify that $\mindp{\psi} = min_{i < \omega}(\mindp{\psi_i})$, observe that $\mindp{\psi} \leq \mindp{\psi_i}$ for all $i$, and consider $n$ such that $\mindp{\psi_n} \leq \mindp{\psi_i}$ for all $i$.
Then we can contradict $\mindp{\psi} < \mindp{\psi_n}$ proceeding as in the previous case, hence $\mindp{\psi} = \mindp{\psi_n}$. Thus we conclude.
\end{proof}
\includeStandardisation{
The view of a \ppterm\ as the concatenation of its components will be used extensively to obtain standardisation results in Section~\ref{sec:peqence}, where the ability of separating a \ppterm\ in head (\ie\ first component) and tail (\ie\ the concatenation of components from the second one on) will be needed as well. This consideration motivates the following formalisation of the concept of tail of a \ppterm.
\begin{definition}
\label{dfn:ppterm-tail}
Let $\psi$ be a proof term. We define the \emph{tail} of $\psi$, notation $\redfrom{\psi}{1}$, as follows: \\
\begin{tabular}{@{$\ \ \bullet\ \ $}p{.9\textwidth}}
If $\psi$ is a one-step, then $\redfrom{\psi}{1} \eqdef tgt(\psi)$. \\
If $\psi = \psi_1 \comp \psi_2$ and $\psi_1$ is a one-step, then $\redfrom{\psi}{1} \eqdef \psi_2$. \\
If $\psi = \psi_1 \comp \psi_2$ and $\psi_1$ is not a one-step, then $\redfrom{\psi}{1} \eqdef \redfrom{\psi_1}{1} \comp \psi_2$. \\
If $\psi = \icomp \psi_i$ and $\psi_0$ is a one-step, then $\redfrom{\psi}{1} \eqdef \icomp \psi_{1+i}$. \\
If $\psi = \icomp \psi_i$ and $\psi_0$ is not a one-step, then $\redfrom{\psi}{1} \eqdef \redfrom{\psi_0}{1} \comp (\icomp \psi_{1+i})$. \\
\end{tabular}
\end{definition}
}
\denotationInOwnChapter{
\subsection{Denotation -- formal definition and proof of existence}
\label{sec:pterm-denotation-defs-existence}
}
\denotationDistributed{
\subsection{Denotation of \redseqs}
\label{sec:pterm-denotation}
}
In this section, we formalise the notion of a \pnpterm\ \emph{denoting} a \redseq, resorting to the definitions of length and $\alpha$-th component of \pnpterms, given in the presentation of such terms.
Then we prove the existence, for any \redseq\ having a countable ordinal length, of a \pnpterm\ which denotes it.
\denotationInOwnChapter{
As we have discussed in the introduction to Section~\ref{sec:pterm-denotation}, denotation of a \redseq\ is not unique.
In the next subsection, we will investigate how to characterise the proof terms denoting the same \redseq.}
\begin{definition}[Denotation for reduction steps]
\label{dfn:redstep-denotation}
Let $a = \langle t, p, \mu \rangle$ be a reduction step, and $\psi$ a one-step.
Then $\psi$ \emph{denotes} $a$ iff all the following apply: $src(\psi) = t$, $tgt(\psi) = tgt(a)$, and $\psi(p) = \mu$, therefore $\sdepth{a} = \mind{\psi}$.
\end{definition}
\begin{definition}[Mapping from one-steps to reduction steps]
\label{dfn:rstepden}
Let \trst\ be a \TRS. We define the mapping $\rstepdenfn$ from the set of one-steps for \trst\ to the set of reduction steps for \trst, as follows:
$\rstepden{\psi} \eqdef \langle src(\psi), \RPos{\psi}, \psi(\RPos{\psi}) \rangle$.
\end{definition}
\begin{lemma}
\label{rsl:rstepden-denotes}
Let $\psi$ be a one-step and $\stepa$ a reduction step.
Then $\psi$ denotes $a$ iff $a = \rstepden{\psi}$.
\end{lemma}
\begin{proof}
We prove each direction of the biconditional.
\noindent
$\Rightarrow )$:
Let us say $a = \langle t, p, \mu \rangle$. Hypotheses imply immediately $t = src(\psi)$, and also $\psi(p) = \mu$, so that $p = \RPos{\psi}$ and $\mu = \psi(\RPos{\psi})$. Thus we conclude.
\noindent
$\Leftarrow )$:
Let us say $\rstepden{\psi} = \langle t, p, \mu \rangle$ and $\mu: l \to h$.
Then it is immediate from Dfn.~\ref{dfn:rstepden} to verify $src(\psi) = t$ and $\psi(p) = \mu$.
In turn, observe that $tgt(\psi) = \repl{\psi}{h[t_1, \ldots, t_m]}{p}$ where $\subtat{\psi}{p} = \mu(t_1, \ldots, t_m)$, and
$t = src(\psi) = \repl{\psi}{l[t_1, \ldots, t_m]}{p}$, so that it is straightforward to verify $tgt(\rstepden{\psi}) = tgt(\psi)$. Thus we conclude.
\end{proof}
\begin{definition}[Denotation for \redseqs]
\label{dfn:redseq-denotation}
Let \reda\ be a \redseq, and $\psi$ a \pnpterm.
We will say that $\psi$ \emph{denotes} \reda\ iff $\ppsteps{\psi} = \redln{\reda}$, $src(\psi) = src(\reda)$ and $\redel{\psi}{\alpha}$ denotes $\redel{\reda}{\alpha}$ for all $\alpha < \redln{\reda}$.
\end{definition}
\begin{lemma}
\label{rsl:redseq-denotation-implications}
Let \reda\ be a \redseq, and $\psi$ a \pnpterm, such that $\psi$ denotes \reda.
Then $\mind{\psi} = \mind{\reda}$, $\psi$ is convergent iff \reda\ is, and in that case, $tgt(\psi) = tgt(\reda)$.
\end{lemma}
\begin{proof}
If $\psi \in \iSigmaTerms$, then the result holds immediately.
Otherwise, the result about $\mindfn$ stems immediately from \reflem{ppterm-seq-mind}.
We prove the result about convergence.
Assume that $\ppsteps{\psi}$ is a limit ordinal, then $\psi = \icomp \psi_i$; \confer\ Lem.~\ref{rsl:steps-ordinal-coherence} and Lem.~\ref{rsl:ptinfC-iff-limit}.
Assume \reda\ convergent, consider some $k < \omega$, and $\alpha$ such that $\sdepth{\redel{\reda}{\beta}} > k$ if $\beta > \alpha$.
\refLem{ordinal-lt-infAdd-then-unique-representation} implies that $\alpha = \sum_{i < n} \ppsteps{\psi_i} + \gamma$ and $\gamma < \ppsteps{\psi_n}$ for some $n$; so that $\alpha < \sum_{i \leq n} \ppsteps{\psi_i}$.
Consider $j > n$, and $\gamma < \ppsteps{\psi_j}$. Observe $\redel{\psi_j}{\gamma} = \redel{\psi}{\beta}$ where $\beta = \sum_{i < j} \ppsteps{\psi_i} + \gamma$, so that $\beta \geq \sum_{i \leq n} \ppsteps{\psi_i} > \alpha$. Therefore $\mind{\redel{\psi_j}{\gamma}} = \mind{\redel{\psi}{\beta}} = \sdepth{\redel{\reda}{\beta}} > k$.
Hence \reflem{ppterm-seq-mind} implies that $\mind{\psi_j} > k$. Consequently, $\psi$ is convergent.
Conversely, assume $\psi$ convergent, let $k < \omega$, consider $n < \omega$ such that $\mind{\psi_j} > k$ if $j > n$.
Let $\alpha \eqdef \sum_{i \leq n} \ppsteps{\psi_i}$, and take $\beta$ such that $ \alpha < \beta < \redln{\reda}$.
Then Lem.~\ref{rsl:ordinal-lt-infAdd-then-unique-representation} implies $\beta = \sum_{i < j} \ppsteps{\psi_i} + \gamma$ and $\gamma < \ppsteps{\psi_j}$, moreover, $\beta > \alpha$ implies $j > n$.
Hence $\sdepth{\redel{\reda}{\beta}} = \mind{\redel{\psi_j}{\gamma}} > k$ by \reflem{ppterm-seq-mind}. Consequently, the requirement about depths in the characterisation of convergent \redseqs, \ie\ condition~(\ref{it:dfn-sred-depth}) in Dfn.~\ref{dfn:sred}, holds for \reda.
To prove the existence of $\lim_{\alpha \to \redln{\reda}} tgt(\redel{\reda}{\alpha})$, \ie\ condition~(\ref{it:dfn-sred-limit-existence}) in Dfn.~\ref{dfn:sred}, it suffices to observe that Lem.~\ref{rsl:mind-big-then-tdist-little}:(\ref{it:convergent-then-has-tgt}) implies that $tgt(\psi)$ is defined, and in turn Lem.~\ref{rsl:ppterm-tgt-limit} implies the desired limit to equal $tgt(\psi)$. Hence $\reda$ is convergent.
If $\ppsteps{\psi}$ is a successor ordinal, then assuming $\reda$ is convergent, a straightforward induction on $\psi$ suffices to prove that $\psi$ is convergent as well; observe that Lem.~\ref{rsl:steps-ordinal-coherence} and Lem~\ref{rsl:ptinfC-iff-limit} imply that only one-step and binary concatenation must be considered. For the other direction, it is enough to observe that $\redln{\reda}$ being a successor ordinal implies immediately convergence of $\reda$.
Finally, the result about targets stems immediately from \reflem{ppterm-tgt-limit} and \reflem{ppterm-tgt-successor}.
\end{proof}
\begin{proposition}
\label{rsl:denotation-existence}
Let $\reda$ be a \redseq\ having a countable length. Then there exists a \pnpterm\ $\psi$ such that $\psi$ denotes $\reda$.
\end{proposition}
\begin{proof}
We proceed by induction on $\redln{\reda}$.
If $\redln{\reda} = 0$, \ie\ $\reda = \redid{t}$, then it suffices to take $\psi \eqdef t$.
Assume that $\redln{\reda} = 1$. Let us say $\redel{\reda}{0} = \langle t, p, \mu \rangle$ where $\mu: l \to h$, implying that $\subtat{t}{p} = l[t_1, \ldots, t_m]$.
Take $\psi \eqdef \repl{t}{\mu(t_1, \ldots, t_m)}{p}$.
It is immediate to verify that $\psi$ is a \ppterm\ verifying $\ppsteps{\psi} = 1$. Moreover, a simple analysis yields $src(\psi) = src(\redel{\reda}{0}) = src(\reda) = t$.
Furthermore, $\psi(p) = \mu$, and $tgt(\psi) = tgt(\redel{\reda}{0}) = \repl{t}{h[t_1, \ldots, t_m]}{p}$; therefore $\redel{\psi}{0} = \psi$ denotes $\redel{\reda}{0}$. Hence $\psi$ denotes $\reda$.
Assume $\redln{\reda} = \alpha+1$ and $\alpha > 0$.
In this case, applying twice \ih\ yields the existence of $\psi_1$, $\psi_2$ such that $\psi_1$ denotes $\redupto{\reda}{\alpha}$ and $\psi_2$ denotes $\redsublt{\reda}{\alpha}{\alpha+1}$.
Then a straightforward analysis allows to obtain that $\psi \eqdef \psi_1 \comp \psi_2$ denotes $\reda$.
Assume $\alpha \eqdef \redln{\reda}$ is a limit ordinal; recall that $\alpha$ is countable. Then Prop.~\ref{rsl:cofinality-omega} implies $\alpha = \sum_{i < \omega} \alpha_i$ where $\alpha_i < \alpha$ for all $i < \omega$.
Therefore, for any $n < \omega$, \ih\ can be applied to obtain some $\psi_n$ denoting $\redsublt{\reda}{\sum_{i < n} \alpha_i}{\sum_{i \leq n} \alpha_i}$.
We take $\psi \eqdef \icomp \psi_i$.
Let $n < \omega$. It is easy to verify that $\redsublt{\reda}{\sum_{i < n} \alpha_i}{\sum_{i \leq n} \alpha_i}$ is convergent, then Lem.~\ref{rsl:redseq-denotation-implications} implies
$tgt(\psi_n)
= tgt(\redsublt{\reda}{\sum_{i < n} \alpha_i}{\sum_{i \leq n} \alpha_i})
= src(\redsublt{\reda}{\sum_{i \leq n} \alpha_i}{\sum_{i \leq n+1} \alpha_i}
= src(\psi_{n+1})$; \confer\ conditions about sources and targets in Dfn.~\ref{dfn:sred}.
Hence $\psi$ is a well-formed proof term.
Recalling that $\redln{\redsublt{\reda}{\sum_{i < n} \alpha_i}{\sum_{i \leq n} \alpha_i}} = \alpha_n$, it is straightforward to obtain $\ppsteps{\psi} = \redln{\reda} = \alpha$.
Moreover, $src(\psi) = src(\psi_0) = src(\redupto{\reda}{\alpha_0}) = src(\reda)$, recall that $\psi_0$ denotes $\redupto{\reda}{\alpha_0}$.
Let $\beta < \alpha$. Then Lem.~\ref{rsl:ordinal-lt-infAdd-then-unique-representation} implies the existence of unique $k$ and $\gamma$ such that $\beta = \sum_{i < k} \alpha_i + \gamma$ and $\gamma < \alpha_k$.
Therefore $\redel{\psi}{\beta} = \redel{\psi_k}{\gamma}$ and $\redel{\reda}{\beta} = \redel{\redsublt{\reda}{\sum_{i<k} \alpha_i}{\sum_{i \leq k} \alpha_i}}{\gamma}$, \confer\ Dfn.~\ref{dfn:ppterm-component} and Dfn.~\ref{dfn:redseq-section}.
Hence $\psi_k$ denoting $\redsublt{\reda}{\sum_{i<k} \alpha_i}{\sum_{i \leq k} \alpha_i}$ implies that $\redel{\psi}{\beta}$ denotes $\redel{\reda}{\beta}$.
Consequently, we conclude.
\end{proof}
\subsection{Uniqueness of denotation}
\label{sec:pterm-denotation-uniqueness}
In this section we will prove the claim we made at the beginning of Section~\ref{sec:pterm-denotation}: \emph{rebracketing equivalence}, which is the result of restricting the \peqence\ relation introduced in Section~\ref{sec:peqence} by allowing only associativity instances as basic equations, is an adequate syntactic counterpart of the relation of ``denoting the same \redseq'', \ie\ \emph{denotational equivalence}, between \ppterms.
In the following we will give formal definitions for the concepts of denotational and rebracketing equivalence, and subsequently prove that the defined relations coincide.
\begin{definition}
\label{dfn:deneq}
Let $\psi$, $\phi$ be \pnpterms.
We say that $\psi$ and $\phi$ are \emph{denotationally equivalent}, notation $\psi \deneq \phi$, iff
either $\ppsteps{\psi} = \ppsteps{\phi} = 0$ and $\psi = \phi$,
or $\ppsteps{\psi} = \ppsteps{\phi} > 0$ and $\redel{\psi}{\alpha} = \redel{\phi}{\alpha}$ for all $\alpha < \ppsteps{\psi}$.
\end{definition}
\begin{definition}
\label{dfn:layer-breqe}
\label{dfn:layer-breq}
Let $\alpha$ be a countable ordinal.
We define the \emph{$\alpha$-th level of base rebracketing equivalence} relation, notation $\layerbreqe{\alpha}$, on the set of \pnpterms, as follows.
Given $\psi$ and $\phi$ \pnpterms, $\psi \layerbreqe{\alpha} \phi$ iff the equation $\psi \layerpeqx{\alpha} \phi$ can be obtained by means of the equational logic system whose basic equations are the instance \peqassoc\ described in Dfn.~\ref{dfn:layer-peqe}, and whose equational rules are \eqlrefl, \eqleqn, \eqlsymm, \eqltrans, \eqlcomp\ and \eqlinfcomp, described also in Dfn.~\ref{dfn:layer-peqe}.
We also define the \emph{$\alpha$-th level of rebracketing equivalence} relation, notation $\layerbreq{\alpha}$, on the set of \pnpterms, analogously, the only difference being that a rule is added, namely the version of the \eqllim\ rule which results from changing, in the premises, the references to the $\layerpeqe{\alpha_k}$ and $\layerpeqe{\beta_k}$ relations, to $\layerbreqe{\alpha_k}$ and $\layerbreqe{\beta_n}$ respectively.
\end{definition}
\begin{definition}
\label{dfn:breqe}
\label{dfn:breq}
Let $\psi$, $\phi$ be \pnpterms. We say that $\psi$ and $\phi$ are (base) rebracketing equivalent, notation ($\psi \breqe \phi$) $\psi \breq \phi$, iff ($\psi \layerbreqe{\alpha} \phi$) $\psi \layerbreq{\alpha} \phi$ for some $\alpha < \omega_1$.
\end{definition}
Observe that all the following inclusions hold where $\alpha$ is any countable ordinal: $\layerbreqe{\alpha} \subseteq \layerbreq{\alpha}$, $\layerbreqe{\alpha} \subseteq \layerpeqe{\alpha}$, $\layerbreq{\alpha} \subseteq \layerpeq{\alpha}$, and consequently $\breqe \subseteq \breq$, $\breqe \subseteq \peqe$ and $\breq \subseteq \peq$.
Therefore, several results stated for \peqence\ hold also for rebracketing equivalence. Particularly, properties proved for the $\peqe$ relation also apply to $\breqe$.
\begin{lemma}
\label{rsl:layerpterm-ppsteps}
Let $\psi$ a \ppterm, and $\alpha$ such that $\psi \in \layerpterm{\alpha}$. Then $\exists n < \omega$ such that $\alpha = \ppsteps{\psi} + n$.
Moreover, if $\alpha$ is a limit ordinal, then $n = 0$, \ie\ $\alpha = \ppsteps{\psi}$.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$.
If $\alpha = 1$ then $\psi$ is a one-step, and then $\ppsteps{\psi} = 1 = \alpha$.
Assume $\alpha$ is a successor ordinal and $\alpha > 1$.
In this case, Lem.~\ref{rsl:ptinfC-iff-limit} and Lem.~\ref{rsl:ptmstep-iff-one} imply that $\psi = \psi_1 \comp \psi_2$, $\psi_i \in \layerpterm{\alpha_i}$ for $i = 1.2$, $\alpha_2$ is successor, and $\alpha = \alpha_1 + \alpha_2 + 1$.
\Ih\ implies $\alpha_1 = \ppsteps{\psi_1} + n_1$ and $\alpha_2 = \ppsteps{\psi_2} + n_2$.
If $\ppsteps{\psi_2} < \omega$, then $\alpha = \ppsteps{\psi} + n_1 + n_2 + 1$, otherwise $\alpha = \ppsteps{\psi} + n_2 + 1$. In either case the conclusion holds, thus we conclude.
Assume that $\alpha$ is a limit ordinal, so that Lem.~~\ref{rsl:ptinfC-iff-limit} implies $\psi = \icomp \psi_i$ and $\alpha = \sum_{i < \omega} \alpha_i$ where $\psi_i \in \layerpterm{\alpha_i}$ for all $i < \omega$.
Observe $\alpha_i < \alpha$ for all $i$.
Then we can apply \ih\ on each $i$ obtaining $\alpha_i = \ppsteps{\psi_i} + n_i$, so that proving $\sum_{i < \omega} \ppsteps{\psi_i} + n_i = \sum_{i < \omega} \ppsteps{\psi_i}$ suffices to conclude.
Let $k < \omega$. Observe $\sum_{i < k} \ppsteps{\psi_i} + n_i \leq \sum_{i < k} \ppsteps{\psi_i} + \sum_{i < k} n_i < \sum_{i < k} \ppsteps{\psi_i} + \omega$.
On the other hand, $\sum_{i < \omega} \ppsteps{\psi_i} = \sum_{i < k} \ppsteps{\psi_i} + \sum_{i < \omega} \ppsteps{\psi_{k+i}} \geq \sum_{i < k} \ppsteps{\psi_i} + \omega$. Then $\sum_{i < k} \ppsteps{\psi_i} + n_i < \sum_{i < \omega} \ppsteps{\psi_i}$. Consequently $\sum_{i < \omega} \ppsteps{\psi_i} + n_i \leq \sum_{i < \omega} \ppsteps{\psi_i}$.
We conclude by observing that it is straightforward to obtain $\sum_{i < \omega} \ppsteps{\psi_i} \leq \sum_{i < \omega} \ppsteps{\psi_i} + n_i$.
\end{proof}
\begin{lemma}
\label{rsl:ptinfC-iff-ppsteps-limit}
Let $\psi$ be a \ppterm. Then $\ppsteps{\psi}$ is a limit ordinal iff $\psi$ is an infinite concatenation.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \in \layerpterm{\alpha}$; \confer\ Dfn.~\ref{dfn:layer-peqe}.
If $\psi$ is a one-step, then we conclude immediately.
If $\psi = \psi_1 \comp \psi_2$ and it is not an infinite concatenation, then $\psi_2$ is neither. Therefore we can apply \ih\ on $\psi_2$ obtaining that $\ppsteps{\psi_2}$ is a successor ordinal. We conclude by recalling that $\ppsteps{\psi} = \ppsteps{\psi_1} + \ppsteps{\psi_2}$.
Finally, if $\psi$ is an infinite concatenation, then Lem.~\ref{rsl:ptinfC-iff-limit} implies that $\psi \in \layerpterm{\alpha}$ where $\alpha$ is a limit ordinal. In turn, Lem.~\ref{rsl:layerpterm-ppsteps} implies that $\ppsteps{\psi} = \alpha$.
\end{proof}
\begin{lemma}
\label{rsl:ppterm-binC-partition}
Let $\psi$ be a \ppterm, $\alpha$ an ordinal verifying $0 < \alpha < \ppsteps{\psi}$, and $\beta$ such that $\psi \in \layerpterm{\beta}$.
Then there exist $\phi$, $\chi$ such that $\psi \breqe \phi \comp \chi$ and $\ppsteps{\phi} = \alpha$.
Moreover, if $\phi \in \layerpterm{\gamma}$ and $\chi \in \layerpterm{\delta}$, then $\gamma < \beta$ and $\delta \leq \beta$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$.
If $\psi \in \iSigmaTerms$ or $\psi$ is a one-step, then no $\alpha$ verifies the hypotheses.
Assume $\psi = \psi_1 \comp \psi_2$, so that $\beta = \beta_1 + \beta_2 + 1$ where $\psi_i \in \layerpterm{\beta_i}$ for $i = 1,2$.
\begin{itemize}
\item
If $\ppsteps{\psi_1} < \alpha$, so that $\alpha = \ppsteps{\psi_1} + \alpha'$, then \ih\ on $\psi_2$ yields the existence of $\phi_2, \chi_2$ satisfying $\psi_2 \breqe \phi_2 \comp \chi_2$, $\ppsteps{\phi_2} = \alpha'$, $\gamma_2 < \beta_2$ and $\delta \leq \beta_2$, where $\phi_2 \in \layerpterm{\gamma_2}$ and $\chi_2 \in \layerpterm{\delta}$.
Therefore, $\psi \breqe \psi_1 \comp (\phi_2 \comp \chi_2) \breqe (\psi_1 \comp \phi_2) \comp \chi_2$ and $\ppsteps{\psi_1 \comp \phi_2} = \ppsteps{\psi_1} + \alpha' = \alpha$. Moreover, $\psi_1 \comp \phi_2 \in \layerpterm{\gamma}$ where $\gamma = \beta_1 + \gamma_2 + 1 < \beta_1 + \beta_2 + 1 = \beta$, and $\delta \leq \beta_2 < \beta$.
\item
If $\ppsteps{\psi_1} = \alpha$ then the result holds trivally.
\item
If $\ppsteps{\psi_1} > \alpha$, then \ih\ on $\psi_1$ yields $\psi_1 \breqe \phi_1 \comp \chi_1$, $\ppsteps{\phi_1} = \alpha$, $\gamma < \beta_1$ and $\delta_1 \leq \beta_1$, where $\phi_1 \in \layerpterm{\gamma}$ and $\chi_1 \in \layerpterm{\delta_1}$.
Therefore $\psi \breqe (\phi_1 \comp \chi_1) \comp \psi_2 \breqe \phi_1 \comp (\chi_1 \comp \psi_2)$.
Moreover, $\gamma < \beta_1 < \beta$, and $\chi_1 \comp \psi_2 \in \layerpterm{\delta}$ where $\delta = \delta_1 + \beta_2 + 1 \leq \beta_1 + \beta_2 + 1 = \beta$.
\end{itemize}
Assume $\psi = \icomp \psi_i$, so that $\ppsteps{\psi} = \sum_{i < \omega} \ppsteps{\psi_i}$.
In this case,
Lem~\ref{rsl:ptinfC-iff-limit} and Lem~\ref{rsl:layerpterm-ppsteps} imply that $\beta$ is a limit ordinal, and therefore $\beta = \ppsteps{\psi}$.
Moreover, Lem~\ref{rsl:ordinal-lt-infAdd-then-unique-representation} implies $\alpha = \sum_{i < n} \ppsteps{\psi_i} + \alpha'$ where $\alpha' < \ppsteps{\psi_n}$, for some $n$ and $\alpha'$.
\Ih\ on $\psi_n$ yields $\psi_n \breqe \phi_n \comp \chi_n$ such that $\ppsteps{\phi_n} = \alpha'$; observe that $\ppsteps{\chi_n} \leq \ppsteps{\psi_n}$.
Therefore \\
$\begin{array}{rcl}
\psi & \breqe & ((\psi_0 \comp \ldots \comp \psi_{n-1}) \comp \psi_n) \comp \icomp \psi_{n+1+i} \\
& \breqe & ((\psi_0 \comp \ldots \comp \psi_{n-1}) \comp (\phi_n \comp \chi_n)) \comp \icomp \psi_{n+1+i} \\
& \breqe & ((\psi_0 \comp \ldots \comp \psi_{n-1} \comp \phi_n) \comp \chi_n) \comp \icomp \psi_{n+1+i} \\
& \breqe & (\psi_0 \comp \ldots \comp \psi_{n-1} \comp \phi_n) \comp (\chi_n \comp \icomp \psi_{n+1+i})
\end{array}$ \\
where $\ppsteps{\psi_0 \comp \ldots \comp \psi_{n-1} \comp \phi_n} = \sum_{i < n} \ppsteps{\psi_i} + \alpha' = \alpha$.
Moreover, if $\psi_0 \comp \ldots \comp \psi_{n-1} \comp \phi_n \in \layerpterm{\gamma}$, then Lem.~\ref{rsl:layerpterm-ppsteps} implies the existence of some $k < \omega$ such that
$\gamma
= \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_{n-1}} + \alpha' + k
< \ppsteps{\psi_0} + \ldots + \ppsteps{\psi_{n-1}} + \ppsteps{\psi_n} + \omega
\leq \ppsteps{\psi} = \beta$.
On the other hand, notice that $\chi_n \comp \icomp \psi_{n+1+i}$ is an infinitary concatenation, so that $\chi_n \comp \icomp \psi_{n+1+i} \in \layerpterm{\delta}$ implies $\delta$ to be a limit ordinal; \confer\ Lem.~\ref{rsl:ptinfC-iff-limit}.
Therefore, recalling that $\ppsteps{\chi_n} \leq \ppsteps{\psi_n}$, Lem.~\ref{rsl:layerpterm-ppsteps} yields
$\delta
= \ppsteps{\chi_n} + \sum_{i < \omega} \ppsteps{\psi_{n+1+i}}
\leq \sum_{i < \omega} \ppsteps{\psi_{n+i}}
\leq \ppsteps{\psi} = \beta$.
\end{proof}
\begin{lemma}
\label{rsl:deneq-then-same-tgt}
Let $\psi \deneq \phi$, such that both are convergent.
Then $tgt(\psi) = tgt(\phi)$.
\end{lemma}
\begin{proof}
Easy, \confer\ Lem.~\ref{rsl:ppterm-tgt-successor} and Lem~\ref{rsl:ppterm-tgt-limit}.
\end{proof}
\begin{lemma}
\label{rsl:deneq-binC-right}
Let $\psi \comp \phi \deneq \psi' \comp \phi'$ and $\psi \deneq \psi'$.
Then $\phi \deneq \phi'$.
\end{lemma}
\begin{proof}
Observe that definition of \ppterms\ implies that $\ppsteps{\phi} > 0$ and $\ppsteps{\phi'} > 0$.
Given $\ppsteps{\psi \comp \phi} = \ppsteps{\psi' \comp \phi'}$ and $\ppsteps{\psi} = \ppsteps{\psi'}$, properties of ordinals yield $\ppsteps{\phi} = \ppsteps{\phi'}$.
We conclude by observing that for any suitable $\alpha$,
$\redel{\phi}{\alpha} =
\redel{(\psi \comp \phi)}{\ppsteps{\psi} + \alpha} =
\redel{(\psi' \comp \phi')}{\ppsteps{\psi'} + \alpha} =
\redel{\phi'}{\alpha}$.
\end{proof}
\begin{proposition}
\label{rsl:breq-then-deneq}
Let $\psi$, $\phi$ be \pnpterms\ such that $\psi \breq \phi$.
Then $\psi \deneq \phi$.
\end{proposition}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \layerbreq{\alpha} \phi$.
We analyse the rule used in the last step of the rebracketing equivalence derivation.
For the rules \eqlrefl, \eqlsymm\ and \eqltrans, the result holds immediately.
Assume that the last used rule in the derivation is \eqleqn, so that
$\psi = (\psi_1 \comp \psi_2) \comp \psi_3$ and $\phi = \psi_1 \comp (\psi_2 \comp \psi_3)$.
In this case we can obtain $\ppsteps{\psi} = \ppsteps{\phi} > 0$ immediately.
Let $\gamma < \ppsteps{\psi}$.
If $\gamma < \ppsteps{\psi_1}$, then $\redel{\psi}{\gamma} = \redel{(\psi_1 \comp \psi_2)}{\gamma} = \redel{\psi_1}{\gamma} = \redel{\phi}{\gamma}$. The other cases, \ie\ $\ppsteps{\psi_1} \leq \gamma < \ppsteps{\psi_1} + \ppsteps{\psi_2}$ and $\ppsteps{\psi_1} + \ppsteps{\psi_2} \leq \gamma$, admit analogous arguments.
Assume that the last used rule is \eqlinfcomp, so that $\psi = \icomp \psi_i$, $\phi = \icomp \phi_i$, and $\psi_n \layerbreq{\beta_n} \phi_n$ where $\beta_n < \alpha$, for all $n < \omega$.
Then \ih\ on each $\beta_n$ implies $\psi_n \deneq \phi_n$. Therefore we obtain $\ppsteps{\psi} = \ppsteps{\phi} > 0$ immediately.
To conclude it is enough to observe, for any $\gamma < \ppsteps{\psi}$, that Lem.~\ref{rsl:ordinal-lt-infAdd-then-unique-representation} implies $\gamma = \sum_{i < n} \ppsteps{\psi_i} + \gamma_0$ where $\gamma_0 < \ppsteps{\psi_n}$, then (given \ih\ on each $\psi_i \layerbreq{\beta_i} \phi_i$) $\redel{\psi}{\gamma} = \redel{\psi_n}{\gamma_0} = \redel{\phi_n}{\gamma_0} = \redel{\phi}{\gamma}$.
If the last used rule is \eqlcomp, then a similar argument applies.
Assume that the rule used in the last derivation step is \eqllim.
Assume for contradiction $\ppsteps{\phi} > \ppsteps{\psi}$, so that the step $\redel{\phi}{\ppsteps{\psi}}$ exists.
Consider $k \eqdef max(\mind{\redel{\phi}{0}}, \mind{\redel{\phi}{\ppsteps{\psi}}})$. Then there exist $\chi_k$, $\phi'_k$, $\psi'_k$ verifying
$\phi \layerbreqe{\alpha_k} \chi_k \comp \phi'_k$, $\psi \layerbreqe{\beta_k} \chi_k \comp \psi'_k$, $\mind{\phi'_k} > k \geq \mind{\redel{\phi}{\ppsteps{\psi}}}$, $\mind{\psi'_k} > k$, $\alpha > \alpha_k$, and $\alpha > \beta_k$.
Recalling that $\layerbreqe{\gamma} \,\subseteq\, \layerbreq{\gamma}$ for any $\gamma$, we can apply \ih\ to $\alpha_k$ obtaining $\phi \deneq \chi_k \comp \phi'_k$, so that $\redel{\phi}{\ppsteps{\psi}} = \redel{(\chi_k \comp \phi'_k)}{\ppsteps{\psi}}$.
Therefore, assuming $\ppsteps{\psi} = \ppsteps{\chi_k} + \gamma$ would imply $\redel{\phi'_k}{\gamma} = \redel{\phi}{\ppsteps{\psi}}$ contradicting $\mind{\phi'_k} > \mind{\redel{\phi}{\ppsteps{\psi}}}$; \confer\ Lem.~\ref{rsl:ppterm-seq-mind}.
Then $\ppsteps{\psi} < \ppsteps{\chi_k}$.
On the other hand, \ih\ can be applied also to $\beta_k$, yielding $\psi \deneq \chi_k \comp \psi'_k$, and therefore $\ppsteps{\psi} \geq \ppsteps{\chi_k}$, \ie\ a contradiction.
Consequently $\ppsteps{\phi} \leq \ppsteps{\psi}$.
A similar argument yields $\ppsteps{\psi} \leq \ppsteps{\phi}$. Thus $\ppsteps{\psi} = \ppsteps{\phi}$.
Let $\gamma < \ppsteps{\psi}$. Then there exists $\chi$, $\psi'$, $\phi'$ such that
$\psi \layerbreqe{\alpha_0} \chi \comp \psi'$, $\phi \layerbreqe{\beta_0} \chi \comp \phi'$, $\mind{\psi'} > \mind{\redel{\psi}{\gamma}}$, $\mind{\phi'} > \mind{\redel{\psi}{\gamma}}$, $\alpha_0 < \alpha$ and $\beta_0 < \alpha$.
Then \ih\ on $\alpha_0$ and $\beta_0$ yields $\psi \deneq \chi \comp \psi'$ and $\phi \deneq \chi \comp \phi'$, so that $\redel{\psi}{\gamma} = \redel{(\chi \comp \psi')}{\gamma}$ and $\redel{\phi}{\gamma} = \redel{(\chi \comp \phi')}{\gamma}$.
Observing that $\gamma = \ppsteps{\chi} + \gamma_0$ would imply $\redel{\psi}{\gamma} = \redel{\psi'}{\gamma_0}$, and then $\mind{\psi'} \leq \mind{\redel{\psi}{\gamma}}$ (\confer\ Lem.~\ref{rsl:ppterm-seq-mind}) thus producing a contradiction, we obtain $\gamma < \ppsteps{\chi}$. Then $\redel{\psi}{\gamma} = \redel{\chi}{\gamma}$, and also $\redel{\phi}{\gamma} = \redel{\chi}{\gamma}$. Hence $\redel{\psi}{\gamma} = \redel{\phi}{\gamma}$.
\end{proof}
\begin{proposition}
\label{rsl:deneq-then-breq}
Let $\psi$, $\phi$ such that $\psi \deneq \phi$.
Then $\psi \breq \phi$.
\end{proposition}
\begin{proof}
We proceed by induction on $\pair{\alpha}{\beta}$ such that $\psi \in \layerpterm{\alpha}$ and $\phi \in \layerpterm{\beta}$.
If $\psi \in \iSigmaTerms$, so that $\ppsteps{\psi} = 0$, then $\psi \deneq \phi$ implies $\psi = \phi$, hence we conclude immediately.
If $\psi$ is a one-step, so that $\ppsteps{\psi} = 1$, then $\psi \deneq \phi$ implies $\psi = \redel{\psi}{0} = \redel{\phi}{0} = \phi$.
Assume $\psi = \psi_1 \comp \psi_2$ and that it is not an infinite concatenation.
In this case, $\ppsteps{\psi} = \ppsteps{\phi} > 1$ is a successor ordinal, so that $\phi = \phi_1 \comp \phi_2$ and it is neither an infinite concatenation; \confer\ Lem.~\ref{rsl:ptinfC-iff-ppsteps-limit}.
Observe that $\alpha = \alpha_1 + \alpha_2 + 1$ and $\beta = \beta_1 + \beta_2 + 1$, where $\psi_i \in \layerpterm{\alpha_i}$ and $\phi_i \in \layerpterm{\beta_i}$ for $i = 1,2$.
We analyse the different cases arising from the comparison between $\ppsteps{\psi_1}$ and $\ppsteps{\phi_1}$.
\begin{itemize}
\item
Assume $\ppsteps{\psi_1} < \ppsteps{\phi_1}$.
In this case we apply Lem.~\ref{rsl:ppterm-binC-partition}, obtaining that $\phi_1 \breq \chi_1 \comp \chi_2$ and $\ppsteps{\chi_1} = \ppsteps{\psi_1}$ for some \ppterms\ $\chi_1 \in \layerpterm{\gamma_1}$ and $\chi_2 \in \layerpterm{\gamma_2}$, and moreover, that $\gamma_1 < \beta_1$ and $\gamma_2 \leq \beta_1$.
Therefore $\phi \ \breq \ (\chi_1 \comp \chi_2) \comp \phi_2 \ \breq \ \chi_1 \comp (\chi_2 \comp \phi_2)$, and hence Prop.~\ref{rsl:breq-then-deneq} and hypotheses yield $\psi = \psi_1 \comp \psi_2 \ \deneq \ \chi_1 \comp (\chi_2 \comp \phi_2) \deneq \phi$.
Observe that for any $\beta < \ppsteps{\psi_1}$, $\redel{\psi_1}{\beta} = \redel{\psi}{\beta} = \redel{\phi}{\beta} = \redel{(\chi_1 \comp (\chi_2 \comp \phi_2))}{\beta} = \redel{\chi_1}{\beta}$; consequently, $\psi_1 \deneq \chi_1$.
In turn, Lem.~\ref{rsl:deneq-binC-right} yields $\psi_2 \deneq \chi_2 \comp \phi_2$.
Observing that $\alpha_i < \alpha$ for $i = 1,2$ suffices to enable the application of \ih\ to both $\psi_1 \deneq \chi_1$ and $\psi_2 \deneq \chi_2 \comp \phi_2$. Therefore, we conclude by \eqlcomp, \eqlsymm\ and \eqltrans.
\item
Assume $\ppsteps{\psi_1} > \ppsteps{\phi_1}$. In this case, an analysis similar to that of the previous case yields $\psi_1 \breqe \chi_1 \comp \chi_2$ such that $\ppsteps{\chi_1} = \ppsteps{\phi_1}$, $\gamma_1 < \alpha_1$ and $\gamma_2 \leq \alpha_1$ where $\chi_i \in \layerpterm{\gamma_i}$ for $i = 1,2$; therefore $\chi_1 \comp (\chi_2 \comp \psi_2) \deneq \psi \deneq \phi = \phi_1 \comp \phi_2$; and consequently $\chi_1 \deneq \phi_1$ and $\chi_2 \comp \psi_2 \deneq \phi_2$.
Observe $\gamma_1 < \alpha_1 < \alpha$.
On the other hand, $\chi_2 \comp \psi_2 \in \layerpterm{\delta}$ where $\delta = \gamma_2 + \alpha_2 + 1 \leq \alpha_1 + \alpha_2 + 1 = \alpha$, and $\beta_2 < \beta$.
Therefore, \ih\ can be applied to both $\chi_1 \deneq \phi_1$ and $\chi_2 \comp \psi_2 \deneq \phi_2$, so that we conclude as in the previous case.
\item
Assume $\ppsteps{\psi_1} = \ppsteps{\phi_1}$. Then a simple analysis of the components of $\psi_1$ and $\phi_1$ yields $\psi_1 \deneq \phi_1$. In turn, this assertion allows to apply Lem.~\ref{rsl:deneq-binC-right} to obtain $\psi_2 \deneq \phi_2$.
Applying \ih\ to both $\psi_i$ we obtain $\psi_1 \breq \phi_1$ and $\psi_2 \breq \phi_2$. Hence we conclude by \eqlcomp.
\end{itemize}
Assume $\psi = \icomp \psi_i$.
In this case, a simple argument based on Lem.~\ref{rsl:ptinfC-iff-ppsteps-limit} yields $\phi = \icomp \phi_i$.
As the verification for this case involves a great number of technical details, we describe the idea first.
We define a \ppterm\ $\chi = \icomp \chi_i$ enjoying the following properties: $\psi \breq \chi$, and $\chi_n \deneq \phi_n$ for all $n < \omega$. The \eqllim\ rule is used in the last step of the derivation $\psi \breq \chi$, verifying that the corresponding premises are valid \wrt\ $\breqe$.
In turn, Lem.~\ref{rsl:layerpterm-ppsteps} allows to apply \ih\ on any $\chi_n$, since $\chi \in \layerpterm{\delta}$ implies $\delta = \ppsteps{\chi} = \ppsteps{\psi} = \alpha$ (\confer\ Prop.~\ref{rsl:breq-then-deneq}).
Therefore we obtain $\chi_n \breq \phi_n$ for all $n < \omega$, implying $\chi \breq \phi$. Then \eqltrans\ yields $\psi \breq \phi$.
A very schematic derivation tree follows:
$
\prooftree
\[
\ldots
\ \
\begin{array}{l}
\psi \breqe \xi_k \comp \psi' \\
\chi \breqe \xi_k \comp \chi'
\end{array}
\ \
\ldots
\justifies
\psi \breq \chi
\using
\eqllim
\]
\quad
\[
\ldots \ \
\[
B_n
\justifies
\chi_n \breq \phi_n
\]
\ \ \ldots
\justifies
\chi \breq \phi
\using
\eqlinfcomp
\]
\justifies
\psi \breq \phi
\using
\eqltrans
\endprooftree
$
\noindent
where we can observe the soundness of the derivation, even if \eqllim\ is applied in some of the $B_n$ derivations.
We define $\chi_k$, by induction on $k$, for all $k < \omega$.
We observe that $\sum_{i < k} \ppsteps{\phi_i} < \ppsteps{\phi} = \ppsteps{\psi}$.
Then we define, along with $\chi_k$, two values $p_k$ and $\beta_k$ as follows:
$p_0 \eqdef 0$, $\beta_0 \eqdef 0$, and if $k > 0$, then $p_k$ and $\beta_k$ are the unique (\confer\ Lem.~\ref{rsl:ordinal-lt-infAdd-then-unique-representation}) values verifying $\sum_{i < k} \ppsteps{\phi_i} = \sum_{i < p_k} \ppsteps{\psi_i} + \beta_k$ and $\beta_k < \ppsteps{\psi_{p_k}}$. We also define $p' \eqdef p_{k+1} - 1$.
Simultaneously with the definiton of $\chi_k$, we will verify the following auxiliary assertion: \\[2pt]
\begin{tabular}{@{$\ \ \bullet \ \ $}p{.93\textwidth}}
$\chi_0 \comp \ldots \comp \chi_k \breqe \psi_0 \comp \ldots \comp \psi_{p'}$ if $\beta_{k+1} = 0$; and \\
there exist
$\chi', \xi$ such that
$\psi_{p_{k+1}} \breqe \chi' \comp \xi$, $\ppsteps{\chi'} = \beta_{k+1}$ and $\chi_0 \comp \ldots \comp \chi_k \breqe \psi_0 \comp \ldots \comp \psi_{p'} \comp \chi'$ (or $\chi_0 \comp \ldots \comp \chi_k \breqe \chi'$ if $p_{k+1} = 0$), if $\beta_{k+1} > 0$.
\end{tabular} \\[2pt]
Therefore, when defining $\chi_n$ for a given $n$, we can consider this assertion to be valid for all $n' < n$.
Let $n < \omega$. Several cases must be analysed to define $\chi_n$.
\begin{itemize}
\item
Assume that either $n = 0$, \ie\ the base case, or $n > 0$ and $\beta_n = 0$.
\begin{itemize}
\item
Assume $p_n = p_{n+1}$, implying $\ppsteps{\phi_n} = \beta_{n+1} > 0$, so that $\ppsteps{\phi_n} < \ppsteps{\psi_{p_n}}$.
In this case we define $\chi_n$ to be some term verifying $\psi_{p_n} \breqe \chi_n \comp \xi$ and $\ppsteps{\chi_n} = \ppsteps{\phi_n}$; \confer\ Lem.~\ref{rsl:ppterm-binC-partition}.
\item
Assume $p_n < p_{n+1}$ and $\beta_{n+1} = 0$, so that $\ppsteps{\phi_n} = \ppsteps{\psi_{p_n}} + \ldots + \ppsteps{\psi_{p'}}$.
In this case we define $\chi_n \eqdef \psi_{p_n} \comp \ldots \comp \psi_{p'}$.
\item
Assume $p_n < p_{n+1}$ and $\beta_{n+1} > 0$, implying $\ppsteps{\phi_n} = \ppsteps{\psi_{p_n}} + \ldots + \ppsteps{\psi_{p'}} + \beta_{n+1}$.
We consider some $\chi', \xi$ verifying $\psi_{p_{n+1}} \breqe \chi' \comp \xi$ and $\ppsteps{\chi'} = \beta_{n+1}$; \confer\ Lem.~\ref{rsl:ppterm-binC-partition}.
Then we define $\chi_n \eqdef \psi_{p_n} \comp \ldots \comp \psi_{p'} \comp \chi'$.
\end{itemize}
In any case, if $n = 0$ then the auxiliary assertion holds immediately; otherwise, it suffices to apply the same assertion on $n-1$ obtaining $\chi_0 \comp \ldots \comp \chi_{n-1} \breqe \psi_0 \comp \ldots \comp \psi_{p_n-1}$, and then \eqlrefl\ and \eqlcomp.
\item
Assume $\beta_n > 0$.
In this case $n > 0$, then the auxiliary assertion on $n-1$ implies the existence of $\chi'$, $\xi$ verifying $\psi_{p_n} \breqe \chi' \comp \xi$, $\ppsteps{\chi'} = \beta_n$ and $\chi_0 \comp \ldots \comp \chi_{n-1} \breqe \psi_0 \comp \ldots \comp \psi_{p_n-1} \comp \chi'$ (or $\chi_0 \comp \ldots \comp \chi_{n-1} \breqe \chi'$ if $p_n = 0$).
\begin{itemize}
\item
Assume $p_{n+1} = p_n$, implying $\beta_{n+1} = \beta_n + \ppsteps{\phi_n} < \ppsteps{\psi_{p_n}} = \beta_n + \ppsteps{\xi}$, implying $\ppsteps{\phi_n} < \ppsteps{\xi}$.
In this case we define $\chi_n$ bo te some term verifying $\xi \breqe \chi_n \comp \xi'$ and $\ppsteps{\chi_n} = \ppsteps{\phi_n}$; \confer\ Lem.~\ref{rsl:ppterm-binC-partition}.
Observe $\psi_{p_n} \breqe (\chi' \comp \chi_n) \comp \xi'$, $\ppsteps{\chi' \comp \chi_n} = \beta_n + \ppsteps{\phi_n} = \beta_{n+1}$ and $\chi_0 \comp \ldots \comp \chi_n \breqe \psi_0 \comp \ldots \comp \psi_{p_n-1} \comp (\chi' \comp \chi_n)$, then the auxiliary statement holds for $n$; recall $\beta_{n+1} > \beta_n \geq 0$.
\item
Assume $p_{n+1} = p_n+1$ and $\beta_{n+1} = 0$, implying $\ppsteps{\psi_{p_n}} = \beta_n + \ppsteps{\phi_n}$.
Observe $\ppsteps{\xi} = \ppsteps{\phi_n}$.
We define $\chi_n \eqdef \xi$.
Then $\chi_0 \comp \ldots \chi_n \breqe \psi_0 \comp \ldots \comp \psi_{p_n-1} \comp \chi' \comp \xi \breqe \psi_0 \comp \ldots \comp \psi_{p_n-1} \comp \psi_{p_n}$, then the auxiliary statement holds for $n$.
\item
Assume $p_{n+1} > p_n+1$ and $\beta_{n+1} = 0$, implying $\ppsteps{\phi_n} = \beta' + \ppsteps{\psi_{p_n+1}} + \ldots + \ppsteps{\psi_{p'}}$, where $\ppsteps{\psi_{p_n}} = \beta_n + \beta'$.
Observe $\ppsteps{\xi} = \beta'$.
We define $\chi_n \eqdef \xi \comp \psi_{p_n+1} \comp \ldots \comp \psi_{p'}$.
We verify the auxiliary statement for $n$ similarly to the previous case.
\item
Assume $p_{n+1} > p_n$ and $\beta_{n+1} > 0$, implying $\ppsteps{\phi_n} = \beta' + \ppsteps{\psi_{p_n+1}} + \ldots + \ppsteps{\psi_{p'}} + \beta_{n+1}$ (or just $\beta' + \beta_{n+1}$ if $p_{n+1} = p_n+1$), where $\ppsteps{\psi_{p_n}} = \beta_n + \beta'$.
Observe $\ppsteps{\xi} = \beta'$.
Let $\chi'', \xi'$ such that $\psi_{p_{n+1}} \breqe \chi'' \comp \xi'$ and $\ppsteps{\chi''} = \beta_{n+1}$.
We define $\chi_n \eqdef \xi \comp \psi_{p_n+1} \comp \ldots \comp \psi_{p'} \comp \chi''$ (or just $\xi \comp \chi''$ if $p_{n+1} = p_n+1$).
We verify the auxiliary statement for $n$ similarly to the previous cases.
\end{itemize}
\end{itemize}
In turn, a simple analysis of each case yields $\ppsteps{\chi_n} = \ppsteps{\phi_n}$ for each $n < \omega$.
We verify $\psi \deneq \chi$, since this assertion is used when obtaining $\psi \breq \chi$.
Given $\ppsteps{\chi_n} = \ppsteps{\phi_n}$ for all $n < \omega$, we obtain immediately $\ppsteps{\chi} = \ppsteps{\phi} = \ppsteps{\psi}$ (recall the hypothesis $\psi \deneq \phi$).
Let $\beta < \ppsteps{\chi}$, let $n$ be a natural number verifying $\beta < \sum_{i \leq n} \ppsteps{\chi_i}$ (\confer\ Lem~\ref{rsl:ordinal-lt-infAdd-then-unique-representation}).
Then $\redel{\chi}{\beta} = \redel{(\chi_0 \comp \ldots \comp \chi_n)}{\beta}$.
Observe $\chi_0 \comp \ldots \comp \chi_n \breqe \psi'$ for some $\psi'$ verifying $\psi \breqe \psi' \comp \psi''$, \confer\ the auxiliary assertion in the definition of $\chi_n$, so that $\ppsteps{\psi'} = \sum_{i \leq n} \ppsteps{\chi_i} > \beta$.
Therefore $\redel{\chi}{\beta} = \redel{(\chi_0 \comp \ldots \comp \chi_n)}{\beta} = \redel{\psi'}{\beta} = \redel{\psi}{\beta}$, \confer\ Prop.~\ref{rsl:breq-then-deneq}. Hence $\psi \deneq \chi$.
We verify $\psi \breq \chi$.
Let $k < \omega$, let $p$ such that $p > 0$ and $\mind{\psi_i} > k$ if $i > p$.
Let $n$ be a natural number verifying $\sum_{i \leq n} \ppsteps{\phi_i} > \sum_{i \leq p} \ppsteps{\psi_i}$.
Observe that $p_{n+1} > p$. We analyse the two possible cases of the auxiliary statement in the definition of $\chi_n$; again, $p' \eqdef p_{n+1}-1$.
If $\beta_{n+1} = 0$, then $\chi_0 \comp \ldots \comp \chi_n \breqe \psi_0 \comp \ldots \comp \psi_{p'}$; observe that also $\chi \breqe \chi_0 \comp \ldots \comp \chi_n \comp (\icomp \chi_{n+1+i})$ and $\psi \breqe \psi_0 \comp \ldots \comp \psi_{p'} \comp (\icomp \psi_{p_{n+1}+i})$.
We obtain immediately $\chi \breqe \psi_0 \comp \ldots \comp \psi_{p'} \comp (\icomp \chi_{n+1+i})$ and $\mind{\icomp \psi_{p_{n+1}+i}} > k$, since $p_{n+1} > p$.
Prop.~\ref{rsl:breq-then-deneq} yields $\chi_0 \comp \ldots \comp \chi_n \deneq \psi_0 \comp \ldots \comp \psi_{p'}$, so that Lem.~\ref{rsl:deneq-binC-right} can be applied to obtain $\icomp \chi_{n+1+i} \deneq \icomp \psi_{p_{n+1}+i}$, and therefore $\mind{\icomp \chi_{n+1+i}} > k$, \confer\ Lem.~\ref{rsl:ppterm-seq-mind}.
Otherwise, there exist some $\chi', \xi$ such that $\chi_0 \comp \ldots \comp \chi_n \breqe \psi_0 \comp \ldots \comp \psi_{p'} \comp \chi'$ and $\psi_{p_{n+1}} \breqe \chi' \comp \xi$.
By an argument analogous to that of the previous case, we obtain $\psi \breqe \psi_0 \comp \ldots \comp \psi_{p'} \comp \chi' \comp (\xi \comp \icomp \psi_{{p_{n+1}}+1+i})$, $\chi \breqe \psi_0 \comp \ldots \comp \psi_{p'} \comp \chi' \comp (\icomp \chi_{n+1+i})$, and $\mind{\xi \comp \icomp \psi_{p_{n+1}+1+i}} = \mind{\icomp \chi_{n+1+i}} > k$.
Consequently we can apply \eqllim\ to obtain $\psi \breq \chi$. Observe that the premises of the \eqllim\ application correspond to the $\breqe$ relation, so that the derivation is sound.
The only element needed to complete the idea described earlier, and then to conclude the proof, is to obtain $\chi_n \deneq \phi_n$ for all $n$.
We have already obtained $\psi \deneq \chi$, so that the hypothesis $\psi \deneq \phi$ implies $\chi \deneq \phi$.
On the other hand, we have also obtained $\ppsteps{\chi_n} = \ppsteps{\phi_n}$ for all $n$. Then a simple induction on $n$ yields $\chi_n \deneq \phi_n$ for all $n$. Thus we conclude.
\end{proof}
\begin{theorem}
\label{rsl:breq-iff-deneq}
Let $\psi$, $\phi$ be \pnpterms. Then $\psi \breq \phi$ iff $\psi \deneq \phi$.
\end{theorem}
\begin{proof}
Immediate corollary of Prop.~\ref{rsl:breq-then-deneq} and Prop.~\ref{rsl:deneq-then-breq}.
\end{proof}
\section{Compression}
\label{sec:compression}
The compression lemma, \cite{orthogonal-itrs-90, orthogonal-itrs-95, terese, KetemaRTA12} established that the full power of strongly convergent reduction can be achieved considering only reductions having length at most $\omega$, \ie\ the first infinite ordinal.
Formally, the lemma states that for any strongly convergent \redseq\ $t \infredx{\reda} u$, there exists another strongly convergent \redseq\ $t \infredx{\redb} u$ and $\redln{\redb} \leq \omega$.
In \cite{orthogonal-itrs-95} a more precise statement is given: for orthogonal \TRSs, then $\redb$ can be chosen such that it is L\'evy-equivalent (\confer\ \cite{huetLevy91}) to $\reda$.
\doNotIncludeStandardisation{
The aim of this section is to present a novel proof of the property of compression for convergent first-order rewriting, based on the characterisation of \peqence\ given in Section~\ref{sec:peqence}.
}
Given that any convergent \redseq\ can be described by means of a proof term, \confer\ Prop.~\ref{rsl:denotation-existence}, compression can be studied within the framework given by proof terms.
In this setting, the compression result can be stated as follows: for any convergent proof term (\confer\ Dfn.~\ref{dfn:ppterm}) $\psi$, there exists a \pnpterm\ $\phi$ such that $\psi \peq \phi$ and $\ppsteps{\phi} \leq \omega$.
Observe that the obtained result is more general than the statements present in the referenced literature, in two ways.
Firstly, the result applies to orthogonal \emph{\redseqs}, even for non-orthogonal \TRSs.
Secondly, the result applies to (the description of) arbitrary contraction activity, independently of whether it is described sequentially. Put in this way, the compression result indicates that any orthogonal contraction activity can be sequentiated in at most $\omega$ steps.
\includeStandardisation{
The compression result is a straightforward consequence of the existence standardisation result, namely Thm.~\ref{rsl:dl-std-peq-existence}.
In this section, an independent proof of compression is given, which does not depend on standardisation. This proof resorts to the \emph{factorisation} results given in Sec.~\ref{sec:factorisation}.
}
\doNotIncludeStandardisation{
This proof resorts to a key technical result, namely the ability of \textbf{factorising} (more precisely, obtaining a factorised version of) any proof term, in a leading part denoting \emph{finite} contraction activity, followed by a tail denoting activity at \emph{arbitrarily big depths}.
The characterisation of \peqence\ shows that the original proof term and its factorised version denote the same contraction activity, while the concatenation symbol included in the signature of proof terms allows to denote the sequential organisation of contraction activity in the factorised version.
Therefore, the main auxiliary result for the compression proof is the existence, for any proof term $\psi$ and $n < \omega$, of two proof terms $\chi$ and $\phi$, such that $\psi \peqe \chi \comp \phi$, $\chi$ is a finite \pnpterm, and $\mind{\phi} > n$.
In the following, we will develop the technical work aiming to obtain the factorisation result, and subsequently we will give a statement of the compression lemma based in proof terms and \peqence, and prove it by resorting to factorisation.
}
\newcommand{collapsing sequence}{collapsing sequence}
\newcommand{collapsing sequences}{collapsing sequences}
\newcommand{Collapsing sequence}{Collapsing sequence}
\newcommand{Collapsing sequences}{Collapsing sequences}
\newcommand{\posseq}[2]{\langle #1 \rangle_{#2}}
\includeStandardisation{\subsubsection{Factorisation for \imsteps}}
\doNotIncludeStandardisation{\subsection{Factorisation for \imsteps}}
\label{scn:factorisation-imsteps}
In this section, a factorisation result for the particular case of \imsteps\ is stated an proved. The proof is based on the concept of \textbf{collapsing sequence of positions} for an \imstep.
Such a sequence indicates that the contraction activity denoted by the \imstep\ includes a series of reduction steps which can be performed consecutively and at the same position, so that all of these steps, except possibly the last one, correspond to collapsing rules.
\Ie, considering the rules $\mu: f(x) \to g(x)$, $\rho: i(x) \to x$ and $\rho': j(x) \to x$, the proof term $h(\rho(\rho'(\mu(a))),\mu(b))$ includes a finite collapsing sequence formed by the occurrences of $\rho$ and $\rho'$ plus the leftmost occurrence of $\mu$. This collapsing sequence indicates that a sequentialisation of the activity denoted by this proof term can include up to three consecutive collapsing steps at the same position.
On the other hand, the proof term $\rho\om$ includes an \emph{infinite} collapsing sequence. Observe that this proof term is \emph{not convergent}.
In the following, a relation between infinite collapsing sequences and non-convergence is shown
\footnote{We conjecture that, in fact, non-convergence of \imsteps, and therefore non-termination of developments of orthogonal sets of redex occurrences in first-order rewriting, can be fully characterised by means of collapsing sequences.
This observation suggests that \imsteps\ could be used as a technical tool to study termination of developments in infinitary rewriting, leading to an approach being alternative to \eg\ the one described in \cite{terese}, Sec.~12.5.
In this work, only the material needed for the factorisation result is developed. Some conjectures follow; further investigation about this subject is left as future work.
Observe that \imsteps\ exist being $\tgtt$-$WN^\infty$ and including infinite collapsing sequences. \Eg, if we add the rule $\tau: h(x,y) \to y$, then $\tau(\rho\om, a)$ has $a$ as $\tgtt$-normal form.
Intuitively, collapsing sequences\ prevent an \imstep\ to be $\tgtt$-$WN^\infty$ are those which cannot be erased. Then we state the following conjecture: an \imstep\ is $\tgtt$-$WN^\infty$ iff it does not include any infinite collapsing sequence\ at a \emph{non-erasable} position, where a position $p$ is erasable for $\psi$ iff $p = p_1 i p_2$, $\psi(p_1) = \mu$, and the $i$-th variable in the left-hand side of $\mu$ does not occur in the corresponding right-hand side.}
, and later exploited in the proof of the factorisation result for \imsteps.
\begin{definition}
\label{dfn:collseq}
Let $\psi$ be an \imstep. A sequence $\posseq{p_i}{i \leq n}$ (resp. $\posseq{p_i}{i < \omega}$) is a \emph{finite} (resp. \emph{infinite}) \emph{collapsing sequence} for $\psi$ iff for all $i < n$ (resp. $i < \omega$), $\psi(p_i) = \mu$ where $\mu: l[x_1, \ldots, x_m] \to x_j$ and $p_{i+1} = p_i \, j$.
\end{definition}
Observe that the length of $\posseq{p_i}{i \leq n}$ is $n + 1$.
Moreover, for any $\posseq{p_i}{i \leq n}$ or $\posseq{p_i}{i < \omega}$, an easy induction (on $k-j$) yields that $j < k < \omega$ implies $p_j < p_k$.
\begin{lemma}
\label{rsl:sub-collseq}
Let $\psi$ be a proof term, $\posseq{p_i}{i \leq n}$ (resp. $\posseq{p_i}{i < \omega}$) a collapsing sequence\ for $\psi$, and $j, k$ such that $j + k \leq n$ (resp $j,k < \omega$).
Then $\posseq{p_{j+i}}{i \leq k}$ is a collapsing sequence\ for $\psi$.
\end{lemma}
\begin{proof}
Easy consequence of Dfn.~\ref{dfn:collseq}.
\end{proof}
Notice that Lem.~\ref{rsl:sub-collseq} implies particularly that $\posseq{p_i}{i \leq k}$ is a collapsing sequence\ if $k \leq n$ (resp. $k < \omega$).
For any $\psi$ \imstep\ and $p \in \Pos{\psi}$, we observe that $\langle p \rangle$ is a collapsing sequence\ for $\psi$ whose length is 1. This is an easy \emph{existence} result. A \emph{uniqueness} result for collapsing sequences\ holds as well, namely:
\begin{lemma}
\label{rsl:collseq-uniqueness}
Let $\psi$ be an \imstep, $p \in \Pos{\psi}$, and $n$ such that $0 < n < \omega$.
Then there is at most one collapsing sequence\ for $\psi$ starting at $p$ and having length $n$.
\end{lemma}
\begin{proof}
We proceed by induction on $n$.
If $n = 1$ then the result holds immediately since the only suitable sequence is $\langle p \rangle$.
Let $n = n'+1$. Let $\posseq{p_i}{i \leq n'}$ and $\posseq{q_i}{i \leq n'}$ two collapsing sequences\ for $\psi$, both starting with $p$.
Lem.~\ref{rsl:sub-collseq} implies that both $\posseq{p_i}{i \leq (n'-1)}$ and $\posseq{q_i}{i \leq (n'-1)}$ are collapsing sequences\ for $\psi$. Then \ih\ on $n'$ implies $p_i = q_i$ if $i < n'$, so that particularly $p_{n'-1} = q_{n'-1}$. Applying Dfn.~\ref{dfn:collseq} on $\posseq{p_i}{i \leq n'}$ and $\posseq{q_i}{i \leq n'}$ yields $\psi(p_{n'-1}) = \psi(q_{n'-1}) = \mu$ such that $\mu: l[x_1, \ldots, x_m] \to x_j$ and $p_{n'} = q_{n'} = p_{n'-1} \, j$.
Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:collseq-prefix}
Let $\psi$ be an \imstep, $p \in \Pos{\psi}$, and $n, k < \omega$ (resp $n < \omega$), such that both $\posseq{p_i}{i \leq n}$ and $\posseq{q_i}{i \leq n+k}$ (resp., and $\posseq{q_i}{i < \omega}$) are collapsing sequences\ for $\psi$ starting with $p$.
Then $i \leq n$ implies $q_i = p_i$.
\end{lemma}
\begin{proof}
Easy consequence of Lem.~\ref{rsl:sub-collseq} and Lem.~\ref{rsl:collseq-uniqueness}.
\end{proof}
We already remarked that any prefix of an infinite collapsing sequence\ is a collapsing sequence\ as well. Conversely, a sequence of growing collapsing sequences\ starting at the same position indicates the presence of an infinite collapsing sequence. The following lemma formalises this idea.
\begin{lemma}
\label{rsl:collseq-infinite}
Let $\psi$ be an \imstep\ and $p \in \Pos{\psi}$, such that for any $n < \omega$, there is a collapsing sequence\ for $\psi$ starting at $p$ and having length $n$.
Then there is an infinite collapsing sequence\ for $\psi$ starting at $p$.
\end{lemma}
\begin{proof}
We define the sequence $\posseq{p_i}{i < \omega}$ as follows: for all $k < \omega$, $p_k \eqdef q_k$ where $\posseq{q_i}{i \leq k}$ is the only (\confer\ Lem.~\ref{rsl:collseq-uniqueness}) collapsing sequence\ for $\psi$ starting at $p$ and having length $k+1$.
Let $j < \omega$, and $\posseq{q_i}{i \leq j}$ and $\posseq{q'_i}{i \leq (j+1)}$ the collapsing sequences\ for $\psi$ starting at $p$ and having lengths $j+1$ and $j+2$ respectively. Observe that Lem.~\ref{rsl:collseq-prefix} implies $p_j = q_j = q'_j$; on the other hand, $p_{j+1} = q'_{j+1}$. Then $\posseq{q'_i}{i \leq (j+1)}$ being a collapsing sequence\ implies that $\psi(p_j) = \psi(q'_j) = \mu$ where $\mu : l[x_1, \ldots, x_m] \to x_i$ and $p_{j+1} = q'_{j+1} = q'_j \, i = p_j \, i$.
Consequently, $\posseq{p_i}{i < \omega}$ is a collapsing sequence. Thus we conclude.
\end{proof}
After this general presentation of collapsing sequences, we will focus on collapsing sequences\ starting with $\epsilon$.
The existence of an infinite collapsing sequence\ starting with $\epsilon$ is invariant \wrt\ partial computation of the target of an \imstep. This implies that an \imstep\ including such a sequence is non-convergent, \ie\ its target cannot be computed, \confer\ Dfn.~\ref{dfn:src-tgt-imstep} and Dfn.~\ref{dfn:imstep-convergence}.
\begin{lemma}
\label{rsl:collseq-infinite-invariant-tgtt}
Let $\psi$ be an \imstep, $\posseq{p_i}{i < \omega}$ a collapsing sequence\ for $\psi$ starting at $\epsilon$, and $\psi \infredxtrs{\reda}{\tgtt} \phi$.
Then there exists some $\posseq{q_i}{i < \omega}$ being a collapsing sequence\ for $\phi$ starting at $\epsilon$.
\end{lemma}
\begin{proof}
We proceed by transfinite induction on $\redln{\reda}$.
If $\redln{\reda} = 0$, so that $\phi = \psi$, then we conclude immediately.
Assume $\redln{\reda} = \alpha + 1$, so that $\psi \infredxtrs{\reda'}{\tgtt} \chi \sstepxtrs{\stepa}{\tgtt} \phi$ where $\redln{\reda'} = \alpha$; let us say $\stepa = \langle \chi, r, \uln{\mu}, \sigma \rangle$, and define $d \eqdef \sdepth{\stepa} = \posln{r}$, where $\uln{\mu}$ is the rule in $\tgtt$ corresponding to a rule $\mu$ in the object \TRS.
\Ih\ can be applied on $\reda'$, obtaining the existence of $\posseq{p'_i}{i < \omega}$, a collapsing sequence\ for $\chi$ starting at $\epsilon$.
Observe that $\phi = \repl{\chi}{\sigma h}{r}$ where $\mu: l[x_1, \ldots, x_m] \to h$, so that $\uln{\mu} : \mu(x_1, \ldots, x_m) \to h$, implying $\sigma = \set{x_i \eqdef \subtat{\chi}{r \,i}}$.
Notice also that $\posln{p'_n} = n$ for all $n < \omega$, implying $\posln{p'_d} = \posln{r}$.
We consider two cases.
\begin{itemize}
\item
Assume $p'_d \disj r$.
Let $n < \omega$. Observe that $n < d$, resp. $n > d$, implies $p'_n < p'_d$, resp. $p'_d < p'_n$. In either case, $r \leq p'_n$ would contradict $p'_d \disj r$, in the former case by transitivity of $<$, in the latter since all prefixes of $p'_n$ form a total order in a tree domain. Hence $r \not\leq p'_n$.
Consequently, for all $n < \omega$, $p'_n \in \Pos{\phi}$ and $\phi(p'_n) = \chi(p'_n)$.
Thus $\posseq{p'_n}{n < \omega}$ is a collapsing sequence\ for $\phi$.
\item
Assume $p'_d = r$.
In this case, $\mu: l[x_1, \ldots, x_m] \to x_j$ and $p'_{d+1} = p'_d \, j$, so that $\phi = \repl{\chi}{\subtat{\chi}{p'_{d+1}}}{p'_d}$. Observe that for any position $p''$, $\subtat{\phi}{p'_d \, p''} = \subtat{\chi}{p'_{d+1} \, p''}$.
Let $\posseq{q_i}{i < \omega}$ be the sequence defined as follows: \\
$q_n \eqdef \left\{
\begin{array}{cl}
p'_n & \textif n \leq d \\
p'_d p'' \textnormal{ where } p'_{n+1} = p'_{d+1} p'' & \textif n > d
\end{array}
\right.$
Let $n < \omega$.
If $n < d$, then $q_n = p'_n < p'_d$, so that $\phi(q_n) = \phi(p'_n) = \chi(p'_n) = \nu$ where $\nu: l[y_1, \ldots, y_m] \to y_i$ and $q_{n+1} = p'_{n+1} = p'_n \,i = q_n \, i$.
Now assume $n \geq d$. Let $p''$ such that $p'_{n+1} = p'_{d+1} p''$, observe that $n = d$ implies $p'' = \epsilon$.
Observe $\chi(p'_{n+1}) = \nu$, $\nu: l[y_1, \ldots, y_m] \to y_i$ and $p'_{n+2} = p'_{n+1} \, i = p'_{d+1} p'' \, i$.
On the other hand, $q_n = p'_d p''$ (if $n = d$, then $q_n = p'_d = p'_d p''$ since in this case $p'' = \epsilon$), $q_{n+1} = p'_d p'' \, i = q_n \, i$, and in turn $\phi(q_n) = \phi(p'_d p'') = \chi(p'_{d+1} p'') = \chi(p'_{n+1}) = \nu$.
Hence $\posseq{q_i}{i < \omega}$ is a collapsing sequence\ for $\phi$. Thus we conclude by observing that $q_0 = p'_0 = \epsilon$.
\end{itemize}
Assume that $\redln{\reda}$ is a limit ordinal.
For any $n < \omega$, we define $\beta_n$, $\chi_n$, $\langle p^n_i \rangle_{i < \omega}$ and $q_n$ as follows: $\beta_n$ is an ordinal such that $\beta_n < \redln{\reda}$ and $\sdepth{\redel{\reda}{\gamma}} > n$ if $\beta_n \leq \gamma < \redln{\reda}$; and $\chi_n$ is the \imstep\ verifying $\psi \infredxtrs{\redupto{\reda}{\beta_n}}{\tgtt} \chi_n \infredxtrs{\redsublt{\reda}{\beta_n}{\redln{\reda}}}{\tgtt} \phi$. Observe that we can assume wlog that $\beta_n \leq \beta_{n+1}$.
In turn, \ih\ on $\redupto{\reda}{\beta_n}$ and Lem.~\ref{rsl:collseq-uniqueness} imply the existence of a unique collapsing sequence\ for $\chi_n$ starting at $\epsilon$; we define $\langle p^n_i \rangle_{i < \omega}$ to be that sequence, and $q_n \eqdef p^n_n$.
Let $n < \omega$.
Then Lem.~\ref{rsl:sub-collseq} implies that $\langle p^n_i \rangle_{i \leq n}$ is a collapsing sequence\ for $\chi_n$.
Moreover, $\beta_n = \beta_{n+1}$ implies $\chi_n = \chi_{n+1}$, and otherwise $\beta_n < \beta_{n+1}$, so that $\psi \infredxtrs{\redupto{\reda}{\beta_n}}{\tgtt} \chi_n \infredxtrs{\redsublt{\reda}{\beta_n}{\beta_{n+1}}}{\tgtt} \chi_{n+1}$ where $\mind{\redsublt{\reda}{\beta_n}{\beta_{n+1}}} > n$.
Furthermore, $\chi_{n+1} \infredxtrs{\redsublt{\reda}{\beta_{n+1}}{\redln{\reda}}}{\tgtt} \phi$ and $\mind{\redsublt{\reda}{\beta_{n+1}}{\redln{\reda}}} > n$.
Therefore $\tdist{\chi_n}{\chi_{n+1}} < 2^{-n}$ and \\ $\tdist{\chi_{n+1}}{\phi} < 2^{-(n+1)}$ by Lem.~\ref{rsl:redseq-mind-big-src-tgt}; in turn Lem.~\ref{rsl:tdist-is-ultrametric} implies $\tdist{\chi_n}{\phi} < 2^{-n}$.
Then for any $j \leq n$, $\chi_n(p^n_j) = \chi_{n+1}(p^n_j) = \phi(p^n_j)$ since $\posln{p^n_j} = j$.
Therefore $\langle p^n_i \rangle_{i \leq n}$ is a collapsing sequence\ for $\chi_{n+1}$, so that Lem.~\ref{rsl:collseq-uniqueness} implies $p^n_j = p^{n+1}_j$ if $j \leq n$.
Hence $q_n = p^{n+1}_n$, so that $\phi(q_n) = \chi_{n+1}(q_n) = \nu$ where $\nu: l[x_1, \ldots, x_m] \to x_i$ and $q_{n+1} = p^{n+1}_{n+1} = p^{n+1}_n \, i = q_n \, i$.
Consequently, $\posseq{q_i}{i < \omega}$ is a collapsing sequence\ for $\phi$. Thus we conclude by observing $q_0 = \epsilon$.
\end{proof}
\begin{lemma}
\label{rsl:infinite-collseq-then-tgtt-non-wn}
Let $\psi$ be an \imstep\ such an infinite collapsing sequence\ for $\psi$ starting at $\epsilon$ exists. Then $\psi$ is not $\tgtt$-weakly normalising.
\end{lemma}
\begin{proof}
Let $\psi \infredtrs{\tgtt} \phi$. Then Lem.~\ref{rsl:collseq-infinite-invariant-tgtt} implies that an infinite collapsing sequence\ for $\phi$ starting at $\epsilon$ exists, so that $\phi$ is not a $\tgtt$-normal form. Thus we conclude.
\end{proof}
On the other hand, the inexistence of arbitrarily large collapsing sequences\ starting at $\epsilon$ allows a finite $\tgtt$-reduction sequence ending in a proof term having a function symbol at the root. In turn, for any finite $\tgtt$-reduction sequence there is a corresponding finite \pnpterm.
\begin{lemma}
\label{rsl:finite-collseq-then-finite-tgtt-redseq}
Let $\psi$ be an \imstep\ and $n$ verifying $1 < n < \omega$, such that there is no collapsing sequence\ for $\psi$ starting at $\epsilon$ and having length $n$.
Then there exists a $\tgtt$-reduction sequence $\reda$ verifying $\psi \sredxtrs{\reda}{\tgtt} \phi$, $\redln{\reda} < n$, $\sdepth{\redel{\reda}{i}} = 0$ for all $i < \redln{\reda}$, and $\phi(\epsilon) \in \Sigma$.
\end{lemma}
\begin{proof}
We proceed by induction on $n$.
Assume $n = 2$. If $\psi(\epsilon) \in \Sigma$ then we conclude immediately.
Otherwise $\psi(\epsilon) = \mu$ where $\mu : l \to f(t_1, \ldots, t_k)$, so that the corresponding rule in $\tgtt$ is $\uln{\mu} : \mu(x_1, \ldots, x_m) \to f(t_1, \ldots, t_k)$, and therefore $\psi \sstepxtrs{(\epsilon, \uln{\mu})}{\tgtt} f(t'_1, \ldots, t'_k)$; thus we conclude by taking $\reda \eqdef \langle (\epsilon, \uln{\mu}) \rangle$.
Assume $n = n' + 1$ and $1 < n' < \omega$.
If $\psi(\epsilon) \in \Sigma$ or $\psi(\epsilon) = \mu$, $\mu: l \to h$ and $h \notin \thevar$, then the argument of the previous case allows to conclude.
Otherwise, \ie\ if $\psi(\epsilon) = \mu$ and $\mu: l[x_1, \ldots, x_m] \to x_k$, then the corresponding rule in $\tgtt$ is $\uln{\mu} : \mu(x_1, \ldots, x_m) \to x_k$, implying that $\psi \sstepxtrs{(\epsilon, \uln{\mu})}{\tgtt} \subtatnarrowleft{\psi}{k}$.
Observe that $\posseq{p_i}{i \leq n'}$ being a collapsing sequence\ for $\subtat{\psi}{k}$ starting at $\epsilon$ would imply $(\langle \epsilon \rangle; \posseq{k \,p_i}{i \leq n'})$ to be a collapsing sequence\ for $\psi$ having length $n$, thus contradicting the lemma hypotheses.
Indeed, if we define $\posseq{q_i}{i \leq n}$ as the given sequence for $\psi$, then $q_0 = \epsilon$ and $q_1 = k$, so that the condition on collapsing sequences\ holds for $j = 0$. If $0 < j < n$, then $q_j = k \, p_{j-1}$, so that $\psi(q_j) = \subtatnarrow{\psi}{k}(p_{j-1}) = \nu$ where $\nu : l[y_1, \ldots, y_m] \to y_i$ and $p_j = p_{j-1} \, i$, implying $q_{j+1} = k \, p_j = k \, p_{j-1} \, i = q_j \, i$.
Therefore \ih\ can be applied to $\subtatnarrowleft{\psi}{k}$, yielding the existence of a reduction sequence $\reda'$ verifying $\subtatnarrowleft{\psi}{k} \sredxtrs{\reda'}{\tgtt} \phi$, $\redln{\reda'} < n'$, $\sdepth{\redel{\reda'}{i}} = 0$ for all $i < n'$, and $\phi(\epsilon) \in \Sigma$.
Thus we conclude by taking $\reda \eqdef (\epsilon,\uln{\mu}); \reda'$.
\end{proof}
\begin{lemma}
\label{rsl:tgtt-step-to-leading-one-step}
Let $\psi$ be an \imstep, and $\psi \sstepxtrs{\stepa}{\tgtt} \phi$. Then there exists a one-step $\chi$ such that $\psi \peqe \chi \comp \phi$ and $\sdepth{\chi} = \sdepth{\stepa}$.
\end{lemma}
\begin{proof}
We proceed by induction on $\sdepth{\stepa}$.
Assume $\stepa = (\epsilon, \uln{\mu})$, say $\mu: l[x_1, \ldots, x_m] \to h[x_1, \ldots, x_m]$ so that the corresponding rule in $\tgtt$ is $\uln{\mu} : \mu(x_1, \ldots, x_m) \to h[x_1, \ldots, x_m]$. Therefore $\psi = \mu(\psi_1, \ldots, \psi_m)$ and $\phi = h[\psi_1, \ldots, \psi_m]$.
We take $\chi \eqdef \mu(src(\psi_1), \ldots, src(\psi_m))$.
Then \peqoutin\ yields exactly $\psi \peqe \chi \comp \phi$. Thus we conclude.
Assume $\stepa = (ip, \uln{\mu})$.
In this case, $\psi = f(\psi_1, \ldots, \psi_i, \ldots, \psi_m)$, $\phi = f(\psi_1, \ldots, \phi_i, \ldots, \psi_m)$, and $\psi_i \sstepxtrs{(p,\uln{\mu})}{\tgtt} \phi_i$. Then \ih\ on $(p, \uln{\mu})$ implies $\psi_i \peqe \chi_i \comp \phi_i$ where $\chi_i$ is a one-step verifying $\sdepth{\chi_i} = \posln{p}$.
We take $\chi \eqdef f(src(\psi_1), \ldots, \chi_i, \ldots, src(\psi_m))$.
Observe that for any $j \neq i$, \peqidleft\ implies $\psi_j \peqe src(\psi_j) \comp \psi_j$, so that \\
\hspace*{1cm}
$\begin{array}{rcl}
\psi & \peqe &
f(src(\psi_1) \comp \psi_1, \ldots, \chi_i \comp \phi_i, \ldots, src(\psi_m) \comp \phi_m) \\
& \peqe &
f(src(\psi_1), \ldots, \chi_i, \ldots, src(\psi_m)) \comp
f(\psi_1, \ldots, \phi_i, \ldots, \psi_m) \\
& = & \chi \comp \phi
\end{array}
$ \\[2pt]
Thus we conclude by noticing that $\sdepth{\chi} = \posln{p} + 1 = \sdepth{\stepa}$.
\end{proof}
\begin{lemma}
\label{rsl:tgtt-redseq-to-leading-pnpterm}
Let $\psi$ be an \imstep\ and $\psi \sredxtrs{\reda}{\tgtt} \phi$.
Then there exists a finite \pnpterm\ $\chi$ such that $\psi \peqe \chi \comp \phi$, $\ppsteps{\chi} = \redln{\reda}$, and $\sdepth{\redel{\chi}{i}} = \sdepth{\redel{\reda}{i}}$ for all $i < \ppsteps{\chi}$.
\end{lemma}
\begin{proof}
Easy induction on $\redln{\reda}$.
If $\reda$ is an empty reduction sequence, then we conclude just by taking $\chi \eqdef src(\psi)$.
Assume $\reda = \stepa; \reda'$, so that $\psi \sstepxtrs{\stepa}{\tgtt} \psi_0 \sredxtrs{\reda'}{\tgtt} \phi$.
Then Lem.~\ref{rsl:tgtt-step-to-leading-one-step} implies that $\psi \peqe \chi_0 \comp \psi_0$ where $\chi_0$ is a one-step verifying $\sdepth{\chi_0} = \sdepth{\stepa}$, and \ih\ on $\reda'$ yields $\psi_0 \peqe \chi' \comp \phi$ where $\chi'$ is a finite \pnpterm\ verifying $\ppsteps{\chi'} = \redln{\reda'} = \redln{\reda} - 1$, and $\sdepth{\redel{\chi'}{i}} = \sdepth{\redel{\reda'}{i}} = \sdepth{\redel{\reda}{i+1}}$ if $i < \ppsteps{\chi'}$.
We take $\chi \eqdef \chi_0 \comp \chi'$. It is straightforward to verify that $\chi$ satisfies the conditions about length and step depth.
Moreover, $\psi_0 \peqe \chi' \comp \phi$ implies $\chi_0 \comp \psi_0 \peqe \chi_0 \comp (\chi' \comp \phi) \peqe \chi \comp \phi$, so that \eqltrans\ yields $\psi \peqe \chi \comp \phi$ (recall $\psi \peqe \chi_0 \comp \psi_0$). Thus we conclude.
\end{proof}
The previous auxiliary results allow to prove the main result of this section, \ie\ factorisation for \imsteps.
\begin{lemma}
\label{rsl:factorisation-imstep}
Let $\psi$ be a convergent \imstep.
Then there exist $\chi$, $\phi$ such that
$\, \psi \peqe \chi \comp \phi \, $,
$\, \chi$ is a finite \pnpterm\ verifying $\sdepth{\redel{\chi}{i}} = 0$ for all $i < \ppsteps{\chi}$, and
$\phi$ is a convergent \imstep\ verifying $\mind{\phi} > 0$.
\end{lemma}
\begin{proof}
We define $A \eqdef \{n \setsthat 0 < n < \omega$ and there is no collapsing sequence\ for $\psi$ starting at $\epsilon$ and having length $n \}$.
Dfn.~\ref{dfn:imstep-convergence} implies that $\psi$ is $\tgtt$-weakly normalising. Then Lem.~\ref{rsl:infinite-collseq-then-tgtt-non-wn} implies that there is no infinite collapsing sequence\ for $\psi$ starting at $\epsilon$, so that Lem.~\ref{rsl:collseq-infinite} implies $A \neq \emptyset$.
Let $n \in A$. Then Lem.~\ref{rsl:finite-collseq-then-finite-tgtt-redseq} implies $\psi \sredxtrs{\reda}{\tgtt} \phi$, where $\redln{\reda} < \omega$, $\sdepth{\redel{\reda}{i}} = 0$ for all suitable $i$, and $\phi$ is an \imstep\ (since it is the target of a $\tgtt$-reduction sequence) verifying $\mind{\phi} > 0$ (since $\phi(\epsilon) \in \Sigma)$.
Moreover, $\psi$ being convergent means that $\psi$ is $\tgtt$-$WN^\infty$, and $\tgtt$ is a convergent \iTRS, so that Lem.~\ref{rsl:orthogonal-leading-head-steps-to-nf} implies that $\phi$ is also $\tgtt$-$WN^\infty$, \ie\ convergent.
We conclude by applying Lem.~\ref{rsl:tgtt-redseq-to-leading-pnpterm} on $\psi \sredxtrs{\reda}{\tgtt} \phi$.
\end{proof}
\subsection{Fixed prefix of contraction activity}
\label{sec:cfpc}
\cfpInsideCompression{
This section introduces a technical tool, in which the extension of the factorisation result from \imsteps\ to arbitrary proof terms is based on.
This tool is a formalisation of a simple observation:
the
}
\cfpInsideStandardisation{The }
contraction activity denoted by a proof term can lie below some \emph{fixed prefix}. \Ie, the contraction activity corresponding to either of the equivalent proof terms
$h(\mu(a) \cdot \nu(a), \pi)$ and $h(\mu(a),a) \cdot h(\nu(a),a) \cdot h(k(a),\pi)$
leaves the context $h(\Box,\Box)$ fixed, so we will say that $h(\Box,\Box)$ is a fixed prefix for these proof terms.
For proof terms involving root activity, the only possible fixed prefix is $\Box$.
In the sequel, we will establish that fixed prefixes are invariant \wrt\ \peqence.
Computing a fixed prefix for a proof term $\psi$ allows to permute it with a one-step (\confer\ Dfn.~\ref{dfn:one-step}) performed on $tgt(\psi)$, whose redex lies in the fixed prefix of $\psi$.
This observation will be crucial in order to
\cfpInsideStandardisation{prove some of the main results of this study.}
\cfpInsideCompression{prove a general factorisation result, since it allows to obtain a proof term in which the (denotation of the) activity near to the root
``shifts to the left as much as possible'', \ie\
lies in the lesser possible positions
\wrt\ the sequentialisation order given by dot occurrences.}
The following definitions and results characterise the common prefix of a proof term in a way allowing to manipulate it.
The
\cfpInsideStandardisation{sets of }
positions mentioned in the statements must be understood as
\cfpInsideStandardisation{sets of \emph{contraction} positions.}
\cfpInsideCompression{being relative to the contraction activity denoted by a proof term, rather than as positions in proof terms themselves.}
We formalise the concept of (the activity denoted by) a proof term having a fixed prefix by defining a relation between proof terms and prefix-closed sets of positions, which we will call \emph{respect}.
Therefore, if $\psi$ respects a set of positions $\Spa$, then $\psi$ has a fixed prefix corresponding to the positions in $\Spa$.
\begin{definition}
\label{dfn:spa-proj}
Let $\Spa$ be a set of positions, and $i \in \Nat$.
Then we define the \emph{projection of $\Spa$ on $i$} as
$\proj{\Spa}{i} \eqdef \set{p \setsthat ip \in \Spa}$.
\end{definition}
\begin{definition}
\label{dfn:term-prefix}
Let $t$ be a term, and $\Spa$ a finite and prefix-closed set of positions such that $\Spa \subseteq \Pos{t}$.
Then we define $\pref{t}{\Spa}$, the \emph{prefix of $t$ \wrt\ $\Spa$}, as follows. \\
If $\Spa = \emptyset$, then $\pref{t}{\Spa} \eqdef \Box$. \\
If $\Spa \neq \emptyset$ and $t \in \thevar$, so that $\Spa = \set{\epsilon}$, then $\pref{t}{\Spa} \eqdef t$. \\
If $\Spa \neq \emptyset$ and $t = f(t_1, \ldots, t_m)$, so that $\Spa = \set{\epsilon} \,\cup\, \bigcup_{1 \leq i \leq m} (i \cdot \proj{\Spa}{i})$, then $\pref{t}{\Spa} \eqdef f(\pref{t_1}{\proj{\Spa}{1}}, \ldots, \pref{t_m}{\proj{\Spa}{m}})$.
\end{definition}
Notice that $C = \pref{t}{\Spa}$ iff $t = C[t_1, \ldots, t_k]$ and $\Spa = \set{p \setsthat p \in \Pos{C} \,\land\, C(p) \neq \Box}$, this can be verified by a simple induction on the cardinal of $\Spa$.
\cfpInsideStandardisation{
\begin{definition}
\label{dfn:respects}
Let $\psi$ be a proof term, $\Spa$ a finite and prefix-closed set of positions.
We say that $\psi$ \emph{respects} $\Spa$ iff $\Spa \subseteq \Pos{src(\psi)}$ and $\frso{\psi}{r}$ is undefined for all $r \in \Spa$.
\end{definition}
}
\cfpInsideCompression{
\begin{definition}
\label{dfn:respects}
Let $\psi$ be a proof term, and $\Spa$ a set of positions.
We say that $\psi$ \emph{respects} $\Spa$ iff $\Spa$ is finite and prefix-closed, and any of the following applies: \\
\begin{tabular}{@{$\ \ \bullet\ \ $}p{.9\textwidth}}
$\psi$ is an \imstep, $\Spa \subseteq \Pos{\psi}$ and $\psi(p) \in \Sigma$ for all $p \in \Spa$. \\
$\psi = \psi_1 \comp \psi_2$ and both $\psi_1$ and $\psi_2$ respect $\Spa$. \\
$\psi = \icomp \psi_i$ and all $\psi_i$ respect $\Spa$. \\
$\psi = f(\psi_1, \ldots, \psi_m)$, at least one of the $\psi_i$ is not an \imstep, and either $\Spa = \emptyset$ or $\psi_i$ respects $\proj{\Spa}{i}$ for all $i \leq m$. \\
$\psi = \mu(\psi_1, \ldots, \psi_m)$, at least one of the $\psi_i$ is not an \imstep, and $\Spa = \emptyset$.
\end{tabular}
\end{definition}
}
The relation just defined enjoys some simple properties.
\cfpInsideCompression{
\begin{lemma}
\label{rsl:respects-then-src}
Let $\psi$ be a proof term and $\Spa$ such that $\psi$ respects $\Spa$. Then $\Spa \subseteq \Pos{src(\psi)}$.
\end{lemma}
\begin{proof}
An easy induction on $\psi$ suffices; \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
\end{proof}
\begin{lemma}
\label{rsl:respects-then-tgt}
Let $\psi$ be a convergent proof term and $\Spa$ such that $\psi$ respects $\Spa$. Then $\Spa \subseteq \Pos{tgt(\psi)}$.
\end{lemma}
\begin{proof}
An easy induction on $\psi$ suffices; \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
If $\psi = \icomp \psi_i$ and $\Spa \subseteq \Pos{tgt(\psi_i)}$ for all $i < \omega$, given $p \in \Spa$, we consider $n$ such that $\tdist{tgt(\psi_i)}{tgt(\psi)} < 2^{-\posln{p}}$ if $i > n$, so that $p \in \Pos{tgt(\psi_{n+1})}$ implies $p \in \Pos{tgt(\psi)}$.
\end{proof}
\begin{lemma}
\label{rsl:respect-fnsymbol-coherence}
Let $\psi = f(\psi_1, \ldots, \psi_m)$, and $\Spa$ a set of positions.
Then $\psi$ respects $\Spa$ iff either $\Spa = \emptyset$ or $\psi_i$ respects $\proj{\Spa}{i}$ for all $i \leq m$.
\end{lemma}
\begin{proof}
If $\psi$ is an \imstep, then a straightforward analysis yields the desired result. If at least one of the $\psi_i$ is not an \imstep, then we conclude immediately. Any other case in Dfn.~\ref{dfn:respects} contradicts the stated form of $\psi$.
\end{proof}
\begin{lemma}
\label{rsl:respects-emptyset}
Let $\psi$ be a proof term. Then $\psi$ respects $\emptyset$.
\end{lemma}
\begin{proof}
A straightforward induction on $\psi$, \confer\ Prop.~\ref{rsl:pterm-induction-principle}, suffices to conclude.
\end{proof}
The \emph{respects} relation can be obtained from conditions on the target and the minimum activity depth of a proof term.
\begin{lemma}
\label{rsl:mind-plus-tgt-then-respects}
Let $\psi$ be a convergent proof term and $\Spa$ a finite, prefix-closed set of positions, such that $\mind{\psi} > n$, $\posln{p} \leq n$ for all $p \in \Spa$, and $\Spa \subseteq \Pos{tgt(\psi)}$.
Then $\psi$ respects $\Spa$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$, \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
Assume that $\psi$ is an \imstep.
If $\Spa = \emptyset$ then Lem.~\ref{rsl:respects-emptyset} allows to conclude immediately.
Otherwise, $\epsilon \in \Spa$, implying $\psi = f(\psi_1, \ldots, \psi_m)$.
We proceed by induction on $n$.
If $n = 0$, then the only set of positions compatible with the lemma hypotheses is $\Spa = \set{\epsilon}$, so that we conclude immediately.
Assume $n = n' + 1$, and let $i$ such that $1 \leq i \leq m$.
It is straightforward to verify that $\mind{\psi_i} > n'$, that $\posln{p} \leq n'$ for all $p \in \proj{\Spa}{i}$, and also that $\proj{\Spa}{i} \subseteq \Pos{tgt(\psi_i)}$ (recall $tgt(\psi) = f(tgt(\psi_1), \ldots tgt(\psi_m)) \,$).
Therefore, we can apply \ih\ on $\psi_i$, obtaining that $\psi_i$ respects $\proj{\Spa}{i}$, so that $\proj{\Spa}{i} \subseteq \Pos{\psi_i}$, and moreover for any $p \in \proj{\Spa}{i}$, $\psi(i p) = \psi_i(p) \in \Sigma$.
Hence the desired result holds immediately.
Assume $\psi = \psi_1 \comp \psi_2$.
In this case, $\mind{\psi_i} > n$ for $i = 1,2$, and $\Spa \subseteq \Pos{tgt(\psi)} = \Pos{tgt(\psi_2)}$.
Then \ih\ applies to $\psi_2$ yielding that $\psi_2$ respects $\Spa$. In turn, Lem.~\ref{rsl:respects-then-src} implies $\Spa \subseteq \Pos{src(\psi_2)} = \Pos{tgt(\psi_1)}$. Then \ih\ applies to $\psi_1$ as well, implying that $\psi_1$ respects $\Spa$. Thus we conclude.
Assume $\psi = \icomp \psi_i$.
Observe that $\mind{\psi} > n$ implies $\mind{\psi_i} > n$ for all $i < \omega$.
Let $k$ such that $\tdist{tgt(\psi_i)}{tgt(\psi)} < 2^{-k}$ for all $i > k$.
Let $j > k$. Then $\Spa \subseteq \Pos{tgt(\psi)}$ implies $\Spa \subseteq \Pos{tgt(\psi_j)}$. Then \ih\ can be applied to $\psi_j$ obtaining that $\psi_j$ respects $\Spa$.
In turn, $\psi_{k+1}$ respecting $\Spa$ implies that $\Spa \subseteq \Pos{src(\psi_{k+1})} = \Pos{tgt(\psi_k)}$.
Therefore \ih\ applies also to $\psi_k$, yielding that $\psi_k$ respects $\Spa$, and then Lem.~\ref{rsl:respects-then-src} implies $\Spa \subseteq \Pos{src(\psi_k)} = \Pos{tgt(\psi_{k-1})}$.
Successive application of an analogous argument yields that $\psi_i$ respects $\Spa$ for all $i \leq k$.
Thus we conclude.
If $\psi = f(\psi_1, \ldots, \psi_m)$, then an argument analogous to that given for \imsteps\ applies.
Finally, $\psi = \mu(\psi_1, \ldots, \psi_m)$ contradicts $\mind{\psi} > n$ for any $n < \omega$.
\end{proof}
}
\cfpInsideStandardisation{
\begin{lemma}
\label{rsl:respects-invariant-proj}
Let $\psi = f(\psi_1, \ldots, \psi_m)$, $\Spa$ such that $\Spa \neq \emptyset$ and $\psi$ respects $\Spa$.
Then for each $i$, $\psi_i$ respect $\proj{\Spa}{i}$.
\end{lemma}
\begin{proof}
Immediate.
\end{proof}
}
The \emph{respects} relation is invariant \wrt\ base \peqence.
\cfpInsideCompression{
\begin{lemma}
\label{rsl:respects-invariant-peqe}
Let $\psi$, $\phi$ be convergent proof terms and $\Spa$ a set of positions, such that $\psi \peqe \phi$.
Then $\psi$ respects $\Spa$ iff $\phi$ respects $\Spa$.
\end{lemma}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \layerpeqe{\alpha} \phi$, analysing the rule used in the last step of that judgement.
If the rule is \eqlrefl, then we conclude immediately.
If the rule is \eqleqn, then we analyse the equation used.
\begin{itemize}
\item
\peqidleft\ or \peqidright, \ie\ $\psi = src(\phi) \comp \phi \ $ or $\ \psi = \phi \comp tgt(\phi)$.
The $\Rightarrow )$ direction is immediate.
For the $\Leftarrow )$ direction, observe that Lem.~\ref{rsl:respects-then-src} and Lem.~\ref{rsl:respects-then-tgt} imply $\Spa \subseteq \Pos{src(\phi)}$ and $\Spa \subseteq \Pos{tgt(\phi)}$ respectively.
Then Dfn.~\ref{dfn:respects} for \imsteps\ implies immediately that both $src(\phi)$ and $tgt(\phi)$ respect $\Spa$. Thus we conclude.
\item
\peqassoc, \ie\ $\psi = \chi_1 \comp (\chi_2 \comp \chi_3)$ and $\phi = (\chi_1 \comp \chi_2) \comp \chi_3$. In this case either $\psi$ or $\phi$ respects $\Spa$ iff $\chi_1$, $\chi_2$ and $\chi_3$ do. Thus we conclude.
\item
\peqstruct, \ie\ $\psi = f(\chi_1, \ldots, \chi_m) \comp f(\xi_1, \ldots, \xi_m)$ and $\phi = f(\chi_1 \comp \xi_1, \ldots, \chi_m \comp \xi_m)$.
If $\Spa = \emptyset$, then both $\psi$ and $\phi$ respect $\Spa$; \confer\ Lem.~\ref{rsl:respects-emptyset}.
Otherwise \\
$\psi$ respects $\Spa$ \\
\begin{tabular}{@{\hspace*{1cm}}l}
iff both $f(\chi_1, \ldots, \chi_m)$ and $f(\xi_1, \ldots, \xi_m)$ do \\
iff for all $j$ such that $1 \leq j \leq m$, both $\chi_j$ and $\xi_j$ respect $\proj{\Spa}{j}$ \\
iff for all $j$ such that $1 \leq j \leq m$, $\chi_j \comp \xi_j$ respects $\proj{\Spa}{j}$ \\
iff $\phi$ respects $\Spa$.
\end{tabular} \\
Thus we conclude.
\item
\peqinfstruct. This case admits an argument analogous to the one used for \peqstruct.
\item
\peqoutin\ and \peqinout. In this case, it is immediate that either $\psi$ or $\phi$ respects $\Spa$ iff $\Spa = \emptyset$.
\end{itemize}
If the rule used in the last step of the judgement $\psi \layerpeqe{\alpha} \phi$ is \eqlsymm, \eqltrans, \eqlfun, \eqlcomp\ or \eqlinfcomp, then a straightforward inductive arguments suffices to obtain the desired result.
Finally, if the rule is \eqlrule, then it is immediate to verify that either $\psi$ or $\phi$ respect $\Spa$ iff $\Spa = \emptyset$.
\end{proof}
}
\cfpInsideStandardisation{
\begin{lemma}
\label{rsl:respects-invariant-peqe}
Let $\psi$, $\phi$ be convergent proof terms and $\Spa$ a set of positions, such that $\psi \peqe \phi$ and $\psi$ respects $\Spa$.
Then $\phi$ respects $\Spa$.
\end{lemma}
\begin{proof}
Lem.~\ref{rsl:peq-then-same-src-mind-tgt} implies immediately $\Spa \subseteq \Pos{\phi}$.
Assume for contradiction that $\frso{\phi}{r}$ is defined for some $r \in \Spa$.
Then L.~\ref{rsl:frso-peqe} implies that $\frso{\psi}{r'}$ is defined for some $r' \leq r$, so that $r' \in \Spa$ (recall $\Spa$ is prefix-closed), contradicting lemma hypotheses. Thus we conclude.
\end{proof}
}
Observe that proof terms whose minimum activity depth is greater than 0 are exactly those which respect $\set{\epsilon}$. Lem.~\ref{rsl:peq-then-same-src-mind-tgt} implies this condition to be stable by \peqence.
For such proof terms, we define their \emph{condensed-to-fixed-prefix-symbol form}, which is a proof term denoting the same activity as the original proof term, and having a function symbol at the root.
\Eg\ the condensed-to-fixed-prefix-symbol form of $f(\mu(a)) \comp f(\nu(a))$ is $f(\mu(a) \comp \nu(a))$. The condensed-to-fixed-prefix-symbol form of an already condensed proof term is itself, so that it is idempotent.
\begin{lemma}
\label{rsl:respects-epsilon-then-stable-root}
Let $\psi$ a convergent proof term which respects $\set{\epsilon}$.
Then $src(\psi)(\epsilon) = tgt(\psi)(\epsilon)$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$, \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
If $\psi = f(\psi_1, \ldots, \psi_m)$ then the result holds immediately, while $\psi = \mu(\psi_1, \ldots, \psi_m)$ contradicts the lemma hypotheses.
If $\psi = \psi_1 \comp \psi_2$ and the result holds for both components, then lemma hypotheses imply that both $\psi_1$ and $\psi_2$ respect $\set{\epsilon}$, so that $src(\psi_j)(\epsilon) = tgt(\psi_j)(\epsilon)$ for $j = 1,2$. Observe $src(\psi) = src(\psi_1)$, $tgt(\psi) = tgt(\psi_2)$, and moreover $tgt(\psi_1) = src(\psi_2)$ (by the coherence condition on the definition of $\psi$). Thus we conclude immediately.
Assume $\psi = \icomp \psi_i$ and the result holds for each $\psi_i$. For any $i < \omega$, lemma hypotheses imply that $\psi_i$ respects $\set{\epsilon}$, and therefore $src(\psi_i)(\epsilon) = tgt(\psi_i)(\epsilon)$.
Given $tgt(\psi_i) = src(\psi_{i+1})$ for all $i < \omega$, an easy inductive argument yields $src(\psi)(\epsilon) = src(\psi_0)(\epsilon) = tgt(\psi_i)(\epsilon)$ for any $i < \omega$. Let $n$ such that $\tdist{tgt(\psi_k)}{tgt(\psi)} < 1$ if $k > n$; recall $tgt(\psi) = \lim_{i \to \omega}(tgt(\psi_i))$.
Then $tgt(\psi)(\epsilon) = tgt(\psi_{n+1})(\epsilon) = src(\psi)(\epsilon)$. Thus we conclude.
\end{proof}
\begin{definition}
\label{dfn:cfps}
Let $\psi$ be a proof term which respects $\set{\epsilon}$.
We define $\cfps{\psi}$, \ie\ the \emph{condensed to fixed prefix symbol form} of $\psi$, as follows. \\[2pt]
\begin{tabular}{@{}l@{}l@{\ \ }l@{\ \ }l}
$\ \ \bullet \ \ $ &
if $\psi = f(\psi_1, \ldots, \psi_n)$ & then &
$\cfps{\psi} \eqdef \psi$. \\
$\ \ \bullet \ \ $ &
if $\psi = \psi_1 \comp \psi_2$ & then &
$\cfps{\psi} \eqdef f(\psi_{11} \comp \psi_{21}, \ldots, \psi_{1m} \comp \psi_{2m})$ \\
& & & where $\cfps{\psi_i} = f(\psi_{i1}, \ldots, \psi_{im})$ for $i = 1,2$ \\
$\ \ \bullet \ \ $ &
if $\psi = \icomp \psi_i$ & then &
$\cfps{\psi} \eqdef f(\icomp \psi_{i1}, \ldots, \icomp \psi_{im})$ \\
& & & where $\cfps{\psi_i} = f(\psi_{i1}, \ldots, \psi_{im})$ for all $i < \omega$. \\
$\ \ \bullet \ \ $ &
\multicolumn{3}{@{}l}{$\psi = \mu(\psi_1, \ldots, \psi_m)$ contradicts $\psi$ respecting $\set{\epsilon}$.}
\end{tabular} \\[2pt]
Lem.~\ref{rsl:respects-epsilon-then-stable-root} implies the soundness of the clauses corresponding to both binary and infinite concatenation.
\end{definition}
Condensed-to-fixed-prefix-symbol forms enjoy some properties related with base \peqence\ and the \emph{respects} relation. In turn, these properties allow a simple proof of the extension of Lem.~\ref{rsl:respects-epsilon-then-stable-root} to arbitrary finite and prefix-closed sets of positions.
\begin{lemma}
\label{rsl:cfps-peqe}
Let $\psi$ be a proof term which respects $\set{\epsilon}$.
Then $\psi \peqe \cfps{\psi}$.
\end{lemma}
\begin{proof}
Easy induction on $\psi$. For the infinitary composition case, resort to the \eqlinfcomp\ rule and the \peqinfstruct\ equation, \confer\ Dfn.~\ref{dfn:peqe}.
\end{proof}
\begin{lemma}
\label{rsl:peqe-then-cfps-proj-peqe}
Let $\psi$, $\phi$ be proof terms such that $\psi \peqe \phi$ and $\psi$, $\phi$ respect $\set{\epsilon}$.
Let $\cfps{\psi} = f(\psi_1, \ldots, \psi_m)$ and $\cfps{\phi} = f'(\phi_1, \ldots, \phi_{m'})$.
Then $f = f' = src(\psi)(\epsilon)$, so that $m = m'$, and $\psi_i \peqe \phi_i$ for each $i$ between 1 and $m$.
\end{lemma}
\begin{proof}
Lem.~\ref{rsl:cfps-peqe} and the hypotheses imply $\psi \peqe \cfps{\psi} \peqe \cfps{\phi}$, then Lem.~\ref{rsl:peq-then-same-src-mind-tgt} yields $f = f' = src(\psi)(\epsilon)$, and therefore $m = m'$.
We prove $\psi_i \peqe \phi_i$ for all $i$ by induction on $\alpha$ where $\psi \layerpeqe{\alpha} \phi$, analysing the rule used in the last step of that judgement.
\begin{itemize}
\item
\eqlrefl: we conclude immediately.
\item
\eqleqn: we analyse each of the equations.
\begin{itemize}
\item
\peqidleft: let $src(\phi) = f(t_1, \ldots, t_m)$ where $t_i = src(\phi_i)$ for all $i$; \confer\ Lem.~\ref{rsl:cfps-peqe} and Lem.~\ref{rsl:peq-then-same-src-mind-tgt}.
Then $\psi = f(t_1, \ldots, t_m) \comp \phi$, so that $\cfps{\psi} = f(t_1 \comp \phi_1, \ldots, t_m \comp \phi_m)$. Thus we conclude.
\item
\peqidright: an analogous argument applies.
\item
\peqassoc: in this case $\psi = \xi \comp (\gamma \comp \chi)$ and $\phi = (\xi \comp \gamma) \comp \chi$.
Let $\cfps{\xi} = f(\xi_1, \ldots, \xi_m)$, $\cfps{\gamma} = f(\gamma_1, \ldots, \gamma_m)$ and $\cfps{\chi} = f(\chi_1, \ldots, \chi_m)$; \confer\ Lem.~\ref{rsl:cfps-peqe} (implying $f = src(\psi)(\epsilon) = src(\xi)(\epsilon) = src(\cfps{\xi})(\epsilon)$) and Lem.~\ref{rsl:respects-epsilon-then-stable-root}.
Then for any $i \leq m$, $\psi_i = \xi_i \comp (\gamma_i \comp \xi_i)$ and $\phi_i = (\xi_i \comp \gamma_i) \comp \chi_i$. Thus we conclude immediately.
\item
\peqstruct\ and \peqinfstruct: in either of these cases Dfn.~\ref{dfn:cfps} allows to conclude immediately.
\item
\peqoutin\ and \peqinout: either of these cases contradict $\psi, \phi$ to respect $\set{\epsilon}$.
\end{itemize}
\item
\eqlsymm\ or \eqltrans: a simple inductive argument applies.
\item
\eqlfun: the hypotheses of the \eqlfun\ rule are enough to conclude immediately.
\item
\eqlrule: this case would imply that neither $\psi$ nor $\phi$ respect $\set{\epsilon}$, thus contradicting lemma hypotheses.
\item
\eqlcomp: in this case,
$\psi = \chi \comp \xi$, $\phi = \gamma \comp \delta$, $\chi \layerpeqe{\alpha_1} \gamma$, $\xi \layerpeqe{\alpha_2} \delta$, $\alpha_1 < \alpha$ and $\alpha_2 < \alpha$.
Let $\cfps{\chi} = f(\chi_1, \ldots, \chi_m)$, $\cfps{\xi} = f(\xi_1, \ldots, \xi_m)$, $\cfps{\gamma} = f(\gamma_1, \ldots, \gamma_m)$ and $\cfps{\delta} = f(\delta_1, \ldots, \delta_m)$.
Let $i$ such that $1 \leq i \leq m$.
Observe $\psi_i = \chi_i \comp \xi_i$ and $\phi_i = \gamma_i \comp \delta_i$.
On the other hand, \ih\ implies $\chi_i \peqe \gamma_i$ and $\xi_i \peqe \delta_i$.
Thus we conclude.
\item
\eqlinfcomp: an analogous argument applies.
In this case, $\psi = \icomp \psi_i$, $\phi = \icomp \phi_i$, and for any $i < \omega$, $\psi_i \layerpeqe{\alpha_i} \phi_i$ where $\alpha_i < \alpha$.
Let $\cfps{\psi_i} = f(\psi^1_i, \ldots, \psi^m_i)$ and $\cfps{\phi_i} = f(\phi^1_i, \ldots, \phi^m_i)$.
Let $j$ such that $1 \leq j \leq m$.
Then $\psi_j = \icomp \psi^j_i$ and $\phi_j = \icomp \phi^j_i$.
\Ih\ on each $\psi_i \layerpeqe{\alpha_i} \phi_i$ yields $\psi^j_i \peqe \phi^j_i$. Thus we conclude.
\end{itemize}
\end{proof}
\begin{lemma}
\label{rsl:cfps-equals-src-tgt}
Let $\psi$ be a proof term such that $\psi$ respects $\set{\epsilon}$.
Then $\cfps{\psi}(\epsilon) = src(\psi)(\epsilon) = tgt(\psi)(\epsilon)$.
\end{lemma}
\begin{proof}
Immediate consequence of Lem.~\ref{rsl:peqe-then-cfps-proj-peqe} and Lem.~\ref{rsl:respects-epsilon-then-stable-root}.
\end{proof}
\begin{lemma}
\label{rsl:cfps-components-respect}
Let $\psi$ be a proof term and $\Spa$ a set of positions such that $\Spa \neq \emptyset$ and $\psi$ respects $\Spa$. Then $\psi_i$ respects $\proj{\Spa}{i}$ for all $i \leq m$, where $\cfps{\psi} = f(\psi_1, \ldots, \psi_m)$.
\end{lemma}
\begin{proof}
Lem.~\ref{rsl:cfps-peqe} implies $\psi \peqe \cfps{\psi}$, then Lem.~\ref{rsl:respects-invariant-peqe} implies $\cfps{\psi}$ respects $\Spa$.
Therefore
\cfpInsideStandardisation{Lem.~\ref{rsl:respects-invariant-proj} }
\cfpInsideCompression{Lem.~\ref{rsl:respect-fnsymbol-coherence} }
allows to conclude.
\end{proof}
\begin{lemma}
\label{rsl:respects-then-invariant-fp}
Let $\psi$ be a convergent proof term and $\Spa$ a set of positions such that $\psi$ respects $\Spa$.
Then $\pref{tgt(\psi)}{\Spa} = \pref{src(\psi)}{\Spa}$.
\end{lemma}
\begin{proof}
We proceed by induction on the cardinal of $\Spa$.
If $\Spa = \emptyset$, then $\pref{tgt(\psi)}{\Spa} = \pref{src(\psi)}{\Spa} = \Box$.
Otherwise, $\Spa = \set{\epsilon} \cup (\bigcup_{1 \leq i \leq m} i \cdot \proj{\Spa}{i})$ where $\cfps{\psi} = f(\psi_1, \ldots, \psi_m)$.
Lem.~\ref{rsl:cfps-peqe} and Lem.~\ref{rsl:peq-then-same-src-mind-tgt} imply $src(\psi) = f(src(\psi_1), \ldots, src(\psi_m))$ and $tgt(\psi) = f(tgt(\psi_1), \ldots, tgt(\psi_m))$, so that $\pref{src(\psi)}{\Spa} = f(\pref{src(\psi_1)}{\proj{\Spa}{1}}, \ldots, \pref{src(\psi_m)}{\proj{\Spa}{m}})$, and $\pref{tgt(\psi)}{\Spa} = f(\pref{tgt(\psi_1)}{\proj{\Spa}{1}}, \ldots, \pref{tgt(\psi_m)}{\proj{\Spa}{m}})$.
On the other hand, Lem.~\ref{rsl:cfps-components-respect} implies that $\psi_i$ respects $\proj{\Spa}{i}$ for all $i$, so that \ih\ can be applied to obtain $\pref{src(\psi_i)}{\proj{\Spa}{i}} = \pref{tgt(\psi_i)}{\proj{\Spa}{i}}$.
Thus we conclude.
\end{proof}
Assume that some proof term, say $\psi$, respects not only the root, but a finite, prefix-closed set of positions $\Spa$.
Then we can define the \emph{condensed-to-fixed-prefix-\textbf{context} form} of $\psi$ \wrt\ $\Spa$, analogously as we have just done with the condensed-to-fixed-prefix-symbol form.
The activity denoted by a condensed-to-fixed-prefix-context form \wrt\ the set of positions $\Spa$ will lie inside a fixed context, \ie\ a context in $\SigmaTerms$, whose set of (non-hole) positions is exactly $\Spa$.
\Eg, the proof term $h(f(g(\mu(a))), \mu(b)) \comp h(f(g(g(\pi))), \nu(b))$ respects $\Spa \eqdef \set{\epsilon,1,11}$. The corresponding condensed-to-fixed-prefix-context is $h(f(g(\mu(a) \comp g(\pi))), \mu(b) \comp \nu(b))$. Observe that the activity of the latter term lies inside the holes of the context $h(f(g(\Box)), \Box)$, whose set of non-hole positions is $\Spa$.
The condensed-to-fixed-prefix-context form of $\psi$ \wrt\ $\Spa$ can be defined in two different ways: either by induction on $\psi$ analogously as the definition of $\cfpsfn$, or by induction on $\Spa$.
The following definition uses the latter option for a pragmatic reason: it leads to simpler proofs of the properties to be stated about these forms.
\begin{definition}
\label{dfn:cfpc}
Let $\psi$ be a proof term and $\Spa$ a prefix-closed set of positions, such that $\psi$ respects $\Spa$.
We define $\cfpc{\psi}{\Spa}$, the \emph{condensed to fixed prefix context form} of $\psi$ \wrt\ $\Spa$, as follows. \\
If $\Spa = \emptyset$, then $\cfpc{\psi}{\Spa} \eqdef \psi$. \\
Otherwise, $\Spa = \set{\epsilon} \,\cup\, ( \bigcup_{1 \leq i \leq m} i \cdot \proj{\Spa}{i} )$, where $src(\psi)(\epsilon) = f/m$.
In this case $\cfpc{\psi}{\Spa} \eqdef f(\cfpc{\psi_1}{\proj{\Spa}{1}}, \ldots \cfpc{\psi_m}{\proj{\Spa}{m}})$, where
$\cfps{\psi} = f(\psi_1, \ldots, \psi_m)$.
\end{definition}
\begin{lemma}
\label{rsl:cfpc-peqe}
Let $\psi$, $\Spa$ such that $\psi$ respects $\Spa$.
Then $\psi \peqe \cfpc{\psi}{\Spa}$.
\end{lemma}
\begin{proof}
We proceed by induction on the cardinal of $\Spa$.
If $\Spa = \emptyset$ then we conclude immediately.
Otherwise, $\Spa = \set{\epsilon} \,\cup\, ( \bigcup_{1 \leq i \leq m} i \cdot \proj{\Spa}{i} )$ where $\cfps{\psi} = f(\psi_1, \ldots, \psi_m)$, and $\cfpc{\psi}{\Spa} = f(\cfpc{\psi_1}{\proj{\Spa}{1}}, \ldots, \cfpc{\psi_m}{\proj{\Spa}{m}})$.
Lem.~\ref{rsl:cfps-components-respect} implies that $\psi_i$ respects $\proj{\Spa}{i}$ for all $i \leq m$.
Therefore \ih\ can be applied on each $\proj{\Spa}{i}$ to obtain $\psi_i \peqe \cfpc{\psi_i}{\proj{\Spa}{i}}$, so that \eqlfun\ rule yields $\cfps{\psi} \peqe \cfpc{\psi}{\Spa}$. On the other hand, Lem.~\ref{rsl:cfps-peqe} implies $\psi \peq \cfps{\psi}$. Thus we conclude by \eqltrans.
\end{proof}
\begin{lemma}
\label{rsl:peqe-then-fp-args-peqe}
Let $\psi$, $\phi$, $\Spa$ such that $\psi$ and $\phi$ are convergent, $\psi \peqe \phi$ and $\psi$, $\phi$ respect $\Spa$.
Then $\cfpc{\psi}{\Spa} = C[\psi_1, \ldots, \psi_k]$, $\cfpc{\phi}{\Spa} = C[\phi_1, \ldots, \phi_k]$ and $\psi_i \peqe \phi_i$ for all $i$, where $C = \pref{src(\psi)}{\Spa}$.
\end{lemma}
\begin{proof}
We proceed by induction on the cardinal of $\Spa$. If $\Spa = \emptyset$ then we conclude immediately. Otherwise $\Spa = \set{\epsilon} \cup (\bigcup_{1 \leq i \leq m} i \cdot \proj{\Spa}{i})$,
$\cfpc{\psi}{\Spa} = f(\cfpc{\psi'_1}{\proj{\Spa}{1}}, $ \\ $\ldots, \cfpc{\psi'_m}{\proj{\Spa}{m}})$, and
$\cfpc{\phi}{\Spa} = f(\cfpc{\phi'_1}{\proj{\Spa}{1}}, \ldots, \cfpc{\phi'_m}{\proj{\Spa}{m}})$, where
$\cfps{\psi} = f(\psi'_1, \ldots, \psi'_m)$ and $\cfps{\phi} = f(\phi'_1, \ldots, \phi'_m)$.
Lem.~\ref{rsl:cfps-peqe} and Lem.~\ref{rsl:peq-then-same-src-mind-tgt} imply that $src(\psi) = f(src(\psi'_1), \ldots, src(\psi'_m))$ and analogously for $\phi$, so that particularly the root symbols of $\cfps{\psi}$ and $\cfps{\phi}$ coincide since $\psi \peqe \phi$.
Let $j$ such that $1 \leq j \leq m$.
Lem.~\ref{rsl:peqe-then-cfps-proj-peqe} implies that $\psi'_j \peqe \phi'_j$, and Lem.~\ref{rsl:cfps-components-respect} implies that both $\psi'_j$ and $\phi'_j$ respect $\proj{\Spa}{j}$.
Then we can apply \ih\ on $\proj{\Spa}{j}$ obtaining that
$\cfpc{\psi'_j}{\proj{\Spa}{j}} = C_j[\psi^j_1, \ldots, \psi^j_{q_j}]$, $\cfpc{\phi'_j}{\proj{\Spa}{j}} = C_j[\phi^j_1, \ldots, \phi^j_{q_j}]$ and $\psi^j_i \peqe \phi^j_i$ for all $i$, where $\pref{src(\psi'_j)}{\proj{\Spa}{j}} = C_j$.
We define $C \eqdef f(C_1, \ldots, C_m)$. It is straightforward to verify that $\pref{src(\psi)}{\Spa} = C$. Moreover, $\cfpc{\psi}{\Spa} = C[\psi_1, \ldots, \psi_k]$ and $\cfpc{\phi}{\Spa} = C[\phi_1 \ldots, \phi_k]$, where $k = \sum_{1 \leq i \leq m} q_i$, and for any $i \leq k$, $\psi_i = \psi^j_l$ and $\phi_i = \phi^j_l$ for some $j \leq m$ and $l \leq q_j$, implying $\psi_i \peqe \phi_i$. Thus we conclude.
\end{proof}
\begin{lemma}
\label{rsl:cfpc-equals-src-tgt}
Let $\psi$, $\Spa$ such that $\psi$ respects $\Spa$.
Then $\pref{\cfpc{\psi}{\Spa}}{\Spa} = \pref{src(\psi)}{\Spa} = \pref{tgt(\psi)}{\Spa}$.
\end{lemma}
\begin{proof}
Straightforward corollary of Lem.~\ref{rsl:peqe-then-fp-args-peqe} and Lem.~\ref{rsl:respects-then-invariant-fp}.
\end{proof}
\cfpInsideStandardisation{
The following lemma states that the definition of $\cfpc{\psi}{\Spa}$ by induction on $\Spa$, as given in Dfn.~\ref{dfn:cfpc}, is equivalent to the result of defining the same concept by induction on $\psi$, for the binary concatenation case.
\begin{lemma}
\label{rsl:cfpc-binc}
Let $\psi = \phi \comp \chi$ a convergent proof term, and $\Spa$ such that $\psi$ respects $\Spa$. Let $C \eqdef \pref{src(\psi)}{\Spa}$.
Then $\cfpc{\phi}{\Spa} = C[\phi_1, \ldots, \phi_k]$, $\cfpc{\chi}{\Spa} = C[\chi_1, \ldots, \chi_k]$, and $\cfpc{\psi}{\Spa} = C[\phi_1 \comp \chi_1, \ldots, \phi_k \comp \chi_k]$.
\end{lemma}
\begin{proof}
We proceed by induction on $\Spa$.
If $\Spa = \emptyset$, so that $C = \Box$, $\cfpc{\phi}{\Spa} = \phi$, $\cfpc{\chi}{\Spa} = \chi$ and $\cfpc{\psi}{\Spa} = \psi$, then we conclude immediately.
Otherwise $\epsilon \in \Spa$. Hypotheses imply that both $\phi$ and $\phi$ respect $\Spa$, so that particularly both respect $\set{\epsilon}$, and $\psi$ being a well-formed proof term implies $tgt(\phi) = src(\chi)$.
We apply Lem.~\ref{rsl:respects-epsilon-then-stable-root} to both $\phi$ and $\chi$ to obtain $f \eqdef src(\psi)(\epsilon) = src(\phi)(\epsilon) = tgt(\phi)(\epsilon) = src(\chi)(\epsilon) = tgt(\chi)(\epsilon)$.
Moreover $\cfpsfn$ is defined for $\psi$, $\phi$ and $\chi$, say $\cfps{\phi} = f(\phi'_1, \ldots, \phi'_m)$ and $\cfps{\chi} = f(\chi'_1, \ldots, \chi'_m)$; \confer\ Lem.~\ref{rsl:cfps-equals-src-tgt}. Therefore $\cfps{\psi} = f(\phi'_1 \comp \chi'_1, \ldots, \phi'_m \comp \chi'_m)$, so that Lem.~\ref{rsl:cfps-peqe} implies $C = \pref{src(\psi)}{\Spa} = f(C_1, \ldots, C_m)$ where $C_j = \pref{src(\phi'_j \comp \chi'_j)}{\proj{\Spa}{j}}$.
Let $j$ such that $1 \leq j \leq m$.
Lem.~\ref{rsl:cfps-components-respect} implies that $\phi'_j \comp \chi'_j$ respects $\proj{\Spa}{j}$, so that \ih\ can be applied, obtaining
$\cfpc{\phi'_j}{\proj{\Spa}{j}} = C_j[\phi^j_1, \ldots, \phi^j_{q_j}]$, $\cfpc{\chi'_j}{\proj{\Spa}{j}} = C_j[\chi^j_1, \ldots, \chi^j_{q_j}]$, and $\cfpc{\phi'_j \comp \chi'_j}{\proj{\Spa}{j}} = C_j[\phi^j_1 \comp \chi^j_1, \ldots, \phi^j_{q_j} \comp \chi^j_{q_j}]$.
Hence: \\
$\begin{array}{rcl}
\cfpc{\phi}{\Spa} & = & f(C_1[\phi^1_1, \ldots, \phi^1_{q_1}], \ldots, C_m[\phi^m_1, \ldots, \phi^m_{q_m}]) \\
\cfpc{\chi}{\Spa} & = & f(C_1[\chi^1_1, \ldots, \chi^1_{q_1}], \ldots, C_m[\chi^m_1, \ldots, \chi^m_{q_m}]) \\
\cfpc{\psi}{\Spa} & = & f(C_1[\phi^1_1 \comp \chi^1_1, \ldots, \phi^1_{q_1} \comp \chi^1_{q_2}], \ldots, C_m[\phi^m_1 \comp \chi^m_1, \ldots, \phi^m_{q_m} \comp \chi^m_{q_m}])
\end{array}
$ \\
Thus we conclude.
\end{proof}
We end this section about fixed prefixes with a result allowing to apply Dfn.~\ref{dfn:respects}, which refers to the \emph{source} of a proof term, to some proof term of which only the \emph{target} is known.
\begin{lemma}
\label{rsl:pos-tgt-respects-then-pos-src}
Let $\psi$ be a convergent proof term and $r$ a position such that $r \in \Pos{tgt(\psi)}$ and $\frso{\psi}{r'}$ is undefined if $r' \leq r$.
Then $r \in \Pos{src(\psi)}$.
\end{lemma}
\begin{proof}
We proceed by induction on $\psi$, \confer\ Prop.~\ref{rsl:pterm-induction-principle}.
If $\psi$ is an \imstep, then we proceed by induction on $\posln{r}$.
If $r = \epsilon$ then we conclude immediately.
If $r = i r_0$, then lemma hypotheses imply $\frso{\psi}{\epsilon}$ to be undefined, so that $\psi = f(\psi_1, \ldots, \psi_m)$.
In turn, for any $r'_0 \leq r_0$, $i r'_0 \leq i r_0 = r$ holds immediately, so that $\frso{\psi}{i r'_0}$ is undefined; therefore so is $\frso{\psi_i}{r'_0}$.
Moreover $i r_0 \in \Pos{tgt(\psi)}$ implies $r_0 \in \Pos{tgt(\psi_i)}$.
Hence \ih\ can be applied on $\psi_i$ and $r_0$, obtaining $r_0 \in \Pos{src(\psi_i)}$. Thus we conclude.
Assume $\psi = \psi_1 \comp \psi_2$.
In this case, for all $r' \leq r$, $\frso{\psi}{r'}$ undefined implies $\frso{\psi_i}{r'}$ undefined for $i = 1,2$.
Then $r \in tgt(\psi) = tgt(\psi_2)$ implies $r \in src(\psi_2) = tgt(\psi_1)$ so that $r \in src(\psi_1) = src(\psi)$, by \ih\ on $\psi_2$ and $\psi_1$ respectively.
Assume $\psi = \icomp \psi_i$.
Let $n$ such that $\tdist{tgt(\psi_i)}{tgt(\psi)} < 2^{-\posln{r}}$ if $i > n$. Then $r \in \Pos{tgt(\psi)}$ implies $r \in \Pos{tgt(\psi_{n+1})}$.
On the other hand, for any $r' \leq r$, $\frso{\psi}{r'}$ undefined implies $\frso{\psi_i}{r'}$ undefined for all $i$.
Therefore, successive application of the \ih\ on each $\psi_i$ where $i \leq n+1$ yields that $r \in tgt(\psi_{n+1})$ then $r \in src(\psi_{n+1}) = tgt(\psi_n)$ then \ldots then $r \in src(\psi_1) = tgt(\psi_0)$ then $r \in src(\psi_0) = src(\psi)$.
Assume $\psi = f(\psi_1, \ldots, \psi_m)$.
If $r = \epsilon$ then we conclude immediately.
Otherwise, \ie\ if $r = i r_0$, then lemma hypotheses imply $r_0 \in \Pos{tgt(\psi_i)}$ and $\frso{\psi_i}{r'_0}$ undefined if $r'_0 \leq r_0$, \confer\ the argument in the \imstep\ case.
Then \ih\ on $\psi_i$ and $r_0$ yields $r_0 \in \Pos{src(\psi_i)}$. Thus we conclude.
Finally, $\psi = \mu(\psi_1, \ldots, \psi_m)$ contradicts lemma hypotheses, since $\frso{\psi}{\epsilon}$ is defined and $\epsilon \leq r$ for any position $r$.
\end{proof}
\begin{lemma}
\label{rsl:respects-pos-tgt}
Let $\psi$ be a proof term and $\Spa$ a prefix-closed set of positions, such that $\Spa \subseteq \Pos{tgt(\psi)}$ and $\frso{\psi}{r}$ is undefined for all $r \in \Spa$.
Then $\psi$ respects $\Spa$.
\end{lemma}
\begin{proof}
By applying Lemma~\ref{rsl:pos-tgt-respects-then-pos-src} on each $r \in \Spa$ we obtain that $\Spa \subseteq \Pos{src(\psi)}$. Then we conclude just by Dfn.~\ref{dfn:respects}.
\end{proof}
}
\includeStandardisation{\subsubsection{General factorisation result}}
\doNotIncludeStandardisation{\subsection{General factorisation result}}
\label{scn:factorisation-general}
In this section we will extend the factorisation result obtained for \imsteps\ in Sec.~\ref{scn:factorisation-imsteps}, to the set of all proof terms.
\doNotIncludeStandardisation{As we have already mentioned, the }
\includeStandardisation{The }
condensed-to-proof-term forms introduced in Sec.~\ref{sec:cfpc} lead to the proof of the main remaining auxiliary result, namely, the ability of obtain proof terms in which activity at lower depths is in low positions \wrt\ the sequentialisation order given by dot occurrences.
\begin{lemma}
\label{rsl:jump-one-step}
Let $\psi$ be a one-step.
Then there exist two numbers $n, n' < \omega$ such that, for any convergent proof term $\xi$ verifying $tgt(\xi) = src(\psi)$ and $\mind{\xi} \geq n + n'$, a one-step $\psi'$ and a convergent proof term $\xi'$ can be found, which verify all the following: $\xi \comp \psi \peqe \psi' \comp \xi'$, $\sdepth{\psi'} = \sdepth{\psi}$, and $\mind{\xi'} \geq \mind{\xi} - n'$.
\end{lemma}
\begin{proof}
We take $n \eqdef \sdepth{\psi}$ and $n' = \Pdepth{\mu} + 1$ where $\mu \eqdef \psi(\RPos{\psi})$.
We consider a convergent proof term $\xi$ verifying $\mind{\xi} \geq n + n'$ and $tgt(\xi) = src(\psi)$.
\cfpInsideCompression{
Let $\Spa_0 \eqdef \set{p \setsthat p \in src(\psi) \land \posln{p} < \sdepth{\psi}}$, $\Spa \eqdef \Spa_0 \cup (\RPos{\psi} \cdot \PPos{\mu})$, and $k \eqdef max \set{\posln{p} \setsthat p \in \Spa}$.
Observe that $p \in \Spa$ implies $\posln{p} \leq \sdepth{\psi} + \Pdepth{\mu}$, so that $k \leq \sdepth{\psi} + \Pdepth{\mu} < \mind{\xi}$.
}
\cfpInsideStandardisation{
Let $\Spa_0 \eqdef \set{p \setsthat p \in src(\psi) \land \posln{p} < \sdepth{\psi}}$ and $\Spa \eqdef \Spa_0 \cup (\RPos{\psi} \cdot \PPos{\mu})$.
Observe that $p \in \Spa$ implies $\posln{p} \leq \sdepth{\psi} + \Pdepth{\mu} < \mind{\xi}$, so that Lem.~\ref{rsl:frso-defined-then-length-geq-mind} implies $\frso{\xi}{p}$ to be undefined.
}
Moreover, it is straightforward to verify that $\Spa \subseteq \Pos{src(\psi)} = \Pos{tgt(\xi)}$.
\cfpInsideCompression{
Therefore Lem.~\ref{rsl:mind-plus-tgt-then-respects} applies \wrt\ $\xi$, $\Spa$ and $k$, implying that $\xi$ respects $\Spa$.
}
\cfpInsideStandardisation{
Therefore Lem.~\ref{rsl:respects-pos-tgt} implies that $\xi$ respects $\Spa$.
}
Then $\xi_F \eqdef \cfpc{\xi}{\Spa}$ can be defined.
In turn, Lem.~\ref{rsl:cfpc-peqe} implies that $\xi \peqe \xi_F$, so that $\xi \comp \psi \peqe \xi_F \comp \psi$, and Lem.~\ref{rsl:cfpc-equals-src-tgt} implies $\pref{\xi_F}{\Spa} = \pref{tgt(\xi)}{\Spa} = \pref{src(\psi)}{\Spa}$.
Let $C \eqdef \pref{src(\psi)}{\Spa_0}$. An easy induction on $\sdepth{\psi}$ yields that $\pref{\psi}{\Spa_0} = C$, so that the comment following Dfn.~\ref{dfn:term-prefix} implies $\psi = C[t_1, \ldots, t_{j-1}, \mu(u_1, \ldots, u_m), t_{j+1}, \ldots, t_k]$ and $\set{p \setsthat p \in \Pos{C} \,\land\, C(p) \neq \Box} = \Spa_0$. Observe that $\posln{\BPos{C}{i}} = \sdepth{\psi}$ for all $i$, and that particularly $\BPos{C}{j} = \RPos{\psi}$ for some $j$.
In turn, the given form of $\psi$ implies that $src(\psi) = tgt(\xi) = C[t_1, \ldots, t_{j-1}, l[u_1, \ldots, u_m], t_{j+1}, \ldots, t_k]$ where $\mu: l \to h$.
Observe that the set of non-hole positions of the context \\ $C[\Box, \ldots, \Box, l[\Box, \ldots, \Box], \Box, \ldots, \Box]$ is exactly $\Spa$, implying that $C = \pref{tgt(\xi)}{\Spa} = \pref{\xi_F}{\Spa}$, and therefore
$\xi_F = C[\xi_1, \ldots, \xi_{j-1}, l[\phi_1, \ldots, \phi_m], \xi_{j+1}, \ldots, \xi_k]$; \confer\ the comment following Dfn.~\ref{dfn:term-prefix}.
Notice that $\xi_F$ is convergent, implying that all the $\xi_i$ and also the $\phi_i$ are; \confer\ Lem.~\ref{rsl:peqe-soundness-convergent} and Lem.~\ref{rsl:ctx-convergence}.
Moreover, $t_i = tgt(\xi_i)$ for any suitable $i$, and also $u_i = tgt(\phi_i)$ for all suitable $i$. Hence \\
$\begin{array}{@{}rl}
\xi_F \comp \psi
\\ \peqe &
C[\xi_1 \comp t_1, \ldots, \xi_{j-1} \comp t_{j-1}, l[\phi_1 \ldots \phi_m] \comp \mu(u_1, \ldots, u_m), \xi_{j+1} \comp t_{j+1}, \ldots, \xi_k \comp t_k]
\\ \peqe &
C[\xi_1, \ldots, \xi_{j-1}, \mu(\phi_1, \ldots, \phi_m), \xi_{j+1}, \ldots, \xi_k]
\\ \peqe &
C[s_1 \comp \xi_1, \ldots, s_{j-1} \comp \xi_{j-1}, \mu(w_1, \ldots, w_m) \comp h[\phi_1, \ldots, \phi_m], s_{j+1} \comp \xi_{j+1}, \ldots, s_k \comp \xi_k]
\\ \peqe &
C[s_1, \ldots, s_{j-1}, \mu(w_1, \ldots, w_m), s_{j+1}, \ldots, s_k] \ \comp \\ &
C[\xi_1, \ldots, \xi_{j-1}, h[\phi_1, \ldots, \phi_m], \xi_{j+1}, \ldots, \xi_k]
\end{array}$ \\[2pt]
where $s_i \eqdef src(\xi_i)$ and $w_i \eqdef src(\phi_i)$, in both cases for all suitable $i$.
To justify the equivalences; \confer\ Lem.~\ref{rsl:struct-ctx}; \peqidright, \peqinout\ and Lem.~\ref{rsl:peqe-compatible-ctx}; \peqidleft, \peqoutin\ and Lem.~\ref{rsl:peqe-compatible-ctx} again; and finally Lem.~\ref{rsl:struct-ctx} again; respectively.
We take $\psi' \eqdef C[s_1, \ldots, s_{j-1}, \mu(w_1, \ldots, w_m), s_{j+1}, \ldots, s_k]$ and \\ $\xi' \eqdef C[\xi_1, \ldots, \xi_{j-1}, h[\phi_1, \ldots, \phi_m], \xi_{j+1}, \ldots, \xi_k]$.
Observe that convergence of all $\xi_i$ and $\phi_i$ imply convergence of $\xi'$, \confer\ Lem.~\ref{rsl:ctx-convergence}.
In order to conclude, we must verify that $\mind{\xi'} \geq \mind{\xi} - n' = \mind{\xi_F} - (\Pdepth{\mu} + 1)$; \confer\ Lem.~\ref{rsl:peq-then-same-src-mind-tgt}.
Let $a$ such that $\mind{\xi_a} \leq \mind{\xi_i}$ for all $i$ such that $1 \leq i \leq k$ and $i \neq j$,
$b$ such that $\mind{\phi_b} + \posln{\BPos{l}{b}} \leq \mind{\phi_i} + \posln{\BPos{l}{i}}$ for all $i$ such that $1 \leq i \leq m$, and
$c$, $k$ such that $\mind{\phi_c} + \posln{\BPos{h}{k}} \leq \mind{\phi_i} + \posln{\BPos{h}{j}}$ if $1 \leq i \leq m$ and $h(\BPos{h}{j}) = x_i$.
In these definitions, $l$ and $h$ are considered as contexts as when we write \eg\ $l[\phi_1, \ldots, \phi_m]$.
Lem.~\ref{rsl:mind-ctx} implies $\mind{\xi_F} = \sdepth{\psi} + min(\mind{\xi_a}, \mind{\phi_b} + \posln{\BPos{l}{b}})$ and $\mind{\xi'} = \sdepth{\psi} + min(\mind{\xi_a}, \mind{\phi_c} + \posln{\BPos{h}{k}})$.
Observe that $\posln{\BPos{l}{i}} \leq \Pdepth{\mu} + 1$ for all $i$.
We show $\mind{\xi_F} - (\Pdepth{\mu} + 1) \leq \mind{\xi'}$.
If $\mind{\xi_a} \leq \mind{\phi_c} + \posln{\BPos{h}{k}}$, then
$\mind{\xi_F} \leq \sdepth{\psi} + \mind{\xi_a} = \mind{\xi'}$ in either case \wrt\ the characterisation of $\mind{\xi_F}$.
Otherwise, \ie\ if $\mind{\phi_c} + \posln{\BPos{h}{k}} < \mind{\xi_a}$,
observe that $\mind{\xi_F} \leq \sdepth{\psi} + \mind{\phi_b} + \posln{\BPos{l}{b}}$ holds in any case. Therefore \\
$\begin{array}{rcl}
\mind{\xi_F} & \leq &
\sdepth{\psi} + \mind{\phi_b} + \posln{\BPos{l}{b}} \\ & \leq &
\sdepth{\psi} + \mind{\phi_c} + \posln{\BPos{l}{c}} \\ & \leq &
\sdepth{\psi} + \mind{\phi_c} + (\Pdepth{\mu} + 1)
\end{array}$ \\
Therefore $\mind{\xi_F} - (\Pdepth{\mu} + 1) \leq
\sdepth{\psi} + \mind{\phi_c} \leq
\sdepth{\psi} + \mind{\phi_c} + \posln{\BPos{h}{k}} = \mind{\xi'}$.
\end{proof}
\begin{lemma}
\label{rsl:jump-ppterm}
Let $\psi$ be a finite \pnpterm.
Then there exist two numbers $n, n' < \omega$ such that, for any convergent proof term $\xi$ verifying $tgt(\xi) = src(\psi)$ and $\mind{\xi} \geq n + n'$, a finite \pnpterm\ $\psi'$ and a convergent proof term $\xi'$ can be found, which verify all the following: $\xi \comp \psi \peqe \psi' \comp \xi'$, $\ppsteps{\psi'} = \ppsteps{\psi}$, $\sdepth{\redel{\psi'}{i}} = \sdepth{\redel{\psi}{i}}$ for all $i$, and $\mind{\xi'} \geq \mind{\xi} - n' \geq n$.
\end{lemma}
\begin{proof}
We proceed by induction on $\ppsteps{\psi}$.
If $\ppsteps{\psi} = 0$, \ie\ $\psi \in \iSigmaTerms$, then $src(\psi) = \psi$.
Therefore we can take $n = n' = 0$, since for any $\xi$ verifying $tgt(\xi) = \psi$, it is straightforward to obtain $\xi \comp \psi \peqe src(\xi) \comp \xi$, and to verify the required properties for $\psi' \eqdef src(\xi)$ and $\xi' \eqdef \xi$ .
Assume $\ppsteps{\psi} = n+1$, \ie\ $\psi = \chi \comp \phi$, where $\chi$ is a one-step and $\phi$ is a \pnpterm\ verifying $\ppsteps{\phi} = n$.
In this case, \ih\ can be applied on $\phi$; let $m$ and $m'$ be the corresponding numbers. Moreover, Lem.~\ref{rsl:jump-one-step} applies to $\chi$; let $p$ and $p'$ be the numbers whose existence is stated by that lemma.
Let $n \eqdef max(m,p)$ and $n' \eqdef m' + p'$.
Let $\xi$ a convergent proof term verifying $\mind{\xi} \geq n + n' = n + m' + p' \geq p + p'$, and $tgt(\xi) = src(\psi) = src(\chi)$.
Then the conclusion of Lem.~\ref{rsl:jump-one-step} implies that $\xi \comp \psi = \xi \comp \chi \comp \phi \peqe \chi' \comp \xi'' \comp \phi$, where $\chi'$ is a one-step verifying $\sdepth{\chi'} = \sdepth{\chi}$ and $\xi''$ is a convergent proof term such that $\mind{\xi''} \geq \mind{\xi} - p' \geq n + m' \geq m + m'$.
In turn, the conclusion of the \ih\ implies that $\chi' \comp \xi'' \comp \phi \peqe \chi' \comp \phi' \comp \xi'$, where $\phi'$ is a \pnpterm\ verifying $\ppsteps{\phi'} = \ppsteps{\phi}$ and $\sdepth{\redel{\phi'}{i}} = \sdepth{\redel{\phi}{i}}$ for all $i$, and $\xi'$ is a convergent proof term such that $\mind{\xi'} \geq \mind{\xi''} - m' \geq n$. We take $\psi' \eqdef \chi' \comp \phi'$, and we conclude by observing that \eqltrans\ implies $\xi \comp \psi \peqe \psi' \comp \xi'$.
\end{proof}
The given auxiliary results allow to prove the statement being the aim of this Section.
\begin{proposition}
\label{rsl:factorisation}
Let $\psi$ be a convergent proof term and $n < \omega$.
Then there exist $\chi$ and $\phi$ such that $\psi \peqe \chi \comp \phi$, $\chi$ is a finite \pnpterm, $\phi$ is convergent and $mind(\phi) > n$.
\end{proposition}
\begin{proof}
We proceed by induction on $\alpha$ where $\psi \in \layerpterm{\alpha}$, analysing the cases in the formation of $\psi$ \wrt\ Dfn.~\ref{dfn:layer-pterm}.
\begin{itemize}
\item
Assume that $\psi$ is an \imstep. In this case we proceed by induction on $n$.
If $n = 0$ then Lem.~\ref{rsl:factorisation-imstep} suffices to conclude.
Assume $n = n' + 1$.
Lem.~\ref{rsl:factorisation-imstep} implies $\psi \peqe \chi_0 \comp \phi'$ where $\chi_0$ is a finite \pnpterm, $\phi'$ is a convergent \imstep\ and $\mind{\phi'} > 0$, so that $\phi' = f(\phi'_1, \ldots, \phi'_m)$.
Observe that $\phi'$ convergent implies $\phi'_i$ convergent for all $i$, \confer\ Lem.~\ref{rsl:imstep-convergent-args}.
Then \ih\ can be applied on all $\phi'_i$ \wrt\ $n'$, yielding
$\phi' \peqe f(\chi_1 \comp \phi_1, \ldots, \chi_m \comp \phi_m)$ where for all $i$, $\chi_i$ is a finite \pnpterm, $\phi_i$ is convergent and $\mind{\phi_i} > n'$.
Hence $\psi \peqe \chi_0 \comp f(\chi_1, \ldots, \chi_m) \comp f(\phi_1, \ldots, \phi_m)$.
Assume that $m = 3$; observe that
$f(\chi_1, \chi_2, \chi_3)
\peqe f(\chi_1 \comp t_1, s_2 \comp \chi_2, s_3 \comp \chi_3)
\peqe f(\chi_1, s_2, s_3) \comp f(t_1, \chi_2, \chi_3)
\peqe f(\chi_1, s_2, s_3) \comp f(t_1 \comp t_1, \chi_2 \comp t_2, s_3 \comp \chi_3)
\peqe f(\chi_1, s_2, s_3) \comp f(t_1, \chi_2, s_3) \comp (t_1, t_2, \chi_3)$.
An analogous reasoning for any $m$ yileds
$f(\chi_1, \chi_2, \ldots, \chi_m)
\peqe f(\chi_1, src(\chi_2), \ldots, src(\chi_m))
\comp f(tgt(\chi_1), \chi_2, \ldots, src(\chi_m))
\comp f(tgt(\chi_1), tgt(\chi_2), \ldots, \chi_m)$.
In turn, it is straightforward to obtain a \ppterm\ $\chi'_k \peqe f(tgt(\chi_1), \ldots, \chi_k, \ldots, src(\chi_m))$, so that $\chi' \eqdef \chi'_0 \comp \ldots \comp \chi'_m$ is a \ppterm\ verifying $\chi' \peqe f(\chi_1, \chi_2, \ldots, \chi_m)$.
Thus we conclude by taking $\chi \eqdef \chi_0 \comp \chi'$ and $\phi \eqdef f(\phi_1, \ldots, \phi_m)$.
\item
Assume $\psi = \psi_1 \comp \psi_2$ and $\psi$ is not an infinite composition.
In this case we can apply \ih\ on $\psi_2$, obtaining $\psi_2 \peqe \chi_2 \comp \phi_2$ where $\chi_2$ is a finite \pnpterm, $\phi_2$ is convergent and $\mind{\phi_2} > n$.
Lem.~\ref{rsl:jump-ppterm} applies to $\chi_2$, implying the existence of two numbers, say $m_0$ and $m'$, which enjoy some properties.
Let $m \eqdef max(n,m_0)$.
Applying \ih\ on $\psi_1$ \uln{\wrt\ $m + m'$}, we obtain $\psi_1 \peqe \chi_1 \comp \phi_1$, where $\chi_1$ is a finite \pnpterm, $\phi_1$ is convergent and $\mind{\phi_1} > m + m' \geq m_0 + m'$. Observe $\psi \peqe \chi_1 \comp \phi_1 \comp \chi_2 \comp \phi_2$, so that $tgt(\phi_1) = src(\chi_2)$.
Therefore, the conclusion of Lem.~\ref{rsl:jump-ppterm} implies $\phi_1 \comp \chi_2 \peqe \chi'_2 \comp \phi'_1$, so that $\psi \peqe \chi_1 \comp \chi'_2 \comp \phi'_1 \comp \phi_2$, where $\chi'_2$ is a finite \pnpterm\ (since $\ppsteps{\chi'_2} = \ppsteps{\chi_2}$), $\phi'_1$ is convergent and $\mind{\phi'_1} \geq \mind{\phi_1} - m' > m \geq n$.
Thus we conclude by taking $\chi \eqdef \chi_1 \comp \chi'_2$ and $\phi \eqdef \phi'_1 \comp \phi_2$.
\item
Assume $\psi = \icomp \psi_i$.
Let $k$ such that $\mind{\psi_i} > n$ if $i > k$; convergence of $\psi$ entails the existence of such $k$.
Then $\psi \peqe \psi_0 \comp \ldots \comp \psi_k \comp (\icomp \psi_{k+1+i})$, and $\mind{\icomp \psi_{k+1+i}} > n$; notice that convergence of $\psi$ implies convergence of $\icomp \psi_{k+1+i}$.
Observe that $\psi_0 \comp \ldots \comp \psi_k \in \layerpterm{\alpha'}$ where $\alpha' < \alpha$. This observation allows to use \ih\ to obtain $\psi_0 \comp \ldots \comp \psi_k \peqe \chi \comp \phi'$ where $\chi$ is a finite \pnpterm, $\phi'$ is convergent and $\mind{\phi'} > n$. Then we conclude by taking $\phi \eqdef \phi' \comp (\icomp \psi_{k+1+i})$.
\item
Assume $\psi = f(\psi_1, \ldots, \psi_m)$ and $\psi$ is not an \imstep.
In this case, we can apply \ih\ on each $\psi_i$ obtaining $\psi_i \peqe \chi_i \comp \phi_i$, where $\chi_i$ is a finite \pnpterm, $\phi_i$ is convergent, and $\mind{\phi_i} > n$.
Then $\psi \peqe f(\chi_1, \ldots, \chi_m) \comp f(\phi_1, \ldots, \phi_m)$.
Hence, an argument about $f(\chi_1, \ldots, \chi_m)$ analogous to that used in the \imstep\ case allows to conclude.
\item
Assume $\psi = \mu(\psi_1, \ldots, \psi_m)$ and $\psi$ is not an \imstep.
Say $\mu: l[x_1, \ldots, x_m] \to h$.
Assume $h = f(h_1, \ldots, h_k)$.
In this case $\psi \peqe \mu(src(\psi_1), \ldots, src(\psi_m)) \comp f(h_1[\psi_1, \ldots, \psi_m], \ldots, h_k[\psi_1, \ldots, \psi_m])$.
Applying \ih\ on each $\psi_i$ yields $\psi_i \peqe \chi_i \comp \phi_i$, where $\chi_i$ is a finite \pnpterm, $\phi_i$ is convergent, and $\mind{\phi_i} > n$. \\
Therefore
$\psi \peqe
\mu(src(\psi_1), \ldots, src(\psi_m)) \comp
f(h_1[\chi_1, \ldots, \chi_m], \ldots, h_k[\chi_1, \ldots, \chi_m])
\comp f(h_1[\phi_1, \ldots, \phi_m], \ldots, h_k[\phi_1, \ldots, \phi_m])$; \confer\ Lem~\ref{rsl:struct-ctx}.
Hence, an argument about $f(h_1[\chi_1, \ldots, \chi_m], \ldots, h_k[\chi_1, \ldots, \chi_m])$ analogous to that used in the \imstep\ case for $f(\chi_1, \ldots, \chi_m)$, \confer\ Lem.~\ref{rsl:struct-ctx}, allows to conclude.
The other possible case is $h = x_j$, implying $\psi \peqe \mu(src(\psi_1), \ldots, src(\psi_m)) \comp \psi_j$. \Ih\ can be applied on $\psi_j$ obtaining $\psi_j \peqe \chi' \comp \phi$, where $\chi'$ is a finite \pnpterm, $\phi$ is convergent and $\mind{\phi} > n$.
Thus we conclude by taking $\chi \eqdef \mu(src(\psi_1), \ldots, src(\psi_m)) \comp \chi'$.
\end{itemize}
\end{proof}
\doNotIncludeStandardisation{
\subsection{Proof of the compression result}
}
\begin{theorem}
\label{rsl:compression}
Let $\psi$ be a convergent proof term. Then there exists some \ppterm\ $\phi$ verifying $\psi \peq \phi$ and $\ppsteps{\phi} \leq \omega$.
\end{theorem}
\begin{proof}
We define the sequences of proof terms $\langle \psi_i \rangle_{i < \omega}$ and $\langle \phi_i \rangle_{i < \omega}$ as follows.
We start defining $\psi_0 \eqdef \psi$.
Then, for each $i < \omega$, we define $\phi_i$ and $\psi_{i+1}$ to be proof terms verifying that $\psi_i \peqe \phi_i \comp \psi_{i+1}$, $\phi_i$ is a finite \pnpterm\ and either $\mind{\psi_{i+1}} > \mind{\psi_i}$ or $\mind{\psi_{i+1}} = \mind{\psi_i} = \omega$; \confer\ Prop.~\ref{rsl:factorisation}.
Observe that $\mind{\psi_i} < \omega$ implies $\mind{\phi_i} = \mind{\psi_i}$ by \ref{rsl:peq-then-same-src-mind-tgt}, so in that case $\phi_i$ is a \ppterm, \ie\ it is not trivial.
Moreover, an easy induction on $n$ yields $\psi \peqe \phi_0 \comp \ldots \comp \phi_n \comp \psi_{n+1}$ for all $n$.
We define $T \eqdef \set{n \setsthat \psi_n \textnormal{ is a trivial proof term}}$.
There are three cases to consider:
\begin{itemize}
\item
If $0 \in T$, \ie\ if $\psi$ is a trivial proof term, then it is enough to take $\phi \eqdef src(\psi)$ and refer to Lem.~\ref{rsl:trivial-pterm-peq-src}.
\item
Assume $0 \notin T$ and $T \neq \emptyset$, let $n$ be the minimal element in $T$.
In this case we take $\phi \eqdef \phi_0 \comp \ldots \comp \phi_{n-1}$.
For any $k < \omega$, observe that $\psi \peqe \phi \comp \psi_n$, $\phi \peqe \phi \comp tgt(\phi)$ (\confer\ \peqidright), and $\mind{\psi_n} = \mind{tgt(\phi)} = \omega > k$, \confer\ Lem.~\ref{rsl:trivial-pterm-mind-omega}.
Then Dfn.~\ref{dfn:peqe} allows to assert $\psi \peq \phi$.
Finally, observe that each $\phi_i$ being finite implies that $\phi$ is also a finite \ppterm, \ie\ it verifies $\ppsteps{\phi} < \omega$.
\item
Assume $T = \emptyset$. In this case, for any $i$ Lem.~\ref{rsl:trivial-pterm-mind-omega} implies that $\mind{\psi_i} < \omega$, so that $\phi_i$ is non-trivial.
We take $\phi \eqdef \icomp \phi_i$.
Let $n < \omega$. We have already verified that $\psi \peqe \phi_0 \comp \ldots \comp \phi_n \comp \psi_{n+1}$, and $\phi \peqe \phi_0 \comp \ldots \comp \phi_n \comp \icomp \phi_{n+1+i}$.
On the other hand, an easy induction on $k$ implies $\mind{\psi_k} = \mind{\phi_k} \geq k$ for all $k$, then $\mind{\psi_{n+1}} > n$, and also $\mind{\icomp \phi_{n+1+i}} > n$.
Hence the rule \eqllim\ can be applied to obtain $\psi \peq \phi$.
We conclude by observing that $\ppsteps{\phi_n} < \omega$ for all $n$ implies that $\ppsteps{\phi} \leq \omega$.
\end{itemize}
\end{proof}
\end{document}
|
\begin{document}
\title{Maximizing Influence-based Group Shapley Centrality}
\author{Ruben Becker \and Gianlorenzo D'Angelo \and Hugo Gilbert}
\date{Gran Sasso Science Institute, L'Aquila, Italy\\[email protected]}
\maketitle
\begin{abstract}
One key problem in network analysis is the so-called influence maximization problem, which consists in finding a set $S$ of at most $k$ seed users, in a social network, maximizing the spread of information from $S$. This paper studies a related but slightly different problem: We want to find a set $S$ of at most $k$ seed users that maximizes the spread of information, when $S$ is added to an already pre-existing -- \emph{but unknown} -- set of seed users $T$. We consider such scenario to be very realistic. Assume a central entity wants to spread a piece of news, while having a budget to influence $k$ users. This central authority may know that some users are already aware of the information and are going to spread it anyhow. The identity of these users being however completely unknown.
We model this optimization problem using the Group Shapley value, a well-founded concept from cooperative game theory. While the standard influence maximization problem is easy to approximate within a factor $1-1/e-\epsilon$ for any $\epsilon>0$, assuming common computational complexity conjectures, we obtain strong hardness of approximation results for the problem at hand in this paper. Maybe most prominently, we show that it cannot be approximated within $1/n^{o(1)}$ under the Gap Exponential Time Hypothesis. Hence, it is unlikely to achieve anything better than a polynomial factor approximation. Nevertheless, we show that a greedy algorithm can achieve a factor of $\frac{1-1/e}{k}-\epsilon$ for any $\epsilon>0$, showing that not all is lost in settings where $k$ is bounded.
\end{abstract}
\section{INTRODUCTION}
Node centrality and propagation of information or influence are two main topics in network analysis. The former regards the problem of determining the most important nodes in a network according to some measure of importance, while the latter studies mathematical models to represent how information propagates in a communication network or how the influence of individuals spreads in a network.
In order to measure the centrality of nodes in a network a real-valued function, called centrality index, associates a real number with each node that reflects its importance or criticality within the network.
Most of the centrality indices defined in the literature are based on graph-theoretical concepts and static graph properties like distance (closeness, harmonic, and degree), spectral (page-rank or Katz), or path-based (betweenness, coverage) properties.
Modeling the spread of influence, instead, requires the combination of a \emph{dynamic} model for influence diffusion and a static model based on the network topology.
Chen and Teng~\cite{chen2017interplay} initiated the study of the interplay between spreading dynamics and network centrality by defining two centrality indices based on dynamic models for influence diffusion: the \emph{single node influence centrality}, which measures the centrality of a node by its capability of spreading influence when acting alone, and the \emph{Shapley centrality}, which uses the Shapley value to measure the capability of a node to increase the spreading capacity of a group of nodes.
In cooperative game theory, the Shapley value assesses the expected relevance of each player within a subset of players (also called coalition), where the expectation is taken over all possible coalitions. More formally, given a characteristic function $\tau$ that maps each coalition to the total payoff that this coalition receives, the Shapley value of a player $i$ can be understood as the expected payoff that $i$ adds to any coalition, w.r.t. function $\tau$.
The Shapley centrality index studied by Chen and Teng~\cite{chen2017interplay} measures the centrality of a node by using the Shapley value and the spreading function $\sigma$ as characteristic function.
Most centrality indices neglect the relevance that coalitions of individuals and their coordination play in social networks. For this reason, many centrality indices have been generalized to \emph{group centrality indices} which are real-valued functions over subsets of nodes instead of single nodes. Typically, a group centrality index is fundamentally different from a combination of the individual centrality indices of the nodes that compose the group, as it captures the relevance of the set as a whole, and not just as a sum of individuals.
This paper extends the notion of influence-based Shapley centrality from single nodes to \emph{groups of nodes} by using the concept of the Group Shapley value.
Our Influence-based Group Shapley (IGS) centrality associates to a set $S$ of nodes, the expected gain in influence that $S$ adds to any pre-existing seed set $T$.
Notably, we investigate the problem of finding a set \(S\) of size at most $k$ with highest IGS value.
Interestingly, we believe that this way of evaluating the importance of a set of seed users is of high interest from a practical viewpoint. Assume a central entity wants to spread a given piece of news, while having a budget to influence a set of $k$ users, at the same time knowing that already some users are aware of the information and are going to spread it anyhow. The central entity, however, may have no knowledge about who these users are. In this case, the central authority should target a set of seed users with large IGS value.
\paragraph{Our contribution.} We formalize the \textsc{Max-Shapley-Group}\xspace problem of finding a set of seed nodes with highest IGS centrality under a cardinality constraint and show how to compute a $(1-\epsilon)$-approximate value for the IGS centrality of a given set of nodes. Unfortunately, assuming common complexity theory conjectures, we obtain strong hardness of approximation results for the \textsc{Max-Shapley-Group}\xspace problem. Maybe most prominently, we show that it cannot be approximated within $1/n^{o(1)}$ under the Gap Exponential Time Hypothesis. Hence, it is unlikely to achieve an approximation factor that is better than a polynomial in $n$. Nevertheless, we show that a greedy algorithm achieves a factor of $\frac{1-1/e}{k}-\epsilon$ for any $\epsilon>0$, showing that not all is lost in settings where $k$ is bounded.
\section{RELATED WORK}
There is a large literature about network centrality indices, see~\cite[Ch.~7]{N10} for an introduction. Centrality indices are usually categorized as distance-based (e.g., closeness centrality~\cite{B50}), path-based (e.g., betweenness centrality~\cite{F77}), or spectral (e.g., page-rank~\cite{BP98}). Most of the literature focuses on defining indices for specific application domains (e.g.,~\cite{B50,BP98,F77}), on the efficient computation of the centrality index of each node or of the top-ranked nodes (e.g.,~\cite{BergaminiBCMM19,RiondatoU18}), or on axiomatic characterization (e.g.,~\cite{AltmanT05,BV14}).
Several centrality indices have been generalized to \emph{group} centrality indices~\cite{AngrimanGBZGM19,BergaminiGM18,ChenWW16,EveretB99,IshakianETB12,MedyaSSBS18,ZhaoLTGX14,ZhaoWLTG17} and to \emph{Shapley} centrality~\cite{GomezGMOPT,MichalakASRJ13,SkibskiMR18,SzczepanskiMR16,TarkowskiMRW18Arxiv,TarkowskiSMHW18}.
In all these papers, the centrality indices are solely based on graph theoretical properties and do not take into account dynamic models for influence spread.
Modeling spread of influence and information diffusion has also been widely investigated in the literature. One of the most studied problems is the so-called \emph{influence maximization problem}: Given a network and a budget $k$, find a set of $k$ nodes, called \emph{seeds}, to be the starters of an influence diffusion process in such a way that the expected number of nodes that have been influenced at the end of the process is maximized.
The influence maximization problem has been introduced by Domingos and Richardson~\cite{DomingosR01,RichardsonD02} and formalized as an optimization problem by Kempe et al.~\cite{DBLP:journals/toc/KempeKT15}. Several work followed these seminal papers, we refer the interest reader to~\cite{ChenCL13} and to the references in~\cite{DBLP:journals/toc/KempeKT15}.
In the literature on influence maximization, the influence capability of a set of seeds is modeled as a function $\sigma$: given a set $S$ of seed nodes, $\sigma(S)$ is the expected number of eventually influenced nodes. The definition of $\sigma$ depends on the model used to represent the spread of influence in the network, the two most popular models being the \emph{Independent Cascade Model (ICM)} and \emph{Linear Threshold Model (LTM)}, see Section~\ref{subsec: Preliminaries on GSV for SI} for more details.
As previously mentioned, Chen and Teng initiated studying the interplay of network centrality and dynamic models for influence spread~\cite{chen2017interplay}. They introduced two centrality indices that are based on the most commonly used models for influence diffusion.
The first index is called Single Node Influence (SNI) and measures the importance of a node $v$ by its capability of influencing other nodes when $v$ is the only seed node, i.e., the SNI of $v$ is $\sigma(\{v\})$.
The second index, called Shapley centrality, is computed as the Shapley value of each node when the payoff function is $\sigma$. The authors presented an axiomatic characterization of the proposed centrality indices, that is they presented five axioms and showed that the Shapley centrality is the only index that satisfies all of them, while the SNI centrality is the only index that satisfies a set of three different axioms. This characterization captures the differences between these two indices: while SNI is suitable to model the centrality of a single node when it acts alone, the Shapley centrality characterizes the additional influence of a single node when acting in a group.
We remark that our IGS centrality extends this latter concept by measuring the added value (in terms of influence) of a group of nodes when it operates within a larger group.
In the same paper, Chen and Teng proposed an efficient algorithm to approximately compute SNI and Shapley centralities and experimentally evaluated it on several real-world networks.
In a follow-up paper, Chen et al~\cite{ChenTZ18Arxiv}, presented a unified framework to extend classical graph-theoretical centrality indices to influence based ones and group centrality indices to their Shapley influence-based counterparts.
They follow an axiomatic approach, that is, they show that the derived influence-based centrality formulations are the unique centrality indices that conform with their corresponding graph-theoretical ones and satisfy the Bayesian axiom.
They also provide scalable algorithms to compute influence-based centrality and Shapley centrality.
We remark that also in this latter paper the aim is to evaluate the centrality of a single node, while the focus of our paper is on evaluating the centrality of a group of nodes.
\section{PRELIMINARIES}
In this section, we aim to connect cooperative game theory with influence maximization. Subsection~\ref{subsec: Preliminaries on GSV} introduces the concept of the Group Shapley value as (arguably most popular) special case of probabilistic generalized values in cooperative game theory.
We then provide some observations on the Group Shapley value that will turn out essential when we turn to its computation later on. Subsection~\ref{subsec: Preliminaries on GSV for SI} recalls the basics needed from the influence maximization literature. Subsection~\ref{subsec: Preliminaries on GSC} explains how the Group Shapley value can be applied to the setting of influence maximization, we refer to it as Influence-based Group Shapley (IGS) centrality in this case.
Throughout the paper, we denote by $[n]$ the set $\{1,\ldots,n\}$, by $2^{S}$ the set of all subsets of $S$, and by \(\binom{S}{k}\) the set of subsets of $S$ of size $k$.
\subsection{The Group Shapley Value} \label{subsec: Preliminaries on GSV}
In cooperative game theory~\cite{chalkiadakis2011computational, myerson2013game}, a game on $n\ge 2$ players
is commonly formalized by a characteristic function $\tau:2^{[n]}\rightarrow \ensuremath{\mathbb{R}}$ that assigns to every subset $S\subseteq [n]$ of players, also called a coalition, a value $\tau(S)$.
Marichal et al.~\cite{DBLP:journals/dam/MarichalKF07} introduced the concepts of \emph{probabilistic generalized values} and \emph{generalized semivalues} as ways of measuring the worth of a coalition $S\subseteq [n]$. Their notions generalize the more classical concepts of probabilistic values and semivalues from individuals to groups of individuals. That is, these values quantify the prospect of groups of players in a game. For fixed $n$, a probabilistic generalized value of a coalition $S\subseteq [n]$ in a game $\tau:2^{[n]}\rightarrow \ensuremath{\mathbb{R}}$ is of the form
$
\phi_\tau(S) := \sum_{T\subseteq [n]\setminus S} p^S_T \cdot (\tau(T \cup S) - \tau(T)),
$
where $p^S$ denotes a probability distribution on the subsets $T$ of $[n]\setminus S$.
That is, generally speaking, a probabilistic generalized value quantifies the average marginal contribution of the set $S$ to any set $T$ of players that is disjoint from $S$. Note that these marginal contributions are assigned different probabilities $p^S_T$.
For fixed $S$, these probabilities can be understood as a-priori likelihoods of sets $T$ to be extended by $S$.
A generalized semivalue is a probabilistic generalized value such that $p^S_T=p^{S'}_{T'}$ if $|S|=|S'|$ and $|T|=|T'|$.
The arguably best known instance of generalized semivalues is the \emph{Group Shapley value}~\cite{DBLP:journals/4or/FloresMT19,DBLP:journals/dam/MarichalKF07}.
For a subset $S\subseteq [n]$ of players in a game $\tau$, the \emph{Group Shapley value} of $S$ is defined as
\[
\phi^{\sh}_\tau(S):=\sum_{T\subseteq [n]\setminus S} \frac{|T|!(n-|S|-|T|)!}{(n-|S|+1)!} \cdot (\tau(T \cup S) - \tau(T)),
\]
i.e., it is the generalized semivalue for which
$p^S_T=\frac{1}{n-|S|+1}/ \binom{n-|S|}{|T|}$. Stated otherwise, the set $T$ can be seen as a random variable chosen by first sampling an integer \(t\in\{0,\ldots,n-|S|\}\) uniformly at random and then picking a set of size $t$ in \([n]\setminus S\) uniformly at random.
The Group Shapley value is a generalization of the well-known Shapley value~\cite{shapley1953value}, which quantifies the contribution of a single player to a coalition in a game. The standard \emph{Shapley value} of a player $i$ w.r.t.\ $\tau$ can simply be defined as the Group Shapley value of the singleton set $\{i\}$, that is $\phi^{\sh}_\tau(i):=\phi^{\sh}_\tau(\{i\})$. It is well known that, using a standard counting argument, this definition is equivalent to
\(
\phi^{\sh}_\tau(i):=\E_{\pi}[\tau(T_{\pi, i} \cup \{i\}) - \tau(T_{\pi, i})],
\)
where $\pi\sim \Pi([n])$ is a permutation of $[n]$ that is picked uniformly at random among all permutations $\Pi([n])$ of $[n]$ and $T_{\pi, i}$ denotes the set of players in $[n]$ ordered before $i$ in $\pi$.
A similar formulation in terms of permutations can also be obtained for the \emph{Group} Shapley value. Let us first introduce the following notation: For a set \(X\subseteq [n]\), let \(X_{\overline{S}} := (X\setminus S) \cup \{ \hat{s}\}\), where \(\hat{s}\) is an auxiliary item representing all the items from \(S\).\footnote{Flores et al.~\cite{DBLP:journals/4or/FloresMT19} define the so-called \emph{merging game} on $V_{\overline{S}}$. Considering games that result by merging players is quite common, see for example the work of Lehrer~\cite{lehrer1988axiomatization}.}
\begin{observation}[Group Shapley formulation using permutations] \label{obs: group shapley with permutation}
The Group Shapley value of a set \(S\) in a game \(\tau\) is equal to
\[
\phi^{\sh}_\tau(S) = \E_{\pi \sim \Pi([n]_{\overline{S}})}[\tau(T_{\pi,\hat{s}}\cup S) - \tau(T_{\pi,\hat{s}})],
\]
where \(T_{\pi,\hat{s}}\) is the subset of $[n]\setminus S$ preceding \(\hat{s}\) in the permutation \(\pi\) of $[n]_{\overline{S}}$ picked uniformly at random from $\Pi([n]_{\overline{S}})$.
\end{observation}
\begin{proof}
Consider the game \(\hat{\tau}\) defined on \([n]_{\overline{S}}\) by \(\hat{\tau}(T) = \tau(T)\) if $\hat{s} \not\in T$ and \(\hat{\tau}(T) = \tau((T\setminus\{\hat{s}\}) \cup S )\) otherwise. Then Observation~\ref{obs: group shapley with permutation} follows from the fact that the Shapley value of $\hat{s}$ in $\hat{\tau}$ coincides with the Group Shapley value of $S$ in \(\tau\).
\end{proof}
We proceed with the following observation on the probability that a given set $R$ intersects with a set $T$ chosen according to probabilities $p^S_T$ for the Group Shapley value. This observation relies on the Group Shapley formulation using the permutations.
\begin{observation}[Intersection Probability for the Group Shapley Value]\label{obs: intersection probability shapley}
Let $S\subseteq [n]$ be a group. For any \(R\subseteq [n]\) with \(R\cap S \neq \emptyset\), it holds that \(\Pr_{\pi\sim \Pi([n]_{\overline{S}})}[R_{\overline{S}} \cap T_{\pi,\hat{s}} = \emptyset] = 1/(|R\setminus S| +1)\), where \(T_{\pi,\hat{s}}\) is the subset of $[n]$ preceding \(\hat{s}\) in the random permutation \(\pi\) of $[n]_{\overline{S}}$.
\end{observation}
\begin{proof}
The event \(R_{\overline{S}}\cap T_{\pi,\hat{s}}=\emptyset\) is equivalent to the permutation \(\pi\) of $[n]_{\overline{S}}$ placing \(\hat{s}\) ahead of all the other nodes in \(R_{\overline{S}}\). Since \(\pi\) is sampled uniformly at random from \(\Pi([n]_{\overline{S}})\), this event happens with probability exactly \(1/|R_{\overline{S}}|=1/(|R\setminus S| +1)\).
\end{proof}
\subsection{Influence Maximization} \label{subsec: Preliminaries on GSV for SI}
We will be interested in Generalized Semivalues for functions that describe influence on information propagation in social networks. Two of the most popular models for describing information propagation in networks are the \emph{Independent Cascade} and \emph{Linear Threshold} models~\cite{DBLP:journals/toc/KempeKT15}. In both of these models, we are given a directed graph \(G=(V, E)\) where $V$ is a set of $n$ nodes, values \(\{p_{uv} \in [0,1]: (u,v) \in E\}\) and an initial node set \(A\subseteq V\) called \emph{seed nodes}. A spread of influence from the set \(A\) is then defined as a randomly generated sequence of node sets $(A_t)_{t\in \mathbb{N}}$, where \(A_0=A\) and \(A_{t-1}\subseteq A_{t}\). These sets represent active users, i.e., we say that a node \(v\) is \emph{active} at time step \(t\) if \(v\in A_t\). This sequence converges as soon as \(A_{t^*}=A_{t^*+1}\), for some time step \(t^*\ge 0\) called the time of quiescence. For a set \(A\), we use \(\sigma(A) = \E[|A_{t^*}|]\) to denote the expected number of nodes activated at the time of quiescence when running the process with seed nodes \(A\). In influence maximization, a common objective is to find a set $A$ maximizing \(\sigma(A)\) under a cardinality constraint.
\paragraph{The Independent Cascade Model.}
In the \emph{Independent Cascade} (IC) model, the values \(\{p_{uv} \in [0,1]: (u,v) \in E\}\) are probabilities. The sequence of node sets $(A_t)_{t\in \mathbb{N}}$, is randomly generated as follows. If \(u\) is active at time step \(t\ge 0\) but was not active at time step \(t-1\), i.e., \(u\in A_t\setminus A_{t-1}\) (with \(A_{-1}=\emptyset\)), it tries to activate each of its neighbors $v$, independently, and succeeds with probability \(p_{uv}\). In case of success, \(w\) becomes active at time step \(t+1\), i.e., \(v\in A_{t+1}\).
\paragraph{The Linear Threshold Model.} In the \emph{Linear Threshold} (LT) model, the values \(\{p_{uv} \in [0,1]: (u,v) \in E\}\) are weights such that for each node $v$, $\sum_{(u,v)\in E} p_{uv} \le 1$. The sequence of node sets $(A_t)_{t\in \mathbb{N}}$, is randomly generated as follows. At time step \(t+1\), every inactive node $v$ such that $\sum_{(u,v)\in E, u\in A_t} p_{uv} \ge \theta_v$ becomes active, i.e., \(v\in A_{t+1}\), where thresholds $\theta_v$ are chosen independently and uniformly at random from the interval $[0, 1]$.
\paragraph{The Triggering Model.} The IC and LT models can be generalized to what is known as the \emph{Triggering Model}, see~\cite[Proofs of Theorem 4.5 and 4.6]{DBLP:journals/toc/KempeKT15}.
For a node \(v\in V\), let \(N_v\) denote all in-neighbors of \(v\). In the Triggering model, every node independently picks a \emph{triggering set} \(T_v \subseteq N_v\) according to some distribution over subsets of its in-neighbors.
For a possible outcome \(X = (T_v)_{v\in V}\) of triggering sets for the nodes in \(V\), let \(G_X = (V,E')\) denote the sub-graph of \(G\) where \(E' = \{(u,v)|v \in V, u\in T_v \}\). Moreover, let \(\rho_X(A)\) be the set of nodes reachable from \(A\) in \(G_X\), then \(\sigma(A)=\E_X[|\rho_X(A)|]\). The IC model is obtained from the Triggering model if for each directed edge $(u,v)$, $u$ is added to $T_v$ with probability $p_{uv}$. Differently, the LT model is obtained if each node $v$ picks at most one of its in-neighbor to be in her triggering set, selecting a node $u$ with probability $p_{uv}$ and selecting no one with probability $1 - \sum_{u \in N_v} p_{uv}$.
Interestingly, the Triggering Model allows for using a concept commonly referred to as reverse reachable sets.
\paragraph{Reverse Reachable (RR) Sets.}
We describe the process of generating so-called \emph{Reverse Reachable (RR) sets}~\cite{DBLP:conf/soda/BorgsBCL14,DBLP:conf/sigmod/TangXS14}. A random RR set $R$ is generated as follows~\cite{chen2017interplay}.
(1) Set $R=\emptyset$.
(2) Uniformly at random select a root node $v\in V$ and add it to $R$.
(3) Until every node in $R$ has a triggering set: Pick a node $u$ from $R$ that does not have a triggering set, sample its triggering set $T_u$ and add it to $R$. A random RR set $R$ can be equivalently generated as all nodes that can reach a uniformly at random sampled root node $v$ in a random graph $G_X$ sampled as in the Triggering Model~\cite{DBLP:conf/soda/BorgsBCL14}. We get the following lemma for the marginal contribution of a set, the lemma generalizes Lemma 22 in~\cite{chen2017interplay} from marginal contribution of nodes to sets of nodes.
\begin{lemma}[Marginal Contribution]\label{lem: marginal contribution}
Let \(R\) be a random RR set. For any \(T \subseteq V\) and \(S\subseteq V\setminus T\):
\begin{align*}
\sigma(T) &= n\cdot \Pr_R[R \cap T \neq \emptyset],\\
\text{ and }\quad
\sigma(T\cup S)-\sigma(T) &= n\cdot \Pr_R[R\cap S \neq \emptyset \land R\cap T=\emptyset].
\end{align*}
\end{lemma}
\begin{proof}
Let \(X\) be a random outcome profile in the triggering model and let \(\rho_X(S)\) denote the set of nodes reachable from \(S\) in $G_X$. Then
\begin{align*}
\sigma(T) &= \E_X[|\rho_X(T)|]
= \E_X[\sum_{v\in V} \mathds{1}_{v\in \rho_X(T)}]\\
&= n\cdot \E_X [\E_{v\sim V} [\mathds{1}_{v \in \rho_X(T)}]]
= n\cdot \Pr_{X,v\sim V} [v \in\rho_X(T)].
\end{align*}
Recall that a random RR set is equivalently generated as all nodes that can reach a uniformly at random sampled root node $v$ in a random outcome graph $G_X$. Hence, the above event is equivalent to \(R\cap T \neq \emptyset\) and the first claim follows.
Similarly, we have
\begin{align*}
\sigma(T\cup S) - \sigma(T) &= \E_X[|\rho_X(T\cup S) \setminus \rho_X(T)|]\\
&= n \cdot \E_X[\E_{u\sim V}[ \mathds{1}_{u\in \rho_X(T\cup S) \setminus \rho_X(T)}]]\\
&= n\cdot \Pr_{X,u\sim V} [u \in \rho_X(T\cup S) \setminus \rho_X(T)].
\end{align*}
By a similar argument, the event \(u \in \rho_X(T\cup S) \setminus \rho_X(S)\) is equivalent to the event \(R \cap S \neq \emptyset \land R\cap T = \emptyset\). This shows the second claim.
\end{proof}
\subsection{The Group Shapley Centrality}\label{subsec: Preliminaries on GSC}
Chen and Teng~\cite{chen2017interplay} consider the Shapley value of nodes w.r.t. the influence spread function $\sigma$ in a social network modeled by the Triggering Model. They use the resulting \emph{Shapley centrality} $\phi^{\sh}_\sigma(i)$ for $i$ being a node in the network as a measure of centrality of node $i$. They furthermore show that this centrality measure satisfies and is uniquely characterized by certain axioms, similar to the axioms characterizing the standard Shapley value. In this work, we consider the Group Shapley value w.r.t. $\sigma$.
For a set $S$ of nodes, we call the Group Shapley value w.r.t.\ $\sigma$ the \emph{Influence-based Group Shapley} (IGS) centrality of $S$, referred to as $\phi^{\sh}(S)$ omitting $\sigma$ as an index:
\begin{equation}
\phi^{\sh}(S)= \E_{\pi \sim \Pi(V_{\overline{S}})}[\sigma(T_{\pi,s}\cup S) - \sigma(T_{\pi,s})]. \label{eq:Shapley Centrality}
\end{equation}
One of the main contributions of Chen and Teng~\cite{chen2017interplay} is an algorithm that approximates the Shapley centrality of every node. The key lemma in their analysis is that for a node $v\in V$, it holds that $\phi^{\sh}_\sigma(v)=n\E_R[\mathds{1}_{v\in R}/|R|]$, where the expected value is over random RR sets generated as described above. For the IGS centrality $\phi^{\sh}(S)$ of a set $S$, we show the following analogous lemma.
\begin{lemma}[IGS centrality via RR sets]\label{lem: shapley value identity}
Let \(S\subseteq V\), it holds that \(\phi^{\sh}(S) = n \cdot \E_R[\frac{\mathds{1}_{R\cap S\neq \emptyset}}{|R\setminus S| + 1}]\).
\end{lemma}
\begin{proof}
For the IGS centrality, it holds that
\begin{align*}
\phi^{\sh}(S)
&= \E_{\pi \sim \Pi(V_{\overline{S}})}[\sigma(T_{\pi,s}\cup S) - \sigma(T_{\pi,s})]\\
&= \E_{\pi \sim \Pi(V_{\overline{S}})}[n\cdot \Pr_R(R\cap S \neq \emptyset \land R \cap T_{\pi,s} = \emptyset)]\\
&=n\cdot \E_R[\E_{\pi \sim \Pi(V_{\overline{S}})}[\mathds{1}_{R\cap S \neq \emptyset \land R\cap T_{\pi,s} = \emptyset}]]\\
&= n\cdot \E_R\Big[\frac{\mathds{1}_{R\cap S\neq \emptyset}}{|R\setminus S| + 1}\Big],
\end{align*}
using Lemma~\ref{lem: marginal contribution} and Observation~\ref{obs: intersection probability shapley}.
\end{proof}
Consequently from this lemma, we obtain that the IGS centrality is a monotonously increasing set function. Furthermore, Lemma~\ref{lem: shapley value identity} provides the following observation on the range of IGS centralities.
\begin{observation}[Range of IGS Centralities]\label{obs: range}
Let $S^*$ be a set of size $k$ maximizing $\phi^{\sh}$. Then $\phi^{\sh}(S^*)\ge 1$. Moreover, $\phi^{\sh}(S)\ge \frac{k}{n}$ for any $S\subseteq V$ of size $k$. Lastly, $\phi^{\sh}(S)\le n$ for any $S$ and $\phi^{\sh}(V)=n$.
\end{observation}
\begin{proof}
According to the normalization axiom of the Shapley value (for single items), we have $\sum_{i\in V}\phi^{\sh}(\{i\})=n$, hence there is a node $i_0$ for which $\phi^{\sh}(\{i_0\})\ge 1$.
Since $\phi^{\sh}$ is a monotonously increasing set function, it holds that $\phi^{\sh}(S)\ge 1$ for any set $S$ of size $k$ containing $i_0$, thus also $\phi^{\sh}(S^*)\ge 1$. In order to show that $\phi^{\sh}(S)\ge\frac{k}{n}$ for any $S\subseteq V\setminus\emptyset$ of size $k$. With $R(u, X)$ we denote the RR set sampled from a node $u$ for an outcome profile $X$. We then observe that, according to Lemma~\ref{lem: shapley value identity}, $\phi^{\sh}(S)$ equals
\[
n \cdot \E_R\Big[\frac{\mathds{1}_{R\cap S\neq \emptyset}}{|R\setminus S| + 1}\Big] \ge n\cdot \frac{1}{n} \cdot \sum_{v\in S}\E_X\Big[\frac{\mathds{1}_{R(v,X)\cap S\neq \emptyset}}{|R(v,X)\setminus S| + 1}\Big]\ge \frac{k}{n},
\]
using that $|R(v,X)\setminus S|\le n-1$ as $v\in S$.
The remaining claims follow analogously using the same equality from Lemma~\ref{lem: shapley value identity}.
\end{proof}
Our ultimate goal would be to find a set \(S\) of size at most $k$ with highest IGS centrality among all such sets. This is formalized below.
\begin{cproblem}{\textsc{Max-Shapley-Group}\xspace }
Input: Influence maximization instance on digraph $G$, integer \(k\).
Find: \(S \subseteq V\) s.t.\ \(|S| \le k\), maximizing \(\phi^{\sh}(S)\).
\end{cproblem}
The naive approach for solving the optimization problem \textsc{Max-Shapley-Group}\xspace would be to evaluate $\phi^{\sh}$ for all subsets of size at most $k$ and pick the one with highest value. Unfortunately, the formula for computing $\phi^{\sh}$ for a single set \(S\) given in Equation~\ref{eq:Shapley Centrality} is already not practical as it requires to compute the difference \(\sigma(S\cup T) - \sigma(T)\) for an exponential number of sets \(T\). Alternatively, one could try to follow an approach similar to the one taken by Chen and Teng~\cite{chen2017interplay} for the Shapley centrality of single nodes. Such approach for IGS centrality however would require updating $O(n^k)$ estimates (one for each candidate set) in every iteration. We will see later on in Section~\ref{sec: approximation algorithm} how to avoid this taking a different route.
In the next section, we focus on overcoming the first difficulty, i.e., we show how to approximate IGS centrality. The approach relies on the representation of $\phi^{\sh}$ given in Lemma~\ref{lem: shapley value identity} and, non-surprisingly, on a Chernoff bound.
\section{EVALUATING IGS CENTRALITY}
This section is concerned with the question of estimating the function $\phi^{\sh}$. We first give a straightforward result that shows how to compute $\phi^{\sh}(S)$ for a given set $S$. Thereafter, we show that by sampling a sufficient number of RR sets, we can give a set function $\hat \phi^{\sh}$ that with high probability approximates $\phi^{\sh}$ in a sense that suffices for obtaining an approximation algorithm for \textsc{Max-Shapley-Group}\xspace . The main tool for this section is the following classical Chernoff bound that can be found in the survey by Chung and Lu~\cite[Theorem 4]{chung2006concentration} or in Appendix C.2 of the full version of Chen and Teng's paper~\cite{chen2017interplay}.
\begin{fact}[Chernoff Bound]\label{fact: chernoff}
Let $Y$ be the sum of $t$ i.i.d. random variables with mean $\mu$ and value range $[0,1]$.
\begin{enumerate}
\item\label{item 01} For any $\alpha\in (0,1)$, we have
$
\Pr[\frac{Y}{t} - \mu \le -\alpha \mu]
\le \exp(-\frac{\alpha^2}{2}t\mu).
$
\item\label{item pos}
For any $\alpha>0$, we have
$
\Pr[\frac{Y}{t} - \mu \ge \alpha \mu]
\le \exp(-\frac{\alpha^2}{2+\frac{2}{3}\alpha}t\mu).
$
\end{enumerate}
\end{fact}
\paragraph{Approximately Evaluating $\phi^{\sh}(S)$.}
In this paragraph, we show that using the above Chernoff bound we can, in a straightforward way, obtain a $(1\pm\epsilon)$-approximation $\tilde \phi^{\sh}(S)$ of $\phi^{\sh}(S)$ for any set $S\subseteq V$ and $\epsilon \in (0,1)$ by sampling $\Theta(n^2\epsilon^{-2}\log n)$ RR sets.
\begin{lemma} \label{lem:evaluate}
Let $S\subseteq V$ and $\epsilon\in (0,1)$. Let $R_1,\ldots, R_t$ be a sequence of $t\ge 6n^2\epsilon^{-2}c\log(n)$ RR sets for some constant $c\ge 2$. Then, with probability at least $1-n^{-c}$, it holds that
$\tilde \phi^{\sh}(S) := \frac{n}{t} \sum_{i=1}^t \frac{ \mathds{1}_{R_i\cap S\neq \emptyset}}{|R_i\setminus S| + 1}$
satisfies $|\tilde \phi^{\sh}(S) - \phi^{\sh}(S)| < \epsilon\cdot \phi^{\sh}(S)$.
\end{lemma}
\begin{proof}
If $S=\emptyset$, the statement trivially holds. Otherwise, define the random variables $Y_i(S):=\frac{\mathds{1}_{R_i\cap S\neq \emptyset}}{|R_i\setminus S| + 1}\in[0,1]$ for $i\in[t]$ and let $Y(S):=\sum_{i=1}^t Y_i(S)$ as well as $\mu(S):=\E[Y_i(S)]$.
Clearly, $\mu(S)=\phi^{\sh}(S)/n$ by Lemma~\ref{lem: shapley value identity} and $\tilde \phi^{\sh}(S)=\frac{n}{t}Y(S)$. Thus $\Pr[|\tilde \phi^{\sh}(S) - \phi^{\sh}(S)| \ge \epsilon \cdot \phi^{\sh}(S)]$ equals
\begin{align*}
\Pr\Big[\Big|\frac{Y(S)}{t} - \mu(S)\Big| \ge \epsilon \cdot \mu(S)\Big]
&\le 2\exp\Big(-\frac{\epsilon^2}{3}\cdot t\mu(S)\Big)
\end{align*}
using Fact~\ref{fact: chernoff} and $\epsilon\le 1$. We lower bound $\mu(S)=\phi(S)/n\ge 1/n^2$ using Observation~\ref{obs: range}. The choice of $t$ leads to the bound of $n^{-c}$.
\end{proof}
\paragraph{An Approximate Characterization $\hat \phi^{\sh}$ of $\phi^{\sh}$.}
Lemma~\ref{lem:evaluate} is unsatisfactory for the following reason. It samples a number of RR sets that is quadratic in the number of nodes $n$ even for evaluating the IGS centrality of a single group. In this paragraph, we show how to circumvent this problem. We show that a near-linear number of RR sets suffices to compute, for any set \(S\), an approximation \(\hat \phi^{\sh}(S)\) of $\phi^{\sh}(S)$ that is good enough for giving an approximation algorithm for \textsc{Max-Shapley-Group}\xspace .
More precisely, we define a function \(\hat \phi^{\sh}\) that meets the following two conditions: (1) For any set \(S\) of size $k$, \(\hat \phi^{\sh}(S)\) does not overestimate \(\phi^{\sh}(S)\) too much and (2) For an optimal set $S^*$, \(\hat \phi^{\sh}(S^*)\) does not underestimate \(\phi^{\sh}(S^*)\) too much. We will show that these conditions suffice for a set \(S\) that is close to being optimal for \(\hat \phi^{\sh}\) to also be close to being optimal for \(\phi^{\sh}\).
\begin{thrm} \label{thm:link shap hmhs}
Let $\epsilon\in (0,1)$ and $R_1,\ldots, R_t$ be a sequence of RR sets of length $t\ge 6n\epsilon^{-2}(c+k)\log(n)$ for some constant $c\ge 2$. Let $S^*$ be a set of size at most $k$ maximizing $\phi^{\sh}$ and let
\[
\hat \phi^{\sh}(S):=\frac{n}{t} \sum_{i=1}^t \frac{ \mathds{1}_{R_i\cap S\neq \emptyset}}{|R_i\setminus S| + 1} \quad \text{ for each set }S\in \binom{V}{k}.
\]
Then, with probability at least $1-n^{-c}$, the following conditions hold.
\begin{enumerate}
\item[(1)] $\hat \phi^{\sh}(S) - \phi^{\sh}(S)< \epsilon\cdot \phi^{\sh}(S^*)$, for all $S\in \binom{V}{k}$.
\item[(2)] $\hat \phi^{\sh}(S^*) - \phi^{\sh}(S^*) > -\epsilon\cdot \phi^{\sh}(S^*)$.
\end{enumerate}
\end{thrm}
\begin{proof}
We first show that (1) holds with probability at least $1-\frac{1}{2}n^{-c}$. Let $S\in \binom{V}{k}$. Define the random variables $Y_i(S):=\frac{\mathds{1}_{R_i\cap S\neq \emptyset}}{|R_i\setminus S| + 1}\in[0,1]$ for every $i\in[t]$ and let $Y(S):=\sum_{i=1}^t Y_i(S)$ as well as $\mu(S):=\E[Y_i(S)]$.
Clearly, $\mu(S)=\phi^{\sh}(S)/n$ by Lemma~\ref{lem: shapley value identity} and $\hat \phi^{\sh}(S)=\frac{n}{t}Y(S)$. Let $\alpha:=\frac{\epsilon \phi^{\sh}(S^*)}{\phi^{\sh}(S)}$. Then, $\Pr[\hat \phi^{\sh}(S) - \phi^{\sh}(S) \ge \epsilon \cdot \phi^{\sh}(S^*)]$ equals
\begin{align*}
\Pr\Big[\frac{Y(S)}{t} - \mu(S) \ge \alpha \cdot \frac{\phi^{\sh}(S)}{n}\Big]
&\le \exp\Big(-\frac{\alpha^2}{2+\frac{2}{3}\alpha}\cdot t\mu(S)\Big),
\end{align*}
using Fact~\ref{fact: chernoff}. Using the definition of $\alpha$, we get that the argument of $\exp$ is equal to $-\frac{\epsilon^2 \phi^{\sh}(S^*)}{2 \phi^{\sh}(S)/\phi^{\sh}(S^*)+2\epsilon/3}\cdot \frac{t}{n}\le -\frac{\epsilon^2 \phi^{\sh}(S^*)}{3n}\cdot t$ using that $\epsilon<1$ and $\phi^{\sh}(S)\le \phi^{\sh}(S^*)$.
Using that $\phi^{\sh}(S^*)\ge 1$ according to Observation~\ref{obs: range} and the definition of $t$ lead to the upper bound of $1/n^{2(c + k)}$.
A union bound over all at most $n^k$ sets in $\binom{[n]}{k}$ shows that $\hat \phi^{\sh}(S) - \phi^{\sh}(S) \ge \epsilon \cdot \phi^{\sh}(S^*)$ holds for every such $S$ with probability at most $n^k\cdot n^{-2(c+k)}\le \frac{1}{2}\cdot n^{-c}$.
We proceed to condition (2). It holds that $\Pr[\hat \phi^{\sh}(S^*) - \phi^{\sh}(S^*) \le -\epsilon \cdot \phi^{\sh}(S^*)]$ is equal to
\begin{align*}
\Pr\Big[\frac{Y(S^*)}{t} - \mu(S^*) \le -\epsilon \cdot \frac{\phi^{\sh}(S^*)}{n}\Big]
&\le \exp\Big(-\frac{\epsilon^2}{2}\cdot t\mu(S^*)\Big)
\end{align*}
using Fact~\ref{fact: chernoff} with $\alpha=\epsilon$.
Furthermore, $\mu(S^*)=\phi^{\sh}(S^*)/n$ and again $\phi^{\sh}(S^*)\ge 1$ as well as the definition of $t$ yield that $\hat \phi^{\sh}(S^*) - \phi^{\sh}(S^*) \le -\epsilon \cdot \phi^{\sh}(S^*)$ holds with probability at most $\frac{1}{2} \cdot n^{-c}$. A union bound over the probabilities that (1) or (2) do not hold, concludes the proof.
\end{proof}
In the next section, we investigate how to find an approximation algorithm for the \textsc{Max-Shapley-Group}\xspace problem without computing the centralities of all sets of size $k$.
\section{FINDING GROUPS OF LARGE IGS CENTRALITY}\label{sec: approximation algorithm}
To address the \textsc{Max-Shapley-Group}\xspace problem, Theorem~\ref{thm:link shap hmhs} suggests the following approach. Sample a near-linear number $t$ of RR sets and compute a set of nodes $S$ that maximizes $\hat \phi^{\sh}(S)$. We formalize this problem as a variant of the well known \textsc{Max-Hitting-Set}\xspace problem, that we call the \textsc{Harmonic-Max-Hitting-Set}\xspace problem.
\begin{cproblem}{\textsc{Harmonic-Max-Hitting-Set}\xspace }
Input: set \(X = \{x_1,\ldots,x_n\}\), set \(Z = \{Z_1, \ldots,Z_m\}\) of subsets of \(X\), integer \(k\).
Find: $S\subseteq X$ s.t.\ $|S|\le k$ maximizing
\(
f_Z(S):=\sum_{i=1}^m \frac{\mathds{1}_{Z_i \cap S \neq \emptyset}}{|Z_i\setminus S|+1}.
\)
\end{cproblem}
It is a non-linear variant of the well-known \textsc{Max-Hitting-Set}\xspace problem (which is itself equivalent to the \textsc{Max-Set-Cover}\xspace problem~\cite{garey2002computers}) in which the objective function is \(\sum_{i=1}^m \mathds{1}_{Z_i \cap S \neq \emptyset}\).
The problem of maximizing the previously defined function \(\hat \phi^{\sh}\) can be stated as a \textsc{Harmonic-Max-Hitting-Set}\xspace problem by letting \(X\) be the set $V$ of nodes in graph $G$ and \(Z\) be the set of generated RR sets. The connection between the \textsc{Harmonic-Max-Hitting-Set}\xspace and \textsc{Max-Shapley-Group}\xspace problems from an approximation algorithm's perspective is made more formal in the next lemma.
\begin{lemma}\label{lem: MGSS HMHS}
Let $\alpha\in(0,1]$, $\epsilon \in (0, 1)$, $c\ge 2$ and $k\in [n]$. Let $S_{\alpha}$ be an \(\alpha\)-approximate solution for the \(\textsc{Harmonic-Max-Hitting-Set}\xspace \) problem with budget $k$, $X=V$ and $Z=\{R_1,\ldots, R_t\}$ s.t. $t\ge 24n\epsilon^{-2}(c+k)\log(n)$ RR sets. Then, $S_{\alpha}$ is an \((\alpha - \epsilon)\)-approximation for \textsc{Max-Shapley-Group}\xspace with probability at least $1-n^{-c}$.
\end{lemma}
\begin{proof}
Let \(\epsilon' = \epsilon/2\). By Theorem~\ref{thm:link shap hmhs}, we have that $\hat \phi^{\sh}(S) - \phi^{\sh}(S)< \epsilon'\cdot \phi^{\sh}(S^*)$ for all $S\in \binom{[n]}{k}$ and $\hat \phi^{\sh}(S^*) - \phi^{\sh}(S^*) > -\epsilon'\cdot \phi^{\sh}(S^*)$ hold with probability at least $1-n^{-c}$,
where $\hat \phi^{\sh}(S):=\sum_{i=1}^t \frac{n\cdot \mathds{1}_{R_i\cap S\neq \emptyset}}{t\cdot (|R_i\setminus S| + 1)}$.
Let \(S^*\) (resp.\ \(\hat{S}^*\)) be a set of size $k$ maximizing \(\phi^{\sh}\) (resp.\ \(\hat \phi^{\sh}\)). Then with probability at least $1-n^{-c}$, it holds that
\begin{align*}
\phi^{\sh}(S_{\alpha}) &+ \epsilon' \cdot \phi^{\sh}(S^*)
\ge \hat \phi^{\sh}(S_{\alpha})
\ge \alpha \cdot \hat \phi^{\sh}(\hat{S}^*)\\
&\ge \alpha \cdot \hat \phi^{\sh}(S^*)
\ge \alpha \cdot (\phi^{\sh}(S^*) - \epsilon'\cdot \phi^{\sh}(S^*)),
\end{align*}
where we have used that \(\hat \phi^{\sh}(S_{\alpha})
\ge \alpha \cdot \hat \phi^{\sh}(\hat{S}^*)\) as \(\hat \phi^{\sh}(\cdot) = \frac{n}{t} f_Z(\cdot)\).
The choice of $\epsilon'$ and $\alpha\le 1$ yield $\alpha - \alpha \epsilon'-\epsilon' \ge \alpha-\epsilon$. Thus $\phi^{\sh}(S_{\alpha}) \ge (\alpha - \epsilon)\cdot \phi^{\sh}(S^*)$ with probability at least $1-n^{-c}$.
\end{proof}
\paragraph{Approximation Algorithm.}
In this section, we describe a \(\frac{1-1/e}{k}\) approximation algorithm for the \textsc{Harmonic-Max-Hitting-Set}\xspace problem. Consider an instance \((X,Z,k)\) and define the following set function
\(
h_{Z}(S) := \sum_{i=1}^m \mathds{1}_{Z_i \cap S \neq \emptyset}/|Z_i|.
\)
Note the similarity between $h_Z$ and $f_Z$. In fact, the approximation algorithm that we propose is to greedily maximize \(h_{Z}\) instead of \(f_{Z}\). Why would this be a good idea?
(1) The set function \(h_{Z}\) is monotone and submodular; thus the greedy algorithm will yield a \(1-1/e\) approximation to maximizing \(h_{Z}\). (2) Given a set \(S\subseteq X\) with \(|S|\le k\), it holds that
\begin{equation}
f_{Z}(S) \ge h_{Z}(S) \ge f_{Z}(S)/k, \label{eq:encadrement}
\end{equation}
that is, the error when considering $h_Z$ instead of $f_Z$ is bounded by $k$.
Hence, if we denote by \(S^{*}_f\) (resp. \(S^{*}_h\)) an optimal solution of size \(k\) for maximizing \(f_{Z}\) (resp. \(h_{Z}\)), we have that \(h_{Z}(S^{*}_h) \ge h_{Z}(S^{*}_f) \ge f_{Z}(S^{*}_f)/k\).
Now let \(S\) be the solution of size $k$ returned by the greedy algorithm. Then,
\(S\) is a \(\frac{1-1/e}{k}\) approximation to maximizing \(f_{Z}\) as
\begin{align*}
f_Z(S) &\ge h_Z(S) \ge \Big(1 - \frac{1}{e}\Big) \cdot h_{Z}(S^{*}_h)
\ge \frac{1 - 1/e}{k} \cdot f_{Z}(S^{*}_f).
\end{align*}
It remains to prove the inequalities in~\eqref{eq:encadrement}.
\begin{proof}[Proof of Inequalities in~\eqref{eq:encadrement}]
We note that, for any $S$, $f_{Z}(S)$ equals
\begin{align}\label{eq:fh}
\sum_{i=1}^m \frac{\mathds{1}_{Z_i \cap S \neq \emptyset}}{|Z_i\setminus S| + 1}
= \sum_{i=1}^m \frac{|Z_i|}{|Z_i|-|Z_i\cap S| + 1} \frac{\mathds{1}_{Z_i \cap S \neq \emptyset}}{|Z_i|}.
\end{align}
The left inequality in~\eqref{eq:encadrement} follows since $|Z_i|\ge |Z_i| - |Z_i\cap S| + 1$, if $Z_i \cap S \neq \emptyset$.
Next, we observe that $\frac{|Z_i|}{|Z_i|-|Z_i\cap S| + 1}=1 + \frac{|Z_i\cap S| - 1}{|Z_i|-|Z_i\cap S| + 1}$ and, if $S$ is of size $k$, $|Z_i\cap S|\le k$. Together with $|Z_i|-|Z_i\cap S|\ge 0$, we obtain $\frac{|Z_i|}{|Z_i|-|Z_i\cap S| + 1} \le k$. The right inequality in~\eqref{eq:encadrement} follows.
\end{proof}
Using Lemma~\ref{lem: MGSS HMHS}, we thus obtain the following theorem.
\begin{thrm}\label{thrm: approximation algorithm}
Let $\epsilon\in (0, 1)$ and $c\ge 2$. Using $\Theta(nk\epsilon^{-2}\log n)$ RR sets, we can obtain a $\frac{1-1/e}{k}-\epsilon$ approximation to the \textsc{Max-Shapley-Group}\xspace problem with probability at least $1-n^{-c}$.
\end{thrm}
We conclude this paragraph with a note on an alternative approach for an approximation algorithm with a similar ratio but worse dependency on the number of generated RR sets. Define the function $h(S):= n \cdot \E_R[\frac{\mathds{1}_{R\cap S\neq \emptyset}}{|R|}]$. Note that $\phi^{\sh}(S)\ge h(S)\ge \phi^{\sh}(S)/k$ holds following a proof analogous to the one of~\eqref{eq:encadrement}. Clearly, $h$ is a monotone and submodular set function (just as $h_Z$ is) and it can be approximated within a $1\pm \epsilon$ factor for any $\epsilon \in (0,1)$ in an analogous way as used in Lemma~\ref{lem:evaluate}. Thus the greedy algorithm can be used in order to maximize $h$ to within a $1 - 1/e - \epsilon$ factor subject to the cardinality constraint. Altogether, we obtain the approximation ratio of $\frac{1 - 1/e - \epsilon}{k}$ which is comparable to what is achieved by Theorem~\ref{thrm: approximation algorithm}. However the required number of RR sets in every step of the $k$ steps of this greedy naive approach is quadratic in the number of nodes.
While Theorem~\ref{thrm: approximation algorithm} provides an interesting result for small \(k\) values, it remains unsatisfactory for large $k$. One could hope for a much stronger result as for example constant-factor approximations. Unfortunately, this is unlikely as we will see in the following section, where we provide several approximation hardness results for the \textsc{Max-Shapley-Group}\xspace problem.
\section{HARDNESS OF APPROXIMATION}
In this section, we show that \textsc{Max-Shapley-Group}\xspace under the IC model is, up to a constant factor, as hard to approximate as \textsc{Densest-\(k\)-Subgraph}\xspace . More precisely, we prove the following theorem.
\begin{thrm} \label{thm: GS hardness}
Let $\alpha\in(0,1]$. If there is an \(\alpha\)-approximation algorithm for \textsc{Max-Shapley-Group}\xspace , then there is an \(\alpha/8\)-approximation algorithm for \textsc{Densest-\(k\)-Subgraph}\xspace .
\end{thrm}
A number of strong hardness of approximation results are known for \textsc{Densest-\(k\)-Subgraph}\xspace . We review some of them:
(1) \textsc{Densest-\(k\)-Subgraph}\xspace cannot be approximated within $1/n^{o(1)}$ if the Gap Exponential Time Hypothesis (Gap-ETH) holds~\cite{manurangsi2017almost}. (2) \textsc{Densest-\(k\)-Subgraph}\xspace cannot be approximated within any constant if the Unique Games with Small Set Expansion conjecture holds~\cite{raghavendra2010graph}. (3) \textsc{Densest-\(k\)-Subgraph}\xspace cannot be approximated within $n^{-(\log\log n)^{-c}}$ for some constant $c$ if the Exponential Time Hypothesis holds~\cite{manurangsi2017almost}.
Using the reduction given in this section, we obtain the same hardness results also for \textsc{Max-Shapley-Group}\xspace . In particular, we would like to stress that, according to (1) and our reduction, it is unlikely to find anything better than an $(n^{-c})$-approximation for \textsc{Max-Shapley-Group}\xspace , where $c$ is a constant. Furthermore, for all settings where $k= O(n^{c})$, such an algorithm is implied by our result in Section~\ref{sec: approximation algorithm}. We proceed by formally defining \textsc{Densest-\(k\)-Subgraph}\xspace .
\begin{cproblem}{\textsc{Densest-\(k\)-Subgraph}\xspace }
Input: Undirected graph \(G=(V, E)\), integer \(k\).
Find: set \(T \subseteq V\) with \(|T| \le k\), s.t.\ \(|E[T]|\) is maximum. Here $E[T]$ are the edges induced by $T$, i.e.\ $E[T]:=\{e \in E: e \subseteq T \}$.
\end{cproblem}
\paragraph{The reduction.} Let us fix an an instance \(\mathcal{P}=(G=(V,E),k)\) of \textsc{Densest-\(k\)-Subgraph}\xspace . Note that, w.l.o.g., we can assume that \(G\) is connected.\footnote{It is not hard to show that from an \(\alpha\)-approximation algorithm for connected graphs, we can obtain an $(\alpha/2)$-approximation algorithm for general graphs by making the graph connected and then applying the approximation algorithm as follows.
Let us assume that $G$ is disconnected, and there are $\nu$ connected components, we construct a graph $\hat{G}=(V,\hat{E})$ by adding $\nu-1$ edges to $E$ in order to make the graph connected.
Let $\hat{T}$ be a solution returned by an approximation algorithm for connected graphs on $\hat{G}$. If $\hat{T}$ contains both nodes at the endpoints of an edge in $\hat{E}\setminus E$, we can compute a solution $T$ for $G$ by iteratively substituting each pair of nodes in $\hat{T}$ that do not induce an edge in $G$ with two nodes that are adjacent in $G$.
It is easy to see that $|E[T]|$ is at least $\frac{1}{2}|\hat{E}[\hat{T}]|$, where $\hat{E}[\hat{T}]$ are the edges induced by $\hat{T}$ in $\hat{G}$: for each edge $e=(u,v)$ in $\hat{E}[\hat{T}]\setminus E$ either there exists an edge in $E[T]$ that is incident to $u$ or $v$, or $u$ and $v$ are substituted with two nodes that are adjacent in $G$.
Moreover, $|\hat{E}[\hat{T}]|\geq\alpha |\hat{E}[\hat{T}^*]|\geq\alpha |E[T^*]|$, where $\hat{T}^*$ and $T^*$ are optimal solution for $G$ and $\hat{G}$, respectively.
}
From \(\mathcal{P}\), we create the following \textsc{Max-Shapley-Group}\xspace instance \(\overline \mathcal{P}=(\overline{G}=(\overline{V},\overline{A}),\{p_a\}_{a\in \overline{A}}, \overline{k})\): (1)~Probabilities \(p_a\) are set to 1 for all $a\in \overline{A}$. (2)~We set the budget to \(\overline{k} := k\cdot t\), where $t := 6|E|$. (3)~The node set is defined as \( \overline{V} := \overline{V}_V \cup \overline{V}_E\), where
$\overline{V}_V := \{u^v_1,\ldots,u^v_t|v \in V\}$
and
$\overline{V}_E := \{u^e_1,\ldots,u^e_{\ell}|e\in E\}$ for $\ell := (2t+1)t|V|+1$.
(4)~The arc set \(\overline{A}\) is defined as follows: for each edge \(\{v,v'\}\in E\), we create a pattern in \(\overline{G}\) as described in Figure~\ref{fig:edge}. This pattern is composed of three layers. Two layers, called $v$-layer and $v'$-layer, gathering all the nodes in $\{u^v_1,\ldots,u^v_t\}$ and $\{u^{v'}_1,\ldots,u^{v'}_t\}$ respectively and one layer called, $\{v,v'\}$-layer, gathering all nodes in $\{u^{\{v,v'\}}_1,\ldots,u^{\{v,v'\}}_{\ell}\}$. There is an arc from each node of the first two layers to each node of the third layer.
\begin{figure}
\caption{Pattern obtained in \(\overline{G}
\label{fig:edge}
\end{figure}
\paragraph{RR sets for $\overline \mathcal{P}$.}
Interestingly, all RR sets that can be generated in \(\overline \mathcal{P}\) are easily described as two different types:
(1) RR-sets that consist of singletons \(\{u^v_p\}\), we call them \emph{Node-RR-sets} and denote the set of all Node-RR-sets by \(\ensuremath{\mathbb{R}}R_V\). There is exactly one Node-RR-set per node \(u^v_p \in \overline{V}_V\).
(2) RR-sets of the form \(\{u^{\{v,v'\}}_p, u^v_1,\ldots,u^v_t,u^{v'}_1,\ldots,u^{v'}_t\}\), we call them \emph{Edge-RR-sets} and denote the set of all Edge-RR-sets by \(\ensuremath{\mathbb{R}}R_E\). There is exactly one Edge-RR-set per node \(u^{\{v,v'\}}_p \in \overline{V}_E\).
We note that each RR-set occurs with the same probability \(1/\overline{V}\). Hence, the IGS centrality of a set \(S\subseteq \overline{V}\), can be written as \(\phi^{\sh}(S) = \phi^{\sh}_V(S) + \phi^{\sh}_E(S)\), where \( \phi^{\sh}_V(S) := \sum_{R \in \ensuremath{\mathbb{R}}R_V} \frac{\mathds{1}_{R \cap S \neq \emptyset}}{|R\setminus S| + 1}\) and \(\phi^{\sh}_E(S) := \sum_{R \in \ensuremath{\mathbb{R}}R_E} \frac{\mathds{1}_{R \cap S \neq \emptyset}}{|R\setminus S| + 1} \).
We proceed with a simple observation. For a node \(u \in \overline{V}\) and a set \(S \subseteq \overline{V}\setminus\{u\}\), we denote by \(I(u,S) := \phi^{\sh}(S \cup \{u\}) - \phi^{\sh}(S)\) the increase in IGS centrality obtained from adding $u$ to $S$.
\begin{observation}
Let \(u\in\overline{V}\) and \(S \subseteq \overline{V}\setminus\{u\}\). If \(u \in \overline{V}_V\), then \(I(u,S) \ge 1\).
If \(u \in \overline{V}_E\), then \(I(u,S) \le 1/2\).
\end{observation}
The proof of the above observation is simple. The first part holds due to the contribution of the Node-RR-set corresponding to \(u\) itself, the second part holds as \(u\) is in only one Edge-RR-set $R$ of size $2t+1$, thus
$\frac{\mathds{1}_{R \cap (S\cup\{u\}) \neq \emptyset}}{|R\setminus (S\cup\{u\})| + 1} - \frac{\mathds{1}_{R \cap S \neq \emptyset}}{|R\setminus S| + 1} \le \frac{1}{2}$.
Hence, we can say that a ``reasonable'' solution for \(\overline \mathcal{P}\) only contains nodes from \(\overline{V}_V\).
\paragraph{Thorough Sets.}
We introduce some notation. For \(S\subseteq \overline{V}_V\) and \(v \in V\), we define \(\nb{S}{v}:=|S\cap\{u_1^v,\ldots,u_t^v\}|\), \(U_S:=\{v \in V : \nb{S}{v}\ge 1\}\), and \(U_S^{(1,t)} := \{v\in U_S : \nb{S}{v} \in (1,t) \}\). Note that $U_S, U_S^{(1,t)} \subseteq V$.
The following notion is central to our analysis.
\begin{definition}
We call a set $S\subseteq \overline V_V$ to be \emph{thorough}, if (1) \(\nb{S''}{v}=t\) for all $v\in U_{S}$ and (2) \(|E[U_{S}]|\ge 1\).
\end{definition}
Now, let \(S^*\) and \(T^*\) be optimal solutions for $\overline\mathcal{P}$ and $\mathcal{P}$, respectively.
We get the following lemma, part (3) of which shows how to transform a thorough set $S$ into a solution of $\mathcal{P}$ in a straightforward way.
\begin{lemma} \label{lemma: hardness inequalities}
It holds that
(1)~\(\phi^{\sh}(S^*) \ge \frac{\ell}{2} |E[T^*]|\),
(2)~for all \( S\subseteq \overline{V}_V\), we have \(\phi^{\sh}_E(S) \ge \phi^{\sh}(S)/2\), and
(3)~if \( S\subseteq \overline{V}_V\) is thorough, we have \(\phi^{\sh}_E(S)\le \ell|E[U_S]|\).
\end{lemma}
\begin{proof}
For~(1), we let \(S_{T^*} := \bigcup_{v\in T^*} \{u^v_i\}_{i\in [t]}\) and get \(\phi^{\sh}(S^*) \ge \phi^{\sh}(S_{T^*}) \ge \phi^{\sh}_E(S_{T^*})\). As each edge in $E[T^*]$ induces \(l\) Edge-RR-sets $R$ with $|R \setminus S_{T^*}| = 1$, we have \(\phi^{\sh}_E(S_{T^*}) \ge \frac{\ell}{2} |E[T^*]|\).
For~(2), fix \(S \subseteq \overline{V}_V\). As $G$ is connected, for any \(u\in \overline{V}_V\), there are at least $l$ Edge-RR-sets containing $u$. Hence, \(\phi^{\sh}_E(S)\ge \frac{\ell}{2t+1}\). On the other hand, note that \(\phi^{\sh}_V(S)\le t|V|\). As \(\ell > (2t+1)t|V|\), we get \(\phi^{\sh}_E(S) \ge \phi^{\sh}_V(S)\) and hence \(\phi^{\sh}_E(S) \ge \frac{\phi^{\sh}(S)}{2}\).
For~(3), let \(S\subseteq \overline{V}_V\) be a thorough set. Define \(E_i := \{e\in E: e\cap U_S = i\}\) for $i=1,2$. Clearly, \(E[U_S]=E_2\) and, for each edge in \(E_1\) (resp.\ \(E_2\)) there are exactly \(\ell\) Edge-RR-sets \(R\) with \(|R\setminus S| = t + 1\) (resp.\ \(|R\setminus S| = 1\)). Hence, we conclude that
\(
\phi^{\sh}_E(S)
= \frac{\ell|E_2|}{2} + \frac{\ell|E_1|}{t+2}
\le \ell |E_2|
\)
by the choice of $t$.
\end{proof}
In Lemmata~\ref{lemma: key hardness 1} and~\ref{lemma: key hardness 2} below, we show that every solution \(S\subseteq \overline{V}_V\) of $\overline \mathcal{P}$ can be transformed in polynomial time into a feasible thorough set $S''$ with \(\phi^{\sh}_E(S'') \ge \phi^{\sh}_E(S)\). This allows us to prove Theorem~\ref{thm: GS hardness}.
\begin{proof}[Proof of Theorem~\ref{thm: GS hardness}]
Let \(S\) be an $\alpha$ approximate solution for $\overline \mathcal{P}$. We can assume, w.l.o.g., that \(S\subseteq \overline{V}_V\). Using (2) and (1) of Lemma \ref{lemma: hardness inequalities}, we get
\(
\phi^{\sh}_E(S) \ge \frac{\phi^{\sh}(S)}{2} \ge \frac{\alpha}{2} \phi^{\sh}(S^*) \ge \frac{\alpha \ell}{4} |E[T^*]|.
\)
We now apply Lemmata~\ref{lemma: key hardness 1} and~\ref{lemma: key hardness 2} to the set \(S\), obtaining a thorough set \(S''\). Together with Lemma~\ref{lemma: hardness inequalities}~(3), we get
$
\phi^{\sh}_E(S) \le \phi^{\sh}_E(S'') \le \ell |E[U_{S''}]|.
$
Thus, $S''$ is an $\frac{\alpha}{4}$-approximaion for $\mathcal{P}$. If $G$ is disconnected, this adds an extra $1/2$ to the approximation ratio. This concludes the proof.
\end{proof}
\paragraph{Transforming \(S\subseteq \overline{V}_V\) into a Thorough Set.}
Let \(S\subseteq \overline{V}_V\) be a set of size $\overline{k}$. We transform $S$ into a thorough set in two steps, the first of which is the following \emph{iterative process} that computes a set \(S'\) with \(|S'| = |S|\) by constructing a sequence \(S_0, \ldots, S_\mu\) with \(S_0=S\) and \(S_\mu=S'\). For a node $v\in V$, let \(I_E(v,S) := \phi^{\sh}_E(S_v) - \phi^{\sh}_E(S)\) where $S_v$ is the set obtained from $S$ by increasing \(\nb{S}{v}\) by one.
While \(U_{S_i}^{(1,t)}\) contains at least two nodes \(v_h\) and \(v_l\) with \(I_E(v_h,S_i) \ge I_E(v_l,S_i)\),
obtain \(S_{i+1}\) from \(S_i\) by increasing (resp.\ decreasing) \(\nb{S_i}{v_h}\) (resp.\ \(\nb{S_i}{v_l}\)) by one until one of the two nodes is not in \(U_{S_i}^{(1,t)}\).
At the end, \(|U_{S_i}^{(1,t)}|\le 1\) and, moreover, the process terminates in polynomial time, since after at most $t$ iterations, one node is removed from \(U_{S_i}^{(1,t)}\).
\begin{lemma} \label{lemma: key hardness 1}
The set \(S'\) satisfies \(\phi^{\sh}_E(S') \ge \phi^{\sh}_E(S)\).
\end{lemma}
\begin{proof}
Let \(\Delta_i:=\phi^{\sh}_E(S_{i+1}) - \phi^{\sh}_E(S_{i})\) for every $i$ and $d(x) := \frac{1}{x(x+1)}$ and note that $d$ is decreasing in $x$. We show that \(I_E(v_h,S_{i})\ge I_E(v_l,S_i)\) implies \(\Delta_i\ge 0\) and \(I_E(v_h,S_{i+1})\ge I_E(v_l,S_{i+1})\).
For a node \(v\in V\), let \(\ensuremath{\mathbb{R}}R(v) := \{R\in \ensuremath{\mathbb{R}}R_E : R\cap\{u^v_1,\ldots,u_t^v\}\neq \emptyset\}\) be all Edge-RR-sets that contain the nodes from $\overline{V}_V$ corresponding to $v$.
Then, we can rewrite \(I_E(v,S_i)\) and \(\Delta_i\) as
\begin{align*}
I_E(v,S_i) &= \sum_{R \in \ensuremath{\mathbb{R}}R(v)} d(|R \setminus S_i|),\\
\Delta_i &= \!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_h) \setminus \ensuremath{\mathbb{R}}R(v_l)} \!\!\!\!\!\!\!\! d(|R\setminus S_{i}|)- \!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_l)\setminus\ensuremath{\mathbb{R}}R(v_h)} \!\!\!\!\!\!\!\! d(|R\setminus S_{i}|+1),
\end{align*}
As $d$ is decreasing,
we have that
\begin{align*}
\Delta_i
\ge\!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_h) \setminus \ensuremath{\mathbb{R}}R(v_l)} \!\!\!\!\!\!\!\! d(|R\setminus S_{i}|)- \!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_l)\setminus\ensuremath{\mathbb{R}}R(v_h)} \!\!\!\!\!\!\!\! d(|R\setminus S_{i}|)
\end{align*}
which equals $I_E(v_h,S_i) - I_E(v_l,S_i)$. The latter is non-negative by choice of $v_h,v_l$.
We turn to showing
\(I_E(v_h,S_{i+1})\ge I_E(v_l,S_{i+1})\). We have that $I_E(v_h,S_{i+1})=\sum_{R \in \ensuremath{\mathbb{R}}R(v_h)}d(|R \setminus S_{i+1}|)$ equals
\begin{align*}
\!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_h)\setminus \ensuremath{\mathbb{R}}R(v_l)} \!\!\!\!\!\!\!\!\!d(|R \setminus S_{i}| - 1) + \!\!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_h)\cap\ensuremath{\mathbb{R}}R(v_l)} \!\!\!\!\!d(|R \setminus S_{i}|),
\end{align*}
which is at least $\sum_{R \in \ensuremath{\mathbb{R}}R(v_h)} d(|R \setminus S_{i}|)=I_E(v_h,S_{i})$ as $d$ is decreasing. Similarly, $I_E(v_l,S_{i+1}) = \sum_{R \in \ensuremath{\mathbb{R}}R(v_l)} d(|R \setminus S_{i+1}|)$ equals
\begin{align*}
\!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_l)\setminus \ensuremath{\mathbb{R}}R(v_h)} \!\!\!\!\!\!\!\!\!d(|R \setminus S_{i}| + 1) + \!\!\!\!\!\!\!\!\!\sum_{R \in \ensuremath{\mathbb{R}}R(v_l)\cap \ensuremath{\mathbb{R}}R(v_h)}\!\!\!\!\! d(|R \setminus S_{i}|),
\end{align*}
which is at most $\sum_{R \in \ensuremath{\mathbb{R}}R(v_l)} d(|R \setminus S_{i}|) = I_E(v_l,S_{i})$.
Hence, \(I_E(v_h,S_{i+1})\ge I_E(v_h,S_{i}) \ge I_E(v_h,S_{i}) \ge I_E(v_l,S_{i+1})\).
\end{proof}
\begin{lemma}\label{lemma: key hardness 2}
The set \(S'\) can be transformed into a thorough set $S''$ with \(|S''| \le |S'|\) and \(\phi^{\sh}_E(S'') \ge \phi^{\sh}_E(S')\).
\end{lemma}
\begin{proof}
We start by treating a few trivial cases. (i) The set $S'$ is thorough. Then, we set $S'':=S'$. (ii)~Property (1) holds for $S'$, but $|E[U_{S'}]| = 0$ and (iii) \(\nb{S'}{v} < t\) for all nodes $v \in U_{S'}$. Recall that \(|U_{S'}^{(1,t)}| \le 1\) by construction of $S'$. Hence, (iii) implies that for all nodes $v\in U_{S'}$ but one, \(\nb{S'}{v} = 1\). In both cases (ii) and (iii), \(\phi^{\sh}_E(S')\le\ell|E|/(t+2)\), as \(|R\setminus S'| \ge t+1\) for each of the \(\ell|E|\) Edge-RR-sets $R$. Thus, by choosing any edge $e=\{v,v'\}$ and setting \(S'' := \{u^v_i,u^{v'}_i : i\in[t] \}\) we obtain a thorough set with $|S''|\le |S'|$ and \(\phi^{\sh}_E(S'') \ge \ell/2 \ge \ell|E|/(t+2) \ge \phi^{\sh}_E(S')\).
If none of (i)-(iii) hold, we can order the nodes in \(U_{S'}\) such that there exists an index $r\in[|U_{S'}|]$ such that \(\nb{S'}{v_i} = t\) for $i\in[1, r-1]$, $\nb{S'}{v_{r}}\in [1, t)$, and \(\nb{S'}{v_{i}} = 1\) for $i\in [r+1, |U_{S'}|]$. Recall that $\overline{k}=kt$, thus \(\sum_{i=r}^{|U_{S'}|} \nb{S'}{v_i}\) is a (non-trivial) multiple of $t$. We conclude by distinguishing two more cases. If \(v_r\) is adjacent to one of \(\{v_i\}_{i\in[r-1]}\), we construct $S''$ from \(S'\) by setting $\nb{S'}{v_r} = t$ and $\nb{S'}{v_i} = 0$ for all $i\in [r+1, |U_{S'}|]$. Otherwise, we find an adjacent node \(v_q \in V\setminus\{v_i\}_{i\in[r-1]}\) (by connectivity of $G$ such node exists) and construct $S''$ from \(S'\) by setting $\nb{S'}{v_q} = t$ and $\nb{S'}{v_i} = 0$ for all \(i \in [r, |U_{S'}|]\).
In both cases $S''$ is thorough and $|S''|\le |S'|$. It remains to show \(\phi^{\sh}_E(S'')\ge \phi^{\sh}_E(S')\).
Indeed, in the first (resp.\ second) case \(v_r\) (resp.\ $v_q$) is in the neighborhood of \(\{v_i\}_{i\in[r-1]}\). By setting $\nb{S'}{v_r}$ (resp.\ $\nb{S'}{v_q}$) to $t$, there exist \(\ell\) Edge-RR-sets $R$ such that $|R\setminus S''| = 1/2$. The total increase of \(\phi^{\sh}_E\) on these RR-sets is at least \(\ell/6 = \ell/2 - \ell/3\). Conversely, the decrease resulting from setting $\nb{S'}{v_i} = 0$ for all \(i \in [r + 1, |U_{S'}|]\) (resp.\ for all \(i \in [r, |U_{S'}|]\)) is at most \(\ell|E|/(t+1)\), since for each of the Edge-RR-sets $R$ that intersect $\{v_{i}\}_{i\in [r + 1, |U_{S'}|]}$ (resp.\ $\{v_{i}\}_{i\in [r, |U_{S'}|]}$), it holds that $|R\setminus S'|\ge t$. As \(\ell/6 \ge \ell|E|/(t+1)\), this concludes the proof.
\end{proof}
\section{CONCLUSION AND FUTURE WORK}
We have formalized the problem of determining a set of $k$ nodes in a social network maximizing an influence-based Group Shapley centrality measure. Assuming common computational complexity conjectures, we have obtained strong hardness of approximation results for the problem at hand in this paper. For instance, this problem cannot be approximated within $1/n^{o(1)}$ under the Gap Exponential Time Hypothesis. On the other hand, we showed that a greedy algorithm achieves a factor of $\frac{1-1/e}{k}-\epsilon$ for any $\epsilon>0$, yielding an interesting result when $k$ is small.
Several directions for future work are conceivable. First, it would be worth investigating an algorithm with an approximation ratio which is sublinear in the number of nodes of the social network. Second, specific properties of the social network could allow more positive approximation results, as, for instance, the connectivity of the graph has a direct impact on the size of the generated reverse reachable sets. Hence, restricting this parameter could have an impact on the complexity of the problem from an approximation viewpoint. Third, it would be interesting to adapt our work to other generalized semivalues as, for instance, the Group Banzhaf value~\cite{DBLP:journals/dam/MarichalKF07}. Lastly, properly engineering and testing the approximation algorithm designed in this paper would be an interesting and complementary work.
\end{document}
|
\begin{document}
\title{Nonlocal quantum correlations under amplitude damping decoherence}
\author{Tanumoy Pramanik}
\email{[email protected]}
\affiliation{Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea}
\author{Young-Wook Cho}
\affiliation{Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea}
\author{Sang-Wook Han}
\affiliation{Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea}
\author{Sang-Yun Lee}
\affiliation{Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea}
\author{Sung Moon}
\affiliation{Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea}
\affiliation{Division of Nano \& Information Technology, KIST School, Korea University of Science and Technology, Seoul 02792, Republic of Korea}
\author{Yong-Su Kim}
\email{[email protected]}
\affiliation{Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea}
\affiliation{Division of Nano \& Information Technology, KIST School, Korea University of Science and Technology, Seoul 02792, Republic of Korea}
\date{\today}
\begin{abstract}
\noindent
Different nonlocal quantum correlations of entanglement, steering and Bell nonlocality are defined with the help of local hidden state (LHS) and local hidden variable (LHV) models. Considering their unique roles in quantum information processing, it is of importance to understand the individual nonlocal quantum correlation as well as their relationship. Here, we investigate the effects of amplitude damping decoherence on different nonlocal quantum correlations. In particular, we have theoretically and experimentally shown that the entanglement sudden death phenomenon is distinct from those of steering and Bell nonlocality. In our scenario, we found that all the initial states present sudden death of steering and Bell nonlocality, while only some of the states show entanglement sudden death. These results suggest that the environmental effect can be different for different nonlocal quantum correlations, and thus, it provides distinct operational interpretations of different quantum correlations.
\end{abstract}
\keywords{Sudden death, Entanglement, Steering, Unsteering, Bell nonlocality, Amplitude damping decoherence}
\maketitle
\section{Introduction}
Nonlocal quantum correlations are not only significant due to their foundational aspects in quantum information theory, but also their applications in various quantum information processing tasks. According to the different local models based on the properties of underlying systems, nonlocal quantum correlations can be categorized into three different forms of entanglement, EPR (Einstein-Podolsky-Rosen) steering, and Bell nonlocality~\cite{E_rev, B_Rev, Jones07_2}. A bipartite quantum system is entangled if it cannot be written as a statistical mixture of products of local states of individual systems. Therefore, for a bipartite entangled state, the correlation cannot be described by local hidden state (LHS)-LHS model. If we weaken the LHS-LHS model to LHS-local hidden variable (LHV) model, i.e., one of the systems is not trusted as a quantum system, then the non-separability becomes EPR steering~\cite{Jones07_1, Jones07_2}. If we further relax the condition to LHV-LHV model, then the non-separability defines Bell nonlocality~\cite{Bell, CHSH, Rev_BN}. Therefore, three forms of nonlocal quantum correlations are interconnected via their definitions. In particular, all Bell nonlocal states are steerable, and all steerable states are entangled. However, there exist some entangled states which are not steerable, and some steerable states are not Bell nonlocal. Therefore, we can explicitly present the relationship between nonlocal quantum correlations as, Bell nonlocality $\subset$ EPR steering $\subset$ Entanglement.
In practice, nonlocal quantum correlations are used as resources of quantum information processing. Entanglement is known as a basic resource for many quantum information processing tasks such as quantum teleportation~\cite{Tele, Tele_Exp1, Tele_Exp2}, quantum communication~\cite{SDC, QKD1,QKD2, QKD3}, and quantum computation~\cite{QCom, Ent_Speed}. However, in order for entanglement to play roles, both systems should be trusted as quantum systems, and there should be no quantum hacking attempts to both systems. On the other hand, EPR steering and Bell nonlocality can play roles in the quantum information processing even when there exist quantum hacking attacks on one of the systems~\cite{1sDIQKD}, and both systems~\cite{QKD2, DIQKD_1, DIQKD_2, DIQKD_3, DIQKD_4}, respectively.
In real world implementation, quantum systems interact with the environment, and it usually causes unavoidable decoherence. As a result, quantum correlations usually gradually decrease with the increasing interaction time, and completely vanish after an infinite time of interaction~\cite{ESD_2, ESD_4, ESD_4_1,ESD_4_2}. Remarkably, the system-environment interaction sometimes causes much faster degradation of quantum correlations, so the quantum system can completely lose quantum correlations in finite time of interaction. This phenomenon is known as the sudden death of quantum correlations~\cite{ESD_2, ESD_1, ESD_3, ESD_4, ESD_5, ESD_5_1}. We also note that the environmental interaction sometimes increases quantum correlations in certain circumstances~\cite{DE_1,DE_2,DE_3, DE_4}.
\begin{figure*}
\caption{(a) Concurrence $C(\theta,D)$, (b) EPR steering (green) and unsteering (yellow) parameters of $T_{16}
\label{3D}
\end{figure*}
It has been widely studied the effect of decoherence on entanglement both in theory and experiment~\cite{ESD_1, ESD_2, ESD_3, ESD_4, ESD_5}. However, there are only a few theoretical studies on other nonlocal quantum correlations~\cite{SSD,SSD_11,SSD_22, BNSD_1}. These studies deal with the entanglement sudden death (ESD)~\cite{ESD_1, ESD_2, ESD_3, ESD_4} and Bell nonlocality sudden death (BNSD)~\cite{BNSD_1}, however, the study of EPR steering sudden death (SSD) is still missing. Moreover, all of these works are limited to one of the nonlocal quantum correlations, and thus they fail to present unified results of the environmental effect on various nonlocal quantum correlations. Considering their relationship and unique roles in quantum information processing, it is of importance to investigate the dynamics of various nonlocal quantum correlations in the presence of decoherence.
In this paper, we theoretically and experimentally investigate entanglement, EPR steering, and Bell nonlocality under an amplitude damping channel (ADC). We found that different quantum correlations present very different environmental effects. For example, in our scenario, all the states present SSD and BNSD, while ESD happens for only some of the initial states. Moreover, we can prepare two different bipartite states with an equal amount of entanglement, but one of them shows sudden death of all nonlocal quantum correlations while the other only shows steering and Bell nonlocality sudden deaths, but not ESD. Therefore, in the presence ADC, entanglement behaves very differently from the other nonlocal quantum correlations, and it provides distinct operational interpretations of different nonlocal quantum correlations.
\section{Theory}
\subsection{Amplitude damping channel}
The interaction between the system $S$ and the environment $E$ via ADC with the interaction strength of $0\le D\le1$ can be modeled as~\cite{lee11,kim12}
\begin{eqnarray}
|0\rangle_S|0\rangle_E & \rightarrow & |0\rangle_S|0\rangle_E, \nonumber \\
|1\rangle_S|0\rangle_E & \rightarrow & \sqrt{1-D} |1\rangle_S|0\rangle_E + \sqrt{D} |0\rangle_S|0\rangle_E.
\label{ADC}
\end{eqnarray}
Here, we assume that the environment is initially in $|0\rangle_E$. Let us consider a two-qubit system is initially prepared in a pure state of $|\psi_\theta\rangle = \cos\theta |0\rangle_A|0\rangle_B + \sin\theta |1\rangle_A|1\rangle_B$,
where $0\le\theta\le\pi/2$ is the biasing parameter. Assuming both qubits $A$ and $B$ are under ADC with an equal interaction strength of $D$, the state becomes~\cite{kim12}
\begin{eqnarray}
\rho_{\theta}^{D} =
\begin{pmatrix}
\alpha_{11} & 0 & 0 & \alpha_{14} \\
0 & \alpha_{21} & 0 & 0 \\
0 & 0 & \alpha_{21} & 0 \\
\alpha_{14} & 0 & 0 & \alpha_{44}
\end{pmatrix},
\label{State_f}
\end{eqnarray}
where $\alpha_{11}=\cos^2\theta + D^2 \sin^2\theta$, $\alpha_{14}=(1-D) \cos\theta\sin\theta$, $\alpha_{21}=(1-D)D \sin^2\theta$, and $\alpha_{44}=(1-D)^2 \sin^2\theta$, respectively.
Now, we study entanglement, EPR steering, and Bell nonlocality of the state $\rho_{\theta}^D$. Here, we quantify the amount of entanglement with concurrence~\cite{Concurrence_1,Concurrence_2}. Bell nonlocality is determined by the Horodecki criterion which provides the necessary and sufficient condition for a $2\otimes 2$ dimensional system~\cite{Horo_Cri,Horo_Cri_2}. We apply the steering criterion developed in Ref.~\citep{Saunders, bennet12} to capture the steerability of a given state. Note that the steering criterion is necessary but not sufficient, and thus it cannot determine the unsteerability of a given state. In order to capture unsteerability, we employ the recently developed sufficient criterion of unsteerability~\cite{PE_St3}. Here, we only provide the results of the theoretical investigation. The detailed estimation procedures can be found in the supplementary materials.
\subsection{Entanglement}
The concurrence of $\rho_{\theta}^D$ is given by
\begin{eqnarray}
C(\theta,D) = \max\left[0,\,2 (1-D) \sin\theta (\cos\theta - D \sin\theta)\right],
\label{C_f}
\end{eqnarray}
and depicted in Fig.~\ref{3D}(a). All the initial states of $D=0$ has non-zero concurrence, and thus are entangled except for $\theta=0$ or $\pi/2$. As the interaction strength $D$ increases, concurrence decreases. One can find that entanglement vanishes, and the state $\rho_{\theta}^D$ becomes separable when $\cot\theta\leq D$. Therefore, the ESD occurs along the red line which corresponds to $D = \cot\theta$.
Note that entanglement of the initial state, $C(\theta,D=0)=\sin2\theta$, is symmetrical with respect to $\theta=\pi/4$. Therefore, the initial states $|\psi_{\phi}\rangle$ and $|\psi_{\frac{\pi}{2}-\phi}\rangle$, where $0\le\phi<\frac{\pi}{4}$, have the same amount of entanglement. This symmetry is broken as $C(\frac{\pi}{2}-\phi,D)<C(\phi,D)$ after the amplitude damping decoherence, $0<D$. This asymmetrical nature becomes more clear for the ESD, i.e., ESD occurs only for $|\psi_{\frac{\pi}{2}-\phi}\rangle$, and never happens for $|\psi_{\phi}\rangle$. It originates from the asymmetrical nature of the ADC where $|1\rangle$ experiences the damping decoherence while $|0\rangle$ is unaffected.
We note that the non-zero concurrence provides the necessary and sufficient condition of the existence of entanglement in a two-qubit system~\cite{Concurrence_1, Concurrence_2}. Therefore, the entanglement sudden death described above is a real physical phenomenon although it has been investigated with the mathematical description of concurrence.
\subsection{EPR steering}
LHS model restricts the correlation $P(a_\mathcal{A},b_{\mathcal{B}})$ between the measurement outcomes $a$ and $b$ of the observables $\mathcal{A}$ and $\mathcal{B}$ on the systems $A$ and $B$, respectively, as
\begin{eqnarray}
P(a_{\mathcal{A}},b_{\mathcal{B}}) = \sum_\lambda P(\lambda) P(a_{\mathcal{A}}|\lambda) P_Q(b_{\mathcal{B}}|\lambda),
\label{LHS}
\end{eqnarray}
where $P(\lambda)$ is the distribution of hidden variables. The subscript $Q$ presents that Bob's probability distribution is obtained from the measurement of observable on the quantum system $B$. The joint probability distribution $P(a_{\mathcal{A}},b_{\mathcal{B}})$ for the shared bipartite state $\rho_\theta^D$ by Alice and Bob can be written as
\begin{eqnarray}
P_{\rho}(a_{\mathcal{A}},b_{\mathcal{B}})=\Tr\Big[\Big(\frac{I+(-1)^a \mathcal{A}}{2}\otimes\frac{I+(-1)^b \mathcal{B}}{2}\Big)\rho_\theta^D\Big]
\label{JPD}
\end{eqnarray}
The experimentally testable steering criterion can be derived with the help of the LHS model of Eq.~(\ref{LHS}). As quantum probability distribution $\big\{P_Q(b_{\mathcal{B}}|\lambda)\big\}$ for the measurement of non-commuting observables are bounded by the uncertainty principle, the correlation $\big\{P(a_{\mathcal{A}},b_{\mathcal{B}})\big\}$ is also bounded by the uncertainty principle. Several steering criteria have been derived based on different forms of uncertainty relation along with the LHS model~\cite{PE_St2,PE_St1, St_C1_2,St_C2,PE_St3}.
Here, we employ the most widely accepted steering criterion of Ref.~\cite{Saunders, bennet12} as
\begin{eqnarray}
T_m=\frac{1}{m} \sum_{k=1}^m \langle \alpha_k (\hat{n}_k\cdot\vec{\sigma}^B)\rangle \leq C_m,
\label{Steer_m}
\end{eqnarray}
where $m$ is the number of the measurement settings of Alice and Bob, and the random variable $\alpha_k\in\{0,1\}$ is Alice's measurement result for $k$-th measurement. Bob's $k$-th measurement corresponds to the spin measurement along the direction $\hat{n}_k$ and $\vec{\sigma}^B\in\{\sigma_x,\sigma_y,\sigma_z\}$, where $\sigma_x,\sigma_y,\sigma_z$ are the Pauli spin operators. $C_m$ is the maximum value of $T_m$ when Bob's system can be described by LHS model. The violation of Eq.~(\ref{Steer_m}) guarantees the steerability of the shared bipartite state $\rho_\theta^D$. The efficiency of the Eq.~(\ref{Steer_m}) increases with $m$, i.e., for a larger $m$, Eq.~(\ref{Steer_m}) captures larger set of steerable states. Here, we follow the technique used in the Refs.~\cite{Saunders, bennet12, Cavalcanti13} to increase the number of measurement settings, $m$. In Refs.~\cite{Saunders, bennet12, Cavalcanti13}, the vertices of the three-dimensional Platonic solids are used to design the measurement directions. There are only five three-dimensional Platonic solids with 4, 6, 8, 12, and 20 vertices. The measurement directions are chosen along the line by joining a vertex with its diametrically opposite vertex, except the Platonic solid with 4 vertices. With that, we can obtain $3,~4,~6,~10$ measurement settings from the Platonic solids with 6, 8, 12, and 20 vertices, respectively. We can increase the number of measurement settings by combining the measurement directions from the four Platonic solids. Here, we have chosen $m=16$ measurement settings
by combining the axes of a dodecahedron (the Platonic solids with 20 vertices) and its dual, the icosahedron (the Platonic solids with 12 vertices). Note that we found that $m=16$ measurement settings capture larger sets of steerable states than other possible combinations using 4 Platonic solids in our scenario. In this case, steerability is guaranteed by the violation of the following inequality~\cite{bennet12,Saunders}.
\begin{eqnarray}
T_{16}(\theta,D)=\frac{1}{16} \sum_{k=1}^{16} \langle \alpha_k (\hat{n}_k\cdot\vec{\sigma}^B)\rangle \leq C_{16}=0.503.
\label{Steer_16}
\end{eqnarray}
Since the steering criterion Eq.~(\ref{Steer_16}) is necessary, but not sufficient, it does not guarantee unsteerability. The unsteerability of the state $\rho_{\theta}^D$ can be verified with the help of the sufficient criterion of unsteerability derived in Ref.~\cite{PE_St3}. According to this criterion, the unsteerability of $\rho^D_{\theta}$ is determined when
\begin{eqnarray}
t_{U}(\theta,D)= \max\left[\alpha, \frac{2 \cos\theta \sqrt{1-D}}{\sqrt{\gamma}}\right] \leq 1,
\label{Unsteer}
\end{eqnarray}
where $\gamma=\cos^2\theta+D\sin^2\theta$ and $\alpha=\{D^2 (\gamma - (1- D) \sin^2\theta)^2 +2 (1-D) \gamma\}/\gamma^2$.
Let us define the normalized unsteering parameter $T_U$ as
\begin{eqnarray}
T_{U}(\theta,D)= 0.503\cdot t_{U}(\theta,D) \leq 0.503,
\label{Unsteer}
\end{eqnarray}
in order to present the steering and unsteering criteria in the same figure, see Fig.~\ref{3D}(b). The green and yellow surfaces show $T_{16}(\theta,D) > 0.503$ and $T_U(\theta,D) > 0.503$, respectively.
Therefore, the states $\rho_{\theta}^D$ lie on the green surface are steerable. Note that, similar to entanglement, the steering parameter $T_{16}(\theta,D)$ becomes asymmetrical with respect to $\theta=\pi/4$ after ADC. The states $\rho_{\theta}^D$ becomes unsteerable when $T_{U}(\theta,D)\leq0.503$. Therefore, the SSD occurs for $T_{U}(\theta,D)=0.503$, and it is presented by a blue curve in the Fig.~\ref{3D}(b). It is remarkable that SSD happens for all the initial states, unlike ESD.
\begin{figure}
\caption{The regions of various nonlocal quantum correlations for the bipartite state $\rho_{\theta}
\label{Fig_Th}
\end{figure}
\subsection{Bell nonlocality}
The Bell nonlocality of a given state can be calculated from the correlation matrix $\lambda_{ij}^{\theta, D} = \Tr[\sigma_i\otimes\sigma_j\cdot\rho_{\theta}^D]$,
where $i,j\in\{x,y,z\}$~\cite{Horo_Cri,Horo_Cri_2}. The eigenvalues of $\left(\lambda_{ij}^{\theta, D}\right)^T\cdot \lambda_{ij}^{\theta, D}$, where the superscript $T$ denotes for transposition, are $\lambda_1=(\cos^2\theta + (1-2 D)^2\sin^2\theta)^2$, and $\lambda_2=(1-D)^2\sin^22\theta$ (with degeneracy). Therefore, the Bell parameter
$S=\langle \alpha_1\beta_1\rangle + \langle \alpha_1\beta_2\rangle + \langle \alpha_2\beta_1\rangle -\langle \alpha_2\beta_2\rangle $, where $\{\alpha_1, \alpha_2\}$ and $\{\beta_1,\beta_2\}$ are sets of Pauli operators for Alice and Bob, respectively,
is given by~\cite{Horo_Cri,Horo_Cri_2}
\begin{eqnarray}
S(\theta, D) = \max\left[2\sqrt{2\lambda_2}, 2 \sqrt{\lambda_1+\lambda_2} \right].
\label{BI_rho}
\end{eqnarray}
The state is Bell nonlocal if $S(\theta, D)>2$. The Bell parameter $S(\theta, D)$ is plotted in the Fig.~\ref{3D}(c). The orange curve shows the Bell nonlocality of the state $\rho_{\theta}^D$ and BNSD occurs along the green line represented by $S(\theta,D) = 2$ where the orange surface touches the horizontal surface. Similar to SSD, BNSD occurs for all the initial states.
\subsection{Sudden death of nonlocal quantum correlations}
The initial state $|\psi_{\theta}\rangle$ is entangled, steerable and Bell nonlocal for all values of $\theta$ chosen from the range of $0<\theta< \pi/2$. As a result of ADC, nonlocal quantum correlations decrease with the increasing interaction strength $D$. In order to compare the sudden death phenomena of various quantum correlations, we present the local-nonlocal boundaries of $C(\theta,D)=0$, $T_{16}(\theta,D)=0.503$, $T_U(\theta,D)=0.503$, and $S(\theta,D)=2$ in Fig.~\ref{Fig_Th}.
The red line corresponds to $C(\theta,D)=0$, and hence, it divides entangled states from separable states. It signifies that the state $|\psi_\theta\rangle$ with $0<\theta\leq \pi/4$ does not show ESD in ADC. The green curve presents $S(\theta,D)=2$, and thus show the BNSD boundary. It has discontinuities at $(\theta,D)\sim(0.35\pi,0.101)$ and $(0.21\pi,0.269)$ due to the maximization over two functions in Eq.~(\ref{BI_rho}). The purple and blue curves correspond to $T_{16}(\theta,D)=0.503$ and $T_U(\theta,D)=0.503$, and thus they are boundaries for steerable and unsteerable states, respectively. Between these two boundaries, there exists a undetermined area in gray where steerability of a given state cannot be concluded with the existing steering and unsteering criteria. As can be seen in the shaded by black region where the steering criterion fails to reveal the EPR steering for Bell nonlocal states, the steering criterion becomes invalid as $\theta\rightarrow0$. This non-ideal presentation can be improved by increasing the number of measurement settings~\cite{Saunders}.
It is interesting to compare the sudden death phenomena among various nonlocal quantum correlations. Although all quantum correlations of the initial state $|\psi_\theta\rangle$ are symmetrical with respect to the parameter $\theta$, they become asymmetrical after ADC. This happens due to the asymmetrical nature of ADC, i.e., ADC does not affect to $|0\rangle$ and $|1\rangle$ symmetrically. As discussed above, while both states $|\psi_\phi\rangle$ and $|\psi_{\phi+\pi/4}\rangle$, where $\phi < \pi/4$, have same amount of entanglement, ESD never happens for states $|\psi_\theta\rangle$. Whereas, all states with $0<\phi < \pi/2$ show SSD and BNSD. These results indicate that different nonlocal quantum correlations are affected by ADC in very different ways.
\section{Experiment}
\subsection{Experimental setup}
\begin{figure}
\caption{ Experimental setup for (a) the initial state preparation, (b) amplitude damping channel, and (c) state measurement for inequality test and quantum state tomography. BD : beam displacer, H : half waveplate, Q : quarter waveplate, BS : beamsplitter, Pol. : Polarizer, SPD : Single-photon detector.}
\label{Setup}
\end{figure}
Figure~\ref{Setup} shows the experimental setup to explore nonlocal quantum correlations affected by ADC. The maximally entangled photon pair of $|\psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)=\frac{1}{\sqrt{2}}(|HH\rangle+|VV\rangle)$ at 780~nm is generated at a sandwich BBO crystal via spontaneous parametric downconversion pumped by a femtosecond laser pulse. Here, $|H\rangle$ and $|V\rangle$ denote horizontal and vertical polarization states, respectively. The sandwich BBO crystal, which is composed of two type-II BBO crystals and a half waveplate in between, is specially designed for efficient generation of two-photon entangled states~\cite{wang16}.
In order to implement the amplitude damping channel (ADC), one needs to keep the probability amplitude of $|0\rangle$ unchanged while that of $|1\rangle$ changes to $|0\rangle$ with the probability $D$. Fig.~\ref{Setup}(b) shows our implementation of ADC with polarization qubits. Two beam displacers (BD) which transmit (reflect) horizontal (vertical) polarization state form a Mach-Zehnder interferometer. With the half waveplates (HWP, H) in the interferometer, one can independently control the ratio between two outputs $|0\rangle_E$ and $|1\rangle_E$ of BD2 for the horizontal and vertical polarization states. In the experiment, we set the HWP at the horizontal polarization path at $45^\circ$ in order to have all horizontal input photons at $|0\rangle_E$. On the other hand, the vertical polarization input state can be found both at $|0\rangle_E$ and $|1\rangle_E$ according to the angle of the HWP at the vertical polarization path. In order to cancel out the effect of the HWP in the interferometer, we position HWP at $45^\circ$ both at $|0\rangle_E$, and $|1\rangle$. The environment qubit is traced out by incoherently mixing $|0\rangle_E$ and $|1\rangle_E$ at a beamsplitter (BS)~\cite{lee11}.
As shown in the Fig.~\ref{Setup}(c), two-qubit quantum state tomography (QST) and various inequality tests are conducted by two-qubit projective measurement and coincidence detection. In the experiment, concurrence $C$ and the unsteering parameter $T_U$ are calculated from the QST result whereas the Bell parameter $S$ and the steering parameter $T_{16}$ are directly obtained from the inequality test data. The details of calculating entanglement and unsteerability as well as measurement settings for Bell nonlocaltity test and steering test can be found in the Appendices.
\subsection{Experimental results}
For experimental verification of the effect of different quantum correlations in the presence of amplitude damping decoherence, we have prepared maximally entangled polarization photon pairs from spontaneous parametric down conversion. To test Bell nonlocality and steerability, we use CHSH and steering inequalities derived in the Ref.~\cite{Saunders, bennet12}. To confirm unsteerability, we experimentally test the sufficient condition of unsteerability of Eq.~(\ref{Unsteer}) via quantum state~tomography~\cite{qst1,qst2}. The details of experiment can be found in the supplemental material.
\begin{figure}
\caption{Experimental results of the parameters of (a) entanglement, (b) Bell nonlocality, (c) EPR steering, and (d) unsteering for the initially maximally entangled state $\rho_{\theta=\pi/4}
\label{Fig_Exp}
\end{figure}
We present parameters of different nonlocal quantum correlations for the initially maximally entangled state $|\psi_{\theta=\pi/4}\rangle$ with respect to the interaction strength $D$ in Fig.~\ref{Fig_Exp}. Figure~\ref{Fig_Exp}(a) shows theoretically and experimentally obtained concurrence $C$. It clearly shows that entanglement gradually degrades as $D$ increases, and the state becomes separable when $D=1$. Therefore, entanglement does not show sudden death phenomenon.
Fig.~\ref{Fig_Exp}(b) represents the Bell parameter $S$. The horizontal straight line corresponds to the upper bound of Bell inequality under LHV model, $S=2$. Similar to the concurrence, $S$ decreases as $D$ increases. More interestingly, $S$ becomes smaller than 2 even for $D<1$, that indicates sudden death of Bell nonlocality happens. In particular, we theoretically found that the sudden death of Bell nonlocality happens at $D\approx0.29$. Our experimental result coincides with the theoretical finding as the Bell nonlocal state at $D=0.2$ becomes Bell local at $D=0.4$. It is notable that unlike entanglement, Fig.~\ref{Fig_Exp}(b) shows the non-monotonous nature of Bell local correlation (i.e., Bell parameter $S$ lies below 2) with respect to the decoherence parameter $D$. The values of $S$ decreases when $D$ increases from 0 to $0.66$, but for further increment of $D$ from $0.66$ to $1$, $S$ increases up to $2$. However, it never exceeds the the classical-quantum boundary of $S=2$. Due to the loss of quantum coherence measured by off-diagonal elements, different nonlocal quantum correlations, entanglement and Bell nonlocal correlation decrease gradually with the strength of decoherence and show monotonic behaviour. However, appearing and disappearing of the diagonal elements due to the effect of ADC is the source of non-monotonic behaviour of the local correlations, Bell local correlation (explained by local hidden variable theory).
The theoretical and experimental results of EPR steering and unsteerability are presented in Figs.~\ref{Fig_Exp}(c) and (d), respectively. The horizontal lines in the Figs.~\ref{Fig_Exp}(c) and (d) are the upper bounds of steering inequality allowed by LHS model, i.e., $T_{16}=0.503$, and the upper bound of sufficient criterion of unsteerability, i.e., $T_U=0.503$, respectively. The vertical red (blue) line denotes the value of $D$ corresponding to the intersection between theoretical $T_{16}$ ($T_U$) and the horizontal line of $T_{16}=0.503$ ($T_{U}=0.503$). The light red shaded regions in both Figs.~\ref{Fig_Exp}(c) and (d) represent the range of $D$ for which the state $\rho_{\pi/4}^D$ is steerable. The light blue shaded region in Fig.~\ref{Fig_Exp}(d) shows the unsteerable region with respect to the parameter $D$. The steerable and unsteerable regions are separated by the gray region of $0.495\leq D\leq 0.6$ where the state cannot be concluded whether steerable or unsteerable with the existing criteria. The existence of unsteerable region verifies the EPR steering sudden death of the state $\rho_{\pi/4}^D$. Similar to the Bell local correlation, non-monotonic behaviour of unsteerability explained by local hidden state model occurs due to the effect of ADC on the diagonal elements.
\section{Conclusion}
We have theoretically and experimentally investigated different nonlocal quantum correlations of entanglement, EPR steering and Bell nonlocality under amplitude damping channel (ADC). Our results also show the dynamics of entanglement is completely different from those of EPR steering and Bell nonlocality in the presence of ADC. For example, in our scenario, entanglement sudden death depends on the preparation of initial entangled states whereas steering and Bell nonlocality sudden deaths happen for all the initial state. Therefore, our findings present clear theoretical and experimental evidences of structural difference between different nonlocal quantum correlations~\cite{tanu19}. They also indicate the operational difference of nonlocal quantum correlations in the presence of decoherence. Considering the fundamental and practical importance of nonlocal quantum correlations in quantum information science, our results not only provide better understanding, but also inspire various applications of quantum information.
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\onecolumngrid
\appendix
\section{Calculation of entanglement}
\label{Apdx_Con}
Entanglement of a bipartite state can be easily verified from its concurrence. If concurrence is positive, then the state is said to be entangled. The concurrence of the state $\rho_{\theta}^D$ can be calculated from the eigenvalues of $\Lambda^C=\rho_\theta^D\cdot\left(\sigma_y\otimes\sigma_y\cdot(\rho_{\theta}^D)^*\cdot\sigma_y\otimes\sigma_y\right)$, where the asterisk `$*$' stands for complex conjugation. For the state $\rho_{\theta}$, the eigenvalues of $\Lambda^C$ in decreasing order becomes
\begin{eqnarray}
\lambda_1 &=& (1-D)^2\sin\theta^2\left(\sqrt{\cos^2\theta + D^2\sin^2\theta} + \cos\theta \right)^2, \nonumber \\
\lambda_2 &=& \lambda_3= (1-D)^2D^2\sin^4\theta, \nonumber \\
\lambda_4 &=& (1-D)^2\sin\theta^2\left(\sqrt{\cos^2\theta + D^2\sin^2\theta} - \cos\theta \right)^2.
\end{eqnarray}
Using the above eigenvalues, the concurrence of the state $\rho_{\theta}^D$ can be calculated as
\begin{eqnarray}
C(\theta,D) &=& \max\left[0,\,\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_3}\right] \nonumber \\
&=& 2 (1-D)\sin\theta(\cos\theta - D\sin\theta).
\end{eqnarray}
\section{Calculation of unsteerability}
\label{Apdx_Unst}
To derive sufficient criterion for an existing local hidden state (LHS) model of the state $\rho_{\theta}^D$, we need to transform it into the canonical form $\varrho=\frac{1}{4}\left(\mathbb{I} + \vec{a}.\vec{\sigma} +\sum_{i=x,y,z} T_i\sigma_i\otimes \sigma_i\right)$,
where $\vec{a}\in\{a_x,a_y,a_z\}$ is Alice's local vector and $\{T_x,T_y,T_z\}$ forms a correlation matrix. $\rho_{\theta}^D$ can be converted to the above canonical form with the help of following transformation
\begin{eqnarray}
\varrho_{\theta} = \mathbb{I}\otimes \left( \left(\rho_{\theta}^D\right)^B \right)^{-1/2} \cdot \rho_{\theta}^D \cdot \mathbb{I}\otimes\left(\left(\rho_{\theta}^D\right)^B\right)^{-1/2},
\end{eqnarray}
where $\left(\rho_{\theta}^D\right)^B=Tr_A[\rho_{\theta}]$. Then the sufficient criterion for unsteerability,
\begin{eqnarray}
T_U(\theta,D)=\max\left[a_z^2+ 2|T_z|, 2|T_x|\right] \leq 1
\end{eqnarray}
becomes
\begin{eqnarray}
\max\left[\alpha, \frac{2 \cos\theta \sqrt{1-D}}{\sqrt{\gamma}}\right] \leq 1,
\end{eqnarray}
where $\gamma=\cos^2\theta+D\sin^2\theta$ and $\alpha=\frac{D^2 (\gamma - (1- D) \sin^2\theta)^2 +2 (1-D) \gamma}{\gamma^2}$.
\section{Calculation of measurement settings for Bell nonlocality}
\label{Apdx_Bell}
Horodecki criterion provides maximum Bell violation of a given state in $2\otimes 2$ dimensional systems~\cite{Horo_Cri, Horo_Cri_2}. The measurement settings for both Alice and Bob corresponding to Bell violation as predicted by Horodecki criterion can be calculated with the help of Ref.~\cite{PE_BN_1,PE_BN_2,PE_BN_3}. To obtain Alice's and Bob's measurement settings corresponding to the Bell violation $S(\theta=\pi/4,\,D)$ of Eq.~(12) in the main text, let us consider two following scenarios. In the first scenario, Alice measures either observable $\mathcal{A}_1=\sigma_x$ or $\mathcal{A}_2=\sigma_y$ on her system $A$. Bob's choice of observables are
\begin{eqnarray}
\mathcal{B}_1 &=&\sigma_x\cos\varphi_1 +\sigma_y\sin\varphi_1, \nonumber \\
\mathcal{B}_2&=&\sigma_x\cos\varphi_2 +\sigma_y\sin\varphi_2.
\label{BN_MS_rho}
\end{eqnarray}
Then, the Bell parameter $S$ becomes
\begin{eqnarray}
S_1(\theta=\pi/4,\,D) &=& (1-D) \left(\cos\varphi_1 + \cos\varphi_2 - \sin\varphi_1 + \sin\varphi_2\right).
\end{eqnarray}
The maximum value of $S_1(\theta=\pi/4,\,D)$ can be found for $\varphi_1=7\pi/4$, and $\varphi_2=\pi/4$. Note that, $S_1(\theta=\pi/4,\,D)=S(\theta=\pi/4,\,D)$ for $0\leq D \leq 0.5$.
In the second scenario, Alice chooses observables from the set $\{\mathcal{A}_1=\sigma_x,\mathcal{A}_2=\sigma_z\}$ and Bob's set is given by
\begin{eqnarray}
\mathcal{B}_1 &=&\sigma_z\cos\chi_1 +\sigma_x\sin\chi_1, \nonumber \\
\mathcal{B}_2&=&\sigma_z\cos\chi_2 +\sigma_x\sin\chi_2.
\end{eqnarray}
In this case, the Bell parameter $S$ is given by
\begin{eqnarray}
S_2(\theta=\pi/4,\,D) &=& (1-2(1-D)D)\cos\chi_1 - (1-2(1-D)D)\cos\chi_2 + (1-D)(\sin\chi_1+\sin\chi_2),
\end{eqnarray}
which becomes maximum for $\chi_1=\arctan\left[(1-D)/(1-2(1-D)D)\right]$ and $\chi_2=\pi + \arctan\left[-(1-D)/(1-2(1-D)D)\right]$. In this scenario, $S_2(\theta=\pi/4,\,D)=S(\theta=\pi/4,\,D)$ for $0.5\leq D\leq 1$. Therefore, when decoherence parameter lies in the range $0\leq D\leq 0.5$, Alice and Bob choose the first scenario, otherwise, they choose the second scenario.
\section{Calculation of Measurement settings for steerability}
\label{Apdx_St}
In order to test the steering inequality with $16$ measurement settings on each subsystem, Bob chooses spin measurement along vertex-to-vertex of dodecahedron and icosahedron, Alice's measurement settings are calculated by maximizing $T_{16}$. Here, Bob's direction of $i$th spin measurement and Alice's direction of corresponding spin measurement settings are given by $\mathcal{B}_i\in\{n_x^i,n_y^i,n_z^i\}$ and $\mathcal{A}_i\in\{\sin\alpha_i \cos\beta_i,\sin\alpha_i \sin\beta_i, \cos\alpha_i\}$, respectively. The above measurement settings $\{\mathcal{A}_i,\mathcal{B}_i\}$ maximize the expectation value $\langle\mathcal{A}\mathcal{B}\rangle$ for the shared state $\rho_\theta^D$. $16$ set of measurement settings are given below
\begin{eqnarray}
\{\mathcal{A}_1,\mathcal{B}_1\} &\equiv & \{\{ \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \},\{\alpha_1=\arctan\left[-\frac{\gamma_1}{\delta_1}\right],\beta_1=\arctan[-1] \}\}, \nonumber\\
\{\mathcal{A}_2,\mathcal{B}_2\} &\equiv & \{\{ - \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \},\{ \alpha_2=\alpha_1 ,\beta_2=\frac{5\pi}{4} \}\}, \nonumber\\
\{\mathcal{A}_3,\mathcal{B}_3\} &\equiv & \{\{ \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \},\{ \alpha_3=\alpha_1 ,\beta_3=\frac{\pi}{4} \}\}, \nonumber\\
\{\mathcal{A}_4,\mathcal{B}_4\} &\equiv & \{\{ \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},- \frac{1}{\sqrt{3}} \},\{ \alpha_4=\pi + \arctan\left[-\frac{\gamma_1}{\delta_4}\right],\beta_4=-\frac{\pi}{4} \}\}, \nonumber\\
\{\mathcal{A}_5,\mathcal{B}_5\} &\equiv & \{\{ 0,\frac{a}{b}, ab \},\{ \alpha_5=\arctan\left[-\frac{\gamma_5}{\delta_5}\right], \beta_5=\frac{\pi}{2} \}\}, \nonumber\\
\{\mathcal{A}_6,\mathcal{B}_6\} &\equiv & \{\{ 0,-\frac{a}{b}, ab \},\{ \alpha_6=\alpha_5, \beta_6=\frac{3\pi}{2} \}\}, \nonumber\\
\{\mathcal{A}_7,\mathcal{B}_7\} &\equiv & \{\{ \frac{a}{b}, ab,0 \},\{ \alpha_7=\frac{\pi}{2},\beta_7=\arctan\left[-\frac{3+\sqrt{5}}{2}\right] \}\}, \nonumber\\
\{\mathcal{A}_8,\mathcal{B}_8\} &\equiv & \{\{ -\frac{a}{b}, ab,0 \},\{ \alpha_8=\frac{\pi}{2},\beta_8=\pi+\arctan\left[\frac{3+\sqrt{5}}{2}\right] \}\}, \nonumber\\
\{\mathcal{A}_9,\mathcal{B}_9\} &\equiv & \{\{ ab,0,\frac{a}{b} \},\{ \alpha_9=\arctan\left[-\frac{\gamma_9}{\delta_4}\right],\beta_9=0 \}\}, \nonumber\\
\{\mathcal{A}_{10},\mathcal{B}_{10}\} &\equiv & \{\{ ab,0,-\frac{a}{b} \},\{ \alpha_{10}=\pi + \arctan\left[\frac{\gamma_9}{\delta_4}\right], \beta_{10}=0 \}\}, \nonumber\\
\{\mathcal{A}_{11},\mathcal{B}_{11}\} &\equiv & \{\{ 0, \frac{c}{d}, -\frac{1}{d} \},\{ \alpha_{11}=\pi+\arctan\left[\frac{\gamma_{11}}{\delta_{4}}\right], \beta_{11}=\frac{3\pi}{2} \}\}, \nonumber\\
\{\mathcal{A}_{12},\mathcal{B}_{12}\} &\equiv & \{\{ 0, \frac{c}{d}, \frac{1}{d} \},\{ \alpha_{12}=\arctan\left[\frac{\gamma_{11}}{\delta_{4}}\right], \beta_{12}=\frac{3\pi}{2} \}\}, \nonumber\\
\{\mathcal{A}_{13},\mathcal{B}_{13}\} &\equiv & \{\{ \frac{c}{d}, \frac{1}{d}, 0 \},\{ \alpha_{13}= \frac{\pi}{2}, \beta_{13}=\arctan\left[-\frac{2}{1+\sqrt{5}}\right] \}\}, \nonumber\\
\{\mathcal{A}_{14},\mathcal{B}_{14}\} &\equiv & \{\{ -\frac{c}{d}, \frac{1}{d}, 0 \},\{ \alpha_{14}= \frac{\pi}{2}, \beta_{14}=\pi + \arctan\left[\frac{2}{1+\sqrt{5}}\right] \}\}, \nonumber\\
\{\mathcal{A}_{15},\mathcal{B}_{15}\} &\equiv & \{\{ \frac{1}{d}, 0, \frac{c}{d} \},\{ \alpha_{15}=\arctan\left[-\frac{\gamma_{5}}{\delta_{15}}\right], \beta_{15}=0 \}\}, \nonumber\\
\{\mathcal{A}_{16},\mathcal{B}_{16}\} &\equiv & \{\{ - \frac{1}{d}, 0, \frac{c}{d} \},\{ \alpha_{16}=\alpha_{15}, \beta_{16}=\pi \}\},
\end{eqnarray}
where
\begin{eqnarray}
\gamma_1 & =& \sqrt{2} (1-D) \sin2\theta, ~~~~ \delta_1= 4 D (1-D) \sin^2\theta - 1, \nonumber\\
\delta_4 &=& \cos^2\theta + (1-2D)^2 \sin^2\theta, \nonumber\\
\gamma_5 & =& 2 (1-D)\sin2\theta, ~~~~\delta_5=(3+\sqrt{5}) (2 D -1-2 D^2 - 2 D (1-D)\cos2\theta ),\nonumber\\
\gamma_9 & =& - (3+\sqrt{5}) (1-D) \sin\theta \cos\theta, \nonumber\\
\gamma_{11} & =& - (1+\sqrt{5}) (1-D) \sin\theta\cos\theta, \nonumber \\
\delta_{15} &=& (1+\sqrt{5}) (2 D -1-2 D^2 - 2 D (1-D)\cos2\theta ), \nonumber \\
a & =& c = \frac{1+\sqrt{5}}{2}, ~~~ b=\frac{1}{\sqrt{3}},~~~~ d=\sqrt{\frac{1}{2}\left(5+\sqrt{5}\right)}. \nonumber \\
\end{eqnarray}
\end{document}
|
\begin{document}
\title{Fundamental Quantum Effects from a Quantum-Optics Perspective}
\author{Ralf Sch\"utzhold}
\affiliation{
Fakult\"at f\"ur Physik, Universit\"at Duisburg-Essen,\\
D-47048 Duisburg, Germany\\
E-mail: [email protected]}
\begin{abstract}
This article provides a brief overview of some fundamental effects of
quantum fields under extreme conditions.
For the Schwinger mechanism, Hawking radiation, and the Unruh effect,
analogies to quantum optics are discussed, which might help to approach
to these phenomena from an experimental point of view.
\end{abstract}
\keywords{quantum fields; quantum optics; Schwinger, Hawking, Unruh effect}
\maketitle
\section{Introduction}\label{intro}
In quantum field theory, the vacuum state id not just empty space,
but a complicated state filled with quantum fluctuations.
If we apply some extreme conditions such as a strong electric field,
these vacuum fluctuations may manifest themselves in the creation
of real particle pairs -- which is a pure quantum effect.
There are several examples for such phenomena --
in the following, we shall focus on three cases:
\begin{tabular}{|c|c|c|}
\hline
Fundamental effect & Extreme condition & Experimental approach? \\
\hline
Schwinger mechanism & electric field & ultra-strong laser field \\
Hawking radiation & gravitational field & black hole analogues \\
Unruh effect & acceleration & electrons in laser field \\
\hline
\end{tabular}
In all of these examples, it will turn out to be interesting to apply
ideas from quantum optics.
Since none of these effects has been (directly) observed yet,
it is also very interesting and desirable to find an experimental
approach.
\section{Schwinger mechanism}\label{Schwinger mechanism}
\subsection{Dirac sea}
Before discussing the Schwinger mechanism, let us briefly review the
concept of the Dirac sea:
As is well known, identifying $i\hbar\partial_t$ with the energy
$\cal E$ and $-\hbar^2\mbox{\boldmath$\nabla$}^2$ with the momentum squared $p^2$, the
Schr\"odinger equation yields the non-relativistic energy momentum
relation
\begin{eqnarray}
i\hbar\frac{\partial}{\partial t}\psi
=
-\frac{\hbar^2}{2m}\,\mbox{\boldmath$\nabla$}^2\psi+V\psi
\quad
\leadsto
\quad
{\cal E}=\frac{p^2}{2m}+V
\,.
\end{eqnarray}
The correct relativistic description of electrons, for example,
is given by the Dirac equation\cite{Dirac}
\begin{eqnarray}
\gamma^\mu
\left(
i\hbar\partial_\mu+qA_\mu\right)\Psi
=mc\Psi
\;
\leadsto
\;
{\cal E}=V\pm\sqrt{c^2p^2+m^2c^4}
\,,
\end{eqnarray}
which reproduces the relativistic energy momentum relation.
However, this relation -- and also the Dirac equation --
always has positive and negative energy solutions, as
indicated by the $\pm$ sign in front of the square root above.
In order to facilitate stable electron solutions at positive
energies, the negative energy levels are completely filled in
the vacuum state according to the Dirac sea picture, see
Fig.~\ref{dirac-schwinger}a.
In this way, the Pauli principle prevents the decay of electrons
into the negative energy levels.
Holes in the Dirac sea then correspond to positrons, whose
existence was in this way foreseen by Dirac even before their
experimental discovery.
\subsection{Tunneling}
Now let us see what happens to the Dirac sea in the presence of a
constant electric field $\f{E}=E\f{e}_x$.
In this case, we get an additional potential $V(x)=qEx$, which
tilts the level spectrum, see Fig.~\ref{dirac-schwinger}b.
As a result, it becomes possible that an electron in the Dirac sea
has the same energy as an empty positive energy level on the
other side of the gap.
Classically, the gap (between $-mc^2$ and $+mc^2$) is forbidden by
energy arguments -- but is quantum theory, particles may tunnel
through this region.
\begin{figure}
\caption{Sketch of the Dirac sea in vacuum (left) and with a constant
electric field (right) where an electron can tunnel from the Dirac sea
to the positive energy levels (Schwinger mechanism).}
\label{dirac-schwinger}
\end{figure}
Such a tunneling process corresponds to the creation of an
$e^+e^-$ pair out of the vacuum due to the electric field.
Let us estimate the probability for such an event.
In quantum mechanics, tunneling is exponentially suppressed.
The exponent can be estimated by the length $L$ of the tunneling
barrier times the corresponding potential difference $\Delta V$
(in the relativistic case).
For a constant electric field $E$, the length can be calculated
easily by energy conservation $qEL=2mc^2$.
The height of the potential barrier, however, is not constant
and thus more complicated -- but we may get a rough estimate by
setting $\Delta V={\cal O}(mc^2)$.
In this way, we get the following rough estimate for the
$e^+e^-$ pair creation probability
\begin{eqnarray}
P_{e^+e^-}
\propto
\exp\left\{-\frac{L\Delta V}{\hbar c}\right\}
\propto
\exp\left\{-\frac{2mc^2}{qE}\,\frac{{\cal O}(mc^2)}{\hbar c}\right\}
\,.
\end{eqnarray}
This result is already rather close to the truth -- an
exact calculation yields the tunneling
exponent\cite{Sauter,Heisenberg+Euler,Schwinger}
\begin{eqnarray}
\label{exponent}
P_{e^+e^-}
\propto
\exp\left\{-\pi\,\frac{c^3}{\hbar}\,\frac{m^2}{qE}\right\}
=
\exp\left\{-\pi\,\frac{E_S}{E}\right\}
\,,
\end{eqnarray}
where we have introduced the Schwinger critical field strength
\begin{eqnarray}
E_S=\frac{c^3}{\hbar}\,\frac{m^2}{q}\approx1.3\times10^{18}\,{\rm V/m}
\,.
\end{eqnarray}
Apart from $e^+e^-$ pair creation, this field strength does also set
the scale where the QED vacuum starts to behave as a non-trivial
medium and shows effects such as birefringence etc.
This critical field strength corresponds to an intensity of
$I_S={\cal O}(10^{29}\rm W/cm^2)$.
Such intensities are currently beyond our experimental
capabilities.
Planned ultra-strong lasers\cite{ELI}
have an envisioned peak intensity of $I={\cal O}(10^{26}\rm W/cm^2)$.
Inserting the maximum electric field achievable with these lasers,
we obtain an exponential suppression of the
$e^+e^-$ pair creation probability of
$\exp\{-\pi E_S/E\}={\cal O}(10^{-61})$.
Unfortunately, this number is too small to be detectable.
Therefore, we are led to the question of whether one might
enhance this probability.
\subsection{Assisted Tunneling}
As we have discussed in the previous Section, the laser intensities
envisioned in the near future are probably not large enough to observe
the Schwinger mechanism directly.
Therefore, we are led to the question of whether (and how) one could
enhance this effect.
To achieve this goal, we borrow an idea known from quantum optics --
assisted tunneling.
Schwinger pair creation is similar to the ionization of an H-atom,
for example, by a constant electric field.
This well-know process can be enhanced by sending additional
electromagnetic waves to the atom.
Applying the same idea to the Schwinger mechanism, we studied the
$e^+ e^-$ pair creation is a constant electric field which is
superimposed by a plane wave x-ray beam.
As one would expect from the analogy to quantum optics,
the x-ray beam can lead to an enhanced pair creation probability
by reducing the tunneling exponent\cite{Dunne}.
\begin{figure}
\caption{Sketch of the Schwinger mechanism with a constant electric
field (left) and an additional x-ray beam of frequency $\omega$ (right).
The x-ray helps the electron to penetrate the classically forbidden
region (mass gap $2m$) such that the remaining way to tunnel is shorter.}
\label{schwinger-assist}
\end{figure}
As an intuitive picture, one can imagine that the x-ray provides
some additional energy and so helps the electron to penetrate the
gap a bit -- such that the remaining way to tunnel is reduced,
see Fig.~\ref{schwinger-assist}.
The maximum enhancement we could achieve -- while still preserving
the non-perturbative nature of the process -- is given by\cite{catalysis}
\begin{eqnarray}
P_{e^+e^-}
\propto
\exp\left\{-\pi\,\frac{E_S}{E}\right\}
\to
\exp\left\{-(\pi-2)\frac{E_S}{E}\right\}
\,.
\end{eqnarray}
Inserting the same values as in the previous Section, we find that
the exponential suppression is no longer $10^{-61}$
(as in the case without x-rays) but now $10^{-22}$.
This drastic enhancement might facilitate an experimental realization.
\subsection{So What?}
In view of the experimental difficulties mentioned above, one could ask the
question of why we should try to observe the Schwinger mechanism.
After all, $e^+ e^-$ pair creation by colliding gamma rays has be observed
already.
The main reason for our interest in the Schwinger mechanism lies in its
purely non-perturbative nature.
The majority of QED effects can be understood from Feynman diagrams
which are based on perturbation theory.
In this approach, the S-matrix, for example, is Taylor expanded in powers
of the coupling (i.e., charge $q$)
\begin{eqnarray}
\bra{{\rm out}}\hat S\ket{{\rm in}}=
a_0+a_1q+a_2q^2+\dots
\end{eqnarray}
In contrast, the Schwinger mechanism is a purely non-perturbative
QED vacuum effect.
This can already be seen from Eq.~(\ref{exponent})
\begin{eqnarray}
P_{e^+e^-}
\propto
\exp\left\{-\pi\,\frac{c^3}{\hbar}\,\frac{m^2}{qE}\right\}
=
\exp\left\{-\pi\,\frac{E_S}{E}\right\}
\,,
\end{eqnarray}
which does not admit a Taylor expansion in $q$ (nor $E$), i.e., we have
an essential singularity at $q=0$.
Therefore, no Feynman diagram can ever describe the Schwinger effect.
In quantum chromo-dynamics (QCD), such non-perturbative effects are
very important for describing experimental data, but direct experimental
tests are complicated by additional properties (e.g., confinement).
Thus, quantum electrodynamics offers the possibility of performing
more controlled experiments -- but non-perturbative QED vacuum effects
such as the Schwinger mechanism have not been observed yet.
\section{Hawking Radiation}\label{Hawking Radiation}
\subsection{Black Hole Evaporation}
In close analogy to the strong electric field which rips apart vacuum
fluctuations and so produces $e^+ e^-$ pairs via the Schwinger mechanism,
a strong gravitational field can also rip apart vacuum fluctuations
leading to the creation of particle pairs.
A prominent example is Hawking radiation as a result of the distortion
of quantum vacuum fluctuations due to event horizon of a black hole.
At the horizon, all wave-packets are ripped apart in the course of
time -- the part inside is trapped and finally hits the singularity
whereas the part of the wave packet outside the horizon can escape
to infinity.
Assuming the the wave packet was initially in its ground state (vacuum),
this process of ripping it apart is a drastic departure from equilibrium
and thus results in an excitation.
It turns out that the so generated occupation of the outgoing part
of the wave packet is thermal with the Hawking temperature\cite{Hawking}
\begin{eqnarray}
T_{\rm Hawking}
=
\frac{1}{8\pi M}
\frac{\hbar\,c^3}{G_{\rm N}k_{\rm B}}
\,.
\end{eqnarray}
Even though the above formula for the Hawking temperature is quite
simple, it combines four (apparently) different areas of physics via
the occurring natural constants:
quantum theory ($\hbar$), relativity ($c$), gravity ($G_{\rm N}$),
and thermodynamics ($k_{\rm B}$).
It almost seems as if nature is trying to give us a hint.
Indeed, understanding the origin of the unexpected thermal nature
of black holes (black hole entropy etc.) will probably be a big step
towards finding the correct laws of nature that unify gravity and
quantum theory.
Unfortunately, for typical astronomical black hole swith a mass of,
say, 30 solar masses, the Hawking temperature is around 2~nK and thus
probably not observable.
\subsection{Black Hole Analogues}
The contrast between the fundamental importance of Hawking radiation
and the difficulty of observing it motivates alternative approaches.
Interestingly, it turns out that phonons propagating in fluids satisfy
the same equation of motion as quantum fields in curved
space-times\cite{Unruh-prl}.
Then, remembering the famous quote of R.~Feynman
{\em ``The same equations have the same solutions.''}, we conclude that
it should (in principle) be possible to achieve the analogue of
Hawking radiation in the laboratory.
Indeed, a de~Laval Nozzle is analogous to a black hole:
At the entrance of the nozzle, the flow is sub-sonic and thus
phonons can propagate in all directions.
This region is analogous to the exterior of the black hole.
When flowing through the nozzle, the fluid speeds up and exits it
with super-sonic flow velocity.
In this region, all phonons are dragged away by the flow and thus
cannot propagate upstream anymore.
Therefore, this is the analogue of the black hole interior where
everything is trapped.
The border between these two regions lies at the most narrow point
of the nozzle, where the flow velocity exceeds the speed of sound.
Consequently, this border is analogous to the event horizon.
Repeating Hawking's derivation of black hole evaporation for this
system, we find that the nozzle emits thermal phonons with the
temperature\cite{Unruh-prl}
\begin{eqnarray}
\label{analogue-Hawking}
T_{\rm Hawking}
=
\frac{\hbar}{2\pi\,k_{\rm B}}\,
\left|\frac{\partial}{\partial r}\left(v_0-c_{\rm s}\right)\right|
\,,
\end{eqnarray}
which is determined by the velocity gradient at the horizon.
Depending on the experimental realization, this temperature could
range from some nano-Kelvin (for Bose-Einstein condensates) up to
fractions of a Kelvin.
It should be mentioned here that the analogy to gravity is not
restricted to phonons, but applies under certain conditions to
other quasi-particles in condensed matter\cite{LNP}.
The described analogy cab be used in basically two ways:
First, as a toy model for the underlying theory
(including quantum gravity).
For example, one can study the dependence of Hawking radiation
on modifications of the dispersion relation at large energies\cite{origin}.
Second, it would be very nice to do experiments and actually measure
(the analogue of) Hawking radiation in the laboratory.
\subsection{Detectability?}
As one may infer from Eq.~(\ref{analogue-Hawking}), there are basically
two ways to make facilitate the detection of (the analogue of) Hawking
radiation in the laboratory.
First, one could try to achieve large velocities (and thus large gradients)
to increase the Hawking temperature in Eq.~(\ref{analogue-Hawking}).
This route is taken in optical or electromagnetic (wave-guide) analogues
of black holes\cite{wave-guide,Philbin}.
The second route is to achieve high measurement accuracy in order to
be able to detect low temperatures.
This is particularly necessary for phonons in Bose-Einstein
condensates\cite{Garay} with a sound speed of order mm/s.
Let us focus on the second case and discuss how to measure a few phonons
with small energies $k_{\rm B}T={\cal O}(10^{-13}\,{\rm eV})$ and a broad
(thermal) spectrum.
One idea is to use doubly detuned optical Raman transitions\cite{Raizen}.
To this end, we assume that the atoms forming the Bose-Einstein condensate
possess an internal electronic structure consisting of two meta-stable
(ground) states plus some excited state(s).
Now, with two laser beams, we can drive Raman transitions between the
two meta-stable states.
However, if we detune one of the laser frequencies a little bit, these
transitions are no longer possible due to energy conservation --
if the Bose-Einstein condensate is in its ground state.
But in the presence of phonon excitations, the transition can be possible
if the missing energy (detuning) of, say $\delta={\cal O}(10^{-13}\,{\rm eV})$
is compensated by the simultaneous absorption of a phonon with this
(or a higher) energy.
In this way, single phonons are transformed into single atoms in the
second meta-stable state.
Fortunately, separating atoms in different electronic states is possible
(optical tweezers) and counting single atoms is possible with present-day
technology.
In summary, observing (the analogue of) Hawking radiation in the
laboratory. is certainly an experimental challenge,
but not completely impossible.
\subsection{Hints for Quantum Gravity?}
As mentioned before, the Hawking effect is very important from a fundamental
point of view.
Let us discuss this aspect in some more detail.
If we start from the formula for the Hawking temperature
\begin{eqnarray}
T_{\rm Hawking}
=
\frac{1}{8\pi M}
\frac{\hbar\,c^3}{G_{\rm N}k_{\rm B}}
\,,
\end{eqnarray}
and try to construct an analogue to the first law of thermodynamics
(which is basically just energy conservation)
\begin{eqnarray}
dE=dMc^2=T_{\rm Hawking}\,dS_{\rm BH}+\dots
\,,
\end{eqnarray}
we obtain the black hole (Bekenstein) entropy\cite{thermo}
\begin{eqnarray}
S_{\rm BH}
=
\frac{A}{4}\,\frac{k_{\rm B}c^3}{G_{\rm N}\hbar}
=
\frac{A}{4}\,\frac{k_{\rm B}}{\ell_{\rm Planck}^2}
\,.
\end{eqnarray}
Apart from Boltzmann's constant $k_{\rm B}$, the entropy is given by
a quarter of the horizon area $A$ in units of the Planck length squared.
This length scale obtained from the natural constants $c$, $G_{\rm N}$,
and $\hbar$ is believed to be a characteristic scale for quantum gravity.
That's why many scientists believe that truly understanding the Hawking
effect will help us to solve puzzles of quantum gravity such as the
black hole information ``paradox''.
\section{Unruh Effect}\label{Unruh Effect}
\subsection{Principle of Equivalence}
As our final example, let us discuss the Unruh effect.
Interestingly, the are deep connections between Hawking radiation and
the Unruh effect:
Similar to Einstein's gedanken experiments with elevators, we may
compare a stationary observer near a black hole and a freely falling
observer on the one hand with an accelerated observer and an
inertial observer in flat space-time on the other hand.
The stationary observer at a fixed distance to the black hole
feels the gravitational attraction and is -- via the equivalence
principle -- locally equivalent to an accelerated observer in
flat space-time.
Similarly, the freely falling observer is locally equivalent to
an inertial observer in flat space-time.
As one would expect from this analogy, the former two observers see a
thermal spectrum whereas the latter two do not observe particles.
In flat space-time, this is the essence of the Unruh effect\cite{Unruh-prd}.
\subsection{Accelerated Observer}
The Unruh effect states that a uniformly accelerated detector
(in flat space-time) experiences the inertial (Minkowski) vacuum state
as a thermal bath with the Unruh temperature\cite{Unruh-prd}
\begin{eqnarray}
T_{\rm Unruh}
=
\frac{\hbar}{2\pi k_{\rm B}c}\,a
=
\frac{\hbar c}{2\pi k_{\rm B}}
\,\frac{1}{d_{\rm horizon}}
\,,
\end{eqnarray}
where $a$ is the acceleration.
Such a uniformly accelerated observer cannot see (nor send signals to)
the full space-time -- it is causally disconnected with a whole region
(the Rindler wedge).
The minimum spatial distance between the observer and this hidden region
is given by the horizon length $d_{\rm horizon}=c^2/a$.
For everyday accelerations such as $a=9.81\,{\rm m/s^2}$, the Unruh
temperature is very small $T_{\rm Unruh}\approx 4\times10^{-20}\,{\rm K}$
and thus probably not observable.
However, electron is a strong laser field, for example, experience
much larger accelerations -- which might lead to observable
signatures\cite{Chen+Tajima}.
\subsection{Accelerated Electrons}
Now, let us consider an electron accelerated by a laser field.
For simplicity, we assume the acceleration to be constant.
Of course, this is not quite correct, but the qualitative picture
remains the same.
An observer co-moving with the accelerated electron would see a
thermal bath of photons.
Since the electron possess a finite cross section due to Thomson
(or Compton) scattering, the observer would conclude that the electron
scatters a photon out of the thermal bath into another mode with a
finite probability (per unit time).
Such a scattering event in the accelerated frame -- after translation
back into inertial (laboratory) frame --
corresponds\cite{Happens,Habs}
to the emission of a real photon {\em pair}.
Thus, we get a conversion of (virtual) quantum vacuum fluctuations
into (real) particle {\em pairs} by non-inertial scattering.
This is a pure quantum effect, which {\em cannot} be explained
within classical electrodynamics.
For example, the emitted photon pairs are entangled
(in energy and polarization).
Thus, they can be understood as signatures of the Unruh Effect.
\subsection{Analogy to Quantum Optics}
For the signatures of the Unruh effect discussed above, there are also
interesting analogies to quantum optics.
Spontaneous parametric down-conversion is a very important process in
this field.
It is basically the main source for entangled photon pairs, which can be
used for many interesting applications, such as teleportation,
quantum cryptography, etc.
Spontaneous parametric down-conversion occurs if a pump beam,
e.g., a blue or UV laser, is sent into a non-linear Kerr medium.
Then a small fraction of the photons of the pump beam are converted
into pairs of red or IR photons (called signal and idler) whose
energies sum up to the energy of the original pump photon,
see Fig.~\ref{down-conversion}.
\begin{figure}
\caption{Sketch of spontaneous parametric down-conversion.}
\label{down-conversion}
\end{figure}
The signatures of the Unruh effect discussed above are very similar
to the process of spontaneous parametric down-conversion:
The laser beam accelerating the electrons
(as discussed in the previous Section) corresponds to the pump beam
and the electrons represent the non-linear medium.
The created pairs of red or IR photons (called signal and idler) are
completely analogous to the photon pairs representing the
signatures of the Unruh effect.
In both cases, we have a spontaneous process driven by quantum
fluctuations and the created photon pairs are entangled.
However, the photon pairs representing the signatures of the Unruh
effect can have much higher energies (e.g., in the keV or even MeV range)
which might enable us to do quantum optics type experiments with
entangled photon pairs in the keV-MeV regime\cite{Habs}.
\section{Conclusions and Outlook}
By means of three examples -- the Schwinger mechanism, Hawking radiation,
and the Unruh effect -- we studied fundamental quantum effects and
established interesting analogies to quantum optics.
Even though the three physical scenarios are quite different, there are
strong similarities:
Under extreme conditions, the quantum vacuum fluctuations can manisfest
themselves as real particles.
These particles are always created in pairs
(for bosons, we have a squeezed state).
All three phenomena are relativistic ($c$) quantum ($\hbar$) field effects.
The analogy to laboratory physics (e.g., quantum optics) might help us
to develop an experimental approach to these fundamental effects.
\section*{Acknowledgments}
The author acknowledges support by the German Research Foundation (DFG)
under grant \# SCHU~1557/1 (Emmy-Noether Programme) and SFB-TR12.
\end{document}
|
\begin{equation}gin{document}
\title{Advantages of multi-copy nonlocality distillation and its application to minimizing communication complexity}
\author{Giorgos Eftaxias}
\email{[email protected]}
\affiliation{Quantum Engineering Centre for Doctoral Training,
University of Bristol, Bristol BS8 1FD, United Kingdom}
\affiliation{Department of Mathematics, University of York, York YO10 5DD, UK}
\author{Mirjam Weilenmann}
\email{[email protected]}
\affiliation{Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria}
\author{Roger Colbeck}
\email{[email protected]}
\affiliation{Department of Mathematics, University of York, York YO10 5DD, UK}
\date{$7^{\text{th}}$ March 2023}
\begin{equation}gin{abstract}
Nonlocal correlations are a central feature of quantum theory, and understanding why quantum theory has a limited amount of nonlocality is a fundamental problem. Since nonlocality also has technological applications, e.g., for device-independent cryptography, it is useful to understand it as a resource and, in particular, whether and how different types of nonlocality can be interconverted. Here we focus on nonlocality distillation which involves using several copies of a nonlocal resource to generate one with more nonlocality. We introduce several distillation schemes which distil an extended part of the set of nonlocal correlations including quantum correlations. Our schemes are based on a natural set of operational procedures known as wirings that can be applied regardless of the underlying theory. Some are sequential algorithms that repeatedly use a two-copy protocol, while others are genuine three-copy distillation protocols. In some regions we prove that genuine three-copy protocols are strictly better than two-copy protocols. By applying our new protocols we also increase the region in which nonlocal correlations are known to give rise to trivial communication complexity. This brings us closer to an understanding of the sets of nonlocal correlations that can be recovered from information-theoretic principles, which, in turn, enhances our understanding of what is special about quantum theory.
\end{abstract}
\maketitle
\ifarxiv\section*{Introduction}\else\noindent{\it Introduction.|}\fi A bound on the strength of correlations realisable between pairs of measurement inputs and outputs in any local theory was first shown by Bell~\cite{Bell,bell1975theory}. This bound is exceeded in quantum theory and there are even stronger correlations theoretically possible without enabling signalling~\cite{tsirelson,PR}. One way to better understand quantum theory is to consider it in light of possible alternative theories, which can be compared in terms of the correlations they can create, and the implications access to such correlations would have. For instance, it is known that theories that permit strong enough correlations have trivial communication complexity~\cite{Brassard}. Furthermore, non-local correlations have found applications in cryptography, where they form a necessary resource for device-independent quantum key distribution~\cite{Ekert91, Mayers1998,BHK,PABGMS} and randomness expansion~\cite{RogerThesis, Pironio2010, Colbeck2011}, for example. Since non-local correlations serve as resources for information processing, it is natural to ask about their interconvertability. In this work we look at non-locality distillation~\cite{Forster2009}, i.e., whether with access to several copies of some non-local resource we can generate stronger ones, which would have implications for the study of device-independent tasks in noisy regimes, for instance.
Non-locality distillation is often analysed in terms of \emph{wirings}~\cite{Forster2009,Brunner,Allcock,Hoyer,Wu2010,Chen2012,Cao2015}, which means interacting with systems by choosing inputs and receiving and processing outcomes from those systems. This has the advantage that, firstly, the distillation procedures apply to non-local quantum correlations no matter how complicated the system these have been obtained from and, secondly, these procedures are applicable beyond quantum theory.
A general theory will prescribe various different ways to measure systems (in quantum theory, for instance, a measurement is described by a POVM). Wirings form an operationally natural sub-class that can be performed in any generalized probabilistic theory (GPT)~\cite{barrett} (including quantum theory).
Previous work on non-locality distillation has focused on specific protocols for the distillation of 2 copies of a non-local resource (see e.g.,~\cite{Forster2009,Brunner,Allcock,Wu2010,Cao2015}). The case of more copies remains largely open, with only few specific results~\cite{Hoyer,Chen2012}. In part, this is because analysing non-locality distillation is challenging: distillation protocols act non-linearly on the correlations and hence cannot be easily optimised. Furthermore, applying a successful 2-copy protocol twice often decreases the non-locality again (see e.g.~\cite{Brunner} for an exception). Hence, understanding 2-copy protocols provides little insight into the $n$-copy case.
In this Letter we describe a sequential adaptive algorithm that uses wirings to distil non-locality. We use this algorithm to explore the distillable region within the set of non-local correlations, and the amount of distillation possible. We demonstrate new wirings that allow distillation of correlations that cannot be distilled with any 2-copy wiring protocol.
Our results have implications for communication complexity. In this problem, Alice with input $x$ and Bob with input $y$ want to enable Alice to compute $f(x,y):\{0,1\}^k\times\{0,1\}^m\to\{0,1\}$. We ask how much communication from Bob to Alice is required to do so. Communication complexity is said to be trivial if any such function (no matter how large $k$ and $m$) can be computed using only one bit of communication. Shared maximally non-local resources are known to make communication complexity trivial in this sense~\cite{vD}. A probabilistic notion of trivial communication complexity was introduced in~\cite{Brassard} in which for any $f$ we require the existence of $p>1/2$ such that Alice can obtain the correct value of $f(x,y)$ with probability at least $p$ for all $x$ and $y$. In this paper, when we talk about trivial communication complexity we mean it in this probabilistic sense. A larger set of shared states that render communication complexity trivial were found in Refs.~\cite{Brassard,Brunner}. Our results further enlarge this set, demonstrating advantages of wirings beyond two copies.
\ifarxiv\section*{Non-locality and wirings}\else\noindent{\it Non-locality and wirings.|}\fi Correlations of inputs $x,y$ and outputs $a,b$ are described by conditional probability distributions $P(ab|xy)$, and we refer to these as a \emph{box} or a \emph{behaviour}. In the context of non-locality, we usually imagine these correlations as generated by two parties, Alice and Bob, who each choose an input ($x$ and $y$ respectively) and obtain an output ($a$ and $b$ respectively). The correlations they can generate according to any theory that is consistent with special relativity have to be \emph{non-signalling}, meaning
$$
\sum_{b} P(ab|xy)=\sum_{b} P(ab|xy') \quad \forall \ a,x,y,y',
$$
and the same holds with the roles of Alice and Bob (i.e., $a,x$ and $b,y$) exchanged. A box is called \emph{local} if it can be written
$$
P(ab|xy)=\sum_{\lambda} P(a|x\lambda)P(b|y\lambda) P(\lambda) \quad \forall \ a,b,x,y\,.
$$
In the language of Bell inequalities, there is a variable ${\cal L}ambda$ that takes the value $\lambda$ with probability $P(\lambda)$. Boxes that cannot be written in this form are \emph{non-local}.
In the case of two binary inputs and outputs, i.e., $a,b,x,y \in \{0,1 \}$, the set of all local boxes is the convex hull of $16$ local deterministic boxes $P^{{\text{r}}m L}_i(ab|xy)=\delta_{a, \mu x \oplus \nu}\, \delta_{b, \sigma y \oplus \tau}$ for $\mu, \nu,\sigma,\tau \in \{0,1\}$, $i=1+\tau+2\sigma+4\nu+8\mu$,
and the set of all non-signalling boxes is the convex hull of these local boxes and $8$ extremal non-local boxes~\cite{Cirelson93,PR} $P^{{\text{r}}m NL}_i(ab|xy)=\frac{1}{2}\delta_{a \oplus b, x y \oplus \mu x \oplus \nu y \oplus \sigma}$ for $\mu, \nu, \sigma \in \{0,1\}$, $i=1+\sigma+2\nu+4\mu$. Up to symmetry, the Clauser-Horne-Shimony-Holt (CHSH) inequality~\cite{CHSH} is the only one that restricts the set of local boxes. Non-locality can hence be quantified in terms of the CHSH value $\operatorname{CHSH}(P(ab|xy))=E_{00} + E_{01} + E_{10} - E_{11}$, with $E_{xy}=P(a=b|xy)-P(a \neq b|xy)$.
Because we work in a black-box picture, the most general operation we consider for each party is a wiring. We describe here the deterministic wirings; the most general wirings are convex combinations of these. Consider a party with access to $n$-boxes with inputs $x_j$ and outputs $a_j$ with $j=1,\ldots,n$. They ``wire'' these together to form a new box with input $x$ and output $a$. The most general deterministic wiring comprises choosing a box to make the first input to and then making a chosen input, then using the output of that box to choose the second box and the input to that second box and so on. We label the $i^{\text{th}}$ box chosen $j_i(x,a_{j_1},\ldots,a_{j_{i-1}})$
and its input $x_{j_i}(x,a_{j_1},\ldots,a_{j_{i-1}})$. The final outcome is chosen depending on the overall input and all previous outcomes $a(x,a_{j_1},\ldots,a_{j_n})$. Thus, if Alice and Bob each do wirings on shares of $n$ boxes, they generate a new box $P(ab|xy)$.
Our main question is then: \emph{given several copies of a non-local box, are there wirings for Alice and for Bob such that the resulting box is more non-local than the original?} In the case of two non-signalling boxes each with binary inputs and outputs, the possible wirings have been fully characterised~\cite{Short2010}. Nevertheless, even in this case, deciding whether these can result in more non-locality for a specific box is computationally intensive: there are $82$ deterministic wirings that each party can perform for each input~\cite{Short2010}, leading to a total of $82^4$ possibilities (one of the $82$ for each input of each party). To make the computation more tractable, we optimise the wirings of one party with a linear program, while iterating over $82^2$ wirings for the other (see Appendix~{\text{r}}ef{sec:app1} for more details). We use this linear programming technique to illustrate the regions in which distillation is possible for various 2-dimensional cross-sections (CSs) of the no-signalling polytope in Figure~{\text{r}}ef{fig:trianlges123_2copy}. In this work we consider three regions:
\begin{equation}gin{align}
\mathrm{CS~I:}&\ \omega P^{\mathrm{NL}}_1+\frac{\eta}{2}( P^{\mathrm{L}}_1+P^{\mathrm{L}}_6)+(1-\omega-\eta)P^{\mathrm{O}} \nonumber \\
\mathrm{CS~II:}&\ \omega P^{\mathrm{NL}}_1+\eta P^{\mathrm{L}}_1+(1-\omega-\eta)P^{\mathrm{O}} \label{CSequations}\\
\mathrm{CS~III:}&\ \omega P^{\mathrm{NL}}_1+\frac{\eta}{2} (P^{\mathrm{L}}_1+P^{\mathrm{L}}_9)+(1-\omega-\eta)P^{\mathrm{O}}\,,\nonumber
\end{align}
where $P^{\mathrm{O}}=3/4P^{\mathrm{NL}}_1+1/4P^{\mathrm{NL}}_2$ is local and $\eta,\omega\geq0$ with $\eta+\omega\leq1$.
\begin{equation}gin{figure}
\centering
\begin{equation}gin{minipage}[t]{0.48\columnwidth}
\includegraphics[width=1\textwidth]{triangle1}
\end{minipage}
\begin{equation}gin{minipage}[t]{0.05\columnwidth}
\end{minipage}
\begin{equation}gin{minipage}[t]{0.47\columnwidth}
\includegraphics[width=1\textwidth]{triangle2}
\end{minipage}
\caption{Protocols sufficient to characterise the two-copy distillability (both the distillable region and the strongest amplification) for two CSs (cf.\ \eqref{CSequations}). The optimal two-copy protocols for CS~II are the two protocols from~\cite{Allcock} (ABL$^+$1,2), while for CS~I the protocol of~\cite{Forster2009} (FWW) is optimal in some cases. The shading indicates where the corresponding protocol is optimal, with the boundary indicated by the black line (see Appendix~{\text{r}}ef{sec:app1} for details of the protocols). The dotted curve indicates the boundary of the set of correlations realisable in quantum theory (computed using the conditions in~\cite{tsirelson,Masanes}).}
\label{fig:trianlges123_2copy}
\end{figure}
We analysed the distillability within these cross sections. Among the optimal protocols we recovered several that were previously known~\cite{Brunner2011,Allcock}. The protocols of~\cite{Allcock} (called ABL$^{+}1,2$) are sufficient to characterize the two-copy distillability in CS~II (see Figure~{\text{r}}ef{fig:trianlges123_2copy}), and CS~III is two-copy non-distillable. The observation that ABL$^{+}$2 achieves no distillation in CS~I shows that optimal protocols depend on the cross-section.
The above analysis is generally not useful for analysing whether repeated distillation of a box can lead to a certain CHSH-value. Applying a wiring that works for two boxes to two copies of the generated box often does not give a further increase in non-locality
, in which case a switch of wirings is needed to distil further.
While there are boxes that cannot be distilled at all with wirings (e.g.\ isotropic boxes~\cite{Beigi}), the maximum CHSH value that can be distilled using multiple copies of a specific resource box is unknown. This means that we do not know how resourceful (multiple copies of) most non-local boxes are for information processing. For instance, shared boxes render communication complexity trivial if their initial CHSH value is greater than $\operatorname{CHSH}(P(ab|xy))= 4 \sqrt{\frac{2}{3}}$~\cite{Brassard}. The complete set of boxes that render commuication complexity trivial is unknown, although an additional region was found with the protocol of~\cite{Brunner}.
\ifarxiv\section*{Sequential algorithms for non-locality distillation and reduction of communication complexity}\else\noindent{\it Sequential algorithms for non-locality distillation and reduction of communication complexity.|}\fi While a repeated application of a successful 2-copy protocol often does not increase the non-locality further, there are various ways to combine different 2-copy wirings (see Appendix~{\text{r}}ef{sec:app2}). Here, we focus on the specific structure illustrated in Figure~{\text{r}}ef{fig:sequential_wiring_architectures}.
\begin{equation}gin{figure}[h]
\centering
\begin{equation}gin{minipage}[t]{0.9\columnwidth}
\includegraphics[width=1\textwidth]{paper_2supereconomicSFSAAimage}
\end{minipage}
\caption{A serial architecture for combining nonlocal resources (gray) in a sequential manner. The first step on the left depicts the usual two-copy distillation scheme. Each subsequent iteration uses another copy of the original box and the previously generated one. Our sequential algorithm optimises the protocol at each round. See Appendix~{\text{r}}ef{sec:app2} for details.}
\label{fig:sequential_wiring_architectures}
\end{figure}
Our serial algorithm consists in optimising the wiring to be applied in every step, which is done in terms of a hybrid procedure of iterating over wirings and linear programming (see Appendix~{\text{r}}ef{sec:app2} for a detailed description of the algorithm). Applying our serial algorithm, we are able to extend the region of non-local boxes known to trivialise communication complexity, see Figure~{\text{r}}ef{fig:trivial2copy}.
\begin{equation}gin{figure}[h]
\centering
\includegraphics[width=0.7\columnwidth]{paper_ComComplColl_Fig_TrI_fitcurve}
\caption{Region of trivial communication complexity in CS~I. The light-gray part was identified in~\cite{Brassard}. The dark-gray region includes boxes that trivialise communication complexity through (up to 4) iterations of ABL$^+$1. The red points (and everything on their right) collapse communication complexity using our serial algorithm. The black solid chord is that of Figure~{\text{r}}ef{fig:trianlges123_2copy} (left) and indicates a change in protocol for the red points -- see Appendix~{\text{r}}ef{sec:app2} for details, including analysis of the black points in the figure.}
\label{fig:trivial2copy}
\end{figure}
Our algorithm furthermore provides us with a way to systematically derive new non-locality distillation protocols for multi-copy non-locality distillation. When performing two-steps of the serial algorithm, we find the three-copy protocol below to be successful.
In the first step, a box is created from two copies of a box $P$ with inputs (outputs) labelled $x_1, y_1$ ($a_1, b_1$) and $x_2, y_2$ ($a_2, b_2$) respectively (first step in Figure~{\text{r}}ef{fig:sequential_wiring_architectures}). Then this is wired to another copy of $P$, $P(a_3 b_3|x_3 y_3)$, using the functions
\begin{equation}gin{align} \label{eq:addon1}
x_1&=x=x',\ x_2=x\oplus \bar{a}_1,\ a=a_1\oplus a_2,\ x_3=x\bar{a} \nonumber \\
y_1&=y=y',\ y_2=y b_1,\ b=b_1\oplus b_2,\ y_3=y\oplus b, \\
a'&= a\oplus a_3 ,\ b'=b\oplus b_3, \nonumber
\end{align}
where $\oplus$ is the logical {\sc xor} and $\bar{z}=z\oplus1$.
This new protocol distils in CS~II a strict superset of non-local boxes compared to the previously known 3-copy distillation protocol of~\cite{Hoyer} (in contrast to CS~I where the protocol of~\cite{Hoyer} is superior). For completeness we introduce the protocol from~\cite{Hoyer} in Appendix~{\text{r}}ef{sec:app3} and we refer to it as HR. The region in which the new protocol distils in CS~II is also shown in Appendix~{\text{r}}ef{sec:app3}.
\ifarxiv\section*{Genuine three-copy distillation protocols}\else\noindent\textit{Genuine three-copy distillation protocols.|}\fi When considering 3-copy distillation, the variety of possible protocols is vastly increased. In this case we can derive new protocols that outperform the previous ones in terms of the boxes for which they offer distillation. For this, we introduce a \emph{genuine three-copy distillation protocol}, which is one that cannot be reduced to a concatenation of 2-copy protocols, i.e., is not of the form of Figure~{\text{r}}ef{fig:sequential_wiring_architectures}. Consider the following wiring, where $\lor$ denotes the logical {\sc or} operation:
\begin{equation}gin{align}
&x_1=x_2=\bar{x},\ x_3=\bar{x}a_1\lor\bar{x}a_2,\ a=a_1a_3\lor a_2a_3\lor\bar{a}_1\bar{a}_2\bar{a}_3,\nonumber\\
&y_1=y_2=y,\ y_3=yb_1\lor yb_2\lor\bar{y}\bar{b}_1\bar{b}_2, \label{eq:3copy_protocol_2} \\
&b=\bar{y}b_1b_3\lor\bar{y}b_2b_3\lor yb_1\bar{b}_3\lor yb_2\bar{b}_3\lor\bar{y}\bar{b}_1\bar{b}_2\bar{b}_3\lor y\bar{b}_1\bar{b}_2b_3. \nonumber
\end{align}
We find larger regions of distillable boxes as compared to the two-copy case, see Figure~{\text{r}}ef{fig:3copy}.
\begin{equation}gin{figure}[h]
\centering
\begin{equation}gin{minipage}[t]{0.48\columnwidth}
\includegraphics[width=1\textwidth]{paper_updatedEWC1VSthepast}
\end{minipage}
\begin{equation}gin{minipage}[t]{0.48\columnwidth}
\includegraphics[width=1\textwidth]{paper_EWC2_trIII}
\end{minipage}
\caption{Region of distillation by means of the 3-copy wiring of \eqref{eq:3copy_protocol_2} bounded by the green lines. The blue and orange lines show the region of optimal 2-copy distillation in CS~I, as in Figure~{\text{r}}ef{fig:trianlges123_2copy} (left). The green shaded area in CS~I depicts where our protocol leads to higher CHSH values than all previously known protocols (i.e., 2-copy and 3-copy FWW, ABL$^{+}$1, HR). In CS~III no 2-copy non-locality distillation is possible and the ability to distil is unlocked only when given access to at least 3 copies of a non-local box where use of a genuine 3-copy protocol is imperative. The dotted curve indicates the boundary of the set of quantum-realizable correlations.}
\label{fig:3copy}
\end{figure}
In CS~III no 2-copy distillation is possible, while with 3 copies it is. Furthermore, the increase in the region of boxes that allow for distillation is considerably larger than that of HR (which is nearly indistinguishable from ABL$^+$1, see also Figure~{\text{r}}ef{fig:HRvsABL1vsStar} in the Appendix).
Additionally we find 3-copy protocols that increase the region where communication complexity is trivial. In particular
\begin{equation}gin{align}
&x_1=x_2=x,\ x_3=xa_2\lor x\bar{a}_1\lor\bar{x}\bar{a}_2a_1,\nonumber \\
&a=a_3a_2\lor a_3\bar{a}_1\lor\bar{a}_3\bar{a}_2a_1,\ y_1=y_2=y,\ y_3=yb_2\lor y \bar{b}_1,\nonumber \\
&b=b_3b_2\lor b_3\bar{b}_1\lor\bar{b}_3\bar{b}_2b_1.\label{eq:3copy_protocol}
\end{align}
We illustrate the use of this protocol for trivialising communication complexity in Figure~{\text{r}}ef{fig:trivialcomm_3}. In addition, we find that in CS~I, starting from any point with $\omega>0$ on the line $\omega=1-\eta$ we can distill arbitrarily close to a PR box by repeatedly iterating this protocol (see Appendix~{\text{r}}ef{sec:app4}). We observe, that as compared to using 2-copy protocols (even sequentially), 3-copy protocols provide further advantages.
Additionally, all the protocols introduced here, i.e., those of~({\text{r}}ef{eq:addon1}), ({\text{r}}ef{eq:3copy_protocol_2}) and~({\text{r}}ef{eq:3copy_protocol}) work in a full dimensional subset of the space of no-signalling correlations. This space is 8 dimensional for bipartite non-signalling boxes with binary inputs and outputs. The form of our distillation protocols (and many others in the literature) implies that the difference between the initial and final CHSH value is a polynomial in the parameters of the initial box $P(ab|xy)$ and hence continuous in these parameters. Thus, for any distillable point not on the boundary of the polytope, there exists an eight-dimensional ball around it that is also distillable.
\begin{equation}gin{figure}
\centering
\begin{equation}gin{minipage}[c]{0.45\columnwidth}
\includegraphics[width=0.9\textwidth]{paper_TRI_3copy_compl_triv_comparison2}
\end{minipage}
\begin{equation}gin{minipage}[c]{0.45\columnwidth}
\includegraphics[width=0.9\textwidth]{paper_ZOOMIN_TRI_trivilization6}
\end{minipage}
\caption{Regions of trivial communication complexity with various protocols. The green region is from repeated use of our genuine 3-copy protocol of \eqref{eq:3copy_protocol}, the blue bounded region is from repeated use of ABL$^+$1 and the dashed gray bounded region is from repeated use of HR. In the magnified view (right) we see a small region where our new 3-copy protocol outperforms HR and any possible 2-copy protocol.}
\label{fig:trivialcomm_3}
\end{figure}
\ifarxiv\section*{Conclusions}\else\noindent\textit{Conclusions.|}\fi We have found a genuine 3-copy protocol that distils nonlocality for boxes in which distillation with two copies is impossible and shown that there are 3-copy protocols that outperform \emph{all} 2-copy protocols (and sequential applications thereof).
For the latter we employed an optimization technique for 2-copy wiring protocols. Although this optimization furthers our understanding, it remains limited to cases with small numbers of inputs and outputs and there remains much more to discover about nonlocality distillation.
Whether the principle of non-trivial communication complexity~\cite{Brassard} defines a closed set of correlations~\cite{Lang} that allows for a simple characterisation and lies well between quantum and non-signalling sets is an open question of interest for the foundations of quantum theory. Indeed, finding a sensible generalised probabilistic theory that leads to a set of correlations between the non-signalling and quantum set with a simple geometric description has been a conundrum. The present work suggests that a better understanding of multi-copy non-locality distillation may give us insights into such a set, namely that of a GPT whose only restriction is imposed by the principle of non-trivial communication complexity. This would further advance the recent research program of experimentally ruling out generalised probabilistic theories due to the correlations they produce in networks~\cite{selftest1, selftest2}.
Some of our distillation protocols work within the set of quantum correlations (see Figure~{\text{r}}ef{fig:3copy}). [See also~\cite{banik22} for recent work aiming to distil quantum correlations.] Being wirings, they are much simpler to perform than entanglement distillation protocols~\cite{BBPS}. It would be interesting to explore applications of these for information processing. We also remark that in recent work we have shown that non-wiring effects can be beneficial for non-locality distillation~\cite{EWC2}.
\ifarxiv\acknowledgements\else\noindent{\it Acknowledgements|}\fi GE is supported by the EPSRC grant EP/LO15730/1. MW is supported by the Lise Meitner Fellowship of the Austrian Academy of Sciences (project number M 3109-N). Some of the preliminary work for this project was performed using the Viking Cluster, a high performance computing facility at the University of York. We are grateful for computational support from the University of York High Performance Computing service, Viking, and the Research Computing team.
\begin{equation}gin{thebibliography}{36}
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\end{thebibliography}
\appendix
\onecolumngrid
\section{Optimising over all two-copy non-locality distillation protocols} \label{sec:app1}
In order to establish whether a non-local box is amenable to 2-copy non-locality distillation, it is convenient (and due to the large number of possible protocols even necessary) to find ways to search and optimise over all such protocols. This can be achieved using Linear Programming. Specifically, while iterating over the extremal wirings of one party, we can optimise the operations of the other this way.
To see how this is possible, notice that the correlations obtained from wiring two boxes $Q_{1}(a_1b_1|x_1 y_1)$, $Q_{2}(a_2b_2|x_2 y_2)$ are
$$P(ab|xy)=\sum_{x_i,y_i,a_i,b_i} Q_{1}(a_1b_1|x_1 y_1)Q_{2}(a_2b_2|x_2 y_2) \chi_{x}(a x_1x_2|a_1 a_2) \xi_{y}(b y_1 y_2|b_1 b_2), $$
where $\chi_{x}(a x_1x_2|a_1 a_2)$ and $\xi_{y}(b y_1 y_2|b_1 b_2)$ describe Alice's and Bob's wirings upon receiving input $x$ and $y$ respectively. For a deterministic wiring, $\chi_{x}(a x_1x_2|a_1 a_2)\in\{0,1\}$ for all $a,\,a_1,\,a_2,\,x_1,\,x_2$, and the wiring $x_1=0$, $x_2=a_1$ and $a=a_1\oplus a_2$ would correspond to $\chi(a x_1x_2|a_1 a_2)=\delta_{x_1,0}\delta_{x_2,a_1}\delta_{a,a_1\oplus a_2}$, for example.
A wiring on Alice's side is made up of $|x|\cdot |a|$ vectors $\chi_x(a)=(\chi_{x}(a x_1x_2|a_1 a_2))_{a_1 a_2 x_1 x_2}$. In the case of 2-inputs and 2-outputs, these are straightforward to characterise since the wirings there are exactly the allowed measurements in a generalised probabilistic theory of non-local boxes~\cite{short2006entanglement}. Specifically, to have a valid wiring in this case, it is necessary and sufficient that the output distribution on any 2-input 2-output non-signalling box returns a valid probability distribution, i.e., for any $ Q \in \{P^{{\text{r}}m L}_i, P^{{\text{r}}m NL}_j\}_{i,j}$
\begin{equation}gin{align}
0 \leq \sum_{a_{1},a_{2}, x_{1}, x_{2}} \chi_{x}(x_{1}x_{2}a|a_{1}a_{2}) Q(a_{1}a_{2}|x_{1}x_{2}) &\leq 1 \qquad \forall x,a , \label{eq:A1}\\
\sum_{a, a_{1},a_{2}, x_{1}, x_{2}} \chi_{x}(x_{1}x_{2}a|a_{1}a_{2}) Q(a_{1}a_{2}|x_{1}x_{2}) &= 1 \qquad \forall x.\label{eq:A2}
\end{align}
These are linear constraints on the vectors $\chi_x(a)$.
Furthermore, $\operatorname{CHSH}(P(ab|xy))$ is a linear function of the $P(ab|xy)$, which in turn is linear in $\chi_x(a)$. Thus, we can optimise the distilled non-locality over Alice's wirings with a linear program. Although this procedure works well when Alice and Bob each hold halves of two 2-input 2-output systems, going beyond this case presents several challenges:
\begin{equation}gin{enumerate}
\item With more than two systems the number of wirings on Bob's side significantly increases.
\item Sticking with two systems but increasing the number of inputs and outputs for each system significantly increases the number of wirings.
\item With more than two systems it is possible that the linear program optimizing over Alice's operations outputs a vector $\chi_x$ that is not a wiring.
\end{enumerate}
The presence of such \emph{non-wirings} for three systems was first noticed in~\cite{Short2010}. In the main text we motivated the use of wirings based on maintaining validity of the results in any GPT. Allowing the non-wirings that come from such a linear program does not significantly alter the theory-independence in the sense that \eqref{eq:A1} and \eqref{eq:A2} are minimal requirements hence if no additional restrictions are placed on the theory any $\chi_x$ output by the linear program should be valid. Nevertheless it may be unnatural to allow non-wirings for Alice while restricting to wirings for Bob. Hence one would either like to add all the non-wirings valid in \emph{any} theory to the set of Bob's possibilities, or remove non-wirings from the set of possible operations of Alice.
In the case of 2 copies of a box, in order to optimise the distilled non-locality over all wirings of Alice \emph{and} Bob, we iterate over the $82^2$ extremal wirings of Bob, as found in~\cite{short2006entanglement} and displayed in Table~{\text{r}}ef{table:couplers}, while optimizing Alice's wiring for each such choice with a linear program as described above.
\begin{equation}gin{table}
\centering
\begin{equation}gin{tabular}{|p{2.4cm}|p{2.6cm}|p{5.3cm}|p{4.0cm}|}
\hline
Wiring class & Number of wirings in class & Elements $\chi(a,a_1,a_2,x_1,x_2)=1$ if the following holds: (otherwise zero) & Label of wiring for each $\mu, \nu, \sigma, \delta, \epsilon \in \{0,1\}$ \\
\hline
Constant & 2 &
$x_1=x_2 , \hspace{0.2cm} a=\mu$
& $\mu+1$ \\
\hline
One-sided & 8 &
$x_1=x_2=\mu , \hspace{0.2cm} a=a_{\nu +1} \oplus \sigma$
& $(4\mu+2\nu+\sigma+1)+2$\\
\hline
XOR-gated & 8 &
$x_1=\mu , \hspace{0.2cm} x_2=\nu ,\hspace{0.2cm} a=a_1 \oplus a_2 \oplus \sigma $
& $(4\mu+2\nu+\sigma+1)+10$ \\
\hline
AND-gated & 32 &
$x_1=\mu , \hspace{0.2cm} x_2=\nu ,$
\newline $ a=(a_1 \oplus \sigma)(a_2 \oplus \delta) \oplus \epsilon $
& $(16\mu+8\nu+4\sigma+2\delta+\epsilon+1)+18$ \\
\hline
Sequential & 32 &
$x_{\mu+1}=\nu , \hspace{0.2cm} x_{(\mu \oplus 1)+1}=a_{\mu+1} \oplus \sigma ,$
\newline $ a=a_{(\mu \oplus 1)+1} \oplus \delta a_{\mu+1} \oplus \epsilon $
& $(16\mu+8\nu+4\sigma+2\delta+\epsilon+1)+50$\\
\hline
\end{tabular}
\caption{Labelling of 2-copy wirings. To iterate over all extremal wirings for Bob, we consider all combinations of $\xi_{0}(b)$, $\xi_{1}(b)$ from the above list, i.e., $82^2$ wirings.}
\label{table:couplers}
\end{table}
Using this technique we can find whether there is a successful protocol for 2-copy non-locality distillation for any non-local box with two inputs and two outputs. In the following we illustrate this on CSs~I and~II (cf.\ \eqref{CSequations}). In both cases, the full optimisation shows that two protocols are sufficient for characterising the region of 2-copy distillation in a CS. None of the points that are not distillable with either of these protocols can be distilled with any other 2-copy wiring there. In both CSs, we can choose non-locality distillation protocols from the literature to achieve this, i.e., known protocols are among the optimal ones when considering the region of distillation. Specifically, the region of distillation of CS~I can be characterised in terms of the protocol from~\cite{Forster2009}, which we call \emph{FWW} here, as well as a protocol from~\cite{Allcock}, called \emph{ABL$^+$1} here, which are both given in the Tables~{\text{r}}ef{tab:sec1} and~{\text{r}}ef{tab:sec2}. The parameters $\omega$ and $\eta$ are chosen like in Figure~{\text{r}}ef{fig:trianlges123_2copy}. Since the boundary of this region can be established as those boxes $P$ for which 2-copy distillation leads to a box $P'$ such that $\operatorname{CHSH}(P(ab|xy))= \operatorname{CHSH}(P'(ab|xy))$, this region can be characterised analytically.
\begin{equation}gin{table}
\centering
\begin{equation}gin{tabular}{ |p{1.6cm}|p{2.3cm}| p{5cm}|p{5cm}| }
\hline
protocol name & wiring & analytic boundary of the region of distillation ($\omega$ as a function of $\eta$) & CHSH value of the distilled box \\
\hline
FWW~\cite{Forster2009} & $x_{1}=x_{2}=x$ \newline $y_{1}=y_{2}=y$ \newline $a=a_{1} \oplus a_{2} $ \newline $b=b_{1} \oplus b_{2} $ &
$\omega=1 - 3 \eta + 2 \sqrt{1 - 3 \eta + 3 \eta^2}$ \newline \newline $\eta \in [1/2,1]$
&
$\frac{1}{2}\Big[(1+\omega)^2-3\eta^2+6\eta(1+\omega)\Big]$\\
\hline
ABL$^{+}$1~\cite{Allcock} & $x_{1}=x$\newline $y_{1}=y$ \newline $x_{2}=x\oplus a_{1} \oplus 1$ \newline $y_{2}=yb_{1}$ \newline $a=a_{1} \oplus a_{2} \oplus 1$ \newline $b=b_{1} \oplus b_{2} \oplus 1$ &
$\omega=-\eta + \frac{1}{\sqrt{3}} \sqrt{3 - 4 \eta + 4 \eta^2}$ \newline \newline $\eta \in [0,1]$
&
$\frac{1}{4} \Big[3 \omega^2+8 \omega-\eta^2+\eta (4+6 \omega)+5\Big]$\\
\hline
\end{tabular}
\caption{Optimal 2-copy distillation protocols for CS~I.}
\label{tab:sec1}
\end{table}
\begin{equation}gin{table}
\centering
\begin{equation}gin{tabular}{ |p{1.6cm}|p{2.3cm}| p{5.1cm}|p{5.5cm}| }
\hline
protocol name & wiring & analytic boundary of the region of distillation ($\omega$ as a function of $\eta$) & CHSH value of the distilled box \\
\hline
ABL$^{+}$2~\cite{Allcock} & $x_{1}=x_{2}=x$ \newline $y_{1}=y_{2}=y$ \newline $a=a_{1}a_{2} $ \newline $b=b_{1}b_{2} $ &
$\omega=3 - 11 \eta + 2 \sqrt{3 - 18 \eta + 31 \eta^2}$ \newline \newline $\eta \in [1/3,1]$
&
$\frac{1}{8}\Big[\omega^2+10\omega-3\eta^2+\eta(6+22\omega)+13\Big]$\\
\hline
ABL$^{+}$1~\cite{Allcock} & $x_{1}=x$\newline $y_{1}=y$ \newline $x_{2}=x\oplus a_{1} \oplus 1$ \newline $y_{2}=yb_{1}$ \newline $a=a_{1} \oplus a_{2}\oplus 1$ \newline $b=b_{1} \oplus b_{2}\oplus 1$ &
$\omega=-\frac{4}{3}\eta + \frac{1}{3} \sqrt{9 - 18 \eta + 25 \eta^2}$ \newline \newline $\eta \in [0,1]$
&
$\frac{1}{4} \Big[3 \omega^2+8 \omega-3\eta^2+\eta (6+8 \omega)+5\Big]$\\
\hline
\end{tabular}
\caption{Optimal 2-copy distillation protocols for CS~II.}
\label{tab:sec2}
\end{table}
The two CSs are displayed in Figure~{\text{r}}ef{fig:trianlges123_2copy}. The black line where the two protocols work equally well is analytically characterised as
\begin{equation}gin{align}
\omega=5\eta-3\,,\hspace{0.5cm} \frac{1}{2}(1+\frac{1}{\sqrt{13}}) \leq \eta \leq \frac{2}{3} \label{chordI}
\end{align}
in CS~I and
\begin{equation}gin{align}
\omega = -\frac{2\sqrt{6}-3}{5}(\eta-1)\,,\hspace{0.7cm}
\frac{1}{25} (9 + \sqrt{6}) \leq \eta \leq 1 \label{chordII}
\end{align}
in CS~II.
We remark here that previously, heuristics to simplify the optimisation over two-copy protocols have been proposed. For instance, the method in~\cite{Brito} suggests to reduce the search over $82^4$ protocols to a manageable number of only $3152$, by only considering protocols that preserve the PR-box, $P^{{\text{r}}m NL}_1$. Using linear programming, as proposed here, has the advantage that it takes all distillation protocols into account. In contrast, the heuristic from~\cite{Brito} discards various protocols, e.g., FWW and ABL$^+$2, that despite not preserving $P^{{\text{r}}m NL}_1$, are useful for non-locality distillation---they are even among the optimal 2-copy distillation protocols in CS~I---so this shortcoming is pertinent.
\section{Sequential non-locality distillation into the region of trivial communication complexity}\label{sec:app2}
In some situations we would like to distil non-locality up to a certain value that is useful for a specific task, e.g.\ because a particular CHSH score is needed in a device-independent scenario, or because we want to draw conclusions about the properties of those correlations, e.g.\ that they are unnatural since they imply that communication complexity is trivial. For this purpose, 2 copies of a non-local box are usually not sufficient. Since the repeated application of a fixed protocol is generally not successful in this respect either, it is natural to combine \emph{different} protocols instead. There are various ``architectures'' that such combinations can take, two of which are displayed in Figure~{\text{r}}ef{fig:architectures2}.
\begin{equation}gin{figure}[h]
\centering
\begin{equation}gin{minipage}[t]{0.48\columnwidth}
\includegraphics[width=0.9\textwidth]{paper_linearSFSAA}
\end{minipage}
\begin{equation}gin{minipage}[t]{0.48\columnwidth}
\includegraphics[width=0.9\textwidth]{paper_exponentialSFSAA}
\end{minipage}
\caption{Two architectures for combining an arbitrary number of resource boxes (gray) in a sequential manner. In each case, the purpose of our sequential algorithm is to find new optimal wirings in each round. Thus, the serial architecture on the left represents the \textit{serial algorithm} introduced in Figure~{\text{r}}ef{fig:sequential_wiring_architectures}. Similarly, the parallel architecture on the right will represent the \textit{parallel algorithm}.}
\label{fig:architectures2}
\end{figure}
Analysing all of the wirings that are possible in such a multi-round procedure is computationally infeasible. We thus propose a sequential algorithm to (partially) optimise these procedures. This algorithm (in either version of Figure~{\text{r}}ef{fig:architectures2}, serial or parallel or some alternatives, analysed more carefully in~\cite{GiorgosThesis}) proceeds as follows:
\begin{equation}gin{enumerate}[(1)]
\item Optimise the wiring step by step using the procedure outlined in Appendix~{\text{r}}ef{sec:app1}. As figure of merit to be optimised we use the CHSH value here.
\item Stop the procedure when either a certain round number is reached or when the CHSH value does not increase any further.
\end{enumerate}
When applying the serial algorithm to the black points from Figure~{\text{r}}ef{fig:trivial2copy}, choosing the serial architecture turned out to be more effective than the parallel (in terms of distilled CHSH values). The tables below compare the findings of the serial algorithm with repeated iterations of other protocols.
\begin{equation}gin{table}
\centering
\begin{equation}gin{tabular}{ |p{0.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{2.5cm}|p{2.5cm}| }
\hline
\multicolumn{7}{|c|}{CS~I, \hspace{0.6cm} point \hspace{0.07cm} $(\eta,\omega)=(0.888, 0.1)$, \hspace{0.6cm} $\text{CHSH}_{init}=2.2$} \\
\hline
\multicolumn{5}{|c|}{$\text{CHSH}_{final}$ \hspace{0.03cm}, \hspace{0.2cm} after \# iterations} &
\multicolumn{2}{|c|}{Serial Algorithm \textbf{STRATEGIES}} \\
\hline
iter \# & two-copy ABL$^{+}$1, blindly repetitive & two-copy FWW, blindly repetitive & two-copy \newline BS, blindly repetitive & Serial Algorithm & Alice's wiring \hspace{0.2cm} $(\chi_{x=0} \hspace{0.1cm}, \hspace{0.1cm} \chi_{x=1}) $ & Bob's wiring \hspace{0.2cm} $(\chi_{y=0} \hspace{0.1cm}, \hspace{0.1cm} \chi_{y=1}) $\\
\hline
1 & 2.2815 &2.3525 & 2.2812 & 2.3525 & (12, 18)& (12, 18)\\
\hline
2& 2.3837 & 2.5546 & 2.3823 & 2.4681 & (12, 18) & (12, 18) \\
\hline
3 & \cellcolor{yellow} 2.4964 & \cellcolor{yellow} 2.7191 & \cellcolor{yellow} 2.4918 & 2.5546 & (12, 18) & (12, 18)\\
\hline
4 & 2.5885 & & 2.5749 & 2.6186 & (12, 18)& (12, 18) \\
\hline
\hline
5 & 2.5927 & & & 2.6729 & (12, 78) & (74, 78) \\
\hline
6 & & & & 2.7236 &(70, 82) & (12, 82) \\
\hline
7 & & & & \cellcolor{yellow} 2.7706 & (12, 78) & (74, 78)\\
\hline
8 & & & &2.8143 & (70, 82) & (12, 82) \\
\hline
9 & & & & ... & $\circlearrowleft$ & $\circlearrowleft$ \\
\hline
10 & & & & ... & $\circlearrowright$ & $\circlearrowright$ \\
\hline
\hline
\hline
36 & & & &3.2683 & (70, 82) & (12, 82) \\
\hline
\hline
\hline
41 & & & &3.2730 & (12, 78) & (74, 78) \\
\hline
\end{tabular}
\caption[$(\eta,\omega)_{I}=(0.888, 0.1)_I$]{Data about the lower black point of Figure~{\text{r}}ef{fig:trivial2copy}. The wirings are described using the labellings of the last column of Table~{\text{r}}ef{table:couplers}. The circular arrows denote the continued switching between the two strategies appearing on each side after the 4th iteration. The distilled CHSH values are recorded here as long as they increase. The yellow shaded entries compare final-CHSH values when each scheme has used 8 identical resource boxes. We observe that (37 copies of) the initial box trivializes communication complexity, a fact that only the serial algorithm reveals.}
\label{(0.888, 0.1)I_strategies_displayed}
\end{table}
\begin{equation}gin{figure}
\centering
\includegraphics[scale=0.5]{paper_ListPlotS}
\caption{Visualization of the data of Table {\text{r}}ef{(0.888, 0.1)I_strategies_displayed} (plus some further iterations that decrease the final-CHSH value). Here, the superiority of the serial algorithm -- as opposed to the independent repetition of a fixed protocol-- makes the initial box surpass the trivial communication complexity threshold (dashed line). The horizontal axis shows CHSH$_2=E_{00}-E_{01}+E_{10}+E_{11}$.}
\label{fig:my_label}
\end{figure}
\begin{equation}gin{table}
\centering
\begin{equation}gin{tabular}{ |p{0.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{2.5cm}|p{2.5cm}| }
\hline
\multicolumn{7}{|c|}{CS~I, \hspace{0.6cm} point \hspace{0.07cm} $(\eta,\omega)=(0.575, 0.375)$, \hspace{0.6cm} $\text{CHSH}_{init}=2.75$} \\
\hline
\multicolumn{5}{|c|}{$\text{CHSH}_{final}$ \hspace{0.03cm}, \hspace{0.2cm} after \# iterations} &
\multicolumn{2}{|c|}{Serial Algorithm \textbf{STRATEGIES}} \\
\hline
iter \# & two-copy ABL$^{+}$1, blindly repetitive & two-copy FWW, blindly repetitive & two-copy \newline BS, blindly repetitive & Serial Algorithm & Alice's wiring \hspace{0.2cm} $(\chi_{x=0} \hspace{0.1cm}, \hspace{0.1cm} \chi_{x=1}) $ & Bob's wiring \hspace{0.2cm} $(\chi_{y=0} \hspace{0.1cm}, \hspace{0.1cm} \chi_{y=1}) $\\
\hline
1 & 2.9212 &2.8212 & 2.9162 & 2.9212 & (12, 78) & (74, 78)\\
\hline
2& \cellcolor{yellow} 3.0294 & \cellcolor{yellow} & \cellcolor{yellow} 3.0096 & 3.0452 & (70, 82) & (12, 82) \\
\hline
3 & & & & \cellcolor{yellow} 3.1327 & $\circlearrowleft$ & $\circlearrowleft$\\
\hline
4 & & & & 3.1930 & $\circlearrowright$ & $\circlearrowright$ \\
\hline
5 & & & & 3.2324 & $\circlearrowleft$ & $\circlearrowleft$\\
\hline
6 & & & & 3.2562 & $\circlearrowright$ & $\circlearrowright$ \\
\hline
7 & & & & 3.2683 & (12, 78) & (74, 78)\\
\hline
8 & & & &3.2718 & (70, 82) & (12, 82) \\
\hline
\end{tabular}
\caption[$(\eta,\omega)_{I}=(0.575,0.375)_I$]{Data for the higher black point of Figure~{\text{r}}ef{fig:trivial2copy}. The wirings are described using the labellings of the last column of Table~{\text{r}}ef{table:couplers}. The circular arrows denote the continued switching between the two strategies appearing on each side after the 4th iteration. The distilled CHSH values are recorded here as long as they increase. The yellow shaded entries compare final CHSH values when each scheme has used 4 identical resource boxes. We observe that (8 copies of) the initial box trivializes communication complexity, and again, this is only revealed using the serial algorithm.}
\label{(0.575,0.375)I_strategies_displayed}
\end{table}
We can furthermore compare the different types of procedure. While we find that in CSs~I and~II, the serial procedure is more successful with respect to the increase in non-locality that is achieved, we have found other CSs where the parallel is favourable. For more details and the analysis of further types of procedures we refer to~\cite{GiorgosThesis}.
Notice also that, after a few iterations, we recover the same iteration of wiring strategies for each party in the two tables. This procedure corresponds to essentially exchanging the roles of the two players between iterations (and some bit-flips):
\begin{equation}gin{align*}
&{\bf ODD \ iterations:} \ \ \ x_2=x, \ x_1=xa_2, \ a=a_1 \oplus {a_2} \oplus 1, \ y_2=y, \ y_1=y \oplus {b_2} \oplus 1, \ b=b_1 \oplus b_2 \oplus 1 \\
&{\bf EVEN \ iterations:} \ x_2=x, \ x_1=x \oplus a_2, \ a=a_1 \oplus {a_2} \oplus 1, \ y_2=y, \ y_1=y ( {b_2} \oplus 1) ,\ b=b_1 \oplus b_2 \oplus 1 .
\end{align*}
\section{3-copy distillation in the literature} \label{sec:app3}
So far, the 3-copy non-locality distillation protocol that was so far known in the literature was introduced in~\cite{Hoyer}. This is specified by the following functions that make up the protocol HR:
{\begin{equation}gin{center}
\begin{equation}gin{tabular}{ |p{5cm}|p{5cm}| }
\hline
Alice's side & Bob's side\\
\hline
$x_{1}=x$ & $y_{1}=y$ \\
$x_{2}=x\oplus a_{1}$ & $y_{2}=y\overline{b}_{1}$ \\ $x_{3}=a_{2}\overline{a}_{1}\oplus x(a_{1}\oplus a_{2} \oplus a_{1}a_{2})$ & $y_{3} =\overline{b}_{1}\oplus b_{2}\overline{b}_{1} \oplus y(\overline{b}_{2}\oplus b_{1}b_{2})$ \\
$a=a_{1} \oplus a_{2}\oplus a_{3} $ & $b =b_{1} \oplus b_{2}\oplus b_{3} $\\
\hline
\end{tabular}
\end{center}
}
In some parts of CS~I, this protocol outperforms the 2-copy distillation protocol ABL$^+$1 (around the point indicated with the star in Figure~{\text{r}}ef{fig:HRvsABL1vsStar}). At the point indicated with the star, HR can distill non-locality while \emph{no} 2-copy protocol can (thus, HR is also a \emph{genuine} 3-copy protocol). However, the region around the starred point where this is possible is extremely small (see Figure~{\text{r}}ef{fig:HRvsABL1vsStar}). This is different for our genuine 3-copy protocols (Equations~({\text{r}}ef{eq:3copy_protocol_2}) and ({\text{r}}ef{eq:3copy_protocol})), for which this increase is considerable. Furthermore, we checked that HR, despite being a genuine 3-copy protocol, distills nothing in CS~III, unlike our genuine 3-copy protocol that unlocks distillation there (Figure~{\text{r}}ef{fig:3copy}).
\begin{equation}gin{figure}
\centering
\includegraphics[scale=0.48]{paper_HR_appendix}
\caption{The blue (gray) boundary includes the boxes that are distillable by the ABL$^{+}$1 (HR). The gray region depicts the set where HR achieves higher distilled CHSH-values than ABL$^{+}$1. The bullet point corresponds to a box that is distillable by ABL$^{+}$1 but not by HR. Interestingly, the star (coordinates $(\eta,\omega)= (\frac{3}{32},\frac{1}{32}(2\sqrt{227}-3))$) corresponds to a box that is not distillable by ABL$^{+}$1 (so, not distillable by any two-copy protocol) but it can be distilled by HR. }
\label{fig:HRvsABL1vsStar}
\end{figure}
\begin{equation}gin{figure}
\centering
\includegraphics[scale=0.45]{paper_overbraceVShr}
\caption{Comparison of our new protocol (Equation~({\text{r}}ef{eq:addon1})), displayed with the purple boundary, to the 3-copy protocol HR from~\cite{Hoyer}, displayed in gray for CS~II. The protocol of Equation~({\text{r}}ef{eq:addon1}) distils a strict superset of the boxes that HR distils and the non-locality increase at each point is also stronger than that of HR.}
\label{fig:add-on}
\end{figure}
\section{Further properties of the novel OR-gated protocols}\label{sec:app4}
In this section we present some extra features of the protocols introduced in Equations ({\text{r}}ef{eq:3copy_protocol_2}) and ({\text{r}}ef{eq:3copy_protocol}).
\begin{equation}gin{table}
\centering
\begin{equation}gin{tabular}{ |p{2cm}|p{9cm}| }
\hline
Cross Section & CHSH value of the distilled box \\
\hline
\hspace{0.5cm} {\cal L}arge I &
{\small $ \frac{1}{16}\Big[\omega^3-5\eta^3+9\omega^2+31\omega+\eta^2(5+7\omega)+\eta(9+22\omega+5\omega^2)+23\Big]$}
\\
\hline
\hspace{0.5cm} {\cal L}arge III &
{\small $ \frac{1}{16}\Big[\omega^3+7\eta^3+9\omega^2+31\omega+\eta^2(5+19\omega)+\eta(-3+18\omega+13\omega^2)+23\Big]$}
\\
\hline
\end{tabular}
\caption{Final CHSH function after one iteration of the protocol of Equation ({\text{r}}ef{eq:3copy_protocol_2}), for the two cross sections of Figure~{\text{r}}ef{fig:3copy}.}
\label{tab:sec2p}
\end{table}
The protocol of Equation~({\text{r}}ef{eq:3copy_protocol}) preserves the line (one dimensional convex combination)
\begin{equation}gin{align*}
\omega P^{\mathrm{NL}}_1+(1-\omega)\frac{P^{\mathrm{L}}_1+P^{\mathrm{L}}_6}{2},
\end{align*}
which is that subset of CS~I corresponding to $\omega=1-\eta$. This means that an operation of the protocol maps any box belonging to that line, back to that line. Each iteration $n$, $n\ge 1$, of the protocol, updates the coordinate $\omega$ according to the recurrence relation
\begin{equation}gin{align}
\omega_n=\frac{1}{4}\omega_{n-1}(7-4\omega_{n-1}+\omega^2_{n-1}) \hspace{0.5cm}, \hspace{0.5cm} \omega_0=\omega. \label{recurrence}
\end{align}
A plot showing the sequence of steps starting at $\omega_0=0.05$ is shown in Figure~{\text{r}}ef{stepsfigure}. From the shape of the curves it is clear that for any initial $\omega\in(0,1)$ repeated iterations allow us to generate a final box arbitrarily close to a PR box.
\begin{equation}gin{figure}[h]
\centering
\includegraphics[scale=0.7]{recurrence_relation}
\caption{The blue curve depicts the function $f_1(\omega)=\frac{1}{4}\omega^2(7-4\omega+\omega^2)$ while the brown the $f_2(\omega)=\omega$, $\omega \in [0,1]$. The black arrows lying in between represent all the steps from $n=1$ to $n=10$ of the recurrence relation ({\text{r}}ef{recurrence}) for the case $\omega_0=0.05$.} \label{stepsfigure}
\end{figure}
\end{document}
|
\begin{document}
\title{A Kolmogorov proof of the Clauser, Horne, Shimony and Holt inequalities }
\author{M. Revzen}
\affiliation {Department of Physics, Technion - Israel Institute of Technology,
Haifa 32000, Israel}
\date{\today}
\begin{abstract}
Boolean logic is used to prove the CHSH inequalities. The proof elucidates the
connection between Einstein elements of reality and quantum non locality.
The violation of the CHSH inequality by quantum theory is discussed and the two stage view of quantum measurement relevance to incompatible observables is outlined.
\end{abstract}
\pacs{XXX}
\maketitle
\section{Introduction}
A convenient specification of Einstein, Podolsky, Rosen and Bohm experimental set up that we adopt is:
An experiment is made of numerous runs. In each run two photons (1,2) in the state $|\psi(1,2)>= \frac{1}{\sqrt{2}}(|+_1+_2>+|-_1-_2>)$ are involved. (Here +(-) means positive (negative) polarization along a common z axis.) The photons propagates to two separated ports
($\alpha, \beta$) therein their polarizations are measured by one of two polarizers: A or A' at
$\alpha$, and B or B' at $\beta$. The experiments are assumed flawless: In each run both photons reach the counters
at $\alpha$ and $\beta$ and record a reading at both: +1 if the photon pass the polarizer and -1 if it doesn't. \\
The issue under study is: can the experimental results of the EPRB set up be accounted for in terms of canonical (Kolmogorov's) probability theory
based on hidden variable, $\lambda$?\\
Two point correlation of the outcomes with the polarizers set at A and B, $<AB>$, is defined by:\\
\begin{equation}\label{th}
<AB>=\frac{1}{N}\sum_{i=1}^{N}a_ib_i,
\end{equation}
Here i enumerates the run and a,b the outcome at the ports $\alpha$, $\beta$ respectively.
An account by probability theory based on hidden variable implies that two point correlations may be expressed by
\begin{equation}\label{hv}
<AB>=\int d\lambda\rho(\lambda)A(\lambda)B(\lambda),
\end{equation}
with a distribution, $\rho(\lambda),$ common to all correlations, $\lambda$ is the "hidden variable".
We note that within such an account the correlations $<AA'>$ and $<BB'>$ are defined though are not measured (perhaps even not measureable
e.g. due to technical difficulties). Thus, e.g., \\
\begin{equation}\label{simul}
<AA'>=\int d\lambda\rho(\lambda)A(\lambda)A'(\lambda).
\end{equation}
The canonical (Kolmogorov's) probability theory involves definition of a sample space wherein each of the points accounts for possible
experimental outcomes. The total sample space, $\Omega$, is a set of points that cover all possible experimental outcomes. The measure assigned to these points is their probability. Observable such as e.g. $A(\lambda)$, are termed events. Our analysis does not require the specification of this space - suffice it to know that such exists for a canonical probability theory. This allows us
to analyse interrelations among events via the algebra of sets aided by intuitively appealing Venn diagrams.\\
In the next section we obtain, assuming the existence of a probability theory and using simple Boolean logic \cite{pitowsky,hess,khren} implied consistency relations among
the two points correlations it allows. These interrelations will then be shown to be the so called Bell's inequalities.\\
\section{Interrelations among Correlations in Probability Theory}
In classical Kolmogorov theory interrelations among probabilities may be analysed
in terms of Venn diagrams pertaining to the corresponding events \cite{zvi},\cite{math}. Thus, e.g., we may consider the "area" in a Venn diagram "occupied" by the event $(AB)_{=}$. This event relates to the probability of observing equal readings, A=B, i.e. polarizer A (at $\alpha$) and B (at $\beta$) both read +1 or both -1. In our notation (using A and B as an example) this means: $(AB)_{=}$ is the set of all the sample space points with A=B. Its measure is the probability of this event, $P((AB)_{=}))$. The complementary event $(\widetilde{AB})_{=}$, containing the sample points with unequal values for A and B, is designated by $(AB)_{\times}$. Its probability is denoted by $P((AB)_{\times})$.\\
Thus,\\
\begin{equation}\label{tilde}
(\widetilde{AB})_{=}=(AB)_{\times}\Rightarrow P((\widetilde{AB})_{=})\equiv P((AB)_{\times}),
\end{equation}
and
\begin{equation}\label{P}
P\big[\big((AB)_{=}\cup (AB')_{=}\big)\big]+P\big[\widetilde{\big(AB)_{=}\cup (AB')_{=}\big)}\big]=1.
\end{equation}
Similar relations hold for the other two points correlations of interest, e.g., $(AB')_{=}, (A'B)_{=}, (A'B')_{=},$ etc.
Boolean logic, i.e. set algebra, dictates \cite{pitowsky},\cite{hess2},\cite{khren},\cite{math},
\begin{equation}\label{tilde2}
\widetilde{\big((AB)_{=}\cup (AB')_{=}\big)}=(\widetilde{AB})_{=}\cap(\widetilde{AB'})_{=}=(AB)_{\times}\cap(AB')_{\times}.
\end{equation}
Where we used Eq.(\ref{tilde}) in the last step.
We have trivially that $(AB)_{\times}\cap(AB')_{\times}\Rightarrow (BB')_{=}$ i.e. :\\
\begin{equation}
\big((AB)_{\times}\cap(AB')_{\times}\big)\subseteq(BB')_{=},
\end{equation}
implying \cite{it},
\begin{equation}\label{b1}
P((AB)_{=})+P((AB')_{=})+P((BB')_{=})\;\ge\; 1.
\end{equation}
Eq.(\ref{b1}) is a consistency requirement stemming from pairing the probabilities of $(AB)_{=}$ with $(AB')_{=}$.\\
\begin{figure}
\caption{Probability Consistency Interrelation: $P((AB)_{=}
\end{figure}
Quite generally this approach allows derivation of consistency relations among probabilities. These are equivalent to
Bell's inequalities which are formulated in terms of correlations. We now proceed to reformulate our consistency relations
in term of correlations.\\
\begin{eqnarray}\label{c1}
P((AB)_{=})+P((AB)_{\times})&=&1, \nonumber \\
P((AB)_{=})-P((AB)_{\times})&=& <AB>, \nonumber \\
\Rightarrow &&\nonumber \\
2P((AB)_{=})\;=1&+&<AB>, \nonumber \\
2P((AB)_{\times})\;=1&-&<AB>.
\end{eqnarray}
Utilizing Eq.(\ref{c1}), Eq.(\ref{b1}) may be written in terms of two point correlations as
\begin{equation}\label{cc1}
\langle AB\rangle +\langle AB'\rangle+\langle BB'\rangle\;\;\ge\; -1
\end{equation}
Going through similar reasoning with the pair $(AB)_{=}\cup(AB')_{\times}$ gives,
\begin{equation}\label{cc2}
\langle AB\rangle-\langle AB'\rangle-\langle BB'\rangle\;\;\ge\; -1
\end{equation}
The inequality for $(A'B)_{=}$ and $(A'B')_{=}$, is of course identical to Eq(\ref{cc1}) with A' replacing A:
\begin{equation}\label{cc3}
\langle A'B\rangle+\langle A'B'\rangle+\langle BB'\rangle\;\;\ge\; -1.
\end{equation}
Combining the inequality, Eq.(\ref{cc2}), with that of Eq.(\ref{cc3}) yields a Bell's inequality:
\begin{equation}\label{B1}
\langle AB\rangle-\langle AB'\rangle+\langle A'B\rangle+\langle A'B'\rangle\;\ge\; -2.
\end{equation}
There are 4 possible pairings:\;$ 1 \;(AB),(AB');\;2.\;(A'B),(A'B');\;3. \;(AB),(A'B);\;4.\; (AB'),(A'B')$.\\
There are 4 combinations for each, e.g.:\;1. \;$(AB)_{=},(AB')_{=};\;2.\;(AB)_{=},(AB')_{\times};\;3.\;(AB)_{\times},(AB')_{=};\;4.\;(AB)_{\times},(AB')_{\times}.$\\
Each implies consistency inequality among the relevant correlations. This gives 16 consistency inequalities.
Eliminating the unmeasured correlations ($<AA'>, <BB'>$) gives the following inequalities
\begin{equation}
|<AB>\mp<A'B>| +|<AB'>\pm<A'B'>|\;\le\;2
\end{equation}
identified as Bell's inequalities. It is accepted that these inequalities are violated experimentally \cite{aspect}.
Thence an account of the EPRB set up by a canonical probability theory fails.
(No violation has been observed of the quantum mechanical (QM) predictions \cite{asher}.)
\section{The Quantum Mechanical State}
The culprit in the failure of an account of the EPRB set up by canonical probability theory appears to be the presence therein of
correlations among observables that do not have values simultaneously i.e. that can be revealed by the same experiment. (These are A with A' and B with B' in the EPRB set up. E.g. Eq.(\ref{simul}) assigns instantaneous relation between values for A and A', within the same distribution.) Such instantaneous relation are, we contend, disallowed by nature. Thence the distribution function
for values of, say, A must be different from that of A' (though both may pertain to the same quantum state) and their measurements must involve distinct experiments.\\
The observed correlations violate the inequalities yet abide by QM. This may seem unexpected in view of the analysis above since
the evaluation of the correlations within QM does involves the same quantum mechanical state, $\rho,$ for
all the correlations. This, seemingly, is equivalent to using the same sample space and measure (i.e probability) in the
canonical probability formulation. Yet the later
led to consistency relations which are violated experimentally indicating thereby its inadequacy. However closer look at the
QM calculations show that {\it mathematical} attributes of the Hilbert space formalism {\it in effect} assigns distinct
( probability) distributions to non commuting observables. Thus in evaluating say the expectation of an arbitrary operator, $\hat{A}=\sum_{a}|a>a<a|$ for a state
specified by an arbitrary $\rho$ the effective $\rho$ is \cite{luo},\cite{r1} in effect (indicated by an arrow) diagonal,
\begin{equation}
\hat{\rho}\rightarrow \rho_{A}\equiv\sum_{a}|a><a|\rho|a><a|.
\end{equation}
Likewise, when evaluating $\hat{A'}$ , it is $\rho_{A'}\equiv\sum_{a'}|a'><a'|\rho|a'><a'|\neq \rho_{A}$, in general.
I.e. in effect we evaluate two non commuting observables with two different distribution functions though the QM state is the same.\\
These considerations were raised elsewhere \cite{r1} and led to viewing QM measurement as made of two stages. The first, termed
unrecorded measurement (URM), involves an unrecorded von Neumann measurement. This is associated with, in general, an actual change
in the QM state. The second stage view the resultant state as amendable for a classical distribution account (for compatible observables) and
the uncovering of the final outcome is handled much like within classical probability theory. Translated to the case at hand, the QM state
is viewed
as an information code encoding the possible distributions for the system \cite{newton}. The relevant distribution for a particular measurement
is attained, by a measurement (this is illustrated below). Thus, whereas the classical state may be viewed as a set of values for attributes - the quantum state may
be viewed as an encoded set of possible distributions for these attributes. It "{\it is (the symbolic representation
of) the ensemble to which it belongs}"\cite{newton}. $\rho$ within a quantum measurement (i.e. Hilbert space formalism) is "projected" to classical like distribution for compatible observables. Once the distribution is determined the measurement's second stage
is purely classical: gaining the numerical value for, say, A within
the distribution $\rho_{A}$. Measurement selects the distribution \cite{lajos}, e.g.,
\begin{equation}
\hat{\rho}\rightarrow \;\rho_{A}\;(\textsf{via \;Unrecorded\;measurement})\;\rightarrow a \;(\textsf{via\; classical\; measurement}.)
\end{equation}
The interpretation has special significance when dealing with
two (or more) particles state that allows separate measurements for each: measurement of one particle discloses partial distribution, in general:
\begin{equation}
\rho\;\rightarrow\;\sum_{a,b,b'}|a>|b><b|<a|\rho|a>|b'><b'|<a|,
\end{equation}
where only the partial distribution of the first particle pertaining to the attribute $\hat{A}$ is revealed. In the special case wherein
$\rho$ relates to a (maximally) entangled state viewing the state as symbolic representation of an ensemble has conceptual advantage.
Thus, e.g., consider measuring
the polarity along some axis, $\theta$, on particle 1 in a two photon maximally entangled state,
\begin{equation}
\rho =\frac{1}{2}\big[(|+>_1|+>_2+|->_1|->_2)(<-|_2<-|_1+<+|_2<+|_1)\big].
\end{equation}
The form of this state is independent of the direction of the polarizers. Thus rotating the polarizers to the $\theta$ direction
leaves the form of state invariant:
\begin{equation}\label{inv}
\rho\Rightarrow \rho_{\theta}=\frac{1}{2}\big[(|+,\theta>_1|+,\theta>_2+|-,\theta>_1|-,\theta>_2)(<+,\theta|_1<+,\theta|_2 + <-,\theta|_1<-,\theta|_2) \big].
\end{equation}
Undertaking a measurement along $\theta$ of one particle, gives:
\begin{equation}\label{dist}
\rho\Rightarrow \rho_{\theta}=\frac{1}{2}\big[|+,\theta>_1|+,\theta>_2<+,\theta|_1<+,\theta|_2 + |-,\theta>_1|-,\theta>_2<-,\theta|_2<-,\theta|_1\big] \big].
\end{equation}
I.e., in the case of a (maximally) entangled state, the one particle measurement (to uncover its distribution) revealed the complete distribution.
If the accessible particle is measured yielding, say, + the experimenter gained the knowledge that the remote particle is in the + state.\\
\section {Concluding Remarks}
A hidden variables account for the Einstein, Podolsky, Rosen and Bohm (EPRB) set up considered in the literature \cite{aspect} is an
account in terms of classical (Kolmogorov's) probability theory. Such an account, necessarily, assigns values to all the EPRB two point
correlations both measured and unmeasured.\\
A mathematical attribute of standard quantum mechanics (QM) (i.e. within its Hilbert space formulation) is an involvement of non commuting
observables. This may be viewed as reflecting nature's disallowance for values of non-commuting observables to be simultaneously (within
the same experimental run) revealed. I.e. their having a defined instantaneous correlations within the same distribution is disallowed - they are incompatible.\\
QM accommodates the above mentioned natures' ruling, classical probability theory in terms of hidden variables does not. This precludes an account via a canonical
probability theory of the EPRB set up: such an account implies, in principle, simultaneous values for some of these (un measured) incompatible observables. Canonical probability formulation entails consistency requirements expressible as interrelation among correlations evaluated within the theory. They were identified as Bell's inequalities. We argued that their (experimental) violation reflects classical probability theory inadequacy in handling the incompatible observables.\\
Phase space formulation of QM aspire to provide it with classical like view. The formulation assigns the Wigner function the role of distribution \cite{leonhardt}.( The role
of hidden variables is played by phase space coordinates.) A consistency condition for viewing the Wigner function as a classical like distribution is it be non negative. Thence the quantal demonstration of "violation" of positivity of the
Wigner function is similar to the quantal violation Bell's inequalities. In either case the direct classical like formalism fails
to uphold nature's disallowance of prescribed values for instantaneous correlations of incompatible observables. Both are
violated within QM that does uphold the disallowance.\\
An appropriate classical like hidden variable account for the EPRB set up will, thus, require intricate information on ensemble of distributions (which is incorporated
within the QM formalism). This is beyond the classical Kolmogorov probability theory. The issue of entanglement enters since for such states the full distribution is revealed via measurement of one of the constituents.\\
To summarize:\\
\noindent 1. A novel concise derivation of Bell's inequalities is presented. The derivation allow better isolation of the reason for their violation which support
viewing quantum measurement as two stage process and quantum states as encoding information on distributions in addition to their probabilistic attributes. \\
\noindent 2. The experimentally observed Bell's inequality violation does not relate in any obvious way to locality attribute of hidden variable account for Einstein, Podolsky, Rosen and Bohm (EPRB) set up. The violation does not rule out EPR contention that QM is an incomplete theory of real physical entities.\\
\noindent 3. Bell's inequalities are consistency conditions for classical probability theory. Their violation indicate inadequacy
of Kolmogorov's classical probability theory to account for the physics involved in EPRB set up. A proper account should require an extended theory, one that could deal with states encoding distributions of classical like distributions.\\
Acknowledgement: Numerous illuminating discussions with Dr. Avi Levi, encouragement and informative comments by
Professors P. Berman, K. Hess and L. Maccone are gratefully acknowledged.\\
\end{document}
|
\begin{document}
\title[Weighted Shifts on Directed Trees]
{An Analytic Model for Left-Invertible \\ Weighted Shifts on Directed Trees}
\author[S. Chavan]{Sameer Chavan}
\address{Department of Mathematics and Statistics\\
Indian Institute of Technology Kanpur, India}
\email{[email protected]}
\author[S. Trivedi]{Shailesh Trivedi}
\address{School of Mathematics \\ Harish-Chandra Research Institute\\
Chhatnag Road, Jhu-nsi, Allahabad 211019, India}
\email{[email protected]}
\subjclass[2010]{Primary 47B37, 47A10; Secondary 46E22, 47B38}
\keywords{weighted shift, directed
tree, multiplication operator, reproducing kernel of finite bandwidth, Hilbert space of holomorphic functions}
\date{}
\maketitle
\begin{abstract}
Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and
let $S_{\lambda} \in B(l^2(V))$ be a left-invertible
weighted shift on ${\mathscr T}$.
We show that
$S_{\lambda}$ can be modelled as a multiplication operator $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on a disc centered at the origin, where $E:=\ker S^*_{\lambda}$.
The reproducing kernel associated with $\mathscr H$ is multi-diagonal and of bandwidth $k_{\mathscr T}.$
Moreover, $\mathscr H$ admits an orthonormal basis consisting of polynomials in $z$ with at most $k_{\mathscr T}+1$ non-zero coefficients.
As one of the applications of this model, we give
a complete spectral picture of $S_{\lambda}.$
Unlike the case $\dim E = 1,$ the approximate point spectrum of $S_{\lambda}$ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed trees with finite branching index.
\end{abstract}
\section{Preliminaries}
The implementation of methods of graph theory into operator theory gives rise to a new class of operators known as {\it weighted shifts on directed trees}. These operators are generalization of adjacency operators of the directed trees. Although, the study of adjacency operators of the directed graphs was initiated by Fujii, Sasaoka and Watatani in \cite{F-S-W}, it was first observed by Jab{\l}o\'nski, Jung and Stochel in \cite{Jablonski} that replacing the directed graphs by directed trees not just gives a successful theory of weighted shifts but also provides a rich source of examples and counter-examples in operator theory \cite{JBS-1}, \cite{JBS-2}.
Several questions related to boundedness, adjoints, normality, subnormality, hyponormality etc. of weighted shifts on directed trees have been studied in depth in \cite{Jablonski}.
In the present paper, we discuss
a rich interplay between the discrete structures (directed trees) and analytic structures (analytic kernels of finite bandwidth).
The starting point of this text is the observation that
any left-invertible weighted shift on a rooted directed tree can be realized as the operator of multiplication by the co-ordinate function on a reproducing kernel Hilbert space $\mathscr H$ of vector-valued holomorphic functions defined on a disc in the complex plane.
In case the directed tree has finite branching index, this analytic model takes a concrete form. In particular, the reproducing kernel associated with $\mathscr H$ turns out to be multi-diagonal. Also, the space $\mathscr H$ may not be obtained by tensoring a Hilbert space of scalar-valued holomorphic functions with another Hilbert space.
In this course, we arrive at a couple of interesting invariants, namely, branching index of a directed tree and radius of convergence for the weighted shift. Importantly, these invariants can be computed explicitly in various situations.
Let $\mathbb Z_+$, $\mathbb Z,$ $\mathbb R$ and $\mathbb C$ stand for
the sets of non-negative integers, integers, real numbers and complex
numbers, respectively.
The complex conjugate of a complex number $w$ will be denoted by $\overline{w}.$
We use $\mathbb D_r$
to denote the open disc $\{z \in \mathbb C : |z| < r\}$ of radius $r > 0.$
In case $r=1,$ we denote the unit disc $\mathbb D_1$ by a simpler notation $\mathbb D.$
For a subset $A$ of a non-empty set $X$, $\mbox{card}(A)$ denotes the cardinality of $A$.
Let $\mathcal H$ be a complex separable Hilbert space.
The inner-product on $\mathcal H$ will be denoted by $\inp{\cdot}{\cdot}_{\mathcal H}$.
If no confusion is likely then we suppress the suffix, and simply write the inner-product as $\inp{\cdot}{\cdot}$.
By a {\it subspace}, we mean a closed linear manifold.
Let $W$ be a subset of $\mathcal H.$ Then $\mbox{span}\,W$ stands for the smallest linear manifold generated by $W.$
In case $W$ is singleton $\{w\},$ we use the convenient notation $\langle w \mbox{ran}gle$ in place of $\mbox{span}\,\{w\}$.
By $\bigvee \{w : w \in W\},$
we understand the subspace generated by $W$.
For a subspace $\mathcal M$ of $\mathcal H,$ we use $P_{\mathcal M}$ to denote the orthogonal
projection of $\mathcal H$ onto $\mathcal M.$ For vectors $x, y \in \mathcal H,$ we use the notation $x \otimes y$ to denote the rank one operator given by
$$x \otimes y (h) = \inp{h}{y}x,~h \in \mathcal H.$$
Unless stated otherwise, all the Hilbert spaces occurring below are complex
infinite-dimensional separable and for any such Hilbert space
$\mathcal H$, ${B}({\mathcal H})$ denotes the Banach algebra of bounded linear operators on $\mathcal H.$
For $T \in B(\mathcal H),$ the symbols $\ker T$ and $\mbox{ran}\,T$ will stand for the kernel and
the range of $T$ respectively.
The Hilbert space adjoint of $T$ will be denoted by $T^*.$
In what follows, we denote the spectrum, approximate point spectrum, essential spectrum and the point spectrum
of $T$ by ${\sigma(T)}$, ${\sigma_{ap}(T)}$, ${\sigma_e(T)}$ and
${\sigma_p(T)}$
respectively.
We reserve the notation $r(T)$ for the spectral radius of $T.$
Let $T \in B(\mathcal{H})$.
We say that $T$ is {\it left-invertible} if there exists $S \in B(\mathcal{H})$ such that $ST=I.$ Note that $T$ is left-invertible if and only if
there exists a constant $\alpha > 0$ such that $T^*T \geq \alpha I.$ In this case, $T^*T$ is invertible and $T$ admits the left-inverse $(T^*T)^{-1}T^*$.
Following \cite{Shimorin}, we refer to the operator $T'$ given by $T':=T(T^*T)^{-1}$ as the {\it Cauchy dual} of the left-invertible operator $T.$
Further, we say that $T$ is {\it analytic} if $\bigcap_{n \geq 0}T^n(\mathcal H)=\{0\}$. If $\mathscr H$ is a reproducing kernel Hilbert space of holomorphic functions defined on a disc in $\mathbb C$, then the multiplication operator $\mathscr M_z$ defined on $\mathscr H$ provides an example of an analytic operator. It is interesting to note that
almost all analytic operators arise in this way.
Indeed, a result of S. Shimorin \cite{Shimorin} asserts that any left-invertible analytic operator is unitarily equivalent to the operator of multiplication by $z$ on a reproducing kernel Hilbert space of vector-valued holomorphic functions defined on a disc.
Since the proof of this fact, as given in \cite[Sections 1 and 2]{Shimorin}, plays a major role in the proof of the main result, we outline it in the following discussion
(cf. \cite[Theorem 2.13]{SV}).
Let $T \in B(\mathcal{H})$ be a left-invertible analytic operator
and
let $E:=\ker T^*$. For each $x \in \mathcal H$, define an $E$-valued holomorphic function $U_x$ as
$$U_x(z)=\displaystyle \sum_{n \geq 0}(P_ET'^{*n}x)z^n,$$
where $T'$ is the {Cauchy dual} of $T$.
A simple application of the spectral radius formula \cite{Conway} shows that the function $U_x(z)=P_E(I-zT'^*)^{-1}x$ is well-defined and holomorphic on the disc $\mathbb D_{r}$, where $r:=\frac{1}{r(T')}.$ Let $\mathscr H$ denote the vector space of $E$-valued holomorphic functions of the form $U_x$, $x \in \mathcal H$. Consider
the map $U:\mathcal H \rightarrow \mathscr H$ defined by $Ux = U_x$. By \cite[Lemma 2.2]{Shimorin}, the kernel of $U$ is precisely
$\bigcap_{n \geq 0}T^n(\mathcal H)$, and hence by the assumption, $U$
is injective.
In particular, we may equip the space $\mathscr H$ with the norm induced from $\mathcal H$, so that $U$ is unitary. It turns out that $\mathscr H$ is a $z$-invariant reproducing kernel Hilbert space with $UT=\mathscr M_zU$, where $\mathscr M_z$ is the operator of multiplication by $z$. Also, the reproducing kernel $\kappa_{\mathscr H} :\mathbb D_{r} \times \mathbb D_{r} \rightarrow {B}(E)$ is given by
\begin{eqnarray} \label{rk} \kappa_{\mathscr H}(z,w)=\displaystyle\sum_{j,k\geq0}P_ET'^{*j}T'^k|_Ez^j\overline{w}^k, \end{eqnarray}
which satisfies the following:
\begin{itemize}
\item[(i)] for any $x \in E$ and $\lambda \in \mathbb D_{r},$
$$\kappa_{\mathscr H}(\cdot,\lambda)x \in \mathscr H;$$
\item[(ii)] for any $x \in E$, $h \in \mathscr H$ and $\lambda \in \mathbb D_{r},$
$$\langle h(\lambda),x \mbox{ran}gle_E=\langle h,\kappa_{\mathscr H}(\cdot,\lambda)x \mbox{ran}gle_{\mathscr H}.$$
\end{itemize}
Conditions (i) and (ii) may be rephrased by saying that the set of bounded point evaluations (for short, bpe) for $\mathscr H$ contains the disc $\mathbb D_r.$
We see in the context of weighted shifts on rooted directed trees that indeed (analytic) bpe contains the disc $\mathbb D_{r_{\lambda}}$ of larger radius $r_{\lambda}$ (see Definition {\rm Re\,}f{defwn}). This occupies the major part of the proof of the main result.
In the remaining part of this section, we invoke some basic concepts from the theory of directed trees which will be frequently used in the rest of this paper. The reader is referred to \cite{Jablonski} for a detailed exposition on directed trees.
A pair $\mathscr T= (V,\mathcal E)$ is called a {\it directed graph} if $V$ is a non-empty set and $\mathcal E$ is a subset of $V \times V \setminus \{(v,v): v \in V\}$. An element of $V$ (resp. $\mathcal E$) is called a {\it vertex} (resp. an {\it edge}) of $\mathscr T$. A finite sequence $\{v_i\}_{i=1}^n$ of distinct vertices is said to be a {\it circuit} of $\mathscr T$ if $n \geq 2$, $(v_i,v_{i+1}) \in \mathcal E$ for all $1 \leq i \leq n-1$ and $(v_n,v_1) \in \mathcal E$. A directed graph $\mathscr T$ is said to be {\it connected} if for any two distinct vertices $u$ and $v$ of $\mathscr T$, there exists a finite sequence $\{v_i\}_{i=1}^n$ of vertices of $\mathscr T$ $(n \geq 2)$ such that $u=v_1$, $v_n=v$ and $(v_i,v_{i+1})$ or $(v_{i+1},v_i) \in \mathcal E$ for all $1 \leq i \leq n-1$. For a
subset $W$ of $V$, define $\child{W} = \bigcup_{u\in W} \{v\in V
\colon (u,v) \in \mathcal E\}.$
One may define inductively $\childn{n}{W}$ for
$n\in \N$ as follows: Set
$\childn{n}{W}=W$ if $n=0$, and
$\childn{n}{W}=\child{\childn{n-1}{W}}$ if $n\geqslant
1$.
Given $v\in V$, we write $\child{v}:=\child{\{v\}}$,
$\childn{n}{v}=\childn{n}{\{v\}}$. A member of $\child{v}$ is called a {\it child} of $v.$ For a given vertex $v\in V$, if there exists a unique vertex
$u \in V$ such that $(u,v)\in \mathcal E$, we say that $v$ has a {\em parent} $u$ and denote it by $\parent{v}$. A vertex $v$ of $\mathscr T$ is called a {\it root} of $\mathscr T$, or $v \in \mathsf{Root}(\mathscr T)$, if there is no vertex $u$ of $\mathscr T$ such that $(u,v)$ is an edge of $\mathscr T$. If $\mathsf{Root}(\mathscr T)$ is a singleton then its unique element is denoted by $\mathsf{root}$. We set $V^\circ:=V \setminus \mathsf{Root}(\mathscr T)$.
A directed graph $\mathscr T= (V,\mathcal E)$ is called a {\it directed tree} if
\begin{itemize}
\item[(i)] $\mathscr T$ has no circuits,
\item[(ii)] $\mathscr T$ is connected and
\item[(iii)] each vertex $v \in V^\circ$ has a parent.
\end{itemize}
\begin{remark}
Any directed tree has at most one root \cite[Proposition 2.1.1]{Jablonski}.
\end{remark}
A directed tree $\mathscr T$ is said to be
\begin{enumerate}
\item[(i)] {\it rooted} if it has a (unique) root.
\item[(ii)] {\it rootless} if it has no root.
\item[(iii)]
{\it locally finite} if $\mbox{card}(\child u)$ is finite for all $u \in V.$
\item[(iv)]
{\it leafless} if every vertex has at least one child.
\end{enumerate}
In what follows, $l^2(V)$ stands for the Hilbert
space of square summable complex functions on $V$
equipped with the standard inner product. Note that
the set $\{e_u\}_{u\in V}$ is an
orthonormal basis of $l^2(V)$, where $e_u \in l^2(V)$
is the indicator function $\chi_{\{u\}}$ of $\{u\}$. Given a system
$\lambdab = \{\lambda_v\}_{v\in V^{\circ}}$ of non-negative real numbers,
we define the {\em weighted shift operator} $S_{\lambda}$ on ${\mathscr T}$
with weights $\lambdab$ by
\begin{align*}
\begin{aligned}
{\mathscr D}(S_{\lambda}) & := \{f \in l^2(V) \colon
\varLambda_{\mathscr T} f \in l^2(V)\},
\\
S_{\lambda} f & := \varLambda_{\mathscr T} f, \quad f \in {\mathscr
D}(S_{\lambda}),
\end{aligned}
\end{align*}
where $\varLambda_{\mathscr T}$ is the mapping defined on
complex functions $f$ on $V$ by
\begin{align*}
(\varLambda_{\mathscr T} f) (v) :=
\begin{cases}
\lambda_v \cdot f\big(\parent v\big) & \text{if } v\in
V^\circ,
\\
0 & \text{if } v \text{ is a root of } {\mathscr T}.
\end{cases}
\end{align*}
Unless stated otherwise, $\{\lambda_v\}_{v\in V^{\circ}}$ consists of positive numbers and $S_{\lambda}$ belongs to $B(l^2(V)).$ It may be concluded from \cite[Proposition 3.1.7]{Jablonski} that $S_\lambda$ is an injective weighted shift on $\mathscr T$ if and only if $\mathscr T$ is leafless.
In what follows, we always assume that all the directed trees considered in this text are countably infinite and leafless.
In the proof of the main result, we frequently use the following elementary facts pertaining to the weighted shifts on directed trees.
\begin{lemma} \label{facts}
If $S_{\lambda} \in B(l^2(V)),$
then
for any $u \in V$ and positive integer $k,$
\begin{enumerate}
\item[(i)]
$ S^k_{\lambda}e_{u} = \displaystyle \sum_{v \in \childn{k}{u}} \lambda_v \lambda_{\parent v}\cdots \lambda_{\parentn{k-1}{v}} e_v$ and \\
$ \|S^k_{\lambda}e_{u}\|^2 = \displaystyle \sum_{v \in \childn{k}{u}} \big(\lambda_v \lambda_{\parent v}\cdots \lambda_{\parentn{k-1}{v}} \big)^2.$
\item[(ii)]
$\displaystyle S^{*k}_{\lambda}e_{u} = \lambda_u \lambda_{\parent u}\cdots \lambda_{\parentn{k-1}{u}} e_{\parentn{k}{u}}$ and \\
$\displaystyle \|S^{*k}_{\lambda}e_{u}\|^2 = \big(\lambda_u \lambda_{\parent u}\cdots \lambda_{\parentn{k-1}{u}} \big)^2,$
where $e_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}$ is understood to be the zero vector in case ${\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)=\emptyset.$
\item[(iii)]
$S^{*k}_{\lambda}S^k_{\lambda}e_u = \|S^k_{\lambda}e_u\|^2e_u$.
\end{enumerate}
\end{lemma}
\begin{proof}
The part (i) has been established in \cite[Lemma 6.1.1]{Jablonski}, whereas (ii) and (iii) can be obtained by a straightforward mathematical induction using \cite[Lemma 3.4.1(iii)]{Jablonski}.
\end{proof}
Let $S_\lambda$ be a left-invertible weighted shift on a rooted
directed
tree with weights
$\{\lambda_v\}_{v\in V^{\circ}}$. It can be easily seen from (i) and (iii) above that the Cauchy dual $S'_{\lambda}$ of
$S_{\lambda}$ is given by
\[S'_{\lambda}e_u:=\displaystyle\sum_{v\in\mathsf{Chi}(u)}\frac{\lambda_v}{
\|S_\lambda
e_{\mathsf{par}(v)}\|^2}e_v~ \text{for all}\ v\in V^{\circ}.\]
Note that $S'_{\lambda} \in B(l^2(V))$ is a weighted shift with weights
$\{\lambda'_v\}_{v\in V^{\circ}},$ where
\begin{eqnarray} \label{weights}
\lambda'_v:=\frac{\lambda_v}{\|S_\lambda
e_{\mathsf{par}(v)}\|^2}~ \text{for all}\ v\in V^{\circ}.
\end{eqnarray}
This also shows that $\{\lambda'_v\}_{v\in V^{\circ}}$ is a bounded subset of positive real line.
Throughout this text, we find it convenient to use the notation $S_{\lambda'}$ in place of $S'_{\lambda}$.
It turns out that any weighted shift $S_{\lambda}$ on a rooted directed tree $\mathscr T$
is analytic (see Lemma {\rm Re\,}f{lem1} below).
Hence by Shimorin's construction as described above, any left-invertible $S_{\lambda}$ admits an analytic model $(\mathscr M_z, \kappa_{\mathscr H}, \mathscr H)$. It turns out that this model can be significantly improved upon provided the underlying directed tree has finite branching index (see Definition {\rm Re\,}f{b-index}). In this case, the analytic model takes a concrete form with {\it multi-diagonal} kernel $\kappa_{\mathscr H}$ defined on a disc $\mathbb D_{r_{\lambda}}$, where $r_{\lambda}$
is a positive number such that $\frac{1}{r(S_{\lambda'})} \leq r_{\lambda} \leq r(S_{\lambda})$ (see \end{equation}ref{radius}). Moreover,
the reproducing kernel Hilbert space admits an orthonormal basis consisting of vector-valued analytic polynomials.
One of the interesting aspects of our model is a handy formula for $r_{\lambda}$ depending on $\mathscr T$ and $S_{\lambda}.$
Although the motivation for the present work comes mainly from the theory of weighted shifts on directed trees as expounded in \cite{Jablonski}, it is closely related to some of the recent developments in the function theoretic operator theory. In particular, the reader is referred to the study of analytic reproducing kernels of finite bandwidth carried out in a series of papers by G. Adams et al \cite{AM}, \cite{AM-1}, \cite{AFM}, \cite{AMSS} (refer also to \cite{Ar} for the general theory of reproducing kernels).
It is also worth noting that the class of weighted shifts on rooted directed trees has some resemblance with the class of adjoints of abstract weighted shifts (in the context of complex Hilbert spaces) \cite{B-1}, \cite{B-2}, \cite{R-1} and also with the class of
operator-valued weighted shifts \cite{L}, \cite{K}, \cite{Ja-0}, \cite{Si}
studied extensively in the literature.
Here is the sketch of the paper. Section 2 is devoted to the statement of the main theorem and some of its immediate consequences. The proof of main theorem is presented in Section 3. In Section 4, we present several examples illustrating the rich interplay between the directed trees and reproducing kernels of finite bandwidth.
In Section 5, we use the main theorem to describe various spectral parts of $S_{\lambda}.$ It turns out that weighted shifts on directed trees with disconnected approximate point spectra are in abundance.
In the final section, we introduce a notion of branching index
for rootless directed trees and use it to obtain an analytic model for a left-invertible weighted shift $S_{\lambda}$ in this setting.
It turns out that $S_{\lambda}$ is an extension of a weighted shift operator on a rooted directed tree.
\section{Main Result: Statement and Consequences}
Let $\mathscr T=(V, \mathcal E)$ be a rooted directed tree with root $\mathsf{root}$. Then
\begin{eqnarray} \label{disjoint}
V = \bigsqcup_{n = 0}^{\infty} \childn{n}{\mathsf{root}}~(\mbox{disjoint union})
\end{eqnarray}
(see \cite[Proposition 2.1.2]{Jablonski}).
For each $u\in V$, let $n_u$ denote the unique non-negative integer
such that
$u \in \mathsf{Chi}^{\langle n_u\mbox{ran}gle}(\mathsf{root})$. We use the convention that
$\mathsf{Chi}^{\langle{j}\mbox{ran}gle}(\mathsf{root})=\emptyset$ if $j < 0.$ Similar convention holds for $\mathsf{par}$.
The statement of the main theorem involves an invariant (to be referred to as the branching index) associated with a rooted directed tree.
\begin{definition} \label{b-index}
Let $\mathscr T$ be a rooted directed tree
and let
$$V_{\prec}:=\{u\in V: \mbox{card}(\mathsf{Chi}(u)) \geq2\}$$ be the set of branching
vertices of $\mathscr T$. Define
\[k_\mathscr{T}:=\begin{cases}
1+\sup\{n_w:w\in V_{\prec}\},& \text{if $V_{\prec}$ is non-empty}\\
0,& \text{if $V_{\prec}$ is empty}.
\end{cases}
\]
We refer to $k_{\mathscr T} \in \mathbb Z_+ \cup \{\infty\}$ as the {\it branching index} of $\mathscr
T.$
\end{definition}
\begin{remark}
If $\mbox{card}(V_{\prec})$ is finite then so is $k_{\mathscr T}$. On the other hand, directed trees $\mathscr T$ with infinite $\mbox{card}(V_{\prec})$ and finite $k_{\mathscr T}$ can be constructed easily.
\end{remark}
The condition (i) in the following proposition says precisely that $\mathscr T$ is Fredholm (refer to \cite[Section 3.6]{Jablonski} for more details related to Fredholm directed trees).
\begin{proposition} \label{probranch}
Let $S_\lambda \in B(l^2(V))$ be a weighted shift on a rooted directed tree $\mathscr T$ with root
$\mathsf{root}$.
Let
$V_{\prec}$ be the set of branching
vertices of $\mathscr T$ and let $k_{\mathscr T}$ be the branching index of $\mathscr T.$
Then the following statements are equivalent:
\begin{enumerate}
\item [(i)] $\mathscr T$ is locally finite such that $\mbox{card}(V_{\prec})$ is finite.
\item [(ii)] $\mathscr T$ is locally finite such that $k_{\mathscr T}$ is finite.
\item [(iii)] The dimension of $E:=\ker S_\lambda^*$ is finite.
\end{enumerate}
\end{proposition}
\begin{proof}
That (i) implies (ii) is obvious. Suppose that (ii) holds. If $V_\prec$ is not finite, then that $k_\mathscr T$ is finite implies
that there exists an infinite subset $W$ of $V_\prec$ such that $n_w$ is
constant for all $w \in W$. Clearly, $W \subseteq \mathsf {Chi}^{\langle
n_w \mbox{ran}gle} (\mathsf {root})$. Therefore, there exists a vertex $v \in V$
with $n_v < n_w$ such that $\mbox{card}(\mathsf{Chi}(v))$ is infinite. This contradicts the assumption that $\mathscr T$ is locally
finite. Thus (ii) implies (i).
By \cite[Proposition
3.5.1(ii)]{Jablonski},
\begin{eqnarray*}
E=\ker S_{\lambda}^*=\langle e_{\mathsf{root}}\mbox{ran}gle \oplus \bigoplus_{v \in
V}\left(l^2(\mathsf{Chi}(v)) \ominus \langle \lambdab^v\mbox{ran}gle \right),
\end{eqnarray*}
where
$\lambdab^v : \mathsf{Chi}(v) \rightarrow \mathbb C$ is defined by
$\lambdab^v(u)=\lambda_u,$ and $\langle f \mbox{ran}gle$ denotes the span of
$\{f\}$.
Observe now that $l^2(\mathsf{Chi}(v)) \ominus \langle\lambdab^v\mbox{ran}gle \neq \{0\}$ if and only if $v \in V_{\prec}$. Therefore,
\begin{eqnarray} \label{kernel}
E=\langle e_{\mathsf{root}}\mbox{ran}gle \oplus \bigoplus_{v \in
V_{\prec}}\left(l^2(\mathsf{Chi}(v)) \ominus \langle \lambdab^v\mbox{ran}gle \right).
\end{eqnarray}
It now follows from \end{equation}ref{kernel} that $\dim E$ is finite if and only if $\mbox{card}(\child v)$ is finite for every $v \in V_{\prec}$ and $\mbox{card}(V_{\prec})$ is finite.
This gives the equivalence of (i) and (iii).
\end{proof}
\begin{remark}
It may happen that $k_{\mathscr T} < \infty$ and $\dim E = \infty$ (see Example {\rm Re\,}f{tree-inf}).
\end{remark}
\begin{definition}\label{defwn}
Let $\mathscr T$ be a rooted directed tree with
root $\mathsf{root}$ and let $k_{\mathscr T}$ be the branching index of $\mathscr T.$
For any integer $n$, consider the set
\begin{eqnarray*}
W_n:= \bigcup_{j=n}^{k_{\mathscr T} + n}
\mathsf{Chi}^{\langle{j}\mbox{ran}gle}(\mathsf{root}).
\end{eqnarray*}
Let $S_\lambda \in B(l^2(V))$ be a left-invertible
weighted shift with weights $\{\lambda_v\}_{v\in V^{\circ}}$
and let $S_{\lambda'}$ be the Cauchy dual of $S_{\lambda}.$
The {\it radius of convergence}
for $S_{\lambda}$ is defined as the non-negative number $r_{\lambda}$ given by
\begin{eqnarray} \label{radius}
r_{\lambda}:=\liminf_{n \rightarrow \infty} \left(\sum_{v \in W_n} \big(\lambda'_v
\lambda'_{\mathsf{par}(v)} \cdots
\lambda'_{{\mathsf{par}}^{\langle n-1
\mbox{ran}gle}(v)}\big)^2 \right)^{-\frac{1}{2n}}.
\end{eqnarray}
\end{definition}
We will see later that $r_{\lambda}$ is positive whenever $k_{\mathscr T}$ is finite (see Lemma {\rm Re\,}f{r-lambda}). Let us compute $r_{\lambda}$ in the case in which $S_{\lambda}$ is a unilateral weighted shift.
\begin{example}[(Diagonal)] \label{diagonal}
Consider the directed tree $\mathscr T_1$ with the set of vertices
$V:=\mathbb{Z}_+$ and $\mathsf{root}=0$. We further require that $\mathsf{Chi}(n)=\{n+1\}$ for
all
$n \geq 0$. For future reference, we note that
$V_\prec=\emptyset$, and hence $k_{\mathscr
T_1}=0$.
The weighted shift $S_\lambda$ on the directed tree $\mathscr T_1$ (to be referred to as {\it unilateral weighted shift})
is given by
\[S_\lambda e_n=\lambda_{n+1}e_{n+1}\ \text{for all}\ n\geq0.\]
(Caution: This differs from the standard definition $S_\lambda e_n=\lambda_{n}e_{n+1}\ \text{for all}\ n\geq0$ of the unilateral weighted shift.)
It is well-known that $S_{\lambda}$ is unitarily equivalent
to the operator ${\mathscr M}_z$ of multiplication by $z$ on the reproducing kernel Hilbert space $\mathscr H$ associated with the kernel
\[\kappa_{\mathscr
H}(z,w)=1 +\displaystyle\sum_{j\geq1}C_{j,j}z^j\overline{w}^j~(z,
w \in \mathbb D_{r}),\]
where $r:=\liminf_{n \rightarrow \infty} \left({\lambda_n}
{\lambda_{n-1}} \cdots
{\lambda_{1}}\right)^{\frac{1}{n}}$ and $\{C_{j, j}\}_{j \geq 0}$ is a sequence of positive numbers
(refer to \cite{S}).
Since $W_n=\{n\}$ for every $n \in \mathbb Z_+,$ the radius of convergence $r_{\lambda}$ for (a left-invertible) $S_{\lambda}$ is precisely
$r.$
Moreover, one can verify that (the rank one operator)
$C_{j,j}$ is (multiplication by) $\frac{1}{\lambda^2_1
\cdots\lambda^2_{j}}$
for all $j\geq1$.
Clearly, the reproducing kernel $\kappa_{\mathscr H}(\cdot,\cdot)$ is
{\it diagonal} in this case.
\end{example}
We are now in a position to state the main result of this paper.
\begin{theorem} \label{thm1}
Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}.$
Let $S_\lambda \in B(l^2(V))$ be a left-invertible weighted shift and let $S_{\lambda'}$ be the Cauchy
dual of $S_{\lambda}.$
Set $E:= \ker S^*_{\lambda}$.
Then there exist a $z$-invariant reproducing kernel Hilbert space
$\mathscr H$ of $E$-valued holomorphic functions defined on the disc
$\mathbb{D}_{r_\lambda}$ and a unitary mapping $U:l^2(V)\longrightarrow\mathscr H$ such
that ${\mathscr M}_zU=US_\lambda,$ where ${\mathscr M}_z$
denotes the operator of multiplication by z on $\mathscr H$ and $r_{\lambda}$ is the radius of convergence for $S_{\lambda}$. Moreover, $r_{\lambda} r(S_{\lambda'}) \geq 1,$ where $r(S_{\lambda'})$ is the spectral radius of $S_{\lambda'}.$
Further, $U$ maps $E$ onto the subspace $\mathscr E$
of $E$-valued constant functions in $\mathscr H$ such that $Ug=g$ for every $g \in E.$
Furthermore,
we have the following:
\begin{itemize}
\item [(i)] The reproducing kernel $\kappa_{\mathscr H} : \mathbb D_{r_{\lambda}}
\times \mathbb D_{r_{\lambda}} \rightarrow B(E)$ associated with
$\mathscr H$ satisfies $\kappa_{\mathscr H}(\cdot,w)g \in \mathscr H$ and
$
\inp{Uf}{\kappa_{\mathscr H}(\cdot,w)g}_{\mathscr H} = \inp{(Uf)(w)}{g}_E$
for every $f \in l^2(V)$ and $g \in E.$
\item [(ii)] $\kappa_{\mathscr H}$ is given by
\begin{equation}\label{eq2}
\kappa_{\mathscr H}(z,w)=I_E+\displaystyle\sum_{\underset{|j-k|\leq
k_\mathscr{T}}{j,k\geq1}}C_{j,k}z^j\overline{w}^k~(z, w \in \mathbb D_{r_{\lambda}}),
\end{equation}
where $I_E$ denotes the identity operator on $E$, and $C_{j,k}$ are bounded
linear operators on $E$ given by
$$C_{j,k}=P_ES^{*j}_{\lambda'}S_{\lambda'}^k|_E~(j, k =1, 2, \cdots)$$ with
$P_E$
being the orthogonal projection of $l^2(V)$ onto $E$.
\item [(iii)] The $E$-valued polynomials in $z$ are dense in $\mathscr H$. In fact,
$$\mathscr H=\bigvee\{z^nf:f\in {\mathscr E},\ n\geq0\}.$$
\item [(iv)] $\mathscr H$ admits an orthonormal basis consisting of polynomials
in $z$ with at most $k_\mathscr{T}+1$ non-zero coefficients.
\end{itemize}
\end{theorem}
\begin{remark}
Let $S_\lambda \in B(l^2(V))$ be a left-invertible weighted shift with non-negative
weights $\{\lambda_v\}_{v\in V^{\circ}}$.
Let $u_0 \in V$ be such that $\parent {u_0} \in V_{\prec}$.
Suppose that $\lambda_{u_0} =0$ and $\lambda_{u} > 0$ for all $u \in V \setminus \{u_0\}$. Then by \cite[Proposition 3.1.6]{Jablonski}, $S_{\lambda}$ can be decomposed as an orthogonal direct sum of two weighted shifts $S_{\lambda,1}, S_{\lambda, 2}$ on directed trees with positive weights. Since $S_{\lambda}$ is left-invertible, so are $S_{\lambda, 1}$ and $S_{\lambda, 2}$.
By the theorem above, there exist multiplication operators $\mathscr M_z^{(i)}$ on reproducing kernel Hilbert spaces
$\mathscr H^{(i)}$ for $i=1, 2$ such that $S_{\lambda}$ is unitarily equivalent to
${\mathscr M_z^{(1)}} \oplus {\mathscr M_z^{(2)}}$. Note that $\mathscr H_1 \oplus \mathscr H_2$ is the reproducing kernel Hilbert space associated with the reproducing kernel $\kappa_{\mathscr H^{(1)}} + \kappa_{\mathscr H^{(2)}}$ (refer to \cite{Ar}).
\end{remark}
The inequality $r_{\lambda} r(S_{\lambda'}) \geq 1$ in Theorem {\rm Re\,}f{thm1} may be strict in general.
\begin{example}
Consider the weighted shift $S_\lambda$ on the directed tree $\mathscr T_1$ (as discussed in Example {\rm Re\,}f{diagonal})
with weights $\lambda_n$ given by
$$\lambda_1=\lambda_2=\lambda_3=\lambda_4=1,\ \mbox{and}\ \lambda_k=\begin{cases}
\frac{1}{2}, & \mbox{if}\ 2^n+1 \leq k \leq 3.2^{n-1},\ n \geq 2\\
1, & \mbox{otherwise.}
\end{cases}$$
Note that $\inf_{n\geq1}\lambda_n=\frac{1}{2}$. Thus $S_\lambda$ is left-invertible and hence $S_{\lambda'}$ is bounded. Further, for any $n \geq 1,$ total number of $\frac{1}{2}$'s occurring in first $2^n$ places is equal to $2^{n-1}-2^{n-2}+2^{n-2}-2^{n-3}+\cdots+4-2=2^{n-1}-2$. Therefore, we get
$$\lambda_1 \lambda_2 \cdots \lambda_{2^n}=\frac{1}{2^{2^{n-1}-2}}=\frac{2^2}{2^{2^{n-1}}}.$$
Let $n$ be any positive integer. Then there is a unique positive integer $m_n$ such that $2^{m_n} \leq n < 2^{m_n+1}$. Let $n=2^{m_n}+k$ for some integer $k$ such that $0 \leq k < 2^{m_n}$. Therefore,
$$\lambda_1 \lambda_2 \cdots \lambda_{n} \geq \frac{1}{2^{2^{m_n-1}-2+k}}=\frac{2^2}{2^{2^{m_n-1}+k}}.$$
Since $k < 2^{m_n}$, $\frac{2^{m_n -1} + k}{n} = 1- \frac{2^{m_n -1}}{2^{m_n}+k} < \frac{3}{4}.$ It follows that
\begin{eqnarray*} \left(\lambda_1 \lambda_2 \cdots \lambda_{n}\right)^{\frac{1}{n}} \geq \left(\frac{2^2}{2^{2^{m_n-1}+k}}\right)^{\frac{1}{n}} > 2^{\frac{2}{n} - \frac{3}{4}}, \end{eqnarray*} and hence $r_{\lambda} = \liminf_{n \rightarrow \infty} \left({\lambda_n}
{\lambda_{n-1}} \cdots
{\lambda_{1}}\right)^{\frac{1}{n}}$ is at least $2^{-\frac{3}{4}}.$
On the other hand, $r(S_{\lambda'})=2.$ This can be seen as follows.
Note that $\lambda'_n=\frac{1}{\lambda_n}$. Therefore, \begin{eqnarray*} r(S_{\lambda'})=\lim_{n \rightarrow \infty}(\sup_{m\geq1}\lambda'_{m+1}\cdots\lambda'_{m+n})^\frac{1}{n}=(2^n)^\frac{1}{n}=2 \end{eqnarray*} (since ${2}$'s occur in $\{\lambda'_n\}$ consecutively at $2^{n-1}$ places for $n \geq 2$). Thus we have $r_{\lambda} r(S_{\lambda'}) \geq 2^{\frac{1}{4}},$ which is obviously bigger than $1.$
\end{example}
Since the proof of Theorem {\rm Re\,}f{thm1} consists of several observations of independent interest, it will be presented in the next section.
In the remaining part of this section, we discuss some immediate consequences of the main theorem.
First a terminology.
Let ${\mathscr M}_z, \kappa_{\mathscr H}$, and $\mathscr H$ be as appearing in
the statement of
Theorem {\rm Re\,}f{thm1}.
For the sake of convenience, we will refer to the triple $({\mathscr M}_z,
\kappa_{\mathscr H},
\mathscr H)$ as the {\it analytic model} of the left-invertible weighted shift $S_{\lambda}$
acting on the directed tree $\mathscr T.$
{\it Except the final section of this paper, we assume that $\mathscr T$ is a leafless, rooted directed tree with finite branching index $k_{\mathscr T}.$}
An operator
$T$ in ${B}({\mathcal H})$ is said to be {\it finitely cyclic} if there are a finite number of vectors $h_1, \cdots, h_m$
in $\mathcal H$ such that
\begin{eqnarray*}
{\mathcal H} = \bigvee {\{T^k h_i : k \geq 0, i=1, \cdots, m \}}.
\end{eqnarray*}
In case $m=1$, we refer to $T$ as {\it cyclic operator} with {\it cyclic vector} $h_1.$
We say that $T$ is {\it infinitely cyclic} if it is not finitely cyclic.
\begin{corollary} \label{cyclic}
Let $S_\lambda \in B(l^2(V))$ be a weighted shift on $\mathscr T$.
If $E:= \ker S^*_{\lambda}$ is finite dimensional then $S_\lambda$ is finitely cyclic.
\end{corollary}
\begin{proof}
Since $\mathscr T$ is leafless, by
\cite[Proposition 3.1.7]{Jablonski}, $S_{\lambda}$ is injective.
If $E:= \ker S^*_{\lambda}$ is finite dimensional then the range of $S_{\lambda}$ is closed, and hence
$S_{\lambda}$ is left-invertible.
Now appeal to Theorem {\rm Re\,}f{thm1}(iii).
\end{proof}
In general, the reproducing kernel Hilbert space $\mathscr H$ as constructed in the proof of Theorem {\rm Re\,}f{thm1} can not be realized as the tensor product $\mathscr K \otimes E$, where $\mathscr K$ is a Hilbert space of scalar-valued holomorphic functions.
We make this explicit in the following result.
\begin{corollary} \label{tensor}
Let $S_\lambda \in B(l^2(V))$ be a left-invertible weighted shift on
$\mathscr T$. Let $({\mathscr M}_z,
\kappa_{\mathscr H},
\mathscr H)$ denote the analytic model of $S_\lambda$ and let $E:= \ker S^*_{\lambda}$.
Suppose that there exist a Hilbert space $\mathscr K$ of scalar-valued holomorphic functions and an isometric isomorphism $\Phi : \mathscr H \rightarrow \mathscr K \otimes E$ such that $\Phi(p f) = p \otimes f$ for every polynomial $p \in \mathscr K$ and $f \in E$.
Then the reproducing kernel $\kappa_{\mathscr H}$ associated with $\mathscr H$ is the diagonal kernel given by
\begin{eqnarray*}
\kappa_{\mathscr H}(z,w)=I_E+\displaystyle\sum_{j=1}^{\infty}
\left(P_ES^{*j}_{\lambda'}S_{\lambda'}^j|_E\right) z^j\overline{w}^j~(z, w \in \mathbb D_{r_{\lambda}}).
\end{eqnarray*}
\end{corollary}
\begin{proof} Note that for any $f, g \in E$ and $m, n \in \mathbb Z_+$,
\begin{eqnarray} \label{orthogonal}
\inp{S^m_{\lambda}f}{S^n_{\lambda} g}_{l^2(V)}=\inp{z^mf}{z^n g}_{\mathscr H} = \inp{\Phi(z^mf)}{\Phi(z^n g)}_{\mathscr K \otimes E}=
\inp{z^m}{z^n}_{\mathscr K}\inp{f}{g}_{E}.
\end{eqnarray}
Since $S^k_{\lambda}e_{\mathsf{root}} \in \bigvee \{e_v : v \in \childn{k}{\mathsf{root}}\}$, by an application of \end{equation}ref{disjoint},
we obtain $\inp{z^m}{z^n}_{\mathscr K}=0$ for $m \neq n$ after letting $f=e_{\mathsf{root}}=g.$
Hence by \end{equation}ref{orthogonal}, we must have $\inp{S^m_{\lambda}f}{S^n_{\lambda} g}_{l^2(V)}=0$
for any $f, g \in E$ and non-negative integers $m \neq n.$
This shows that the sequence $\{S^k_{\lambda}E\}_{k \geq 0}$ of subspaces of $l^2(V)$ is mutually orthogonal.
It follows immediately that for any $f, g \in E$ and non-negative integers $j \neq k,$ \begin{eqnarray*} \inp{P_ES^{*j}_{\lambda'}S_{\lambda'}^kf}{g}_E= \inp{S^{*j}_{\lambda'}S_{\lambda'}^kf}{g}_{l^2(V)}=\inp{S_{\lambda'}^kf}{S^{j}_{\lambda'}g}_{l^2(V)}=0. \end{eqnarray*}
In particular, $P_ES^{*j}_{\lambda'}S_{\lambda'}^k|_E=0$ for all non-negative integers $j \neq k.$ The desired conclusion now follows from Theorem {\rm Re\,}f{thm1}(ii).
\end{proof}
\begin{remark}
In view of Shimorin's model (as discussed in Section 1), after replacing $r_{\lambda}$ by $\frac{1}{r(S_{\lambda'})}$, one may obtain the conclusion of Corollary {\rm Re\,}f{tensor} for any directed tree with infinite branching index $k_{\mathscr T}$. Thus, even for directed trees $\mathscr T$ with infinite $k_{\mathscr T}$, the associated reproducing kernel $k_{\mathscr H}$ could be multi-diagonal.
\end{remark}
Recall that $T \in B(\mathcal H)$ is an {\it isometry} if $T^*T=I.$
\begin{corollary}
Consider the analytic model $({\mathscr M}_z,
\kappa_{\mathscr H},
\mathscr H)$
of a left-invertible weighted shift $S_{\lambda}$ on $\mathscr T$ and let $E:= \ker S^*_{\lambda}$. If $S_{\lambda}$ is an isometry then $\kappa_{\mathscr H}$ is the $B(E)$-valued Cauchy kernel given by
\begin{eqnarray*}
\kappa_{\mathscr H}(z, w) = \frac{I_E}{1- z \overline{w}}~(z, w \in \mathbb D).
\end{eqnarray*}
In particular, $\mathscr H$ is the $E$-valued Hardy space of the open unit disc.
\end{corollary}
\begin{proof}
Assume that $S_{\lambda}$ is an isometry.
Note that $S_{\lambda'}$ is also isometry in view of hypothesis and $S^*_{\lambda'}S_{\lambda'} =(S^*_{\lambda}S_{\lambda})^{-1}.$ By the uniqueness of the reproducing kernel, it suffices to check that $C_{j, k}=\delta_{j, k}I_E$, where $$C_{j,k}:=P_ES^{*j}_{\lambda'}S_{\lambda'}^k|_E ~(j, k =1, 2, \cdots)$$ and $\delta_{j, k}$ denotes the Kronecker delta. If $j=k$ then
obviously $C_{j, k} = I_E.$ If $j < k$ then $C_{j, k} =
P_ES_{\lambda'}^{k-j}|_E = 0$ since $S_{\lambda'}E \subseteq \mbox{ran}\, S_\lambda = E^{\perp}.$
\end{proof}
One rather striking consequence of the preceding corollary is as follows:
{\it If $S_{\lambda}$ is an isometry then $r_{\lambda}=1$}. Note that this observation is irrespective of the structure of the directed tree $\mathscr T$. On the other hand, the definition of $r_{\lambda}$ relies on the weight sequence $\{\lambda_v\}_{v \in V^{\circ}}$ and of course on the structure of $\mathscr T.$ To see this fact, assume that $S_{\lambda}$ is an isometry.
By Theorem
{\rm Re\,}f{thm1}, we must have $r_{\lambda} r(S_{\lambda'}) \geq 1.$ However,
$S_{\lambda'}$ being isometry, $r(S_{\lambda'})=1$, and hence $r_{\lambda} \geq 1.$
Since $\frac{I_E}{1- z \overline{w}}$ is not defined on $\mathbb D_r \times \mathbb D_r$ for any $r > 1,$ by Theorem {\rm Re\,}f{thm1}(ii), $r_{\lambda}$ can not exceed $1.$
\section{Proof of the Main Theorem}
The proof of Theorem {\rm Re\,}f{thm1} involves several lemmas. The first of which collects some facts related to the set $W_n$. Recall from Definition {\rm Re\,}f{defwn} that for any integer $n$, the set $W_n$ is given by
\begin{eqnarray} \label{Wn}
W_n:= \bigcup_{j=n}^{k_{\mathscr T} + n}
\mathsf{Chi}^{\langle{j}\mbox{ran}gle}(\mathsf{root}).
\end{eqnarray}
\begin{lemma} \label{branch}
Let
$S_\lambda \in B(l^2(V))$ be a weighted shift on $\mathscr T$ and let $E:= \ker S^*_{\lambda}$.
Then we have the following statements:
\begin{enumerate}
\item[(i)] $E$ is a subspace of the (possibly infinite dimensional) space $\bigvee \{e_v: v \in W_0\}$.
\item[(ii)] $\mbox{card}(\mathsf{Chi}^{\langle n \mbox{ran}gle
}(\mathsf{root}))$ (possibly countably infinite) is constant for $n\geq k_\mathscr T$. In particular, $\mbox{card}\,(W_n)$ is constant for $n \geq k_{\mathscr T}$.
\item[(iii)] For
every $v
\notin W_n$, $e_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}$ belongs to the orthogonal complement of $E,$
where $e_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}$ is understood to be the zero vector in case ${\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)=\emptyset.$
\item[(iv)] For non-negative integers $m$ and $n,$ $W_n \cap W_m \neq \emptyset$ if and only if $|n-m| \leq k_{\mathscr T}.$
\end{enumerate}
\end{lemma}
\begin{proof}
Note that $\child{V_{\prec}} \subseteq W_0.$ Hence by \end{equation}ref{kernel},
\begin{eqnarray} \label{ker-W0} E \subseteq \langle e_{\mathsf {root}}\mbox{ran}gle
\oplus \bigoplus_{v \in
V_{\prec}} l^2(\mathsf{Chi}(v)) \subseteq \bigvee \{e_v : v \in W_0\}.\end{eqnarray}
This yields (i). To see (ii), recall that $n_u$ is the unique non-negative integer
such that $u \in \mathsf{Chi}^{\langle n_u\mbox{ran}gle}(\mathsf{root})$.
Note that $\mbox{card}(\mathsf{Chi}(u))=1$ if
$n_u\geq k_\mathscr T$, where we used the assumption that $\mathscr T$ is leafless.
Thus $\mbox{card}(\mathsf{Chi}^{\langle n \mbox{ran}gle
}(\mathsf{root}))$ is constant for $n\geq k_\mathscr T$.
This proves (ii).
We now check (iii).
Let $v \notin W_n.$ Since
$E$ is a subspace of $\bigvee \{e_v : v \in W_0\}$ by part
(i), it suffices to check that $e_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}$ is orthogonal to $\{e_v : v
\in W_0\}$. Note that $n_v < n$ or $n_v > k_{\mathscr T} + n,$
If $n_v < n$ then $e_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}=e_{\emptyset} =0$ by convention. Otherwise,
${\mathsf{par}}^{\langle
n \mbox{ran}gle}(v) \notin W_0,$ and hence $e_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}$ is orthogonal to $\{e_v : v
\in W_0\}$. To see (iv), let $n, m$ be two non-negative integers such that $n < m.$ If $n + k_{\mathscr T} < m$ then
clearly, $W_n \cap W_m = \emptyset.$ So suppose that $n + k_{\mathscr T} \geq m.$
Then $W_n \cap W_m = \cup_{k=m}^{n+k_{\mathscr T}} \childn{k}{\mathsf {root}},$ which is obviously non-empty.
\end{proof}
Next we prove that the radius of convergence for any left-invertible weighted shift with finite dimensional cokernel is positive.
\begin{lemma} \label{r-lambda}
Let $S_\lambda \in B(l^2(V))$ be a left-invertible
weighted shift on $\mathscr T$ and
let $S_{\lambda'}$ be the Cauchy dual of $S_{\lambda}.$
If $r(S_{\lambda'})$ denotes the spectral radius of $S_{\lambda'}$ then the radius of convergence $r_{\lambda}$ for $S_{\lambda}$ satisfies $r_{\lambda} r(S_{\lambda'}) \geq 1.$ In particular, $r_{\lambda}$ is
positive.
\end{lemma}
\begin{proof}
By Lemma {\rm Re\,}f{facts}(i), for any integer $k \geq 0,$
$$\|S^k_{\lambda'}e_{\mathsf {root}}\|^2 = \sum_{v \in \childn{k}{\mathsf {root}}} \big(\lambda'_v \lambda'_{\parent v}\cdots \lambda'_{\parentn{k-1}{v}} \big)^2.$$
It follows from \end{equation}ref{disjoint} that
\begin{eqnarray*}
\sum_{v \in W_n}\big(\lambda'_v \lambda'_{\parent v}\cdots \lambda'_{\parentn{k-1}{v}} \big)^2 &=& \sum_{k=n}^{n+k_{\mathscr T}}
\sum_{v \in \childn{k}{\mathsf {root}}} \big(\lambda'_v \lambda'_{\parent v}\cdots \lambda'_{\parentn{k-1}{v}} \big)^2 \\ &=&
\sum_{k=n}^{n+k_{\mathscr T}} \|S^k_{\lambda'}e_{\mathsf {root}}\|^2
= \sum_{k=0}^{k_{\mathscr T}} \|S^{n+k}_{\lambda'}e_{\mathsf {root}}\|^2 \\
&\leq & \|S^{n}_{\lambda'}\|^2 \sum_{k=0}^{k_{\mathscr T}} \|S^k_{\lambda'}e_{\mathsf {root}}\|^2.
\end{eqnarray*}
If we set $M:=\sum_{k=0}^{k_{\mathscr T}}\|S^k_{\lambda'}e_{\mathsf {root}}\|^2$ (which is finite since $k_{\mathscr T} < \infty$) then
by the definition of $r_{\lambda},$
\begin{eqnarray*}
r_{\lambda} \geq \liminf_{n \rightarrow \infty} \Big(M\|S^{n}_{\lambda'}\|^2 \Big)^{-\frac{1}{2n}} =
\Big(\lim_{n \rightarrow \infty} M^{-\frac{1}{2n}} \Big) \Big(\liminf_{n \rightarrow \infty} \|S^{n}_{\lambda'}\|^{-\frac{1}{n}}\Big) = \frac{1}{r(S_{\lambda'})},
\end{eqnarray*}
which completes the proof of the lemma.
\end{proof}
\begin{remark}
If $S_{\lambda}$ is an expansion (that is, $S^*_{\lambda} S_{\lambda} \geq I)$ then $S_{\lambda'}$ is a contraction (that is, $S^*_{\lambda} S_{\lambda} \leq I)$. In this case, $r_{\lambda}$ is at least $1.$
\end{remark}
We need one more fact in the proof of the main result (cf. \cite[Proposition 4.5]{ACJS}).
\begin{lemma}\label{lem1}
Let $S_\lambda \in B(l^2(V))$ be a
weighted shift on $\mathscr T$. Then $S_\lambda$ is analytic.
\end{lemma}
\begin{proof}
Put $V_0:=V\ \text{and}\ V_k:=\displaystyle V \setminus
\cup_{j=0}^{k-1}\mathsf{Chi}^{\langle j\mbox{ran}gle}(\mathsf{root})~(k \geq 1).$
Note that $\{V_k\}_{k\geq0}$ is a strictly decreasing
sequence of sets such that $\displaystyle \cap_{k\geq0}V_k=\emptyset$. Now,
for all $u\in V$ and all integers $k\geq0$, by Lemma {\rm Re\,}f{facts}(i),
\[S_\lambda^k e_u=\displaystyle
\sum_{v \in \childn{k}{u}} \lambda_v \lambda_{\parent v}\cdots \lambda_{\parentn{k-1}{v}}
e_v.\]
It follows that
\[\mbox{ran}\,S_\lambda^k \subseteq\bigvee\{e_u:u\in V_k\}:=M_k,\ \text{say}.\]
Also, if $f\in M_k$, then $f(u)=0$ for $u\in V\setminus V_k =
\cup_{j=0}^{k-1}\mathsf{Chi}^{\langle j\mbox{ran}gle}(\mathsf{root})$. Thus, if
$f \in \displaystyle\cap_{k=0}^{\infty} M_k$, then $f(u)=0$ for
$u \in \displaystyle \cup_{j=0}^{\infty} \mathsf{Chi}^{\langle
j\mbox{ran}gle}(\mathsf{root}) =V$. That is, $f=0$. Hence
\begin{eqnarray*} \{0\} \subseteq \displaystyle \cap_{k = 0}^{\infty}\, \mbox{ran}\,
S_\lambda^k \subseteq \cap_{k=0}^{\infty}\, M_k =\{0\}. \end{eqnarray*} This shows that
$S_\lambda$ is
analytic.
\end{proof}
\begin{proof}[{Proof of Theorem {\rm Re\,}f{thm1}}]
As mentioned earlier, the proof relies on the
ideas developed in \cite[Sections 1 and
2]{Shimorin}.
Let
$f = \sum_{v \in V}f(v)e_v
\in l^2(V).$ By Lemmas {\rm Re\,}f{facts}(ii) and {\rm Re\,}f{branch}(iii),
\begin{eqnarray*}
P_ES_{\lambda'}^{*n}f
&=& \sum_{v \in V} f(v)\lambda'_v
\lambda'_{\mathsf{par}(v)} \cdots
\lambda'_{{\mathsf{par}}^{\langle n-1 \mbox{ran}gle}(v)}P_Ee_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)} \\
&=& \sum_{v \in W_n} f(v)\lambda'_v
\lambda'_{\mathsf{par}(v)} \cdots
\lambda'_{{\mathsf{par}}^{\langle n-1 \mbox{ran}gle}(v)}P_Ee_{{\mathsf{par}}^{\langle
n \mbox{ran}gle}(v)}.
\end{eqnarray*}
We claim that the $E$-valued series
\begin{eqnarray} \label{Uf}
U_f(z):=\displaystyle \sum_{n\geq0}(P_ES_{\lambda'}^{*n}f)z^n \end{eqnarray} converges
absolutely in $E$ on the disc $\mathbb D_{r_{\lambda}}$ for every $f \in l^2(V)$.
By Lemma {\rm Re\,}f{branch}(iv), for non-negative integers $m$ and $n,$ $W_n \cap W_m \neq \emptyset$ if and only if $|n-m| \leq k_{\mathscr T}.$
It follows that
\begin{eqnarray} \label{estimate}
\sum_{\underset{n \geq 0}{v \in W_n}} |f(v)|^2 &=& \sum_{v \in W_0} |f(v)|^2 + \sum_{v \in W_1} |f(v)|^2 + \cdots \nonumber \\ &\leq &
\sum_{v \in V} (k_{\mathscr T}+1)|f(v)|^2 = (k_{\mathscr T}+1)\|f\|^2.
\end{eqnarray}
Now
by the Cauchy-Schwarz inequality, for any integer $k \geq 0,$
\begin{eqnarray*}
\Big\|\sum_{n = 0}^k (P_ES_{\lambda'}^{*n}f)z^n \Big\|
&\leq & \sum_{\underset{n \geq 0}{v \in W_n}} |f(v)|\lambda'_v
\lambda'_{\mathsf{par}(v)} \cdots
\lambda'_{{\mathsf{par}}^{\langle n-1 \mbox{ran}gle}(v)}
|z|^n \\ &\leq & \Big(\sum_{\underset{n \geq 0}{v \in W_n}} |f(v)|^2
\Big)^{\frac{1}{2}} \Big(
\sum_{\underset{n \geq 0}{v \in W_n}} \big(\lambda'_v
\lambda'_{\mathsf{par}(v)} \cdots
\lambda'_{{\mathsf{par}}^{\langle n-1 \mbox{ran}gle}(v)}\big)^2|z|^{2n}\Big)^{\frac{1}{2}}
\\
&\overset{\end{equation}ref{estimate}} \leq & \sqrt{k_{\mathscr T}+1}~\|f\|\Big(
\sum_{\underset{n \geq 0}{v \in W_n}} \big(\lambda'_v
\lambda'_{\mathsf{par}(v)} \cdots
\lambda'_{{\mathsf{par}}^{\langle n-1 \mbox{ran}gle}(v)}\big)^2|z|^{2n}\Big)^{\frac{1}{2}}.
\end{eqnarray*}
Since the series on the right hand side converges absolutely on $\mathbb D_{r_{\lambda}}$,
the claim stands verified.
Thus $U_f$
is holomorphic in the disk $\mathbb{D}_{r_{\lambda}}.$ This allows us to define the map $U : l^2(V) \rightarrow \mathscr H$ by $Uf = U_f,$ where
$\mathscr H$ denotes the complex vector space of $E$-valued holomorphic
functions of the form $U_f$.
By Lemma {\rm Re\,}f{lem1}, $S_{\lambda}$ is analytic, and hence by \cite[Lemma 2.2]{Shimorin}, $U$ is injective. Thus the inner-product given by
\[\langle U_f,U_g\mbox{ran}gle=\langle f,g\mbox{ran}gle_{l^2(V)}\ \text{for all}\ f,g\in
l^2(V)\]
makes $\mathscr H$ an inner-product space. Also, the very definition of the inner-product on
$\mathscr H$ shows that $U$ is unitary, and hence $\mathscr H$ is a Hilbert
space.
Note that for each $f\in E$,
$U_f(z)=f$.
We now show that $\mathscr H$ is $z$-invariant. Let
$U_f\in\mathscr H$. Since $S^*_{\lambda'}S_{\lambda} = I$ and
$P_ES_{\lambda}=0,$ we get
\begin{eqnarray*} zU_f(z)
&=& \displaystyle\sum_{n\geq0}(P_ES_{\lambda'}^{*n}f)z^{n+1} =\sum_{n\geq1}
(P_ES_{\lambda'}^{*n-1}f)z^n \\ &=& \sum_{n\geq0}(P_ES_{\lambda'}^{*n}S_\lambda
f)z^n=U_{S_\lambda f}(z)\in\mathscr H.\end{eqnarray*}
Above expression also verifies that ${\mathscr M}_zU=US_\lambda$, where
${\mathscr M}_z$ is the
operator of multiplication by z on $\mathscr H$.
Part (i) has been recorded on \cite[Pg 154]{Shimorin} (see the discussion following \end{equation}ref{rk}).
To see (ii), recall from \end{equation}ref{rk} that
\[\kappa_{\mathscr
H}(z,w)=\displaystyle\sum_{j,k\geq0}C_{j,k}z^j\overline{w}^k,\]
where $C_{j,k}$ is a bounded linear operator on $E$ given by
$C_{j,k}=P_ES_{\lambda'}^{*j}S_{\lambda'}^k|_E$. Since ${\ker}
S_{\lambda'}^*={\ker}
S_\lambda^*=E$, it follows that $P_ES_{\lambda'}^{*j}|_E=0$ for all $j\geq1$.
Since $C^*_{j, k} = C_{k, j}$, we get
$C_{j,0}=0=C_{0,j}$ for all $j\geq1$. Hence the above expression for
$\kappa_{\mathscr H}(z,w)$ reduces to
\[\kappa_{\mathscr
H}(z,w)=I_E+\displaystyle\sum_{j,k\geq1}C_{j,k}z^j\overline{w}^k.\]
As recorded earlier in \end{equation}ref{ker-W0}, $E \subseteq \bigvee \{e_v : v \in W_0\}$ (see also \end{equation}ref{Wn} for the definition of $W_n$).
It follows that
\[S_{\lambda'}^{*j}S_{\lambda'}^kE\subseteq \bigvee\big\{
e_v:v \in W_{k-j}\big\},\] and
therefore
$S_{\lambda'}^{*j}S_{\lambda'}^kE$ is orthogonal to $E$ if
$|j-k|>k_\mathscr T$. Thus $C_{j,k}=0$
if
$|j-k|>k_\mathscr T$. This proves (ii).
To prove (iii), note that by Lemma {\rm Re\,}f{lem1}, $S_{\lambda'}$ is
analytic. Therefore, by \cite[Proposition 2.7]{Shimorin},
\begin{eqnarray*}
l^2(V) = \bigvee_{n \geq 0} S^n_{\lambda}(E).
\end{eqnarray*}
Since ${\mathscr M}_z$ is
unitarily equivalent to
$S_\lambda$ and $\ker S^*_{\lambda}=E$, it follows that
\begin{eqnarray*}
\mathscr H = \bigvee_{n \geq 0} {\mathscr M}^n_{z}(\mathscr E).
\end{eqnarray*}
This is precisely (iii).
Finally, since $U$ is unitary and
$\{e_u:u\in V\}$ is an orthonormal basis of $l^2(V)$, $\{U_{e_u}:u\in V\}$ is
an
orthonormal basis of $\mathscr H$. Note that
\begin{eqnarray} \label{onb} U_{e_u}(z)=\displaystyle\sum_{k\geq0}(P_ES_{\lambda'}^{*k}e_u)z^k=\sum_{
0\leq k\leq n_u}(P_ES_{\lambda'}^{*k}e_u)z^k\end{eqnarray}
(see the discussion following \end{equation}ref{disjoint} for the definition of $n_u$).
If $n_u \leq k_{\mathscr T}$ then clearly $U_{e_u}$ has at most $k_{\mathscr
T}+1$ number of non-zero coefficients. Suppose now that $n_u>k_\mathscr T$.
By Lemma {\rm Re\,}f{facts}(ii), $S_{\lambda'}^{*k}e_u = \alpha_{\lambda, k} e_{\mathsf{par}^{\langle
k \mbox{ran}gle}(u)}$ for some scalar $\alpha_{\lambda, k}.$ Let $k$ be an integer
such that $0\leq k \leq n_u-k_\mathscr T-1$.
Then $n_{\mathsf{par}^{\langle
k \mbox{ran}gle}(u)} = n_u - k \geq k_{\mathscr T} + 1,$ and hence
${\mathsf{par}^{\langle
k \mbox{ran}gle}(u)} \notin W_0.$
By Lemma
{\rm Re\,}f{branch}(i),
$$P_ES_{\lambda'}^{*k}e_u=\alpha_{\lambda, k} P_Ee_{\mathsf{par}^{\langle
k \mbox{ran}gle}(u)}=0.$$
Hence total
number of possible non-zero coefficients in above expression of $U_{e_u}$ are
$n_u+1-(n_u-k_\mathscr T)=k_\mathscr T+1$. Thus for each $u\in V$, $U_{e_u}$
is a polynomial in $z$ with at most $k_\mathscr T+1$ non-zero coefficients.
This
completes the proof of the theorem.
\end{proof}
\begin{remark}
The proof above actually shows that the set of analytic bounded point evaluation of $\mathscr H$ contains the disc $\mathbb D_{r_{\lambda}}$ (refer to \cite[Chapter II, Section 7]{Co}).
\end{remark}
We conclude this section with a brief discussion on a possible line of investigation. Note that the proof of Theorem {\rm Re\,}f{thm1} relies on the notion of Cauchy dual operator, present already in Shimorin's construction of an analytic model for a left inertible analytic operator \cite{Shimorin}. In case $\dim E=1$, the analytic model can be replaced by the model in which the weighted shift operator can be realized as the operator of multiplication by $z$ on a Hilbert space of formal power series \cite{S} (refer also to \cite{K}). It would be interesting to find a counter-part of the later model in case $\dim E > 1.$ In this regard, the authors would like to draw reader's attention to \cite[Theorems 2.12 and 2.13]{SV} in which it is shown that the adjoint of an arbitrary cyclic operator can be modelled as a backward shift on a reproducing kernel Hilbert space.
\section{Examples}
In this section, we illustrate Theorem {\rm Re\,}f{thm1} with the help of several interesting examples. In particular, we see that various directed trees (discrete structures) render to analytic multi-diagonal kernels (analytic structures). These include mainly tridiagonal and pentadiagonal kernels (the reader is referred to \cite{AM}, \cite{AM-1}, \cite{AFM}, \cite{AMSS} for a systematic study of scalar-valued and matrix-valued kernels of finite bandwidth; refer
also to \cite{KM} for a new class of matrix-valued kernels on the unit disc arising in the classification problem of homogeneous operators).
All important examples are summarized in the form of a table at the end of this section.
\begin{example}[(Tridiagonal)] \label{tridiagonal}
Consider the directed tree $\mathscr T_2$ with set of vertices
$$V:=\{(0,0)\}\cup\{(1,i), (2,i):i\geq1\}$$ and
$\mathsf{root}=(0,0)$. We further require that $\mathsf{Chi}(0,0)=\{(1,1),(2,1)\}$ and
\[\mathsf{Chi}(1,i)=\{(1,i+1)\},\
\mathsf{Chi}(2,i)=\{(2,i+1)\},\ \text{for all}\ i\geq1. \]
Let $S_{\lambda}$ be a left-invertible weighted shift on $\mathscr T_2.$
It is easy to see from \end{equation}ref{kernel} that
$$E:= \ker S_\lambda^* = \{\alpha e_{(0,0)} + \beta(\lambda_{(2, 1)}e_{(1,1)}-
\lambda_{(1, 1)}e_{(2,1)}) : \alpha, \beta \in \mathbb C\}.$$
Also, $V_\prec=\{(0,0)\}$ and $k_{\mathscr T_2}=1$. Therefore, by Theorem
{\rm Re\,}f{thm1}, $\kappa_{\mathscr H}(\cdot,\cdot)$ takes the form
\[\kappa_{\mathscr H}(z,w)=I_E+\displaystyle\sum_{\underset{|j-k|\leq
1}{j,k\geq1}}C_{j,k}z^j\overline{w}^k~(z, w \in \mathbb D_{r_{\lambda}}),\]
where $C_{j,k}$ is given by
$C_{j,k}=P_ES^{*j}_{\lambda'}S_{\lambda'}^k|_E~(j, k =1, 2, \cdots)$.
Let us find an explicit expression for the radius of convergence $r_{\lambda}$ for $S_{\lambda}$.
Note that $$W_n = \{(1, n), (2, n), (1, n+1), (2, n+1)\}$$ for $n \geq 1,$ and
hence by \end{equation}ref{radius},
\begin{eqnarray} \label{tri-rad}
r_{\lambda}=\liminf_{n \rightarrow \infty} \left(\sum_{j=1}^2 \left[\big(\lambda'_{(j, n)} \cdots
\lambda'_{(j, 1)}\big)^2 + \big(\lambda'_{(j, n+1)} \cdots
\lambda'_{(j, 2)}\big)^2 \right]\right)^{-\frac{1}{2n}},
\end{eqnarray}
where the sequence
$\{\lambda'_{(j, n)}\}_{n \geq 1}$ for $j=1, 2$ is given by
\begin{eqnarray} \label{seq-dual}
\lambda'_{(j, n)} = \begin{cases}
\frac{\lambda_{(j, n)}}{\lambda^2_{(1, n)} + \lambda^2_{(2, n)}}~\mbox{if~}n=1 \\
\frac{1}{\lambda_{(j, n)}}~\mbox{if~}n \geq 2.\end{cases}
\end{eqnarray}
In this case, the reproducing kernel $\kappa_{\mathscr H}(\cdot,\cdot)$ is
{\it tridiagonal}. Finally, we note that the weight sequence $\lambda$ can be chosen so that $\kappa_{\mathscr H}(\cdot,\cdot)$ is not diagonal. In fact, a routine calculation shows that \begin{eqnarray} \label{C21} C_{2, 1}(\lambda_{(2, 1)}e_{(1,1)}-
\lambda_{(1, 1)}e_{(2,1)}) = \frac{\lambda_{(1, 1)}\lambda_{(2, 1)}}{\lambda^2_{(1, 1)} + \lambda^2_{(2, 1)}}\left(\frac{1}{\lambda^2_{(1, 2)}} - \frac{1}{\lambda^2_{(2, 2)}}\right)e_{(0, 0)},\end{eqnarray} which is clearly non-zero in case $\lambda_{(1, 2)} \neq \lambda_{(2, 2)}.$
\end{example}
The tridiagonal kernel $\kappa_{\mathscr H}$ appearing in Example {\rm Re\,}f{tridiagonal} takes a concrete form for a family of weighted shifts $S_{\lambda}$.
\begin{proposition} Let $\mathscr T_2$ and $S_{\lambda}$ be as discussed in Example {\rm Re\,}f{tridiagonal}. Let $x:=e_{(0, 0)}, y:=\lambda_{(2, 1)}e_{(1,1)}-
\lambda_{(1, 1)}e_{(2,1)}.$ Assume that the weight sequence $\{\lambda_{(j, i)} : i \geq 1, j=1, 2\}$ of $S_{\lambda}$ satisfies the following:
\begin{enumerate}
\item[(i)] $\lambda_{(1, 1)} = \lambda_{(2, 1)}$,
\item[(ii)] $\lambda_{(1, 2)} \neq \lambda_{(2, 2)}$,
\item[(iii)]
$\lambda_{(1, n)} \cdots \lambda_{(1, 2)} = \lambda_{(2, n)} \cdots \lambda_{(2, 2)}$ for every integer $n \geq 3.$
\end{enumerate}
Then the reproducing kernel $\kappa_{\mathscr H}$ takes the form
\begin{eqnarray*}
\kappa_{\mathscr H}(z,w) &=& I_E + \alpha(x \otimes y\, z^2\overline{w} + y \otimes x\, z\overline{w}^2) \\ &+& \sum_{k=1}^{\infty} \Big({\alpha_k}\, x \otimes x + {\alpha_{k+1}}\, y \otimes y \Big) z^k \overline{w}^k
~(z, w \in \mathbb D_{r_{\lambda}}),
\end{eqnarray*}
where $\alpha:=\frac{\lambda^2_{(1, 1)}}{\|y\|^4}\left({\lambda^{-2}_{(1, 2)}} - {\lambda^{-2}_{(2, 2)}}\right)$ is a non-zero real number, and
\begin{eqnarray*} \alpha_{k}:= \begin{cases}
{
\|y\|^{-2}}~\mbox{if~}k=1,\\
{\lambda^2_{(1, 1)}}{\|y\|^{-4}}\Big( {\lambda^{-2}_{(1, 2)}} + {\lambda^{-2}_{(2, 2)}} \Big)~\mbox{if~}k=2,\\
{\|y\|^{-2}} \big({\lambda_{(1, k)} \cdots \lambda_{(1, 2)}}\big)^{-2}~\mbox{if~} k \geq 3.\end{cases}
\end{eqnarray*}
\end{proposition}
\begin{proof}
As seen in Example {\rm Re\,}f{tridiagonal}, $\kappa_{\mathscr H}(\cdot,\cdot)$ is given by
\[\kappa_{\mathscr H}(z,w)=I_E+\displaystyle\sum_{\underset{|j-k|\leq
1}{j,k\geq1}}C_{j,k}z^j\overline{w}^k~(z, w \in \mathbb D_{r_{\lambda}}).\]
Since $C_{2,1}e_{(0, 0)}=0$, by \end{equation}ref{C21} and (i), $C_{21}$ is the rank one operator $\alpha\,x \otimes y.$
Note that $\alpha \neq 0$ in view of (ii).
Also, $C_{1, 2} = C^*_{2, 1} = \alpha\,y \otimes x.$
We claim that $C_{k, k+1}=0=C_{k+1, k}$ for all integers $k \geq 2.$ Let us first compute the diagonal operator $S^{*k}_{\lambda'}S^k_{\lambda'}$. Fix an integer $k \geq 2.$ It may be concluded from \cite[Lemma 6.1.1]{Jablonski} that
\begin{eqnarray} \label{power} S^{*k}_{\lambda'}S^k_{\lambda'}e_{u} = \sum_{v \in \childn{k}{u}} \big(\lambda'_v \lambda'_{\parent v}\cdots \lambda'_{\parentn{k-1}{v}} \big)^2e_u.\end{eqnarray}
Since $S^*_{\lambda'}x=0,$ it follows from \end{equation}ref{power} that $C_{k+1, k}x = 0$. Note further that
\begin{eqnarray*}
S^{*k}_{\lambda'}S^k_{\lambda'}y &=& \lambda_{(2, 1)}S^{*k}_{\lambda'}S^k_{\lambda'}e_{(1, 1)} - \lambda_{(1, 1)}
S^{*k}_{\lambda'}S^k_{\lambda'}e_{(2, 1)}\\
&=&
\lambda_{(2, 1)}\big(\lambda'_{(1, k+1)} \cdots \lambda'_{(1, 2)}\big)^2 e_{(1, 1)} - \lambda_{(1, 1)}
\big(\lambda'_{(2, k+1)} \cdots \lambda'_{(2, 2)}\big)^2e_{(2, 1)} \\
&\overset{\mbox{(iii)}}=& \frac{1}{\big(\lambda_{(1, k+1)} \cdots \lambda_{(1, 2)}\big)^2}\, y,
\end{eqnarray*}
where in the last step we used \end{equation}ref{seq-dual}.
This immediately yields that $C_{k+1, k}y = 0$. This completes the verification of the claim.
The above calculation also shows that
$C_{k, k}y = \alpha_{k+1} \|y\|^2 y$ for all integers $k \geq 2.$
Since $S^{*}_{\lambda'}S_{\lambda'}y = \frac{\lambda_{(2, 1)}}{\lambda^2_{(1, 2)}} e_{(1, 1)} - \frac{\lambda_{(1, 1)}}{\lambda^2_{(2, 2)}} e_{(2, 1)},$ we have
\begin{eqnarray*}
C_{1, 1}y &=& P_{E}\left(\frac{\lambda_{(2, 1)}}{\lambda^2_{(1, 2)}} e_{(1, 1)} - \frac{\lambda_{(1, 1)}}{\lambda^2_{(2, 2)}} e_{(2, 1)}\right)\\ &=& \frac{\lambda_{(2, 1)}}{\lambda^2_{(1, 2)}} \inp{e_{(1, 1)}}{{y}}\frac{y}{\|y\|^2} - \frac{\lambda_{(1, 1)}}{\lambda^2_{(2, 2)}}\inp{e_{(2, 1)}}{y}\frac{y}{\|y\|^2} \\
&=& \left(\frac{\lambda^2_{(2, 1)}}{\lambda^2_{(1, 2)}} + \frac{\lambda^2_{(1, 1)}}{\lambda^2_{(2, 2)}}\right)\frac{y}{\|y\|^2} \overset{\mbox{(i)}}= {\alpha_2}{\|y\|^2} y.
\end{eqnarray*}
To compute the diagonal entry $C_{k, k}$, by \end{equation}ref{power}, for any integer $k \geq 1,$
\begin{eqnarray*} C_{k,k}x = \sum_{j=1}^2 \big(\lambda'_{(j, k)} \cdots \lambda'_{(j, 1)}\big)^2 x =\alpha_k x,
\end{eqnarray*}
where we used \end{equation}ref{seq-dual} and (iii).
Now it is easy to see that the rank two operator $C_{k, k}$ is given by
$C_{k,k}=
\Big({\alpha_k}\, x \otimes x + {\alpha_{k+1}}\, y \otimes y \Big)
$ for all integers $k \geq 1.$
\end{proof}
\begin{example}
The preceding proposition is applicable to $S_{\lambda}$ with weights
$\lambda_{(1, 1)}=\lambda_{(2, 1)}= \lambda_{(1, 2)}=1, \lambda_{(2, 2)}=\sqrt{2}=\lambda_{(1, 3)},$ and $\lambda_{(2, 3)}=1=\lambda_{(j, i)}$ for $i \geq 4$ and for $j=1, 2.$
In this case, $\alpha =\frac{1}{8}, \alpha_1 = \frac{1}{2}$, $\alpha_2 = \frac{3}{8}$ and $\alpha_k = \frac{1}{8}$ for all integers $k \geq 3.$
Thus the reproducing kernel $\kappa_{\mathscr H}$ takes the form
\begin{eqnarray*}
\kappa_{\mathscr H}(z,w) &=& I_E + \frac{1}{8}\big(x \otimes y\, z^2\overline{w} + y \otimes x\, z\overline{w}^2 \big) +
\frac{1}{2}\Big(x \otimes x + \frac{3}{4}y \otimes y\Big) z \overline{w} \\ &+& \frac{1}{8}\Big( 3 x \otimes x + y \otimes y \Big) z^2 \overline{w}^2 + \frac{1}{8} \sum_{k=3}^{\infty} \big(x \otimes x + y \otimes y\big) z^k \overline{w}^k
~(z, w \in \mathbb D_{r_{\lambda}}),
\end{eqnarray*}
where, in view of \end{equation}ref{tri-rad}, $r_{\lambda}$ can be easily seen to be equal to $1$.
Also, one may easily deduce from the proof of Theorem {\rm Re\,}f{thm1}(iv) that for all integers $i \geq 1$,
\begin{eqnarray*} U_{e_{(j, i)}}(z)=(P_ES_{\lambda'}^{*i-1}e_{(j, i)})z^{i-1} + (P_ES_{\lambda'}^{*i}e_{(j, i)})z^{i},~j=1, 2.\end{eqnarray*}
It is now easy to see that the orthonormal basis for the reproducing kernel Hilbert space $\mathscr H$ associated with $\kappa_{\mathscr H}$ is given by
\begin{eqnarray*}
\{x, p(z), zp(z)\} \cup \left \{\frac{1}{\sqrt{2}}z^{k-1}p(z)\right \}_{k \geq 3} \cup \{q(z)\} \cup \left \{\frac{1}{\sqrt{2}}z^{k-1}q(z)\right \}_{k \geq 2},
\end{eqnarray*}
where $p(z) = \frac{1}{2}(y+ x z)$ and $q(z) = \frac{1}{2}(xz -y)$ are linear $E$-valued polynomials.
\end{example}
\begin{example}[(Pentadiagonal)] \label{pentadiagonal}
Consider the directed tree $\mathscr T_3$ with set of vertices $V=\{(0,0),
(1,1)\}\cup\{(2,i), (3,i):i\geq1\}$ and $\mathsf{root}=(0,0)$.
We further require that $\mathsf{Chi}(0,0)=\{(1,1)\}$, $\mathsf{Chi}(1,1)=\{(2,1),(3,1)\}$ and
\[\mathsf{Chi}(2,i)=\{(2,i+1)\},\
\mathsf{Chi} (3,i)=\{(3,i+1)\},\ \text{for all}\ i\geq1. \]
Let $S_{\lambda}$ be a left-invertible weighted shift on $\mathscr T_3.$
As in the preceding example, one can see that
$$E:= \ker S_\lambda^* = \{\alpha e_{(0,0)} + \beta(\lambda_{(3, 1)}e_{(2,1)}-
\lambda_{(2, 1)}e_{(3,1)}) : \alpha, \beta \in \mathbb C\}.$$
Also $V_\prec=\{(1,1)\}$ and $k_{\mathscr T_3}=2$. Therefore, from Theorem
{\rm Re\,}f{thm1}, $\kappa_\mathcal{H}(\cdot,\cdot)$ takes the form
\[\kappa_\mathcal{H}(z,w)=I_E+\displaystyle\sum_{\underset{|j-k|\leq
2}{j,k\geq1}}C_{j,k}z^j\overline{w}^k~(z, w \in \mathbb D_{r_{\lambda}}).\]
Moreover, for $j\geq1$, $S_{\lambda'}^jE \subseteq \mbox{span}~ \Big\{e_v : v
\in \mathsf{Chi}^{\langle j \mbox{ran}gle} \big\{(0,0), (2,1), (3,1) \big\}\Big\}$.
Therefore,
$$S_{\lambda'}^{*j+1} S_{\lambda'}^j E \subseteq \mbox{span}~
\Big\{e_v : v \in \mathsf{par} \big\{(0,0), (2,1), (3,1)
\big\}\Big\}= \mbox{span}~\{e_{(1,1)}\},$$
which gives that
$P_ES^{*j+1}_{\lambda'}S_{\lambda'}^j|_E=0.$
Thus $C_{j,k}=0$ if $|j-k|=1$. Therefore from above,
$\kappa_\mathcal{H}(\cdot,\cdot)$ becomes
\[\kappa_\mathcal{H}(z,w)=I_E+\displaystyle\sum_{\underset{|j-k|=0,
2}{j,k\geq1}}C_{j,k}z^j\overline{w}^k~(z, w \in \mathbb D_{r_{\lambda}}).\]
Since $W_n = \{(2, n-1), (3, n-1), (2, n), (3, n), (2,
n+1), (3, n+1)\}$ for $n
\geq 2,$ the radius of convergence $r_{\lambda}$ for $S_{\lambda}$ is given by \begin{eqnarray*}
\liminf_{n \rightarrow \infty} \left(\sum_{j=2}^3 \left[\big(\lambda'_{(j, n-1)} \cdots
\lambda'_{(1, 1)}\big)^2 + \big(\lambda'_{(j, n)} \cdots
\lambda'_{(j, 1)}\big)^2 + \big(\lambda'_{(j, n+1)} \cdots
\lambda'_{(j, 2)}\big)^2 \right]\right)^{-\frac{1}{2n}}.
\end{eqnarray*}
In this case, the reproducing kernel $\kappa_\mathcal{H}(\cdot,\cdot)$ is
{\it pentadiagonal}.
\end{example}
The invariant $k_{\mathscr T}$ may be bigger than $\dim E$ as shown below.
\begin{example}[(Septadiagonal)] Consider the directed tree $\mathscr T_4$ with set of vertices $V=\{(0,0),
(1, 1), (2, 2)\}
\cup\{(3,i), (4,i):i\geq1\}$ and $\mathsf{root}=(0,0)$.
We further require that
$\mathsf{Chi}(0,0)=\{(1,1)\}$, $\mathsf{Chi}(1,1)=\{(2,2)\}$, $\mathsf{Chi}(2,2)=\{(3,1), (4, 1)\}$, and
\[\mathsf{Chi}(3,i)=\{(3,i+1)\},\
\mathsf{Chi} (4,i)=\{(4,i+1)\},\ \text{for all}\ i\geq1. \]
It is easy to see that $\dim\, \ker S^*_{\lambda} = 2, k_{\mathscr T_4}=3$.
The kernel $\kappa_{\mathscr H}$ in this example is septadiagonal.
We leave the details to the reader.
\end{example}
The main result also applies to a directed tree which is not locally finite.
\begin{example} \label{tree-inf}
Consider the directed tree $\mathscr T_\infty$ with set of vertices
$V=\{(i,j): i, j \geq 0\},$ and $\mathsf{root}=(0,0)$. We further require that
\begin{center}
$\mathsf{Chi}(i,j)=\begin{cases}
\{(1,k) : k \geq 0\}& \text{if}\ (i,j) = \mathsf{root},\\
\{(i+1, j)\}& \text{otherwise}.
\end{cases}$
\end{center}
Let $S_\lambda$ be a bounded left-invertible weighted shift on $\mathscr
T_\infty$. Then $E=\ker S_\lambda^*$ is of infinite dimension. Also
$V_\prec=\{\mathsf{root}\}$, and hence $k_{\mathscr T_\infty}=1$.
By Theorem {\rm Re\,}f{thm1}, the reproducing kernel
$\kappa_\mathscr
H$ is tridiagonal.
\end{example}
\begin{remark} \label{direct-sum}
In general, $S_{\lambda}$ is not unitarily equivalent to orthogonal direct sum of unilateral weighted shifts. To see this, consider the weighted shift $S_{\lambda}$ on $\mathscr T_{\infty},$ and
suppose that $S_{\lambda}$ is unitarily equivalent to direct sum $T:=\oplus_{i=1}^{\infty}T_i$ of unilateral weighted shifts $T_i.$ Choose weights of $S_{\lambda}$ such that $\lambda_{(2, 0)} \neq \lambda_{(2, 1)}.$ In this case,
$$\inp{S^2_{\lambda}g_1}{S_{\lambda}g_2} = \lambda_{(1, 0)}\lambda_{(1, 1)}(\lambda^2_{(2, 0)} - \lambda^2_{(2, 1)}) \neq 0,$$
where $g_1 = e_{\mathsf{root}}$ and $g_2 = \lambda_{(1, 1)}e_{(1, 0)} - \lambda_{(1, 0)}e_{(1, 1)}$ belong to $\ker S^*_{\lambda}$.
However, $\inp{T^mX}{T^nY} = 0$ for any $X, Y \in \ker T^*$ and for any positive integers $m, n$ such that $m \neq n.$
\end{remark}
{\small
\begin{table}[H]
\caption{}
\begin{center}
\begin{tabular}{| c | c | c | c |}
\hline
Directed Tree $\mathscr T$ & Dimension of $\ker S^*_{\lambda}$ &
$k_{\mathscr T}$ & Form of $\kappa_{\mathscr H}(z, w)$ \\ \hline
$\mathscr T_1$ & $1$ & $0$ & \mbox{diagonal} \\ \hline
$\mathscr T_2$ & $2$ & $1$ & \mbox{tridiagonal} \\ \hline
$\mathscr T_{3}$ & $2$ & $2$ & \mbox{pentadiagonal} \\ \hline
$\mathscr T_{4}$ & $2$ & $3$ & \mbox{septadiagonal} \\ \hline
$\mathscr T_{\infty}$ & $\infty$ & $1$ & \mbox{tridiagonal}\\ \hline
\end{tabular}
\end{center}
\end{table}
}
\section{Spectral Picture of $S_{\lambda}$}
In this section, we use analytic model constructed in Sections 2 and 3 to discuss spectral theory of weighted shifts $S_{\lambda}$ on rooted directed trees. This part has an overlap with \cite[Theorems 2.1 and 2.3]{C},
where the spectral picture of certain weighted composition operators is described. However,
the conclusion of (i)-(iii) of Theorem {\rm Re\,}f{spectral} can not be deduced from the aforementioned results of \cite{C} as the directed trees considered in this part need not be locally finite. On the other hand, in the context of rooted directed trees, weighted shifts always have connected spectrum. This is in contrast with \cite[Example 5]{C}, where a composition operator with disconnected spectrum has been constructed.
Positively, the power of analytic model comes into the picture while computing the point spectra of $S_{\lambda}$ and $S^*_{\lambda}$. In this regard, the rather technical proof of \cite[Theorem 2.1]{C}
should be compared with that of (i) and (ii) of Theorem {\rm Re\,}f{spectral}.
Before we state the main result of this section, we recall a couple of known facts about $S_{\lambda}$.
Any weighted shift $S_{\lambda}$ on a directed tree is {\it circular} \cite[Theorem 3.3.1]{Jablonski}:
{\it For every $\theta \in \mathbb R$, there exists a unitary $U_\theta$ on $l^2(V)$ such that $U_\theta S_{\lambda} = e^{i \theta} S_{\lambda} U_\theta$.} An immediate consequence of this shows that all spectral parts of $S_{\lambda}$ have circular symmetry about $0$ \cite[Corollary 3.3.2]{Jablonski}.
Here is the statement of the main result of this section.
\begin{theorem} \label{spectral}
Let $S_\lambda \in B(l^2(V))$ be a left-invertible weighted shift on $\mathscr T$ and let $E:={\ker} S_\lambda^*$.
Then we have the following.
\begin{enumerate}
\item[(i)] The point spectrum $\sigma_p(S_\lambda)$ of $S_{\lambda}$ is empty.
\item[(ii)] If $r_\lambda$ is the radius of convergence for $S_{\lambda}$ then $$\mathbb D_{r_\lambda} \subseteq \sigma_p(S^*_{\lambda}) \subseteq \sigma(S_{\lambda}) =
\overline{\mathbb D}_{r(S_{\lambda})}.$$
\item[(iii)]
$\bigvee\{\ker (S_\lambda^*- w): w \in \mathbb D_\epsilonilon\}=l^2(V)$ for every positive number $\epsilonilon$.
\end{enumerate}
If, in addition, $E$ is finite dimensional then
\begin{enumerate}
\item[(iv)] $\sigma_{ap}(S_{\lambda})=\sigma_e(S_{\lambda})$ is a union of at most $\dim E$ number of annuli centered at the origin.
\item[(v)]
the Fredholm index $\mbox{ind}\, (S_{\lambda}-w)$ of $S_{\lambda}-w$ is at least $-\dim E$ on any connected component of $\mathbb C \setminus \sigma_e(S_{\lambda})$.
Moreover, $\mbox{ind}\, (S_{\lambda}-w)$ is exactly $-\dim E$ on the connected component of $\mathbb C \setminus \sigma_e(S_{\lambda})$ that contains $0$.
\item[(vi)] for any positive integer $k,$ $$\dim \big(\ker S^{*k}_\lambda/\ker S^{*k-1}_\lambda \big)=\dim E.$$
\end{enumerate}
\end{theorem}
\begin{remark}
Since
$r_\lambda r(S_{\lambda'})\geq1$ (Theorem {\rm Re\,}f{thm1}),
by the inclusion in (ii), $r(S_\lambda)r(S_{\lambda'})\geq1$.
This inequality is sharp. In fact, if $S_{\lambda}$ is an isometry then $r(S_\lambda)=1=r(S_{\lambda'}),$ so that equality holds in $r(S_\lambda)r(S_{\lambda'})\geq1$. Also, if $r(S_\lambda)=1=r(S_{\lambda'})$ then $r_{\lambda}$ is necessarily equal to $1.$ Finally, since $S_{\lambda}$ is analytic, the part (vi) above precisely says that $S^*_{\lambda}$ is an abstract backward shift in the sense of \cite{B-1} and \cite{R-1}.
\end{remark}
In the proof of Theorem {\rm Re\,}f{spectral}, we need the analytic model as well as a number of general facts about $S_{\lambda}$.
The first of which generalizes a well-known fact that the spectrum of a weighted shift is connected \cite[Theorem 4]{S}(see also \cite[Theorem 8]{G}, \cite[Theorem 3.5]{R-1}).
\begin{lemma} \label{connected}
The spectrum of an analytic operator is connected.
\end{lemma}
\begin{proof}
Let $T \in B(\mathcal H)$ be analytic. We adapt the technique
of \cite[Lemma 3.8]{Ch-Ya} to the present situation. Since
$T$ is analytic,
\begin{equation}\label{eq1}
\displaystyle\bigvee_{k\geq0}{\ker}\, T^{*k}=\mathcal H.
\end{equation}
Therefore, $0 \in \sigma(T).$
Let $K_1$ be the connected component of $\sigma(T^*)$ containing 0 and
$K_2=\sigma(T^*)\setminus K_1$.
If possible, suppose that $K_2$ is non-empty. Then by Riesz
Decomposition Theorem \cite[Chapter VII, Proposition 4.11]{Conway}, there are closed subspaces $\mathcal H_1$ and
$\mathcal H_2$ invariant under $T^*$ such that
$\mathcal H=\mathcal H_1 \oplus \mathcal H_2$,
$\sigma({T^*}|_{\mathcal H_1})=K_1$ and $\sigma({T^*}|_{\mathcal H_2})=K_2$.
Let
$h\in {\ker} {T^*}^{k}$. Then
$h=x+y$ for $x\in \mathcal H_1$ and $y\in \mathcal H_2$. Since $T^{*k}h=0$, it follows
that $T^{*k}x=0=T^{*k}y$. If $y$ is
non-zero, then
$0\in \sigma_p({T^{*k}}|_{\mathcal H_2})
\subseteq\sigma({T^{*k}}|_{\mathcal H_2}),
$ and hence
by spectral mapping property,
$0\in\sigma({T^*}|_{\mathcal H_2})=K_2$, which is a contradiction.
So $y$ must
be
zero. Therefore, $\mathcal H_1$ contains $\ker
T^{*k}$ for all $k \geq 0$. Hence from \end{equation}ref{eq1}, we get
$\mathcal H_1=\mathcal H$, and hence $K_2$ must be empty. This is contrary to the assumption that $K_2 \neq \emptyset$. This shows that $\sigma(T^*)$ is connected. Since $\sigma(T) = \{\overline{z} : z \in \sigma(T^*)\}$ and $z \rightsquigarrow \overline{z}$ is continuous, $\sigma(T)$ is connected.
\end{proof}
\begin{lemma}
\label{decomposition}
Let $S_{\lambda}$ be a weighted shift on $\mathscr T$ and let $d:=\mbox{card}(\childn{k_{\mathscr T}}{\mathsf{root}})$ (possibly infinite).
Then there exist subspaces $\mathcal M$ and $\mathcal H_i~(i=1, \cdots, d)$ such that \begin{eqnarray} \label{deco}
S_{\lambda}=\left[\begin{array}{ccccc}
A & 0 & 0 & \cdots & 0 \\
A_1 & S_1 & 0 & \cdots & 0 \\
A_2 & 0 & S_2 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
A_d & 0 & \cdots & 0 & S_d
\end{array}\right] ~\mbox{on~}l^2(V) = \mathcal M \oplus \mathcal H_1 \oplus \cdots \oplus \mathcal H_d,
\end{eqnarray}
where $A:=P_{\mathcal M}S_{\lambda}|_{\mathcal M},$ $A_i:=P_{\mathcal H_i}S_{\lambda}|_{\mathcal M},$ and $S_i:=S_{\lambda}|_{\mathcal H_i}$ for $i=1, \cdots, d.$ Moreover, the following statements hold.
\begin{enumerate}
\item[(i)] Each $S_i$ is unitarily equivalent to a unilateral weighted shift.
\item[(ii)]
Let $\mathscr T$ possess the property that
$v \in \childn{k_{\mathscr T}-1}{\mathsf{root}}$ whenever $\mbox{card}(\child v)$ is infinite for some $v \in W_0.$ Then
$S_{\lambda}$ is a finite rank perturbation of $S_1 \oplus \cdots \oplus S_d.$
\end{enumerate}
\end{lemma}
\begin{proof}
Note that for all $v \in \childn{k_{\mathscr T}}{\mathsf{root}}$, $\mbox{card}(\child v)=1.$ Let $W_{-1}$ be as defined in \end{equation}ref{Wn}.
We relabel the set $V$ of vertices as follows:
\begin{eqnarray*} \label{label}
V= W_{-1} \sqcup \{v_{i, n} : n \geq 0,\ i=1, \cdots, d\}
\end{eqnarray*}
such that $\childn{k_{\mathscr T}}{\mathsf{root}}=\{v_{i, 0} : i=1, \cdots, d\}$, and
$\child {v_{i, n}} = \{v_{i, n+1}\}$ for all $n \geq 0,$ $i=1, \cdots, d.$
Now consider the subspaces $\mathcal M$ and $\mathcal H_i$ of $l^2(V)$ given by
\begin{eqnarray*}
\mathcal M:=\bigvee\,\{e_v : v \in W_{-1}\},\ \mathcal H_i:=\bigvee \{e_{v_{i, n}} : n \geq 0\},\ i=1, \cdots, d.
\end{eqnarray*}
Note that
the subspaces $\mathcal H_1, \cdots, \mathcal H_d$ are invariant under $S_{\lambda}.$
Then $S_{\lambda}$ admits the decomposition as given by \end{equation}ref{deco}.
Since $S_ie_{v_{i, n}} = \lambda_{v_{i, n+1}} e_{v_{i, n+1}}~(n \geq 0),$ it is clear that each $S_i$ is unitarily equivalent to a unilateral weighted shift.
To see (ii), note that if $\mbox{card}(W_{-1})$ is infinite then for some $v \in W_{-1} \subseteq W_0$,
we must have $\child v \subseteq W_{-1}$ and $\mbox{card}(\child v)$ is infinite. But then by hypothesis, $v \in \childn{k_{\mathscr T}-1}{\mathsf{root}}$, which implies that $\child v \cap W_{-1} = \emptyset.$ Thus we arrive at a contradiction. This shows that $\mbox{card}(W_{-1})$ is finite, and hence $\mathcal M$ is finite dimensional. Thus $A, A_1, \cdots, A_d$ are finite rank operators, and the conclusion in (ii) is now immediate.
\end{proof}
The following is certainly known. We include it for the sake of completeness.
\begin{lemma} \label{f-cyclic} Let $T \in B(\mathcal H)$ be finitely cyclic.
If $\sigma_p(T)$ is empty then $\sigma_{ap}(T)=\sigma_e(T)$.
\end{lemma}
\begin{proof}
By \cite[Proposition 1(i)]{H}, $\dim \ker(T^*-w)$ is finite for every $w \in \mathbb C.$ If $\sigma_p(T)=\emptyset$ then it is easy to see that $\sigma_{ap}(T)=\sigma_e(T)$.
\end{proof}
We also need exact description of the kernel of positive integral powers of $S^*_{\lambda}$ in the proof of Theorem {\rm Re\,}f{spectral}.
\begin{lemma}\label{lemABS}
Let $S_{\lambda} \in B(l^2(V))$ be a weighted shift on $\mathscr T=(V, \mathcal E)$. Then, for all integers $k \geq 1$,
\begin{eqnarray} \label{kernelk}
\ker S^{*k}_\lambda = \bigvee\Big \{e_v : v \in \cup_{i=0}^{k-1} \childn{i}{\mathsf{root}}\Big \} \oplus \bigoplus_{v \in W_{-1}}\Big(l^2 \big(\childn{k}{v} \big) \ominus \langle \lambdab^v_k \mbox{ran}gle\Big),
\end{eqnarray}
where $\lambdab^v_k : \childn{k}{v} \rightarrow \mathbb C~\mbox{is defined by~}
\lambdab^v_k(u)= \lambda_u \lambda_{\parent u} \cdots \lambda_{\parentn{k-1}{u}},$ and $W_{-1}$ is given by \end{equation}ref{Wn}. Consequently,
\begin{eqnarray}\label{dim-kernelk}
\dim \ker S^{*k}_\lambda = \displaystyle \sum_{i=0}^{k-1}\mbox{card}\big(\childn{i}{\mathsf{root}}\big) + \sum_{v \in W_{-1}}\Big(\mbox{card}\big(\childn{k}{v}\big)-1\Big).
\end{eqnarray}
\end{lemma}
\begin{proof}
Following the lines of the proof of \cite[Proposition 3.5.1]{Jablonski}, one can easily deduce that for all integers $k \geq 1$,
$$\ker S^{*k}_\lambda = \bigvee\Big \{e_v : v \in \cup_{i=0}^{k-1} \childn{i}{\mathsf{root}}\Big \} \oplus \bigoplus_{v \in V}\Big(l^2 \big(\childn{k}{v} \big) \ominus \langle \lambdab^v_k \mbox{ran}gle\Big).$$
From Lemma {\rm Re\,}f{branch}(ii), we know that for a vertex $v \in V$, $\mbox{card}(\child v)=1$ if $n_v \geq k_\mathscr T$. This is equivalent to the fact that $\mbox{card}(\childn{m}{v})=1$ for all $m \geq 1$ if $v \notin W_{-1}$. Hence, $l^2 \big(\childn{k}{v} \big) \ominus \langle \lambdab^v_k \mbox{ran}gle=\{0\}$ if $v \notin W_{-1}$. This proves \end{equation}ref{kernelk}. The proof of \end{equation}ref{dim-kernelk} is obvious in view of \end{equation}ref{kernelk}.
\end{proof}
\begin{proof}[Proof of Theorem {\rm Re\,}f{spectral}]
In
view of Theorem {\rm Re\,}f{thm1}, it is sufficient to work with the analytic model $({\mathscr
M}_z,
\kappa_{\mathscr H},\mathscr H)$ of $S_{\lambda},$ where the reproducing kernel
Hilbert space $\mathscr H$ consists of
$E$-valued holomorphic functions $U_f$ on the disc
$\mathbb D_{r_{\lambda}}$ given by \begin{eqnarray*}
U_f(z):=\displaystyle \sum_{n\geq0}(P_ES_{\lambda'}^{*n}f)z^n. \end{eqnarray*}
We check that $\sigma_p(\mathscr M_z)=\emptyset.$
Let $w \in \mathbb C$ and $h = \sum_{k=0}^{\infty} a_n z^n \in \mathscr H$ be such that
$(\mathscr M_z - w) h = 0,$ where $\{a_n\}_{n \geq 0} \subseteq E.$
Then for any $g \in E,$
\begin{eqnarray*}
(z-w)\sum_{k=0}^{\infty} \inp{a_n}{g} z^n = 0~\mbox{for all~}z \in \mathbb D_{r_{\lambda}}.
\end{eqnarray*}
It follows that $\inp{a_n}{g}=0$ for $g \in E,$ and hence $a_n=0$ every $n \geq 0$. This shows that $h=0,$ which gives (i).
To see (ii),
note that for $f\in l^2(V)$, $g\in E$ and $w \in\mathbb{D}_{r_{\lambda}}$, by Theorem
{\rm Re\,}f{thm1}(i),
\[\langle U_f, {\mathscr M}_z^*\kappa_{\mathscr H}(\cdot,w)g\mbox{ran}gle=\langle
{\mathscr M}_zU_f, \kappa_{\mathscr H}(\cdot, w)g\mbox{ran}gle=\langle w U_f(w),
g\mbox{ran}gle
=\langle U_f, \overline{w}\kappa_{\mathscr H}(\cdot,w)g\mbox{ran}gle.\]
Thus ${\mathscr M}_z^*\kappa_{\mathscr H}(\cdot, w)g=\overline{w}k_\mathscr
H(\cdot,w)g$ for
all $w \in\mathbb{D}_{r_{\lambda}}$ and $g\in E$. Hence the point spectrum of ${\mathscr
M}_z^*$ contains $\mathbb D_{r_{\lambda}}$.
As recorded earlier, $S_{\lambda}$ is circular, so that
$\sigma(S^*_{\lambda})=\sigma(S_{\lambda}).$
The second inclusion in (ii) now follows from $\sigma_p(S^*_{\lambda}) \subseteq \sigma(S^*_{\lambda}).$ To complete the proof of (ii), it is only left to check
that $\sigma(S_{\lambda}) =
\overline{\mathbb D}_{r(S_{\lambda})}$. By Lemmas {\rm Re\,}f{lem1} and {\rm Re\,}f{connected}, $\sigma(S^*_{\lambda})$ is
connected.
Now, suppose that
$\sigma(S_\lambda)\neq\overline{\mathbb{D}}_{r(S_\lambda)}$. Since
$r(S_{\lambda}) \in \sigma(S_{\lambda}),$
there is a $w_0 \in {\mathbb{D}}_{r(S_\lambda)}$ such that
$w_0 \in \rho(S_\lambda):=\mathbb C \setminus \sigma(S_{\lambda}).$ Since $\rho(S_\lambda)$
is open, there is
an $\epsilonilon > 0$ such that
$\mathbb{D}_\epsilonilon(w_0):=\{w \in\mathbb{C}:|w-w_0|< \epsilonilon\}
\subseteq\rho(S_\lambda) \cap {\mathbb{D}}_{r(S_\lambda)}$. Since
$S_{\lambda}$ is circular and $\sigma(S_{\lambda})$ is connected, we arrive at
a contradiction.
Thus, the
spectrum of $S_\lambda$ is a disk
of radius $r(S_\lambda)$ centred at the origin.
Suppose that
$U_f$ is orthogonal to $\bigvee\{\ker ({\mathscr
M}_z^*-\overline{w}):w\in\mathbb{D}_\epsilonilon\}$. By the preceding paragraph,
$\kappa_{\mathscr H}(\cdot,w)g$ belongs to $\ker ({\mathscr
M}_z^*-\overline{w})$ for every $w \in \mathbb{D}_{r_{\lambda}}.$ Hence
$$
\sum_{n\geq0}\langle
P_ES_{\lambda'}^{*n}f,g\mbox{ran}gle w^n = \langle
U_f(w),g\mbox{ran}gle = \langle
U_f,\kappa_{\mathscr H}(\cdot,w)g\mbox{ran}gle=0$$
$\mbox{for all~} w \in \mathbb
D_\epsilonilon \cap \mathbb D_{r_{\lambda}}~\mbox{and~}
g\in E.$
This implies that $\langle P_ES_{\lambda'}^{*n}f,g\mbox{ran}gle=0$ for all
$n\geq0$ and $g\in E$. In particular, $\langle
P_ES_{\lambda'}^{*n}f,P_ES_{\lambda'}^{*n}f\mbox{ran}gle=0$ for all $n\geq0$. Thus
$U_f=0$. Therefore,
$\bigvee\{{\ker}({\mathscr M}_z^*-\overline{w}):w\in\mathbb
D_\epsilonilon\}=\mathscr
H$.
Assume now that $E$ is finite dimensional. By Corollary {\rm Re\,}f{cyclic}, $S_{\lambda}$ is finitely cyclic.
By (i) above and Lemma {\rm Re\,}f{f-cyclic}, $\sigma_{ap}(S_{\lambda}) = \sigma_e(S_{\lambda}).$
Thus to see (iv), it suffices to check that $\sigma_{e}(S_{\lambda})$ is a finite union of annuli centered at the origin.
Let $d:=\mbox{card}(\childn{k_{\mathscr T}}{\mathsf{root}}),$ which is finite in view of Proposition {\rm Re\,}f{probranch}.
Also, since $E$ is finite dimensional, by the same proposition $\mathscr T$ is locally finite.
Hence by Lemma {\rm Re\,}f{decomposition}(ii), there exist unilateral weighted shifts $S_1, \cdots, S_d$ such that
$S_{\lambda}$ is a finite rank perturbation of $S_1 \oplus \cdots \oplus S_d.$
In particular, the essential spectrum of $S_{\lambda}$ equals the union of
essential spectrum of $S_1, \cdots, S_d$ \cite{Conway}. Another application of Lemma {\rm Re\,}f{f-cyclic} shows that $\sigma_{ap}(S_i)=\sigma_e(S_i)$. However, the approximate point spectrum of a unilateral weighted shift is necessarily an annulus centered at the origin \cite[Theorem 1]{R}. The desired conclusion in (iv) is now immediate.
Let us now see part (v). For any $w \in \mathbb C,$ note that \begin{eqnarray*} \mbox{ind}\, (S_{\lambda} - w) = \mbox{ind}\, \oplus_{i=1}^d (S_{i}-w)
\overset{\mbox{(i)}}= - \oplus_{i=1}^d \dim \ker (S^*_{i}-\overline{w}). \end{eqnarray*}
Since $\dim \ker (S^*_{i}-\overline{w})$ at most one, $\mbox{ind}\, (S_{\lambda} - w)$ is at least $-\dim E.$ However,
$\dim \ker S^*_{i}=1$ for all $i$, and hence $-\dim E = \mbox{ind}\, S_{\lambda}= -d.$ Note that the proof above shows that $\mbox{card}(\childn{k_{\mathscr T}}{\mathsf{root}}) = \dim E$, and hence by Lemma {\rm Re\,}f{branch}(ii),
\begin{eqnarray} \label{chi-ker} \mbox{card}(\childn{k}{\mathsf{root}}) = \dim E~\mbox{for all integers~}k \geq k_{\mathscr T}.\end{eqnarray}
To see (vi), fix an integer $k \geq 1$ and let
$\mathscr Q_k:=\ker S^{*k}_\lambda/\ker S^{*k-1}_\lambda$.
Then using \end{equation}ref{dim-kernelk}, we get
\begin{eqnarray} \label{quotnt}
\dim \mathscr Q_k
&=& \mbox{card}\big(\childn{k-1}{\mathsf{root}}\big)+ \sum_{v \in W_{-1}}\mbox{card}\big(\childn{k}{v}\big)
-\sum_{v \in W_{-1}}\mbox{card}\big(\childn{k-1}{v}\big)\nonumber\\
&=&\sum_{v \in W_{-1}}\mbox{card}\big(\childn{k}{v}\big)-\sum_{v \in W_{-1}\setminus \{\mathsf{root}\}}\mbox{card}\big(\childn{k-1}{v}\big)
\end{eqnarray}
Since $\childn{l}{\mathsf{root}} = \childn{l-1}{\child{\mathsf{root}}}$, it follows that $$\mbox{card}\big(\childn{l}{\mathsf{root}}\big) = \displaystyle \sum_{v \in \child{\mathsf{root}}} \mbox{card}\big(\childn{l-1}{v}\big)$$ for any positive integer $l.$
Therefore, $\sum_{v \in W_{-1}}\mbox{card}\big(\childn{k}{v}\big)$ is equal to
\begin{eqnarray*}
\mbox{card} \big(\childn{k}{\mathsf{root}}\big) + \displaystyle \sum_{v \in \child{\mathsf{root}}} \mbox{card}\big(\childn{k}{v}\big)+ \cdots + \displaystyle \sum_{v \in \childn{k_\mathscr T-1}{\mathsf{root}}} \mbox{card}\big(\childn{k}{v}\big)\\
= \mbox{card}\big(\childn{k}{\mathsf{root}}\big) + \mbox{card}\big(\childn{k+1}{\mathsf{root}}\big)+ \cdots + \mbox{card}\big(\childn{k+k_\mathscr T-1}{\mathsf{root}}\big).
\end{eqnarray*}
Similarly, $\sum_{v \in W_{-1}\setminus \{\mathsf{root}\}}\mbox{card}\big(\childn{k-1}{v}\big)$ is equal to
\begin{eqnarray*}
\mbox{card}\big(\childn{k}{\mathsf{root}}\big)+\mbox{card}\big(\childn{k+1}{\mathsf{root}}\big)+ \cdots + \mbox{card}\big(\childn{k-1+k_\mathscr T-1}{\mathsf{root}}\big).
\end{eqnarray*}
Substituting last two identities in \end{equation}ref{quotnt}, we get
$$\dim \mathscr Q_k= \mbox{card}\big(\childn{k+k_\mathscr T-1}{\mathsf{root}}\big) \overset{\end{equation}ref{chi-ker}}= \dim E.$$
This completes the proof of the theorem.
\end{proof}
\begin{remark}
The identity \end{equation}ref{chi-ker}, as established in the proof of Theorem {\rm Re\,}f{spectral}(v), comes surprisingly as a consequence of index theory.
As evident, this identity is otherwise difficult to disclose. Note that the left hand side of \end{equation}ref{chi-ker} is a variant dependent on $\mathscr T$ while the right hand side of \end{equation}ref{chi-ker} depends solely on $S_{\lambda}$.
Further, since $\dim E$ is finite, by Proposition {\rm Re\,}f{probranch}, $\mathscr T$ is locally finite and $\mbox{card}(V_\prec) < \infty.$ Therefore, using \end{equation}ref{kernel} and \end{equation}ref{chi-ker}, one gets the following.
\begin{eqnarray*}
\mbox{card}(\childn{k_{\mathscr T}}{\mathsf{root}}) = 1- \mbox{card}(V_\prec)+\sum_{v \in V_\prec}\mbox{card}(\child v).
\end{eqnarray*}
\end{remark}
One particular consequence of Theorem {\rm Re\,}f{spectral}(iv) is that $\sigma_{ap}(S_{\lambda})$ (resp. $\sigma_{e}(S_{\lambda})$) of a weighted shift $S_{\lambda}$ on a directed tree could be {\it disconnected}.
For instance, in case $\dim E=2,$ by choosing the weight sequence $\lambda$ appropriately (so that the approximate point spectra of $S_1$ and $S_2$, as appearing in the proof of Theorem {\rm Re\,}f{spectral}, are disjoint annuli), we can have two connected components of $\sigma_{ap}(S_{\lambda})$. Moreover, the index of $S_{\lambda}- w$ may vary from $-2$ to $0$ on different components of $\mathbb C \setminus \sigma_e(S_{\lambda})$. Again, in the above situation,
\begin{eqnarray*}
\mbox{ind}\, (S_{\lambda} - w) =\begin{cases} -2~\mbox{on a bounded component of}~\mathbb C \setminus \sigma_e(S_{\lambda})~\mbox{containing}~0 \\
-1~\mbox{on a bounded component of}~\mathbb C \setminus \sigma_e(S_{\lambda})~\mbox{not containing}~0.
\end{cases}
\end{eqnarray*}
This is not possible in case $\dim E =1$ in view of \cite[Theorem 1]{R}.
The conclusion of Theorem {\rm Re\,}f{spectral}(iv) need not be true in case $\dim E$ is infinite.
\begin{example} Let $\mathscr T_{\infty}$ be the directed tree as discussed in Example {\rm Re\,}f{tree-inf} and let $S_{\lambda}$ be a left-invertible weighted shift on $\mathscr T_{\infty}.$
For a given $\mu > 0,$ choose the weight sequence of $S_{\lambda}$ such that for each $j \geq 0,$ the sequence $\{\lambda_{(i+1, j)}\}_{i \geq 0}$ converges to $\mu.$ As seen in the proof of Lemma {\rm Re\,}f{decomposition}, $S_{\lambda}$ is a rank one perturbation of the direct sum of unilateral weighted shifts $S_j$ on $\mathcal H_j$. Thus $\sigma_e(S_{\lambda}) = \sigma_e(\oplus_{j=1}^{\infty}S_j).$
Note that $\sigma_e(S_j)=\sigma_{ap}(S_j)$ is the circle of radius $\mu$ centered at the origin \cite{S}. Since
$\sigma_p(S^*_j)=\mathbb D_{\mu}$,
the the essential spectrum of $\oplus_{j=1}^{\infty}S_j$ contains $\mathbb D_{\mu}$. As essential spectrum is always closed, $\overline{\mathbb D}_\mu \subseteq \sigma_e(\oplus_{j=1}^{\infty}S_j).$
Also, $$\sigma_e(\oplus_{j=1}^{\infty}S_j) \subseteq \sigma(\oplus_{j=1}^{\infty}S_j) = \overline{\mathbb D}_\mu.$$
This shows that $\sigma_e(S_{\lambda})=\overline{\mathbb D}_\mu.$
On the other hand, $0 \notin \sigma_{ap}(S_{\lambda})$ since $S_{\lambda}$ is left-invertible. In particular,
$\sigma_{ap}(S_{\lambda}) \neq \sigma_{e}(S_{\lambda}).$
\end{example}
We see below that $S_{\lambda}$ belongs to the Cowen-Douglas class (refer to \cite{C-D}; refer also to \cite{C-S} for the extended definition of $B_n(\Omega)$ in case $n$ is not finite).
\begin{corollary}\label{CD}
Let $S_\lambda \in B(l^2(V))$ be a left-invertible weighted shift on $\mathscr T$ and let $S_{\lambda'}$ denote the Cauchy dual of
$S_{\lambda}$. Let $E:=\ker S_\lambda^*$ and $\delta :=\frac{1}{\|S_{\lambda'}\|}$. Then $S_\lambda^*$ belongs to Cowen-Douglas class
$B_{\dim E}(\mathbb{D}_\delta)$.
\end{corollary}
\begin{proof}
Since $S_\lambda$ is left-invertible,
$$(S_\lambda^*S_\lambda)^{-1}
=S_{\lambda'}^*S_{\lambda'}\leq\|S_{\lambda'}^*S_{\lambda'}\|I.$$
That is, $S_\lambda^*S_\lambda\geq\frac{1}{\|S_{\lambda'}\|^2}I= \delta^2 I$, which gives
$\|S_\lambda f\|\geq\delta\|f\|~\mbox{for~ all~} f\in
l^2(V).$
Therefore, $\sigma_{ap}(S_{\lambda}) \cap \mathbb D_{\delta} = \emptyset.$
It follows that
for
all $w \in \mathbb{D}_\delta$, $\ker (S_\lambda-w)=\{0\}$ and
$\mbox{ran}\, (S_\lambda-w)$ is closed. Hence
$\mbox{ran}\, (S^*_\lambda-w)$ is dense in $\mathscr
H$ for all
$w\in\mathbb{D}_\delta$. Since $\mbox{ran}\, (S_\lambda-w)$ is closed, it
follows
that $\mbox{ran}\, (S^*_\lambda-\overline{w})$ is closed
\cite[Chapter XI, Section 6]{Conway}, and hence
$\mbox{ran}\, (S^*_\lambda-\overline{w})=l^2(V)$ for all
$w\in\mathbb{D}_\delta$. In case $\dim E < \infty$,
the desired conclusion follows from (iii) and (v) of Theorem {\rm Re\,}f{spectral}.
Suppose now the case in which $\dim E$ is not finite.
Consider the analytic model $({\mathscr
M}_z,
\kappa_{\mathscr H},\mathscr H)$ of $S_{\lambda}.$
We show that
$$\{\kappa_{\mathscr H}(\cdot,w)g_i:i=1,\cdots,\ k\}$$
is linearly independent in $\ker (\mathscr M^*_z - \overline{w})$ whenever $\{g_i:i=1,\cdots,\ k\}$ is linearly independent in $E$ for every integer $k \geq 1$ and $w \in \mathbb D_{r_{\lambda}}$. To this end, suppose that
$\kappa_{\mathscr H}(\cdot,w)g=0$ for some $g\in E$. Then
$\langle
U_g(w),g\mbox{ran}gle = \langle U_g,\kappa_{\mathscr H}(\cdot,w)g\mbox{ran}gle=0$. However, by \end{equation}ref{Uf}, $U_g=g$ for any $g \in E.$ It follows that $g=0$, and $\dim \ker
({\mathscr M}_z^*-\overline{w})=\dim E$ for all $w \in \mathbb{D}_{r_{\lambda}}$.
\end{proof}
\section{A model for weighted shifts on rootless directed trees}
In this short section, we show a way to generalize the main result of this paper to the setting of rootless directed trees.
One interest in the theory of weighted shifts on rootless directed trees is due to the fact that these are composition operators in disguise (see \cite[Lemma 4.3.1]{JBS-1}).
We begin with a counter-part of branching index for rootless directed trees.
\begin{definition}
Let $\mathscr T=(V, \mathcal E)$ be a rootless directed tree and let $V_{\prec}$ be the set of branching vertices of $\mathscr T$.
We say that $\mathscr T$ has {\it finite branching index} if there exists
a smallest non-negative integer $m_{\mathscr T}$ such that
$$\childn{k}{V_{\prec}} \cap V_{\prec} = \emptyset~\mbox{for every integer~} k \geq m_{\mathscr T}.$$
\end{definition}
The role of $\mathsf{root}$ in the notion of the branching index of a rooted directed tree is taken by a special vertex in the context of rootless directed trees with finite branching index as shown below.
\begin{lemma} \label{root}
Let $\mathscr T=(V, \mathcal E)$ be a rootless directed tree with finite branching index $m_{\mathscr T}$.
Then there exists a vertex $\rot \in V$ such that \begin{eqnarray} \label{rot} \mbox{card}(\child{\parentn{k}{\rot}})=1~\mbox{for all integers}~k \geq 1.\end{eqnarray} Moreover, if $V_{\prec}$ is non-empty then there exists a unique $\rot \in V_{\prec}$ satisfying \end{equation}ref{rot}.
\end{lemma}
\begin{proof}
In case $V_{\prec}=\emptyset$, then every vertex of $V$ satisfies \end{equation}ref{rot}. Therefore, we may assume that $V_{\prec}$ contains at least one vertex, say, $u_0$.
On contrary, assume that for every $u \in V_{\prec}$ there exists a positive integer $k_u$ (depending on $u$) such that $$\mbox{card}(\child{\parentn{k_u}{u}})=0~\mbox{or~}\mbox{card}(\child{\parentn{k_u}{u}}) \geq 2.$$ Since $\mathscr T$ is rootless, the first case can not occur. Hence $\mbox{card}(\child{\parentn{k_u}{u}}) \geq 2,$ that is,
$\parentn{k_u}{u} \in V_{\prec}.$
Define inductively $\{u_n\}_{n \geq 0} \subseteq V_{\prec}$ as follows. By assumption, there exists an integer $k_{u_0} \geq 1$ such that $u_1:={\parentn{k_{u_0}}{u_0}} \in V_{\prec}.$
By finite induction, there exist integers $k_{u_1}, \cdots, k_{u_{n-1}} \geq 1$ such that $$u_n:={\parentn{k_{u_0} + k_{u_1} \cdots + k_{u_{n-1}}}{u_0}} \in V_{\prec}.$$
In case $n > m_{\mathscr T},$ $u_0 \in \childn{k_{u_0} + k_{u_1} \cdots + k_{u_{n-1}}}{V_{\prec}} \cap V_{\prec}.$ This is not possible since $\childn{k}{V_{\prec}} \cap V_{\prec} = \emptyset$ for all integers $k \geq m_{\mathscr T}.$
To see the uniqueness part, suppose that there exist distinct vertices $\{\rot_i\}_{i=1}^{N}$ in $V_{\prec}$ satisfying \end{equation}ref{rot}, where either $N$ is a positive integer bigger than $1$ or $N$ is infinite.
It is easy to see with the help of \end{equation}ref{rot} that for integers $i \neq j,$
$\parentn{k_1}{\rot_i} \neq \parentn{k_2}{\rot_j}$ for any non-negative integers $k_1$ and $k_2.$ One may now easily verify that $\mathscr T$ has the separation $\mathscr T = \sqcup_{i=1}^N \mathscr T_i,$ where
\begin{eqnarray*}
\mathscr T_i = \big(\cup_{ k \geq 1} \childn{k}{\rot_i}\big) \cup \big\{\parentn{k}{\rot_i} : k \geq 0 \big\}.
\end{eqnarray*}
Since $\mathscr T$ is connected, we arrive at a contradiction.
\end{proof}
Note that a rootless directed tree $\mathscr T_0$ with empty $V_{\prec}$ is isomorphic to the directed tree with set of vertices $\mathbb Z$ and $\child n = \{n+1\}$ for $n \in \mathbb Z.$ As it is well-known that any weighted shift on $\mathscr T_0$ (to be referred to as {\it bilateral weighted shift}) can be modelled as the operator of multiplication by $z$ on a Hilbert space of formal Laurent series \cite[Proposition 7]{S}, we assume in
the remaining part of this section that $V_{\prec}$ is non-empty.
We refer to the vertex $\rot \in V_{\prec}$ appearing in the statement of
Lemma {\rm Re\,}f{root} as the {\it generalized root} of $\mathscr T.$
The generalized root may not exist in general. For example,
consider the directed tree $\mathscr T$ with set of vertices
$V=\mathbb Z \times \mathbb Z$ such that
\begin{center}
$\mathsf{Chi}(i,j)=\begin{cases}
\{(i,j+1)\}& \text{if}\ j \neq 0,\\
\{(i,j+1), (i+1, j)\}& \text{if}\ j=0
\end{cases}$
\end{center}
(cf. \cite[Example 4.4]{Ja}).
In this case, $V_{\prec}= \{(i, 0) : i \in \mathbb Z\},$ and hence the set $\child{\parentn{k}{(i, 0)}}$ contains precisely two vertices for any integer $k \geq 1$.
With the notion of generalized root, we immediately obtain the following.
\begin{lemma} \label{rootless}
Let $\mathscr T=(V, \mathcal E)$ be a rootless directed tree with finite branching index $m_{\mathscr T}$ and generalized root $\rot.$
Let $S_{\lambda} \in B(l^2(V))$ be a weighted shift on $\mathscr T.$
Let $V^{(2)}:=\{v_k:=\parentn{k}{\rot} : k \geq 1\}$ and let $V^{(1)}:=V \setminus V^{(2)}.$ Let $\mathscr T^{(1)}$ and ${\mathscr T^{(2)}}$ be the directed subtrees corresponding to the sets of vertices $V^{(1)}$ and $V^{(2)}$ respectively.
Then $S_{\lambda}$ admits the following decomposition:
\begin{eqnarray} \label{r-less-deco}
S_{\lambda}=\left[\begin{array}{cc}
T_{\lambda} & \lambda_{\rot} e_{\rot} \otimes e_{v_1} \\
0 & B_{\lambda}
\end{array}\right] ~\mbox{on~}l^2(V) = l^2(V^{(1)}) \oplus l^2(V^{(2)}),
\end{eqnarray}
where $T_{\lambda} \in B(l^2(V^{(1)}))$ is a weighted shift on the rooted directed tree $\mathscr T^{(1)}$ with root $\rot$ and finite branching index $k_{\mathscr T^{(1)}}=m_{\mathscr T}$, and $B_{\lambda} \in B(l^2(V^{(2)})$ is the backward unilateral weighted shift given by \begin{eqnarray*} B_{\lambda} e_{v_k}=\begin{cases}
0&~\mbox{if~}k=1\\
\lambda_{v_{k-1}}e_{v_{k-1}}&~\mbox{if~}k \geq 2.
\end{cases}
\end{eqnarray*}
\end{lemma}
\begin{proof}
Note that $\mathscr T^{(1)}$ is a rooted directed tree with root $\rot.$ Also,
the set $V_{\prec}$ of branching vertices of $\mathscr T$ is contained in $V^{(1)}$ as
$V^{(2)} \cap V_{\prec} = \emptyset.$
It follows that $k_{\mathscr T^{(1)}}=1+\sup\{n_v : v \in V_{\prec}\}.$ Since $m_{\mathscr T}$ is the smallest integer such that
$\childn{m_{\mathscr T}}{V_{\prec}} \cap V_{\prec} = \emptyset,$ we must have $\sup\{n_v : v \in V_{\prec}\}=m_{\mathscr T}-1.$
This shows that $\mathscr T^{(1)}$ has branching index precisely $m_{\mathscr T}$.
Since $S^*_{\lambda}e_{v_k} = \lambda_{v_k}e_{v_{k+1}}$,
$l^2(V^{(2)})$ is invariant under $S^*_{\lambda}$. This gives us the decomposition \begin{eqnarray*}
S_{\lambda}=\left[\begin{array}{cc}
S_{\lambda}|_{l^2(V^{(1)})} & P_1S_{\lambda}|_{l^2(V^{(2)})}\\
0 & P_{2}S_{\lambda}|_{l^2(V^{(2)})}
\end{array}\right] ~\mbox{on~}l^2(V) = l^2(V^{(1)}) \oplus l^2(V^{(2)}),
\end{eqnarray*}
where $P_i$ denotes the orthogonal projection of $l^2(V)$ onto ${l^2(V^{(i)})}$ for $i=1, 2.$
It is easy to see that $P_{1}S_{\lambda}|_{l^2(V^{(2)})}$ is the rank one operator $\lambda_{\rot} e_{\rot} \otimes e_{v_1}.$ That $P_{2}S_{\lambda}|_{l^2(V^{(2)})} =B_{\lambda}$ is also a routine verification.
\end{proof}
\begin{remark}
Note that every weighted shift on a rootless directed tree with finite branching index is an extension of a weighted shift on a rooted direct tree with finite branching index.
\end{remark}
We illustrate the result above with the help of the following simple example.
\begin{example}
Consider the directed tree $\mathscr T$ with set of vertices
$$V:=\{(1,i), (2,i):i\geq1\} \cup \{-k : k \geq 0\}.$$ We further require that $
\mathsf{Chi}(-k) = -(k-1)$ if $k \geq 1,$
$\mathsf{Chi}(0)=\{(1,1),(2,1)\}$ and
\[\mathsf{Chi}(1,i)=\{(1,i+1)\},\
\mathsf{Chi}(2,i)=\{(2,i+1)\},\ \text{for all}\ i\geq1. \]
In this case, the branching index $m_{\mathscr T}=1$ and the generalized root $\rot$ is $0.$ Also, $V^{(1)} =\{(1,i), (2,i):i\geq1\} \cup \{0\}$ and $V^{(2)} = \{-k : k \geq 1\}.$
The weighted shift $T_{\lambda}$ on $\mathscr T^{(1)}$ as defined in the last lemma can be identified with the weighted shift on the directed tree $\mathscr T_2$ (with root $0$) as discussed in Example {\rm Re\,}f{tridiagonal}. Further, the rank one operator $\lambda_{\rot} e_{\rot} \otimes e_{v_1}$ is precisely
$\lambda_0 e_0 \otimes e_{-1}$. Finally,
the backward unilateral weighted shift $B_{\lambda}$ can be identified with the adjoint of the weighted shift on the directed tree $\mathscr T_1$ (with root $-1$) as discussed in Example {\rm Re\,}f{diagonal}.
\end{example}
We now present a counter-part of Theorem {\rm Re\,}f{thm1} for rootless directed trees.
\begin{theorem} \label{deco-rootless}
Let $\mathscr T=(V, \mathcal E)$ be a rootless directed tree with finite branching index and generalized root $\rot.$
Let $S_{\lambda} \in B(l^2(V))$ be a left-invertible weighted shift on $\mathscr T.$ Then there exist a Hilbert space $\mathscr H$ of vector-valued Holomorphic functions in $z$ defined on a disc in $\mathbb C$, and a Hilbert space $\mathcal H$ of scalar-valued holomorphic functions in $t$ defined on a disc in $\mathbb C$ such that $S_{\lambda}$ is unitarily equivalent to
\begin{eqnarray*}
\left[\begin{array}{cc}
{\mathscr M}_z & f \otimes g \\
0 & M^*_t
\end{array}\right] ~\mbox{on~} \mathscr H \oplus \mathcal H,
\end{eqnarray*}
where ${\mathscr M}_z$ is the operator of multiplication by $z$ on $\mathscr H,$ $f \otimes g$ is a rank one operator with
$f \in \ker \mathscr M^*_z \setminus \{0\},$ $g \in \ker M^*_t \setminus \{0\},$ and
$M_t$ is the operator
of multiplication by the co-ordinate function $t$ on $\mathcal H.$
\end{theorem}
\begin{proof}
By Lemma {\rm Re\,}f{rootless}, $S_{\lambda}$ admits the decomposition \end{equation}ref{r-less-deco}. Since $S_{\lambda}$ is left-invertible, so are $T_{\lambda}$ and $B^*_{\lambda}.$ The desired decomposition
now follows immediately from Theorem {\rm Re\,}f{thm1}.
\end{proof}
\begin{remark}
A routine calculation shows that the self-commutator $[S^*_{\lambda}, S_{\lambda}]:=S^*_{\lambda}S_{\lambda} - S_{\lambda}S^*_{\lambda}$ of $S_{\lambda}$ (upto unitary equivalence) is equal to
\begin{eqnarray*}
\left[\begin{array}{cc}
[{\mathscr M}^*_z, \mathscr M_z] - f\otimes f & 0 \\
0 & g\otimes g - [M^*_t, M_t]
\end{array}\right].
\end{eqnarray*}
In particular, $[S^*_{\lambda}, S_{\lambda}]$ is compact if and only if so are $[\mathscr M^*_z, \mathscr M_z]$ and $[M^*_t, M_t].$
\end{remark}
We conclude this paper with one application to the spectral theory of weighted shifts on rootless directed trees (cf. \cite[Theorem 2.3]{C}).
\begin{corollary} With the hypotheses and notations of Theorem {\rm Re\,}f{deco-rootless}, we have \begin{eqnarray*} \sigma_e(S_{\lambda}) = \sigma_e(\mathscr M_z) \cup \sigma_e(M^*_t).\end{eqnarray*}
If, in addition, $S_{\lambda}$ is Fredholm then so are $\mathscr M_z$ and $M_t.$ In this case, \begin{eqnarray*} \mbox{ind}\,S_{\lambda} = \mbox{ind}\,\mathscr M_z + 1.\end{eqnarray*}
\end{corollary}
\textit{Acknowledgment}. \
We express our sincere thanks to Jan Stochel and Zenon Jan Jab{\l}o\'nski for many helpful suggestions.
In particular, we acknowledge drawing our attention to the work \cite{C} on the spectral theory of composition operators. Further, the first author is thankful to the faculty and the administrative unit of School of Mathematics,
Harish-Chandra Research Institute, Allahabad for their warm hospitality during the preparation of this paper.
\end{document}
|
\begin{document}
\mbox{}
\title{Irreducibility and factorizations \\ in monoid rings}
\author{Felix Gotti}
\address{Department of Mathematics\\UC Berkeley\\Berkeley, CA 94720 \newline \indent Department of Mathematics\\Harvard University\\Cambridge, MA 02138}
\email{[email protected]}
\email{[email protected]}
\subjclass[2010]{Primary: 20M25, 13F15; Secondary: 13A05}
\date{\today}
\begin{abstract}
For an integral domain $R$ and a commutative cancellative monoid $M$, the ring consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$ is called the monoid ring of $M$ over $R$. An integral domain is called atomic if every nonzero nonunit element can be written as a product of irreducibles. In the investigation of the atomicity of integral domains, the building blocks are the irreducible elements. Thus, tools to prove irreducibility are crucial to study atomicity. In the first part of this paper, we extend Gauss's Lemma and Eisenstein's Criterion from polynomial rings to monoid rings. An integral domain $R$ is called half-factorial (or an HFD) if any two factorizations of a nonzero nonunit element of $R$ have the same number of irreducible elements (counting repetitions). In the second part of this paper, we determine which monoid algebras with nonnegative rational exponents are Dedekind domains, Euclidean domains, PIDs, UFDs, and HFDs. As a side result, we characterize the submonoids of $(\mathbb{Q}_{\ge 0},+)$ satisfying a dual notion of half-factoriality known as other-half-factoriality.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
Given an integral domain $R$ and a commutative cancellative monoid $M$, the ring of all polynomial expressions with coefficients in $R$ and exponents in $M$ is known as the monoid ring of $M$ over $R$ (cf. group rings). Although the study of group rings dates back to the first half of the twentieth century, it was not until the 1970s that the study of monoid rings gained significant attention. A systematic treatment of ring-theoretical properties of monoid rings was initiated by R.~Gilmer and T.~Parker~\cite{rG74,GP74,GP75} in 1974.
Since then monoid rings have received a substantial amount of consideration and have permeated through many fields under active research, including algebraic combinatorics~\cite{BCMP98}, discrete geometry~\cite{BG02}, and functional analysis~\cite{mA04}. During the last decades, monoid rings have also been studied from the point of view of factorization theory; see, for instance,~\cite{AJ15,AS97,hK01}. Gilmer in \cite{rG84} offers a comprehensive exposition on the advances of commutative semigroup ring theory until mid 1980s.
An integral domain is called atomic if every nonzero nonunit element it contains can be written as a product of irreducibles. Irreducible elements (sometimes called atoms) are the building blocks of atomicity and factorization theory. As a result, techniques to argue irreducibility are crucial in the development of factorization theory. Gauss's Lemma and Eisenstein's Criterion are two of the most elementary but effective tools to prove irreducibility in the context of polynomial rings. After reviewing some necessary terminology and background in Section~\ref{sec:background}, we dedicate Section~\ref{sec:Irreducibility Criteria} to extend Gauss's Lemma and Eisenstein's Criterion from the context of polynomial rings to that one of monoid rings.
An atomic monoid $M$ is called half-factorial provided that for all $x \in M$, any two factorizations of $x$ have the same number of irreducibles (counting repetitions). In addition, an integral domain is called half-factorial (or an HFD) if its multiplicative monoid is half-factorial. The concept of half-factoriality was first investigated by L.~Carlitz in the context of algebraic number fields; he proved that an algebraic number field is half-factorial if and only if its class group has size at most two~\cite{lC60}. Other-half-factoriality, on the other hand, is a dual version of half-factoriality, and it was introduced by J.~Coykendall and W.~Smith in~\cite{CS11}.
Additive monoids of rationals have a wild atomic structure~\cite{fG19,fG17} and a complex arithmetic of factorizations~\cite{fG18a,GO19}. The monoid rings they determine have been explored in~\cite{ACHZ07}. In addition, examples of such monoid rings have also appeared in the past literature, including \cite[Section~1]{aG74}, \cite[Example~2.1]{AAZ90}, and more recently, \cite[Section~5]{CG19}. In the second part of this paper, which is Section~\ref{sec:factorization in monoid algebras}, we study half-factoriality and other-half-factoriality in the context of additive monoids of rationals and the monoid algebras they induce. We also determine which of these monoid algebras are Dedekind domains, Euclidean domains, PIDs, UFDs, and HFDs.
\section{Notation and Background}
\label{sec:background}
\subsection{General Notation} Throughout this paper, we let $\mathbb{N}_0$ denote the set of all nonnegative integers, and we set $\mathbb{N} := \mathbb{N}_0 \setminus \{0\}$. If $a,b \in \mathbb{Z}$ and $a \le b$, then we let $\ldb a,b \rdb$ denote the interval of integers from $a$ to $b$, i.e.,
\[
\ldb a,b \rdb := \{j \in \mathbb{Z} \mid a \le j \le b\}.
\]
For a subset $X$ of $\mathbb{R}$, we set $X^\bullet := X \setminus \{0\}$. In addition, if $r \in \mathbb{R}$, we define
\[
X_{> r} := \{x \in X \mid x > r\} \quad \text{and} \quad X_{\ge r} := \{x \in X \mid x \ge r\}.
\]
If $q \in \mathbb{Q}_{> 0}$, then we denote the unique $m,n \in \mathbb{N}$ such that $q = m/n$ and $\gcd(m,n)=1$ by $\mathsf{n}(q)$ and $\mathsf{d}(q)$, respectively.
\subsection{Monoids} Within the scope of our exposition, a \emph{monoid} is defined to be a commutative and cancellative semigroup with an identity element. In addition, monoids here are written multiplicatively unless we specify otherwise. Let $M$ be a monoid. We let $U(M)$ denote the set of units (i.e., invertible elements) of $M$. When $U(M)$ consists of only the identity element, $M$ is said to be \emph{reduced}. On the other hand, $M$ is called \emph{torsion-free} if for all $x,y \in M$ and $n \in \mathbb{N}$, the equality $x^n = y^n$ implies $x = y$. For $S \subseteq M$, we let $\langle S \rangle$ denote the submonoid of $M$ generated by $S$. Further basic definitions and concepts on commutative cancellative monoids can be found in~\cite[Chapter~2]{pG01}.
If $y,z \in M$, then $y$ \emph{divides} $z$ \emph{in} $M$ provided that there exists $x \in M$ such that $z = xy$; in this case we write $y \mid_M z$. Also, the elements $y$ and $z$ are called \emph{associates} if $y \mid_M z$ and $z \mid_M y$; in this case we write $y \simeq z$. An element $p \in M \setminus U(M)$ is said to be \emph{prime} when for all $x,y \in M$ with $p \mid_M xy$, either $p \mid_M x$ or $p \mid_M y$. If every element in $M \setminus U(M)$ can be written as a product of primes, then $M$ is called \emph{factorial}. In a factorial monoid every nonunit element can be uniquely written as a product of primes (up to permutation and associates). In addition, an element $a \in M \setminus U(M)$ is called an \emph{atom} if for any $x,y \in M$ such that $a = xy$ either $x \in U(M)$ or $y \in U(M)$. The set of all atoms of $M$ is denoted by $\mathcal{A}(M)$, and $M$ is said to be \emph{atomic} if every nonunit element of $M$ is a product of atoms. Since every prime element is clearly an atom, every factorial monoid is atomic.
\subsection{Factorizations} Let $M$ be a monoid, and take $x \in M \setminus U(M)$. Suppose that for $m \in \mathbb{N}$ and $a_1, \dots, a_m \in \mathcal{A}(M)$,
\begin{equation} \label{eq:factorization definition}
x = a_1 \cdots a_m.
\end{equation}
Then the right-hand side of~(\ref{eq:factorization definition}) (treated as a formal product of atoms) is called a \emph{factorization} of $x$, and $m$ is called the \emph{length} of such a factorization. Two factorizations $a_1 \cdots a_m$ and $b_1 \cdots b_n$ of $x$ are considered to be equal provided that $m=n$ and that there exists a permutation $\sigma \in S_m$ such that $b_i \simeq a_{\sigma(i)}$ for every $i \in \ldb 1,m \rdb$. The set of all factorizations of $x$ is denoted by $\mathsf{Z}_M(x)$ or, simply, by $\mathsf{Z}(x)$. We then set
\[
\mathsf{Z}(M) := \bigcup_{x \in M \setminus U(M)} \mathsf{Z}(x).
\]
For $z \in \mathsf{Z}(x)$, we let $|z|$ denote the length of $z$.
\subsection{Monoid Rings} For an integral domain $R$, we let $R^\times$ denote the group of units of~$R$. We say that $R$ is \emph{atomic} if every nonzero nonunit element of $R$ can be written as a product of irreducibles (which are also called atoms).
Let $M$ be a reduced torsion-free monoid that is additively written. For an integral domain $R$, consider the set $R[X;M]$ comprising all maps $f \colon M \to R$ satisfying that
\[
\{s \in M \mid f(s) \neq 0 \}
\]
is finite. We shall conveniently represent an element $f \in R[X;M]$ by
\[
f = \sum_{s \in M} f(s)X^s = \sum_{i=1}^n f(s_i)X^{s_i},
\]
where $s_1, \dots, s_n$ are those elements $s \in M$ satisfying that $f(s) \neq 0$. Addition and multiplication in $R[X;M]$ are defined as for polynomials, and we call the elements of $R[X;M]$ \emph{polynomial expressions}. Under these operations, $R[X;M]$ is a commutative ring, which is called the \emph{monoid ring of} $M$ \emph{over} $R$ or, simply, a \emph{monoid ring}. Following Gilmer~\cite{rG84}, we will write $R[M]$ instead of $R[X;M]$. Since $R$ is an integral domain, $R[M]$ is an integral domain~\cite[Theorem~8.1]{rG84} with set of units $R^\times$~\cite[Corollary~4.2]{GP74}. If~$F$ is a field, then we say that $F[M]$ is a \emph{monoid algebra}. Now suppose that the monoid $M$ is totally ordered. For $k \in \mathbb{N}$, we say that
\[
f = \alpha_1X^{q_1} + \dots + \alpha_k X^{q_k} \in R[M] \setminus \{0\}
\]
is written in \emph{canonical form} when the coefficient $\alpha_i$ is nonzero for every $i \in \ldb 1,k \rdb$ and $q_1 > \dots > q_k$. Observe that there is only one way to write $f$ in canonical form. We call $\deg(f) := q_1$ the \emph{degree} of $f$. In addition, $\alpha_1$ is called the \emph{leading coefficient} of $f$, and $\alpha_k$ is called the \emph{constant coefficient} of~$f$ provided that $q_k = 0$. As it is customary for polynomials, $f$ is called a \emph{monomial} when $k = 1$.
Suppose that $\psi \colon M \to M'$ is a monoid homomorphism, where $M$ and $M'$ are reduced torsion-free monoids. Also, let $\psi^* \colon R[M] \to R[M']$ be the ring homomorphism determined by the assignment $X^s \mapsto X^{\psi(s)}$. It follows from \cite[Theorem~7.2(2)]{rG84} that if $\psi$ is injective (resp., surjective), then $\psi^*$ is injective (resp., surjective). Let us recall the following easy observation.
\begin{remark} \label{rem:isomophism of monoid algebras}
If $R$ is an integral domain and the monoids $M$ and $M'$ are isomorphic, then the monoid rings $R[M]$ and $R[M']$ are also isomorphic.
\end{remark}
\section{Irreducibility Criteria for Monoid Rings}
\label{sec:Irreducibility Criteria}
\subsection{Extended Gauss's Lemma} Our primary goal in this section is to offer extended versions of Gauss's Lemma and Eisenstein's Criterion for monoid rings.
Let $R$ be an integral domain and take $r_1, \dots, r_n \in R \setminus \{0\}$ for some $n \in \mathbb{N}$. An element $r \in R$ is called a \emph{greatest common divisor} of $r_1, \dots, r_n$ if $r$ divides $r_i$ in $R$ for every $i \in \ldb 1,n \rdb$ and $r$ is divisible by each common divisor of $r_1, \dots, r_n$. Any two greatest common divisors of $r_1, \dots, r_n$ are associates in $R$. We let $\text{GCD}(r_1, \dots, r_n)$ denote the set of all greatest common divisors of $r_1, \dots, r_n$.
\begin{definition}
An integral domain $R$ is called a \emph{GCD-domain} if any finite subset of $R \setminus \{0\}$ has a greatest common divisor in $R$.
\end{definition}
Let $M$ be a reduced torsion-free monoid, and let $R$ be an integral domain. Suppose that for the polynomial expression
\[
f = \alpha_1 X^{q_1} + \dots + \alpha_k X^{q_k} \in R[M] \setminus \{0\}
\]
the exponents $q_1, \dots, q_k$ are pairwise distinct. Then GCD$(\alpha_1, \dots, \alpha_k)$ is called the \emph{content} of $f$ and is denoted by $\mathsf{c}(f)$. If $\mathsf{c}(f) = R^\times$, then $f$ is called \emph{primitive}. Notice that if $R$ is not a GCD-domain, then $\mathsf{c}(f)$ may be the empty set. It is clear that $\mathsf{c}(rf) = r \mathsf{c}(f)$ for all $r \in R \setminus \{0\}$ and $f \in R[M] \setminus \{0\}$. As for the case of polynomials, the following lemma holds.
\begin{lemma} \label{lem:content identity}
Let $M$ be a reduced torsion-free monoid, and let $R$ be a GCD-domain. If $f$ and $g$ are elements of $R[M] \setminus \{0\}$, then $\mathsf{c}(fg) = \mathsf{c}(f) \mathsf{c}(g)$.
\end{lemma}
\begin{proof}
Since $R$ is a GCD-domain, there exist primitive polynomial expressions~$f_1$ and~$g_1$ in $R[M]$ such that $f = \mathsf{c}(f) f_1$ and $g = \mathsf{c}(g) g_1$. Because $M$ is a torsion-free monoid, it follows from~\cite[Proposition~4.6]{GP74} that the element $f_1 g_1$ is primitive in $R[M]$. Therefore $\mathsf{c}(f_1 g_1) = R^\times$. As a consequence, we find that
\[
\mathsf{c}(fg) = \mathsf{c} \big( \mathsf{c}(f) f_1 \mathsf{c}(g) g_1 \big) = \mathsf{c}(f) \mathsf{c}(g) \mathsf{c}(f_1 g_1) = \mathsf{c}(f) \mathsf{c}(g),
\]
as desired.
\end{proof}
Let $F$ denote the field of fractions of a GCD-domain $R$. Gauss's Lemma states that a non-constant polynomial $f$ with coefficients in $R$ is irreducible in $R[X]$ if and only if it is irreducible in $F[X]$ and primitive in $R[X]$. Now we extend Gauss's Lemma to the context of monoid rings.
\begin{theorem}[Extended Gauss's Lemma]
Let $M$ be a reduced torsion-free monoid, and let $R$ be a GCD-domain with field of fractions $F$. Then an element $f \in R[M] \setminus R$ is irreducible in $R[M]$ if and only if $f$ is irreducible in $F[M]$ and primitive in $R[M]$.
\end{theorem}
\begin{proof}
For the direct implication, suppose that $f$ is irreducible in $R[M]$. If $r \in \mathsf{c}(f)$, then there exists $g \in R[M] \setminus R$ such that $f = rg$. Because $R[M]^\times \subset R$, the element $g$ is not a unit of $R[M]$. As $f$ is irreducible in $R[M]$, one finds that $r \in R[M]^\times = R^\times$. So $\mathsf{c}(f) = R^\times$, which implies that $f$ is primitive in $R[M]$. To argue that $f$ is irreducible in $F[M]$, take $g_1, g_2 \in F[M]$ such that $f = g_1 g_2$. Since $R$ is a GCD-domain, there exist nonzero elements $a_1, a_2, b_1, b_2 \in R$ such that both
\[
h_1 := \frac{a_1}{b_1} g_1 \quad \text{and} \quad h_2 := \frac{a_2}{b_2} g_2
\]
are primitive elements of $R[M]$. Clearly, $a_1 a_2 f = b_1 b_2 h_1 h_2$. This, along with Lemma~\ref{lem:content identity}, implies that
\[
a_1 a_2 R^\times = a_1 a_2 \mathsf{c}(f) = \mathsf{c}(a_1 a_2 f) = \mathsf{c}(b_1 b_2 h_1 h_2) = b_1 b_2 \mathsf{c}(h_1) \mathsf{c}(h_2) = b_1 b_2 R^\times.
\]
Then $\frac{a_1 a_2}{b_1 b_2} \in R^\times$ and, as a consequence, $\frac{a_1 a_2}{b_1 b_2} f = h_1 h_2$ is irreducible in $R[M]$. Thus, either $h_1 \in R[M]^\times = R^\times$ or $h_2 \in R[M]^\times = R^\times$. This, in turn, implies that either~$g_1$ or~$g_2$ belongs to $F^\times = F[M]^\times$. Hence $f$ is irreducible in $F[M]$.
Tor argue the reverse implication, suppose that $f$ is irreducible in $F[M]$ and primitive in $R[M]$. Then take elements $g_1$ and $g_2 \in R[M]$ such that $f = g_1 g_2$. Since $f$ is irreducible in $F[M]$, either $g_1 \in F[M]^\times = F^\times$ or $g_2 \in F[M]^\times = F^\times$. This, along with the fact that $R[M] \cap F^\times = R \setminus \{0\}$, implies that either $g_1 \in \mathsf{c}(f)$ or $g_2 \in \mathsf{c}(f)$. As $\mathsf{c}(f) = R^\times = R[M]^\times$, either $g_1$ or $g_2$ belongs to $R[M]^\times$. As a result, $f$ is irreducible in $R[M]$, which concludes the proof.
\end{proof}
\subsection{Extended Eisenstein's Criterion} It is hardly debatable that Eisenstein's Criterion is one of the most popular and useful criteria to argue the irreducibility of certain polynomials. Now we proceed to offer an extended version of Eisenstein's Criterion for monoid rings.
\begin{prop}[Extended Eisenstein's Criterion] \label{prop:Eisenstein Criterion for PA}
Let $M$ be a reduced totally-ordered torsion-free monoid, and let $R$ be an integral domain. Suppose that the element
\[
f = \alpha_n X^{q_n} + \dots + \alpha_1 X^{q_1} + \alpha_0 \in R[M] \setminus \{0\},
\]
written in canonical form, is primitive. If there exists a prime ideal $P$ of $R$ satisfying the conditions
\begin{enumerate}
\item $\alpha_n \notin P$,
\item $\alpha_j \in P$ for every $j \in \ldb 0,n-1 \rdb$, and
\item $\alpha_0 \notin P^2$,
\end{enumerate}
then $f$ is irreducible in $R[M]$.
\end{prop}
\begin{proof}
We let $\bar{R}$ denote the quotient $R/P$ and, for any $h \in R[M]$, we let~ $\bar{h}$ denote the image of $h$ under the natural surjection $R[M] \to \bar{R}[M]$, i.e., $\bar{h}$ is the result of reducing the coefficients of $h$ modulo $P$. To argue that $f$ is irreducible suppose, by way of contradiction, that $f = g_1 g_2$ for some nonzero nonunit elements $g_1$ and $g_2$ of $R[M]$. As~$f$ is primitive, $g_1 \notin R$ and $g_2 \notin R$. By the condition~(2) in the statement, one obtains that $\bar{g}_1 \bar{g}_2 = \bar{f} = \bar{\alpha}_n X^{q_n}$. Thus, both $\bar{g}_1$ and $\bar{g}_2$ are monomials. This, along with the fact that none of the leading coefficients of $g_1$ and $g_2$ are in $P$ (because $\alpha_n \notin P$), implies that the constant coefficients of both $g_1$ and $g_2$ are in $P$. As a result, the constant coefficient $\alpha_0$ of $f$ must belong to $P^2$, which is a contradiction.
\end{proof}
\begin{cor} \label{cor:irreducible polynomials of any degree}
Let $M$ be a reduced totally-ordered torsion-free monoid, and let $R$ be an integral domain containing a prime element. Then for each $q \in M^\bullet$, there exists an irreducible polynomial expression in $R[M]$ of degree $q$.
\end{cor}
\begin{proof}
Let $p$ be a prime element of $R$. It suffices to verify that, for any $q \in M^\bullet$, the element $f := X^q + p \in R[M]$ is irreducible. Indeed, this is an immediate consequence of Proposition~\ref{prop:Eisenstein Criterion for PA} once we take $P := (p)$.
\end{proof}
In Corollary~\ref{cor:irreducible polynomials of any degree}, the integral domain $R$ is required to contain a prime element. This condition is not superfluous, as the next example illustrates.
\begin{example}
For a prime number $p$, consider the monoid algebra $\mathbb{F}_p[M]$, where $M$ is the submonoid $\langle 1/p^n \mid n \in \mathbb{N} \rangle$ of $(\mathbb{Q}_{\ge 0},+)$ and $\mathbb{F}_p$ is a finite field of characteristic $p$. It is clear that $M$ is a reduced totally-ordered torsion-free monoid. Now let
\[
f := \alpha_1 X^{q_1} + \dots + \alpha_n X^{q_n}
\]
be an element of $\mathbb{F}_p[M] \setminus \mathbb{F}_p$ written in canonical form. As $\mathbb{F}_p$ is a perfect field of characteristic $p$, the Frobenius homomorphism $x \mapsto x^p$ is surjective and, therefore, for each $i \in \ldb 1,n \rdb$ there exists $\beta_i \in \mathbb{F}_p$ with $\alpha_i = \beta_i^p$. On the other hand, it is clear that $q_i/p \in M$ for every $i \in \ldb 1,n \rdb$. As
\[
f = \alpha_1 X^{q_1} + \dots + \alpha_n X^{q_n} = \big( \beta_1 X^{q_1/p} + \dots + \beta_n X^{q_n/p} \big)^p,
\]
the polynomial expression $f$ is not irreducible in $\mathbb{F}_p[M]$. Hence the monoid algebra $\mathbb{F}_p[M]$ does not contain irreducible elements. Clearly, the field $\mathbb{F}_p$ is an integral domain containing no prime elements.
\end{example}
\section{Factorizations in Monoid Algebras}
\label{sec:factorization in monoid algebras}
A \emph{numerical semigroup} is a submonoid $N$ of $(\mathbb{N}_0,+)$ whose complement is finite, i.e., $|\mathbb{N}_0 \setminus N| < \infty$. Numerical semigroups are finitely generated and, therefore, atomic. However, the only factorial numerical semigroup is $(\mathbb{N}_0,+)$. For an introduction to numerical semigroups, see \cite{GR09}, and for some of their many applications, see~\cite{AG16}. A \emph{Puiseux monoid}, on the other hand, is an additive submonoid of $(\mathbb{Q}_{\ge 0},+)$. Albeit Puiseux monoids are natural generalizations of numerical semigroups, the former are not necessarily finitely generated or atomic; for example, consider $\langle 1/2^n \mid n \in \mathbb{N} \rangle$. The factorization structure of Puiseux monoids have been compared with that of other well-studied atomic monoids in~\cite{fG18} and, more recently, in~\cite{CGG19}. In this section, we determine the Puiseux monoids whose monoid algebras are Dedekind domains, Euclidean domains, PIDs, UFDs, or HFDs.
\begin{definition}
An atomic monoid $M$ is \emph{half-factorial} (or an \emph{HF-monoid}) if for all $x \in M \setminus U(M)$ and $z, z' \in \mathsf{Z}(x)$ the equality $|z| = |z'|$ holds. An integral domain is \emph{half-factorial} (or an \emph{HFD}) if its multiplicative monoid is an HF-monoid.
\end{definition}
Clearly, half-factoriality is a relaxed version of being a factorial monoid or a UFD. Although the concept of half-factoriality was first considered by Carlitz in his study of algebraic number fields~\cite{lC60}, it was A.~Zaks who coined the term ``half-factorial domain"~\cite{aZ76}.
\begin{definition}
An atomic monoid $M$ is \emph{other-half-factorial} (or an \emph{OHF-monoid}) if for all $x \in M \setminus U(M)$ and $z, z' \in \mathsf{Z}(x)$ the equality $|z| = |z'|$ implies that $z = z'$.
\end{definition}
Observe that other-half-factoriality is somehow a dual version of half-factoriality. Although an integral domain is a UFD if and only if its multiplicative monoid is an OHF-monoid~\cite[Corollary~2.11]{CS11}, OHF-monoids are not always factorial or half-factorial, even in the class of Puiseux monoids.
\begin{prop} \label{prop:HF and OHF PM characterization}
For a nontrivial atomic Puiseux monoid $M$, the following conditions hold.
\begin{enumerate}
\item $M$ is an HF-monoid if and only if $M$ is factorial.
\item $M$ is an OHF-monoid if and only if $|\mathcal{A}(M)| \le 2$.
\end{enumerate}
\end{prop}
\begin{proof}
For the direct implication of~(1), suppose that $M$ is an HF-monoid. Since $M$ is an atomic nontrivial Puiseux monoid, $\mathcal{A}(M)$ is not empty. Let $a_1$ and $a_2$ be two atoms of $M$. Then $z_1 := \mathsf{n}(a_2) \mathsf{d}(a_1) a_1$ and $z_2 := \mathsf{n}(a_1) \mathsf{d}(a_2) a_2$ are two factorizations of the element $\mathsf{n}(a_1) \mathsf{n}(a_2) \in M$. Because $M$ is an HF-monoid, $|z_1| = |z_2|$ and so
\[
\mathsf{n}(a_2) \mathsf{d}(a_1) = \mathsf{n}(a_1) \mathsf{d}(a_2).
\]
Therefore $a_1 = a_2$, and then $M$ contains only one atom. Hence $M \cong (\mathbb{N}_0,+)$ and, as a result, $M$ is factorial. The reverse implication of~(1) is trivial.
To prove the direct implication of~(2), assume that $M$ is an OHF-monoid. If $M$ is factorial, then $M \cong (\mathbb{N}_0,+)$, and we are done. Then suppose that $M$ is not factorial. In this case, $|\mathcal{A}(M)| \ge 2$. Assume, by way of contradiction, that $|\mathcal{A}(M)| \ge 3$. Take $a_1, a_2, a_3 \in \mathcal{A}(M)$ satisfying that $a_1 < a_2 < a_3$. Let $d = \mathsf{d}(a_1) \mathsf{d}(a_2) \mathsf{d}(a_3)$, and set $a'_i = d a_i$ for each $i \in \ldb 1,3 \rdb$. Since $a'_1, a'_2$, and $a'_3$ are integers satisfying that $a'_1 < a'_2 < a'_3$, there exist $m,n \in \mathbb{N}$ such that
\begin{equation} \label{eq:OHF}
m(a'_2 - a'_1) = n(a'_3 - a'_2).
\end{equation}
Clearly, $z_1 := ma_1 + na_3$ and $z_2 := (m+n)a_2$ are two distinct factorizations in $\mathsf{Z}(M)$ satisfying that $|z_1| = m+n = |z_2|$. In addition, after dividing both sides of the equality~(\ref{eq:OHF}) by $d$, one obtains that
\[
ma_1 + na_3 = (m+n)a_2,
\]
which means that $z_1$ and $z_2$ are factorizations of the same element. However, this contradicts that $M$ is an OHF-monoid. Hence $|\mathcal{A}(M)| \le 2$, as desired. For the reverse implication of~(2), suppose that $|\mathcal{A}(M)| \le 2$. By~\cite[Proposition~3.2]{fG17}, $M$ is isomorphic to a numerical semigroup $N$. As $N$ is generated by at most two elements, either $N = (\mathbb{N}_0,+)$ or $N = \langle a, b \rangle$ for $a, b \in \mathbb{N}_{\ge 2}$ with $\gcd(a,b) = 1$. If $N = (\mathbb{N}_0,+)$, then~$N$ is factorial and, in particular, an OHF-monoid. On the other hand, if $N = \langle a, b \rangle$, then it is an OHF-monoid by~\cite[Example~2.13]{CS11}.
\end{proof}
In~\cite[Theorem~8.4]{GP74} Gilmer and Parker characterize the monoid algebras that are Dedekind domains, Euclidean domains, or PIDs. We conclude this section extending such a characterization in the case where the exponent monoids are Puiseux monoids.
\begin{theorem}
For a nontrivial Puiseux monoid $M$ and a field $F$, the following conditions are equivalent:
\begin{enumerate}
\item $F[M]$ is a Euclidean domain;
\item $F[M]$ is a PID;
\item $F[M]$ is a UFD;
\item $F[M]$ is an HFD;
\item $M \cong (\mathbb{N}_0,+)$;
\item $F[M]$ is a Dedekind domain.
\end{enumerate}
\end{theorem}
\begin{proof}
It is well known that every Euclidean domain is a PID, and every PID is a UFD. Therefore condition~(1) implies condition~(2), and condition~(2) implies condition~(3). In addition, it is clear that every UFD is an HFD, and so condition~(3) implies condition~(4). As Puiseux monoids are torsion-free,~\cite[Proposition~1.4]{hK01} ensures that $M$ is an HF-monoid when $F[M]$ is an HFD. This, along with Proposition~\ref{prop:HF and OHF PM characterization}(1), guarantees that $M \cong (\mathbb{N}_0,+)$ provided that $F[M]$ is an HFD. Thus, condition~(4) implies condition~(5). Now notice that if condition~(5) holds, then $F[M] \cong F[\mathbb{N}_0] = F[X]$ (by Remark~\ref{rem:isomophism of monoid algebras}) is a Euclidean domain, which is condition~(1). Then we have argued that the first five conditions are equivalent.
To include~(6) in the set of already-established equivalent conditions, observe that condition~(2) implies condition~(6) because every PID is a Dedekind domain. On the other hand, suppose that the monoid algebra $F[M]$ is a Dedekind domain. Then the fact that $M$ is torsion-free, along with~\cite[Theorem~8.4]{GP74}, implies that $M \cong (\mathbb{N}_0,+)$. Hence condition~(6) implies condition~(5), which completes the proof.
\end{proof}
\section*{Acknowledgments}
\noindent While working on this paper, the author was supported by the NSF-AGEP Fellowship and the UC Dissertation Year Fellowship. The author would like to thank an anonymous referee, whose suggestions help to simplify and improve the initially-submitted version of this paper.
\end{document}
|
\begin{document}
\author{{\sc R. Aharoni}$^*$ \and {\sc E. Berger}$^*$ \and
{\sc R. Meshulam}{\tilde H}anks{Department of Mathematics, Technion,
Haifa 32000, Israel.~~ e-mails:
\newline [email protected],
[email protected], [email protected]} }
\date{}
\insert\footins{\footnotesize\rule{0pt}{\footnotesep}
\\ {\it Math Subject Classification:} 13F55, 05C69.
\\ {\it Keywords:} Flag complexes, homology, domination in graphs.\\}
\title{Eigenvalues and Homology of Flag Complexes \
and Vector Representations of Graphs}
\pagestyle{plain}
\begin{abstract}
The flag complex of a graph $G=(V,E)$ is the simplicial complex
$X(G)$ on the vertex set $V$ whose simplices are subsets of $V$
which span complete subgraphs of $G$. We study relations between
the first eigenvalues of successive higher Laplacians of $X(G)$.
One consequence is the following \ \\ \\{\bf Theorem:} Let
$\lambda_2(G)$ denote the second smallest eigenvalue of the
Laplacian of $G$. If $\lambda_2(G)
> \frac{k}{k+1} |V|$ then ${\tilde H}^{k}(X(G);{\fam\bbfam\twelvebb R})=0$. \ \\ \\
Applications include a lower bound on the homological connectivity
of the independent sets complex ${\rm I}(G)$, in terms of a new graph
domination parameter $\Gamma(G)$ defined via certain vector
representations of $G$. This in turns implies a Hall type theorem
for systems of disjoint representatives in hypergraphs.
\end{abstract}
\section{Introduction}
Let $G=(V,E)$ be a graph with $|V|=n$ vertices. The {\it
Laplacian} of $G$ is the $V \times V$ positive semidefinite matrix
$L_G$ given by $$L_G(u,v) = \left\{
\begin{array}{ll}
\deg(u) & u=v \\
-1 & uv \in E \\
0 & {\rm otherwise}
\end{array}
\right.~~ $$ Let $0=\lambda_1(G) \leq \lambda_2(G) \leq \mathcal Dots
\leq \lambda_n(G)$ denote the eigenvalues of $L_G$. The second
smallest eigenvalue $\lambda_2(G)$, called the {\it spectral gap},
is a parameter of central importance in a variety of problems. In
particular it controls the expansion properties of $G$ and the
convergence rate of a random walk on $G$ (see e.g. {\rm I}te{Boll98}).
The {\it Flag Complex} of $G$ is the simplicial complex $X(G)$ on
the vertex set $V$ whose simplices are all subsets $\sigma \subset
V$ which form a complete subgraph of $G$. Topological properties
of $X(G)$ play key roles in recent results in matching theory (see
below).
\\
In this paper we study relations between $\lambda_2(G)$, the
cohomology of $X(G)$, and a new graph domination parameter
$\Gamma(G)$ which is defined via certain vector representations of
$G$. As an application we obtain a Hall type theorem for systems
of disjoint representatives in families of hypergraphs. \ \\ \\
For $k \geq -1$ let $C^k(X(G))$ denote the space of real valued
simplicial $k$-cochains of $X(G)$ and let $d_k:C^k(X(G))
\rightarrow C^{k+1}(X(G))$ denote the coboundary operator. For $k
\geq 0$ define the reduced $k$-dimensional Laplacian of $X(G)$ by
$\Delta_k=d_{k-1}d_{k-1}^*+d_k^*d_k$ (see section \ref{hodge} for
details). Let $\mu_k(G)$ denote the minimal eigenvalue of
$\Delta_k$. Note that $\mu_0(G)=\lambda_2(G)$.
Our main result is
the following
\begin{theorem}
\label{eigenv} For $k \geq 1$
\begin{equation}
\label{maineq} k\mu_k(G) \geq (k+1)\mu_{k-1}(G) -n~~.
\end{equation}
\end{theorem}
As a direct consequence of Theorem \ref{eigenv} we
obtain
\begin{theorem} \label{ei} If $\lambda_2(G) >
\frac{kn}{k+1}$ then ${\tilde H}^{k}(X(G),{\fam\bbfam\twelvebb R})=0$.
\end{theorem}
{\bf Remarks:}
\\
1. Theorem \ref{ei} is related to a well-known result
of Garland
(Theorem 5.9 in {\rm I}te{G73}) and its extended version
by Ballmann and \'{S}wi\c{a}tkowski
(Theorem 2.5 in {\rm I}te{BS97}).
Roughly speaking, these results (in their simplest untwisted
form) guarantee the vanishing of ${\tilde H}^k(X;{\fam\bbfam\twelvebb R})$
provided that for {\it each} $(k-1)$-simplex $\tilde{A}u$ in $X$,
the spectral gap of the $1$-skeleton of the link of $\tilde{A}u$
is sufficiently large. Theorem \ref{ei} is, in a sense, a global
counterpart of this statement for flag complexes.
\\
2. Let $n=r \ell$ where $r \geq 1,\ell \geq 2$, and let $G$
be the Tur\'an graph $T_r(n)$, i.e. the complete $r$-partite graph
on $n$ vertices with all sides equal to $\ell$. The flag complex
$X(T_r(n))$ is homotopy equivalent to the wedge of $(\ell-1)^r$
$(r-1)$-dimensional spheres. It can be checked that
$\mu_k(T_r(n))=\ell(r-k-1)$ for all $0 \leq k \leq r-1$, hence
(\ref{maineq}) is satisfied with equality.
Furthermore, $\lambda_2(G)=\ell(r-1)=\frac{r-1}{r}n$ while
${\tilde H}^{r-1}(X(G)) \neq 0$. Therefore the assumption in Theorem
\ref{ei} cannot be replaced by $\lambda_2(G) \geq \frac{kn}{k+1}$.
\ \\ \\ We next study some graph theoretical consequences of
Theorem \ref{ei}. The {\it Independence Complex} ${\rm I}(G)$ of $G$
is the simplicial complex on the vertex set $V$ whose simplices
are all independent sets $\sigma \subset V$. Thus
${\rm I}(G)=X(\overline{G})$ where $\overline{G}$ denotes the complement of $G$.
Recent work on hypergraph matching, starting in {\rm I}te{AH00} with
later developments in {\rm I}te{Aharoni01,M01,ABZ02,ACK02,M03}, has
utilized topological properties of ${\rm I}(G)$ to derive new Hall
type theorems for hypergraphs. The main ingredient in these
developments are lower bounds on the homological connectivity of
${\rm I}(G)$. For a simplicial complex $Z$ let
$\eta(Z)=\min\{i:{\tilde H}^i(Z,{\fam\bbfam\twelvebb R}) \neq 0\}+1~.$ It turns out that
various domination parameters of $G$ may be used to provide lower
bounds on $\eta({\rm I}(G))$. For a subset of vertices $S \subset V$
let $N(S)$ denote all vertices that are adjacent to at least one
vertex of $S$ and let $N'(S)=S \mathcal Up N(S)$. $S$ is a {\it
dominating set} if $N'(S) = V$. $S$ is a {\it totally dominating
set} if $N(S) = V$. Here are a few domination parameters:
\begin{itemize}
\item
The {\it domination number} $\gamma(G)$ is the minimal size of a
dominating set.
\item
The {\it total domination number} $\tilde{\gamma}(G)$ is the minimal size of
a totally dominating set.
\item
The {\it independent domination number} $i\gamma(G)$ is the maximum,
over all independent sets $I$ in $G$, of the minimal size of a set
$S$ such that $N(S) \supset I$.
\item The {\it strong fractional domination number},
$\gamma^*_s(G)$ is the minimum of $\sum_{v\in V}f(v)$, over all
nonnegative functions $f:V \to \mathbb{R}$ such that \\ $\sum_{uv \in
E}f(u)+\deg(v)f(v) \ge 1$ for every vertex $v$.
\end{itemize}
Some known lower bounds on $\eta$ are: $\eta({\rm I}g) \ge \tilde{\gamma}(G)/2$
{\rm I}te{M01}, $\eta({\rm I}g) \ge i\gamma(G)$ {\rm I}te{AH00}, $\eta({\rm I}g) \ge
\gamma^*_s(G)$ {\rm I}te{M03}. \ \\ \\ Here we introduce a new
domination parameter, defined by vector representations. It is
similar in spirit to the $\Theta$ function defined by Lov\'asz
{\rm I}te{Lovasz79}. It uses vectors to mimick domination, in a
way similar to that in which the $\Theta$ function mimicks independence of sets
of vertices. It is defined as follows. A {\em vector
representation} of a graph $G=(V,E)$ is an assignment $P$ of a
vector $P(v) \in {\fam\bbfam\twelvebb R}^\ell$ for some fixed $\ell$ to every vertex
$v$ of the graph, such that the inner product $P(u) \mathcal Dot P(v) \ge
1$ whenever $u,v$ are adjacent in $G$ and $P(u) \mathcal Dot P(v) \geq 0$
if they are not adjacent. We shall identify the representation
with the matrix $P$ whose $v$-th row is the vector $P(v)$.
\\
Let ${\bf 1}$ denote the all $1$ vector in ${\fam\bbfam\twelvebb R}^V$. A non-negative
vector ${\bf \alpha}$ on $V$ is said to be {\em dominating for $P$} if
$\sum_{v \in V} \alpha(v) P(v) \mathcal Dot P(u) \ge 1$ for every vertex
$u$, namely ${\bf \alpha} PP^T \ge {\bf 1}$. (Note that taking ${\bf \alpha}$
to be the characteristic function of some totally dominating set
satisfies this condition regardless of the representation.) The
{\it value} of $P$ is $$|P|=\min\{ {\bf \alpha} \mathcal Dot {\bf 1}~:~{\bf \alpha}
\geq 0~,~{\bf \alpha} PP^T \ge {\bf 1}~\}~~.$$ The supremum of $|P|$ over
all vector representations $P$ of $G$ is denoted by $\Gamma(G)$.
Our main application of Theorem \ref{ei} is the following
\begin{theorem}\label{etagegamma}
$ \eta({\rm I}g) \ge \Gamma(G)~.$
\end{theorem}
{\bf Remark:} One natural vector representation of $G$ is obtained
by taking $P(v) \in {\fam\bbfam\twelvebb R}^E$ to be the edge incidence vector of the
vertex $v$. For this representation $|P|=\gamma^*_s(G)$ hence
$\Gamma(G) \geq \gamma^*_s(G)$. The bound $\eta({\rm I}g) \geq
\gamma^*_s(G)$ was previously obtained in {\rm I}te{M03}. Theorem
\ref{etagegamma} is however stronger and often gives much sharper
estimates for $\eta({\rm I}g)$, see e.g. the case of cycles described
in Section \ref{s:etag}. \ \\ \\ We next use Theorem
\ref{etagegamma} to derive a new Hall type result for hypergraphs.
Let ${\mathcal Al F} \subset 2^V$ be a hypergraph on a finite ground set $V$.
The {\it width} $w({\mathcal Al F})$ of ${\mathcal Al F}$ is the minimal $t$ for which
there exist $F_1,\ldots,F_t \in {\mathcal Al F}$ such that for any $F \in
{\mathcal Al F}$, $F_i \mathcal Ap F \neq \emptyset$ for some $1 \leq i \leq t$.
\\
The {\it fractional width} $w^*({\mathcal Al F})$ of ${\mathcal Al F}$ is the minimum of
$\sum_{E \in {\mathcal Al F}} f(E)$ over all non-negative functions $f : {\mathcal Al F}
\rightarrow \mathbb{R}$ with the property that for every edge $E
\in {\mathcal Al F}$ the sum $\sum_{F \in {\mathcal Al F}} f(F) |E \mathcal Ap F|$ is at least 1.
A {\it matching} in ${\mathcal Al F}$ is a subhypergraph $\mathcal M \subset {\mathcal Al F}$
such that $F \mathcal Ap F' = \emptyset$ for all $F \neq F' \in \mathcal M$. Let
${\mathcal Al F}i$ be a family of hypergraphs. A {\it system of disjoint
representatives (SDR)} of ${\mathcal Al F}i$ is a matching $F_1,\ldots,F_m$
such that $F_i \in {\mathcal Al F}_i$ for $1 \leq i \leq m$. Haxell
{\rm I}te{Haxell95} proved the following
\begin{theorem}{{\rm I}te{Haxell95}}
\label{hax95} If ${\mathcal Al F}i$ satisfies $w(\mathcal Up_{i\in I} {\mathcal Al F}_i) \geq
2|I|-1$ for all $\emptyset \neq I \subset [m]$, then ${\mathcal Al F}i$ has an
SDR.
\end{theorem}
Here we use Theorem \ref{etagegamma} to show
\begin{theorem}
\label{wstar} If ${\mathcal Al F}i$ satisfies $w^*(\mathcal Up_{i\in I} {\mathcal Al F}_i) >
|I|-1$ for all $\emptyset \neq I \subset [m]$, then ${\mathcal Al F}i$ has an
SDR.
\end{theorem}
The paper is organized as follows. In section \ref{hodge} we
recall some topological terminology and the simplicial Hodge
theorem. Theorems \ref{eigenv} and \ref{ei} are proved in section
\ref{s:eandc}. The proofs utilize the approach of Garland
{\rm I}te{G73} and its exposition by
Ballmann and \'{S}wi\c{a}tkowski
{\rm I}te{BS97}. In section \ref{s:etag} we relate the $\Gamma$
parameter to homological connectivity and prove Theorem
\ref{etagegamma}. In section \ref{s:hall} we recall a homological
Hall type condition (Proposition \ref{hom}) for the existence of
colorful simplices in a colored complex. Combining this condition
with Theorem \ref{etagegamma} then completes the proof of Theorem
\ref{wstar}.
\section{Topological Preliminaries}
\label{hodge} Let $X$ be a finite simplicial complex on the vertex
set $V$. Let $X(k)$ denote the set of $k$-dimensional simplices in
$X$, each taken with an arbitrary but fixed orientation. A
simplicial $k$-cochain is a real valued skew-symmetric function on
all ordered $k$-simplices of $X$. For $k \geq 0$ let $C^k(X)$
denote the space of $k$-cochains on $X$. The $i$-face of an
ordered $(k+1)$-simplex $\sigma=[v_0,\ldots,v_{k+1}]$ is the
ordered $k$-simplex
$\sigma_i=[v_0,\ldots,\widehat{v_i},\ldots,v_{k+1}]$. The
coboundary operator $d_k:C^k(X) \rightarrow C^{k+1}(X)$ is given
by $$d_k \phi (\sigma)=\sum_{i=0}^{k+1} (-1)^i \phi
(\sigma_i)~~.$$ It will be convenient to augment the cochain
complex $\{C^i(X)\}_{i=0}^{\infty}$ with the $(-1)$-degree term
$C^{-1}(X)={\fam\bbfam\twelvebb R}$ with the coboundary map $d_{-1}:C^{-1}(X)
\rightarrow C^0(X)$ given by $d_{-1}(a)(v)=a$ for $a \in {\fam\bbfam\twelvebb R}~,~v
\in V$. Let $Z^k(X)= \ker (d_k)$ denote the space of $k$-cocycles
and let $B^k(X)={\rm Im}(d_{k-1})$ denote the space of
$k$-coboundaries. For $k \geq 0$ let ${\tilde H}^k(X)=Z^k(X)/B^k(X)~$
denote the $k$-th reduced cohomology group of $X$ with real
coefficients. For each $k \geq -1$ endow $C^k(X)$ with the
standard inner product $(\phi,\psi)=\sum_{\sigma \in X(k)}
\phi(\sigma)\psi(\sigma)~~$ and the corresponding $L^2$ norm
$||\phi||=(\sum_{\sigma \in X(k)} \phi(\sigma)^2)^{1/2}$.
\\ Let $d_k^*:C^{k+1}(X) \rightarrow
C^k(X)$ denote the adjoint of $d_k$ with respect to these standard
inner products. The reduced $k$-Laplacian of $X$ is the mapping
$$\Delta_k=d_{k-1}d_{k-1}^*+d_k^*d_k : C^k(X) \rightarrow
C^k(X)~~.$$ Note that if $G$ denotes the $1$-skeleton of $X$ and
$J$ is the $V \times V$ all ones matrix, then the matrix $J+L_G$
represents $\Delta_0$ with respect to the standard basis. In
particular, the minimal eigenvalue of $\Delta_0$ equals
$\lambda_2(G)$.
\\
The space of harmonic $k$-cochains $\tilde{\mathcal Al H}^k(X) = \ker
\Delta_k$ consists of all $\phi \in C^k(X)$ such that both
$d_k\phi$ and $d_{k-1}^*\phi$ are zero. The simplicial version of
Hodge Theorem is the following well-known
\begin{proposition}
\label{hte} $\tilde{\mathcal Al H}^k(X) \cong {\tilde H}^k(X)~$ for $k \geq 0$.
\end{proposition}
In particular, ${\tilde H}^k(X)=0$ iff the minimal eigenvalue of
$\Delta_k$ is positive.
\section{Eigenvalues of Higher Laplacians}
\label{s:eandc}
Let $X=X(G)$ be the flag complex of a graph $G=(V,E)$ on $|V|=n$
vertices. For an $i$-simplex $\eta \in X$ let $\deg(\eta)$ denote
the number of $(i+1)$-simplices in $X$ which contain $\eta$. The
{\it link} of a simplex $\sigma \in X$ is the complex
$${\rm lk}(\sigma)=\{\tilde{A}u \in X ~:~\sigma \mathcal Up \tilde{A}u \in X~,~ \sigma
\mathcal Ap \tilde{A}u = \emptyset~\}~.$$ For two ordered simplices $\sigma \in
X~,~\tilde{A}u \in {\rm lk}(\sigma)$ let $\sigma\tilde{A}u$ denote their ordered
union.
\begin{claim}
\label{dkpnorm} For $\phi \in C^k(X)$
$$
||d_k \phi||^2 =\sum_{\sigma \in X(k)} \deg(\sigma)
\phi(\sigma)^2 - 2 \sum_{\eta \in X(k-1)}\sum_{vw \in {\rm lk}(\eta)}
\phi(v \eta) \phi(w \eta)~~.
$$
\end{claim}
{\bf Proof:} Recall that for $\tilde{A}u \in X(k+1)$ we denoted by
$\tilde{A}u_i$ the ordered $k$-simplex obtained by removing the $i$-th
vertex of $\tilde{A}u$. Thus $$||d_k \phi||^2=\sum_{\tilde{A}u \in X(k+1)} d_k
\phi(\tilde{A}u)^2=\sum_{\tilde{A}u \in X(k+1)} \sum_{i=0}^{k+1} (-1)^i
\phi(\tilde{A}u_i) \sum_{j=0}^{k+1} (-1)^j \phi(\tilde{A}u_j)= $$ $$\sum_{\tilde{A}u
\in X(k+1)} \sum_{i=0}^{k+1} \phi(\tilde{A}u_i)^2 + \sum_{\tilde{A}u\in
X(k+1)} \sum_{i \neq j} (-1)^{i+j} \phi(\tilde{A}u_i) \phi(\tilde{A}u_j)=$$ $$
\sum_{\sigma \in X(k)} \deg(\sigma) \phi(\sigma)^2 - 2 \sum_{\eta
\in X(k-1)}\sum_{vw \in {\rm lk}(\eta)} \phi(v \eta) \phi(w \eta)~~. $$
{\begin{flushright} $\Box$ \end{flushright}} For $\phi \in C^k(X)$ and a vertex $u \in V$ define $\phi_u
\in C^{k-1}(X)$ by $$\phi_u(\tilde{A}u)= \left\{
\begin{array}{ll}
\phi(u\tilde{A}u) & u \in {\rm lk}(\tilde{A}u) \\
0 & {\rm otherwise}
\end{array}
\right.~~ $$
\begin{claim}
\label{dphun}
For $\phi \in C^k(X)$
$$\sum_{u \in V} ||d_{k-1} \phi_u
||^2=$$
$$\sum_{\sigma \in X(k)} (\sum_{\tilde{A}u \in \sigma(k-1)}
\deg(\tilde{A}u)) \phi(\sigma)^2 - 2k\sum_{\tilde{A}u \in X(k-1)} \sum_{vw \in
{\rm lk}(\tilde{A}u)}\phi(v \tilde{A}u)\phi(w \tilde{A}u)~.
$$
\end{claim}
{\bf Proof:} Applying Claim \ref{dkpnorm} with $\phi_u \in
C^{k-1}(X)$ we obtain $$||d_{k-1} \phi_u ||^2= \sum_{\tilde{A}u \in
X(k-1)} \deg(\tilde{A}u) \phi_u(\tilde{A}u)^2-2 \sum_{\eta \in X(k-2)}\sum_{vw
\in {\rm lk}(\eta)} \phi_u(v \eta)\phi_u(w \eta)~.$$ Hence $$\sum_{u \in V}
||d_{k-1} \phi_u ||^2=$$ $$\sum_{u \in V} \sum_{\tilde{A}u \in X(k-1)}
\deg(\tilde{A}u) \phi_u(\tilde{A}u)^2-2 \sum_{u \in V}\sum_{\eta \in
X(k-2)}\sum_{vw \in {\rm lk}(\eta)} \phi_u(v \eta)\phi_u(w \eta)=$$ $$
\sum_{\sigma \in X(k)} (\sum_{\tilde{A}u \in \sigma(k-1)} \deg(\tilde{A}u))
\phi(\sigma)^2 - 2\sum_{\eta \in X(k-2)}\sum_{vw \in {\rm lk}(\eta)}
\sum_{u \in {\rm lk}(v \eta) \mathcal Ap {\rm lk}(w \eta)} \phi(vu \eta)\phi(wu
\eta)=$$ $$\sum_{\sigma \in X(k)} (\sum_{\tilde{A}u \in \sigma(k-1)}
\deg(\tilde{A}u)) \phi(\sigma)^2 - 2k \sum_{\tilde{A}u \in X(k-1)}\sum_{vw \in
{\rm lk}(\tilde{A}u)} \phi(v \tilde{A}u)\phi(w \tilde{A}u)~.$$
The last equality follows from
the fact that since $X$ is a flag complex, if $\eta \in
X(k-2)$~,~$vw \in {\rm lk}(\eta)$ and $u \in {\rm lk}(v \eta) \mathcal Ap {\rm lk}(w
\eta)~$, then $vw \in {\rm lk}(u\eta)$.
{\begin{flushright} $\Box$ \end{flushright}} Claims
\ref{dkpnorm} and \ref{dphun} imply $$
k(||d_k \phi||^2 - \sum_{\sigma \in X(k)}
\deg(\sigma) \phi(\sigma)^2)=$$
\begin{equation}
\label{onee} \sum_{u \in V} ||d_{k-1}\phi_u||^2- \sum_{\sigma \in
X(k)}( \sum_{\tilde{A}u \in \sigma(k-1)}\deg(\tilde{A}u)) \phi(\sigma)^2~~.
\end{equation}
\begin{claim}
\label{phtn}
For $\phi \in C^k(X)$
\begin{equation}
\label{twoe}
\sum_{u \in V} ||d_{k-2}^* \phi_u||^2= k ||d_{k-1}s
\phi||^2~~.
\end{equation}
\end{claim}
{\bf Proof:} For $\tilde{A}u \in X(k-1)$ $$d_{k-1}s\phi(\tilde{A}u)= \sum_{v \in
{\rm lk}(\tilde{A}u)} \phi(v \tilde{A}u)~.$$ Therefore $$ ||d_{k-1}s
\phi||^2=\sum_{\tilde{A}u \in X(k-1)}d_{k-1}s \phi(\tilde{A}u)^2=$$
\begin{equation}
\label{midt}
\sum_{\tilde{A}u
\in X(k-1)}(\sum_{v \in {\rm lk}(\tilde{A}u)} \phi(v \tilde{A}u)) (\sum_{w \in
{\rm lk}(\tilde{A}u)} \phi(w \tilde{A}u))= \sum_{\tilde{A}u \in X(k-1)}\sum_{(v,w) \in
{\rm lk}(\tilde{A}u)^2} \phi(v \tilde{A}u)\phi(w \tilde{A}u)~.
\end{equation}
Substituting $\phi_u$ in (\ref{midt}) we obtain
$$\sum_{u \in V} ||d_{k-2}^*\phi_u||^2=
\sum_{u \in V}\sum_{\eta \in X(k-2)}\sum_{(v,w) \in {\rm lk}(\eta)^2}
\phi_u(v \eta)\phi_u(w \eta)~=$$ $$ \sum_{\eta \in X(k-2)}\sum_{u \in
{\rm lk}(\eta)} \sum_{(v,w) \in {\rm lk}(u\eta)^2} \phi(v u\eta)\phi(w u
\eta)~= $$ $$ k\sum_{\tilde{A}u \in X(k-1)}\sum_{(v,w) \in {\rm lk}(\tilde{A}u)^2}
\phi(v \tilde{A}u)\phi(w \tilde{A}u)=k||d_{k-1}s \phi||^2~.$$ {\begin{flushright} $\Box$ \end{flushright}} Let $\phi
\in C^k(X)$. Summing (\ref{onee}) and (\ref{twoe}) we obtain the
following key identity: $$k(\Delta_k \phi,\phi)=$$
\begin{equation}
\label{thre}\sum_{u \in V}(\Delta_{k-1} \phi_u,\phi_u) - \sum_{\sigma
\in X(k)} (\sum_{\tilde{A}u \in \sigma(k-1)} \deg(\tilde{A}u)-k \deg(\sigma))
\phi(\sigma)^2~~.
\end{equation}
To estimate the righthand side of (\ref{thre}) we need the
following
\begin{claim}
\label{comb} For $\sigma \in X(k)$
\begin{equation}
\label{combi} \sum_{\tilde{A}u \in \sigma(k-1)} \deg(\tilde{A}u)-k
\deg(\sigma) \leq n~.
\end{equation}
\end{claim}
{\bf Proof:} Recall that $N(v)$ is the set of neighbors of $v$ in
$G$. Let $\sigma=[v_0,\ldots,v_k]$ then for any $I \subset
\{0,\ldots,k\}$ $$\deg([v_i:i \in I])=|\bigcap_{i \in I}
N(v_i)|~~.$$ Therefore
\begin{equation}
\label{clear} \sum_{\tilde{A}u \in \sigma(k-1)} \deg(\tilde{A}u)-k
\deg(\sigma)=\sum_{i=0}^k |\bigcap_{j \neq i} N(v_j)| -k
|\bigcap_{j=0}^k N(v_j)|~~.
\end{equation}
The Claim now follows since
each $v \in V$ is counted at most once on the righthand
side of (\ref{clear}). {\begin{flushright} $\Box$ \end{flushright}} {\bf Proof of
Theorem \ref{eigenv}:} Let $0 \neq \phi \in C^k(X)$ be an
eigenvector of $\Delta_k$ with eigenvalue $\mu_k(G)~$. By double
counting
\begin{equation}
\label{nor} \sum_{u \in V} ||\phi_u||^2=(k+1)||\phi||^2~~.
\end{equation}
Combining (\ref{thre}),(\ref{combi}) and (\ref{nor}) we obtain $$
k \mu_k(G) ||\phi||^2= k(\Delta_k \phi,\phi) \geq \sum_{u \in
V}(\Delta_{k-1} \phi_u,\phi_u) - n \sum_{\sigma \in X(k)}
\phi(\sigma)^2~~ \geq $$ $$\mu_{k-1}(G) \sum_{u \in V} ||\phi_u||^2
- n ||\phi||^2 = ((k+1)\mu_{k-1}(G)-n) ||\phi||^2~~. $$ {\begin{flushright} $\Box$ \end{flushright}} \
\\
\\ {\bf Proof of Theorem \ref{ei}:} Inequality (\ref{maineq})
implies by induction on $k$ that $\mu_k(G) \geq (k+1) \mu_0(G)
-kn$. Therefore, if $\mu_0(G)=\lambda_2(G)>\frac{kn}{k+1}$ then
$\mu_k(G)>0$ and ${\tilde H}^{k}(X(G),{\fam\bbfam\twelvebb R})=0$ follows from the
simplicial Hodge Theorem. {\begin{flushright} $\Box$ \end{flushright}}
\section{Vector Domination and Homology}
\label{s:etag} Let $G=(V,E)$ be a graph with $|V|=n$. We first
reformulate Theorem \ref{ei} in terms of the independence complex
${\rm I}g$.
\begin{theorem}
\label{indeta} $\eta({\rm I}g) \geq \frac{n}{\lambda_n(G)}.$
\end{theorem}
{\bf Proof:} Let $\ell=\lceil\frac{n}{\lambda_n(G)}\rceil$. Since
$\lambda_n(G)=n-\lambda_2(\overline{G})$ it follows that
$\lambda_2(\overline{G})>\frac{\ell-2}{\ell-1}n$. Therefore by Theorem
\ref{ei}, ${\tilde H}^i({\rm I}g)={\tilde H}^i(X(\overline{G}))=0$ for $i \leq \ell-2$.
Hence $\eta({\rm I}g) \geq \ell$. {\begin{flushright} $\Box$ \end{flushright}} The proof of Theorem
\ref{etagegamma} depends on Theorem \ref{indeta} and the following
\begin{claim}
\label{lambdan} Let $P$ be a vector representation of $G=(V,E)$.
Then
$$
\lambda_n(G) \leq \max_{u \in V}~ P(u) \mathcal Dot
\sum_{v \in V} P(v)~~.
$$
\end{claim}
{\bf Proof:} Let $x=(x(v):v \in V)$ be a vector in ${\fam\bbfam\twelvebb R}^V$. Then
$$x^T L_G x =\sum_{uv \in E} (x(u)-x(v))^2 \leq $$ $$\frac{1}{2}
\sum_{(u,v) \in V \times V} (x(u)-x(v))^2 P(u) \mathcal Dot P(v) =$$ $$
\sum_{u \in V}x(u)^2P(u) \mathcal Dot \sum_{v \in V} P(v) ~-~ ||
\sum_{v\in V} x(v)P(v)||^2 \leq $$ $$ ||x||^2~ \max_{u \in V}~
P(u) \mathcal Dot \sum_{v \in V} P(v)~.$$ The Claim follows since
$\lambda_n(G)= \max~\{\frac{x^TL_Gx}{||x||^2}~:~0 \neq x \in
{\fam\bbfam\twelvebb R}^V\}.$ {\begin{flushright} $\Box$ \end{flushright}} Let ${\fam\bbfam\twelvebb Z}_+$ denote the positive integers and
let ${\fam\bbfam\twelvebb Q}_+$ denote the positive rationals. For a vector
${\bf a}=(a(v):v \in V) \in {\fam\bbfam\twelvebb Z}_+^V$ let $G_{\va}$ denote the graph
obtained by replacing each $v \in V$ by an independent set of size
$a(v)$. Formally $V(G_{\va})=\{(v,i):v \in V~,~1 \leq i \leq a(v)\}$
and $\{(u,i),(v,j)\} \in E(G_{\va})$ if $\{u,v\} \in E$. The projection
$(v,i) \rightarrow v$ induces a homotopy equivalence between
${\rm I}(G_{\va})$ and ${\rm I}(G)$. In particular $
\eta({\rm I}(G_{\va}))=\eta({\rm I}(G))$.
\
\\
\\ {\bf Proof of Theorem \ref{etagegamma}:} Let $P$ be a
representation of $G$. By linear programming duality $$ |P|=\min\{
{\bf \alpha} \mathcal Dot {\bf 1}~:~{\bf \alpha} \geq 0~,~{\bf \alpha} PP^T \ge
{\bf 1}~\}=$$ $$\max\{ {\bf \alpha} \mathcal Dot {\bf 1}~:~{\bf \alpha} \geq 0~,~{\bf \alpha}
PP^T \leq {\bf 1}~\}=$$
$$
{\rm Sup}\{{\bf \alpha} \mathcal Dot
{\bf 1}~:~{\bf \alpha} \in {\fam\bbfam\twelvebb Q}_+^V~~,~~{\bf \alpha} PP^T \leq {\bf 1}~\}~.
$$ Let ${\bf \alpha} \in {\fam\bbfam\twelvebb Q}_+^V$ such that ${\bf \alpha} PP^T \leq {\bf 1}$.
Write ${\bf \alpha}= \frac{{\bf a}}{k}$ where $k \in {\fam\bbfam\twelvebb Z}_+$ and
${\bf a}=(a(v):v \in V) \in {\fam\bbfam\twelvebb Z}_+^V$. Let $N=|V(G_{\va})|=\sum_{u \in
V}a(u)$. Consider the representation $Q$ of $G_{\va}$ given by
$Q((u,i))=P(u)$ for $(u,i) \in V(G_{\va})$. By Claim \ref{lambdan}
$$\lambda_N(G_{\va}) \leq \max_{(u,i) \in V(G_{\va})}~ Q((u,i)) \mathcal Dot
\sum_{(v,j) \in V(G_{\va})} Q((v,j))= $$ $$ \max_{u \in V}~P(u) \mathcal Dot
\sum_{v \in V} a(v)P(v) \leq k~~.$$ Hence by Theorem \ref{indeta}
$${\bf \alpha} \mathcal Dot {\bf 1}= \frac{1}{k}\sum_{v \in V} a(v) =
\frac{N}{k} \leq$$ $$ \frac{N}{\lambda_N(G_{\va})} \leq
\eta({\rm I}(G_{\va}))=\eta({\rm I}(G))~~.$$ {\begin{flushright} $\Box$ \end{flushright}} \noindent {\bf Remarks:}
\\ 1. Let $C_n$ denote the $n$-cycle on the vertex
set $V=\{0,\ldots, n-1\}$. For $n=3k$ define a representation $P$
of $C_{3k}$ by $$ P(\ell)= \left\{
\begin{array}{ll}
e_{2j} & \ell=3j \\
e_{2j}+e_{2j+1} & \ell=3j+1 \\
e_{2j+1}+e_{2j+2} & \ell=3j+2
\end{array}
\right. $$ where $e_0,\ldots,e_{2k-1}$ are orthogonal unit vectors
and the indices are cyclic modulo $2k$. Let $\alpha
\in {\fam\bbfam\twelvebb R}^V$ be given by $~\alpha(\ell)=1~$ if $3$ divides $\ell$
and zero otherwise. Since $\alpha PP^T = {\bf 1}$, it follows by
linear programming duality that $\Gamma(C_{3k}) \geq \alpha \mathcal Dot
{\bf 1}=k$. On the other hand (see Claim 3.3 in {\rm I}te{M03})
$\eta(I(C_n))=\lfloor\frac{n+1}{3}\rfloor$. Therefore
$\eta({\rm I}(C_{3k}))=\Gamma(C_{3k})=k$. For $n=3k+1$ it can
similarly be shown that $\eta({\rm I}(C_{3k+1}))=\Gamma(C_{3k+1})=k$.
The case $n=3k-1$ is more involved and we only have the bounds
$k-\frac{1}{2} \leq \Gamma(C_{3k-1}) \leq \eta({\rm I}(C_{3k-1}))=k$.
Note that for cycles the bound $\eta({\rm I}(G)) \geq \gamma^*_s(G)$
is weaker since $\gamma^*_s(C_n)=\frac{n}{4}$.
\\ 2.
It can be shown that for any graph $\Gamma(G) \geq {\rm
Sup}\{\gamma^*_s(G_{\va})~:~{\bf a} \in {\fam\bbfam\twelvebb Z}_+^V\}. $ We do not know of
examples with strict inequality.
\section{A Hall Type Theorem for Fractional Width}
\label{s:hall}
Let $Z$ be a simplicial complex on the vertex set $W$ and let
$\bigcup_{i=1}^m W_i$ be a partition of $W$. A simplex $\tilde{A}u \in
Z$ is {\it colorful} if $|\tilde{A}u \mathcal Ap W_i|=1$ for all $1 \leq i \leq
m$. For $W' \subset W$ let $Z[W']$ denote the induced subcomplex
on $W'$. The following Hall's type sufficient condition for the
existence of colorful simplices appears in {\rm I}te{AH00} and in
{\rm I}te{M01}.
\begin{proposition}
\label{hom} If for all $\emptyset \neq I \subset [m]$
$$\eta(Z[\bigcup_{i \in I} W_i]) \geq |I|$$ then $Z$ contains a
colorful simplex.
\end{proposition}
Let $G$ be a graph on the vertex set $W$ with a partition
$W=\bigcup_{i=1}^m W_i$. A set $S \subset W$ is {\it colorful} if
$S \mathcal Ap W_i \neq \emptyset$ for all $1 \leq i \leq m$. The induced
subgraph on $W' \subset W $ is denoted by $G[W']$. Combining
Theorem \ref{etagegamma} and Proposition \ref{hom} we obtain the
following
\begin{theorem}
\label{gammahall} If $\Gamma(G[\bigcup_{i\in I} W_i]) > |I|-1$
for all $\emptyset \neq I \subset [m]$ then $G$ contains a
colorful independent set.
\end{theorem}
Let ${\mathcal Al F} \subset 2^V$ be a hypergraph, possibly with multiple
edges. The {\it line graph} $G_{{\mathcal Al F}}=(W,E)$ associated with ${\mathcal Al F}$
has vertex set $W={\mathcal Al F}$ and edge set $E$ consisting of all
$\{F,F'\} \subset {\mathcal Al F}$ such that $F \mathcal Ap F' \neq \emptyset$. A
matching in ${\mathcal Al F}$ corresponds to an independent set in $G_{{\mathcal Al F}}$.
For each $F \in {\mathcal Al F}$ let $P(F) \in {\fam\bbfam\twelvebb R}^V$ denote the incidence
vector of $F$. $P$ is clearly a vector representation of $G_{{\mathcal Al F}}$
and satisfies $|P|=w^*({\mathcal Al F})$. Thus $\Gamma(G_{{\mathcal Al F}}) \geq
w^*({\mathcal Al F})$. \ \\
\\ {\bf Proof of
Theorem \ref{wstar}:} Let ${\mathcal Al F}$ denote the disjoint union of the
${\mathcal Al F}_i$'s, and consider the graph $G_{{\mathcal Al F}}=(W,E)$ with the partition
$W=\mathcal Up_{i=1}^m W_i$ where $W_i={\mathcal Al F}_i$. Then for any $\emptyset
\neq I \subset [m]$ $$\Gamma(G_{{\mathcal Al F}}[\mathcal Up_{i \in I}
W_i])=\Gamma(G_{\mathcal Up_{i \in I} {\mathcal Al F}_i}) \geq $$ $$w^*(\mathcal Up_{i \in
I} {\mathcal Al F}_i) > |I|-1~~.$$ Theorem \ref{gammahall} implies that
$G_{{\mathcal Al F}}$ contains a colorful independent set, hence ${\mathcal Al F}i$
contains an SDR. {\begin{flushright} $\Box$ \end{flushright}}
\end{document}
|
\begin{document}
\title{Non-conservative Forces via Quantum Reservoir Engineering}
\author{Shanon L. Vuglar}
\affiliation{University of Melbourne, Parkville, VIC 3010, Australia }
\affiliation{Princeton University, Princeton, NJ 08544, USA}
\author{Dmitry V. Zhdanov}
\affiliation{Northwestern University, Evanston, IL 60208, USA}
\author{Renan Cabrera}
\affiliation{Princeton University, Princeton, NJ 08544, USA}
\author{Tamar Seideman}
\affiliation{Northwestern University, Evanston, IL 60208, USA}
\author{Christopher Jarzynski}
\affiliation{University of Maryland, College Park, MD, 20742, USA}
\author{Denys I. Bondar}
\affiliation{Princeton University, Princeton, NJ 08544, USA}
\date{\today}
\begin{abstract}
A systematic approach is given for engineering dissipative environments that steer quantum wavepackets along desired trajectories. The methodology is demonstrated with several illustrative examples: environment-assisted tunneling, trapping, effective mass assignment and pseudo-relativistic behavior. Non-conservative stochastic forces do not inevitably lead to decoherence -- we show that purity can be well-preserved.
These findings highlight the flexibility offered by non-equilibrium open
quantum dynamics.
\end{abstract}
\pacs{03.65.Ta, 03.65.Ca, 03.63.Yz}
\maketitle
\section{Introduction}
Throughout its short history, the control of quantum systems has predominantly been implemented using conservative forces, e.g., manipulating quantum phenomena in Hamiltonian systems via dipole coupling with laser or microwave pulses.
This may seem surprising given the widespread use of non-conservative forces in other control applications -- consider the wind (sailing vessels, windmills) and friction (mechanical brakes).
The historical focus on conservative forces is, perhaps, best explained
by the widely held belief that immersing a quantum system into a complex environment inevitably destroys its quantum dynamical features.
The monopoly of conservative forces in quantum control is now being challenged by quantum reservoir engineering (QRE)
\cite{poyatos1996quantum,verstraete2009quantum,fedortchenko2014finite,kurizki2015thermal,kienzler2015quantum,pan2016ground,rouchon2014models}. In particular, it has been shown that it is possible to preserve and even enhance the quantum dynamical features of a system by judiciously coupling the system to a dissipative environment.
Applications of quantum reservoir engineering
include amplification \cite{metelmann2014quantum},
nonreciprocal photon transmission \cite{metelmann2015nonreciprocal,metelmann2017nonreciprocal},
photon blockade \cite{miranowicz2014state},
efficient photoinduced charge separation in solar energy conversion \cite{Zhdanov2015quantum},
binding of atoms \cite{lemeshko2013dissipative,wuster2017quantum},
inducing phase transitions \cite{kaczmarczyk2016dissipative,weimer2016tailored,overbeck2017multicritical},
implementation of quantum gates \cite{albert2016holonomic,ticozzi2017quantum,arenz2016universal,arenz2017lindbladian},
and the generation of entangled
\cite{cheng2016preservation,liu2016comparing,zippilli2015steady,yang2015generation,mirza2015controlling,arenz2013generation},
squeezed \cite{kronwald2013arbitrarily,woolley2014two,grimsmo2016quantum},
and other exotic
\cite{koga2012dissipation,holland2015single,asjad2014reservoir,chestnov2016permanent} quantum states.
In this Letter, we provide a systematic approach for engineering dissipative environments that steer quantum wavepackets along desired trajectories as defined by the following equations:
\begin{subequations}\label{__problem_gen}
\begin{gather}
\der{}{t}{\midop{\hat x}}{=}\midop{G(\hat p)}, \label{eqn:Ehrenfest1}\\
\der{}{t}{\midop{\hat p}}{=}\midop{F(\hat x)}. \label{eqn:Ehrenfest2}
\end{gather}
\end{subequations}
Here, $\midop{\hat x}$ and $\midop{\hat p}$ denote the wavepacket's mean position and momentum.
The environments obtained not only enhance desired quantum properties,
but can also be made to preserve the purity of the underlying quantum system.
Equations~\eqref{__problem_gen} with various functions $G$ and $F$ embrace a plethora of quantum behaviors; we provide several illustrative examples.
We first consider compensating for a potential barrier in the case of quantum tunneling and then mimicking a potential to trap a wave packet at a desired location.
We also consider more exotic applications such as changing the effective mass of a quantum particle and emulating relativistic effects.
The scope of our analysis is restricted to Markovian environments modeled within the Lindblad formalism.
We also discuss possible laboratory realizations of the Lindblad operators for specific examples.
\section{Formal analysis}
For definiteness, assume that the system of interest is a one-dimensional particle of mass $m$ moving in a potential $U(x)$. Our objective is to dissipatively couple the system to $K+N$ baths in such a way that the average particle localization in phase space will follow Eqs.~\eqref{__problem_gen} for given, desirable, $G(\hat p)$ and $F(\hat x)$. Assuming
Markovian system-bath interactions, the system state (described by the density matrix $\hat\rho$) evolves according to the Lindblad master equation
\begin{align}
\frac{\,\mathrm{d} \hat{\rho}}{\,\mathrm{d} t} &=
- \frac{i}{\hbar} [ \hat{H}, \hat{\rho} ]
+ \sum_{k{=}1}^{K}\mathcal{D}_{\hat{A}_k}[\hat{\rho}]
+ \sum_{n{=}1}^{N}\mathcal{D}_{\hat{B}_n}[\hat{\rho}]
, \label{eqn:dW}
\end{align}
where $\hat H$ is a given system Hamiltonian
\begin{align}
\hat{H} &= \frac{1}{2m}{\hat{p}}^2 + U(\hat{x}), \label{eqn:H}
\end{align}
and the effect of the bath is represented via the operators
$\hat A_k$, $\hat B_n$ as
\begin{align}
\mathcal{D}_{\hat{A}}[\hat{\rho}] &=
\frac{1}{\hbar} \Big( \hat{A} \hat{\rho} \hat{A}^\dagger
- \frac{1}{2} \hat{\rho} \hat{A}^\dagger \hat{A}
- \frac{1}{2} \hat{A}^\dagger \hat{A} \hat{\rho}
\Big). \label{eqn:D}
\end{align}
Under these assumptions, the control problem reduces to determining suitable forms for
the operators $\hat A_k$, $\hat B_n$
and providing physical evidence that the corresponding environments can be engineered
in the laboratory.
Using Operational Dynamical Modeling \cite{Bondar2011c,Bondar2014wigner} the following expressions for $\hat{A}_k = A_k(\hat{x})$ and $\hat{B}_n = B_n(\hat{p})$ are obtained:
\newcommand{{\cal K}}{{\cal K}}
\begin{subequations}\label{eqn:sol}
\begin{align}
A_k(x) = \: &R_k(x) \exp \left(
{i\int \frac{f_k(x)}{R_k^2(x)} \,\mathrm{d} x}
\right), \label{eqn:A} \\
B_n(p) = \: &S_n(p) \exp \left(
{-i \int \frac{g_n(p)}{S_n^2(p)} \,\mathrm{d} p}
\right). \label{eqn:B}
\end{align}
Here, $f_k(x)$, $g_k(p)$, $R_k(x)$, and $S_k(p)$
denote arbitrary real valued functions such that
\begin{gather}\label{eqn:F_k}
\sum_{k=1}^K f_k(x){=}F(x){+}\der{U(x)}{x};~~\sum_{n{=}1}^{N} g_n(p){=}G(p){-}\frac{p}{m}.
\end{gather}
\end{subequations}
Note that Eqs.~\eqref{__problem_gen} are satisfied regardless of the initial state.
To provide insight into the physical nature of environments that implement (5), we now consider several illustrative examples.
Unless stated otherwise, atomic units (a.u.), $\hbar = m_e = |e| = 1$, are used throughout.
\section{Environmentally assisted quantum tunneling}
It is common knowledge that cycling uphill is much easier with assistance from a tailwind.
Similarly, a ``polarized electron wind'' can be used to enhance tunneling rates for an atomic wavepacket approaching a potential barrier $U(\hat x)$ (see Fig.~\ref{@FIG.01}).
If non-conservative forces are engineered so as to cancel the potential forces of the system, then dynamics similar to those of a free particle can be obtained.
Consider Eqs.~\eqref{__problem_gen} and choose $G(p){=}\frac{p}{m}$ and $F(\hat x){=}0$.
These dynamics can be obtained with the following choice of environmental operators $A_{\pm}$, which satisfy \eqref{eqn:sol} for the case
$K=2$, $N=0$, and $R_1 = R_2 = C$ where $C$ is a constant:
\begin{subequations}\label{eqn:sol_tun}
\begin{gather}
A_{\pm}{=}C e^{\pm\frac{2i}{\hbar}\int{{\tilde p_{\pm}(x)d x}}}, \label{eqn:A_tun}
\end{gather}
where the functions $\tilde p_{\pm}(x)$ obey the relation
\begin{gather}
\tilde p_{+}(x){-}\tilde p_{-}(x){=}\tfrac{\hbar}{2C^2}\tfrac{\,\mathrm{d} U(x)} {\,\mathrm{d} x}. \label{eqn:p_pm_tun}
\end{gather}
\end{subequations}
\begin{figure}
\caption{Environment assisted quantum tunneling resembles cycling with an umbrella: The environment action is qualitatively similar to tailwind (headwind) when going uphill (downhill). The net effect is a reduction of the back-scattering probability with minimal side effects on the wavepacket parameters.
\label{@FIG.01}
\label{@FIG.01}
\end{figure}
Inspired by the wind analogy, we now propose a physical implementation of the environment~\eqref{eqn:sol_tun}.
Consider a quantum probe that is an atom of mass $m$ in the non-degenerate ground electronic state with electric polarizability $\alpha$, zero angular momentum, and negligible magnetic polarizability.
Suppose that the motion of the probe along the $\vec\epsilon_x$-axis is impeded by an effective barrier $U(x)=-\alpha {\cal E}(x)^2/4$
created by an off-resonant, blue-detuned ($\alpha{<}0$) laser field ${\vec\epsilon}_x \mathcal{E}(x)\cos(\omega(t-{z}/{c}))$.
In the presence of a static magnetic field of the form ${\vec\epsilon}_z{\cal B}(x)$, the desired dissipative environment can be created by two counterpropagating electron jets, in which the electrons have opposite magnetic moments
$\hat{\mu }_s{=}{\pm}\hat{\sigma }_z \mu_{\mathrm{B}}$, incident velocities ${\pm}{\vec\epsilon }_x\frac{p_0}{m_{\mathrm{e}}}$, and fluxes ${\pm}{\vec\epsilon }_x j$ (here $\mu _{\mathrm{B}}$ is the Bohr magneton).
The resulting electron recoils create an effective pressure on the probe.
Note that without a magnetic field, the mean impacts of both jets would mutually compensate each other.
However, when a magnetic field is applied, the opposite electron spin polarizations of the jets break this symmetry resulting in a nonzero net force on the probe.
To quantitatively describe this effect, we assume that~i) the electron flux $j$ is low enough to neglect multiple scattering of electrons,~ii) all interactions of electrons with the probe can be modeled as ideal elastic backscattering events,~iii) the incident electron velocity ${p_0}/{m_{\mathrm{e}}}$ is much larger than the characteristic velocities of the probe, and~iv)
\begin{equation}
p_0{\gg}\sqrt{2\mu_{\mathrm{B}}m_{\mathrm{e}}|{\cal B}(x)|} \label{eqn:pgg}.
\end{equation}
The inequality \eqref{eqn:pgg} allows the wavefunctions of incident electrons in the jets to
be modeled semiclassically as
\begin{equation}
\psi_{\pm}{\propto}\frac{e^{\frac{\pm i}{\hbar}\int \tilde p_{\pm}(x)\,\mathrm{d} x}}{\tilde p_{\pm}(x)},
~~\tilde p_{\pm}(x){=}\sqrt{p_0^2\pm 2\mu_{\mathrm{B}}m_{\mathrm{e}}{\cal B}(x)}. \label{eqn:ptilde}
\end{equation}
In the case of $C = \sqrt{\hbar \tilde{\sigma} j}$ where $\tilde\sigma$ is the scattering cross section, Eqs. (\ref{eqn:dW}) and \eqref{eqn:A_tun}
describe the ``wind effect'' of the electron jets on the probe.
Note that $C^2$ is proportional to the number of electron scatterings in a given time interval. Under the assumption of Poissonian statistics, the standard deviation over the same time interval of the force exerted by the collisions is expected to be proportional to $Cp_0$.
This parameter will be used below to elucidate physical mechanisms.
Finally, Eqs.~\eqref{eqn:p_pm_tun} and \eqref{eqn:ptilde} determine the magnetic field profile required for effectively barrierless propagation:
\begin{align}
{\cal B}(x) &= \frac{\hbar}{16 \mu_\mathrm{B}m_\mathrm{e}C^4}
{\frac{\,\mathrm{d} U(x)}{\,\mathrm{d} x}
\sqrt{16 {p_0}^2 C^4 - \left( \hbar \frac{\,\mathrm{d} U(x)}{\,\mathrm{d} x}\right)^2}}.
\label{eqn:calB}
\end{align}
The character of the system-environment coupling is determined by the momenta $p_{\pm}$ of the incident electrons.
For small magnitudes of $|p_{\pm}|$, large collision rates are required to create sufficient non-conservative forces to oppose the potential forces.
In this case, the overall effect of the collisions can be represented as an effective pressure, and the dissipative term in \eqref{eqn:dW} in the limit $C{\to}\infty$, $|p_{\pm}|{\to}0$ can be represented as an effective Hamiltonian,
$\hat{H}_{\mathrm{eff}} = -U(\hat{x})$,
which cancels the potential barrier $U(x)$ and results in entirely coherent (essentially free-particle) dynamics.
On the other hand, large $|p_{\pm}|$ corresponds to the shot noise limit where strong but rare collisions produce highly fluctuating stochastic environmental forces. This leads to rapid wavepacket decoherence and a reduction in tunneling probabilities.
These effects can be seen in Fig.~\ref{fig:line_plots},
which depicts simulation results for a hydrogen-like atom ($m{=}1837m_e$ where $m_e$ is electron mass) tunneling through a Gaussian potential barrier in the presence of an engineered environment as described by Eqs. (\ref{eqn:dW}), \eqref{eqn:sol_tun}, and \eqref{eqn:ptilde}.
In all cases, Eqs.~\eqref{__problem_gen} are satisfied.
For small values of $|p_{\pm}|$, high tunneling rates and purity are achieved
for the atomic quantum state after interaction with the barrier.
However, above a critical $|p_{\pm}|$
(which depends on the standard deviation of the environmental force $Cp_0$),
the tunneling probability and purity dramatically degrade;
this corresponds to the shot noise regime.
\begin{figure}
\caption{The transmission probability (a) and purity (b) for a Gaussian atomic wavepacket tunneling through a potential barrier in the presence of electron jets [Eqs.~\eqref{eqn:dW}
\label{fig:line_plots}
\end{figure}
One can observe slight increases in the purity prior to the rapid falling away in each of the curves in Fig.~\ref{fig:line_plots}(b). These peaks correspond to a transitional regime wherein the tunneling rates are starting to degrade [see Fig.~\ref{fig:line_plots}(a)] and reflection from the barrier becomes noticeable ($\propto 10\%$).
Furthermore, the inequality \eqref{eqn:pgg} is only marginally satisfied; the observed purity increase may be an artifact of the semi-classical approximation \eqref{eqn:ptilde}.
The ability of environmental coupling to enhance tunneling rates has been previously recognized.
Under certain physical conditions, a metastable quantum system submerged into a low temperature environment decays, exciting directional bath modes such that the quantum system acquires kinetic energy which in turn assists under the barrier motion~\cite{leggett1984quantum,grabert1984quantum, pollak1986transition,leggett1996effect}.
In particular, an atom can acquire an extra momentum kick, facilitating tunneling by spontaneously emitting a photon.
This mechanism has been systematically explored in Refs. \cite{japha1996, Schaufler1999keyhole} and yielded Zeno and anti-Zeno quantum control schemes \cite{barone2004}.
In these schemes the incident wavepackets undergo destructive spontaneous dissipative changes. However, in our example the enhanced tunneling is achieved without destroying the state purity, as can be seen from Fig.~\ref{fig:line_plots}.
It is noteworthy that environmentally assisted tunneling was recently
experimentally demonstrated in
lithium niobate \cite{somma2014high}.
\section{Dissipative traps}
The same strategy can also be used to trap an atom;
by setting $U(x){=}{-}U_{\text{eff}}(x)$ in Eq.~\eqref{eqn:calB}
the environment will mimic the potential $U_{\text{eff}}(x)$.
Figure.~\ref{fig:trapping} depicts simulation results for a hydrogen atom immersed in
trapping environments with different standard deviations of the environmental force $Cp_0$.
As in the tunneling case, larger values for $Cp_0$ for a given $p_0$ cause additional heating.
This deteriorates the trapping via purity losses and wavepacket spreading.
Nevertheless, one can see that for each of the
cases depicted the environmentally trapped wavepacket remains more spatially localized than the free wavepacket.
These results suggest that the optimal strategy for trapping a particle is to use jets with the smallest $Cp_0$ for which~\eqref{eqn:pgg} is satisfied.
\begin{figure}
\caption{The spreading in position (a),
increase in energy (b),
spreading in momentum (c), and
decrease in purity (d)
for an atomic wavepacket initially in the ground state of a harmonic oscillator [$U(x) = \frac{1}
\label{fig:trapping}
\end{figure}
It was shown in Ref.~\cite{lemeshko2013dissipative} that non-conservative forces between atoms can lead to binding,
even when the potential interaction is repulsive.
\section{Exotic applications}
We have demonstrated that non-conservative forces can effectively mimic desired conservative interactions, however, the utility of such forces is much wider.
Non-conservative forces can also be used to obtain modifications $G(p)$ to the dispersion relationship~(\ref{eqn:Ehrenfest1}) --
note that such modifications cannot be implemented via conservative forces.
We consider two applications for such modifications:
tuning the effective mass of a quantum particle and emulating relativistic effects.
Consider a quantum particle of mass $m$ in a potential $U(x)$. The particle will exhibit an effective mass $M$ when immersed in an environment
described by the dissipator
$\mathcal{D}_{B}$
[as in Eq.~(\ref{eqn:dW})]
with
\begin{equation} \label{eqn:env3}
B(p) = C \exp \left[ - \frac{i (m - M) p^2}{ 2mMC^2} \right].
\end{equation}
That is, the system dynamics will satisfy the constraints
\begin{equation}
\frac{\,\mathrm{d} \;}{\,\mathrm{d} t} \langle \hat{x} \rangle
= \frac{1}{M} \langle \hat{p} \rangle, \quad
\frac{\,\mathrm{d}}{\,\mathrm{d} t} \langle \hat{p} \rangle
=
- \left\langle \frac{\,\mathrm{d} U(\hat{x})} {\,\mathrm{d} \hat{x}} \right\rangle
. \label{eqn:Ehrenfest3}
\end{equation}
Figure~\ref{fig:ex3} depicts simulation results:
the particle of mass $m$ in the environment \eqref{eqn:env3} evolves in excellent agreement with an environment-free particle of mass $M$.
\begin{figure}
\caption{
The expectation value of the position as a function of time for a hydrogen atom in a ramp potential $U(x) = 3.2 \times 10^{-3}
\label{fig:ex3}
\end{figure}
The effective mass approximation is ubiquitously used to describe the motion of a quantum particle in the periodic field of a solid.
Recently, a negative effective mass was experimentally achieved \cite{khamehchi2017negative}.
An atom interacting with the standing wave of a single photon in the cavity also acquires an effective mass \cite{larson2005effective}.
We conjecture that environmentally induced mass can emerge for an atom elastically scattering off incoherent light seeded into a cavity.
We now turn our attention to environmentally induced quasi-relativistic behaviour.
Once again, consider a quantum particle of mass $m$ in a potential $U(x)$.
Suppose we wish the system dynamics to satisfy the constraints
\begin{equation}
\frac{\,\mathrm{d} \;}{\,\mathrm{d} t} \langle \hat{x} \rangle
= \left\langle \frac{\hat{p}}{\sqrt{m^2 + \hat{p}^2 / c^2}} \right\rangle, \quad
\frac{\,\mathrm{d}}{\,\mathrm{d} t} \langle \hat{p} \rangle
= - \left\langle \frac{\,\mathrm{d} U(\hat{x})} {\,\mathrm{d} \hat{x}} \right\rangle.
\label{eqn:Ehrenfest4}
\end{equation}
This can be achieved with an environment described by the dissipator
$\mathcal{D}_{B}$
with
\begin{equation} \label{eqn:env4}
B(p) = C \exp \left[ \frac{i}{C^2} \left( \frac{p^2}{2m} - c \sqrt{m^2c^2 + p^2} \right) \right].
\end{equation}
\begin{figure}
\caption{The expectation value of the velocity as a function of time for an electron in a ramp potential $U(x) = - 10^3 x$ a.u.
The dotted blue curve depicts the environment-free case.
The solid green curve depicts the electron in an environment \eqref{eqn:env4}
\label{fig:ex4}
\end{figure}
Figure~\ref{fig:ex4} depicts simulation results
confirming that the chosen environment induces quasi-relativistic behaviour for an arbitrarily small speed of light. In particular, the environment mimics the effect of time dilation
as the particle velocity approaches the chosen speed of light.
The dispersion relation emerges as an effective description of the self-interaction of a bare quantum particle with a larger system with some characteristic symmetry. Generalizing the logic of Ref. \cite{larson2005effective}, we conjecture that tailoring the spectral transmission characteristics of a cavity
and employing multi-color electromagnetic radiation with specific photon statistics
should provide access to a large class of dispersion relations.
\textit{Outlook.}
Physicists, chemists, and engineers are increasingly looking for new ways to
manipulate quantum systems -- non-conservative environments provide one such resource.
We give a systematic approach
for designing such environments to steer wavepackets along desired trajectories.
The method is demonstrated via several examples:
enhancing quantum tunneling,
trapping particles,
inducing effective mass, and
emulating relativistic effects.
The proposed dissipators not only enhance desired quantum properties,
they can be engineered to do so while preserving the purity of the underlying system.
A distinct feature of our method is that for a given $F$ and $G$,
the resulting dynamics always satisfy
Eqs.~\eqref{__problem_gen},
irrespective of the initial state.
Finally, note that $F(x)$ in Eq.~(\ref{eqn:Ehrenfest2}) is the sum of the potential force $-\,\mathrm{d} U(x) /\,\mathrm{d} x$
and the environmentally induced forces $f_k(x)$ [Eq.~\eqref{eqn:F_k}].
Despite being of different physical origins [Eqs.~(\ref{eqn:H}) and (\ref{eqn:A}) respectively],
these forces contribute to $F(x)$ on an equal footing.
This observation may help shed light on the discussion regarding the entropic interpretation of the gravitational force \cite{verlinde2011origin}.
In this regard, it would be beneficial to find a dynamical signature that could efficiently discriminate between potential and statistical interactions.
\textit{Acknowledgments.}
S.L.V. was supported by the Australian Research Council
(DP130104510). D.I.B., R.C. respectively acknowledge financial support
from NSF CHE 1464569 and DOE DE-FG-02-ER-15344.
T. S. thanks the National Science Foundation (Award No. CHE-1465201) for support.
D. I. B. is also supported by Humboldt Research Fellowship for Experienced Researchers and AFOSR Young Investigator Research Program (No. FA9550-16-1-0254).
\end{document}
|
\begin{document}
\title[Wave breaking for Whitham-type equations]{The wave breaking for Whitham-type equations revisited}
\author{Jean-Claude Saut}
\author{Yuexun Wang}
\address{ Universit\' e Paris-Saclay, CNRS, Laboratoire de Math\' ematiques d'Orsay, 91405 Orsay, France.}
\email{[email protected]}
\address{
School of Mathematics and Statistics,
Lanzhou University, 370000 Lanzhou, China.}
\address{Universit\' e Paris-Saclay, CNRS, Laboratoire de Math\' ematiques d'Orsay, 91405 Orsay, France.}
\email{[email protected]}
\thanks{}
\subjclass[2010]{76B15, 76B03, 35S30, 35A20}
\keywords{weak dispersion, shock formation}
\begin{abstract} We prove wave breaking (shock formation) for some Whitham-type
equations which include the Burgers-Hilbert equation, the fractional Korteweg-de Vries equation, and the classical Whitham equation. The result seems to be new for the Burgers-Hilbert equation. In the other cases we provide simpler proofs than the known ones.
\end{abstract}
\maketitle
\section{Introduction}
We consider nonlocal dispersive perturbations of the Burgers equation
\begin{equation}\label{Whitype}
u_t+uu_x+\int_{-\infty}^{\infty}k(x-y)u_x(y,t)dy=0,
\end{equation}
where $k$ is a real-valued kernel measuring the (weak) dispersive effects.
This equation can also be written in the form
\begin{equation*}\label{Whibis}
u_t+uu_x-Lu_x=0,
\end{equation*}
where the Fourier multiplier operator $L$ is defined by
$$\widehat{Lf}(\xi)=p(\xi)\hat{f}(\xi),$$
where $p=\hat{k}.$
A particular case is the fractional KdV equation (fKdV)
\begin{align}\label{eq:main-1}
\partial_t u+u\partial_xu-|D|^\alpha \partial_xu=0,
\end{align}
where \(u\) maps \(\mathbb{R}_t\times\mathbb{R}_x\) to \(\mathbb{R}\) and \(|D|^\alpha\) is the usual Fourier multiplier operator with symbol \(|\xi|^\alpha\).
We will restrict for the fKdV equation to the {\it weakly dispersive} case $-1<\alpha<0$, and refer to \cite{LPS2, MPV} for a study of the case $0<\alpha<1$ which displays quite different (dispersive) properties.
When \(\alpha=-1\), \eqref{eq:main-1} is the Burgers-Hilbert equation introduced in \cite{BH} as a model for waves with constant nonzero linearized frequency providing an effective equation for the motion of a vorticity discontinuity in a two-dimensional flow of an inviscid, incompressible fluid:
\begin{align}\label{eq:BH}
u_t+uu_x-\mathcal H u=0,
\end{align}
where $\mathcal H= \text{p.v.}\; \frac{1}{x}$ is the Hilbert transform with Fourier symbol $-i\text{sgn}\;\xi.$
We will also consider the Whitham equation introduced in \cite{Whi}
\begin{equation}\label{Whit}
\partial_tu+u\partial_xu+\int_\mathbb{R} K(x-y)\partial_y u(y,t)dy=0,
\end{equation}
where
\begin{align*}
K(x)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R} e^{\mathrm{i}x\xi}\sqrt{\frac{\tanh \xi}{\xi}}\,\mathrm{d} \xi,
\end{align*}
and its rescaled version
\begin{align}\label{Whitresc}
\partial_t u+\epsilon u\partial_x u+\int_\mathbb{R} K_\epsilon(x-y)\partial_y u(y,t)\,\mathrm{d} y=0,
\end{align}
where \(K_\epsilon\) is defined by
\begin{align}\label{res-kernal}
K_\epsilon(x)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R} e^{\mathrm{i}x\xi}\sqrt{\frac{\tanh \sqrt{\epsilon}\xi}{\sqrt{\epsilon}\xi}}\,\mathrm{d} \xi=\frac{1}{\sqrt{\epsilon}}K\left(\frac{x}{\sqrt{\epsilon}}\right),
\end{align}
$\epsilon$ being a positive parameter, the long wave limit $\epsilon\to 0$ making the link with the KdV equation, see \cite{KLPS} and Section \ref{sec7} below.
Although of not clear physical relevance (see however \cite{KLPS} for a rigorous connection of the Whitham equation to the modeling of weakly nonlinear water waves), the fKdV equation \eqref{eq:main-1} when $-1<\alpha<0$ and the Whitham equation \eqref{Whit} (likes the fKdV equation with $\alpha=-\frac{1}{2}$ for high frequencies) are very rich toy models to investigate the effects of a weak dispersive term on the dynamics of a conservation law such as the Burgers equation
\begin{equation}\label{Bur}
\partial_t u+\epsilon u\partial_x u=0, \quad u(x,0)=\phi(x).
\end{equation}
It is well known that for the Burgers equation \eqref{Bur} any non trivial non increasing $H^2$ initial data $\phi$ of size $O(1)$ will lead to a shock formation at a finite time of order $O\left(\frac{1}{\epsilon ||u_0||_{H^2}}\right).$
The aim of the present paper is to investigate the effect of adding a weak dispersion on this phenomenon, in particular to see if the shock formation persists. Such a question was already raised for the first time by Whitham in \cite{W} and this issue has been considered in previous works that we describe now.
Naumkin and Shishmarev \cite{NS} and Constantin and Escher \cite{CE} have proven a wave breaking phenomena for a Whitham type equation such as \eqref{Whitype} with a kernel $k$ satisfying
\begin{equation*}
k\in C(\mathbb{R})\cap L^1(\mathbb{R}), \;\text{symmetric and monotonically decreasing on}\; \mathbb{R}_+.
\end{equation*}
This result does not apply to the Whitham equation \eqref{Whit} since the Whitham kernel satisfies
$K(0)=\infty.$
When $-1<\alpha<0,$ Castro, C\' ordoba and Gancedo have proven for the fKdV equation \eqref{eq:main-1} with some initial data in $L^2(\mathbb{R})\cap C^{1+\delta}(\mathbb{R})$ with $0<\delta<1,$ a finite time blow-up of the $C^{1+\delta}(\mathbb{R})$ norm, without proving the occurrence of a wave breaking, that is blow-up of the sup norm of gradient of the solution, the solution itself remaining bounded.
To our knowledge no rigorous proof of shock formation for the Burgers-Hilbert equation \eqref{eq:BH} has been established although the numerical simulations in \cite{BH} strongly suggest its existence.
Finally, the existence of a wave breaking for the fKdV equation \eqref{eq:main-1} when
$-1<\alpha<-\frac{1}{3}$ and for the Whitham equation \eqref{Whit} has been established in \cite{HT, Hur}.
We refer to \cite{MR3317254, KLPS} for various numerical simulations of the fKV equation and Whitham equations, in particular for a description of the blow-up solutions.
We aim in this paper to provide a simple proof of wave breaking for the fKdV equation in the all range $-1\leq \alpha<-\frac{2}{5}$ (including thus the Burgers-Hilbert equation) and for the Whitham equation.
Contrary to \cite{HT,Hur} our proof does not use an infinite number of ODE's and hence less assumptions on the initial data are needed. The wave breaking for the Burgers-Hilbert equation \eqref{eq:BH} is not covered in \cite{HT,Hur} whose argument causes a logarithmic loss in estimating the term \(K_1(t,x)\) (see \eqref{BH-kernal}) which prevents the proof to work. We overcome this difficulty by using the cancellation property of Hilbert transform and Morrey's inequality to replace the integration by parts.
For the Whitham equation \eqref{Whit}, our new observation is that one can use interpolation between \(\|\partial_x^3u\|_{L^2}\) (the singularity is too high which can not be used directly) and \(\|\partial_xu\|_{L^\infty}\) (the singularity is low) to control \(\|\partial_x^2u\|_{L^\infty}\), this balance allows us to give a very simple proof. This idea also works for the fKdV equation \eqref{eq:main-1} in the range $-\frac{1}{2}\leq \alpha<-\frac{2}{5}$. We remark that one could push \(\alpha\) forward up to $\alpha =0$ if one could obtain better estimates on higher derivatives of the solution.
For the rescaled Whitham equation \eqref{Whitresc} we show that its wave breaking time has the order \(\mathcal{O}\big(\epsilon^{-1}[-\inf_{\mathbb{R}}\phi^\prime(x)]^{-1}\big)\) which confirms that the long-time existence in \cite{KLPS} is optimal.
\section{The main results}
To present our main results, we will use the
best constants from Gagliardo-Nirenberg interpolation, Sobolev embedding and Morrey embedding inequalities
\begin{equation*}
\begin{aligned}
&C_{\mathrm{GN}}:=\inf_{f\neq 0} \frac{\|\partial_x^2u\|_{L^\infty(\mathbb{R})}}{\|\partial_xu\|_{L^\infty(\mathbb{R})}^{\frac{1}{3}}\|\partial_x^3u\|
_{L^2(\mathbb{R})}^{\frac{2}{3}}},\\
&C_{\mathrm{Sob}}:=\inf_{f\neq 0} \frac{\|f\|_{L^\infty(\mathbb{R})}}{\|f\|_{H^1(\mathbb{R})}},\quad
C_{\mathrm{Mor}}:=\inf_{f\neq 0}\frac{|f|_{C^{0,\frac{1}{2}}(\mathbb{R})}}{\|f_x\|_{L^2(\mathbb{R})}},
\end{aligned}
\end{equation*}
where the semi norm \(|\cdot|_{C^{0,\gamma}(\mathbb{R})}\) is defined by
\begin{equation*}
\begin{aligned}
|f|_{C^{0,\gamma}(\mathbb{R})}=\colon\sup_{x,y\in\mathbb{R},\ x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\gamma}.
\end{aligned}
\end{equation*}
We say that the solution of \eqref{eq:main-1} (\eqref{eq:BH} or \eqref{Whit} or \eqref{Whitresc}) exhibits wave breaking
if there exists some \(T>0\) such that
\begin{equation*}
\begin{aligned}
|u(x,t)|<\infty,\quad x\in\mathbb{R},\ t\in[0,T),
\end{aligned}
\end{equation*}
but
\begin{equation*}
\begin{aligned}
\inf_{\mathbb{R}}\partial_xu(x,t)\longrightarrow -\infty,\quad \text{as}\ t\longrightarrow T-.
\end{aligned}
\end{equation*}
Our first result can be stated precisely as follows:
\begin{theorem}[Burgers-Hilbert equation]\label{th:BH} Let \(\delta\in(0,1-\frac{\sqrt{3}}{2}]\). Assume that \(\phi\in H^2(\mathbb{R})\) satisfies
\begin{align}
&\delta^2(\inf_{\mathbb{R}}\phi^\prime(x))^2> C_{\mathrm{Sob}}\|\phi\|_{H^2}+4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2},\label{t1c1}\\
&-(1-\delta)^2\inf_{\mathbb{R}}\phi^\prime(x)> 6\bigg(\frac{\|\phi\|_{L^2}}{C_0}\bigg)+ 24C_{\mathrm{Mor}}\bigg(\frac{\|\phi^{\prime}\|_{L^2}}{C_0}\bigg),\label{t1c2}\\
&-(1-\delta)^3\inf_{\mathbb{R}}\phi^\prime(x)> 8\bigg(\frac{\|\phi^{\prime}\|_{L^2}}{C_1}\bigg)+ 128C_{\mathrm{Mor}}\bigg(\frac{\|\phi^{\prime\prime}\|_{L^2}}{C_1}\bigg),\label{t1c3}
\end{align}
where the constant \(C_0\) and \(C_1\) satisfy
\begin{equation}\label{t1c4}
\begin{aligned}
\|\phi\|_{L^\infty}\leq\frac{C_0}{2},\quad
\|\phi^\prime\|_{L^\infty}\leq\frac{C_1}{2}.
\end{aligned}
\end{equation}
Then the solution to the Cauchy problem \eqref{eq:BH} with the initial data \(u(0,x)=\phi(x)\) exhibits wave breaking at some time \(T>0\). Moreover
\begin{equation}\label{wb}
\begin{aligned}
-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{1+\delta}<T<-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{(1-\delta)^2}.
\end{aligned}
\end{equation}
\end{theorem}
In order to deal with the Whitham equation we first collect the following property of \(K(x)\) \cite{Hur}:
\begin{lemma}\label{le:Hur} There exist constants \(L_0, L_\infty>0\) such that
\begin{equation*}
\begin{aligned}
K(x)\leq \frac{L_0}{\sqrt{|x|}}\quad \mathrm{and}\ |K^\prime(x)|\leq \frac{L_0}{\sqrt{|x|^3}},\quad \mathrm{for}\ 0<|x|\leq 1,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\int_1^\infty |K^\prime(x)|\,\mathrm{d} x\leq L_\infty.
\end{aligned}
\end{equation*}
\end{lemma}
Our result on the Whitham equation is as follows:
\begin{theorem}[Whitham equation]\label{th:W} Let \(\delta\in(0,1-\frac{2\sqrt{2}}{3}]\). Assume that \(\phi\in H^3(\mathbb{R})\) satisfies
\begin{align}
&\delta^2(\inf_{\mathbb{R}}\phi^\prime(x))^2> 4L_0C_{\mathrm{Sob}}\|\phi\|_{H^3}+2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}},\label{t2c1}\\
&-(1-\delta)^2\inf_{\mathbb{R}}\phi^\prime(x)> 8(3L_0+L_\infty)+16L_0\bigg(\frac{C_1}{C_0}\bigg),\label{t2c2}\\
&-(1-\delta)^3\inf_{\mathbb{R}}\phi^\prime(x)> 4(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}\bigg(\frac{\|\phi^{\prime\prime\prime}\|_{L^2}}{C_1}\bigg)^{\frac{2}{3}},\label{t2c3}
\end{align}
where the constant \(C_0\) and \(C_1\) satisfy
\begin{equation}\label{t2c4}
\begin{aligned}
\|\phi\|_{L^\infty}\leq \frac{C_0}{2},\quad
\|\phi^\prime\|_{L^\infty}\leq\frac{C_1}{2}.
\end{aligned}
\end{equation}
Then the solution of the Cauchy problem \eqref{Whit} with initial data \(u(0,x)=\phi(x)\) exhibits wave breaking at some time \(T>0\). Moreover
\begin{equation*}
\begin{aligned}
-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{1+\delta}<T<-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{(1-\delta)^2}.
\end{aligned}
\end{equation*}
\end{theorem}
\begin{theorem}[fKdV equation: \(\alpha\in(-1,-\frac{2}{5})\)]\label{th:fKdV} Let \(\delta>0\) be sufficiently small and \(\alpha\in\big(-1,\frac{5(1-\delta)^2-7}{7-2(1-\delta)^2}\big)\). Assume that \(\phi\in H^3(\mathbb{R})\) satisfies
\begin{align}
&\delta^2(\inf_{\mathbb{R}}\phi^\prime(x))^2> C_{\mathrm{Sob}}\|\phi\|_{H^2}+\frac{4C_1}{1+\alpha}+\frac{18C_{\mathrm{GN}}}{-\alpha}\bigg(C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\bigg),\label{t3c1}\\
&-(1-\delta)^2\inf_{\mathbb{R}}\phi^\prime(x)> \frac{8}{-\alpha(1+\alpha)}+\frac{2}{\alpha^2 }\bigg(\frac{C_1}{C_0}\bigg),\label{t3c2}\\
&-(1-\delta)^3\inf_{\mathbb{R}}\phi^\prime(x)> \frac{8}{1+\alpha}+\frac{36C_{\mathrm{GN}}}{-\alpha}\bigg(\frac{\|\phi^{\prime\prime\prime}\|_{L^2}}{C_1}\bigg)^{\frac{2}{3}},\label{t3c3}
\end{align}
where the constant \(C_0\) and \(C_1\) satisfy
\begin{equation}\label{t3c4}
\begin{aligned}
\|\phi\|_{L^\infty}\leq \frac{C_0}{2},\quad
\|\phi^\prime\|_{L^\infty}\leq \frac{C_1}{2}.
\end{aligned}
\end{equation}
Then the solution of the Cauchy problem \eqref{eq:main-1} with the initial data \(u(0,x)=\phi(x)\) exhibits wave breaking at some time \(T>0\). Moreover
\begin{equation*}
\begin{aligned}
-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{1+\delta}<T<-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{(1-\delta)^2}.
\end{aligned}
\end{equation*}
\end{theorem}
\begin{remark}
\rm{It is easy to see that there exists some \(\phi\in H^2(\mathbb{R})\) satisfying \eqref{t1c1}-\eqref{t1c3} and \eqref{t1c4} in Theorem \ref{th:BH}. Indeed, given any \(\phi_0\in H^2(\mathbb{R})\) with \(\inf_{\mathbb{R}}\phi_0^\prime(x)<0\), let \(\phi=\lambda\phi_0\)
with \(\lambda>0\) and \(C_0=2\lambda\|\phi_0\|_{L^\infty},C_1=2\lambda\|\phi_0^\prime\|_{L^\infty}\) that obviously satisfy \eqref{t1c4}, then
choosing \(\lambda\) sufficiently large, one checks that \(\phi\) satisfies \eqref{t1c1}-\eqref{t1c3} by comparing the powers of $\lambda$ in both sides of each inequality. One can similarly analyze the assumptions in Theorem \ref{th:W} and \ref{th:fKdV}.}
\end{remark}
\begin{remark}
\rm{The result in Theorem \ref{th:BH} (Theorem \ref{th:fKdV}) does not contradict the existence of smooth solutions of the Burgers-Hilbert equation (fKdV equation) with initial data of size $\mathcal{O}(\epsilon)$ on the enhanced time scale $\mathcal{O}(1/\epsilon^2)$ which has been established in \cite{HI, HITW} (\cite{EW}).}
\end{remark}
We finally give a very simple blowup result on the Burgers-Hilbert equation which reads as:
\begin{theorem}\label{th:main-2} Assume that \(\phi\in H^2(\mathbb{R})\) satisfies
\begin{align}\label{b1}
F(0)=\colon-\int_0^\infty \big(\phi(x)-\phi(0)\big)\exp(-x)\, \mathrm{d} x\geq
4\|\phi\|_{L^2}^{\frac{1}{2}}.
\end{align}
Then, the lifespan \(T^*\) of the solution \(u\in C\big([0,T^*);H^2(\mathbb{R})\big)\) to the Cauchy problem \eqref{eq:BH} with the initial data \(u(0,x)=\phi(x)\) is bounded above by
\begin{align}\label{b2}
T^*\leq\frac{4}{F(0)}=\colon T^{**},
\end{align}
and
\begin{align}\label{b2.5}
\lim_{t\rightarrow T^{**}-}\|u_x\|_{L^\infty}=\infty.
\end{align}
\end{theorem}
\begin{remark}
\rm{One can relax the assumption \(\phi\in H^2(\mathbb{R})\) in Theorem \ref{th:main-2} as \(\phi\in L^2(\mathbb{R})\cap C^{0,1}(\mathbb{R})\) if there exists a solution in \(u\in L^2(\mathbb{R})\cap C\big([0,T^*);C^{0,1}(\mathbb{R})\big)\).}
\end{remark}
\begin{remark}\rm{To prove Theorem \ref{th:main-2}, we use a functional (see \eqref{half line}) with a smooth, fast decay weight defined on {\it half line} inspired by \cite{LW} which uses a similar functional to study the blowup of Euler and Euler-Poisson equations. The choice of this functional makes our proof very simple.
One may refer to \cite{CCG,Hur1} for the use of functionals with singular weights on the whole line to prove blowup of dispersive equations.}
\end{remark}
\section{Proof of Theorem \ref{th:BH}}\label{sec3}
\begin{proof}[Proof of Theorem \ref{th:BH}]
It is standard that the Cauchy problem of \eqref{eq:BH} with \(u(0,x)=\phi(x)\) is well-posed in the class \(C\big([0,T):H^2(\mathbb{R})\big)\) for some \(T>0\). We assume that \(T\) is the maximal time of existence hereafter.
We define the particle path
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d} }{\mathrm{d} t}X(t,x)=u(X(t,x),t),\quad X(0,x)=x.
\end{aligned}
\end{equation*}
Since \(u(x,t)\in C\big([0,T):H^2(\mathbb{R})\big)\), the ODE theory shows that \(X(\cdot;x)\) exists throughout the interval \(t\in[0,T)\) for all \(x\in\mathbb{R}\).
We denote
\begin{equation*}
\begin{aligned}
v_0(t,x)=u(X(t,x),t),\quad v_1(t,x)=\partial_xu(X(t,x),t),
\end{aligned}
\end{equation*}
and
\begin{equation}\label{1}
\begin{aligned}
m(t)=\inf_{x\in\mathbb{R}}v_1(t;x)=\inf_{x\in\mathbb{R}}\partial_xu(x,t)=:m(0)q^{-1}(t).
\end{aligned}
\end{equation}
It is easy to see that
\begin{align}
&m(t)<0,\quad t\in [0,T),\label{2}\\
&q(0)=1,\quad
q(t)>0,\quad t\in [0,T).\label{3}
\end{align}
It follows from \eqref{eq:BH} that
\begin{align}
&\frac{\mathrm{d} v_0}{\mathrm{d} t}+K_0(t,x)=0,\label{4}\\
&\frac{\mathrm{d} v_1}{\mathrm{d} t}+v_1^2+K_1(t,x)=0,\label{5}
\end{align}
where
\begin{equation}\label{BH-kernal}
\begin{aligned}
&K_0(t,x)=\mathcal{H}u(X(t,x),t)
=\int_\mathbb{R} \frac{\mathrm{sgn}(y)}{|y|}u(X(t,x)-y,t)\,\mathrm{d} y,\\
&K_1(t,x)=\mathcal{H}\partial_xu(X(t,x),t)
=\int_\mathbb{R} \frac{\mathrm{sgn}(y)}{|y|}\partial_xu(X(t,x)-y,t)\,\mathrm{d} y.
\end{aligned}
\end{equation}
The main ingredient in proving Theorem \ref{th:BH} is to show that
\begin{equation}\label{8}
\begin{aligned}
|K_1(t,x)|<\delta^2 m^2(t),\quad \mathrm{for\ all}\ t\in[0,T)\ \mathrm{and} \ x\in\mathbb{R}.
\end{aligned}
\end{equation}
Once \eqref{8} is shown, we may easily finish the proof. Indeed,
for \(t \in [0, T)\), and any \(x\in \Sigma_{\delta}(t)=\Sigma_{\delta,1}(t)=\{x\in\mathbb{R}:v_1(t,x)\leq (1-\delta)m(t)\}\), we deduce by applying Lemma \ref{le:a1} from Appendix with $t_1=0, t_2=t$ that
\begin{equation*}
\begin{aligned}
m(0)\leq v_1(0,x)\leq (1-\delta)m(0),
\end{aligned}
\end{equation*}
and then combining this with \eqref{a3} and \eqref{a6.5} one sees that
\begin{equation*}
\begin{aligned}
r(t,x)\leq m(0)(v_1^{-1}(0,x)+(1-\delta)t)\leq \frac{1}{1-\delta}+m(0)(1-\delta)t.
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
r(t,x)\geq m(0)(v_1^{-1}(0,x)+(1+\delta)t)\geq (1-\delta)+m(0)(1-\delta^2)t.
\end{aligned}
\end{equation*}
These two inequalities together with \eqref{a4} give
\begin{equation*}
\begin{aligned}
(1-\delta)+m(0)(1-\delta^2)t\leq q(t)\leq \frac{1}{1-\delta}+m(0)(1-\delta)t,
\end{aligned}
\end{equation*}
that is
\begin{equation*}
\begin{aligned}
(1-\delta)+ \inf_{x\in\mathbb{R}}\phi^\prime(x)(1-\delta^2)t\leq q(t)\leq \frac{1}{1-\delta}+\inf_{x\in\mathbb{R}}\phi^\prime(x)(1-\delta)t.
\end{aligned}
\end{equation*}
Hence \eqref{wb} follows.
In the rest of this section, we turn to prove \eqref{8}.
First observe that \eqref{8} holds at \(t=0\):
\begin{equation*}
\begin{aligned}
|K_1(0,x)|=|\mathcal{H}\phi^\prime(x)|\leq C_{\mathrm{Sob}}\|\phi\|_{H^2}<\delta^2 m^2(0),\quad x\in\mathbb{R},
\end{aligned}
\end{equation*}
where we have used the Sobolev embedding and the assumption \eqref{t1c1}.
We now prove \eqref{8} by contradiction for \(t\neq 0\). Suppose that \(|K_1(T_1,x_0)|=\delta^2 m^2(T_1)\) for some \(T_1\in(0,T)\) and some \(x_0\in\mathbb{R}\).
By continuity, without loss of generality, we may assume that
\begin{equation}\label{9}
\begin{aligned}
|K_1(t,x)|\leq \delta^2 m^2(t),\quad \text{for\ all}\ t\in[0,T_1]\ \mathrm{and} \ x\in\mathbb{R}.
\end{aligned}
\end{equation}
We claim that
\begin{align}\label{10}
\|v_0(t)\|_{L^\infty}=\|u(t)\|_{L^\infty}<C_0,\quad \text{for\ all}\ t\in [0,T_1],
\end{align}
and
\begin{align}\label{11}
\|v_1(t)\|_{L^\infty}=\|\partial_xu(t)\|_{L^\infty}<C_1q^{-1}(t),\quad \text{for\ all}\ t\in [0,T_1],
\end{align}
where \(C_0,C_1\) satisfy \eqref{t1c4}. First observe by \eqref{t1c4} and \eqref{3} that
\begin{equation*}
\begin{aligned}
\|v_0(0)\|_{L^\infty}=\|\phi\|_{L^\infty}<C_0,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|v_1(0)\|_{L^\infty}=\|\phi^\prime\|_{L^\infty}<C_1q^{-1}(0).
\end{aligned}
\end{equation*}
We will use a contradiction argument to show \eqref{10} and \eqref{11}.
Suppose that \eqref{10} and \eqref{11} hold for all \(t\in [0, T_2)\), but fails for either \eqref{10} or \eqref{11} at \(t = T_2\) for some \(T_2\in (0, T_1]\). Hence, by continuity, it holds
\begin{align}\label{12}
\|v_0(t)\|_{L^\infty}=\|u(t)\|_{L^\infty}<C_0,\quad \text{for\ all}\ t\in [0,T_2],
\end{align}
and
\begin{align}\label{13}
\|v_1(t)\|_{L^\infty}=\|\partial_xu(t)\|_{L^\infty}<C_1q^{-1}(t),\quad \text{for\ all}\ t\in [0,T_2].
\end{align}
To bound \(K_0(t;x)\),
we split, for \(\eta\in(0,1]\), the integral into two parts as follows:
\begin{equation*}
\begin{aligned}
K_0(t,x)=\underbrace{\int_{|y|<\eta}\frac{\mathrm{sgn}(y)}{|y|}u(X(t,x)-y,t)\,\mathrm{d} y}_{I_1}
+\underbrace{\int_{|y|\geq\eta}\frac{\mathrm{sgn}(y)}{|y|}u(X(t,x)-y,t)\,\mathrm{d} y}_{I_2}.
\end{aligned}
\end{equation*}
The term \(I_2\) can be easily estimated as:
\begin{equation}\label{14}
\begin{aligned}
|I_2|
\leq \bigg(\int_{|y|\geq\eta} \frac{1}{|y|^2}\,\mathrm{d} y\bigg)^{1/2}\|u\|_{L^2}\leq 2\eta^{-\frac{1}{2}}\|\phi\|_{L^2},
\end{aligned}
\end{equation}
due to the conservation of \(\|u\|_{L^2}\).
For the term \(I_1\), we estimate
\begin{equation}\label{15}
\begin{aligned}
|I_1|
&=\bigg|\int_{|y|<\eta}\frac{\mathrm{sgn}(y)}{|y|}[u(X(t,x)-y,t)-u(X(t,x),t)]\,\mathrm{d} y\bigg|\\
&\leq |u|_{C^{0,\frac{1}{2}}(\mathbb{R})} \int_{|y|<\eta}\frac{1}{|y|}|y|^{\frac{1}{2}}\,\mathrm{d} y
\leq 4C_{\mathrm{Mor}}\eta^{\frac{1}{2}}\|\partial_xu\|_{L^2(\mathbb{R})},
\end{aligned}
\end{equation}
where we have used Morrey's inequality
\begin{equation*}
\begin{aligned}
|u|_{C^{0,\frac{1}{2}}(\mathbb{R})}\leq
C_{\mathrm{Mor}}\|\partial_xu\|_{L^2(\mathbb{R})}.
\end{aligned}
\end{equation*}
It remains to control \(\|\partial_xu\|_{L^2}\).
Taking the first derivative \(\partial_x\) on \eqref{eq:BH} with respect to \(x\), multiplying it by \(\partial_xu\) and integrating it on \(x\) over \(\mathbb{R}\), one finally gets
\begin{equation*}
\begin{aligned}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int_\mathbb{R}(\partial_xu)^2\,\mathrm{d} x
&=-\int_\mathbb{R}\partial_xu\mathcal{H}\partial_xu\,\mathrm{d} x-
\int_\mathbb{R} [(\partial_xu)^3+u\partial_x^2u\partial_xu]\,\mathrm{d} x\\
&=-\frac{1}{2}\int_\mathbb{R} (\partial_xu)^3\,\mathrm{d} x,
\end{aligned}
\end{equation*}
where on the right hand side we have used the fact that the first term vanishes due to the anti-symmetry of \(\mathcal{H}\) and integration by parts in the second term.
Hence, one obtains
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d} t}\int_\mathbb{R}(\partial_xu)^2\,\mathrm{d} x
&=-\int_\mathbb{R}\partial_xu(\partial_xu)^2\,\mathrm{d} x
\leq - m(t)\int_\mathbb{R}(\partial_xu)^2\,\mathrm{d} x\\
&=- m(0)q^{-1}(t)\int_\mathbb{R}(\partial_xu)^2\,\mathrm{d} x,
\end{aligned}
\end{equation*}
which combines with \eqref{a8} implies for all \(t\in [0, T_2 ]\) that
\begin{equation}\label{16}
\begin{aligned}
\|\partial_xu(t)\|_{L^2}
&\leq \|\phi^{\prime}\|_{L^2}(1-\delta)^{-\frac{1}{2(1-\delta)^2}}
q(t)^{-\frac{1}{2(1-\delta)^2}}\\
&\leq 2\|\phi^{\prime}\|_{L^2}
q(t)^{-\frac{1}{2(1-\delta)^2}},
\end{aligned}
\end{equation}
where we have used \(\delta\in(0,1-\frac{\sqrt{3}}{2}]\).
One finally obtains from \eqref{15} and \eqref{16} that
\begin{equation}\label{17}
\begin{aligned}
|I_1|
\leq 8C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2}\eta^{\frac{1}{2}}
q(t)^{-\frac{1}{2(1-\delta)^2}}.
\end{aligned}
\end{equation}
By choosing \(\eta=q(t)^{\frac{1}{2(1-\delta)^2}}\), for all \(t\in[0,T_2]\) and \(x\in\mathbb{R}\), we see from \eqref{14} and \eqref{17} that
\begin{equation}\label{K_0}
\begin{aligned}
|K_0(t,x)|&\leq 8C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2}\eta^{\frac{1}{2}}
q(t)^{-\frac{1}{2(1-\delta)^2}}+2\eta^{-\frac{1}{2}}\|\phi\|_{L^2}\\
&\leq (2\|\phi\|_{L^2}+8C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2})q(t)^{-\frac{1}{4(1-\delta)^2}}\\
&\leq (2\|\phi\|_{L^2}+8C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2})q(t)^{-\frac{1}{3}},
\end{aligned}
\end{equation}
where we have used \(\delta\in(0,1-\frac{\sqrt{3}}{2}]\) and \eqref{a5}.
In view of \eqref{4}, \eqref{K_0} and \eqref{a7}, we may now control \(v_0(t,x)\) for all \(t\in[0,T_2]\) and for all \(x\in\mathbb{R}\) as follows:
\begin{equation}\label{18}
\begin{aligned}
&\quad|v_0(t,x)|\leq \|\phi\|_{L^\infty}+\int_0^{t}|K_0(\tau,x)|\,\mathrm{d} \tau\\
&\leq \frac{1}{2}C_0+(2\|\phi\|_{L^2}+8C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2})\int_0^{t}
q^{-\frac{1}{3}}(\tau)\,\mathrm{d} \tau\\
&\leq \frac{1}{2}C_0-(3\|\phi\|_{L^2}+12C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2})m^{-1}(0)(1-\delta)^{-\frac{4}{3}}[(1-\delta)^{-\frac{2}{3}}-q^{\frac{2}{3}}(t)]\\
&\leq \frac{1}{2}C_0-(3\|\phi\|_{L^2}+12C_{\mathrm{Mor}}\|\phi^{\prime}\|_{L^2})(1-\delta)^{-2}m^{-1}(0)\\
&< C_0,
\end{aligned}
\end{equation}
where we have used
\begin{equation*}
\begin{aligned}
\|\phi\|_{L^\infty}\leq \frac{1}{2}C_0\leq \frac{1}{2}C_0q^{-1}(t),
\end{aligned}
\end{equation*}
due to the assumption \eqref{t1c4} and \eqref{a5} in the first inequality, and \eqref{t1c2} in the last inequality.
To estimate \(K_1(t;x)\), we again, for \(\eta\in(0,1]\), split the integral into two parts as follows:
\begin{equation*}
\begin{aligned}
K_1(t,x)=\underbrace{\int_{|y|<\eta}\frac{\mathrm{sgn}(y)}{|y|}\partial_xu(X(t,x)-y,t)\,\mathrm{d} y}_{I_3}
+\underbrace{\int_{|y|\geq\eta}\frac{\mathrm{sgn}(y)}{|y|}\partial_xu(X(t,x)-y,t)\,\mathrm{d} y}_{I_4}.
\end{aligned}
\end{equation*}
With the same manipulation as in \(I_2\), one can estimate
\begin{equation}\label{19}
\begin{aligned}
|I_4|
\leq 2\eta^{-\frac{1}{2}}\|\partial_xu\|_{L^2}\leq 4\|\phi^{\prime}\|_{L^2}\eta^{-\frac{1}{2}}
q(t)^{-\frac{1}{2(1-\delta)^2}},
\end{aligned}
\end{equation}
where we have invoked \eqref{16}.
In a similar fashion to \(I_1\), we have
\begin{equation}\label{20}
\begin{aligned}
|I_3|
\leq 4C_{\mathrm{Mor}}\eta^{\frac{1}{2}}\|\partial_x^2u\|_{L^2(\mathbb{R})}.
\end{aligned}
\end{equation}
It remains to control \(\|\partial_x^2u\|_{L^2}\). Similarly to treatment of the first derivative \(\partial_xu\), we have
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d} t}\int_\mathbb{R}(\partial_x^2u)^2\,\mathrm{d} x
&=-5\int_\mathbb{R}\partial_xu(\partial_x^2u)^2\,\mathrm{d} x
\leq -5 m(t)\int_\mathbb{R}(\partial_x^2u)^2\,\mathrm{d} x\\
&=-5 m(0)q^{-1}(t)\int_\mathbb{R}(\partial_x^2u)^2\,\mathrm{d} x,
\end{aligned}
\end{equation*}
which gives for all \(t\in [0, T_2 ]\) that
\begin{equation}\label{21}
\begin{aligned}
\|\partial_x^2u(t)\|_{L^2}
&\leq \|\phi^{\prime\prime}\|_{L^2}(1-\delta)^{-\frac{5}{2(1-\delta)^2}}
q(t)^{-\frac{5}{2(1-\delta)^2}}\\
&\leq 16 \|\phi^{\prime\prime}\|_{L^2}
q(t)^{-\frac{5}{2(1-\delta)^2}},
\end{aligned}
\end{equation}
where we have used \(\delta\in(0,1-\frac{\sqrt{3}}{2}]\).
It follows from \eqref{20} and \eqref{21}
\begin{equation}\label{22}
\begin{aligned}
|I_3|
\leq 64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2}\eta^{\frac{1}{2}}
q(t)^{-\frac{5}{2(1-\delta)^2}}.
\end{aligned}
\end{equation}
In view of \eqref{19} and \eqref{22}, taking \(\eta=q(t)^{\frac{2}{(1-\delta)^2}}\), we conclude for all \(t\in [0, T_2 ]\) and \(x\in\mathbb{R}\) that
\begin{equation}\label{23}
\begin{aligned}
|K_1(t,x)|&\leq 64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2}\eta^{\frac{1}{2}}
q(t)^{-\frac{5}{2(1-\delta)^2}}+4\|\phi^{\prime}\|_{L^2}\eta^{-\frac{1}{2}}
q(t)^{-\frac{1}{2(1-\delta)^2}}\\
&\leq (4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2})q(t)^{-\frac{3}{2(1-\delta)^2}}\\
&\leq (4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2})q(t)^{-2},
\end{aligned}
\end{equation}
where we have used \eqref{a5} and
\begin{equation*}
\begin{aligned}
-\frac{3}{2(1-\delta)^2}\geq -2,
\end{aligned}
\end{equation*}
which follows from
\(\delta\in(0,1-\frac{\sqrt{3}}{2}]\). Recalling \eqref{5} that
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d} v_1}{\mathrm{d} t}=-v_1^2-K_1(t,x)\leq |K_1(t,x)|,
\end{aligned}
\end{equation*}
one uses \eqref{23} and \eqref{a7} to estimate for all \(t\in [0, T_2 ]\) and \(x\in\mathbb{R}\) that
\begin{equation}\label{24}
\begin{aligned}
&\quad v_1(t,x)
\leq\|\phi^\prime\|_{L^\infty}+(4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2})\int_0^tq^{-2}(\tau)\,\mathrm{d} \tau\\
&\leq\frac{1}{2}C_1q^{-1}(t)-(1-\delta)^{-3}m^{-1}(0)(4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2})[q^{-1}(t)-(1-\delta)]\\
&\leq\frac{1}{2}C_1q^{-1}(t)-(1-\delta)^{-3}m^{-1}(0)(4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2})q^{-1}(t)\\
&<C_1q^{-1}(t),
\end{aligned}
\end{equation}
where we have used
\begin{equation*}
\begin{aligned}
\|\phi^\prime\|_{L^\infty}\leq \frac{1}{2}C_1\leq \frac{1}{2}C_1q^{-1}(t),
\end{aligned}
\end{equation*}
due to the assumption \eqref{t1c4} and \eqref{a5} in the first inequality, and the assumption \eqref{t1c3} in the last inequality.
On the other hand, \eqref{t1c4} and \eqref{1} imply that
\begin{equation}\label{25}
\begin{aligned}
v_1(t,x)\geq m(t)=m(0)q^{-1}(t)
\geq-\frac{1}{2}C_1q^{-1}(t),
\end{aligned}
\end{equation}
for all \(t\in [0, T_2 ]\) and \(x\in\mathbb{R}\).
A contradiction to \eqref{12}-\eqref{13} occurs following from \eqref{18}, \eqref{24} and \eqref{25}.
Now we go back to \eqref{23} and use \eqref{t1c1} to find that
\begin{equation*}
\begin{aligned}
|K_1(t,x)|\leq (4\|\phi^{\prime}\|_{L^2}+64C_{\mathrm{Mor}}\|\phi^{\prime\prime}\|_{L^2})m^{-2}(0)m^2(t)<\delta^2m^2(t),
\end{aligned}
\end{equation*}
for all \(t\in [0, T_1 ]\) and all \(x\in\mathbb{R}\). We get a contradiction to \eqref{9}! This means we have shown \eqref{8} for all \(t\in [0, T)\) and all \(x\in\mathbb{R}\).
\end{proof}
\section{Proof of Theorem \ref{th:W}}\label{sec4}
\begin{proof}[Proof of Theorem \ref{th:W}]
We first note that the Cauchy problem of \eqref{Whit} with \(u(0,x)=\phi(x)\) is well-posed in the class \(C\big([0,T):H^3(\mathbb{R})\big)\) for some \(T>0\) and we now assume that \(T\) is the maximal time of existence. Using the same notations \(X(t,x),v_0(t,x),v_1(t,x),m(t)\) and \(q(t)\) as Section \ref{sec3}, it then
follows from \eqref{Whit} that
\begin{align}
&\frac{\mathrm{d} v_0}{\mathrm{d} t}+K_0(t,x)=0,\label{29}\\
&\frac{\mathrm{d} v_1}{\mathrm{d} t}+v_1^2+K_1(t,x)=0,\label{30}
\end{align}
where
\begin{equation}\label{Whit-kernal}
\begin{aligned}
K_0(t,x)=\int_\mathbb{R} K(y)\partial_xu(X(t,x)-y,t)\,\mathrm{d} y,\\
K_1(t,x)=\int_\mathbb{R} K(y)\partial_x^2u(X(t,x)-y,t)\,\mathrm{d} y.
\end{aligned}
\end{equation}
To prove Theorem \ref{th:W}, it suffices to show that
\begin{equation}\label{31}
\begin{aligned}
|K_1(t,x)|<\delta^2 m^2(t),\quad \mathrm{for\ all}\ t\in[0,T)\ \mathrm{and} \ x\in\mathbb{R}.
\end{aligned}
\end{equation}
We first check that \eqref{31} holds at \(t=0\). To estimate \(K_1(0;x)\), we split the integral as follows:
\begin{equation*}
\begin{aligned}
&K_1(0,x)=\int_\mathbb{R} K(y)\phi^{\prime\prime}(x-y)\,\mathrm{d} y\\
&=\int_{|y|< 1} K(y)\phi^{\prime\prime}(x-y)\,\mathrm{d} y+\int_{|y|\geq 1} K(y)\phi^{\prime\prime}(x-y)\,\mathrm{d} y.
\end{aligned}
\end{equation*}
In view of Lemma \ref{le:Hur}, one has
\begin{equation}\label{32}
\begin{aligned}
\bigg|\int_{|y|< 1} K(y)\phi^{\prime\prime}(x-y)\,\mathrm{d} y\bigg|
\leq \|\phi^{\prime\prime}\|_{L^\infty}\bigg|\int_{|y|< 1} K(y)\,\mathrm{d} y\bigg|
\leq 4L_0C_{\mathrm{Sob}}\|\phi\|_{H^3}.
\end{aligned}
\end{equation}
where we have used Sobolev embedding.
We use integration by parts to get
\begin{equation}\label{33}
\begin{aligned}
&\bigg|\int_{|y|\geq 1} K(y)\phi^{\prime\prime}(x-y)\,\mathrm{d} y\bigg|\\
&\leq \big|K(1)[\phi^{\prime}(-1-y)-\phi^{\prime}(1-y)]\big|
+\bigg|\int_{|y|\geq 1} K^{\prime}(y)\phi^{\prime}(x-y)\,\mathrm{d} y\bigg|\\
&\leq 2L_0\|\phi^{\prime}\|_{L^\infty}+\|\phi^{\prime}\|_{L^\infty}\bigg|\int_{|y|\geq 1} K^{\prime}(y)\,\mathrm{d} y\bigg|\\
&\leq 2(L_0+L_\infty)\|\phi^{\prime}\|_{L^\infty}\leq C_1(L_0+L_\infty),
\end{aligned}
\end{equation}
where we have used Lemma \ref{le:Hur} in the third inequality and \eqref{t2c4} in the last inequality.
It follows from \eqref{32} and \eqref{33} that
\begin{equation*}
\begin{aligned}
|K_1(0,x)|\leq 4L_0C_{\mathrm{Sob}}\|\phi\|_{H^3}+C_1(L_0+L_\infty)<\delta^2 m^2(0),\quad x\in\mathbb{R},
\end{aligned}
\end{equation*}
where we have used \eqref{t2c1}.
We now turn to prove \eqref{31} by contradiction for \(t\neq 0\). Suppose that \(|K_1(T_1,x_0)|=\delta^2 m^2(T_1)\) for some \(T_1\in(0,T)\) and some \(x_0\in\mathbb{R}\).
By continuity, without loss of generality, we may assume that
\begin{equation}\label{34}
\begin{aligned}
|K_1(t,x)|\leq \delta^2 m^2(t),\quad \text{for\ all}\ t\in[0,T_1]\ \mathrm{and} \ x\in\mathbb{R}.
\end{aligned}
\end{equation}
We claim that
\begin{align}\label{35}
\|v_0(t)\|_{L^\infty}=\|u(t)\|_{L^\infty}<C_0,\quad \text{for\ all}\ t\in [0,T_1],
\end{align}
and
\begin{align}\label{36}
\|v_1(t)\|_{L^\infty}=\|\partial_xu(t)\|_{L^\infty}<C_1q^{-1}(t),\quad \text{for\ all}\ t\in [0,T_1],
\end{align}
where \(C_0,C_1\) satisfy \eqref{t2c4}. First observe that
\begin{equation*}
\begin{aligned}
\|v_0(0)\|_{L^\infty}=\|\phi\|_{L^\infty}<C_0,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|v_1(0)\|_{L^\infty}=\|\phi^\prime\|_{L^\infty}<C_1q^{-1}(0).
\end{aligned}
\end{equation*}
A contradiction argument will be used to show \eqref{35} and \eqref{36}.
Suppose that \eqref{35} and \eqref{36} hold for all \(t\in [0, T_2)\), but fails for either \eqref{35} or \eqref{36} at \(t = T_2\) for some \(T_2\in (0, T_1]\). Hence, by continuity, it holds
\begin{align}\label{37}
\|v_0(t)\|_{L^\infty}=\|u(t)\|_{L^\infty}<C_0,\quad \text{for\ all}\ t\in [0,T_2],
\end{align}
and
\begin{align}\label{38}
\|v_1(t)\|_{L^\infty}=\|\partial_xu(t)\|_{L^\infty}<C_1q^{-1}(t),\quad \text{for\ all}\ t\in [0,T_2].
\end{align}
To control \(K_0(t,x)\), we split the integral with \(\eta\in(0,1]\) as follows:
\begin{equation*}
\begin{aligned}
K_0(t,x)=\underbrace{\int_{|y|\leq\eta}K(y)\partial_xu(X(t,x)-y,t)\,\mathrm{d} y}_{I_1}
+\underbrace{\int_{|y|>\eta}K(y)\partial_xu(X(t,x)-y,t)\,\mathrm{d} y}_{I_2}.
\end{aligned}
\end{equation*}
For the term \(I_1\), using Lemma \ref{le:Hur} and \eqref{38}, we have
\begin{equation}\label{39}
\begin{aligned}
|I_1|
\leq 2\int_{|y|\leq\eta} \frac{L_0}{\sqrt{|y|}}\,\mathrm{d} y\cdot\|v_1\|_{L^\infty}\leq 4L_0\eta^{\frac{1}{2}}\|v_1\|_{L^\infty}\leq 4L_0C_1\eta^{\frac{1}{2}}q^{-1}(t).
\end{aligned}
\end{equation}
Considering the term \(I_2\), we use integration by parts to bound it as follows:
\begin{equation}\label{40}
\begin{aligned}
|I_2|
&\leq\big|K(\eta)[u(X(t,x)-\eta,t)-u(X(t,x)+\eta,t)]\big|\\
&\quad+\bigg|\int_{\eta<|y|\leq 1} K^\prime(y)u(X(t,x)-y,t)\,\mathrm{d} y\bigg|\\
&\quad+\bigg|\int_{|y|>1} K^\prime(y)u(X(t,x)-y,t)\,\mathrm{d} y\bigg|\\
&\leq 2L_0\eta^{-\frac{1}{2}}\|v_0\|_{L^\infty}+4L_0(\eta^{-\frac{1}{2}}-1)\|v_0\|_{L^\infty}
+2L_\infty\|v_0\|_{L^\infty}\\
&\leq 2(3L_0\eta^{-\frac{1}{2}}+L_\infty)\|v_0\|_{L^\infty}
\leq 2C_0(3L_0+L_\infty)\eta^{-\frac{1}{2}},
\end{aligned}
\end{equation}
where we have used Lemma \ref{le:Hur} in the third inequality and \eqref{37} in the last inequality.
In view of \eqref{39} and \eqref{40}, one chooses \(\eta=q(t)\) to get
\begin{equation}\label{41}
\begin{aligned}
|K_0(t,x)|\leq 2\big[C_0(3L_0+L_\infty)+2L_0C_1\big]q^{-\frac{1}{2}}(t),
\end{aligned}
\end{equation}
for all \(t\in[0,T_2]\) and for all \(x\in\mathbb{R}\).
By \eqref{29}, \eqref{41} and \eqref{a7}, we may now control \(v_0(t;x)\) for all \(t\in[0,T_2]\) and for all \(x\in\mathbb{R}\) as follows:
\begin{equation}\label{42}
\begin{aligned}
&\quad|v_0(t,x)|\leq \|\phi\|_{L^\infty}+\int_0^{t}|K_0(\tau,x)|\,\mathrm{d} \tau\\
&\leq \frac{1}{2}C_0+2\big[C_0(3L_0+L_\infty)+2L_0C_1\big]\int_0^{t}
q^{-\frac{1}{2}}(\tau)\,\mathrm{d} \tau\\
&\leq \frac{1}{2}C_0-4\big[C_0(3L_0+L_\infty)+2L_0C_1\big]m^{-1}(0)(1-\delta)^{-\frac{3}{2}}[(1-\delta)^{-\frac{1}{2}}-q^{\frac{1}{2}}(t)]\\
&\leq \frac{1}{2}C_0-4\big[C_0(3L_0+L_\infty)+2L_0C_1\big](1-\delta)^{-2}m^{-1}(0)\\
&< C_0.
\end{aligned}
\end{equation}
where we have used \eqref{t2c2} in the last inequality.
To bound \(K_1(t,x)\), we proceed as:
\begin{equation*}
\begin{aligned}
K_1(t,x)=\underbrace{\int_{|y|\leq\eta}K(y)\partial_x^2u(X(t,x)-y,t)\,\mathrm{d} y}_{I_3}
+\underbrace{\int_{|y|>\eta}K(y)\partial_x^2u(X(t,x)-y,t)\,\mathrm{d} y}_{I_4}.
\end{aligned}
\end{equation*}
Similar to \(I_2\), by integration by parts, one has
\begin{equation}\label{43}
\begin{aligned}
|I_4|
&\leq 2L_0\eta^{-\frac{1}{2}}\|v_1\|_{L^\infty}+4L_0(\eta^{-\frac{1}{2}}-1)\|v_1\|_{L^\infty}+2L_\infty\|v_1\|_{L^\infty}\\
&\leq 2C_1(3L_0+L_\infty)\eta^{-\frac{1}{2}}q^{-1}(t),
\end{aligned}
\end{equation}
where we have used Lemma \ref{le:Hur} in the second inequality and \eqref{38} in the last inequality.
For the term \(I_3\), one estimates
\begin{equation}\label{44}
\begin{aligned}
|I_3|
\leq 2\int_{|y|\leq\eta} \frac{L_0}{\sqrt{|y|}}\,\mathrm{d} y \cdot\|\partial_x^2u\|_{L^\infty}\leq 4L_0\eta^{\frac{1}{2}}\|\partial_x^2u\|_{L^\infty},
\end{aligned}
\end{equation}
where we have used Lemma \ref{le:Hur} again.
In order to control \(\|\partial_x^2u\|_{L^\infty}\),
we need to estimate \(\|\partial_x^3u\|_{L^2}\).
We differentiate \eqref{Whit} three times with respect to \(x\), multiply by \(\partial_x^3u\) and integrate on \(x\) over \(\mathbb{R}\) to get
\begin{equation}\label{45}
\begin{aligned}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int_\mathbb{R}(\partial_x^3u)^2\,\mathrm{d} x
&=-\int_\mathbb{R}\partial_x^3u\int_\mathbb{R} K(x-y)\partial_y^4u(y)\,\mathrm{d} y\,\mathrm{d} x\\
&\quad-
\int_\mathbb{R} [4\partial_xu(\partial_x^3u)^2+3(\partial_x^2u)^2\partial_x^3u+u\partial_x^4u\partial_x^3u]\,\mathrm{d} x.
\end{aligned}
\end{equation}
Obviously, the first term on the right hand side of \eqref{45} vanishes since \(K(\cdot)\) is even. On the other hand, one uses integration by parts to see that
\begin{equation}\label{46}
\begin{aligned}
&\int_\mathbb{R}(\partial_x^2u)^2\partial_x^3u\,\mathrm{d} x=0,\\
&\int_\mathbb{R} u\partial_x^4u\partial_x^3u\,\mathrm{d} x=-\frac{1}{2}\int_\mathbb{R}\partial_xu(\partial_x^3u)^2\,\mathrm{d} x.
\end{aligned}
\end{equation}
We substitute \eqref{46} into \eqref{45} to deduce
\begin{equation}\label{46.5}
\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d} t}\int_\mathbb{R}(\partial_x^3u)^2\,\mathrm{d} x
=-7\int_\mathbb{R}\partial_xu(\partial_x^3u)^2\,\mathrm{d} x\leq -7 m(0)q^{-1}(t)\|\partial_x^3u\|_{L^2}^2.
\end{aligned}
\end{equation}
Solving \eqref{46.5} by using \eqref{a8} gives
\begin{equation}\label{47}
\begin{aligned}
\|\partial_x^3u\|_{L^2}
&\leq \|\phi^{\prime\prime\prime}\|_{L^2}(1-\delta)^{-\frac{7}{2(1-\delta)^2}}
q(t)^{-\frac{7}{2(1-\delta)^2}}\\
&\leq 16\|\phi^{\prime\prime\prime}\|_{L^2}
q(t)^{-\frac{7}{2(1-\delta)^2}}
\end{aligned}
\end{equation}
for all \(t\in [0, T_2 ]\), where we have used \(\delta\in(0,1-\frac{2\sqrt{2}}{3}]\). However the bound \eqref{47} for \(\|\partial_x^3u\|_{L^2}\) is too bad to control \(\|\partial_x^2u\|_{L^\infty}\) by Sobolev embedding directly. To get a better bound,
we use Gagliardo-Nirenberg interpolation to deduce
\begin{equation}\label{47.5}
\begin{aligned}
\|\partial_x^2u\|_{L^\infty}\leq C_{\mathrm{GN}} \|\partial_xu\|_{L^\infty}^{\frac{1}{3}}\|\partial_x^3u\|_{L^2}^{\frac{2}{3}}\leq 9
C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}q(t)^{-\frac{1}{3}-\frac{7}{3(1-\delta)^2}},
\end{aligned}
\end{equation}
where we have used \eqref{38} and \eqref{47}.
We then may estimate in view of \eqref{44} and \eqref{47.5} that
\begin{equation}\label{48}
\begin{aligned}
|I_3|
\leq 36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\eta^{\frac{1}{2}}
q(t)^{-\frac{1}{3}-\frac{7}{3(1-\delta)^2}}.
\end{aligned}
\end{equation}
In light of \eqref{43} and \eqref{48}, we take \(\eta=q(t)^{-\frac{2}{3}+\frac{7}{3(1-\delta)^2}}\) to obtain
\begin{equation}\label{49}
\begin{aligned}
|K_1(t,x)|
&\leq \big[2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\big]q(t)^{-\frac{2}{3}-\frac{7}{6(1-\delta)^2}}\\
&\leq \big[2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\big]q^{-2}(t),
\end{aligned}
\end{equation}
for all \(t\in[0,T_2]\) and for all \(x\in\mathbb{R}\), where we have used
\begin{equation*}
\begin{aligned}
-\frac{2}{3}-\frac{7}{6(1-\delta)^2}\geq -2,
\end{aligned}
\end{equation*}
which follows from
\(\delta\in(0,1-\frac{2\sqrt{2}}{3}]\) and \eqref{a5}.
Recalling \eqref{30} that
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d} v_1}{\mathrm{d} t}=-v_1^2-K_1(t,x)\leq |K_1(t,x)|,
\end{aligned}
\end{equation*}
one uses \eqref{49} and \eqref{a7} to estimate
\begin{equation}\label{50}
\begin{aligned}
&v_1(t,x)\\
&\leq\|\phi^\prime\|_{L^\infty}+\big[2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\big]\int_0^tq^{-2}(\tau)\,\mathrm{d} \tau\\
&\leq\frac{1}{2}C_1q^{-1}(t)-(1-\delta)^{-3}m^{-1}(0)\big[2C_1(3L_0+L_\infty)
+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\big][q^{-1}(t)-(1-\delta)]\\
&\leq\frac{1}{2}C_1q^{-1}(t)-(1-\delta)^{-3}m^{-1}(0)\big[2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\big]q^{-1}(t)\\
&<C_1q^{-1}(t),
\end{aligned}
\end{equation}
where we have used \eqref{t2c3} in the last inequality.
On the other hand, one has
\begin{equation}\label{51}
\begin{aligned}
v_1(t,x)\geq m(t)=m(0)q^{-1}(t)
\geq-\frac{1}{2}C_1q^{-1}(t).
\end{aligned}
\end{equation}
for all \(t\in [0, T_2 ]\) and \(x\in\mathbb{R}\).
A contradiction to \eqref{37}-\eqref{38} occurs following from \eqref{42}, \eqref{50} and \eqref{51}.
Now we go back to \eqref{49} and use \eqref{t2c1} to find that
\begin{equation*}
\begin{aligned}
|K_1(t,x)|\leq [2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}\big]m^{-2}(0)m^2(t)
<\delta^2m^2(t),
\end{aligned}
\end{equation*}
for all \(t\in [0, T_1 ]\) and all \(x\in\mathbb{R}\). We get a contradiction to \eqref{34}! This means that we have shown \eqref{31} for all \(t\in [0, T)\) and all \(x\in\mathbb{R}\).
\end{proof}
\section{Proof of Theorem \ref{th:fKdV}}\label{sec5}
\begin{proof}[Proof of Theorem \ref{th:fKdV}]
We first note that the Cauchy problem of \eqref{eq:main-1} with \(u(0,x)=\phi(x)\) is well-posed in the class \(C\big([0,T):H^3(\mathbb{R})\big)\) for some \(T>0\) and we now assume that \(T\) is the maximal time of existence. Using the same notations \(X(t,x),v_0(t,x),v_1(t,x),m(t)\) and \(q(t)\) as in Section \ref{sec3},
it then follows from \eqref{eq:main-1} that
\begin{align}
&\frac{\mathrm{d} v_0}{\mathrm{d} t}+K_0(t,x)=0,\label{54}\\
&\frac{\mathrm{d} v_1}{\mathrm{d} t}+v_1^2+K_1(t,x)=0,\label{55}
\end{align}
where
\begin{equation}\label{fKdV-kernal}
\begin{aligned}
&K_0(t,x)
=\int_\mathbb{R}\frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[u(X(t,x),t)-u(X(t,x)-y,t)]\,\mathrm{d} y,\\
&K_1(t,x)
=\int_\mathbb{R} \frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[\partial_xu(X(t,x),t)-\partial_xu(X(t,x)-y,t)]\,\mathrm{d} y.
\end{aligned}
\end{equation}
We are done if we show
\begin{equation}\label{56}
\begin{aligned}
|K_1(t,x)|<\delta^2 m^2(t),\quad \mathrm{for\ all}\ t\in[0,T)\ \mathrm{and} \ x\in\mathbb{R}.
\end{aligned}
\end{equation}
In view of \eqref{t3c1}, one easily checks that \eqref{56} holds at \(t=0\).
We will prove \eqref{56} by contradiction. Suppose that \(|K_1(T_1,x_0)|=\delta^2 m^2(T_1)\) for some \(T_1\in(0,T)\) and some \(x_0\in\mathbb{R}\).
By continuity, without loss of generality, we may assume that
\begin{equation*}
\begin{aligned}
|K_1(t,x)|\leq \delta^2 m^2(t),\quad \text{for\ all}\ t\in[0,T_1]\ \mathrm{and} \ x\in\mathbb{R}.
\end{aligned}
\end{equation*}
We claim that
\begin{align}\label{57}
\|v_0(t)\|_{L^\infty}=\|u(t)\|_{L^\infty}<C_0,\quad \text{for\ all}\ t\in [0,T_1],
\end{align}
and
\begin{align}\label{58}
\|v_1(t)\|_{L^\infty}=\|\partial_xu(t)\|_{L^\infty}<C_1q^{-1}(t),\quad \text{for\ all}\ t\in [0,T_1],
\end{align}
where \(C_0,C_1\) satisfy \eqref{t3c4}. First observe that
\begin{equation*}
\begin{aligned}
\|v_0(0)\|_{L^\infty}=\|\phi\|_{L^\infty}<C_0,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|v_1(0)\|_{L^\infty}=\|\phi^\prime\|_{L^\infty}<C_1q^{-1}(0).
\end{aligned}
\end{equation*}
We then proceed by contradiction in order to show \eqref{57} and \eqref{58}.
Suppose that \eqref{57} and \eqref{58} hold for all \(t\in [0, T_2)\), but fails for either \eqref{57} or \eqref{58} at \(t = T_2\) for some \(T_2\in (0, T_1]\). Hence, by continuity, it holds
\begin{align}\label{59}
\|v_0(t)\|_{L^\infty}=\|u(t)\|_{L^\infty}<C_0,\quad \text{for\ all}\ t\in [0,T_2],
\end{align}
and
\begin{align}\label{60}
\|v_1(t)\|_{L^\infty}=\|\partial_xu(t)\|_{L^\infty}<C_1q^{-1}(t),\quad \text{for\ all}\ t\in [0,T_2].
\end{align}
Let \(\eta\in(0,1]\), we split the integral into two parts:
\begin{equation*}
\begin{aligned}
K_0(t,x)&=\underbrace{\int_{|y|<\eta}\frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[u(X(t,x),t)-u(X(t,x)-y,t)]\,\mathrm{d} y}_{I_1}\\
&\quad+\underbrace{\int_{|y|\geq\eta}\frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[u(X(t,x),t)-u(X(t,x)-y,t)]\,\mathrm{d} y}_{I_2}.
\end{aligned}
\end{equation*}
We then estimate
\begin{equation}\label{61}
\begin{aligned}
|I_1|
&=\bigg|\int_{|y|<\eta}\frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[u(X(t,x)-y,t)-u(X(t,x),t)]\,\mathrm{d} y\bigg|\\
&\leq |u|_{C^{0,1}(\mathbb{R})} \int_{|y|<\eta}\frac{1}{|y|^{2+\alpha}}|y|\,\mathrm{d} y\\
&\leq \frac{2}{-\alpha}\eta^{-\alpha}\|\partial_xu\|_{L^\infty}\leq \frac{2C_1}{-\alpha}\eta^{-\alpha}q(t)^{-1},
\end{aligned}
\end{equation}
and
\begin{equation}\label{61.5}
\begin{aligned}
|I_2|
\leq \frac{4}{1+\alpha}\eta^{-(1+\alpha)}\|v_0\|_{L^\infty}\leq \frac{4C_0}{1+\alpha}\eta^{-(1+\alpha)}.
\end{aligned}
\end{equation}
Choosing \(\eta=q(t)\), for all \(t\in[0,T_2]\) and for all \(x\in\mathbb{R}\), one obtains from \eqref{61} and \eqref{61.5} that
\begin{equation*}
\begin{aligned}
|K_0(t,x)|\leq \bigg(\frac{4C_0}{1+\alpha}+\frac{2C_1}{-\alpha}\bigg)q(t)^{-(1+\alpha)}.
\end{aligned}
\end{equation*}
This together with \eqref{54} and \eqref{a7} yields
\begin{equation}\label{62}
\begin{aligned}
&|v_0(t,x)|\leq \|\phi\|_{L^\infty}+\int_0^{t}|K_0(t,x)|\,\mathrm{d} t\\
&\leq \frac{1}{2}C_0+\bigg(\frac{4C_0}{1+\alpha}+\frac{2C_1}{-\alpha}\bigg)\int_0^{t}
q^{-(1+\alpha)}(\tau)\,\mathrm{d} \tau\\
&\leq \frac{1}{2}C_0-\bigg(\frac{4C_0}{1+\alpha}+\frac{2C_1}{-\alpha}\bigg)m^{-1}(0)(1-\delta)^{-(\alpha+2)}(-\alpha)^{-1}[(1-\delta)^{\alpha}-q^{-\alpha}(t)]\\
&\leq \frac{1}{2}C_0+\frac{1}{\alpha}\bigg(\frac{4C_0}{1+\alpha}+\frac{2C_1}{-\alpha}\bigg)(1-\delta)^{-2}m^{-1}(0)\\
&< C_0,
\end{aligned}
\end{equation}
for all \(t\in[0,T_2]\) and for all \(x\in\mathbb{R}\), where we have used \eqref{t3c2}.
To estimate \(K_1(t,x)\), we also split the integral into two parts:
\begin{equation*}
\begin{aligned}
K_1(t,x)&=\underbrace{\int_{|y|<\eta}\frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[\partial_xu(X(t,x),t)-\partial_xu(X(t,x)-y,t)]\,\mathrm{d} y}_{I_3}\\
&\quad+\underbrace{\int_{|y|\geq\eta}\frac{\mathrm{sgn}(y)}{|y|^{2+\alpha}}[\partial_xu(X(t,x),t)-\partial_xu(X(t,x)-y,t)]\,\mathrm{d} y}_{I_4}.
\end{aligned}
\end{equation*}
The term \(I_4\) can be simply estimated as
\begin{equation}\label{63}
\begin{aligned}
|I_4|
\leq \frac{4C_1}{1+\alpha}\eta^{-(1+\alpha)}q^{-1}(t).
\end{aligned}
\end{equation}
For the term \(I_3\), we estimate
\begin{equation}\label{64}
\begin{aligned}
|I_3|
\leq |\partial_xu|_{C^{0,1}(\mathbb{R})} \int_{|y|<\eta}\frac{1}{|y|^{2+\alpha}}|y|\,\mathrm{d} y
\leq \frac{2}{-\alpha}\eta^{-\alpha}\|\partial_x^2u\|_{L^\infty}.
\end{aligned}
\end{equation}
Following the same line as in the proof of \eqref{45}-\eqref{47.5} and using the assumption that \(\delta>0\) is sufficient small (to make the last inequality of \eqref{47} to be true), one has
\begin{equation}\label{65}
\begin{aligned}
\|\partial_x^2u\|_{L^\infty}\leq C_{\mathrm{GN}} \|\partial_xu\|_{L^\infty}^{\frac{1}{3}}\|\partial_x^3u\|_{L^2}^{\frac{2}{3}}\leq 9
C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}q(t)^{-\frac{1}{3}-\frac{7}{3(1-\delta)^2}}.
\end{aligned}
\end{equation}
We then may estimate in view of \eqref{64} and \eqref{65} that
\begin{equation}\label{66}
\begin{aligned}
|I_3|
\leq \frac{18C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}}{-\alpha}
\eta^{-\alpha}
q(t)^{-\frac{1}{3}-\frac{7}{3(1-\delta)^2}}.
\end{aligned}
\end{equation}
By choosing \(\eta=q(t)^{-\frac{2}{3}+\frac{7}{3(1-\delta)^2}}\), we conclude from \eqref{63} and \eqref{66} that
\begin{equation}\label{67}
\begin{aligned}
|K_1(t,x)|
&\leq \bigg(\frac{4C_1}{1+\alpha}+\frac{18C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}}{-\alpha}\bigg)q(t)^{-\big[{\frac{1}{3}+\frac{7}{3(1-\delta)^2}}\big](1+\alpha)+\alpha}\\
&\leq \bigg(\frac{4C_1}{1+\alpha}+\frac{18C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}}{-\alpha}\bigg)q^{-2}(t),
\end{aligned}
\end{equation}
for all \([0, T_2 ]\) and all \(x\in\mathbb{R}\), where we have used
\begin{equation*}
\begin{aligned}
-\bigg[{\frac{1}{3}+\frac{7}{3(1-\delta)^2}}\bigg](1+\alpha)+\alpha\geq -2,
\end{aligned}
\end{equation*}
which follows from the assumptions that \(\delta>0\) is sufficient small and \(\alpha\in(-1,\frac{5(1-\delta)^2-7}{7-2(1-\delta)^2})\).
In view of \eqref{55}, \eqref{67} and \eqref{a7}, one has
\begin{equation}\label{68}
\begin{aligned}
&v_1(t,x)
\leq\|\phi^\prime\|_{L^\infty}+\bigg(\frac{4C_1}{1+\alpha}+\frac{18C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}}{-\alpha}\bigg)\int_0^tq^{-2}(\tau)\,\mathrm{d} \tau\\
&\leq \frac{1}{2}C_1q^{-1}(t)-(1-\delta)^{-3}m^{-1}(0)\bigg(\frac{4C_1}{1+\alpha}+\frac{18C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}}{-\alpha}\bigg)
[q^{-1}(t)-(1-\delta)]\\
&\leq \frac{1}{2}C_1q^{-1}(t)-(1-\delta)^{-3}m^{-1}(0)\bigg(\frac{4C_1}{1+\alpha}+\frac{18C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}}}{-\alpha}\bigg)q^{-1}(t)\\
&<C_1q^{-1}(t),
\end{aligned}
\end{equation}
where we have used \eqref{t3c3}.
On the other hand, one also has
\begin{equation}\label{69}
\begin{aligned}
v_1(t,x)\geq m(t)=m(0)q^{-1}(t)
\geq-\frac{1}{2}C_1q^{-1}(t).
\end{aligned}
\end{equation}
With \eqref{62}, \eqref{68} and \eqref{69} at hand, the same argument in Section \ref{sec3} can be used to complete the proof.
\end{proof}
\section{Proof of Theorem \ref{th:main-2}}\label{sec6}
\begin{proof}[Proof of Theorem \ref{th:main-2}] It is trivial that the Cauchy problem of \eqref{eq:BH} with \(u(0,x)=\phi(x)\) is well-posed in the class \(C\big([0,T^*):H^2(\mathbb{R})\big)\) for some \(T^*>0\). We assume that \(T^*\) is the maximal time of existence hereafter.
Let \(Y(t)\) solve the ODE
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d} t}Y(t)=u(t,Y(t)),\quad Y(0)=0.
\end{align*}
It is easy to see that \(Y(t)\) is well-defined over \([0,T^*)\). We define
\begin{align*}
v(t,x)=u(t,x+Y(t)).
\end{align*}
A small calculation shows that \(v\) satisfies the equation
\begin{align}\label{eq:BH-new}
\big(v(t,x)-v(t,0)\big)_t+\big(v(t,x)-v(t,0)\big)v_x(t,x)-\mathcal{H}\big(v(t,x)-v(t,0)\big)=0.
\end{align}
We introduce the weighted velocity with a smooth, fast decay weight on {\it half line} as follows:
\begin{align}\label{half line}
F(t)=-\int_0^\infty \big(v(t,x)-v(t,0)\big)\exp(-x)\, \mathrm{d} x.
\end{align}
Suppose \(T^*=\infty\).
It results from \eqref{eq:BH-new} that
\begin{equation}\label{b3}
\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d} t}F(t)
&=\int_0^\infty\big[\big(v(t,x)-v(t,0)\big)\partial_xv-\mathcal{H}\big(v(t,x)-v(t,0)\big)\big]\exp(-x)\,\mathrm{d} x.
\end{aligned}
\end{equation}
Since \(v(t,x)-v(t,0)\) vanishes at \(x=0\), by integrating by parts, one has
\begin{equation}\label{b4}
\begin{aligned}
&\int_0^\infty \big(v(t,x)-v(t,0)\big)\partial_xv\exp(-x)\,\mathrm{d} x\\
&=\int_0^\infty \big(v(t,x)-v(t,0)\big)\partial_x\big(v(t,x)-v(t,0)\big)\exp(-x)\,\mathrm{d} x\\
&=\frac{1}{2}\int_0^\infty \big(v(t,x)-v(t,0)\big)^2\exp(-x)\,\mathrm{d} x=\colon Q(t).
\end{aligned}
\end{equation}
By the property of Hilbert transform, one obtains
\begin{equation}\label{b5}
\begin{aligned}
&\int_0^\infty\mathcal{H}\big(v(t,x)-v(t,0)\big)\exp(-x)\,\mathrm{d} x\\
&\leq \bigg(\int_0^\infty[\mathcal{H}\big(v(t,x)-v(t,0)\big)]^2\,\mathrm{d} x\bigg)^{\frac{1}{2}}\bigg(\int_0^\infty\exp(-2x)\,\mathrm{d} x\bigg)^{\frac{1}{2}}\\
&\leq \frac{\sqrt{2}}{2}\bigg(\int_\mathbb{R}[\mathcal{H}\big(v(t,x)-v(t,0)\big)]^2\,\mathrm{d} x\bigg)^{\frac{1}{2}}\leq\frac{\sqrt{2}}{2}\|v(t,x)-v(t,0)\|_{L^2}\\
&=\frac{\sqrt{2}}{2}\|u(t,x+y(t))-u(t,x)\|_{L^2}\leq \sqrt{2}\|u_0\|_{L^2}.
\end{aligned}
\end{equation}
On the other hand, we have
\begin{equation}\label{b6}
\begin{aligned}
F^2(t)
\leq\int_0^\infty \big(v(t,x)-v(t,0)\big)^2\exp(-x)\,\mathrm{d} x\int_0^\infty\exp(-x)\,\mathrm{d} x=2Q(t).
\end{aligned}
\end{equation}
We conclude from \eqref{b3}-\eqref{b6} that
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d} t}F(t)\geq \frac{1}{2}F^2(t)-\sqrt{2}\|u_0\|_{L^2}.
\end{align*}
In view of \eqref{b1}, it follows that
\begin{align*}
F(t)\geq \frac{F(0)}{1-\frac{F(0)}{4}t},
\end{align*}
which means that \(F(t)\) will blow up no later than the time \(\frac{4}{F(0)}=\colon T^{**}\) that confirms \eqref{b2}. On the other hand
\begin{align*}
F(t)\leq \sup_{x\in\mathbb{R}}\bigg|\frac{v(t,x)-v(t,0)}{x}\bigg|\int_0^\infty x\exp(-x)\, \mathrm{d} x\leq C\|u_x\|_{L^\infty},
\end{align*}
which leads to \eqref{b2.5}.
This contradiction completes the proof.
\end{proof}
\section{The rescaled Whitham equation}\label{sec7}
We consider here the rescaled Whitham equation \eqref{Whitresc}. We will revisit Theorem \ref{th:W} and its proof by keeping the small parameter $\epsilon,$ our goal being to estimate the blow-up time which should of course be at least of order $\mathcal{O}(1/\epsilon)$ since the local solution of the Cauchy problem exists at least on this time scale.
We first need a "rescaled" version of Lemma \ref{le:Hur} :
\begin{lemma} \label{Hur2}There exist constants \(L_0, L_\infty>0\) and \(\eta_0\in(0,1)\) such that
\begin{equation*}
\begin{aligned}
K_\epsilon(x)\leq \epsilon^{-1/4}\frac{L_0}{\sqrt{|x|}}\quad \mathrm{and}\ |K^\prime(x)|\leq \epsilon^{-1/4}\frac{L_0}{\sqrt{|x|^3}},\quad \mathrm{for}\ 0<|x|<\epsilon^{1/2}\eta_0,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\int_{\epsilon^{1/2}\eta_0}^\infty |K_\epsilon^\prime(x)|\,\mathrm{d} x\leq \epsilon^{-1/2}L_\infty.
\end{aligned}
\end{equation*}
\end{lemma}
Theorem \ref{th:W} is reformulated as follows:
\begin{theorem}[rescaled Whitham equation]\label{rescWhit} Let \(\delta\in(0,1-\frac{2\sqrt{2}}{3}]\) and \(\epsilon^{-1}\delta<1/2\). Assume that \(\phi\in H^3(\mathbb{R})\) satisfies
\begin{equation*}
\begin{aligned}
&\delta^2\epsilon^{1/4}(\inf_{\mathbb{R}}\phi^\prime(x))^2> 4L_0C_{\mathrm{Sob}}\|\phi\|_{H^3}+2C_1(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}C_1^{\frac{1}{3}}\|\phi^{\prime\prime\prime}\|_{L^2}^{\frac{2}{3}},\\
&-(1-\delta)^2\epsilon^{1/4}\inf_{\mathbb{R}}\phi^\prime(x)> 8(3L_0+L_\infty)+16L_0\bigg(\frac{C_1}{C_0}\bigg),\\
&-(1-\delta)^3\epsilon^{1/4}\inf_{\mathbb{R}}\phi^\prime(x)> 4(3L_0+L_\infty)+36L_0C_{\mathrm{GN}}\bigg(\frac{\|\phi^{\prime\prime\prime}\|_{L^2}}{C_1}\bigg)^{\frac{2}{3}},
\end{aligned}
\end{equation*}
where the constant \(C_0\) and \(C_1\) satisfy
\begin{equation*}
\begin{aligned}
\|\phi\|_{L^\infty}\leq\frac{C_0}{2},\quad
\|\phi^\prime\|_{L^\infty}\leq\frac{C_1}{2}.
\end{aligned}
\end{equation*}
Then the solution of the Cauchy problem \eqref{Whitresc} with the initial data \(u(0,x)=\phi(x)\) exhibits wave breaking at some time \(T>0\), namely
\begin{equation*}
\begin{aligned}
|u(x,t)|<\infty,\quad x\in\mathbb{R}, t\in[0,T),
\end{aligned}
\end{equation*}
but
\begin{equation*}
\begin{aligned}
\inf_{\mathbb{R}}\partial_xu(x,t)\longrightarrow -\infty,\quad \text{as}\ t\longrightarrow T-.
\end{aligned}
\end{equation*}
Moreover
\begin{equation}\label{30.5}
\begin{aligned}
-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{\epsilon(1+\epsilon^{-1}\delta)}<T<-\frac{1}{\inf_{\mathbb{R}}\phi^\prime(x)}\frac{1}{\epsilon(1-\epsilon^{-1}\delta)^2}.
\end{aligned}
\end{equation}
\end{theorem}
It is worth relating Theorem \ref{rescWhit} to the results in \cite {KLPS} that compare the solution of the rescaled Whitham equation and that of the KdV equation
\begin{equation}\label{KdV}
\partial_t w+\partial_x w+\epsilon w\partial_x w+ \frac{1}{6}\partial_x^3 w=0.
\end{equation}
Actually the next result is proven in \cite{KLPS} (Theorem 2).
\begin{theorem} \label{compare}
Let $\phi \in H^{\infty}(\mathbb R)$ and let $u$ and $w$ be the respective solutions of \eqref{Whitresc} and \eqref{KdV} with initial data $\phi.$ Then, for all $j \in \mathbb N$, $j \ge 0$, there exists
$M_j=M_j(\|\phi\|_{H^{j+8}})>0$ such that
\begin{equation} \label{maintheo.1}
\|(u-w)(t)\|_{H^j_x} \le M_j \epsilon^2t,
\end{equation}
for all $0 \le t \lesssim \epsilon^{-1}$.
\end{theorem}
\begin{remark}
\rm{The implicit constant in the notation $t \lesssim \epsilon^{-1}$ depends on $\|\phi\|_{H^2}^{-1}$ for $j=0$, $1$ and $\|\phi\|_{H^{j+1}}^{-1}$ for $j\geq 2.$}
\end{remark}
By Theorem \ref{compare}, the solution \(u\) of the rescaled Whitham equation \eqref{Whitresc} cannot
blow up before a time of order \(\mathcal{O}\big(\epsilon^{-1}\|\phi\|_{H^3}^{-1}\big)\), which is indeed confirmed by \eqref{30.5} since the obtained wave breaking time is \(\mathcal{O}\big(\epsilon^{-1}[-\inf_{\mathbb{R}}\phi^\prime(x)]^{-1}\big)\).
The proof of Theorem \ref{rescWhit} follows that of Theorem \ref{th:W} and Lemma \ref{le:a1}-Lemma \ref{le:a3} by keeping the small parameter $\epsilon$ and we omit it.
\section{Final remarks}
The results in this paper suggest that the fKdV when $-1\leq \alpha<0$ and the Whitham equation share some properties of the nonlinear hyperbolic Burgers equation. One may ask for instance if they possess global weak (entropy) solutions. This has been proven for the Burgers-Hilbert equation in \cite{BN}.
On the other hand those equations have a (weak) dispersive part, with the possibility of existence of global strong small solutions. This is suggested by some numerical simulations in \cite{KLPS} and has been proven for the {\it cubic} fKdV equation with $-1<\alpha<0$ in the work \cite{SW} of the Authors.
\section{Appendix}
In the proofs of Theorem \ref{th:BH}, \ref{th:W}, \ref{th:fKdV}, and \ref{rescWhit}, we need to handle the following ODE
\begin{align}\label{a0}
\frac{\mathrm{d} v_1}{\mathrm{d} t}+\epsilon v_1^2+K_{1,\epsilon}(t,x)=0.
\end{align}
For the Burgers-Hilbert equation \eqref{eq:BH}, the Whitham equation \eqref{Whit}, and the fKdV equation \eqref{eq:main-1}, one shall take \(\epsilon=1\) in \eqref{a0}, and \(K_{1,\epsilon}(t,x)\\
=\colon K_1(t,x)\) being defined as \eqref{BH-kernal}, \eqref{Whit-kernal}, and \eqref{fKdV-kernal} respectively. For the rescaled Whitham equation \eqref{Whitresc}, one shall keep \(\epsilon\) being a small parameter in \eqref{a0} and define \(K_{1,\epsilon}(t,x)\) by
\begin{align*}
K_{1,\epsilon}(t,x)=\int_\mathbb{R} K_\epsilon(y)\partial_x^2u(X(t,x)-y,t)\,\mathrm{d} y,
\end{align*}
where \(K_\epsilon(\cdot)\) is given by \eqref{res-kernal}.
In the following we always assume \(\delta\in(0,\frac{1}{2})\), \(\epsilon\in (0,1]\), \(\epsilon^{-1}\delta<1/2\), and \(t\in[0,T_1]\), and then let
\begin{equation*}
\begin{aligned}
\Sigma_{\delta,\epsilon}(t)=\{x\in\mathbb{R}:v_1(t,x)\leq (1-\epsilon^{-1}\delta)m(t)\},
\end{aligned}
\end{equation*}
and also define
\[v_1(t,x)=:m(0)r^{-1}(t,x).\]
We denote \(\Sigma_{\delta,1}(t)=\colon \Sigma_\delta(t)\) (\(\epsilon=1\)) for simplicity.
The following technical lemmas for \(\epsilon=1\) were proved in \cite{NS,HT,Hur}, which can be extended to all \(\epsilon\in (0,1]\) with slight modifications.
We include the proofs here for the sake of completeness and readers' convenience.
\begin{lemma}\label{le:a1} One has \(\Sigma_{\delta,\epsilon}(t_2)\subset \Sigma_{\delta,\epsilon}(t_1) \) whenever \(0\leq t_1\leq t_2\leq T_1\).
\end{lemma}
\begin{proof} Suppose that there exists some \(x_1\in\mathbb{R}\) such that
\(x_1\notin\Sigma_{\delta,\epsilon}(t_1)\) but \(x_1\in\Sigma_{\delta,\epsilon}(t_2)\) for some \(0\leq t_1\leq t_2\leq T_1\), that is
\begin{equation}\label{a1}
\begin{aligned}
v_1(t_1,x_1)> (1-\epsilon^{-1}\delta)m(t_1)\quad \text{and}\quad v_1(t_2,x_1)\leq (1-\epsilon^{-1}\delta)m(t_2)<\frac{1}{2}m(t_2).
\end{aligned}
\end{equation}
One can choose \(t_1\) and \(t_2\) close so that
\begin{equation*}
\begin{aligned}
v_1(t,x_1)\leq \frac{1}{2}m(t),\quad \text{for\ all}\ t\in [t_1,t_2].
\end{aligned}
\end{equation*}
Indeed since \(v_1(\cdot,x_1)\) and \(m\) are uniformly continuous on \([0,T_1]\), let
\begin{equation}\label{a2}
\begin{aligned}
v_1(t_1,x_2)=m(t_1)\leq \frac{1}{2}m(t_1).
\end{aligned}
\end{equation}
We may necessarily choose \(t_2\) close to \(t_1\) so that
\begin{equation*}
\begin{aligned}
v_1(t,x_2)\leq \frac{1}{2}m(t),\quad \text{for\ all}\ t\in [t_1,t_2].
\end{aligned}
\end{equation*}
According to \eqref{9} (\eqref{31} or \eqref{56}), one has
\begin{equation*}
\begin{aligned}
|K_{1,\epsilon}(t,x_j)|\leq \delta^2 m^2(t)\leq 4\delta^2v_1^2(t,x_j)<\frac{1}{2}\delta v_1^2(t,x_j),
\end{aligned}
\end{equation*}
for all \(t\in[t_1,t_2],\ j=1,2\).
This together with \eqref{a0} yields
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d} v_1}{\mathrm{d} t}(\cdot,x_1)=- \epsilon v_1^2(\cdot,x_1)-K_{1,\epsilon}(t,x_1)\geq (-\epsilon-\frac{\delta}{2})v_1^2(\cdot,x_1)
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\frac{\mathrm{d} v_1}{\mathrm{d} t}(\cdot,x_2)=- \epsilon v_1^2(\cdot,x_2)-K_{1,\epsilon}(t,x_2)\leq (-\epsilon+\frac{\delta}{2})v_1^2(\cdot,x_2),
\end{aligned}
\end{equation*}
for \(t\in [t_1,t_2]\). Solving the resulting two inequalities above gives
\begin{equation}\label{a2.1}
\begin{aligned}
v_1(t_2,x_1)\geq\frac{v_1(t_1,x_1)}{1+(\epsilon+\frac{\delta}{2})v_1(t_1,x_1)(t_2-t_1)},
\end{aligned}
\end{equation}
and
\begin{equation}\label{a2.2}
\begin{aligned}
v_1(t_2,x_2)\leq\frac{v_1(t_1,x_2)}{1+(\epsilon-\frac{\delta}{2})v_1(t_1,x_2)(t_2-t_1)}.
\end{aligned}
\end{equation}
Applying \eqref{a2} to \eqref{a2.2}, one obtains
\begin{equation}\label{a2.3}
\begin{aligned}
m(t_2)\leq\frac{m(t_1)}{1+(\epsilon-\frac{\delta}{2})m(t_1)(t_2-t_1)}.
\end{aligned}
\end{equation}
In view of \eqref{a1} and \eqref{a2.1}, one estimates
\begin{equation*}
\begin{aligned}
v_1(t_2,x_1)&>\frac{(1-\epsilon^{-1}\delta)m(t_1)}{1+(\epsilon+\frac{\delta}{2})(1-\epsilon^{-1}\delta)m(t_1)(t_2-t_1)}\\
&>\frac{(1-\epsilon^{-1}\delta)m(t_1)}{1+(\epsilon-\frac{\delta}{2})m(t_1)(t_2-t_1)}\\
&>(1-\epsilon^{-1}\delta)m(t_2),
\end{aligned}
\end{equation*}
where we have used \eqref{a2.3} in the last inequality. We get a contradiction!
\end{proof}
\begin{lemma}\label{le:a2} We have
\begin{equation}\label{a3}
\begin{aligned}
\epsilon(1+\epsilon^{-1}\delta)m(0)\leq \frac{\mathrm{d} r}{\mathrm{d} t}\leq \epsilon(1-\epsilon^{-1}\delta)m(0),
\end{aligned}
\end{equation}
\begin{equation}\label{a4}
\begin{aligned}
q(t)\leq r(t)\leq (1-\epsilon^{-1}\delta)^{-1}q(t),
\end{aligned}
\end{equation}
and
\begin{equation}\label{a5}
\begin{aligned}
0<q(t)\leq 1.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
Let \(x\in\Sigma_{\delta,\epsilon}(T_1)\), it then follows from Lemma \ref{le:a1} that
\begin{equation}\label{a6}
\begin{aligned}
m(t)\leq v_1(t,x)\leq (1-\epsilon^{-1}\delta)m(t), \quad \text{for\ all}\ t\in [0,T_1].
\end{aligned}
\end{equation}
The solution of \eqref{a0} can be expressed
\begin{equation}\label{a6.5}
\begin{aligned}
v_1(t,x)=\frac{v_1(0,x)}{1+ \epsilon v_1(0,x)\int_0^t\big(1+\epsilon^{-1}v_1^{-2}K_{1,\epsilon}(\tau,x)\big)\,\mathrm{d} \tau}=m(0)r^{-1}(t,x).
\end{aligned}
\end{equation}
It follows from \eqref{a6} and \eqref{9} (\eqref{31} or \eqref{56}) that
\begin{equation*}
\begin{aligned}
|v_1^{-2}K_{1,\epsilon}(t,x)|\leq (1-\delta)^{-2}\delta^2<\delta,\quad \text{for\ all}\ t\in [0,T_1].
\end{aligned}
\end{equation*}
This together with \eqref{a6.5} implies \eqref{a3}. The inequality \eqref{a4} is a consequence of \eqref{a6} and \eqref{a6.5}. It is easy to see that \(r(t,x)\) is decreasing for all \(t\in [0,T_1]\) from \eqref{a3}, and hence \(v_1(t,x)\) too.
Furthermore, by \eqref{1}, \(q(t)\) is also decreasing for all \(t\in [0,T_1]\)， which implies \eqref{a5} by \eqref{3}.
\end{proof}
\begin{lemma}\label{le:a3} It holds that
\begin{equation}\label{a7}
\begin{aligned}
\int_0^tq^{-s}(\tau)\,\mathrm{d} \tau\leq -(1-\epsilon^{-1}\delta)^{-(s+1)}(1-s)^{-1}m^{-1}(0)[(1-\epsilon^{-1}\delta)^{s-1}-q^{1-s}(t)],
\end{aligned}
\end{equation}
where\(s>0, s\neq 1\), and
\begin{equation}\label{a8}
\begin{aligned}
\int_0^tq^{-1}(\tau)\,\mathrm{d} \tau\leq -(1-\epsilon^{-1}\delta)^{-2}m^{-1}(0)[-\log (1-\epsilon^{-1}\delta)-\log q(t)].
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
Let \(s>0, s\neq 1\), we use \eqref{a3} and \eqref{a4} to deduce that
\begin{equation*}
\begin{aligned}
&\int_0^tq^{-s}(\tau)\,\mathrm{d} \tau
\leq (1-\epsilon^{-1}\delta)^{-s}\int_0^tr^{-s}(\tau,x)\,\mathrm{d} \tau\\
&\leq (1-\epsilon^{-1}\delta)^{-(s+1)}m^{-1}(0)\int_0^tr^{-s}(\tau,x)\frac{\mathrm{d} }{\mathrm{d} t}r(\tau,x)\,\mathrm{d} \tau\\
&=(1-\epsilon^{-1}\delta)^{-(s+1)}(1-s)^{-1}m^{-1}(0)[r^{s-1}(t,x)-r^{1-s}(0,x)],
\end{aligned}
\end{equation*}
which combines \eqref{a4} implies \eqref{a7}. One can verify \eqref{a8} similarly.
\end{proof}
\noindent {\bf Acknowledgments.} The work of both authors was partially supported by the ANR project ANuI.
\end{document}
|
{{\rm \rangle}m \mathfrak{b}}egin{document}
\title{\textbf{Simpson Filtration and Oper Stratum Conjecture}}
{{\rm \rangle}m \mathfrak{a}}uthor{Zhi Hu}
{{\rm \rangle}m \mathfrak{a}}ddress{ \textsc{School of Science, Nanjing University of Science and Technology, Nanjing 210094, China}{\rm \epsilon}ndgraf \textsc{Department of Mathematics, Mainz University, 55128 Mainz, Germany}}
{\rm \epsilon}mail{[email protected]; [email protected]}
{{\rm \rangle}m \mathfrak{a}}uthor{Pengfei Huang}
{{\rm \rangle}m \mathfrak{a}}ddress{ \textsc{School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China}{\rm \epsilon}ndgraf
\textsc{Laboratoire J.A. Dieudonn\'e, Universit\'e C\^ote d'Azur, CNRS, 06108 Nice, France}}
{\rm \epsilon}mail{[email protected]; [email protected]}
\subjclass[2010]{14D20, 14D22, 14J60, 57N80}
\keywords{${{\rm \rangle}m \lambda}ambda$-flat bundle, Chain, System of Hodge bundles, Simpson filtration, Oper stratification, Complex variations of Hodge structure, Moduli spaces}
\date{}
{{\rm \rangle}m \mathfrak{b}}egin{abstract}
In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum.
{\rm \epsilon}nd{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Many moduli spaces admit stratifications, even arising from different points of view. For example, Atiyah and Bott studied the Morse stratification on an infinite dimensional affine space $\mathcal{A}$ of unitary connections on a $C^\infty$-vector bundle over a Riemann surface $X$ of genus $g\geq 2$, which is defined by the gradient flow of the norm square of moment map with respect to the gauge group {{\rm \rangle}m \mathfrak{c}}ite{AB}, and coincides with a stratification defined by the Harder--Narasimhan types of algebraic vector bundles over $X$ {{\rm \rangle}m \mathfrak{c}}ite{D}. Hitchin introduced the moduli space of (stable) Higgs bundle over $X$ {{\rm \rangle}m \mathfrak{c}}ite{hi}, it admits two stratifications: one is also the Harder--Narasimhan (HN) stratification defined by the Harder--Narasimhan types of the underlying vector bundles, the other one is the Bialynicki-Birula (BB) stratification defined by the fixed points of $\mathbb{C}^*$-action on the moduli space. The latter one also has the Morse theoretic interpretation. Indeed, the norm square of Higgs field is a Morse function on the moduli space, and is a moment map with respect to the Hamiltonian $S^1$-action, by Kirwan's result {{\rm \rangle}m \mathfrak{c}}ite{K}, the stratification defined by the upwards Morse flow of that Morse function coincides with the BB stratification. In general, the HN stratification does not coincide with the BB stratification, however, they are the same for the case of rank two due to Hausel {{\rm \rangle}m \mathfrak{c}}ite{Ha}. For the moduli space of flat bundles over $X$, the HN stratification can be obviously well-defined. In {{\rm \rangle}m \mathfrak{c}}ite{CS6}, Simpson constructed a stratification of this moduli space by embedding it into the moduli space of ${{\rm \rangle}m \lambda}ambda$-flat bundles (varying ${{\rm \rangle}m \lambda}ambda\in \mathbb{C}$) and showing the natural $\mathbb{C}^*$-action on the bigger moduli space defines a BB stratification. Simpson called this stratification the {\rm \epsilon}mph{oper stratification} since the ``minimal'' stratum is the moduli space of opers. So far, we do not know whether the oper stratification can be viewed as a certain Morse stratification.
Let $\mathbb{M}_{\mathrm{Hod}}(X,r)$ be the coarse moduli space of semistable ${{\rm \rangle}m \lambda}ambda$-flat bundles over $X$ of fixed rank $r$ with vanishing first Chern class, which is called the {\rm \epsilon}mph{Hodge moduli space}. This space has a natural fibration $\mathbb{M}_{\mathrm{Hod}}(X,r)\to\mathbb{C}, (E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda){\rm \hspace{1mm} \rightarrow \hspace{1mm}}sto{{\rm \rangle}m \lambda}ambda$ such that the fiber over 0 is the {\rm \epsilon}mph{Dolbeault moduli space} $\mathbb{M}_{\mathrm{Dol}}(X,r)$, namely the moduli space of semistable Higgs bundles over $X$ of rank $r$ with vanishing first Chern class; and the fiber over 1 is the {\rm \epsilon}mph{de Rham moduli space} $\mathbb{M}_{\mathrm{dR}}(X,r)$, namely the moduli space of flat bundles over $X$ of rank $r$. The natural $\mathbb{C}^*$-action on $\mathbb{M}_{\mathrm{Dol}}(X,r)$ by rescaling the Higgs field can be extended to an action on $\mathbb{M}_{\mathrm{Hod}}(X,r)$ via $t{{\rm \rangle}m \mathfrak{c}}dot(E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda):=(E,tD^{{\rm \rangle}m \lambda}ambda,t{{\rm \rangle}m \lambda}ambda)$ for $t\in\mathbb{C}^*$.
Simpson showed in {{\rm \rangle}m \mathfrak{c}}ite{CS6} that for each $(E, D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda)\in \mathbb{M}_{\mathrm{Hod}}(X,r)$, the limit ${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda)$ exists uniquely as a $\mathbb{C}^*$-fixed point lying in $\mathbb{M}_{\mathrm{Dol}}(X,r)$. If we divide the set $V(X,r)$ of $\mathbb{C}^*$-fixed points into the connected components $V(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha V_{{\rm \rangle}m \mathfrak{a}}lpha$, then we will have a stratification $\mathbb{M}_{\mathrm{Hod}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha G_{{\rm \rangle}m \mathfrak{a}}lpha$, where the locally closed subset $G_{{\rm \rangle}m \mathfrak{a}}lpha$ is defined as $G_{{\rm \rangle}m \mathfrak{a}}lpha=\{(E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda)\in\mathbb{M}_{\mathrm{Hod}}(X,r): {{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda)\in V_{{\rm \rangle}m \mathfrak{a}}lpha\}$. In particular, the restriction to the fiber over 0 leads to the BB stratification of $\mathbb{M}_{\mathrm{Dol}}(X,r)$, and the restriction to the fiber over 1 gives rises to the oper stratification: $\mathbb{M}_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{{\rm \rangle}m \mathfrak{a}}lpha}\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha$, or $M_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{{\rm \rangle}m \mathfrak{a}}lpha}S_{{\rm \rangle}m \mathfrak{a}}lpha$ for smooth locus.
In order to obtain the $\mathbb{C}^*$-limit points of flat bundles, the {\rm \epsilon}mph{Simpson filtration} plays a crucial role, which is a filtration of subbundles satisfying two extra conditions: Griffiths transversality for the flat connection and semistability for the associated graded Higgs bundle. Such filtration always exists by iterated destabilizing modifications.
For the oper stratification, Simpson proposed many interesting conjectures, part of them were already solved by Simpson himself and others, but many of them are still open questions.
{{\rm \rangle}m \mathfrak{b}}egin{introconjecture} [\textbf{Foliation Conjecture}]
The Lagrangian fibers of the projections $\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha{\rm \rangle}ightarrow V_{{\rm \rangle}m \mathfrak{a}}lpha$ fit together into a smooth Lagrangian foliation with closed leaves.
{\rm \epsilon}nd{introconjecture}
{{\rm \rangle}m \mathfrak{b}}egin{introconjecture}[\textbf{Nestedness Conjecture}]
The oper stratification $\mathbb{M}_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{{\rm \rangle}m \mathfrak{a}}lpha}\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha$ is nested, namely, there is a partial order on the index set $\{{{\rm \rangle}m \mathfrak{a}}lpha\}$ such that
$$
\overline{\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha}={{\rm \rangle}m \mathfrak{c}}oprod_{{{\rm \rangle}m \mathfrak{b}}eta{{\rm \rangle}m \lambda}eq {{\rm \rangle}m \mathfrak{a}}lpha}\mathbb{S}_{{\rm \rangle}m \mathfrak{b}}eta.
$$
{\rm \epsilon}nd{introconjecture}
{{\rm \rangle}m \mathfrak{b}}egin{introconjecture}[\textbf{Oper Stratum Conjecture}]
The oper stratum is the unique closed stratum and the unique stratum of minimal dimension.
{\rm \epsilon}nd{introconjecture}
{{\rm \rangle}m \mathfrak{b}}egin{rmk}
For the first conjecture, the part of Lagrangian property is already known {{\rm \rangle}m \mathfrak{c}}ite{C,CS6} (see also Lemma {\rm \rangle}ef{halfdim}), the closedness of fibers is quite obvious, since these fibers are affine spaces by applying BB theory to Hodge moduli space (see Proposition 2.1 of {{\rm \rangle}m \mathfrak{c}}ite{H-H}, or Corollary 1.5 of {{\rm \rangle}m \mathfrak{c}}ite{C}) and the de Rham moduli space is an affine analytic variety. The whole foliation conjecture for the moduli space of rank two connections over four punctured projective line was proved by the authors in {{\rm \rangle}m \mathfrak{c}}ite{LSS}. As pointed out by Simpson, if this conjecture is affirmative, it could be useful for the context of geometric Langlands, where the full moduli stack of vector bundles might be replaced by the algebraic space of leaves of foliations. What we should mention is that the analogous statement of closedness on the side of the moduli space of Higgs bundles is not true. For example, one picks up $u=(E_0,0)$ as a stable vector bundle $E_0$ of rank $r$ with trivial Higgs field, then $Y^0_u:=\{(E,\theta)\in M_{\mathrm{Dol}}(X,r): {{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\theta)=u\}\simeq H^0(X,{\rm End}(E_0)\otimes {{\rm \rangle}m \mathcal{O}}mega^1_X)$, it is closed in $ M_{\mathrm{Dol}}(X,r)$ if and only if $E_0$ is very stable, i.e. there is no non-zero nilpotent Higgs field on $E_0$ {{\rm \rangle}m \mathfrak{c}}ite{PP}. For the second conjecture, we only know very few, in {{\rm \rangle}m \mathfrak{c}}ite{CS6}, Simpson showed the nestedness conjecture for the case of rank two by a beautiful deformation theory argument, which in particular, implies the oper stratum conjecture.
{\rm \epsilon}nd{rmk}
In this paper, we mainly focus on the oper stratum conjecture. In particular, we will prove the following theorem which partially confirms Simpson's oper stratum conjecture (the part on dimension).
{{\rm \rangle}m \mathfrak{b}}egin{introtheorem}[= Corollary {\rm \rangle}ef{mainthm}] For the oper stratification $M_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha S_{{\rm \rangle}m \mathfrak{a}}lpha$, we have
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item the open dense stratum $N(X,r)$ consisting of irreducible flat bundles such that the underlying vector bundles are stable is the unique maximal stratum with dimension $2r^2(g-1)+2$,
\item the closed oper stratum $S_{\mathrm{oper}}$ is the unique minimal stratum with dimension $r^2(g-1)+g+1$.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{introtheorem}
The paper is organized as follows. In the following section, we collect some basic materials on the theory of ${{\rm \rangle}m \lambda}ambda$-flat bundles, and holomorphic chains that can be used to describe the $\mathbb{C}^*$-fixed points (i.e. $\mathbb{C}$-VHSs). In the third section, we introduce the Simpson filtrations of flat bundles and the oper stratification of de Rham moduli space, we also give an explicit description of the Simpson filtrations on flat bundles of rank 3, as well as an upper bound on the degree of subbundles. In the last section, we will give the proof of our main theorem, which reduces to the study of the connected components $V_{{\rm \rangle}m \mathfrak{a}}lpha$ since each stratum $S_{{\rm \rangle}m \mathfrak{a}}lpha$ is a fibration over $V_{{\rm \rangle}m \mathfrak{a}}lpha$ with fibers as Lagrangian submanifolds of $M_{\mathrm{dR}}(X,r)$.
{{\rm \rangle}m \mathfrak{b}}igskip
\noindent\textbf{Acknowledgements}.
The authors would like to express their deep gratitude to Prof. Brian Collier, Prof. Peter Gothen, Prof. Jochen Heinloth and Prof. Richard Wentworth for communications on various occasions, and to the anonymous referee for many valuable suggestions. The author P. Huang would like to thank Prof. Carlos Simpson for kind help and useful discussions, and thank Prof. Jiayu Li for his continuous encouragement.
\section{Preliminaries}
\subsection{Flat ${{\rm \rangle}m \lambda}ambda$-Connections}
The notion of flat ${{\rm \rangle}m \lambda}ambda$-connection as the interpolation of usual flat connection and Higgs field was suggested by Deligne {{\rm \rangle}m \mathfrak{c}}ite{D}, illustrated by Simpson in {{\rm \rangle}m \mathfrak{c}}ite{CS4} and further studied in {{\rm \rangle}m \mathfrak{c}}ite{CS5,CS6}. We recall the definition here.
Throughout the paper, $X$ is always assumed to be a compact Riemann surface of genus $g\geq 2$, and let $K_X={{\rm \rangle}m \mathcal{O}}mega^1_X$ be the canonical line bundle over $X$.
{{\rm \rangle}m \mathfrak{b}}egin{definition} [{{\rm \rangle}m \mathfrak{c}}ite{CS4}]
Assume ${{\rm \rangle}m \lambda}ambda\in\mathbb{C}$.
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item Let $E$ be a holomorphic vector bundle over $X$, a {\rm \epsilon}mph{${{\rm \rangle}m \lambda}ambda$-connection} on $E$ is a $\mathbb{C}$-linear operator $D^{{\rm \rangle}m \lambda}ambda: E\to E\otimes{{\rm \rangle}m \mathcal{O}}mega_X^{1}$ that satisfies the following ${{\rm \rangle}m \lambda}ambda$-twisted Leibniz rule:
$$
D^{{\rm \rangle}m \lambda}ambda(fs)=fD^{{\rm \rangle}m \lambda}ambda s+{{\rm \rangle}m \lambda}ambda s\otimes df,
$$
where $f$ and $s$ are holomorphic sections of $\mathcal{O}_X$ and $E$, respectively. If $D^{{\rm \rangle}m \lambda}ambda{{\rm \rangle}m \mathfrak{c}}irc D^{{\rm \rangle}m \lambda}ambda=0$ under the natural extension $D^{{\rm \rangle}m \lambda}ambda: E\otimes{{\rm \rangle}m \mathcal{O}}mega_X^{p}\to E\otimes{{\rm \rangle}m \mathcal{O}}mega_X^{p+1}$ for any integer $p\geq0$, we call $D^{{\rm \rangle}m \lambda}ambda$ a {\rm \epsilon}mph{flat ${{\rm \rangle}m \lambda}ambda$-connection}, and the pair $(E,D^{{\rm \rangle}m \lambda}ambda)$ is called a {\rm \epsilon}mph{${{\rm \rangle}m \lambda}ambda$-flat bundle}.
\item A ${{\rm \rangle}m \lambda}ambda$-flat bundle $(E,D^{{\rm \rangle}m \lambda}ambda)$ over $X$ is called {\rm \epsilon}mph{stable} (resp. {\rm \epsilon}mph{semistable}) if for any ${{\rm \rangle}m \lambda}ambda$-flat subbundle $(F,D^{{\rm \rangle}m \lambda}ambda|_F)$ of $0<{{\rm \rangle}m rank}(F)<{{\rm \rangle}m rank}(E)$, we have the following inequality
$$
\mu(F)< (\text{resp.} {{\rm \rangle}m \lambda}eq) \,\mu(F),
$$
where $\mu({{\rm \rangle}m \mathfrak{b}}ullet)={{\rm \rangle}m \mathfrak{f}}rac{\deg({{\rm \rangle}m \mathfrak{b}}ullet)}{{{\rm \rangle}m rank}({{\rm \rangle}m \mathfrak{b}}ullet)}$ denotes the {\rm \epsilon}mph{slope} of bundle. And we call $(E,D^{{\rm \rangle}m \lambda}ambda)$ is {\rm \epsilon}mph{polystable} if it decomposes as a direct sum of stable ${{\rm \rangle}m \lambda}ambda$-flat bundles with the same slope.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{definition}
Let $\mathcal{M}_{\mathrm{Hod}}(X,r)$ be the moduli stack of
${{\rm \rangle}m \lambda}ambda$-flat bundles (varying ${{\rm \rangle}m \lambda}ambda\in\mathbb{C}$) over $X$ of rank $r$ with vanishing first Chern class, and let $\mathbb{M}_{\mathrm{Hod}}(X,r)$ be the coarse moduli space of semistable ${{\rm \rangle}m \lambda}ambda$-flat bundles of this stack, called the {\rm \epsilon}mph{Hodge moduli space}. It's known that $\mathbb{M}_{\mathrm{Hod}}(X,r)$ is a quasi-projective variety and parameterizes the isomorphism classes of polystable ${{\rm \rangle}m \lambda}ambda$-flat bundles {{\rm \rangle}m \mathfrak{c}}ite{CS4}, let $M_{\mathrm{Hod}}(X,r)$ be the smooth locus of $\mathbb{M}_{\mathrm{Hod}}(X,r)$, which is a Zariski dense open subset and parameterizes the isomorphism classes of stable ${{\rm \rangle}m \lambda}ambda$-flat bundles. There is a natural fibration ${{\rm \rangle}m \mathfrak{p}}i: \mathbb{M}_{\mathrm{Hod}}(X,r){\rm \rangle}ightarrow\mathbb{C}, (E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda){\rm \hspace{1mm} \rightarrow \hspace{1mm}}sto{{\rm \rangle}m \lambda}ambda$ such that the fiber ${{\rm \rangle}m \mathfrak{p}}i^{-1}({{\rm \rangle}m \lambda}ambda)=:\mathbb{M}_{\mathrm{Hod}}^{{\rm \rangle}m \lambda}ambda(X,r)$ is the coarse moduli space of semistable ${{\rm \rangle}m \lambda}ambda$-flat bundles over $X$ of rank $r$ with vanishing first Chern class, in particular,
{{\rm \rangle}m \mathfrak{b}}egin{itemize}
\item[${{\rm \rangle}m \mathfrak{b}}ullet$] ${{\rm \rangle}m \mathfrak{p}}i^{-1}(1)=\mathbb{M}_{\mathrm{dR}}(X,r)$, the coarse moduli space of flat bundles over $X$ of rank $r$, called the {\rm \epsilon}mph{de Rham moduli space}, which is algebraically isomorphic to $\mathbb{M}_{\mathrm{dR}}(X,r)$ for any ${{\rm \rangle}m \lambda}ambda\neq0$;
\item[${{\rm \rangle}m \mathfrak{b}}ullet$] ${{\rm \rangle}m \mathfrak{p}}i^{-1}(0)=\mathbb{M}_{\mathrm{Dol}}(X,r)$, the coarse moduli space of semistable Higgs bundles over $X$ of rank $r$ with vanishing first Chern class, called the {\rm \epsilon}mph{Dolbeault moduli space}.
{\rm \epsilon}nd{itemize}
The natural $\mathbb{C}^*$-action on $\mathbb{M}_{\mathrm{Dol}}(X,r)$ via $t{{\rm \rangle}m \mathfrak{c}}dot(E,\theta):=(E,t\theta)$ can be generalized to the $\mathbb{C}^*$-action on $\mathbb{M}_{\mathrm{Hod}}(X,r)$ via $t{{\rm \rangle}m \mathfrak{c}}dot (E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda) := (E,tD^{{\rm \rangle}m \lambda}ambda,t{{\rm \rangle}m \lambda}ambda)$. As this action change the fibers over ${{\rm \rangle}m \lambda}ambda\neq0$, therefore, all the fixed points must lie in the fiber over 0, that is, in $\mathbb{M}_{\mathrm{Dol}}(X,r)$. It is known that a such $\mathbb{C}^*$-fixed point is a polystable Higgs bundle that has a structure of system of Hodge bundles ({{\rm \rangle}m \mathfrak{c}}ite[Lemma 4.1]{CS3}), along the phraseology of {{\rm \rangle}m \mathfrak{c}}ite{C}, we will call it a complex variation of Hodge structure (briefly as $\mathbb{C}$-VHS). For the Dolbeault moduli space $\mathbb{M}_{\mathrm{Dol}}(X,r)$, the properness and the $\mathbb{C}^*$-equivariant property of the Hitchin map preserve that, for each Higgs bundle $(E,\theta)$, the limit point ${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t\to0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\theta)$ exists as a $\mathbb{C}$-VHS. There is no analogue of Hitchin map for the Hodge moduli space $\mathbb{M}_{\mathrm{Hod}}(X,r)$, however in {{\rm \rangle}m \mathfrak{c}}ite{CS6}, Simpson showed that for the $\mathbb{C}^*$-action on $\mathbb{M}_{\mathrm{Hod}}(X,r)$, the limit point ${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot (E,D^{{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda)$ also exists as a $\mathbb{C}$-VHS (see Section {\rm \rangle}ef{sec3}).
\subsection{Chains and $\mathbb{C}$-VHSs}
A $\mathbb{C}$-VHS has the form $(E={{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^lE_i,\theta={{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^{l-1}\theta_i)$ with each $\theta_i: E_i\to E_{i+1}\otimes K_X$, ${{\rm \rangle}m rank}(E_i)=r_i$, and $\deg(E_i)=d_i$. The pair $(\overrightarrow{r},\overrightarrow{d}):=(r_1,{{\rm \rangle}m \mathfrak{c}}dots,r_l; d_1,{{\rm \rangle}m \mathfrak{c}}dots,d_l)$ is called the {\rm \epsilon}mph{type} of the $\mathbb{C}$-VHS. This can be characterized by a chain of holomorphic bundles with certain stability parameter.
{{\rm \rangle}m \mathfrak{b}}egin{definition}[{{\rm \rangle}m \mathfrak{c}}ite{A,B}]
\mbox{}
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item Let $\overrightarrow{r}=(r_1,{{\rm \rangle}m \mathfrak{c}}dots, r_l), \overrightarrow{d}=(d_1,{{\rm \rangle}m \mathfrak{c}}dots,d_l)$ and $|\overrightarrow{r}|=\sum_{i=1}^lr_i, |\overrightarrow{d}|=\sum_{i=1}^ld_i$. A {\rm \epsilon}mph{chain} $Ch_{\overrightarrow{r},\overrightarrow{d}}$ of length $l$ is a tuple $$(\mathcal{E}_i,i=1,{{\rm \rangle}m \mathfrak{c}}dots,l; {\rm va}rphi_i, i=1,{{\rm \rangle}m \mathfrak{c}}dots i-1),$$ consisting of holomorphic bundles $\mathcal{E}_i$ on $X$ with ${{\rm \rangle}m rank}(\mathcal{E}_i)=r_i, \deg(\mathcal{E}_i)=d_i (i=1,{{\rm \rangle}m \mathfrak{c}}dots,l)$, and holomorphic morphisms ${\rm va}rphi_i:\mathcal{E}_i{\rm \rangle}ightarrow \mathcal{E}_{i+1} (i=1,{{\rm \rangle}m \mathfrak{c}}dots,l-1)$. We write a chain as
$$
Ch_{\overrightarrow{r},\overrightarrow{d}}:\mathcal{E}_1\xrightarrow{{\rm va}rphi_1}\mathcal{E}_2\xrightarrow{{\rm va}rphi_2}{{\rm \rangle}m \mathfrak{c}}dots\xrightarrow{{\rm va}rphi_{l-1}}\mathcal{E}_l,
$$
the pair $(\overrightarrow{r},\overrightarrow{d})$ is called the {\rm \epsilon}mph{type} of the chain. If each ${\rm va}rphi_i$ dose not vanish, the chain is called {\rm \epsilon}mph{indecomposable}.
\item Let $\overrightarrow{{{{\rm \rangle}m \mathfrak{a}}lpha}}=({{\rm \rangle}m \mathfrak{a}}lpha_1,{{\rm \rangle}m \mathfrak{c}}dots,{{\rm \rangle}m \mathfrak{a}}lpha_l)\in\mathbb{R}^l$, and call it a {\rm \epsilon}mph{stability parameter}. For a chain $Ch_{\overrightarrow{r},\overrightarrow{d}}$, we introduce its {\rm \epsilon}mph{$\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}$-slope} as
$$
\mu_{\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}}(Ch_{\overrightarrow{r},\overrightarrow{d}})={{\rm \rangle}m \mathfrak{f}}rac{|\overrightarrow{d}|+\sum_{i=1}^l{{\rm \rangle}m \mathfrak{a}}lpha_ir_i}{|\overrightarrow{r}|}.
$$
A chain $Ch_{\overrightarrow{r},\overrightarrow{d}}$ is called {\rm \epsilon}mph{$\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}$-stable} (resp., {\rm \epsilon}mph{$\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}$-semistable}) if for any non-zero proper subchain $Ch_{\overrightarrow{r^{{\rm \rangle}m \mathfrak{p}}rime},\overrightarrow{d^{{\rm \rangle}m \mathfrak{p}}rime}}$ we have
$$
\mu_{\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}}(Ch_{\overrightarrow{r^{{\rm \rangle}m \mathfrak{p}}rime},\overrightarrow{d^{{\rm \rangle}m \mathfrak{p}}rime}})<(\mathrm{resp.},\ {{\rm \rangle}m \lambda}eq)\ \mu_{\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}}(Ch_{\overrightarrow{r},\overrightarrow{d}}).
$$
A chain $Ch_{\overrightarrow{r},\overrightarrow{d}}$ is called {\rm \epsilon}mph{$\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}$-polystable} if it can be decomposed into the direct sum of two subchains $Ch_{\overrightarrow{r},\overrightarrow{d}}=Ch_{\overrightarrow{r_1},\overrightarrow{d_1}}{{\rm \rangle}m \mathfrak{b}}igoplus Ch_{\overrightarrow{r_2},\overrightarrow{d_2}}$ such that $Ch_{\overrightarrow{r_i},\overrightarrow{d_i}} (i=1,2)$ is $\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha_i}$-stable with $\mu_{\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha_i}}(Ch_{\overrightarrow{r_i},\overrightarrow{d_i}})=\mu_{\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}}(Ch_{\overrightarrow{r},\overrightarrow{d}})$, where $\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha_i}=\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}|_{Ch_{\overrightarrow{r_i},\overrightarrow{d_i}}}$.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{definition}
In {{\rm \rangle}m \mathfrak{c}}ite{PH}, the authors considered the necessary conditions for the existence of $\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}$-semistable chains:
{{\rm \rangle}m \mathfrak{b}}egin{proposition} [{{\rm \rangle}m \mathfrak{c}}ite{PH}]{{\rm \rangle}m \lambda}abel{1}
Let $Ch_{\overrightarrow{r},\overrightarrow{d}}$ be a $\overrightarrow{{{{\rm \rangle}m \mathfrak{a}}lpha}}$-semistable chain of length $l$, where $\overrightarrow{{{{\rm \rangle}m \mathfrak{a}}lpha}}=({{\rm \rangle}m \mathfrak{a}}lpha_1,{{\rm \rangle}m \mathfrak{c}}dots,{{\rm \rangle}m \mathfrak{a}}lpha_l)$ is a stability parameter satisfying ${{\rm \rangle}m \mathfrak{a}}lpha_1>{{\rm \rangle}m \mathfrak{c}}dots>{{\rm \rangle}m \mathfrak{a}}lpha_l$, and let $\mu=\mu_{\overrightarrow{{{{\rm \rangle}m \mathfrak{a}}lpha}}}(Ch_{\overrightarrow{r},\overrightarrow{d}})$, then
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item for all $j\in{2,{{\rm \rangle}m \mathfrak{c}}dots,l}$, we have$${{\rm \rangle}m \mathfrak{f}}rac{\sum_{i=j}^l(d_i+{{\rm \rangle}m \mathfrak{a}}lpha_i r_i)}{\sum_{i=j}^lr_i}{{\rm \rangle}m \lambda}eq \mu;$$
\item for all $j$ such that $r_j=r_{j+1}$, we have $$d_j{{\rm \rangle}m \lambda}eq d_{j+1};$$
\item for all $1{{\rm \rangle}m \lambda}eq k<j{{\rm \rangle}m \lambda}eq l$ such that $r_k<\min\{r_{k+1},{{\rm \rangle}m \mathfrak{c}}dots, r_j\}$, we have
$$
{{\rm \rangle}m \mathfrak{f}}rac{\sum_{i\notin[k,j]}(d_i+{{\rm \rangle}m \mathfrak{a}}lpha_ir_i)+(j-k+1)d_k+(\sum_{i=k}^j{{\rm \rangle}m \mathfrak{a}}lpha_i)r_k}{\sum_{i\notin[k,j]}r_i+(j-k+1)r_k}{{\rm \rangle}m \lambda}eq \mu;
$$
\item for all $1{{\rm \rangle}m \lambda}eq k<j{{\rm \rangle}m \lambda}eq l$ such that $r_k>\max\{r_{k+1},{{\rm \rangle}m \mathfrak{c}}dots, r_j\}$, we have
$$
{{\rm \rangle}m \mathfrak{f}}rac{\sum_{i=k}^{j-1}(d_i-d_j+{{\rm \rangle}m \mathfrak{a}}lpha_i(r_i-r_j))}{\sum_{i=k}^{j-1}(r_i-r_j)}{{\rm \rangle}m \lambda}eq \mu.
$$
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{proposition}
Given a $\mathbb{C}$-VHS of the form $(E={{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^lE_i,\theta={{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^{l-1}\theta_i)$ of type $(\overrightarrow{r},\overrightarrow{d}):=(r_1,{{\rm \rangle}m \mathfrak{c}}dots,r_l; d_1,{{\rm \rangle}m \mathfrak{c}}dots,d_l)$, we can obtain a chain with a parameter $\delta\in\mathbb{Z}$ as
$$
Ch_{\overrightarrow{r},\overrightarrow{d'}}:\mathcal{{E}}_1\xrightarrow{{\rm va}rphi_1}\mathcal{{E}}_2\xrightarrow{{\rm va}rphi_2}{{\rm \rangle}m \mathfrak{c}}dots\xrightarrow{{\rm va}rphi_{l-1}}\mathcal{{E}}_l,
$$
where each $\mathcal{{E}}_i={E}_i\otimes K_X^{-(l-i+\delta)}$, ${\rm va}rphi_i=\theta_i\otimes \mathrm{Id}$ and $d'_i=d_i-r_i(l-i+\delta)(2g-2)$. We assign each $\mathcal{E}_i$ an integer
$$
{{\rm \rangle}m \mathfrak{a}}lpha_i=(l-i+\delta)(2g-2)
$$
to form a stability parameter $\overrightarrow{{{{\rm \rangle}m \mathfrak{a}}lpha}}_{\mathrm{Higgs}}$, then $Ch_{\overrightarrow{r},\overrightarrow{d'}}$ is
$\overrightarrow{{{{\rm \rangle}m \mathfrak{a}}lpha}}_{\mathrm{Higgs}}$-semistable.
{{\rm \rangle}m \mathfrak{b}}egin{proposition}{{\rm \rangle}m \lambda}abel{11}
Denote by $V(X,r)$ the set of all $\mathbb{C}$-VHSs lying in $\mathbb{M}_{\mathrm{Dol}}(X,r)$.
Let $V^{\overrightarrow{r},\overrightarrow{d}}\subset V(X,r)$ be the subset that consists of $\mathbb{C}$-VHSs of type $(\overrightarrow{r},\overrightarrow{d})$ with each component $\theta_i$ of the Higgs field $\theta$ non-zero. If $V^{\overrightarrow{r},\overrightarrow{d}}$ is non-empty, then $(\overrightarrow{r},\overrightarrow{d})$ should satisfy the following conditions:
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item for all $1<j{{\rm \rangle}m \lambda}eq l$, we have
{{\rm \rangle}m \mathfrak{b}}egin{align}{{\rm \rangle}m \lambda}abel{2.1}
\sum_{i=j}^ld_i<0,
{\rm \epsilon}nd{align}
\item for all $j$ such that $r_j=r_{j+1}$, we have
{{\rm \rangle}m \mathfrak{b}}egin{align}{{\rm \rangle}m \lambda}abel{2.2}
{{\rm \rangle}m \mathfrak{f}}rac{d_j}{r_j}-{{\rm \rangle}m \mathfrak{f}}rac{d_{j+1}}{r_{j+1}}{{\rm \rangle}m \lambda}eq 2g-2,
{\rm \epsilon}nd{align}
\item for all $1{{\rm \rangle}m \lambda}eq k<j{{\rm \rangle}m \lambda}eq l$ such that $r_k<\min\{r_{k+1},{{\rm \rangle}m \mathfrak{c}}dots, r_j\}$, we have
{{\rm \rangle}m \mathfrak{b}}egin{align}{{\rm \rangle}m \lambda}abel{2.4.3}
-\sum_{i=k+1}^jd_i+(j-k)(d_k-(j-k+1)(g-1)r_k){{\rm \rangle}m \lambda}eq0,
{\rm \epsilon}nd{align}
\item for all $1{{\rm \rangle}m \lambda}eq k<j{{\rm \rangle}m \lambda}eq l$ such that $r_k>\max\{r_{k+1},{{\rm \rangle}m \mathfrak{c}}dots, r_j\}$, we have
{{\rm \rangle}m \mathfrak{b}}egin{align}{{\rm \rangle}m \lambda}abel{2.4}
\sum_{i=k}^{j-1}d_i-(j-k)(d_j+(j-k+1)(g-1)r_j){{\rm \rangle}m \lambda}eq 0,
{\rm \epsilon}nd{align}
{\rm \epsilon}nd{enumerate}
Conversely, if the type $(\overrightarrow{r},\overrightarrow{d})$ satisfies the conditions {\rm \epsilon}qref{2.1}-{\rm \epsilon}qref{2.4}, then $V^{\overrightarrow{r},\overrightarrow{d}}$ is non-empty.
{\rm \epsilon}nd{proposition}
{{\rm \rangle}m \mathfrak{b}}egin{proof}
Viewing a $\mathbb{C}$-VHS as a chain of certain type with the stability parameter $\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}_{\mathrm{Higgs}}=({{\rm \rangle}m \mathfrak{a}}lpha_1,{{\rm \rangle}m \mathfrak{c}}dots,{{\rm \rangle}m \mathfrak{a}}lpha_l)$, since ${{\rm \rangle}m \mathfrak{a}}lpha_1>{{\rm \rangle}m \mathfrak{c}}dots>{{\rm \rangle}m \mathfrak{a}}lpha_l$, then we can apply Proposition {\rm \rangle}ef{1} to obtain the four inequalities in proposition.
Conversely, given $(\overrightarrow{r},\overrightarrow{d})$, by a result of {{\rm \rangle}m \mathfrak{c}}ite{BGG}, the conditions {\rm \epsilon}qref{2.1}-{\rm \epsilon}qref{2.4} implies that there is an $\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}_{\mathrm{Higgs}}$-semistable chain $Ch_{\overrightarrow{r},\overrightarrow{d_0}}$ of type $(\overrightarrow{r},\overrightarrow{d_0}=\overrightarrow{d}-\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}_{\mathrm{Higgs}})$ for $\overrightarrow{{{\rm \rangle}m \mathfrak{a}}lpha}_{\mathrm{Higgs}}=((l-1)(2g-2),{{\rm \rangle}m \mathfrak{c}}dots,2g-2,0)$, hence produces a $\mathbb{C}$-VHS via the suitable semisimplification of the Jordan--H\"older filtration. Moreover, the condition {\rm \epsilon}qref{2.1} guarantees such chain is indecomposable, namely each $\theta_i$ is non-zero, thus the last statement is shown.
{\rm \epsilon}nd{proof}
\section{Simpson Filtration and Oper Stratification}{{\rm \rangle}m \lambda}abel{sec3}
{{\rm \rangle}m \mathfrak{b}}egin{definition} [{{\rm \rangle}m \mathfrak{c}}ite{CS6}]
Let $E$ be a holomorphic vector bundle over $X$ with a holomorphic flat connection $\nabla:E{\rm \rangle}ightarrow E\otimes_{\mathcal{O}_X}K_X$. A decreasing filtration $\mathcal{F}=\{F^{{\rm \rangle}m \mathfrak{b}}ullet\}$ of $E$ by strict subbundles
$$
E=F^0\supset F^1\supset{{\rm \rangle}m \mathfrak{c}}dots\supset F^k=0
$$
is called a {\rm \epsilon}mph{Simpson filtration} if it satisfies the following two conditions:
{{\rm \rangle}m \mathfrak{b}}egin{itemize}
\item Griffiths transversality: $\nabla: F^p{\rm \rangle}ightarrow F^{p-1}\otimes_{\mathcal{O}_X}K_X$,
\item graded-semistability: the associated graded Higgs bundle $(\mathrm{Gr}_{\mathcal{F}}(E),\mathrm{Gr}_{\mathcal{F}}(\nabla))$, where $\mathrm{Gr}_{\mathcal{F}}(E)={{\rm \rangle}m \mathfrak{b}}igoplus_pE^p$ with $E^p=F^p/F^{p+1}$ and $\mathrm{Gr}_{\mathcal{F}}(\nabla)={{\rm \rangle}m \mathfrak{b}}igoplus_p\theta^p$ with $\theta^p: E^p{\rm \rangle}ightarrow E^{p-1}\otimes_{\mathcal{O}_X}K_X$ induced from $\nabla$, is a semistable Higgs bundle.
{\rm \epsilon}nd{itemize}
{\rm \epsilon}nd{definition}
In {{\rm \rangle}m \mathfrak{c}}ite{CS6}, Simpson studied the $\mathbb{C}^*$-action on flat bundles and obtained the following nice theorem:
{{\rm \rangle}m \mathfrak{b}}egin{theorem} [{{\rm \rangle}m \mathfrak{c}}ite{CS6}]{{\rm \rangle}m \lambda}abel{22}
Let $(E,\nabla)$ be a flat bundle over $X$.
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item There exist Simpson filtrations $\mathcal{F}$ on $(E,\nabla)$.
\item Let $\mathcal{F}_1$, $\mathcal{F}_2$ be two Simpson filtrations on $(E,\nabla)$, then the associated graded Higgs bundles $(\mathrm{Gr}_{\mathcal{F}_1}(E),\mathrm{Gr}_{\mathcal{F}_1}(\nabla))$ and $(\mathrm{Gr}_{\mathcal{F}_2}(E),\mathrm{Gr}_{\mathcal{F}_2}(\nabla))$ are $S$-equivalent{{\rm \rangle}m \mathfrak{f}}ootnote{We say two semistable Higgs bundles are {\rm \epsilon}mph{$S$-equivalent} if their associated graded Higgs bundles defined by the Jordan-H\"older filtrations are isomorphic.}.
\item $(\mathrm{Gr}_\mathcal{F}(E),\mathrm{Gr}_\mathcal{F}(\nabla))$ is a stable Higgs bundle iff the Simpson filtration is unique.
\item ${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(\mathrm{Gr}_\mathcal{F}(E),\mathrm{Gr}_\mathcal{F}(\nabla))$.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{theorem}
The existence of Simpson filtrations is obtained via a beautiful iterating process, here we sketch how it works.
Suppose $(E,\nabla)$ admits a filtration
$$
\mathcal{F}: E=F^0\supset F^1\supset{{\rm \rangle}m \mathfrak{c}}dots\supset F^k=0
$$
that satisfies the Griffiths transversality $\nabla(F^p)\subset F^{p-1}\otimes K_X$, and such that the associated Higgs bundle $(V,\theta):=(\mathrm{Gr}_{\mathcal{F}}(E),\mathrm{Gr}_{\mathcal{F}}(\nabla))$ is not semistable. To see the existence of such filtration, we can begin with the trivial filtration $E=F^0\supset F^1=0$, the graded Higgs bundle will be $(\mathrm{Gr}_{\mathcal{F}}(E),\mathrm{Gr}_{\mathcal{F}}(\nabla))=(E,0)$. Take $H\subset(V,\theta)$ to be the maximal destabilizing subsheaf, which is known being unique and a subbundle of $V$, and the quotient $V/H$ is also a subbundle of $E$. As a sub-Higgs bundle of a $\mathbb{C}$-VHS, $H$ is also a $\mathbb{C}$-VHS, and is a sub-$\mathbb{C}$-VHS of $(V,\theta)$, that is, $H={{\rm \rangle}m \mathfrak{b}}igoplus H^p$ with each $H^p=H{{\rm \rangle}m \mathfrak{b}}igcap \mathrm{Gr}_{\mathcal{F}}^p(E)\subset F^p/F^{p+1}$ being a strict subbundle.
The new filtration $\mathcal{G}=\{G^{{\rm \rangle}m \mathfrak{b}}ullet\}$ is defined as
$$
G^p:={\rm Ker}{{\rm \rangle}m \lambda}eft(E\to{{\rm \rangle}m \mathfrak{f}}rac{E/F^p}{H^{p-1}}{\rm \rangle}ight).
$$
It satisfies the Griffiths traversality since $\theta(H^p)\subset H^{p-1}\otimes K_X$, and it fits into the short exact sequence
$$
0{{\rm \rangle}m \lambda}ongrightarrow \mathrm{Gr}_{\mathcal{F}}^p(E)/H^p{{\rm \rangle}m \lambda}ongrightarrow\mathrm{Gr}_{\mathcal{G}}^p(E){{\rm \rangle}m \lambda}ongrightarrow H^{p-1}{{\rm \rangle}m \lambda}ongrightarrow0.
$$
If the new resulting graded Higgs bundle $(\mathrm{Gr}_{\mathcal{G}}(E),\mathrm{Gr}_{\mathcal{G}}(\nabla))$ is still not semistable, then we continue this process to obtain a new graded Higgs bundle. By introducing three bounded invariants, Simpson showed that the iteration process will strictly decrease these invariants in lexicographic order{{\rm \rangle}m \mathfrak{f}}ootnote{Details on these invariants can be found in Simpson's paper {{\rm \rangle}m \mathfrak{c}}ite{CS6} (see also the second named author's thesis {{\rm \rangle}m \mathfrak{c}}ite{Hua}). }. Therefore, after a finite step, we will find a filtration such that the associated graded Higgs bundle is semistable.
This can be concluded as the following algorithm flowchart:
\tikzstyle{startstop} = [rectangle, rounded corners, minimum width = 2cm, minimum height=0.6cm,text centered, draw = black]
\tikzstyle{io} = [trapezium, trapezium left angle=70, trapezium right angle=110, minimum width=2cm, minimum height=0.8cm, text centered, draw=black]
\tikzstyle{process} = [rectangle, minimum width=3cm, minimum height=0.8cm, text centered, draw=black]
\tikzstyle{decision} = [diamond, aspect = 4, text centered, draw=black]
\tikzstyle{arrow} = [->,>=stealth]
\vspace*{-15pt}
\[
{{\rm \rangle}m \mathfrak{b}}egin{tikzpicture}[node distance=0.8cm]
\node[startstop](start){Start};
\node[io, below of = start, yshift = -0.5cm](in1){$(E,\nabla,\mathcal{F})$};
\node[process, below of = in1, yshift = -0.6cm](dec1){$H\subset(\mathrm{Gr}_{\mathcal{F}}(E),\mathrm{Gr}_{\mathcal{F}}(\nabla))$ maximal destabilizing subsheaf};
\node[process, below of = dec1, yshift = -0.8cm](dec2){$\mathcal{G}: G^p=\mathrm{Ker}\Big(E\to {{\rm \rangle}m \mathfrak{f}}rac{E/F^p}{H^{p-1}}\Big)$};
\node[decision, below of = dec2, yshift = -1.2cm](pro1){ $\overset{(\mathrm{Gr}_{\mathcal{G}}(E),\mathrm{Gr}_{\mathcal{G}}(\nabla))}{\mathrm{semistable} ?}$};
\node[process, below of = dec1, yshift = -0.8cm,xshift=-6cm](dec3){$\mathcal{F}:=\mathcal{G}$};
\node[io, below of = pro1, yshift = -1.2cm](out1){$(\mathrm{Gr}_{\mathcal{G}}(E),\mathrm{Gr}_{\mathcal{G}}(\nabla))$};
\node[startstop, below of = out1, yshift = -0.6cm](stop){Stop};
\draw [arrow] (start) -- (in1);
\draw [arrow] (in1) -- (dec1);
\draw [arrow] (dec1) -- (dec2);
\draw [arrow] (dec2) -- (pro1);
\draw [arrow] (pro1) -| node [right] {{{\rm \rangle}m \mathfrak{c}}olor{red}{No}} (dec3);
\draw [arrow] (dec3) |- (dec1);
\draw [arrow] (pro1) -- node [right] {{{\rm \rangle}m \mathfrak{c}}olor{red}{Yes}} (out1);
\draw [arrow] (out1) -- (stop);
{\rm \epsilon}nd{tikzpicture}
\]
Generally speaking, calculating the Simpson filtration for a given flat bundle is quite hard. For rank 2 case, the Simpson filtration is exactly given by the Harder--Narasimhan filtration of the bundle itself. But for the flat bundles of higher rank, it's very complicated. Here we give an explicit description for rank 3 case (details especially the proof can be found in {{\rm \rangle}m \mathfrak{c}}ite{Hua1}). An analogous result for the Higgs bundles of rank 3 can be found in {{\rm \rangle}m \mathfrak{c}}ite{GZ}.
{{\rm \rangle}m \mathfrak{b}}egin{example}
Let $(E,\nabla)$ be a flat bundle of rank 3 over a $X$, and we assume $E$ is not a stable bundle. The Simpson filtration on $(E,\nabla)$ is described as follows.
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item Assume the Harder-Narasimhan filtration on $E$ is given by $H^1\subset E$ with ${{\rm \rangle}m rank} (H^1)=1, \deg(H^1)=d$, then $0<d{{\rm \rangle}m \lambda}eq {{\rm \rangle}m \mathfrak{f}}rac{2}{3}(2g-2)$. The flat connection $\nabla$ induces a nonzero morphism
$\theta: H^1 {\rm \rangle}ightarrow E/H^1\otimes K_X$. Denote by $I$ the line subbundle of $E/H^1$ obtained by saturating the subsheaf $\theta(H^1)\otimes K_X^{-1}$.
{{\rm \rangle}m \mathfrak{b}}egin{description}
\item[1.1] If $d-2g+2{{\rm \rangle}m \lambda}eq \deg(I)<-d$, then the Simpson filtration on $(E,\nabla)$ coincides with the Harder-Narasimhan filtration and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(H^1\oplus E/H^1, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
\theta& 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
).$$
\item[1.2] If $\deg(I)=-d$, although the Simpson filtration on $(E,\nabla)$ is not unique, the limiting polystable Higgs bundle is given by
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(H^1\oplus I, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
\theta & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
)\oplus ({{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I},0).$$
\item[1.3] If $-d< \deg(I){{\rm \rangle}m \lambda}eq -{{\rm \rangle}m \mathfrak{f}}rac{d}{2}$, one defines $$ F^1:={\rm Ker} (E{\rm \rangle}ightarrow{{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I}),$$
then the Simpson filtration is given by $H^1\subset F^1\subset E$, and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(H^1\oplus I\oplus {{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I},{{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{ccc}
0 & 0 & 0 \\
\theta & 0 & 0 \\
0 & {\rm va}rphi & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
),$$
where the nonzero morphism ${\rm va}rphi: I{\rm \rangle}ightarrow{{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I}\otimes K_X$ is induced by $\nabla: F^1{\rm \rangle}ightarrow E\otimes K_X$.
{\rm \epsilon}nd{description}
\item Assume the Harder-Narasimhan filtration on $E$ is given by $G^1\subset E$ with ${{\rm \rangle}m rank} (G^1)=2, \deg(G^1)=l$, then $0<l{{\rm \rangle}m \lambda}eq{{\rm \rangle}m \mathfrak{f}}rac{2}{3}(2g-2)$. The flat connection $\nabla$ induces a nonzero morphism
${\rm va}rtheta: G^1 {\rm \rangle}ightarrow E/G^1\otimes K_X$. Denote by $N$ the line subbundle of $G^1$ obtained by saturating the subsheaf ${\rm Ker}({\rm va}rtheta)$.
{{\rm \rangle}m \mathfrak{b}}egin{description}
\item[2.1]If $2l-2g+2{{\rm \rangle}m \lambda}eq\deg(N)<0$, then the Simpson filtration on $(E,\nabla)$ coincides with the Harder-Narasimhan filtration and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(G^1\oplus E/G^1, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
{\rm va}rtheta& 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
).$$
\item[2.2] If $\deg(N)=0$, the limiting polystable Higgs bundle is given by
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(N,0)\oplus(G^1/N\oplus E/G^1, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
{\rm va}rtheta & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
).$$
\item[2.3] If $0<\deg(N){{\rm \rangle}m \lambda}eq {{\rm \rangle}m \mathfrak{f}}rac{l}{2}$, then the Simpson filtration on $(E,\nabla)$ is given by $N\subset G^1\subset E$, and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(N\oplus G^1/N\oplus E/G^1,{{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{ccc}
0 & 0 & 0 \\
{{\rm \rangle}m \mathfrak{p}}hi & 0 & 0 \\
0 & {\rm va}rtheta & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
),$$
where the nonzero morphism ${{\rm \rangle}m \mathfrak{p}}hi: N{\rm \rangle}ightarrow G^1/N\otimes K_X$ is induced by $\nabla:N{\rm \rangle}ightarrow G^1\otimes K_X$.
{\rm \epsilon}nd{description}
\item Assume the Harder-Narasimhan filtration on $E$ is given by $A^1\subset A^2\subset E$ with ${{\rm \rangle}m rank} (A^i)=i, \deg(A^i)=a_i, i=1,2$, then $0<2a_1-a_2{{\rm \rangle}m \lambda}eq 2g-2, 0<2a_2-a_1{{\rm \rangle}m \lambda}eq 2g-2$. The flat connection $\nabla$ induces nonzero morphisms ${{\rm \rangle}m \mathfrak{p}}si: A^1{\rm \rangle}ightarrow E/A^1\otimes K_X$ and ${{\rm \rangle}m \mathfrak{c}}hi: A^2{\rm \rangle}ightarrow E/A^2\otimes K_X$, then define $J={{\rm \rangle}m \mathfrak{p}}si(A^1)\otimes K_X^{-1}\subset E/A^1$ and $M={\rm Ker}({{\rm \rangle}m \mathfrak{c}}hi)\subset A^2$ viewing as line subbundles by saturating, and define $L^1={\rm Ker}(E{\rm \rangle}ightarrow{{\rm \rangle}m \mathfrak{f}}rac{E/A^1}{J})$.
{{\rm \rangle}m \mathfrak{b}}egin{description}
\item[3.1] When $-a_1<\deg(J){{\rm \rangle}m \lambda}eq a_2-a_1$, the Simpson filtration on $(E,\nabla)$ is given by $A^1\subset L^1\subset E$, and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(A^1\oplus J\oplus {{\rm \rangle}m \mathfrak{f}}rac{E/A^1}{J},{{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{ccc}
0 & 0 & 0 \\
{{\rm \rangle}m \mathfrak{p}}si & 0 & 0 \\
0 & {\rm \rangle}ho & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
),$$ where the nonzero morphism ${\rm \rangle}ho: J{\rm \rangle}ightarrow{{\rm \rangle}m \mathfrak{f}}rac{E/A^1}{J}\otimes K_X$ is induced by $\nabla: L^1{\rm \rangle}ightarrow E\otimes K_X$. In particular, if $\deg(J)=a_2-a_1$, the Simpson filtration coincides with the Harder-Narasimhan filtration.
\item[3.2] When $\deg(J)=-a_1$, the limiting polystable Higgs bundle is given by
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(A^1\oplus J, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
{{\rm \rangle}m \mathfrak{p}}si & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
)\oplus ({{\rm \rangle}m \mathfrak{f}}rac{E/A^1}{J},0).$$
\item[3.3]
When $a_1-2g+2{{\rm \rangle}m \lambda}eq \deg(J)<-a_1$,
{{\rm \rangle}m \mathfrak{b}}egin{description}
\item [3.3.1]if $a_2-a_1<0$, the Simpson filtration is given by $A_1\subset E$, and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(A^1\oplus E/A^1, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
{{\rm \rangle}m \mathfrak{p}}si & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
).$$
\item [3.3.2] if $a_2-a_1\geq0$, {{\rm \rangle}m \mathfrak{b}}egin{description}
\item[3.2.2.1] for $2a_2-2g+2{{\rm \rangle}m \lambda}eq\deg(M)<0$, the Simpson filtration is given by $A^2\subset E$, and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(A^2\oplus E/A^2, {{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
{{\rm \rangle}m \mathfrak{c}}hi & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
).$$
\item[3.3.2.2] for $a_2-a_1\geq0,\deg(M)=0$, the limiting polystable Higgs bundle is given by
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(M,0)\oplus ( A^2/M\oplus E/A^2,{{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{cc}
0 & 0 \\
{{\rm \rangle}m \mathfrak{c}}hi & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
).$$
\item[3.3.2.3] for $a_2-a_1>0, 0<\deg(M){{\rm \rangle}m \lambda}eq a_2-a_1$, the Simpson filtration is given by
$M\subset A^2\subset E$, and
$${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,\nabla)=(M\oplus A^2/M\oplus E/A^2,{{\rm \rangle}m \lambda}eft(
{{\rm \rangle}m \mathfrak{b}}egin{array}{ccc}
0 & 0 & 0 \\
{\rm va}rrho & 0 & 0 \\
0 & {{\rm \rangle}m \mathfrak{c}}hi & 0 \\
{\rm \epsilon}nd{array}
{\rm \rangle}ight)
),$$ where the nonzero morphism ${\rm va}rrho: M{\rm \rangle}ightarrow A^2/M$ is induced by $\nabla: M{\rm \rangle}ightarrow A^2\otimes K_X$. In particular, as $\deg(M)=a_2-a_1$, the underlying vector bundle of the limiting Higgs bundle coincides with graded vector bundle from the Harder-Narasimhan filtration.
{\rm \epsilon}nd{description}
{\rm \epsilon}nd{description}
{\rm \epsilon}nd{description}
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{example}
The following proposition is an application of the above description of Simpson filtration.
{{\rm \rangle}m \mathfrak{b}}egin{proposition}
Let $E$ be a vector bundle of rank 3 and degree 0 over $X$, and let $W$ be any subbundle of $E$. If $E$ admits a flat ${{\rm \rangle}m \lambda}ambda$-connection (${{\rm \rangle}m \lambda}ambda\neq 0$), then $\deg(W){{\rm \rangle}m \lambda}eq 4g-4$.
{\rm \epsilon}nd{proposition}
{{\rm \rangle}m \mathfrak{b}}egin{proof}
We only need to consider the case of ${{\rm \rangle}m \lambda}ambda=1$, then we can apply the the above example.
Firstly, we consider the case 1.1.
There is a short exact sequence
$$0{\rm \rangle}ightarrow W{{\rm \rangle}m \mathfrak{b}}igcap H^1{\rm \rangle}ightarrow W{\rm \rangle}ightarrow W/(W{{\rm \rangle}m \mathfrak{b}}igcap H^1){\rm \rangle}ightarrow 0.$$
Since $ W/(W{{\rm \rangle}m \mathfrak{b}}igcap H^1)$ is a subsheaf of $E/H^1$, it follows from the stability of Higgs bundle that
$$\deg(W)=\deg( W{{\rm \rangle}m \mathfrak{b}}igcap H^1)+\deg (W/(W{{\rm \rangle}m \mathfrak{b}}igcap H^1)){{\rm \rangle}m \lambda}eq \deg(H^1)=d<g-1.$$
Next, we consider the cases 1.2 and 1.3. Again by stability of Higgs bundle, we have
{{\rm \rangle}m \mathfrak{b}}egin{align*}
\deg(W/(W{{\rm \rangle}m \mathfrak{b}}igcap F^1))&{{\rm \rangle}m \lambda}eq0,\\
\deg(W{{\rm \rangle}m \mathfrak{b}}igcap H^1\oplus I\oplus {{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I})&{{\rm \rangle}m \lambda}eq0,\\
\deg ({{\rm \rangle}m \mathfrak{f}}rac{W{{\rm \rangle}m \mathfrak{b}}igcap F^1}{W{{\rm \rangle}m \mathfrak{b}}igcap H^1}\oplus {{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I})&{{\rm \rangle}m \lambda}eq0.
{\rm \epsilon}nd{align*}
Therefore we arrive at
{{\rm \rangle}m \mathfrak{b}}egin{align*}
\deg(W){{\rm \rangle}m \lambda}eq \deg(W{{\rm \rangle}m \mathfrak{b}}igcap F^1)&{{\rm \rangle}m \lambda}eq\deg(W{{\rm \rangle}m \mathfrak{b}}igcap H^1)-\deg( {{\rm \rangle}m \mathfrak{f}}rac{E/H^1}{I})\\
&{{\rm \rangle}m \lambda}eq2\deg(H^1)+\deg(I)\\
&{{\rm \rangle}m \lambda}eq{{\rm \rangle}m \mathfrak{f}}rac{3}{2}d{{\rm \rangle}m \lambda}eq 2g-2.
{\rm \epsilon}nd{align*}
Similar arguments show that
$$\deg(W){{\rm \rangle}m \lambda}eq{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
l<g-1, & \hbox{case 2.1;} \\
{{\rm \rangle}m \mathfrak{f}}rac{3}{2}l{{\rm \rangle}m \lambda}eq2g-2, & \hbox{cases 2.2 and 2.3;} \\
a_1+a_2{{\rm \rangle}m \lambda}eq4g-4, & \hbox{cases 3.1 and 3.2;}\\
a_1<g-1, & \hbox{case 3.3.1;}\\
a_2<g-1, & \hbox{case 3.3.2.1;}\\
2a_2-a_1{{\rm \rangle}m \lambda}eq2g-2 & \hbox{cases 3.3.2.2 and 3.3.2.3}.{\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
The conclusion follows.
{\rm \epsilon}nd{proof}
For a ${{\rm \rangle}m \lambda}ambda$-flat bundle $(E,D^{{\rm \rangle}m \lambda}ambda) ({{\rm \rangle}m \lambda}ambda\neq0)$, Simpson showed the limit ${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t\to0}t{{\rm \rangle}m \mathfrak{c}}dot(E,D^{{\rm \rangle}m \lambda}ambda)$ can be obtained by taking the grading of the Simpson filtration on the associated flat bundle $(E,{{\rm \rangle}m \lambda}ambda^{-1}D^{{\rm \rangle}m \lambda}ambda)$, hence it is a $\mathbb{C}$-VHS, namely, we have
{{\rm \rangle}m \mathfrak{b}}egin{theorem}[{{\rm \rangle}m \mathfrak{c}}ite{CS6}]{{\rm \rangle}m \lambda}abel{limit}
Let $(E,D^{{\rm \rangle}m \lambda}ambda)\in\mathbb{M}_{\mathrm{Hod}}(X,r)$ be a ${{\rm \rangle}m \lambda}ambda$-flat bundle (${{\rm \rangle}m \lambda}ambda\neq 0$), then we have
$$
{{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t{\rm \rangle}ightarrow 0}t{{\rm \rangle}m \mathfrak{c}}dot(E,D^{{\rm \rangle}m \lambda}ambda)=(\mathrm{Gr}_{\mathcal{F}_{{\rm \rangle}m \lambda}ambda}(E),\mathrm{Gr}_{\mathcal{F}_{{\rm \rangle}m \lambda}ambda}({{\rm \rangle}m \lambda}ambda^{-1}D^{{\rm \rangle}m \lambda}ambda)),
$$
where ${\mathcal{F}_{{\rm \rangle}m \lambda}ambda}$ is the Simpson filtration on the associated flat bundle $(E,{{\rm \rangle}m \lambda}ambda^{-1}D^{{\rm \rangle}m \lambda}ambda)$.
{\rm \epsilon}nd{theorem}
With previous notation, divide $V(X,r)$ into connected components $V(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha V_{{\rm \rangle}m \mathfrak{a}}lpha$. Define the subset $\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha\subset \mathbb{M}_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{p}}i^{-1}(1)$ via
$$
\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha:={{\rm \rangle}m \lambda}eft\{y\in\mathbb{M}_{\mathrm{dR}}(X,r)\ \Big|\ {{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t\to0}t{{\rm \rangle}m \mathfrak{c}}dot y\in V_{{\rm \rangle}m \mathfrak{a}}lpha{\rm \rangle}ight\},
$$
then each $\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha$ is locally closed, and these subsets partition the de Rham moduli space into the {\rm \epsilon}mph{oper stratification} {{\rm \rangle}m \mathfrak{c}}ite{CS6}
$$
\mathbb{M}_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha,\quad M_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha S_{{\rm \rangle}m \mathfrak{a}}lpha,
$$
where $S_{{\rm \rangle}m \mathfrak{a}}lpha:=\mathbb{S}_{{\rm \rangle}m \mathfrak{a}}lpha{{\rm \rangle}m \mathfrak{b}}igcap M_{\mathrm{dR}}(X,r)$.
{{\rm \rangle}m \mathfrak{b}}egin{definition}
For the oper stratification $M_{\mathrm{dR}}(X,r)={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha S_{{\rm \rangle}m \mathfrak{a}}lpha$, a (non-empty) stratum $S_{{\rm \rangle}m \mathfrak{a}}lpha$ is called
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item the {\rm \epsilon}mph{maximal stratum} if it has maximal dimension among all the (non-empty) strata,
\item the {\rm \epsilon}mph{minimal stratum} if it has minimal dimension among all the (non-empty) strata.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{definition}
{{\rm \rangle}m \mathfrak{b}}egin{example}
A {\rm \epsilon}mph{uniformizing Higgs bundle} $(E,\theta)$ of rank $r\geq 2$ is given by
$$
E=L\oplus L\otimes K_X^{-1}\oplus{{\rm \rangle}m \mathfrak{c}}dots\oplus L\otimes K_X^{-r+1}
$$
and $\theta$ is determined by the natural isomorphisms $L\otimes K_X^i\xrightarrow{\simeq} (L\otimes K_X^{i-1})\otimes K_X$,
where $L$ is a line bundle of degree $(r-1)(g-1)$. The moduli space $V_{\mathrm{uni}}$ of uniformizing Higgs bundles is isomorphic to $\mathrm{Jac}^{(r-1)(g-1)}(X)$, which is an irreducible component of $V(X,r)$. The stratum $S_{\mathrm{oper}}$ corresponding to $V_{\mathrm{uni}}$ is the moduli space of $\mathrm{GL}(r,\mathbb{C})$-opers, called the {\rm \epsilon}mph{oper stratum}. It is shown that the stratum $S_{\mathrm{oper}}$ is closed in $M_{\mathrm{dR}}(X,r)$ {{\rm \rangle}m \mathfrak{c}}ite{BB}.
{\rm \epsilon}nd{example}
\section{Oper Stratum Conjecture}
Let $u\in V(X,r)$ be a $\mathbb{C}$-VHS, we define the following {\rm \epsilon}mph{Hodge fiber} over $u$ in $\mathbb{M}_{\mathrm{Hod}}(X,r)$ as
$$
Y_u :={{\rm \rangle}m \lambda}eft\{(E, D^{{\rm \rangle}m \lambda}ambda)\in\mathbb{M}_{\mathrm{Hod}}(X,r)\ \Big|\ {{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t\to0}t{{\rm \rangle}m \mathfrak{c}}dot(E,D^{{\rm \rangle}m \lambda}ambda)=u{\rm \rangle}ight\},
$$
and define $Y_u^{{\rm \rangle}m \lambda}ambda:=Y_u{{\rm \rangle}m \mathfrak{b}}igcap{{\rm \rangle}m \mathfrak{p}}i^{-1}({{\rm \rangle}m \lambda}ambda)$, called the {\rm \epsilon}mph{${{\rm \rangle}m \lambda}ambda$-Hodge fiber}. It's clear $Y_u^{{\rm \rangle}m \lambda}ambda\simeq Y_u^{{{\rm \rangle}m \lambda}ambda'}$ as analytic varieties whenever ${{\rm \rangle}m \lambda}ambda,{{\rm \rangle}m \lambda}ambda'\in\mathbb{C}^*$. Moreover, when $u$ is stable, then all fibers $Y_u^{{\rm \rangle}m \lambda}ambda, {{\rm \rangle}m \lambda}ambda\in\mathbb{C}$ are analytic isomorphic ({{\rm \rangle}m \mathfrak{c}}ite{C,H-H}), in particular, these isomorphisms can be realized by conformal limits ({{\rm \rangle}m \mathfrak{c}}ite{C}).
In the following, we show that each ${{\rm \rangle}m \lambda}ambda$-Hodge fiber $Y_u^{{\rm \rangle}m \lambda}ambda$ over a stable $\mathbb{C}$-VHS $u$ is of half-dimension of the moduli space $\mathbb{M}_{\mathrm{Hod}}^{{\rm \rangle}m \lambda}ambda(X,r)$. This property has been shown by several authors (cf. {{\rm \rangle}m \mathfrak{c}}ite{CS6} for ${{\rm \rangle}m \lambda}ambda=1$, and {{\rm \rangle}m \mathfrak{c}}ite{C} for ${{\rm \rangle}m \lambda}ambda=0$).
{{\rm \rangle}m \mathfrak{b}}egin{lemma}{{\rm \rangle}m \lambda}abel{halfdim}
Let $u\in V(X,r)$ be a stable $\mathbb{C}$-VHS, then the tangent space $T_{v}Y^{{\rm \rangle}m \lambda}ambda_u$ at $v\in Y^{{\rm \rangle}m \lambda}ambda_u$ is of half-dimension of $T_v\mathbb{M}_{\mathrm{Hod}}^{{\rm \rangle}m \lambda}ambda(X,r)$.
{\rm \epsilon}nd{lemma}
{{\rm \rangle}m \mathfrak{b}}egin{proof}
For our purpose, here we just show the case ${{\rm \rangle}m \lambda}ambda\neq0$.
Write $u=(\mathcal{E},\theta)=({{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^{k}E_{i},{{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^{k-1}\theta_i:E_i{\rm \rangle}ightarrow E_{i+1}\otimes K_X)$, and let $v=(E,D^{{\rm \rangle}m \lambda}ambda) \in Y_u^{{\rm \rangle}m \lambda}ambda$, namely there is a Simpson filtration $\mathcal{F}=\{F^{{\rm \rangle}m \mathfrak{b}}ullet\}: E=F^0\supset F^1\supset{{\rm \rangle}m \mathfrak{c}}dots F^k=0$ on the flat bundle $(E,{{\rm \rangle}m \lambda}ambda^{-1}D^{{\rm \rangle}m \lambda}ambda)$ such that $(\mathrm{Gr}_{\mathcal{F}}(E),\mathrm{Gr}_{\mathcal{F}}({{\rm \rangle}m \lambda}ambda^{-1}D^{{\rm \rangle}m \lambda}ambda))=u$. The Simpson filtration $\{F^{{\rm \rangle}m \mathfrak{b}}ullet\}$ produces a filtration $\widetilde{\mathcal{F}}$ on ${\rm End}(E)$ by
$$
F^p({\rm End}(E))=\{{\rm va}rphi\in {\rm End}(E): {\rm va}rphi: F^q{\rm \rangle}ightarrow F^{p+q}\textrm{ for any }q\},
$$
hence the graded objects are given by
$\mathfrak{E}^p:=\mathrm{Gr}_{\widetilde{\mathcal{F}}}^p({\rm End}(E))={{\rm \rangle}m \mathfrak{b}}igoplus_{i=1}^{k}{{\rm \rangle}m \mathfrak{H}}om(E_i,E_{i+p})$.
Then there is an filtration on ${\rm End}(E)\otimes {{\rm \rangle}m \mathcal{O}}mega^1_X$ by
$$
F^p({\rm End}(E)\otimes {{\rm \rangle}m \mathcal{O}}mega^1_X)=F^{p-1}({\rm End}(E))\otimes{{\rm \rangle}m \mathcal{O}}mega^1_X,
$$
which induces a filtration on the hypercohomology $\mathbb{H}^i({{\rm \rangle}m \mathcal{O}}mega_X^{{\rm \rangle}m \mathfrak{b}}ullet({\rm End}(E)),{{\rm \rangle}m \lambda}ambda^{-1}D^{{{\rm \rangle}m \lambda}ambda})$.
By Lemma 7.1 in {{\rm \rangle}m \mathfrak{c}}ite{CS6}, we have
{{\rm \rangle}m \mathfrak{b}}egin{align*}
T_{v}Y^{{\rm \rangle}m \lambda}ambda_u\simeq&F^1(\mathbb{H}^1({{\rm \rangle}m \mathcal{O}}mega_X^{{\rm \rangle}m \mathfrak{b}}ullet({\rm End}(E)),{{\rm \rangle}m \lambda}ambda^{-1}D^{{{\rm \rangle}m \lambda}ambda}))\\
\simeq&{{\rm \rangle}m \mathfrak{b}}igoplus_{p= 1}^{k-1}\mathbb{H}^1(Gr^p_{\widetilde{\mathcal{F}}}({\rm End}(E))\xrightarrow{\mathrm{Gr}_{\widetilde{\mathcal{F}}}({{\rm \rangle}m \lambda}ambda^{-1}D^{{{\rm \rangle}m \lambda}ambda})}\mathrm{Gr}^{p-1}_{\widetilde{\mathcal{F}}}({\rm End}(E))\otimes K_X)\\
\simeq&{{\rm \rangle}m \lambda}eft\{({{\rm \rangle}m \mathfrak{a}}lpha,{{\rm \rangle}m \mathfrak{b}}eta)\in{{\rm \rangle}m \mathcal{O}}mega^{0,1}_X({{\rm \rangle}m \mathfrak{b}}igoplus_{p=1}^{k-1}\mathfrak{E}^p)\oplus{{\rm \rangle}m \mathcal{O}}mega^{1,0}_X({{\rm \rangle}m \mathfrak{b}}igoplus_{p=0}^{k-1}\mathfrak{E}^p) :{{\rm \rangle}m \mathfrak{b}}ar{{\rm \rangle}m \mathfrak{p}}artial_{\mathcal{E}}{{\rm \rangle}m \mathfrak{b}}eta+[\theta,{{\rm \rangle}m \mathfrak{a}}lpha]
={{\rm \rangle}m \mathfrak{p}}artial_{\mathcal{E},h}{{\rm \rangle}m \mathfrak{a}}lpha+[\theta^\dagger_h,{{\rm \rangle}m \mathfrak{b}}eta]=0{\rm \rangle}ight\},
{\rm \epsilon}nd{align*}
where $h$ is the pluri-harmonic metric on $(\mathcal{E},\theta)$.
The dimension of the last space can be calculated by Riemann-Roch formula as done in Lemma 3.6 of {{\rm \rangle}m \mathfrak{c}}ite{C}.
{\rm \epsilon}nd{proof}
The following properties of $V^{\overrightarrow{r},\overrightarrow{d}}$ are crucial in the proof of our main theorem.
{{\rm \rangle}m \mathfrak{b}}egin{lemma} [{{{\rm \rangle}m \mathfrak{c}}ite[Theorem 4.1]{BGG}}, {{{\rm \rangle}m \mathfrak{c}}ite[Theorem 3.8 (iv)]{A}}]{{\rm \rangle}m \lambda}abel{irred}
Given type $\overrightarrow{r}=(r_1,{{\rm \rangle}m \mathfrak{c}}dots,r_l),\overrightarrow{d}=(d_1,{{\rm \rangle}m \mathfrak{c}}dots,d_{l})$, we have
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item if $V^{\overrightarrow{r},\overrightarrow{d}}$ is not empty, then it is irreducible;
\item if $V^{\overrightarrow{r},\overrightarrow{d}}$ consists of stable $\mathbb{C}$-VHSs, then its dimension is given by
$$
\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}=(g-1)\sum_{i=1}^lr_i(r_i+r_{i+1})+\sum_{i=1}^lr_i(d_{i+1}-d_{i-1})+1,
$$
where one assigns $r_{l+1}=d_0=d_{l+1}=0$.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{lemma}
Applying Lemma {\rm \rangle}ef{irred}, we can obtain the following positivity of codimension of $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}$.
{{\rm \rangle}m \mathfrak{b}}egin{proposition}{{\rm \rangle}m \lambda}abel{ff}
Let $V^{\overrightarrow{r},\overrightarrow{d}}_\mathrm{s}\subset V^{\overrightarrow{r},\overrightarrow{d}}$ be the subset consisting of stable $\mathbb{C}$-VHSs, and let $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}=V^{\overrightarrow{r},\overrightarrow{d}}{{\rm \rangle}m \mathfrak{b}}ackslash V^{\overrightarrow{r},\overrightarrow{d}}_\mathrm{s}$. Assume $V^{\overrightarrow{r},\overrightarrow{d}}_\mathrm{s}$ and $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}$ are both non-empty, then
then the codimension of $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}$ in $V^{\overrightarrow{r},\overrightarrow{d}}$ is positive.
{\rm \epsilon}nd{proposition}
{{\rm \rangle}m \mathfrak{b}}egin{proof}
Given a partition $\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}_{a=1}^k$ with
{{\rm \rangle}m \mathfrak{b}}egin{align*}
\sum_{a=1}^k\overrightarrow{r}^{(a)}&=\overrightarrow{r},\quad \sum_{a=1}^k\overrightarrow{d}^{(a)}=\overrightarrow{d},\\
|\overrightarrow{d}^{(a)}|&=0, \quad a=1,{{\rm \rangle}m \mathfrak{c}}dots, k,
{\rm \epsilon}nd{align*}
one denotes $V^{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}_{\mathrm{ss}}=V^{\{\overrightarrow{r}^{(1)},\overrightarrow{d}^{(1)}\}}_{\mathrm{ss}}\times {{\rm \rangle}m \mathfrak{c}}dots \times V^{\{\overrightarrow{r}^{(k)},\overrightarrow{d}^{(k)}\}}_{\mathrm{ss}}$, and defines an injective map
$$
f_{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}: V^{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}_{\mathrm{ss}}{{\rm \rangle}m \lambda}ongrightarrow V^{\overrightarrow{r},\overrightarrow{d}}
$$
by taking direct sum of each component on the left hand side. Since each point in $V^{\overrightarrow{r},\overrightarrow{d}}$ represents a polystable $\mathbb{C}$-VHS, we have
$$
\dim_\mathbb{C}V_{\mathrm{ss}}^{\overrightarrow{r},\overrightarrow{d}}=\dim_\mathbb{C}{{\rm \rangle}m \mathfrak{b}}igcup_{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}\mathrm{Im}(f_{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}})
=\max_{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}\dim_\mathbb{C}V^{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}_{\mathrm{ss}}.
$$
Since $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}$ is non-empty and our chains are indecomposable, the set $\{i:r_i>1\}$ is non-empty. We only care about the lower bound of $\mathrm{codim}_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}$, then it is clear that we can take $\overrightarrow{r}=\overrightarrow{r^{{\rm \rangle}m \mathfrak{p}}rime}+\overrightarrow{r^{{{\rm \rangle}m \mathfrak{p}}rime{{\rm \rangle}m \mathfrak{p}}rime}},\overrightarrow{d}=\overrightarrow{d^{{\rm \rangle}m \mathfrak{p}}rime}+\overrightarrow{d^{{{\rm \rangle}m \mathfrak{p}}rime{{\rm \rangle}m \mathfrak{p}}rime}}$ with $\overrightarrow{r^{{\rm \rangle}m \mathfrak{p}}rime}=(r_1,{{\rm \rangle}m \mathfrak{c}}dots,r_{m-1},r_m-1,r_{m+1},{{\rm \rangle}m \mathfrak{c}}dots,r_l), \overrightarrow{r^{{{\rm \rangle}m \mathfrak{p}}rime{{\rm \rangle}m \mathfrak{p}}rime}}=(0,{{\rm \rangle}m \mathfrak{c}}dots,0,1,0,{{\rm \rangle}m \mathfrak{c}}dots,0),\overrightarrow{d^{{\rm \rangle}m \mathfrak{p}}rime}=\overrightarrow{d},\overrightarrow{d^{{{\rm \rangle}m \mathfrak{p}}rime{{\rm \rangle}m \mathfrak{p}}rime}}=0$. From the dimension formula in Lemma {\rm \rangle}ef{irred} (2) it follows that
{{\rm \rangle}m \mathfrak{b}}egin{align*}
\mathrm{codim}_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}&=\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}-\dim_\mathbb{C}{{\rm \rangle}m \mathfrak{b}}igg(\max_{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}\dim_\mathbb{C}V^{\{\overrightarrow{r}^{(a)},\overrightarrow{d}^{(a)}\}}_{\mathrm{ss}}{{\rm \rangle}m \mathfrak{b}}igg)\\
&=(g-1)(2r_m+r_{m+1}+r_{m-1})+d_{m+1}-d_{m-1}+1.
{\rm \epsilon}nd{align*}
By Proposition {\rm \rangle}ef{11}, we have
{{\rm \rangle}m \mathfrak{b}}egin{align*}
d_{m-1}-d_m&{{\rm \rangle}m \lambda}eq(2g-2)\min
\{r_{m-1},r_m\},\\
d_{m}-d_{m+1}&{{\rm \rangle}m \lambda}eq(2g-2)\min
\{r_{m},r_{m+1}\},
{\rm \epsilon}nd{align*}
then we arrive at
$$\mathrm{codim}_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}\geq{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
(g-1)(r_{m+1}+r_{m-1}-2r_m)+1 , & \hbox{$r_{m-1}\geq r_m, r_{m+1}\geq r_m$;} \\
(g-1)(2r_m-r_{m+1}-r_{m-1})+1 , & \hbox{$r_{m-1}{{\rm \rangle}m \lambda}eq r_m, r_{m+1}{{\rm \rangle}m \lambda}eq r_m$;}\\
(g-1)(r_{m+1}-r_{m-1})+1 , & \hbox{$ r_{m+1}\geq r_m\geq r_{m-1}$;} \\
(g-1)(r_{m-1}-r_{m+1})+1 , & \hbox{$ r_{m+1}{{\rm \rangle}m \lambda}eq r_m{{\rm \rangle}m \lambda}eq r_{m-1}$,}
{\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
which implies $\mathrm{codim}_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{ss}}\geq1$.
{\rm \epsilon}nd{proof}
Now we give the lower and upper bounds of the dimension of non-empty irreducible components $V^{\overrightarrow{r},\overrightarrow{d}}$, and discuss when these bounds reach.
{{\rm \rangle}m \mathfrak{b}}egin{theorem} {{\rm \rangle}m \lambda}abel{m}
If $V^{\overrightarrow{r},\overrightarrow{d}}$ is not empty, then we have the inequalities
$$
g{{\rm \rangle}m \lambda}eq\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}{{\rm \rangle}m \lambda}eq r^2(g-1)+1.
$$
In particular, the equality on the left hand side holds only when
$$
\overrightarrow{r}=(1,{{\rm \rangle}m \mathfrak{c}}dots,1),\overrightarrow{d}=((r-1)(g-1),(r-3)(g-1),{{\rm \rangle}m \mathfrak{c}}dots,(-r+1)(g-1)),
$$
and the equality on the right hand side holds only when $\overrightarrow{r}=(r), \overrightarrow{d}=(0)$.
{\rm \epsilon}nd{theorem}
{{\rm \rangle}m \mathfrak{b}}egin{proof}
Firstly we show the inequality $\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}\geq g$, and the equality holds if and only if
$
\overrightarrow{r}=(1,{{\rm \rangle}m \mathfrak{c}}dots,1),\overrightarrow{d}=((r-1)(g-1),(r-3)(g-1),{{\rm \rangle}m \mathfrak{c}}dots,(-r+1)(g-1)),
$.
Consider the case $\overrightarrow{r}=(1,{{\rm \rangle}m \mathfrak{c}}dots,1)$, under which, the $\mathbb{C}$-VHSs lying in $V^{\overrightarrow{r},\overrightarrow{d}}$ are stable. By above half-dimension property (Lemma {\rm \rangle}ef{halfdim}), we have {{\rm \rangle}m \mathfrak{b}}egin{align*}
\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}&=(g-1)(2r-1)+\sum_{i=1}^{r-1}(d_{i+1}-d_i)+1\\
&\geq(g-1)(2r-1)-(r-1)(2g-2)+1=g,
{\rm \epsilon}nd{align*}
where we have used Proposition {\rm \rangle}ef{11} for the second inequality. In particular, the equality holds if and only if $\overrightarrow{d}=((r-1)(g-1),(r-3)(g-1),{{\rm \rangle}m \mathfrak{c}}dots,(-r+1)(g-1))$.
For the case when there exists some $r_i>1$, since stability is an open condition, one can consider our problem at the level of stack. Let $\mathcal{V}^{\overrightarrow{r},\overrightarrow{d}}$ be the corresponding moduli stack, and let $\mathcal{V}^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ be the substack consisting of stable objects. It follows from $\mathcal{V}^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ being a $\mathbb{G}_m$-gerbe over its coarse moduli space $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ {{\rm \rangle}m \mathfrak{c}}ite{PHS} that $\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}\geq\dim_\mathbb{C}\mathcal{V}'+1$, where $\mathcal{V}'$ is a substack of $\mathcal{V}^{\overrightarrow{r},\overrightarrow{d}}$ containing $\mathrm{Bun}^{r_{i_0},d_{i_0}}(X)$ for some $r_{i_0}>1, d_{i_0}$. Here $\mathrm{Bun}^{1,d_{i_0}}(X)\times\mathrm{Bun}^{1,d_{i_1}}(X)$ for some $d_{i_0}, d_{i_1}$, where $\mathrm{Bun}^{r_i,d_i}(X)$ denotes the moduli stack of vector bundles of rank $r_i$ and degree $d_i$ over $X$. Therefore, we have
$$
\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}\geq\max{{\rm \rangle}m \mathfrak{b}}ig\{r_{i_0}^2(g-1)+1,2g-1{{\rm \rangle}m \mathfrak{b}}ig\}>g.
$$
Now we show the inequality $\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}{{\rm \rangle}m \lambda}eq r^2(g-1)+1$, and the equality holds only when $\overrightarrow{r}=(r),\overrightarrow{d}=(0)$.
Note that all $V^{\overrightarrow{r},\overrightarrow{d}}$ are contained in the nilpotent cone, so by Lemma {\rm \rangle}ef{irred}, each $V^{\overrightarrow{r},\overrightarrow{d}}$ is contained in certain irreducible component of the nilpotent cone. It is known that each such irreducible component is a Lagrangian submanifold of $\mathbb{M}_{\mathrm{Dol}}(X,r)$, this is due to Laumon {{\rm \rangle}m \mathfrak{c}}ite{Lau}, Faltings {{\rm \rangle}m \mathfrak{c}}ite{Fal}, Beilinson--Drinfeld {{\rm \rangle}m \mathfrak{c}}ite{BB2}, and Ginzburg {{\rm \rangle}m \mathfrak{c}}ite{Gin}. So the inequality $\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}{{\rm \rangle}m \lambda}eq r^2(g-1)+1$ holds. On the other hand, by Lemma 11.9 of {{\rm \rangle}m \mathfrak{c}}ite{CSaa}, the dimension of $V^{\overrightarrow{r},\overrightarrow{d}}$ is strictly less than $r^2(g-1)+1$, the dimension of the nilpotent cone, if $V^{\overrightarrow{r},\overrightarrow{d}}$ contains $\mathbb{C}$-VHSs with non-zero Higgs fields. Therefore, the equality holds only when $V^{\overrightarrow{r},\overrightarrow{d}}=V^{(r),(0)}=i(\mathbb{U}(X,r))$, where $\mathbb{U}(X,r)$ is the moduli space of semistable vector bundles over $X$ of rank $r$ and degree $0$, and $i: \mathbb{U}(X,r)\hookrightarrow\mathbb{M}_{\mathrm{Dol}}(X,r)$ is the natural embedding.
{\rm \epsilon}nd{proof}
{{\rm \rangle}m \mathfrak{b}}egin{example}
Here we give an explicit description of $V^{\overrightarrow{r},\overrightarrow{d}}$ for the cases $|\overrightarrow{r}|=\sum_{i=1}^lr_i=3, 4$.
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item When $|\overrightarrow{r}|=\sum_{i=1}^lr_i=3$, then $V^{\overrightarrow{r},\overrightarrow{d}}$ can be described as the following:
Case I: $\overrightarrow{r}=(3),\overrightarrow{d}=(0)$. In this case, $V^{\overrightarrow{r},\overrightarrow{d}}=i(\mathbb{U}(X,3))$, which is known to be irreducible of dimension $9g-8$.
Case II: $\overrightarrow{r}=(1,1,1),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$. We have the isomorphism {{\rm \rangle}m \mathfrak{c}}ite{P1,G}
{{\rm \rangle}m \mathfrak{b}}egin{align*}
V^{\overrightarrow{r},\overrightarrow{d}}&\simeq \mathrm{Jac}^{d_1}(X)\times S^{d_2-d_1+2g-2}X\times S^{-d_1-2d_2+2g-2}X\\
(E_1,E_2,E_3;\theta_1,\theta_2)&{\rm \hspace{1mm} \rightarrow \hspace{1mm}}sto(E_1,\mathrm{ div}(\theta_1),\mathrm{div}(\theta_2)),
{\rm \epsilon}nd{align*}
where $\mathrm{Jac}^{d_1}(X)$ is the moduli space of line bundles over $X$ of degree $d_1$, $S^pX$ denotes the space of effective divisors of degree $p$ in $X$ which is isomorphic to $X^p/S_p$ for the symmetry group $S_p$, and $\mathrm{ div}(\theta_i)$ stands for the effective divisor of the morphism $\theta_i$.
Case III: $\overrightarrow{r}=(1,2),\overrightarrow{d}=(d_1,-d_1)$. If $0<d_1<g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a $\mathbb{P}^N$-fibration over $\mathrm{Jac}^{-2d_1+2g-2}(X)\times \mathrm{Jac}^{d_1-2g+2}(X)$ for $N=-3d_1+5g-6$ {{\rm \rangle}m \mathfrak{c}}ite{B,P1}. If $d_1=g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and $V^{\overrightarrow{r},\overrightarrow{d}}\simeq \mathrm{Jac}^{0}(X)\times\mathrm{Jac}^{-g+1}(X)$ {{\rm \rangle}m \mathfrak{c}}ite{B}.
Case IV: $\overrightarrow{r}=(2,1),\overrightarrow{d}=(d_1,-d_1)$. If $0<d_1<g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a $\mathbb{P}^N$-fibration over $\mathrm{Jac}^{2d_1-2(2g-2)}(X)\times \mathrm{Jac}^{-d_1}(X)$ {{\rm \rangle}m \mathfrak{c}}ite{B,P1}. If $d_1=g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and $V^{\overrightarrow{r},\overrightarrow{d}}\simeq \mathrm{Jac}^{-2g+2}(X)\times\mathrm{Jac}^{-g+1}(X)$ {{\rm \rangle}m \mathfrak{c}}ite{B}.
Therefore, we have the dimension formula
$$
\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}
={{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
9g-8, & \hbox{$\overrightarrow{r}=(3), \overrightarrow{d}=(0) $;}\\
5g-4-2d_1-d_2, & \hbox{$\overrightarrow{r}=(1,1,1),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$;} \\
7g-6-3d_1& \hbox{$\overrightarrow{r}=(1,2)$ or $(2,1)$,$\overrightarrow{d}=(d_1,-d_1)$, $0<d_1<g-1$;}\\
2g, & \hbox{$\overrightarrow{r}=(1,2)$ or $(2,1)$,$\overrightarrow{d}=(g-1,-g+1)$}.
{\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
Then it follows from Proposition {\rm \rangle}ef{11} that we have the inequalities
$$0<2d_1+d_2{{\rm \rangle}m \lambda}eq{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
4g-4, & \hbox{$\overrightarrow{r}=(1,1,1),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$;} \\
g-1, & \hbox{$\overrightarrow{r}=(1,2)$ or $(2,1)$,$\overrightarrow{d}=(d_1,d_2=-d_1)$,}
{\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
which lead to the inequalities on $\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}$ as
$$
{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
\qquad \qquad\ \dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}=9g-8, \ \ & \hbox{$\overrightarrow{r}=(3),\overrightarrow{d}=(0) $;}\\
\quad\quad\ g{{\rm \rangle}m \lambda}eq \dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}<5g-4, \ \ & \hbox{$\overrightarrow{r}=(1,1,1),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$;} \\
4g-3{{\rm \rangle}m \lambda}eq \dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}<7g-6, \ \ & \hbox{$\overrightarrow{r}=(1,2)$ or $(2,1)$,$\overrightarrow{d}=(d_1,-d_1)$.}
{\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
\item Similarly when $|\overrightarrow{r}|=\sum_{i=1}^lr_i=4$, then $V^{\overrightarrow{r},\overrightarrow{d}}$ can be described as the following:
Case I: $\overrightarrow{r}=(4),\overrightarrow{d}=(0)$. In this case, $V^{\overrightarrow{r},\overrightarrow{d}}=i(\mathbb{U}(X,4))$, which is known to be irreducible of dimension $16g-15$.
Case II: $\overrightarrow{r}=(1,1,1,1),\overrightarrow{d}=(d_1,d_2,d_3,d_4=-d_1-d_2-d_3)$. We have the isomorphism
{{\rm \rangle}m \mathfrak{b}}egin{align*}
V^{\overrightarrow{r},\overrightarrow{d}}&\simeq \mathrm{Jac}^{d_1}(X)\times S^{d_2-d_1+2g-2}X\times S^{d_3-d_2+2g-2}X\times S^{-d_1-d_2-2d_3+2g-2}X\\
(E_1,E_2,E_3,E_4;\theta_1,\theta_2,\theta_3)&{\rm \hspace{1mm} \rightarrow \hspace{1mm}}sto(\mathcal{E}_1,\mathrm{ div}(\theta_1),\mathrm{div}(\theta_2),\mathrm{div}(\theta_3)).
{\rm \epsilon}nd{align*}
Case III: $\overrightarrow{r}=(1,3),\overrightarrow{d}=(d_1,-d_1)$. If $0<d_1<g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a $\mathbb{P}^N$-fibration over $U(X,2,-2d_1+2g-2)\times\mathrm{ Jac}^{d_1-2g+2}(X)$ for $N=-4d_1+8g-9$ {{\rm \rangle}m \mathfrak{c}}ite{B}, where $U(X,r,d)$ denotes the moduli space of stable vector bundles over $X$ of rank $r$ and degree $d$. If $d_1=g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and
$V^{\overrightarrow{r},\overrightarrow{d}}\simeq \mathbb{U}(X,2)\times\mathrm{ Jac}^{-g+1}(X)$ {{\rm \rangle}m \mathfrak{c}}ite{B}.
Case IV: $\overrightarrow{r}=(3,1),\overrightarrow{d}=(d_1,-d_1)$. If $0<d_1<g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a $\mathbb{P}^N$-fibration over $U(X,2,2d_1-6g+6)\times \mathrm{Jac}^{-d_1}(X)$ {{\rm \rangle}m \mathfrak{c}}ite{B}. If $d_1=g-1$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and
$V^{\overrightarrow{r},\overrightarrow{d}}\simeq \mathbb{U}(X,2,-4g+4)\times \mathrm{Jac}^{-g+1}(X)$ {{\rm \rangle}m \mathfrak{c}}ite{B}, where $\mathbb{U}(X,r,d)$ denotes the moduli space of semistable vector bundles over $X$ of rank $r$ and degree $d$.
Case V: $\overrightarrow{r}=(2,2),\overrightarrow{d}=(d_1,-d_1)$. In this case, $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is birationally equivalent to a $\mathbb{P}^{N'}$-fibration over $U(X,2,d_1-4g+4)\times S^{-2d_1+4g-4}X$ for $N'=-2d_1+4g-4$ {{\rm \rangle}m \mathfrak{c}}ite{B}.
Case VI: $\overrightarrow{r}=(1,1,2),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$. If $d_1-d_2{{\rm \rangle}m \lambda}eq 2g-2, 2d_2+d_1<2g-2$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a $\mathbb{P}^{N''}$-fibration over $\mathrm{ Jac}^{d_2-2g+2}(X)\times \mathrm{ Jac}^{-d_1-2d_2+2g-2}(X)\times S^{d_2-d_1+2g-2}X$ for $N''=-d_1-3d_2+5g-6$ {{\rm \rangle}m \mathfrak{c}}ite{A}. If $d_1-d_2{{\rm \rangle}m \lambda}eq 2g-2, 2d_2+d_1=2g-2$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and $V^{\overrightarrow{r},\overrightarrow{d}}\simeq \mathrm{ Jac}^{d_2-2g+2}(X)\times \mathrm{ Jac}^{0}(X)\times S^{d_2-d_1+2g-2}X$.
Case VII: $\overrightarrow{r}=(2,1,1),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$. If $d_1-d_2<2g-2, 2d_2+d_1{{\rm \rangle}m \lambda}eq 2g-2$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a smooth irreducible variety of dimension $9g-8-2d_1$ {{\rm \rangle}m \mathfrak{c}}ite{A}. If $d_1-d_2=2g-2, 2d_2+d_1{{\rm \rangle}m \lambda}eq 2g-2$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and $V^{\overrightarrow{r},\overrightarrow{d}}\simeq\mathrm{ Jac}^0(X)\times \mathrm{ Jac}^{d_1-4g+4}(X)\times S^{-3d_1+6g-6}(X)$.
Case VIII: $\overrightarrow{r}=(1,2,1),\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$. If $d_1-d_2{{\rm \rangle}m \lambda}eq 2g-2, 2d_2+d_1{{\rm \rangle}m \lambda}eq 2g-2, \overrightarrow{d}\neq (2g-2,0,-2g+2)$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is non-empty, and it is birationally equivalent to a $\mathbb{P}^{N'''}$-fibration over $\mathrm{ Jac}^{d_1-4g+4}(X)\times \mathrm{ Jac}^{d_2-d_1}(X)\times S^{-2d_1-d_2+4g-4}X$ for $ N^{{{\rm \rangle}m \mathfrak{p}}rime{{\rm \rangle}m \mathfrak{p}}rime{{\rm \rangle}m \mathfrak{p}}rime}=-2d_1-d_2+4g-5$ {{\rm \rangle}m \mathfrak{c}}ite{A}. If $ \overrightarrow{d}=(2g-2,0,-2g+2)$, then $V^{\overrightarrow{r},\overrightarrow{d}}_{\mathrm{s}}$ is empty, and $V^{\overrightarrow{r},\overrightarrow{d}}\simeq \mathrm{ Jac}^{-2g+2}(X)\times \mathrm{ Jac}^{-g+1}(X)$.
Therefore, the dimension of each irreducible component is given by
$$
\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}=
{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
16g-15, & \hbox{$\overrightarrow{r}=(4),\overrightarrow{d}=(0) $;}\\
7g-6-2d_1-d_2-d_3, & \hbox{$\overrightarrow{r}=(1,1,1,1),\overrightarrow{d}=(d_1,d_2,d_3,-d_1-d_2-d_3)$;} \\
13g-12-4d_1, & \hbox{$\overrightarrow{r}=(1,3)$ or $(3,1)$,$\overrightarrow{d}=(d_1,-d_1),d_1\neq g-1$;}\\
5g-3, & \hbox{$\overrightarrow{r}=(1,3)$ or $(3,1)$,$\overrightarrow{d}=(g-1,-g+1)$;}\\
12g-11-4d_1, & \hbox{$\overrightarrow{r}=(2,2),\overrightarrow{d}=(d_1,-d_1)$;}\\
9g-8-2d_1-2d_2, & \hbox{$\overrightarrow{r}=(1,1,2)$,$\overrightarrow{d}=(d_1,d_2,-d_1-d_2),2d_2+d_1\neq 2g-2$;}\\
5g-3-{{\rm \rangle}m \mathfrak{f}}rac{3}{2}d_1, & \hbox{$\overrightarrow{r}=(1,1,2)$,$\overrightarrow{d}=(d_1,g-1-{{\rm \rangle}m \mathfrak{f}}rac{d_1}{2},-g+1-{{\rm \rangle}m \mathfrak{f}}rac{d_1}{2})$;}\\
9g-8-2d_1, & \hbox{$\overrightarrow{r}=(2,1,1)$,$\overrightarrow{d}=(d_1,d_2,-d_1-d_2),d_1-d_2\neq 2g-2$;}\\
8g-6-3d_1, & \hbox{$\overrightarrow{r}=(2,1,1)$,$\overrightarrow{d}=(d_1,-2g+2+d_1,2g-2-2d_1)$;}\\
10g-9-4d_1-2d_2, & \hbox{$\overrightarrow{r}=(1,2,1)$, $\overrightarrow{d}=(d_1,d_2,-d_1-d_2)\neq(2g-2,0,-2g+2)$;} \\ 2g, & \hbox{$\overrightarrow{r}=(1,2,1)$, $\overrightarrow{d}=(2g-2,0,-2g+2)$.}
{\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
Then it again follows from Proposition {\rm \rangle}ef{11} that we have the inequalities
{{\rm \rangle}m \mathfrak{b}}egin{align*}
0&<2d_1+d_2+d_3{{\rm \rangle}m \lambda}eq{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
6g-6, & \hbox{$\overrightarrow{r}=(1,1,1,1),\overrightarrow{d}=(d_1,d_2,d_3,-d_1-d_2-d_3)$;} \\
g-1, & \hbox{$\overrightarrow{r}=(1,3)$ or $(3,1)$,$\overrightarrow{d}=(d_1,d_2=-d_1)$,$d_3=0$;}\\
2g-2, & \hbox{$\overrightarrow{r}=(2,2)$,$\overrightarrow{d}=(d_1,d_2=-d_1)$,$d_3=0$;}\\
2g-2, & \hbox{$\overrightarrow{r}=(2,1,1)$,$\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$;}
{\rm \epsilon}nd{array}
{\rm \rangle}ight. \\
0&<d_1+d_2{{\rm \rangle}m \lambda}eq2g-2, \overrightarrow{r}=(1,1,2),\overrightarrow{d}=(d_1,d_2,-d_1-d_2); \\
0&<2d_1+d_2{{\rm \rangle}m \lambda}eq 4g-4, \overrightarrow{r}=(1,2,1), \overrightarrow{d}=(d_1,d_2,-d_1-d_2).
{\rm \epsilon}nd{align*}
Hence, we arrive at
$$
{{\rm \rangle}m \lambda}eft\{
{{\rm \rangle}m \mathfrak{b}}egin{array}{ll}
\qquad\qquad \dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}=16g-15, \ \ & \hbox{$\overrightarrow{r}=(4),\overrightarrow{d}=(0) $;}\\
\quad\quad\ g{{\rm \rangle}m \lambda}eq\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}<7g-6, \ \ & \hbox{$\overrightarrow{r}=(1,1,1,1),\overrightarrow{d}=(d_1,d_2,d_3,-d_1-d_2-d_3)$;} \\
5g-3{{\rm \rangle}m \lambda}eq\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}<13g-12, \ \ & \hbox{$\overrightarrow{r}=(1,3)$ or $(3,1)$,$\overrightarrow{d}=(d_1,-d_1)$;}\\
5g-4{{\rm \rangle}m \lambda}eq\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}<12g-11, \ \ & \hbox{$\overrightarrow{r}=(2,2),\overrightarrow{d}=(d_1,-d_1)$;}\\
\quad\ \ 2g{{\rm \rangle}m \lambda}eq\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}{{\rm \rangle}m \lambda}eq9g-8, \ \ & \hbox{$\overrightarrow{r}=(2,1,1)$ or $(1,1,2)$,$\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$;}\\
2g-1{{\rm \rangle}m \lambda}eq\dim_\mathbb{C}V^{\overrightarrow{r},\overrightarrow{d}}<10g-9, \ \ & \hbox{$\overrightarrow{r}=(1,2,1)$, $\overrightarrow{d}=(d_1,d_2,-d_1-d_2)$.} {\rm \epsilon}nd{array}
{\rm \rangle}ight.
$$
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{example}
{{\rm \rangle}m \mathfrak{b}}igskip
Finally, we complete the proof of our main result.
{{\rm \rangle}m \mathfrak{b}}egin{corollary} {{\rm \rangle}m \lambda}abel{mainthm}
For the oper stratification $M_{\mathrm{dR}}={{\rm \rangle}m \mathfrak{c}}oprod_{{\rm \rangle}m \mathfrak{a}}lpha S_{{\rm \rangle}m \mathfrak{a}}lpha$, we have
{{\rm \rangle}m \mathfrak{b}}egin{enumerate}
\item the open dense stratum $N(X,r)$ consisting of irreducible flat bundles such that the underlying vector bundles are stable is the unique maximal stratum with dimension $2r^2(g-1)+2$,
\item the closed oper stratum $S_{\mathrm{oper}}$ is the unique minimal stratum with dimension $r^2(g-1)+g+1$.
{\rm \epsilon}nd{enumerate}
{\rm \epsilon}nd{corollary}
{{\rm \rangle}m \mathfrak{b}}egin{proof}
We have seen that $S_{\mathrm{oper}}$ is a fibration over $V_{\mathrm{uni}}$ with fibers as Lagrangian submanifolds of $M_{\mathrm{dR}}(X,r)$, so the dimension is calculated as follows
{{\rm \rangle}m \mathfrak{b}}egin{align*}
\dim_\mathbb{C}S_{\mathrm{oper}}&=\dim_\mathbb{C}\mathrm{Jac}^{(r-1)(g-1)}(X)+{{\rm \rangle}m \mathfrak{f}}rac{1}{2}\dim_\mathbb{C}M_{\mathrm{dR}}(X,r)\\
&=r^2(g-1)+g+1.
{\rm \epsilon}nd{align*}
It follows from Theorem {\rm \rangle}ef{limit} that if $u$ is a stable $\mathbb{C}$-VHS, then $Y_u^1\subset M_{\mathrm{dR}}(X,r)$, and if $v\in M_{\mathrm{dR}}(X,r)$, then ${{\rm \rangle}m \lambda}im{{\rm \rangle}m \lambda}imits_{t\to0}t{{\rm \rangle}m \mathfrak{c}}dot v$ is a stable $\mathbb{C}$-VHS. For a general stratum $S_{{\rm \rangle}m \mathfrak{a}}lpha$, the dimension is given by $\dim_{\mathbb{C}}V_{{\rm \rangle}m \mathfrak{a}}lpha+{{\rm \rangle}m \mathfrak{f}}rac{1}{2}\dim_\mathbb{C}M_{\mathrm{dR}}(X,r)$, which reaches the maximal or minimum value only when the dimension of $V_{{\rm \rangle}m \mathfrak{a}}lpha$ reaches the maximal or minimum value. Note that in two extreme cases when $(\overrightarrow{r},\overrightarrow{d})=((1,{{\rm \rangle}m \mathfrak{c}}dots,1),((r-1)(g-1),(r-3)(g-1),{{\rm \rangle}m \mathfrak{c}}dots,(-r+1)(g-1))$ and $(\overrightarrow{r},\overrightarrow{d})=((r),(0))$, $V^{\overrightarrow{r},\overrightarrow{d}}$ are both the connected components of $V(X,r)$, then by Lemma {\rm \rangle}ef{irred} and Theorem {\rm \rangle}ef{m} the conclusion follows.
{\rm \epsilon}nd{proof}
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{{\rm \rangle}m \mathfrak{b}}ibitem{P1} O. Garc\'{i}a-Prada, P. Gothen, V. Mu\~{n}oz, Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, {\rm \epsilon}mph{Memoirs Amer. Math. Soc.}, \textbf{187} (2007), viii+80 pp.
{{\rm \rangle}m \mathfrak{b}}ibitem{PH} O. Garc\'{i}a-Prada, J. Heinloth, The $y$-genus of the moduli space of $PGL_n$-Higgs bundles on a curve (for degree coprime to $n$), {\rm \epsilon}mph{Duke Math. J.}, \textbf{162} (2013), 2731-2749.
{{\rm \rangle}m \mathfrak{b}}ibitem{PHS} O. Garc\'{i}a-Prada, J. Heinloth, A. Schmitt, On the motives of moduli of chains and Higgs bundles, {\rm \epsilon}mph{J. Euro. Math. Soc.}, \textbf{12} (2014), 2617-2668.
{{\rm \rangle}m \mathfrak{b}}ibitem{Gin} V. Ginzburg, The global nilpotent variety is Lagrangian, {\rm \epsilon}mph{Duke Math. J.}, \textbf{109} (2001), 511-519.
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{{\rm \rangle}m \mathfrak{b}}ibitem{GZ} P. Gothen, R. Z\'u\~niga-Rojas, Stratifications on the moduli space of Higgs bundles, {\rm \epsilon}mph{Portugal. Math.}, \textbf{74} (2017), 127-148.
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{{\rm \rangle}m \mathfrak{b}}ibitem{CS} C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, {\rm \epsilon}mph{Inst. Hautes \'Etudes Sci. Publ. Math.}, \textbf{79} (1994), 47-129.
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{{\rm \rangle}m \mathfrak{b}}ibitem{CS4} C. Simpson, The Hodge filtration on nonabelian cohomology. In {\rm \epsilon}mph{Algebraic geometry-Santa Cruz 1995}, 217-281.
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{\rm \epsilon}nd{document}
|
\begin{document}
\setcounter{section}{0}
\renewcommand{\arabic{section}.\arabic{equation}}{\arabic{section}.\arabic{equation}}
\renewcommand{\bf}{\bf}
\title{f On gauge transformations of B\"acklund type and higher
order nonlinear Schr\"odinger equations}
\begin{abstract}
We introduce a new, more general type of nonlinear gauge transformation in
nonrelativistic quantum mechanics that
involves derivatives of the wave function and belongs
to the class of B\"acklund transformations. These transformations satisfy
certain reasonable, previously proposed requirements for gauge transformations.
Their application to the Schr\"odinger equation
results in higher order partial differential equations.
As an example,
we derive a general family of 6th-order
nonlinear Schr\"odinger equations, closed under our nonlinear gauge
group. We also introduce a new gauge invariant current
${\bf \sigma}=\rho {\bf \nabla}\triangle \ln \rho $, where $\rho=\bar\psi \psi$. We
derive gauge invariant quantities, and characterize the subclass of the 6th-order equations
that is gauge equivalent to the free
Schr\"odinger equation. We relate our development to nonlinear equations
studied by Doebner and Goldin, and by Puszkarz.
\end{abstract}
{\bf PACS}: 11.30N Nonlinear symmetries, 03.65 Quantum mechanics,
11.15 Gauge field theories \\
\section{Introduction}
The notion of nonlinear gauge transformation, introduced in quantum mechanics by
Doebner and Goldin,
extends the usual group of unitary gauge transformations.$^{1-3}$ The resulting
nonlinear transformations act on a parameterized family of nonlinear Schr\"odinger equations
(NLSEs) that includes the linear Schr\"odinger equation as a special case. They are called gauge
transformations because they leave invariant the outcomes of all physical measurements.
In this paper we extend the notion of gauge transformation further to include
transformations that depend explicitly on derivatives of the wave function.
The result is a group of transformations of B\"acklund type.$^4$
As described in earlier work,$^3$ a (nonlinear) gauge transformation is
implemented by a transformation $\psi' = {\mathcal{N}}[\psi]$, assumed to satisfy the following conditions:
\vspace{
amount}
\begin{itemize}
\item 1. The {\it principle of gauge-independence of positional
measurements:} Invariance is required of all quantities describing outcomes of positional measurements,
including {\it sequences} of measurements performed successively at different times. In particular,
$\rho({\bf x},t)=|\psi({\bf x},t)|^2$ should be invariant under $\mathcal{N}$ for the single-particle
wave function $\psi$.
\item 2. {\it Strict locality:} If $\psi$ is a single-particle function, the value
of $\psi'$ at $({\bf x},t)$ is assumed to depend
only on the value of ${\bf x}$, the value of $t$, and the value of $\psi$ at $({\bf x},t)$.
\item 3. A {\it separation condition:} If $\psi^{(N)}$ is a wave function describing a set of $N$
noninteracting particles (i.e., a product state), then $\psi^{(N)'}$ is well defined as
the product of gauge transformed single particle wave functions. This condition ensures that
gauge transformations extend to the whole $N$-particle hierarchy of wave functions in a way that
subsystems that are uncorrelated remain so in the gauge-transformed theory.
\end{itemize}
Here we modify the condition of strict locality, allowing $\psi'({\bf x},t)$ to depend not
only on the values of $\psi({\bf x},t)$, ${\bf x}$, and $t$,
but also on finitely many spatial derivatives
of $\psi$ evaluated at $({\bf x}, t)$. Thus our transformations are local, in that $\psi'({\bf x},t)$
does not depend on space-time points any distance from $({\bf x}, t)$, but they are no longer ``strictly"
local, since derivative terms are allowed. We shall call this property {\it weak locality}.
One motivation for introducing this generalization
is to explore the relation between the resulting nonlinear gauge
generalization of
the Schr\"odinger equation
and the equations proposed by Puszkarz.$^5$
The condition that our set of transformations forms a group (i.e., that it is closed under composition
and includes all inverse transformations) while the number of derivatives of $\psi$ remains bounded,
imposes an additional restriction. This {\it group property} is automatically satisfied in the
strictly local theory, but
here it requires explicit discussion. Thus, we shall add it to the conditions already mentioned.
We then call the transformations that obey the
following four conditions {\it weakly local gauge transformations}:
1. the principle of gauge-independence of positional measurements;
$2'$. weak locality;
3. the separation condition; and
4. the group property.
In Sec. 2 of this paper, we first consider a general class of nonlinear, single particle
Schr\"odinger equations
that are equivalent to the free Schr\"odinger equation
under the assumption that condition 1 is satisfied. Using the other three conditions,
we obtain a particularly simple form for weakly local gauge transformations.
Following the method of ``gauge generalization,"$^3$ we then derive a general family of 6th-order
nonlinear Schr\"odinger equations, closed under our nonlinear gauge
group, which are not all equivalent to the free 2nd-order Schr\"odinger
equation. In Sec. 3 we construct a complete set of
gauge invariant quantities. As particular cases, we use these to characterize the subclass
of the 6th-order equations that are gauge equivalent to the
Schr\"odinger equation, and those equivalent to the wider class of nonlinear equations
studied by Doebner and Goldin.
We further relate our development to the nonlinear equations proposed by Puszkarz
based on additional quantum currents that involve higher derivatives of $\psi$.
\section{Gauge Transformations and NLSEs}
\setcounter{equation}{0}
Consider the transformation
\begin{equation}
\psi'({\bf x},t)\,=\,e^{i\varphi}\psi({\bf x},t),
\end{equation}
where $\varphi$ is a real-valued functional that depends on $\psi, {\bf x}$, and $t$.
By this we mean that $\varphi$ can depend explicitly on $\psi$, $\bar \psi$, derivatives
of $\psi$ and $\bar \psi$ of arbitrary order, integrals or integral transforms
of $\psi$ and $\bar \psi$, etc., as well as directly on ${\bf x}$
and $t$. Eq. (2.1)
preserves the probability density
$\rho({\bf x},t)=\bar\psi({\bf x},t)\psi({\bf x},t)$, as required by the first condition in Sec. 1,
but if nonlocal it does not generally respect sequences of positional measurements.
The following then describes the general class of NLSEs that are
equivalent via (2.1) to the free Schr\"odinger equation:
if $\psi'$
satisfies
\begin{equation}
i\frac{\partial\psi'}{\partial t} + \frac{\hbar}{2m} \triangle \psi'\,=\,
i\frac{\partial\psi'}{\partial t} - \nu_1' \triangle \psi'=0,
\end{equation}
then $\psi$ satisfies the NLSE
\begin{equation}
i\frac{\partial\psi}{\partial t} -\nu_1' \triangle \psi + iI[\psi, {\bf x}, t] \psi +
R[\psi, {\bf x}, t] \psi=0,
\end{equation}
where
\begin{equation}
R[\psi, {\bf x}, t]\,=\,\frac{\partial\varphi}{\partial
t}-2\nu_1'(\frac{{\bf \nabla} \varphi\cdot\hat{ \bf j}}
{\rho}+\frac{1}{2}({\bf \nabla}\varphi)^2)
\end{equation}
and
\begin{equation}
I[\psi, {\bf x}, t] \,=\, \nu_1' (\triangle \varphi+\frac{{\bf \nabla}
\varphi\cdot{\bf \nabla}\rho}{\rho})\,=\,
\nu_1'[\frac{1}{\rho}({\bf \nabla}\cdot(\rho{\bf \nabla}\varphi)],
\end{equation}
with
\begin{equation}
\hat{ {\bf j}}=\frac{m}{\hbar}{\bf j}=\frac{1}{2i}[\bar\psi{\bf\nabla}\psi-({\bf \nabla}\bar\psi)\psi].
\end{equation}
The verification is by direct substitution of (2.1) into (2.2).
As was shown by Doebner and Goldin$^1$, a general form for strictly local
gauge transformations (that satisfy all the initial
requirements discussed in Sec. 1) corresponds to the choice
\begin{equation}
\varphi=\frac{1}{2}\gamma(t)\ln\rho + [\Lambda(t) -1]S
+ \theta({\bf x},t), \quad \Lambda \neq 0 ,
\end{equation}
where $\psi\,=\, \sqrt{\rho}\,e^{iS}$. For simplicity, we
consider $\theta({\bf x},t) \equiv 0$.
The family of NLSEs with arbitrary coefficients that directly generalizes (2.3)
and is invariant (as a family) under gauge transformations (2.1)
with $\varphi$ as in (2.7),
then has the form$^1$
\begin{equation}
i\frac{\partial\psi}{\partial t}\,=\,\{i\sum_{j=1}^{2}\nu_j(t)R_j+\sum_{j=1}^{5}\mu_j(t)R_j\}\psi,
\end{equation}
where
\begin{equation}
R_1=\frac{{\bf \nabla}\cdot \hat {\bf j}}{\rho}, \quad
R_2=\frac{\triangle \rho}{\rho}, \quad R_3=\frac{{\hat {\bf
j}}^2}{\rho^2}, \quad
R_4=\frac{{\hat {\bf j}} \cdot {\bf \nabla}\rho}{\rho^2}, \quad R_5=\frac{{({\bf \nabla} \rho)}^2}{\rho^2}.
\end{equation}
In obtaining (2.8), one uses the identity $\triangle\psi/\psi\,=\,iR_1 + \frac{1}{2}R_2
- R_3 - \frac{1}{4}R_5$.
Invariance of the family (2.8) under (2.1) and (2.7) means that if $\psi$
satisfies an equation in this family with coefficients
$\nu_j$ and $\mu_j$, then $\psi'$ satisfies another equation in the family
with coefficients $\nu_j'$ and $\mu_j'$; thus our choice of the primed
coefficient $\nu_1'$ in
writing Eq. (2.2).
Now the class of nonlinear gauge transformations in quantum mechanics can be essentially extended if
we replace strict locality by weak locality, thus allowing the gauge functional
$\varphi$ to depend on derivatives of $\psi$.
Under this assumption the gauge transformation is no longer simply a point
transformation; it is a {\it B\"acklund transformation}.$^4$
Here we consider gauge transformations of B\"acklund type that form a
group, satisfying the physically motivated
requirements discussed in Sec. 1, with strict locality replaced by weak locality.
We observe that if $\varphi$ is permitted to depend on derivatives of $S$ as well as
derivatives of $\rho$, then the set of gauge transformations in general does not respect the
group property. However, if the derivatives of $S$ are excluded from $\varphi$, then
the transformations do respect this property.
One way to see this
is to write nonlinear gauge transformations as they act on logarithmic
coordinates $T$ and $S$, with $\ln \psi = T + iS$ (so that $T = \frac{1}{2} \ln \rho)$,
omitting for simplicity the explicit ${\bf x}$ and $t$ dependence:
\begin{equation}
\mathbf{}\left(\begin{array}{c}S'\\T'\end{array}\right)=
\mathbf{}\left(\begin{array}{cc}L&G\\0&1\end{array}\right)
\mathbf{}\left(\begin{array}{c}S\\T\end{array}\right),
\end{equation}
where $L$ is a linear or nonlinear functional of $S$ and its derivatives,
and $G$ is a linear or nonlinear functional of $T$ and its derivatives.
In the strictly local case, we have $L[S] = \Lambda S$ and $F[T] = \gamma \,T$.
If we perform two transformations (2.10) successively, $T'' = T' = T$
and $S'' = L_2[L_1[S] + G_1[T]\,] + G_2[T]$. Then derivatives present
in the form of $G$ never act successively, so that their order
does not increase; but derivatives in the form of $L$ do act successively.
Thus the group property, with the condition that the number of derivatives
of $\psi$ remains bounded, rules out derivative terms in $L$---but not in $G$.
Now a simple gauge transformation that is no longer strictly local,
but satisfies the four requirements discussed in Sec. 1, has the form (2.1) with
\begin{equation}
\varphi\,=\,\frac{1}{2}\gamma\ln\rho + (\Lambda -1) S+\eta\triangle\ln\rho\,=\,
\frac{1}{2}\gamma\ln\rho+(\Lambda -1) S +\eta(R_2-R_5),
\end{equation}
where $\eta$ is a real parameter that, like $\gamma$ and $\Lambda$,
can in principle depend on $t$.
This corresponds to the choice $G[T] = \gamma\,T + \eta\triangle T$ in (2.10).
Thus we have a group of nonlinear gauge transformations
modeled on three (in general time-dependent) parameters,
obeying the group law
\begin{equation}
{\mathcal N}_{(\gamma_2,\Lambda_2,\eta_2)} \circ {\mathcal N}_{(\gamma_1,\Lambda_1,\eta_1)}
= {\mathcal N}_{(\gamma_2 + \Lambda_2\gamma_1, \Lambda_2\Lambda_1, \eta_2 + \Lambda_2\eta_1)}.
\end{equation}
But we note further that $G[T]$ need not be linear in $T$. Indeed,
while the linear term $\triangle \ln \rho \,=\, R_2-R_5$ satisfies the
separation condition, its nonlinear parts $R_2$ and $R_5$ do so separately!
Considering a two-particle product wave function $\psi^{(2)}({\bf x_1}, \ {\bf x_2}, \ t)\,=\,
\psi_1({\bf x_1}, t)\psi_2({\bf x_2}, t)$, and defining
$\rho^{(2)}=\overline{\psi^{(2)}}\psi^{(2)}$,
$\rho_1=\bar\psi_1\psi_1$, and $\rho_2=\bar\psi_2\psi_2$, we have
$$
R^{(2)}_2[\psi^{(2)}]= \frac{\triangle^{(2)}\rho^{(2)}}{\rho^{(2)}}=
\frac{\triangle^{(2)} (\rho_1\rho_2)}{\rho_1\rho_2}= \frac{\triangle_1\rho_1}
{\rho_1} \frac{\triangle_2 \rho_2}{\rho_2}= R_2[\psi_1]R_2[\psi_2],
$$
where $\triangle^{(2)} =\triangle_1 + \triangle_2$. Similarly for $R_5$:
$$
R^{(2)}_5 [\psi^{(2)}]=\frac{[{\bf
\nabla}^{(2)}\rho^{(2)}]^2}{{(\rho^{(2)})}^2}=
\frac{[({\bf \nabla}_1,
{\bf \nabla}_2)\rho_1\rho_2]^2}{(\rho_1\rho_2)^2}=R_5[\psi_1]R_5[\psi_2].
$$
Thus a further generalization of (2.11) that gives weakly local
nonlinear gauge transformations is to
allow the derivative terms to enter with different coefficients:
\begin{equation}
\varphi\,=\,\frac{1}{2}\gamma\ln\rho + (\Lambda -1) S +
\eta_1 R_2 + \eta_2 R_5.
\end{equation}
Let us next write the gauge generalized family of NLSEs
derived from (2.11).
Beginning with the standard, free Schr\"odinger equation in the form
\begin{equation}
i\frac{\partial \psi'}{\partial t}\,=\,-\frac{\hbar}{2m}[iR_1' +(\frac{1}{2}R_2' -R_3' -\frac{1}{4}R_5')] \psi',
\end{equation}
where $R'_j$ means $R_j[\psi']$,
we transform by (2.1) with $\varphi$ as in (2.11), and from (2.3)-(2.5)
we find the form of the resulting NLSEs for $\psi$. We generalize, following Ref. 3, by
allowing arbitrary coefficients for the nonlinear functionals,
maintaining the invariance of the family of NLSEs
under the nonlinear gauge group.
In this fashion, we obtain the following equations:
\begin{equation}
i\frac{\partial\psi}{\partial t}\,=\,\{i\sum_{j=1,2,6}\nu_jR_j+\sum_{j=1}^{12}\mu_jR_j\}\psi
\,=\, \{i\hat I + \hat R\}\psi,
\end{equation}
where $R_1,...,R_5$ are as in (2.9), and where the
new functionals $R_6,...,R_{12}$ are given by:
\begin{equation}
R_6=\frac{{\bf \nabla}\cdot {\bf \sigma}}{\rho},\qquad
R_7=\frac{{\hat {\bf j}} \cdot {\bf \sigma}}{\rho^2},\qquad
R_8=\frac{{\bf \sigma} \cdot {\bf \nabla}\rho}{\rho^2},
\end{equation}
$$
R_9=\frac{{ {\bf \sigma}}^2}{\rho^2}, \quad
R_{10}=\triangle R_1,\quad
R_{11}=\triangle R_2,\quad
R_{12}=\triangle R_6,
$$
with
\begin{equation}
{\bf \sigma}=\rho {\bf \nabla}\triangle \ln \rho =
\rho {\bf \nabla} (R_2-R_5).
\end{equation}
Note that the functionals $R_6,...,R_{11}$ involve no higher than fourth derivatives
of $\psi$, but the presence of the term $R_{12}$ in (2.15) makes it in general
of 6th order. If we use (2.13) in place of (2.11), we shall need
separately the new currents $\rho{\bf \nabla}R_2$ and $\rho{\bf
\nabla}R_5$. These give rise to additional nonlinear functionals in $\psi$.
Equation (2.15) still conserves the quantum probability $\bar\psi\psi$
It gives rise to the gauge invariant current
\begin{equation}
{\bf J}^{gi}=-2(\nu_1\hat{{\bf j}}+\nu_2{\bf \nabla}\rho +\nu_6{\bf \sigma})
\end{equation}
that enters the continuity equation
\begin{equation}
\frac{\partial\rho}{\partial t} =-{\bf \nabla}\cdot {\bf J}^{gi}= 2\hat I\rho.
\end{equation}
\section{Gauge transformations and invariants for the family of
6th-order NLSEs}
\setcounter{equation}{0}
Under the gauge transformations (2.1), with $\varphi$ given by (2.11)
the coefficients $\nu_j ,\ \mu_j$ of (2.15) transform as follows:
\begin{equation}
\nu_1'=\frac{\nu_1}{\Lambda},\qquad
\nu_2'=\nu_2-\frac{1}{2}\gamma\frac{\nu_1}{\Lambda},\qquad
\nu_6'=\nu_6-\frac{\eta}{\nu_1\Lambda}\qquad (\Lambda=\lambda+1);
\end{equation}
\begin{equation}
\mu_1'=\mu_1 - \frac{\gamma\nu_1}{\Lambda}, \qquad
\mu_2'=\Lambda\mu_2 -\frac{1}{2}\gamma\mu_1
+\frac{\gamma^2}{2\Lambda}\nu_1 -\gamma\nu_2, \qquad
\mu_3'=\frac{\mu_3}{\Lambda}
\end{equation}
$$
\mu_4'=\mu_4 - \frac{\gamma \mu_3}{\Lambda}, \qquad
\mu_5'=\Lambda\mu_5 -\frac{1}{2}\gamma\mu_4
+\frac{\gamma^2}{4\Lambda}\mu_3 ,
$$
$$
\mu_6'=\Lambda\mu_6 -\gamma\nu_6
-\eta\mu_1 + \frac{\eta \gamma}{\Lambda}\nu_1, \qquad
\mu_7'=\mu_7 - \frac{2\eta \mu_3}{\Lambda}
$$
$$
\mu_8'=\Lambda\mu_8 -\eta \mu_4 - \frac{1}{2}\gamma\mu_7 + \frac{\gamma\eta \mu_3}{\Lambda},\qquad
\mu_9'=\Lambda\mu_9 -\eta \mu_7 + \frac{\eta^2 \mu_3}{\Lambda},\qquad
\mu_{10}'=\mu_{10} - \frac{2\eta \nu_1}{\Lambda},
$$
$$
\mu_{11}'=\Lambda\mu_{11}- 2\eta \nu_2 -\frac{1}{2}\gamma\mu_{10} + \frac{\gamma\eta \nu_1}{\Lambda},\qquad
\mu_{12}'=\Lambda\mu_{12}- 2\eta \nu_6 -\eta \mu_{10}+ \frac{2\eta^2 \nu_1}{\Lambda}.
$$
Note that as expected, $\eta$ does not enter
the transformation laws for $\nu_1$, $\nu_2$, or $\mu_1, ..., \mu_5$, which are the same as in Refs. 1-3.
Note also that if we begin with $\mu_{12} = 0$, then $\eta \not= 0$
leads to $\mu_{12}' \not= 0$; thus we cannot have an invariant
family of 4th-order partial differential equations for these
transformations.
We now write functionally independent gauge invariants $\tau_j \
(j=1,2,...,12)$ as follows:
\begin{equation}
\tau_1= \nu_2-\frac{\mu_1}{2},\,\,\,
\tau_2=\nu_1\mu_2-\mu_1\nu_2,\,\,\,
\tau_3=\frac{\mu_3}{\nu_1},\,\,\,
\tau_4=\mu_4-\mu_1\frac{\mu_3}{\nu_1},\,\,\,
\hat\tau_5=\mu_5\mu_3-(1/4)\mu_4^2,
\end{equation}
$$
\tau_6=\mu_6\nu_1-\mu_1\nu_6,\qquad
\tau_7=\mu_7-2\nu_6\frac{\mu_3}{\nu_1}, \qquad
\tau_8=\mu_8\nu_1-\mu_4\nu_6+\mu_6\mu_3-(1/2)\mu_7\mu_1, \qquad
$$
$$
\tau_9=\mu_9\mu_3-(1/4)\mu_7^2, \,\,\,
\tau_{10}=\mu_{10}-2\nu_6, \,\,\,
\tau_{11}=\mu_{11}\nu_1-\mu_{10}\nu_2, \,\,\,
\tau_{12}=\mu_{12}\nu_1-\nu_6^2-(1/4)\mu_{10}^2.
$$
In this list of gauge invariants, we have
included a new quantity $\hat\tau_5$ instead of the original
$\tau_5=\nu_1\mu_5-\nu_2\mu_4+ \nu_2^2(\mu_3/\nu_1)$ that was used
in Refs. 1-3,
since the expression
for $\hat\tau_5$ is simpler.
The relation between these two
gauge invariants is, of course,
wholly gauge invariant:
$\hat\tau_5=\tau_3\tau_5 +\tau_1\tau_3(\tau_4- \tau_1\tau_3)-(1/4)\tau_4^2=
\tau_3\tau_5 - (\tau_1\tau_3-\frac{1}{2}\tau_4)^2$.
It should be noted that (2.15) is invariant under Galilean transformations
\begin{equation}
{\tilde{\bf x}}={\bf x} - {\bf v}t, \ \tilde{t}=t, \
\tilde{\psi}({\tilde{\bf x}}, \tilde{t})\,=\,\psi({\bf x}, t)\,e^{\,\frac{i}{2\nu_1}({\bf x}\cdot{\bf v} +
\frac{1}{2}v^2t)}
\end{equation}
when
\begin{equation}
\frac{\mu_3}{\nu_1}=-1, \ \mu_1+\mu_4=0, \ \mu_7+\mu_{10}=0,
\end{equation}
and consequently, the gauge invariants $\tau_1,$... $\tau_{12}$ must satisfy the conditions
\begin{equation}
\tau_3=-1, \quad \tau_4=0, \quad \tau_7 + \tau_{10}=0.
\end{equation}
Under time reversal, all the coefficients $\nu_j, \mu_j$
change sign. Thus time reversal invariance requires
\begin{equation}
\tau_1 = 0, \quad \tau_4 = 0, \quad \tau_7 = 0, \quad \tau_{10} = 0.
\end{equation}
In particular, when (2.15) is the
Schr\"odinger equation, we have
\begin{equation}
\nu_1=-\frac{\hbar}{2m},\quad \mu_2=-\frac{\hbar}{4m}, \quad \mu_3=\frac{\hbar}{2m}, \quad
\mu_5=\frac{\hbar}{8m},
\end{equation}
and all other coefficients are zero. Eqs. (3.7) then give
\begin{equation}
\tau_2=\frac{\hbar^2}{8m^2}, \quad \tau_3=-1, \quad \tau_5=\frac{\hbar^2}{16m^2},
\end{equation}
with all other $\tau$'s equal to zero. For the equations
studied by Doebner and Goldin, $\tau_1, ..., \tau_5$ are arbitrary,
but $\tau_6, ..., \tau_{12}$ are zero.
Some of the equations discussed by Puszkarz,$^5$
belong to the class (2.15), when $\mu_{12}=0$.
Puszkarz's modification of the Schr\"odinger equation
is the formal extension of the equations
of Doebner and Goldin obtained by modifying
the current (2.6), adding to it any or all
of the following terms with higher derivatives:
$$\rho \triangle (\frac{{\bf j}}{\rho}), \quad
\rho {\bf \nabla} (\frac{{\bf j}\cdot {\bf \nabla}\rho}{\rho^2}), \quad
\rho {\bf \nabla} (\frac{{\bf j}^2}{\rho^2}), \quad
\rho {\bf \nabla} R_2, \quad \rho {\bf \nabla}R_5.$$
Since Puszkarz's modification directly affects only the
imaginary part
of the nonlinear functional for $i\frac{\partial \psi}{\partial t} / \psi$,
namely $(-1/2\rho){\bf \nabla}\cdot{\bf J}$ where $\bf J$ is the current
that appears in the equation of continuity,
and does not change the real part, the
resulting equation is 4th-order. Our equations are
in general 6th-order because
of the term with $R_{12}$,
which is needed in order to maintain invariance under
the nonlinear gauge group. The equations of Puszkarz with the first three currents
do not belong to any family that is closed under
a group of weakly local nonlinear gauge transformations,
since the transformations
giving rise to those currents involve derivatives of the phase $S$. His equations
with the latter two currents belong to the family obtained from (2.15) through
gauge generalization.
In short, we have obtained a natural family of 6th-order partial
differential equations invariant (as a family) for nonlinear gauge transformations
of B\"acklund type, that includes a subclass gauge equivalent to the linear
Schr\"odinger equation, a wider subclass gauge equivalent to the
equations that Doebner and Goldin studied, and another subclass that
intersects the family of equations proposed by Puszkarz. Given a particular
equation in our family, we can calculate the $12$ gauge-invariant parameters,
and from these immediately determine whether the equation is physically
equivalent to the free Schr\"odinger equation or an equation of
Doebner-Goldin type, and whether it is Galilean and/or time-reversal
invariant.
\end{document}
|
\begin{document}
\title[Classification of terminal quartics and rationality]{A classification of terminal quartic $3$-folds and applications to rationality questions}
\date{}
\author{Anne-Sophie Kaloghiros}
\address{Department of Pure Mathematics and Mathematical Statistics,
Uni\-ver\-si\-ty of Cambridge, Wilberforce Road, Cambridge CB3 0WB, Uni\-ted
Kingdom}
\email{[email protected]}
\setcounter{tocdepth}{1}
\maketitle
\begin{abstract}
This paper studies the birational geometry of terminal Gorenstein Fano $3$-folds. If $Y$ is not $\Q$-factorial, in most cases, it is possible to describe explicitly the divisor class group $\Cl Y$ by running a Minimal Model Program (MMP) on $X$, a small $\Q$-factorialisation of $Y$. In this case, the generators of $\Cl Y/ \Pic Y$ are ``topological traces " of $K$-negative extremal contractions on $X$. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano $3$-folds are rational. In particular, I give some examples of rational quartic hypersurfaces $Y_4\subset \PS^4$ with $\rk \Cl Y=2$ and show that when $\rk \Cl Y\geq 6$, $Y$ is always rational.
\end{abstract}
\tableofcontents
\section*{Introduction}
\label{sec:introduction}
Let $Y_4 \subset \PS^4$ be a quartic hypersurface in $\PS^4$ with terminal singularities.
The Grothendieck-Lefschetz theorem states that every Cartier divisor on $Y$ is the restriction of a Cartier
divisor on $\PS^4$. Recall that a variety $Y$ is $\Q$-factorial when a multiple of every Weil divisor is Cartier. There is no analogous statement for the group of Weil divisors: \cite{Kal07b} bounds the rank
of the divisor class group $\Cl Y$ of $Y_4\subset \PS^4$, but when $Y_4$ is not factorial, $\Cl Y$ remains poorly understood.
Terminal quartic hypersurfaces in $\PS^4$
are terminal Gorenstein Fano $3$-folds.
Any terminal Gorenstein Fano $3$-fold $Y$ is a $1$-parameter
flat deformation of a nonsingular Fano $3$-fold $\mathcal{Y}_{\eta}$ with
$\rho(Y)= \rho(\mathcal{Y}_{\eta})$ \cite{Nam97a}. Nonsingular Fano $3$-folds are classified in \cite{Isk77, Isk78} and
\cite{MM82, MM03}; there are $17$ deformation families with Picard rank $1$. Terminal Fano $3$-folds with
$\Q$-factorial singularities play a central role in Mori theory: they are one of the possible end products of the
Minimal Model Program (MMP) on nonsingular varieties. In \cite{T02}, Takagi develops a method to classify such $\Q$-Fano $3$-folds under some mild assumptions; his techniques rely in an essential way on the sudy of Weil non-Cartier divisors on some canonical Gorenstein Fano $3$-folds.
By definition,
$\Q$-factoriality is a global topological property: it depends on the prime divisors lying on $Y$ rather than on
the local analytic type of its singular points alone. The divisor class group of a terminal Gorenstein $3$-fold $Y$ is
torsion free \cite{Kaw88}, so that $Y$ is $\Q$-factorial precisely when it is is factorial. If $Y$ is a
terminal Gorenstein Fano $3$-fold, by Kawamata-Viehweg Vanishing $\Pic Y\simeq H^2(Y, \Z)$, and by \cite[Theorem 3.2]{NS95}$, \Cl Y\simeq H_4(Y,\Z)$. Hence $Y$ is factorial if and only if
\[
H_4(Y, \Z)\simeq H^2(Y, \Z).
\]
Birational techniques can be used to bound the rank of the divisor class group of terminal Gorenstein Fano $3$-folds \cite{Kal07b}. More precisely, under the assumption that $Y$ does not contain a plane, Weil
non-Cartier divisors on $Y$ are precisely the divisors that are contracted by the MMP on a small factorialisation
$X\to Y$. The assumption that $Y$ does not contain a plane guarantees that each step of the MMP on $X$ is a
Gorenstein weak Fano $3$-fold, and hence that this MMP can be studied explicitly.
In \cite{Kal07b}, I use numerical constraints associated to the extremal contractions of this MMP to bound the
Picard rank of $X$, or equivalently the rank of $\Cl Y$.
These methods can be refined to describe explicitly the possible extremal divisorial contractions that occur in
this MMP.
For instance, if $Y_4 \subset \PS^4$ a terminal quartic hypersurface, one can state a geometric ``motivation'' of
non-factoriality as follows:
\begin{thm}[Main Theorem]
\label{thm:1}
Let $Y_4^3 \subset \PS^4$ be a terminal Gorenstein quartic $3$-fold. Then one of the following holds:
\begin{enumerate}
\item[1.] $Y$ is factorial.
\item[2.] $Y$ contains a plane $\PS^2$.
\item[3.] $Y$ contains an anticanonically embedded del Pezzo surface of
degree $4$ and $\rk \Cl Y=2$.
\item[4.] A small factorialisation of $Y$ has a structure of Conic Bundle over $\PS^2$, $\F_0$
or $\F_2$ and $\rk \Cl Y= 2$ or $3$.
\item[5.] $Y$ contains a rational scroll $E \to C$ over a curve $C$ whose
genus and degree appear in Table~\ref{table1} (see page \pageref{table1}).
\end{enumerate}
\end{thm}
Analogous results can be stated for any terminal Gorenstein Fano $3$-fold (see Section~\ref{tables}).
When $Y$ is not factorial, either all small factorialisations of $Y$ are Mori fibre spaces or at least one of the surfaces listed in Theorem~\ref{thm:1} is a generator of
$\Cl Y/\Pic Y$; observe that these surfaces have relatively low degree (see Remark~\ref{degree}).
Further, when $Y$ does not contain a plane, the rank of $\Cl Y$ can only be large when $Y$ contains many
independent surfaces of low degree.
The divisor class group is generated by prime divisors that are ``topological traces'' on $Y$ of the extremal
divisorial contractions that occur when running the MMP on $X$.
The explicit study of the MMP on $X$ exhibits birational models of $Y$ that are small modifications of terminal
Gorenstein Fano $3$-folds with Picard rank $1$ and higher anticanonical degree. Questions on rationality of $Y$,
or at the other end of the spectrum, questions on rigidity, can be easier to answer on these models than on
$Y$ itself.
A nonsingular quartic hypersurface $Y=Y_4\subset \PS^4$ is nonrational \cite{IM}; more precisely, $Y$ is the unique Mori fibre space in its birational equivalence class (up to birational automorphisms), i.e.~ $Y_4$ is \emph{birationally rigid}. This remains true for a quartic hypersurface with ordinary double points: \cite{Me04} shows that if $Y$ has ordinary double points but is factorial, $Y$ remains birationally rigid. When the factoriality assumption is dropped, many examples of rational quartic hypersurfaces with ordinary double points are known. It is a notoriously difficult question to determine which mildly singular quartic hypersurfaces are rational. The methods of this paper yield a partial answer and a byproduct of Theorem~\ref{thm:1} is :
\begin{cor}
\label{rat}
Let $Y_4\subset \PS^4$ be a non-factorial quartic hypersurface with no worse than terminal singularities. Assume that $Y$ contains a rational scroll as in Theorem~\ref{thm:1}. Then $Y$ is rational except possibly when $\rk \Cl Y=2$ and $Y$ is the midpoint of a Sarkisov link of type $15,17,25,29,35$ or $36$ in Table~\ref{table1}.
\end{cor}
There are some rationality criteria for strict Mori fibre spaces \cite{Sh83,Al87, Sh07}, so that partial answers are known when a small $\Q$-factorialisation of $Y$ is a strict Mori fibre space.
Terminal quartic hypersurfaces that contain a plane may in some cases be studied directly.
The main open case is therefore the one addressed by Corollary~\ref{rat}. The behaviour of the $6$ cases left out by Corollary~\ref{rat} and that of factorial terminal quartic hypersurfaces is unclear. Some of these cases would be settled by the following conjecture.
\begin{con}
\label{con:rig}
A factorial quartic hypersurface $Y_4\subset \PS^4$ (resp.~ a generic complete intersection $Y_{2,3}\subset \PS^5$) with no worse than terminal singularities has a finite number of models as
Mori fibre spaces, i.e.~ the pliability of $Y$ is finite.
\end{con}
Whereas nodal quartic hypersurfaces are birationally rigid, \cite{CM04} constructs an example of a ``bi-rigid'' terminal factorial quartic hypersurface $Y_4\subset \PS^4$. A nonsingular general complete intersection $Y_{2,3}$ of a quadric and a cubic in $\PS^3$ is birationally rigid \cite{IP96}.However, \cite{ChGr} constructs a small deformation of a (factorial) rigid $Y_{2,3}\subset \PS^5$ with one ordinary double point to a bi-rigid complete intersection of the same type. This example relies on an appropriate deformation of a Sarkisov link between two complete intersections $Y_{2,3}$ (compare with case $35$ in Table~\ref{table1}). Hence, the notion of finite pliability-- rather than that of birational rigidity-- is the one that might behave well in (suitable) families.
Let $Y$ be a terminal quartic hypersurface and $X\to Y$ a small factorialisation. Assume that $X$ is not a strict Mori fibre space. Assuming Conjecture~\ref{con:rig}, $Y$ has finite pliability when $Y$ is factorial or has $\rk \Cl Y=2$ and $Y$ is one of cases $35$ or $36$; $Y$ is rational in all other cases except possibly when $Y$ has $\rk \Cl Y= 2$ and $Y$ is one of cases $15,17,25$ or $29$. In particular, the question of rationality or of finite pliability of $Y$ would be of a topological nature and would be determined by $\Cl Y$.
\subsection*{Outline}
I sketch the proofs of Theorem~\ref{thm:1} and of Corollary~\ref{rat} and present an outline of this paper.
In Section~\ref{weak-star}, I recall the definition of weak-star Fano $3$-folds introduced in \cite{Kal07b}.
If $Y$ is a terminal Gorenstein Fano $3$-fold that does not contain a plane, a small factorialisation $X \to Y$
is a weak-star Fano $3$-fold. The category of weak-star Fano $3$-folds is preserved by the birational operations
of the MMP. If $X$ is weak-star Fano, then each birational step of the MMP on $X$ is either a flop or an
extremal divisorial contraction for which the geometric description of Cutkosky-Mori \cite{Cut88} holds.
The end product of the MMP on $X$ is well understood. This approach then yields a complete description
of $\Cl Y$: $\Cl Y/\Pic Y$ is generated by the proper transforms of the exceptional divisors of the divisorial
contractions of the MMP on $X$.
If $X$ has Picard rank $2$ and if $\phi \colon X \to X'$ is a divisorial contration, then $\phi$ is one side of
a Sarkisov link with centre along $Y$. A \emph{$2$-ray game} on $X$ as in \cite{Tak89} determines a
(finite number of) possibilities for the contraction $\phi$; by construction, $\Cl Y$ is then generated by
$\mathcal{O}_Y(1)$ and by the image of $\Exc \phi$ on $Y$.
In the general case, if $\phi \colon X\to X'$ is a divisorial extremal contraction with $\Exc \phi=E$, there is
an extremal contraction $\varphi \colon Z \to Z_1$ where $Z$ is a Picard rank $2$ small modification of $Y$ that
sits under $X$ and such that $\Exc \varphi$ is the image of $E$ on $Z$. Since $Z$ is not factorial, there is a
priori no ``sensible'' Sarkisov link with centre along $Y$ involving $Z$. The proof will rely on
exhibiting a natural link.
In Section~\ref{weak-star}, I show that the explicit geometric description of extremal
divisorial contractions of \cite{Cut88} holds on non-factorial terminal Gorenstein $3$-folds so long as the exceptional divisor is Cartier.
Section~\ref{deformation} recalls results on the deformation theory of (small modifications of) terminal Gorenstein Fano $3$-folds and on the deformation of extremal contractions. Following the above notation, there
is a $1$-parameter proper flat deformation $Z\hookrightarrow\mathcal{Z}$, where
$\mathcal{Z}_{\eta}$ is a nonsingular Picard rank $2$ small modification of a terminal Gorenstein Fano $3$-fold
$\mathcal{Y}_{\eta}$ that is a $1$-parameter proper flat deformation of $Y$. The extremal contraction $\varphi$
deforms to an extremal contraction on $\mathcal{Z}_{\eta}$ that is one side of a Sarkisov link with centre along
$\mathcal{Y}_{\eta}$. Each possible Sarkisov link with centre along $\mathcal{Y}_{\eta}$ obtained by the $2$-ray
game on $\mathcal{Z}_{\eta}$ can then be specialised to the central fibre. The specialisation to the central fibre is a Sarkisov link with centre
along $Y$, one side of which is $\varphi$. The divisor class group $\Cl \mathcal{Y}_{\eta}$ is isomorphic to a rank $2$
sublattice of $\Cl Y$.
Section~\ref{motivation} presents the systems of Diophantine equations used in the $2$-ray game.
Roughly, to each extremal contraction one associates numerical constraints on some intersection numbers in
cohomology. If a Sarkisov link involves two extremal contractions $\varphi$ on $Z$ and $\alpha$ on
$\widetilde{Z}$, since $Z$ and $\widetilde{Z}$ are connected by a flop, the constraints on each side
give
rise to systems of Diophantine equations. All possible Sarkisov links with centre along $Y$ are solutions of
these systems. The solutions to all systems associated to Sarkisov links with centre along a terminal Gorenstein
Fano $3$-fold with Picard rank $1$ are listed in Section~\ref{tables}.
Section~\ref{motivation} classifies terminal quartic hypersurfaces according to their divisor class group. The
case when there is no extremal divisorial contraction on $X$, where the previous arguments do not apply, is
treated separately.
When $Y$ does not contain a plane, a consequence of the explicit xstudy of the MMP on a small factorialisation $X \to Y$ is
that the bound on the rank of $\Cl Y$ stated in \cite{Kal07b} is not optimal (see Remark~\ref{bound}).
Section~\ref{rationality} first states some easy consequences of the previous explicit study on rationality of
non-factorial Fano $3$-folds. I conjecture that a terminal quartic hypersurface $Y_4\subset \PS^4$ either has
finite pliability or is rational; and that this is determined by $\Cl Y$. I show that most non-factorial terminal quartic hypersurfaces that do not contain a plane are rational. I then study explicitly quartic hypersurfaces that contain a plane and state some partial results.
Last, Section~\ref{examples} gives some examples of non-factorial terminal Gorenstein Fano $3$-folds and gathers some observations and remarks.
\subsection*{Notations and conventions}
All varieties considered in this paper are normal, projective and
defined over
$\C$. Let $Y$ be a terminal Gorenstein Fano $3$-fold, $A_Y=-K_Y$ denotes the anticanonical divisor of $Y$.
The \emph{Fano index} of $Y$ is the maximal integer such that $A_Y=i(Y) H$ with $H$ Cartier.
As I only consider Fano $3$-folds
with terminal Gorenstein singularities in this paper, the term index always stands for Fano index.
The \emph{degree} of $Y$ is $H^3$ and the \emph{genus} of $Y$ is $g(Y)=h^0(X,A_Y)-2$.
I denote by $Y_{2g-2}$ for $2 \leq g \leq 10$ or $g=12$ (resp.~
$V_d$ for $1\leq d \leq 5$) terminal Gorenstein Fano $3$-folds of Picard rank $1$, index $1$ (resp.~ $2$) and
genus $g$ (resp.~ degree $d$).
Finally, $\F_m= \PS(\mathcal{O}_{\PS^1}\oplus \mathcal{O}_{\PS^1}(-m))$ denotes the
$m$th Segre-Hirzebruch surface.
\section{Birational Geometry of weak Fano $3$-folds}
\label{weak-star}
In this section, I recall the definition of weak-star Fano $3$-folds and some of their properties.
Most terminal Gorenstein Fano $3$-folds have a weak-star small factorialisation $X\to Y$.
If $X$ is weak-star Fano, the MMP on $X$ is well behaved, i.e.~ there is an explicit geometric description of each step, it terminates and its end product is either a terminal factorial
Fano $3$-fold or a simple Mori fibre space.
Since $\Cl Y \simeq \Cl X \simeq \Pic X$, the MMP on $X$ yields much information on the divisor class group of
$Y$. I also gather some easy results on elementary contractions on small modifications of terminal
Gorenstein Fano $3$-folds: these will be used in the following Sections.
\subsection{Weak-star Fano $3$-folds}
\begin{dfn}\label{dfn:1}
\mbox{}\begin{enumerate}
\item[1.]
A $3$-fold $Y$ with terminal Gorenstein singularities is \emph{Fano} if its anticanonical divisor
$A_Y={-}K_Y$ is ample.
\item[2.]
A $3$-fold $X$ with terminal Gorenstein singularities is \emph{weak Fano} if $A_X$ is
nef and big.
\item[3.]The morphism $X \to Y$ defined by $\vert {-}nK_X \vert$ for $n>\!\!>0$ is the
\emph{(pluri-)anticanonical map} of $X$, $R=R(X, A)$ is the \emph{anticanonical ring} of $X$ and $Y= \Proj R$
is the \emph{anticanonical model} of $X$.
\item[4.] A weak Fano $3$-fold $X$ is a \emph{weak-star Fano} if, in addition:
\begin{enumerate}
\item[(i)] $A_X$ is ample outside of a finite set of curves, so that $h\colon X \to Y$ is a small modification,
\item[(ii)] $X$ is factorial, and in particular $X$ is Gorenstein,
\item[(iii)] $X$ is inductively Gorenstein, that is $(A_X)^2\cdot S >1$ for every irreducible divisor $S$
on $X$,
\item[(iv)] $\vert A_X \vert$ is basepoint free, so that $\varphi_{\vert A \vert}$ is generically finite.
\end{enumerate}
\end{enumerate}
\end{dfn}
\begin{rem}\label{rem:1} Let $Y$ be a terminal Gorenstein Fano $3$-fold and $X$ a small factorialisation of $Y$ as in
\cite{Kaw88}. \cite[Lemma 2.3]{Kal07b} shows that when $Y$ has Picard rank $1$ and genus $g\geq 3$,
$X$ is weak-star unless $Y$ contains a plane $\PS^2$ with ${A_Y}_{\vert \PS^2}=\mathcal{O}_{\PS^2}(1)$.
\end{rem}
\begin{nt}
I call a surface $S \subset Y_{2g-2}$ a plane (resp.~ a quadric) when the image of $S$ by the anticanonical map
is a plane (resp.~ a quadric) in $\PS^{g+1}$, that is when $(A_Y)^2\cdot S=1$ (resp.~ $2$).
\end{nt}
\begin{thm}
\label{thm:3}\cite[Theorem 3.2, Lemma 3.3]{Kal07b}
The category of weak-star Fano $3$-folds is preserved by the
birational operations of the MMP.
More precisely, if $X := X_0$ is a weak-star Fano $3$-fold whose anticanonical
model $Y_0$ has Picard rank $1$, there is a sequence of extremal contractions:
\begin{equation*}
\xymatrix{
X_0 \ar@{-->}[r]^-{\varphi_0} \ar[d]& X_1 \ar@{-->}[r]^-{\varphi_1}\ar[d] & \cdots
& X_{n-1}\ar@{-->}[r]^-{\varphi_{n-1}} \ar[d]& X_n \ar[d] \\
Y_0 & Y_1 & \cdots & Y_{n-1} & Y_n
}
\end{equation*}
where for each $i$, $X_i$ is a weak-star Fano $3$-fold, $Y_i$ is
its anticanonical model, and each
$\varphi_i$ is either a divisorial contraction or a flop. The Picard rank of
$Y_i$, $\rho(Y_i)$, is equal to $1$ for all $i$. The final $3$-fold
$X_n$ is either a Fano $3$-fold with $\rho(X_n)=1$ or a strict Mori fibre space.
In that latter case, $X_n$ is a del Pezzo fibration over $\PS^1$ or a conic bundle over $\PS^2, \F_0$ or
$\F_2$ and $\rho(X_n)=2$ or $3$.
\end{thm}
\subsection{Elementary contractions of terminal Gorenstein $3$-folds}
Let $h\colon Z\to Y$ be a small modification of a terminal Gorenstein Fano $3$-fold $Y$ with $\rho(Y)=1$ and
$g\geq 3$.
Suppose that $\varphi \colon Z \to Z_1$ is an extremal contraction such that $E=\Exc \varphi$ is Cartier.
If $g \colon X \to Z$ is a small factorialisation, and if $\widetilde{E}=g^{\ast} E$, there is an extremal
contraction $\phi \colon X \to X_1$ that makes the diagram
\begin{eqnarray}
\label{eq:40}
\xymatrix{\widetilde{E} \subset X \ar[r]^{\phi} \ar[d]_g & X_1\ar[d]^{g_1}\\
E \subset Z \ar[d]_h \ar[r]^{\varphi} & Z_1\\
\overline{E} \subset Y &
}
\end{eqnarray}
commutative, where $\widetilde{E}=\Exc \phi$ and $g_1$ is an isomorphism in codimension $1$
(See the proof of \cite[Lemma 3.3]{Kal07b} for details).
\begin{rem}
Note that
$\overline{E}= h(E)$ is not Cartier: since $\rho(Y)=1$, $E$ would be ample if it were Cartier.
\end{rem}
Cutkosky extended Mori's geometric description of extremal contractions to terminal Gorenstein factorial
$3$-folds \cite[Theorems 4 and 5]{Cut88}. The next Lemma is an easy generalisation of his results to divisorial
contractions with Cartier exceptional divisor on terminal Gorenstein Fano $3$-folds that are not necessarily
factorial.
\begin{lem}
\label{lem:10}\cite[Lemma 3.1]{Kal07b}
Let $Z$ be a small modification of a terminal Gorenstein Fano $3$-fold $Y$. Assume that $\vert A_Z \vert$ is
basepoint free.
Denote by $\varphi \colon Z \to Z'$ an extremal
divisorial contraction with centre a curve $\Gamma$ and assume that $\Exc \varphi=E$ is a Cartier divisor.
Then $\Gamma \subset Z'$ is locally a complete intersection and has planar singularities. The contraction
$\varphi$ is locally the blow up of the ideal sheaf $\mathcal{I}_{\Gamma}$. In addition, the following relations
hold :
\begin{align}
A_Z^3&=(A_{Z'})^3-2(A_{Z'})\cdot \Gamma-2+2p_a (\Gamma) \\
A_Z^2 \cdot E&=
A_{Z'}\cdot \Gamma+2-2 p_a(\Gamma)\\
A_{Z} \cdot E^{2}&=-2+2p_a(\Gamma) \\
E^{3}&=-(A_{Z'})\cdot \Gamma +2 -2 p_a(\Gamma)
\end{align}
\end{lem}
\begin{lem}
\label{lem:7}\mbox{}
Assume that $\varphi \colon Z \to Z_1$ contracts $E$ to a point $P$, then one of the
following holds:
\begin{enumerate}
\item[E$2$:] $(E,\mathcal{O}_{E}(E))\simeq ( \PS^2,
\mathcal{O}_{\PS^2}(-1))$ and $P$ is nonsingular.
\item[E$3$:] $(E,\mathcal{O}_{E}(E))\simeq (\PS^1 \times \PS^1,
\mathcal{O}_{\PS^1 \times \PS^1}(-1,-1))$ and $P$ is an ordinary double point.
\item[E$4$:] $(E, \mathcal{O}_{E}(-E))\simeq (Q, \mathcal{O}_{Q}(-1))$, where $Q\subset \PS^3$ is an irreducible reduced
singular quadric surface, and $P$ is a cA$_{n-1}$ point.
\item[E$5$:] $(E,\mathcal{O}_{E}(E))\simeq ( \PS^2,
\mathcal{O}_{\PS^2}(-1))$, and $P$ is a point of Gorenstein
index $2$.
\end{enumerate}
\end{lem}
\begin{proof}
Diagram~\eqref{eq:40} shows that $g$ maps the centre of $\phi$ to the centre of $\varphi$; in particular $\phi$
also contracts a divisor to a point unless $\phi$ has centre along a curve $C$ such that $A_{X_1} \cdot C=0$. In
this case, by Lemma~\ref{lem:10}, $\widetilde{E}\simeq \F_2$ or $\widetilde{E}\simeq \PS^1 \times \PS^1$ and
$\varphi$ is of type E$3$ or E$4$.
I now assume that the centre of $\phi$ is a point. The divisor $E\subset Z$ is Cartier by assumption and
$A_E=(A_Z-E)_{\vert E}$ is ample: $E$ is a Gorenstein, possibly nonnormal, del Pezzo surface.
The birational morphism $g_{\vert E}\colon \widetilde{E} \to E$ induced by $g$
is an isomorphism outside a finite set of curves. Since $g$ preserves the anticanonical degree of $E$,
Cutkosky's classification \cite{Cut88} shows that this degree is $1,2$ or $4$ and that the normalisation of $E$
is a plane or a quadric.
Since $E$ is Cartier and $Z_1$ is Cohen Macaulay, the Serre criterion shows that $E$ is nonnormal if and only if
it is not regular in codimension $1$. As the centre of $g_{\vert E}$ is at worst a finite number of points,
$E$ is normal and $E \simeq \widetilde{E}$; the result follows from \cite{Cut88}.
\end{proof}
\begin{lem}
\label{lem:8}
Let $Y_4\subset \PS^4$ be a non-factorial terminal quartic hypersurface
that does not contain a plane. Let $Z\to Y$ be a small
modification such that $\rho(Z/Y)=1$. Assume that $\varphi
\colon Z \to Z_1$ is an
extremal contraction such that $E=\Exc \varphi$ is Cartier and that $E$ is
mapped to a curve $\Gamma$.
Then $Z_1$ is a terminal Gorenstein Fano $3$-fold with $\rho(Z_1)=1$
and the following relations hold:
\begin{enumerate}
\item[1.] If $i(Z_1)=1$ and $A_{Z_1}^3=2g_1-2$, then $\deg(\Gamma)=g_1-4+p_a(\Gamma)$ and $p_a(\Gamma) \leq g_1-1$,
\item[2.] If $i(Z_1)=2$ and
$A_{Z_1}^3= 8d$, then
$2\deg(\Gamma)= 4d-3+p_a(\Gamma)$ and $p_a(\Gamma)=2k+1$, for some $0\leq k\leq 2d-1 $
\item[3.] If $Z_1$ is a quadric in $\PS^4$, then $3
\deg(\Gamma)= 24+p_a(\Gamma)$ and $p_a(\Gamma)= 3k$, for some $0 \leq k \leq 9$,
\item[4.] If $Z_1= \PS^3$, then $4 \deg(\Gamma)=29+p_a(\Gamma) $
and $p_a(\Gamma)= 4k-1$, for some $0 \leq k \leq 7$.
\end{enumerate}
\end{lem}
\begin{rem}
\label{rem:35} Note that the bound obtained on the genus of $\Gamma$ is
sharper than the Castelnuovo bound when $\Gamma$ is a nonsingular curve.
\end{rem}
\begin{proof}
Since $\vert A_Z\vert = \vert \varphi^{\ast}A_{Z_1}-E\vert$ is basepoint free, $\Gamma$ is a scheme theoretic
intersection of members of $\vert A_{Z_1} \vert$, and hence $A_{Z_1}\cdot \Gamma\leq A_{Z_1}^3$. The Lemma
then follows from standard manipulation of the relations of Lemma~\ref{lem:10}.
\end{proof}
\section{Deformation theory}
\label{deformation}
This Section first recalls results on the deformation theory of terminal Gorenstein Fano $3$-folds and
of their small modifications. I then state an easy extension of results of \cite{Mo82, KM92} on
deformations of extremal contractions.
As is explained in the Introduction, if $X \to Y$ is a small
factorialisation of a terminal Gorenstein Fano $3$-fold and if $\rho(X)=2$, a \emph{$2$-ray game} on $X$ as
in \cite{Tak89} determines all possible Sarkisov links with centre along $Y$ and hence
all possible $K$-negative extremal contractions $\varphi \colon X \to X'$ and all possible generators of $\Cl Y/\Pic Y$.
In the general case, I show that a similar $2$-ray game can be played on $Z$, a small partial
factorialisation of $Y$ with $\rho(Z)=2$. This procedure is delicate because $Z$ is not factorial. However,
$Z$ can be smoothed and the $2$-ray game on the generic fibre yields Sarkisov links that specialise to
appropriate Sarkisov links involving $Z$ with centre along $Y$. All
possible generators of $\Cl Y/\Pic Y$ arise in that way.
\subsection{Deformation Theory of weak Fano $3$-folds}
\begin{dfn}
\label{kura}
Let $X$ be a projective variety.
The \emph{Kuranishi space} $\Def(X)$ of $X$ is the semi-universal
space of flat deformations of $X$. When the functor of flat deformations of $X$ is pro-representable,
the Kuranishi family $\mathcal{X}$ is the universal deformation object.
\end{dfn}
\begin{thm}\cite{Nam97a}
\label{thm:7}
Let $X$ be a small modification of a terminal Gorenstein Fano $3$-fold.
There is a $1$-parameter flat deformation of $X$
\[
\xymatrix{
X\ar[r] \ar[d]& \mathcal{X} \ar[d]\\
\{0\} \ar[r]& \Delta
}
\]
such that the generic fibre $\mathcal{X}_{\eta}$ is a nonsingular small modification of a terminal Gorenstein
Fano $3$-fold. The Picard ranks, the anticanonical
degrees and the indices of $X$ and $\mathcal{X}_{\eta}$ are equal.
\end{thm}
Let $f\colon X \to Y$ be a small modification of a terminal Gorenstein Fano $3$-fold.
Let $E$ be a Cartier divisor such that $\overline{E}=f(E)$ is not
Cartier, and denote by $Z$ the symbolic blow up of
$\overline{E}$ on $Y$, i.e.~ $Z= \Proj_Y \bigoplus_{n \geq 0}\mathcal{O}_Y(n\overline{E})$. Then $f$ naturally decomposes as:
\[
f \colon X \stackrel{h}\to Z \stackrel{g}\to Y.
\]
Note that if $\rho(X/Y)=1$, $h$ is the identity and $Z=X$. I recall some
results that relate the deformations of $X, Z$ and $Y$.
\begin{pro}\cite[11.4,11.10]{KM92}
\label{pro:1}
Let $X$ be a normal projective $3$-fold and $f \colon X \to Y$ a proper map with connected fibres such that
$R^1f_{\ast}\mathcal{O}_X=0$.
\begin{enumerate}
\item[1.]
There are natural morphisms $F$ and $\mathcal{F}$ that make the diagram
\[
\xymatrix{
\mathcal {X} \ar[r]^{\mathcal{F}} \ar[d] & \mathcal{Y} \ar[d]\\
\Def(X) \ar[r]^{F} & \Def(Y)
}
\]
commutative. In addition, $\mathcal{F}$ restricts to $f$ on $X$.
\item[2.]
Assume further that $X$ has terminal Gorenstein singularities and that
$f$ contracts a curve $C \subset X$ with $A_{X}$-trivial
components to a point $\{Q\} \in Y$.
Let $X_S \to S$ be a flat deformation of $X$ over the germ of a
complex space $0 \in S$.
Then, $f$ extends to a contraction $F_S \colon X_S \to Y_S$,and the
flop $F_S^+ \colon X_S^+ \to Y_S $ exists and commutes with any base change.
\end{enumerate}
\end{pro}
\begin{thm}\cite[12.7.3-12.7.4]{KM92}
\label{thm:2}
Let $f\colon X \to Y$ be a small factorialisation of a terminal Gorenstein $3$-fold $Y$.
Then, $F\colon \Def(X)\to \Def(Y)$ is finite and $\im [\Def(X) \to \Def(Y)]$
is closed and independent of the choice of $f$.
\end{thm}
\begin{rem}
\label{stratification}
By Proposition~\ref{pro:1}, there are maps
$\mathcal{G}$ and $\mathcal{H}$ that
restrict to $g$ and $h$ on the central fibre and that make the diagram
\[
\xymatrix{
\mathcal{X}\ar[r]^{\mathcal{H}}\ar[d] &
\mathcal{Z}\ar[r]^{\mathcal{G}}\ar[d]& \mathcal{Y}\ar[d] \\
\Def(X)\ar[r]^{H} & \Def(Z)\ar[r]^{G} & \Def(Y)
}
\]
commutative.
The Kuranishi space of $X$ thereby acquires a natural stratification by sublattices of $\Cl Y$;
by Theorem~\ref{thm:2}, there is an inclusion of
closed subspaces \[
\Def(X) \subset \Def(Z) \subset \Def(Y).
\]
As the Picard rank is constant in any $1$-parameter deformation of
$Z$, $\Def(Z) \subset \Def(Y)$ corresponds to the
locus of the Kuranishi space where the algebraic cycle representing $E$ is preserved.
Further, these inclusions are strict because a smoothing of $Y$ does
not sit under any $1$-parameter flat deformation of $Z$.
\end{rem}
\subsection{Deformation of extremal rays}
For future reference, I state a mild generalisation of the results on
deformation of extremal rays in \cite{Mo82, KM92}.
\begin{thm}[Extension of extremal contractions] \mbox{}
\label{thm:5}
Let $Z \to Y$ be a small modification of a terminal Gorenstein Fano $3$-fold $Y$.
Consider a projective flat deformation $\mathcal{Z} \to S$
of $Z$, where $S$ is a smooth affine complex curve with closed point $\{0\}$ and generic point
$\eta$.
Let $\varphi \colon Z \to Z_1$ be the contraction of an extremal ray $R
\subset Z$ and assume that if $\Exc \varphi$ is a
divisor, it is Cartier.
The contraction $\varphi$ extends to an $S$-morphism $f \colon
\mathcal{Z} \stackrel{\Phi} \to \mathcal{Z}_1$, where $\mathcal{Z}_1\to S$
is a projective $1$-parameter flat deformation of $Z_1$, and
\begin{enumerate}
\item[1.] $\Phi_{\eta}$ is the contraction of an extremal ray,
\item[2.] If $\varphi= \Phi_{0}$ contracts a subset
of $\codim \geq 2$ (resp.~ a divisor, resp.~ is a fibre space of generic
relative dimension $k$), so does $\Phi_\eta$,
\item[3.] If $\Exc \varphi$ is a Cartier divisor, in the notation of Lemma~\ref{lem:7},
either $\Phi_{\eta}$ and $\varphi$ are of the same type, or $\Phi_{\eta}$
and $\varphi$ are of types E$3$ and E$4$.
\end{enumerate}
\end{thm}
\begin{proof}
The assumption that $E$ is Cartier ensures that the proof of
\cite[Theorem 3.47]{Mo82} can be extended to this case.
See \cite{Kal07a} for a complete proof.
\end{proof}
\begin{thm}[The $2$-ray game]
\label{thm:2ray}
Let $Z \to Y$ be a small modification of a terminal Gorenstein Fano $3$-fold $Y$ with $\rho(Y)=1$ and
$\rho(Z/Y)=1$.
Assume that $\varphi \colon Z \to Z_1$ is a divisorial contraction and that $E=\Exc \varphi$ is Cartier.
There is a diagram:
\begin{eqnarray}
\label{eq:41}
\xymatrix{ \quad & Z \ar[dl]_{\varphi} \ar[dr]^{g}
\ar@{<-->}[rr]^{\Phi} &\quad
& \widetilde{Z} \ar[dr]^{\alpha} \ar[dl]_{\tilde{g}} &\quad\\
Z_1 &\quad & Y & \quad & \widetilde{Z_1}}
\end{eqnarray}
where:
\begin{enumerate}
\item[1.] $Z$ and $\widetilde{Z}$ are small modifications of $Y$ with Picard rank
$2$,
\item[3.] $\Phi$ is a composition of flops that is not an isomorphism,
\item[4.] $\alpha$ is a $K$-negative extremal contraction,
\item[5.] $Z_1$ (resp~$\widetilde{\mathcal{Z}}_1$) is one of:
\begin{enumerate}
\item[(i)] a terminal Gorenstein Fano $3$-fold with Picard rank $1$ if $\varphi$ (resp.~$\alpha$) is
birational,
\item[(ii)] $\PS^2$ if $\varphi$ (resp.~$\alpha$) is a conic bundle,
\item[(iii)] $\PS^1$ if $\varphi$ (resp.~$\alpha$) is a del Pezzo fibration.
\end{enumerate}
\end{enumerate}
If $\alpha$ is birational, then $\Exc \alpha$ is Cartier.
\end{thm}
\begin{rem}
I want to stress that since $Z$ is not factorial, such a diagram does not automatically exist, and in particular,
$\Exc \alpha$ is not necessarily Cartier when it is a divisor.
\end{rem}
\begin{proof}
By Theorem~\ref{thm:7}, there is a $1$-parameter smoothing $\mathcal{Z}\to \Delta$ of $Z$. For all
$t\in \Delta\smallsetminus \{0\}$, $\mathcal{Z}_t$ is a nonsingular small modification of a terminal Gorenstein
Fano $3$-fold with $\rho(\mathcal{Z}_t)=2$. Let $g_t \colon \mathcal{Z}_t \to \mathcal{Y}_t$ be the
anticanonical map.
Note that Proposition~\ref{pro:1} ensures that $\mathcal{Y}_t$ is a $1$-parameter flat deformation of $Y$;
in particular $\mathcal{Y}_t$ is a terminal Gorenstein Fano $3$-fold with $\rho(\mathcal{Y}_t)=1$ and
$A_{\mathcal{Y}_t}^3=A_Y^3$, .
Theorem~\ref{thm:5} shows that there is an extremal contraction $\varphi_t $ of $\mathcal{Z}_t$ that specialises
to $\varphi$ on the central fibre. As
$\mathcal{Z}_t$ is factorial, a $2$-ray game as in \cite{Tak89} yields a diagram:
\[
\xymatrix{ \quad & \mathcal{Z}_t \ar[dl]_{\varphi_t} \ar[dr]^{h_t}
\ar@{<-->}[rr]^{\Phi_t} &\quad
& \widetilde{Z}_t \ar[dr]^{\alpha_t} \ar[dl]_{\widetilde{h}_t} &\quad\\
\mathcal{Z}_{1,t} &\quad & \mathcal{Y}_t & \quad &
\widetilde{\mathcal{Z}_{1,t}},}
\]
where
\begin{enumerate}
\item[1.] $\mathcal{Z}_t$ and $\widetilde{\mathcal{Z}_t}$ are
nonsingular small modifications of $\mathcal{Y}_t$ with Picard rank
$2$,
\item[2.] $\Phi_t$ is a composition of flops that is not an isomorphism,
\item[3.] $\alpha_t$ is a $K$-negative extremal contraction,
\item[4.] $\mathcal{Z}_{1,t}$ (resp.~ $\widetilde{\mathcal{Z}_{1,t}}$) is one of:
\begin{enumerate}
\item[(i)] a terminal Gorenstein Fano $3$-fold with Picard rank $1$ if $\varphi_t$ (resp.~ $\alpha_t$) is
birational,
\item[(ii)] $\PS^2$ if $\varphi_t$ (resp.~ $\alpha_t$) is a conic bundle,
\item[(iii)] $\PS^1$ if $\varphi_t$ (resp.~ $\alpha_t$) is a del Pezzo fibration.
\end{enumerate}
\end{enumerate}
The theorem then follows from Lemma~\ref{lem:spe}.
\end{proof}
\begin{lem}[Specialisation of a $2$-ray game]
\label{lem:spe}
The elementary Sarkisov link on $\mathcal{Z}_t$, $t \neq 0$, induces
an elementary Sarkisov link on the central fibre of $\mathcal{Z} \to \Delta$.
\end{lem}
\begin{proof}
This lemma is standard, see \cite{Kal07a} for a proof; it follows from the more general
\cite[Theorem 4.1]{dFH09}.
\end{proof}
Let $Y$ be a terminal Gorenstein Fano $3$-fold with $\rho(Y)=1$. Assume that $X$, a small factorialisation of
$Y$, is weak-star Fano. Theorem~\ref{thm:3} shows that there is a sequence of contractions:
\begin{equation}
\label{eq:7}
\xymatrix{
X_0 \ar@{-->}[r]^-{\phi_0} \ar[d]& X_1 \ar@{-->}[r]^-{\phi_1}\ar[d] & \cdots
& X_{n-1}\ar@{-->}[r]^-{\phi_{n-1}} \ar[d]& X_n \ar[d] \\
Y_0 & Y_1 & \cdots & Y_{n-1} & Y_n
}
\end{equation}
I assume that at least one of the contractions $\varphi_i$ is divisorial.
Then, for a suitable small factorialisation $X_0$, $\varphi_{0}= \varphi$ is divisorial.
Let $\widetilde{E}= \Exc \phi$ and $Z_0$ be a small modification of $Y$ such that $X_0\to Y_0$ factors through
$Z_0$, $\rho(Z_0/Y_0)=1$ and such that $E$, the image of $\widetilde{E}$ on $Z_0$, is Cartier. Then there is an
extremal contraction $\varphi \colon Z_0\to Z_1=Y_1$, such that the diagram
\begin{eqnarray}
\label{eq:1}
\xymatrix{\widetilde{E} \subset X \ar[r]^{\phi} \ar[d]_g & X_1\ar[d]^{g_1}\\
E \subset Z \ar[d]_h \ar[r]^{\varphi} & Z_1\\
\overline{E} \subset Y &
}
\end{eqnarray}
commutes.
Theorem~\ref{thm:2ray} shows that $Z_0,Y_0$ and $Z_1$ fit in an elementary Sarkisov link as in \eqref{eq:41}.
To each such elementary link, one can associate systems of Diophantine equations that reflect the numerical
constraints imposed by contractions of extremal rays on intersection of classes in cohomology.
These constraints can be made explicit when $\Exc \varphi$ (and $\Exc \alpha$, if it is a divisor) is Cartier,
as is explained in Lemma~\ref{lem:10} and in Section~\ref{motivation}.
This procedure can be carried out at each divisorial step of the MMP on $X_0$.
\section{A geometric motivation of non-factoriality}
\label{motivation}
In this section, I write down explicitly systems of Diophantine equations associated to elementary Sarkisov
links as in \eqref{eq:41}. As Lemma~\ref{lem:10} shows, one can associate to each extremal contraction numerical
constraints. These systems of Diophantine equations reflect the relationships between the constraints associated
to the extremal contractions $\varphi$ and $\alpha$ on both sides of the link.
I then list all possible divisorial contractions that can occur when running the MMP on weak-star Fano $3$-folds
$X$ whose anticanonical model $Y$ have $\rho(Y)=1$. Not all links listed in this Section are geometrically realizable, I call them \emph{numerical links} in order to stress this fact.
\subsection{Systems of Diophantine equations associated to elementary links}
\label{2raygame}
In this section, I use the notation set in \eqref{eq:41}. Let $\widetilde{E}$ be the proper transform of
$E=\Exc\varphi$ on $\widetilde{Z}$, $\widetilde{E}$ is a Cartier divisor.
In what follows, I assume that $i(\widetilde{Z})= i(Z)=i(Y)=1$. This is a convenience, Remark~\ref{rem:hif}
explains how to recover the general case.
By construction, $H$ and $\widetilde{E}$
are generators of $\Pic \widetilde{Z}$.
Let $g$ denote the genus of $Y,Z$ and $\widetilde{Z}$.
Since $\Phi$ is a sequence of flops,
\begin{eqnarray}
\label{eq:17}
A_Z^2 {\cdot} E & = A_{\widetilde{Z}}^2 {\cdot} \widetilde{E} \nonumber\\
A_Z {\cdot} E^{2} & = A_{\widetilde{Z}}{\cdot} \widetilde{E}^{2}\\
\widetilde{E}^{3} & =E^{3}-e.\nonumber
\end{eqnarray}
\begin{lem}\cite{T02}
\label{lem:4}
The correction term $e$ in \eqref{eq:17} is a strictly positive integer.
\end{lem}
\begin{proof}
This is standard, I include the argument for clarity of exposition.
As $E$ is Cartier and $\varphi$-negative, for any effective curve $\gamma \in \Exc \Phi$,
$E\cdot\gamma$ is strictly positive.
Since $\widetilde{E}$ is also Cartier, $e$ is an integer.
Consider a common resolution of $Z$ and $\widetilde{Z}$:
\[
\xymatrix{
\quad & W\ar[dr]^{q}\ar[dl]_{p}& \quad \\
Z \ar@{<-->}[rr]^{\Phi} &\quad & \widetilde{Z},
}
\]
Since $Z$ and $\widetilde{Z}$ are terminal, the Negativity Lemma shows that
every $p$-exceptional divisor is also $q$-exceptional and that $p^{\ast}A_Z= q^{\ast}A_{\widetilde{Z}}$.
Then,
\[p_{\ast}^{-1}E= p^{\ast} E-R=
q^{\ast}(\widetilde{E})-R',\]
where $R$ and $R'$ are effective exceptional divisors for $p$ and $q$.
In particular:
\[
-p^{\ast}(E)=-q^{\ast}(\widetilde{E})+R'-R.
\]
By the construction of the $E$-flop, $-q^{\ast}(\widetilde{E})$ is
$p$-nef. The Negativity
Lemma shows that $R'-R$ is strictly effective because $\Phi$ is not
an isomorphism, and its pushforward $p_{\ast}(R'-R)$ is effective.
Hence,
$-p_{\ast}(R'-R)^2$ is a non-zero
effective $1$-cycle contained in the indeterminacy locus of $\Phi$, and $e=-p_{\ast}(R'-R)\cdot E>0$.
\end{proof}
I now write down numerical constraints associated to the extremal
contraction $\alpha$; this is similar to what is done in Lemma~\ref{lem:10} . These constraints and \eqref{eq:17} yield the systems of Diophantine equations that underlie the
$2$-ray game.
\subsubsection{$\alpha$ is divisorial}
Let $D$ be the exceptional divisor of $\alpha$ and $C$ its centre. Since $D$ is Cartier, there are integers $x, y$ such that:
\begin{equation}
\label{eq:21}
D=x A_{\widetilde{Z}} - y \widetilde{E}.
\end{equation}
If $\alpha$ is of type E$1$,
\begin{equation}
\label{eq:20}
A_{\widetilde{Z}}= \alpha^{\ast}(A_{\widetilde{Z}_1})-D.
\end{equation}
Since $\widetilde{Z}_1$ is Gorenstein, $\alpha(\widetilde{E})$ is Cartier because it is $\Q$-Cartier;
\eqref{eq:21} and \eqref{eq:20} show that
$y$ divides $x+1$. Note that $y$ is the index of $\widetilde{Z}_1$ and define $k$ by $x+1=yk$.
By Lemma~ \ref{lem:10},
\[
\left \{
\begin{array}{c}
(A_{\widetilde{Z}}+D)^3=(A_{\widetilde{Z}}+D)^2 A_{\widetilde{Z}}=(A_{\widetilde{Z}_1})^3\\
(A_{\widetilde{Z}}+D)^2 D=0\\
(A_{\widetilde{Z}}+D)DA_{\widetilde{Z}}=A_{\widetilde{Z}_1}\cdot C=i(\widetilde{Z}_1) \deg(C)\\
A_{\widetilde{Z}}D^2=2p_a(C)-2
\end{array}
\right .
\]
These relations and \eqref{eq:17} yield the system of equations associated to the
configuration $(\varphi, \alpha)$:
\[
\left \{
\begin{array}{c}
y^2[A_Z^3k^2-2(A_{Z_1}\cdot
\Gamma+2-2p_a(\Gamma))k+2p_a(\Gamma)-2]\nonumber
\\=A_{\widetilde{Z_1}}^3 \nonumber \\
A_Z^3k^2(yk-1)+(A_{Z_1}\cdot
\Gamma+2-2p_a(\Gamma))(2k-3k^2y)\nonumber \\+ (2p_a(\Gamma)-2)(3ky-1)+(A_{Z_1}\cdot
\Gamma-2+2p_a(\Gamma)+e)y=0 \nonumber\\
A_Z^3k(yk-1)-(A_{Z_1}\cdot
\Gamma+2-2p_a(\Gamma))(2yk-1)\nonumber \\+ (2p_a(\Gamma)-2)y=
\deg(C) \nonumber\\
A_Z^3(yk-1)^2-2(A_{Z_1}\cdot
\Gamma+2-2p_a(\Gamma))y(yk-1)\nonumber \\
+(2p_a(\Gamma)-2)y^2=2p_a(C)-2 \nonumber
\end{array}
\right .
\]
\begin{rem}
Assume that the degree of $Z$ is fixed.
Since $Z_1$ and $\widetilde{Z_1}$ are terminal Gorenstein Fano $3$-folds with Picard rank $1$, there are
finitely many possible values for ${A_{Z_1}}^3$ and ${A_{\widetilde{Z_1}}}^3$. Further, once ${A_{Z_1}}^3$ and ${A_{\widetilde{Z_1}}}^3$ are fixed, Lemma~\ref{lem:8} shows that there are only finitely many
possibilities for $(p_aC, \deg C)$. As a result, there is a finite number of Diophantine systems to consider to determine all numerical Sarkisov links $(\varphi, \alpha)$ with centre along $Z$.
\end{rem}
\subsubsection{$\alpha$ is a conic bundle}
Let $L$ be the pull back of
an ample generator of $\Pic\widetilde{ Z_1}$. Since $\rho(\widetilde{Z_1})=2$,
$\widetilde{Z_1}=\PS^2$ \cite[Lemma 3.6]{Kal07b} and $L=\alpha^{\ast}\mathcal{O}_{\PS^2}(1)$.
There are integers $x, y$ such that:
\begin{eqnarray}
\label{eq:22}
L=x A_{\widetilde{Z}} - y \widetilde{E}.
\end{eqnarray}
\begin{cla}
The integers $x$ and $y$ are positive and coprime; $y$ is
equal to $1$ or $2$.
\end{cla}
This is similar to the argument in \cite{Tak89}.
Since $E$ is fixed on $Z$,
$\widetilde{E}$ is fixed, and hence $x\geq 0$.
If $y\leq 0$, $\vert L \vert \supset \vert x
A_{\widetilde{Z}}\vert$, and $L$ is big. This contradicts
$\alpha$ being of fibering type. The integers $x, y$ are
coprime because $A_{\widetilde{Z}}$ and $\widetilde{E}$ form a
$\Z$-basis of $\Pic \widetilde{Z}$, $L$ is prime and $L$ is not an
integer multiple of either of them.
Denote by $l$ an effective nonsingular curve that is contracted by $\alpha$.
Then $A_{\widetilde{Z}}\cdot l \leq 2$, and since
$x(A_{\widetilde{Z}}\cdot l)=y\widetilde{E}\cdot
l$, the claim follows.
Let $\Delta\sim-\alpha_{\ast}(A_{\widetilde{Z}/\widetilde{Z}_1})^2$ be the discriminant curve of $\alpha$.
\[
\left \{ \begin{array}{c}
L^3=0\\
L^2 {\cdot} A_{\widetilde{Z}}=2 \\
L {\cdot} A_{\widetilde{Z}}^2 =12-\deg(\Delta)
\end{array} \right.
\]
The system of equations associated to the configuration $(\varphi, \alpha)=(E1, CB)$ writes:
\[
\left \{
\begin{array}{c}
A_Z^3x^3
-3(A_{Z_1}{\cdot} \Gamma+2-2p_a(\Gamma))x^2y
\nonumber \\+3(2p_a(\Gamma)-2)xy^2
+(A_{Z_1}{\cdot}
\Gamma-2+2p_a(\Gamma)+e)y^3=0\\
A_Z^3x^2-2(A_{Z_1}{\cdot} \Gamma+2-2p_a(\Gamma))xy\nonumber\\+(2p_a(\Gamma)-2)y^2=2\\
A_Z^3x-(A_{Z_1}{\cdot} \Gamma+2-2p_a(\Gamma))y= 12-\deg(\Delta)
\end{array}
\right .
\]
\subsubsection{$\alpha$ is a del Pezzo fibration}
Let $L$ be the pullback of an ample generator of $\Pic \widetilde{Z_1}$.
As $\widetilde{Z_1}=\PS^1$, $L=\alpha^{\ast}\mathcal{O}_{\PS^1}(1)$. Let $d$ be the degree of the generic
fibre. There are integers $x,y$ such that \eqref{eq:22} holds.
\begin{cla}
The integers $x$ and $y$ are positive and coprime; $y$ can only be
equal to $1,2$ or $3$.
\end{cla}
This is proved as in the conic bundle case.
\[
\left \{
\begin{array}{c}
L^2{\cdot} A_{\widetilde{Z}}=0\\
L^2 {\cdot} \widetilde{E}=0\\
L{\cdot}A_{\widetilde{Z}}^2=d
\end{array}
\right .
\]
The system of equations associated to $(\varphi, \alpha)$ writes:
\[
\left \{
\begin{array}{c}
A_Z^3x^2-2(A_{Z_1}{\cdot}
\Gamma+2-2p_a(\Gamma))xy+\nonumber\\(2p_a(\Gamma)-2)y^2=0 \nonumber
\\
(A_{Z_1}{\cdot} \Gamma+2-2p_a(\Gamma))x^2-2(2p_a(\Gamma)-2)xy \nonumber\\-(A_{Z_1}{\cdot}
\Gamma-2+2p_a(\Gamma)+e)y^2=0 \nonumber\\
A_Z^3x-(A_{Z_1}{\cdot} \Gamma+2-2p_a(\Gamma))y= d \nonumber
\end{array}
\right .
\]
\begin{rem}
\label{rem:hif}
\cite{Sh89} shows that when $i(Y)=4$, $Y$ is isomorphic to $\PS^3$.
If $i(Y)=2$,
both $\alpha$ and $\varphi$ are either E$2$ contractions, \'etale conic bundles or quadric bundles.
If $i(Y)=3$, then $\alpha$ and $\varphi$ are $\PS^2$-bundles over $\PS^1$.
The MMP on small modifications of higher index Fano $3$-folds is therefore very simple.
If $\widetilde{H}$ is such that $A_{\widetilde{Z}}= i(\widetilde{Z})H$, the systems written above hold
for any index after replacing $A_{\widetilde{Z}}$ by $\widetilde{H}$ in \eqref{eq:21} and \eqref{eq:22}.
\end{rem}
\begin{table}
\caption{Numerical Sarkisov Links, $g=3$}
\label{table1}
\begin{tabular}{rclclccc}
\hline
&$(\varphi, \alpha)$ &$Z_1$& $ \varphi$ & $\widetilde{Z_1}$ & $\alpha$ & $e$ & $R$\\
\hline
$1$ & E$1$-E$1$ &$X_{22}$ &$(g,d)= (0,8)$ & $X_{22}$ &
$(h,e)=(0, 8)$ & $268$ & $+$\\
$2$ & E$1$-E$1$& $X_{22}$ & $(g,d)= (1,9)$ & $V_5$ &
$(h,e)= (1,9)$ & $171$ & $+$\\
$3$ & E$1$-E$1$& $X_{22} $ & $(g,d)=(2,10)$ & $X_{22}$ &
$(h,e)= (2,10)$ & $80$& $+$\\
$4$ & E$1$-CB & $X_{22}$ & $(g,d)=(2,10)$ & $ \PS^{2}$ &
$\deg \Delta= 4$ &$92$& $+$\\
$5$& E$1$-E$1$ & $X_{22}$ & $(g,d)=(3,11)$ & $X_{12}$ &
$(h,e)=(0,3) $ & $29$& $+$\\
$6$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(0,6)$ & $X_{18}$&
$(h,e)= (0,6)$ & $144$ & $+$\\
$7$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(1,7)$ & $V_4$&
$(h,e)=(1,7)$ & $77$ & $+$\\
$8$& E$1$-E$1$ & $X_{18}$ & $(g,d)=(2,8)$& $X_{18}$ &
$(h,e)= (2,8)$ & $16$& $+$\\
$\times 9$ & E$1$-CB& $X_{18}$ & $(g,d)=(2,8)$ & $\PS^{2}$&
$\deg \Delta = 6$ & $26$ & $+$ \\
$10$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(0,5)$ & $Q $ &
$(h,e)= (3,9)$ & $103$& $+$\\
$11$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(1,6)$ & $X_{16}$ &
$(h,e)=(1,6)$ & $42$& $+$\\
$12$ & E$1$-dP& $X_{16}$ & $(g,d)=(1,6)$ & $\PS^{1}$& $k=6$ & $48$& $+$\\
$13$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(2,7)$ & $X_{2,2,2}$ &
$(h,e)= (0,1)$ & $8$ & $+$\\
$14$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(2,7)$ & $V_4$&
$(h,e)= (5,9)$ & $4$& $+$\\
$15$ & E$1$-E$1$& $X_{14}$ & $(g,d)=(0,4)$ & $X_{14}$ &
$(h,e)= (0,4)$ & $68$ & ?\\
$16$ & E$1$-E$1$& $X_{14}$ & $(g,d)=(1,5)$ & $Q$&
$(h,e)=(9, 11)$ & $24$ & $+$\\
$\bullet17$ & E$1$-E$1$& $X_{14}$ & $(g,d)=(1,5)$& $V_3$ &
$(h,e)= (1,5)$ & $25$& ?\\
$18$ & E$1$-E$1$& $X_{12}$ & $(g,d)=(0,3)$ & $X_{22}$&
$(h,e)= (3,11)$ &$29$ & $+$\\
$\bullet19$ & E$1$-E$1$& $X_{12}$ & $(g,d)=(0,3)$& $\PS^{3}$ &
$(h,e)=(7, 9)$ & $45$ & $+$\\
$20$ & E$1$-E$1$& $X_{12}$& $(g,d)=(1,4)$ & $X_{12}$ &
$(h,e)= (1,4)$ & $8$& $+$\\
$21$ & E$1$-dP& $X_{12}$ & $(g,d)=(1,4)$ & $\PS^1$& $k=4$ & $12$ & $+$\\
$\bullet22$& E$2$-E$2$& $X_{12}$ && $X_{12}$ && $30$ & $+$\\
$\times 23$& E$2$-E$1$& $X_{12}$& & $X_{10}$ &
$(h,e)= (0,2)$ & $29$& $+$\\
$\times 24$& E$2$-E$1$& $X_{12}$& & $V_{5}$ &
$(h,e)= (7,12)$ & $24$& $+$\\
$\bullet25$ & E$1$-E$1$& $X_{10}$ & $(g,d)=(0,2)$ & $X_{10}$ &
$(h,e)= (0,2)$ & $28$ & ?\\
$\times 26$ & E$1$-E$1$& $X_{10}$ & $(g,d)=(0,2)$ & $V_5$&
$(h,e)= (7,12)$ & $23$& $+$\\
$\times27$ & E$1$-E$2$& $X_{10}$ & $(g,d)=(0,2)$ & $X_{12}$&& $29$ & $+$ \\
$\bullet28$ & E$1$-E$1$& $X_{10}$ & $(g,d)=(1,3)$ & $\PS^3$&
$(h,e)= (15,11)$ & $2$ & $+$\\
$\times 29$ & E$1$-E$1$& $X_{10}$ & $(g,d)=(1,3)$& $V_2$ & $(h,e)= (1,3)$ & $3$ & ?\\
$\bullet30$ & E$1$-CB& $X_{2,2,2}$ & $(g,d)=(0,1)$ & $\PS^{2}$ & $\deg
\Delta = 7$ & $17$& \\
$\times31$ & E$1$-E$1$& $X_{2,2,2}$ & $(g,d)=(0,1)$ &$X_{16}$ &
$(h,e)=(2,7)$ & $8$ & $+$\\
$32$ & E$1$-dP& $V_2$ & $(g,d)=(1,3)$ & $\PS^1$ & $k=6$ & $48$& $+$\\
$33$& E$1$-E$1$ & $V_2$ & $(g,d)=(1,3)$ & $X_{16}$ & $(h,e)=(1,6)$ & $42$& $+$\\
$\bullet34$ & E$1$-E$1$& $V_3$ & $(g,d)=(3,6)$ & $\PS^{3}$ & $(h,e)=(3,8)$ & $65$ & $+$
\\
$\bullet35$ &E$3$-E$3$& $X_{2,3}$ & $(g,d)=(0,0)$ & $X_{2,3}$ & $(h,e)=(0,0)$ & $12$ & $?$ \\
$36$ &E$3$-E$1$& $X_{2,3}$ & $(g,d)=(0,0)$ & $V_3$ & $(h,e)=(3,6)$ & $9$ & $?$ \\
$37$ &E$3$-E$1$& $X_{2,3}$ & $(g,d)=(0,0)$ & $Q$ & $(h,e)=(12,12)$ & $8$ & $+$ \\
\hline
\end{tabular}
\end{table}
\begin{nt} Most of the notation used in Tables~\ref{table1} and \ref{table2} is self explanatory.
The column labelled $R$ gathers results from Section~\ref{rationality} on rationality. The symbol $\bullet$ indicates that the link is a known geometric constructions (e.g.~ Cases $17,19$ and $26$), see Section~\ref{examples} for examples and details. The symbol $\times$ indicates that the link is not geometrically realizable. Every numerical Sarkisov link that involves a contraction of type E$1$ with centre along a curve $\Gamma$ such that $(p_a(\Gamma), \deg \Gamma)=(0,0)$ also appears with that contraction replaced by a contraction of type E$3$ or E$4$. I do not repeat these solutions in the tables.
\end{nt}
\begin{rem}
\label{degree}
Observe that the possible generators of $\Cl Y/\Pic Y$ have relatively low degree. When $Y$ is the midpoint of a Sarkisov link of type $(\varphi, \alpha)$, one can choose as a generator of $\Cl Y/\Pic Y$ either $\Exc \varphi$ or $\Exc \alpha$ when both $\varphi$ and $\alpha$ are divisorial. When $\varphi$ (resp.~ $\alpha$) is a strict fibration, consider the pullback of $\mathcal{O}_{Z_1}(1)$ (resp.~ $\mathcal{O}_{\widetilde{Z}_1}(1)$) instead of $\Exc \varphi$ (resp.~ $\Exc \alpha$). The anticanonical degree of the generator of $\Cl Y/\Pic Y$ is then given by Lemma~\ref{lem:10} or by the systems in Section~\ref{2raygame}. For $Y\subset \PS^4$ a quartic $3$-fold, the degree of a generator of $\Cl Y/ \Pic Y$ is at most $10$.
\end{rem}
\begin{rem}{(Exclusion of Cases)}
It is known that if $\widetilde{X}\to \PS^2$ is a standard Conic Bundle whose
discriminant has degree at least $6$, then it is not rational. This shows, using Lemma~\ref{ratio}, that the (deformed numerical) Sarkisov
link $9$ in Table~\ref{table1} is not geometrically realizable.
The other excluded cases correspond to deformed numerical Sarkisov links that are not geometrically realizable \cite{Tak89, IP99}.
\end{rem}
\subsection{Numerical Sarkisov links with centre along higher degree Fano $3$-folds}
\label{tables}
The numerical Sarkisov links with centres along terminal Gorenstein Fano $3$-folds of index $1$ and genus $g\geq 3$ are listed in Table~\ref{table2}.
\begin{center}
\begin{longtable}[p]{ rclclccc}
\label{table2}\\
\caption{Numerical Sarkisov links, $g\geq4$}\\
\hline
& $(\varphi, \alpha)$ & $Z_1$ &$\varphi$ & $\widetilde{Z_1} $& $\alpha$ & $e$ & $R$
\\[2mm] \hline
\endfirsthead
\caption[]{Continued}\\
\hline
& $(\varphi, \alpha)$ & $Z_1$ &$\varphi$ & $\widetilde{Z_1} $& $\alpha$ & $e$ & $R$
\\[2mm] \hline
\endhead
\multicolumn{8}{r}{{Continued on Next Page\ldots}} \\
\endfoot
\\[2mm] \hline
\endlastfoot
\multicolumn{8}{c}{$g=4$}\\ \hline
$1$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(0,7)$ & $X_{22}$ & $(h,e)=(0,7)$ & $89$ & $+$ \\
$2$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(1,6)$ & $Q$ & $(h,e)=(1,8)$ & $48$ & $+$\\
$3$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(2,9)$ & $X_{14}$ & $(h,e)=(0,3)$ & $12$ & $+$\\
$4$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(1,6)$ & $X_{18}$ & $(h,e)=(1,6)$ & $12$ & $+$ \\
$5$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(0,5)$ & $V_5$ & $(h,e)=(2,9)$ & $47$ & $+$ \\
$6$ & E$1$-dP& $X_{18}$ & $(g,d)=(1,6)$ & $\PS^1$ & $k=6$ & $18$& $+$ \\
$7$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(0,4)$ & $X_{16}$ & $(h,e)=(0,4)$ & $32$ & $+$\\
$8$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(1,5)$ & $\PS^3$ &$(h,e)=(8,9)$ & $8$ &$+$\\
$9$ & E$1$-E$1$& $X_{14}$ & $(g,d)=(0,3)$ & $X_{22}$ & $(h,e)=(2,9)$ & $12$ & $+$\\
$10$ & E$1$-E$1$& $X_{14}$ & $(g,d)=(1,4)$ & $X_{2,2,2}$ & $(h,e)=(0,0)$ & $4$ & ?\\
$11$ & E$1$-CB& $X_{14}$ & $(g,d)=(0,3)$ & $\PS^2$ &$\deg \Delta= 5$ & $23$ &$+$ \\
$12$&E$1$-E$3$& $X_{14}$ & $(g,d)=(1,4)$ & $V_1$ & & $4$ & $?$ \\
$\bullet 13$& E$2$-E$1$& $X_{14}$& & $V_{3}$ & $(h,e)= (0,4)$ & $16$& ?\\
$\times14$& E$2$-E$1$& $X_{14}$& & $Q$ & $(h,e)= (7,10)$ & $15$& $+$\\
$\bullet15$ & E$1$-E$1$& $X_{12}$ & $(g,d)=(0,2)$ & $Q$ &$(h,e)=(7,10)$ & $14$ & $+$ \\
$\times16$ & E$1$-E$1$& $X_{12}$ & $(g,d)=(0,2)$ & $V_3$ &$(h,e)=(0,4)$ & $15$ & $+$ \\
$\bullet17$ & E$1$-E$1$& $X_{10}$ & $(g,d)=(0,1)$ & $X_{10}$ & $(h,e)=(0,1)$ & $11$& ? \\
$\times18$& E$1$-E$1$ & $X_{10}$ & $(g,d)=(0,1)$ & $V_5$ & $(h,e)=(6,11)$ & $6$ & $+$\\
$19$ & E$3$-E$1$& $X_{2,2,2}/V_1$ & $(g,d)=(0,0)$ & $X_{14}$ & $(h,e)=(1,4)$ & $4$ & ?\\
$20$ & E$3$-dP& $X_{2,2,2}/V_1$ & $(g,d)=(0,0)$ & $\PS^1$ & $k=4$ & $8$ & ? \\
$21$ & E$1$-E$1$& $V_5$ & $(g,d)=(6, 11)$ & $V_5$ & $(h,e)=(0,8)$ & $45$ & $+$ \\
$22$ & E$1$-E$1$& $V_4$ & $(g,d)=(4,8) $ & $\PS^1$ & $k=8$ & $32$ & $+$\\
$23$ & E$1$-E$1$& $V_4$ & $(g,d)=(4,8)$ & $X_{22}$ & $(h,e)=(1,8)$ & $24$ & $+$ \\
$24$ & E$1$-E$1$& $V_3$ & $(g,d)=(2,5)$ & $V_4$ & $(h,e)=(0,6)$ & $28$ & $+$\\
$\times25$ & E$1$-E$1$& $V_2$ & $(g,d)=(0,2)$ & $X_{16}$ & $(h,e)=(0,4)$ & $32$ & $+$\\ \hline
\multicolumn{8}{c}{$g=5$}\\
\hline
$1$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(0,6)$ & $X_{22}$ & $(h,e)=(0,6)$ & $36$ & $+$ \\
$\bullet2$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(1,7)$ & $\PS^3$ & $(h,e)=(1,7)$ & $14$ & $+$ \\
$3$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(1,5)$ & $X_{12}$ & $(h,e)=(0,1)$ & $3$ & $+$ \\
$4$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(0,4)$ & $Q$ & $(h,e)=(2,8)$ & $20$ & $+$ \\
$5$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(0,3)$ & $V_5$ & $(h,e)=(3,9)$ & $12$ & $+$ \\
$\times 6$& E$2$-E$1$& $X_{16}$& & $X_{14}$ & $(h,e)= (0,2)$ & $11$& $+$ \\
$\bullet7$& E$2$-E$2$ & $X_{16}$ &&$X_{16}$ && $12$ & $+$\\
$\times 8$& E$2$-E$2$ & $X_{16}$ &&$V_{2}$ && $12$ & $+$\\
$\bullet9$ & E$1$-E$1$& $X_{14}$ & $(g,d)=(0,2)$ & $X_{14}$ & $(h,e)=(0,2)$ & $10$ & ? \\
$\times 10$& E$1$-E$2$& $X_{14}$ & $(g,d)=(0,2)$ & $X_{16}$&& $11$ & $+$ \\
$\times11$& E$1$-E$2$& $X_{14}$ & $(g,d)=(0,2)$ & $V_2$ && $11$ & $+$ \\
$\times12$ & E$1$-E$1$& $X_{12}$ & $(g,d)=(0,1)$ & $X_{18}$ & $(h,e)=(1,5)$ & $3$ & $+$\\
$\bullet13$ & E$1$-dP& $X_{12}$ & $(g,d)=(0,1)$ & $\PS^1$ & $k=5$ & $8$ & $+$\\
$14$ & E$3$-CB& $X_{10}$ & $(g,d)=(0,0)$ & $\PS^2$ & $\deg \Delta= 6$ & $6$ & ?\\
$15$ & E$1$-dP& $V_3$ & $(g,d)=(1,4)$ & $\PS^1$ & $k=8$ & $24$ & $+$\\
$16$ & E$1$-dP & $Q$ & $(g,d)=(30,23)$ & $\PS^1$ & $k=2$ & $4464$ & $+$\\
\hline
\multicolumn{8}{c}{$g=6$}\\
\hline
$1$ & E$1$-E$1/2$& $X_{22}$ & $(g,d)=(1,6)$ & $X_{16}$ & $(h,e)=(0,2)$ & $2$ & $+$ \\
$2$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(0,5)$ & $V_5$ & $(h,e)=(0,7)$ & $18$& $+$ \\
$3$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(0,3)$ & $X_{18}$ & $(h,e)=(0,3)$ & $9$ & $+$\\
$\times4$& E$2$-E$1$& $X_{18}$& & $X_{22}$ & $(h,e)= (1,6)$ & $3$& $+$\\
$\bullet5$& E$2$-dP & $X_{18} $&& $\PS^1$ & $k=6$ & $9$ & $+$\\
$\times6$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(0,2)$ & $X_{22}$ & $(h,e)=(1,6)$ & $2$ & $+$\\
$\bullet7$ & E$1$-dP& $X_{16}$ & $(g,d)=(0,2)$ & $\PS^1$ & $k=6$ & $8$ & $+$ \\
$\bullet8$ & E$1$-CB& $X_{14}$ & $(g,d)=(0,1)$ & $\PS^2$ & $\deg \Delta=5$ & $6$ & $+$ \\
$9$& E$3$-E$1$& $X_{12}$ & $(g,d)=(0,0)$ & $\PS^3$ & $(h,e)=(6,8)$ & $5$& $+$ \\
$10$& E$1$-CB & $V_4$ & $(g,d)=(2,6)$ & $\PS^2$ & $\deg \Delta=2$ & $14$ & $+$ \\
$\times11$ & E$1$-dP& $V_2$ & $(g,d)=(0,1)$ & $\PS^1$ & $k=6$ & $8$ & $+$\\
$\times12$ & E$1$-E$1$& $V_2$ & $(g,d)=(0,1)$ & $X_{22}$ & $(h,e)=(1,6)$ & $2$ & $+$ \\
$13$& E$1$-dP & $Q$ & $(g,d)=(36,33)$ & $\PS^1$ & $k=2,6,8$ & $1620$ & $+$\\
\hline
\multicolumn{8}{c}{$g=7$}\\
\hline
$1$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(0,4)$ & $X_{22}$ & $(h,e)=(0,4)$ & $8$ & $+$ \\
$2$ & E$1$-CB& $X_{18}$ & $(g,d)=(0,2)$ & $\PS^2$ &$\deg \Delta=4$ & $6$ & $+$\\
$\bullet3$ & E$1$-E$1$& $X_{16}$ & $(g,d)=(0,1)$ & $\PS^3$ & $(h,e)=(3,7)$ & $5$ & $+$ \\
$4$ & E$3$-E$1$& $X_{14}$ & $(g,d)=(0,0)$ & $Q$ & $(h,e)=(4,8)$ & $4$ & $+$\\
$5$& E$1$-dP & $Q$ & $(g,d)=(1,14)$ & $\PS^1$ & $k=6$ & $2016$ & $+$\\
$6$ & E$1$-dP & $Q$ & $(g,d)=(37,38)$ & $\PS^1$ & $k=6$ & $462$ & $+$\\
\hline
\multicolumn{8}{c}{$g=8$}\\
\hline
$1$ & E$1$-CB& $X_{22}$ & $(g,d)=(0,3)$ & $\PS^2$ & $\deg \Delta=3$ & $6$ & $+$ \\
$\bullet2$& E$2$-E$1$& $X_{22}$& & $\PS^3$ & $(h,e)= (0,6)$ & $6$& $+$\\
$\bullet3$ & E$1$-E$1$& $X_{18}$ & $(g,d)=(0,1)$ & $Q$ & $(h,e)=(2,7)$ & $4$ & $+$\\
$4$& E$3$-E$1$ & $X_{16}/V_2$ & $(g,d)=(0,0)$ & $V_4$ & $(h,e)=(0,4)$ & $4$ & $+$ \\
$5$ & E$1$-dP& $V_3$ & $(g,d)=(0,2)$ & $\PS^1$ & $k=8$ & $8$ & $+$\\
$6$& E$1$-dP & $Q$ & $(g,d)=(26,30)$ & $\PS^1$ & $k=2$ & $360$ & $+$\\
\hline
\multicolumn{8}{c}{$g=9$}\\
\hline
$\bullet1$ & E$1$-E$1$& $X_{22}$ & $(g,d)=(0,2)$ & $Q$ & $(h,e)=(0,6)$ & $4$ & $+$ \\
$2$ & E$3$-E$1$& $X_{18}$ & $(g,d)=(0,0)$ & $V_{5}$ & $(h,e)=(1,6)$ & $3$ & $+$\\
\hline
\multicolumn{8}{c}{$g=10$}\\
\hline
$\bullet1$& E$1$-E$1$ & $X_{22}$ & $(g,d)=(0,1)$ & $V_5$ & $(h,e)=(0,5)$ & $3$ & $+$\\
\end{longtable}
\end{center}
\section{A classification of non-factorial terminal Gorenstein Fano $3$-folds}
\label{classification}
Let $Y_4^3 \subset \PS^4$ be a terminal non-factorial quartic $3$-fold.
Well-known examples of non-factorial quartic
$3$-folds contain planes or quadrics. Yet, a very
general determinantal quartic hypersurface $Y'$ is not factorial and it contains neither a plane nor a quadric. However, $Y'$
does contain a degree $6$ \emph{Bordigo surface}, i.e.~ a surface whose ideal is generated by the $3\times 3$ minors of the matrix defining $Y'$. In the general case, I show that $Y$ contains
some surface of relatively low degree. In other words, the degree of
the surface lying on $Y$ that breaks factoriality cannot be
arbitrarily large.
\subsection{Quartic $3$-folds}
I now prove Theorem~\ref{thm:1}.
\begin{proof}
Let $Y$ be a non-factorial terminal Gorenstein Fano $3$-fold and $X\to Y$ a small factorialisation of $X$. I assume that $Y$ does not contain a plane: $X$ is weak-star Fano by Remark~\ref{rem:1}.
We may run a MMP on $X$ as in Theorem~\ref{thm:3}.
If the MMP on $X$ involves at least one divisorial contraction, then up to a different choice of factorialisation $X\to Y$, we may assume that $X \to X_1$ is divisorial; let $E$ be its exceptional divisor. The solutions of the systems of Diophantine equations in Section~\ref{2raygame} determine
all the possible contractions $X\to X_1$. To each configuration is associated a Weil non-Cartier divisor $F$ on $Y$.
By Section~\ref{motivation}, $E$ is a rational scroll over a curve $\Gamma$ as in Table~\ref{table1}.
I now assume that the MMP on $X$ involves no divisorial contraction.
If any small factorialisation $X \to Y$ is a Conic bundle over $\PS^2, \F_0$ or $\F_1$, we are in Case $5.$ of the Theorem. Hence, it suffices to prove that if $Y$ is the midpoint of a link between two del Pezzo fibrations, then $Y$ contains one of the surfaces listed in the Theorem.
Vologodsky shows that if $Y$ is the midpoint of a link between two nonsingular weak Fano $3$-folds that are extremal del Pezzo
fibrations of degrees $d,d'$, then $d=d'=2$ or $4$ \cite{Vo01}. \cite[Lemma 3.4]{Kal07b} shows that $d\neq2$ because $A_Y$ is very ample.
\begin{cla}
If $Y$ is the midpoint of a link between two weak-star Fano dP$4$ fibrations $X$ and $\widetilde{X}$, $Y$
contains an anticanonically embedded del Pezzo surface of degree $4$,
and the equation of $Y$ can be written:
\[
Y=\{a_2q +b_2q'=0 \}\subset \PS^4
\]
where $a_2, b_2, q$ and $q'$ are homogeneous forms of degree $2$ on $\PS^4$.
\end{cla}
Let $F$ be a general fibre of $X \to \PS^1$; $F$ is a
nonsingular del Pezzo surface of degree $4$ and $A_F= {A_X}_{\vert
F}$.
Since $\vert A_X \vert_{\vert F}\subset \vert A_F \vert$, the restriction of the anticanonical map of $Y$ to $F$ factors as $g_{\vert F} = \nu \circ
\Phi_{\vert A_F \vert}$, where $\nu$ is the projection from a
(possibly empty) linear subspace
\[
\xymatrix{\PS(H^0(F, A_F))\simeq \PS^4\ar@{-->}[r] & \PS(H^0(F,
\vert A_{X} \vert_{\vert F}))}.
\]
If $\nu_{\vert F}$ is not the identity, as $h^0(A_F)=h^0(A_X)=5$, the
map $i$ in
\begin{align*}
0 \to H^{0}(X, A_X-F)\to H^{0}(X, A_X) \stackrel{i} \to H^{0}(F,
A_F)\to \\ \to H^{1}(X, A_X-F)\to 0
\end{align*}
is not surjective, and $H^0(X, A_X-F)\neq (0)$: there is a hyperplane section of $Y$ that contains
$\Phi_{\vert A_X \vert}(F)$.
As this holds for the general fibre $F$, the fibration $X \to \PS^1$
is induced by a pencil of hyperplanes on $Y$.
Without loss of generality, we may assume that
$\xymatrix{Y \ar@{-->}[r]& \PS^1}$ is determined by the pencil of
hyperplanes $\mathcal{H}_{(\lambda{:} \mu)}= \{\lambda x_0+\mu
x_1=0\}$ for $(\lambda{:} \mu)\in \PS^1$.
The map $X \to Y$ is a resolution of the base locus of $\mathcal{H}$ on $Y$ and therefore $\Pi= \{x_0{=}x_1{=}0\}= \Bs\mathcal{H}$ lies on $Y$: this contradicts $X$ being weak-star Fano.
As $H^0(X, A_X-F)=(0)$, $\nu$ is the identity and $Y$ contains an anticanonically embedded
nonsingular del Pezzo surface $S$ of degree $4$, i.e.~ the intersection
of two quadric hypersurfaces in $\PS^4$.
Since $S =\{
q{=}q'{=}0\} \subset \PS^4$ lies on $Y$, where $q$ and $q'$ are
homogeneous quadric forms, the equation of $Y$ writes:
\begin{equation}\label{eq:dp4}
Y=\{a_2q +b_2q'=0 \}\subset \PS^4
\end{equation}
with $a_2$ and $b_2$ homogeneous forms of degree $2$.
Geometrically, the two structures of del Pezzo fibrations on small factorialisations of $Y$ arise as
the maps induced by the
pencils of quadrics (eg $\mathcal{L}=\{ q,q'\}$ and $\mathcal{M}=\{
a_2,b_2\}$) after blowing up their base locus on $Y$, which are
anticanonically embedded del Pezzo surfaces of degree $4$.
Conversely, if the equation of $Y$ is of the form \eqref{eq:dp4} and if $\rk \Cl Y=2$, let $X$ (resp.~ $X'$) be the blow up of $X$ along $S$ (resp.~ along $S'=
\{a_2{=}b_2{=}0\}$), there is a diagram
\[
\xymatrix{
\quad & X \ar[dl] \ar[dr] \ar@{-->}[rr] & \quad & X'\ar[dl] \ar[dr] \\
\PS^1 & \quad & Y & \quad & \PS^1
}
\]
The $3$-fold $X$ (resp.~ $X'$) lies on $Q \times \PS^1$ (resp.~ $Q'
\times \PS^1$) for $Q \subset
\PS^4$ (resp.~ $Q'$) a quadric that is the proper transform of $\{a_2{=}0 \}$ under
the blow up of $\PS^4$ along $S$ (resp.~ $S'$).
The $3$-fold $X$
(resp.~ $X'$) is the
section of a linear system $\vert 2M +2F \vert$ on $ Q \times \PS^1$
(resp.~ $Q'\times\PS^1$), where $M=p_1^{\ast}\mathcal{O}_Q(1)$
(resp.~ $M=p_1^{\ast}(\mathcal{O}_{Q'}(1))$) and $F=p_2^{\ast}\mathcal{O}_{\PS^1}(1)$.
The map $\xymatrix{X \ar@{-->}[r] & X'}$ is a flop in the curves lying above the points
$\{q{=}q'{=}a_2{=}b_2{=}0\}$.
\end{proof}
\begin{rem}
\label{bound}
The bound on the rank of the divisor class group of quartic $3$-folds given in \cite{Kal07b} is too high: if $Y_4\subset \PS^4$ does not contain a plane, $\rk \Cl Y\leq 6$. Fujita classifies all polarised del Pezzo $3$-folds $(V, L)$ with Cohen-Macaulay Gorenstein singularities \cite{F90}. It is possible that the application of his results would yield an even finer bound.
\end{rem}
\subsection{Non-factorial terminal Gorestein Fano $3$-folds with $g\geq 4$}
By the same methods as above, one obtains the following theorem for non-factorial terminal Gorenstein Fano $3$-folds of index $1$ and higher genus.
\begin{thm}
Let $Y=Y_{2g-2}\subset \PS^{g+1}$ be a terminal Gorenstein Fano $3$-fold with $\rho(Y)=1$ and $g(Y)=g$. Then one of the following holds:
\begin{enumerate}
\item[1.] $Y$ is factorial.
\item[2.]$Y$ contains a plane $\PS^2$ and $g\leq 8$.
\item[3.] $Y$ is the midpoint of a link between two weak-star Fano del Pezzo fibrations of degree $g+1$ and $g \leq 8$, $g\neq 6$.
\item[4.] $Y$ has a structure of Conic Bundle over $\PS^2$, $\F_0$ or $\F_2$.
\item[5.] $Y$ contains a rational scroll $E \to C$ over a curve $C$ whose
genus and degree appear in the appropriate section of Table~\ref{table2} (see page \pageref{table2}).
\end{enumerate}
\end{thm}
\begin{proof}
This is entirely similar to what is done in the previous subsection. See \cite{Vo01} for $3$.
\end{proof}
\section{Rationality}
\label{rationality}
Classically, it was known that del Pezzo surfaces are rational over any algebraically closed field.
Understanding whether Fano varieties are rational or not was one of the early problems of higher dimensional birational geometry. Intuitively, Fano varieties can be thought of as being close to $\PS^n$: they are covered by rational curves and, in some sense, these curves should govern their birational geometry. However, the rationality question proved very difficult and it was not until the early seventies that it was settled for nonsingular Fano hypersurfaces in $\PS^4$ \cite{IM, CG72}.
\cite{IM} developed the Noether-Fano method and proved that any smooth quartic hypersurface is \emph{birationally rigid}-- i.e.~ that every rational map from a smooth quartic hypersurface to a Mori fibre space is a birational automorphism-- and in particular, that quartic hypersurfaces are very far from being rational. This approach was further developed and applied to a number of cases; it yielded surprising rigidity results-- see \cite{S82, Co95, Co00,CPR, Me04, IP99} or the survey \cite{Puk07}. The Noether-Fano method works in principle in any dimension and for singular varieties, but the technical difficulties are considerable. This section presents some results related to the rationality question for terminal Gorenstein Fano $3$-folds.
\subsection{Rationality, Rational connectivity and ruledness for mildly singular $3$-folds}
\cite{P04} shows that most canonical Gorenstein Fano $3$-folds with Picard rank $1$ that have at least one non-cDV point are rational. These results concern $3$-folds that are strictly canonical. However, one could argue that singularities make Fano $3$-folds ``more rational''. From the point of view of the Noether-Fano method, the valuations with centre at a singular point give rise to infinitely more complex divisorial extractions--even in the case of isolated hypersurface singularities \cite{Kawk01,Kawk02,Kawk03}-- and hence potentially to many more Sarkisov links and birational maps to other Mori fibre spaces. The following results do not require anything that technical but they do formalise this idea.
\begin{thm} [Matsusaka's Theorem] \cite[IV.1.6]{Kol96}
\label{mat}
Let $R$ be a DVR with quotient field $K$ and residue field $k$ and denote $T= \Spec R$. Let $f\colon X\to T$ be
a morphism where $X$ is normal and irreducible.
\begin{enumerate}
\item[1.]
If $X_K$ is ruled over $K$, then $X_k$ has ruled components over $k$.
\item[2.]
If $X_K$ is geometrically ruled, then every reduced irreducible component of $X_k$ is geometrically ruled.
\end{enumerate}
\end{thm}
\begin{thm}\cite{KMM92a}
\label{rc}
Let $X$ be a normal projective weak Fano $3$-fold. If $X$ is klt, $X$ is rationally connected.
\end{thm}
\begin{lem}
\label{ratio}
Let $f\colon \mathcal{Y}\to \Delta$ be a $1$-parameter smoothing of a terminal Gorenstein Fano $3$-fold $Y$. If $\mathcal{Y}_{\eta}$ is geometrically rational then so is $Y$.
\end{lem}
\begin{proof}
This is a direct consequence of Theorem~\ref{mat}. Indeed, $Y$ is rationally connected by Theorem~\ref{rc}, so that $Y$ is rational if and only if $Y$ is ruled.
\end{proof}
In now recall and discuss Conjecture~\ref{con:rig}.
\setcounter{con}{0}
\begin{con}
A factorial quartic hypersurface $Y_4\subset \PS^4$ (resp.~ a generic complete intersection $Y_{2,3}\subset \PS^5$) with no worse than terminal singularities has a finite number of models as
Mori fibre spaces, i.e.~ the pliability of $Y$ is finite.
\end{con}
\begin{rem}\mbox{}
\label{rem11}
\begin{enumerate}
\item[1.]
Conjecture~\ref{con:rig} is supported by some evidence. \cite{Me04} shows that a factorial quartic $3$-fold $Y_4\subset \PS^4$ with ordinary double points is rigid, while \cite{IP96} shows that the same is true for a general non-singular $Y_{2,3}$. Mella's proof is based on the Noether-Fano/maximal singularity method of Iskovskikh-Manin as formulated in \cite{Co00, CPR}. It is difficult to extend these results to terminal Gorenstein singularities, because these methods require a careful analysis of $3$-fold divisorial extractions with centre along (possibly singular) points or curves. While divisorial extractions centred at nonsingular or ordinary double points are reasonably tractable, there is an a priori infinite number of divisorial extractions centred on slightly more complicated singularities \cite{Kawk01,Kawk02,Kawk03}.
\item[2.] Conjecture~\ref{con:rig} does not hold for some other rigid Fano $3$-folds with Picard rank $1$. For instance, a cubic $3$-fold with a single ordinary double point is both rational and factorial.
Since several Sarkisov links exist between a nonsingular cubic $3$-fold and a nonsingular Fano $3$-fold $X_{14}\subset \PS^9$ of genus $8$ \cite{IP99,Tak89}, the same phenomenon can be expected on $X_{14}$.
\item[3.] \cite{ChGr} shows that birational rigidity is not preserved under small deformations, and exhibits a small deformation from a rigid $Y_{2,3}$ with one ordinary double point to a bi-rigid $Y_{2,3}$. Similarly, \cite{CM04} gives an example of a bi-rigid terminal factorial quartic hypersurface. As I mention in the Introduction, in known examples where a birationally rigid Fano $3$-fold $V$ of genus $3$ or $4$ degenerates to a non-rigid and nonrational $3$-fold $V'$, $V'$ has finitely many models as a Mori fibre space, i.e~ $V'$ has finite \emph{pliability}. I believe that the correct notions to consider are rationality on the one hand, and finite pliability on the other.
\end{enumerate}
\end{rem}
\subsection{Rationality of terminal quartic $3$-folds}
\subsubsection{Quartic $3$-folds that do not contain a plane}
Let $Y$ be a non-factorial terminal Gorenstein Fano $3$-fold. Theorem~\ref{thm:1} shows that when $Y$ does not contain a plane, $Y$ has a structure of Conic Bundle, $Y$ is the midpoint of a link between two del Pezzo fibrations of degree $4$, or $Y$ contains a scroll as in Table~\ref{table1}.
Let $X$ be a small factorialisation of $Y$.
\begin{lem}
Let $Y$ be a non-factorial terminal quartic $3$-fold and denote $X\to Y$ a small factorialisation. Assume that the MMP on $X$ involves at least one divisorial contraction. Then $Y$ is rational except possibly if the first divisorial contraction $\varphi$ is one of cases $15,17,25, 29,35$ or $36$ in Table~\ref{table1}.
\end{lem}
\begin{proof}
This is an immediate consequence of the classification of Tables~\ref{table1} and \ref{table2} and of Lemma~\ref{ratio}.
\end{proof}
\begin{lem}
If $\varphi$ is one of cases $15,17,25$ and $29$, and if the MMP on $X$ involves at least another divisorial contraction or a del Pezzo fibration, $Y$ is rational.
In particular, if $\rk \Cl Y \geq 5$, $Y$ is rational.
\end{lem}
\begin{rem}\mbox{}
\begin{enumerate}
\item[1.]
Note that if $\varphi$ is as in cases $17$ or $36$ and if $\widetilde{Z}_1$ has a singular point, $Y$ is rational.
\item[2.] In Case $29$, when $\rk \Cl Y=2$, the Conic bundle on the deformed Sarkisov link is nonrational \cite{Sh83}. However, it is not clear whether the same is true for $Y$.
\item[3.] According to Conjecture~\ref{con:rig}, one can expect that when $\rk \Cl Y=2$, Case $36$ is impossible, and that when Case $35$ occurs, $Y$ is birationally rigid.
\item[4.] It is unlikely that these methods would lead to any conclusion when $\rk \Cl Y=2$ and $Y$ is one of Cases $15,17,25$ or $29$ (see Rem~\ref{rem11}).
\end{enumerate}
\end{rem}
When $X\to \PS^1$ is an extremal del Pezzo fibration, recall the following rationality criteria.
\begin{thm}\cite[Section III.3]{Kol96}
Let $S_k$ be a nonsingular, proper and geometrically irreducible del Pezzo surface of degree $d\geq 5$ over an arbitrary field $k$. Assume that $S(k)\neq \emptyset$, then $S_k$ is rational.
\end{thm}
\begin{thm}\cite{CT87,KMM92b}
Let $C$ be an algebraic curve defined over an algebraically closed field and let $K=k(C)$ be its field of rational functions. If $X$ is a del Pezzo surface over $K$, then $X(K)\neq \emptyset$ is dense in the Zariski topology of $X$.
In particular, if $X\to \PS^1$ is a del Pezzo fibration of degree $d\geq 5$, then $X$ is rational.
\end{thm}
\begin{thm}[\cite{Al87, Sh07}]
Let $V \to \PS^1$ be a standard fibration by del Pezzo surfaces of degree $4$. The topological Euler characteristic $\chi(V)$ equals $-8,-4$ or $0$ precisely when $V$ is rational.
\end{thm}
\begin{rem}
In particular, rationality of a del Pezzo fibration $V\to \PS^1$ of degree $4$ is a topological question and depends only on the Hodge numbers of $V$. \cite{Ch06} shows that if $V\to \PS^1$ is the small factorialisation of a terminal quartic $3$-fold and is nonsingular, then $V$ is nonrational.
\end{rem}
Last, recall the following rationality criterion for standard Conic Bundles over minimal surfaces.
\begin{thm}\cite{Sh83}
Let $X\to S$ be a standard Conic Bundle over $S= \PS^2$ or $\F_n$. Assume that $\Delta$, the discriminant curve, is connected. If one of the following holds:
\begin{enumerate}
\item[1.] $\Delta+2K_S$ is not effective,
\item[2.] $\Delta\subset \PS^2$ has degree $5$ and the associated double cover $\overline{\Delta}\to \Delta$ has even theta characteristic,
\end{enumerate}
$X$ is rational.
\end{thm}
\subsubsection{Quartic $3$-folds that contain a plane.}
Assume that $Y\subset \PS^4$ contains a plane $\Pi=\{x_0{=}x_1{=}0\}$ and let $X$ be the blow up of $Y$ along $\Pi$; $X$ has a natural structure of dP$3$ fibration $\pi\colon X \to \PS^1$ induced by the pencil of hyperplanes that contains $\Pi$ on $\PS^4$ (see \cite[Section 4]{Kal07b} for details).
Write the equation of $Y$ as:
\begin{equation}
\label{eq:2}
\{x_0a_3(x_0, x_1,x_2, x_3,x_4)+ x_1 b_3(x_0, x_1,x_2, x_3,x_4)=0\} \subset \PS^4
\end{equation}
so that $X$ is given by:
\begin{eqnarray}
\label{eq:3}
\{t_0a_3(t_0x, t_1x,x_2, x_3,x_4)+ t_1 b_3(t_0x, t_1x,x_2, x_3,x_4)=0\}
\\
\subset \PS_{(t_0{:}t_1)}\times \PS(x, x_2, x_3,x_4) .\nonumber
\end{eqnarray}
\begin{lem}\cite[Lemma 4.1]{Kal07b}
The divisor class group $\Cl Y$ is generated by $\pi^{\ast}\mathcal{O}_{\PS^1}(1)$, by the completion of divisors that generate $\Pic X_{\eta}$ and by irreducible components of the reducible fibres of $X$.
\end{lem}
As $X$ has terminal Gorenstein singularities, \cite{Co96} shows that there is a birational map
\[\xymatrix{X \ar@{-->}[r]^{\Phi} \ar[d] & X' \ar[d]\\
\PS^1 & \PS^1
}\]
where $\Phi$ is the composition of projections from planes contained in reducible fibres and $X'$ has irreducible and reduced fibres. Note that $X_{\eta}\simeq X'_{\eta}$ because $\Phi$ is an isomorphism outside of the reducible fibres of $X$. In particular, if $X_{\eta}$ is rational, $X\to \PS^1$ is geometrically rational, i.e.~ is birational to $\PS^2\times \PS^1$ .
I recall some results on rationality of cubic surfaces over arbitrary fields.
Let $X_{\eta}$ be a nonsingular cubic surface defined over a field $\eta$ and let $K/\eta$ be a field extension over which the $27$ lines of $X$ are geometric.
Denote $S_n$ any subset of the $27$ lines on $X_{\eta}\otimes K$ that consists of $n$ skew lines and that is defined over $X_{\eta}$, i.e.~ if $S_n$ contains a line $L$, then it contains all its conjugates under the action of $\Gal(K/\eta)$. Note that, by the geometry of the configuration of the $27$ lines on $X_K$, any $S_n$ has $n\leq 6$.
\begin{thm}\cite{Se42,SD70}
\label{cubics}
\begin{enumerate}\item[1.]
$\overline{NS}(X_{\eta})\otimes_{\Z}\Q$ is generated as a $\Q$-vector space by the class of a hyperplane section of $X_{\eta}$ and by the classes of the $S_n$, when there are any.
\item[2.] If $X_{\eta}$ has an $S_4$ or an $S_5$, $X_{\eta}$ has an $S_2$ or an $S_6$.
\item[3.] If $X_{\eta}$ has an $S_2$, $X_{\eta}$ is rational over $\eta$.
\item[4.] If $X_{\eta}$ has an $S_3$ or an $S_6$ and $X_{\eta}(\eta) \neq \emptyset$, $X_{\eta}$ is rational over $\eta$.
\end{enumerate}
\end{thm}
Here, $X_{\eta}$ is the generic fibre of $X \to \PS^1$, $X_{\eta}$ is a nonsingular cubic surface embedded in $\PS^3$ over $\C(t)$, with coordinates $x, x_2,x_3,x_4$ (see \eqref{eq:3}).
\begin{cla}
Assume that $X_{\eta}$ contains a Cartier divisor of type $S_n$ and denote
$D_n$ the completion of $S_n$ to a (Weil) divisor on $X$. The proper transform of $D_n$ on a small factorialisation of $X$ has anticanonical degree $n$; the image of $D_n$ on $Y$ is Weil non-Cartier.
\end{cla}
In the light of Theorem~\ref{cubics}, it is then natural to consider the following cases:
\setcounter{case}{0}
\begin{case}{$X$ is an extremal Mori fibre space, i.e.~ $\rk \Cl Y =2$, $X\to \PS^1$ has irreducible and reduced fibres and $\rho(X_{\eta})=1$.}
\end{case}
It is known that $X$ admits another model as a Mori fibre space \cite{BCZ}. Indeed, $Y$ is the midpoint of a link
\[
\xymatrix{
\quad & X \ar@{-->}[rr] \ar[dl]\ar[dr] &&\widetilde{X}\ar[dr]\ar[dl]& \quad\\
\PS^1 & \quad& Y & \quad & Z}
\]
where $Z=Y_{3,3}\subset \PS(1^5,2)$ is a codimension $2$ terminal Fano $3$-fold with one point of Gorenstein index $2$ at $P=(0{:}0{:}0{:}0{:}0{:}0{:}1)$; $Y\dashrightarrow Z$ can be described as follows. Introduce a variable of weight $2$ \[y= \frac{a_3}{x_1}=\frac{b_3}{x_0},\] then $Z$ is the complete intersection:
\[\left \{ \begin{array}{c}
a_3-yx_1=0\\
b_3-yx_0=0
\end{array}\right.
\]
The contraction $\widetilde{X}\to Z$ contracts the preimage of the plane $\{x_0{=}x_1{=}0\}$ to the point $P$, the map $X\dashrightarrow \widetilde{X}$ is the flop of the rational curves lying above the locus $\{ x_0{=}x_1{=}a_3{=}b_3{=}0\}$, and $\widetilde{X} \to Y$ is the blow up of the surface $\{a_3{=}b_3{=}0\}$.
Recall that $X$ is a section of the linear system $ \vert 3M+L\vert $ on the scroll $\F(0,0,1)$ (see \cite{BCZ} for notation conventions on scrolls); \cite{Ch08} shows that if $X$ is a general member of $\vert 3M+L \vert$, $X$ is nonrational. I make the following conjecture:
\begin{con}
If $X$ is a standard dP3 fibration, $X$ is bi-rigid.
\end{con}
\begin{case}{$X$ is not an extremal Mori fibre space, i.e.~ $\rk \Cl Y \geq 3$, and $\rho(X_{\eta})=1$.}
\end{case}
\begin{lem}{\cite{Kol96}}
\label{3planes}
Let $Y_4\subset \PS^4$ be a quartic hypersurface. If $Y$ contains three planes $\Pi_0, \Pi_1, \Pi_2$ such that $\Pi_0\cap\Pi_1\cap\Pi_2=\emptyset$, $Y$ is rational.
\end{lem}
\begin{cor} Let $X\to Y$ be as above. Assume that $\rho(X_{\eta})=1$,
if there are at least $3$ planes lying in at least $2$ distinct reducible fibres of $X$, $Y$ is rational. More precisely, if $X$ has either at least two reducible fibres, one of which is the union of $3$ planes or if $X$ has at least $3$ reducible fibres, $Y$ is rational.
\end{cor}
\begin{proof}
This follows from the possible configurations of planes lying in reducible fibres obtained as in \cite[Section 4]{Kal07b}.
\end{proof}
Assume that $X\to \PS^1$ has $\rho(X_{\eta})=1$, and that $X\to \PS^1$ has $1$ or $2$ reducible fibres, each containing a quadric ($\rk \Cl Y=3$ or $4$).
Among the generators of $\Cl Y/\Pic Y$, there is a surface $S$ such that $A_Y^2\cdot S= 2$, i.e.~ there is a quadric lying on $Y$. Denote $f\colon \widetilde{X}\to X \to Y$ a small factorialisation of $X$ and $Y$ and note that there is an extremal divisorial contraction $\varphi \colon \widetilde{X} \to \widetilde{X}_1$ such that $\widetilde{S}=f_{\ast}^{-1}S=\Exc \varphi$ (possibly after flops of $\widetilde{X}$).
Observe that $\widetilde{X}_1$ is the small modification of a terminal Gorenstein Fano $3$-fold $Y_1=Y_{2,3}\subset \PS^5$. For any divisor $D_1\subset \widetilde{X_1}$, the proper transform $D$ of $D_1$ on $\widetilde{X}$ is such that $A_{Y}^2\cdot D\leq A_{Y_1}^2\cdot D_1$ and the inequality is strict when $D$ intersects the quadric $S$ (see the proof of \cite[Theorem 3.2]{Kal07b}).
Note that $\Pi$ and all planes contained in reducible fibres of $X\to \PS^1$ do intersect the quadric $S$ and since $\rho(X_{\eta})=1$, $\widetilde{X_1}$ is weak-star Fano: the methods of the previous subsection apply.
More precisely, as $Y_1$ is terminal Gorenstein and has $\rk \Cl(Y_1)\geq 2$, unless $Y_1$ has a structure of Conic Bundle or the MMP on $\widetilde{X_1}$ consists of one divisorial contraction of type $10,11,12,13,14,18,20$ or $21$ in Table~\ref{table2}, $Y$ is rational.
\begin{exa}
In particular, this gives potential examples of rational cubic fibrations that are not geometrically rational.
\end{exa}
\begin{case} {$X$ is not an extremal Mori fibre space, i.e.~ $\rk \Cl Y \geq 3$, and $\rho(X_{\eta})>1$.}
\end{case}
\begin{pro}
If $\rho(X_{\eta})\geq 3$, $X$ is rational. If $\rho(X_{\eta})=2$ and either $\Cl Y/\Pic Y$ is not generated by planes or $X$ has at least one reducible fibre, $X$ is rational.
\end{pro}
\begin{proof}
Theorem~\ref{cubics} shows that unless $\Pic X_{\eta}$ is generated by the class of a hyperplane section and divisors of type $S_1$, $X_{\eta}$ is rational. But then, as \cite{Co96} shows that $X \to \PS^1$ is birational to a cubic fibration $X'\to \PS^1$ with reduced and irreducible fibres and $X_{\eta}\simeq X'_{\eta}$, $X\to \PS^1$ is rational.
We now turn to the case when $\rho(X_{\eta})>1$ and $X_{\eta}$ does not contain any $S_n$ for $n\geq 2$.
The proposition follows from the following Claims.
\begin{cla}
If $\rho(X_{\eta})\geq 3$ and if $\Pi', \Pi''$ are two planes on $X\to \PS^1$ that arise as completions of divisors of type $S_1$ on $X_{\eta}$, then $\Pi\cap\Pi'\cap\Pi''= \emptyset$.
\end{cla}
Any $S_1$ lying on $X_{\eta}$ completes to a plane $\Pi'$ that meets $\Pi$ in a point.
Indeed, if $\Pi$ and $\Pi'$ met in a line, the image of $\Pi'$ on $Y$ would be contained in a hyperplane section of the original quartic $Y$, and $\Pi'$ would have to be contained in a reducible fibre. If $X_{\eta}$ contains two distinct $S_1$, these cannot be skew (otherwise they would form an $S_2$) and therefore up to coordinate change on $\PS(x, x_2, x_3,x_4)$, $X_{\eta}$ contains the lines
\begin{eqnarray*}
L=\{x_2{=}x_3{=}0\}\\
L'=\{x_2{=}x_4{=}0\}
\end{eqnarray*}
so that $Y$ contains the planes $\{x_0{=}x_1{=}0\}$, $\{x_2{=}x_3{=}0\}$ and $\{x_2{=}x_4{=}0\}$ and by Lemma~\ref{3planes}, $Y$ is rational.
\begin{cla}
If there are at least $3$ planes lying in reducible fibres of $X\to \PS^1$ then we may choose $\Pi''$ lying in a reducible fibre of $X$ such that $\Pi\cap\Pi'\cap\Pi''= \emptyset$.
\end{cla}
Since any plane contained in a fibre of $X\to \PS^1$ intersect $\Pi$ in a line and that given any $3$ such planes, \cite{Kal07b} shows that the $3$ associated lines are distinct and non-concurrent, we may choose one plane that does not contain $\Pi\cap\Pi'$.
\end{proof}
We have proved the following.
\begin{pro}
Let $Y_4\subset \PS^4$ be a quartic hypersurface that contains a plane. If $6\leq \rk \Cl Y\leq 16$, $Y$ is rational.
\end{pro}
\section{Examples and geometric realizability of numerical Sarkisov links}
\label{examples}
\subsection{Examples}
In this section, I construct examples of non-factorial Fano $3$-folds with terminal Gorenstein singularities. I use the Tables of numerical Sarkisov links to recover some known examples and construct some new ones.
\begin{exa}
Let $X_{2g-2}\subset \PS^{g+1}$ be a nonsingular Fano $3$-fold of genus $g \geq 7$ and let $P\in X$ be a point that does not lie on any line of $X$ (such a point exists by \cite{Isk78}). Let $\widetilde{X}\to X$ be the blow up of $P$. Then $\widetilde{X}$ is a weak-star Fano $3$-fold with Picard rank $2$. The anticanonical model $Y$ of $\widetilde{X}$ is a terminal Gorenstein non-factorial Fano $3$-fold of genus $g-4$. The map $\widetilde{X}\to Y$ is small and contracts the preimages of conics through $P$ to points (\cite{Tak89} proves that there are finitely many such conics).
\begin{enumerate}
\item[1.] When $g=7$, $Y$ is a quartic $3$-fold that is the midpoint of a link where both contractions of the Sarkisov link are of type E$2$. The link is a self-map of $X_{12}\subset \PS^9$; the centre of the link is a rational quartic $3$-fold $Y$ with $\rk \Cl Y=2$.
\item[2.] When $g\geq 8$, \cite{Tak89} lists all possible constructions starting with a nonsingular Fano $3$-fold $X_{2g-2}\subset \PS^{g+1}$. Takeuchi uses Hodge theoretical computations to show that some numerical Sarkisov links are not realizable. Here, since I allow terminal Gorenstein singularities, it is not clear that these links can be excluded (see Remark~\ref{Hodge}).
\end{enumerate}
\end{exa}
Let $X$ be a nonsingular Fano $3$-fold with $\rho(X)=1$ and $\Gamma \subset X$ a curve such that $X=Z_1$ and $\Gamma$ is the centre of $\varphi$ for one case appearing in Table~\ref{table1} (resp.~ of Table~\ref{table2}).
Let $\widetilde{X} \to X$ be the blow up of $X$ along $\Gamma$. By construction, $A_{\widetilde{X}}^3=4$ (resp.~ $2g-2$ for $g\geq 4$), so that if $A_{\widetilde{X}}$ is nef, it is big and $\widetilde{X}$ is a Picard rank $2$ weak Fano $3$-fold. Observe that when $\Gamma$ is an intersection of members of $\vert A_X\vert$, then $A_{\widetilde{X}}$ is nef and big.
The anticanonical map $f \colon \widetilde{X} \to Y$ maps to a Gorenstein Fano $3$-fold with canonical singularities. If, in addition, $(A_{\widetilde{X}})^2\cdot D>0$ for every effective divisor $D$, $Y$ has terminal singularities and $f$ is small. Still by construction, in this case, $f$ is not an isomorphism because $e\neq 0$, and $Y$ is a non-factorial terminal Gorenstein Fano $3$-fold with $\rho(Y)=1$, $\rk \Cl Y=2$.
\begin{thm}\cite{Sh79, Re80, Tak89, IP99}\label{lineconic}
Let $X_{2g-2}\subset \PS^{g+1}$ be a nonsingular anticanonically embedded Fano $3$-fold of index $1$. If $g\geq 5$ (resp.~ $g\geq 6$), there exists a line (resp.~ a smooth conic) on $X$.
For any $g\geq 5$, if $X$ contains a line and a smooth conic, it also contains a rational normal cubic curve.
\end{thm}
\begin{exa}\cite{ Isk78, IP99}
Let $X=X_{2g'-2}\subset \PS^{g'+1}$ be a nonsingular (or more generally terminal Gorenstein factorial) Fano $3$-fold of index $1$ such that $A_X$ is very ample and let $\Gamma$ be a line lying on $X$. As above, let $\widetilde{X}\to X$ be the blow up along $\Gamma$ and let $\widetilde{X}\to Y$ be the anticanonical map of $\widetilde{X}$. Recall the following result of Iskovskikh's:
\begin{thm}\cite{Isk78}
If $\Gamma \subset X_{2g'-2}$ is a line on a nonsingular Fano $3$-fold of genus $g'$ and Picard rank $1$, and if $\widetilde{X}\to X$ is the blow up of $X$ along $\Gamma$, then $\widetilde{X}$ is a small modification of a terminal Gorenstein Fano $3$-fold $Y_{2g-2}$ of index $1$, Picard rank $1$ and genus $g= g'-2$.
\end{thm}
By construction, $Y$ is not factorial and its divisor class group is generated by the hyperplane section and by a surface $\overline{E}=f(\PS(\mathcal{N}^v_{\Gamma/X}))$, which is the image by the anticanonical map of a cubic scroll.
The blow up $f$ is one side of a Sarkisov link with midpoint along $Y$. Note that the rational map between the two sides of the Sarkisov link $Z_1=X_{2g'-2}\dashrightarrow \widetilde{Z}_1$ is Iskovskikh's double projection from a line \cite{Isk78}, that enabled him to classify Fano $3$-folds of the first species.
\begin{enumerate}
\item[1.]Case $30$ in Table~\ref{table1} is a geometric construction that was known classically \cite{Be77, BCZ}.
Let $X=X_{2,2,2}\subset \PS^6$ be a codimension $3$ complete intersection of quadrics in $\PS^3$ and $l\subset X$ be a line. Then the other contraction in the link starting with the projection from $l$ is a conic bundle with discriminant of degree $7$. Conversely, given a plane curve $\Delta\subset \PS^2 $ of degree $7$, \cite{BCZ} constructs standard conic bundles with ramification data a $2$-to-$1$ admissible cover $N \to \Delta$. When $\deg \Delta=7$, there are $4$ deformation families of standard conic bundles and Case $30$ corresponds to the generic even theta characteristic case.
By \cite{Sh83}, the standard conic bundle $\widetilde{X}$ is non rational.
\item[2.]\cite{Isk78}
When $X$ is nonsingular, the link that occurs is Case $17, g=4$ in Table $2$ for $g'=6$, Case $13, g=5$ for $g'=7$, Case $8,g=6$ for $g'=8$, Case $3,g=7$ for $g'=9$, Case $3, g=8$ for $g'=10$ and Case $1,g=10$ for $g'=12$.
\end{enumerate}
Note that Case $31$ in Table~\ref{table1}, and Cases $18, g=4$, $12, g=5$, and Cases $11,12, g=6 $do not occur \cite{Isk78, IP99} if $Z_1$ is nonsingular.
One can describe explicitly the inverse rational map $\widetilde{Z}_1 \dashrightarrow Z_1$ for $g \geq 7$ by choosing the curve $C$ carefully on an appropriate $\widetilde{Z_1}$ \cite{IP99}.
\end{exa}
\begin{exa}\cite{Tak89}
Let $X=X_{2g'-2}\subset \PS^{g'+1}$ be a nonsingular (or more generally terminal Gorenstein factorial) Fano $3$-fold of index $1$ such that $A_X$ is very ample and let $\Gamma$ be a smooth conic lying on $X$. As above, let $\widetilde{X}\to X$ be the blow up along $\Gamma$ and let $\widetilde{X}\to Y$ be the anticanonical map of $\widetilde{X}$.
\begin{thm}\cite{Tak89}
Let $\Gamma \subset X_{2g'-2}$ be a conic on a nonsingular Fano $3$-fold of genus $g'$ and Picard rank $1$, and $\widetilde{X}\to X$ the blow up of $X$ along $\Gamma$. If $g'\geq 7$ and $\Gamma$ is general, then $\widetilde{X}$ is a small modification of a terminal Gorenstein Fano $3$-fold $Y_{2g-2}$ of index $1$, Picard rank $1$ and genus $g= g'-3$. If $g'\geq 9$, the same holds for any conic $\Gamma \subset X$.
\end{thm}
Note that when $\widetilde{X}$ is a small modification of a terminal Gorenstein Fano $3$-fold $Y$ with $\rho(Y)=1$, the divisor class group of $Y$ is generated by a hyperplane section and by a surface $\overline{E}= f(\PS(\mathcal{N}_{\Gamma/X}^v))$ which has degree $4$. More precisely, $\mathcal{N}_{\Gamma/X}= \mathcal{O}_{\PS^1}(d)\oplus\mathcal{O}_{\PS^1}(-d)$, for $d=0,1$ or $2$.
If $d=0$, $f_{\vert E}\colon \PS^1 \times \PS^1\to \overline{E}$ is induced by a divisor of bidegree $(1,2)$.
If $d=1$, $f_{\vert E}\colon \F_2 \to \overline{E}$ is induced by $\vert s+3f\vert$ and if $d=2$, $f_{\vert E}\colon \F_4 \to \overline{E}$ is induced by $\vert s+4f\vert$.
The blow up $f$ is one side of a Sarkisov link with midpoint along $Y$ and corresponds to one of Cases $25$ or $26$ in Table~\ref{table1}, or Cases $15,16$ for $g=4$, $9,10,11$ for $g=5$, $6,7$ for $g=6$, $2$ for $g=7$ or $1$ for $g=9$. \cite{Tak89} shows that for nonsingular Fano $3$-folds $X_{2g-2}=Z_1$, the Cases indicated by $\bullet$ in Tables~\ref{table1} and \ref{table2} are the only geometrically realizable constructions.
\end{exa}
\begin{exa} Let $X_{2g'-2}\subset \PS^{g+1}$ be a nonsingular Fano $3$-fold with $g'\geq 6$. By Theorem~\ref{lineconic}, there is a rational normal cubic curve $\Gamma$ lying on $X$. If $\widetilde{X}\to Y$ is small, then if $g'=7$, we are in case $18$ or $19$ of Table~\ref{table1}, $Y$ is a terminal Gorentsein factorial quartic $3$-fold that is rational; and if $g'\geq 8$, we are in one of Cases $9$ or $11, g=4$, $5, g=5$, $3, g=6$ or $1,g=8$ of Table~\ref{table2}.
\end{exa}
\begin{thm}\cite{Mo84, MM83}
\label{exi}
Let $k=\overline{k}$ be a field of characteristic $0$ and let $d>0$ and $g\geq 0$ be integers.
There exists a nonsingular curve $C$ lying on a nonsingular quartic surface $S_4\subset \PS^3_k$ with $(p_a C, \deg C)=(g,d)$ if and only if
\[g=d^2/8+1\mbox{ or } g<d^2/8\] and $(g,d)\neq (3,5)$.
\end{thm}
\begin{exa}
I now use Theorem~\ref{exi} to show that some constructions that appear in Tables~\ref{table1} and \ref{table2} may be geometrically realizable.
\begin{enumerate}
\item[1.]
Let $C\subset \PS^3$ be a nonsingular curve that is an intersection of nonsingular quartic surfaces with $(p_a C, \deg C)= (15,11)$. Let $X$ be the blow up of $\PS^3$ along $C$, assume that $X$ is the small modification of a terminal quartic hypersurface $Y \subset \PS^4$. The linear system $\vert \mathcal{O}_{\PS^3} \vert $ determines a rational map $\PS^3 \dashrightarrow X_{10}\subset \PS^7$ that corresponds to the inverse of Case $28$ in Table~\ref{table1}. The midpoint $Y_4\subset \PS^4$ is a non-factorial rational quartic $3$-fold; $\Cl Y$ is generated by the hyperplane section and the image in $Y$ of $E$ or $D$. Note that this rational map provides an example of a rational Fano $3$-fold of genus $6$.
\item[2.]\cite{IP99}
Let $C\subset \PS^3$ be a nonsingular curve that is an intersection of nonsingular quartic surfaces with $(p_a C, \deg C)= (7,9)$. Let $X$ be the blow up of $\PS^3$ along $C$, then since $C$ is an intersection of nonsingular quartic surfaces, $X$ is the small modification of a terminal quartic hypersurface $Y \subset \PS^4$. The linear system $\vert \mathcal{O}_{\PS^3}(15)-4C \vert$ determines a rational map $\PS^3 \dashrightarrow X_{12}\subset \PS^8$ that corresponds to the inverse of Case $19$ in Table~\ref{table1}. The midpoint $Y_4\subset \PS^4$ is a non-factorial rational quartic $3$-fold; $\Cl Y$ is generated by the hyperplane section and the image in $Y$ of $E$ or $D$.
\item[3.]
Let $C\subset \PS^3$ be a nonsingular curve that is an intersection of nonsingular quartic surfaces with $(p_a C, \deg C)= (3,8)$. Let $X$ be the blow up of $\PS^3$ along $C$, assume that $X$ is the small modification of a terminal quartic hypersurface $Y \subset \PS^4$. The linear system $\vert \mathcal{O}_{\PS^3}\vert$ determines a rational map $\PS^3 \dashrightarrow V_3\subset \PS^4$ that corresponds to the inverse of Case $34$ in Table~\ref{table1}. The midpoint $Y_4\subset \PS^4$ is a non-factorial rational quartic $3$-fold; $\Cl Y$ is generated by the hyperplane section and the image in $Y$ of $E$ or $D$. Note that in this case $V_3\subset \PS^4$ would necessarily be singular, because it would be rational.
\item[4.]
Let $C\subset \PS^3$ be a nonsingular curve that is an intersection of nonsingular quartic surfaces with $(p_a C, \deg C)= (1,7)$. Let $X$ be the blow up of $\PS^3$ along $C$, assume that $X$ is the small modification of a terminal Gorenstein Fano $3$-fold $Y_{2,2,2} \subset \PS^6$ that is non-factorial and rational. The linear system $\vert \mathcal{O}_{\PS^3}\vert$ determines a rational map $\PS^3 \dashrightarrow X_{22}$ that corresponds to the inverse of Case $2, g=5$ in Table~\ref{table2}.
\item[5.]
Let $C\subset \PS^3$ be a nonsingular curve lying that is an intersection of nonsingular quartic surfaces with $(p_a C, \deg C)= (6,8)$. Let $X$ be the blow up of $\PS^3$ along $C$, assume that $X$ is the small modification of a terminal Gorenstein Fano $3$-fold $Y_{10} \subset \PS^7$ that is non-factorial and rational. The linear system $\vert \mathcal{O}_{\PS^3}\vert$ determines a rational map $\PS^3 \dashrightarrow X_{22}$ that corresponds to the inverse of Case $9,g=6$ in Table~\ref{table2}.
\end{enumerate}
\end{exa}
\begin{exa}\cite{IP99}
Let $C\subset \PS^3$ be a nonsingular non hyperelliptic curve lying on a nonsingular quartic surface with $(p_a C, \deg C)= (3,7)$. Let $X$ be the blow up of $\PS^3$ along $C$, $X$ is the small modification of a terminal Gorenstein Fano $3$-fold $Y_{12} \subset \PS^8$,= that is non-factorial and rational. The linear system $\vert \mathcal{O}_{\PS^3}\vert$ determines a rational map $\PS^3 \dashrightarrow X_{16}$ that corresponds to the inverse of Case $3,g=7$ in Table~\ref{table2}.
\end{exa}
\subsection{Some remarks on geometric realizability}
Classically, it has been shown that numerical Sarkisov links were not geometrically realizable by using constraints on the Hodge numbers of blow ups of nonsingular varieties along smooth centres or constraints on Euler characteristics of fibrations. In the case of divisorial contractions of factorial terminal Gorenstein $3$-folds, I have been unable to extend these results so as to use them to rule out some numerical Sarkisov links.
It is easy to show the following weakened version:
\begin{lem}
\label{htexc}
Let $Z$ be a nonsingular weak Fano $3$-fold and $\varphi \colon Z\to Z_1$ an extremal divisorial contraction with centre along a curve $\Gamma$ and such that $Z_1$ is a terminal Gorenstein Fano $3$-fold.
Then
\[
h^{1,2}(Z)\leq h^{1,2}(\mathcal{Z}_{1, \eta})+ p_a(\Gamma),
\]
where $p_a(\Gamma)$ denotes the arithmetic genus of $\Gamma$ and $\mathcal{Z}_1$ a smoothing of $Z_1$.
\end{lem}
\begin{rem}
\label{Hodge}
\cite{Kol89} shows that $Z$ and $\widetilde{Z}$ have the same analytic type of singularities. Since $h^{1,2}(Z)$ and $h^{1,2}(\widetilde{Z})$ can be expressed only in terms of local invariants of singularities and of $\rk \Cl Y$, where $Y$ is the anticanonical model of $Z$ and $\widetilde{Z}$, $h^{1,2}(Z)=h^{1,2}(\widetilde{Z})$ . In order to exclude some numerical Sarkisov links, I would need to find a lower bound for $h^{1,2}(Z)$ (resp.~ $h^{1,2}(\widetilde{Z})$).
This would follow if the following question could be answered.
\begin{que}
Is it possible to relate $W_3H^4(Z)$ and $W_2H^3(Z)$ when $Z$ has terminal Gorenstein singularities? What if $Z$ is factorial?
\end{que}
\end{rem}
\begin{rem}
Observe that in order to determine that a numerical Sarkisov link is not realizable, it is enough to observe that no deformed (nonsingular) link exists between a Fano in the deformation family of $Z_1$ and a Fano in the deformation family of $\widetilde{Z}_1$. This has been used in the previous subsections.
\end{rem}
Another question of interest would be to understand the geometric meaning of the correction term $e$ that appears in the tables of numerical Sarkisov links. The proof of Lemma~\ref{lem:4} shows that $e$ is the intersection of $E$, the exceptional divisor of the left hand side contraction, with the flopping locus of $Z\dashrightarrow \widetilde{Z}$. The large values of $e$ that appear in the table suggest the following question.
\begin{que}
Let $f \colon Z\to Y$ be a small factorialisation and assume that $f^{-1}(P)$ is a chain of rational curves $\cap \Gamma_i$. If $E$ is the proper transform on $X$ of a Weil non-Cartier divisor passing through the singular point $P$, is it possible to have $E\cdot \Gamma_i>1$? The surface $E$ is a priori not Cohen Macaulay at $P$, but is it possible to bound this intersection number?
\end{que}
\end{document}
|
\begin{document}
\flushbottom
\title{The quantum Zeno and anti-Zeno effects with driving fields in the weak and strong coupling regimes}
\thispagestyle{empty}
\section{Introduction}
Unlike classical measurements, quantum measurements in general disturb the state of the quantum system. This back-action of measurements is a peculiar concept of quantum
mechanics which gives rise to striking phenomena such as the quantum Zeno effect (QZE).
In the QZE, repeated measurements hinder the time evolution of the quantum system\cite{Sudarshan1977,FacchiPhysLettA2000,FacchiPRL2002,
FacchiJPA2008,WangPRA2008,ManiscalcoPRL2008,FacchiJPA2010,MilitelloPRA2011,RaimondPRA2012,SmerziPRL2012,
WangPRL2013,McCuskerPRL2013,StannigelPRL2014,ZhuPRL2014,SchafferNatCommun2014,SignolesNaturePhysics2014,
DebierrePRA2015,AlexanderPRA2015,QiuSciRep2015,HePRA2018,HanggiNJP2018,HePRA2019,MullerAnnPhys}. However, it has also been observed that if the measurements are not rapid enough, a reverse effect, known as a quantum anti-Zeno effect (QAZE), can occur whereby the measurements accelerate the
quantum evolution \cite{KurizkiNature2000,RaizenPRL2001,BaronePRL2004,KoshinoPhysRep2005,BennettPRB2010,YamamotoPRA2010,ChaudhryPRA2014zeno,Chaudhryscirep2017b,HePRA2017,WuPRA2017,Chaudhryscirep2018,WuAnnals2018,ChaudhryEJPD2019a}. Both the QZE
and the QAZE have gained considerable interest theoretically and experimentally due
to their huge importance in the foundations of quantum mechanics as well as possible
applications in quantum technologies. For example, the effect of repeated measurements can be used to infer properties of the environment of a quantum system, that is, noise sensing \cite{sakuldee2020,mullerPLA2020,sakuldeePRA2020}. Generally speaking, so far, the main focus of the studies performed on QZE and the QAZE have been on the population decay model \cite{KurizkiNature2000,RaizenPRL2001,BaronePRL2004,KoshinoPhysRep2005,ManiscalcoPRL2006,SegalPRA2007,ZhengPRL2008,BennettPRB2010,YamamotoPRA2010,AiPRA2010,ThilagamJMP2010,ThilagamJCP2013} and the dephasing model \cite{ChaudhryPRA2014zeno} (notable exceptions include Refs.~{\text{Re}newcommand{}\cite[]{Chaudhryscirep2016} to time-dependent Hamiltonians. We first write the system-environment Hamiltonian as $H(t) = H_F(t) + H_{SB}$, where $H_F(t) = H_S(t) + H_B$ is the sum of the free system and environment Hamiltonians and $H_{SB}$ describes the system-environment interaction. Using the standard perturbation approach, we set $U(t)=U_F(t)U_I(t)$, where ${U}_F(t)$ describes the free time evolution of driven system and environment (this may be non-trivial and involve time-ordering due to the possible time dependence of $H_S$), and ${U}_{I}{}\cite[]{Chaudhryscirep2016,Chaudhryscirep2017a}).
What is lacking is a rigorous general study of the QZE and QAZE in the presence of coherent driving fields. The idea of controlling the coherent dynamics of a quantum system by an external time-dependent field has found
widespread theoretical and experimental interest in many areas of physics and chemistry \cite{Hanggidrivenquantumtunneling}. For example, driving fields are a commonly used tool to manipulate qubits as well as to control chemical reactions by external laser fields \cite{kofmanPRL2001,kofmanPRL2004,gordonJPB2007,gordonPRL2008}. In quantum optics, it has been shown that
a frequency-modulated excitation of a two-level atom significantly modifies the time-evolution of the system \cite{noelPRA1998} . In quantum tunneling systems, it has been demonstrated that an appropriately designed coherent drive can bring the tunneling to an (almost) complete standstill - this is known as coherent destruction of tunneling \cite{grossmannPRL1991,shaoPRA1997}. Driving fields can even effectively remove the interaction of the quantum system with its environment, which is precisely the idea behind dynamical decoupling \cite{ViolaPRA1998,LloydPRL1999,FanchiniPRA12007,ChaudhryPRA12012,ChaudhryPRA22012,ChaudhryPRA2019}. Driving fields have also been recently used in noise sensing \cite{doNJP2019}.
\begin{comment}In fact, the coherent dynamics of the two-level system is destroyed by contact with an environment \cite{Weissbook}. It is worth noting that the CDT effect survives to some extent \cite{dittrich1993driven} in the presence of the environment. Hence, it can be used to slow down the relaxation of a quantum system as well as the decoherence of a system by a suitably tuned external field \cite{eckel2009coherent,thorwartJMO2000}. Apart from that, cavity-induced spontaneous emission of two-level
the system can be suppressed either by manipulating the system-environment interaction by a fast frequency-modulation \cite{agarwalPRA1999} or controlling the cavity mode by a strong field \cite{agarwalPRA1993}. Other interesting examples of controlling the system dynamics include the long time coherent oscillations \cite{eckel2009coherent}, and enhancement of two-level system response to a weak coherent field for some optimal value of system-environment coupling strength i.e, quantum stochastic resonance (QSR) \cite{grifoni1996driven}. Such driven open systems have gained revived interest in
superconducting qubits, where all relevant parameters of the system are tunable \cite{magazzunature2018} and thus different dynamical behaviors can be
studied. Various methods like Rabi model \cite{Rabimodelpaper} and
the related Jaynes-Cummings model \cite{JaynesCummingspaper}, Landau-Zener problem \cite{ZenerLZtransition}, formation of dressed \cite{nakamuraPRL2001,wilsonPRL2007,wilsonPRB2010} and Floquet states \cite{dengPRL2015} (see also Refs \cite{XiePRA2010,BarnesPRL2012}) are proposed to control the coherent dynamics of quantum systems.
\end{comment}
It is then clear that driving fields, repeated measurements as well as the environment can all drastically influence the temporal evolution of a quantum system. Consequently, in this work, we study the QZE and the QAZE when driving fields are applied to the quantum system as well. We start by deriving a general expression of the effective decay rate for the driven quantum system, provided that the system-environment coupling is weak, thereby extending the formalism of Ref.~{\text{Re}newcommand{}\cite[]{Chaudhryscirep2016} to time-dependent Hamiltonians. We first write the system-environment Hamiltonian as $H(t) = H_F(t) + H_{SB}$, where $H_F(t) = H_S(t) + H_B$ is the sum of the free system and environment Hamiltonians and $H_{SB}$ describes the system-environment interaction. Using the standard perturbation approach, we set $U(t)=U_F(t)U_I(t)$, where ${U}_F(t)$ describes the free time evolution of driven system and environment (this may be non-trivial and involve time-ordering due to the possible time dependence of $H_S$), and ${U}_{I}{}\cite[]{Chaudhryscirep2016}. We then obtain general expressions for the decay rate of a two-level system subjected to arbitrary driving fields. In particular, we consider in detail both the population decay model and the pure dephasing model in the presence of different driving fields. For example, we show that the effective decay rate for the driven population decay model can no longer be obtained using the usual sinc-squared function (as can be done in the absence of any driving fields \cite{KurizkiNature2000}). Moreover, counter-rotating terms in the system-environment interaction Hamiltonian can become important in the presence of the driving fields, in contrast with the undriven case. We then extend our results to more than one two-level system by modeling the multiple two-level systems as a single large spin \cite{VorrathPRL2005}. We also demonstrate that our results can be extended to the strong system-environment coupling regime via the well-known polaron transformation \cite{SilbeyJCP1984,Vorraththesis,LeeJCP2012,changJCP2013,jang2008theory,ChinPRL2011,GuzikJPCL2015} along with perturbation theory. All in all, our results generally indicate that the effective decay rate is very significantly influenced by the driving fields.
\section{Results}
\subsection*{Effective rate of an arbitrary driven quantum system in the weak coupling regime}
We start by writing the total Hamiltonian of a quantum system, in the presence of driving fields, interacting with its environment as
\begin{equation}\label{D1}
\hat{H}(t) = \hat{H}_S(t) + \hat{H}_B + \hat{H}_{SB}.
\end{equation}
Here, the first term $\hat{H}_S(t)$ describes the central quantum system Hamiltonian. This carries explicit time-dependence due to the application of external driving fields on the system; consequently, we write it as the sum of a time-independent part $\hat{H}_S$ and a time-dependent part $\hat{H}_c(t)$ describing the effect of the external fields. The second term $\hat{H}_B$ corresponds to environment, whereas the last term $\hat{H}_{SB}$ is the coupling between them, which, for later convenience, we write in the diagonal form $\hat{H}_{SB} = \sum_\mu \hat{F}_\mu \otimes \hat{B}_\mu$, with the $\hat{F}_\mu$ operators belonging to the system Hilbert space and the $\hat{B}_\mu$ operators living in the environment Hilbert space. From now on, we will be suppressing the `hats' on the operators - the context should make it clear whether or not we are dealing with an operator. Keeping in mind our objective of investigating the quantum Zeno and anti-Zeno effects in such driven open quantum systems, our primary quantity of interest is the effective decay rate of the system when repeated projective measurements are performed on the system with time interval $\tau$. To calculate the effective decay rate, we assume that the system is initially prepared in the pure state $\ket{\psi}$. We then find the system density matrix at time $\tau$, that is, $\rho_S(\tau)$, and then use this to find the survival probability $s(\tau)$ that the system is still in state $\ket{\psi}$ at time $\tau$. Thereafter, we can find the effective decay rate $\Gamma(\tau)$ via $\Gamma(\tau)=-\text{ln}\,s(\tau)/\tau$. The system density matrix at time $\tau$ is obtained via $\rho_S(\tau)=\text{Tr}_B[U(\tau)\rho(0)U^{\dagger}(\tau)]$, where $\rho(0)$ is the state of total system plus environment, $\text{Tr}_B$ is the partial trace with respect to environment states and $U(\tau)$ is the total unitary time-evolution operator corresponding to the total Hamiltonian $H(t)$. Generally speaking, for the time-dependent system-environment models considered here, it is usually impossible to calculate the time-evolution operator exactly. However, for weakly coupled system-environment models, we can find the time-evolution operator $U(\tau)$ using time-dependent perturbation theory \cite{Sakuraibook}. We assume that the system-environment state is initially of a simple product form, that is, $\rho(0)=\ket{\psi}\bra{\psi}\otimes\rho_B$, where $\rho_B=e^{-\beta H_B}/Z_B$ is the thermal equilibrium state of the environment with $Z_B=\text{Tr}_B[e^{-\beta H_B}]$. Extending the treatment of Ref.~{\text{Re}newcommand{}\cite[]{Chaudhryscirep2016} to time-dependent Hamiltonians. We first write the system-environment Hamiltonian as $H(t) = H_F(t) + H_{SB}$, where $H_F(t) = H_S(t) + H_B$ is the sum of the free system and environment Hamiltonians and $H_{SB}$ describes the system-environment interaction. Using the standard perturbation approach, we set $U(t)=U_F(t)U_I(t)$, where ${U}_F(t)$ describes the free time evolution of driven system and environment (this may be non-trivial and involve time-ordering due to the possible time dependence of $H_S$), and ${U}_{I}{}\cite[]{Chaudhryscirep2016} to time-dependent Hamiltonians in a straightforward manner, we find that the decay rate of the quantum state $\ket{\psi}$, in the presence of projective measurements and for weak system-environment coupling, is given by `overlap integral' of two functions - the generalized filter function $Q(\omega,\tau)$ and the spectral density of the environment $J(\omega)$ (see the Methods section), that is,
\begin{equation}\label{Decayrate}
\Gamma(\tau) = \int_0^\infty \, d\omega \,Q(\omega,\tau) J(\omega).
\end{equation}\\
Here, the generalized filter function is given by
\begin{equation}\label{Filterfunction}
Q(\omega,\tau) =\dfrac{2}{\tau}\text{Re}\left(
\sum_{\mu \nu}\int_0^\tau dt \int_0^{t} dt' f_{\mu \nu}(\omega,t')
\text{Tr}_S\bigr[P_{\perp}\widetilde{F}_\nu(t-t')\rho_S(0)\widetilde{F}_\mu(t) \bigr]\right),
\end{equation}\\
where $F_\mu(t) = U_S^\dagger (t) H_S(t) U_S(t)$, with $U_S(t)$ being the unitary time-evolution operators corresponding to $H_S(t)$ only, and $P_{\perp}$ is the projector onto the system subspace orthogonal to $\ket{\psi}\bra{\psi}$. The environment correlation function is $C_{\mu\nu}(t) = \text{Tr}_B [\rho_B e^{iH_B t} B_\mu e^{-iH_B t} B_\nu]$, which can generally be simplified to the form $C_{\mu\nu}(t) = \sum_k |g_k|^2f_{\mu\nu}(\omega_k,t)$, where $g_k$ is the
coupling between the system and the $k^{\text{th}}$ mode of the environment. The function $f_{\mu\nu}(\omega_k,t)$ then contains the remaining information about $C_{\mu\nu}(t)$. The sum over the modes is typically converted to an integral over the environment frequencies via the substitution $\sum_k
|g_k|^2(\hdots) \rightarrow \int_0^\infty \, d\omega \, J(\omega) (\hdots.)$, thereby introducing the spectral density function $J(\omega)$ of the environment. It should be noted that $Q(\omega,\tau)$ depends not only on the frequency of the measurements, the way that system is coupled to its environment, the state of system that is repeatedly prepared, and part of the environment correlation function $f_{\mu \nu}(\omega,t)$; most importantly for us, it also depends on the driving fields applied. Similar analytical expressions to account for the effect of driving fields have been considered before \cite{kofmanPRL2001,kofmanPRL2004,gordonJPB2007,gordonPRL2008}. However, our expression takes into account the effect of both measurements as well as the concurrent application of driving fields for arbitrary system-environment models, and we do not make any assumptions regarding the driving fields such as the adiabatic approximation \cite{MilitelloPRA2019a,MilitelloPRA2019b}. We also note that there are different ways to define the survival probability and hence the decay rate, as well as different ways of identifying the Zeno and anti-Zeno regimes. For example, one can also look at the history of measurements \cite{Halliwellhistoriesreview, DankoPRA2018} when calculating the survival probability \cite{Chaudhryscirep2018}. Similarly, we identify the Zeno and anti-Zeno regimes by looking at when the decay rate $\Gamma(\tau)$ is an increasing function (the Zeno regime) or a decreasing function (the anti-Zeno regime) \cite{Chaudhryscirep2016}; an alternative approach is to compare the measurement modified decay rate with the decay rate without measurement \cite{FacchiPRL2001}.
\subsection*{General expression of the decay rate for a driven two-level system}
To apply our formalism to a two-level system, we first note that, without loss of generality, we can assume the initial state to be $\ket{e}$, where $\sigma_z\ket{e}=\ket{e}$, since we can always choose our coordinate system in this manner. We then check, with time interval $\tau$, whether or not the system is still in this state or not. The projector onto the orthogonal subspace is $\ket{g}\bra{g}$, where $\sigma_z \ket{g} = -\ket{g}$. Now comes a key insight. No matter what the external driving fields are, the system unitary time-evolution operator corresponding to $U_S(t)$ can always be written in the form
\begin{equation}\label{Er1}
U_S(t)=e^{-i \alpha(t) \sigma_z/2}e^{-i \beta(t) \sigma_y/2}e^{-i\gamma(t) \sigma_z/2},
\end{equation}
where $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ are time-dependent Euler angles. These are arbitrary functions of time with the constraint that $U_S(t = 0) = \mathds{1}$. The corresponding Hamiltonian $H_S(t)$ can be worked out from Schrodinger's equation. We find that
\begin{align}\label{EH}
H_S(t)&=\dfrac{\sigma_z}{2}\dfrac{\partial \alpha(t)}{\partial t}+\dfrac{1}{2}\dfrac{\partial \beta(t)}{\partial t}\bigr(\cos[\alpha(t)]\sigma_y-\sin[\alpha(t)]\sigma_x\bigr)+\dfrac{1}{2}\dfrac{\partial \gamma(t)}{\partial t}\cos[\beta(t)]\sigma_z \, \notag\\
&+\dfrac{1}{2}\dfrac{\partial \gamma(t)}{\partial t}\bigr(\sin[\beta(t)]\cos[\alpha(t)]\sigma_x+\sin[\beta(t)]\sin[\alpha(t)]\sigma_y\bigr).
\end{align}
In other words, by choosing the functions $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ appropriately, we can work backwards to find the corresponding system Hamiltonian.
With this form of $U_S(t)$, we can work out the effective decay rate. Using our previous general expression given in Eq.~\eqref{Filterfunction}, we find that the filter function $Q(\omega,\tau)$ is
\begin{equation}\label{filter}
Q(\omega,\tau) =\frac{2}{\tau}\text{Re} \left(\sum_{\mu\nu}\int_0^\tau dt \int_0^t dt'f_{\mu\nu}(\omega,t')G_\mu(t)\bar{G}_\nu(t-t')\right),
\end{equation}
where
\begin{align}\label{Gmu}
G_\mu(t) = e^{i(\alpha(t)+\gamma(t))}\cos^2[\dfrac{\beta(t)}{2}]F_{\mu eg}-e^{-i(\alpha(t)-\gamma(t))}\sin^2[\dfrac{\beta(t)}{2}]F_{\mu ge}+e^{i\gamma(t)}\dfrac{\sin[\beta(t)]}{2}(F_{\mu gg}-F_{\mu ee}),
\end{align}
and
\begin{align}\label{Gnu}
\bar{G}_\nu(t) =e^{-i(\alpha(t)+\gamma(t))}\cos^2[\dfrac{\beta(t)}{2}]F_{\nu ge}-e^{i(\alpha(t)-\gamma(t))}\sin^2[\dfrac{\beta(t)}{2}]F_{\nu eg}+e^{-i\gamma(t)}\dfrac{\sin[\beta(t)]}{2}(F_{\nu gg}-F_{\nu ee}),
\end{align}
with $F_{\mu ij}=\bra{i}F_{\mu}\ket{j}$. In the following sections, we use this form of the filter function with various $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ to evaluate the effective decay rate for different system-environment models.
\begin{comment}Throughout we will use ohmic spectral density $J(\omega)=G\omega e^{-\omega/\omega_c}$, where $G$ stands for the dimensionless coupling strength between system and its environment and $\omega_c$ is the cutoff frequency. Note
that behavior of $\Gamma(\tau)$ as a function of
$\tau$, allows us to identify Zeno and the anti-Zeno regimes. When $\Gamma(\tau)$ is an increasing
function of $\tau$, we are in the Zeno regime, since in this case, decreasing the measurement interval decreases the
effective decay rate. If the opposite is true, then we are in the anti-Zeno regime.
$H_S(t)$ can also be written as $H_S+H_\text{ext}(t)$, where $H_S$ is the system Hamiltonian and $H_\text{ext}(t)$ is the interaction term between the quantum system and external driving field.
Consider the interaction Hamiltonian $H_{SB}$ between system and environment of diagonal form
\end{comment}
\subsection*{Application to the driven population decay model in weak coupling regime}
To illustrate our formalism, we first consider a single two-level system interacting with an environment of harmonic oscillators in the presence of external driving fields. The total system-environment Hamiltonian is written as (we set $\hbar=1$ throughout)
\begin{equation}\label{CT}
H(t) = \frac{\varepsilon_0}{2}\sigma_z +H_c(t)+ \sum_k \omega_k b_k^\dagger b_k + \sum_k (g_k^*b_k \sigma_+ + g_k b_k^\dagger \sigma_-),
\end{equation}
where $H_{S}(t)=\frac{\varepsilon_0}{2}\sigma_z+H_c(t)$ is the system Hamiltonian with $\varepsilon_0$ representing the energy spacing of the two-level system, while $H_c(t)$ is a time-dependent external driving field acting on the system. $H_{B} =\sum_k \omega_k b_k^\dagger b_k$ is the environment Hamiltonian with $b_k$ and $b_k^\dagger$ representing the usual annihilation and creation operators, and $H_{SB} =\sum_k (g_k^*b_k \sigma_+ + g_k b_k^\dagger \sigma_-)$ is the system-environment interaction Hamiltonian with $g_k$ denoting the coupling strength between the central two-level system and the environment oscillators. As usual, $\sigma_z$ is the standard Pauli spin-1/2 matrix, and $\sigma_+(\sigma_-)$ are the raising (lowering) operators. Note that we have made the rotating-wave approximation (RWA) here for the system-environment interaction, which means that we have neglected those processes which do not conserve energy \cite{DebierrePRA2015,VegaRMP2017}.
\begin{figure}
\caption{\textbf{Filter function for the driven population decay model with the RWA}
\label{Qab}
\end{figure}
With the model specified, we now move to find the effective decay rate of the two-level system using the formalism described before. As is usually the case in studies of the quantum Zeno and anti-Zeno effects, we initially prepare our system in the excited state $\ket{e}$ such that $\sigma_z\ket{e}=\ket{e}$, and we then repeatedly check, with time interval $\tau$, whether or not the system is still in the excited state. To calculate the decay rate using our formalism, we note that $F_1 = \sigma_+$, $F_2 =\sigma_-$, $C_{\mu\nu}(t)=\text{Tr}_B[\rho_B\widetilde{B}_\mu(t)B_\nu]$, $\widetilde{B}_\mu(t)=e^{iH_Bt}B_\mu e^{-iH_Bt}$, $B_1 = \sum_k g_k^* b_k$, and $B_2 = \sum_k g_k b_k^\dagger$. In the limit of zero temperature, we find that $f_{12}(\omega,t) = e^{-i\omega t}$, while $f_{11} = f_{22} = f_{21} = 0$. Moreover, we find $
G_1(t)=e^{i(\alpha(t)+\gamma(t))} \cos^2[\beta(t)/2] $,
and
$\bar{G}_2(t-t')= e^{-i(\alpha(t-t')+\gamma(t-t'))} \cos^2[\beta(t-t')/2]$. Using these results, we obtain
\begin{equation}\label{rotating}
Q(\omega,\tau) =\frac{2}{\tau} \int_0^\tau dt \int_0^t dt'\cos[\alpha(t)-\alpha(t-t')+\gamma(t)-\gamma(t-t')-\omega t'] \cos^2[\beta(t)/2] \cos^2[\beta(t-t')/2].
\end{equation}\\
In general, this can be a very complicated function. Therefore, we first consider the simplest case where $\alpha(t)=\varepsilon_0 t$, while $\beta(t) = \gamma(t) = 0$. This corresponds to [see Eq.~\eqref{EH}] $H = \frac{\varepsilon_0}{2}\sigma_z$, that is, the usual population decay model with no driving field. After performing the integrals, we get $Q(\omega,\tau)=\tau \text{sinc}^2[\dfrac{(\varepsilon_0-\omega)\tau}{2}]$, thereby reproducing the well-known sinc-squared function for $Q(\omega,\tau)$. Our formalism, on the other hand, allows us to go much further. The next case that we can consider is $\alpha(t)=\varepsilon_0 t+(V_0/\Omega)\sin(\Omega t)$ with $\beta(t) = \gamma(t) = 0$, which corresponds to $H_S = \frac{\varepsilon_0}{2}\sigma_z$ and $H_c(t) = V_0 \cos(\Omega t)\sigma_z/2$, with $V_0$ the amplitude of the applied sinusoidal field and $\Omega$ its frequency. Using the Jacobi-Auger identity $e^{ix \sin(\Omega t)}=\sum_{l=-\infty}^{\infty}J_l(x)e^{il\Omega t}$, with $J_l(x)$ being the Bessel functions of the first kind \cite{Gradshteynbook}, we find that now
\begin{equation}\label{Bessel}
Q(\omega,\tau)=\sum_{m,n=-\infty}^{\infty}\frac{\tau}{\varepsilon_0-\omega+m\Omega}A_{mn}\bigr(\text{sinc}^2[\dfrac{(m-n)\Omega\tau}{2}](m-n)\Omega+\text{sinc}^2[\dfrac{(\varepsilon_0-\omega+n\Omega)\tau}{2}](\varepsilon_0-\omega+n\Omega)\bigr),
\end{equation}
with $A_{mn}=J_m(V_0/\Omega)J_n(V_0/\Omega)$. It is clear that the filter function is no longer a simple sinc-squared function - although the average value of the sinusoidal applied field is zero, the filter function changes in a very non-trivial manner. In particular, it is clear that the filter function $Q(\omega, \tau)$ is no longer generally peaked at $\omega = \epsilon_0$, even for changing measurement interval $\tau$. Rather, the second term in Eq.~\eqref{Bessel} makes it particularly clear that such a simple conclusion no longer holds in the driven case, and in fact the peak of the filter function changes as the measurement interval changes. Carrying on, we can also consider $\alpha(t)=\varepsilon_0 t$, with non-zero $\beta(t)$ (while $\gamma(t) = 0$). This corresponds to the driving field $H_c(t)=\dfrac{1}{2}\dfrac{\partial \beta(t)}{\partial t}\bigr(\cos[\varepsilon_0 t]\sigma_y-\sin[\varepsilon_0 t]\sigma_x\bigr)$. In these cases, $Q(\omega,\tau)$ needs to be calculated numerically, but the point is that in all such cases, the filter function, and hence the decay rate, changes in a very non-trivial manner. Similarly, we can also study non-zero values of $\gamma(t)$; the part of the Hamiltonian which contributes in $H_c(t)$ due to $\gamma(t)$ is of form $\dfrac{1}{2}\dfrac{\partial \gamma(t)}{\partial t}\cos[\beta(t)]\sigma_z+\dfrac{1}{2}\dfrac{\partial \gamma(t)}{\partial t}\bigr(\sin[\beta(t)]\cos[\alpha(t)]\sigma_x+\sin[\beta(t)]\sin[\alpha(t)]\sigma_y\bigr)$.
We now illustrate the change in the filter function as a result of these driving fields. Our results are shown in Fig. \text{Re}f{Qab} where we demonstrate the behavior of the filter function $Q(\omega,\tau)$ as a function of oscillator frequency $\omega$ for two different measurement intervals $\tau$ both with and without driving fields. We have set $\beta(t) = \gamma(t) = 0$, while $\alpha(t) = \varepsilon_0 t$ for the solid, black curve (the undriven case) and $\alpha(t) = \varepsilon_0 t +( V_0/\Omega) \sin(\Omega t)$ for the other curves (the driven cases). For the small measurement interval case illustrated in Fig.~\text{Re}f{Qab}(a), the different filter functions practically overlap - this is simply a manifestation of the convergence to the Zeno limit in the small measurement interval scenario even in the presence of the driving fields. However, for relatively large measurement interval $\tau$, the filter function for the population decay model (given by the usual sinc-squared function) is qualitatively different from the cases where we place the central system in a time-dependent external field [see Fig.~\text{Re}f{Qab}(b)]. It can be seen that for the solid, black curve (the no driving case), the filter function $Q(\omega, \tau)$ is sharply peaked at $\varepsilon_0=\omega$ for $\tau=1$, and changes very appreciably in the presence of strong driving fields (the dot-dashed green and long-dashed magenta curves). The long-dashed magenta curve corresponds to relatively lower frequency ($V_0 = 5$ and $\Omega = 1$), and to a first approximation, this filter function can be obtained by considering that the peak of the usual sinc-squared filter function is now shifted to $\varepsilon_0 + V_0$. However, for strong driving fields with higher frequencies (the dot-dashed green curve), such a naive picture is no longer applicable. Looking at Eq.~\eqref{Bessel} and using the fact that for higher frequencies, the Bessel functions are rapidly decaying so that only a few terms in the sum are important, it is clear that not only is the frequency $\omega = \varepsilon_0$ important in the filter function, but also other frequencies such as $\omega = \varepsilon_0 + \Omega$, $\omega = \varepsilon_0 - \Omega$, and so on. This leads to a much richer and complicated filter function, whose peak in fact also changes as the measurement interval $\tau$ is varied. As a result, we can expect that the effective decay rate is non trivially modified.
\begin{figure}
\caption{ \textbf{Filter function for the driven population decay model with the RWA}
\label{Q1ab}
\end{figure}
Let us now consider more complicated driving fields such that we have non-zero values of $\beta(t)$ and $\gamma(t)$. As an example, we consider $\alpha(t)=\varepsilon_0 t$ and $\beta(t)=\upsilon t$, while $\gamma(t) = 0$. This introduces oscillatory fields in the system Hamiltonian [see Eq.~\eqref{EH}], and the filter function now changes as shown in Fig.~\text{Re}f{Q1ab}(a). It is clear that adding in the control fields now greatly reduces the filter function (see the dot-dashed orange curve), and is thus expected to lead to a decrease in the effective decay rate. We can also consider what happens if these oscillating control fields are `damped' - this is illustrated by the dashed blue curve. We have checked as that as the fields become more damped, the filter function starts to coincide with the undriven scenario (the solid black curve). Proceeding along these lines, we can also work out the filter function when $\gamma(t)$ is also non-zero, further illustrating the drastic effect of the driving fields on the filter function [see Fig.~\text{Re}f{Q1ab}(b)].
\begin{figure}
\caption{ \textbf{Effective decay rate for the driven population decay model with the RWA}
\label{ab}
\end{figure}
Having illustrated the effect of the driving fields on the filter function, we now demonstrate how this translates to a change in the effective decay rate and thereby the quantum Zeno and anti-Zeno behavior. The behavior of the effective decay rate $\Gamma(\tau)$ versus measurement interval $\tau$ is shown in Fig.~\text{Re}f{ab} for different driving fields. Let us note how the behavior of $\Gamma(\tau)$ helps us to identify the quantum Zeno and anti-Zeno regimes. If the effective decay rate $\Gamma(\tau)$ decreases by shortening the measurement interval $\tau$, we are in the quantum Zeno regime while if it increases, then we are in the quantum anti-Zeno regime \cite{SegalPRA2007, ThilagamJMP2010, ChaudhryPRA2014zeno, Chaudhryscirep2016}. Moreover, to actually compute the effective decay rate, we need to specify the spectral density of the environment. Throughout this work, we will use an Ohmic spectral density $J(\omega)=G\omega e^{-\omega/\omega_c}$, where $G$ stands for the dimensionless coupling strength between the system and its environment, and $\omega_c$ is the cutoff frequency. As we have discussed before, for weak system-environment coupling, the effective decay rate is the overlap integral of the spectral density of the environment $J(\omega)$ and the generalized filter function $Q(\omega,t)$. If the peak value of the filter function is near the cutoff frequency of spectrum of environment then there will be a significant overlap between $J(\omega)$ and $Q(\omega, t)$, giving an enhanced decay rate. On the other hand, if the peak value of generalized filter function is well beyond the $\omega_c$, then that minimizes the overlap between $J(\omega)$ and $Q(\omega, t)$ , leading to a reduced decay rate. The dynamically modified filter function in the presence of driving fields [see Figs.~\text{Re}f{Qab} and \text{Re}f{Q1ab}] affects the overlap of $J(\omega)$ with $Q(\omega,t)$, and thus can either accelerate or inhibit the decay rate as compared to the undriven scenario. In particular, it is clear from Fig.~\text{Re}f{ab}(a) that for the simple population decay case with no driving fields (the solid black curve), there is a single local crossover between quantum Zeno and anti-Zeno regime, meaning that, in short time regime, effective decay rate $\Gamma(\tau)$ decreases by decreasing the measurement interval $\tau$ while for large measurement interval, it increases by decreasing the measurement interval. In the presence of driving fields, not only is the effective decay rate greatly affected (see the long-dashed magenta curve), but also there are multiple Zeno and anti-Zeno regimes (see the dashed red curve and the dot-dashed green curve). For long-dashed magenta curve, the peak value of the filter function is at approximately $\omega=5.6$ [see Fig.~\text{Re}f{Qab}(b)] which is more near to the peak of the spectrum of the environment as compared to the solid black curve (for which the peak is at $\omega=1$). As a result, this gives maximum overlap of $Q(\omega,\tau)$ with $J(\omega) $ for the magenta curve, which consequently enhances the effective decay rate compared to the undriven case. For the dot-dashed green curve, we observed previously that a large driving field frequency means that the peak of the filter function keeps changing as the measurement interval changes, and this leads to multiple Zeno and anti-Zeno regimes. However, at short measurement intervals, all the curves agree. This is expected, since, as we have seen before, with very fast measurements, the filter function becomes the same, leading to the same decay rate. We have also looked at what happens with $\beta(t) \neq 0$. Having seen how the filter function is influenced by such driving fields in Fig.~\text{Re}f{Q1ab}(a), we illustrate what happens to the corresponding decay rate in Fig.~\text{Re}f{ab}(b) for the same $\beta(t)$ as used in Fig.~\text{Re}f{Q1ab}(a), where $\beta(t)$ is a damped function for the dashed blue curve, and it is a linear function of $t$ for the dot-dashed orange curve. Since a linear function of $t$ in the presence of non-zero $\varepsilon_0$ leads to a reduction in the peak value of the filter function [see Fig.~\text{Re}f{Q1ab}(a)], the overlap between $Q(\omega,\tau)$ and $J(\omega)$ reduces for $\omega_c=10$, leading to a reduction in the effective decay rate as compared to the solid black curve. On the other hand, if $\beta(t)$ is a damped function, effects of the oscillating fields are suppressed, meaning that the effective decay rate is increased for the dashed blue curve as compared to dot-dashed orange curve. Once again, it is clear that the driving fields greatly influence the decay rate both quantitatively and qualitatively.
We next discuss the effect of the non-rotating terms of the system-environment interaction on the dynamics of the central system in the presence of a driving field. The total Hamiltonian is now given by
\begin{equation}
H(t) = \frac{\varepsilon_0}{2}\sigma_z +H_c(t)+ \sum_k \omega_k b_k^\dagger b_k +\sigma_x \sum_k (g_k^*b_k + g_k b_k^\dagger).
\end{equation}\\
Notice the different form of the system-environment coupling as compared to before - the system-environment Hamiltonian now contains the `non-rotating' terms $\sigma_+ b_k^\dagger$ and $\sigma_- b_k$. To calculate filter function $Q(\omega,t)$ now, we first evaluate the environment correlation function. With $F=\sigma_x$, we find $G(t)=- e^{-i(\alpha(t)-\gamma(t))} \sin^2[\beta (t)/2]+e^{i(\alpha(t)+\gamma(t))}\cos^2[\beta (t)/2]$ and $\bar{G}(t-t')=- e^{i(\alpha(t-t')-\gamma(t-t'))} \sin^2[\beta(t-t')/2]+e^{-i(\alpha(t-t')+\gamma(t-t'))}\cos^2[\beta(t-t')/2]$. Also, $f(\omega,t) = e^{-i\omega t}$ at zero temperature. Putting all this together,
\begin{align}\label{nonrotating}
Q(\omega,\tau) =\frac{2}{\tau} \int_0^\tau dt \int_0^t dt'\bigr( D_1(t,t')+D_2(t,t')+D_3(t,t')+D_4(t,t')\bigr),
\end{align}\\
where
\begin{align}\label{JQ1}
D_1(t,t')=\cos[\gamma(t)-\gamma(t-t')-\omega t']\cos[\alpha(t)]\cos[\beta(t)]\cos[\alpha(t-t')]\cos[\beta(t-t')],
\end{align}
\begin{equation}\label{JQ2}
D_2(t,t')=-\sin[\gamma(t)-\gamma(t-t')-\omega t']\cos[\alpha(t-t')]\cos[\beta(t-t')]\sin[\alpha(t)],
\end{equation}
\begin{equation}\label{JQ3}
D_3(t,t')=\sin[\gamma(t)-\gamma(t-t')-\omega t']\cos[\alpha(t)]\cos[\beta(t)]\sin[\alpha(t-t')],
\end{equation}
\begin{equation}\label{JQ4}
D_4(t,t')=\cos[\gamma(t)-\gamma(t-t')-\omega t']\sin[\alpha(t)]\sin[\alpha(t-t')].
\end{equation}
Compared with Eq.~\eqref{rotating}, we can see that the two filter functions agree for $\beta(t)=0$. This means that in the absence of driving fields (where $\alpha(t) = \varepsilon_0 t$ and $\beta(t) = 0$), the counter-rotating terms do not affect the decay rate. However, in the presence of driving fields with $\beta(t) \neq 0$, the counter-rotating terms become important, even in the weak system-environment coupling regime we are dealing with. The influence of the non-rotating terms is shown in Fig.~\text{Re}f{Rab}, where the behavior of the filter function as a function of the frequency $\omega$ is shown in Fig.~\text{Re}f{Rab}(a) and the behavior of the effective decay rate as a function of measurement interval is illustrated in Fig.~\text{Re}f{Rab}(b). Comparing Figs.~\text{Re}f{Q1ab}(a) and Fig.~\text{Re}f{Rab}(a), it is clear that when there are no driving fields, the filter function does not change since the solid, black curve is the same in both figures. However, as shown by the dashed blue and dot-dashed orange curves, in the presence of driving fields with $\beta(t) \neq 0$, the filter function does change. This correspondingly changes the effective decay rate by modifying the overlap of $J(\omega)$ with $Q(\omega, t)$ as can be seen by comparing Figs.~\text{Re}f{ab}(b) and \text{Re}f{Rab}(b). The counter-rotating terms help to enhance the peak value of the filter function, leading to an increase in the effective decay rate.
\begin{figure}
\caption{\textbf{Filter function and effective decay rate for the driven population decay model without the RWA}
\label{Rab}
\end{figure}
\subsection*{Application to the driven dephasing model in weak coupling regime}
We now consider the pure dephasing model given by the system-environment Hamiltonian
\begin{equation}\label{ddr}
H = \frac{\varepsilon_0}{2} \sigma_z+ \sum_k \omega_k b_k^\dagger b_k +\sigma_z \sum_k (g_k^*b_k + g_k b_k^\dagger).
\end{equation}
Notice the different form of the system-environment interaction. With this model, the populations of the system energy eigenstates cannot change - only the off-diagonal coherences can change, which is why this is referred to as a pure dephasing model\cite{ChaudhryPRA2014zeno}. The initial state usually considered in this model is $\ket{\psi}=\dfrac{1}{\sqrt{2}}(\ket{e}+\ket{g})$, with $\bra{e}g \rangle=0$. However, with the formalism we have developed, the initial state we considered was $\ket{e}$. To use our formalism, we consequently perform a unitary operation $U_R = e^{i\pi \sigma_y/4}$. The initial state then again becomes $\ket{e}$, while the Hamiltonian is transformed to
\begin{equation}\label{rotated}
H^{(R)} = U^{(R)}HU^{(R)\dagger} = -\frac{\varepsilon_0}{2} \sigma_x+ \sum_k \omega_k b_k^\dagger b_k -\sigma_x \sum_k (g_k^*b_k + + g_k b_k^\dagger).
\end{equation}
To find the filter function now, we look at Eq.~\eqref{EH} and find that $\alpha(t) = \pi/2$, $\beta(t) = \varepsilon_0 t$ and $\gamma(t) = -\pi/2$ gives the same Hamiltonian as Eq.~\eqref{rotated}. Then, using our developed formalism, we find that $G(t)=1$ and $\bar{G}(t-t')=1$. Consequently, assuming that the environment is at zero temperature, we get $Q(\omega,\tau)=\frac{2}{\tau}\frac{1-\cos(\omega \tau)}{\omega^2}$, which agrees with the filter function obtained using the exact solution \cite{Chaudhryscirep2016}. Next, we add in the effect of the driving fields. To this end, we look at more complicated time-dependent functions $\alpha(t)$, $\beta(t)$, and $\gamma(t)$. We write the corresponding system-environment Hamiltonian as
\begin{equation}\label{PN}
H(t) = -\frac{\varepsilon_0}{2} \sigma_x+ H_c(t)+\sum_k \omega_k b_k^\dagger b_k -\sigma_x \sum_k (g_k^*b_k + + g_k b_k^\dagger).
\end{equation}
To take into account the additional control fields given by $H_c(t)$, we write $\alpha(t) = \pi/2+ \widetilde{\alpha}(t)$ and $\gamma(t)=-\pi/2+\widetilde{\gamma}(t)$, while $\beta(t)$ remains $\varepsilon_0 t$. Simple calculations then lead to the filter function
\begin{align}
Q(\omega,\tau) =\frac{2}{\tau} \int_0^\tau dt \int_0^t dt'\bigr( D_1(t,t')+D_2(t,t')+D_3(t,t')+D_4(t,t')\bigr),
\end{align}\\
where now
\begin{align}
D_1(t,t')=\cos[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t')-\omega t']\sin[\widetilde{\alpha}(t)]\cos[\beta(t)]\sin[\widetilde{\alpha}(t-t')]\cos[\beta(t-t')],
\end{align}
\begin{equation}
D_2(t,t')=\sin[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t')-\omega t']\sin[\widetilde{\alpha}(t-t')]\cos[\beta(t-t')]\cos[\widetilde{\alpha}(t)],
\end{equation}
\begin{equation}
D_3(t,t')=-\sin[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t')-\omega t']\sin[\widetilde{\alpha}(t)]\cos[\beta(t)]\cos[\widetilde{\alpha}(t-t')],
\end{equation}
\begin{equation}
D_4(t,t')=\cos[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t')-\omega t']\cos[\widetilde{\alpha}(t)]\cos[\widetilde{\alpha}(t-t')].
\end{equation}
Using these expressions, we have plotted the filter function (for $\tau = 1$) in Fig.~\text{Re}f{dab}(a) for different control fields. Once again, it is clear that the driving fields greatly influence the filter function in general. For instance, with a sinusoidal driving field ($\widetilde{\alpha}(t)=(V_0/\Omega) \sin(\Omega t)$, $\beta(t)=\varepsilon_0 t$, and $\widetilde{\gamma}(t)=0$), the filter function is very different as compared with the undriven case (compare the solid black curve with the dot-dashed green and long-dashed magenta curves), with the difference becoming smaller as the driving field strength is reduced (see the dashed red curve). Consequently, the behavior of effective decay rate and the corresponding quantum Zeno and anti-Zeno phenomena is expected to be greatly modified due to the different overlap of the filter function with the spectrum of the environment. That this is indeed the case can be seen in Fig.~\text{Re}f{dab}(b). We see that there is a single peak in the case of the undriven pure dephasing model. On the other hand, due to the application of external fields, the effective decay rate increases with increasing $V_0$ and the effective decay rate shows multiple Zeno and anti-Zeno regimes for fast oscillating external fields.
\begin{figure}
\caption{ \textbf{Filter function and effective decay rate for the driven dephasing model}
\label{dab}
\end{figure}
\subsection*{Application to the driven large spin-boson model in weak coupling regime}
We now briefly show how we can extend our formalism to more general systems in which a large spin greater than spin-1/2
is coupled to a harmonic oscillator environment. Such a model can describe, for instance, a collection of $N_S$ two-level systems coupled to a common environment\cite{ChaudhryPRA2014zeno,VorrathPRL2005,KurizkiPRL2011}. We first define $J_k=\dfrac{1}{2} \sum_i\sigma_{k}^{(i)}$, where $J_k$ ($k = x, y, z$) are the large spin operators. As a concrete example, we consider the driven population decay model given by the system-environment Hamiltonian
\begin{equation}\label{NTWA}
H(t) = {\varepsilon_0}J_z +H_c(t) +\sum_k \omega_k b_k^\dagger b_k + 2 J_x \sum_k (g_k^*b_k + g_k b_k^\dagger),
\end{equation}
where $\varepsilon_0$ is the energy level spacing for each spin-1/2 particle, and $H_c(t)$ is the control field Hamiltonian. Analogous to what we did for the single spin-1/2 case, we consider the free system unitary time evolution operator to be $U_S(t) = e^{-i\alpha(t) J_z} e^{-i\beta(t) J_y} e^{-i\gamma(t)J_z}$. We take the initial state to be $\ket{j}$ with $J_z \ket{j} = j\ket{j}$ and $j = N_S/2$. Performing the calculation for the filter function using our formalism, we find that the filter function is exactly the same as for the single spin-1/2 case [see Eq.~\eqref{nonrotating}] except for an additional multiplicative factor of $N_S$ (see the Methods section for details). That is, the effective decay rate is now enhanced by a factor of $N_S$, reminiscent of the superradiance effect \cite{Dicke1954}. A similar calculation shows that if the pure dephasing model is extended to the large spin case in an analogous manner, the effective rate is again enhanced by a factor of $N_S$.
\subsection*{Application to the driven spin-boson model in the strong system-environment coupling regime}
Finally, let us extend our treatment of the driven population decay model to the strong system-environment case. We start from the system-environment Hamiltonian $H(t) = H_S(t) + H_B + H_{SB}$ with $H_{S}(t)=\dfrac{\varepsilon(t)}{2}\sigma_z+\dfrac{\Delta}{2}\sigma_x$, $H_{B}=\sum_k\omega_kb_k^\dagger b_k$, and $H_{SB}=\sigma_z\sum_k(g^*_kb_k+g_kb^\dagger_k)$. The driving fields are contained in $\varepsilon(t)$. Now, if the system and the environment are interacting strongly, we cannot treat the interaction term between system and its environment perturbatively. Instead, we can consider performing a polaron transformation \cite{Vorraththesis,ChinPRL2011,LeeJCP2012,GuzikJPCL2015,Chaudhryscirep2017a}, which transforms our Hamiltonian to (see the supplementary material)
\begin{equation}\label{PT}
H^{(P)}(t)=e^{\chi\sigma_z/2}H(t)e^{-\chi\sigma_z/2} = \dfrac{\varepsilon(t)}{2}\sigma_z+\sum_k\omega_kb^\dagger_kb_k+\dfrac{\Delta}{2}(\sigma_+Y+\sigma_-Y^\dagger),
\end{equation}\\
where $P$ denotes the so-called polaron frame, $\chi=\sum_k[\dfrac{2g_k}{\omega_k}b_k^\dagger-\dfrac{2g^*_k}{\omega_k}b_k]$, and $Y = e^\chi$.
We see that in the polaron frame, the form of the system-environment interaction is different. Now, if the tunneling parameter $\Delta$ is small, we can apply time-dependent perturbation theory, treating the system-environment coupling in the polaron frame perturbatively. As before, at the initial time, we prepare our system in the excited state $\ket{e}$, and then perform projective measurements on the system with time interval $\tau$ to check whether the system is still present in excited state $\ket{e}$ or not. It is also important to note that initial state of system and environment in the untransformed frame cannot be written in the simple usual product form $\rho(0)=\ket{e}\bra{e}\otimes e^{-\beta H_B}/Z_B$, with $\rho_B=e^{-\beta H_B}/Z_B$ and $Z_B=\text{Tr}_B[e^{-\beta H_B}]$, since the system and the environment are strongly interacting in that frame and consequently, the initial system-environment correlations are significant \cite{PollakPRE2008,ChaudhryPRA2013a,ChaudhryPRA2013b,ChaudhryEJPD2019b,ChaudhryPRA2020}. However, since the system and its environment are effectively weakly interacting in the polaron frame, the initial state in the polaron frame can be taken as a simple product state. The rest of the calculation, performed in the polaron frame, proceeds in a similar way as the weak coupling case using perturbation theory. We eventually arrive at (see the supplementary Material)
\begin{equation}\label{PRI}
\Gamma(\tau)=\dfrac{\Delta^2}{2\tau}\int_0^\tau dt\int_0^t dt'e^{-\Phi_R(t')}\cos[\zeta(t)-\zeta(t-t')-\Phi_I(t')],
\end{equation}
where $\Phi_I(t)=\int_0^\infty d\omega J(\omega)\dfrac{\sin(\omega t)}{\omega^2}$, $\Phi_R(t)=\int_0^\infty d\omega J(\omega)\dfrac{1-\cos(\omega t)}{\omega^2}\coth(\dfrac{\beta\omega}{2})$, and $\zeta(t) = \int_0^t \varepsilon(t')\, dt'$.
Assuming, as before, an Ohmic spectral density of the form $J(\omega)=G\omega e^{-\omega/\omega_c}$, $\Phi_I(t)$ and $\Phi_R(t)$ are found to be (at zero temperature) $\Phi_I(t)=G\tan^{-1}(\omega_ct)$ and $\Phi_R(t)=\frac{G}{2}\ln(1+\omega_c^2t^2)$.
\begin{figure}
\caption{ \textbf{Effective decay rate for the strongly coupled driven spin-boson model}
\label{Pab}
\end{figure}
We now have everything we need to calculate the effective decay rate. It should be obvious from our expressions above that for the strong system-environment coupling regime that we are considering here, the effective decay rate $\Gamma(\tau)$ no longer depends on an overlap integral of the generalized filter function with the spectral density of environment. Rather, there is now a non-linear dependence on the spectral density, leading to very different qualitative behavior as compared with the usual weak system-environment coupling regime. For instance, as
the system-environment coupling strength increases, $e^{-\Phi_R(t)}$ decreases, and thus we anticipate $\Gamma(\tau)$ to decrease. Most importantly for us, we expect that the driving fields have a drastic, non-trivial effect not only the value of the effective decay rate, but also quantum Zeno and anti-Zeno behavior since the integrand in Eq.~\eqref{PRI} obviously depends on the driving fields. To make these claims concrete, let us show the quantitative behavior of effective decay rate $\Gamma(\tau)$.
It is clear from Fig.~\text{Re}f{Pab} that not only the driving fields affect the decay rate very significantly, but also that increasing the system-environment coupling strength $G$ reduces the effective decay rate [compare Figs.~\text{Re}f{Pab}(a) and (b)], in contrast with the weak coupling regime. Here again, we observe multiple oscillations in quantum Zeno
to anti-Zeno regimes, as we increase $\Omega$.
Interestingly, Fig.~\text{Re}f{ab}(a) looks very similar with Fig.~\text{Re}f{Pab}(b) despite having coupling strength $G$ in different regimes. This similarity can be understood noting that in both cases, the pointer states are the eigenstates of the operator $\sigma_z$, and that in the strong coupling regime, we have a population decay model in the polaron frame (see Eq.~\eqref{PT}). In both cases, the driving fields modulate the energy-level splitting. These similarities leads to the same qualitative form of Fig.~\text{Re}f{ab}(a) and Fig.~\text{Re}f{Pab}(b).
\section{Discussion}
In this paper, we started off by introducing a general formalism to calculate the effective decay rate of a quantum system subjected to both periodic projective measurements and driving fields, valid for weak system-environment coupling strength. We then applied this formalism to derive a general expression for the decay rate of a driven two-level system. The decay rate is an overlap integral of the spectral density of environment and a filter function expressed using time-dependent Euler angles. These results were illustrated using the population decay model as well as the pure dephasing model. In both cases, the application of the driving fields very significantly changes the form of filter function of the central system, which then modifies (either enhances or minimizes) the effective decay rate and, consequently, the quantum Zeno and anti-Zeno regimes are altered. Interestingly, for the population decay model, the driving fields can lay bare the effect of the counter-rotating terms, even for the weak system-environment coupling regime that we are dealing with. These results were then generalized to large spin systems to show a possible amplification of the decay rate. Finally, we also looked at driven two-level systems strongly coupled to an environment of harmonic oscillators, where the effective decay rate shows a non-linear
dependence on the spectral density of the environment. We showed once again that the effective decay rate is modified by the application of driving fields. Our general expressions and insights should of interest in the broad areas of quantum control and quantum state engineering, such as quantum noise sensing, as well as in fundamental studies of the quantum Zeno and anti-Zeno effects. For example, a quantum system can be put into the Zeno regime, thereby protecting it from decoherence, by applying suitable control fields. On the other hand, the decay rate can be enhanced in the anti-Zeno regime via the applied control fields, and this can be useful for cooling the quantum system \cite{Kurizki2015}.
\section{Methods}
\subsection*{Effective decay rate using perturbation theory}
We first discuss how to obtain effective decay rate of system under frequent measurements, extending the treatment in Ref.~{\text{Re}newcommand{}\cite[]{Chaudhryscirep2016} to time-dependent Hamiltonians. We first write the system-environment Hamiltonian as $H(t) = H_F(t) + H_{SB}$, where $H_F(t) = H_S(t) + H_B$ is the sum of the free system and environment Hamiltonians and $H_{SB}$ describes the system-environment interaction. Using the standard perturbation approach, we set $U(t)=U_F(t)U_I(t)$, where ${U}_F(t)$ describes the free time evolution of driven system and environment (this may be non-trivial and involve time-ordering due to the possible time dependence of $H_S$), and ${U}_{I}{}\cite[]{Chaudhryscirep2016} to time-dependent Hamiltonians. We first write the system-environment Hamiltonian as $H(t) = H_F(t) + H_{SB}$, where $H_F(t) = H_S(t) + H_B$ is the sum of the free system and environment Hamiltonians and $H_{SB}$ describes the system-environment interaction. Using the standard perturbation approach, we set $U(t)=U_F(t)U_I(t)$, where ${U}_F(t)$ describes the free time evolution of driven system and environment (this may be non-trivial and involve time-ordering due to the possible time dependence of $H_S$), and ${U}_{I}(t)$ is the leftover part that can be expanded perturbatively (assuming the system-environment interaction to be weak). Expanding up to second order in system-environment
coupling, we can write ${U}_{I}(t)=\mathds{1}+{G}_{1}+{G}_{2}$, where $G_1$ and $G_2$ are the first, and second order corrections respectively. Using this expansion, the density matrix at time $\tau$ can be written as
\begin{align*}\label{NTS6}
{\rho}_{S}(\tau)\approx\text{Tr}_{B}\bigr[{U}_{F}(\tau)\bigr({\rho}(0)+{\rho}(0){G}^{\dagger}_{1}+{G}_{1}{\rho}(0)+{\rho}(0){G}^{\dagger}_{2}+{G}_{2}{\rho}(0)+{G}_{1}{\rho}(0){G}^{\dagger}_{1}\bigr){U}^{\dagger}_{F}(\tau)\bigr].
\end{align*}
Perturbation theory tells us that ${G}_{1}=-i\int^{\tau}_{0}dt_{1}\widetilde{H}_{SB}(t_{1})$, and ${G}_{2}=-\int^{\tau}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\widetilde{H}_{SB}(t_{1})\widetilde{H}_{SB}(t_{2})$, where $\widetilde{H}_{SB}(t)={U}^{\dagger}_{F}(t){{H}}_{SB} {U}_{F}(t)=\sum_\mu{U}_{S}^{\dagger}(t)F_\mu {U}_{S}(t)\otimes{{U}}^{\dagger}_{B}(t)B_\mu{{U}}_{B}(t)={\widetilde{F}_\mu}(t){\widetilde{B}}_\mu(t)$. Each term can then be simplified to eventually arrive at
\begin{equation}
\rho_S(\tau) = U_S(\tau) \biggl( \rho_S(0) + \sum_{\mu \nu}\int_0^\tau dt_1 \int_0^{t_1} dt' C_{\mu \nu}(t')
[\widetilde{F}_\nu(t_1-t')\rho_S(0),\widetilde{F}_\mu(t_1)] + \text{h.c.} \biggr) U_S^\dagger (\tau).
\label{densitymatrixattau}
\end{equation}
where $C_{\mu \nu}(t') = \text{Tr}_B [\frac{e^{-\beta H_B}}{Z_B} \widetilde{B}_\mu(t')\widetilde{B}_\nu(0)]$ are the environment correlation functions, and $\text{h.c.}$ denotes hermitian conjugate.
Once we have the expression for the system density matrix at time $\tau$, we can evaluate the survival probability of the system in the initial state. Since we are interested to investigate the system evolution due to system-environment interaction only, we first apply the free driven system unitary operator on both sides of Eq.~\eqref{densitymatrixattau} to remove the system evolution due to free driven system Hamiltonian itself. Note that this may not be necessary if $U_S(\tau)$ commutes with $\rho_S(0)$. We thereafter find the probability that the system is still in the initial state $\ket{\psi}$ after a projective measurement given by the projector $\ket{\psi}\bra{\psi}$ to be given by
\begin{align}
s(\tau) = 1-2\text{Re}\,\biggr(
\sum_{\mu \nu}\int_0^\tau dt \int_0^{t} dt' C_{\mu \nu}(t')
\text{Tr}_S\bigr[P_{\perp}\widetilde{F}_\nu(t-t')\rho_S(0)\widetilde{F}_\mu(t) \bigr]\biggr),
\end{align}\\
where $P_{\perp}$ is the projector onto the subspace orthogonal to $\ket{\psi}\bra{\psi}$. After performing a sequence of $M$ repeated projective measurements, we find the survival probability that the system state is still present in the initial state is $S(M\tau)=[s(\tau)]^M$ if the system-environment correlations are ignored during the evolution. We can then find the effective decay rate $\Gamma(\tau)$ by $S(M\tau)=e^{-\Gamma(\tau)M\tau}$ allowing us to write $\Gamma(\tau)=-\ln s(\tau)/\tau$. In the weak system-environment coupling regime, we can further write
\\\begin{align*}\label{EDRT}
\Gamma(\tau) =\frac{2}{\tau}\text{Re}\biggr(
\sum_{\mu \nu}\int_0^\tau dt \int_0^{t} dt' C_{\mu \nu}(t')
\text{Tr}_S\bigr[P_{\perp}\widetilde{F}_\nu(t-t')\rho_S(0)\widetilde{F}_\mu(t) \bigr]\biggr).
\end{align*}\\
This can be cast into the form
\begin{equation*}
\Gamma(\tau) = \int_0^\infty \, d\omega \,Q(\omega,\tau) J(\omega),
\end{equation*}\\
with $Q(\omega,\tau)$ given in Eq.~\eqref{Filterfunction}.
\subsection*{Finding the filter function for the driven large-spin population decay model}
In this case, using the standard angular momentum relations $[J_i,J_j]=i\varepsilon_{ijk}J_k$, we evaluate
$$\widetilde{F}(t) =2 \bigr(J_xc_x(t)+J_yc_y(t)+J_zc_z(t)\bigr),$$
where
\begin{align}
c_x(t)&= \cos[\alpha(t)]\cos[\beta(t)\cos[\gamma(t)]-\sin[\alpha(t)]\sin[\gamma(t)], \notag \\ c_y(t)&=\cos[\alpha(t)]\cos[\beta(t)\sin[\gamma(t)]+\sin[\alpha(t)]\cos[\gamma(t)], \notag \\ c_z(t) &=-\cos[\alpha(t)]\sin[\beta(t)].
\end{align}
For $\rho_S(0)=\ket{j,j}\bra{j,j}$ and $P_{\perp}=\sum_{m=1}^{j-1} \ket{j,m}\bra{j,m}$ with $J_z\ket{j,m}=m\ket{j,m}$, we have $$\text{Tr}_S\bigr[P_{\perp}\widetilde{F}(t-t')\rho_S(0)\widetilde{F}(t) \bigr]=\sum_{m=1}^{j-1} \bra{j,m}\widetilde{F}(t-t')\ket{j,j}\bra{j,j}\widetilde{F}(t)\ket{j,m}.$$
We next note that $\bra{j,j}\widetilde{F}(t)\ket{j,m}= \sqrt{2j}\delta_{j-1,m}\bigr(c_x(t)-ic_y(t)\bigr)$. This leads to the generalized filter function
\begin{equation*}
Q(\omega,t)=(2j)\frac{2}{\tau} \int_0^\tau dt \int_0^t dt'\bigr( D_1(t,t')+D_2(t,t')+D_3(t,t')+D_4(t,t')\bigr),
\end{equation*} \\
where expressions of $D_1(t,t')$, $D_2(t,t')$, $D_3(t,t')$ and $D_4(t,t')$ are defined in Eqs.~\eqref{JQ1}-\eqref{JQ4}.
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\section*{Author contributions statement}
A.~Z.~C. came up with the basic idea behind this work. M.~M.~ carried out the calculations and plotted the graphs. Both authors contributed towards the writing of the manuscript.
\section*{Additional information}
\textbf{Competing interests:} The authors declare no competing interests.
\section*{Supplemental Material for `The quantum Zeno and anti-Zeno effects with driving fields in the weak and strong coupling regimes'}
\noindent In this supplemental Material, we use same symbols as introduced in our main text. For completeness, we present
the analytical expressions for the Hamiltonian in the polaron frame. We also outline the calculation of the decay rate $\Gamma(\tau)$ for a single two-level system interacting strongly with an environment of harmonic oscillators.
\subsection*{Spin-boson Hamiltonian in polaron frame}
To transform spin-boson Hamiltonian to polaron frame, we need to find
\begin{equation*}\label{PF}
H^{(P)}(t)=e^{\chi\sigma_z/2}H(t)e^{-\chi\sigma_z/2}.
\end{equation*}
We use the Hadamard lemma
\begin{equation}\label{BCH}
e^AOe^{-A}=O+[A,O]+\dfrac{1}{2!}[A,[A,O]]+....
\end{equation}\\
where $A=\chi\sigma_z/2$, with $\chi=\sum_k[\dfrac{2g_k}{\omega_k}b_k^\dagger-\dfrac{2g^*_k}{\omega_k}b_k]$ and $O=H(t)=\dfrac{\varepsilon(t)}{2}\sigma_z+\dfrac{\Delta}{2}\sigma_x+\sum_k\omega_kb_k^\dagger b_k+\sigma_z\sum_k(g^*_kb_k+g_kb^\dagger)$. \\
We find that
\begin{equation*}
e^{\chi\sigma_z/2}\sigma_ze^{-\chi\sigma_z/2}=\sigma_z,
\end{equation*}
and
\begin{align*}
e^{\chi\sigma_z/2}\sigma_xe^{-\chi\sigma_z/2}=\sigma_+e^{\chi}+\sigma_-e^{-\chi}.
\end{align*}
$\sigma_-$ and $\sigma_+$ are the standard spin-1/2 lowering and raising operators. Carrying on further, we find
\begin{equation*}
e^{\chi\sigma_z/2}(\sum_k\omega_kb^\dagger_kb_k)e^{-\chi\sigma_z/2}=\sum_k\omega_kb^\dagger_kb_k-\sigma_z\sum_k(g^*_kb_k+g_kb^\dagger)+\sum_k\dfrac{|g_k|^2}{\omega_k}.
\end{equation*}
Similarly
\begin{equation*}
e^{\chi\sigma_z/2}\biggr(\sigma_z\sum_k(g^*_kb_k+g_kb^\dagger)\biggr)e^{-\chi\sigma_z/2}=\sigma_z\sum_k(g^*_kb_k+g_kb^\dagger)-2\sum_k\dfrac{|g_k|^2}{\omega_k},
\end{equation*}
The third term of Eq. (\text{Re}f{BCH}) in the above expression is a constant number, so higher order commutators are zero. Now putting all these terms back together, the required Hamiltonian in polaron frame takes the following form\\
\begin{equation*}
H^{(P)}(t)=\dfrac{\varepsilon(t)}{2}\sigma_z+\sum_k\omega_kb^\dagger_kb_k+\dfrac{\Delta}{2}(\sigma_+Y+\sigma_-Y^\dagger)-\sum_k\dfrac{|g_k|^2}{\omega_k},
\end{equation*}\\
with $ Y=e^\chi$, and $\sum_k\dfrac{|g_k|^2}{\omega_k}$ is a constant number term that gives a constant shift in transformed Hamiltonian, and can thus be dropped.
\subsection*{Effective decay rate of spin-boson model in polaron frame}
Since system-environment coupling is weak in the polaron frame, we can use expression [see the main text] \\
$$
\Gamma(\tau) =\frac{2}{\tau}\text{Re}\biggr(
\sum_{\mu \nu}\int_0^\tau dt \int_0^{t} dt' C_{\mu \nu}(t')
\text{Tr}_S\bigr[P_{\perp}\widetilde{F}_\nu(t-t')\rho_S(0)\widetilde{F}_\mu(t) \bigr]\biggr),$$\\
to calculate effective decay rate . Here $\rho_S(0)=\ket{e}\bra{e}$ and $P_\perp=\ket{g}\bra{g}$. We identity $F_1=\dfrac{\Delta}{2}\sigma_+$, $F_2=\dfrac{\Delta}{2}\sigma_-$, $B_1=Y$, $B_2=Y^\dagger$, $\widetilde{F}_1(t)=\dfrac{\Delta}{2}\sigma_+e^{i\zeta(t)}$ and $\widetilde{F}_2(t)=\dfrac{\Delta}{2}\sigma_-e^{-i\zeta(t)}$ with $\zeta(t)=\int_0^tdt'\varepsilon(t')$ leading us to
\begin{align}\label{C12}
\Gamma(\tau) =\frac{2}{\tau}\text{Re}\biggr(
\int_0^\tau dt \int_0^{t} dt' C_{12}(t') e^{i(\zeta(t)-\zeta(t-t'))} \biggr).
\end{align}\\
To get expression of effective decay rate, the environment correlation function $C_{12}(t)$ needs to be worked out. We now show the details how to find $C_{12}$. As we know $C_{12}(t)=\text{Tr}_B[\rho_B\widetilde{B}_1(t)B_2]$, with $B1=Y$, $B_2=Y^\dagger$, $\widetilde{B}_1(t)=e^{iH_B^{(P)}t}Y e^{-iH_B^{(P)t}}$, $H^{(P)}_B=\sum_k\omega_kb^\dagger_kb_k$, $Y=e^\chi$ and $\chi=\sum_k[\dfrac{2g_k}{\omega_k}b_k^\dagger-\dfrac{2g^*_k}{\omega_k}b_k]$. Next, we calculate
\begin{equation*}
\widetilde{B}_1(t)=e^{\sum_k\biggr(\frac{g_k}{\omega_k}b^\dagger_ke^{i\omega_kt}-\frac{g^*_k}{\omega_k}b_ke^{-i\omega_kt}\biggr)},
\end{equation*}
\\using the fact $U^\dagger(t)e^AU(t)=e^{U^\dagger(t)AU(t)}$, and then find
\begin{equation*}
\widetilde{B}_1(t)B_2=e^{-i\sum_k\dfrac{|g_k|^2}{\omega_k^2}\sin(\omega_kt)}e^{\sum_k\biggr(\frac{g_k}{\omega_k}b^\dagger_k(e^{i\omega_kt}-1)+\frac{g^*_k}{\omega_k}b_k(e^{-i\omega_kt}-1)\biggr)}.
\end{equation*} In order to convert double exponential in a single exponential to use useful fact $\text{Tr}_B[\rho_Be^C]=e^{\langle C^2\rangle/2}$, where operator $C$ is a linear combination of annihilation and creation operators; we use the identity $e^Xe^Y=e^{X+Y+\frac{1}{2}[X,Y]}$. Fortunately in this case, first commutator is a constant number, so higher order commutators are zero. Finally we have
\begin{equation*}
C_{12}(t)=e^{-i\Phi_I(t)}e^{-\Phi_R(t)}.
\end{equation*}
where $\Phi_I(t)$ and $\Phi_R(t)$ have been defined in the main text. Carrying on further,we have
\begin{equation}\label{PRI}
\Gamma(\tau)=\dfrac{\Delta^2}{2\tau}\int_0^\tau dt\int_0^t dt'e^{-\Phi_R(t')}\cos[\zeta(t)-\zeta(t-t')-\Phi_I(t')].
\end{equation}
\end{document}
|
\begin{document}
\title{On the Verge of Solving Rocket League using Deep Reinforcement Learning and Sim-to-sim Transfer}
\author{
\IEEEauthorblockN{Marco Pleines\footnote{Correspondence to [email protected]}, Konstantin Ramthun, Yannik Wegener, Hendrik Meyer, Matthias Pallasch, Sebastian Prior\\
Jannik Dr{\"o}gem{\"u}ller, Leon B{\"u}ttinghaus, Thilo R{\"o}themeyer, Alexander Kaschwig,\\
Oliver Chmurzynski, Frederik Rohkr{\"a}hmer, Roman Kalkreuth, Frank Zimmer\IEEEauthorrefmark{1}, Mike Preuss\IEEEauthorrefmark{2}}
\IEEEauthorblockA{
\textit{Department of Computer Science},
\textit{TU Dortmund University},
Dortmund, Germany \\
\IEEEauthorrefmark{1}\textit{Department of Communication and Environment, Rhine-Waal University of Applied Sciences}, Kamp-Linfort, Germany \\
\IEEEauthorrefmark{2}\textit{LIACS Universiteit Leiden}, Leiden, Netherlands \\
}
}
\maketitle
\begin{abstract}
Autonomously trained agents that are supposed to play video games reasonably well rely either on fast simulation speeds or heavy parallelization across thousands of machines running concurrently.
This work explores a third way that is established in robotics, namely sim-to-real transfer, or if the game is considered a simulation itself, sim-to-sim transfer.
In the case of Rocket League, we demonstrate that single behaviors of goalies and strikers can be successfully learned using Deep Reinforcement Learning in the simulation environment and transferred back to the original game.
Although the implemented training simulation is to some extent inaccurate, the goalkeeping agent saves nearly $100\%$ of its faced shots once transferred, while the striking agent scores in about $75\%$ of cases.
Therefore, the trained agent is robust enough and able to generalize to the target domain of Rocket League.
\end{abstract}
\begin{IEEEkeywords}
rocket league, sim-to-sim transfer, deep reinforcement learning, proximal policy optimization
\end{IEEEkeywords}
\section{Introduction}
\makeatletter
\newcommand*{\rom}[1]{\expandafter\@slowromancap\romannumeral #1@}
\makeatother
The spectacular successes of agents playing considerably difficult games, such as StarCraft~\rom{2} \cite{DBLP:journals/nature/VinyalsBCMDCCPE19} and DotA~2 \cite{DBLP:journals/corr/abs-1912-06680}, have been possible only because the employed algorithms were able to train on huge numbers of games on the order of billions or more.
Unfortunately, and despite many improvements achieved in AI in recent years, the utilized Deep Learning methods are still relatively sample inefficient.
To deal with this problem, fast running environments or high amounts of computing resources are vital.
OpenAI~Five for DotA~2 \cite{DBLP:journals/corr/abs-1912-06680} is an example of the utilization of hundreds of thousands of computing cores in order to achieve high throughput in terms of played games.
However, this way is closed for games that run only on specific platforms and are thus very hard to parallelize.
Moreover, not many research groups have such resources at their disposal.
Video games that suffer from not being able to be sped up significantly, risk minimal running times and hence repeatability.
Therefore it makes sense to look for alternative ways to tackle difficult problems.
\IEEEpubid{\begin{minipage}{\textwidth}\ \\ \\[12pt] Accepted to IEEE CoG 2022 \end{minipage}}
\begin{figure}
\caption{The game of Rocket League (top) and the contributed simulation (bottom), which notably advances its ancestor project \emph{RoboLeague}
\label{fig:both_sims}
\end{figure}
Sim-to-real transfer offers such an alternative way and is well established in robotics, and it follows the general idea that robot behavior can be learned in a very simplified simulation environment and the trained agents can then be successfully transferred to the original environment.
If the target platform is a game as well, we may speak of sim-to-sim transfer because the original game is also virtual, just computationally much more costly.
This approach is applicable to current games, even if they are not parallelizable, and makes them available for modern Deep Reinforcement Learning (DRL) methods.
There is of course a downside of this approach, namely that it may be difficult or even infeasible to establish a simulation that is similar enough to enable transfer later on, but still simple enough to speed up learning significantly.
A considerable amount of effort has to be invested in establishing this simulation environment before we can make any progress on the learning task.
To our knowledge, the sim-to-sim approach has not yet been applied to train agents for a recent game. Therefore we aim to explore the possibilities of this direction in order to detect how simple the simulation can be, and how good the transfer to the original game works.
The game we choose as a test case of the sim-to-sim approach is Rocket League (Figure \ref{fig:both_sims}), which basically resembles indoor football with cars and teams of 3.
Rocket league is freely available for Windows and Mac, possesses a bot API (RLBot \cite{RLBotWiki2021}) and a community of bot developers next to a large human player base.
As the 3 members of each team control car avatars with physical properties different from human runners, the overall tactics are the one of rotation without fixed roles.
Thereby, large parts of the current speed can be conserved and players do not have to accelerate from zero when ball possession changes \cite{Verhoefen2020}.
Next to basic abilities attempting to shoot towards the goal and to move the goalie in order to prevent a goal, Rocket League is a minimal \emph{team AI} setting \cite{MozgovoyPB21} where layers of team tactics and strategy can be learned.
The first step of our work re-implements not all, but multiple physical gameplay mechanics of Rocket League using the game engine Unity, which results in a slightly inaccurate simulation.
We then train an agent in a relatively easy goalie and striker environment using Proximal Policy Optimization (PPO) \cite{Schulman2017}.
The learned behaviors are then transferred to Rocket League for evaluation.
Even though the training simulation is imperfect, the transferred behaviors are robust enough to succeed at their tasks by generalizing to the domain of Rocket League.
The goalkeeping agent saves nearly $100\%$ of the shots faced, while the striking agent scores about $75\%$ of its shots.
The sim-to-sim transfer is further examined by ablating physical adaptations that were added to the training simulation.
This paper proceeds with elaborating on related work.
Then, the physical gameplay mechanics of Rocket League are shown.
After illustrating the trained goalie and striker environment, PPO and algorithmic details are presented.
Section \ref{sec:sim-to-sim} examines the sim-to-sim transfer.
Before concluding our work, a discussion is provided.
\section{Related Work}
Sim-to-sim transfer on a popular multiplayer team video game touches majorly on two different areas, namely multi-agent and sim-to-real transfer.
DotA~2 and StarCraft~\rom{2} are the already mentioned prominent examples in the field of multi-agent environments.
As this work focuses on single-agent environments, namely the goalkeeper and striker environments, related work on sim-to-real transfer is focused next.
Given the real world, a considered prime example for multi-agent scenarios is \emph{RoboCup}.
RoboCup is an annual international competition~\cite{DBLP:conf/agents/KitanoAKNO97} that offers a publicly effective open challenge for the intersection of robotics and AI research.
The competition is known for the robot soccer cup but also includes other challenges.
Reinforcement Learning (RL) has been successfully applied to simulated robot soccer in the past~\cite{DBLP:journals/corr/HausknechtS15a} and has been found a powerful method for tackling robot soccer.
A recent survey \cite{AntonioniSRN21} provides insights into robot soccer and highlights significant trends, which briefly mention the transfer from simulation to the real world.
In general, sim-to-real transfer is a well-established method for robot learning and is widely used
in combination with RL.
It allows the transition of an RL agent's behavior, which has been trained in simulations, to real-world environments.
Sim-to-real transfer has been predominantly applied to RL-based robotics~\cite{DBLP:conf/ssci/ZhaoQW20} where the robotic agent has been trained with state-of-the-art RL techniques like PPO~\cite{Schulman2017}.
Popular applications for sim-to-real transfer in robotics have been autonomous racing~\cite{DBLP:conf/icra/BalajiMGGDKRSTT20}, Robot Soccer~\cite{DBLP:journals/corr/abs-1911-01529}, navigation~\cite{DBLP:journals/corr/abs-1906-04452}, and control tasks~\cite{DBLP:conf/icra/PedersenMC20}.
To address the inability to exactly match the real-world environment, a challenge commonly known as sim-to-real gap, steps have also been taken towards generalized sim-to-real transfer for robot learning~\cite{DBLP:conf/cvpr/RaoHILIK20,DBLP:conf/icra/HoRXJKB21}.
The translation of synthetic images to realistic ones at the pixel level is employed by a method called GraspGAN~\cite{DBLP:conf/icra/BousmalisIWBKKD18} which utilizes a generative adversarial network (GAN)~\cite{DBLP:conf/nips/GoodfellowPMXWOCB14}.
GANs are able to generate synthetic data with good generalization ability.
This property can be used for image synthesis to model the transformation between simulated and real images. GraspGAN provides a method called \textit{pixel-level domain adaptation}, which translates synthetic images to realistic ones at the pixel level.
The synthesized pseudo-real images correct the sim-to-real gap to some extent.
Overall, it has been found that the respective policies learned with simulations execute more successfully on real robots when GraspGAN is used~\cite{DBLP:conf/icra/BousmalisIWBKKD18}.
Another approach to narrow the sim-to-real gap is domain randomization \cite{DomainRandomizationTobin2017}.
Its goal is to train the agent in plenty of randomized domains to generalize to the real domain.
By randomizing all physical properties and visual appearances during training in the simulation, a trained behavior was successfully transferred to the real world to solve the Rubik's cube \cite{RubikCube2019}.
\input{tables/physics}
\section{Rocket League's Physical Game Mechanics}
The implementation of the Unity simulation originates from the so called \emph{RoboLeague} repository \cite{Roboleague2021}.
As this version of the simulation is by far incomplete and inaccurate, multiple fundamental aspects and concepts are implemented, which are essentially based on the physical specifications of Rocket League.
These comprise, for example, the velocity and acceleration of the ball and the car, as well as the concept of boosting.
Jumps, double jumps as well as dodge rolls are now possible and also collisions and interactions.
There is friction caused by the interaction of a car with the ground, but also friction caused by the air is taken into account.
However, some further adjustments are necessary.
Therefore, table \ref{tab:physics} provides an overview on all the material that was considered during implementing vital physical components, while highlighting distinct adjustments, which are different to the ones found in the references.
It has to be noted that most measures are provided in \emph{unreal units} (uu).
To convert them to Unity's scale, these have to be divided by $100$.
Some adjustments are based on empirical findings by comparing the outcome of distinct physical maneuvers inside both simulations.
A completely custom solution for realizing the car's suspension is implemented.
Both concepts are detailed next.
\subsection{Empirical Approach}
\begin{figure}
\caption{Dogde Roll: Comparison with Rocket League's Ground Truth}
\label{fig:dodge_alignment}
\end{figure}
\subsection{Suspension}
\label{sec:suspension}
The physical suspension of cars consists of a compression area above the wheel's rest position, in which the suspension can be compressed (up to max. $3 \text {uu}$), and an extension area below the wheel's rest position, in which the suspension is extended (up to max. $ 12 \text {uu}$).
As long as the wheel is in the extension area, it does not apply any force to the car, but it is capable of registering the contact. The wheels not collide with the hitbox of the car.
Only when a wheel is pushed into the compression zone a force will be applied upwards along the suspension to the car, which is 0 in the rest position and becomes stronger while pushing the wheel into the compression zone.
When the vehicle falls to the ground, the wheel is pushed into the compression zone of the suspension and the suspension applies a force to the vehicle that is opposite to the force of gravity, thus decelerating the vehicle.
\section{Rocket League Environment}
This section starts out by providing an overview of vital components of Rocket League's physical gameplay mechanics, which are implemented in the training simulation based on the game engine Unity and the ML-Agents Toolkit~\cite{Juliani2019}.
RLBot~\cite{RLBotWiki2021} provides the interface to Rocket League where the training situations can be reproduced.
Afterward, the DRL environments, designated for training, and their properties are detailed.
The code is open source\footnote{\url{https://github.com/PG642}}.
\subsection{Implementation of the Training Simulation}
\label{sec:env}
\input{tables/physics}
\begin{figure}
\caption{The physical maneuver of a dodge roll is executed to exemplary show the alignment of the Unity simulation to the ground truth by using different max angular velocities.}
\label{fig:dodge_alignment}
\end{figure}
The implementation of the Unity simulation originates from the so called \emph{RoboLeague} repository \cite{Roboleague2021}.
As this version of the simulation is by far incomplete and inaccurate, multiple fundamental aspects and concepts are implemented, which are essentially based on the physical specifications of Rocket League.
These comprise, for example, the velocity and acceleration of the ball and the car, as well as the concept of boosting.
Jumps, double jumps as well as dodge rolls are now possible, and also collisions and interactions.
There is friction caused by the interaction of a car with the ground, but also friction caused by the air is taken into account.
However, further adjustments are necessary.
Therefore, table \ref{tab:physics} provides an overview of all the material that was considered during implementing essential physical components, while highlighting distinct adjustments that differ from the information provided by the references.
It has to be noted that most measures are given in \emph{unreal units} (uu).
To convert them to Unity's scale, these have to be divided by $100$.
Some adjustments are based on empirical findings by comparing the outcome of distinct physical maneuvers inside the implemented training simulation and the ground truth provided by Rocket League.
A physical maneuver simulates several player inputs over time, such as applying throttle and steering left or right.
While the simulation is conducted in both simulations, multiple relevant game state variables like positions, rotations, and velocities are monitored for later evaluation.
Figure \ref{fig:dodge_alignment} is an example where the physical maneuver orders the car to execute a dodge roll.
Whereas the original max angular velocity of $5.5\rs$ does not compare well to the ground truth, a more suitable value of $7.3 \rs$ is found by analyzing the observed data.
The speed of the training simulation is about $950~steps/second$, while RLBot is constrained to the real-time, where only $120~steps/second $ are possible.
This simulation performance is measured on a Windows Desktop utilizing a GTX 1080 and a AMD Ryzen 7 3700X.
\subsection{Goalie Environment}
In the goalie environment, the agent is asked to save shots.
1000 different samples of shots, which uniformly vary in speed, direction, and origin, are faced by the agent during training.
In every episode, one shot is fired towards the agent's goal.
The agent's position is reset to the center of the goal at the start of each episode.
Every save rewards the agent with $+1$.
A goalkeeping episode terminates if the ball hits the goal or is deflected by the agent.
\subsection{Striker Environment}
To score a goal is the agent's task inside the striker environment.
The ball moves bouncy, slowly, close, and in parallel to the goal.
Its speed and origin are sampled uniformly from 1000 samples during the agent's training.
The agent's position is farther away from the goal while being varied as well.
$+1$ is the only reward signal that the agent receives upon scoring.
Once the ball hits the goal or a time limit is reached, the episode terminates and the environment is reset.
\subsection{Observation and Action Space}
\begin{figure}
\caption{The contents of the agent's observation.}
\label{img:obs_space}
\end{figure}
Both environments share the same observation and action space.
The agent perceives 23 normalized game state variables to fully observe its environment as illustrated by figure \ref{img:obs_space}.
The agent's action space is multi-discrete and contains the following 8 dimensions:
\begin{itemize}
\begin{multicols}{2}
\item Throttle (5 actions)
\item Steer (5 actions)
\item Yaw (5 actions)
\item Pitch (5 actions)
\item Roll (3 actions)
\item Boost (2 actions)
\item Drift or Air Roll (2 actions)
\item Jump (2 actions)
\end{multicols}
\end{itemize}
Rocket League is usually played by humans using a gamepad as input device.
Some of the inputs (e.g. thumbstick) are thus continuous and not discrete.
To simplify the action space, the continuous actions throttle, steer, yaw, and pitch are discretized using buckets as suggested by Pleines et al. \cite{Pleines2019}.
By this means, the agent picks one value from a bucket containing the values $-1$, $-0.5$, $0$, $0.5$ and $1$.
The roll action is also discretized using the values $-1$, $0$ and $1$.
All other actions determine whether the concerned discrete action is executed or not.
The action dimension that is in charge of drifting and air rolling is another special case.
Both actions can be boiled down to one because drifting is limited to being on the ground, whereas air rolling can be done in the air only.
Moreover, multi-discrete action spaces allow the execution of concurrent actions.
One discrete action dimension could achieve the same behavior.
This would require defining actions that feature every permutation of the available actions.
As a consequence, the already high-dimensional action space of Rocket League would be much larger and therefore harder to train.
\section{Deep Reinforcement Learning}
The actor-critic, on-policy algorithm PPO~\cite{Schulman2017} and its clipped surrogate objective (Equation \ref{eq:ppo}) is used to train the agent's policy $\pi$, with respect to its model parameters $\theta$, inside the Unity simulation.
PPO, algorithmic details, and the model architecture are presented next.
\subsection{Proximal Policy Optimization}
$L_t^C(\theta)$ denotes the policy objective, which optimizes the probability ratio of the current policy $\pi_\theta$ and the old one $\pi_{\theta old}$:
\small
\begin{equation}
L^{C}_t(\theta) = \hat{\mathbb{E}}_t [min(q_t(\theta)\hat{A}_t,clip(q_t(\theta),1- \epsilon,1+\epsilon)\hat{A}_t)]
\label{eq:ppo}
\end{equation}
\begin{equation*}
\textnormal{with the surrogate objective}~q_t(\theta) = \frac{\pi _\theta(a_t|s_t)}{\pi _{\theta \text{old}}(a_t|s_t)}
\end{equation*}
\normalsize
$s_t$ is the environment's state at step $t$.
$a_t$ is an action tuple, which is executed by the agent, while being in $s_t$.
The clipping range is stated by $\epsilon$ and $\hat{A}_t$ is the advantage, which is computed using generalized advantage estimation~\cite{Schulman2015GAE}.
While computing the squared error loss $L_t^V$ of the value function, the maximum between the default and the clipped error loss is determined.
\small
\begin{equation}
V_t^{C} = V_{\theta old}(s_t) + clip(V_\theta(s_t) - V_{\theta old}(s_t), -\epsilon, \epsilon)
\end{equation}
\begin{equation}
L_t^V = max((V_\theta(s_t) - G_t)^2, (V_t^C-G_t)^2)
\end{equation}
\begin{equation*}
\textnormal{with the sampled return}~G_t = V_{\theta old}(s_t) + \hat{A}_t
\end{equation*}
\normalsize
The final objective is established by $L^{CVH}_t(\theta)$:
\small
\begin{equation}
L^{CVH}_t(\theta)=\hat{\mathbb{E}}_t [L^{C}_t(\theta)-c_1L^{V}_t(\theta)+c_2\mathcal{H}[\pi_\theta](s_t)]
\end{equation}
\normalsize
To encourage exploration, the entropy bonus $\mathcal{H}[\pi_\theta](s_t)$ is added and weighted by the coefficient $c_2$.
Weighting is also applied to the value loss using $c_1$.
\subsection{Algorithmic Details and Model Architecture}
PPO starts out by sampling multiple trajectories of experiences, which may contain multiple completed and truncated episodes, from a constant number of concurrent environments (i.e. workers).
The model parameters are then optimized by conducting stochastic gradient descent for several epochs of mini-batches, which are sampled from the collected data.
Before computing the loss function, advantages are normalized across each mini-batch.
The computed gradients are clipped based on their norm.
\begin{figure}
\caption{The policy and the value function share gradients and several parameters.
After feeding 23 game states variables as input to the model and processing a shared fully connected layer, the network is split into a policy and value stream starting with their own fully connected layer.
The policy stream outputs action probabilities for each available action dimension, whereas the value stream exposes its estimated state-value.}
\label{img:model}
\end{figure}
A relatively shallow neural net (model) is shared by the value function and the policy (Figure \ref{img:model}).
To support multi-discrete actions, the policy head of the model outputs 8 categorical action probability distributions.
During action selection, each distribution is used to sample actions, which are provided to the agent as a tuple.
The only adjustment to the policy's loss computation is that the probabilities of the selected actions are concatenated.
Concerning the entropy bonus, the mean of the action distributions' entropies is used.
\input{tables/alignment}
\section{Sim-to-sim Transfer}
\label{sec:sim-to-sim}
Two major approaches are considered to examine learned behaviors inside the Unity simulation and its transfer to Rocket League.
The first one runs various handcrafted scenarios (like seen in section \ref{sec:env}) in both simulations to directly compare their alignment.
This way, it can be determined whether the car or the ball behave similarly or identically concerning their positions and velocities.
The second approach trains the agent in Unity given the goalie and the striker environment, while all implemented physics components are included.
We further conduct an ablation study on the implemented physics where each experiment turns off one or all components.
Turning off may also refer to use the default physics of Unity.
If not stated otherwise, each training run is repeated 5 times and undergoes a thorough evaluation.
Each model checkpoint is evaluated in Unity and Rocket League by 10 training and 10 novel shots, which are repeated 3 times.
Therefore, each data point aggregates 150 episodes featuring one shot.
Result plots show the interquartile mean of the cumulative reward and a confidence interval of 95\% as recommended by Agarwal et al. \cite{agarwal2021statistics}.
The hyperparameters are detailed in Table \ref{tab:hyperparameters}.
At last, we describe some of the learned behaviors that are also retrieved from training in a more difficult striker environment.
\input{tables/hyperparameters_horizontal}
\subsection{Alignment Comparison using Handcrafted Scenarios}
\label{sec:alginment}
To directly compare the alignment between both simulations, six physical maneuvers are assessed by 3 different handcrafted scenarios:
\begin{enumerate}
\item Acceleration
\begin{itemize}
\item Car drives forward and steers left and right
\item Car drives backward and steers left and right
\item Car uses boost and steers left and right
\end{itemize}
\item Air Control
\begin{itemize}
\item Car starts up in the air, looks straight up, boosts shortly and boosts while rolling in the air
\item Car starts up in the air, has an angle of $45^{\circ}$, boosts shortly and boosts while rolling in the air
\item Car starts up in the air, looks straight up and concurrently boosts, yaws, and air rolls
\end{itemize}
\item Drift
\begin{itemize}
\item Car drives forward for a bit and then starts turning and drifting while moving forward
\item Car drives backward for a bit and then starts turning and drifting while moving forward
\item Car uses boost and then starts turning and drifting while using boost
\end{itemize}
\item Jump
\begin{itemize}
\item Car makes a short jump, then a long one and at last a double jump
\item Car makes a front flip, a back flip and a dodge roll
\item Car drives forward, does a diagonal front flip and at last a back flip
\end{itemize}
\item Ball Bounce
\begin{itemize}
\item Ball falls straight down
\item Ball falls down with an initial force applied on its x-axis
\item Ball falls down with an initial force applied on its x-axis and an angular velocity
\end{itemize}
\item Shot
\begin{itemize}
\item Car drives forward and hits the motionless ball
\item Car drives forward and the ball rolls to the car
\item Ball jumps, the car jumps while boosting and hits the ball using a front flip
\end{itemize}
\end{enumerate}
Each scenario tracks the position of the ball and the car during each frame.
As both simulations end up monitoring the incoming data with slight time differences, the final data is interpolated to match in shape.
Afterward, the error for each data point between both simulations is measured.
The final results are described by Table \ref{tab:align}, which comprises the mean, max, and standard deviation (Std) error across each run scenario.
Letting the ball bounce for some time shows the least error, while a significant one is observed when examining the scenarios where the car shoots the ball.
Note that slight inaccuracies during acceleration may cause a strongly summed error when considering a different hit location on the ball.
The other scenarios, where the error is based on the car's position, also indicate that the Unity simulation suffers from inaccuracies.
\subsection{Physics Ablation Study based on PPO Training}
\begin{figure*}
\caption{Results of training the goalie environment under different ablations and transferring it to Rocket League. The agent is evaluated on training shots and ones, which were not seen during training. The agent easily solves the goalie task under all circumstances. Both, training and unseen shots, behave identically in Rocket League.}
\label{fig:goalie_results}
\end{figure*}
\begin{figure*}
\caption{Results of training the striker environment under different ablations and transferring it to Rocket League. The agent is evaluated on training situations and ones, which were not seen during training. The agent scores in about 75\% of the played episodes given all physical adaptations, while any ablation turns out catastrophic. Both, training and unseen situations, behave identically in Rocket League.}
\label{fig:striker_results}
\end{figure*}
The previously shown imperfections of the Unity simulation may lead to the impression that successfully transferring a trained behavior is rather unlikely.
This assumption can be negated by considering the results retrieved from training the agent in the goalie environment (Figure \ref{fig:goalie_results}).
Even though each experiment ablates all, single or no physical adaptations, the agent is still capable of saving nearly every ball once transferred to Rocket League.
A drawback of the goalie environment lies in its simplicity because the agent only has to somehow hit the ball to effectively deflect it.
The next step of complexity is posed by the striker environment, where the agent has to land a more accurate hit on the ball to score a goal.
Figure \ref{fig:striker_results} illustrates the results of the striker training.
Notably, when all physical adaptations are present, the transferred behavior manages to score in about 75\% of the played episodes.
Catastrophic performances emerge in Rocket League once single physical adaptations are turned off.
\subsection{Learned Policies}
During the performed experiments, several intriguing agent behaviors emerged\footnote{\url{https://www.youtube.com/watch?v=WXMHJszkz6M&list=PL2KGNY2Ei3ix7Vr_vA-ZgCyVfOCfhbX0C}}.
When trained as a goalkeeper, the agent tries to hit the ball very early, while making its body as big as possible towards the ball.
This is achieved by simultaneously jumping and rolling forward or executing a forward flip.
Concerning the striker environment, the agent usually approaches the ball using its boost.
To get a better angle to the ball, the agent steers left and right or vice versa.
Drifting is sometimes used to aid this purpose.
Jumping is always used when needed.
This is usually the case if the agent is close to the ball, which is located above the agent.
Otherwise, the agent's preference is to stay on the ground.
Further training experiments were conducted in a more difficult striker environment.
The ball is not anymore simply passed in parallel and close to the goal.
Instead, the ball bounces higher and farther away from the goal, which increases the challenge of making a good touch on the ball to score.
Given this setting, two different policies were achieved.
One policy approaches the ball as fast as possible while using a diagonal dodge roll to make the final touch to score.
However, this behavior fails a few shots.
The other emerged behavior can be considered as the opposite.
Depending on the distance and the height of the ball, the agent waits some time or even backs up to ensure that it will hit the ball while being on the ground.
Therefore, the agent avoids jumping.
This is surprising because the agent should maximize its discounted cumulative reward and therefore finish the episode faster.
Although the increased difficulty led to different behaviors, the agent may struggle a lot to get there.
Usually, 2 out of 5 training runs succeeded, while the other ones utterly failed.
\section{Discussion}
In this work, the agent is trained on isolated tasks, which are quite apart from a complete match of Rocket League.
To train multiple cooperative and competitive agents, the first obstacle that comes to mind is the tremendously high computational complexity, which might be infeasible for smaller research groups.
But before going this far, several aspects need to be considered that can be treated in isolation as well.
At last, the difficulties of training the more difficult striker environments are discussed.
\subsection{On Improving the Sim-to-sim Transfer}
At first, the Unity simulation is still lacking the implementation of physical concepts like the car-to-car interaction and suffers from the reported (Section \ref{sec:alginment}) inaccuracies.
These can be further improved by putting more work into the simulation, but also other approaches are promising.
At the cost of more computational resources, domain randomization \cite{DomainRandomizationTobin2017} could achieve a more robust agent, potentially comprising an improved ability to generalize to the domain of Rocket League.
As the ground truth is provided by Rocket League, approaches from the field of supervised learning can be considered as well.
\subsection{Training under Human Conditions}
Once the physical domain gap is narrowed, the Unity simulation still does not consider training under human conditions.
Notably, the current observation space provides perfect information on the current state of the environment, whereas players in Rocket League have to cope with imperfect information due to solely perceiving the rendered image of the game.
Thus, the Unity simulation has to implement Rocket League's camera behavior as well.
However, one critical concern is that the RLBot API does not reveal the rendered image of Rocket League and therefore makes a transfer impossible as of now.
However, even if that information is made available by Psyonix, both simulations' visual appearances are very different.
The Unity simulation's aesthetics are very abstract, whereas Rocket League impresses with multiple arenas featuring many details concerning lighting, geometry, shaders, textures, particle effects, etc..
To overcome this gap of visual appearance, approaches of the previously described related work, like GraspGAN \cite{DBLP:conf/icra/BousmalisIWBKKD18}, can be considered.
Another challenge arises once the environment is partially observable.
It should be considered that the agent will probably need memory to be able to compete with human players.
Otherwise, the agent might not be able to capture the current affairs of its teammates and opponents.
For this purpose, multiple memory-based approaches might be suitable, like using a recurrent neural network or a transformer architecture.
Moreover, the multi-discrete action space used in this paper is a simplification of the original action space that features concurrent continuous and discrete actions.
Initially, the training was done using the PPO implementation of the ML-Agents toolkit \cite{Juliani2019}, which supports mixed (or hybrid) concurrent action spaces.
However, these experiments were quite unstable and hindered progress.
Therefore, Rocket League presents an interesting challenge for exploring such action spaces, of which other video games or applications are likely to take advantage.
\input{content/discussion_difficulties_striker}
\section{Conclusion}
Towards solving Rocket League by the means of Deep Reinforcement Learning, a fast simulation is crucial, because the original game cannot be sped up and neither parallelized on Linux-based clusters.
Therefore, we advanced the implementation of a Unity project that mimics the physical gameplay mechanics of Rocket League.
Although the implemented simulation is not perfectly accurate, we remarkably demonstrate that transferring a trained behavior from Unity to Rocket League is robust and generalizes when dealing with a goalkeeper and striker task.
Hence, the sim-to-sim transfer is a suitable approach for learning agent behaviors in complex game environments.
After all, Rocket League still poses further challenges when targeting a complete match under human circumstances.
Based on our findings, we believe that Rocket League and its Unity counterpart will be valuable to various research fields and aspects, comprising: sim-to-sim transfer, partial observability, mixed action-spaces, curriculum learning, competitive and cooperative multi-agent settings.
\end{document}
|
\begin{document}
\title{A generic framework for genuine multipartite entanglement detection}
\author{Xin-Yu Xu}
\author{Qing Zhou}
\author{Shuai Zhao}
\author{Shu-Ming Hu}
\affiliation{Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China}
\affiliation{CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China}
\author{Li Li}
\email{[email protected]}
\author{Nai-Le Liu}
\email{[email protected]}
\author{Kai Chen}
\email{[email protected]}
\affiliation{Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China}
\affiliation{CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China}
\affiliation{Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China}
\maketitle
\begin{abstract}
Design of detection strategies for multipartite entanglement stands as a central importance on our understanding of fundamental quantum mechanics and has had substantial impact on quantum information applications. However, accurate and robust detection approaches are severely hindered, particularly when the number of nodes grows rapidly like in a quantum network. Here we present an exquisite procedure that generates novel entanglement witness for arbitrary targeted state via a generic and operational framework. The framework enjoys a systematic and high-efficient character and allows to substantiate genuine multipartite entanglement for a variety of states that arise naturally in practical situations, and to dramatically outperform currently standard methods. With excellent noise tolerance, our framework should be broadly applicable to witness genuine multipartite entanglement in various practically scenarios, and to facilitate making the best use of entangled resources in the emerging area of quantum network.
\end{abstract}
\section{Introduction} \label{sec:introduction}
As a unique property in quantum theory, entanglement \cite{RevModPhys.81.865} is recognized as a kind of quantum resource \cite{RevModPhys.91.025001} and plays a central role in numerous quantum computing and quantum communication tasks \cite{bennett2000quantum,PhysRevLett.70.1895,RevModPhys.81.1301,feynman1982simulating,deutsch1985quantum}. The ability to generate an increasing number of entangled particles is an essential benchmark for quantum information processing. In past decades, considerable efforts have been made to prepare larger and more complex entangled states in various platforms \cite{Luo620,arute2019quantum,PhysRevLett.105.210504,yao2012,doi:10.1126/science.abg7812,yokoyama2013ultra,PhysRevLett.112.155304}, which experimental systems are currently evolving from several qubits to noisy intermediate scale quantum system (NISQ) \cite{Preskill2018quantumcomputingin}.
The developments of quantum technologies raise immediately important questions regarding characterization of quantum entanglement of underlying systems. In bipartite systems, various theoretical works have been contributed, such as separability criterions \cite{PhysRevLett.77.1413,HORODECKI1997333,chen205017quantum,rudolph2005further} and entanglement measures \cite{PhysRevLett.95.040504,PhysRevLett.95.210501,plenio2014introduction}, which provide standard tools for characterizing bipartite entanglement. For good reviews, please refer to Refs.\ \cite{RevModPhys.81.865,GUHNE20091,friis2019entanglement}. When it comes to multipartite systems, the problem is much more complicated. The entanglement structure becomes much richer for multipartite systems \cite{PhysRevLett.108.110501,zhou2019detecting}, since the number of possible divisions grows exponentially with the system size \cite{RevModPhys.81.865}. This leads to many types of multipartite entanglement, ranging from non-fully-separable to genuine multipartite entanglement (GME). In the following, we focus on the detection of genuine multipartite entanglement, which is an essential task for multipartite quantum communication and quantum computing tasks. For the detection of GME, many standard tools in the bipartite case, such as separability criterions, become infeasible since they only detect entanglement between two partitions. Meanwhile, a tomographic reconstruction of quantum state required in these methods becomes time-consuming and computationally difficult in the multipartite case.
For genuine multipartite entanglement detection, entanglement witness (EW) \cite{HORODECKI19961,terhal2000bell,lewenstein2000optimization,hyllus2005relations,lewenstein2001characterization} provides an elegant solution both theoretically and experimentally without need of having full tomographic knowledge about the state. Moreover, it is also known that witness operator can also be used to estimate entanglement measures \cite{PhysRevLett.98.110502}. On account of simplicity and efficiency of entanglement witness, it has been widely used for experimental certification of GME in many platforms, such as trapped ions \cite{PhysRevX.8.021012,PhysRevLett.106.130506}, photonic qubits \cite{PhysRevLett.95.210502,gao2010experimental,PhysRevX.8.021072,PhysRevLett.124.160503}, and superconducting qubits\cite{PhysRevLett.122.110501}. Most available GME witnesses are tailored towards some specific states, for instance, the Greenberger-Horne-Zeilinger (GHZ) states \cite{Greenberger1989}, W-states \cite{PhysRevA.62.062314}, graph states \cite{PhysRevA.69.062311,hein2006entanglement}, and so on. Despite few general methods for the construction of GME witness have been proposed \cite{PhysRevLett.92.087902,PhysRevLett.106.190502,PhysRevLett.113.100501,PhysRevLett.111.110503}, their performance is very limited, especially as the size of system grows. One major drawback is the limited scope of noise resistance. For example, the fidelity-based method \cite{PhysRevLett.92.087902} is a canonical witness construction and widely used nowadays. Its noise tolerance decreases dramatically as the system size increases. In realistic NISQ systems, however, the noise always inevitably grows with the system size. In fact, it has been shown that the fidelity witnesses fail to detect a large amount of mixed entangled states \cite{PhysRevLett.124.200502}. To find more robust GME witnesses, numerical methods have been introduced \cite{PhysRevLett.106.190502},which, however, suffer from expensive computational costs as the system size grows. Hence, although it is known that for any entangled state there exists some EW to detect it \cite{HORODECKI19961}, how to construct a desirable EW to recognize a GME state is still a formidable challenge.
In this work, we propose a generic framework to design robust GME witnesses by analytical and systematic construction. We start by introducing an exquisite method for GME witness with a novel lifting from any set of bipartite EWs. This establishes the link between the standard tools developed in the bipartite case and the GME witness construction. We then provide a well-designed class of optimal bipartite EWs that allows the design of robust GME witnesses for arbitrary pure GME states with our method. The performance of this framework on many typical classes of GME states is further evaluated in terms of white noise tolerance. It can be shown that the framework outperforms the most widely used fidelity-based method with certainty, and outperforms much better than the best known EWs in many cases. Finally, benefiting from the high robustness of the resulting witnesses, we also demonstrate further applications of the framework, such as to provide a tighter lower bounds on the genuine multipartite entanglement measures and detecting unfaithful GME states \cite{PhysRevLett.124.200502}.
\section{Results}
\subsection{Preliminaries}
To start with, we first give the precise definition of biseparable, genuine multipartite entanglement and entanglement witness. A pure state is called \textit{biseparable} if it can be written as a tensor product of two state vectors, i.e., $|\psi_A\rangle\otimes|\psi_{\bar{A}}\rangle$. Then a mixed state is called biseparable if it can be decomposed into a mixture of pure biseparable states, formally,
\begin{equation}
\rho_{bs}=\sum_{A|\bar{A},i}p_{A|\bar{A},i} |\psi_A^i\rangle\langle\psi_A^i|\otimes|\psi_{\bar{A}}^i\rangle\langle\psi_{\bar{A}}^i|,
\end{equation}
where the summation can be performed over different bipartitions $A|\bar{A}$ of the whole system. A state that is not biseparable is referred to as genuine multipartite entangled. To detect the GME states, the most widely used method is to find an observable $\mathcal{W}_{GME}$ that is nonnegative for all separable states and has negative expectation value on at least one GME state. Then for some multipartite quantum state $\rho$, the fact $Tr(\mathcal{W}_{GME}\rho) < 0$ will reveal the existence of genuine multipartite entanglement, and the $\mathcal{W}_{GME}$ is called a \textit{GME witness}. Moreover, given two EWs $\mathcal{W}_1$ and $\mathcal{W}_2$, if there exists $\lambda > 0$ such that $\mathcal{W}_1 - \lambda \mathcal{W}_2$ is positive semidefinite, i.e., $\mathcal{W}_1 \succeq \lambda \mathcal{W}_2$, one says that $\mathcal{W}_2$ is \textit{finer} than $\mathcal{W}_1$ \cite{lewenstein2000optimization}. The finer witness operator $\mathcal{W}_2$ detects more entangled states than $\mathcal{W}_1$. An EW is \textit{optimal} if no finer EW exists.
\subsection{Design GME witness from a complete set of bipartite EWs}
Due to its non-negativity over all biseparable states, a GME witness $\mathcal{W}_{GME}$ also serves as bipartite EW with respect to each possible bipartition of the whole system. In other words, there exists a complete set of bipartite EWs $\{\mathcal{W}_{A|\bar{A}}\}$ satisfying $\mathcal{W}_{GME} \succeq \mathcal{W}_{A|\bar{A}}$ for each bipartition $A|\bar{A}$. This fact, from the opposite point of view, indicates that the GME witness $\mathcal{W}_{GME}$ is designed based on the set $\{\mathcal{W}_{A|\bar{A}}\}$ according to the constraint $\mathcal{W}_{GME} \succeq \mathcal{W}_{A|\bar{A}}$. This naturally provides a general framework for constructing GME witnesses from a complete set of bipartite EWs. Remarkably, the set $\{\mathcal{W}_{A|\bar{A}}\}$ itself cannot be directly used to detect GME states, as there exist biseparable states that are entangled with respect to every possible bipartition \cite{GUHNE20091}. While there are two crucial issues with such a framework. The first one is how to find the operator satisfying $\mathcal{W}_{GME} \succeq \mathcal{W}_{A|\bar{A}}$, and the second one is to decide which set of bipartite EWs should be used. Optimal solutions to these two problems is hard in general, and there have been only a few previous related studies on these issues. In Ref.\ \cite{PhysRevLett.113.100501}, an alternative solution was proposed to establish a connection between positive maps and multipartite EWs, where EWs detecting multipartite bound entangled state have been obtained. While in the following, we present a novel alternatively solution which is capable of constructing robust GME witnesses.
\subsection{An operational framework for constructing robust GME witness}
Any mixed GME state contains at least one pure GME state as a component, while the remaining components can be treated as noises. In order to detect mixed GME states with linear EW, it is natural to employ a witness operator for the pure GME component that is sufficiently robust to noise from the other components. In fact, the set of all optimal GME witnesses for all pure GME states will be sufficient to detect all GME states. However, finding all optimal GME witnesses is naturally a formidable task. Therefore, to advance a solution to this problem, we propose an operational framework to construct a class of robust GME witnesses for all pure GME states.
To address the problem of lifting any given set of bipartite EWs to multipartite, one can accomplish it in two steps: (1) For the first step, each bipartite EW $\mathcal{W}_{A|\bar{A}}$ is decomposed into some projectors. Note that the entanglement witness is designed for some pure entangled state $|\psi\rangle$. Hence we extract a term $-|\psi\rangle\langle\psi|$ before the decomposition. That is, the bipartite EWs are rewritten as $\mathcal{W}_{A|\bar{A}} = \mathcal{O}_{A|\bar{A}} - |\psi\rangle\langle\psi|$, and a spectral decomposition of $\mathcal{O}_{A|\bar{A}} = \mathcal{W}_{A|\bar{A}}+|\psi\rangle\langle\psi|$ is performed
\begin{equation}\label{bEW}
\mathcal{O}_{A|\bar{A}}=\sum_{|\vec{v}_{i,A|\bar{A}}\rangle \in \mathcal{S}_{A|\bar{A}}}
c_{i,A|\bar{A}} |\vec{v}_{i,A|\bar{A}}\rangle\langle \vec{v}_{i,A|\bar{A}}|,
\end{equation}
with $\mathcal{S}_{A|\bar{A}}$ being the set of eigenvectors and $c_{i,A|\bar{A}}$ being the corresponding eigenvalues. All these eigenvectors are collected into a set $\mathcal{S} = \cup_{A|\bar{A}} \mathcal{S}_{A|\bar{A}}$. (2). For the second step, the obtained set $\mathcal{S}$ is divided into $m$ subsets $\{\mathcal{S}_k\}_{k=1}^m$, such that the vectors from different subsets are orthogonal with each other. Denote $\tilde{I}_k$ as the identity operator on the subspace $V_k$ spanned by the state vectors from subset $\mathcal{S}_k$, and $c_k=max_{|\vec{v}_{i,A|\bar{A}}\rangle \in \mathcal{S}_k} c_{i,A|\bar{A}} $ as the maximal coefficient attached to the state vectors in $\mathcal{S}_k$. With the above preparation and notation, we proceed to the following Theorem:
\begin{theorem}
Given any pure GME state $|\psi\rangle$ and a set of bipartite EWs $\{\mathcal{W}_{A|\bar{A}}\}$ detecting $|\psi\rangle$ for all possible $A|\bar{A}$, the following operator $\mathcal{\hat{W}}$
\begin{equation}
\mathcal{\hat{W}}=\sum_{k=1}^m c_k \tilde{I}_k -|\psi\rangle\langle\psi|,
\end{equation}
is nonnegative over all biseparable states, where the $c_k$ and $\tilde{I}_k$ have been defined above.
\end{theorem}
\begin{proof}
To prove the statement, it suffices to observe
\begin{equation}
\begin{aligned}
\mathcal{\hat{W}}-\mathcal{W}_{A|\bar{A}}=&\sum_{k=1}^m c_k \tilde{I}_k -\mathcal{O}_{A|\bar{A}} \\
=&\sum_{k=1}^m \left(c_k \tilde{I}_k-\sum_{|v_{i,A|\bar{A}}\rangle \in \mathcal{S}_k \cap \mathcal{S}_{A|\bar{A}}} c_{i,A|\bar{A}}
|\vec{v}_{i,A|\bar{A}}\rangle\langle \vec{v}_{i,A|\bar{A}}|\right) \\
\ge& \sum_{k=1}^m c_k\left(\tilde{I}_k-\sum_{|v_{i,A|\bar{A}}\rangle \in \mathcal{S}_k \cap \mathcal{S}_{A|\bar{A}}}
|\vec{v}_{i,A|\bar{A}}\rangle\langle \vec{v}_{i,A|\bar{A}}|\right) \\
\ge& 0,
\end{aligned}
\end{equation}
where the inequalities can be derived directly from the definitions of $c_k$ and $\tilde{I}_k$.
\end{proof}
The above construction can be interpreted geometrically. That is, noise from different subspaces has different degrees of influence on the entanglement properties of the target state. The influence is characterized by the coefficients $c_k$, and a small $c_k$ indicates that noise from this subspace hardly affects the entanglement property of target state. Therefore, Theorem 1 can be seen as robust GME witness construction with the help of some prior knowledge of the target state, which comes from the set of bipartite EWs $\{\mathcal{W}_{A|\bar{A}}\}$.
Remarkably, Theorem 1 itself cannot be used as an operational framework for GME witness construction, since the resulting operators can be positive semidefinite and fail to detect any GME state. In fact, one can hardly expect a nontrivial result when the set of bipartite EWs $\mathcal{W}_{A|\bar{A}}$ are chosen randomly. Fortunately, standard tools exist for constructing bipartite EWs based on positive maps. In the following, in order to obtain an operational and generic framework for GME witness construction, we provide a promising choice on the set of bipartite EWs, which are designed for the target states based on partial transposition.
Under any given bipartition $A|\bar{A}$, the target state $|\psi\rangle$ can be written in a Schmidt decomposition form $|\psi\rangle=\sum_{i=0}^{r_A-1} \sqrt{\lambda_{i,~A|\bar{A}}}|i_Ai_{\bar{A}}\rangle$, with $r_A$ being the corresponding Schmidt rank. Note that here the local dimension of the Hilbert space need not be fixed. Then we introduce a class of bipartite EWs $\mathcal{W}_{o,A|\bar{A}}$ in order to use them in the construction of GME witness.
\begin{equation}\label{eq:obEW}
\mathcal{W}_{o,A|\bar{A}}= \sum_{i,j=0}^{r_A-1} \sqrt{\lambda_{i,~A|\bar{A}}\lambda_{j,~A|\bar{A}}}
|i_Aj_{\bar{A}}\rangle\langle i_Aj_{\bar{A}}|-|\psi\rangle\langle\psi|.
\end{equation}
The choice of $\mathcal{W}_{o,A|\bar{A}}$ is mainly based on two considerations. Firstly, $\mathcal{W}_{o,A|\bar{A}}+|\psi\rangle\langle\psi|$ naturally takes the decomposition form in the Eq.\ (\ref{bEW}). Secondly, the above $\mathcal{W}_{o,A|\bar{A}}$ are a class of optimal bipartite EWs. For a detailed illustration and discussion on the $\mathcal{W}_{o,A|\bar{A}}$, please refer to Appendix.\ \ref{sec:appendix bipartite EW}.
These bipartite EWs, together with Theorem 1, promise a generic framework to construct GME witnesses with certainty. The explicit procedure is as follows:
\begin{enumerate}[(1).]
\item Firstly, find the set $\mathcal{S}$. For each bipartition $M|\bar{M}$, calculate the Schmidt decomposition of $|\psi\rangle$ with respect to $M|\bar{M}$,
\begin{equation}
|\psi\rangle=\sum_{i=0}^{r_{M|\bar{M}}-1} \lambda_{i,M|\bar{M}}|\varphi_{i,M}\rangle|\varphi_{i,\bar{M}}\rangle,
\end{equation}
with $r_{M|\bar{M}}$ being the Schmidt rank under this bipartition. A total of $r^2_{M|\bar{M}}$ vectors will be added to the set $\mathcal{S}$, and each of them has a corresponding coefficient. This is denoted by
\begin{equation}
\{(\sqrt{\lambda_{i,M|\bar{M}}\lambda_{j,M|\bar{M}}},~|\varphi_{i,M}\rangle|\varphi_{j,\bar{M}}\rangle)\}_{i,j=0}^{r_{M|\bar{M}}-1}.
\end{equation}
After traversing all possible bipartitions, one will end up with a set of vectors $\mathcal{S}$ as well as their corresponding coefficients, that is, $\{(c_k,~|\psi_k\rangle)\}_{k=1}^{|\mathcal{S}|}$.
\item Secondly, find the finest division of $\mathcal{S}$ such that vectors from different subsets are orthogonal with each other. This can be achieved with the following steps:
\begin{enumerate}[(i)]
\item
Put the first element $|\psi_1\rangle$ of $\mathcal{S}$ into an empty subset $\mathcal{S}_1$.
\item
For every other vector in $\mathcal{S}-\mathcal{S}_1$, if it is not orthogonal with all vectors in the set $\mathcal{S}_1$, it is added into $\mathcal{S}_1$. Repeat this step until no new vector can be added to $\mathcal{S}_1$.
\item
For the rest vectors in $\mathcal{S}-\mathcal{S}_1$, repeat the above two steps to obtain $\mathcal{S}-\mathcal{S}_1-\mathcal{S}_2$, $\mathcal{S}-\mathcal{S}_1-\mathcal{S}_2-\mathcal{S}_3$, $\cdots$, until one has classified all the elements of $\mathcal{S}$.
\item
One obtain a division $\mathcal{S}=\sum_{k=1}^{m}\mathcal{S}_k$.
\end{enumerate}
\item Thirdly, calculate the subspace spanned by the vectors in subset $\mathcal{S}_k$. By performing Schmidt orthogonalization of the vectors in $\mathcal{S}_k$, one can derive the subspace spanned by these vectors and obtain the identity operator $\tilde{I}_k$ on this subspace.
\item Finally, for each subset $\mathcal{S}_k$, find the maximal coefficients $c_k$ attached to the vectors in it, and construct a GME witness using Theorem 1.
\end{enumerate}
There are two remarks to note about this method. Firstly, the resulting witness from the above procedure is always finer than the commonly used GME fidelity witness $\mathcal{W}_F =\lambda I-|\psi\rangle\langle\psi| $ for $|\psi\rangle$, with $\lambda=\max_{A|\bar{A}} \lambda_{0,A|\bar{A}}$ (Here it is assumed that the Schmidt coefficients $\lambda_{i,A|\bar{A}}$ are in decreasing order). To illustrate this, note that if the bipartite EWs are chosen as the bipartite fidelity witness $\mathcal{W}_{F,A|\bar{A}} = \lambda_{0,A|\bar{A}} I-|\psi\rangle\langle\psi|$, by applying Theorem 1, the obtained operator is nothing but the $\mathcal{W}_F$. Whereas by checking $\mathcal{W}_{F,A|\bar{A}}-\mathcal{W}_{o,A|\bar{A}} \succeq 0$, it is straightforward to verify that the above $\mathcal{W}_{o,A|\bar{A}}$ is finer than the bipartite fidelity witness $\mathcal{W}_{F,A|\bar{A}}$. Therefore, when Theorem 1 is applied to the set of $\mathcal{W}_{o,A|\bar{A}}$, the resulting GME witness strictly outperforms the corresponding fidelity witness $\mathcal{W}_{F}$. Secondly, one starts from a complete set of bipartite EWs in the above construction, leading to EWs that detect genuine multipartite entanglement. While if one starts from a smaller set of bipartite EWs, the method allows also for flexible applications in verifying other kinds of multipartite entanglement, e.g., characterizing the entanglement depth.
\section{Examples}
To help a better understanding as well as quantitatively investigating the robustness of the framework, we proceed to some explicit examples, where the white noise tolerance is employed as a figure of merit to evaluate its performance in practice. The white noise tolerance of some witness operator $\mathcal{W}$ for $|\psi\rangle$ is the critical value of $p$ such that the mixed state $pI/d+(1-p)|\psi\rangle\langle\psi|$ is not detected by $\mathcal{W}$.
\subsection{$W$-state}
To investigate the asymptotic behavior of this framework with an increasing system size, we start with the $n$-qubit $W$-state $|W_n\rangle=(|00\cdots01\rangle$$+|00\cdots10\rangle+$$\cdots+|10\cdots00\rangle)/\sqrt{n}$, which is widely used in quantum information processing tasks. For the $W$-state, we find the GME witness (see Appendix.\ \ref{sec:appendix w state} for a proof.)
\begin{equation}
\mathcal{W}_{|W_n\rangle}=\frac{n-1}{n}\mathcal{P}^n_1+\frac{\sqrt{\lfloor n/2\rfloor(n-\lfloor n/2\rfloor)}}{n}
(\mathcal{P}^n_0 +\mathcal{P}^n_2)-|W_n\rangle\langle W_n|,
\end{equation}
with $\mathcal{P}^n_i=\sum_m\pi_m(|0\rangle^{\otimes n-i}|1\rangle^{\otimes i})\pi_m(\langle 0|^{\otimes n-i}\langle 1|^{\otimes i})$, where the summation $m$ is over all possible permutation of $|0\rangle^{\otimes n-i}|1\rangle^{\otimes i}$. The $\mathcal{W}_{|W_n\rangle}$ recovers a class of EWs presented in Ref.\ \cite{Bergmann_2013}, which are the most powerful ones for the $W$-state presently. Its white noise tolerance also tends to 1 for an increasing number of qubits. While for the fidelity witness, its white noise tolerance is $1/(n(1-1/2^n))$, tending to $1/n$ for large $n$.
\subsection{Graph state}
Graph states are a class of genuine multipartite entangled states that are of great importance for measurement-based quantum computation \cite{briegel2009measurement} and quantum error correction \cite{PhysRevA.65.012308}, etc. In Refs.\ \cite{PhysRevLett.106.190502,PhysRevA.84.032310}, the authors have developed powerful entanglement witnesses for graph states. While our framework suggests that there is still much room for improvement in the robustness of these existing results.
More specifically, we focus on a typical class of graph state---the $n$-qubit ($n\ge4$) linear cluster states $|Cl_n\rangle$ in this example. The $|Cl_n\rangle$ can be expressed by a set of stabilizers $\{g_i\}_{i=1}^n$, with $g_i=Z_{i-1}X_iZ_{i+1}$ ($2\le i \le n-1$), $g_1=X_1Z_2$ and $g_n=Z_{n-1}X_n$ respectively, where the $X$ and $Z$ are Pauli operators. All the common eigenstates of these stabilizers introduce a complete basis, i.e., the graph state basis. This basis can be denoted by $|\vec{a}\rangle_{Cl_n}$, with $\vec{a}=a_1a_2\cdots a_n\in \{0,1\}^n$, such that $g_i|\vec{a}\rangle_{Cl_n}=(-1)^{a_i}|\vec{a}\rangle_{Cl_n}$ for $i=1,\cdots,n$. Specially, the $|Cl_n\rangle$ corresponds to $|00\cdots0\rangle_{Cl_n}$. When applied to the linear cluster state, our framework results in a GME witness which is diagonal under the graph state basis, (For the explicit construction process, we refer to Appendix.\ \ref{sec:appendix graph state}.)
\begin{equation}
\mathcal{W}_{Cl_n}=\sum_{k=1}^{\lceil n/3 \rceil}
\sum_{\vec{a}\in V_k}\frac{1}{2^k-1}|\vec{a}\rangle_{Cl_n}\langle\vec{a}|-|Cl_n\rangle\langle Cl_n|.
\end{equation}
Here a vector $\vec{a}$ belongs to $V_k$ if there exist at most $k$ for the number of `$1$'s in $\vec{a}$, such that their distance with each other are larger than $2$ at the same time (for instance, $1101100$ belongs to $V_2$ while $1001011$ belongs to $V_3$).
\begin{figure*}
\caption{
In this figure we illustrate the performance of $\mathcal{W}
\label{fig:fig1}
\end{figure*}
Its white noise tolerance $p_{Cl_n}$ of the $\mathcal{W}_{Cl_n}$ is presented in Fig.\ \ref{fig:fig1}. It is observed that the $\mathcal{W}_{Cl_n}$ can outperform the best known class of EWs provided in the Ref.\ \cite{PhysRevLett.106.190502} for $n>5$. Meanwhile, the white noise tolerance $p_{Cl_n}$ exhibits a similar asymptotic behavior as in the first example, that is, tending to $1$ for large $n$. We remark that while the resulting EWs are quite robust, they are not optimal. In fact, the optimality of the bipartite EWs employed in the construction is not sufficient to guarantee the optimality of the resulting GME witness. For some explicit target states, one may either analytically or numerically optimize the result. While a systematic and operational improvement of this framework remains an open question. A brief discussion on this issue is provided at the end of Appendix.\ \ref{sec:appendix graph state}.
\subsection{Multipartite states admitting Schmidt decomposition}
In the above examples, we benchmark our method with some well studied states. And now we turn to other less investigated states, where this method remains powerful. A typical class is the multipartite states admitting Schmidt decomposition. Without loss of generality, such state takes the form $|\phi_s\rangle=\sum_{i=0}^{d-1}\sqrt{\lambda_i}|i\rangle^{\otimes n}$, where the $\lambda_i$ are in decreasing order. This class of states includes high-dimensional GHZ states $ |GHZ_n^d\rangle = \sum_{i=0}^{d-1} |i\rangle^{\otimes n}/\sqrt{d}$ as a typical case when all the Schmidt coefficients are equal. For the multipartite states admitting Schmidt decomposition, our method leads to a class of optimal EWs (see Appendix.\ \ref{sec:appendix GHZ state} for a proof.)
\begin{equation}\label{eq:eg3}
\mathcal{W}_{|\phi_s\rangle}=\sum_{\substack{i,j=0,\\i< j}}^{d-1} \sum_{r=1}^{n-1}\sum_m \sqrt{\lambda_i\lambda_j}\pi_m(|i\rangle^{\otimes r}|j\rangle^{\otimes n-r})\pi_m(\langle i^{\otimes r}|\langle j|^{\otimes n-r})+\sum_{i=0}^{d-1}\lambda_i|i\rangle\langle i|^{\otimes n}-|\phi_s\rangle\langle\phi_s|,
\end{equation}
where $\pi_m(|i\rangle^{\otimes r}|j\rangle^{\otimes n-r})$ is a permutation of $|i\rangle^{\otimes r}|j\rangle^{\otimes n-r}$
and the summation of $m$ is over all possible permutations.
\begin{figure*}
\caption{
In sub-figure (a),(b),(c) and (d), we show how the white noise tolerance of different EWs varies with an increasing qudit number $n$, with $d=3,~4,~5,~6$ respectively. The target state is $d$-dimensional GHZ states, which belongs to the class of states in Example 1. In each sub-figure, the white noise tolerance of the GME witness in the Eq.\ (\ref{eq:eg3}
\label{fig:ghz}
\end{figure*}
The white noise tolerance of $\mathcal{W}_{|\phi_s\rangle}$ is $p_{\mathcal{W}_{|\phi_s\rangle}}=(1-\sum_{i=0}^{d-1}\lambda_i^2)/(1-\sum_{i=0}^{d-1}\lambda_i^2+\frac{2^{n-1}-1}{d^n}((\sum_{i=0}^{d-1}\sqrt{\lambda_i})^2-1)))$. The $p_{\mathcal{W}_{|\phi_s\rangle}}$ tends to $1$ for large $n$ when $d > 2$. As a comparison, the best-known GME witness for this kind of states comes from the fidelity-based method, with $\mathcal{W}_F^{|\psi_s\rangle} = \lambda_0 I- |\phi_s\rangle\langle\phi_s|$. The white noise tolerance of $\mathcal{W}_F^{|\psi_s\rangle}$ is $(1-\lambda_0)/(1-1/d^n)$, which tends to $1-\lambda_0\le 1-1/d$ with an increasing system size. For the special case of $n$-qudit GHZ states $|GHZ_n^d\rangle$, the performance of our construction and the fidelity-based method is compared in Fig.\ \ref{fig:ghz}, where a significant improvement is demonstrated. Note that for $n$-qubit GHZ states $|GHZ_n\rangle$, the fidelity witness is already optimal, and hence we start from the local dimension $d=3$ in Fig.\ \ref{fig:ghz}.
\subsection{The four-qubit singlet state}
Multi-qubit singlet states are another interesting family of multi-qubit states. They are invariant under a simultaneous unitary rotation on all qubits ($U^{\otimes n}|\psi\rangle\langle \psi| {U^{\dag}}^{\otimes n} =|\psi\rangle\langle \psi|$). In the four-qubit case, all four-qubit singlet states live in a two-dimensional subspace of the whole Hilbert space. Without loss of generality, it can be denoted as
\begin{equation}
|\varphi_4\rangle = a |\psi_{12}^-\rangle\otimes |\psi_{34}^-\rangle + e^{i\theta}b |\psi_{13}^-\rangle\otimes |\psi_{24}^-\rangle,
\end{equation}
with the constraint $a^2+b^2+cos(\theta)ab=1$ and $|\psi_{12}^-\rangle$ being the two-qubit singlet state $(|01\rangle-|10\rangle)/\sqrt{2}$ on the first two qubits. Specially, for the choice of $\theta = \pi/2$, one arrives at a class of four-qubit singlet states decided by a single parameter $|\varphi_4(a)\rangle = a |\psi_{12}^-\rangle\otimes |\psi_{34}^-\rangle + \sqrt{1-a^2} |\psi_{13}^-\rangle\otimes |\psi_{24}^-\rangle$ with $a \in [-1,1]$. For this class of state $|\varphi_4(a)\rangle$, our framework results in the following GME witness
\begin{equation}
\mathcal{W}_4(a)= \alpha \mathcal{P}^4_2 +\frac{1}{2}(\mathcal{P}^4_1 + \mathcal{P}^4_3) +\frac{1}{4}(\mathcal{P}^4_0 + \mathcal{P}^4_4)- |\varphi_4\rangle\langle \varphi_4|,
\end{equation}
where $\alpha = \max\{(1+3(1-a^2))/4, (1+3a^2)/4\} \ge 5/8$. While the fidelity based witness for such state is $\mathcal{W}_4'(a) = \alpha I - |\varphi_4\rangle\langle \varphi_4|$. In Appendix.\ \ref{sec:appendix singlet}, a further discussion of the entanglement detection for multi-qubit singlet states is proposed based on our framework.
Consequently, we have provided a generic framework for detecting arbitrary target GME state in a noisy systems by constructing robust GME witnesses. Firstly, by benchmarking its performance on some well-studied states, it is observed that this framework results in robust GME witnesses that perform comparable to the current best witnesses for these states. For other less investigated states, the most widely used method to construct EW for them is the fidelity-based method. As shown in these examples, our framework can provide a significant improvement compared with the fidelity-based method. This also leads to the conjecture that a large amount of pure GME states become fairly robust to noise as the system size increases. Secondly, the advantage of our framework against the fidelity-based method comes with no experimental overheads. This benefits from the fact that the $\sum_k c_k \tilde{I}_k$ term in this construction is usually diagonal in some well-defined basis, such as the graph state basis and the computational basis. Finally, it should be stressed that Theorem 1 can be applied not only to the class of bipartite EWs shown in Eq.\ (\ref{eq:obEW}), but also to other classes of bipartite EWs. This potentially results in some different GME witnesses. Further example is provided in Appendix.\ \ref{sec:appendix generalization}.
\section{Applications of the resulting GME witnesses}
\subsection{Detection of unfaithfulness}
The unfaithful entangled states are a large class of states that cannot be recognized with any fidelity witness and have been attracted both theoretical and experimental interests \cite{PhysRevLett.126.140503,zhan2020detecting,PhysRevA.103.042417,PhysRevLett.127.220501}. Therefore, given that we have already gained access to construct finer GME witnesses than the fidelity-based method, it is natural to investigate their ability on the detection of unfaithful GME states.
In general, deciding whether an entangled state is unfaithful is a nontrivial task, since one has to prove that the state is not detected by all fidelity witnesses, rather than a certain one. In bipartite case, a necessary and sufficient criterion for a state $\rho_{AB}$ to be unfaithful has been proposed \cite{PhysRevLett.126.140503}, while for multipartite case, it remains an open question on characterization of unfaithfulness. To avoid this difficulty and verify an EW indeed detects unfaithfulness, we limit our attention to a special class of states $\rho(p)=p I/d^n+(1-p) \rho_0$. that there is an upper bound on the white noise tolerance of any fidelity witness for arbitrary state. Denote $\mathcal{W}_F=\alpha I-\rho'$ as an arbitrary fidelity witness, then one can derive its white noise tolerance $p_F$ for arbitrary $\rho(p)$ by solving $Tr(\mathcal{W}_F\rho(p))=0$, which leads to
\begin{equation}
p_F=\max\{\frac{Tr(\rho_0\rho')-\alpha}{Tr(\rho_0\rho')-1/d^n},0\}.
\end{equation}
Then it is straightforward to see that $p_F\le (1-1/d)/(1-1/d^n)$, due to the fact that $Tr(\rho_0\rho')\le 1$ and $\alpha \ge 1/d$. Hence it can be concluded that an EW can be employed to detect some unfaithful entangled states, as long as its white noise tolerance for some state is higher than $(1-1/d)/(1-1/d^n)$. This is precisely the case for many GME witnesses constructed with our framework. For example, in an $n$-qubit case, this upper bound is $1/(2(1-1/2^n))$ and decreases to $1/2$ as $n$ grows. While our framework provides large amount of EWs with white noise tolerance converging to $1$, allowing for the certification of unfaithfulness of many states in $n$-qubit case.
\subsection{Estimating entanglement measures}
Moreover, a witness operator is useful not only for entanglement certification, but also for entanglement quantification. To start with, we briefly review the method developed in Ref.\ \cite{PhysRevLett.98.110502} for optimally estimating some entanglement measure $E$ given the expectation value of some witness operator $\mathcal{W}$. The task can be described as finding the lower bound
\begin{equation}
\epsilon(w)=\inf_{\rho} \{E(\rho)|Tr(\rho \mathcal{W})=w\},
\end{equation}
where the infimum is taken over all states compatible with the data $w=Tr(\rho \mathcal{W})$. Note that $\epsilon(w)$ is a convex function, and thus there exist bounds of the type
\begin{equation}
\epsilon(w) \ge r\cdot w-c,
\end{equation}
for an arbitrary $r$. By inserting $w=Tr(\rho \mathcal{W})$ and $E(\rho) \ge \epsilon(w)$, it is observed that
\begin{equation}
c \ge r \cdot Tr(\rho \mathcal{W})-E(\rho),
\end{equation}
should be satisfied for any $\rho$. Hence given a "slope" $r$, the optimal constant $c$ is
\begin{equation}
c= \hat{E}(r\cdot \mathcal{W}):=\sup_{\rho} \{ r\cdot Tr(\rho \mathcal{W})-E(\rho)\}.
\end{equation}
Finally, an optimal lower bound is obtained after optimizing $r$:
\begin{equation}
\epsilon(w)=\sup_r \{r\cdot w - \hat{E}(r\cdot \mathcal{W})\}.
\end{equation}
It should be remarked that we limit our discussions into the nontrivial case where a negative expectation $w$ of a witness operator is observed in the following. In this case, the optimal "slope" $r$ should always be negative.
Now, suppose that the $\mathcal{W}_2$ is a finer EW than the $\mathcal{W}_1$, satisfying $\mathcal{W}_2 \preceq \mathcal{W}_1$. It is straightforward to see that
\begin{equation}
\hat{E}(r\cdot \mathcal{W}_1) \ge \hat{E}(r\cdot \mathcal{W}_2).
\end{equation}
Therefore, when these two operators $\mathcal{W}_1$ and $\mathcal{W}_2$ have the same expectation value $w_0$,
\begin{equation}
\begin{aligned}
\epsilon_2(w_0)= &\sup_r \{r\cdot w_0 - \hat{E}(r\cdot \mathcal{W}_2)\} \\
\ge &\sup_r \{r\cdot w_0 - \hat{E}(r\cdot \mathcal{W}_1)\} \\
=& \epsilon_1(w_0).
\end{aligned}
\end{equation}
For the same target state $\rho$, the expectations $w_1$ and $w_2$ of these two witness operators always satisfy $w_1 \ge w_2$, which leads to $\epsilon_2(w_2) \ge \epsilon_2(w_1) \ge \epsilon_1(w_1)$. That is, a finer EW provides a tighter lower bound on the entanglement measure for the same state. Hence, our framework enables a better estimation of the entanglement measures of genuine multipartite entanglement than the fidelity-based method.
To quantitatively investigate the improvement from these new GME witness, we discuss the estimation on the geometry measure of genuine multipartite entanglement for noisy $n$-partite $d$-dimensional GHZ states $\rho_{n,d}(p)=pI/d^n+(1-p) |GHZ_n^d\rangle\langle GHZ_n^d|$, with $ |GHZ_n^d\rangle = \sum_{i=0}^{d-1} |i\rangle^{\otimes n}/\sqrt{d}$. For arbitrary multipartite pure state $|\psi\rangle$, its geometric measurement of GME is defined by $E_G(|\psi\rangle) = 1-\max_{|\phi_{bs}\rangle}|\langle\phi_{bs}|\psi\rangle|^2$, with $|\phi_{bs}\rangle$ being arbitrary pure biseparable state. The geometric measure of GME is extended to mixed states by the convex roof construction
\begin{equation}
E_G(\rho)=\inf \limits_{p_i,|\psi_i\rangle}\sum_i p_i E_G(|\psi_i\rangle),
\end{equation}
where the minimization runs over all possible decompositions $\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$.
Based on the result in Ref.\ \cite{PhysRevApplied.13.054022}, one can derive a lower bound $\epsilon_f^{n,d}(p)$ for $E_G(\rho_{n,d}(p))$
\begin{equation}
E_G(\rho_{n,d}(p)) \ge \epsilon_f^{n,d}(p) :=1-\gamma(S),
\end{equation}
where $\gamma(S)=[\sqrt{S}+\sqrt{(d-1)(d-S)}]^2/d$ with $S=\max \{1, d(1-p)+p/d^{n-1}\}$. This is just the lower bound related to the fidelity witness $\mathcal{W}_F=I/d-|GHZ_n^d\rangle\langle GHZ_n^d|$. Whereas it has been proved in the previous section that finer EW is accessible with our method, that is,
\begin{equation}
\begin{aligned}
\mathcal{W}_{o,|GHZ_n^d\rangle}=&\sum_{\substack{i,j=0,\\i< j}}^{d-1} \sum_{r=1}^{n-1} \sum_m \frac{1}{d}\pi_m(|i\rangle^{\otimes r}|j\rangle^{\otimes n-r})\pi_m(\langle i^{\otimes r}|\langle j|^{\otimes n-r}) \\
&+\sum_{i=0}^{d-1}\frac{1}{d}|i\rangle\langle i|^{\otimes n}-|GHZ_n^d\rangle\langle GHZ_n^d|.
\end{aligned}
\end{equation}
With the expectation value $w_{n,d}(p)=Tr(\rho_{n,d}(p)\mathcal{W}_{o,|GHZ_n^d\rangle})$ from this finer EW, a lower bound $\epsilon_o^{n,d}(p)$ can be derived by employing the technique developed in Ref.\ \cite{PhysRevLett.98.110502}:
\begin{equation} \label{lowerbd}
E_G(\rho_{n,d}(p)) \ge \epsilon_o^{n,d}(p) :=\sup \limits_{r}\left\{r\cdot w_{n,d}(p)-\hat{E}_G(r\mathcal{W}_{o,|GHZ_n^d\rangle})\right\},
\end{equation}
with $r$ being a real number, and
\begin{equation}\label{hatE}
\hat{E}_G(r\mathcal{W}_{o,|GHZ_n^d\rangle})=\sup\limits_{|\psi\rangle}\sup\limits_{|\phi_{bs}\rangle} \left\{\langle\psi|(r\mathcal{W}_{o,|GHZ_n^d\rangle}+|\phi_{bs}\rangle\langle\phi_{bs}|)|\psi\rangle-1\right\},
\end{equation}
where the maximization runs over all pure state $|\psi\rangle$ and biseparable state $|\phi_{bs}\rangle$. Furthermore, in this special case, it can be verified that one has to choose $|\phi_{bs}\rangle$ as a state having the largest overlap with $|GHZ_n^d\rangle$, which results in
\begin{equation}
\hat{E}_G(r\mathcal{W}_{o,|GHZ_n^d\rangle})=\frac{1-r}{2}+\frac{1}{2}\sqrt{(1-r)^2+4r\frac{d-1}{d}}+\frac{r}{d}-1.
\end{equation}
By inserting this equation into Eq.\ (\ref{lowerbd}), the lower bound $\epsilon_o^{n,d}(p)$ can be solved directly.
\begin{figure*}
\caption{
In this figure, we choose $d=3$ and $n=3,~5,~7,~9$ as examples to compare the lower bound $\epsilon_o^{n,d}
\label{fig:measure}
\end{figure*}
In Fig.\ \ref{fig:measure}, we have shown the results for $d=3$ and $n=3,~5,~7,~9$ as examples, to illustrate the performance of our method with an increasing system size. As the number of subsystems grows, the critical value of $p$ tends to $1$, when the lower bound $\epsilon_o^{n,d}(p)$ vanishes. Meanwhile, the $\epsilon_o^{n,d}(p)$ is always larger than the $\epsilon_f^{n,d}(p)$ above, which vanishes at $p=1-1/d$ for large $n$. That is, the new EWs $\mathcal{W}_{o,|GHZ_n^d\rangle}$ are able to provide a better estimation on the geometric measure of GME for $\rho_{n,d}(p)$. It remains open whether $\epsilon_o^{n,d}(p)$ equals $E_G(\rho_{n,d}(p))$. However, it is still reasonable to expect that such new GME witnesses can provide faithful estimations on entanglement measures without the need for quantum tomography, as they are already robust enough.
\section{Conclusion and outlook}
In summary, we have developed a exquisite framework and scheme for genuine multipartite entanglement detection, and demonstrated its operability and universality by applying it on typical GME states that arise in practice. In particular, this is achieved using a novel method to bring any complete set of bipartite EWs to a single GME witness. This method allows to make full use of some prior information about the target state to improve the noise resistance. In fact, the resulting GME witnesses turn out to be quite robust, whose white noise tolerance converge to $1$ in many cases. As a consequence, this framework holds great practical potential in real-life situations, especially for detecting entanglement in noisy multipartite or high-dimensional systems. This will play a certain role in facilitating the solution of the very challenging problem of genuine multipartite entanglement detection.
In addition to genuine multipartite entanglement, we remark that our method is highly flexible and admits natural generalizations for detecting other types of entanglement. A relevant case is entanglement detection in quantum networks, which is currently under active investigations. In quantum networks, multipartite entanglement exhibits novel features due to the complex network topology \cite{PhysRevLett.125.240505,PhysRevA.103.L060401,PhysRevLett.128.220501}, and better techniques are urgently needed for the characterization of genuine network multipartite entanglement. Finally, it will also be interesting to seek for further extension of the framework in high-order entanglement detection \cite{PhysRevLett.105.210504} as well as bound entanglement detection.
\section*{Acknowledgments}
We thank Yi-Zheng Zhen for very valuable discussion. This work has been supported by the National Natural Science Foundation of China (Grants No. 62031024, 11874346, 12174375), the National Key R$\&$D Program of China (2019YFA0308700), the Anhui Initiative in Quantum Information Technologies (AHY060200), and the Innovation Program for Quantum Science and Technology (No. 2021ZD0301100).
\appendix
\section*{Appendix}
\section{Proof and discussions of the bipartite EW in Eq. (5)}\label{sec:appendix bipartite EW}
\subsection {A class of bipartite entanglement witness}
Let $|\phi\rangle$ be an arbitrary pure entangled state in the $d\times d$ dimensional Hilbert space $\mathcal{H}_d\otimes\mathcal{H}_d$. Without loss of generality, one can assume $|\phi\rangle=\sum_{i=0}^{d-1}\sqrt{\lambda_i}|ii\rangle$, where all $\lambda_i\ge 0$ are Schmidt coefficients in decreasing order and $\sum_i \lambda_i=1$. One can define a positive operator $Q$ as
\begin{equation}\label{eq:Q}
Q=\sum_{\substack{i,j=0,\\i<j}}^{d-1} \sqrt{\lambda_i\lambda_j}(|ij\rangle-|ji\rangle)(\langle ij|-\langle ji|),
\end{equation}
which can be used for constructing an EW for $|\phi\rangle$.
\begin{lemma}
The partial transpose of $Q$ provides an optimal EW $\mathcal{W}_o^{|\phi\rangle}$, which $\mathcal{W}_o^{|\phi\rangle}$ reads
\begin{equation}
\mathcal{W}_o^{|\phi\rangle}=Q^{\Gamma}=\sum_{i,j=0}^{d-1} \sqrt{\lambda_i\lambda_j}|ij\rangle\langle ij|-|\phi\rangle\langle\phi|
\end{equation}
\end{lemma}
\begin{proof}
To prove that the $\mathcal{W}_o^{|\phi\rangle}$ is an EW, note that it is of the form $Q^{\Gamma}$ with $Q$ being positive semidefinite ($Q\succeq 0$). Thus for all separable states $Tr(\rho_{sep}\mathcal{W}_o^{|\phi\rangle}) = Tr(\rho_{sep}^{\Gamma}Q)\ge 0$. Meanwhile, $Tr(\mathcal{W}_o^{|\phi\rangle}|\phi\rangle\langle\phi|)=\sum_{i}\lambda_i^2-1=-\sum_{i\ne j}\lambda_i\lambda_j<0$. Then $\mathcal{W}_o^{|\phi\rangle}$ is an EW by definition.
To show the optimality of $\mathcal{W}_o^{|\phi\rangle}$, it is sufficient to prove that the set of pure separable states $\{|\phi_1\rangle\otimes|\phi_2\rangle\}$ satisfying $\langle\phi_1|\langle\phi_2|\mathcal{W}_o|\phi_1\rangle|\phi_2\rangle=0$ span the whole Hilbert space $\mathcal{H}_d\otimes\mathcal{H}_d$ \cite{PhysRevA.62.052310}. For qubit case, one has $\mathcal{W}_o^{(2)}=\sqrt{\lambda_0\lambda_1}(|01\rangle-|10\rangle)(\langle 01|-\langle10|)^{\Gamma}$. It is easy to verify that the set of separable states $\{|00\rangle,~(|0\rangle+|1\rangle)(|0\rangle+|1\rangle)/2,~(|0\rangle+i|1\rangle)(|0\rangle-i|1\rangle)/2,~|11\rangle\}$ satisfying $Tr(\rho_{sep}\mathcal{W}_o^{(2)}) = 0$. This set of states span the whole Hilbert space $\mathcal{H}_2\otimes\mathcal{H}_2$. In fact, it has been shown that any decomposable EW acting on $\mathcal{H}_2\otimes\mathcal{H}_d$ is optimal iff it takes the form $\mathcal{W}=Q^{\Gamma}$ for some $Q\succeq 0$ \cite{Augusiak_2011}.
Similarly, in the qudit case ($d>2$), there exist separable states $\{|ee\rangle,~(|e\rangle+|f\rangle)(|e\rangle+|f\rangle)/2,~(|e\rangle+i|f\rangle)(|e\rangle-i|f\rangle)/2,~|ff\rangle\}$ satisfying $Tr(\rho_{sep}\mathcal{W}_o^{|\phi\rangle})=0$, for each pair $0\le e< f \le d-1$. These states span the same space with $\{|ee\rangle,~|ef\rangle,~|fe\rangle,~|ff\rangle\}$. By iterating over all $e<f$, one ends up with a set of separable states spanning the whole space $\mathcal{H}_d\otimes\mathcal{H}_d$. Thus the EW $\mathcal{W}_o^{|\phi\rangle}$ is optimal. This finishes the proof.
\end{proof}
\subsection{Detection of bipartite unfaithful state}
Remarkably, for the $|\phi\rangle$, the most widely used fidelity witness reads $\mathcal{W}_F^{|\phi\rangle} = \lambda_0 I -|\phi\rangle\langle\phi|$. It is straightforward to observe that $\mathcal{W}_F^{|\phi\rangle} -\mathcal{W}_o^{|\phi\rangle}\succeq 0$, which means that the $\mathcal{W}_o^{|\phi\rangle}$ is finer than the $\mathcal{W}_F^{|\phi\rangle}$. This leads to a byproduct that the $\mathcal{W}_o^{|\phi\rangle}$ can detect unfaithful states. Unfaithful states are entangled states which can not be detected by all fidelity witnesses \cite{PhysRevLett.124.200502}, namely, an entangled state $\rho$ is unfaithful if and only if $Tr(\rho W_F^{|\psi\rangle}) \ge 0$ for all $|\psi\rangle$. Therefore, the relationship $\mathcal{W}_F^{|\phi\rangle} -\mathcal{W}_o^{|\phi\rangle} \succeq 0$ itself is not sufficient to demonstrate that the extra entangled states detected by $\mathcal{W}_o^{|\phi\rangle}$ is unfaithful. And a further clarification is required to justify the statement that $\mathcal{W}_o^{|\phi\rangle}$ detects unfaithful state.
Now we would like to provide qualitative and quantitative characterization on the ability to detect unfaithfulness of the $\mathcal{W}_{o}^{|\phi\rangle}$. Consider the class of states $\rho_{|\phi\rangle}(p)=pI/d^2+(1-p)|\phi\rangle\langle\phi|$. From the Observation 1 in Ref.\ \cite{PhysRevLett.126.140503}, it is known that such states are faithful if and only if it is detected by $\mathcal{W}_m=I/d-|\phi_d^+\rangle\langle\phi_d^+|$, with $|\phi_d^+\rangle$ being the maximally entangled state $\sum_{i=0}^{d-1}1/\sqrt{d}|ii\rangle$. By solving $Tr(\rho_{|\phi\rangle}(p)\mathcal{W}_{m})=0$, we obtain that the white noise tolerance of $\mathcal{W}_m$ for $|\phi\rangle$ is
\begin{equation}
p_f^{|\phi\rangle}=\frac{\sum_{i,j=0}^{d-1}\sqrt{\lambda_i\lambda_j}-1}{\sum_{i,j=0}^{d-1}\sqrt{\lambda_i\lambda_j}-\frac{1}{d}}.
\end{equation}
That is, $\rho_{|\phi\rangle}(p)$ is faithful when $p<p_f^{|\phi\rangle}$.
Similarly, one can obtain the white noise tolerance of $\mathcal{W}_o^{|\phi\rangle}$ for $|\phi\rangle$, which is
\begin{equation}
p_o^{|\phi\rangle}=\frac{1-\sum_{i=0}^{d-1}\lambda_i^2}{1-\sum_{i=0}^{d-1}\lambda_i^2+\frac{1}{d^2}(\sum_{i,j=0}^{d-1}\sqrt{\lambda_i\lambda_j}-1)}.
\end{equation}
It can be observed that
\begin{equation}
\begin{aligned}
\frac{1/p_{o}^{|\phi\rangle}-1}{1/p_{f}^{|\phi\rangle}-1}=&\frac{(\sum_{i,j=0}^{d-1}\sqrt{\lambda_i\lambda_j}-1)^2}{d(d-1)(1-\sum_i \lambda_i^2)} \\
=&1+\frac{(\sum_{i\ne j}\sqrt{\lambda_i\lambda_j})^2-d(d-1)(1-\sum_i \lambda_i^2)}{d(d-1)(1-\sum_i \lambda_i^2)} \\
\le&1+\frac{d(d-1)(\sum_{i\ne j}\lambda_i\lambda_j)-d(d-1)(1-\sum_i \lambda_i^2)}{d(d-1)(1-\sum_i \lambda_i^2)} \\
=&1+\frac{d(d-1)((\sum_{i=0}^{d-1}\lambda_i)^2-1)}{d(d-1)(1-\sum_i \lambda_i^2)}=1.
\end{aligned}
\end{equation}
In other words, $p_{f}^{|\phi\rangle} \le p_{o}^{|\phi\rangle}$, where the inequality comes from the Cauchy–Schwarz inequality, and takes equality if $d=2$ or $\lambda_i=1/d$ for all $i$. Therefore, the $\mathcal{W}_o^{|\phi\rangle}$ can always detect unfaithful state $\rho_{|\phi\rangle}(p)$ for $p\in[p_f^{|\phi\rangle},~p_o^{|\phi\rangle})$, unless $|\phi\rangle$ being a two-qubit state or maximally entangled.
\begin{table}[t]
\centering
\begin{tabular}{|c |c |c |c |c |c |}
\hline
d &3 &4&5&6&7 \\
\hline
$l_d$&0.2679&0.4202&0.5195&0.5896&0.6624\\
\hline
\end{tabular}
\caption{
Maximal unfaithful length $l_d$ from the class of EWs $\mathcal{W}_o^{|\phi\rangle}$. We remark that the optimization of $l_d$ may arrive at a local maximum. We use enough random starting points to support the claim that we arrive at the global maximum.
}
\label{tab:unfaithful_max}
\end{table}
As a quantitative investigation, we numerically maximize the interval length $l_d$ of $[p_f^{|\phi\rangle},~p_o^{|\phi\rangle})$ over all $|\phi\rangle$ for different local dimension $d$. We name $l_d$ the maximal unfaithful length from the class of EWs $\mathcal{W}_o^{|\phi\rangle}$, and the results are listed in Table. \ref{tab:unfaithful_max} for $d=3,4,\cdots,7$. It can be seen that $l_d$ grows significantly with an increasing dimension $d$, indicating that the $\mathcal{W}_o^{|\phi\rangle}$ can greatly outperform the fidelity witness. This is also in agreement with the statement that most states are unfaithful as claimed in Ref.\ \cite{PhysRevLett.124.200502}.
Except for the $l_d$, one may be also interested in the average performance of this new class of EWs on unfaithfulness detection. As a comparison, it is natural to consider two interval $[p_e^{|\phi\rangle}, p_f^{|\phi\rangle})$ and $[p_o^{|\phi\rangle}, p_f^{|\phi\rangle})$, where the $p_e^{|\phi\rangle}$ is the critical value such that $\rho_{|\phi\rangle}(p)$ becomes separable. The former interval contains all unfaithful $\rho_{|\phi\rangle}(p)$, while the latter contains the part that can be detected by the class of $\mathcal{W}_o^{|\phi\rangle}$. Then one can use $avg_{|\phi\rangle} (p_o^{|\phi\rangle}-p_f^{|\phi\rangle})/(p_e^{|\phi\rangle}-p_f^{|\phi\rangle})$ to evaluate the average performance of $\mathcal{W}_o^{|\phi\rangle}$ for detecting unfaithfulness, as shown in Table. \ref{tab:unfaithful_avg}. It is observed that a large percentage of unfaithful states have been detected. This is also the premise that GME witnesses constructed from this class of bipartite EWs can detect multipartite unfaithful state.
\begin{table}[hbtp]
\centering
\begin{tabular}{|c |c |c |c |c |c |}
\hline
d &3 &4&5&6&7 \\
\hline
$avg_{|\phi\rangle} (p_o^{|\phi\rangle}-p_f^{|\phi\rangle})$ &0.0804&0.0969&0.0963&0.0909&0.0848\\
\hline
$avg_{|\phi\rangle} (p_e^{|\phi\rangle}-p_f^{|\phi\rangle})$ &0.1190&0.1460&0.1457&0.1379&0.1286\\
\hline
$avg_{|\phi\rangle} \frac{p_o^{|\phi\rangle}-p_f^{|\phi\rangle}}{p_e^{|\phi\rangle}-p_f^{|\phi\rangle}}$ &0.5605&0.5937&0.6089&0.6181&0.6248\\
\hline
\end{tabular}
\caption{
Average performance of $\mathcal{W}_o^{|\phi\rangle}$ for detecting unfaithfulness. For different local dimension $d$, the average is taken by randomly generating $10^7$ pure states in $\mathcal{H}_d \otimes \mathcal{H}_d$. Since any pure bipartite state admits a Schmidt decomposition $|\phi\rangle = \sum_i \sqrt{\lambda_i} |ii\rangle$, we replace the randomly generated pure bipartite states with random vectors $(\sqrt{\lambda_0},\cdots,\sqrt{\lambda_{d-1}})$ uniformly distributed on the $d$-dimensional unit sphere. Moreover, the critical value $p_e^{|\phi\rangle}$ is $\frac{d^2\sqrt{\lambda_0\lambda_1}}{1+d^2\sqrt{\lambda_0\lambda_1}}$ according to the results in Ref.\ \cite{PhysRevA.59.141}, assuming that the Schmidt coefficients $\lambda_i$ are in decreasing order.}
\label{tab:unfaithful_avg}
\end{table}
\subsection{Generalization of Lemma 1}\label{sec:appendix generalization}
Finally, we provide a generalization of Lemma 1. For the above entangled state$|\phi\rangle$, one can construct another positive operator
\begin{equation}\label{eq:general_Q}
\tilde{Q}=\sum_{\substack{i,j=0,\\i<j}}^{d-1} (\alpha_{ij}|ij\rangle-\beta_{ij}|ji\rangle)(\alpha_{ij}\langle ij|-\beta_{ij}\langle ji|),
\end{equation}
instead of $Q$, where $\alpha_{ij}\beta_{ij}=\sqrt{\lambda_i\lambda_j}$ and the $\alpha_{ij},\beta_{ij}$ are all positive. The operator $\tilde{\mathcal{W}}_o=\tilde{Q}^{\Gamma}$ is also optimal EW and applicable in our framework for GME witness construction. Here, the proof of the optimality of $\tilde{\mathcal{W}}_o$ is similar to the case in Lemma 1. It is sufficient to verify that the set of state $\{|ee\rangle,~|ff\rangle,~(\sqrt{\beta_{ef}}|e\rangle + \sqrt{\alpha_{ef}}|f\rangle)\otimes(\sqrt{\alpha_{ef}}|e\rangle + \sqrt{\beta_{ef}}|f\rangle),~(\sqrt{\beta_{ef}}|e\rangle + i\sqrt{\alpha_{ef}}|f\rangle)\otimes(\sqrt{\alpha_{ef}}|e\rangle - i\sqrt{\beta_{ef}}|f\rangle)\}_{e,f=0}^{d-1}$ have zero expectation value when measured with $\tilde{\mathcal{W}}_o$, and span the whole $d^2$-dimensional Hilbert space.
\section{Proof of the examples}
In this section, we will show explicitly how this construction can be applied to some commonly used multipartite entangled states, and make further discussions on the results.
\subsection{$W$-state}\label{sec:appendix w state}
The $W$-state is an important class of multiqubit entangled states. A class of EWs for $W$-state which can outperform significantly than the fidelity witness has been proposed in Ref.\ \cite{Bergmann_2013}. In Ref.\ \cite{Bergmann_2013}, the authors construct an operator at first, and then prove that this operator is decomposable bipartite EW with respect to all possible bipartitions. While our construction is in the opposite direction. We construct a complete set of bipartite EWs for $W$-state, and lift them to a GME witness. Although different method has been used, our construction recovers the result in Ref.\ \cite{Bergmann_2013}.
We start with the simplest 3-qubit case, where the target state is
\begin{equation}
|W_3\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle).
\end{equation}
For the bipartition $1|23$, the EW constructed from Lemma 1 is of the form
\begin{equation}
\begin{aligned}
\mathcal{W}_{1|23}^{|W_3\rangle}
&=\frac{\sqrt{2}}{3}(|000\rangle\langle000|+|1\psi^+\rangle\langle1\psi^+|) \\
&~~~~+\frac{2}{3}|0\psi^+\rangle\langle0\psi^+|+\frac{1}{3}|100\rangle\langle100|-|W_3\rangle\langle W_3|,
\end{aligned}
\end{equation}
with $|\psi^+\rangle=(|01+10\rangle)/\sqrt{2}$. And for the other two bipartitions, the $\mathcal{W}_{2|13}$ and $\mathcal{W}_{3|12}$ can be obtained after permutation of qubits. Then for $|W_3\rangle$, the set $\mathcal{S}$ reads
\begin{equation}
\begin{aligned}
\mathcal{S}=\{&|000\rangle\langle000|,|001\rangle\langle001|,~|010\rangle\langle010|,~|100\rangle\langle100|,\\
&|0\psi^+\rangle\langle0\psi^+|,~|0_2\psi_{13}^+\rangle\langle0_2\psi_{13}^+|,~|\psi^+0\rangle\langle\psi^+0|,\\
&|1\psi^+\rangle\langle1\psi^+|,~|1_2\psi_{13}^+\rangle\langle1_2\psi_{13}^+|,~|\psi^+1\rangle\langle\psi^+1|\}.
\end{aligned}
\end{equation}
These states in $\mathcal{S}$ can be grouped into 3 subsets according to the procedure in the main text:
\begin{equation}
\begin{aligned}
\mathcal{S}_1&=\{|000\rangle\langle000|\},\\
\mathcal{S}_2&=\{|001\rangle\langle001|,~|010\rangle\langle010|,~|100\rangle\langle100|,\\
&~~~~~~~|0\psi^+\rangle\langle0\psi^+|,~|0_2\psi_{13}^+\rangle\langle0_2\psi_{13}^+|,~|\psi^+0\rangle\langle\psi^+0|\}\\
\mathcal{S}_3&=\{|1\psi^+\rangle\langle1\psi^+|,~|1_2\psi_{13}^+\rangle\langle1_2\psi_{13}^+|,~|\psi^+1\rangle\langle\psi^+1|\},
\end{aligned}
\end{equation}
and the corresponding $\alpha_k$ by Theorem 1 is
\begin{equation}
\alpha_1=\sqrt{2}/3, ~\alpha_2=2/3,~\alpha_3=\sqrt{2}/3
\end{equation}
respectively. This result in a GME witness
\begin{equation}
\begin{aligned}
\mathcal{W}_{|W_3\rangle}=&\frac{\sqrt{2}}{3}(|000\rangle\langle000|+|101\rangle\langle101|+|011\rangle\langle011|+|110\rangle\langle110|)\\
&+\frac{2}{3}(|001\rangle\langle001|+|010\rangle\langle010|+|100\rangle\langle100|)-|W_3\rangle\langle W_3|.
\end{aligned}
\end{equation}
Moreover, by employing the generalization of Lemma 1 in Eq.\ (\ref{eq:general_Q}), one obtains
\begin{equation}
\mathcal{W}'_{1|23}=\left[(a|0\rangle|00\rangle-b|1\rangle|\psi^+\rangle)(a\langle0|\langle00|-b\langle1|\langle\psi^+|)\right]^{\Gamma_1},
\end{equation}
where $a$, $b$ are positive numbers and satisfy $ab=\sqrt{2}/3$. The other two bipartite EWs are obtained immediately after rearrangement of the qubits. For this set of bipartite EWs, the EW $\mathcal{W}_{|W_3\rangle}$ can be generalized into
\begin{equation}
\begin{aligned}
\mathcal{W}_{|W_3\rangle}'=&a^2|000\rangle\langle000|+b^2(|101\rangle\langle101|+|011\rangle\langle011|+|110\rangle\langle110|)\\
&+\frac{2}{3}(|001\rangle\langle001|+|010\rangle\langle010|+|100\rangle\langle100|)-|W_3\rangle\langle W_3|.
\end{aligned}
\end{equation}
In $n$-qubit cases, if a subsystem $A$ contains $m$ qubits, the corresponding bipartite EW from the Lemma 1 is of the form ($1\le m\le n-1$)
\begin{equation}
\mathcal{W}_{m|n-m}=\sqrt{\frac{m(n-m)}{n^2}}|\psi\rangle_{m|n-m}\langle\psi|^{\Gamma_A},
\end{equation}
where $|\psi\rangle_{m|n-m}={|0\rangle^{\otimes m}}_A{|0\rangle^{\otimes n-m}}_{\bar{A}}-|W_m\rangle_A|W_{n-m}\rangle_{\bar{A}}$. Then the set $\mathcal{S}$ for $|W_n\rangle$ can still be grouped into 3 subsets:
\begin{equation}
\{|0^{\otimes n}\rangle\},~\{|\pi_m(0^{\otimes n-1}1)\rangle\},~\{|\pi_m'(0^{\otimes n-2}1^{\otimes 2})\rangle\},
\end{equation}
with the corresponding coefficients $\alpha_k$ being
\begin{equation}
\begin{aligned}
\alpha_1&=\max_m \sqrt{\frac{m(n-m)}{n^2}}=\frac{\sqrt{\lfloor n/2\rfloor(n-\lfloor n/2\rfloor)}}{n},\\
\alpha_2&=\max_m \frac{n-m}{n}=\frac{n-1}{n}, \\
\alpha_3&=\max_m \sqrt{\frac{m(n-m)}{n^2}}=\frac{\sqrt{\lfloor n/2\rfloor(n-\lfloor n/2\rfloor)}}{n}.
\end{aligned}
\end{equation}
Therefore we arrive at the following $\mathcal{W}_{|W_n\rangle}$,
\begin{equation} \label{eq:w_state}
\mathcal{W}_{|W_n\rangle}=\frac{n-1}{n}\mathcal{P}_1+\frac{\sqrt{\lfloor n/2\rfloor(n-\lfloor n/2\rfloor)}}{n}
(|0\rangle\langle 0|^{\otimes n} +\mathcal{P}_2)-|W_n\rangle\langle W_n|,
\end{equation}
with $\mathcal{P}_i=\sum_m\pi_m(|0\rangle^{\otimes n-i}|1\rangle^{\otimes i})\pi_m(\langle 0|^{\otimes n-i}\langle 1|^{\otimes i})$, where the summation $m$ is over all possible permutation of $|0\rangle^{\otimes n-i}|1\rangle^{\otimes i}$.
The EW $\mathcal{W}_{|W_n\rangle}$ can also be generalized in a similar manner with the $\mathcal{W}_{|W_3\rangle}$, so as to recover the results of Ref.\ \cite{Bergmann_2013}. Although ending up with the same witness operator, our construction provides a different insight on why the $\mathcal{W}_{|W_n\rangle}$ takes such a form.
\subsection{Graph states}\label{sec:appendix graph state}
Before discussing the construction of GME witnesses for the graph states, we first give a brief introduction to the graph states. A graph is a pair $G=(V,E)$ of sets, where the elements of $V$ are called vertices, and the elements of $E$ are edges connecting the vertices. For example, $(1,2) $ represents the edge connecting vertex $1$ and $2$. Two vertices are called neighboring if they are connected by an edge. A graph can also be represented by the adjacency matrix $\Gamma$ with
\begin{equation}
\Gamma_{ij} = \left\{
\begin{aligned}
&1, \quad if ~(v_i,~v_j) \in E,\\
&0, \quad otherwise.
\end{aligned}
\right.
\end{equation}
Then, an $n$-qubit graph state $|G\rangle$ is defined with an $n$-vertex graph $G$ whose vertices correspond to qubits and edges correspond to control-Z (C-Z) gate between two qubits. Graph state can be expressed with a set of stabilizers
\begin{equation}
g_i = X_i\prod_{j \in N(i)} Z_j, i = 1,\cdots,n,
\end{equation}
where $X_i$ and $Z_i$ are Pauli operators on qubit (vertex) $i$, and $N(i)$ is the neighborhood of $i$ (\textit{i.e.} the set of vertices directly connected to $i$ by edges). These operators $g_i$ commute with each other and $|G\rangle$ is the common eigenstate of them such that
\begin{equation}
\forall i=1,\cdots,n,~g_i|G\rangle=|G\rangle,
\end{equation}
Moreover, all the $2^n$ common eigenstates of these $g_i$ form a basis named graph state basis. Each term in this basis is uniquely decided by the eigenvalues of $g_i$. As the eigenvalues of $g_i$ are either $1$ or $-1$, the graph state basis can be denoted by a vector $\vec{a} \in \{0,1\}^n$ such that
\begin{equation}\label{eq:graph_state_basis}
\forall i=1,\cdots,n,~g_i|a_1\cdots a_n\rangle_G = (-1)^{a_i}|a_1\cdots a_n\rangle_G.
\end{equation}
And the density matrix of $|\vec{a}\rangle_G$ is
\begin{equation}
|a_1\cdots a_n\rangle_G\langle a_1\cdots a_n|=\prod_{i=1}^{n}\frac{(-1)^{a_i}g_i +I}{2}.
\end{equation}
Specially, the graph state $|G\rangle$ is denoted as $|0\cdots0\rangle_G$.
Remarkably, by choosing the graph state basis instead of the computational basis, the calculation of GME witness construction can be greatly simplified, without needing to perform the Schmidt decomposition. Firstly, the partial transposition of a graph state is diagonal under the graph state basis, namely, $|\vec{a}_0\rangle_G\langle \vec{a}_0|^{T_A}$ is of the form $\sum_{\vec{a}} c_{\vec{a}} |\vec{a}\rangle_G\langle \vec{a}|$. Meanwhile, the operator $Q$ in Eq.\ (\ref{eq:Q}) can be seen as a linear combination of all the eigenstates with negative eigenvalue of $|G\rangle\langle G|^{T_A}$. Therefore, when Lemma 1 is applied to the graph state $|G\rangle$, the resulting bipartite EW $\mathcal{W}_{o,A|\bar{A}}^{|G\rangle}$ is diagonal in the graph state basis. In this case, the vectors in the set $\mathcal{S}$ can be taken as the base vectors $|\vec{a}\rangle_G$, such that the construction in Theorem 1 is easy to achieve. In the following, we propose an explicit procedure for finding the decomposition of $\mathcal{W}_{o,A|\bar{A}}^{|G\rangle}$ in the graph state basis.
Firstly, for the given bipartition $A|\bar{A}$, the adjacency matrix $\Gamma$ can be decomposed into following blocks
\begin{equation}\label{eq:adj_matrix}
\left(
\begin{aligned}
&G_A~~\Gamma_{A|\bar{A}} \\
&\Gamma_{A|\bar{A}}~~G_{\bar{A}}
\end{aligned}
\right).
\end{equation}
We denote $k=rank(\Gamma_{A|\bar{A}})$ as the rank of the submatrix $\Gamma_{A|\bar{A}}$. It is known that a graph state can be transformed into tensor product of $k$ Bell states across the partitions $A$ and $\bar{A}$, using C-Z gates within each partition and local complementation operations \cite{PhysRevA.69.062311}. Here the local complementation $\tau_a$ on a vertex $a$ is defined as follows: $\tau_a : G \to \tau_a(G)$, such that the edge set $E'$ of the new graph $\tau_a(G)$ is $E'= E\cup E\left(N\left(a\right),N\left(a\right)\right)-E\cap E\left(N\left(a\right),N\left(a\right)\right)$. The local complementation $\tau_a(G)$ can be implemented with the following local unitary operation \cite{PhysRevA.69.062311}:
\begin{equation}
U_a(G) = (-iX_a)^{1/2}\prod_{b\in N(a)}(iZ_b)^{1/2}.
\end{equation}
After this operation, $|G\rangle$ is turned into $|\tau_a(G)\rangle$ and the stabilizers of $|G\rangle$ transform according to the following equations:
\begin{equation}\label{eq:stabizer_transfer}
\begin{aligned}
&U_a(G)g_b^GU_a(G)=g_a^{\tau_a(G)}g_b^{\tau_a(G)},~if ~b\in N(a); \\
&U_a(G)g_b^GU_a(G)=g_b^{\tau_a(G)}, ~~~~~~~~~if ~b\notin N(a).
\end{aligned}
\end{equation}
Meanwhile, we remark that a bipartite EW $\mathcal{W}_{A|\bar{A}}$ for $|G\rangle$ has been transformed into another bipartite EW $\mathcal{W}_{A|\bar{A}}'$ for $|G'\rangle$ after some local unitary operation with respect to $A|\bar{A}$. Hence our task for constructing bipartite EW of the initial graph state $|G\rangle$ has been turned into finding a bipartite EW for $|Bell\rangle^{\otimes k}$ by employing Lemma 1, a much easier task compared with the initial one.
Secondly, after reversing the above transformation process from $|G\rangle$ to $|Bell\rangle^{\otimes k}$, the EW for $|Bell\rangle^{\otimes k}$ which is diagonal in Bell state basis will be turned back into a bipartite EW for $|G\rangle$ which is diagonal in graph state basis.
\begin{figure}
\caption{Representation of $n$-qubit linear cluster state with graph, where $n$ qubits are connected one by one with C-Z gates.}
\label{fig:appendix_fig1}
\end{figure}
With the above foundation, we move on to a explicit discussion on a typical class of graph states: linear cluster state $|Cl_n\rangle$. Linear cluster state is represented with the graph in Fig.\ \ref{fig:appendix_fig1}.
We call the bipartition $A|\bar{A}$ a rank-$k$ bipartition if $rank(\Gamma_{A|\bar{A}})=k$, with the $\Gamma_{A|\bar{A}}$ defined in Eq.\ (\ref{eq:adj_matrix}). All rank-$1$ bipartitions of linear cluster state have only two possible types of the subgraph on the boundary $\Gamma_{A|\bar{A}}$ (Fig.\ \ref{fig:appendix_fig2}).
\begin{figure}
\caption{Possible subgraphs on the boundary across rank-1 bipartition. For a given bipartition $A|\bar{A}
\label{fig:appendix_fig2}
\end{figure}
Any other edge is deleted by C-Z gates within each partition. For the type-1 subgraph $G_{i,i+1}$, the bipartite EW reads
\begin{equation}
\begin{aligned}
\mathcal{W}_{G_{i,i+1}}=&\frac{1}{2}\left(|0_i0_{i+1}\rangle_{G_{i,i+1}}\langle 0_i0_{i+1}|+|0_i1_{i+1}\rangle_{G_{i,i+1}}\langle 0_i1_{i+1}|+|1_i0_{i+1}\rangle_{G_{i,i+1}}\langle 1_i0_{i+1}|\right. \\
&\left. +|1_i1_{i+1}\rangle_{G_{i,i+1}}\langle 1_i1_{i+1}|\right)
-|G_{i,i+1}\rangle\langle G_{i,i+1}|.
\end{aligned}
\end{equation}
by employing Lemma 1, where the state vectors like $|0_i1_{i+1}\rangle_{G_{i,i+1}}$ are graph state basis defined in the Eq.\ (\ref{eq:graph_state_basis}), and the `$0$'s on other vertices are omitted for simplicity here and after. Note that the $g_j^{G_{i,i+1}}$ can be transformed back to the $g_j^{Cl_n}$ by employing C-Z gates without disturbing the eigenvalue of the state vector. Therefore the bipartite EW for the original state $|Cl_n\rangle$ is
\begin{equation}
\begin{aligned}
\mathcal{W}_{A|\bar{A}}=&\frac{1}{2}\left(|Cl_n\rangle\langle Cl_n|+|0_i1_{i+1}\rangle_{Cl_n}\langle 0_i1_{i+1}|+|1_i0_{i+1}\rangle_{Cl_n}\langle 1_i0_{i+1}|\right. \\
&\left. +|1_i1_{i+1}\rangle_{Cl_n}\langle 1_i1_{i+1}|\right)-|Cl_n\rangle\langle Cl_n|,
\end{aligned}
\end{equation}
when formulated in the graph state basis. After normalizing the $\mathcal{W}_{A|\bar{A}}$ to meet the constraint $Tr(\mathcal{W}_{A|\bar{A}}|Cl_n\rangle\langle Cl_n|)=-1$, we obtain
\begin{equation}
\mathcal{W}_{A|\bar{A}}'=|0_i1_{i+1}\rangle_{Cl_n}\langle 0_i1_{i+1}|+|1_i0_{i+1}\rangle_{Cl_n}\langle 1_i0_{i+1}|+|1_i1_{i+1}\rangle_{Cl_n}\langle 1_i1_{i+1}|-|Cl_n\rangle\langle Cl_n|,
\end{equation}
as the bipartite EW used in our construction. This bipartite EW contributes the following terms to the set $\mathcal{S}$:
\begin{equation}
\{|0_i1_{i+1}\rangle_{Cl_n},~|1_i0_{i+1}\rangle_{Cl_n},~|1_i1_{i+1}\rangle_{Cl_n}\}.
\end{equation}
This set is denoted as $\mathcal{S}_{i,type-1}=\{01,~10,~11\}_{i,i+1}$ for short.
Meanwhile, the type-2 subgraph $G_{i,i+1,i+2}$ in Fig.\ \ref{fig:appendix_fig2} can be transformed into the type-1 subgraph after applying $C_{Z,(i,i+2)}$, $U_i(G_{i,i+1,i+2})$ and $C_{Z,(i,i+2)}$ sequentially. We remark that the local complementation operation $U_i(G_{i,i+1,i+2})$ may change the corresponding eigenvalue when $g_j^{G_{i,i+1,i+2}}$ turns into $g_j^{G_{i,i+1}}$, which is decided by the Eq.\ (\ref{eq:stabizer_transfer}). Therefore, this kind of bipartitions contribute the following set of states to the set $\mathcal{S}$:
\begin{equation}
\mathcal{S}_{i,type-2}=\{010,~101,~111\}_{i,i+1,i+2}.
\end{equation}
In summary, all the rank-$1$ bipartitions contribute an operator $R_1$ by our construction. If one denotes the $V_1$ as the set of vectors from $\{0,1\}^n$ such that the maximal distance between the `$1$'s appearing in each vector is smaller than 3, the $R_1$ can be formulated as
\begin{equation}
R_1=\sum_{\vec{a}\in V_1}|\vec{a}\rangle_{Cl_n}\langle\vec{a}|.
\end{equation}
For rank-$2$ bipartitions, their boundaries are composed of two rank-$1$ boundaries. For example, if there are two type-$1$ parts, the bipartite EW takes the form
\begin{equation}
\mathcal{W}_{A|\bar{A}}=\frac{1}{4}\sum_{\vec{a}\in\{0,1\}^4}|\vec{a}_{i,i+1,j,j+1}\rangle_{Cl_n}\langle \vec{a}_{i,i+1,j,j+1}|-|Cl_n\rangle\langle Cl_n|.
\end{equation}
After normalization, the bipartite EW reads
\begin{equation}
\mathcal{W}_{A|\bar{A}}=\frac{1}{3}\sum_{\substack{\vec{a}\in\{0,1\}^4,\\\vec{a}\ne\vec{0}}}|\vec{a}_{i,i+1,j,j+1}\rangle_{Cl_n}\langle \vec{a}_{i,i+1,j,j+1}|-|Cl_n\rangle\langle Cl_n|.
\end{equation}
Such bipartite EW contribute the following new terms to the set $S$:
\begin{equation}
S_{i,type-1}\otimes S_{j,type-1}=\{0101,~0110,~0111,~1001,~1010,~1011,~1101,~1110,~1111\}_{i,i+1,j,j+1}.
\end{equation}
The contribution of other possibilities can be decided in a similar manner as above for the type-2 subgraph. All these rank-$2$ bipartitions contribute a set $V_2$ to $\mathcal{S}$. Here a vector from $\{0,1\}^n$ belongs to $V_2$ if there exist at most two `$1$'s whose distance is larger than $2$ at the same time in the vector. Finally, all the rank-$2$ bipartitions introduce an operator $R_2$ to our construction, with
\begin{equation}
R_2=\sum_{\vec{a}\in V_2}\frac{1}{3}|\vec{a}\rangle_{Cl_n}\langle\vec{a}|.
\end{equation}
For a rank-$k$ bipartition, the subgraph on the boundary is nothing but a combination of $k$ rank-$1$ part. After repeating the above process, it is shown that all the rank-$k$ bipartitions contribute the following operator $R_k$:
\begin{equation}
R_k=\sum_{\vec{a}\in V_k}\frac{1}{2^k-1}|\vec{a}\rangle_{Cl_n}\langle\vec{a}|.
\end{equation}
A vector $\vec{a}$ belongs to $V_k$ if there exist at most $k$ for the number of `$1$'s in $\vec{a}$, such that their distance with each other are larger than $k$ at the same time. It can be observed immediately that $k\le \lceil n/3 \rceil$, indicating that the partition whose rank is higher than $\lceil n/3 \rceil$ gives no extra contribution.
After considering all the bipartitions, we end up with the GME witness $\mathcal{W}_{Cl_n}$ introduced in the main text, namely,
\begin{equation}
\mathcal{W}_{Cl_n}=\sum_{k=1}^{\lceil n/3 \rceil} R_k-|Cl_n\rangle\langle Cl_n|.
\end{equation}
As an example, for $4$-qubit cluster state,
\begin{equation}
\mathcal{W}_{Cl_4}=\sum_{\vec{a}\in V_1}|\vec{a}\rangle_G\langle \vec{a}|+\frac{1}{3}\sum_{\vec{a}\in V_2}|\vec{a}\rangle_G\langle \vec{a}|-|G\rangle\langle G|,
\end{equation}
where $V_1$ is the set $\{0001,~0010,~0011,~0100,~0101,~0110,~0111,~1000,~1010,~1100,~1110\}$, and $V_2$ is the set $\{1001,~1011,~1101,~1111\}$.
Remarkably, in $4$-qubit case, the best known EW is \cite{PhysRevLett.106.190502}
\begin{equation}
\mathcal{W}_{Cl_4}^{opt}=\sum_{\vec{a}\in V_1}|\vec{a}\rangle_G\langle \vec{a}|-|G\rangle\langle G|.
\end{equation}
It is finer than the $\mathcal{W}_{Cl_4}$ above. That is, while our approach is already quite powerful, there is still room for improvement. In this particular case, the improvement can be achieved by an elaborate choice of the set of bipartite EWs, instead of using Lemma 1 only. If the bipartite EWs for $13|24$
and $14|23$ in the above construction are replaced by
\begin{equation}
\begin{aligned}
\mathcal{W}_{13|24}=&|0001\rangle_{Cl_4}\langle0001| + |0100\rangle_{Cl_4}\langle0100|+|0101\rangle_{Cl_4}\langle0101| +|0011\rangle_{Cl_4}\langle0011| \\
&+|0110\rangle_{Cl_4}\langle0110| +|0111\rangle_{Cl_4}\langle0111| -|Cl_4\rangle\langle Cl_4|, \\
\mathcal{W}_{14|23}=&|0001\rangle_{Cl_4}\langle0001| + |0010\rangle_{Cl_4}\langle0010|+|0101\rangle_{Cl_4}\langle0101| +|0011\rangle_{Cl_4}\langle0011| \\
&+|0110\rangle_{Cl_4}\langle0110| +|0111\rangle_{Cl_4}\langle0111| -|Cl_4\rangle\langle Cl_4|,
\end{aligned}
\end{equation}
respectively, one can recover the $\mathcal{W}_{Cl_4}^{opt}$ with Theorem 1. With this example on $4$-qubit cluster state, we highlight that Lemma 1 is just an alternative choice which ends up with robust GME witnesses. Our construction in fact allows a flexible choice on the set of EWs to be lifted to multipartite case, and a suitable choice can further improve its performance. Moreover, it should be remarked that our discussion was based on the partial transposition throughout this paper, to obtain higher noise resistance. If bipartite EWs in the construction are designed by other positive maps (e.g., the Choi's map), different classes of GME witness can be found. This may help to harness the full potential of Theorem 1 in future work.
\subsection{Multipartite states admitting Schmidt decomposition.}\label{sec:appendix GHZ state}
A special case of multipartite entangled states is the multipartite states admitting Schmidt decomposition. Without loss of generality, we can assume that such states are of the form $|\phi_s\rangle=\sum_{i=0}^{d-1}\sqrt{\lambda_i}|i\rangle^{\otimes n}$ with $\lambda_i\ge 0 $ in decreasing order. Then the Lemma 1 gives a set of bipartite EWs $\mathcal{W}_{A|\bar{A}}^{|\phi_{SD}\rangle}$:
\begin{equation}
\mathcal{W}_{A|\bar{A}}^{|\phi_{SD}\rangle}=\sum_{i,j=0}^{d-1} \sqrt{\lambda_i\lambda_j}{|i\rangle^{\otimes k}}_A{|j\rangle^{\otimes n-k}}_{\bar{A}}{\langle i|^{\otimes k}}_A {\langle j|^{\otimes n-k}}_{\bar{A}}-|\phi_s\rangle\langle\phi_s|,
\end{equation}
where $k=|A|$ is the number of qudits in subsystem $A$. For these bipartite EWs, the set $S$ is
\begin{equation}
\{\pi_m(|i\rangle^{\otimes r}|j\rangle^{\otimes n-r})\}_{r,i,j,\pi_m}\cup\{|l\rangle^{\otimes n}\}_{l=0}^{d-1},
\end{equation}
with $r=1,2,\cdots,n-1$, $i,j=0,1,\cdots,d-1$ ($i<j$) and $\pi_m$ being all possible permutations of $|i\rangle^{\otimes r}|j\rangle^{\otimes n-r}$. Note that all state vectors in $S$ are orthogonal with each other, thus our construction ends up with the following multipartite EW
\begin{equation}
\begin{aligned}
\mathcal{W}_{|\phi_s\rangle}=&\sum_{\substack{i,j=0,\\i< j}}^{d-1} \sum_{r=1}^{n-1} \sum_m \sqrt{\lambda_i\lambda_j}\pi_m(|i\rangle^{\otimes r}|j\rangle^{\otimes n-r})\pi_m(\langle i^{\otimes r}|\langle j|^{\otimes n-r}) \\
&+\sum_{i=0}^{d-1}\lambda_i|i\rangle\langle i|^{\otimes n}-|\phi_s\rangle\langle\phi_s|,
\end{aligned}
\end{equation}
where the summation of $m$ is over all possible permutations $\pi_m(|i\rangle^{\otimes r}|j\rangle^{\otimes n-r})$ of $|i\rangle^{\otimes r}|j\rangle^{\otimes n-r}$.
Moreover, similar to the case of proving the optimality of $\mathcal{W}_o^{|\phi\rangle}$ in the first section, one can verify the optimality of ${W}_{|\phi_s\rangle}$ by checking that all the biseparable states satisfying $Tr(\mathcal{W}_{|\phi_s\rangle}\rho_{bs})=0$ span the whole Hilbert space $\mathcal{H}_d^{\otimes n}$.
\subsection{GME witness for multi-qubit singlet states}\label{sec:appendix singlet}
Multi-qubit singlet states are of particular experimental interest, while the GME witness for them is less investigated. In this example, it is shown that our framework works well for the multi-qubit singlet states. In the main text, we provide the result for a specific class of four-qubit singlet states. While here we begin with the discussion on general four-qubit singlet states
\begin{equation}
|\varphi_4\rangle = a |\psi_{12}^-\rangle\otimes |\psi_{34}^-\rangle + e^{i\theta}b |\psi_{13}^-\rangle\otimes |\psi_{24}^-\rangle,
\end{equation}
with the constraint $a^2+b^2+cos(\theta)ab=1$ and $|\psi_{12}^-\rangle$ being the two-qubit singlet state $(|01\rangle-|10\rangle)/\sqrt{2}$ on the first two qubits. By performing our construction procedure for all four-qubit singlet states, it is observed that the set $\mathcal{S}$ is always divided into $5$ subsets and the identity operators on the corresponding subspaces are just $\{\mathcal{P}_i^4\}_{i=0}^4$ (The $\mathcal{P}_i^4$ has been defined below the Eq.\ (\ref{eq:w_state})). More specifically, the resulting witness is
\begin{equation}
\mathcal{W}_4 = c_2 \mathcal{P}_2^4 + c_1 (\mathcal{P}_1^4 + \mathcal{P}_3^4) + c_0 (\mathcal{P}_0^4 + \mathcal{P}_4^4) - |\varphi_4\rangle\langle\varphi_4|,
\end{equation}
with the coefficients decided by
\begin{equation}
\begin{aligned}
c_2 &= \max\{1-\frac{3}{4}a^2, 1-\frac{3}{4}b^2, \frac{3}{4}(a^2+b^2) -\frac{1}{2}\}, \\
c_1 &= \frac{1}{2}, \\
c_0 &= \max\{\frac{1}{2}-\frac{1}{4}(a^2+b^2),\frac{1}{4}a^2,\frac{1}{4}b^2\}.
\end{aligned}
\end{equation}
Specially, with a choice of $\theta = \pi/2$, this recovers the EW in the main text. While if $a=-1$, $b=1$ and $\theta=0$, $|\varphi_4\rangle$ becomes a biseparable state $|\psi_{14}^-\rangle\otimes |\psi_{23}^-\rangle$ and the corresponding EW become positive semidefinite.
When the number of qubit grows, achieving a generic expression becomes more complicated. To investigate the GME witness construction in this case, we consider the following six-qubit singlet state
\begin{equation}
\begin{aligned}
|\varphi_6\rangle = &\frac{1}{2}\left(|\psi_{12}^-\rangle\otimes |\psi_{34}^-\rangle \otimes |\psi_{56}^-\rangle + i |\psi_{13}^-\rangle\otimes |\psi_{24}^-\rangle \otimes |\psi_{56}^-\rangle \right. \\
& \left. + i |\psi_{12}^-\rangle\otimes |\psi_{35}^-\rangle \otimes |\psi_{46}^-\rangle - |\psi_{13}^-\rangle\otimes |\psi_{25}^-\rangle \otimes |\psi_{46}^-\rangle\right),
\end{aligned}
\end{equation}
for which we arrive at the GME witness
\begin{equation}
\mathcal{W}_6= \frac{5}{8} \mathcal{P}_3^6 + \frac{1}{2} (\mathcal{P}_2^6 + \mathcal{P}_4^6) + \frac{1}{4} (\mathcal{P}_1^6 + \mathcal{P}_5^6) + \frac{1}{8} (\mathcal{P}_0^6 + \mathcal{P}_6^6) - |\varphi_6\rangle\langle\varphi_6|.
\end{equation}
Based on these results, it is reasonable to conjecture that for some $2n$-qubit singlet state $|\varphi_{2n}\rangle$, there exists a GME witness taking the form
\begin{equation}
\mathcal{W}_{2n}= c_n \mathcal{P}_n^{2n} + \sum_{i=0}^{n-1} c_i(\mathcal{P}_i^{2n} + \mathcal{P}_{2n-i}^{2n})- |\varphi_{2n}\rangle\langle\varphi_{2n}|.
\end{equation}
with $c_i \ge c_{i-1} \ge 0$ for $i=1,\cdots, n$ and $c_n$ is the maximal squared overlap between $|\varphi_6\rangle$ and biseparable states. Moreover, if $c_i$ scales with $(1/2)^{-(n-i+1)}$ as in the four- and six-qubit case, a high white noise tolerance tending to $1$ can be expected for a large number of qubit.
\end{document}
|
\begin{document}
\title{Twisted partially pure spinors}
\author{
Rafael Herrera\footnote{Centro de
Investigaci\'on en Matem\'aticas, A. P. 402,
Guanajuato, Gto., C.P. 36000, M\'exico. E-mail: [email protected]}
\footnote{Partially supported by
grants of CONACyT, LAISLA (CONACyT-CNRS), and the IMU Berlin Einstein Foundation
Program} \kern1pt\kern1pt and
Ivan Tellez\footnote{Centro de
Investigaci\'on en Matem\'aticas, A. P. 402,
Guanajuato, Gto., C.P. 36000, M\'exico. E-mail: [email protected]}
\footnote{Partially supported by a
CONACYT scholarship}
}
\date{}
\hat laketitle
{
\abstract{
Motivated by the relationship between orthogonal complex structures and spure spinors,
we define twisted partially pure spinors
in order to characterize spinorially subspaces of Euclidean space endowed with a complex structure.
}
}
\section{Introduction}
In this paper, we characterize subspaces of Euclidean space $\hat lathbb{R}^{n}$ endowed with an orthogonal
complex structure by
means of twisted spinors, which
is a generalization of the relation between classical pure spinors and orthogonal complex
structures on Euclidean space $\hat lathbb{R}^{2m}$.
Recall that a classical pure spinor $\phi\mbox{\ns i}n\Delta_{2m}$ is a spinor such
that the (isotropic) subspace of complexified vectors $X-iY\mbox{\ns i}n\hat lathbb{R}^{2m}\otimes \hat lathbb{C}$,
$X,Y\mbox{\ns i}n\hat lathbb{R}^{2m}$, which
annihilate $\phi$ under Clifford multiplication
\[(X-iY)\cdot \phi =0\]
is of maximal dimension, where $m\mbox{\ns i}n\hat lathbb{N}$ and $\Delta_{2m}$ is
the standard complex representation of the Spin group $Spin(2m)$ (cf. \cite{Lawson}). This means that
for every $X\mbox{\ns i}n\hat lathbb{R}^{2m}$ there exists a $Y\mbox{\ns i}n\hat lathbb{R}^{2m}$ satisfying
\[X\cdot \phi = i Y\cdot \phi. \]
By setting $Y=J(X)$, one can see that a pure spinor determines a complex
structure on $\hat lathbb{R}^{2m}$.
Geometrically, the two subspaces $TM\cdot\phi$ and $i\kern1ptTM\cdot\phi$ of
$\Delta_{2m}$ coincide, which means $TM\cdot \phi$ is a complex subspace of $\Delta_{2m}$, and
the effect of multiplication by the number $i=\sqrt{-1}$ is transferred to the tangent space $TM$ in
the form of $J$.
The authors of \cite{Charlton,Trautman} investigated (the classification of) non-pure classical spinors
by means of their isotropic subspaces. In \cite{Trautman}, the authors noted that there may be many spinors
(in different orbits under the action of the Spin group) admitting isotropic subspaces of the same dimension,
and that there is a gap in the possible dimensions of such isotropic subspaces.
In our Euclidean/Riemannian
context, such isotropic subspaces correspond to subspaces of
Euclidean space endowed with orthogonal complex structures.
In this paper, we define twisted partially pure spinors (cf. Definition
\mbox{$\mathbb R$}f{def:twisted-partially-pure-spinor}) in order to establish a one-to-one
correspondence between subspaces of Euclidean space (of a fixed codimension) endowed with orthogonal complex
structures (and oriented orthogonal complements), and orbits of such spinors under a particular subgroup
of the twisted spin group (cf. Theorem \mbox{$\mathbb R$}f{theo:characterization}). By using spinorial twists we avoid having
different orbits under the full twisted spin group and also the aforementioned gap in the dimensions.
The need to establish such a correspondence arises from our interest in developing a spinorial
setup to study the geometry of manifolds admitting (almost) CR structures (of arbitrary codimension) and
elliptic structures. Since such manifolds are not necessarily Spin nor Spin$^c$, we are led to consider
spinorially twisted spin groups, representations, structures, etc.
Geometric and topological considerations regarding such manifolds will be presented
in \cite{Herrera-Nakad}.
The paper is organized as follows. In Section \mbox{$\mathbb R$}f{sec:preliminaries} we recall basic material on Clifford
algebras, spin groups and representations; we define the twisted spin groups and representations that will be
used, and the space of anti-symmetric 2-forms and endomorphims associated to twisted spinors; we also present
some results on subgroups and branching of representations.
In Section \mbox{$\mathbb R$}f{sec:twisted-partially-pure-spinors}, we define partially pure spinors, deduce their
basic properties and prove the main theorem, Theorem \mbox{$\mathbb R$}f{theo:characterization}, which establishes the
aforementioned one-to-one correspondence.
{\bf Acknowledgments}.
The first named author would like to thank Helga Baum for her hospitality and support, as well as the
following
institutions: Humboldt University, the
International Centre for Theoretical Physics and the Institut des Hautes \'Etudes Scientifiques.
\section{Preliminaries}\label{sec:preliminaries}
In this section, we briefly recall basic facts about Clifford algebras, the Spin
group and the standard Spin representation \cite{Friedrich}. We also define the twisted spin groups and
representations, and the antisymmetric 2-forms and endomorphisms associated to a twisted spinor, and
describe various inclusions of groups into (twisted) spin groups.
\subsection{Clifford algebras}
Let $Cl_n$ denote the Clifford algebra generated by the orthonormal vectors
$e_1, e_2, \ldots, e_n\mbox{\ns i}n \hat lathbb{R}^n$
subject to the relations
\begin{eqnarray*}
e_j e_k + e_k e_j &=& -2\left< e_j,e_k\right>,
\end{eqnarray*}
where $\big< , \big>$ denotes the standard inner product in $\hat lathbb{R}^n$.
Let
\[\hat lathbb{C}l_n=Cl_n\otimes_{\hat lathbb{R}}\hat lathbb{C}\]
denote the complexification of $Cl_n$. The Clifford algebras are isomorphic to matrix algebras. In particular,
\[\hat lathbb{C}l_n\cong \left\{
\begin{array}{ll}
{\rm End}(\hat lathbb{C}^{2^k}), & \hat lbox{if $n=2k$,}\\
{\rm End}(\hat lathbb{C}^{2^k})\opluslus{\rm End}(\hat lathbb{C}^{2^k}), & \hat lbox{if $n=2k+1,$}
\end{array}
\right.
\]
where
\[\Delta_n:=\hat lathbb{C}^{2^k}=\underbrace{\hat lathbb{C}^2\otimes \ldots \otimes \hat lathbb{C}^2}_{k\kern1pt\kern1pt\kern1pt\kern1pt
{\rm times}}\]
is the tensor product of $k=[{n\over 2}]$ copies of $\hat lathbb{C}^2$.
The map
\[\kappa:\hat lathbb{C}l_n \longrightarrow {\rm End}(\hat lathbb{C}^{2^k})\]
is defined to be either the above mentioned isomorphism if $n$ is even, or the isomorphism followed by
the projection onto the first summand if $n$ is odd.
In order to make $\kappa$ explicit, consider the following matrices
\[Id = \left(\begin{array}{ll}
1 & 0\\
0 & 1
\end{array}\right),\quad
g_1 = \left(\begin{array}{ll}
i & 0\\
0 & -i
\end{array}\right),\quad
g_2 = \left(\begin{array}{ll}
0 & i\\
i & 0
\end{array}\right),\quad
T = \left(\begin{array}{ll}
0 & -i\\
i & 0
\end{array}\right).
\]
In terms of the generators $e_1, \ldots, e_n$, $\kappa$ can be
described explicitly as follows,
\begin{eqnarray}
e_1&\hat lapsto& Id\otimes Id\otimes \ldots\otimes Id\otimes Id\otimes g_1,\nonumber\\
e_2&\hat lapsto& Id\otimes Id\otimes \ldots\otimes Id\otimes Id\otimes g_2,\nonumber\\
e_3&\hat lapsto& Id\otimes Id\otimes \ldots\otimes Id\otimes g_1\otimes T,\nonumber\\
e_4&\hat lapsto& Id\otimes Id\otimes \ldots\otimes Id\otimes g_2\otimes T,\nonumber\\
\vdots && \dots\nonumber\\
e_{2k-1}&\hat lapsto& g_1\otimes T\otimes \ldots\otimes T\otimes T\otimes T,\nonumber\\
e_{2k}&\hat lapsto& g_2\otimes T\otimes\ldots\otimes T\otimes T\otimes T,\nonumber
\end{eqnarray}
and, if $n=2k+1$,
\[ e_{2k+1}\hat lapsto i\kern1pt\kern1pt T\otimes T\otimes\ldots\otimes T\otimes T\otimes T.\]
The vectors
\[u_{+1}={1\over \sqrt{2}}(1,-i)\quad\quad\hat lbox{and}\quad\quad u_{-1}={1\over \sqrt{2}}(1,i),\]
form a unitary basis of $\hat lathbb{C}^2$ with respect to the standard Hermitian product.
Thus,
\[\{u_{\varepsilon_1,\ldots,\varepsilon_k}=u_{\varepsilon_1}\otimes\ldots\otimes
u_{\varepsilon_k}\kern1pt\kern1pt|\kern1pt\kern1pt \varepsilon_j=\pm 1,
j=1,\ldots,k\},\]
is a unitary basis of $\Delta_n=\hat lathbb{C}^{2^k}$
with respect to the naturally induced Hermitian product.
We will denote inner and Hermitian products by the same symbol $\left<\cdot,\cdot\right>$ trusting that the
context will make clear
which product is being used.
Clifford multiplication is defined by
\begin{eqnarray*}
\hat lu_n:\hat lathbb{R}^n\otimes \Delta_n &\longrightarrow&\Delta_n\\
x \otimes \psi &\hat lapsto& \hat lu_n(x\otimes \psi)=x\cdot\psi :=\kappa(x)(\psi) .
\end{eqnarray*}
A quaternionic structure $\alpha$ on $\hat lathbb{C}^2$ is given by
\[\alpha\left(\begin{array}{c}
z_1\\
z_2
\end{array}
\right) = \left(\begin{array}{c}
-\overline{z}_2\\
\overline{z}_1
\end{array}\right),\]
and a real structure $\beta$ on $\hat lathbb{C}^2$ is given by
\[\beta\left(\begin{array}{c}
z_1\\
z_2
\end{array}
\right) = \left(\begin{array}{c}
\overline{z}_1\\
\overline{z}_2
\end{array}\right).\]
Real and quaternionic structures $\gamma_n$ on $\Delta_n=(\hat lathbb{C}^2)^{\otimes
[n/2]}$ are built as follows
\[
\begin{array}{cclll}
\gamma_n &=& (\alpha\otimes\beta)^{\otimes 2k} &\hat lbox{if $n=8k,8k+1$}& \hat lbox{(real),} \\
\gamma_n &=& \alpha\otimes(\beta\otimes\alpha)^{\otimes 2k} &\hat lbox{if $n=8k+2,8k+3$}&
\hat lbox{(quaternionic),} \\
\gamma_n &=& (\alpha\otimes\beta)^{\otimes 2k+1} &\hat lbox{if $n=8k+4,8k+5$}&\hat lbox{(quaternionic),} \\
\gamma_n &=& \alpha\otimes(\beta\otimes\alpha)^{\otimes 2k+1} &\hat lbox{if $n=8k+6,8k+7$}&\hat lbox{(real).}
\end{array}
\]
\subsection{The Spin group and representation}
The Spin group $Spin(n)\subset Cl_n$ is the subset
\[Spin(n) =\{x_1x_2\cdots x_{2l-1}x_{2l}\kern1pt\kern1pt|\kern1pt\kern1ptx_j\mbox{\ns i}n\hat lathbb{R}^n, \kern1pt\kern1pt
|x_j|=1,\kern1pt\kern1ptl\mbox{\ns i}n\hat lathbb{N}\},\]
endowed with the product of the Clifford algebra.
It is a Lie group and its Lie algebra is
\[\hat lathfrak{spin}(n)=\hat lbox{span}\{e_ie_j\kern1pt\kern1pt|\kern1pt\kern1pt1\leq i< j \leq n\}.\]
Recall that the Spin group $Spin(n)$ is the universal double cover of $SO(n)$, $n\ge 3$. For $n=2$
we consider $Spin(2)$ to be the connected double cover of $SO(2)$.
The covering map will be denoted by
\[\lambda_n:Spin(n)\rightarrow SO(n).\]
Its differential is given
by $\lambda_{n_*}(e_ie_j) = 2E_{ij}$, where $E_{ij}=e_i^*\otimes e_j - e_j^*\otimes e_i$ is the
standard basis of the skew-symmetric matrices, and $e^*$ denotes the metric dual of the vector $e$.
Furthermore, we will abuse the notation and also denote by $\lambda_n$ the induced representation on
$\raise1pt\hbox{$\ts\bigwedge$}^*\hat lathbb{R}^n$.
The restriction of $\kappa$ to $Spin(n)$ defines the Lie group representation
\[
Spin(n)\longrightarrow GL(\Delta_n),\]
which is, in fact, special unitary.
We have the corresponding Lie algebra representation
\[
\hat lathfrak{spin}(n)\longrightarrow \hat lathfrak{gl}(\Delta_n).\]
{\bf Remark}.
For the sake of notation we will set
\[SO(0)=\{1\},\quad\quad SO(1)=\{1\},\]
\[Spin(0)=\{\pm1\},\quad\quad Spin(1)=\{\pm1\},\]
and
\[\Delta_0 = \Delta_1 =\hat lathbb{C}\]
a trivial $1$-dimensional representation.
Clifford multiplication $\hat lu_n$ has the following properties:
\begin{itemize}
\mbox{\ns i}tem It is skew-symmetric with respect to the Hermitian product
\begin{equation}
\left<x\cdot\psi_1 , \psi_2\right>
=-\left<\psi_1 , x\cdot \psi_2\right>. \label{clifford-skew-symmetric}
\end{equation}
\mbox{\ns i}tem $\hat lu_n$ is an equivariant map of $Spin(n)$ representations.
\mbox{\ns i}tem $\hat lu_n$ can be extended to an equivariant map
\begin{eqnarray*}
\hat lu_n:\raise1pt\hbox{$\ts\bigwedge$}^*(\hat lathbb{R}^n)\otimes \Delta_n &\longrightarrow&\Delta_n\\
\omega \otimes \psi &\hat lapsto& \omega\cdot\psi,
\end{eqnarray*}
of $Spin(n)$ representations.
\end{itemize}
At this point we will make the following convention. Consider the involution
\begin{eqnarray*}
F_{2m}: \Delta_{2m}&\longrightarrow& \Delta_{2m}\\
\phi &\hat lapsto& (-i)^m e_1e_2\cdots e_{2m} \cdot \phi,
\end{eqnarray*}
and let
\[\Delta_{2m}^\pm = \{\phi \kern1pt\kern1pt|\kern1pt\kern1pt F_{2m}(\phi)=\pm \phi \}.\]
This definition of positive and negative Weyl spinors differs from the one in \cite{Friedrich} by a factor
$(-1)^m$. Nevertheless, we have chosen this convention so that the spinor $u_{1,\ldots,1}$ is
always
positive and corresponds to the standard (positive) complex structure on $\hat lathbb{R}^{2m}$.
\subsection{Spinorially twisted Spin groups}
Consider the following groups:
\begin{itemize}
\mbox{\ns i}tem By using the unit complex numbers $U(1)$,
the Spin group can be twisted \cite{Friedrich}
\[Spin^c(n) = (Spin(n) \times U(1))/\{\pm (1,1)\} =
Spin(n) \times_{\hat lathbb{Z}_2} U(1),\]
with Lie algebra
\[\hat lathfrak{spin}^c(n)=\hat lathfrak{spin}(n)\opluslus i\hat lathbb{R}.\]
\mbox{\ns i}tem In \cite{Espinosa-Herrera} we have considered
the twisted Spin group $Spin^r(n)$, $r\mbox{\ns i}n\hat lathbb{N}$, defined as follows
\[
Spin^{r}(n) = (Spin(n) \times Spin(r))/\{\pm (1,1)\} =
Spin(n) \times_{\hat lathbb{Z}_2} Spin(r).
\]
The Lie algebra of $Spin^r(n)$ is
\[\hat lathfrak{spin}^r(n) = \hat lathfrak{spin}(n) \opluslus \hat lathfrak{spin}(r).\]
\mbox{\ns i}tem Here, we will also consider the following group
\begin{eqnarray*}
Spin^{c,r}(n) &=& (Spin(n) \times Spin^c(r))/\{\pm (1,1)\} \\
&=& Spin(n) \times_{\hat lathbb{Z}_2} Spin^c(r),
\end{eqnarray*}
where $r\mbox{\ns i}n\hat lathbb{N}$, whose
Lie algebra is
\[\hat lathfrak{spin}^c(n)=\hat lathfrak{spin}(n)\opluslus \hat lathfrak{spin}(r)\opluslus i\hat lathbb{R}.\]
It fits into the exact sequence
\[1\longrightarrow \hat lathbb{Z}_2\longrightarrow Spin^{c,r}(n)\hbox{{\footnotesize $\#$}}rightarrow{\lambda_n\times\lambda_r\times \lambda_2} SO(n)\times SO(r)
\times U(1)\longrightarrow 1,\]
where
\begin{eqnarray*}
(\lambda_n\times\lambda_r\times \lambda_2)([g,[h,z]]) &=& (\lambda_n(g),\lambda_r(h),z^2).
\end{eqnarray*}
\end{itemize}
{\bf Remark}. For $r=0,1$, $Spin^{c,r}(n)=Spin^c(n)$.
\subsection{Twisted spin representations}
Consider the following twisted representations:
\begin{itemize}
\mbox{\ns i}tem The Spin representation $\Delta_n$ extends to a representation of $Spin^c(n)$ by letting
\begin{eqnarray*}
Spin^c(n)&\longrightarrow& GL(\Delta_n)\\
\kern1pt[g,z] &\hat lapsto& z\kappa_n(g)=:zg.
\end{eqnarray*}
\mbox{\ns i}tem The twisted $Spin^{c,r}(n)$ representation
\begin{eqnarray*}
Spin^{c,r}(n)&\longrightarrow& GL(\Delta_r\otimes
\Delta_n)\\
\kern1pt[g,[h,z]] &\hat lapsto& z \kern1pt \kappa_r(h)\otimes \kappa_n(g)=:zh\otimes g.
\end{eqnarray*}
which is also unitary with respect to the natural Hermitian metric.
\mbox{\ns i}tem For $r=0,1$, the twisted spin representation is simply the $Spin^c(n)$ representation $\Delta_n$.
\end{itemize}
We will also need the map
\begin{eqnarray*}
\hat lu_r\otimes\hat lu_n:\left(\raise1pt\hbox{$\ts\bigwedge$}^*\hat lathbb{R}^r\otimes_\hat lathbb{R} \raise1pt\hbox{$\ts\bigwedge$}^*\hat lathbb{R}^n\right)
\otimes_\hat lathbb{R} (\Delta_r\otimes \Delta_n) &\longrightarrow& \Delta_r\otimes\Delta_n\\
(w_1 \otimes w_2)\otimes (\psi\otimes \varphi) &\hat lapsto&
(w_1\otimes w_2)\cdot (\psi\otimes \varphi)
= (w_1\cdot\psi) \otimes (w_2\cdot \varphi).
\end{eqnarray*}
As in the untwisted case,
$\hat lu_r\otimes\hat lu_n$ is an equivariant homomorphism of $Spin^{c,r}(n)$ representations.
\subsection{Skew-symmetric 2-forms and endomorphisms associated to twisted spinors}
We will often write $f_{kl}$ for the Clifford product $f_kf_l$.
\begin{defi}
{\rm \cite{Espinosa-Herrera}}
Let $r\geq 2$, $\phi\mbox{\ns i}n\Delta_r\otimes\Delta_n$, $X,Y\mbox{\ns i}n\hat lathbb{R}^n$, $(f_1\ldots,f_r)$
an orthonormal basis of $\hat lathbb{R}^r$ and $1\leq k,l\leq r$.
\begin{itemize}
\mbox{\ns i}tem
Define the real $2$-forms associated to the spinor $\phi$ by
\[\eta_{kl}^{\phi} (X,Y) = {\rm Re}\left< X\wedge Y\cdot f_kf_l\cdot \phi,\phi\right>.\]
\mbox{\ns i}tem Define the antisymmetric endomorphisms
$\hat\eta_{kl}^\phi\mbox{\ns i}n{\rm End}^-(\hat lathbb{R}^n)$ by
\[X\hat lapsto \hat\eta_{kl}^\phi(X):=(X\lrcorner \kern1pt\eta_{kl}^{\phi})^\sharp,\]
where $X\mbox{\ns i}n\hat lathbb{R}^n$, $\lrcorner$ denotes contraction and $^\sharp$ denotes metric dualization from
$1$-fomrs to vectors.
\end{itemize}
\end{defi}
\begin{lemma}
Let $r\geq 2$, $\phi\mbox{\ns i}n\Delta_r\otimes\Delta_n$, $X,Y\mbox{\ns i}n\hat lathbb{R}^n$, $(f_1\ldots,f_r)$
an orthonormal basis of $\hat lathbb{R}^r$ and $1\leq k,l\leq r$. Then
\begin{eqnarray}
{\rm Re}\left< f_kf_l\cdot
\phi,\phi\right>&=&0,\label{vanishing2}\nonumber\\
{\rm Re}\left< X\wedge Y\cdot
\phi,\phi\right>&=&0,\label{vanishing3}\\
{\rm Im}\left< X\wedge Y\cdot f_kf_l\cdot
\phi,\phi\right>&=&0,\label{vanishing4}\\
{\rm Re} \left< X\cdot \phi,Y\cdot\phi \right>
&=& \left<X,Y\right>|\phi|^2, \label{real-part}
\end{eqnarray}
\end{lemma}
{\em Proof}.
By using \rf{clifford-skew-symmetric} twice
\begin{eqnarray*}
\left< f_kf_l\cdot\phi,\phi\right>
&=& -\overline{\left< f_kf_l\phi,\phi\right>}.
\end{eqnarray*}
For identity \rf{vanishing3}, recall that for $X, Y\mbox{\ns i}n \hat lathbb{R}^n$
\[X\wedge Y = X\cdot Y + \left<X,Y\right>.\]
Thus
\begin{eqnarray*}
\left< X\wedge Y\cdot \phi,\phi\right>
&=& -\overline{\left< X\wedge Y \cdot\phi,\phi\right>}.
\end{eqnarray*}
Identities \rf{vanishing4} and \rf{real-part} follow similarly.
\qd
{\bf Remarks}.
\begin{itemize}
\mbox{\ns i}tem For $k\not= l$,
\[\eta_{kl}^\phi = (\delta_{kl}-1)\eta_{lk}^\phi.\]
\mbox{\ns i}tem By \rf{vanishing4}, if $k\not= l$,
\[\eta_{kl}^{\phi} (X,Y) =\left< X\wedge Y\cdot f_kf_l\cdot
\phi,\phi\right>.\]
\end{itemize}
\begin{lemma}{\rm \cite{Espinosa-Herrera}}
Any spinor $\phi\mbox{\ns i}n\Delta_r\otimes\Delta_n$, $r\geq 2$, defines two maps (extended by linearity)
\begin{eqnarray*}
\raise1pt\hbox{$\ts\bigwedge$}^2 \hat lathbb{R}^r&\longrightarrow& \raise1pt\hbox{$\ts\bigwedge$}^2 \hat lathbb{R}^n\\
f_{kl} &\hat lapsto& \eta_{kl}^{\phi}
\end{eqnarray*}
and
\begin{eqnarray*}
\raise1pt\hbox{$\ts\bigwedge$}^2 \hat lathbb{R}^r&\longrightarrow& {\rm End}(\hat lathbb{R}^n)\\
f_{kl} &\hat lapsto& \hat\eta_{kl}^{\phi},
\end{eqnarray*}
\end{lemma}
\qd
\subsection{Subgroups, isomorphisms and decompositions}
In this section we will describe various inclusions of groups into (twisted) spin groups.
\begin{lemma}\label{lemma-subgroup1}
There exists a monomorphism $h:Spin(2m)\times_{\hat lathbb{Z}_2} Spin(r) \longrightarrow Spin(2m+r)$
such that the following diagram commutes
\[
\begin{array}{ccc}
Spin(2m)\times_{\hat lathbb{Z}_2} Spin(r) & \hbox{{\footnotesize $\#$}}rightarrow{h} & Spin(2m+r)\\
\downarrow & & \downarrow\\
SO(2m)\times SO(r) & \hookrightarrow & SO(2m+r)
\end{array}
\]
\end{lemma}
{\em Proof}.
Consider the decomposition
\[\hat lathbb{R}^{2m+r}=\hat lathbb{R}^{2m}\opluslus\hat lathbb{R}^{r},\]
and let
\begin{eqnarray*}
Spin(2m) &=& \left\{\mbox{$\mathbb P$}od_{i=1}^{2s} x_i \mbox{\ns i}n Cl_{2m+r} \kern1pt\kern1pt | \kern1pt\kern1pt
x_i\mbox{\ns i}n\hat lathbb{R}^{2m}, |x_i|=1, s\mbox{\ns i}n\hat lathbb{N} \right\}\quad\subset\quad Spin(2m+r),\\
Spin(r) &=& \left\{\mbox{$\mathbb P$}od_{j=1}^{2t} y_j \mbox{\ns i}n Cl_{2m+r} \kern1pt\kern1pt | \kern1pt\kern1pt
y_j\mbox{\ns i}n\hat lathbb{R}^{r}, |y_j|=1, t\mbox{\ns i}n\hat lathbb{N} \right\}\quad\subset\quad Spin(2m+r).
\end{eqnarray*}
It is easy to prove that
\[Spin(2m)\cap Spin(r) = \{1,-1\}.\]
Define the homomorphism
\begin{eqnarray*}
h:Spin(2m)\times_{\hat lathbb{Z}_2} Spin(r) &\longrightarrow& Spin(2m+r)\\
{[g,g']} &\hat lapsto & gg'.
\end{eqnarray*}
If $[g,g']\mbox{\ns i}n Spin(2m)\times_{\hat lathbb{Z}_2} Spin(r)$ is such that
\[gg'=1\mbox{\ns i}n Spin(2m+r),\]
then
\[g'=g^{-1} \mbox{\ns i}n Spin(2m)\subset Spin(2m+r),\]
so that
\[g,g'\mbox{\ns i}n Spin(2m)\cap Spin(r)=\{1,-1\}.\]
Hence $[g,g']=[1,1]$
and $h$ is injective.
\qd
\begin{lemma}\label{lemma:subgroup2}
Let $r\mbox{\ns i}n \hat lathbb{N}$.
There exists an monomorphism $h:U(m)\times SO(r) \hookrightarrow
Spin^{c,r}(2m+r)$
such that the following diagram commutes
\[
\begin{array}{ccc}
& & Spin^{c,r}(2m+r)\\
& \nearrow & \downarrow\\
U(m)\times SO(r) & \longrightarrow & SO(2m+r)\times SO(r)\times U(1)
\end{array}
\]
\end{lemma}
{\em Proof}.
Suppose we have an orthogonal complex structure on $\hat lathbb{R}^{2m}\subset\hat lathbb{R}^{2m+r}$
\[J:\hat lathbb{R}^{2m} \longrightarrow \hat lathbb{R}^{2m}, \quad J^2=-{\rm Id}_{2m}, \quad
\langle\cdot,\cdot\rangle=\langle J\cdot,J\cdot\rangle.\]
The subgroup of $SO(2m+r)$ that respects both the orthogonal decomposition
$\hat lathbb{R}^{2m+r}=\hat lathbb{R}^{2m}\opluslus\hat lathbb{R}^{r}$
and $J$ is
\[U(m)\times SO(r)\quad\subset\quad SO(2m)\times SO(r)\quad\subset\quad SO(2m+r).\]
There exists a lift \cite{Friedrich}
\[\begin{array}{ccc}
& & Spin^c(2m)\\
& \nearrow & \downarrow\\
U(m) & \rightarrow & SO(2m)\times U(1)\\
&&\\
A & \hat lapsto &(A_{\hat lathbb{R}},\det_{\hat lathbb{C}}(A))
\end{array}
\]
and we can consider the diagram \cite{Espinosa-Herrera}
\[
\begin{array}{ccc}
& & Spin(r)\times_{\hat lathbb{Z}_2} Spin(r)\\
& \nearrow & \downarrow\\
SO(r) & \hbox{{\footnotesize $\#$}}rightarrow{\rm diagonal} & SO(r)\times SO(r)
\end{array}.
\]
We can put them together as follows
{\footnotesize
\[
\begin{array}{ccccccc}
& & Spin^c(2m)\times_{\hat lathbb{Z}_2} Spin^r(r) \cong Spin^r(2m)\times_{\hat lathbb{Z}_2}
Spin^c(r)&\hookrightarrow& Spin(2m+r)\times_{\hat lathbb{Z}_2} Spin^c(r)\\
&\nearrow & \downarrow & &\\
U(m)\times SO(r) & \hookrightarrow & SO(2m)\times U(1)\times SO(r)\times SO(r)
&&
\end{array}
\]
}
where the last inclusion is due to Lemma \mbox{$\mathbb R$}f{lemma-subgroup1}.
It is easy to prove that
the lift monomorphism $U(m)\times SO(r)\longrightarrow Spin^c(2m)\times_{\hat lathbb{Z}_2} Spin^r(r)$
exists and
there is a natural isomorphism
\[Spin^c(2m)\times_{\hat lathbb{Z}_2} Spin^r(r) \cong Spin^r(2m)\times_{\hat lathbb{Z}_2} Spin^c(r).\]
\qd
\begin{lemma}\label{factorization}
Let $r\mbox{\ns i}n\hat lathbb{N}$.
The standard representation $\Delta_{2m+r}$ of $Spin(2m+r)$ decomposes as follows
\[\Delta_{2m+r} = \Delta_r\otimes\Delta_{2m}^+ \kern1pt\kern1pt\opluslus\kern1pt\kern1pt\Delta_r\otimes\Delta_{2m}^-,\]
with respect to the subgroup
$Spin(2m)\times_{\hat lathbb{Z}_2} Spin(r) \subset Spin(2m+r)$.
\end{lemma}
{\em Proof}. Consider the restriction of the standard representation of $Spin(2m+r)$ to
\[Spin(2m)\times_{\hat lathbb{Z}_2} Spin(r)\subset Spin(2m+r) \longrightarrow Gl(\Delta_{2m+r}).\]
By using the explicit description of a unitary basis of $\Delta_{2m+r}$, we see that
the elements of $Spin(2m)$ act on the last $m$ factors of
\[\Delta_{2m+r} = \underbrace{\hat lathbb{C}^2 \otimes \cdots\otimes \hat lathbb{C}^2}_{[r/2] \: {\rm times}} \otimes
\underbrace{\hat lathbb{C}^2 \otimes\cdots\otimes\hat lathbb{C}^2 }_{m \:
{\rm times}}, \]
as they do on $\Delta_{2m}=\Delta_{2m}^+\opluslus\Delta_{2m}^-$. The elements of $Spin(r)$
act as usual on the first $[r/2]$ factors of $\Delta_r$,
act trivially on $\Delta_{2m}^+$,
and act by multiplication by $(-1)$ on the factor $\Delta_{2m}^-$.
\qd
\section{Twisted partially pure spinors}\label{sec:twisted-partially-pure-spinors}
In order to simplify the statements, we will
consider the twisted spin representation
\[
\Sigma_r\otimes\Delta_n \subseteq \Delta_r\otimes\Delta_n.
\]
where
\[
\Sigma_r=
\left\{
\begin{array}{ll}
\Delta_r & \hat lbox{if $r$ is odd,}\\
\Delta_r^+ & \hat lbox{if $r$ is even,}
\end{array}
\right.
\]
$n,r\mbox{\ns i}n\hat lathbb{N}$.
\begin{defi}\label{def:twisted-partially-pure-spinor}
Let $(f_1,\ldots,f_r)$ be an orthonormal frame of $\hat lathbb{R}^r$.
A unit-length spinor $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$, $r<n$, is called a {\em twisted partially pure
spinor} if
\begin{itemize}
\mbox{\ns i}tem there exists a $(n-r)$-dimensional subspace $V^\phi\subset\hat lathbb{R}^n$ such that
for every $X\mbox{\ns i}n V^\phi$, there exists a $Y\mbox{\ns i}n V^\phi$ such that
\[X\cdot \phi = i\kern1pt\kern1ptY\cdot \phi. \]
\mbox{\ns i}tem it satisfies the equations
\begin{eqnarray*}
(\eta_{kl}^\phi + f_kf_l)\cdot \phi&=&0,\\
\left<f_kf_l\cdot \phi,\phi\right>&=&0,
\end{eqnarray*}
for all $1\leq k<l\leq r$.
\mbox{\ns i}tem If $r=4$, it also satisfies the condition
\[\left<f_1f_2f_3f_4\cdot \phi,\phi\right>=0.\]
\end{itemize}
\end{defi}
{\bf Remarks}.
\begin{enumerate}
\mbox{\ns i}tem The requirement $|\phi|=1$ is made in order to avoid renormalizations later on.
\mbox{\ns i}tem The extra condition for the case $r=4$ is fulfilled for all other ranks.
\mbox{\ns i}tem From now on we will drop the adjective twisted since it will be clear from the context.
\end{enumerate}
\subsection{Example of partially pure spinor}
\begin{lemma}\label{lemma:existence}
Given $r,m\mbox{\ns i}n\hat lathbb{N}$, there exists a partially pure spinor in
$\Sigma_r\otimes\Delta_{2m+r}$.
\end{lemma}
{\em Proof}.
Let $(e_1,\ldots,e_{2m},e_{2m+1},\ldots,e_{2m+r})$ and
$(f_1,\ldots,f_r)$ be orthonormal frames
of $\hat lathbb{R}^{2m+r}$ and $\hat lathbb{R}^r$ respectively.
Consider the decomposition of Lemma \mbox{$\mathbb R$}f{factorization}
\[\Delta_{2m+r}=\Delta_r\otimes\Delta_{2m}^+\kern1pt\kern1pt\opluslus\kern1pt\kern1pt\Delta_r\otimes\Delta_{2m}^-,\]
corresponding to the decomposition
\[\hat lathbb{R}^{2m+r}={\rm span}\{e_1,\ldots,e_{2m}\}
\opluslus {\rm span}\{e_{2m+1},\ldots,e_{2m+r}\}.\]
Let
\[\varphi_0= u_{1,\ldots,1}\mbox{\ns i}n \Delta_{2m}^+,\]
and
\[\{v_{\varepsilon_1,\ldots,\varepsilon_{[r/2]}}\kern1pt|\kern1pt (\varepsilon_1,\ldots,\varepsilon_{[r/2]}) \mbox{\ns i}n
\{\pm1\}^{[r/2]}\}\]
be the unitary basis of the twisting factor $\Delta_r=\Delta({\rm span}(f_1,\ldots,f_r))$ which contains
$\Sigma_r$.
Let us define the standard twisted partially pure spinor
$\phi_0\mbox{\ns i}n\Sigma_r\otimes\Delta_r\otimes\Delta_{2m}^+$ by
\begin{equation}
\phi_0 = \left\{
\begin{array}{ll}
{1\over \sqrt{2^{[r/2]}}}\kern1pt\kern1pt \left(\sum_{I\mbox{\ns i}n\{\pm1\}^{\times [r/2]}}
v_I\otimes
\gamma_{r}(u_I)\right)\otimes \varphi_0 & \hat lbox{if $r$ is
odd,}\\
{1\over \sqrt{2^{[r/2]-1}}}\kern1pt\kern1pt \left(\sum_{I\mbox{\ns i}n\left[\{\pm1\}^{\times [r/2]}\right]_+}
v_I\otimes
\gamma_{r}(u_I)\right)\otimes \varphi_0 &\hat lbox{if $r$ is even},
\end{array} \right. \nonumber
\end{equation}
where the elements of $\left[\{\pm1\}^{\times [r/2]}\right]_+$
contain an even number of $(-1)$.
Checking the conditions in the definition of partially pure spinor for $\phi_0$ is done by a (long) direct
calculation as in \cite{Espinosa-Herrera}.
For instance, taking $n=7$, $r=3$, we have
$$\phi_0=\frac{1}{\sqrt{2}}(v_{1}\otimes\gamma_3(u_{1})\otimes u_{1}\otimes
u_{1}+v_{-1}\otimes\gamma_3(u_{-1})\otimes u_{1}\otimes
u_{1})$$
where $\gamma_3$ is a quaternionic structure. We check that this $\phi_0$ is a partially pure spinor. Putting
$C=1/\sqrt{2}$ and remembering that $\gamma_3(u_{\epsilon})=-i\epsilon
u_{-\epsilon}$, we get
$$\phi_0=iC(v_{-1}\otimes u_{1}\otimes u_{1}\otimes
u_{1}-v_{1}\otimes u_{-1}\otimes u_{1}\otimes
u_{1}),$$
which has unit length.
Let $\{e_i\}$ be the standard basis of
$\hat lathbb{R}^7$, so that
\begin{align*}
e_1\cdot\phi_0 & = iC(v_{-1}\otimes u_{1}\otimes u_{1}\otimes
g_1(u_{1})-v_{1}\otimes u_{-1}\otimes u_{1}\otimes
g_1(u_{1}))\\
& = ie_2\cdot \phi_0,
\end{align*}
and, similarly,
\begin{align*}
e_3\cdot\phi_0
& = ie_4\cdot \phi_0.
\end{align*}
So, $\phi_0$ induces the standard complex structure on
$V^{\phi_0}=\langle e_1,\kern1pte_2,\kern1pte_3,\kern1pte_4\rangle$.
Let $\{f_i\}$ be the
standard basis of $\hat lathbb{R}^3$.
Similar calculations give
\[\eta_{kl}^{\phi_0}=e_{4+k}\wedge e_{4+l},\]
\[(\eta_{kl}^{\phi_0}+f_{kl})\cdot\phi_0=0,\]
and
\[\left<f_{kl}\cdot\phi_0,\phi_0\right>=0.\]
\qd
\subsection{Properties of partially pure spinors}\label{sec: basic properties}
\begin{lemma}
The definition of partially pure spinor does not depend on the choice of orthonormal basis of $\hat lathbb{R}^r$.
\end{lemma}
{\em Proof}.
If $r=0,1$, a partially pure spinor is a classical pure spinor for $n$ even or the straightforward
generalization of pure spinor for $n$ odd \cite[p. 336]{Lawson}.
Suppose $(f_1',\ldots, f_r')$ is another orthonormal frame of
$\hat lathbb{R}^r$, then
\[f_i'=\alpha_{i1}f_1+\cdots + \alpha_{ir}f_r,\]
so that the matrix $A=(\alpha_{ij})\mbox{\ns i}n SO(r)$.
Let us denote
\[\eta_{kl}'^{\phi} (X,Y):={\rm Re}\left<X\wedge Y\cdot f_k'f_l'\cdot\phi,\phi\right>\]
Thus,
\begin{eqnarray*}
\eta_{kl}'^{\phi}\cdot\phi
&=& \sum_{1\leq a<b\leq n}\eta_{kl}'^{\phi}(e_a,e_b)e_ae_b
\cdot\phi\\
&=& \sum_{1\leq a<b\leq n}{\rm Re}\left< e_ae_b
\cdot\left(\sum_{s=1}^r \alpha_{ks}f_s\right)\left(\sum_{t=1}^r
\alpha_{lt}f_t\right)\cdot\phi,\phi\right>e_ae_b\cdot\phi\\
&=& \sum_{1\leq a<b\leq n}\sum_{s=1}^r \sum_{t=1}^r
\alpha_{ks}\alpha_{lt}{\rm Re}\left< e_ae_b
\cdot f_sf_t\cdot\phi,\phi\right>e_ae_b\cdot\phi\\
&=& \sum_{1\leq a<b\leq n}\sum_{s=1}^r \sum_{t=1}^r
\alpha_{ks}\alpha_{lt}\eta_{st}^\phi(e_a,e_b) e_ae_b\cdot\phi\\
&=& \sum_{s=1}^r \sum_{t=1}^r
\alpha_{ks}\alpha_{lt}\eta_{st}^\phi\cdot\phi\\
&=& -\sum_{s=1}^r \sum_{t=1}^r
\alpha_{ks}\alpha_{lt}f_sf_t\cdot\phi\\
&=& -\left(\sum_{s=1}^r \alpha_{ks}f_s\right)\left(\sum_{t=1}^r\alpha_{lt}f_t\right)\cdot\phi\\
&=& -f_k'f_l'\cdot\phi.
\end{eqnarray*}
For the third part of the definition, note that
\begin{eqnarray*}
\left<f_k'f_l'\cdot\phi , \phi \right>
&=&
\left<
\left(\sum_{s=1}^r \alpha_{ks}f_s\right)
\left(\sum_{t=1}^r\alpha_{lt}f_t\right)\cdot\phi,\phi\right>\\
&=& \sum_{s=1}^r \sum_{t=1}^r
\alpha_{ks}\alpha_{lt}\left<f_sf_t\cdot\phi,\phi\right>\\
&=& 0.
\end{eqnarray*}
For $r=4$,
the volume form is invariant under $SO(4)$,
$ f_1'f_2'f_3'f_4'
=f_1f_2f_3f_4$,
and
\[
\left<f_1'f_2'f_3'f_4'\cdot \phi,\phi\right>
=
\left<f_1f_2f_3f_4\cdot \phi,\phi\right>\\
=
0.
\]
\qd
\begin{lemma}
Given a partially pure spinor
$\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$,
there exists an orthogonal complex structure on $V^\phi$ and $n-r \equiv 0$ {\rm (mod 2)} .
\end{lemma}
{\em Proof}.
By definition, for every $X\mbox{\ns i}n V^\phi$, there exists $Y\mbox{\ns i}n V^\phi$ such that
\[X\cdot\phi = i \kern1pt Y\cdot \phi,\]
and
\[Y\cdot\phi = i\kern1pt (-X)\cdot\phi.\]
If we set
\[J^\phi(X) :=Y,\]
we get a linear transformation
$J^\phi:V^\phi\rightarrow V^\phi$, such that $(J^\phi)^2=-{\rm Id}_{V^\phi}$,
i.e. $J^\phi$ is a complex structure on the vector space $V^{\phi}$ and
$\dim_{\hat lathbb{R}}(V^\phi)$ is even.
Furthermore, this complex structure is orthogonal. Indeed, for every $X\mbox{\ns i}n V^\phi$,
\begin{eqnarray*}
X\cdot JX\cdot\phi &=& -i \vert X\vert^2\phi,\\
JX\cdot X\cdot\phi &=& i \vert JX\vert^2\phi,
\end{eqnarray*}
and
\[ (-2 \left<X, JX\right> + i(
\vert JX\vert^2-\vert X\vert^2))\kern1pt\kern1pt\phi=0,\]
i.e.
\begin{eqnarray*}
\left<X, JX\right> &=&0 \\
\vert X\vert &=& \vert JX\vert.
\end{eqnarray*}
\qd
\begin{lemma} Let $r\geq2$ and $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$ be a
partially pure spinor.
The forms $\eta_{kl}^\phi$ are non-zero, $1\leq k<l\leq r$.
\end{lemma}
{\em Proof}.
Since $(f_kf_l)^2=-1$, the equation
\begin{equation}
\eta_{kl}^\phi\cdot\phi=-f_kf_l\cdot \phi\label{eq:despeje1}
\end{equation}
implies
\begin{equation}
\eta_{kl}^\phi\cdot f_kf_l\cdot \phi=\phi.\label{eq:despeje2}
\end{equation}
By taking an orthonormal frame $(e_1,\ldots,e_n)$ of $\hat lathbb{R}^n$ we can write
\[\eta_{kl}^\phi = \sum_{1\leq i<j\leq n} \eta_{kl}^\phi(e_i,e_j)e_ie_j.\]
By \rf{eq:despeje2}, and taking hermitian product with $\phi$
\begin{eqnarray*}
1&=& |\phi|^2 \\
&=& \left< \eta_{kl}^\phi\cdot f_kf_l\cdot \phi,\phi\right> \\
&=& \left< \sum_{1\leq i<j\leq n} \eta_{kl}^\phi(e_i,e_j)e_ie_j\cdot f_kf_l\cdot \phi,\phi\right> \\
&=& \sum_{1\leq i<j\leq n} \eta_{kl}^\phi(e_i,e_j)\left< e_ie_j\cdot f_kf_l\cdot \phi,\phi\right> \\
&=& \sum_{1\leq i<j\leq n} \eta_{kl}^\phi(e_i,e_j)^2.
\end{eqnarray*}
\qd
\begin{lemma}\label{lemma:lie-algebra}
Let $r\geq2$.
The image of the map associated to a partially pure spinor $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$,
\begin{eqnarray*}
\raise1pt\hbox{$\ts\bigwedge$}^2 \hat lathbb{R}^r&\longrightarrow& {\rm End}(\hat lathbb{R}^n)\\
f_{kl} &\hat lapsto& \hat\eta_{kl}^{\phi},
\end{eqnarray*}
forms a Lie algebra of endomorphisms isomorphic to $\hat lathfrak{so}(r)$.
\end{lemma}
{\em Proof}.
Let $(e_1,\ldots,e_n)$ be an orthonormal frame of $\hat lathbb{R}^n$.
First, let us consider the following calculation for $i\not=j$, $k\not =l$, $s\not=t$:
\begin{eqnarray*}
{\rm Re}\left<e_s e_t \cdot \eta_{ij}^\phi\cdot f_kf_l\cdot\phi,\phi\right>
&=& {\rm Re}\left<e_s e_t \cdot \left(\sum_{a<b}\eta_{ij}^\phi(e_a,e_b)e_a e_b\right)\cdot
f_kf_l\cdot\phi,\phi\right>\\
&=& {\rm Re}\sum_{a<b}\eta_{ij}^\phi(e_a,e_b)\left<e_s\cdot e_t \cdot e_a\cdot e_b\cdot
f_kf_l\cdot\phi,\phi\right>\\
&=& -\sum_{b}\eta_{ij}^\phi(e_s,e_b)\eta_{kl}^\phi(e_b,e_t)
+\sum_{b}\eta_{kl}^\phi(e_s,e_b)\eta_{ij}^\phi(e_b,e_t)\\
&=& -\sum_{b}[\hat\eta_{kl}^\phi]_{tb}[\hat\eta_{ij}^\phi]_{bs}
+\sum_{b}[\hat\eta_{ij}^\phi]_{tb}[\hat\eta_{kl}^\phi]_{bs}\\
&=&[\hat\eta_{ij}^\phi,\hat\eta_{kl}^\phi]_{ts}\label{eq:long-from-bracket-calculation}
\end{eqnarray*}
is the entry in row $t$ and column $s$ of the matrix $[\hat\eta_{ij}^\phi,\hat\eta_{kl}^\phi]$.
Secondly, we prove that the endomorphisms $\hat\eta_{kl}^\phi$ satisfy the commutation
relations of $\hat lathfrak{so}(r)$:
\begin{enumerate}
\mbox{\ns i}tem If $1\leq i,j,k,l\leq r$ are all different,
\begin{equation}
[\hat\eta_{kl}^\phi,\hat\eta_{ij}^\phi]=0.\label{eq:[kl,ij]=0}
\end{equation}
\mbox{\ns i}tem If $1\leq i,j,k\leq r$ are all different,
\begin{equation}
[\hat\eta_{ij}^\phi,\hat\eta_{jk}^\phi]= -\hat\eta_{ik}^\phi.\label{[ij,jk]=-ik}
\end{equation}
\end{enumerate}
To prove \rf{eq:[kl,ij]=0}, note that by \rf{eq:despeje1},
\begin{eqnarray}
\eta_{ij}^\phi\cdot f_kf_l\cdot \phi
&=&\eta_{kl}^\phi\cdot f_if_j\cdot \phi,\label{eq:identity1}
\end{eqnarray}
by \rf{eq:long-from-bracket-calculation}
\begin{eqnarray*}
{\rm Re}\left<e_s e_t \cdot \eta_{ij}^\phi\cdot f_kf_l\cdot\phi,\phi\right>
&=&[\hat\eta_{ij}^\phi,\hat\eta_{kl}^\phi]_{ts},\\
{\rm Re}\left<e_se_t \cdot \eta_{kl}^\phi\cdot f_if_j\cdot\phi,\phi\right>
&=&[\hat\eta_{kl}^\phi,\hat\eta_{ij}^\phi]_{ts},
\end{eqnarray*}
and by \rf{eq:identity1} and the anticommutativity of the bracket,
\[[\hat\eta_{ij}^\phi,\hat\eta_{kl}^\phi]=0.\]
To prove \rf{[ij,jk]=-ik}, note that by \rf{eq:despeje1}
\begin{eqnarray*}
f_if_j\cdot\eta_{jk}^\phi\cdot \phi
&=& f_if_k\cdot\phi
\end{eqnarray*}
and
\begin{eqnarray*}
f_jf_k\cdot\eta_{ij}^\phi\cdot \phi
&=& -f_if_k\cdot\phi
\end{eqnarray*}
so that
\begin{eqnarray*}
f_jf_k\cdot\eta_{ij}^\phi\cdot \phi
&=&f_if_j\cdot\eta_{jk}^\phi\cdot \phi - 2 f_if_k\cdot \phi.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
{\rm Re}\left<e_s e_t \cdot \eta_{ij}^\phi\cdot f_jf_k\cdot\phi,\phi\right>
&=& {\rm Re}\left<e_s e_t \cdot \eta_{jk}^\phi\cdot
f_if_j\cdot\phi,\phi\right>- 2 \eta_{ik}^\phi(e_s,e_t)
\end{eqnarray*}
and by \rf{eq:long-from-bracket-calculation}
\begin{eqnarray*}
[\hat\eta_{ij}^\phi,\hat\eta_{jk}^\phi] &=& [\hat\eta_{jk}^\phi,\hat\eta_{ij}^\phi ] - 2\hat\eta_{ik}^\phi,
\end{eqnarray*}
i.e.
\begin{eqnarray*}
[\hat\eta_{ij}^\phi,\hat\eta_{jk}^\phi] &=& -\hat\eta_{ik}^\phi.
\end{eqnarray*}
Thirdly, we will prove, in five separate cases, that the set of endomorphisms $\{\hat\eta_{kl}^\phi\}$ is
linearly
independent.
For $r=0,1$ there are no endomorphisms.
For $r=2$ it is obvious since there is only one non-zero endomorphism.
For $r=3$, suppose
\begin{eqnarray*}
0 &=& \alpha_{12}\hat\eta_{12}^\phi + \alpha_{13}\hat\eta_{13}^\phi+\alpha_{23}\hat\eta_{23}^\phi,
\end{eqnarray*}
where $\alpha_{12}\not=0$.
Take the Lie bracket with $\hat\eta_{13}^\phi$ to get
\begin{eqnarray*}
0
&=& \alpha_{12}\hat\eta_{23}^\phi
-\alpha_{23}\hat\eta_{12}^\phi ,
\end{eqnarray*}
i.e.
\[\hat\eta_{23}^\phi ={\alpha_{23}\over \alpha_{12}}\hat\eta_{12}^\phi.\]
We can also consider the bracket with $\hat\eta_{23}^\phi$,
\begin{eqnarray*}
0
&=& -\alpha_{12}\hat\eta_{13}^\phi
+\alpha_{13}\hat\eta_{12}^\phi ,
\end{eqnarray*}
so that
\[\hat\eta_{13}^\phi ={\alpha_{13}\over \alpha_{12}}\hat\eta_{12}^\phi.\]
By substituting in the original equation we get
\begin{eqnarray*}
0
&=& (\alpha_{12}^2 + \alpha_{13}^2+\alpha_{23}^2)\hat\eta_{12}^\phi,
\end{eqnarray*}
which gives a contradiction.
Now suppose $r\geq 5$ and that there is a linear combination
\[0=\sum_{k<l} \alpha_{kl}\hat\eta_{kl}^\phi .\]
Taking succesive brackets with $\hat\eta_{13}^\phi$, $\hat\eta_{12}^\phi$, $\hat\eta_{34}^\phi$ and
$\hat\eta_{45}^\phi$ we get the identity
\[\alpha_{12}\hat\eta_{15}^\phi=0,\]
i.e. $\alpha_{12}=0$. Similar arguments give the vanishing of every $\alpha_{kl}$.
For $r=4$, suppose there is a linear combination
\[0=
\alpha_{12}\eta_{12}^\phi +
\alpha_{13}\eta_{13}^\phi +
\alpha_{14}\eta_{14}^\phi +
\alpha_{23}\eta_{23}^\phi +
\alpha_{24}\eta_{24}^\phi +
\alpha_{34}\eta_{34}^\phi.
\]
Multiply by $-\phi$
\begin{eqnarray*}
0
&=&
(\alpha_{12}f_{12} +
\alpha_{13}f_{13} +
\alpha_{14}f_{14} +
\alpha_{23}f_{23} +
\alpha_{24}f_{24} +
\alpha_{34}f_{34})\cdot\phi.
\end{eqnarray*}
Multiply by $-f_{12}$
\begin{eqnarray*}
0
&=&
(\alpha_{12}
-\alpha_{13}f_{23}
-\alpha_{14}f_{24}
+\alpha_{23}f_{13}
+\alpha_{24}f_{14}
-\alpha_{34}f_{1234})\cdot\phi.
\end{eqnarray*}
Now, take hermitian product with $\phi$
\begin{eqnarray*}
0
&=&
\left<(\alpha_{12}
-\alpha_{13}f_{23}
-\alpha_{14}f_{24}
+\alpha_{23}f_{13}
+\alpha_{24}f_{14}
-\alpha_{34}f_{1234})\cdot\phi,\phi\right>\\
&=&
\alpha_{12}|\phi|^2
-\alpha_{34}\left<f_{1234}\cdot\phi,\phi\right>\\
&=&
\alpha_{12}.
\end{eqnarray*}
Similar arguments give the vanishing of the other coefficients.
\qd
\begin{lemma}\label{lemma:kernel}
Let $r\geq2$ and $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$ be a partially pure spinor. Then
\[V^\phi\subseteq \bigcap_{1\leq k<l\leq r}\ker\hat\eta_{kl}^\phi.\]
\end{lemma}
{\em Proof}. Let $1\leq k<l \leq r$ be fixed
and $X\mbox{\ns i}n V^\phi$.
Since $\hat lathbb{R}^n=V^\phi \opluslus (V^\phi)^\perp$
and $J^\phi$ is a complex structure on $V^\phi$, there exists a basis
$\{e_1,e_2,\ldots,e_{2m-1},e_{2m}\}\cup\{e_{2m+1},\ldots,e_{2m+r}\}$
such that
\begin{eqnarray*}
V^\phi &=& {\rm span}(e_1,e_2,\ldots,e_{2m-1},e_{2m}),\\
(V^\phi)^\perp &=& {\rm span}(e_{2m+1},\ldots,e_{2m+r}),\\
J^\phi(e_{2j-1}) &=& e_{2j},\\
J^\phi(e_{2j}) &=& -e_{2j-1},
\end{eqnarray*}
where $m=(n-r)/2$ and $1\leq j \leq m$.
Note that
\begin{eqnarray*}
\hat\eta_{kl}^\phi(e_{2j-1})
&=& \sum_{a=1}^n{\rm Re}\left< e_{2j-1}\wedge e_a\cdot f_{kl}\cdot\phi,\phi\right>e_a\\
&=& -\sum_{a\not= 2j-1}^n{\rm Re}\left< f_{kl}\cdot e_a e_{2j-1} \cdot\phi,\phi\right>e_a\\
&=& -\sum_{a\not= 2j-1}^n{\rm Re}\left< f_{kl}\cdot e_a (iJ^\phi (e_{2j-1})) \cdot\phi,\phi\right>e_a\\
&=& \sum_{a\not= 2j-1}^n{\rm Im} \left< e_a e_{2j} \cdot f_{kl}\cdot\phi,\phi\right>e_a\\
&=& -{\rm Im}\left<f_{kl}\cdot\phi,\phi\right>e_{2j}\\
&=&0.
\end{eqnarray*}
\qd
\begin{lemma}
Let $r\geq2$ and $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$ be a partially pure spinor.
Then $(V^\phi)^\perp$ carries a standard representation of $\hat lathfrak{so}(r)$, and an orientation.
\end{lemma}
{\em Proof}.
By Lemma \mbox{$\mathbb R$}f{lemma:lie-algebra}, $\hat lathfrak{so}(r)$ is represented non-trivially on $\hat lathbb{R}^n =
V^\phi \opluslus (V^\phi)^\perp$ and, by Lemma \mbox{$\mathbb R$}f{lemma:kernel}, it acts trivially on $V^\phi$.
Thus $(V^\phi)^\perp$ is a nontrivial representation of $\hat lathfrak{so}(r)$ of dimension $r$.
\qd
{\bf Remark}.
The existence of a partially pure spinor implies $r\equiv n$ (mod $2$).
In this case, let
$(e_1,\ldots, e_n)$ and $(f_1,\ldots, f_r)$ be orthonormal frames for $\hat lathbb{R}^n$ and
$\hat lathbb{R}^r$ respectively,
\[{\rm vol}_n= e_1\cdots e_n, \quad\quad {\rm vol}_r= f_1\cdots f_r,\]
and
\begin{eqnarray*}
F:\Sigma_r\otimes\Delta_n &\longrightarrow& \Sigma_r\otimes\Delta_n\\
\phi &\hat lapsto& (-i)^{n/2}i^{r/2}{\rm vol}_n\cdot {\rm vol}_r \cdot\phi.
\end{eqnarray*}
Note that $i^{r/2}{\rm vol}_r$ acts as $(-1)^{r/2} {\rm Id}_{\Sigma_r}$ on $\Sigma_r$ and that
$(-i)^{n/2}{\rm vol}_n$ determines the decomposition $\Delta_n=\Delta_n^+\opluslus \Delta_n^-$.
Thus we have that
\[\Sigma_r\otimes\Delta_n = (\Sigma_r\otimes\Delta_n)^+ \opluslus (\Sigma_r\otimes\Delta_n)^-,\]
and we will call elements in $(\Sigma_r\otimes\Delta_n)^+$ and $(\Sigma_r\otimes\Delta_n)^-$
positive and negative twisted spinors respectively.
\begin{defi} Let $n$ be even, $\hat lathbb{R}^n$ be endowed with the standard inner product and orientation, and
${\rm vol}_n$ denote the volume form. Let
$V$, $W$ be two orthogonal oriented subspaces such that $\hat lathbb{R}^n = V\opluslus W$. Furthermore, assume $V$
admits a complex structure inducing the given orientation on $V$. The oriented triple $(V,J,W)$ will be called
{\em positive} if given (oriented) orthonormal frames $(v_1,J(v_1),\ldots,v_m,J(v_m))$ and $(w_1,\ldots,w_r)$
of $V$ and $W$ respectively,
\[v_1\wedge J(v_1)\wedge \ldots\wedge v_m\wedge J(v_m)\wedge w_1\wedge\ldots\wedge w_r = {\rm vol}_n,\]
and {\em negative} if
\[v_1\wedge J(v_1)\wedge \ldots\wedge v_m\wedge J(v_m)\wedge w_1\wedge\ldots\wedge w_r = -{\rm vol}_n.\]
\end{defi}
\begin{lemma}
If $r$ is even, a partially pure spinor $\phi$ is either positive or negative. Furthermore,
a partialy pure spinor $\phi$ is positive (resp. negative) if and only if
the corresponding oriented triple $(V^\phi, J^\phi, (V^\phi)^\perp)$ is positive (resp. negative).
\end{lemma}
{\em Proof}.
We must prove that either
$\phi\mbox{\ns i}n(\Sigma_r\otimes\Delta_n)^+$ or
$\phi\mbox{\ns i}n(\Sigma_r\otimes\Delta_n)^-$.
Since $\phi$ is a partially pure spinor,
there exist frames
$(e_1',\ldots, e_{2m}')$ and $(e_{2m+1}',\ldots,e_{2m+r}')$
of $V^\phi$ and $(V^\phi)^\perp$ respectively such that
\[e_{2j}'= J(e_{2j-1}')\quad\hat lbox{and}\quad
\eta_{kl}^\phi = e_{2m+k}'\wedge e_{2m+l}',\]
where $1\leq j\leq m$ and $1\leq k<l\leq r$.
Now,
\[e_1'\wedge e_2'\wedge \ldots \wedge e_{2m}'\wedge e_{2m+1}'\wedge\ldots\wedge e_{2m+r}'=\pm {\rm vol}_n.\]
Then,
\begin{eqnarray*}
(-i)^{n/2}i^{r/2}{\rm vol}_n\cdot {\rm vol}_r \cdot\phi
&=&
\pm (-i)^{n/2}i^{r/2} e_1' e_2' \cdots e_{2m}' e_{2m+1}'\cdots e_{2m+r}'\cdot f_1\cdots f_r\cdot \phi\\
&=&
\pm (-i)^{n/2}i^{r/2} e_1' J(e_1') \cdots e_{2m-1}'J(e_{2m-1}') \eta_{12}^\phi\cdots
\eta_{r-3,r-2}^\phi\cdot
f_{12}\cdots f_{r-1,r}\cdot \eta_{r-1,r}^\phi\cdot\phi\\
&=&
\pm (-i)^{n/2}i^{r/2} e_1' J(e_1') \cdots e_{2m-1}'J(e_{2m-1}') \eta_{12}^\phi\cdots
\eta_{r-3,r-2}^\phi\cdot
f_{12}\cdots f_{r-3,r-2}\cdot\phi\\
&=&
\pm (-i)^{n/2}i^{r/2} e_1' J(e_1') \cdots e_{2m-1}'J(e_{2m-1}') \cdot\phi\\
&=&
\pm (-i)^{n/2}i^{r/2} e_1' J(e_1') \cdots e_{2m-3}'J(e_{2m-3}')e_{2m-1}'(-ie_{2m-1}') \cdot\phi\\
&=&
\pm (-1)^m(-i)^{n/2+m}i^{r/2} \phi\\
&=&
\pm \phi,
\end{eqnarray*}
i.e.
$\phi\mbox{\ns i}n(\Sigma_r\otimes\Delta_n)^\pm$.
\qd
\subsection{Orbit of a partially pure spinor}
\begin{lemma} \label{lemma:g(phi)-partially-pure-spinor}
Let $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$ be a
partially pure spinor. If $g\mbox{\ns i}n Spin^{c,r}(n)$, then $g(\phi)$ is also a partially pure spinor.
\end{lemma}
{\em Proof}.
Let $g\mbox{\ns i}n Spin^{c,r}(n)$ and
$\lambda_n^{c,r}(g)=(g_1,g_2,g_3)\mbox{\ns i}n SO(n)\times SO(r)\times U(1)$.
First, suppose $X,Y\mbox{\ns i}n V^\phi$,
\[X\cdot \phi = i \kern1pt\kern1pt Y\cdot \phi.\]
Apply $g$ on both sides
\begin{eqnarray*}
g_1(X) \cdot g(\phi)
&=& i\kern1pt\kern1ptg_1(Y)\cdot g(\phi).
\end{eqnarray*}
which means that $g_1$ maps $V^\phi$ into $V^{g(\phi)}$ injectively.
On the other hand, any pair of
vectors $\tilde X, \tilde Y\mbox{\ns i}n V^{g(\phi)}$ such that
\[\tilde X\cdot g(\phi) = i \kern1pt\kern1pt \tilde Y\cdot g(\phi),\]
are the image under $g_1$ of some vectors $X,Y\mbox{\ns i}n\hat lathbb{R}^n$, i.e.
\[g_1(X)\cdot g(\phi) = i \kern1pt\kern1pt g_1(Y)\cdot g(\phi).\]
Apply $g^{-1}$ on both sides to get
\[X\cdot \phi = i\kern1pt\kern1pt Y\cdot \phi,\]
so that $X,Y\mbox{\ns i}n V^\phi$, i.e.
$V^{g(\phi)} = g_1(V^\phi)$.
Moreover,
\[ J^{g(\phi)} = g_1|_{V^\phi}\raise1.6pt\hbox{\footnotesize$\circ$}c J^\phi \raise1.6pt\hbox{\footnotesize$\circ$}c (g_1|_{v^\phi})^{-1}.\]
Now, let $e_a'=g_1^{-1}(e_a)$ and $f_k'=g_2^{-1}(f_k)$, so that
\begin{eqnarray*}
\eta_{kl}^{g(\phi)}\cdot g(\phi)
&=& \sum_{1\leq a<b\leq n} \eta_{kl}^{g(\phi)}(e_a,e_b)e_ae_b\cdot g(\phi) \\
&=& \sum_{1\leq a<b\leq n} \left<g_1(e_a')g_1(e_b') \cdot g_2(f_k')g_2(f_l')\cdot g(\phi),
g(\phi)\right>g_1(e_a')g_1(e_b')\cdot g(\phi) \\
&=& \sum_{1\leq a<b\leq n} \left<e_a'e_b' \cdot f_k'f_l'\cdot \phi, \phi\right>g(e_a'e_b'\cdot \phi)\\
&=& g\left(\sum_{1\leq a<b\leq n} \eta_{kl}'^{\phi}(e_a',e_b')e_a'e_b'\cdot\phi\right)\\
&=& g\left(\eta_{kl}'^{\phi}\cdot\phi\right)\\
&=& g\left(-f_k'f_l'\cdot\phi\right)\\
&=& -f_kf_l\cdot g(\phi),
\end{eqnarray*}
and
\begin{eqnarray*}
\left< f_kf_l\cdot g(\phi),g(\phi)\right>
&=& \left< g(f_k'f_l'\cdot \phi), g(\phi)\right>\\
&=& \left< f_k'f_l'\cdot \phi, \phi\right>\\
&=& 0.
\end{eqnarray*}
For $r=4$, note that the volume form is invariant under
$SO(4)$
\[\left<f_1f_2f_3f_4\cdot g(\phi),g(\phi)\right> = \left<f_1f_2f_3f_4\cdot \phi,\phi\right> = 0.\]
\qd
\begin{lemma}
Let $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$ be a partially pure spinor.
The stabilizer of $\phi$ is isomorphic to $U(m)\times SO(r)$.
\end{lemma}
{\em Proof}. Let $g\mbox{\ns i}n Spin^{c,r}(n)$ be such that
$g(\phi)=\phi$ and $\lambda_n^{c,r}(g)=(g_1,g_2,g_3)\mbox{\ns i}n SO(n)\times SO(r)\times U(1)$.
It can be checked easily that
\begin{eqnarray*}
[g_1,J^\phi] &=& 0 \\
g_1(V^\phi) &=& V^\phi \\
g_1|_{V^\phi} &\mbox{\ns i}n& U(V^\phi,J^\phi) \cong U(m).
\end{eqnarray*}
Clearly, $g_1((V^\phi)^\perp)= (V^\phi)^\perp$.
As in Lemma \mbox{$\mathbb R$}f{lemma:kernel}, one can prove
\[\eta_{kl}^\phi = \sum_{2m+1\leq a< b \leq 2m+r} \eta_{kl}^\phi(e_a,e_a)e_ae_b\mbox{\ns i}n \raise1pt\hbox{$\ts\bigwedge$}^2(V^\phi)^\perp,\]
where $(e_1,\ldots,e_{2m+r})$ is an oriented frame of $V^\phi \opluslus (V^\phi)^\perp$.
Furthermore,
\begin{eqnarray*}
g_1(\eta_{kl}^\phi)
&=&
\eta_{kl}'^\phi,
\end{eqnarray*}
where
$f_k'=g_2(f_k)$.
Now, we have that
\[
\begin{array}{ccc}
f_kf_l & \hbox{{\footnotesize $\#$}}rightarrow{g_2} & f_k'f_l'\\
\downarrow & & \downarrow\\
\eta_{kl}^\phi & \hbox{{\footnotesize $\#$}}rightarrow{h_2} & \eta_{kl}'^\phi
\end{array}
\]
for the diagram
\[
\begin{array}{ccccccc}
\hat lathfrak{so}(r)&\cong&\raise1pt\hbox{$\ts\bigwedge$}^2\hat lathbb{R}^r & \hbox{{\footnotesize $\#$}}rightarrow{g_2} & \raise1pt\hbox{$\ts\bigwedge$}^2\hat lathbb{R}^r&\cong&\hat lathfrak{so}(r)\\
&&\downarrow & & \downarrow&&\\
\hat lathfrak{so}(r)&\cong&\raise1pt\hbox{$\ts\bigwedge$}^2(V^\phi)^\perp & \hbox{{\footnotesize $\#$}}rightarrow{h2} & \raise1pt\hbox{$\ts\bigwedge$}^2(V^\phi)^\perp &\cong& \hat lathfrak{so}(r)
\end{array}
\]
where the vertical arrows are Lie algebra isomorphisms and the horizontal arrows correspond to $g_2$ and $h_2$
acting
via the adjoint representation of $SO(r)$. Thus, $h_2$ and $g_2$ correspond to each other under the
isomorphism
$\raise1pt\hbox{$\ts\bigwedge$}^2(V^\phi)^\perp\cong \raise1pt\hbox{$\ts\bigwedge$}^2\hat lathbb{R}^r$ given by
$f_{kl}\hat lapsto \eta_{kl}^\phi$.
Since $h_1$ is unitary with respect to $J$, there is a frame $(e_1,\ldots, e_{2m})$ of $V^\phi$ such that
\[e_{2j}= J(e_{2j-1})\]
and $h_1$ is diagonal with respect to the unitary basis
$\{e_{2j-1}-ie_{2j}\kern1pt|\kern1pt j=1,\ldots,m\}$,
i.e.
\[h_1(e_{2j-1}-ie_{2j})=e^{i\theta_j}(e_{2j-1}-ie_{2j})\]
where $0\leq \theta_j<2\pi$.
On the other hand,
there is a frame $(f_1,\ldots, f_r)$ of $\hat lathbb{R}^r$ such that
\[g_2= R_{\varphi_1}\raise1.6pt\hbox{\footnotesize$\circ$}c \cdots \raise1.6pt\hbox{\footnotesize$\circ$}c R_{\varphi_{[r/2]}}
\]
where $R_{\varphi_k}$ is a rotation by an angle $\varphi_k$ on the plane generated by
$f_{2k-1}$ and $f_{2k}$, $1\leq k\leq [r/2]$. Now, since the endomorphisms $\hat\eta_{kl}^\phi$ span an
isomorphic
copy of $\hat lathfrak{so}(r)$, there is a frame
$(e_{2m+1},\ldots,e_{2m+r})$ of $(V^\phi)^\perp$ such that
\[\eta_{kl}^\phi = e_{2m+k}\wedge e_{2m+l},\]
$1\leq k<l\leq r$.
Since the adjoint representation of $SO(r)$ is faithful
\[h_2= R_{\varphi_1}'\raise1.6pt\hbox{\footnotesize$\circ$}c \cdots \raise1.6pt\hbox{\footnotesize$\circ$}c R_{\varphi_{[r/2]}}'
\]
where $R_{\varphi_k}'$ is a rotation by an angle $\varphi_k$ on the plane generated by
$e_{2m+2k-1}$ and $e_{2m+2k}$, $1\leq k\leq [r/2]$.
Thus,
\begin{eqnarray*}
g
&=&
\pm\left[\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)- \sin(\theta_j/2)e_{2j-1}e_{2j})\cdot
\mbox{$\mathbb P$}od_{k=1}^{[r/2]} (\cos(\varphi_k/2)- \sin(\varphi_k/2)\eta_{2k-1,2k}^\phi),\right.\\
&&
\left.
\mbox{$\mathbb P$}od_{k=1}^{[r/2]} (\cos(\varphi_k/2)- \sin(\varphi_k/2)f_{2k-1}f_{2k})
, e^{i\theta/2}
\right] .
\end{eqnarray*}
Now,
\begin{eqnarray*}
\phi
&=&
g(\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)- \sin(\theta_j/2)e_{2j-1}e_{2j})
\\
&&
\cdot\mbox{$\mathbb P$}od_{k=1}^{[r/2]} (\cos(\varphi_k/2)- \sin(\varphi_k/2)\eta_{2k-1,2k}^\phi)\cdot\mbox{$\mathbb P$}od_{k=1}^{[r/2]}
(\cos(\varphi_k/2)- \sin(\varphi_k/2)f_{2k-1}f_{2k}) (\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)- \sin(\theta_j/2)e_{2j-1}e_{2j})
\cdot\mbox{$\mathbb P$}od_{k=1}^{[r/2]-1} (\cos(\varphi_k/2)- \sin(\varphi_k/2)\eta_{2k-1,2k}^\phi)\\
&&
\cdot\mbox{$\mathbb P$}od_{k=1}^{[r/2]}
(\cos(\varphi_k/2)- \sin(\varphi_k/2)f_{2k-1}f_{2k})
(\cos(\varphi_{[r/2]}/2)-
\sin(\varphi_{[r/2]}/2)\eta_{2{[r/2]}-1,2{[r/2]}}^\phi)\cdot (\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)- \sin(\theta_j/2)e_{2j-1}e_{2j})
\\
&&
\cdot\mbox{$\mathbb P$}od_{k=1}^{[r/2]-1} (\cos(\varphi_k/2)- \sin(\varphi_k/2)\eta_{2k-1,2k}^\phi)\cdot
\mbox{$\mathbb P$}od_{k=1}^{[r/2]-1}
(\cos(\varphi_k/2)- \sin(\varphi_k/2)f_{2k-1}f_{2k})\\
&&
(\cos(\varphi_{[r/2]}/2)- \sin(\varphi_{[r/2]}/2)f_{2{[r/2]}-1}f_{2{[r/2]}})\cdot
(\cos(\varphi_{[r/2]}/2)+\sin(\varphi_{[r/2]}/2)f_{2{[r/2]}-1}f_{2{[r/2]}})\cdot (\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)- \sin(\theta_j/2)e_{2j-1}e_{2j})
\\
&&
\cdot\mbox{$\mathbb P$}od_{k=1}^{[r/2]-1} (\cos(\varphi_k/2)- \sin(\varphi_k/2)\eta_{2k-1,2k}^\phi)\cdot
\mbox{$\mathbb P$}od_{k=1}^{[r/2]-1}
(\cos(\varphi_k/2)- \sin(\varphi_k/2)f_{2k-1}f_{2k}) (\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)- \sin(\theta_j/2)e_{2j-1}e_{2j})
(\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)+i \sin(\theta_j/2)e_{2j-1}(iJ(e_{2j-1})))
(\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)+i \sin(\theta_j/2)e_{2j-1}e_{2j-1})
(\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m (\cos(\theta_j/2)-i \sin(\theta_j/2))
(\phi)\\
&=&
\pm e^{i\theta/2}\mbox{$\mathbb P$}od_{j=1}^m e^{-i\theta_j/2}
(\phi)\\
&=&
\pm e^{{i\over 2}(\theta-\sum_{j=1}^m\theta_j)}
(\phi).
\end{eqnarray*}
This means
\[e^{{i\over 2}(\theta-\sum_{j=1}^m\theta_j)}=\pm1\]
i.e.
\begin{eqnarray*}
{\det}_{\hat lathbb{C}}(h_1)
&=&
e^{i\sum_{j=1}^m \theta_j} \\
&=&
e^{i\theta} .
\end{eqnarray*}
Thus we have found that
\[\lambda_{n}^{c,r}(g) = ((h_1,h_2),h_2,{\det}_{\hat lathbb{C}}(h_1)) ,\]
which is in the image of the horizontal row in the diagram of Lemma \mbox{$\mathbb R$}f{lemma:subgroup2}
\[
\begin{array}{ccc}
& & Spin^{c,r}(n)\\
& \nearrow & \downarrow\\
U(m)\times SO(r) & \rightarrow & SO(n)\times SO(r)\times U(1)
\end{array}
\]
\qd
{\bf Remark}.
Note that for any spinor $\phi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$, $g\mbox{\ns i}n Spin^{c,r}(n)$, $\lambda_n^{c,r}(g)\mbox{\ns i}n
SO(n)\times SO(r)\times U(1)$,
\begin{eqnarray*}
\eta_{kl}^{g(\phi)}(X,Y)
&=&
\left<X\wedge Y\cdot f_kf_l \cdot g(\phi),g(\phi)\right> \\
&=& \left<g_1(X')\wedge g_1(Y')\cdot g_2(f_k')g_2(f_l') \cdot g(\phi),g(\phi)\right> \\
&=& \left<g(X'\wedge Y'\cdot f_k'f_l' \cdot \phi),g(\phi)\right> \\
&=& \left<X'\wedge Y'\cdot f_k'f_l' \cdot \phi,\phi\right> \\
&=:& \eta_{kl}'^{\phi} (X',Y'),
\end{eqnarray*}
for $X'=g_1^{-1}(X),Y'=g_1^{-1}(Y)\mbox{\ns i}n\hat lathbb{R}^n$,
$f_k'=g_2^{-1}(f_k)$. Thus, the matrices representing
$\eta_{kl}^{g(\phi)}$ (with respect to some basis) are conjuagte to the matrices representing
$\eta_{kl}'^\phi$.
\begin{lemma}\label{lemma: little orbit}
Let $\phi,\psi\mbox{\ns i}n\Sigma_r\otimes\Delta_n$ be partially pure spinors and
$Spin^c(r)$ the standard copy of this group in $Spin^{c,r}(n)$.
Then, $\psi\mbox{\ns i}n Spin^{c}(r)\cdot\phi$ if and only if they generate the same oriented tiple
$(V^\phi,J^\phi,(V^\phi)^\perp)=(V^\psi,J^\psi,(V^\psi)^\perp)$.
\end{lemma}
{\em Proof}. Suppose $\psi = g(\phi)$ for some $g\mbox{\ns i}n Spin^c(r)\subset Spin^{c,r}(n)$, and
let $\lambda_n^{c,r}(g)=(1,g_2,e^{i\theta})$.
Such an element induces
\begin{eqnarray*}
\left<X\wedge Y\cdot f_kf_l \cdot g(\phi),g(\phi)\right>
&=& \left<X\wedge Y\cdot f_k'f_l' \cdot \phi,\phi\right>
\end{eqnarray*}
for $f_k'=g_2^{-1}(f_k)$, i.e.
\[\eta_{kl}^{g(\phi)}(X,Y) = \eta_{kl}'^{\phi}(X,Y),
\]
so that they span the same copy of $\hat lathfrak{so}(r)$ in ${\rm End}^-(\hat lathbb{R}^n)$,
\[ {\rm span}(\eta_{kl}^{g(\phi)}) = {\rm span}(\eta_{kl}'^\phi)\cong
\hat lathfrak{so}(r)\subset{\rm End}^-(\hat lathbb{R}^n).
\]
Thus, by Lemma \mbox{$\mathbb R$}f{lemma:g(phi)-partially-pure-spinor},
the partially pure spinors $\phi$ and $g(\phi)$ determine the same oriented triple
$(V^{g(\phi)}, J^{g(\phi)},(V^{g(\phi)})^\perp ) =(V^\phi,V^\phi, (V^\phi)^\perp)$.
Conversely, assume $(V^\phi,J^\phi,(V^\phi)^\perp)=(V^\psi,J^\psi,(V^\psi)^\perp)$, and
consider the subalgebras of
\begin{eqnarray*}
\hat lathfrak{so}(r)^\phi &=& {\rm span}(\eta_{kl}^\phi + f_{kl})\\
\hat lathfrak{so}(r)^\psi &=& {\rm span}(\eta_{kl}^\psi + f_{kl}).
\end{eqnarray*}
There exist frames $(e_{2m+1},\ldots,e_{2m+r})$ and $(e_{2m+1}',\ldots,e_{2m+r}')$ of
$(V^\phi)^\perp$ and $ (V^\psi)^\perp$ respectively, such that
\begin{eqnarray*}
\eta_{kl}^\phi &=& e_{2m+k}\wedge e_{2m+l},\\
\eta_{kl}^\psi &=& e_{2m+k}'\wedge e_{2m+l}'.
\end{eqnarray*}
Let $A=(a_{kl})\mbox{\ns i}n SO(r)$ the matrix such that
\[A:\kern1pt\kern1pte_{2m+k}'\hat lapsto a_{k1}e_{2m+1}' + \cdots + a_{kr}e_{2m+r}' = e_{2m+k}\]
$1\leq k < l \leq r$.
The induced transformation maps
\[A:\kern1pt\kern1pte_{2m+k}'\wedge e_{2m+l}' \hat lapsto e_{2m+k}\wedge e_{2m+l},\]
and set
\[A^T:\kern1pt\kern1pt f_k\hat lapsto a_{1k}f_{1} + \cdots + a_{rk}f_{r}=f_k',\]
and
\[A^T:\kern1pt\kern1ptf_k\wedge f_l\hat lapsto f_k'\wedge f_l'.\]
Consider
\begin{eqnarray*}
\left<e_{2m+p}\wedge e_{2m+q}\cdot f_k'f_l'\cdot \psi,\psi\right>
&=&
\left<\left(\sum_{s=1}^r a_{ps}e_{2m+s}'\right)\wedge \left(\sum_{t=1}^r a_{qt}e_{2m+t}'\right)\cdot
f_k'f_l'\cdot \psi,\psi\right>\\
&=&
\left<\left(\sum_{s<t} (a_{ps}a_{qt}-a_{pt}a_{qs})e_{2m+s}'\wedge e_{2m+t}'\right)\cdot
f_k'f_l'\cdot \psi,\psi\right>\\
&=&
\sum_{s<t} (a_{ps}a_{qt}-a_{pt}a_{qs})\left<e_{2m+s}'\wedge e_{2m+t}'\cdot
f_k'f_l'\cdot \psi,\psi\right>\\
&=&
\sum_{s<t} (a_{ps}a_{qt}-a_{pt}a_{qs})\left<e_{2m+s}'\wedge e_{2m+t}'\cdot
\left(\sum_{i=1}^r a_{ik}f_i\right)\left(\sum_{j=1}^r a_{jl}f_j\right)\cdot \psi,\psi\right>\\
&=&
\sum_{s<t} (a_{ps}a_{qt}-a_{pt}a_{qs})\left<e_{2m+s}'\wedge e_{2m+t}'\cdot
\left(\sum_{i<j} (a_{ik}a_{jl}-a_{il}a_{jk})f_if_j\right)\cdot \psi,\psi\right>\\
&=&
\sum_{s<t}\sum_{i<j} (a_{ps}a_{qt}-a_{pt}a_{qs})(a_{ik}a_{jl}-a_{il}a_{jk})\left<e_{2m+s}'\wedge
e_{2m+t}'\cdot
f_if_j\cdot \psi,\psi\right>\\
&=&
\sum_{s<t}\sum_{i<j} (a_{ps}a_{qt}-a_{pt}a_{qs})(a_{ik}a_{jl}-a_{il}a_{jk}) \delta_{si}\delta_{tj}\\
&=&
\sum_{s<t} (a_{ps}a_{qt}-a_{pt}a_{qs})(a_{sk}a_{tl}-a_{sl}a_{tk}) \\
&=&
\delta_{pk}\delta_{ql} ,
\end{eqnarray*}
since the induced tranformation by $A$ on $\raise1pt\hbox{$\ts\bigwedge$}^2\hat lathbb{R}^r$ is orthogonal.
This means
\[\eta_{kl}'^\psi = \eta_{kl}^\phi= e_{2m+k}\wedge e_{2m+l}.\]
Now consider $g\mbox{\ns i}n Spin^c(r)\subset Spin^{c,r}(n)$ such that $\lambda_n^{c,r}(g)=(1,A,1)\mbox{\ns i}n SO(n)\times
SO(r)\times U(1)$.
Then
\begin{eqnarray*}
\delta_{pk}\delta_{ql}
&=&
\left<e_{2m+p}\wedge e_{2m+q}\cdot f_k'f_l'\cdot \psi,\psi\right> \\
&=&
\left<g(e_{2m+p}\wedge e_{2m+q}\cdot f_k'f_l'\cdot \psi),g(\psi)\right> \\
&=&
\left<e_{2m+p}\wedge e_{2m+q}\cdot A(f_k')A(f_l')\cdot g(\psi),g(\psi)\right> \\
&=&
\left<e_{2m+p}\wedge e_{2m+q}\cdot f_kf_l\cdot g(\psi),g(\psi)\right> ,
\end{eqnarray*}
i.e.
\[\eta_{kl}^{g(\psi)} = e_{2m+k}\wedge e_{2m+l} = \eta_{kl}^\phi,\]
so that
\[\hat lathfrak{so}(r)^{g(\psi)} = {\rm span}(\eta_{kl}^{g(\psi)} + f_{kl}) = {\rm span}(\eta_{kl}^{\phi} +
f_{kl})=\hat lathfrak{so}(r)^\phi.\]
This implies that $g(\psi)$ and $\phi$ share the same stabilizer
\[ U(V^\phi,J^\phi)\times exp(\hat lathfrak{so}(r)^\phi)= U(V^{g(\psi)},J^{g(\psi)})\times
exp(\hat lathfrak{so}(r)^{g(\psi)}) \cong U(m)\times SO(r).\]
But there is only a 1-dimensional summand in the decomposition of $\Sigma_r\otimes \Delta_n$
under this subgroup. More precisely, under this subgroup
\begin{eqnarray*}
\Sigma_r\otimes \Delta_n
&=&
\Sigma_r\otimes\Delta_r\otimes\Delta_{2m},
\end{eqnarray*}
where $\Delta_{2m}$ decomposes under $U(m)$ and contains only a $1$-dimensional trivial summand
\cite{Friedrich}, while
$\Sigma_r\otimes\Delta_r$ is isomorphic to a subspace of the complexified space of alternating forms on
$\hat lathbb{R}^r$ which also contains only a $1$-dimensional trivial summand.
Thus, $g(\psi)= e^{i\theta} \phi$ for some $\theta\mbox{\ns i}n[0,2\pi)\subset\hat lathbb{R}$.
\qd
\begin{lemma}
\begin{itemize}
\mbox{\ns i}tem If $r$ is odd, the group $Spin^{c,r}(n)$ acts transtitively on the set of partially pure spinors in
$\Sigma_r\otimes\Delta_n $.
\mbox{\ns i}tem If $r$ is even, the group $Spin^{c,r}(n)$ acts transtitively on the set of positive partially pure
spinors in
$(\Sigma_r\otimes\Delta_n)^+ $.
\end{itemize}
\end{lemma}
{\em Proof}.
Suppose that $r$ is odd.
Note that the standard partially pure spinor $\phi_0$ satisfies the conditions
\begin{equation}
\left\{
\begin{array}{rcl}
e_{2j-1}e_{2j}\cdot\phi_0 & = & i\phi_0, \\
e_{2m+k}e_{2m+l}\cdot\phi & = & -f_{kl}\cdot\phi, \\
\left<f_{kl}\cdot\phi,\phi\right> & = & 0,
\end{array}
\right.\label{eq: equations standard partially pure spinor}
\end{equation}
where $(e_1,\ldots, e_{n})$ and $(f_1,\ldots,f_r)$ are the standard oriented frames of $\hat lathbb{R}^n$ and
$\hat lathbb{R}^r$ respectively.
There exist frames
$(e_1',\ldots, e_{2m}')$ and $(e_{2m+1}',\ldots,e_{2m+r}')$ of $V^\phi$ and $(V^\phi)^\perp$ respectively such
that
\[e_{2j}'= J(e_{2j-1}')\quad
\hat lbox{and}
\quad\eta_{kl}^\phi = e_{2m+k}'\wedge e_{2m+l}',\]
$1\leq k<l\leq r$, $1\leq j\leq m$.
Call $g_1'\mbox{\ns i}n O(n)$
the transformation of $\hat lathbb{R}^n$ taking the new frame to the standard one.
Define $g_1\mbox{\ns i}n SO(n)$ as follows
\[\left\{
\begin{array}{ll}
g_1 = g_1', & \hat lbox{if $e_1'\wedge\ldots\wedge e_{2m+r}'= {\rm vol}_n$,} \\
g_1 = -g_1', & \hat lbox{if $e_1'\wedge\ldots\wedge e_{2m+r}'= -{\rm vol}_n$.} \\
\end{array}
\right.
\]
Then $(g_1,1,1)\mbox{\ns i}n SO(n)\times SO(r) \times U(1)$ has two preimages $\pm\tilde g\mbox{\ns i}n Spin^{c,r}(n)$.
By Lemma \mbox{$\mathbb R$}f{lemma:g(phi)-partially-pure-spinor}, $\tilde{g}(\phi)$ is a partially pure spinor.
We will check that $\tilde{g}(\phi)$ satisfies \rf{eq: equations standard partially pure
spinor} as $\phi_0$ does.
Indeed,
\begin{eqnarray*}
e_{2j-1}e_{2j}\cdot \tilde{g}(\phi)
&=&
g_1'(e_{2j-1}')g_1'(e_{2j}')\cdot \tilde{g}(\phi)\\
&=&
(\pm g_1(e_{2j-1}'))(\pm g_1(e_{2j}'))\cdot \tilde{g}(\phi)\\
&=&
g_1(e_{2j-1}') g_1(e_{2j}')\cdot \tilde{g}(\phi)\\
&=&
\tilde{g}(e_{2j-1}'e_{2j}'\cdot \phi)\\
&=&
\tilde{g}(i \phi)\\
&=&
i\tilde{g}( \phi),
\end{eqnarray*}
and
\begin{eqnarray*}
e_{2m+k}e_{2m+l}\cdot \tilde{g}(\phi)
&=&
g_1'(e_{2m+k}')g_1'(e_{2m+l}')\cdot \tilde{g}(\phi)\\
&=&
(\pm g_1(e_{2m+k}'))(\pm g_1(e_{2m+l}'))\cdot \tilde{g}(\phi)\\
&=&
g_1(e_{2m+k}')g_1(e_{2m+l}')\cdot \tilde{g}(\phi)\\
&=&
\tilde{g}(e_{2m+k}'e_{2m+l}'\cdot \phi)\\
&=&
\tilde{g}(-f_{k}f_{l}\cdot \phi)\\
&=&
-\lambda_2(\tilde{g})(f_{k})\lambda_2(\tilde{g})(f_{l})\cdot \tilde{g}(\phi)\\
&=&
-f_{k}f_{l}\cdot \tilde{g}(\phi),
\end{eqnarray*}
since $\lambda_2(\tilde{g})=1$.
Similarly,
\begin{eqnarray*}
\left< f_kf_l\cdot \tilde{g}(\phi),\tilde{g}(\phi)\right>
&=&
\left< \lambda_2(\tilde{g})(f_k)\lambda_2(\tilde{g})(f_l)\cdot \tilde{g}(\phi),\tilde{g}(\phi)\right> \\
&=&
\left< \tilde{g}(f_kf_l\cdot\phi),\tilde{g}(\phi)\right> \\
&=&
\left< f_kf_l\cdot\phi,\phi\right> \\
&=&
0.
\end{eqnarray*}
Thus, $\tilde{g}(\phi)$ generates the same oriented triple $(V^{\tilde g(\phi)},J^{\tilde g(\phi)},
(V^{\tilde g(\phi)})^\perp)=(V^{\phi_0},J^{\phi_0},
(V^{\phi_0})^\perp)$ as $\phi_0$ which, by Lemma \mbox{$\mathbb R$}f{lemma: little orbit}, concludes the proof for $r$ odd.
The case for $r$ even is similar.
\qd
\begin{theo} \label{theo:characterization}
Let $\hat lathbb{R}^n$ be endowed with the standard inner product and orientation.
Given $r\mbox{\ns i}n\hat lathbb{N}$ such that $r< n$, the following objects are equivalent:
\begin{enumerate}
\mbox{\ns i}tem A (positive) triple consisting of a codimension $r$ vector subspace endowed with an orthogonal complex
structure
and an oriented orthogonal complement.
\mbox{\ns i}tem An orbit $Spin^c(r)\cdot\phi$ for some (positive) twisted partially pure spinor $\phi\mbox{\ns i}n
\Delta_n\otimes
\Sigma_r$.
\end{enumerate}
\end{theo}
{\em Proof}. Given a codimension $r$ vector subspace $D$ endowed with an orthogonal complex structure,
$\dim_{\hat lathbb{R}}(D)=2m$, $n=2m+r$. By Lemma \mbox{$\mathbb R$}f{factorization}
\[\Delta_n \cong \Delta(D^\perp)\otimes\Delta(D) .\]
Let us define
\begin{eqnarray*}
\Sigma_r&\cong& \left\{
\begin{array}{ll}
\Delta(D^\perp) & \hat lbox{if $r$ is odd,}\\
\Delta(D^\perp)^+ & \hat lbox{if $r$ is even,}
\end{array}
\right.
\end{eqnarray*}
so that
\[\Sigma_r\otimes\Delta_n \]
contains the standard twisted partially pure spinor $\phi_0$ of Lemma \mbox{$\mathbb R$}f{lemma:existence}.
The proof of the converse is the content of Subsection \mbox{$\mathbb R$}f{sec: basic properties}.
\qd
Let $\tilde{\hat lathcal{S}}$ denote the set of all partially pure spinors of rank $r$
\[\tilde{\hat lathcal{S}} = {Spin^{c,r}(n) \over U(m)\times SO(r)}.\]
Consider
\[ \hat lathcal{S}={\tilde{\hat lathcal{S}}\over Spin^c(r) } \]
where $Spin^c(r)$ is the canonical copy of such a group in $Spin^{c,r}(n)$.
Thus we have the following expected result.
\begin{corol}
The space parametrizing subspaces with orthogonal complex structures of codimension $r$ in
$\hat lathbb{R}^n$ with oriented orthogonal complements is
\[\hat lathcal{S}\cong{SO(n)\over U(m)\times SO(r)}.\]
\end{corol}
\qd
{\small
\mbox{$\mathbb R$}newcommand{0.5}{0.5}
\newcommand{10}{10}
\newcommand{Berlin\kern3pt Heidelberg\kern3pt New York}{Berlin\kern3pt Heidelberg\kern3pt New York}
\bi{Charlton} P. Charlton, {\mbox{\ns i}t The Geometry of Pure Spinors, with Applications}. PhD
thesis, Department of Mathematics, The University of Newcastle, Australia, 1998.
\bi{Espinosa-Herrera} Espinosa, M.; Herrera, R.: Spinorially twisted Spin structures, I: curvature
identities and eigenvalue estimates. Preprint (2014), arXiv:1409.6246
\bi{Friedrich} Friedrich, Th.: Dirac operators in Riemannian geometry,
Graduate studies in mathematics, Volume 25, American Mathematical Society.
\bi{Herrera-Nakad} Herrera, R.; Nakad, R.:
Spinorially twisted spin structures, III: CR structures. (in progress)
\bi{Lawson} Lawson, H. B., Jr.; Michelsohn, M.-L.: Spin geometry.
Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.
xii+427 pp. ISBN: 0-691-08542-0
\bi{Trautman} Trautman A.; Trautman, K.: Generalized pure spinors. Journal of Geometry and Physics
{\bf 15} (1994), 1–21.
\end{thebibliography} }
\end{document}
|
\begin{document}
\newgeometry{bottom=1.5in}
\begin{center}
\LARGE{\textbf{nlive: an R Package to facilitate the application of the sigmoidal and random changepoint mixed models}}
\title{}
\maketitle
\thispagestyle{empty}
\begin{tabular}{cc}
\normalsize{
Ana W. Capuano\upstairs{\affilone} and
Maude Wagner\upstairs{\affilone}}
\\ [0.25ex]
\small{\affilone~Rush Alzheimers Disease Center, Rush University Medical Center, Chicago, IL, USA}
\end{tabular}
\emails{~~~~~~~~~~~~ Corresponding author: ana\[email protected]}
\vspace*{0.1in}
\begin{abstract}
\footnotesize{The use of mixed effect models with a specific functional form such as the Sigmoidal Mixed Model and the Piecewise Mixed Model (or Changepoint Mixed Model) with abrupt or smooth random change allow the interpretation of the defined parameters to understand longitudinal trajectories. Currently, there are no interface R packages that can easily fit the Sigmoidal Mixed Model allowing the inclusion of covariates or incorporate recent developments to fit the Piecewise Mixed Model with random change. To facilitate the modeling of the Sigmoidal Mixed Model, and Piecewise Mixed Model with abrupt or smooth random change, we have created an R package called nlive. All needed pieces such as functions, covariance matrices, and initials generation were programmed. The package was implemented with recent developments such as the polynomial smooth transition of piecewise mixed model with improved properties over Bacon-Watts, and the stochastic approximation expectation-maximization (SAEM) for efficient estimation. It was designed to help interpretation of the output by providing features such as annotated output, warnings, and graphs. Functionality, including time and convergence, was tested using simulations. We provided a data example to illustrate the package use and output features and interpretation. The package implemented in the R software is available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=nlive. The nlive package for R fits the Sigmoidal Mixed Model and the Piecewise Mixed: abrupt and smooth. The nlive allows fitting these models with only five mandatory arguments that are intuitive enough to the less sophisticated users.}
\end{abstract}
\end{center}
\small
\section{BACKGROUND}
\noindent
Continuous longitudinal data may have a trajectory that is not linear. This is the case in the study of cognitive aging, which presents a faster decline close to death, as well as the process in many other fields such as agriculture [1], pharmacology [2] and marketing [3]. Although some less parsimonious models have been proposed to model such longitudinal data, the use of models with a specific functional form such as the Sigmoidal Mixed Model (SMM) [4] and the Piecewise Mixed Model (PMM) [5] with abrupt or smooth change allow the interpretation of the defined parameters.
\\
The SMM is currently implemented in SAS using PROC NLMIXED [4], which maximizes the marginal likelihood by using an adaptive Gaussian quadrature [6] or other approximation methods, such as the first-order method [7]. However, none of these packages can fit the SMM allowing the inclusion of covariates for all 4 parameters. The PMM is commonly fitted using Bayesian inference and implemented in OpenBugs or WinBUGS, but is also commonly fit in R using the lme4 [8], which maximizes the marginal likelihood by using a Laplace approximation. A recently developed Stochastic Approximation Expectation Maximization (SAEM) algorithm was shown to be more successful [9] and faster [10] to identify the maximum likelihood estimators of non-linear mixed models. This can be implemented directly using the package saemix [11] (version 3.0). The limitation is that both lme4 and saemix are very flexible packages that can fit a wide range of models and because of that they require more analytical skills as they are not a one-line of code. It is worth noting that there are some simple-to-use packages in R that can fit the abrupt PMM, including segmented [12] and rcpm [13]. However, these packages also do not use SAEM and with them, it is not possible to (i) include covariates for all 4 parameters, (ii) consider a smooth polynomial transition, and/or (iii) estimate directly the last level (e.g. level close to death).
\\
In this work, we present the $nlive$ package implemented within R software. The main objective of the package is to facilitate and broaden the application and interpretation of the SMM and PMM for longitudinal data. All needed elements to fit the models have been programmed, including the computation of the structural model and the automatic generation of initials for the main parameters. As such, less experienced R users only need to specify the model to fit via a single intuitive line of code, with only five mandatory arguments. The package was implemented with the most recent and efficient algorithms for non-linear models. Implementation was also performed with the most interpretable parameterization and was based on the most recent developments in each type of model. For example, for the smooth PMM, instead of using the Bacon–Watts [14] which can create an artificial increase in the trajectory right after the changepoint [15], we considered the most recently developed polynomial smooth transition [16]. In the following, we reintroduce these models, describe the implementation of the package, and provide a simulation study to demonstrate the performance of the package. We also demonstrate the use of the model and interpretation of the output using a made-up illsutrative sample dataset with trajectories similar to those observed in cohorts such as the Religious Order Study and the Rush Memory and Aging Project [17].
\section{MODEL SPECIFICATIONS}
\noindent
As a prelude to the introduction and demonstration of the new $nlive$ package, we first describe the general formulation of the nonlinear mixed models implemented in the package. The simplified general form of nonlinear mixed models can be written in terms of a known nonlinear function $f$ given by:
\begin{equation}\label{1}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y_{ij}=f(t_{ij},\psi_i)+\epsilon_{ij}
\end{equation}
\noindent where $y_{ij}$ denotes the longitudinal outcome value of subject $i$ $(i=1, ..., N)$ collected at the observation time $t_{ij}$ $(j=1, ..., n_i)$; $\psi_i$ is a vector of normally distributed person-specific parameters function of fixed effects and individual random effects; and $\epsilon_{ij}$ are random error, with $\epsilon_{ij} \sim N(0, \sigma_{\epsilon}^2)$.
\\
Motivated by the application on late-life cognitive decline, the $nlive$ package implements two main classes of nonlinear mixed models: the Sigmoidal Mixed Model (SMM) [18,4] with four parameters and the Piecewise Linear Mixed Model (PMM) [5] with two linear phases and a single changepoint. In the following sub-sections, we provide a brief introduction of these models. For simplicity, some annotations can be similar from one model to another, while the interpretation of the parameters remains specific to each of them.
\subsection{The Sigmoidal Mixed Model}
The SMM introduced by Capuano and colleagues [4] is based on the four-parameter logistic that allows the inclusion of covariates related to four parametric quantities. The non-linear trajectory of the outcome $Y$ can be formulated as follows:
\begin{equation}
\label{2}
~~~~~~~~~~~~~~~~~~~~~~~~f(t_{ij}, \psi_i) = \psi_{1i} + \frac{\psi_{2i} - \psi_{1i}}{1+(t_{ij}/\psi_3)^{\psi_4}}
\end{equation}
\\
\noindent where the first parameter, $\psi_{1i}$, represents the person-specific initial level of the outcome before the onset of decline. The second parameter, $\psi_{2i}$, represents the person-specific level of the outcome at time equal to zero (e.g., death), or the intercept. We will call it the last level although the meaning of time may differ depending on the application. $\psi_{3}$ represents the marginal time when half of the total decline occurred. We will call it the midpoint. $\psi_{4}$ represents the marginal Hill slope and will define the nonlinear pattern of the trajectory (e.g. determining the steepness, earlier versus later acceleration of change). These two latter parameters are kept as marginal for convergence purposes [4].
The four parameters are assumed to obey the following equations:
\begin{equation}
\label{3}
~~~~~~~~~\mbox{initial level}: \psi_{1i} = \alpha_1 + \beta_1 X_{1i} + \eta_{1i}
\end{equation}
\begin{equation}
\label{4}
~~~~~~~~~\mbox{last level (intercept)}: \psi_{2i} = \alpha_2 + \beta_2 X_{2i} + \eta_{2i}
\end{equation}
\begin{equation}
\label{5}
~~~~~~~~~\mbox{midpoint or time of half decline}: \psi_{3} = \alpha_3 + \beta_3 X_{3i}
\end{equation}
\begin{equation}
\label{6}
~~~~~~~~~\mbox{Hill slope}: \psi_{4} = \alpha_4 + \beta_4 X_{4i}
\end{equation}
\\
\noindent where $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the mean values for the last level, initial level, midpoint, and Hill slope, respectively; $X_{1i}$, $X_{2i}$, $X_{3i}$, and $X_{4i}$ are vectors of covariates associated with the vector of fixed effects $\beta_1$, $\beta_2$, $\beta_3$, and $\beta_4$, respectively; and $\eta_{1i}$ and $\eta_{2i}$ are random effects with $(\eta_{1i}, \eta_{2i})^\top \sim MVN(0, B)$ and $B$ assuming correlations between $\eta_{1i}$ and $\eta_{2i}$.
\subsection{The Piecewise Linear Mixed Model with a Random Changepoint}
The PMM model [5] assumes that the stochastic process of the longitudinal outcome is characterized by two or more different phases. Under this class of models, the $nlive$ package implements two PMM models with an abrupt change (PMM-abrupt) [19] and a smooth polynomial transition (PMMs-mooth) [16] between the two linear phases. These models provide an appealing statistical approach to detect the time when the onset of accelerated decline occurs.
\subsubsection{PMM with abrupt change}
The PMM-abrupt model (also known as the linear-linear or the broken-stick mixed model), consists of an intercept at time zero, a slope close to the intercept, a change point at which the slope changes, and a slope after this change point. The non-linear trajectory of the outcome $Y$ can be formulated as follows:
\begin{equation}\label{7}
~~~~~~f(t_{ij},\psi_i)=
\left\{
\begin{array}{l}
\psi_{1i} + \psi_{2i}\psi_{4i} + \psi_{3i}(t_{ij}-\psi_{4i}) ~~~ \mbox{ if } t_{ij}<\psi_{4i}\\
\psi_{1i} + \psi_{2i}t_{ij} ~~~~~~~~~~~~~~~~~~~~~~~~~ \mbox{ if }t_{ij}\geq\psi_{4i}
\end{array}
\right.
\end{equation}
\\
\noindent where the first parameter, $\psi_{1i}$, represents the person-specific level of the outcome at time zero, or the intercept; $\psi_{2i}$ represents the person-specific slope before the changepoint; $\psi_{3i}$ represents the person-specific slope after the changepoint; and $\psi_{4i}$ represents the person-specific changepoint time parameter.
\\
Assuming an alignment at death for example (for interpretation purposes), the parameters $\psi_{1i}$ to $\psi_{4i}$ are supposed to obey the following equations:
\begin{equation}
\label{8}
~~~~~~~~~\mbox{last level (intercept)}: \psi_{1i} = \alpha_1 + \beta_1 X_{1i} + \eta_{1i},
\end{equation}
\begin{equation}
\label{9}
~~~~~~~~~\mbox{slope before the changepoint}: \psi_{2i} = \alpha_2 + \beta_2 X_{2i} + \eta_{2i},
\end{equation}
\begin{equation}
\label{10}
~~~~~~~~~\mbox{slope after the changepoint}: \psi_{3i} = \alpha_3 + \beta_3 X_{3i} + \eta_{3i},
\end{equation}
\begin{equation}
\label{11}
~~~~~~~~~\mbox{changepoint time}: \psi_{4i} = \alpha_4 + \beta_4 X_{4i} + \eta_{4i}
\end{equation}
\\
where $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the mean values for the last level, the slope before the change point, the slope after the changepoint, and the changepoint time, respectively; $X_{1i}$, $X_{2i}$, $X_{3i}$, and $X_{4i}$ are vectors of covariates associated with the vector of fixed effects $\beta_1$, $\beta_2$, $\beta_3$, and $\beta_4$, respectively; and $\eta_{1i}$ to $\eta_{4i}$ are random effects with $(\eta_{1i}, \eta_{2i}, \eta_{3i}, \eta_{4i})^\top \sim MVN(0, B)$ and $B$ assuming correlations only between $\eta_{2i}$ and $\eta_{3i}$.
\subsubsection{PMM with smooth polynomial transition}
The PMM-smooth model is an extension of the abrupt PMM. The initial smooth PMM was proposed by Bacon and Watts [14] and included a hyperbolic tangent transition. In this work, however, we consider a more recent development that considers a smooth polynomial transition introduced in Van den Hout, Muniz-Terrera, and Matthews [16]. Unlike the Bacon-Watts model, the PMM-smooth enables a direct interpretation of the parameters associated with the two linear phases and allows to model the beginning of the smooth transition – that is the onset of the changepoint – ather than the middle of the smooth transition in the Bacon-Watts model.
\\
In PMM-smooth, the transition is modeled using a third-degree polynomial function fitted between the two straight lines. In the original work [16,20], the intercept parameter cannot be interpreted directly as it reflects the level parameter projection using the early slope at time zero. To allow direct interpretation of the intercept, we re-formulated the PMM-smooth model as:
\begin{equation}\label{12}
f(t_{ij},\psi_i) =
\left\{
\begin{array}{l}
\psi_{1i} + \psi_{2i}t_{ij} + (\psi_{3i}-\psi_{2i})(t_{ij}-{\psi_{4i}}+\frac{v}{2}) ~~~~ \mbox{ if } t_{ij}<{\psi}_{4i}\\
g_{transition} (t_{ij} \mbox{\textbar} \psi_{1i},\psi_{2i},\psi_{3i},v) ~~~~~~~~~~~~~~~~~~~ \mbox{ if } {\psi}_{4i}\leq t_{ij}\leq {\psi}_{4i}+v\\
\lambda_{i}+\psi_{2i}t_{ij} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \mbox{ if }t_{ij}>{\psi}_{4i}+v
\end{array}
\right.
\end{equation}
where $\psi_{1i}$, $\psi_{2i}$, and $\psi_{3i}$ have been previously defined for Equation \eqref{7}. ${\psi}_{4i}$ is the person-specific time when the smooth transition phase of length $v$ begins. $v$ is a value representing the time interval where the polynomial curve occurs between $t_{ij}$ = ${\psi}_{4i}$ and $t_{ij}$ = ${\psi}_{4i}$ + $v$. In order to be closer to the PMM-abrupt, the two linear parts should intersect at the middle of the transition phase and the constraint $\lambda_i=\psi_{1i}+\psi_{2i}({\psi_{4i}}+\frac{v}{2})-\psi_{3i} ({\psi}_{4i}+\frac{v}{2})$ is imposed. Note that $v$ set to 0 reduces to a PMM-abrupt model.
\\
The smoothness of the transition function involves four linear equations with four parameters:
\begin{equation}\label{13}
~~~~~~~~~~~~~~g_{transition}({\psi}_{4i}) = \lambda_i+\psi_{3i} {\psi}_{4i}
\end{equation}
\begin{equation}\label{14}
~~~~~~~~~~~~~~g_{transition}({\psi}_{4i}+v) = \psi_{1i}+ \psi_{2i} ({\psi}_{4i}+v)
\end{equation}
\begin{equation}\label{15}
~~~~~~~~~~~~~~(\frac{\partial}{\partial{t_{ij}}}g_{transition})({\psi}_{4i}) = \psi_{3i}
\end{equation}
\begin{equation}\label{16}
~~~~~~~~~~~~~~(\frac{\partial}{\partial{t_{ij}}}g_{transition})({\psi}_{4i}+v) = \psi_{2i}
\end{equation}
\\
where $g_{transition}$ is obtained by solving the system of four linear equations with four unknown parameters. The derivatives of $g_{transition}$ at the times $t_{ij}={\psi}_{4i}$ and $t_{ij}={\psi}_{4i}+v$ are respectively $\psi_{3i}$ and $\psi_{2i}$.
\section{IMPLEMENTATION}
\subsection{Software}
To facilitate the application and interpretation of the SMM, PMM-abrupt, and PMM-smooth models for a broader audience, who is not necessarily familiar with statistical programming, we developed a user-friendly R package called “nlive” (\textbf{n}on-\textbf{l}inear mixed models with \textbf{i}nitial \textbf{v}alues \textbf{e}stimated) with R version 4.0.3. All needed elements to fit the models, including the definition of the structural model and the generation of initials for the four main parameters, have been programmed so that the user only need to specify a single intuitive line of code to obtain estimations in both tabular and graphical formats. A variety of options can also be specified.
\\
The $nlive$ package was built on top of several existing R packages and is freely available via the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=nlive. A made-up dataframe, under the name “dataset”, is provided with the package and was generated to mimic cognitive decline patterns observed before death in the Religious Order Study and the Memory and Aging Project (ROSMAP) from the Rush Alzheimer’s Disease Center [17]; this dataset is used in the Example Section.
\subsection{Estimation}
The SMM, PMM-abrupt, and PMM-smooth previously described were all estimated using the $saemix$ package (version 3.0) developed by Comets and colleagues [11]. The $saemix$ package, among other things, implies the definition of the structural model; thus, first, we programmed the structures of the models. For SMM, we relied on the $SSlogis5()$ function of the $nlraa$ package [21] (version 1.2), which initially defines a 5-parameter logistic curve but can be reduced to a 4-parameter logistic when the $5^{th}$ parameter is fixed to 1. For PMM-abrupt and PMM-smooth, since the specification of these models was not available in any existing package, we coded it explicitly. Note that, as $saemix$ only provides p-values for the effects of the covariates, and not for the main parameters, the calculation of these p-values has also been programmed.
\subsection{The SAEM algorithm}
The computational technique for maximum likelihood estimation implemented in $saemix$ is the Stochastic Approximation Expectation Maximization (SAEM) algorithm, which is a stochastic approximation version of the standard EM algorithm proposed by Khuhn and Lavielle [22]. The SAEM algorithm showed to be efficient in the context of non-linear mixed models, converging quickly to the maximum likelihood estimators [10] and achieving better performance than linearization-based algorithms [9]. In preliminary testing during the algorithm coding process, in line with the literature, $saemix$ showed convergence to the adequate solution more often than two main competing software package [11]: $nlme$ [6] and $lme4$ [8].
\subsection{Initial values}
The SAEM algorithm requires that the four main parameters of the models are specified, namely $\hat{\alpha_1}, \hat{\alpha_2}, \hat{\alpha_3}$, and $\hat{\alpha_4}$. By choosing values close to the maximizer, the number of iterations needed to reach convergence should be reduced. Thus, in $nlive$, for each model, we have designed an algorithm that generates informative initial values informed by the input dataset.
\\
For SMM, the four main parameters to be specified reflect the last level, first level, midpoint, and Hill slope. These starting values were automatically computed building upon the algorithm developed in SAS by Capuano and colleagues [4] (algorithm hosted and accessible at github.com/AWCapuano/sigmoidal). This SAS algorithm was based on the original motivating case of longitudinal cognitive data and we generalized it here for other types of data (e.g., different time scales). Briefly, the initial and final levels of the outcome are informed by the average levels observed at the $5^{th}$ and $95^{th}$ percentiles of the time distribution. The time of half decline is set to 300 if the curve is nearly linear, and to 2 otherwise. Finally, the Hill slope is set to a high and low value based (0.5 and 1.05).
\\
Similarly, for PMM-abrupt and PMM-smooth, estimation of the models requires the specification of four starting values related to the four main parameters: last level, slope before the changepoint, slope after the changepoint, and the changepoint time. The last level is informed by the mean level observed at the 95th percentile of the time distribution. For the other three parameters, we first estimate where the acceleration of decline (i.e., changepoint) occurred approximately over time by estimating five separate standard linear mixed models, each considering a subset of cognitive measures collected every 20 percentiles of the time distribution (i.e., [0, 20th percentile[, [20th percentile[, etc.). Then, the changepoint time was defined as the lower bound of the time interval where the fastest slope occurred. Lastly, the early and final slopes are informed by the slope of cognitive decline estimated using a linear mixed model considering the subsets of cognitive measures collected before and after the approximated changepoint, respectively. The linear mixed models used were implemented using the hlme function from the {lcmm} [23] package (version 1.9.5) to fit mixed effect models on segments of the longitudinal data. Although one of the great advantages of the nlive package is that users do not need to enter initials values, the option $start$ allows for manual entry of initials values in a very user-friendly way.
\section{OVERVIEW OF THE PACKAGE}
\noindent
The principal function of the package is $nlive()$. This function fits the model requested through the option "model=1" for SMM, "model=2" for PMM-abrupt, and "model=3" for PMM-smooth, and require to take as input a dataset that provides information on the longitudinal outcome of interest, participant ID, time, and predictors (if any). The call to the $nlive$ function is:
\begin{lstlisting}
nlive(model, dataset, ID, outcome, time,
predictor.all = NULL,
predictor.par1 = NULL,
predictor.par2 = NULL,
predictor.par3 = NULL,
predictor.par4 = NULL,
start = NULL,
traj.marg = FALSE,
traj.marg.group = FALSE)
\end{lstlisting}
\noindent The first five arguments are mandatory, while all the others have default values. Below is a brief description of the arguments:
\begin{itemize}
\item {$model$}: indicator of the model to fit (1=SMM, 2=abrupt PMM, 3=smooth PMM).
\item {$dataset$}: data frame containing the variables named in {$ID$}, {$outcome$}, {$time$}, {$predictor.all$}, and {$predictor.par1$} to {$predictor.par4$}.
\item {$ID$}: name of the variable representing the grouping structure specified with " (e.g., “ID” representing the unique identifier of participants).
\item {$outcome$}: name of the time-varying variable representing the longitudinal outcome specified with " (e.g., “outcome”)
\item {$time$}: name of the variable representing the timescale specified with " (e.g., “time”). Can be negative or positive. Note that model 1, SMM, will always report a positive value under the midpoint parameter (e.g. if the time in the data goes from 0 down to -10 and the time of the midpoint is -2, the model will report a midpoint of 2).
\item {$predictor.all$}: optional vector indicating the name of the variable(s) that the four main parameters of the model will be adjusted to (e.g. {$predictor.all$ = $c("X1","X2")$}). Default to NULL.
\item {$predictor.par1$}: optional vector indicating the name of the variable(s) that the first main parameter of the model will be adjusted to (e.g. {$predictor.all$ = $c("X1","X2")$}). For model 1, the first parameter = last level. For models 2 and 3, first parameter = intercept. Default to NULL.
\item {$predictor.par2$}: optional vector indicating the name of the variable(s) that the second main parameter of the model will be adjusted to (e.g. {$predictor.all$ = $c("X1","X2")$}). For model 1, the second parameter = initial level. For models 2 and 3, second parameter =~slope1 (slope before the change-point). Default to NULL.
\item {$predictor.par3$}: optional vector indicating the name of the variable(s) that the third main parameter of the model will be adjusted to (e.g. {$predictor.all$ = $c("X1","X2")$}). For model 1, the third parameter = midpoint. For models 2 and 3, third parameter = slope2 (slope between intercept and changepoint). Default to NULL.
\item {$predictor.par4$}: optional vector indicating the name of the variable(s) that the fourth main parameter of the model will be adjusted to (e.g. {$predictor.all$ = $c("X1","X2")$}). For model 1, the fourth parameter is the Hill slope. For models 2 and 3, the fourth parameter is the changepoint. Default to NULL.
\item {$start$}: optional vector to override the specification of the four initial values for the main parameters. For model 1, the values must be included in the following order: last level, initial level, midpoint, Hill slope. For models 2 and 3, the values must be included in the following order: intercept, slope1 (slope before the changepoint), slope2 (between intercept and the changepoint), and changepoint. Default to NULL.
\item {$traj.marg$}: logical indicating if the marginal estimated trajectories should be plotted. Default to FALSE.
\item {$traj.marg.group$}: name of the grouping variable listed in one of the predictor arguments to plot and contrast the estimated marginal trajectories between two specific groups. If the variable is binary, the trajectories are contrasted between the two groups of interest. If the variable is continuous, the $10^{th}$ and $90^{th}$ percentile values will automatically be considered. Default to NULL.
\end{itemize}
\noindent The list of options that can be set is described in detail in the help files provided with the package (https://CRAN.R-project.org/package=nlive).
\section{RESULTS}
\subsection{Performance}
We employed a simulation study to evaluate the convergence adequacy of the SMM and PMM models fitted using the SAEM algorithm. For all the models, we examined the convergence rate and the average computation times, after considering an increasing number of individuals (i.e., n=100, 200, and 500) and an increasing number of predictors for each of the 4 parameters (i.e., zero, one, and two covariates). We simulated data close to the longitudinal cognitive trajectories observed before death in ROSMAP [17]. Decedent-specific visit times were generated using a uniform distribution in [–2, 2] months around theoretical annual visits from –24 years to death (year 0) over an average follow-up of 10 years (SD=5); individuals had to have at least 4 cognitive observations.
\\
To evaluate the SMM estimation, the longitudinal cognitive response was generated using a flexible sigmoidal structure. The marginal cognitive trajectory was characterized by an early, progressive decline over time and a moderate acceleration close to time 0, with $\psi_1$ (last level) = -1.03, $\psi_2$ (initial level) = 0.37, $\psi_3$ (midpoint) = -4.0, and $\psi_4$ (Hill slope) = 1.69; a variance of 2.13 ($\sigma_{\eta_1}$=1.46) and 0.26 ($\sigma_{\eta_2}$=0.51) for the random effects $\eta_1$ and $\eta_2$, respectively; a correlation of zero between random effects; and an error variance of 0.08 ($\sigma_{\epsilon}$=0.28). For the PMM models, the longitudinal cognitive response was generated using a piecewise linear structure with $\psi_1$ (last level) = -1.21, $\psi_2$ (slope before changepoint) = -0.03, $\psi_3$ (slope between intercept and changepoint) = -0.32, and $\psi_4$ (changepoint) = -3; a variance of 2.13 ($\sigma_{\eta_1}$= 1.70), 0.26 ($\sigma_{\eta_2}$=0.01), 2.13 ($\sigma_{\eta_3}$=0.05), and 0.26 ($\sigma_{\eta_4}$=6.91) for the random effects $\eta_1$, $\eta_2$, $\eta_3$, and $\eta_4$, respectively; a correlation of -0.07 between the random effects $\eta_2$ and $\eta_3$; and an error variance of 0.08 ($\sigma_{\epsilon}$=0.28). For each Scenario, we generated 500 datasets.
\\
For each model, we found that the convergence rate was excellent (100\%). Mean Squared Errors from the average of the estimated marginal cognitive trajectories ranged from 0.07 to 0.02. Figure \ref{Figure1} displays the computation time for a varying sample size stratified by the number of predictors considered for all parameters. We observed that run times generally increased with the sample size and that there is a gap between the models with no covariates and those adjusted. Additionally, we found that compared with PMMs, the SMM model generally required more time to converge, although this time was reasonable on average (see Fig. 1). Of note, the models were fitted on a HP ProBook 400 G6 containing an i7-8565U processor and 16 gigabytes of RAM running R version 4.0.3. Together, these profiling results support that the application of the SAEM algorithm to fit SMM, PMM-abrupt, and PMM-smooth is efficient.
\begin{figure}
\caption{Evolution of computation times with the number of individuals, stratified by the number of covariates considered for all the 4 parameters.}
\label{Figure1}
\end{figure}
\subsection{Example}
In this section, we show how the $nlive()$ function can be used to fit the SMM, PMM-abrupt, and PMM-smooth models, and we present the main outputs provided by the package. In the context of our motivating application, late-life cognitive decline, each model were fitted using the made-up illustrative sample $dataset$ available in the package. Thus, the first step consists in loading $nlive$, which will automatically load $dataset$.
\begin{footnotesize}
\begin{lstlisting}
R > library(nlive)
\end{lstlisting}
\end{footnotesize}
\subsubsection{Data}
The $dataset$ contains 1200 individuals with annual cognitive testing for at least 4 years until death (mean follow-up=7 [SD=5] years); description of the data can be accessed via the command $summary(dataset)$. On each line, we can read the unique participant identifier ($ID$), the negative retrospective time before death in years ($time$), the repeated values of the composite score of global cognition collected over time ($cognition$), and the age at death of individuals in years; in the natural scale ($ageDeath$) and centered at its mean ($ageDeath$ -- 90 = $ageDeath90$) for interpretation purposes. The following lines create the continuous $ageDeath90$ variable and display the first lines of $dataset$:
\begin{footnotesize}
\begin{lstlisting}
R > dataset$ageDeath90 <- dataset$ageDeath - 90
R > head(dataset)
ID time cognition ageDeath ageDeath90
1 1000 -10.00 0.45 91 1
2 1000 -9.08 0.27 91 1
3 1000 -8.04 0.19 91 1
4 1000 -6.82 0.15 91 1
5 1000 -5.99 0.05 91 1
6 1000 -4.98 0.15 91 1
\end{lstlisting}
\end{footnotesize}
\subsubsection{Modeling the SMM}
For demonstration purposes, we fit a relatively simple SMM model with all main parameters adjusted for $ageDeath90$. The user only needs to specify the name of the dataframe and the columns containing the participant ID, the response, the timescale, and predictor. We also include arguments to plot the marginal estimated trajectories before death.
\begin{footnotesize}
\begin{lstlisting}
R > smm.fit <- nlive(model = 1, dataset = dataset, ID = "ID",
+ outcome = "cognition",
+ time = "time",
+ predictor.all = c("ageDeath90"),
+ traj.marg = TRUE,
+ traj.marg.group = c("ageDeath90"))
\end{lstlisting}
\end{footnotesize}
\noindent The first output automatically provided by the package is a spaghetti plot of observed individual cognitive trajectories before death for 70 individuals randomly selected (see Fig. 2). This Figure is produced with the $ggplot2$ package [24] and allows to better appreciate the variability of the trajectories within each individual and between individuals over time. Label of axes corresponds to the raw names of covariates. The options $plot.xlabel$ and $plot.ylabel$ allows the user to specify a character string to define axes x and y, respectively. For example: $plot.xlabel$ = $c("Years$ $before$ $death")$. The list of options that can be set is described in detail in the help files provided with the package.
\begin{figure}
\caption{Observed individual trajectories of global cognition in the 20 years before death for 70 individuals randomly selected in the made-up illustrative sample $dataset$ available in the $nlive$ package.}
\label{Figure2}
\end{figure}
\noindent The second output automatically generated corresponds to the general output from $saemix$, which is a summary of the data and the model, followed by the numerical results, including parameter estimates, their standard errors and several statistical criteria [11]. We also added a line that specifies the processing time of the program.
\begin{footnotesize}
\begin{lstlisting}
...
----------------------------------------------------
----------- Variance of random effects -----------
----------------------------------------------------
Parameter Estimate SE CV(
last.level omega2.last.level 1.283 0.0556 4.3
first.level omega2.first.level 0.146 0.0071 4.9
covar cov.last.level.first.level 0.049 0.0143 28.9
----------------------------------------------------
------ Correlation matrix of random effects ------
----------------------------------------------------
omega2.last.level omega2.first.level
omega2.last.level 1.00 0.11
omega2.first.level 0.11 1.00
----------------------------------------------------
--------------- Statistical criteria -------------
----------------------------------------------------
Likelihood computed by linearisation
-2LL= 9732.152
AIC = 9756.152
BIC = 9817.233
Likelihood computed by importance sampling
-2LL= 9731.84
AIC = 9755.84
BIC = 9816.921
----------------------------------------------------
Parameter Estimate SE p-value
1 last.level -1.088 0.035 P<.0001
2 beta_ageDeath90(last.level) -0.061 0.004 P<.0001
3 first.level 0.24 0.015 P<.0001
4 beta_ageDeath90(first.level) -0.044 0.002 P<.0001
5 midpoint 2.567 0.034 P<.0001
6 beta_ageDeath90(midpoint) 0.031 0.004 P<.0001
7 hill.slope 1.789 0.04 P<.0001
8 beta_ageDeath90(hill.slope) 0.007 0.005 0.081
9 error 0.279 0.002 P<.0001
----------------------------------------------------
The program took 346.51 seconds
\end{lstlisting}
\end{footnotesize}
\noindent The fitted SMM model indicates that higher age at death was associated with lower cognitive level at baseline (see term {beta\_ageDeath90(first.level)}) and close to death (see term {beta\_ageDeath90(last.level)}). In addition, higher age at death was associated with an earlier half of cognitive decline (see term {beta\_ageDeath90(midpoint)}). However, age at death was not associated with the Hill slope(see term {beta\_ageDeath90(hill.slope)}). It is important to note here that the "midpoint" and "hill.slope" parameters have an interpretation in terms of the positive number of years before death as SMM models can only be fitted considering a positive timescale. Negative timescales, like in this example, are automatically converted in the function to positive time for model fitting purposes.
\\
To facilitate the interpretation of the estimated parameters, it is convenient to visualize the estimated average trajectories over time. In $nlive()$, users can easily plot two types of marginal estimated trajectories. First, by setting up the argument $traj.marg=T$, the function can provide a graph of the estimated marginal trajectory of global cognition before death in the whole study sample, for the most common profile of covariates (see Fig. 3). In this example, this would represent the most common average age at death (i.e., 90 years). Second, by specifying $traj.marg.group=c("ageDeath90")$, the function can provide a plot of estimated marginal trajectories of global cognition contrasted between two groups corresponding to participants in the $10^{th}$ versus $90^{th}$ percentile of the $ageDeath90$ distribution, for the most common profile of covariates (see Fig. 4). Users can manually specify the percentile values using the $traj.marg.group.val$ option. For example, $traj.marg.group.val=c(0.25,0.75)$ will plot trajectories for the $25^{th}$ and $75^{th}$ percentiles, respectively.
\noindent Lastly, since $nlive()$ fits the models based on the $saemix$ package, users can take advantage of many $saemix$ functions available to extract specific information from the model object $smm.fit$. Generic functions include, for example, $psi(smm.fit, type="mean")$, which extracts the subject-specific predictions; $eta(smm.fit, type="mean")$, which extracts the subject-specific random effects; $plot(smm.fit, plot.type="convergence")$, which provides the convergence plots; and $plot(pmm.abrupt.fit, plot.type="individual.fit", ilist=c(1:9)$, which plots the observed and predicted values of the first nine subject. All the $saemix$ functions are available in the online help.
\begin{figure}
\caption{Estimated marginal trajectory of global cognition before death, using the Sigmoidal Mixed Model.}
\label{Figure3}
\end{figure}
\begin{figure}
\caption{Estimated marginal trajectory of global cognition before death, according to age at death, using the Sigmoidal Mixed Model.}
\label{Figure4}
\end{figure}
\subsection{Modeling the abrupt PMM}
For PMM-abrupt, the call of $nlive()$ is:
\begin{footnotesize}
\begin{lstlisting}
R > pmm.abrupt.fit <- nlive(model = 2, dataset = dataset, ID = "ID",
+ outcome = "cognition",
+ time = "time",
+ predictor.all = c("ageDeath90"),
+ traj.marg = TRUE,
+ traj.marg.group = c("ageDeath90"))
\end{lstlisting}
\end{footnotesize}
\noindent The general summary output is:
\begin{footnotesize}
\begin{lstlisting}
...
----------------------------------------------------
----------- Variance of random effects -----------
----------------------------------------------------
Parameter Estimate SE CV(
last.level omega2.last.level 1.07196 4.7e-02 4.4
slope1 omega2.slope1 0.00062 7.4e-05 11.9
slope2 omega2.slope2 0.03830 2.0e-03 5.2
changepoint omega2.changepoint 0.58980 7.9e-02 13.4
covar cov.slope1.slope2 0.00378 3.2e-04 8.4
----------------------------------------------------
------ Correlation matrix of random effects ------
----------------------------------------------------
omega2.last.level omega2.slope1 omega2.slope2
omega2.last.level 1 0.00 0.00
omega2.slope1 0 1.00 0.78
omega2.slope2 0 0.78 1.00
omega2.changepoint 0 0.00 0.00
omega2.changepoint
omega2.last.level 0
omega2.slope1 0
omega2.slope2 0
omega2.changepoint 1
----------------------------------------------------
--------------- Statistical criteria -------------
----------------------------------------------------
Likelihood computed by linearisation
-2LL= 12349.9
AIC = 12377.9
BIC = 12449.16
Likelihood computed by importance sampling
-2LL= 12296.83
AIC = 12324.83
BIC = 12396.1
----------------------------------------------------
Parameter Estimate SE p-value
1 last.level -1.103 0.031 P<.0001
2 beta_ageDeath90(last.level) -0.062 0.004 P<.0001
3 slope1 -0.017 0.002 P<.0001
4 beta_ageDeath90(slope1) -0.0003 0.0004 0.082
5 slope2 -0.249 0.007 P<.0001
6 beta_ageDeath90(slope2) -0.001 0.001 0.159
7 changepoint -4.25 0.048 P<.0001
8 beta_ageDeath90(changepoint) -0.059 0.006 P<.0001
9 error 0.281 0.002 P<.0001
----------------------------------------------------
The program took 168.58 seconds
\end{lstlisting}
\end{footnotesize}
\noindent In this example, for the PMM-abrupt model, we found that each additional year of age at death was associated with worse mean cognitive level close to death (see term {beta\_ageDeath90(last.level)}). In addition, each increment in the age at death was related to an earlier onset of accelerated decline (see term {beta\_ageDeath90(changepoint)}). However, age at death was not related to the preterminal decline (see term {beta\_ageDeath90(slope1)}) or terminal decline (see term {beta\_ageDeath90(slope2)}). The marginal estimated trajectories in the whole study sample and in the $10^{th}$ versus $90^{th}$ percentiles of the age at death distribution are displayed in Figure 5 and Figure 6, respectively.
\begin{figure}
\caption{Estimated marginal trajectory of global cognition before death, using the Piecewise Mixed Model with abrupt change.}
\label{Figure5}
\end{figure}
\begin{figure}
\caption{Estimated marginal trajectory of global cognition before death, according to age at death, using the Piecewise Mixed Model with abrupt change.}
\label{Figure6}
\end{figure}
\subsection{Modeling the smooth PMM}
For PMM-smooth, the call of $nlive()$ is:
\begin{footnotesize}
\begin{lstlisting}
R > pmm.smooth.fit <- nlive(model = 3, dataset = dataset, ID = "ID",
+ outcome = "cognition",
+ time = "time",
+ predictor.all = c("ageDeath90"),
+ traj.marg = TRUE,
+ traj.marg.group = c("ageDeath90"))
\end{lstlisting}
\end{footnotesize}
\noindent The general summary output is:
\begin{footnotesize}
\begin{lstlisting}
...
----------------------------------------------------
----------- Variance of random effects -----------
----------------------------------------------------
Parameter Estimate SE CV(
last.level omega2.last.level 1.0699 4.7e-02 4.4
slope1 omega2.slope1 0.0006 7.3e-05 12.1
slope2 omega2.slope2 0.0377 2.0e-03 5.2
changepoint omega2.changepoint 0.6037 8.1e-02 13.3
covar cov.slope1.slope2 0.0038 3.1e-04 8.3
----------------------------------------------------
------ Correlation matrix of random effects ------
----------------------------------------------------
omega2.last.level omega2.slope1 omega2.slope2
omega2.last.level 1 0.00 0.00
omega2.slope1 0 1.00 0.79
omega2.slope2 0 0.79 1.00
omega2.changepoint 0 0.00 0.00
omega2.changepoint
omega2.last.level 0
omega2.slope1 0
omega2.slope2 0
omega2.changepoint 1
----------------------------------------------------
--------------- Statistical criteria -------------
----------------------------------------------------
Likelihood computed by linearisation
-2LL= 12357.39
AIC = 12385.39
BIC = 12456.65
Likelihood computed by importance sampling
-2LL= 12293.15
AIC = 12321.15
BIC = 12392.41
----------------------------------------------------
Parameter Estimate SE p-value
1 last.level -1.099 0.031 P<.0001
2 beta_ageDeath90(last.level) -0.062 0.004 P<.0001
3 slope1 -0.017 0.002 P<.0001
4 beta_ageDeath90(slope1) -0.0003 0.0004 0.082
5 slope2 -0.246 0.007 P<.0001
6 beta_ageDeath90(slope2) -0.001 0.001 0.159
7 changepoint -5.3 0.049 P<.0001
8 beta_ageDeath90(changepoint) -0.058 0.006 P<.0001
9 error 0.281 0.002 P<.0001
----------------------------------------------------
The program took 134.25 seconds
\end{lstlisting}
\end{footnotesize}
\noindent In this example, as expected, findings are generally similar to those obtained for the PMM-abrupt model. The main difference is that the estimated changepoint parameter represents here the beginning of the transition period. Marginal estimated trajectories in the whole study sample and according to age at death are displayed in Figure 7 and Figure 8, respectively.
\begin{figure}
\caption{Estimated marginal trajectory of global cognition before death, using the Piecewise Mixed Model with smooth polynomial transition.}
\label{Figure7}
\end{figure}
\begin{figure}
\caption{Estimated marginal trajectory of global cognition before death, according to age at death, using the Piecewise Mixed Model with smooth polynomial transition.}
\label{Figure8}
\end{figure}
\section{CONCLUDING REMARKS}
\noindent
In this work, we introduce a newly developed R package $nlive$ to fit three non-linear mixed models for Gaussian longitudinal data: the sigmoidal mixed model (SMM) and two piecewise linear mixed models with a random changepoint (PMM-abrupt and PMM-smooth). The SMM includes 4 parameters, which allow for estimation of early level, half of the decline, Hill slope (the steepness of the curve), and final level of the longitudinal outcome of interest. The two PMM separate the trajectory into two linear phases and allow for estimation of the early slope, changepoint, final slope, and final level. These models were chosen for the implementation as they currently cannot be easily implemented in R and are of importance, especially in aging research. All needed pieces such as functions, covariance matrices, and initials generation were programmed. The $nlive()$ function allows fitting these models with one line of code that is intuitive enough to the less sophisticated users. The yielding product has only five mandatory arguments. Options are available to readily accommodate user preferences, including manual specification of starting values or diagnostic plots. It was also designed to help interpretation of the output by providing features such as annotated output, warnings (e.g. small sample, number of covariates), and graphs.
\\
This package is the first to provide a seamless user interface to fit the Sigmoidal Mixed Effect Model. Some packages in R can fit the Sigmoid curve but not the mixed effect model. Most of these packages focus on dose-response optimization and curve-fitting [25] such as $qpcR$ [26], $grofit$ [27], $FlexParamCurve$ [28], $drfit$ [29], and $MCPMod$ [30] or aim to automated fitting and classify multiple curves [25]. Although this is not the first package to provide an interface to fit the PMM, the $nlive$ includes the more recent developments in the model structure and the likelihood maximization algorithm. The smooth PMM implemented is based on the polynomial transition that was demonstrated to have improved properties over to the Bacon-Wats. PMM models were also reparameterized which allows the interpretation based on the estimated value at time zero and not a projection to zero from the first and more distant slope. Here we build upon recently developed tools in R such as the $saemix$ package that utilizes the Stochastic Approximation EM-based algorithm, shown in several tests to have a better convergence rate than the Maximum Likelihood. All models fit with the same algorithm. Extensive testing of functionality was already performed for $saemix$ development. In this interface, however, convergence adequacy was tested given the particular complexity of these models. Overall the convergence rate was high, the time was reasonable, and the bias was low.
\\
The motivation of this package was aging research including biomarkers of the Alzheimer’s pathological cascade (a.k.a. Jack curves) [31], natural history of cognition [4,32,33], retesting effect [34,35], and terminal decline [36,37]. These models, however, are non-specific and the {nlive} can be used in a wide variety of fields. Many processes were demonstrated to follow a sigmoid trajectory over time (a.k.a. 3 to 5 parameters logistic, Hill, Langmuir, Langmuir–Hill, and Hill–Langmuir equation). Such processes are found in agriculture [1], pharmacology [2] and marketing [3], to cite a few. Similarly, many processes that are initially linear may have an unknown change that may modify the trajectory. Such processes are found in a wide variety of fields from environmental sciences [38] to engineering [39].
\\
In conclusion, we hope that this very user-friendly package will encourage the adoption of more sophisticated models for longitudinal data by the R community, with varying degrees of experience. Although illustrated in the context of cognitive aging, the package can be used in a wide variety of applications.
\section*{Ethics approval and consent to participate}
\noindent Not applicable.
\section*{Consent for publication}
\noindent Not applicable.
\section*{Availability of data and materials}
\noindent The R package nlive can be installed directly using install.packages("nlive") in an R console. Archived versions are available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=nlive. The made-up illustrative dataset analysed during the current study is bundled with the R package nlive, and can be accessed by running the command data(dataset, package = "nlive").
\section*{Competing interests}
\noindent The authors declare that they have no competing interests.
\section*{Funding}
\noindent The study was supported by NIA grants Grants (R01AG 022018, P30AG072975, R01AG17917) and the Illinois Department of Public Health. Dr. Maude Wagner is supported by a post-doctoral fellowship from the French Foundation for Alzheimer’s Research (alzheimer-recherche.org). The funding organizations had no role in the design or conduct of the study; the collection, management, analysis, or interpretation of the data; or the writing of the report or the decision to submit it for publication.
\section*{Abbreviations}
\noindent CRAN, the comprehensive R archive newtwork
\\PMM, piecewise mixed model
\\SAEM, stochastic approximation expectation maximization
\\SMM, sigmoidal mixed model
\section*{Authors' contributions}
\noindent AWC was responsible for the study conception and design. AWC and MW were involved in the analysis and interpretation of the data, drafting of the manuscript and critically revised the manuscript. MW performed the simulations. All authors take responsibility for the integrity of the data and the accuracy of the data analysis. All authors read and approved the final manuscript.
\section*{REFERENCES}
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10. Delyon B, Lavielle M, Moulines E. Convergence of a stochastic approximation version of the EM algorithm. Annals of statistics. 1999:94-128.
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\end{document}
|
\begin{document}
\title{Triply-resonant Optical Parametric Oscillator by Four-wave Mixing with Rubidium Vapor inside an Optical Cavity}
\author{Xudong Yu$^{1}$, Min Xiao$^{2}$, and Jing Zhang$^{1\dagger}$}
\affiliation{$^{1}$The State Key Laboratory of Quantum Optics
Quantum Optics Devices, Institute of Opto-Electronics, Shanxi
University, Taiyuan 030006, P.R. China}
\affiliation{$^{2}$ Department of Physics, University of Arkansas,
Fayetteville, Arkansas 72701, USA}
\begin{abstract}
We present an experimental demonstration of simultaneous
above-threshold oscillations of the Stokes and anti-Stokes fields
together with the single pumping beam with rubidium atoms inside an
optical standing-wave cavity. The triple resonant conditions can be
achieved easily by making use of the large dispersions due to
two-photon transitions in the three-level atomic system. This work
provides a way to achieve high efficient nonlinear frequency
conversion and the generated bright Stokes and anti-Stokes cavity
output beams are potential resource for applications in quantum
information science.
\end{abstract}
\maketitle
Various atomic systems have been used as intracavity gain media to
realize cavity oscillations
\cite{one,two,three,four,five,six,seven}. Especially, multi-level
atomic systems have more interesting characteristics and can be more
efficient in managing the absorption and dispersion properties of
the intracavity gain medium for building up resonance simultaneously
of different cavity modes. It is easy to have an off-resonant pump
beam to make the Stokes or anti-Stakes field be on resonance with
one of the cavity modes in a three-level $\Lambda$-type atomic
system, and get it to oscillate with a large pump power
\cite{seven}. When the pump beam is tuned near one of the atomic
transitions in a three-level $\Lambda$ system inside an optical
cavity, cavity field oscillation (or lasing without population
inversion) can occur in the frequency corresponding to the other
atomic transition \cite{Wu}. This system can be considered as an
optical parametric oscillator (OPO) since the atomic variables can
be adiabatically eliminated in treating the atom-field interactions
\cite{Guzman,Xiong}.
In this Letter, we present our experimental demonstration of a
triply-resonant atom-cavity system, as shown in Fig.1. With one pump
laser beam (frequency $\omega_{p}$) tuned to the cavity resonance as
the cavity input, two optical fields, both the Stokes field
(frequency $\omega_{b}$) and anti-Stokes field (frequency
$\omega_{a}$), are generated simultaneously. The frequency of the
pump beam is detuned from the atomic resonances
$|1\rangle\leftrightarrow|0\rangle$ and
$|2\rangle\leftrightarrow|0\rangle$ by the amount of
$\Delta_{b}=\omega_{p}-\omega_{01}$ and
$\Delta_{a}=\omega_{p}-\omega_{02}$ (thus
$\Delta_{a}-\Delta_{b}=\omega_{12}$), which generates two sidebands
at frequencies of $\pm\omega_{12}$ from the pump beam frequency,
respectively. Typically, the generated Stokes and anti-Stokes fields
are difficult to be made on resonance with the cavity modes at the
same time, since the cavity mode has already been tuned to be on
resonance with the pump field. However, since we work with the
naturally mixed $^{87}Rb$ and $^{85}Rb$ vapor cell as the
intracavity medium, there are several broad absorption bands in the
transmission spectrum, as shown in Fig. 2(a). The weak field cavity
transmission spectrum is given in Fig. 2(b), which shows the cavity
transmission peaks with modified peak separations caused by the
enhanced dispersions associated with the tails of the absorption
lineshape at high atomic density \cite{Yu}. By manipulating the
large intracavity dispersions via pump laser frequency detuning, the
Stokes and anti-Stokes fields can be made to be simultaneously on
resonance with the modified cavity modes together with the pump
field, which is very similar to the case of a triple-resonant OPO
with nonlinear crystals
\cite{Fabre,Villar,Villar1,Villar2,Coelho,Dauria,Su,Laurat}.
\begin{figure}
\caption{ (Color online). Schematic of the experimental setup of the
coupled three-level atoms-cavity system. $\lambda /4$: quarter-wave
plate; D1, D2: detectors; HV-AMP: high voltage amplifier; PZT:
piezoelectric transducer; PBS: polarized beam splitter; $L$: optical
lens; F-P: Febry-Perot cavity. Bottom: relevant energy levels and
field configurations. $\omega_{a}
\label{Fig1}
\end{figure}
The experiment is done by placing a naturally mixed rubidium vapor
cell inside an optical standing-wave cavity of length 17.7 $cm$. The
cavity is composed of two curved mirrors with the same radius of
curvature of 100 $mm$. The reflectivity is $90\%$ at 780 $nm$ for
the input coupler $M1$ (on right), which is mounted on a PZT to
adjust the cavity length. The left mirror $M2$ has a reflectivity of
$99.5\%$ at 780 $nm$. The finesse of the cavity (including the
losses of two faces of the atomic cell) is about $F=20$. The length
of the vapor cell is 7.5 $cm$. Thus we may obtain that the cavity
bandwidth (the half-width at half maximum for the cavity) is about
21 $MHz$ and the OPO efficiency (output coupling over total losses)
about $20\%$. The temperature of the vapor cell can be controlled by
a heater. A beam from a grating-stabilized diode laser is injected
into a tapered amplifier (TA). The high power output from the TA
then passes a standard polarization maintaining single-mode fiber,
which is used as the cavity pump beam with an input power of 100
$mW$. The pump laser beam with the spatial mode filter by the
optical fiber is easy to be mode-matched to the $TEM_{00}$ mode of
the optical cavity. This atom-cavity system is studied by monitoring
the cavity reflection spectra using a scanned F-P cavity. We explore
two different input and output configurations of the cavity to
detect the different polarizations of the generated Stokes and
anti-Stokes fields. One is that the pump field first passes through
a polarized beamsplitter (PBS) and is injected into the cavity with
horizontal polarization. The vertically-polarized component of the
cavity reflection field is reflected by the same PBS, which mainly
contains the generated Stokes and anti-Stokes fields with only a
little pump field. The total reflected field then passes through an
optical isolator and is monitored by a scanned F-P cavity. The other
configuration is to use the pump field to inject into the cavity
with circular polarization by passing through a $\lambda/4$
waveplate. The output pump field from cavity reflection is reflected
by the PBS when passes through the $\lambda/4$ waveplate again.
Thus, the total reflected field from the PBS contains the reflected
pump field from the cavity and also the generated Stokes and
anti-Stokes fields with the same polarization as the pump field
inside the cavity.
\begin{figure}
\caption{(a) The saturated absorption spectrum is shown for the
frequency reference of the pump light. (b) The weak-field cavity
transmission spectrum with atoms inside the cavity (red line). The
empty cavity transmission spectrum without atoms (green line).
\label{Fig2}
\label{Fig2}
\end{figure}
Figure 3 presents the F-P cavity transmission spectra when the pump
beam frequency is set at different positions as indicated in Fig.
2(a) and the pump, Stokes and anti-Stokes fields build up on
resonance in the cavity simultaneously. First, as the pump frequency
$\omega_{p}$ is tuned to the frequency as indicated by the arrow
$a$, the F-P cavity transmission spectrum is given by Fig. 3(a). The
pump beam power is set at 100 $mW$. The two nearby side peaks
(separated from the large middle peak of the pump light by $\pm6.8$
$GHz$) are from the generated Stokes and anti-Stokes fields of the
$^{87}Rb$ atoms, which oscillate above thresholds. The central peak
and the free-spectral range (FSR) are for the pump beam. The
different peak heights (corresponding to the output power) for the
Stokes and anti-Stokes fields come from several mechanisms including
the cavity detuning, the dependence of gain on the detuning of the
pump field, and the variation of intracavity absorption by other
atomic transitions (as can be easily seen from Fig. 2). The two
outer small peaks on both sides are also the Stokes and anti-Stokes
fields due to the periodicity of the scanned F-P cavity (i.e. FSR).
As the pump frequency is tuned to the position marked by arrow $b$,
the F-P cavity transmission spectrum given by Fig. 3(b) presents the
generated Stokes field to be on resonance with the cavity and above
threshold, where the three-level atomic system is $^{85}Rb$ atoms
with the ground hyperfine state separation of about 3.035 $GHz$. The
anti-Stokes field is absorbed inside the cavity below threshold.
When the pump frequency is further tuned to the red side (arrow c),
both F-P cavity transmission peaks for the Stokes and anti-Stokes
fields appear simultaneously, showing triple-resonant oscillations
for the atom-cavity system. Under both different polarization
configurations of the cavity input field, we can detect the
generated Stokes and anti-Stokes fields with different polarizations
above thresholds.
\begin{figure}
\caption{The scanned F-P cavity transmission spectra. (a) , (b) and
(c) show the generated Stokes and anti-Stokes fields when the pump
frequency is set at different positions a, b, and c respectively as
indicated in Fig. 2(a). The temperature of the vapor cell is set at
$105^{0}
\label{Fig3}
\end{figure}
The measurement of the oscillation threshold for the anti-Stokes
field (shown in Fig. 3(a)) is presented in Fig.4. The cavity
oscillation starts gradually at low input pump power, and saturates
at high pump power as expected in a laser-like system. The threshold
behaviors for the Stokes and anti-Stokes fields in other cases (for
examples Figs. 3(b) and (c)) are similar. The total output powers of
the Stokes and anti-Stokes fields can reach more than 1 $mW$ when
the pump power is about 100 $mW$, which indicates that such OPO
system with multi-level atoms can be very efficient. The current
double-$\Lambda$ atomic configuration is an ideal system to generate
correlated Stokes and anti-Stokes photon pairs \cite{Lett2}. With
such atomic medium inside an optical cavity and driven above
oscillation thresholds, bright correlated twin beams can be
obtained, similar to the triple-resonant nondegenerate OPOs above
thresholds with $\chi^{(2)}$ nonlinear crystal
\cite{Fabre,Villar,Villar1,Villar2,Coelho,Dauria,Su,Laurat}. The
bright Stokes and anti-Stokes cavity output beams above threshold
will possess the different quantum characteristics comparing with
that produced from single-pass four-wave mixing process with
coherent signal injection \cite{Lett2}. Many interesting phenomena
can be realized using this demonstrated triple-resonant OPO system
in double-$\Lambda$ atomic system, especially with easily managed
large intracavity dispersions.
\begin{figure}
\caption{The measured threshold behavior of the emitted anti-Stokes
light in Fig. 3(b) as a function the pump beam power. The
temperature of the Rb cell is $T=105^{0}
\label{Fig4}
\end{figure}
In conclusion, we have experimentally demonstrated simultaneous
resonances of the generated Stokes and anti-Stokes fields, together
with the single pump field, in an optical cavity. The modified large
dispersions due to the intracavity dense atomic medium are very
important in allowing such triple-resonant conditions to be
simultaneously satisfied. Such experimental system can be useful to
study atom-cavity interaction, especially for quantum entangled
beams in quantum information processing \cite{app}.
$^{\dagger} $Corresponding author's email address:
[email protected], [email protected]
\acknowledgments
We thanks K. Peng, C. Xie and T. Zhang for the helpful discussions.
This research was supported in part by NSFC for Distinguished Young
Scholars (Grant No. 10725416), National Basic Research Program of
China (Grant No. 2006CB921101), NSFC Project for Excellent Research
Team (Grant No. 60821004), and NSFC (Grant No. 60678029). M. X.
acknowledges the funding support from NSF (US).
\end{document}
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\newcommand{\Large On the implied weights of linear regression for causal inference}{\Large On the implied weights of linear regression for causal inference}
\if00
{\Large On the implied weights of linear regression for causal inferencele{\Large On the implied weights of linear regression for causal inference\thanks{For comments and conversations, we thank Peter Aronow, Eric Cohn, Avi Feller, David Hirshberg, Winston Lin, Bijan Niknam, Jamie Robins, Paul Rosenbaum, and Dylan Small. This work was supported through a Patient-Centered
Outcomes Research Institute (PCORI) Project Program Award (ME-2019C1-16172) and grants from the Alfred P. Sloan Foundation (G-2018-10118, G-2020-13946).}}
\author{Ambarish Chattopadhyay\thanks{Department of Statistics, Harvard University, 1 Oxford Street
Cambridge, MA 02138; email: \url{[email protected]}.} \and Jos\'{e} R. Zubizarreta\thanks{Departments of Health Care Policy, Biostatistics, and Statistics, Harvard University, 180 A Longwood Avenue, Office 307-D, Boston, MA 02115; email: \url{[email protected]}.}
}
\date{}
\title{ it hanks{For comments and conversations, we thank Peter Aronow, Eric Cohn, Avi Feller, David Hirshberg, Winston Lin, Bijan Niknam, Jamie Robins, Paul Rosenbaum, and Dylan Small. This work was supported through a Patient-Centered
Outcomes Research Institute (PCORI) Project Program Award (ME-2019C1-16172) and grants from the Alfred P. Sloan Foundation (G-2018-10118, G-2020-13946).}
}\fi
\if10
\Large On the implied weights of linear regression for causal inferencele{\bf \Large On the implied weights of linear regression for causal inference}
\date{}
\title{ it hanks{For comments and conversations, we thank Peter Aronow, Eric Cohn, Avi Feller, David Hirshberg, Winston Lin, Bijan Niknam, Jamie Robins, Paul Rosenbaum, and Dylan Small. This work was supported through a Patient-Centered
Outcomes Research Institute (PCORI) Project Program Award (ME-2019C1-16172) and grants from the Alfred P. Sloan Foundation (G-2018-10118, G-2020-13946).}
\fi
\begin{abstract}
A basic principle in the design of observational studies is to approximate the randomized experiment that would have been conducted under ideal circumstances.
In practice, linear regression models are commonly used to analyze observational data and estimate causal effects.
How do linear regression adjustments in observational studies emulate key features of randomized experiments, such as covariate balance, self-weighted sampling, and study representativeness?
In this paper, we provide answers to this and related questions by analyzing the implied (individual-level data) weights of various linear regression methods, bringing new insights at the intersection of regression modeling and causal inference.
We derive new closed-form expressions of these implied weights and examine their properties in finite and large samples.
Among others, in finite samples we characterize the implied target population of linear regression and in large samples demonstrate the multiply robust properties of regression estimators from the perspective of their implied weights.
We show that the implied weights of general regression methods can be equivalently obtained by solving a convex optimization problem.
This equivalence allows us to bridge ideas from the regression modeling and causal inference literatures.
As a result, we propose novel regression diagnostics for causal inference that are part of the design stage of an observational study.
We implement the weights and diagnostics in the new \texttt{lmw} package for \texttt{R}.
\end{abstract}
\begin{center}
\noindent Keywords:
{Causal inference; Linear regression; Observational studies}
\end{center}
\doublespacing
\singlespacing
\pagebreak
\tableofcontents
\pagebreak
\doublespacing
\section{Introduction}
\label{sec_introduction}
\subsection{Regression and experimentation}
In a landmark paper in 1965, Cochran recommended that ``the planner of an observational study should always ask himself the question, `How would the study be conducted if it were possible to do it by controlled experimentation?''' (\perp\!\!\!\perptealt{cochran1965planning}).
Some key features of randomized experiments are: (a) covariate balance, i.e., pre-treatment variables are balanced in expectation; (b) study representativeness, i.e., the target population for inference is the experimental sample itself or a broader population under a known sample selection mechanism; (c) self-weighted sampling, i.e., randomization produces unweighted samples of treated and control units, and analyses that impose differential weights on the units reduce the efficiency of estimators; and (d) sample boundedness, i.e., since covariates are balanced in expectation, covariate adjustments and effect estimates are mostly an interpolation and not an extrapolation beyond the support of the observed data.
At present, linear regression models are extensively used to analyze observational data and estimate average causal effects.
But to what extent does regression emulate these key features of a randomized experiment?
More concretely, how does regression adjust for or balance the covariates included as regressors in the model?
What is the population that regression adjustments actually target?
And what is the connection between regression and other methods for statistical adjustment, such as matching and weighting?
In this paper, we answer these and related questions.
In particular, we show how linear regression acts on the individual-level data to approximate a randomized experiment and produce an average treatment effect estimate.
To this end, we examine how regression implicitly weights the treatment and control individual observations by finding its implied weights.
\subsection{Contribution and related works}
In this paper, we derive and analyze the implied weights of various linear regression estimators.
We obtain new closed-form, finite sample expressions of the weights.
The regression estimators we consider include: (1) uni-regression imputation (URI), which arguably is the most common regression estimator in practice and is equivalent to estimating the coefficient of the treatment indicator in a linear regression model on the covariates and the treatment without including their interactions, (2) multi-regression imputation (MRI) or g-computation, which is equivalent to estimating the coefficient of the treatment indicator in a similar model including interactions, (3) weighted least squares analogs of URI and MRI, and (4) augmented inverse probability weighting (\perp\!\!\!\perptealt{robins1994estimation}).
In relation to URI, previous works have obtained weighting representations of the regression estimand (\perp\!\!\!\perptealt{angrist2008mostly}, Chapter 3) or the regression estimators using asymptotic approximations (\perp\!\!\!\perptealt{aronow2016does}).
Our work differs from them in that we provide a weighting representation of the URI estimator in closed forms and in finite samples.
This representation clarifies how URI acts on each individual-level observation in the study sample to produce an average treatment effect estimate.
In relation to MRI, previous works have derived closed-form expressions of the MRI weights in specific settings such as univariate regression (\perp\!\!\!\perptealt{imbens2015matching}), synthetic controls (\perp\!\!\!\perptealt{abadie2015comparative}, \perp\!\!\!\perptealt{ben2021augmented}), and regression discontinuity designs (\perp\!\!\!\perptealt{gelman2018high}). Some of these results are analogous to those of regression estimation in sample survey (\perp\!\!\!\perptealt{huang1978nonnegative}, \perp\!\!\!\perptealt{deville1992calibration}).
Our work builds on and generalizes these important contributions to general multivariate MRI estimators for several estimands of interest.
We derive the properties of the implied weights and the corresponding estimators in both finite and large sample regimes.
In finite samples, we analyze the weights in terms of (a) covariate balance, (b) study representativeness, (c) dispersion, (d) sample-boundedness, and (e) optimality.
In particular, we characterize the implied target population of linear regression in finite samples, which to our knowledge has not been done in causal inference.
Moreover, using the implied weights, we provide an alternative view of regression adjustments from the standpoint of mathematical optimization, and in turn, connect regression to modern matching and weighting methods in causal inference.
In large samples, we show that under certain conditions URI and MRI are equivalent to inverse probability weighting.
Leveraging this equivalence, we show that the URI and MRI estimators are multiply robust.
Our results generalize previous results on double robustness for regression estimators in survey sampling (\perp\!\!\!\perptealt{robins2007comment}) and causal inference (\perp\!\!\!\perptealt{kline2011oaxaca}) to multiple robustness and other types of regression estimators such as URI.
Finally, this implied weighting framework allows us to propose new regression diagnostics for causal inference.
These diagnostics assess covariate balance, model extrapolation, dispersion of the weights and effective sample size, and influence of a given observation on an estimate of the average treatment effect.
Conventionally, regression adjustments are viewed as part of the analysis stage of an observational study, but as we discuss in this paper, they can be conducted as part of the design stage \perp\!\!\!\perptep{rubin2008objective}. In this sense,
our first three proposed diagnostics can be regarded as design-based diagnostics for regression adjustments in observational studies.
\section{Notation, estimands, and assumptions}
\label{sec_notation}
We operate under the potential outcomes framework for causal inference (\perp\!\!\!\perpteauthor{neyman1923application} 1923, 1990, \perp\!\!\!\perptealt{rubin1974estimating}) and consider a sample of $n$ units randomly drawn from a population.
For each unit $i = 1, ..., n$, $Z_i$ is a treatment assignment indicator with $Z_i = 1$ if the unit is assigned to treatment and $Z_i = 0$ otherwise; $\bm{X}_i \in \mathbb{R}^k$ is a vector of observed covariates; and $Y^{\text{obs}}_i$ is the observed outcome variable. Under the Stable Unit Treatment Value Assumption (SUTVA; \perp\!\!\!\perptealt{rubin1980randomization}), let $\{Y_i(1), Y_i(0) \}$ be the potential outcomes under treatment and control, respectively, where only one of them is observed in the sample: $Y^{\text{obs}}_i = Z_i Y_i(1) + (1-Z_i)Y_i(0)$.
We focus on estimating the Average Treatment Effect (ATE), defined as $\text{ATE} = {E}\{Y_i(1) - Y_i(0)\}$, and the Average Treatment Effect on the Treated (ATT), given by $\text{ATT} = {E}\{Y_i(1) - Y_i(0)\mid Z_i=1\}$.
We also consider the Conditional Average Treatment Effect (CATE).
The CATE for a population with a fixed covariate profile $\bm{x}^* \in \mathbb{R}^k$ is given by $\text{CATE}(\bm{x}^*) = {E}\{Y_i(1) - Y_i(0)\mid \bm{X}_i = \bm{x}^*\}$; i.e., the CATE is the ATE in the subpopulation of units with covariate vector equal to $\bm{x}^*$.
For identification of these estimands, we assume that the treatment assignment satisfies the unconfoundedness and positivity assumptions: $Z_i \perp\!\!\!\perp \{Y_i(0),Y_i(1)\} \mid \bm{X}_i$ and $0<\text{pr}(Z_i = 1\mid \bm{X}_i = \bm{x})<1$ for all $\bm{x} \in \text{supp}(\bm{X}_i)$, respectively \perp\!\!\!\perptep{rosenbaum1983central}.
For conciseness, we adopt the following additional notation.
Denote the conditional mean functions of the potential outcomes under treatment and control as $m_1(\bm{x}) = {E}\{Y_i(1)\mid \bm{X}_i=\bm{x}\}$ and $m_0(\bm{x}) = {E}\{Y_i(0)\mid \bm{X}_i=\bm{x}\}$, respectively.
Let $n_t = \sum_{i=1}^{n}Z_i$ and $n_c = \sum_{i=1}^{n}(1-Z_i)$ be the treatment and control group sizes.
Write $\bm{\underline{X}}$ for the $n\times k$ matrix of covariates in the full sample that pools the treatment and control groups, and let $\bar{\bm{X}}_t = \sum_{i:Z_i=1} \bm{X}_i/n_t$ and $\bar{\bm{X}}_c = \sum_{i:Z_i=0} \bm{X}_i/n_c$.
The average of the covariate vectors $\bm{X}_i$ in the full sample is given by $\bar{\bm{X}} = n^{-1}(n_t \bar{\bm{X}}_t + n_c \bar{\bm{X}}_c)$.
Also, let $\bm{S}_t = \sum_{i:Z_i=1}(\bm{X}_i - \bar{\bm{X}}_t)(\bm{X}_i - \bar{\bm{X}}_t)^\top$ and $\bm{S}_c = \sum_{i:Z_i=0}(\bm{X}_i - \bar{\bm{X}}_c)(\bm{X}_i - \bar{\bm{X}}_c)^\top$ be the scaled covariance matrices in the treatment and control group respectively. Throughout the paper, we assume that $\bm{S}_t$ and $\bm{S}_c$ are invertible.
Finally, let $\bar{Y}_t$ and $\bar{Y}_c$ be the mean observed outcomes in the treatment and control group, respectively.
\section{Implied weights of linear regression}
\label{sec_implied}
A widespread approach to estimate the ATE goes as follows.
On the entire sample, use ordinary least squares (OLS) to fit a linear regression model of the observed outcome $Y^{\text{obs}}_i$ on the baseline covariates $\bm{X}_i$ and the treatment indicator $Z_i$, and compute the coefficient associated with $Z_i$ (see, e.g., Chapter 3 of \perp\!\!\!\perptealt{angrist2008mostly} and Section 12.2.4 of \perp\!\!\!\perptealt{imbens2015causal}).
Under mean unconfoundedness, this approach can be motivated by the structural model $Y_i(z) = \beta_0 + \bm{\beta}^\top_1 \bm{X}_i + \tau z + \epsilon_{iz} , \hspace{0.1cm}{E}(\epsilon_{iz}\mid \bm{X}_i) = 0$, $z \in \{0,1\}$.
Here the CATE is constant and equal to $\tau$ across the space of the covariates; i.e., $m_1(\bm{x}) - m_0(\bm{x}) = \tau$.
Thus $\text{ATE} = {E}\{m_1(\bm{X}_i) - m_0(\bm{X}_i)\} = \tau$, and by unconfoundedness, ${E}\{Y_i(z)\mid \bm{X}_i = \bm{x}\} = {E}\{Y^{\text{obs}}_i\mid \bm{X}_i = \bm{x},Z_i = z\} = \beta_0 + \bm{\beta}^\top_1 \bm{x} + \tau z$.
By standard linear model theory, if the model for $Y_i(z)$ is correct, then the OLS estimator $\hat{\tau}^{{\scriptscriptstyle \text{OLS}}}$ is the best linear unbiased and consistent estimator for the ATE.
Now, since $\text{ATE} = {E}\{ m_1(\bm{X}_i) - m_0(\bm{X}_i)\}$, a natural way to estimate this quantity is to compute its empirical analog $n^{-1}\sum_{i=1}^{n} \{ {m}_1(\bm{X}_i) - {m}_0(\bm{X}_i) \}$. Therefore, a broad class of imputation estimators of the ATE has the form
$\widehat{\text{ATE}} = n^{-1}\sum_{i=1}^{n} \{ \hat{m}_1(\bm{X}_i) - \hat{m}_0(\bm{X}_i) \}$, where $\hat{m}_1(\bm{x})$ and $\hat{m}_0(\bm{x})$ are some estimators of ${m}_1(\bm{x})$ and ${m}_0(\bm{x})$, respectively.
Such imputation estimators are popular in causal inference (see, e.g., Chapter 13 of \perp\!\!\!\perptealt{hernan2020causal}).
Clearly, $\hat{\tau}^{{\scriptscriptstyle \text{OLS}}}$ is also an imputation estimator.
Henceforth, we term this approach uni-regression imputation (URI), because the potential outcomes are imputed using a single (uni) regression model.
In Proposition \ref{prop_uri} we show that $\hat{\tau}^{{\scriptscriptstyle \text{OLS}}}$ can be represented as a difference of weighted means of the treated and control outcomes.
We also provide closed form expressions for the implied regression weights.
\begin{proposition}
\label{prop_uri}
The URI estimator of the ATE can be expressed as $\hat{\tau}^{{\scriptscriptstyle \text{OLS}}} = \sum_{i:Z_i=1}w^{{\scriptscriptstyle \text{URI}}}_i Y^{\text{obs}}_i - \sum_{i:Z_i=0}w^{{\scriptscriptstyle \text{URI}}}_i Y^{\text{obs}}_i$ where $w^{{\scriptscriptstyle \text{URI}}}_i = n^{-1}_t + n n^{-1}_c (\bm{X}_i -\bar{\bm{X}}_t)^\top (\bm{S}_t + \bm{S}_c)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t)$ for each unit in the treatment group and $w^{{\scriptscriptstyle \text{URI}}}_i = n^{-1}_c + n n^{-1}_t (\bm{X}_i -\bar{\bm{X}}_c)^\top (\bm{S}_t + \bm{S}_c)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_c)$ for each unit in the control group.
Moreover, within each group the weights add up to one, $\sum_{i:Z_i=0} w^{{\scriptscriptstyle \text{URI}}}_i =1$ and $\sum_{i:Z_i=1} w^{{\scriptscriptstyle \text{URI}}}_i =1$.
\end{proposition}
A proof for Proposition \ref{prop_uri}, as well as for all the other results in the paper, is presented in the Supplementary Material.
According to Proposition \ref{prop_uri}
the regression estimator $\hat{\tau}^{{\scriptscriptstyle \text{OLS}}}$ is a H\'{a}jek estimator with weights $w^{{\scriptscriptstyle \text{URI}}}_i$.
Furtheremore, we see that the URI weights depend on the treatment indicators and the covariates but not on the observed outcomes.
Therefore, although typical software implementations of URI require the outcomes and simultaneously adjust for the covariates and produce effect estimates, the weighting representation in Proposition \ref{prop_uri} shows that the linear regression model can be ``fit'' without the outcomes.
In other words, using \perp\!\!\!\perpte{rubin2008objective}'s classification of the stages of an observational study, the URI weights can be obtained as a part of the `design stage' of the study, as opposed to its `analysis stage,' helping to preserve the objectivity of the study and bridge ideas from matching and weighting to regression modeling.
Another type of imputation estimator obtains $\hat{m}_1(\bm{x})$ and $\hat{m}_0(\bm{x})$ by fitting two separate linear regression models on the treatment and control samples, given by $Y^{\text{obs}}_i = \beta_{0t} + \bm{\beta}^\top_{1t}\bm{X}_i + \epsilon_{it}$ and $Y^{\text{obs}}_i = \beta_{0c} + \bm{\beta}^\top_{1c}\bm{X}_i + \epsilon_{ic}$, respectively.
This approach is more flexible than the former since it allows for treatment effect modification.
In particular, under mean unconfoundedness, the conditional average treatment effect is linear in the covariates, i.e., $\text{CATE}(\bm{x}) = m_1(\bm{x}) - m_0(\bm{x}) = (\beta_{0t} - \beta_{0c}) + (\bm{\beta}_{1t} - \bm{\beta}_{1c})^\top \bm{x}$.
We call this approach multi-regression imputation (MRI).
Clearly, MRI and URI are equivalent if the model used in the URI approach includes all possible interaction terms between the treatment indicator and the mean-centered covariates.
With the MRI approach, it is convenient to estimate a wide range of estimands, including the ATE, ATT, and the CATE.
The following proposition shows the implied form of weighting of the treated and control units under the MRI approach.
\begin{proposition}
\label{prop_mri}
The MRI estimators of the ATE, ATT, and CATE can be expressed as
\begin{enumerate}[label=(\alph*)]
\item $\widehat{\text{ATE}} = \sum_{i:Z_i = 1}w^{{\scriptscriptstyle \text{MRI}}}_i(\bar{\bm{X}}) Y^{\text{obs}}_i - \sum_{i:Z_i = 0}w^{{\scriptscriptstyle \text{MRI}}}_i(\bar{\bm{X}}) Y^{\text{obs}}_i$,
\item $\widehat{\text{ATT}} = \bar{Y}_t - \sum_{i:Z_i = 0}w^{{\scriptscriptstyle \text{MRI}}}_i(\bar{\bm{X}}_t) Y^{\text{obs}}_i$, and
\item $\widehat{\text{CATE}}(\bm{x}^*) = \sum_{i:Z_i = 1}w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x}^*)Y^{\text{obs}}_i - \sum_{i:Z_i = 0}w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x}^*) Y^{\text{obs}}_i$,
\end{enumerate}
where $w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x}) = n^{-1}_t + (\bm{X}_i -\bar{\bm{X}}_t)^\top \bm{S}_t^{-1} (\bm{x} - \bar{\bm{X}}_t)$ if unit $i$ is in the treatment group and $w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x}) = n^{-1}_c + (\bm{X}_i -\bar{\bm{X}}_c)^\top \bm{S}_c^{-1} (\bm{x} - \bar{\bm{X}}_c)$ if unit $i$ is in the control group.
Moreover, $\sum_{i:Z_i=1}w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x}) = \sum_{i:Z_i=0}w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x}) = 1$ for all $\bm{x} \in \mathbb{R}^k$.
\end{proposition}
We observe that, similar to the URI weights, the MRI weights do not depend on the outcomes and hence can be part of the design stage of the study.
Propositions \ref{prop_uri} and \ref{prop_mri} highlight how the implied weights depart from uniform weights as a function of covariate balance before adjustments. In particular, both URI and MRI weights become uniform if the covariates in the treatment and control groups are exactly mean balanced a priori.
These weighting expressions can show when a particular observation has a large impact on the analysis via its implied weight (see Section \ref{sec_regression} for related diagnostics).
Finally, propositions \ref{prop_uri} and \ref{prop_mri} imply that $w^{{\scriptscriptstyle \text{URI}}}_i = w^{{\scriptscriptstyle \text{MRI}}}_i(\bm{x^*})$, where $\bm{x^*} = \bm{S}_c(\bm{S}_t + \bm{S}_c)^{-1} \bar{\bm{X}}_t + \bm{S}_t (\bm{S}_t + \bm{S}_c)^{-1} \bar{\bm{X}}_c $. This means that the URI weights are a special case of the MRI weights, where we impute the potential outcomes of a unit with $\bm{x} = \bm{x^*}$.
In particular, a sufficient condition for $w^{{\scriptscriptstyle \text{URI}}}_i = w^{{\scriptscriptstyle \text{MRI}}}_i(\bar{\bm{X}})$ is that $n_t\bm{S}_t = n_c\bm{S}_c$, which holds if the treatment groups are of equal size and have the same sample covariance matrix.
Indeed, another sufficient condition for the weights to be equal is that $\bar{\bm{X}}_t = \bar{\bm{X}}_c$, which implies that both weights are uniform.
\section{Properties of the implied weights}
\label{sec_properties}
\subsection{Finite sample properties}
\label{sec_finite}
In this section, we study the finite sample properties of the implied weights in regards to: (a) covariate balance, (b) representativeness of the weighted sample, (c) dispersion or variability, (d) extrapolation, and (e) optimality from a mathematical programming standpoint.
The following proposition summarizes these properties for both the URI and the MRI weights for the ATE estimation problem. Henceforth, we denote the MRI weights for the ATE as $w^{\scriptscriptstyle \text{MRI}}_i$.
\begin{proposition}
\label{prop_finite_sample_properties}
\begin{enumerate}[label =(\alph*)]
\item[]
\item {Balance}: The URI and MRI weights exactly balance the means of the covariates included in the model, with respect to different profiles $\bm{X}^{*{\scriptscriptstyle \text{URI}}}$ and $\bm{X}^{*{\scriptscriptstyle \text{MRI}}}$:
$
\sum_{i:Z_i=1} w^{{\scriptscriptstyle \text{URI}}}_i \bm{X}_i = \sum_{i:Z_i=0} w^{{\scriptscriptstyle \text{URI}}}_i \bm{X}_i = \bm{X}^{*{\scriptscriptstyle \text{URI}}}, \quad \sum_{i:Z_i=1} w^{{\scriptscriptstyle \text{MRI}}}_i \bm{X}_i = \sum_{i:Z_i=0} w^{{\scriptscriptstyle \text{MRI}}}_i \bm{X}_i = \bm{X}^{*{\scriptscriptstyle \text{MRI}}}.
$
\item {Representativeness}: With the URI and MRI weights, the covariate profiles are
$
\bm{X}^{*{\scriptscriptstyle \text{URI}}} = \bm{S}_c(\bm{S}_t + \bm{S}_c)^{-1} \bar{\bm{X}}_t + \bm{S}_t (\bm{S}_t + \bm{S}_c)^{-1} \bar{\bm{X}}_c$ and $\bm{X}^{*{\scriptscriptstyle \text{MRI}}} = \bar{\bm{X}},
$ respectively.
\item {Dispersion}: The variances of the URI weights in the treatment and control groups are given by
$n^2 n^{-1}_t n^{-2}_c (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top (\bm{S}_t + \bm{S}_c)^{-1}\bm{S}_t (\bm{S}_t + \bm{S}_c)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t)
$ and $n^2n^{-1}_c n^{-2}_t (\bar{\bm{X}} - \bar{\bm{X}}_c)^\top (\bm{S}_t + \bm{S}_c)^{-1}\bm{S}_c (\bm{S}_t + \bm{S}_c)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_c)
$, respectively.
Similarly, the variances of the MRI weights in the treatment and control groups are
$n^{-1}_t (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \bm{S}_t^{-1}(\bar{\bm{X}} - \bar{\bm{X}}_t)
$ and $n^{-1}_c (\bar{\bm{X}} - \bar{\bm{X}}_c)^\top \bm{S}_c^{-1}(\bar{\bm{X}} - \bar{\bm{X}}_c)$, respectively.
\item {Extrapolation}: The URI and MRI weights can both take negative values and produce average treatment effect estimators that are not sample bounded.
\item {Optimality}: The URI and MRI weights are the weights of minimum variance that add up to one and satisfy the corresponding covariate balance constraints in (a).
\end{enumerate}
\end{proposition}
One of our motivating questions was, to what extent does regression emulate the key features of a randomized experiment?
Proposition \ref{prop_finite_sample_properties} provides answers to this question.
Part (a) says that linear regression, both in its URI and MRI variants, exactly balances the means of the covariates included in the model, but with respect to different profiles, $\bm{X}^{*{\scriptscriptstyle \text{URI}}}$ and $\bm{X}^{*{\scriptscriptstyle \text{MRI}}}$.
Part (b) provides closed form expressions for these profiles, which in turn characterize the implied target populations of regression.
While MRI exactly balances the means of the covariates at the overall study sample mean, URI balances them elsewhere. In this sense, the URI weights may distort the structure of the original study sample.
Under linearity of the potential outcomes models, if treatment effects are homogeneous, both URI and MRI produce unbiased estimators of the ATE. However, if treatments effects are heterogeneous, then even under linearity, the URI estimator is biased for the ATE, whereas the MRI estimator is unbiased. See the Supplementary Material for details.
Part (c) characterizes the variances of the weights.
For instance, the variance of both the URI and MRI weights in the treatment group are a scaled distance between $\bar{\bm{X}}$ and $\bar{\bm{X}}_t$, multiplied by positive definite matrices.
Since $(\bar{\bm{X}} - \bar{\bm{X}}_t) = n^{-1}n_c (\bar{\bm{X}}_c - \bar{\bm{X}}_t)$, the variance of both the URI and MRI weights can also be interpreted as a distance similar to the Mahalanobis distance between the treatment and the control groups.
Therefore, for fixed $\bm{S}_t$ and $\bm{S}_c$, using URI or MRI on an a priori well-balanced sample
will lead to weights that are less variable than that on an imbalanced sample. The variance of the weights is important because it directly impacts the variance of a weighted estimator.
Part (d) establishes that both the URI and MRI weights can take negative values, so the corresponding estimators are not sample bounded in the sense of \perp\!\!\!\perpte{robins2007comment} and their estimates can lie outside the support, or convex hull, of the observed outcome data.
This property has been noted in instances of MRI, for example, in simple regression estimation of the ATT by \perp\!\!\!\perpte{imbens2015matching} and in synthetic control settings by \perp\!\!\!\perpte{abadie2015comparative}, but not in general.
We refer the reader to Section \ref{sec_extrapolation} for a discussion on the implications of this property.
Finally, part (e) groups these results and states that the URI and MRI weights are the least variable weights that add up to one and exactly balance the means of the covariates included in the models with respect to given covariate profiles.
This result helps to establish a connection between the implied linear regression weights and existing matching and weighting methods. For example, part (e) shows that URI and MRI can be viewed as weighting approaches with moment-balancing conditions on the weights, such as entropy balancing (\perp\!\!\!\perptealt{hainmueller2012balancing}) and the stable balancing weights (\perp\!\!\!\perptealt{zubizarreta2015stable}).
We also note the connection of the implied weights to matching approaches, e.g., cardinality matching \perp\!\!\!\perptep{zubizarreta2014matching} where the weights are constrained to be constant integers representing a matching ratio and an explicit assignment between matched units.
In connection to sample surveys, Part (e) establishes URI and MRI as two-step calibration weighting methods (\perp\!\!\!\perptealt{deville1992calibration}), where the weights are calibrated separately in the treatment and control groups.
See the Supplementary Material for results analogous to Proposition \ref{prop_finite_sample_properties} when the estimand is the ATT or the CATE($\bm{x}$).
\subsection{Asymptotic properties}
\label{sec_asymptotic_properties}
In this section, we study the large-sample behavior of the URI and MRI weights and their associated estimators.
This analysis reveals a connection between regression imputation and inverse probability weighting (IPW).
In particular, we show that under a given functional form for the true propensity score model, the MRI weights converge pointwise to the corresponding true inverse probability weights.
Moreover, the convergence is uniform if the $L_2$ norm of the covariate vector is bounded over its support.
Theorem \ref{thm_mri_convergence} formalizes this result for the ATE estimation problem.
An analogous result holds for the ATT.
\begin{theorem}
\label{thm_mri_convergence}
Suppose we wish to estimate the ATE.
Let $w^{{\scriptscriptstyle \text{MRI}}}_{\bm{x}}$ be the MRI weight of a unit with covariate vector $\bm{x}$. Then
\begin{enumerate}[label = (\alph*)]
\item For each treated unit, $n w^{{\scriptscriptstyle \text{MRI}}}_{\bm{x}} \xrightarrow[n \to \infty]{P} 1/e(\bm{x})$ for all $\bm{x} \in \text{supp}(\bm{X}_i)$ if and only if the propensity score is an inverse linear function of the covariates; i.e., $e(\bm{x}) = 1/(\alpha_0 + \bm{\alpha}^\top_1 \bm{x})$, $\alpha_0 \in \mathbb{R}$, $\bm{\alpha}_1 \in \mathbb{R}^k$. Moreover, if $\sup_{\bm{x} \in \text{supp}(\bm{X}_i)}\norm{\bm{x}}_2 < \infty$, then $\sup_{\bm{x} \in \text{supp}(\bm{X}_i)}\mid nw^{{\scriptscriptstyle \text{MRI}}}_{\bm{x}} - \{1/e(\bm{x})\}\mid \xrightarrow[n \to \infty]{P} 0$.
\item Similarly, for each control unit, $n w^{{\scriptscriptstyle \text{MRI}}}_{\bm{x}} \xrightarrow[n \to \infty]{P} 1/\{1-e(\bm{x})\}$ if and only if $1-e(\bm{x})$ is an inverse linear function of the covariates, and the convergence is uniform if $\sup_{\bm{x} \in \text{supp}(\bm{X}_i)}\norm{\bm{x}}_2 < \infty$.
\end{enumerate}
\end{theorem}
Theorem \ref{thm_mri_convergence} says that, by fitting a linear regression model of the outcome in the treatment group, we implicitly estimate the propensity score.
Moreover, it says that if the true propensity score model is inverse linear, then the implied scaled weights converge pointwise and uniformly in the supremum norm to the true inverse probability weights.
This implies that the MRI estimator for the treated units $\sum_{i:Z_i=1}w^{{\scriptscriptstyle \text{MRI}}}_i Y^{\text{obs}}_i$ of ${E}\{Y_i(1)\}$ can be viewed as a Horvitz-Thompson IPW estimator $\frac{1}{n}\sum_{i:Z_i=1} Y^{\text{obs}}_i/\hat{e}(\bm{X}_i)$ where $\hat{e}(\bm{X}_i) = (nw^{{\scriptscriptstyle \text{MRI}}}_i)^{-1} = \{n n^{-1}_t + n (\bm{X}_i - \bar{\bm{X}}_t)^\top \bm{S}^{-1}_t (\bar{\bm{X}} - \bar{\bm{X}}_t) \}^{-1}$.
A similar algebraic equivalence between the IPW estimator and the MRI estimator holds when propensity scores and conditional means are estimated using nonparametric frequency methods (\perp\!\!\!\perptealt{hernan2020causal}, Section 13.4).
Part (b) of Theorem \ref{thm_mri_convergence} provides an analogous result for the MRI weights of the control units.
However, instead of $e(\bm{x})$, now $1-e(\bm{x})$ needs to be inverse-linear on the covariates.
Therefore, the linear regression model in the control group implicitly assumes a propensity score model different from the one assumed in the treatment group, since $e(\bm{x})$ and $1-e(\bm{x})$ cannot be inverse linear simultaneously, unless $e(\bm{x})$ is constant.
This also means that, unless the propensity score is a constant function of the covariates, the MRI weights for both treated and control units cannot converge simultaneously to their respective true inverse probability weights.
This condition of constant propensity scores can hold by design in randomized experiments, but is less likely in observational studies.
We now focus on the convergence of the MRI estimator of the ATE.
By standard OLS theory, the MRI estimator is consistent for the ATE if both $m_1(\bm{x})$ and $m_0(\bm{x})$ are linear in $\bm{x}$.
The convergence of the MRI weights to the true inverse probability weights in Theorem \ref{thm_mri_convergence} unveils other paths for convergence of the MRI estimator.
In fact, we obtain five non-nested conditions under which the MRI estimator $\sum_{i:Z_i=1} w^{{\scriptscriptstyle \text{MRI}}}_i Y^{\text{obs}}_i - \sum_{i:Z_i=0} w^{{\scriptscriptstyle \text{MRI}}}_i Y^{\text{obs}}_i$ is consistent for the ATE.
\begin{theorem}
\label{thm_mri_consistency}
The MRI estimator is consistent for the ATE if any of the following conditions holds:
(i) $m_0(\bm{x})$ is linear and $e(\bm{x})$ is inverse linear; (ii) $m_1(\bm{x})$ is linear and $1-e(\bm{x})$ is inverse linear;
(iii) $m_1(\bm{x})$ and $m_0(\bm{x})$ are linear; (iv) $e(\bm{x})$ is constant; (v) $m_1(\bm{x})- m_0(\bm{x})$ is a constant function, $e(\bm{x})$ is linear, and $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$ where $p$ is the probability limit of $n_t/n$.
\end{theorem}
Conditions (i), (ii), and (iv) follow from the convergence of the weights described in Theorem \ref{thm_mri_convergence}. Condition (iii) follows from standard OLS theory. Condition (v) relies on an asymptotic equivalence condition between MRI and URI, which we discuss in the Supplementary Material.
Theorem \ref{thm_mri_consistency} shows that the MRI estimator is multiply robust, extending the results of \perp\!\!\!\perpte{robins2007comment} and \perp\!\!\!\perpte{kline2011oaxaca}. The conditions for multiple robustness are characterized by a combination of conditions on $\{m_1(\cdot), m_0(\cdot), e(\cdot)\}$ jointly, as opposed to conditions on either $\{m_1(\cdot), m_0(\cdot)\}$ or $e(\cdot)$ separately.
While intriguing, this multiple robustness property of the MRI estimator needs to be understood in an adequate context.
In principle, Theorem \ref{thm_mri_consistency} seems to suggests that many estimators can be doubly robust; the question is under what conditions of the true treatment and outcome models.
For instance, an inverse linear model for $e(\bm{x})$ or $1-e(\bm{x})$ (as in conditions (i) and (ii)) is not very realistic, since the probabilities under an inverse-linear model are not guaranteed to lie inside the (0, 1) range, as noted by \perp\!\!\!\perpte{robins2007comment}. Also, even if an inverse-linear model for the treatment is plausible, conditions (i)--(v) may be more stringent in practice than correct specification of either the treatment model or the potential outcome models separately.
Finally, we discuss the asymptotic properties of the URI weights and its associated estimator.
We present these additional results and derivations to the Supplementary Material.
We find that, similar to the MRI weights, the URI weights also converge to the true inverse probability weights, albeit under additional conditions to those in Theorem \ref{thm_mri_convergence}.
A sufficient additional condition for consistency of the URI weights is $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$ where $p$ is the probability limit of the proportion of treated units. This condition appears in condition (v) of Theorem \ref{thm_mri_consistency}.
Accordingly, we show that the URI estimator is multiply robust for the ATE, which to our knowledge has not been noted in the literature.
As we show, the required conditions for multiple robustness of URI are even stronger than those for MRI.
\section{Regression diagnostics using the implied weights}
\label{sec_regression}
\subsection{Overview}
The implied weights help us to connect the linear models and observational studies literatures and devise new diagnostics for causal inference using regression.
In this section, we discuss diagnostics based on the implied weights for (i) covariate balance, (ii) model extrapolation, (iii) dispersion and effective sample size, and (iv) influence of a given observation on an estimate of the average treatment effect. We note that the diagnostics in (i), (ii), and (iii) are solely based on the implied weights and do not involve any outcome information.
In this sense, (i), (ii), and (iii) are part of the design stage of the study \perp\!\!\!\perptep{rubin2008objective}.
In contrast, (iv) requires information from outcomes in addition to the weights, and hence are part of the analysis stage.
We illustrate these weight diagnostics using as a running example the well-known Lalonde study (\perp\!\!\!\perptealt{lalonde1986evaluating}) on the impact of a labor training program on earnings.
The study consists of $n_t = 185$ treated units (enrolled in the program), $n_c = 2490$ control units (not enrolled in program), and $k = 8$ covariates.
For illustration, here we consider the problem of estimating the ATE.
We implement the implied weights and regression diagnostics in the new \texttt{lmw} package for \texttt{R} available at \texttt{https://github.com/ngreifer/lmw}.
\begin{figure}
\caption{Diagnostics for the URI and MRI weights for the Lalonde observational dataset.}
\label{fig:diagnostics}
\end{figure}
\subsection{Covariate balance}
\label{sec_balance_diag}
The implied weights can be used to check balance of the distributions of the covariates in the treatment and control groups relative to a target population.
As discussed in Section \ref{sec_finite}, although both the URI and MRI weights exactly balance the means of the covariates included in the model, they target different covariate profiles.
Moreover, neither the URI nor the MRI weights are guaranteed to balance the covariates (or transformations thereof) not included in the model.
Therefore, it is advisable to check balance on transformations that are not balanced relative to the target by construction.
A suitable measure for this task is the Target Absolute Standardized Mean Difference (TASMD, \perp\!\!\!\perptealt{chattopadhyay2020balancing}), which is defined as the absolute value of the standardized difference between the mean of the covariate transformation in the weighted sample and the corresponding mean in the target population.
We recommend using the TASMD for balance diagnostics as opposed to the more commonly used Absolute Standardized Mean Difference (ASMD), since it provides a flexible measure of imbalance of a weighted sample relative to arbitrary target profiles, which in principle can represent a single individual.
The upper left panel of Figure \ref{fig:diagnostics} shows the ASMDs and TASMDs of the eight covariates in the Lalonde study with both URI and MRI.
The plots illustrate the results in Proposition \ref{prop_finite_sample_properties}.
In the figure, the first ASMD plot demonstrates the exact mean balancing property of both the URI and the MRI weights, but not relative to a target profile.
The TASMD plots, on the other hand, provide a more complete picture of the representativeness of URI and the MRI in terms of the first moments of the covariates.
Since the target profile in this case are the mean of the covariates in the full sample, the MRI weights yield a TASMD of zero for each covariate by construction.
However, the URI weights do not achieve exact balance relative to the target.
In fact, in the last TASMD plot we see that the URI weights actually exacerbate the initial imbalances in the control group relative to the target.
\subsection{Extrapolation}
\label{sec_extrapolation}
An important feature of both URI and MRI is that their implied weights can be negative and extreme in magnitude.
Negative weights are difficult to interpret and, moreover, they can produce estimates that are an extrapolation outside of the support of the available data.
In other words, negative weights can produce H\'{a}jek estimators of average treatment effects that are not sample bounded in the sense of \perp\!\!\!\perpte{robins2007comment} (see also \perp\!\!\!\perptealt{chattopadhyay2020balancing}). On the other hand, H\'{a}jek estimators with non-negative weights are, by construction, sample bounded.
In some settings, there is no alternative to using negative weights in order to adjust for or balance certain features of the distributions of the observed covariates.
That is, one can only balance the means of such features with negative weights.
If the model behind the adjustments is correctly specified, then extrapolation is not detrimental; however, if the model is misspecified, then it is possible that other features or transformations of the covariates are severely imbalanced and that the estimators are highly biased if these other
transformations determine $m_1(\cdot)$ and $m_0(\cdot)$.
The bottom left panel of Figure \ref{fig:diagnostics} presents bubble plots of the URI and MRI weights within each treatment group for two covariates, `Black' and `Earnings `75'.
Each bubble represents an observation.
The size of each bubble is proportional to the absolute value of the corresponding weight.
A red (respectively, black) bubble indicates a negative (positive) weight.
The asterisk is the target value of a covariate and the black vertical line represents the weighted average of that covariate in the corresponding treatment group.
We observe that both MRI and URI produce negative weights.
Moreover, some of the negative weights are also extreme in magnitude with respect to a particular covariate profile.
For instance, a control unit with `Earnings `75' greater than $\$$150,000 receives a large negative weight under URI.
\perp\!\!\!\perpte{imbens2015matching} illustrated this phenomenon in the Lalonde study for the ATT using an MRI regression with a single covariate.
As explained by \perp\!\!\!\perpte{imbens2015matching}, OLS linear regression takes linearity very seriously and thus it can render observations with extremely different values from the target profile as highly informative.
\subsection{Weight dispersion and effective sample size}
The variance of the weights is another helpful diagnostic as it directly impacts the variance of the resulting H\'{a}jek estimator under homoscedasticity of the potential outcome models (see, e.g., \perp\!\!\!\perptealt{chattopadhyay2020balancing}). However, the variance of the weights often lacks a clear physical interpretation and thus is not directly suitable as a standalone diagnostic in practice. A related but more practical diagnostic is the effective sample size (ESS) of the weighted sample, which provides an intuitive and palpable measure of the number of units that effectively contribute in a weighted sample.
A standard measure of the ESS of a generic weighted sample with non-negative and normalized weights $\{w_1,...,w_{\tilde{n}}\}$ is given by \perp\!\!\!\perpte{kish1965survey} as $\tilde{n}_{\text{eff}} = 1/\sum_{i=1}^{\tilde{n}}w^2_i$.
However, for negative weights, this measure may take fractional values or values greater than $\tilde{n}$, which are difficult to interpret.
To incorporate negative weights, we propose the following modified definition for the ESS: $\tilde{n}_{\text{eff}} = {(\sum_{i=1}^{\tilde{n}}|w_i|)^2}/{\sum_{i=1}^{\tilde{n}}w_i^2}$.
Intuitively, the magnitude of a unit's weight determines its dominance over the other units in the sample.
Instead of the original weights $w_i$, our definition of the ESS uses Kish's formula on the $|w_i|$.
The above definition ensures that $\tilde{n}_{\text{eff}} \in [1,\tilde{n}]$.
When all the weights are non-negative, this definition boils down to Kish's definition of the ESS.
For both URI and MRI, we recommend computing and reporting the ESS separately for the treatment group and the control group.
The top right panel of Figure \ref{fig:diagnostics} plots the densities of the URI and MRI weights in each treatment group with their corresponding effective sample sizes.
While the ESS of URI in the treatment group is almost $98\%$ of the original treatment group size, the ESS of URI in the control group is only $48\%$ of the original control group size.
This connects to the diagnostics in the previous section where URI proved to have a few units in the control group with extremely large values.
In contrast, the MRI yields a comparatively high and low ESS in the control and treatment groups, respectively.
\subsection{Influence of a given observation}
Finally, we can characterize the influence of each observation on the regression estimator of the ATE by computing its Sample Influence Curve (SIC, \perp\!\!\!\perptealt{cook1982residuals}).
In general, consider an estimator $T(\hat{F})$ of a functional $T(F)$, where $F$ is a distribution function and $\hat{F}$ is its corresponding empirical distribution based on a random sample of size $\tilde{n}$.
The SIC of the $i$th unit in the sample is defined as $\text{SIC}_i = (\tilde{n} - 1) \{T(\hat{F}_{(i)}) - T(\hat{F})\} $, where $\hat{F}_{(i)}$ is the empirical distribution function when the $i$th unit is excluded from the sample.
$\text{SIC}_i$ is thus proportional to the change in the estimator if the $i$th unit is removed from the data.
High values of $\text{SIC}_i$ imply a high influence of the $i$th unit on the resulting estimator.
In Proposition \ref{SIC}, we compute the SIC of the $i$th unit for the URI and MRI estimators of the ATE.
\begin{proposition} \label{SIC}
For the URI and MRI estimators of the ATE, the Sample Influence Curves of unit $i$ in treatment group $g \in \{t,c\}$ are
$\text{SIC}_i = (n-1)(2Z_i-1)e_iw^{\scriptscriptstyle \textrm{URI}}_i/(1-h_{ii,\bm{D}})$ and $\text{SIC}_i = (n_g-1)e_i w^{\scriptscriptstyle \textrm{MRI}}_i/(1-h_{ii,g})$, respectively, where $e_i$ is the residual for unit $i$ under the corresponding regression model, and $h_{ii}$ and $h_{ii,g}$ are the corresponding leverages of unit $i$.
\end{proposition}
Proposition \ref{SIC} says that the SIC of a unit is a function of its residual, leverage, and implied regression weight.
A unit can be influential if either its residual, leverage, or weight are large in magnitude.
In particular, for two units in the same treatment group with the same leverages and residuals, one unit will be more influential than the other if it has a larger weight. However, a large weight alone does not necessarily imply that the unit will have high influence on the corresponding URI or MRI estimator.
We recommend plotting the absolute values of the SIC in Proposition \ref{SIC} for each observation versus its index as a simple graphical diagnostic of influence.
In the lower-right panel of Figure \ref{fig:diagnostics}, we plot the absolute-SIC of MRI and URI.
The values are scaled so that the maximum of the absolute-SIC among the units is one.
The plot for URI indicates the presence of three highly influential units, whereas, the plot for MRI indicates the presence of one highly influential unit.
In such cases, one can also plot the SIC versus each covariate to identify which areas of the covariate space lead to these influential units.
\section{Weighted regression and doubly robust estimation}
\label{sec_weighted}
In this section, we extend the results of sections \ref{sec_implied} and \ref{sec_properties} to weighted least squares (WLS) regression.
In causal inference and sample surveys, WLS can be used to construct doubly robust estimators (e.g., \perp\!\!\!\perptealt{kang2007demystifying}). Here we consider extensions of the URI and MRI approaches to WLS, which we call WURI and WMRI respectively. In both WURI and WMRI, a set of base weights $w^\textrm{base}_i$, $i\in \{1,2,...,n\}$, is used in the WLS step to estimate the coefficients of the respective regression models.
Without loss of generality, we assume that $\sum_{i=1}^{n} w^\textrm{base}_i = 1$ for WURI and that $\sum_{i:Z_i=1} w^\textrm{base}_i = \sum_{i:Z_i=0} w^\textrm{base}_i = 1$ for WMRI.
In addition to WURI and WMRI, here we analyze the widely used bias-corrected doubly robust (DR) estimator or the augmented inverse probability weighted (AIPW) estimator \perp\!\!\!\perptep{robins1994estimation}.
For the DR estimator, a set of base weights $w^\textrm{base}_i$ with $\sum_{i:Z_i=1} w^\textrm{base}_i = \sum_{i:Z_i=0} w^\textrm{base}_i = 1$ is used in a bias-correction term for the MRI estimator.
In particular, for $w^\textrm{base}_i$ equal to the inverse probability weights normalized within each treatment group, we can write the DR estimator of the ATE as $\widehat{\text{ATE}}_{\textrm{DR}} = \big[ n^{-1}\sum_{i=1}^{n} \hat{m}_1(\bm{X}_i) + \sum_{i:Z_i=1} w^\textrm{base}_i \{ Y^{\textrm{obs}}_i - \hat{m}_1(\bm{X}_i) \} \big] - \big[ n^{-1}\sum_{i=1}^{n} \hat{m}_0(\bm{X}_i) + \sum_{i:Z_i=0} w^\textrm{base}_i \{ Y^{\textrm{obs}}_i - \hat{m}_0(\bm{X}_i) \} \big]$.
In sections \ref{sec_implied} and \ref{sec_properties}, we showed that the URI and MRI estimators under OLS admit a weighting representation with weights that can be equivalently obtained by solving a quadratic programming problem that minimizes the variance of the weights subject to a normalization constraint and exact mean balancing constraints for the covariates included in the model.
Here we show that the WURI, WMRI, and DR estimators of the ATE also admit a weighting representation and obtain closed-form expression of their implied weights.
\begin{theorem}
\label{thm_general}
Consider the following quadratic programming problem in the control group
\begin{equation*}\label{calibration}
\begin{aligned}
& \underset{\bm{w}}{\text{minimize}}
& & \sum_{i: Z_i=0}\frac{(w_i - \tilde{w}^\textrm{base}_i)^2}{w^\textrm{scale}_i}\\
& \text{subject to}
& & \mid\sum_{i: Z_i=0} w_i \bm{X}_i - \bm{X}^*\mid \leq \bm{\delta} \\
& & & \sum_{i: Z_i=0} w_i=1\\
\end{aligned}
\end{equation*}
where $\tilde{w}^\textrm{base}_i$ are normalized base weights in the control group, $w^\textrm{scale}_i$ are scaling weights, and $\bm{X}^* \in \mathbb{R}^k$ is a covariate profile, all of them determined by the investigator.
Then, for $\bm{\delta} = \bm{0}$ the solution to this problem is
$$
w_i = \tilde{w}^\textrm{base}_i + w^\textrm{scale}_i(\bm{X}_i - \bar{\bm{X}}_c^\textrm{scale})^\top ( \bm{S}_c^\textrm{scale}/n_c)^{-1} (\bm{X}^* - \bar{\bm{X}}_c^\textrm{base}),
$$
where $\bar{\bm{X}}_c^\textrm{scale} = (\sum_{i:Z_i=0} w^\textrm{scale}_i \bm{X}_i)/(\sum_{i:Z_i=0} w^\textrm{scale}_i) $, $\bar{\bm{X}}_c^\textrm{base} = \sum_{i:Z_i=0} \tilde{w}^\textrm{base}_i \bm{X}_i$, and $\bm{S}_c^\textrm{scale} = n_c \sum_{i:Z_i=0} w^\textrm{scale}_i (\bm{X}_i - \bar{\bm{X}}_c^\textrm{scale}) (\bm{X}_i - \bar{\bm{X}}_c^\textrm{scale})^\top $.
Further, as special cases the implied weights of the WURI, WMRI, and DR estimators for the ATE are
\begin{enumerate}[label=(\alph*)]
\item WURI: $\tilde{w}^\textrm{base}_i = w^\textrm{base}_i/(\sum_{j: Z_{j} = 0}w^\textrm{base}_j)$, $w^\textrm{scale}_i = w^\textrm{base}_i$, $\bm{X}^* = n^{-1}_c\bm{S}_c^\textrm{scale} (n^{-1}_t\bm{S}_t^\textrm{scale} + n^{-1}_c\bm{S}_c^\textrm{scale})^{-1} \bar{\bm{X}}_t^\textrm{scale} + n^{-1}_t \bm{S}_t^\textrm{scale} \left(n^{-1}_t\bm{S}_t^\textrm{scale} + n^{-1}_c \bm{S}_c^\textrm{scale}\right)^{-1} \bar{\bm{X}}_c^\textrm{scale}$.
\item WMRI: $\tilde{w}^\textrm{base}_i = w^\textrm{scale}_i = w^\textrm{base}_i$, $\bm{X}^* = \bar{\bm{X}}$.
\item DR: $\tilde{w}^\textrm{base}_i = w^\textrm{base}_i = \{1-\hat{e}(\bm{X}_i)\}^{-1}/\sum_{j:Z_j=0}\{1-\hat{e}(\bm{X}_j)\}^{-1}$, $w^\textrm{scale}_i = 1$, $\bm{X}^* = \bar{\bm{X}}$.
\end{enumerate}
Here, $\bar{\bm{X}}_t^\textrm{scale} = (\sum_{i:Z_i=1} w^\textrm{scale}_i \bm{X}_i)/(\sum_{i:Z_i=1} w^\textrm{scale}_i)$ and $\bm{S}_t^\textrm{scale} = n_t \sum_{i:Z_i=1} w^\textrm{scale}_i (\bm{X}_i - \bar{\bm{X}}_t^\textrm{scale}) (\bm{X}_i - \bar{\bm{X}}_t^\textrm{scale})^\top$.
The weights for the treated units are obtained analogously. \end{theorem}
Theorem \ref{thm_general} shows that the implied weights of WURI, WMRI, and DR estimators can all be obtained by solving one quadratic programming problem.
Coming back to question about the connection between experiments and regression, this theorem also characterizes the covariate adjustments of WURI, WMRI, and DR estimators in terms of covariate balance and representativeness, weight dispersion, and sample boundedness.
First, the implied weights of all the estimators exactly balance the means of the covariates, but towards different target profiles. While both WMRI and DR estimators balance covariates towards the overall sample mean, WURI balances them towards a target that is harder to interpret.
Second, the weights minimize a general measure of weight dispersion, e.g., the WURI and WMRI weights minimize a Chi-square-type distance from the base weights, whereas the DR weights minimize the Euclidean distance from the base weights.
For uniform base weights, these measures of dispersion boil down to the variance of the weights.
Finally, the weights can be negative and hence can extrapolate, since the underlying quadratic programs do not impose any non-negativity constraint on the weights.
In terms of the above features, the quadratic programming problem in Theorem \ref{thm_general} also clarifies similarities and differences of the WURI, WMRI, and DR weights with the stable balancing weights (\perp\!\!\!\perptealt{zubizarreta2015stable}).
The stable balancing weights are the weights of minimum variance that sum to one, are non-negative, and approximately balance the means of functions of the covariates towards a pre-specified profile.
Here, the non-negativity constraints are used to produce a sample bounded estimator and the approximate balance constraints help to trade bias for variance.
Methodologically, imposing these two types of constraints may correspond to fitting a new type of regularized and sample-bounded regression.
\section{Extensions to other settings}
\label{sec_extensions}
\subsection{Multi-valued treatments}
In this section, we let $Z_i$ be a multi-valued treatment indicator with values $v \in \{1,2,...,\text{V}\}$.
For simplicity in exposition, we consider the average treatment effect of treatment $v$ relative to $1$, $\text{ATE}_{v,1} := \mathbb{E}[Y_i(v) - Y_i(1)]$, $v \in \{2,...,\text{V} \}$.
We can identify $\text{ATE}_{v,1}$ under multi-valued versions of the positivity and unconfoundedness assumptions in Section \ref{sec_notation} (\perp\!\!\!\perptealt{imbens2000role}).
In this case, the positivity assumption is given by $0 < \text{pr}(Z_i = v | \bm{X}_i = \bm{x}) < 1$ for all $\bm{x} \in \text{supp}(\bm{X}_i)$ and all $v \in \{1,...,\text{V}\}$.
The unconfoundedness assumption states that $\mathbbm{1}(Z_i = v) \perp\!\!\!\perp Y_i(v) |\bm{X}_i$ for all $v \in \{1,...,\text{V}\}$.
It is straightforward to extend the URI and MRI approaches to this setting.
In the MRI approach, we fit separate linear models of the outcome on the observed covariates in each treatment group and estimate the conditional mean functions $m_v(\bm{x}) := \mathbb{E}[Y_i(v)|\bm{X}_i = \bm{x}] = \mathbb{E}[Y^{\textrm{obs}}_i|\bm{X}_i = \bm{x},Z_i=v]$.
The estimator we consider is $\widehat{\text{ATE}}_{v,1} = \frac{1}{n}\sum_{i=1}^{n} \hat{m}_v(\bm{X}_i) - \frac{1}{n}\sum_{i=1}^{n} \hat{m}_1(\bm{X}_i) = \sum_{i:Z_i=v} w^{{\scriptscriptstyle \textrm{MRI}}}_i Y^{\textrm{obs}}_i - \sum_{i:Z_i=1} w^{{\scriptscriptstyle \textrm{MRI}}}_iY^{\textrm{obs}}_i$, where the $w^{\scriptscriptstyle \textrm{MRI}}_i$ has the same form as given in Proposition \ref{prop_mri}. As a result, the properties of the MRI weights discussed in Section \ref{sec_properties} carry over to this case. We observe that $\widehat{\text{ATE}}_{v,1}$ only uses information from treatment groups $v$ and $1$ and not from other groups.
This is an important distinction with the multi-valued version of the URI approach, which we will discuss next.
Arguably, the URI approach is the more common approach to estimating the causal effects of multi-valued treatments.
This approach is widely used in related problems such as hospital quality measurement (e.g., \perp\!\!\!\perptealt{krumholz2011administrative}).
Here, we consider the linear regression model $Y^{\textrm{obs}}_i = \beta_0 + \bm{\beta}^\top_1 \bm{X}_i + \sum_{v=2}^{\text{V}}\tau_{v,1} \mathbbm{1}(Z_i=v) + \epsilon_i $.
We denote the URI estimator of $\text{ATE}_{v,1}$ by $\hat{\tau}^{\text{OLS}}_{v,1}$.
By Proposition \ref{prop_uri}, $\hat{\tau}^{\text{OLS}}_{v,1} = \sum_{i:Z_i=v}w^{{\scriptscriptstyle \textrm{URI}}}_{i,v,1} Y^{\textrm{obs}}_i - \sum_{Z_i \neq v} w^{{\scriptscriptstyle \textrm{URI}}}_{i,v,1} Y^{\textrm{obs}}_i$.
The additional subscripts $v,1$ in $w^{{\scriptscriptstyle \textrm{URI}}}_{i,v,1}$ highlight the active treatment group and the reference treatment group respectively, since, unlike the MRI approach, the URI approach uses information from (i.e., puts non-zero weights on) all the treatment groups besides groups $v$ and $1$.
In other words, for estimating the average treatment effect of level $v$ compared to level $1$, the URI approach borrows strength from other treatment groups through linearity.
However, Proposition \ref{prop_finite_sample_properties}(a) implies $\sum_{i:Z_i=r} w^{{\scriptscriptstyle \textrm{URI}}}_{i,v,1} = 0$ for $r \in \{2,...,\text{V}\}, r\neq v$.
Also, the URI weights achieve exact mean balance on $\bm{X}_i$ between group $v$ and the combined group $\{r\in\{1,2,...,V\}: r \neq v\}$, but not necessarily towards the full sample.
We state an implication of this last fact when we include a rare (sparse) covariate in the regression model.
Let $k = 1$ and $\bm{X}_i$ be an indicator variable, e.g., for an under represented minority.
Suppose there are no minority persons in treatment group $v$ (i.e., $\bm{X}_i = 0$ for all $i: Z_i = v$), but they are present in other treatment groups.
By Proposition \ref{prop_finite_sample_properties}(a), $\sum_{i:Z_i \neq v}w^{{\scriptscriptstyle \textrm{URI}}}_{i,v,1} \bm{X}_i = \sum_{i:Z_i = v}w^{{\scriptscriptstyle \textrm{URI}}}_{i,v,1} \bm{X}_i = 0$.
Here, the treatment groups different to $v$ are weighted in such a way that effectively results in zero proportion of minority persons in the combined weighted group.
Therefore, if one uses the URI approach to estimate the effect of treatment $v$ in the whole population, one ends up comparing the treatment groups in a weighted sample that \textit{zeroes out} the minority.
This is appropriate if the estimand of interest is the effect of treatment $v$ on those who received treatment $v$.
However, even in that case, the implied URI weights may not balance the other covariates relative to treatment group $v$.
Now, let $\underline{\tilde{\bm{X}}}_v$ and $\underline{\tilde{\bm{X}}}$ be the design matrices in treatment group $v$ and the full sample, respectively. In the above example, $\tilde{\underline{\bm{X}}}^{\top}_v \tilde{\underline{\bm{X}}}_v$ is not invertible, which provides a warning about the infeasibility of estimating the effect of treatment $v$ using MRI.
However, here URI can still be feasible as the matrix $\tilde{\underline{\bm{X}}}^{\top} \tilde{\underline{\bm{X}}}$ may be invertible.
By borrowing strength from linearity, URI estimates the effect of treatment $v$ on a population that includes the minority, despite not having any data for them in treatment group $v$.
Since the pooled design matrix $\tilde{\underline{\bm{X}}}$ masks the singularity of $\tilde{\underline{\bm{X}}}_v$, this strong functional form assumption of the outcome can fail to be noticed by an investigator using URI.
Therefore, as a diagnostic for URI, we recommend checking the invertibility of $\tilde{\underline{\bm{X}}}^{\top}_v \tilde{\underline{\bm{X}}}_v$ for all $v \in \{1,2,...,V\}$.
This is equivalent to checking multicollinearity of the design matrix within each treatment group, which can be done using measures such as condition numbers of the design matrices and variance inflation factors (see, e.g., Chapter 9 of \perp\!\!\!\perptealt{chatterjee2015regression}).
\subsection{Regression adjustment after matching}
\perp\!\!\!\perpte{rubin1979using} studied the ATT estimation problem using regression after matching.
He considered pair matching without replacement and an estimator of the form $\widehat{\text{ATT}} = (\bar{Y}_{mt} - \bar{Y}_{mc}) - (\bar{\bm{X}}_{mt} - \bar{\bm{X}}_{mc})^\top \hat{\bm{\beta}}$, where the subindices $mt$ and $mc$ denote the matched treated and control groups.
If $\hat{\bm{\beta}} = 0$, the estimator reduces to the simple difference in outcome means between the matched treated and control groups.
Another choice of $\hat{\bm{\beta}}$ arises from a two-group analysis of covariance model which ignores the matched pair structure and is equivalent to the URI method.
Following \perp\!\!\!\perpte{rubin1979using}, we are interested in a third choice of $\hat{\bm{\beta}}$, which corresponds to regressing the matched-pair differences of the outcome on the matched-pair differences of the covariates.
We examine this approach through the lens of its implied weights.
In Proposition \ref{matching_reg} we describe some finite sample properties of this approach.
\begin{proposition}\label{matching_reg}
For a matched sample of size $\tilde{n}$, let $Y^{\textrm{obs}}_{ti}$ and $Y^{\textrm{obs}}_{ci}$ (likewise, $\bm{X}_{ti}$ and $\bm{X}_{ci}$) be the observed outcomes (observed covariate vectors) of the $i$th pair of matched treated and control units, $i \in \{1,2,...,\tilde{n}\}$.
Let $\hat{\bm{\beta}}$ be the OLS estimator of $\bm{\beta}$ in the regression model $Y_{di} = \alpha + \bm{\beta}^\top \bm{X}_{di} + \epsilon_i$, $i = 1,...,\tilde{n}$, where $\bm{X}_{di} = \bm{X}_{ti} - \bm{X}_{ci}$, $Y_{di} = Y_{ti} - Y_{ci}$. Then the regression adjusted matching estimator can be written as
$$
\widehat{\emph{ATT}} = (\bar{Y}_{mt} - \bar{Y}_{mc}) - (\bar{\bm{X}}_{mt} - \bar{\bm{X}}_{mc})^\top \hat{\bm{\beta}} = \sum_{i=1}^{\tilde{n}} w_i (Y_{ti} - Y_{ci}),
$$
where $w_i = \frac{1}{\tilde{n}} - \bar{\bm{X}}_d^\top \bm{S}^{-1}_d (\bm{X}_{di} - \bar{\bm{X}}_d)$, $\bar{\bm{X}}_d = \bar{\bm{X}}_{mt} - \bar{\bm{X}}_{mc}$, $\bm{S}_d = \sum_{i=1}^{\tilde{n}}(\bm{X}_{di} - \bar{\bm{X}}_d)(\bm{X}_{di} - \bar{\bm{X}}_d)^\top$.
Furthermore, the weights satisfy (a) $\sum_{i=1}^{\tilde{n}} w_i = 1$ and (b) $\sum_{i=1}^{\tilde{n}}w_i \bm{X}^\top_{ti} = \sum_{i=1}^{\tilde{n}}w_i \bm{X}^\top_{ci} = \bar{\bm{X}}^\top_t \big\{\bm{S}^{-1}_d(\bm{S}_{mc} - \bm{S}_{mtc})\big\} + \bar{\bm{X}}^{\top}_c \big\{\bm{S}^{-1}_d(\bm{S}_{mt} - \bm{S}_{mct})\big\}$, where
$\bm{S}_{mtc} = \sum_{i=1}^{\tilde{n}}(\bm{X}_{ti} - \bar{\bm{X}}_{mt})(\bm{X}_{ci} - \bar{\bm{X}}_{mc})^\top$ and $\bm{S}_{mct} = \sum_{i=1}^{\tilde{n}}(\bm{X}_{ci} - \bar{\bm{X}}_{mc})(\bm{X}_{ti} - \bar{\bm{X}}_{mt})^\top$.
\end{proposition}
Proposition \ref{matching_reg} shows that the regression adjusted matching estimator can be expressed as a H\'{a}jek estimator and provides a closed-form expression for its implied weights.
It reveals a special structure of the weights, namely, that the weight of a treated unit is the same as the weight of its matched control.
The proposition also shows that the implied weights exactly balance the means of the covariates, albeit toward a covariate profile that does not correspond to the intended target group.
In other words, while regression adjustment after matching successfully reduces the residual imbalances between the treated and control groups after matching, it can move the two groups away from the target covariate profile $\bar{\bm{X}_t}$.
If the treated and control groups are well-balanced on the mean of $\bm{X}_i$ after the matching step, the implied weights of the regression adjustment step tend to be close to uniform, leading to small mean-imbalance on $\bm{X}_i$ relative to the target.
We illustrate these results in the Lalonde study. Here we consider the problem of estimating the ATT. We first obtain 135 pairs of matched treated and control units using cardinality matching on the means of the 8 original covariates (\perp\!\!\!\perptealt{zubizarreta2014matching}). Subsequently, we fit a linear regression model of the treatment-control difference of the outcome within each matched pair on the treatment-control difference of the 8 covariates within that pair. We compute the TASMD of each of the 8 covariates in both treatment and control groups before matching, after matching, and after regression adjustment on the matched-pair differences. Since the estimand is the ATT, the TASMDs are calculated relative to the unmatched treatment group. The Love plots of the TASMDs are shown in Figure \ref{fig:TASMD_match}.
\begin{figure}
\caption{\footnotesize TASMDs before matching, after matching, and after regression adjustment on the matched-pair differences for the Lalonde data set.}
\label{fig:TASMD_match}
\end{figure}
The right panel of Figure $\ref{fig:TASMD_match}$ shows that the control group is heavily imbalanced relative to the treatment group prior to matching.
Matching reduces these imbalances, but leaves scope for further reductions relative to the treatment group.
Finally, the regression adjustment step following the matching step reduces the imbalance on some covariates (e.g. `black', `nodegree'), but more importantly, worsens the balance achieved by matching on a few covariates (e.g., `re74',`age'). A more prominent repercussion of regression adjustment on the matched sample occurs for the treatment group, as shown in the left panel of Figure \ref{fig:TASMD_match}. Here the initial matching step trims some treated units, leading to a modest increase in imbalances relative to the unmatched treated group. The regression adjustment step, however, substantially increases these imbalances for almost all the covariates.
Figure \ref{fig:TASMD_match} thus reiterates the result of Proposition \ref{matching_reg} in that here the mean covariate profiles of both the weighted treatment and control groups after regression, although equal, are shifted away from the target profile of interest. As seen in Section \ref{sec_finite}, a similar phenomenon occurs if URI is used to adjust for covariates after matching.
From the perspective of bias, unless the treatment effects are homogeneous (which was one of the assumptions in \perp\!\!\!\perptealt{rubin1979using}), this shift from the target profile leads to bias in the treatment effect estimate.
\section{Conclusion}
\label{sec_conclusion}
Across the health and social sciences, linear regression is extensively used in observational studies to adjust for covariates and estimate the effects of treatments.
In this paper, we demonstrated how linear regression adjustments in observational studies emulate key features of randomized experiments.
For this, we represented regression estimators as weighting estimators and derived their implied weights.
We obtained new closed-form, finite-sample expressions for the weights for various types of estimators (URI, MRI, AIPW) based on multivariate linear regression models under both ordinary and weighted least squares.
Among others, we obtained a closed-form expression of the covariate profile targeted by the implied weights, which, in turn, characterizes the implied target population of regression.
In this regard, we showed that URI estimators may distort the structure of the study sample in such a way that the resulting estimator is biased for the average treatment effect in the target population.
In contrast, MRI estimators preserve the structure of this population in terms of its first moments, albeit with negative weights.
This implied weighting framework allowed us to connect ideas from the regression modeling and the causal inference literature, and leveraging this connection, to propose a set of new regression diagnostics for causal inference.
These diagnostics allow researchers to (i) evaluate the degree of covariate balance of each weighted treatment group relative to the target population, (ii) compute the effective sample size of each treatment group after regression adjustments, (iii) assess the degree of extrapolation and sample-boundedness of the regression estimates, and (iv) measure the influence of a given observation on an average treatment effect estimate.
We also showed that the URI and MRI estimators are both multiply robust for the average treatment effect, albeit under conditions that impose stringent functional forms on the treatment model, the potential outcome models, or a combination of both.
Extensions of this work include the implied weights of regression with continuous and multi-valued treatments, and regression after matching.
As future work, we plan to extend the implied weights framework to analyze instrumental variables methods \perp\!\!\!\perptep{abadie2003semiparametric} and fixed-effects regressions in difference-in-differences settings \perp\!\!\!\perptep{ding2019bracketing}.
\pagebreak
\section{Supplementary materials}
\section{Additional theoretical results}
\subsection{Derivation of the WURI weights}
\label{wuri_derive}
Here we use the notations of Theorem 3. In WURI, we fit the model $\bm{y} = \mu \bm{1} + \underline{\bm{X}}\bm{\beta} + \tau \bm{Z} + \bm{\epsilon} $ using WLS with the base weights $(w^{\text{base}}_1,...,w^{\text{base}}_n)$. Without loss of generality, assume $\sum_{i=1}^{n}w^{\text{base}}_i =1$. Also, for all $i \in \{1,...,n\}$, let $w^{\text{scale}}_i = w^{\text{base}}_i$. Denote the design matrix based on the full sample, treatment group, and control group as $\underline{\tilde{\bm{X}}}$, $\underline{\tilde{\bm{X}}}_t$, and $\underline{\tilde{\bm{X}}}_c$ respectively. Also, let $\bm{W}^{\frac{1}{2}} = diag((w^{\text{base}}_1)^{0.5},...,(w^{\text{base}}_n)^{0.5})$, $\bm{W} = \bm{W}^{\frac{1}{2}}\bm{W}^{\frac{1}{2}}$, $\dbar{\bm{X}} = \bm{W}^{\frac{1}{2}} \underline{\tilde{\bm{X}}} = \bm{W}^{\frac{1}{2}}
(\bm{1} , \bm{X})$, $\dbar{\bm{y}} = \bm{W}^{\frac{1}{2}} \bm{y}$, $\dbar{\bm{Z}} = \bm{W}^{\frac{1}{2}}\bm{Z}$. The objective function under WLS is given by
\begin{align}
\mathop{\mathrm{argmin}}_{\mu,\bm{\beta}, \tau} \Big(\bm{y} - \underline{\tilde{\bm{X}}}\begin{psmallmatrix}
\mu\\
\bm{\beta}
\end{psmallmatrix} - \tau \bm{Z} \Big)^\top \bm{W} \Big(\bm{y} - \underline{\tilde{\bm{X}}}\begin{psmallmatrix}
\mu\\
\bm{\beta}
\end{psmallmatrix} - \tau \bm{Z} \Big) \nonumber\\
= \mathop{\mathrm{argmin}}_{\mu,\bm{\beta}, \tau} \Big(\dbar{\bm{y}} - \dbar{\bm{X}}\begin{psmallmatrix}
\mu\\
\bm{\beta}
\end{psmallmatrix} - \tau \dbar{\bm{Z}} \Big)^\top \Big(\dbar{\bm{y}} - \dbar{\bm{X}}\begin{psmallmatrix}
\mu\\
\bm{\beta}
\end{psmallmatrix} - \tau \dbar{\bm{Z}} \Big).
\label{s1.1.1}
\end{align}
Therefore, the objective function under WLS is equivalent to that under OLS for a linear regression of $\dbar{\bm{y}}$ on $\dbar{\bm{X}}$ and $\dbar{\bm{Z}}$. Let $\underline{\bm{I}}$ be the identity matrix of order $k+1$ and $\mathcal{P}_{\dbar{\bm{X}}}$ be the projection matrix onto the column space of $\dbar{\bm{X}}$. Using the Frisch-Waugh-Lovell Theorem (\perp\!\!\!\perptealt{frisch1933partial, lovell1963seasonal}), we can write the corresponding estimator of $\tau$ as
\begin{equation}
\hat{\tau} = (\dbar{\bm{Z}}^\top (\underline{\bm{I}} - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{y}})/(\dbar{\bm{Z}}^\top (\underline{\bm{I}} - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{Z}}) = \bm{l}^\top \bm{y},
\label{s1.1.2}
\end{equation}
where $\bm{l} = (\bm{W}^{\frac{1}{2}}(I - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{Z}})/(\dbar{\bm{Z}}^\top (I - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{Z}})$. Denote $m_t = \sum_{i:Z_i=1}w^{\text{base}}_i $ and $m_c = \sum_{i:Z_i=0}w^{\text{base}}_i $. By assumption, $m_t + m_c = 1$.
Now, the denominator $\dbar{\bm{Z}}^\top (\underline{\bm{I}} - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{Z}} = m_t - \bm{Z}^\top \bm{W} \underline{\tilde{\bm{X}}}(\underline{\tilde{\bm{X}}}^\top \bm{W}\underline{\tilde{\bm{X}}})^{-1} \underline{\tilde{\bm{X}}}^\top \bm{W} \bm{Z} = m_t \big\{1 - m_t (1, \bar{\bm{X}}^{\text{scale}\top}_{t})(\underline{\tilde{\bm{X}}}^\top \bm{W}\underline{\tilde{\bm{X}}})^{-1} (1, \bar{\bm{X}}^{\text{scale}\top}_{t})^\top \big\} $. Similarly, the numerator is
\begin{equation}
\bm{W}^{\frac{1}{2}}(\underline{\bm{I}} - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{Z}} = \bm{WZ} - \bm{W}\underline{\tilde{\bm{X}}}(\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} \underline{\tilde{\bm{X}}}^{\top} \bm{W} \bm{Z}
\label{s1.1.3}
\end{equation}
To simplify the expression, let us assume, without loss of generality, that the first $n_t$ units in the sample are in the treatment group and the rest are in the control group. Moreover, let $\bm{w}_t = (w^{\text{base}}_1,....,w^{\text{base}}_{n_t})$ and $\bm{w}_c = (w^{\text{base}}_{n_t+1},....,w^{\text{base}}_{n})$ be the vector of base weights for the treatment and control group respectively. This implies
\begin{align}
\bm{W}^{\frac{1}{2}}(I - \mathcal{P}_{\dbar{\bm{X}}})\dbar{\bm{Z}} & = \begin{psmallmatrix}
\bm{w}_t\\
\bm{0}
\end{psmallmatrix} - \bm{W} \begin{psmallmatrix}
\underline{\tilde{\bm{X}}}_t (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} \underline{\tilde{\bm{X}}}^\top_t \bm{w}_t\\
\underline{\tilde{\bm{X}}}_c (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} \underline{\tilde{\bm{X}}}^\top_t \bm{w}_t
\end{psmallmatrix}
\label{s1.1.4}
\end{align}
From Equation \ref{s1.1.2} we get that we can write $\hat{\tau}$ as $\hat{\tau} = \sum_{i:Z_i=1}w_i Y^{\text{obs}}_i - \sum_{i:Z_i=0}w_i Y^{\text{obs}}_i $. Let $\bar{\bm{X}}^{\text{scale}} = \sum_{i=1}^{n} w^{\text{scale}}_i\bm{X}_i$.
From Equation \ref{s1.1.4}, it follows that if the $i$th unit belongs to the treatment group then
\begin{align}
w_i &= \frac{w^{\text{base}}_i \big\{ 1 - (1,\bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} \underline{\tilde{\bm{X}}}_t^\top \bm{w}_t \big\}}{ m_t \big(1 - m_t (1, \bar{\bm{X}}^{\text{scale}\top}_{t})(\underline{\tilde{\bm{X}}}^\top \bm{W}\underline{\tilde{\bm{X}}})^{-1} (1, \bar{\bm{X}}^{\text{scale}\top}_{t})^\top \big)} \nonumber\\
& = \frac{w^{\text{base}}_i \big(1 - m_t(0,(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top) (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} (1, \bar{\bm{X}}^{\text{scale}\top}_{t})^\top - m_t (1,\bar{\bm{X}}^{\text{scale}\top}_{t}) (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} (1,\bar{\bm{X}}^{\text{scale}\top}_{t})^\top \big)}{ m_t \big(1 - m_t (1, \bar{\bm{X}}^{\text{scale}\top}_{t})(\underline{\tilde{\bm{X}}}^\top \bm{W}\underline{\tilde{\bm{X}}})^{-1} (1, \bar{\bm{X}}^{\text{scale}\top}_{t})^\top \big)} \nonumber\\
& = w^{\text{base}}_i \Big[ \frac{1}{m_t} + \frac{(0,(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top) (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} (0,(\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t})^\top) }{1 - m_t (1, \bar{\bm{X}}^{\text{scale}\top}_{t})(\underline{\tilde{\bm{X}}}^\top \bm{W}\underline{\tilde{\bm{X}}})^{-1} (1, \bar{\bm{X}}^{\text{scale}\top}_{t})^\top } \Big] \nonumber\\
& = w^{\text{base}}_i \Big[ \frac{1}{m_t} + \frac{(0,(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top) (\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} (0,(\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t})^\top) }{1 - m_t \big( 1 + (0, (\bar{\bm{X}}^{\text{scale}}-\bar{\bm{X}}^{\text{scale}}_{t})^\top)(\underline{\tilde{\bm{X}}}^\top \bm{W}\underline{\tilde{\bm{X}}})^{-1} (0, (\bar{\bm{X}}^{\text{scale}}-\bar{\bm{X}}^{\text{scale}}_{t})^\top)^\top \big)} \Big] \nonumber\\
& = w^{\text{base}}_i \Big[ \frac{1}{m_t} + \frac{(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top (\frac{\bm{S}^{\text{scale}}}{n})^{-1} (\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t}) }{1 - m_t \big( 1 + (\bar{\bm{X}}^{\text{scale}}-\bar{\bm{X}}^{\text{scale}}_{t})^\top(\frac{\bm{S}^{\text{scale}}}{n})^{-1} (\bar{\bm{X}}^{\text{scale}}-\bar{\bm{X}}^{\text{scale}}_{t}) \big)} \Big],
\label{s1.1.5}
\end{align}
where $\bm{S}^{\text{scale}} = n\big(\sum_{j=1}^{n} w^{\text{scale}}_{j} \bm{X}_{j} \bm{X}^\top_{j} - \bar{\bm{X}}^{\text{scale}} \bar{\bm{X}}^{\text{scale}\top}\big) $. The second equality holds since $\underline{\tilde{\bm{X}}}_t^\top \bm{w}_t = m_t(1, \bar{\bm{X}}^{\text{scale}\top}_{t})^\top$. The third and fourth equality hold since $(1, \bar{\bm{X}}^{\text{scale}\top})^\top = \underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}} \bm{e}_1$, where $\bm{e}_1 = (1,0,0,...,0)$ is the first standard basis vector of $\mathbb{R}^{k+1}$. To see the fifth equality, we first see that $\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}} = \begin{pmatrix}
1 & \bar{\bm{X}}^{\text{scale}\top} \\
\bar{\bm{X}}^{\text{scale}} & \underline{\bm{X}}^\top \bm{W} \underline{\bm{X}}
\end{pmatrix}$. Let $(\underline{\tilde{\bm{X}}}^\top \bm{W} \underline{\tilde{\bm{X}}})^{-1} = \begin{pmatrix}
{B_{11}}^{(1\times 1)} & {B_{12}}^{(1\times k))} \\
{B_{21}}^{(k \times 1)} & {B_{22}}^{(k \times k)}
\end{pmatrix}$. Using the formula for the inverse of a partitioned matrix, we get $B_{22} = \big[\underline{\bm{X}}^\top \bm{W} \underline{\bm{X}} - \bar{\bm{X}}^{\text{scale}} \bar{\bm{X}}^{\text{scale}\top} \big]^{-1} = \big[ \sum_{j=1}^{n} w^{\text{scale}}_{j} \bm{X}_{j} \bm{X}^\top_{j} - \bar{\bm{X}}^{\text{scale}} \bar{\bm{X}}^{\text{scale}\top} \big]^{-1} = (\frac{\bm{S}^{\text{scale}}}{n})^{-1} $. Now, we observe that $\bm{S}^{\text{scale}} = n(\frac{\bm{S}^{\text{scale}}_t}{n_t} + \frac{\bm{S}^{\text{scale}}_c}{n_c} + m_t m_c (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)(\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)^\top )$. This implies,
\begin{align}
&\big(\frac{\bm{S}^{\text{scale}}_t}{n_t} + \frac{\bm{S}^{\text{scale}}_c}{n_c} \big)\big(\frac{\bm{S}^{\text{scale}}}{n}\big)^{-1} (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c) \nonumber\\ &= \big(\frac{\bm{S}^{\text{scale}}}{n} - m_t m_c (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)(\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)^\top \big)\big(\frac{\bm{S}^{\text{scale}}}{n}\big)^{-1} (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c) \nonumber\\
&= (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c) - \chi (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c) \nonumber\\
&\implies \big(\frac{\bm{S}^{\text{scale}}_t}{n_t} + \frac{\bm{S}^{\text{scale}}_c}{n_c} \big)^{-1} (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c) = \big(\frac{\bm{S}^{\text{scale}}}{n}\big)^{-1} (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)/(1-\chi) ,
\label{s1.1.6}
\end{align}
where $\chi = m_t m_c (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)^\top \big(\frac{\bm{S}^{\text{scale}}}{n}\big)^{-1} (\bar{\bm{X}}^{\text{scale}}_{t} - \bar{\bm{X}}^{\text{scale}}_c)$. From Equations \ref{s1.1.5} and \ref{s1.1.6}, we get
\begin{align}
w_i & = w^{\text{base}}_i \Big[ \frac{1}{m_t} + \frac{(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top (\frac{\bm{S}^{\text{scale}}}{n})^{-1} (\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t}) }{m_c (1-\chi)} \Big] \nonumber\\
&= w^{\text{base}}_i \Big[ \frac{1}{m_t} + \frac{1}{m_c}(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top \big(\frac{\bm{S}^{\text{scale}}_t}{n_t} + \frac{\bm{S}^{\text{scale}}_c}{n_c} \big)^{-1} (\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t}) \Big] \nonumber\\
&= \tilde{w}^{\text{base}}_i + \frac{w^{\text{scale}}_i}{m_c}(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_{t})^\top \big(\frac{\bm{S}^{\text{scale}}_t}{n_t} + \frac{\bm{S}^{\text{scale}}_c}{n_c} \big)^{-1} (\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t}),
\label{s1.1.7}
\end{align}
where in the last equality, we have used the fact that the base weights are same as the scaling weights. Now, let $\bm{X}^* = \frac{\bm{S}_c^\text{scale}}{n_c} \left(\frac{\bm{S}_t^\text{scale}}{n_t} + \frac{\bm{S}_c^\text{scale}}{n_c}\right)^{-1} \bar{\bm{X}}_t^\text{scale} + \frac{\bm{S}_t^\text{scale}}{n_t} \left(\frac{\bm{S}_t^\text{scale}}{n_t} + \frac{\bm{S}_c^\text{scale}}{n_c}\right)^{-1} \bar{\bm{X}}_c^\text{scale}$. By simple substitution and using the fact that $\bar{\bm{X}}^{\text{base}} = \bar{\bm{X}}^{\text{scale}}$, we get
\begin{equation}
\Big(\frac{\bm{S}^{\text{scale}}_t}{n_t} \Big)^{-1} (\bm{X}^* - \bar{\bm{X}}^{\text{base}}_t) = \frac{1}{m_c} \big(\frac{\bm{S}^{\text{scale}}_t}{n_t} + \frac{\bm{S}^{\text{scale}}_c}{n_c} \big)^{-1} (\bar{\bm{X}}^{\text{scale}} - \bar{\bm{X}}^{\text{scale}}_{t}).
\label{s1.1.8}
\end{equation}
This implies,
\begin{equation}
w_i = \tilde{w}^\text{base}_i + w^\text{scale}_i(\bm{X}_i - \bar{\bm{X}}_t^\text{scale})^\top \left( \frac{\bm{S}_t^\text{scale}}{n_t} \right)^{-1} (\bm{X}^* - \bar{\bm{X}}_t^\text{base}).
\end{equation}
Using the structural symmetry between the treatment and control group, it follows that, if the $i$th unit belongs to the control group, then
\begin{equation*}
w_i = \tilde{w}^\text{base}_i + w^\text{scale}_i(\bm{X}_i - \bar{\bm{X}}_c^\text{scale})^\top \left( \frac{\bm{S}_c^\text{scale}}{n_c} \right)^{-1} (\bm{X}^* - \bar{\bm{X}}_c^\text{base}).
\end{equation*}
\subsection{Derivation of WMRI weights}
\label{wmri_derive}
Here we use the notations of Theorem 3. Without loss of generality, let us assume that the first $n_c$ units in the sample belong to the control group. In WMRI, we fit the linear model $\bm{y}_c = \beta_{0c} \bm{1} + \underline{\bm{X}}_c \bm{\beta}_{1c} + \bm{\epsilon}_c$ in the control group using WLS with the base weights $(w^{\text{base}}_1,...,w^{\text{base}}_{n_c})$, where $\sum_{i=1}^{n_c}w^{\text{base}}_i = 1$. For all $i\in \{1,...,n_c\}$, let $w^{\text{scale}}_i = w^{\text{base}}_i$. Also, let $\bm{W}_c = diag(w^{\text{base}}_1,...,w^{\text{base}}_{n_c})$.
The WLS estimator of the parameter vector $\bm{\beta}_c = (\beta_{0c}, \bm{\beta}^\top_{1c})^\top$ is given by $\hat{\bm{\beta}}_{c} = (\underline{\tilde{\bm{X}}}^\top \bm{W}_c \underline{\tilde{\bm{X}}})^{-1} \underline{\tilde{\bm{X}}}^\top \bm{W}_c \bm{y}_c $. The estimated mean function of the control potential outcome for any unit with a generic covariate profile $\bm{x}$ is $\hat{m}_0(\bm{x}) = \hat{\beta}_{0c} + \hat{\bm{\beta}}^\top_{1c} \bm{x} = (\bm{w}_c)^\top \bm{y}_c$, where $\bm{w}_c = (w_1,...,w_{n_c})^\top$ is given by
\begin{align}
\bm{w}_c &= \bm{W}_c^\top \underline{\tilde{\bm{X}}}_c (\underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c)^{-1} (1, \bm{x}^{\top})^\top \nonumber \\
&= \bm{W}_c \underline{\tilde{\bm{X}}}_c (\underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c)^{-1} (1, \bar{\bm{X}}^{\text{scale}\top}_c)^\top + \bm{W}_c \underline{\tilde{\bm{X}}}_c (\underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c)^{-1} (0, (\bm{x} - \bar{\bm{X}}^{\text{scale}}_c)^\top)^\top \nonumber\\
&= \bm{W}_c \bm{1} + \bm{W}_c \underline{\tilde{\bm{X}}}_c (\underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c)^{-1} (0, (\bm{x} - \bar{\bm{X}}^{\text{scale}}_c)^\top)^\top
\label{s1.2.1}
\end{align}
The second inequality is obtained by noting that $(1, \bar{\bm{X}}^{\text{scale}\top}_c)^\top = \underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \bm{1} = \underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c \bm{e}_1$, where $\bm{e}_1 = (1,0,...,0)^\top$ is the first standard basis vector of $\mathbb{R}^{k+1}$. Therefore, $\hat{m}_0(\bm{x}) = \sum_{i:Z_i=0} w_i Y^{\text{obs}}_i$, where
\begin{align}
w_i &= w^{\text{base}}_i \big\{ 1 + (1,\bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c)^{-1} (0, (\bm{x} - \bar{\bm{X}}^{\text{scale}}_c)^\top)^\top \big\} \nonumber\\
&= w^{\text{base}}_i \big\{ 1 + (0,(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_c)^\top) (\underline{\tilde{\bm{X}}}^\top_c \bm{W}_c \underline{\tilde{\bm{X}}}_c)^{-1} (0, (\bm{x} - \bar{\bm{X}}^{\text{scale}}_c)^\top)^\top \big\} \nonumber\\
&= w^{\text{base}}_i + w^{\text{base}}_i(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_c)^\top \big(\frac{\bm{S}^{\text{scale}}_c}{n_c}\big)^{-1} (\bm{x} - \bar{\bm{X}}^{\text{scale}}_c).
\label{s1.2.2}
\end{align}
The last equality holds by similar arguments as in the WURI case. Using the fact that the base weights and the scaling weights are the same and that the base weights are normalized, we get
\begin{equation}
w_i = \tilde{w}^{\text{base}}_i + w^{\text{scale}}_i(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_c)^\top \big(\frac{\bm{S}^{\text{scale}}_c}{n_c}\big)^{-1} (\bm{x} - \bar{\bm{X}}^{\text{base}}_c).
\label{s1.2.3}
\end{equation}
Similarly, if we fit the linear model $\bm{y}_t = \beta_{0t} \bm{1} + \bm{X}_c \bm{\beta}_{1t} + \bm{\epsilon}_c $ in the treatment group using WLS then, by similar steps, the estimated mean function of the treatment potential outcome for any unit with covariate profile $\bm{x}^*$ is given by $\hat{m}_1(\bm{x}^*) = \hat{\beta}_{0t} + \hat{\bm{\beta}}^\top_{1t} \bm{x}^* = \sum_{i=1}^{n_t} w_i Y^{\text{obs}}_{i,t} $, where
\begin{equation}
w_i = \tilde{w}^{\text{base}}_i + w^{\text{scale}}_i(\bm{X}_i - \bar{\bm{X}}^{\text{scale}}_t)^\top \big(\frac{\bm{S}^{\text{scale}}_t}{n_t}\big)^{-1} (\bm{x} - \bar{\bm{X}}^{\text{base}}_t).
\label{s1.2.4}
\end{equation}
Therefore, we have the following results.
\begin{enumerate}
\item By linearity, he WMRI estimator of the ATE is $\widehat{\text{ATE}} = \hat{m}_1(\bar{\bm{X}}) - \hat{m}_0(\bar{\bm{X}}) = \sum_{i:Z_i=1} w_i Y^{\text{obs}}_i - \sum_{i:Z_i=0} w_i Y^{\text{obs}}_i$, where $w_i$ has the form as given in Equations \ref{s1.2.3} and \ref{s1.2.4} with $\bm{x}$ replaced by $\bar{\bm{X}}$.
\item By linearity, he WMRI estimator of the ATT is $\widehat{\text{ATT}} = \bar{Y}_t - \hat{m}_0(\bar{\bm{X}}_t) = \bar{Y}_t - \sum_{i:Z_i=0} w_i Y^{\text{obs}}_i$, where $w_i$ has the form as given in Equation \ref{s1.2.3} with $\bm{x}$ replaced by $\bar{\bm{X}}_t$.
\item The WMRI estimator of the CATE at covariate profile $\bm{x}^*$ is given by $\widehat{\text{CATE}}(\bm{x}^*) = \hat{m}_1(\bm{x}^*) - \hat{m}_0(\bm{x}^*) = \sum_{i:Z_i=1} w_i Y^{\text{obs}}_i - \sum_{i:Z_i=0} w_i Y^{\text{obs}}_i$, where $w_i$ has the form as given in Equations \ref{s1.2.3} and \ref{s1.2.4} with $\bm{x}$ replaced by $\bm{x}^*$.
\end{enumerate}
\subsection{Derivation of the DR weights}
\label{drest_derive}
The DR estimator is given by $\widehat{\text{ATE}}_{\text{DR}} = \big[ \frac{1}{n}\sum_{i=1}^{n} \hat{m}_1(\bm{X}_i) + \sum_{i:Z_i=1} w^\text{base}_i \{ Y^{\text{obs}}_i - \hat{m}_1(\bm{X}_i) \} \big] - \big[ \frac{1}{n}\sum_{i=1}^{n} \hat{m}_0(\bm{X}_i) + \sum_{i:Z_i=0} w^\text{base}_i \{ Y^{\text{obs}}_i - \hat{m}_0(\bm{X}_i) \} \big]$, where $\hat{m}_1$ and $\hat{m}_0$ are obtained using MRI and the base weights are normalized. We prove that the second term of $\widehat{\text{ATE}}_{\text{DR}}$, i.e. $\frac{1}{n}\sum_{i=1}^{n} \hat{m}_0(\bm{X}_i) + \sum_{i:Z_i=0}w^{\text{base}}_i \{Y^{\text{obs}}_i - \hat{m}_0(\bm{X}_i)\}$ is of the form $\sum_{i:Z_i=0}w_i Y^{\text{obs}}_i$, where $w_i$ has the form given in Theorem 3. Now, from Proposition 2, we know that for a generic covariate profile $\bm{x}$, $\hat{m}_0(\bm{x}) = \sum_{i:Z_i=0} w^{\scriptscriptstyle \text{MRI}}_i(\bm{x}) Y^{\text{obs}}_i$, where $w^{\scriptscriptstyle \text{MRI}}_i(\bm{x}) = \frac{1}{n_c} + (\bm{x} - \bar{\bm{X}}_c)^\top \bm{S}^{-1}_c (\bm{X}_i - \bar{\bm{X}}_c)$. By linearity, we have
\begin{align}
\frac{1}{n}\sum_{i=1}^{n} \hat{m}_0(\bm{X}_i) + \sum_{i:Z_i=0}w^{\text{base}}_i (Y^{\text{obs}}_i - \hat{m}_0(\bm{X}_i)) &= \hat{\beta}_{0c} + \bm{\beta}^\top_{1c} \bar{\bm{X}} + \sum_{i:Z_i=0}w^{\text{base}}_i Y^{\text{obs}}_i - (\hat{\beta}_{0c} + \bm{\beta}^\top_{1c} \bar{\bm{X}}^{\text{base}}_{c}) \nonumber\\
&= \sum_{i:Z_i=0} \big\{ w^{\scriptscriptstyle \text{MRI}}_i(\bar{\bm{X}}) - w^{\scriptscriptstyle \text{MRI}}_i(\bar{\bm{X}}^{\text{base}}_c) + w^{\text{base}}_i \big\} Y^{\text{obs}}_i.
\label{s1.3.1}
\end{align}
Note that $w^{\scriptscriptstyle \text{MRI}}_i(\bar{\bm{X}}) - w^{\scriptscriptstyle \text{MRI}}_i(\bar{\bm{X}}^{\text{base}}_{c}) + w^{\text{base}}_i = w^{\text{base}}_i + (\bar{\bm{X}} - \bar{\bm{X}}^{\text{base}}_{c})^\top \bm{S}^{-1}_c (\bm{X}_i - \bar{\bm{X}}_c)$. Since $w^{\text{base}}_i$s are normalized, we get
\begin{equation}
w_i = \tilde{w}^{\text{base}}_i + (\bm{X}_i - \bar{\bm{X}}_c)^\top \bm{S}^{-1}_c (\bar{\bm{X}} - \bar{\bm{X}}^{\text{base}}_{c}).
\end{equation}
\subsection{Variance of the URI and MRI weights}
\label{var_urimri}
We first derive the variance of the URI weights in the treatment group. The variance of the weights in the control group can be derived analogously.
\begin{align}
&\frac{1}{n_t}\sum_{i:Z_i=1}(w^{\scriptscriptstyle \text{URI}}_i - \frac{1}{n_t})^2 \nonumber\\
&= \frac{1}{n_t}\sum_{i:Z_i=1} \big\{\frac{n}{n_c} (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top (\bm{S}_t + \bm{S}_c)^{-1}(\bm{X}_i -\bar{\bm{X}}_t) \big\}^2 \nonumber\\
&= \frac{1}{n_t}\frac{n^2}{n^2_c}(\bar{\bm{X}} - \bar{\bm{X}}_t)^\top (\bm{S}_t + \bm{S}_c)^{-1} \big\{ \sum_{i:Z_i=1} (\bm{X}_i -\bar{\bm{X}}_t) (\bm{X}_i -\bar{\bm{X}}_t)^\top \big\} (\bm{S}_t + \bm{S}_c)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t)\nonumber\\
&= \frac{1}{n_t}\frac{n^2}{n^2_c}(\bar{\bm{X}} - \bar{\bm{X}}_t)^\top (\bm{S}_t + \bm{S}_c)^{-1} \bm{S}_t (\bm{S}_t + \bm{S}_c)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t).
\label{s1.4.1}
\end{align}
Similarly, the variance of the MRI weights (for the ATE case) in the treatment group can be derived as follows.
\begin{align}
\frac{1}{n_t}\sum_{i:Z_i=1}(w^{\scriptscriptstyle \text{MRI}}_i - \frac{1}{n_t})^2 &= \frac{1}{n_t}\sum_{i:Z_i=0} \big\{ (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \bm{S}_t^{-1} (\bm{X}_i -\bar{\bm{X}}_t)\big\}^2 \nonumber\\
&= \frac{1}{n_t}(\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \bm{S}_t^{-1} \big\{ \sum_{i:Z_i=1}(\bm{X}_i -\bar{\bm{X}}_t)(\bm{X}_i -\bar{\bm{X}}_t)^\top \big\} \bm{S}_t^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t) \nonumber\\
&= \frac{1}{n_t}(\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \bm{S}_t^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t).
\label{s1.4.2}
\end{align}
Now, from Equations \ref{s1.4.1} and \ref{s1.4.2}, we get
\begin{align}
&\sum_{i:Z_i=1}(w^{\scriptscriptstyle \text{URI}}_i - \frac{1}{n_t})^2 - \sum_{i:Z_i=1}(w^{\scriptscriptstyle \text{MRI}}_i - \frac{1}{n_t})^2 \nonumber\\
&= (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \big\{\frac{n^2}{n^2_c}(\bm{S}_t + \bm{S}_c)^{-1}\bm{S}_t (\bm{S}_t + \bm{S}_c)^{-1} - \bm{S}^{-1}_t \big\} (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \nonumber\\
&= (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \bm{S}^{-\frac{1}{2}}_t \big\{\frac{n^2}{n^2_c}\bm{S}^{\frac{1}{2}}_t(\bm{S}_t + \bm{S}_c)^{-1}\bm{S}^{\frac{1}{2}}_t \bm{S}^{\frac{1}{2}}_t (\bm{S}_t + \bm{S}_c)^{-1} \bm{S}^{\frac{1}{2}}_t - \underline{\bm{I}} \big\}\bm{S}^{-\frac{1}{2}}_t (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top,
\label{s1.4.18}
\end{align}
where $\bm{S}^{\frac{1}{2}}_t$ is the symmetric square root matrix of $\bm{S}_t$, and $\bm{S}^{-\frac{1}{2}}_t := (\bm{S}^{\frac{1}{2}}_t)^{-1}$. Now, let us assume $n_t\bm{S}_t \succcurlyeq n_c \bm{S}_c$. We get,
\begin{align}
n_t\bm{S}_t \succcurlyeq n_c \bm{S}_c &\implies n\bm{S}_t \succcurlyeq n_c(\bm{S}_t + \bm{S}_c)\nonumber\\
& \implies (\bm{S}_t + \bm{S}_c)^{-1} \succcurlyeq \frac{n_c}{n}\bm{S}^{-1}_t \nonumber\\
& \implies \frac{n}{n_c}\bm{S}^{\frac{1}{2}}_t(\bm{S}_t + \bm{S}_c)^{-1}\bm{S}^{\frac{1}{2}}_t \succcurlyeq \underline{\bm{I}} \nonumber\\
& \implies \big\{\frac{n}{n_c}\bm{S}^{\frac{1}{2}}_t(\bm{S}_t + \bm{S}_c)^{-1}\bm{S}^{\frac{1}{2}}_t\}^2 \succcurlyeq \underline{\bm{I}}
\label{s1.4.19}
\end{align}
Equation \ref{s1.4.19} implies that $\frac{n^2}{n^2_c}\bm{S}^{\frac{1}{2}}_t(\bm{S}_t + \bm{S}_c)^{-1}\bm{S}^{\frac{1}{2}}_t \bm{S}^{\frac{1}{2}}_t (\bm{S}_t + \bm{S}_c)^{-1} \bm{S}^{\frac{1}{2}}_t - \underline{\bm{I}}$ is non-negative definite, and hence from Equation \ref{s1.4.18} we get, $\sum_{i:Z_i=1}(w^{\scriptscriptstyle \text{URI}}_i - \frac{1}{n_t})^2 \geq \sum_{i:Z_i=1}(w^{\scriptscriptstyle \text{MRI}}_i - \frac{1}{n_t})^2$. Thus, when $n_t\bm{S}_t \succcurlyeq n_c \bm{S}_c$, the variance of the URI weights in the treatment group is no less than that of the MRI weights. By similar calculations it follows that in this case, the variance of the URI weights in the treatment group is no greater than that of the MRI weights. The inequalities are reversed when $n_t\bm{S}_t \preccurlyeq n_c \bm{S}_c$.
We now compare the total variance of the URI and MRI weights across all $n$ units in the sample. By similar calculations, we get
\begin{align}
\sum_{i=1}^{n}(w^{\scriptscriptstyle \text{MRI}}_i - \frac{2}{n})^2 - \sum_{i=1}^{n}(w^{\scriptscriptstyle \text{URI}}_i - \frac{2}{n})^2 &= (\bar{\bm{X}}_t - \bar{\bm{X}}_c)^\top \big\{\frac{1}{n^2}(n^2_c \bm{S}^{-1}_t + n^2_t \bm{S}^{-1}_c) - (\bm{S}_t + \bm{S}_c)^{-1} \big\} (\bar{\bm{X}}_t - \bar{\bm{X}}_c)
\end{align}
We will now show that $\frac{1}{n^2}(n^2_c \bm{S}^{-1}_t + n^2_t \bm{S}^{-1}_c) \succcurlyeq (\bm{S}_t + \bm{S}_c)^{-1}$, which is equivalent to showing $\frac{1}{n^2}(n^2_c \bm{S}^{-1}_t + n^2_t \bm{S}^{-1}_c) (\bm{S}_t + \bm{S}_c) \succcurlyeq \underline{\bm{I}}$.
We now use the following lemma.
\begin{lemma}
For a $k \times k$ non-negative definite matrix $\bm{A}$, $\frac{1}{2}(\bm{A}+\bm{A}^{-1}) \succcurlyeq \underline{\bm{I}}$.
\label{lemma_1.1}
\end{lemma}
\noindent \textit{Proof of Lemma \ref{lemma_1.1}.} Using the spectral decomposition of $\bm{A}$, we can write $\bm{A} = \bm{P}\bm{\Lambda}\bm{P}^\top$, where $\bm{P}$ is an orthogonal matrix and $\Lambda$ is the diagonal matrix of the (ordered) eigenvalues of $\bm{A}$. Therefore, $\frac{1}{2}(\bm{A} + \bm{A}^{-1}) = \bm{P}\frac{1}{2}(\bm{\Lambda} + \bm{\Lambda}^{-1})\bm{P}^\top$. Therefore, if $(\lambda_1,...,\lambda_k)$ are the eigenvalues of $A$, the eigenvalues of $\frac{1}{2}(\bm{A} + \bm{A}^{-1})$ are $(\lambda_1 + \frac{1}{\lambda_1},...,\lambda_k+ \frac{1}{\lambda_k})$. By AM-GM inequality, $\forall i \in \{1,...,k\}$, $\frac{1}{2}(\lambda_i + \frac{1}{\lambda_i}) \geq 1$. So the eigenvalues of $\frac{1}{2}(\bm{A} + \bm{A}^{-1}) - \underline{\bm{I}}$ are non-negative, which completes the proof.
Now, let $A = n_t^{-1}n_c \bm{S}_t^{-\frac{1}{2}}\bm{S}_c\bm{S}_t^{-\frac{1}{2}}$, which is non-negative definite. Lemma \ref{lemma_1.1} implies $\frac{1}{2}(n_t^{-1}n_c \bm{S}_t^{-\frac{1}{2}}\bm{S}_c\bm{S}_t^{-\frac{1}{2}} + n_c^{-1}n_t \bm{S}_t^{\frac{1}{2}}\bm{S}^{-1}_c\bm{S}_t^{\frac{1}{2}}) \succcurlyeq \underline{\bm{I}}$. Finally, we note that
\begin{align}
\frac{1}{2}(n_t^{-1}n_c \bm{S}_t^{-\frac{1}{2}}\bm{S}_c\bm{S}_t^{-\frac{1}{2}} + n_c^{-1}n_t \bm{S}_t^{\frac{1}{2}}\bm{S}^{-1}_c\bm{S}_t^{\frac{1}{2}}) \succcurlyeq \underline{\bm{I}} &\implies \frac{1}{2}(n_t^{-1}n_c \bm{S}_c\bm{S}_t^{-1} + n_c^{-1}n_t \bm{S}_t\bm{S}_c^{-1}) \succcurlyeq \underline{\bm{I}} \nonumber\\
& \implies n^2_c \bm{S}_c\bm{S}^{-1}_t + n^2_t \bm{S}_t\bm{S}^{-1}_c \succcurlyeq 2n_tn_c \nonumber\\
& \implies \frac{1}{n^2}(n^2_c \bm{S}^{-1}_t + n^2_t \bm{S}^{-1}_c) (\bm{S}_t + \bm{S}_c) \succcurlyeq \underline{\bm{I}}
\end{align}
This proves that the variance of the MRI weights in the full-sample is no less than that of the URI weights.
\subsection{Bias of the URI and MRI estimators}
Consider a generic H\'{a}jek estimator $T = \sum_{i:Z_i=1}w_i Y^{\text{obs}}_i - \sum_{i:Z_i=0}w_i Y^{\text{obs}}_i$ of the ATE, where the weights are normalized within each treatment group. Under unconfoundedness, the bias of $T$ due to imbalances on the observed covariates is completely removed if the weights satisfy $\sum_{i:Z_i=1}w_i m_1(\bm{X}_i) = n^{-1}\sum_{i=1}^{n} m_1(\bm{X}_i)$ and $\sum_{i:Z_i=0}w_i m_0(\bm{X}_i) = n^{-1}\sum_{i=1}^{n} m_0(\bm{X}_i)$.
As a special case, when both $m_1(\cdot)$ and $m_0(\cdot)$ are linear in $\bm{X}_i$, balancing the mean of $\bm{X}_i$ relative to $\bar{\bm{X}}$ suffices to remove the bias of $T$.
Now, if $\sum_{i:Z_i=1}w_i\bm{X}_i = \sum_{i:Z_i=0}w_i\bm{X}_i = \bm{X}^*$, then ${E}(T\mid \bm{Z},\underline{\bm{X}}) = \text{CATE}(\bm{X}^*)$. This implies that the URI estimates the average treatment effect on a population characterized by the profile $\bm{X}^{*{\scriptscriptstyle \text{URI}}}$, whereas MRI estimates the average treatment effect on a population characterized by the profile $\bm{X}^{*{\scriptscriptstyle \text{MRI}}}$. Now, treatment effect homogeneity implies $\text{CATE}(\bm{x}) = \text{ATE}$ for all $\bm{x} \in \text{supp}(\bm{X}_i)$.
Thus, under linearity and treatment effect homogeneity, the disparity between the implied target population under URI and the intended target population does not matter and the URI estimator is unbiased for the ATE.
However, if CATE($\bm{x}$) is a non-trivial function of $\bm{x}$, then ${E}\{\text{CATE}(\bm{X}^{*{\scriptscriptstyle \text{URI}}})\} \neq \text{ATE}$ in general and the URI estimator is biased for the ATE, despite balancing the mean of $\bm{X}_i$ exactly.
On the other hand, as long as $m_0(\cdot)$ and $m_1(\cdot)$ are linear in $\bm{X}_i$, ${E}\{\text{CATE}(\bar{\bm{X}})\} = \text{ATE}$ and the MRI estimator is unbiased for the ATE.
However, if $m_0(\cdot)$ and $m_1(\cdot)$ are linear on some other transformations of $\bm{X}_i$, both URI and MRI weights can produce biased estimators, since the implied weights are not guaranteed to yield exact mean balance on these transformations.
\subsection{URI and MRI weights under no-intercept model}
Let $\bm{y}$, $\bm{y}_t$, and $\bm{y}_c$ be the vector of observed outcomes in the full-sample, treatment group, and control group, respectively. In the URI approach with a no-intercept model, we fit the regression model $Y^{\text{obs}}_i = \bm{\beta}^\top \bm{X}_i + \tau Z_i + \epsilon_i$. Let $\bm{\mathcal{P}}$ be the projection matrix onto the column space of $\underline{\bm{X}}$. By the Frisch–Waugh–Lovell theorem (\perp\!\!\!\perptealt{frisch1933partial},\perp\!\!\!\perptealt{lovell1963seasonal}), the OLS estimator of $\tau$ can be written as,
\begin{equation}
\hat{\tau}^{\text{OLS}} = \frac{\bm{Z}^\top (\underline{\bm{I}} - \bm{\mathcal{P}})\bm{y}}{\bm{Z}^\top (\underline{\bm{I}} - \bm{\mathcal{P}})\bm{Z}} = \bm{l}^\top \bm{y}.
\label{s1.5.1}
\end{equation}
where $\bm{l} = (l_1,...,l_n)^\top = \frac{(\underline{\bm{I}} - \bm{\mathcal{P}})\bm{Z}}{\bm{Z}^\top (\underline{\bm{I}} - \bm{\mathcal{P}})\bm{Z}}$. So, $\hat{\tau}^{\text{OLS}}$ can be written as $\hat{\tau} = \sum_{i:Z_i=1}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i - \sum_{i:Z_i=0}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i$, where $w^{\scriptscriptstyle \text{URI}}_i = (2Z_i-1)l_i$. We observe that $\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{URI}}_i = \bm{l}^\top \bm{Z} = 1$. So in this case, the URI weights are normalized in the treatment group. However, $\sum_{i:Z_i=0}w^{\scriptscriptstyle \text{URI}}_i$ may not be equal to 1 in general. Also, we note that $\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{URI}}_i\bm{X}_i - \sum_{i:Z_i=0}w^{\scriptscriptstyle \text{URI}}_i\bm{X}_i = \underline{\bm{X}}^\top \bm{l} = \bm{0}$, since $\underline{\bm{X}}^\top \bm{\mathcal{P}} = \underline{\bm{X}}^\top$. So, the weighted \textit{sums} of the covariates are the same in the treatment and the control group. However, the weighted \textit{means} of the covariates are not guaranteed to be the same.
Similarly, in the MRI approach, we fit the model $Y^{\text{obs}}_i = \bm{\beta}_t\bm{X}_i + \epsilon_{it}$ in the treatment group, and $Y^{\text{obs}}_i = \bm{\beta}^\top_c\bm{X}_i + \epsilon_{ic}$ in the control group.
For a fixed profile $\bm{x} \in \mathbb{R}^k$, $\hat{m}_1(\bm{x}) = \hat{\bm{\beta}}^\top_{t}\bm{x} = \bm{w}^\top_t \bm{y}_t$, where $\bm{w}_t = \underline{\bm{X}}_t(\underline{\bm{X}}^\top_t \underline{\bm{X}}_t)^{-1} \bm{x}$. Thus, $\hat{m}_1(\bm{x})$ can be written as $\hat{m}_1(\bm{x}) = \sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i(\bm{x}) Y^{\text{obs}}_i$. Note that here the weights do not necessarily sum to one. However, $\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i(\bm{x}) \bm{X}_i = \underline{\bm{X}}^\top_t \bm{w}_t = \bm{x}$. Therefore, the weighted \textit{sum} of the covariates in the treatment group is balanced relative to the target profile, but the corresponding weighted \textit{mean} of the covariates is imbalanced in general.
\subsection{Asymptotic properties of URI}
\begin{theorem}{\label{uri_conv}}
Let $w^{{\scriptscriptstyle \text{URI}}}_{\bm{x}}$ be the URI weight of a unit with covariate vector $\bm{x}$. Assume that $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$, where $p = P(Z_i=1)$. Then
\begin{enumerate}[label = (\alph*)]
\item For each treated unit, $n w^{{\scriptscriptstyle \text{URI}}}_{\bm{x}} \xrightarrow[n \to \infty]{P} \frac{1}{e(\bm{x})}$ for all $\bm{x} \in \text{supp}(\bm{X}_i)$ if and only if the propensity score is an inverse-linear function of the covariates; i.e., $e(\bm{x}) = \frac{1}{\alpha_0 + \bm{\alpha}^\top_1 \bm{x}}$, $\alpha_0 \in \mathbb{R}$, $\bm{\alpha}_1 \in \mathbb{R}^k$. Moreover, if $\sup\limits_{\bm{x} \in \text{supp}(\bm{X}_i)}\norm{\bm{x}}_2 < \infty$, $\sup \limits_{\bm{x} \in \text{supp}(\bm{X}_i)}\mid nw^{{\scriptscriptstyle \text{URI}}}_{\bm{x}} - \frac{1}{e(\bm{x})}\mid \xrightarrow[n \to \infty]{P} 0$.
\item Similarly, for each control unit, $n w^{{\scriptscriptstyle \text{URI}}}_{\bm{x}} \xrightarrow[n \to \infty]{P} \frac{1}{1-e(\bm{x})}$ if and only if $1-e(\bm{x})$ is inverse linear function of the covariates, and the convergence is uniform if $\sup\limits_{\bm{x} \in \text{supp}(\bm{X}_i)}\norm{\bm{x}}_2 < \infty$.
\end{enumerate}
\end{theorem}
\textit{Proof of Theorem \ref{uri_conv}.}
Let $\bm{\mu}_t = {E}(\bm{X}_i\mid Z_i=1)$, $\bm{\mu}_c = {E}(\bm{X}_i\mid Z_i=0)$, $\bm{\mu} = {E}(\bm{X}_i)$, and $\bm{\Sigma}_t = \text{var}(\bm{X}_i\mid Z_i=1)$, $\bm{\Sigma}_c = \text{var}(\bm{X}_i\mid Z_i=0)$. By WLLN and Slutsky's theorem, we have $\bar{\bm{X}}_t = \frac{\frac{1}{n}\sum_{i=1}^{n}Z_i\bm{X}_i}{\frac{1}{n}\sum_{i=1}^{n} Z_i} \xrightarrow[n \to \infty]{P} \frac{{E}(Z_i \bm{X}_i)}{p} = \bm{\mu}_t$. Similarly, we have $\frac{\bm{S}_t}{n_t} \xrightarrow[n \to \infty]{P} \bm{\Sigma}_t$ and $\frac{\bm{S}_c}{n_c} \xrightarrow[n \to \infty]{P} \bm{\Sigma}_c$. Now, we consider a treated unit with covariate vector $\bm{x} \in \text{supp}(\bm{X}_i)$. By continuous mapping theorem, we have
\begin{align}
nw^{\scriptscriptstyle \text{URI}}_{\bm{x}} &= \frac{n}{n_t} + \frac{n}{n_c}(\bm{x} - \bar{\bm{X}}_t)^\top \big(\frac{\bm{S}_t}{n_t}\frac{n_t}{n} + \frac{\bm{S}_c}{n_c}\frac{n_c}{n}\big)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t) \nonumber\\
&\xrightarrow[n \to \infty]{P} \frac{1}{p} \Big[ 1 + \frac{p}{1-p}(\bm{x} - \bm{\mu}_t)^\top \{p\bm{\Sigma}_t + (1-p)\bm{\Sigma}_c\}^{-1} (\bm{\mu} - \bm{\mu}_t) \Big].
\label{s1.6.1}
\end{align}
Under the assumption that $p^2 \bm{\Sigma}_t = (1-p)^2 \bm{\Sigma}_c$, the RHS of Equation \ref{s1.6.1} boils down to $\frac{1}{p} \big\{ 1 + (\bm{x} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t (\bm{\mu} - \bm{\mu}_t) \big\}$, which is same as the probability limit of $nw^{\scriptscriptstyle \text{MRI}}_{\bm{x}}$ (see the proof of Theorem 1). Thus, when $p^2 \bm{\Sigma}_t = (1-p)^2 \bm{\Sigma}_c$, the URI and MRI weights are asymptotically equivalent. The rest of the proof follows from the proof of Theorem 1.
\begin{lemma}
Let the true propensity score be linear on the covariates, i.e., $e(\bm{x}) = a_0 + \bm{a}^\top_1\bm{x}$ for some constants $a_0 \in \mathbb{R}$, $\bm{a}_1 \in \mathbb{R}^k$. Then $$ \bm{a}_1 = \frac{p(1-p)}{1+ p(1-p)c} \bm{A}^{-1}(\bm{\mu}_t - \bm{\mu}_c),\hspace{0.1cm} a_0 = p - \bm{a}_1^\top \bm{\mu},$$
where $\bm{A} = p\bm{\Sigma}_t + (1-p)\bm{\Sigma}_c$, and $c = (\bm{\mu}_t - \bm{\mu}_c)^\top \bm{A}^{-1} (\bm{\mu}_t - \bm{\mu}_c)$. Here $p,\bm{\mu}, \bm{\mu}_t, \bm{\Sigma}_t, \bm{\Sigma}_c$ are the same as in the proof of Theorem \ref{uri_conv}.
\label{linear_propensity}
\end{lemma}
\textit{Proof of Lemma \ref{linear_propensity}.} Since ${E}(e(\bm{X}_i)) = p$, we have $a_0 = p - \bm{a}^\top \bm{\mu}$. Next, expanding the identity ${E}((\bm{X}_i - \bm{\mu}_t)e(\bm{X}_i)) = 0$, it is straightforward to show that
\begin{equation}
\bm{a}_1 = p(1-p) \bm{\Sigma}^{-1}(\bm{\mu}_t - \bm{\mu}_c),
\label{s1.6.2}
\end{equation}
where $\bm{\Sigma} = \text{var}(\bm{X}_i)$. Moreover, by conditioning on $Z_i$, we can decompose $\text{var}(\bm{X}_i)$ as
\begin{equation}
\bm{\Sigma} = (p\bm{\Sigma}_t + (1-p)\bm{\Sigma}_c) + p(1-p)(\bm{\mu}_t - \bm{\mu}_c)(\bm{\mu}_t - \bm{\mu}_c)^\top = \bm{A} + p(1-p)(\bm{\mu}_t - \bm{\mu}_c)(\bm{\mu}_t - \bm{\mu}_c)^\top.
\label{s1.6.3}
\end{equation}
Applying the Sherman-Morrison-Woodbury formula (\perp\!\!\!\perptealt{sherman1950adjustment}, \perp\!\!\!\perptealt{woodbury1950inverting}), we can write the inverse of $\bm{\Sigma}$ as
\begin{equation}
\bm{\Sigma}^{-1} = \bm{A}^{-1} - \frac{p(1-p) \bm{A}^{-1}(\bm{\mu}_t - \bm{\mu}_c)(\bm{\mu}_t - \bm{\mu}_c)^\top \bm{A}^{-1}}{1+p(1-p)c}.
\label{s1.6.4}
\end{equation}
Substituting the expression of $\bm{\Sigma}^{-1}$ in \ref{s1.6.2}, it follows that
\begin{equation}
\bm{a}_1 = \frac{p(1-p)}{1+ p(1-p)c} \bm{A}^{-1}(\bm{\mu}_t - \bm{\mu}_c).
\end{equation}
This completes the proof of the Lemma.
\begin{theorem}{\label{dr_uri}}
The URI estimator for the ATE is consistent if any of the following conditions holds.
\begin{enumerate}[label=(\roman*)]
\item $m_0(\bm{x})$ is linear, $e(\bm{x})$ is inverse linear, and $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$.
\item $m_1(\bm{x})$ is linear, $1-e(\bm{x})$ is inverse linear, and $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$.
\item Both $m_1(\bm{x})$ and $m_0(\bm{x})$ are linear and $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$.
\item $e(\bm{x})$ is a constant function of $\bm{x}$.
\item Both $m_1(\bm{x})$ and $m_0(\bm{x})$ are linear and $m_1(\bm{x}) - m_0(\bm{x})$ is a constant function.
\item $m_1(\bm{x})- m_0(\bm{x})$ is a constant function and $e(\bm{x})$ is linear in $\bm{x}$.
\end{enumerate}
\end{theorem}
\noindent \textit{Proof of Theorem \ref{dr_uri}.}
Let $p,\bm{\mu}, \bm{\mu}_t, \bm{\Sigma}_t, \bm{\Sigma}_c$ be defined as in the proof of Theorem \ref{uri_conv}. By similar calculations as in the proof of Theorem \ref{uri_conv} we have,
\begin{align}
\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i &= \bar{Y}_t + \frac{n}{n_c} (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \big(\frac{\bm{S}_t + \bm{S}_c}{n} \big)^{-1} \big\{ \frac{1}{n} \sum_{i:Z_i=1} (\bm{X}_i - \bar{\bm{X}}_t)Y^{\text{obs}}_i \big\} \nonumber \\
&\xrightarrow[n \to \infty]{P} {E} \Big(m_1(\bm{X}_i)e(\bm{X}_i) \big\{\frac{1}{p} + \frac{1}{1-p} (\bm{\mu} - \bm{\mu}_t)^\top (p\bm{\Sigma}_t + (1-p)\bm{\Sigma}_c)^{-1}(\bm{X}_i - \bm{\mu}_t)\big\} \Big) .
\label{1.6.2}
\end{align}
First, if $p^2 \bm{\Sigma}_t = (1-p)^2 \bm{\Sigma}_c$, the right hand side of Equation 1 becomes ${E}\Big(\frac{m_1(\bm{X}_i)e(\bm{X}_i)}{p} \big\{1+ (\bm{\mu} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t(\bm{x} - \bm{\mu}_t) \big\} \Big)$, which is same as the probability limit of $\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i$. Similarly, when $p^2 \bm{\Sigma}_t = (1-p)^2 \bm{\Sigma}_c$, we can show that $\sum_{i:Z_i=0}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i$ has the same probability limit as $\sum_{i:Z_i=0} w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i$. This observation, along with conditions Theorem 2, proves consistency of the URI estimator under parts (i), (ii) and (iii) of Theorem \ref{dr_uri}. Second, if the propensity score is constant, $\bm{\mu} = \bm{\mu}_t$ and the right hand side of Equation \ref{1.6.2} becomes ${E}(m_1(\bm{X}_i))$, which equals ${E}(Y_i(1))$. Similarly, in this case, the probability limit of $\sum_{i:Z_i=0} w^{\scriptscriptstyle \text{URI}}Y^{\text{obs}}_i$ becomes ${E}(Y_i(0))$. This proves consistency of the URI estimator under \textit{(iv)}. Third, by consistency of OLS estimators of regression coefficients under well-specified model, the URI estimator is consistent for the ATE under part (v).
Finally, let $e(\bm{x})= a_0 + \bm{a}_1^\top \bm{x}$. Using the notation in Lemma \ref{linear_propensity}, we know that $ \bm{a}_1 = \frac{p(1-p)}{1+ p(1-p)c} \bm{A}^{-1}(\bm{\mu}_t - \bm{\mu}_c)$, $a_0 = p - \bm{a}_1^\top \bm{\mu}$. Let $d = {E}(e(\bm{X}_i)(1-e(\bm{X}_i))$. Using the expressions of $a_0$ and $\bm{a}_1$, it is straightforward to show that
\begin{equation}
d = \frac{p(1-p)}{1+p(1-p)c}.
\label{1.6.3}
\end{equation}
This implies,
\begin{equation}
1-e(\bm{x}) = d \big\{ \frac{1}{p} + \frac{1}{1-p}(\bm{\mu} - \bm{\mu}_t) \bm{A}^{-1}(\bm{x} - \bm{\mu}_t) \big\}
\label{1.6.4}
\end{equation}
Equations \ref{1.6.2} and \ref{1.6.4} imply,
\begin{equation}
\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i \xrightarrow[n \to \infty]{P} \frac{{E}(e(\bm{X}_i)\{1-e(\bm{X}_i)\}m_1(\bm{X}_i))}{{E}(e(\bm{X}_i)\{1-e(\bm{X}_i)\}]}.
\label{1.6.5}
\end{equation}
By similar calculations for the control group, we get
\begin{equation}
\sum_{i:Z_i=0}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i \xrightarrow[n \to \infty]{P} \frac{{E}(e(\bm{X}_i)\{1-e(\bm{X}_i)\}m_0(\bm{X}_i))}{{E}(e(\bm{X}_i)\{1-e(\bm{X}_i)\}]}.
\label{1.6.6}
\end{equation}
Equations \ref{1.6.5} and \ref{1.6.6} imply that, when the propensity score is linear on the covariates,
\begin{equation}
\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i - \sum_{i:Z_i=0}w^{\scriptscriptstyle \text{URI}}_i Y^{\text{obs}}_i \xrightarrow[n \to \infty]{P} \frac{{E}(e(\bm{X}_i)\{1-e(\bm{X}_i)\}\{m_1(\bm{X}_i) - m_0(\bm{X}_i)\} )}{{E}(e(\bm{X}_i)\{1-e(\bm{X}_i)\})}.
\label{1.6.7}
\end{equation}
Note that this limiting representation of the URI estimator is equivalent to that in \perp\!\!\!\perpte{aronow2016does}. Now, the consistency of the URI estimator under condition (vi) follows from Equation \ref{1.6.7} by noting that if $m_1(\bm{x}) - m_0(\bm{x}) = \tau$ for all $\bm{x} \in \text{supp}(\bm{X}_i)$, the RHS of Equation \ref{1.6.7} equals $\tau$.
\section{Proofs of propositions and theorems}
\subsection{Proof of Theorem 3}
For $\bm{\delta} = 0$, the Lagrangian of the optimization problem is given by
\begin{equation}
L(\bm{w}, \lambda_1, \bm{\lambda}_2) = \sum_{i: Z_i=0}\frac{(w_i - \tilde{w}^\text{base}_i)^2}{w^\text{scale}_i} + \lambda_1 (\sum_{i:Z_i=0}w_i - 1) + \bm{\lambda}_2^\top (\sum_{i:Z_i=0} w_i \bm{X}_i - \bm{X}^*).
\label{6.1_1}
\end{equation}
Computing the partial derivatives $\frac{\partial L}{\partial \bm{w}}$, $\frac{\partial L}{\partial \lambda_1}$, $\frac{\partial L}{\partial \bm{\lambda}_2}$ and equating them to zero, we get the following equations:
\begin{equation}
w_i = \tilde{w}^\text{base}_i - w^\text{scale}_i \frac{\lambda_1 + \bm{\lambda}_2^\top \bm{X}_i}{2} \hspace{0.2cm} \text{for all $i:Z_i=0$.}
\label{6.1_2}
\end{equation}
\begin{equation}
\sum_{i: Z_i=0} w_i=1
\label{6.1_3}
\end{equation}
\begin{equation}
\sum_{i: Z_i=0} w_i \bm{X}_i^\top = \bm{X}^{*\top}
\label{6.1_4}
\end{equation}
Substituting the expression of $w_i$ from Equation \ref{6.1_2} in Equation \ref{6.1_3}, we get,
\begin{equation}
\lambda_1 + \bm{\lambda}^\top_2 \bar{\bm{X}}^{\text{scale}}_c = 0\\
\iff \lambda_1 = - \bm{\lambda}^\top_2 \bar{\bm{X}}^{\text{scale}}_c
\end{equation}
Substituting the expression of $w_i$ from Equation \ref{6.1_2} in Equation \ref{6.1_4}, we get,
\begin{align}
&\bar{\bm{X}}^{\text{base}\top}_c - \frac{1}{2} \Big[(\sum_{i:Z_i=0}w^\text{scale}_i)\lambda_1 \bar{\bm{X}}^{\text{scale} \top}_c +
\bm{\lambda}^\top_2 \{\frac{\bm{S}^{\text{scale}}_c}{n_c} + \bar{\bm{X}}^{\text{scale}}_c\bar{\bm{X}}^{\text{scale} \top}_c (\sum_{i:Z_i=0}w^\text{scale}_i) \} \Big] = \bm{X}^{*\top} \nonumber\\
&\iff \bar{\bm{X}}^{\text{base}\top}_c - \frac{1}{2} \bm{\lambda}_2^\top \frac{\bm{S}^{\text{scale}}_c}{n_c} = \bm{X}^{*\top} \nonumber\\
&\iff \bm{\lambda}_2 = 2 \Big(\frac{\bm{S}^{\text{scale}}_c}{n_c} \Big)^{-1} (\bar{\bm{X}}^{\text{base}}_c - \bm{X}^*)
\end{align}
Substituting $\lambda_1$ and $\bm{\lambda}_2$ in Equation \ref{6.1_2}, we get the resulting expression of $w_i$.
\noindent The corresponding results for the WURI, WMRI, and DR weights follow from the derivations in Sections \ref{wuri_derive}, \ref{wmri_derive}, and \ref{drest_derive} of the Supplementary Materials, respectively.
\subsection{Proof of Proposition 1}
The proof follows from setting $w^{\text{base}}_i = \frac{1}{n}$ in the derivation of WURI weights in Section \ref{wuri_derive} of the Supplementary Materials.
\subsection{Proof of Proposition 2}
The proof follows from setting $w^{\text{base}}_i = \frac{1}{n_t}$ for all $i:Z_i=1$, $w^{\text{base}}_i = \frac{1}{n_c}$ for all $i:Z_i=0$ in the derivation of WMRI weights in Section \ref{wmri_derive} of the Supplementary Materials.
\subsection{Proof of Proposition 3}
Parts (a), (b), and (e) of Proposition 3 directly follows from Theorem 3. Part (d) is a direct consequence of the closed form expression of the weights, given in Propositions 3.1 and 3.2 (see also Section 5.2 of the paper). Part (b) follows from Section \ref{var_urimri} of the Supplementary Materials.
\subsection{Proof of Theorem 1}
Let $p = P(Z_i=1)$, $\bm{\mu}_t = {E}(\bm{X}_i\mid Z_i=1)$, $\bm{\mu} = {E}(\bm{X}_i)$, and $\bm{\Sigma}_t = \text{var}(\bm{X}_i\mid Z_i=1)$. We first show that when $e(\bm{x})$ is of the form $\frac{1}{\alpha_0 + \bm{\alpha}^\top_1 \bm{x}}$, then
\begin{equation}
\alpha_0 + \bm{\alpha}^\top_1 \bm{x} = \frac{1}{p} \big\{1 + (\bm{x} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t (\bm{\mu} - \bm{\mu}_t)\big\}
\label{4.2.1}
\end{equation}
Denoting $\bm{b}_1 = p \bm{\Sigma}_t \bm{\alpha}_1$ and $b_0 = \alpha_0 + \frac{1}{p}\bm{b}^\top_1 \bm{\Sigma}^{-1}_t \bm{\mu}_t$, we have
\begin{equation}
\frac{1}{e(\bm{x})} = b_0 + \frac{1}{p}\bm{b}^\top_1 \bm{\Sigma}^{-1}_t (\bm{x} - \bm{\mu}_t)
\label{4.2.2}
\end{equation}
It is enough to show $b_0 = \frac{1}{p}$ and $\bm{b}_1 = \bm{\mu} - \bm{\mu}_t$. Now,
\begin{align}
\bm{\mu}_t &= \frac{{E}(\bm{X}_i e(\bm{X}_i))}{p}\label{4.2.3}\\
\implies \frac{1}{p} \bm{b}^\top_1 \bm{\Sigma}^{-1} \bm{\mu}_t &= \frac{1}{p} + \frac{1}{p} {E} \big[ \frac{\frac{1}{p}\bm{b}^\top_1 \bm{\Sigma}^{-1}_t \bm{\mu}_t - b_0}{b_0 + \frac{1}{p}\bm{b}^\top_1 \bm{\Sigma}^{-1}_t (\bm{X}_i - \bm{\mu}_t )} \big] \label{4.2.4}\\
\implies \frac{1}{p} \bm{b}^\top_1 \bm{\Sigma}^{-1} \bm{\mu}_t &= \frac{1}{p} + \frac{1}{p} \big(\frac{1}{p}\bm{b}^\top_1 \bm{\Sigma}^{-1}_t \bm{\mu}_t - b_0 \big) p \label{4.2.5}\\
\implies b_0 &= \frac{1}{p}.
\label{4.2.6}
\end{align}
Here Equation \ref{4.2.3} holds by definition of conditional expectation and law of iterated expectations. Equation \ref{4.2.4} is obtained by multiplying both sides of Equation \ref{4.2.3} by $\frac{1}{p}\bm{b}^\top_1 \bm{\Sigma}^{-1}_t$ and applying Equation \ref{4.2.2}. Equation \ref{4.2.5} holds since $p = {E}(e(\bm{X}_i))$.
Similarly,
\begin{align}
\bm{\Sigma}_t &= \frac{{E}((\bm{X}_i - \bm{\mu}_t)(\bm{X}_i - \bm{\mu}_t)^\top e(\bm{X}_i))}{p} \label{4.2.7}\\
\implies \frac{\bm{b}^\top_1}{p} &= \frac{1}{p}{E}((\bm{X}_i - \bm{\mu}_t)^\top) - \frac{1}{p^2}{E}((\bm{X}_i - \bm{\mu}_t)^\top e(\bm{X}_i)) \label{4.2.8}\\
\implies \bm{b}_1 &= \bm{\mu} - \bm{\mu}_t.
\label{4.2.9}
\end{align}
Here Equation \ref{4.2.7} holds by definition of conditional expectation and law of iterated expectation. Equation \ref{4.2.8} is obtained by multiplying both sides of Equation \ref{4.2.7} by $\frac{\bm{b}^\top_1 \bm{\Sigma}^{-1}_t}{p}$ and applying Equation \ref{4.2.2}. Equation \ref{4.2.9} holds since ${E}((\bm{X}_i - \bm{\mu}_t)^\top e(\bm{X}_i)) = 0$. This proves Equation \ref{4.2.1}.
Now, by WLLN and Slutsky's theorem, we have $\bar{\bm{X}}_t = \frac{\frac{1}{n}\sum_{i=1}^{n}Z_i\bm{X}_i}{\frac{1}{n}\sum_{i=1}^{n} Z_i} \xrightarrow[n \to \infty]{P} \frac{{E}(Z_i \bm{X}_i)}{p} = \bm{\mu}_t$. Similarly, we have $\frac{\bm{S}_t}{n_t} \xrightarrow[n \to \infty]{P} \bm{\Sigma}_t$. Now, we consider a treated unit with covariate vector $\bm{x} \in \text{supp}(\bm{X}_i)$. By continuous mapping theorem, we have
\begin{equation}
nw^{\scriptscriptstyle \text{MRI}}_{\bm{x}} = \frac{n}{n_t} + \frac{n}{n_t}(\bm{x} - \bar{\bm{X}}_t)^\top \big(\frac{\bm{S}_t}{n_t}\big)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t) \xrightarrow[n \to \infty]{P} \frac{1}{p} \big\{ 1 + (\bm{x} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t (\bm{\mu} - \bm{\mu}_t) \big\}.
\label{4.2.10}
\end{equation}
This proves pointwise convergence of the MRI weights for a treated unit. To prove uniform convergence, we assume $\sup\limits_{\bm{x} \in Supp(\bm{X}_i)}\mid \mid \bm{x}\mid \mid _2 <\infty$.
\begin{align}
\sup\limits_{\bm{x} \in \text{supp}(\bm{X}_i)}\mid nw^{\scriptscriptstyle \text{MRI}}_{\bm{x}} - \frac{1}{e(\bm{x})}\mid &\leq \mid \frac{n}{n_t} - \frac{1}{p}\mid + \mid \frac{n}{n_t}\bar{\bm{X}}^\top_t \big(\frac{\bm{S}_t}{n_t} \big)^{-1} (\bar{\bm{X}} - \bar{\bm{X}}_t) - \frac{1}{p} \bm{\mu}^\top_t\bm{\Sigma}^{-1}_t(\bm{\mu} - \bm{\mu}_t)\mid \nonumber\\
& + \sup\limits_{\bm{x} \in \text{supp}(\bm{X}_i)}\mid \big\{\frac{n}{n_t}(\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \big(\frac{\bm{S}_t}{n_t} \big)^{-1} - \frac{1}{p} (\bm{\mu} - \bm{\mu}_t)^\top\bm{\Sigma}^{-1}_t\big\} \bm{x}\mid
\label{4.2.11}
\end{align}
The first term on the right hand side converges in probability to zero by WLLN. The second term converges in probability to zero by WLLN, Slutsky's theorem and continuous mapping theorem. By Cauchy-Schwarz inequality, the third term is bounded above by $|| \big\{\frac{n}{n_t}(\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \big(\frac{\bm{S}_t}{n_t} \big)^{-1} - \frac{1}{p} (\bm{\mu} - \bm{\mu}_t)^\top\bm{\Sigma}^{-1}_t\big\}||_2 \Big\{\sup\limits_{\bm{x} \in \text{supp}(\bm{X}_i)}||\bm{x}||_2 \Big\}$. Since $||\bm{x}||_2$ is bounded, this term converges in probability to zero. This proves part (a) of the Theorem. Part (b) can be proved similarly by switching the role of treatment and control group.
\subsection{Proof of Theorem 2}
Consider the first term of the MRI estimator $\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i$. By standard OLS theory, when $m_1(\bm{x})$ is linear on $\bm{x}$ we have
\begin{equation}
\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i = \hat{\beta}_{0t} + \hat{\bm{\beta}}^\top_{1t} \bar{\bm{X}} \xrightarrow[n \to \infty]{P} {E}(m_1(\bm{X}_i)) = {E}(Y_i(1)).
\label{4.3.1}
\end{equation}
Similarly, when $m_0(\bm{x})$ is linear on $\bm{x}$,
\begin{equation}
\sum_{i:Z_i=0}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i = \hat{\beta}_{0c} + \hat{\bm{\beta}}^\top_{1c} \bar{\bm{X}} \xrightarrow[n \to \infty]{P} {E}(m_0(\bm{X}_i)) = {E}(Y_i(0)).
\label{4.3.2}
\end{equation}
Equations \ref{4.3.1} and \ref{4.3.2} prove part (iii) of the Theorem.
Now, let $p,\bm{\mu}, \bm{\mu}_t, \bm{\Sigma}_t$ be as in the proof of Theorem 1.
\begin{align}
\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i &= \bar{Y}_t + (\bar{\bm{X}} - \bar{\bm{X}}_t)^\top \big( \frac{\bm{S}_t}{n_t}\big)^{-1} \Big\{\frac{1}{n_t}\sum_{i:Z_i=1}(\bm{X}_i- \bar{\bm{X}}_t)Y^{\text{obs}}_i \Big\} \nonumber\\
& \xrightarrow[n \to \infty]{P} {E}(Y^{\text{obs}}_i\mid Z_i=1) + (\bm{\mu} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t \text{cov}(\bm{X}_i, Y^{\text{obs}}_i\mid Z_i=1) ,
\label{4.3.3}
\end{align}
where the above convergence holds by a combination of WLLN, Slutsky's theorem and continuous mapping theorem. Under unconfoundedness, ${E}(Y^{\text{obs}}_i\mid Z_i=1) = \frac{1}{p}{E}(m_1(\bm{X}_i)e(\bm{X}_i))$, Similarly, $\text{cov}(\bm{X}_i, Y^{\text{obs}}_i\mid Z_i=1)= \frac{1}{p} \big( {E}(\bm{X}_i m_1(\bm{X}_i)e(\bm{X}_i)) - \bm{\mu}_t {E}(m_1(\bm{X}_i)e(\bm{X}_i)) \big)$. This implies,
\begin{align}
{E}(Y^{\text{obs}}_i\mid Z_i=1) + (\bm{\mu} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t \text{cov}(\bm{X}_i, Y^{\text{obs}}_i\mid Z_i=1) \nonumber\\
= {E}\Big(\frac{m_1(\bm{X}_i)e(\bm{X}_i)}{p} \big\{1+ (\bm{\mu} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t(\bm{X}_i - \bm{\mu}_t) \big\} \Big)
\label{4.3.4}
\end{align}
Now, if $e(\bm{x})$ is inverse-linear on $\bm{x}$, by Equation \ref{4.2.1} in the proof of Theorem 1, we have $\frac{1}{e(\bm{x})} = \frac{1}{p} \big\{1 + (\bm{x} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t (\bm{\mu} - \bm{\mu}_t) \big\}$. From Equation \ref{4.3.4}, we get ${E}(Y^{\text{obs}}_i\mid Z_i=1) + (\bm{\mu} - \bm{\mu}_t)^\top \bm{\Sigma}^{-1}_t \text{cov}(\bm{X}_i, Y^{\text{obs}}_i\mid Z_i=1) = {E}(m_1(\bm{X}_i)) = {E}(Y_i(1))$. Therefore, if $e(\bm{x})$ is inverse-linear, we have
\begin{equation}
\sum_{i:Z_i=1}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i \xrightarrow[n \to \infty]{P} {E}(Y_i(1)).
\label{4.3.5}
\end{equation}
Similarly, if $1-e(\bm{x})$ is inverse-linear, we have
\begin{equation}
\sum_{i:Z_i=0}w^{\scriptscriptstyle \text{MRI}}_i Y^{\text{obs}}_i \xrightarrow[n \to \infty]{P} {E}(Y_i(0)).
\label{4.3.6}
\end{equation}
Equations \ref{4.3.2} and \ref{4.3.5} prove consistency of the MRI estimator under condition (i) of the Theorem. Equations \ref{4.3.1} and \ref{4.3.6} prove consistency under condition (ii). When $e(\bm{x})$ is constant, then both $e(\bm{x})$ and $1-e(\bm{x})$ can be regarded as inverse-linear on $\bm{x}$ and hence consistency under condition (iv) follows from Equations \ref{4.3.5} and \ref{4.3.6}. Finally, when $p^2 \text{var}(\bm{X}_i\mid Z_i=1) = (1-p)^2 \text{var}(\bm{X}_i\mid Z_i=0)$, the URI and MRI estimator are asymptotically equivalent. Hence, consistency under (v) holds by similar argument as in the URI case (see the proof of Theorem \ref{dr_uri}.
\subsection{Proof of Proposition 4}
For $g \in \{t,c\}$, let $\underline{\tilde{\bm{X}}}_g$ be the design matrix in treatment group $g$. Also, let $\underline{\tilde{\bm{X}}}$ be the design matrix in the full-sample.
We consider the MRI approach first. Without loss of generality, we compute the sample influence curve for a treated unit $i$. Since the two regression models in MRI are fitted separately, the SIC for unit $i$ for the MRI estimator of the ATE is the same as that for the MRI estimator of $\hat{{E}}(Y(1))$.
Let $\hat{\bm{b}}_t: = (\hat{\beta}_{0t},\hat{\bm{\beta}}^\top_{1t})^\top$ be the estimated vector of coefficients in the regression model in the treatment group. Also, let $\hat{\bm{b}}_{(i)t}$ be the corresponding estimated vector of coefficients when the model is fitted excluding unit $i$. It follows that (see \perp\!\!\!\perptealt{cook1982residuals}, Chapter 3),
\begin{equation}
\hat{\bm{b}}_t - \hat{\bm{b}}_{(i)t} = (\underline{\tilde{\bm{X}}}^\top_t \underline{\tilde{\bm{X}}}_t)^{-1} \tilde{\bm{X}_i}\frac{e_i}{1-h_{ii,t}},
\label{t5.1.1}
\end{equation}
where $\tilde{\bm{X}_i} = (1,\bm{X}^\top_i)^\top$. Denote $\tilde{\bar{\bm{X}}} = (1,\bar{\bm{X}}^\top)^\top$. Since $\hat{{E}}(Y(1)) = \tilde{\bar{\bm{X}}}^\top \hat{\bm{b}}_t$. Therefore, the SIC of unit $i$ is given by
\begin{equation}
\text{SIC}_i = (n_t-1)(\tilde{\bar{\bm{X}}}^\top \hat{\bm{b}}_t - \tilde{\bar{\bm{X}}}^\top\hat{\bm{b}}_{(i)t}) = (n_t-1)\tilde{\bar{\bm{X}}}^\top(\underline{\tilde{\bm{X}}}^\top_t \underline{\tilde{\bm{X}}}_t)^{-1} \tilde{\bm{X}_i}\frac{e_i}{1-h_{ii,t}}
\label{t5.1.2}
\end{equation}
We observe that $\sum_{i:Z_i=1} w^{{\scriptscriptstyle \text{MRI}}}_i Y^{\text{obs}}_i = \tilde{\bar{\bm{X}}}^\top \hat{\bm{b}}_t = \tilde{\bar{\bm{X}}}^\top (\underline{\tilde{\bm{X}}}^\top_t \underline{\tilde{\bm{X}}}_t)^{-1} \underline{\tilde{\bm{X}}}^\top_t \bm{y}_t$, where $\bm{y}_t$ is the vector of outcomes in the treatment group. So we can alternatively express the MRI weights in the treatment group as $w^{{\scriptscriptstyle \text{MRI}}}_i = \tilde{\bar{\bm{X}}}^\top(\underline{\tilde{\bm{X}}}^\top_t \underline{\tilde{\bm{X}}}_t)^{-1} \tilde{\bm{X}_i}$. It follows from Equation \ref{t5.1.2} that,
\begin{equation}
\text{SIC}_i = (n_t-1)\frac{e_i}{1-h_{ii,t}}w^{{\scriptscriptstyle \text{MRI}}}_i.
\label{t5.1.3}
\end{equation}
This completes the proof for MRI.
Let $\bm{l} = (0,0,...0,1) \in \mathbb{R}^{k+2}$. Consider the URI regression model $Y^{\text{obs}}_i = \beta_0 + \bm{\beta}^\top_1 \bm{X}_i + \tau Z_i + \epsilon_i$. Similar to the MRI case, let $\hat{\bm{b}}$ (respectively, $\hat{\bm{b}}_{(i)}$) be the vector of regression coefficients when the regression model is fitted using all the units (respectively, all excluding the $i$th unit). By similar calculations as before, it follows that,
\begin{equation}
\hat{\bm{b}} - \hat{\bm{b}}_{(i)} = (\underline{\tilde{\bm{X}}}^\top \underline{\tilde{\bm{X}}})^{-1} \tilde{\bm{X}_i}\frac{e_i}{1-h_{ii}},
\label{t5.1.4}
\end{equation}
Now the URI estmimator $\hat{\tau}^{\text{OLS}}$ can be expressed as,
\begin{equation}
\hat{\tau}^{\text{OLS}} = \bm{l}^\top \hat{\bm{b}} = \bm{l}^\top (\underline{\tilde{\bm{X}}}^\top \underline{\tilde{\bm{X}}})^{-1} \tilde{\bm{X}}^\top\bm{y}.
\label{t5.1.5}
\end{equation}
Since $\hat{\tau}^{\text{OLS}} = \sum_{i:Z_i=1} w^{{\scriptscriptstyle \text{URI}}}_i Y^{\text{obs}}_i - \sum_{i:Z_i=0} w^{{\scriptscriptstyle \text{URI}}}_i Y^{\text{obs}}_i$, we can alternatively express the URI weight of unit $i$ as $w^{{\scriptscriptstyle \text{URI}}}_i = (2Z_i-1)\tilde{\bm{X}}_i (\underline{\tilde{\bm{X}}}^\top \underline{\tilde{\bm{X}}})^{-1} \bm{l}$. Therefore, the sample influence curve of unit $i$ is given by
\begin{align}
\text{SIC}_i = (n-1)(\bm{l}^\top \hat{\bm{b}} - \bm{l}^\top\hat{\bm{b}}_{(i)}) = (n-1)\bm{l}^\top(\underline{\tilde{\bm{X}}}^\top \underline{\tilde{\bm{X}}})^{-1} \tilde{\bm{X}_i}\frac{e_i}{1-h_{ii}} \nonumber\\
= (n-1) \frac{e_i}{(1-h_{ii})}(2Z_i-1)w^{\scriptscriptstyle \text{URI}}_i.
\label{t5.1.6}
\end{align}
This completes the proof for URI.
\end{document}
|
\begin{document}
\title{The Error in Multivariate Linear Extrapolation with Applications to Derivative-Free Optimization}
\begin{abstract}
We study in this paper the function approximation error of multivariate linear extrapolation.
The sharp error bound of linear interpolation already exists in the literature.
However, linear extrapolation is used far more often in applications such as derivative-free optimization, while its error is not well-studied.
We introduce in this paper a method to numerically compute the sharp bound on the error, and then present several analytical bounds along with the conditions under which they are sharp.
We analyze in depth the approximation error achievable by quadratic functions and the error bound for the bivariate case.
All results are under the assumptions that the function being interpolated has Lipschitz continuous gradient and is interpolated on an affinely independent sample set.
\mathbf{e}nd{abstract}
\section{Introduction} \label{sec:intro}
Polynomial interpolation is one of the most basic techniques for approximating functions and plays an essential role in applications such as finite element methods and derivative-free optimization.
This led to a large amount of literature concerning its approximation error.
This paper contributes to this area of study by analyzing the function approximation error of linear interpolation and extrapolation.
Specifically, given a function $f: \R^n \rightarrow \R$ and an affinely independent sample set $\Theta:= \{\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_{n+1}\} \subset \R^n$, one can find a unique affine function $\hat{f}: \R^n \rightarrow \R$ such that $\hat{f}(\mathbf{x}_i) = f(\mathbf{x}_i)$ for all $i \in \{1,\dots,n+1\}$.
We investigate in this paper the (sharp) upper bound on the approximation error $|\hat{f}(\mathbf{x}) - f(\mathbf{x})|$ when the sample set $\Theta$ and the point where the error is measured $\mathbf{x}$ are given, and $f$ is assumed to belong to $C_\nu^{1,1}(\R^n)$.
The class $C_\nu^{1,1}(\R^n)$ represents the differentiable functions defined on $\R^n$ with their first derivative $Df$ being $\nu$-Lipschitz continuous, i.e.,
\begin{equation} \label{eq:Lipschitz}
\|Df(\mathbf{u}) - Df(\mathbf{v})\| \le \nu \|\mathbf{u} - \mathbf{v}\| \quad \text{for all } \mathbf{u},\mathbf{v} \in \R^n,
\mathbf{e}nd{equation}
where $\nu>0$ is the Lipschitz constant, and the norms are Euclidean.
The sharp bound on $|\hat{f}(\mathbf{x}) - f(\mathbf{x})|$ is already discovered and proved in \cite{waldron1998error} for linear interpolation, but only for the case when the word ``interpolation'' is used in its narrow sense, i.e., when $\mathbf{x} \in \conv(\Theta)$, the convex hull of $\Theta$.
In this paper, we make no assumption on the location of $\mathbf{x}$ relative to $\Theta$, and the word ``interpolation'' is typically used to refer to this general case.
The function approximation error of univariate ($n=1$) interpolation using polynomials of any degree is already well-studied, and the results can be found in classical literature such as \cite{davis1975book}.
If a $(d+1)$-times differentiable function $f$ defined on $\R$ is interpolated by a polynomial of degree $d$ on $d+1$ unique points $\{x_1, x_2, \dots, x_{d+1}\} \subset \R$, then the resulting polynomial has the approximation error
\begin{equation} \label{eq:Cauchy remainder}
\frac{(x-x_1)(x-x_2)\cdots(x-x_{d+1})}{(d+1)!} D^{n+1} f(\mathbf{x}i) \quad \text{for all } x \in \R
\mathbf{e}nd{equation}
for some $\mathbf{x}i$ with $\min(x,x_1,,\dots,x_{d+1}) < \mathbf{x}i < \max(x,x_1,\dots,x_{d+1})$.
Unfortunately this result cannot be extended to the multivariate ($n>1$) case directly, even if the polynomial is linear ($d=1$).
The function approximation error of multivariate polynomial interpolation has been studied by researchers from multiple research fields.
Motivated by their application in finite element methods, formulae for the errors in both Lagrange and Hermite interpolation with polynomials of any degree were derived in \cite{ciarlet1972general}.
As a part of an effort to develop derivative-free optimization algorithms, a bound on the error of quadratic interpolation was provided in \cite{powell2001lagrange}.
The sharp error bound for linear interpolation was found by researchers of approximation theory for the case when $\mathbf{x} \in \conv(\Theta)$ using the unique Euclidean sphere that contains $\Theta$ in \cite{waldron1998error}.
Following \cite{waldron1998error}, a number of sharp error bounds were derived in \cite{stampfle2000optimal} for linear interpolation under several different smoothness or continuity assumptions in addition to \mathbf{e}qref{eq:Lipschitz}.
While the sharp error bound for the $\mathbf{x} \in \conv(\Theta)$ case is already established, in applications like model-based derivative-free optimization (DFO), where linear interpolation is employed to approximate the black-box objective function \cite{powell1994direct, DFO_book}, the approximation model $\hat{f}$ is used more often than not to estimate the function value at a point outside $\conv(\Theta)$.
As illustrated in Figure~\ref{fig:DFO-TR}, these optimization algorithms attempt to minimize the objective function by alternately constructing a linear interpolation model and minimizing the model inside a trust region, where the trust region is typically a ball around the point with the lowest known function value.
The minimizer of the model inside the trust region would then have its function value evaluated and become part of the sample set for constructing the linear interpolation model in the next iteration.
In practice, this minimizer rarely locates inside $\conv(\Theta)$.
There is also another class of DFO methods known as the simplex methods.
One example is the famous Nelder-Mead method \cite{nelder1965simplex}.
As illustrated in Figure~\ref{fig:NelderMead}, the main routine of these algorithms involves taking a set of $n+1$ affinely independent points $\Theta$ (the vertices of a simplex) and reflecting the one with the largest function value through the hyperplane defined by the rest.
While linear interpolation is not used in these algorithms, the range of the function value at this reflection point ($\mathbf{x}_4$ in Figure~\ref{fig:NelderMead} and is always outside $\conv(\Theta)$) can be determined by the sum of the value estimated by interpolation model and the error of the estimation.
\begin{figure}[tbhp]
\centering
\subfloat[\raggedright Two consecutive iterations of a DFO algorithm based on linear interpolation and trust region method. The circle represents the trust region, which changes center and expands after finding the minimizer $\mathbf{x}_4$ that has a lower function value than the current center $\mathbf{x}_3$. ]{\label{fig:DFO-TR}\resizebox{0.7\linewidth}{!}{\input{figures/DFO-TR.tex}}}
\subfloat[\raggedright One iteration of the Nelder-Mead method. The next simplex will be formed by $\{\mathbf{x}_2,\mathbf{x}_3,\mathbf{x}_4\}$. ]{\label{fig:NelderMead}\resizebox{0.25\linewidth}{!}{\input{figures/NelderMead.tex}}}
\caption{An illustration of two DFO algorithm when minimizing a bivariate function, where $f(\mathbf{x}_1) >$ $f(\mathbf{x}_2) >$ $f(\mathbf{x}_3)$. The vertices of the triangles represents $\Theta$. This figure only illustrates the algorithms' behavior when the trial point $\mathbf{x}_4$ satisfies $f(\mathbf{x}_4) < f(\mathbf{x}_3)$.}
\label{fig:DFO}
\mathbf{e}nd{figure}
To further the design and analysis of these DFO algorithms, we use both numerical and analytical approaches to investigate the sharp upper bound on the function evaluation error of linear interpolation.
The results of this investigation provides a theoretical basis to the analysis of numerical methods that use linear interpolation including the DFO methods mentioned above.
Furthermore, it can also be directly applied to improve certain DFO algorithms.
For example, the model-based algorithms, which are usually designed to optimize functions that are computationally expensive to evaluate, typically request a function evaluation for one of two purposes: to check a point predicted by the model to have an improvement in function value (as shown in Figure~\ref{fig:DFO-TR}) or to explore a point that can contribute to the construction of a more accurate approximation model.
Being able to estimate the magnitude of the approximation error at a given point in the former case allows the algorithm to compare it to the predicted improvement in function value and make an informed decision on whether the point is worth evaluating.
By prioritizing spending the function evaluation to improve the model rather than check the point when the error is relatively large, the algorithm's overall efficiency can be improved.
The applications of this paper's results in DFO will be further discussed later, but please keep in mind that our analysis is for linear interpolation in general and can be applied wherever this approximation technique is used.
Our main contributions are as follows.
\begin{enumerate}
\item We formulate the problem of finding the sharp error bound as a nonlinear programming problem and show that it can be solved numerically to obtain the desired bound.
\item An analytical bound on the function approximation error is derived and proved to be sharp for interpolation and, under certain conditions, for extrapolation.
\item The largest function approximation error that is achievable by quadratic functions in $C_\nu^{1,1}(\R^n)$ is derived, and the condition under which it is an upper bound on the error achievable by all functions in $C_\nu^{1,1}(\R^n)$ is determined.
\item For bivariate ($n=2$) linear extrapolation, we analyze the case when neither of the two previous results equals to the sharp bound on the function approximation error and provide the formula for the actual sharp bound. We also show piecewise quadratic functions can achieve the approximation error indicated by the sharp bound.
\mathbf{e}nd{enumerate}
The paper is organized as follows.
Our notation and the preliminary knowledge are introduced in Section~\ref{sec:preliminaries}.
The nonlinear programming problem is present in Section~\ref{sec:numerical}.
In Section~\ref{sec:phase1}, we generalize an existing analytical bound and then improve it.
In Section~\ref{sec:phase2}, we study the error in approximating quadratic functions.
In Section~\ref{sec:phase3}, we show how to calculate the sharp bound on function approximation error of bivariate linear interpolation.
We conclude the paper in Section~\ref{sec:discussion} by discussing our findings and some open questions.
\section{Notation and Preliminaries} \label{sec:preliminaries}
Since the research in this paper involves approximation theory and optimization, to appeal to audiences from both research fields, we provide a detailed introduction to our notation and the preliminary knowledge.
Throughout the paper, vectors are denoted by boldface letters and matrices by capital letters.
We denote by $\|\cdot\|$ the Euclidean norm.
The dot product between vectors or matrices of the same size, $\mathbf{u}\cdot\mathbf{v}$ or $U \cdot V$, is the summation of the entry-wise product, which are customarily denoted by $\mathbf{u}^T\mathbf{v}$ and Tr$(U^T V)$ in optimization literature.
Let $\mathbf{e}_i$ be the vector that is all 0 but have 1 as its $i$th entry.
Let $Y \in \R^{(n+1)\times n}$ be the matrix such that its $i$th row $\displaystyle Y^T \mathbf{e}_i = \mathbf{x}_i-\mathbf{x}$ for all $i = 1,2,\dots,n+1$.
We define $\phi:\R^n \rightarrow \R^{n+1}$ as the {\it basis function} such that $\phi(\mathbf{u}) = \begin{bmatrix} 1 &\mathbf{u}^T \mathbf{e}nd{bmatrix}^T$ for all $\mathbf{u}\in\R^n$,
and $\Phi$ as the $(n+1)$-by-$(n+1)$ matrix $\begin{bmatrix} \mathbf{1} &Y \mathbf{e}nd{bmatrix}$, where $\mathbf{1}$ is the all one vector.
Notice the affine independence of $\Theta$ implies the nonsigularity of $\Phi$.
Let $\mathbf{e}ll_1, \dots, \mathbf{e}ll_{n+1}$ be the {\it Lagrange polynomials}, i.e. the unique set of polynomials such that $\mathbf{e}ll_i(\mathbf{x}_j) = 1$ if $i=j$, and $\mathbf{e}ll_i(\mathbf{x}_j) = 0$ if $i\neq j$.
The values of these polynomials at $\mathbf{x}$ coincides with the set of barycentric coordinates of $\mathbf{x}$ with respect to $\Theta$ and have the following properties:
\begin{align}
\sum_{i=1}^{n+1} \mathbf{e}ll_i(\mathbf{x}) f(\mathbf{x}_i) &= \hat{f}(\mathbf{x}), \label{eq:Lagrange m} \\
\sum_{i=1}^{n+1} \mathbf{e}ll_i(\mathbf{x}) &= 1, \label{eq:Lagrange 0} \\
\text{and } \sum_{i=0}^{n+1} \mathbf{e}ll_i(\mathbf{x}) \mathbf{x}_i &= \mathbf{0}, \label{eq:Lagrange Y}.
\mathbf{e}nd{align}
The concepts of basis functions and Lagrange polynomials are fundamental to approximation theory.
The book \cite{DFO_book} offers a comprehensive introduction to them in the context of derivative-free optimization.
For the ease of exposition, we abbreviate $\mathbf{e}ll_i(\mathbf{x})$ to $\mathbf{e}ll_i$ and define $\mathbf{x}_0 = \mathbf{x}$ and $\mathbf{e}ll_0 = -1$.
Another reason for the artificially defined $\mathbf{x}_0$ and $\mathbf{e}ll_0$ will be made clear in Section~\ref{sec:numerical}.
Without loss of generality, we assume the set $\Theta = \{\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_{n+1}\}$ is ordered in a way such that $\mathbf{e}ll_1 \mathbf{g}e \mathbf{e}ll_2 \mathbf{g}e \cdots \mathbf{g}e \mathbf{e}ll_{n+1}$.
We define the following two sets of indices:
\begin{subequations} \begin{align}
\cI_+ &= \{i\in \{0,1,\dots,n+1\}:~ \mathbf{e}ll_i>0\} = \{1,2,\dots, |\cI_+|\}, \\
\cI_- &= \{i\in \{0,1,\dots,n+1\}:~ \mathbf{e}ll_i<0\} = \{0, n+3-|\cI_-|, \dots, n+1\}.
\mathbf{e}nd{align} \mathbf{e}nd{subequations}
Notice \mathbf{e}qref{eq:Lagrange 0} implies $\cI_+ \neq \mathbf{e}mptyset$, and $\mathbf{e}ll_0=-1$ implies $\cI_- \neq \mathbf{e}mptyset$.
It is possible for $n+3-|\cI_-| > n+1$, in which case $\cI_- = \{0\}$.
We define the following matrix $G\in\R^{n\times n}$:
\begin{equation} \label{eq:G}
G = \sum_{i=0}^{n+1} \mathbf{e}ll_i \mathbf{x}_i \mathbf{x}_i^T,
\mathbf{e}nd{equation}
which will be used frequently in our analysis.
The notation $\mathbf{x}_i \mathbf{x}_i^T$ is the outer product of $\mathbf{x}_i$ and is sometimes denoted by $\mathbf{x}_i^2$ or $\mathbf{x}_i \otimes \mathbf{x}_i$ otherwise.
The matrix $G$ has the property that for any $\mathbf{u},\mathbf{v} \in \R^n$,
\begin{equation} \label{eq:G recenter} \begin{aligned}
\sum_{i=0}^{n+1} \mathbf{e}ll_i [\mathbf{x}_i-\mathbf{u}] [\mathbf{x}_i-\mathbf{v}]^T
&= \sum_{i=0}^{n+1} \mathbf{e}ll_i \left[\mathbf{x}_i\mathbf{x}_i^T - \mathbf{u}\mathbf{x}_i^T - \mathbf{x}_i\mathbf{v}^T + \mathbf{u}\mathbf{v}^T \right] \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange Y}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i \left[\mathbf{x}_i\mathbf{x}_i^T + \mathbf{u}\mathbf{v}^T\right]
\stackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i \mathbf{x}_i\mathbf{x}_i^T = G.
\mathbf{e}nd{aligned} \mathbf{e}nd{equation}
The class of functions $C_\nu^{1,1}(\R^n)$ is ubiquitous in the research of nonlinear optimization.
It is well-known (see, e.g., section 1.2.2 of the textbook \cite{Nesterov_book}) that the inclusion $f \in C_\nu^{1,1}(\R^n)$ implies
\begin{equation} \label{eq:Lipschitz quadratic}
|f(\mathbf{v}) - f(\mathbf{u}) - Df(\mathbf{u}) \cdot (\mathbf{v} - \mathbf{u})| \le \frac{\nu}{2} \|\mathbf{v} - \mathbf{u}\|^2 \text{ for all } \mathbf{u},\mathbf{v} \in \R^n,
\mathbf{e}nd{equation}
and that if $f$ is twice differentiable on $\R^n$, \mathbf{e}qref{eq:Lipschitz} and \mathbf{e}qref{eq:Lipschitz quadratic} are equivalent to
\begin{equation} \label{eq:Lipschitz Hessian}
-\nu I \preceq D^2 f(\mathbf{u}) \preceq \nu I \text{ for all } \mathbf{u} \in \R^n,
\mathbf{e}nd{equation}
where the condition \mathbf{e}qref{eq:Lipschitz Hessian} is often written as $\|~|D^2 f|~\|_{L_\infty(\R^n)} \le \nu$ in approximation theory literature.
What is less well-known about the class $C_\nu^{1,1}(\R^n)$ is that $f \in C_\nu^{1,1}(\R^n)$ also implies
\begin{equation} \label{eq:Lipschitz stronger} \begin{aligned}
f(\mathbf{v}) \le &f(\mathbf{u}) + \frac{1}{2} (Df(\mathbf{u}) + Df(\mathbf{v})) \cdot (\mathbf{v} - \mathbf{u}) \\
&+ \frac{\nu}{4} \|\mathbf{v}-\mathbf{u}\|^2 - \frac{1}{4\nu} \|Df(\mathbf{v}) - Df(\mathbf{u})\|^2 \text{ for all } \mathbf{u},\mathbf{v} \in \R^n.
\mathbf{e}nd{aligned} \mathbf{e}nd{equation}
For differentiable functions, \mathbf{e}qref{eq:Lipschitz}, \mathbf{e}qref{eq:Lipschitz quadratic}, and \mathbf{e}qref{eq:Lipschitz stronger} are equivalent.
\section{Error Estimation Problem} \label{sec:numerical}
In this section, we formulate the problem of finding the sharp error bound as a numerically solvable nonlinear optimization problem.
We first make the important observation that the problem of finding the sharp upper bound on the error is the same as asking for the largest error that a function from $C_\nu^{1,1}(\R^n)$ can achieve.
Thus, it can be formulated as the following problem of maximizing the approximation error over the functions in $C_\nu^{1,1}(\R^n)$:
\begin{equation} \label{prob:D} \tag{EEP} \mathbf{e}verymath{\displaystyle} \begin{array}{ll}
\max_f &|\hat{f}(\mathbf{x}) - f(\mathbf{x})| \\
\text{s.t. } &f \in C_\nu^{1,1}(\R^n),
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
where $\hat{f}$ is the affine function that interpolates $f$ on a given set of $n+1$ affinely independent points $\Theta = \{\mathbf{x}_1,\dots,\mathbf{x}_{n+1}\}$.
We call this problem the \textit{error estimation problem} (EEP), a name inspired by the \textit{performance estimation problem} (PEP).
First proposed in \cite{drori2014performance}, a PEP is a nonlinear programming formulation of the problem of finding an optimization algorithm's worst-case performance over a set of possible objective functions.
It involves maximizing a performance measure of the given algorithm (the larger the measure, the worse the performance) over the objective functions and, similar to \mathbf{e}qref{prob:D}, is an infinite-dimensional problem.
However, with some algorithms and functions, particularly first-order nonlinear optimization methods and convex functions, the PEP is shown to have finite-dimensional equivalents that can be solved numerically \cite{taylor2017smooth,taylor2017exact}, thus providing a computer-aided analysis tool for estimating an algorithm's worst-case performance.
Using these theories developed for PEP, we can process the functional constraint $f \in C_\nu^{1,1}(\R^n)$ and turn \mathbf{e}qref{prob:D} into a finite-dimensional problem.
Particularly, we use the following theorem from \cite{taylor2017exact}, which states $f \in C_\nu^{1,1}(\R^n)$ can be replaced by \mathbf{e}qref{eq:Lipschitz stronger} for every pair of points in $\Theta \cup \{\mathbf{x}\}$.
\begin{proposition}[Theorem 3.10 \cite{taylor2017exact}]\label{prop:functional}
Let $\nu > 0$ and $\cI$ be an index set, and consider a set of triples $\{(\mathbf{x}_i,\mathbf{g}_i,y_i)\}_{i\in\cI}$ where $\mathbf{x}_i\in\R^n$, $\mathbf{g}\in\R^n$, and $y_i\in\R$ for all $i\in\cI$.
There exists a function $f\in C_\nu^{1,1}(\R^n)$ such that both $\mathbf{g}_i = Df(\mathbf{x}_i)$ and $y_i = f(\mathbf{x}_i)$ hold for all $i\in\cI$ if and only if the following inequality holds for all $i,j\in\cI$:
\begin{equation} \label{eq:Lipschitz stronger ij} \begin{aligned}
y_j \le y_i + \frac{1}{2} (\mathbf{g}_i + \mathbf{g}_j) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j - \mathbf{g}_i\|^2.
\mathbf{e}nd{aligned} \mathbf{e}nd{equation}
\mathbf{e}nd{proposition}
The above proposition allows us to replace the functional variable $f$ with the function values $\{y_i\}$ and gradients $\{\mathbf{g}_i\}$ at $\Theta$ and $\mathbf{x}$.
Before applying this proposition, we first substitute the approximated function value $\hat{f}(\mathbf{x})$ in \mathbf{e}qref{prob:D} with $\sum_{i=1}^{n+1} \mathbf{e}ll_i f(\mathbf{x}_i)$ using \mathbf{e}qref{eq:Lagrange m} and drop the absolute sign in the objective function.
The absolute sign can be dropped thanks to the symmetry of \mathbf{e}qref{eq:Lipschitz}, that is, $-f \in C_\nu^{1,1}(\R^n)$ for any $f \in C_\nu^{1,1}(\R^n)$, and the approximation error on the two functions $f$ and $-f$ are negatives of each other.
Finally, by applying Proposition~\ref{prop:functional}, we arrive at \mathbf{e}qref{prob:f-D}, a finite-dimensional equivalent to \mathbf{e}qref{prob:D}:
\begin{equation} \label{prob:f-D} \tag{f-EEP} \mathbf{e}verymath{\displaystyle} \begin{array}{ll}
\max_{y_i,\mathbf{g}_i} &\sum_{i=0}^{n+1}\mathbf{e}ll_i y_i \\
\text{s.t. } &y_j \le y_i + \frac{1}{2} (\mathbf{g}_i + \mathbf{g}_j) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 \\
&\qquad - \frac{1}{4\nu} \|\mathbf{g}_j - \mathbf{g}_i\|^2 \quad \forall i,j\in\{0,\dots,n+1\}.
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
The optimization problem \mathbf{e}qref{prob:f-D} is a convex quadratically constrained quadratic program (QCQP).
This type of problem can be solved by standard nonlinear optimization solvers.
However, \mathbf{e}qref{prob:f-D} contains $n+1$ redundant degrees of freedom, which means it has infinitely many optimal solutions, and the solvers can sometimes have difficulty solving it.
It is best to eliminate these degrees of freedom first.
The elimination can be done in many ways.
For example, one can fix $\{y_i\}_{i=1}^{n+1}$ in \mathbf{e}qref{prob:f-D} to their observed values.
Indeed, these function values are needed for constructing the affine approximation $\hat{f}$, so it is natural to assume they are known.
However, we note that the optimal value of \mathbf{e}qref{prob:D} and \mathbf{e}qref{prob:f-D} is affected by the locations of the sample points $\Theta$ in the input space but is invariant to the observed function values at these points.
Thus, for the purpose of solving \mathbf{e}qref{prob:f-D}, it is also justified to simply set $y_i = 0$ for all $i=1,\dots,n+1$.
Alternatively, one can also fix $(\mathbf{g}_i,y_i)$ to $(\mathbf{0}, 0)$ for any $i \in \{0,1,\dots,n+1\}$.
We formally prove in the following proposition the $n+1$ degrees of freedom can be removed in these two ways.
\begin{proposition}
The following statements are true.
\begin{enumerate}
\item If any function $f$ is optimal to \mathbf{e}qref{prob:D}, then the function $f'(\mathbf{u}) = f(\mathbf{u}) + c + \mathbf{g}\cdot\mathbf{u}$ is also optimal with any $c\in\R$ and $\mathbf{g}\in\R^n$.
\item The optimal value of \mathbf{e}qref{prob:f-D} does not change if $\{y_i\}_{i=1}^{n+1}$ are fixed to any arbitrary values.
\item The optimal value of \mathbf{e}qref{prob:f-D} does not change if $\mathbf{g}_k$ and $y_k$ are fixed to any arbitrary values for some $k \in \{0,1,\dots,n+1\}$.
\mathbf{e}nd{enumerate}
\mathbf{e}nd{proposition}
\begin{proof}
By the definition \mathbf{e}qref{eq:Lipschitz}, it is easy to see $f' \in C_\nu^{1,1}(\R^n)$ whenever $f\in C_\nu^{1,1}(\R^n)$. The two objective values can also be shown to be the same using \mathbf{e}qref{eq:Lagrange m}, \mathbf{e}qref{eq:Lagrange 0}, and \mathbf{e}qref{eq:Lagrange Y}:
\[ \sum_{i=0}^{n+1} \mathbf{e}ll_i f'(\mathbf{x}_i) = \sum_{i=0}^{n+1} \mathbf{e}ll_i [f(\mathbf{x}_i) + c + \mathbf{g}\cdot\mathbf{x}_i] = \sum_{i=0}^{n+1} \mathbf{e}ll_i f(\mathbf{x}_i). \]
The first statement is thus true.
To prove the second statement, we first assume \mathbf{e}qref{prob:f-D} has an optimal solution $\{y_i^\star, \mathbf{g}_i^\star\}_{i=0}^{n+1}$.
Now suppose the problem has an additional set of constraints that fixes the function values of the points in $\Theta$ to some arbitrary values $\{y_i\}_{i=1}^{n+1}$.
Then, this new problem has the exact same optimal value as the original \mathbf{e}qref{prob:f-D}, and an optimal solution satisfies $\mathbf{g}_i = \mathbf{g}_i^\star + \mathbf{g}$ for all $i=0,1,\dots,n+1$ and $y_0 = y_0^\star + c + \mathbf{g}\cdot\mathbf{x}_0$, where $(\mathbf{g},c)$ is the unique solution to the linear system $c + \mathbf{g}\cdot\mathbf{x}_i = y_i - y_i^\star, i=1,\dots,n+1$.
Indeed, the constraints of this new problem are satisfied as
\[ \begin{aligned}
&- y_j + y_i + \frac{1}{2} (\mathbf{g}_i + \mathbf{g}_j) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j - \mathbf{g}_i\|^2 \\
&= - y_j + y_i + \frac{1}{2} (\mathbf{g}_i^\star + \mathbf{g}_j^\star + 2\mathbf{g}) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j^\star - \mathbf{g}_i^\star\|^2 \\
&= - y_j^\star + y_i^\star + \frac{1}{2} (\mathbf{g}_i^\star + \mathbf{g}_j^\star) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j^\star - \mathbf{g}_i^\star\|^2
\mathbf{g}e 0
\mathbf{e}nd{aligned} \]
for all $i,j = 0,1,\dots,n+1$,
where the second equality is true because $\mathbf{g}\cdot(x_j-x_i) = (y_j - y_j^\star - c) - (y_i - y_i^\star - c)$, and the objective function
\[ \sum_{i=0}^{n+1} \mathbf{e}ll_i y_i
= y_0^\star + c + \mathbf{g}\cdot\mathbf{x}_0 + \sum_{i=1}^{n+1} \mathbf{e}ll_i y_i
\stackrel{\mathbf{e}qref{eq:Lagrange 0}\mathbf{e}qref{eq:Lagrange Y}}{=} y_0^\star + \sum_{i=0}^{n+1} \mathbf{e}ll_i [y_i + c + \mathbf{g}\cdot\mathbf{x}_i]
= \sum_{i=0}^{n+1} \mathbf{e}ll_i y_i^\star.
\]
Similarly, \mathbf{e}qref{prob:f-D} with $(\mathbf{g}_k,y_k)$ fixed for some $k\in\{0,1,\dots,n+1\}$ also has the same optimal value as \mathbf{e}qref{prob:f-D}, and its optimal solution satisfies $\mathbf{g}_i = \mathbf{g}_i^\star - \mathbf{g}_k^\star + \mathbf{g}_k$ and $y_i = y_i^\star - y_k^\star + y_k + (\mathbf{g}_k-\mathbf{g}_k^\star)\cdot(\mathbf{x}_i-\mathbf{x}_k)$ for all $i = 0,1,\dots,n+1$.
The constraints are satisfied as
\[ \begin{aligned}
&- y_j + y_i + \frac{1}{2} (\mathbf{g}_i + \mathbf{g}_j) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j - \mathbf{g}_i\|^2 \\
&= - [y_j^\star - y_k^\star + y_k + (\mathbf{g}_k-\mathbf{g}_k^\star)\cdot(\mathbf{x}_j-\mathbf{x}_k)] + [y_i^\star - y_k^\star + y_k + (\mathbf{g}_k-\mathbf{g}_k^\star)\cdot(\mathbf{x}_i-\mathbf{x}_k)] \\
&\quad + \frac{1}{2} (\mathbf{g}_i^\star + \mathbf{g}_j^\star + 2\mathbf{g}_k - 2\mathbf{g}_k^\star) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j^\star - \mathbf{g}_i^\star\|^2 \\
&= - y_j^\star + y_i^\star + \frac{1}{2} (\mathbf{g}_i^\star + \mathbf{g}_j^\star) \cdot (\mathbf{x}_j - \mathbf{x}_i) + \frac{\nu}{4} \|\mathbf{x}_j-\mathbf{x}_i\|^2 - \frac{1}{4\nu} \|\mathbf{g}_j^\star - \mathbf{g}_i^\star\|^2
\mathbf{g}e 0
\mathbf{e}nd{aligned} \]
for all $i,j = 0,1,\dots,n+1$,
and the objective function
\[ \sum_{i=0}^{n+1} \mathbf{e}ll_i y_i
= \sum_{i=0}^{n+1} \mathbf{e}ll_i [y_i^\star - y_k^\star + y_k + (\mathbf{g}_k-\mathbf{g}_k^\star)\cdot(\mathbf{x}_i-\mathbf{x}_k)]
\stackrel{\mathbf{e}qref{eq:Lagrange 0}\mathbf{e}qref{eq:Lagrange Y}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i y_i^\star.
\]
\mathbf{e}nd{proof}
Apart from its application in model-based derivative-free optimization as introduced in Section~\ref{sec:intro}, \mathbf{e}qref{prob:f-D} also offers us insight into the approximation error and guidance in seeking the analytical form of the sharp bound.
Particularly, it can be used to visualize the sharp error bound for bivariate linear interpolation.
We do this by first selecting a fixed set of three affinely independent sample points $\Theta \subset \R^2$ and a $100\times100$ grid.
Then, \mathbf{e}qref{prob:f-D} is solved repeatedly while $\mathbf{x}$ is set to each point on the grid.
The result of one instance of this numerical experiment is shown in Figure~\ref{fig:numerical}.
It can be observed that this bound is a piecewise smooth function of $\mathbf{x}$, and the boundaries between the smooth pieces align with the edges of the triangle defined by $\Theta$.
It will be shown in Section~\ref{sec:phase2} that this piecewise smooth function, at least in the case shown in Figure~\ref{fig:numerical} where $\conv(\Theta)$ is an acute triangle, can the represented by a single formula.
\begin{figure}[tbhp]
\centering
\includegraphics[width=0.9\linewidth, trim={0 1cm 0 0.9cm}, clip]{figures/numerical2D.eps}
\caption{The sharp error bound on $|\hat{f}(\mathbf{x}) - f(\mathbf{x})|$ for each $\mathbf{x}$ on the $100\times100$ grid that covers the area $[-2.5,2.5]\times[-1.5,2.5]$ evenly. The sample set and the Lipschitz constant are chosen as $\Theta = \{(-0.3,1), (-1.1,-0.5), (1,0)\}$ and $\nu = 1$.}
\label{fig:numerical}
\mathbf{e}nd{figure}
In \mathbf{e}qref{prob:f-D}, the point $\mathbf{x}$ and its derivative and function value are represented by $(\mathbf{x}_0, \mathbf{g}_0, y_0)$, whereas $(\mathbf{x}_i, \mathbf{g}_i,y_i)$ are used for the points $\mathbf{x}_i\in\Theta$ with $i=1,\dots,n+1$.
If we ignore what these points represent in linear interpolation and look at the optimization problem \mathbf{e}qref{prob:f-D} as it is, we can see that, in \mathbf{e}qref{prob:f-D}, the point $\mathbf{x}$ is not special comparing to the points in $\Theta$, with the only difference being the coefficient of $y_0$ in the objective is fixed to $\mathbf{e}ll_0 = -1$.
Therefore, to symbolize the point's ordinary status and simplify the expressions, we index $\mathbf{x}$ the zeroth point and sometimes use $\mathbf{x}_0$ in place of the customary $\mathbf{x}$.
This observation also leads us to the following proposition, which shows how the sharp error bound changes when $\mathbf{x}$ is swapped with a point in $\Theta$ and will be used to greatly simplify the analysis in Section~\ref{sec:phase3}.
\begin{proposition} \label{thm:swap}
Assume there is an affinely independent sample set $\Theta=\{\mathbf{x}_1,\dots,\mathbf{x}_{n+1}\}$ and a point $\mathbf{x}\in\R^n$ such that $\Theta\setminus\{\mathbf{x}_k\}\cup\{\mathbf{x}\}$ is also affinely independent for a given $k\in\{1,\dots,n+1\}$.
Let $\mathbf{e}ll_k$ be the Lagrange polynomial (with respect to $\Theta$ not $\Theta\setminus\{\mathbf{x}_k\}\cup\{\mathbf{x}\}$) corresponding to $\mathbf{x}_k$.
Let $\hat{f}$ and $\hat{f}'$ be the affine functions that interpolates some $f:\R^n \rightarrow \R$ on $\Theta$ and $\Theta\setminus\{\mathbf{x}_k\}\cup\{\mathbf{x}\}$, respectively.
The following two statements hold.
\begin{enumerate}
\item The function approximation error of $\hat{f}'$ at $\mathbf{x}_k$ is the error of $\hat{f}$ at $\mathbf{x}$ divided by $-\mathbf{e}ll_k(\mathbf{x})$, i.e., $\hat{f}'(\mathbf{x}_k) - f(\mathbf{x}_k) = (\hat{f}(\mathbf{x}) - f(\mathbf{x})) / (-\mathbf{e}ll_k(\mathbf{x}))$.
\item If $f\in C_\nu^{1,1}(\R^n)$ and $|\hat{f}(\mathbf{x}) - f(\mathbf{x})|$ is the largest error achievable by any function in $C_\nu^{1,1}(\R^n)$, then $f$ also achieves the largest $|\hat{f}'(\mathbf{x}_k) - f(\mathbf{x}_k)|$.
\mathbf{e}nd{enumerate}
\mathbf{e}nd{proposition}
\begin{proof}
If we divide $\hat{f}(\mathbf{x}) - f(\mathbf{x}) = \sum_{i=0}^{n+1} \mathbf{e}ll_i(\mathbf{x}) y_i$ by $-\mathbf{e}ll_k(\mathbf{x})$, the coefficient before $y_i$ becomes $\alpha_i = -\mathbf{e}ll_i(\mathbf{x})/\mathbf{e}ll_k(\mathbf{x})$ for all $i = 0,1,\dots,n+1$.
Since $\alpha_k = -1$, $\sum_{i=0}^{n+1} \alpha_i = 0$, and $\sum_{i=0}^{n+1} \alpha_i \mathbf{x}_i = 0$, the coefficients $\{\alpha_i\}_{i=0,i\neq k}^{n+1}$ are the values of the Lagrange polynomials with respect to $\Theta\setminus\{\mathbf{x}_k\}\cup\{\mathbf{x}\}$ at $\mathbf{x}_k$.
Thus, the quotient is exactly $\hat{f}'(\mathbf{x}_k) - f(\mathbf{x}_k)$.
The premise of the second statement assumes $f$ is an optimal solution to \mathbf{e}qref{prob:D}.
The same $f$ must also be an optimal solution to the problem of finding the largest $|\hat{f}'(\mathbf{x}_k) - f(\mathbf{x}_k)|$, since this optimization problem is simply \mathbf{e}qref{prob:D} with its objective function divided by the constant $-\mathbf{e}ll_k(\mathbf{x}_k)$, and, as discussed before, the absolute sign can be ignored due to symmetry.
\mathbf{e}nd{proof}
\section{An Improved Upper Bound} \label{sec:phase1}
We now begin our attempt at finding the analytical form of the bound.
The theoretical results in \cite{ciarlet1972general} and \cite{powell2001lagrange} are obtained by comparing $f$ against its Taylor expansion at $\mathbf{x}$.
We generalize their approach in Theorem~\ref{thm:phase1} by using the Taylor expansion of $f$ at an arbitrary $\mathbf{u} \in \R^n$.
\begin{theorem}\label{thm:phase1}
Assume $f \in C^{1,1}_\nu(\R^n)$.
Let $\hat{f}$ be the linear function that interpolates $f$ at any set of $n+1$ affinely independent vectors $\Theta = \{\mathbf{x}_1,\dots,\mathbf{x}_{n+1}\}\subset \R^n$.
The function approximation error of $\hat{f}$ at any $\mathbf{x}\in\R^n$ is bounded as
\begin{equation} \label{eq:phase1 u}
|\hat{f}(\mathbf{x}) - f(\mathbf{x})| \le \frac{\nu}{2} \left(\|\mathbf{x}-\mathbf{u}\|^2 + \sum_{i=1}^{n+1} |\mathbf{e}ll_i(\mathbf{x})| \|\mathbf{x}_i-\mathbf{u}\|^2\right),
\mathbf{e}nd{equation}
where $\mathbf{u}$ can be any vector in $\R^n$.
\mathbf{e}nd{theorem}
\begin{proof}
By \mathbf{e}qref{eq:Lipschitz quadratic}, we have for any $\mathbf{u} \in \R^n$
\begin{subequations} \label{phase1 set of inequalities} \begin{align}
\mathbf{e}ll_i [f(\mathbf{x}_i) - f(\mathbf{u}) - Df(\mathbf{u})\cdot (\mathbf{x}_i-\mathbf{u})] &\le \mathbf{e}ll_i \frac{\nu}{2} \|\mathbf{x}_i-\mathbf{u}\|^2 \text{ for all } i\in \cI_+, \\
-\mathbf{e}ll_i [-f(\mathbf{x}_i) + f(\mathbf{u}) + Df(\mathbf{u})\cdot(\mathbf{x}_i-\mathbf{u})] &\le -\mathbf{e}ll_i \frac{\nu}{2} \|\mathbf{x}_i-\mathbf{u}\|^2 \text{ for all } i\in \cI_-.
\mathbf{e}nd{align} \mathbf{e}nd{subequations}
Now add all inequalities above together. The sum of the left-hand sides is
\[ \begin{aligned}
&\sum_{i=0}^{n+1} \mathbf{e}ll_i [f(\mathbf{x}_i) - f(\mathbf{u})] + Df(\mathbf{u}) \cdot \sum_{i=0}^{n+1} \mathbf{e}ll_i [\mathbf{u}-\mathbf{x}_i] \\
&\stackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i f(\mathbf{x}_i) + Df(\mathbf{u}) \cdot \sum_{i=1}^{n+1} \mathbf{e}ll_i \mathbf{x}_i
\stackrel{\mathbf{e}qref{eq:Lagrange m}\mathbf{e}qref{eq:Lagrange Y}}{=} \hat{f}(\mathbf{x}) - f(\mathbf{x}),
\mathbf{e}nd{aligned} \]
while the sum of the right-hand sides is $\nu/2 \sum_{i=0}^{n+1} |\mathbf{e}ll_i| \|\mathbf{x}_i-\mathbf{u}\|^2$.
Thus the sum of the inequalities in \mathbf{e}qref{phase1 set of inequalities} is \mathbf{e}qref{eq:phase1 u} when $\hat{f}(\mathbf{x})-f(\mathbf{x}) \mathbf{g}e 0$.
If the inequalities in \mathbf{e}qref{phase1 set of inequalities} have their left-hand sides multiplied by $-1$, they would still hold according to \mathbf{e}qref{eq:Lipschitz quadratic}, and their summation would be \mathbf{e}qref{eq:phase1 u} for the $\hat{f}(\mathbf{x})-f(\mathbf{x}) < 0$ case.
\mathbf{e}nd{proof}
The existing bounds from \cite{ciarlet1972general} is similar to \mathbf{e}qref{eq:phase1 u} but has $\mathbf{u}$ fixed to $\mathbf{x}$.
In comparison, the new bound provides more convenience in analyzing DFO algorithms that use trusting region methods, since the free point $\mathbf{u}$ can be set to the center of the trust region.
Another advantage of the new bound is that it can be minimized with respect of $\mathbf{u}$, especially considering the right-hand side of \mathbf{e}qref{eq:phase1 u} is a convex function of $\mathbf{u}$ defined on $\R^n$.
This results in the improved bound \mathbf{e}qref{eq:phase1}.
\begin{corollary} \label{cor:phase1}
Under the setting of \mathbf{e}qref{thm:phase1}, the function approximation error of $\hat{f}$ at any $\mathbf{x}\in\R^n$ is bounded as
\begin{equation} \label{eq:phase1}
|\hat{f}(\mathbf{x}) - f(\mathbf{x})| \le \frac{\nu}{2} \left(\|\mathbf{x}-\mathbf{w}\|^2 + \sum_{i=1}^{n+1} |\mathbf{e}ll_i| \|\mathbf{x}_i-\mathbf{w}\|^2\right),
\mathbf{e}nd{equation}
where
\[ \mathbf{w} = \frac{\mathbf{x} + \sum_{i=1}^{n+1} |\mathbf{e}ll_i| \mathbf{x}_i}{1 + \sum_{i=1}^{n+1} |\mathbf{e}ll_i|}.
\]
\mathbf{e}nd{corollary}
To check the sharpness of the bound \mathbf{e}qref{eq:phase1}, we compare it against the optimal value of \mathbf{e}qref{prob:f-D} numerically.
The comparison shows that \mathbf{e}qref{eq:phase1} is sharp if and only if $\mathbf{x}$ is located in $\conv(\Theta)$ or in one of the cones
\begin{equation} \label{eq:cone}
\left\{\mathbf{x}_i + \sum_{j=1}^{n+1} \alpha_j(\mathbf{x}_i - \mathbf{x}_j):~ \alpha_j \mathbf{g}e 0 \text{ for all } j = 1,2,\dots,n+1 \right\}
\mathbf{e}nd{equation}
for some $i\in\{1,\dots,n+1\}$.
We illustrate the geometric meaning of this observation in Figure~\ref{fig:phase1}, which shows the three sets of areas in which $\mathbf{x}$ can locate relative to the sample set $\Theta$ from Figure~\ref{fig:numerical}.
Figure~\ref{fig:phase1 hull} shows the convex hull of $\Theta$, and Figure~\ref{fig:phase1 negative} shows the cones.
In all the remaining areas, as shown in Figure~\ref{fig:phase1 not covered}, the bound \mathbf{e}qref{eq:phase1} is observed to be smaller than the solution of \mathbf{e}qref{prob:f-D}.
Additionally, we want to mention that these areas can also be classified using the signs of the values of the Lagrange functions at $\mathbf{x}$.
The point $\mathbf{x} \in \conv(\Theta)$ if and only if $\mathbf{e}ll_i \mathbf{g}e 0$ for all $i=1,\dots,n+1$; and $\mathbf{x}$ is in the cone \mathbf{e}qref{eq:cone} if and only if $\mathbf{e}ll_i$ is the only positive one among $\{\mathbf{e}ll_i\}_{i=1}^{n+1}$.
\begin{figure}[tbhp]
\centering
\subfloat[\raggedright The convex hull covered by Theorem~\ref{thm:phase1 convex hull}]{\label{fig:phase1 hull}\resizebox{0.3\linewidth}{!}{\input{figures/phase1hull}}}
\subfloat[\raggedright The cones covered by Theorem~\ref{thm:phase1 negative}]{\label{fig:phase1 negative}\resizebox{0.3\linewidth}{!}{\input{figures/phase1negative}}}
\subfloat[The areas where \mathbf{e}qref{eq:phase1} holds but is not sharp]{\label{fig:phase1 not covered} \resizebox{0.3\linewidth}{!}{\input{figures/phase1notcovered}}}
\caption{A visualization of results in Section~\ref{sec:phase1} for bivariate interpolation. The ordering of the points in $\Theta = \{\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3\}$ in this figure and all figures hereafter is arbitrary and not determined by the values of the Lagrange polynomials at $\mathbf{x}$. }
\label{fig:phase1}
\mathbf{e}nd{figure}
When $f \in C^{1,1}_\nu(\R^n)$, the proof of Theorem 3.1 in \cite{waldron1998error} essentially shows that
\begin{equation} \label{eq:Waldron}
|\hat{f}(\mathbf{x}) - f(\mathbf{x})| \le \frac{\nu}{2} \left(\sum_{i=1}^{n+1} \mathbf{e}ll_i \|\mathbf{x}_i\|^2 - \|\mathbf{x}\|^2\right),
\mathbf{e}nd{equation}
holds for all $\mathbf{x} \in \conv(\Theta)$ and is a sharp upper bound, as linear interpolation makes an error equal to this upper bound when approximating the quadratic function $f(\mathbf{u}) = \nu \|\mathbf{u}\|^2/2$.
We show in Theorem~\ref{thm:phase1 convex hull} that \mathbf{e}qref{eq:phase1} is indeed the same as \mathbf{e}qref{eq:Waldron} in this case.
\begin{theorem} \label{thm:phase1 convex hull}
When $\mathbf{x} \in \conv(\Theta)$, the bound \mathbf{e}qref{eq:phase1} has $\mathbf{w}=\mathbf{x}$ and is identical to \mathbf{e}qref{eq:Waldron}.
\mathbf{e}nd{theorem}
\begin{proof}
This theorem is a direct result of the properties of the Lagrange functions \mathbf{e}qref{eq:Lagrange 0} and \mathbf{e}qref{eq:Lagrange Y}.
\mathbf{e}nd{proof}
In Theorem~\ref{thm:phase1 negative}, we verify mathematically that the improved bound \mathbf{e}qref{eq:phase1} is sharp for linear extrapolation when $\mathbf{x}$ is in one of the cones indicated by \mathbf{e}qref{eq:cone} and depicted in Figure~\ref{fig:phase1 negative}.
\begin{theorem} \label{thm:phase1 negative}
Assume the sample points are ordered such that $\mathbf{e}ll_1\mathbf{g}e\mathbf{e}ll_2\mathbf{g}e\cdots\mathbf{g}e\mathbf{e}ll_{n+1}$ and $\mathbf{e}ll_1$ is the only positive one, then the bound \mathbf{e}qref{eq:phase1} is sharp with $\mathbf{w}=\mathbf{x}_1$.
\mathbf{e}nd{theorem}
\begin{proof}
Since $\mathbf{e}ll_i \le 0$ for all $i=0,2,3,\dots,n+1$,
\[ \mathbf{w} = \frac{2\mathbf{e}ll_1 \mathbf{x}_1 - \sum_{i=0}^{n+1} \mathbf{e}ll_i \mathbf{x}_i}{2\mathbf{e}ll_1 -\sum_{i=0}^{n+1} \mathbf{e}ll_i}
\stackrel{\mathbf{e}qref{eq:Lagrange 0}\mathbf{e}qref{eq:Lagrange Y}}{=} \frac{2\mathbf{e}ll_1 \mathbf{x}_1}{2\mathbf{e}ll_1}
= \mathbf{x}_1.
\]
The bound \mathbf{e}qref{eq:phase1} equals $\nu/2$ multiplies
\[ \begin{aligned}
\sum_{i=0}^{n+1} |\mathbf{e}ll_i| \|\mathbf{x}_i-\mathbf{w}\|^2
&= - \sum_{i=0}^{n+1} \mathbf{e}ll_i \|\mathbf{x}_i-\mathbf{x}_1\|^2
= \text{Tr}\left( - \sum_{i=0}^{n+1} \mathbf{e}ll_i [\mathbf{x}_i-\mathbf{x}_1][\mathbf{x}_i-\mathbf{x}_1]^T \right) \\
&\leftstackrel{\mathbf{e}qref{eq:G recenter}}{=} \text{Tr}\left( - \sum_{i=0}^{n+1} \mathbf{e}ll_i \mathbf{x}_i\mathbf{x}_i^T \right)
=-\sum_{i=0}^{n+1} \mathbf{e}ll_i \|\mathbf{x}_i\|^2.
\mathbf{e}nd{aligned} \]
Consider the function $f(\mathbf{u}) = -\frac{\nu}{2} \|\mathbf{u}\|^2 \stackrel{\mathbf{e}qref{eq:Lipschitz Hessian}}{\in} C^{1,1}_\nu(\R^n)$.
We have
\[\hat{f}(\mathbf{x}) - f(\mathbf{x})
\stackrel{\mathbf{e}qref{eq:Lagrange m}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i f(\mathbf{x}_i)
= -\sum_{i=0}^{n+1} \mathbf{e}ll_i \frac{\nu}{2}\|\mathbf{x}_i\|^2,
\]
which matches \mathbf{e}qref{eq:phase1}.
\mathbf{e}nd{proof}
\section{The Worst Quadratic Function} \label{sec:phase2}
We have derived an improved error bound in the previous section and showed when it is sharp.
In this section, we try to find the mathematical formula for the piecewise smooth function in the remaining areas indicated in Figure~\ref{fig:phase1 not covered}.
Instead of attempting to improve another existing upper bound, we take the opposite approach by trying to find the function that can achieve the maximum error.
Considering quadratic functions are easier to analyze as they share a general closed-form formula and, under the settings of both Theorem~\ref{thm:phase1 convex hull} and Theorem~\ref{thm:phase1 negative}, the optimal set of \mathbf{e}qref{prob:D} contains at least one quadratic function, we investigate whether \mathbf{e}qref{prob:D} has an analytical solution when $f$ is restricted to be quadratic.
Let $f$ be a quadratic function of the form $f(\mathbf{u}) = c + \mathbf{g} \cdot \mathbf{u} + \frac{1}{2} H\mathbf{u} \cdot \mathbf{u}$ with $c\in\R, \mathbf{g}\in\R^n$, and symmetric $H \in \R^{n \times n}$.
Because of \mathbf{e}qref{eq:Lipschitz Hessian} and
\[ \begin{aligned}
\hat{f}(\mathbf{x}) - f(\mathbf{x}) ~
&\leftstackrel{\mathbf{e}qref{eq:Lagrange m}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i f(\mathbf{x}_i)
= \sum_{i=0}^{n+1} \mathbf{e}ll_i \left[c + \mathbf{g} \cdot \mathbf{x}_i + \frac{1}{2} H\mathbf{x}_i \cdot \mathbf{x}_i \right] \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange Y}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i \left[c + \frac{1}{2} H\mathbf{x}_i \cdot \mathbf{x}_i \right]
\stackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} \sum_{i=0}^{n+1} \mathbf{e}ll_i \left[\frac{1}{2} H\mathbf{x}_i \cdot \mathbf{x}_i \right] \\
&= \frac{1}{2} H \cdot \sum_{i=0}^{n+1} \mathbf{e}ll_i \mathbf{x}_i \mathbf{x}_i^T
\stackrel{\mathbf{e}qref{eq:G}}{=} \frac{1}{2} G \cdot H,
\mathbf{e}nd{aligned} \]
the problem of maximizing linear interpolation's approximation error over quadratic functions in $C_\nu^{1,1}(\R^n)$ can be formulated as
\begin{equation} \label{prob:quadratic} \mathbf{e}verymath{\displaystyle} \begin{array}{ll}
\max_H &G \cdot H / 2 \\
\text{s.t.} &-\nu I \preceq H \preceq \nu I.
\mathbf{e}nd{array} \mathbf{e}nd{equation}
The absolute sign in the objective function is again dropped due to symmetry.
It turns out the problem \mathbf{e}qref{prob:quadratic} can be solved analytically.
Since $G$ is real and symmetric, it must have eigendecomposition $G = P \Lambda P^T$, where $\Lambda \in \R^{n \times n}$ is the diagonal matrix of the eigenvalues $\lambda_1,\dots,\lambda_n$, and $P \in \R^{n \times n}$ is the orthonormal matrix whose columns are the corresponding eigenvectors.
The objective function $G\cdot H/2 = (P\Lambda P^T)\cdot H/2 = \Lambda \cdot (P^T H P)/2$.
Since $P$ is orthonormal, the constraint in \mathbf{e}qref{prob:quadratic} is equivalent to $-\nu I \preceq P^T H P \preceq \nu I$, indicating all diagonal elements of $P^T H P$ are bounded between $-\nu$ and $\nu$.
Since $\Lambda$ is diagonal, only the diagonal elements of $P^T H P$ would affect the objective function value.
Therefore, a solution to \mathbf{e}qref{prob:quadratic}, denoted by $H^\star$, has the property $P^T H^\star P = \nu \text{sign}(\Lambda)$.
This optimal solution is
\begin{equation} \label{eq:Hstar}
H^\star = \nu P \text{sign}(\Lambda) P^T.
\mathbf{e}nd{equation}
Solution \mathbf{e}qref{eq:Hstar} indicates the maximum approximation error by quadratic functions of
\begin{equation} \label{eq:phase2}
G \cdot H^\star/2 = \frac{\nu}{2} \sum_{i=1}^n |\lambda_i|.
\mathbf{e}nd{equation}
We again compare this new bound to the optimal value of \mathbf{e}qref{prob:f-D} numerically.
Our results show these two are exactly the same in all three cases in Figure~\ref{fig:phase1}, and \mathbf{e}qref{eq:phase2} is a formula of the piecewise smooth function in Figure~\ref{fig:numerical}.
However, this does not mean \mathbf{e}qref{eq:phase2} is a formula to the optimal value of \mathbf{e}qref{prob:f-D}.
For example, for bivariate linear interpolation, it is observed that when the triangle $\conv(\Theta)$ is obtuse and $\mathbf{x}$ locates in one of the four shaded areas indicated in Figure~\ref{fig:phase2}, the optimal value of \mathbf{e}qref{prob:f-D} is larger than \mathbf{e}qref{eq:phase2}.
These shaded areas are open subsets of $\R^2$ and do not include their boundaries.
From left to right, they can be described as
\begin{itemize}
\item $\mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1]\cdot[\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_2 [\mathbf{x}_3-\mathbf{x}_2]\cdot[\mathbf{x}_1-\mathbf{x}_2] > 0$ and $\mathbf{e}ll_2>0$;
\item $\mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1]\cdot[\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_2 [\mathbf{x}_3-\mathbf{x}_2]\cdot[\mathbf{x}_1-\mathbf{x}_2] < 0$, $\mathbf{e}ll_3>0$, and $\mathbf{e}ll_2<0$;
\item $\mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1]\cdot[\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3[\mathbf{x}_2-\mathbf{x}_3]\cdot[\mathbf{x}_1-\mathbf{x}_3] < 0$, $\mathbf{e}ll_2>0$, and $\mathbf{e}ll_3<0$;
\item $\mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1]\cdot[\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3]\cdot[\mathbf{x}_1-\mathbf{x}_3] > 0$ and $\mathbf{e}ll_3>0$.
\mathbf{e}nd{itemize}
In the remaining parts of this section, we will investigate analytically when \mathbf{e}qref{eq:phase2} is the sharp error bound.
\begin{figure}[tbhp]
\centering
\resizebox{0.4\textwidth}{!}{\input{figures/phase3}}
\caption{The areas to which if $\mathbf{x}$ belongs, \mathbf{e}qref{eq:phase2} is not an upper bound on the function approximation error for bivariate interpolation.
The dashed line on the left is perpendicular to the line going through $\mathbf{x}_1$ and $\mathbf{x}_2$; and the one on the right is perpendicular to the line going through $\mathbf{x}_3$ and $\mathbf{x}_1$. }
\label{fig:phase2}
\mathbf{e}nd{figure}
\subsection{Certification of Upper Bound}
The maximum error \mathbf{e}qref{eq:phase2} provides a lower bound to the optimal value of \mathbf{e}qref{prob:D}, while \mathbf{e}qref{eq:phase1} provides an upper bound.
By evaluating both \mathbf{e}qref{eq:phase1} and \mathbf{e}qref{eq:phase2}, one can have a reasonable estimation of sharp error bound without having to solve the QCQP \mathbf{e}qref{prob:f-D}.
However, the formula \mathbf{e}qref{eq:phase2} would be a lot more useful if there is an efficient way to check whether $\mathbf{x}$ is in one of those areas where \mathbf{e}qref{eq:phase2} is not an upper bound on the approximation error.
The existence of these areas appears to be influence by the existence of obtuse angles at the vertices of the simplex $\conv(\Theta)$.
Unlike triangles, which can only have up to one obtuse angle, simplices in higher dimension can have obtuse angles in many ways.
They can have $(\mathbf{x}_j-\mathbf{x}_i)\cdot(\mathbf{x}_k-\mathbf{x}_i) < 0$ at multiple vertices $\mathbf{x}_i$ and at the same time for multiple $(j,k)$ for each $\mathbf{x}_i$.
While there can only be up to four disconnected subset of $\R^2$ where \mathbf{e}qref{eq:phase2} is not an upper bound on the approximation error, our numerical experiments show this number can go up to at least twenty for trivariate ($n=3$) linear interpolation.
Considering a precise description of the four shaded areas in Figure~\ref{fig:phase2} already requires four unintuitive inequalities or some wordy explanation, any description of these areas would almost certainly be extremely complicated, especially in higher dimension.
Regardless, we have found an efficient way to check whether $\mathbf{x}$ is in one of these areas without having to describe any of them.
The theoretical proof that validates our approach is extremely technical and will be presented later in section~\ref{sec:phase2 proofs}.
Our approach relies on a set of parameters $\{\mu_{ij}\}_{(i,j)\in\cI_+\times\cI_-}$ that can be computed as follows.
Remember $\Theta$ is assumed to be ordered in a way so that $\mathbf{e}ll_1 \mathbf{g}e \mathbf{e}ll_2 \mathbf{g}e \cdots \mathbf{g}e \mathbf{e}ll_{n+1}$, and let $\diag(\mathbf{e}ll) \in \R^{(n+1)\times(n+1)}$ be the diagonal matrix containing $\mathbf{e}ll_1, \dots, \mathbf{e}ll_{n+1}$.
We now partition $\diag(\mathbf{e}ll), G$, and $H^\star$ with respect to $\cI_+$ and $\cI_-$.
Let $\diag(\mathbf{e}ll_+)\in\R^{|\cI_+|\times|\cI_+|}$ be the diagonal matrix containing $\{\mathbf{e}ll_i\}_{i\in\cI_+}$, and $\diag(\mathbf{e}ll_-) \in \R^{(|\cI_-|-1)\times(|\cI_-|-1)}$ be the diagonal matrix containing $\{\mathbf{e}ll_i\}_{i\in\cI_-\setminus\{0\}}$.
Let $Y_+ \in \R^{|\cI_+| \times n}$ and $Y_- \in \R^{(|\cI_-|-1) \times n}$ be the first $|\cI_+|$ and the last $|\cI_-|-1$ rows of $Y$, respectively.
The matrix $G$ has $|\cI_+|-1$ positive eigenvalues and $|\cI_-|-1$ negative eigenvalues, as will be proved later.
Let $\Lambda_+ \in \R^{(|\cI_+|-1) \times (|\cI_+|-1)}$ and $\Lambda_- \in \R^{(|\cI_-|-1) \times (|\cI_-|-1)}$ respectively be the the diagonal matrices that contain the positive and negative eigenvalues of $G$, and $P_+ \in \R^{n \times (|\cI_+|-1)}$ and $P_- \in \R^{n \times (|\cI_-|-1)}$ their corresponding eigenvector matrices.
Then we have
\begin{equation} \label{eq:G+-} \begin{aligned}
G~ &\leftstackrel{\mathbf{e}qref{eq:G recenter}}{=} Y^T \diag(\mathbf{e}ll) Y = Y_+^T \diag(\mathbf{e}ll_+) Y_+ + Y_-^T \diag(\mathbf{e}ll_-) Y_- \\
&= P \Lambda P^T
= P_+ \Lambda_+ P_+^T + P_- \Lambda_- P_-^T
\mathbf{e}nd{aligned} \mathbf{e}nd{equation}
and
\begin{equation} \label{eq:Hstar+-}
H^\star = \nu P \text{sign}(\Lambda) P^T
= \nu (P_+ P_+^T - P_- P_-^T).
\mathbf{e}nd{equation}
We now present the definition of $\{\mu_{ij}\}_{(i,j)\in\cI_+\times\cI_-}$ and the main theorem of this section.
\begin{theorem} \label{thm:phase2}
Consider the matrix $M \stackrel{\rm def}{=} \diag(\mathbf{e}ll_+) Y_+ P_- (Y_- P_-)^{-1}$.
Let $\mu_{ij} = \mathbf{e}_i^T M \mathbf{e}_{j - n-1+|\cI_-|}$ for all $i\in\cI_+$ and $j\in\cI_-\setminus\{0\}$, and $\mu_{i0} = \mathbf{e}ll_i - \sum_{j \in \cI_-\setminus\{0\}} \mu_{ij}$ for all $i\in\cI_+$.
Assume $f \in C^{1,1}_\nu(\R^n)$.
If $\mu_{ij} \mathbf{g}e 0$ for all $(i,j) \in \cI_+\times\cI_-$, then \mathbf{e}qref{eq:phase2} is a sharp upper bound on the function approximation error $|\hat{f}(\mathbf{x}) - f(\mathbf{x})|$ for linear interpolation.
\mathbf{e}nd{theorem}
\begin{remark}
We note that $\{j-n-1+|\cI_-|\}_{j\in\cI_-\setminus\{0\}} = \{1,2,\dots,|\cI_-|\}$.
The matrix $M$ is of size $|\cI_+| \times (|\cI_-|-1)$.
Each of its row corresponds to a sample point with positive Lagrange polynomial values at $\mathbf{x}$, while each of its column corresponds to a sample point with negative Lagrange polynomial values at $\mathbf{x}$.
\mathbf{e}nd{remark}
\subsection{Technical Proofs} \label{sec:phase2 proofs}
In the remaining of this section, we provide the complete proof to Theorem~\ref{thm:phase2}.
We start with the number of positive and negative eigenvalues in the matrix $G$.
\begin{lemma} \label{lem:Sylvester}
The numbers of positive and negative eigenvalues in $G$ are $|\cI_+| - 1$ and $|\cI_-|-1$, respectively.
\mathbf{e}nd{lemma}
\begin{proof}
Let $\diag(\mathbf{e}ll)$ be the diagonal matrix containg $\mathbf{e}ll_1, \dots, \mathbf{e}ll_{n+1}$.
Consider the matrix $\bar{G} = \sum_{i=1}^{n+1} \mathbf{e}ll_i \phi(\mathbf{x}_i-\mathbf{x}) \phi(\mathbf{x}_i-\mathbf{x})^T = \Phi^T \diag(\mathbf{e}ll) \Phi$.
The first element of the first column is $\sum_{i=1}^{n+1} \mathbf{e}ll_i \stackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} 1$, while the rest of the column is $\sum_{i=1}^{n+1} \mathbf{e}ll_i [\mathbf{x}_i-\mathbf{x}] \stackrel{\mathbf{e}qref{eq:Lagrange Y}}{=} \mathbf{x} - \sum_{i=1}^{n+1} \mathbf{e}ll_i \mathbf{x} \stackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} \mathbf{0}$.
The bottom-right $n\times n$ submatrix of $\bar{G}$ is
\[ \sum_{i=1}^{n+1} \mathbf{e}ll_i [\mathbf{x}_i-\mathbf{x}][\mathbf{x}_i-\mathbf{x}]^T
= \sum_{i=0}^{n+1} \mathbf{e}ll_i [\mathbf{x}_i-\mathbf{x}][\mathbf{x}_i-\mathbf{x}]^T
\stackrel{\mathbf{e}qref{eq:G recenter}}{=} G.
\]
Thus, $\bar{G}$ and its eigendecomposition should be
\[ \bar{G} = \begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &G \mathbf{e}nd{bmatrix}
= \begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &P \mathbf{e}nd{bmatrix}
\begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &\Lambda \mathbf{e}nd{bmatrix}
\begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &P^T \mathbf{e}nd{bmatrix}.
\]
Then we have
\[ \bar\Lambda \stackrel{\rm def}{=} \begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &\Lambda \mathbf{e}nd{bmatrix}
= \begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &P^T \mathbf{e}nd{bmatrix}
\Phi^T \diag(\mathbf{e}ll) \Phi
\begin{bmatrix} 1 &\mathbf{0}^T\\ \mathbf{0} &P \mathbf{e}nd{bmatrix},
\]
which shows $\bar\Lambda$ is congruent to $\diag(\mathbf{e}ll)$.
Then by Sylvester's law of inertia \cite{sylvester1852xix} (or Theorem 4.5.8 of \cite{horn2012matrix}), the number of positive and negative eigenvalues in $\bar\Lambda$ are $|\cI_+|$ and $|\cI_-|-1$, respectively.
Since $\bar{G}$ shares the same eigenvalues as $G$ except an additional one that is 1, the lemma is proven.
\mathbf{e}nd{proof}
The next lemma shows that $\{\mu_{ij}\}_{(i,j)\in\cI_+\times\cI_-}$ is well-defined by proving the invertibility of $Y_- P_-$.
\begin{lemma} \label{lem:invertible}
The matrix $Y_- P_-$ is invertible.
\mathbf{e}nd{lemma}
\begin{proof}
For the purpose of contradiction, assume $Y_- P_-$ is singular.
That means there is a non-zero vector $\mathbf{u} \in \R^{|\cI_-|-1}$ such that $Y_- P_- \mathbf{u} = \mathbf{0}$.
Let $\mathbf{v} = P_- \mathbf{u}$.
We have $Y_- \mathbf{v} = \mathbf{0}$, $P_+ \mathbf{v} = P_+ P_- \mathbf{u} = \mathbf{0}$ and $P_-^T \mathbf{v} = P_-^T P_- \mathbf{u} = \mathbf{u}$.
Then we have the contradiction
\[ \begin{aligned}
\mathbf{v}^T G \mathbf{v} &= (Y_+\mathbf{v})^T \diag(\mathbf{e}ll_+) Y_+\mathbf{v} + (Y_-\mathbf{v})^T \diag(\mathbf{e}ll_-) Y_-\mathbf{v} = (Y_+\mathbf{v})^T \diag(\mathbf{e}ll_+) Y_+\mathbf{v} \mathbf{g}e 0 \\
\mathbf{v}^T G \mathbf{v} &= (P_+^T \mathbf{v})^T \Lambda_+ P_+^T \mathbf{v} + (P_-^T \mathbf{v})^T \Lambda_- P_-^T \mathbf{v} = (P_-^T \mathbf{v})^T \Lambda_- P_-^T \mathbf{v} = \mathbf{u}^T \Lambda_- \mathbf{u} < 0.
\mathbf{e}nd{aligned} \]
\mathbf{e}nd{proof}
We develop in the following lemma the essential properties of $\{\mu_{ij}\}$.
\begin{lemma} \label{lem:mu}
The following properties hold:
\begin{align}
\sum_{j \in \cI_-} \mu_{ij} &= \mathbf{e}ll_i &&\text{for all } i \in \cI_+, \label{eq:mu0 +}\\
\sum_{i \in \cI_+} \mu_{ij} &= - \mathbf{e}ll_j &&\text{for all } j \in \cI_-, \label{eq:mu0 -}\\
(\nu I-H^\star) \sum_{j \in \cI_-} \mu_{ij} \mathbf{x}_j &= (\nu I-H^\star) \mathbf{e}ll_i \mathbf{x}_i &&\text{for all } i \in \cI_+, \label{eq:mu1 +}\\
(\nu I+H^\star) \sum_{i \in \cI_+} \mu_{ij}\mathbf{x}_i &= -(\nu I+H^\star) \mathbf{e}ll_j \mathbf{x}_j &&\text{for all } j \in \cI_-. \label{eq:mu1 -}
\mathbf{e}nd{align}
\mathbf{e}nd{lemma}
\begin{proof}
The equations \mathbf{e}qref{eq:mu0 +} are true by their definition.
Since
\[ \begin{aligned}
\diag(\mathbf{e}ll_-) \mathbf{1} + M^T \mathbf{1}
&= \diag(\mathbf{e}ll_-) \mathbf{1} + (P_-^T Y_-^T)^{-1} P_-^T Y_+^T \diag(\mathbf{e}ll_+) \mathbf{1} \\
&= (P_-^T Y_-^T)^{-1} P_-^T [Y_-^T \diag(\mathbf{e}ll_-) \mathbf{1} + Y_+^T \diag(\mathbf{e}ll_+) \mathbf{1}]
\stackrel{\mathbf{e}qref{eq:Lagrange Y}}{=} \mathbf{0},
\mathbf{e}nd{aligned} \]
the equations \mathbf{e}qref{eq:mu0 -} are also true.
Notice $P_-^T (Y_-^T M^T - Y_+^T \diag(\mathbf{e}ll_+)) = \mathbf{0}$ by the definition of $M$, and $\nu I - H^\star \stackrel{\mathbf{e}qref{eq:Hstar+-}}{=} \nu (P_+P_+^T + P_-P_-^T) - \nu (P_+P_+^T - P_-P_-^T) = 2\nu P_-P_-^T$.
Following these two equations, we have for all $i \in \cI_+$,
\[ \begin{aligned}
(\nu I-H^\star) \left[\sum_{j \in \cI_-} \mu_{ij} \mathbf{x}_j - \mathbf{e}ll_i \mathbf{x}_i\right]
&\leftstackrel{\mathbf{e}qref{eq:mu0 -}}{=} (\nu I-H^\star) \left[\sum_{j \in \cI_-} \mu_{ij} (\mathbf{x}_j-\mathbf{x}) - \mathbf{e}ll_i [\mathbf{x}_i-\mathbf{x}]\right] \\
&= (\nu I-H^\star) (Y_-^T M^T - Y_+^T \diag(\mathbf{e}ll_+)) \mathbf{e}_i \\
&= 2\nu P_- P_-^T (Y_-^T M^T - Y_+^T \diag(\mathbf{e}ll_+)) \mathbf{e}_i \\
&= 2\nu P_- \mathbf{0} \mathbf{e}_i = \mathbf{0},
\mathbf{e}nd{aligned} \]
which proves \mathbf{e}qref{eq:mu1 +}.
To prove \mathbf{e}qref{eq:mu1 -}, we use $G$ and its eigendecomposition.
The diagonal matrix of the eigenvalues $\Lambda$ is
\[ \begin{bmatrix} \Lambda_- &\mathbf{0}\\ \mathbf{0} &\Lambda_+ \mathbf{e}nd{bmatrix}
= \begin{bmatrix} P_-^T Y_-^T &P_-^T Y_+^T\\ P_+^T Y_-^T &P_+^T Y_+^T \mathbf{e}nd{bmatrix}
\begin{bmatrix} \diag(\mathbf{e}ll_-) &~\\ ~ &\diag(\mathbf{e}ll_+) \mathbf{e}nd{bmatrix}
\begin{bmatrix} Y_- P_- &Y_- P_+\\ Y_+ P_- &Y_+ P_+ \mathbf{e}nd{bmatrix},
\]
which contains two equivalent block equalities with zero left-hand side.
They are $P_+^T Y_-^T \diag(\mathbf{e}ll_-) Y_- P_- + P_+^T Y_+^T \diag(\mathbf{e}ll_+) Y_+ P_- = \mathbf{0}$, so
\[ P_+^T Y_-^T \diag(\mathbf{e}ll_-) + P_+^T Y_+^T \diag(\mathbf{e}ll_+) Y_+ P_- (Y_- P_-)^{-1} = P_+^T Y_-^T \diag(\mathbf{e}ll_-) + P_+^T Y_+^T M = \mathbf{0}.
\]
Then with $\nu I+H^\star \stackrel{\mathbf{e}qref{eq:Hstar+-}}{=} \nu(P_+P_+^T + P_-P_-^T) + \nu(P_+P_+^T - P_-P_-^T) = 2\nu P_+ P_+^T$, we obtain
\[ (\nu I + H^\star) (Y_-^T \diag(\mathbf{e}ll_-) + Y_+^T M)
= 2\nu P_+ P_+^T (Y_-^T \diag(\mathbf{e}ll_-) + Y_+^T M)
= 2\nu P_+ \mathbf{0} = \mathbf{0},
\]
which proves \mathbf{e}qref{eq:mu1 -} for all $j \in \cI_- \setminus \{0\}$; and
\[ \begin{aligned}
(\nu I+H^\star) \left( \mathbf{e}ll_0 \mathbf{x} + \sum_{i\in\cI_+} \mu_{i0} \mathbf{x}_i \right)
&\leftstackrel{\mathbf{e}qref{eq:mu0 -}}{=} 2\nu P_+P_+^T \sum_{i\in\cI_+} \mu_{i0} (\mathbf{x}_i-\mathbf{x}) \\
&= 2\nu P_+P_+^T \sum_{i\in\cI_+} \left(\mathbf{e}ll_i - \sum_{j\in\cI_-\setminus\{0\}} \mu_{ij}\right) (\mathbf{x}_i-\mathbf{x}) \\
&= 2\nu P_+P_+^T Y_+^T (l_+ - M\mathbf{1}) \\
&= 2\nu P_+P_+^T (Y_+^T l_+ + Y_-^T \mathbf{e}ll_-)
\stackrel{\mathbf{e}qref{eq:Lagrange Y}}{=} \mathbf{0},
\mathbf{e}nd{aligned} \]
which proves \mathbf{e}qref{eq:mu1 -} for $j = 0$.
\mathbf{e}nd{proof}
The function $\psi$ is defined and proved non-positive in Lemma~\ref{lem:psi}.
It will be used to prove Theorem~\ref{thm:phase2} in conjunction with the parameters $\{\mu_{ij}\}$.
\begin{lemma} \label{lem:psi}
Assume $f \in C^{1,1}_\nu(\R^n)$.
For any $\mathbf{u}, \mathbf{v} \in \R^n$ and any matrix $H \in \R^{n \times n}$, we have
\begin{equation} \label{eq:Lipscthiz stronger H} \begin{aligned}
\psi(\mathbf{u},\mathbf{v},H) \stackrel{\rm def}{=} &f(\mathbf{u}) - f(\mathbf{v}) - \frac{1}{2\nu} [(\nu I-H)(\mathbf{u}-\mathbf{v})] \cdot Df(\mathbf{u}) \\
&- \frac{1}{2\nu} [(\nu I+H) (\mathbf{u}-\mathbf{v})] \cdot Df(\mathbf{v})
- \frac{1}{4\nu} \|H (\mathbf{u} - \mathbf{v})\|^2 - \frac{\nu}{4} \|\mathbf{u} - \mathbf{v}\|^2 \le 0.
\mathbf{e}nd{aligned} \mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
For the purpose of contradiction, assume \mathbf{e}qref{eq:Lipscthiz stronger H} is false. Then we have
\[ \begin{aligned}
- f(\mathbf{u}) <& - f(\mathbf{v}) - \frac{1}{2\nu} [(\nu I+H) (\mathbf{u}-\mathbf{v})] \cdot Df(\mathbf{u}) \\
&- \frac{1}{2\nu} [(\nu I-H)(\mathbf{u}-\mathbf{v})] \cdot Df(\mathbf{v}) - \frac{1}{4\nu} \|H (\mathbf{u} - \mathbf{v})\|^2 - \frac{\nu}{4} \|\mathbf{u} - \mathbf{v}\|^2.
\mathbf{e}nd{aligned} \]
Add this inequality to \mathbf{e}qref{eq:Lipschitz stronger} and we arrive at
\[ \frac{1}{4\nu} \|H (\mathbf{u} - \mathbf{v}) - (Df(\mathbf{u}) - Df(\mathbf{v}))\|^2 < 0,
\]
which leads to contradiction.
\mathbf{e}nd{proof}
Finally, we prove the main result of this section, Theorem~\ref{thm:phase2}, which states \mathbf{e}qref{eq:phase2} is a sharp bound when $\{\mu_{ij}\}$ are all non-negative.
\begin{proof}[proof of Theorem~\ref{thm:phase2}]
We only provide the proof for the case when $\hat{f}(\mathbf{x}) - f(\mathbf{x}) \mathbf{g}e 0$.
When $\mu_{ij} \mathbf{g}e 0$ for all $(i,j) \in \cI_+\times\cI_-$, the following inequality holds
\begin{equation} \label{eq:phase2 summation}
\sum_{i \in \cI_+} \sum_{j \in \cI_-} \mu_{ij} \psi(\mathbf{x}_i,\mathbf{x}_j,H^\star) \stackrel{\mathbf{e}qref{eq:Lipscthiz stronger H}}{\le} 0.
\mathbf{e}nd{equation}
The zeroth-order term in the summation \mathbf{e}qref{eq:phase2 summation} is
\[ \begin{aligned}
\sum_{i \in \cI_+} \sum_{j \in \cI_-} \mu_{ij} (f(\mathbf{x}_i) - f(\mathbf{x}_j))
&= \left[\sum_{i \in \cI_+} \sum_{j \in \cI_-} \mu_{ij} f(\mathbf{x}_i)\right] - \left[\sum_{i \in \cI_+} \sum_{j \in \cI_-} \mu_{ij} f(\mathbf{x}_j)\right] \\
&\leftstackrel{\mathbf{e}qref{eq:mu0 +}\mathbf{e}qref{eq:mu0 -}}{=} \left[ \sum_{i \in \cI_+} \mathbf{e}ll_i f(\mathbf{x}_i)\right] + \left[\sum_{j \in \cI_-} \mathbf{e}ll_j f(\mathbf{x}_j)\right] \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange m}}{=} \hat{f}(\mathbf{x}) - f(\mathbf{x}).
\mathbf{e}nd{aligned} \]
The sum of the first-order terms is $-1/(2\nu)$ multiplies
\[ \begin{aligned}
&\sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} \big( [(\nu I-H^\star)(\mathbf{x}_i-\mathbf{x}_j)] \cdot Df(\mathbf{x}_i) + [(\nu I+H^\star)(\mathbf{x}_i-\mathbf{x}_j)] \cdot Df(\mathbf{x}_j) \big) \\
&= \left[\sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} [(\nu I-H^\star)\mathbf{x}_i] \cdot Df(\mathbf{x}_i) \right]
- \left[\sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} [(\nu I+H^\star)\mathbf{x}_j] \cdot Df(\mathbf{x}_j) \right] \\
&\quad -\left[\sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} [(\nu I-H^\star)\mathbf{x}_j] \cdot Df(\mathbf{x}_i) \right] + \left[\sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} [(\nu I+H^\star)\mathbf{x}_i] \cdot Df(\mathbf{x}_j) \right] \\
&= \left[\sum_{i \in \cI_+} \mathbf{e}ll_i [(\nu I-H^\star)\mathbf{x}_i] \cdot Df(\mathbf{x}_i) \right]
+ \left[\sum_{j\in\cI_-} \mathbf{e}ll_j [(\nu I+H^\star)\mathbf{x}_j] \cdot Df(\mathbf{x}_j) \right] \\
&\quad -\left[\sum_{i \in \cI_+} \mathbf{e}ll_i [(\nu I-H^\star)\mathbf{x}_i] \cdot Df(\mathbf{x}_i) \right] - \left[\sum_{j\in\cI_-} \mathbf{e}ll_j [(\nu I+H^\star)\mathbf{x}_j] \cdot Df(\mathbf{x}_j) \right]
= \mathbf{0},
\mathbf{e}nd{aligned} \]
where the second equality holds because of \mathbf{e}qref{eq:mu0 +}, \mathbf{e}qref{eq:mu0 -}, \mathbf{e}qref{eq:mu1 +}, and \mathbf{e}qref{eq:mu1 -} respectively for the four terms.
Notice $H^{\star T} H^\star = \nu^2I$.
The constant term in the summation \mathbf{e}qref{eq:phase2 summation} is $-1/2$ multiplies
\[ \begin{aligned}
&\sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} \left(\frac{1}{2\nu}\|H^\star(\mathbf{x}_i-\mathbf{x}_j)\|^2 + \frac{\nu}{2} \|\mathbf{x}_i-\mathbf{x}_j\|^2\right) \\
&= \nu \left[ \sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} (\mathbf{x}_i-\mathbf{x}_j) \cdot \mathbf{x}_i \right] - \nu \left[ \sum_{i \in \cI_+} \sum_{j\in\cI_-} \mu_{ij} (\mathbf{x}_i-\mathbf{x}_j) \cdot \mathbf{x}_j \right] \\
&\stackrel{\mathmakebox[\mathbf{w}idthof{=}]{\scriptsize \begin{array}{c}\mathbf{e}qref{eq:mu0 +}\\\mathbf{e}qref{eq:mu0 -}\mathbf{e}nd{array}}}{=} \sum_{i \in \cI_+} \nu \left(\mathbf{e}ll_i \mathbf{x}_i- \sum_{j\in\cI_-} \mu_{ij} \mathbf{x}_j\right) \cdot \mathbf{x}_i - \sum_{j\in\cI_-} \nu \left(\sum_{i \in \cI_+} \mu_{ij} \mathbf{x}_i + \mathbf{e}ll_j \mathbf{x}_j\right) \cdot \mathbf{x}_j \\
&\leftstackrel{\mathmakebox[\mathbf{w}idthof{=}]{\scriptsize \begin{array}{c}\mathbf{e}qref{eq:mu1 +}\\ \mathbf{e}qref{eq:mu1 -}\mathbf{e}nd{array}}}{=} \sum_{i \in \cI_+} \left[H^\star \left(\mathbf{e}ll_i \mathbf{x}_i- \sum_{j\in\cI_-} \mu_{ij} \mathbf{x}_j\right)\right] \cdot \mathbf{x}_i + \sum_{j\in\cI_-} \left[H^\star \left(\sum_{i \in \cI_+} \mu_{ij} \mathbf{x}_i + \mathbf{e}ll_j \mathbf{x}_j\right)\right] \cdot \mathbf{x}_j \\
&= \left[ \sum_{i \in \cI_+} \mathbf{e}ll_i [H^\star \mathbf{x}_i] \cdot \mathbf{x}_i \right] + \left[ \sum_{j\in\cI_-} \mathbf{e}ll_j [H^\star \mathbf{x}_j] \cdot \mathbf{x}_j \right]
= G \cdot H^\star.
\mathbf{e}nd{aligned} \]
Thus the summation \mathbf{e}qref{eq:phase2 summation} is \mathbf{e}qref{eq:phase2} when $\hat{f}(\mathbf{x}) - f(\mathbf{x}) \mathbf{g}e 0$.
\mathbf{e}nd{proof}
\section{Sharp Error Bounds for Bivariate Extrapolation} \label{sec:phase3}
We investigate in this section the sharp error bounds when $\mathbf{x}$ is in the four areas shown in Figure~\ref{fig:phase2}.
This investigation is not just for the completeness of our analysis of the sharp error bound, but also to understand what type of function can be more difficult for linear interpolation to approximate than the quadratics.
We first notice the case where $\mathbf{x}$ is in the shaded triangle on the left in Figure~\ref{fig:phase2} is symmetric to the case where $\mathbf{x}$ is in the triangle on the right, and they are essentially the same.
The same argument applies the two shaded cones.
This reduces the cases that need to be studied to the two in Figure~\ref{fig:phase3}.
Furthermore, after we obtain a formula for the sharp error bound for the case in Figure~\ref{fig:phase3 triangle}, a formula for the case in Figure~\ref{fig:phase3 cone} can be obtained by switching the roles of $\mathbf{x}$ and $\mathbf{x}_2$ and apply Proposition~\ref{thm:swap}.
Therefore, the only case that needs to be studied is the one in Figure~\ref{fig:phase3 triangle}.
\begin{figure}[tbhp]
\centering
\subfloat[\raggedright When $\mathbf{x}$ is in the open triangle such that $\mathbf{e}ll_1[\mathbf{x}_2-\mathbf{x}_1{]} \cdot [\mathbf{x}_3-\mathbf{x}_1{]} - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3{]} \cdot [\mathbf{x}_1-\mathbf{x}_3{]} < 0$, $\mathbf{e}ll_2>0$, and $\mathbf{e}ll_3<0$]{\label{fig:phase3 triangle}\resizebox{0.4\linewidth}{!}{\input{figures/phase3triangle}}}
\hspace{0.1\linewidth}
\subfloat[\raggedright When $\mathbf{x}$ is in the open cone such that $\mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1{]} \cdot [\mathbf{x}_3-\mathbf{x}_1{]} - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3{]} \cdot [\mathbf{x}_1-\mathbf{x}_3{]} > 0$ and $\mathbf{e}ll_3>0$]{\label{fig:phase3 cone}\resizebox{0.4\linewidth}{!}{ \input{figures/phase3cone}}}
\caption{Two configurations of $\Theta$ and $\mathbf{x}$ where \mathbf{e}qref{eq:phase2} is an invalid error bound for bivariate extrapolation.}
\label{fig:phase3}
\mathbf{e}nd{figure}
The case in Figure~\ref{fig:phase3 triangle} can be defined mathematically as $\mathbf{e}ll_2>0, \mathbf{e}ll_3<0$, and $\mathbf{e}ll_1[\mathbf{x}_2-\mathbf{x}_1] \cdot [\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3[\mathbf{x}_2-\mathbf{x}_3] \cdot [\mathbf{x}_1-\mathbf{x}_3] < 0$.
The following lemma shows the point $\mathbf{w}$, as defined in \mathbf{e}qref{eq:phase3 w}, is the intersection of the line going through $\mathbf{x}_1$ and $\mathbf{x}_3$ and the line going through $\mathbf{x}$ and $\mathbf{x}_2$.
\begin{lemma}
Assume $-\mathbf{e}ll_0-\mathbf{e}ll_2 \stackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} \mathbf{e}ll_1+\mathbf{e}ll_3\neq 0$ for some affinely independent $\Theta\subset\R^2$ and $\mathbf{x}\in\R^2$.
Let
\begin{equation} \label{eq:phase3 w}
\mathbf{w} = \frac{-\mathbf{e}ll_0\mathbf{x}+\mathbf{e}ll_1\mathbf{x}_1-\mathbf{e}ll_2\mathbf{x}_2+\mathbf{e}ll_3\mathbf{x}_3}{-\mathbf{e}ll_0+\mathbf{e}ll_1-\mathbf{e}ll_2+\mathbf{e}ll_3}.
\mathbf{e}nd{equation}
Then
\[ \mathbf{w} = \frac{\mathbf{e}ll_1\mathbf{x}_1+\mathbf{e}ll_3\mathbf{x}_3}{\mathbf{e}ll_1+\mathbf{e}ll_3} = \frac{\mathbf{e}ll_0\mathbf{x}+\mathbf{e}ll_2\mathbf{x}_2}{\mathbf{e}ll_0+\mathbf{e}ll_2},
\]
and
\begin{subequations} \label{eq:phase3 w y} \begin{align}
\mathbf{e}ll_0[\mathbf{x}-\mathbf{w}] + \mathbf{e}ll_2[\mathbf{x}_2-\mathbf{w}] &= 0, \\
\mathbf{e}ll_1[\mathbf{x}_1-\mathbf{w}] + \mathbf{e}ll_3[\mathbf{x}_3-\mathbf{w}] &= 0.
\mathbf{e}nd{align} \mathbf{e}nd{subequations}
\mathbf{e}nd{lemma}
\begin{proof}
These equalities are direct results of \mathbf{e}qref{eq:Lagrange 0} and \mathbf{e}qref{eq:Lagrange Y}.
\mathbf{e}nd{proof}
We define in the following lemma an $H^\star$, which is different from the one defined in \mathbf{e}qref{eq:Hstar} and is asymmetric.
\begin{lemma}
Assume for some affinely independent $\Theta\subset\R^2$ and $\mathbf{x}\in\R^2$ that $\mathbf{e}ll_2>0, \mathbf{e}ll_3<0$, and $\mathbf{e}ll_1[\mathbf{x}_2-\mathbf{x}_1] \cdot [\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3] \cdot [\mathbf{x}_1-\mathbf{x}_3] < 0$.
Let
\begin{equation} \label{eq:Hstar 3}
H^\star = P \begin{bmatrix} +\nu &0\\ 0 &-\nu \mathbf{e}nd{bmatrix} P^{-1} \text{ with } P = \begin{bmatrix} \mathbf{x}_2-\mathbf{x} &\mathbf{x}_1-\mathbf{x}_3 \mathbf{e}nd{bmatrix}.
\mathbf{e}nd{equation}
Let $\mathbf{w}$ be defined as \mathbf{e}qref{eq:phase3 w}.
Then
\begin{equation} \label{eq:phase3 Hstar eigvector} \begin{aligned}
H^\star(\mathbf{x}_i-\mathbf{w}) &= \nu(\mathbf{x}_i-\mathbf{w}) \text{ for } i\in\{0,2\}, \\
H^\star(\mathbf{x}_i-\mathbf{w}) &= -\nu(\mathbf{x}_i-\mathbf{w}) \text{ for } i\in\{1,3\}.
\mathbf{e}nd{aligned} \mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
It is clear from Figure~\ref{fig:phase3 triangle} that the assumption guarantees the invertibility of $P$ and $-\mathbf{e}ll_0-\mathbf{e}ll_2 = \mathbf{e}ll_1+\mathbf{e}ll_3\neq 0$.
Notice by the definition of $H^\star$, we have $H^\star(\mathbf{x}_2-\mathbf{x}) = \nu(\mathbf{x}_2-\mathbf{x})$ and $H^\star(\mathbf{x}_1-\mathbf{x}_3) = -\nu(\mathbf{x}_1-\mathbf{x}_3)$.
The lemma holds true because $\mathbf{x}_i-\mathbf{w}$ is parallel to $\mathbf{x}_2-\mathbf{x}$ for $i\in\{0,2\}$ and to $\mathbf{x}_1-\mathbf{x}_3$ for $i\in\{1,3\}$.
\mathbf{e}nd{proof}
Now we are ready to show $G \cdot H^\star/2$, with $H^\star$ defined in \mathbf{e}qref{eq:Hstar 3}, is an upper bound on the function approximation error for the case in Figure~\ref{fig:phase3 triangle}.
\begin{theorem} \label{thm:phase3}
Assume $f \in C^{1,1}_\nu(\R^2)$.
Let $\hat{f}$ be the affine function that interpolates $f$ at any set of three affinely independent vectors $\Theta = \{\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3\}\subset \R^2$ such that $(\mathbf{x}_2-\mathbf{x}_1) \cdot (\mathbf{x}_3-\mathbf{x}_1) < 0$.
Let $\mathbf{x}$ be any vector in $\R^2$ such that its barycentric coordinates satisfies $\mathbf{e}ll_2>0, \mathbf{e}ll_3<0$, and $\mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1] \cdot [\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3] \cdot [\mathbf{x}_1-\mathbf{x}_3] < 0$.
Let $G$ and $H^\star$ be the matrices defined in \mathbf{e}qref{eq:G} and \mathbf{e}qref{eq:Hstar 3}.
Then the function approximation error of $\hat{f}$ at $\mathbf{x}$ is bounded as
\begin{equation} \label{eq:phase3}
|\hat{f}(\mathbf{x}) - f(\mathbf{x})| \le \frac{1}{2} G \cdot H^\star.
\mathbf{e}nd{equation}
\mathbf{e}nd{theorem}
\begin{proof}
We only provide the proof for the case when $\hat{f}(\mathbf{x})-f(\mathbf{x}) \mathbf{g}e 0$.
We use the function $\psi$ defined in \mathbf{e}qref{eq:Lipscthiz stronger H} again.
Since $\mathbf{e}ll_3<0$, $(\mathbf{x}_2-\mathbf{x}_1) \cdot (\mathbf{x}_3-\mathbf{x}_1) < 0$, and
\[ \begin{aligned}
0 &> \mathbf{e}ll_1 [\mathbf{x}_2-\mathbf{x}_1] \cdot [\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3] \cdot [\mathbf{x}_1-\mathbf{x}_3] \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} (1-\mathbf{e}ll_2-\mathbf{e}ll_3) [\mathbf{x}_2-\mathbf{x}_1] \cdot [\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3 [\mathbf{x}_2-\mathbf{x}_3] \cdot [\mathbf{x}_1-\mathbf{x}_3] \\
&= (1-\mathbf{e}ll_2) [\mathbf{x}_2-\mathbf{x}_1] \cdot [\mathbf{x}_3-\mathbf{x}_1] - \mathbf{e}ll_3 \|\mathbf{x}_1-\mathbf{x}_3\|^2,
\mathbf{e}nd{aligned} \]
we have $1-\mathbf{e}ll_2 > 0$, and thus the following inequalities hold:
\begin{subequations} \label{eq:phase3 sum} \begin{align}
(1-\mathbf{e}ll_2) \psi(\mathbf{x}_1, \mathbf{x}, H^\star) &\le 0, \\
\mathbf{e}ll_2 \psi(\mathbf{x}_2, \mathbf{x}, H^\star) &\le 0, \\
-\mathbf{e}ll_3 \psi(\mathbf{x}_1, \mathbf{x}_3, H^\star) &\le 0.
\mathbf{e}nd{align} \mathbf{e}nd{subequations}
Similar to the previous proofs, we add these inequalities together.
The sum of their zeroth-order terms is
\[ \begin{aligned}
&\hspace{-1em} (1-\mathbf{e}ll_2) [f(\mathbf{x}_1) - f(\mathbf{x})] + \mathbf{e}ll_2 [f(\mathbf{x}_2) - f(\mathbf{x})] - \mathbf{e}ll_3 [f(\mathbf{x}_1) - f(\mathbf{x}_3)] \\
&= (1-\mathbf{e}ll_2-\mathbf{e}ll_3) f(\mathbf{x}_1) + \mathbf{e}ll_2 f(\mathbf{x}_2) + \mathbf{e}ll_3 f(\mathbf{x}_3) - f(\mathbf{x})
\stackrel{\mathbf{e}qref{eq:Lagrange m}\mathbf{e}qref{eq:Lagrange 0}}{=} \hat{f}(\mathbf{x}) - f(\mathbf{x}).
\mathbf{e}nd{aligned} \]
The sum of their first-order terms is $-1/(2\nu)$ multiplies
\[ \begin{aligned}
&\hspace{-2em} (1-\mathbf{e}ll_2) \left\{ [(\nu I-H^\star) (\mathbf{x}_1-\mathbf{x})] \cdot Df(\mathbf{x}_1) + [(\nu I+H^\star) (\mathbf{x}_1-\mathbf{x})] \cdot Df(\mathbf{x}) \right\} \\
&\hspace{-2em} + \mathbf{e}ll_2 \left\{ [(\nu I-H^\star) (\mathbf{x}_2-\mathbf{x})] \cdot Df(\mathbf{x}_2) + [(\nu I+H^\star) (\mathbf{x}_2-\mathbf{x})] \cdot Df(\mathbf{x}) \right\} \\
&\hspace{-2em} -\mathbf{e}ll_3 \left\{ [(\nu I-H^\star) (\mathbf{x}_1-\mathbf{x}_3)] \cdot Df(\mathbf{x}_1) + [(\nu I+H^\star) (\mathbf{x}_1-\mathbf{x}_3)] \cdot Df(\mathbf{x}_3) \right\} \\
&= \{(\nu I-H^\star) [(1-\mathbf{e}ll_2)(\mathbf{x}_1-\mathbf{x}) - \mathbf{e}ll_3(\mathbf{x}_1-\mathbf{x}_3)]\} \cdot Df(\mathbf{x}_1) \\
&\quad + \mathbf{e}ll_2 [(\nu I-H^\star) (\mathbf{x}_2-\mathbf{x})] \cdot Df(\mathbf{x}_2) - \mathbf{e}ll_3 [(\nu I+H^\star) (\mathbf{x}_1-\mathbf{x}_3)] \cdot Df(\mathbf{x}_3) \\
&\quad + \left\{ (\nu I+H^\star) [(1-\mathbf{e}ll_2)(\mathbf{x}_1-\mathbf{x}) + \mathbf{e}ll_2(\mathbf{x}_2-\mathbf{x})] \right\} \cdot Df(\mathbf{x}) \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange 0}\mathbf{e}qref{eq:Lagrange Y}}{=} \mathbf{e}ll_2 [(\nu I-H^\star) (\mathbf{x} - \mathbf{x}_2)] \cdot Df(\mathbf{x}_1) + \mathbf{e}ll_2 [(\nu I-H^\star) (\mathbf{x}_2-\mathbf{x})] \cdot Df(\mathbf{x}_2) \\
&\quad - \mathbf{e}ll_3 [(\nu I+H^\star) (\mathbf{x}_1-\mathbf{x}_3)] \cdot Df(\mathbf{x}_3) + \mathbf{e}ll_3 [(\nu I+H^\star) (\mathbf{x}_1 - \mathbf{x}_3)] \cdot Df(\mathbf{x}) \\
&\leftstackrel{\mathbf{e}qref{eq:phase3 Hstar eigvector}}{=} \mathbf{0}.
\mathbf{e}nd{aligned} \]
Let $\mathbf{w}$ be defined as \mathbf{e}qref{eq:phase3 w}.
The sum of the constant terms is $-1/2$ times
\[ \begin{aligned}
&\hspace{-1em} (1-\mathbf{e}ll_2) \left[\frac{1}{2\nu} \|H^\star (\mathbf{x}_1-\mathbf{x})\|^2 + \frac{\nu}{2} \|\mathbf{x}_1-\mathbf{x}\|^2 \right] + \mathbf{e}ll_2 \left[\frac{1}{2\nu} \|H^\star (\mathbf{x}_2-\mathbf{x})\|^2 \right. \\
&\hspace{-1em} \left.+ \frac{\nu}{2} \|\mathbf{x}_2-\mathbf{x}\|^2 \right] - \mathbf{e}ll_3 \left[\frac{1}{2\nu} \|H^\star (\mathbf{x}_1-\mathbf{x}_3)\|^2 + \frac{\nu}{2} \|\mathbf{x}_1-\mathbf{x}_3\|^2 \right] \\
&\leftstackrel{\mathbf{e}qref{eq:phase3 Hstar eigvector}}{=} (1-\mathbf{e}ll_2)\left\{ -H^\star(\mathbf{x}_1-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{w}) + H^\star(\mathbf{x}-\mathbf{w}) \cdot (\mathbf{x}-\mathbf{w}) \right\} \\
&\qquad + \mathbf{e}ll_2 H^\star(\mathbf{x}_2-\mathbf{x}) \cdot (\mathbf{x}_2-\mathbf{x}) + \mathbf{e}ll_3 H^\star(\mathbf{x}_1-\mathbf{x}_3) \cdot (\mathbf{x}_1-\mathbf{x}_3) \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange 0}}{=} H^\star[\mathbf{e}ll_3(\mathbf{x}_1-\mathbf{x}_3)-(\mathbf{e}ll_1+\mathbf{e}ll_3)(\mathbf{x}_1-\mathbf{w})] \cdot (\mathbf{x}_1-\mathbf{w}) - \mathbf{e}ll_3 H^\star(\mathbf{x}_1-\mathbf{x}_3) \cdot (\mathbf{x}_3-\mathbf{w}) \\
&\qquad + H^\star[(1-\mathbf{e}ll_2)(\mathbf{x}-\mathbf{w}) - \mathbf{e}ll_2(\mathbf{x}_2-\mathbf{x})] \cdot (\mathbf{x}-\mathbf{w}) + \mathbf{e}ll_2 H^\star(\mathbf{x}_2-\mathbf{x}) \cdot (\mathbf{x}_2-\mathbf{w}) \\
&\leftstackrel{\mathbf{e}qref{eq:Lagrange 0}\mathbf{e}qref{eq:Lagrange Y}}{=} 0 - \mathbf{e}ll_3 [H^\star(\mathbf{x}_1-\mathbf{w}) - H^\star(\mathbf{x}_3-\mathbf{w})] \cdot (\mathbf{x}_3-\mathbf{w}) \\
&\quad + 0 + \mathbf{e}ll_2 [H^\star(\mathbf{x}_2-\mathbf{w}) - H^\star(\mathbf{x}-\mathbf{w})] \cdot (\mathbf{x}_2-\mathbf{w}) \\
&\leftstackrel{\mathbf{e}qref{eq:phase3 w y}}{=} \sum_{i=0}^{3} \mathbf{e}ll_i H^\star (\mathbf{x}_i-\mathbf{w}) \cdot (\mathbf{x}_i-\mathbf{w})
\stackrel{\mathbf{e}qref{eq:G recenter}}{=} G \cdot H^\star.
\mathbf{e}nd{aligned} \]
Thus, the sum of the inequalities in \mathbf{e}qref{eq:phase3 sum} is \mathbf{e}qref{eq:phase3} when $\hat{f}(\mathbf{x}) - f(\mathbf{x}) \mathbf{g}e 0$.
\mathbf{e}nd{proof}
We show in Theorem~\ref{thm:phase3 sharp} the upper bound \mathbf{e}qref{eq:phase3} can be achieved by a piecewice quadratic, and therefore \mathbf{e}qref{eq:phase3} is sharp.
\begin{theorem} \label{thm:phase3 sharp}
Under the setting of Theorem~\ref{thm:phase3}, the bound \mathbf{e}qref{eq:phase3} is sharp and can be achieved by
\[ f(\mathbf{u}) = \left\{ \begin{aligned}
&\frac{\nu}{2} \|\mathbf{u}-\mathbf{w}\|^2 - \frac{\nu[(\mathbf{x}_1-\mathbf{x}_3) \cdot (\mathbf{u}-\mathbf{w})]^2}{\|\mathbf{x}_1-\mathbf{x}_3\|^2} &&\text{if } (\mathbf{u}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) \le 0, \\
&\frac{\nu}{2} \|\mathbf{u}-\mathbf{w}\|^2 &&\text{if } (\mathbf{u}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) \mathbf{g}e 0.
\mathbf{e}nd{aligned} \right.,
\]
where $\mathbf{w}$ is defined in \mathbf{e}qref{eq:phase3 w}.
\mathbf{e}nd{theorem}
\begin{proof}
The function approximation error for this piecewise quadratic function is
\[ \begin{aligned}
\hat{f}(&\mathbf{x}) - f(\mathbf{x})
= \sum_{i=0}^{n+1} \mathbf{e}ll_i \mathbf{x}_i \\
&= \frac{\nu}{2} \sum_{i=0}^3 \mathbf{e}ll_i \|\mathbf{x}_i-\mathbf{w}\|^2 - \frac{\nu\mathbf{e}ll_1 [(\mathbf{x}_1-\mathbf{x}_3) \cdot (\mathbf{x}_1-\mathbf{w})]^2}{\|\mathbf{x}_1-\mathbf{x}_3\|^2} - \frac{2\nu\mathbf{e}ll_3 [(\mathbf{x}_1-\mathbf{x}_3) \cdot (\mathbf{x}_3-\mathbf{w})]^2}{\|\mathbf{x}_1-\mathbf{x}_3\|^2} \\
&= \frac{\nu}{2} \sum_{i=0}^3 \mathbf{e}ll_i \|\mathbf{x}_i-\mathbf{w}\|^2 - \nu\mathbf{e}ll_1\|\mathbf{x}_1-\mathbf{w}\|^2 - 2\nu\mathbf{e}ll_3\|\mathbf{x}_3-\mathbf{w}\|^2 \\
&= \frac{\nu}{2} \left(\mathbf{e}ll_0\|\mathbf{x}-\mathbf{w}\|^2 - \mathbf{e}ll_1\|\mathbf{x}_1-\mathbf{w}\|^2 + \mathbf{e}ll_2\|\mathbf{x}_2-\mathbf{w}\|^2 - \mathbf{e}ll_3\|\mathbf{x}_3-\mathbf{w}\|^2\right) \\
&\leftstackrel{\mathbf{e}qref{eq:phase3 Hstar eigvector}}{=} \frac{1}{2} \sum_{i=0}^3 \mathbf{e}ll_i \|\mathbf{x}_i-\mathbf{w}\|_{H^\star}
\stackrel{\mathbf{e}qref{eq:G recenter}}{=} \frac{1}{2} G \cdot H^\star.
\mathbf{e}nd{aligned} \]
Now we prove $f \in C_\nu^{1,1}(\R^n)$.
Firstly, it is clear that $f$ is continuous on $\R^2$ and differentiable on the two half spaces $\{\mathbf{u}:~ (\mathbf{u}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) < 0\}$ and $\{\mathbf{u}:~ (\mathbf{u}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) > 0\}$.
Then given any $\mathbf{u}$ such that $(\mathbf{u}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) = 0$, it can be calculated for any $\mathbf{v}\in\R^2$ that
\begin{multline*}
|f(\mathbf{u}+\mathbf{v}) - f(\mathbf{u}) - \nu (\mathbf{u}-\mathbf{w})\cdot\mathbf{v}| \\
= \left\{ \begin{aligned}
&-\frac{\nu}{2}\|\mathbf{v}\|^2 - \frac{\nu [(\mathbf{x}_1-\mathbf{x}_3)\cdot \mathbf{v}]^2}{\|\mathbf{x}_1-\mathbf{x}_3\|^2} &&\text{if } (\mathbf{u}+\mathbf{v}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) \le 0, \\
&-\frac{\nu}{2}\|\mathbf{v}\|^2 &&\text{if } (\mathbf{u}+\mathbf{v}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) \mathbf{g}e 0.
\mathbf{e}nd{aligned} \right.
\mathbf{e}nd{multline*}
Thus
\[ \lim_{v\rightarrow\mathbf{0}} \frac{|f(\mathbf{u}+\mathbf{v}) - f(\mathbf{u}) - \nu(\mathbf{u}-\mathbf{w})\cdot\mathbf{v}|}{\|\mathbf{v}\|} = 0,
\]
which shows $f$ is differentiable with gradient $\nu(\mathbf{u}-\mathbf{w})$ on $\{\mathbf{u}:~ (\mathbf{u}-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3) = 0\}$.
The condition \mathbf{e}qref{eq:Lipschitz} is clearly satisfied if $\mathbf{u}_1$ and $\mathbf{u}_2$ are in the same half space.
Now assume $(\mathbf{u}_1-\mathbf{w})\cdot(\mathbf{x}_1-\mathbf{x}_3) < 0$ and $(\mathbf{u}_2-\mathbf{w})\cdot(\mathbf{x}_1-\mathbf{x}_3) > 0$.
Then, we have
\[ \begin{aligned}
&\|Df(\mathbf{u}_1) - Df(\mathbf{u}_2)\|^2 \\
&= \|\nu(\mathbf{u}_1-\mathbf{w}) - 2\nu\left[(\mathbf{x}_1-\mathbf{x}_3) \cdot (\mathbf{u}_1-\mathbf{w})/\|\mathbf{x}_1-\mathbf{x}_3\|^2\right] (\mathbf{x}_1-\mathbf{x}_3)- \nu(\mathbf{u}_2-\mathbf{w})\|^2 \\
&= \nu^2\|\mathbf{u}_1-\mathbf{u}_2\|^2 + 4\nu^2 [(\mathbf{u}_1-\mathbf{w}) \cdot (\mathbf{x}_1-\mathbf{x}_3)] [(\mathbf{u}_2-\mathbf{w})\cdot(\mathbf{x}_1-\mathbf{x}_3)] /\|\mathbf{x}_1-\mathbf{x}_3\|^2 \\
&< \nu^2\|\mathbf{u}_1-\mathbf{u}_2\|^2,
\mathbf{e}nd{aligned} \]
which shows \mathbf{e}qref{eq:Lipschitz} always holds.
Therefore $f \in C_\nu^{1,1}(\R^n)$.
\mathbf{e}nd{proof}
\section{Discussion} \label{sec:discussion}
We presented a numerical approach to calculate the sharp bound on the function approximation error of linear interpolation and extrapolation and proved several conditionally sharp analytical bound along with their conditions for sharpness.
These analytically bounds include one that improves the existing ones to better cover the extrapolation case \mathbf{e}qref{eq:phase1}, a sharp bound for quadratic functions \mathbf{e}qref{eq:phase2}, and one for bivariate extrapolation \mathbf{e}qref{eq:phase3}.
The two bounds \mathbf{e}qref{eq:phase2} and \mathbf{e}qref{eq:phase3} together provide the sharp error bound for bivariate linear interpolation under any configuration of $\mathbf{x}$ and an affinely independent $\Theta$.
These bounds can provide an important theoretical foundation for the design and analysis of derivative-free optimization methods and any other numerical methods that utilizes linear interpolation.
While our results are developed under the condition that $f\in C_\nu^{1,1}(\R^n)$, they can stand under weaker conditions (but would require more complicated analysis).
In existing literature, the condition often used is that $\| |D^2 f| \|_{L_\infty(Q)} \le \nu$, where $Q$, for example, is the star-shaped set that connects $\mathbf{x}$ to each point in $\Theta$ in \cite{ciarlet1972general} and $\conv(\Theta)$ in \cite{waldron1998error}.
Our results do not necessarily require the twice-differentiability of $f$ and only need $f\in C_\nu^{1,1}(Q)$ for some $Q\subset\R^n$.
For \mathbf{e}qref{eq:phase1}, $Q$ at least needs to cover (almost everywhere, same hereafter) the star-shaped set $\cup_{i=0}^{n+1} \{\alpha \mathbf{x}_i + (1-\alpha)\mathbf{w}:~ 0\le\alpha\le1\}$.
For \mathbf{e}qref{eq:phase2}, we need $Q$ to cover
\[ \bigcup_{(i,j)\in\cI_+\times\cI_-}
\left( \begin{aligned}
&\{\alpha \mathbf{x}_i + (1-\alpha)[(\mathbf{u}_i+\mathbf{u}_j)/2+H^\star(\mathbf{u}_i-\mathbf{u}_j)/(2\nu)]:~ 0\le\alpha\le1 \} \\
&\quad \cup \{\alpha \mathbf{x}_j + (1-\alpha)[(\mathbf{u}_i+\mathbf{u}_j)/2+H^\star(\mathbf{u}_i-\mathbf{u}_j)/(2\nu)]:~ 0\le\alpha\le1 \}
\mathbf{e}nd{aligned} \right),
\]
where $H^\star$ is defined as \mathbf{e}qref{eq:Hstar}.
For \mathbf{e}qref{eq:phase3}, we need
\[ Q \supseteq \{\alpha \mathbf{x}_2 + (1-\alpha)\mathbf{w}:~ 0\le\alpha\le1 \} \cup \{\alpha \mathbf{x}_3 + (1-\alpha)\mathbf{w}:~ 0\le\alpha\le1 \},
\]
where $\mathbf{w}$ is defined as \mathbf{e}qref{eq:phase3 w}.
We proposed to compute $\{\mu_{ij}\}$ and check their signs to determine whether \mathbf{e}qref{eq:phase2} is a sharp bound and proved in Theorem~\ref{thm:phase2} that $\{\mu_{ij}\}$ being all non-negative is a sufficient condition.
We want to mention that one of our numerical experiments seems to indicate that it is also a necessary condition.
This experiment involves generating many different $\Theta$ and $\mathbf{x}$ with various $n$ and calculated the corresponding $\{\mu_{ij}\}$.
From this experiment, we also observed some geometric pattern of the signs of $\{\mu_{ij}\}$, which we present in the following conjecture.
\begin{conjecture}
Assume $f \in C^{1,1}_\nu(\R^n)$.
Let $\hat{f}$ be the linear function that interpolates $f$ at any set of $n+1$ affinely independent vectors $\Theta = \{\mathbf{x}_1,\dots,\mathbf{x}_{n+1}\}\subset \R^n$.
Let $\mathbf{x}$ be any vector in $\R^n$.
Let $\{\mu_{ij}\}_{i \in \cI_+,~ j \in \cI_-}$ be the set of parameters defined in Theorem~\ref{thm:phase2}. Then the following statements are true.
\begin{enumerate}
\item When there is no obtuse angle at the vertices of the simplex $\conv(\Theta)$, that is, when
\begin{equation} \label{eq:acute simplex}
(\mathbf{x}_j-\mathbf{x}_i) \cdot (\mathbf{x}_k-\mathbf{x}_i) \mathbf{g}e 0 \text{ for all } i,j,k = 1,2,\dots,n+1,
\mathbf{e}nd{equation}
the parameters $\{\mu_{ij}\}$ are all non-negative for any $\mathbf{x} \in \R^n$.
\item If there is at least one obtuse angle at the vertices of the simplex $\conv(\Theta)$, then there is a non-empty subset of $\R^n$ to which if $\mathbf{x}$ belongs, there is at least one negative element in $\{\mu_{ij}\}$.
\mathbf{e}nd{enumerate}
\mathbf{e}nd{conjecture}
A general formula for the sharp bound on the function approximation error of linear interpolation and extrapolation remains an open question.
It would appear $G\cdot H^\star/2$ is a good candidate, since all the bounds developed in this paper can be written in this form, but the matrix $H^\star$ depends on the geometry of $\Theta$ and $\mathbf{x}$.
Using $G\cdot H^\star/2$ as the general formula, we would need five different definition of $H^\star$ even for the bivariate case (\mathbf{e}qref{eq:Hstar} and four variants of \mathbf{e}qref{eq:Hstar 3} that corresponds to the four shaded areas in Figure~\ref{fig:phase2}).
Note that the matrix $H^\star$ is tied to $\{\mu_{ij}\}$ in \mathbf{e}qref{eq:mu1 +} and \mathbf{e}qref{eq:mu1 -}, and we believe even when there are negatives in $\{\mu_{ij}\}$, they are still tied in the same manner to a version of $\{\mu_{ij}\}$ that is modified to be all non-negative.
In fact, \mathbf{e}qref{eq:mu0 +} - \mathbf{e}qref{eq:mu1 -} all hold true under the setting of Theorem~\ref{thm:phase3} if $H^\star$ is defined as \mathbf{e}qref{eq:Hstar 3} and $\{\mu_{ij}\}$ is defined as
\[ \begin{aligned}
\mu_{10} &= 1-\mathbf{e}ll_2,
&\mu_{13} &= -\mathbf{e}ll_3,
&\mu_{20} &= \mathbf{e}ll_2,
&\mu_{23} &= 0,
\mathbf{e}nd{aligned} \]
which are the coefficients in \mathbf{e}qref{eq:phase3 sum}.
Considering the difficulty in analyzing the signs of $\{\mu_{ij}\}$, it is unlikely for $G\cdot H^\star/2$ to be suitable for this general formula.
Whether there even exists a concise analytical form to the sharp error bound that can fit all the geometric configurations of $\Theta$ and $\mathbf{x}$ is still unclear to us.
\section*{Acknowledgment}
We would like to acknowledge the help from Dr. Xin Shi and Yunze Sun in solving \mathbf{e}qref{prob:quadratic}.
We would like to thank Dr. Shuonan Wu and Dr. Katya Scheinberg for carefully reading this paper and providing their suggestions.
\mathbf{e}nd{document}
|
\begin{document}
\title[A Liouville theorem for the vectorial Allen-Cahn energy]
{A Liouville theorem for minimizers with finite potential energy for the vectorial Allen-Cahn equation}
\author{Christos Sourdis} \address{Department of Mathematics and Applied Mathematics, University of
Crete.}
\email{[email protected]}
\maketitle
\begin{abstract}
We prove that if a globally minimizing solution to the vectorial
Allen-Cahn equation has finite potential energy, then it is a
constant.
\end{abstract}
Consider the semilinear elliptic system
\begin{equation}\label{eqEq}
\Delta u=\nabla W(u)\ \ \textrm{in}\ \ \mathbb{R}^n,\ \ n\geq 1,
\end{equation}
where $W:\mathbb{R}^m\to \mathbb{R}$, $m\geq 1$, is sufficiently
smooth and nonnegative. It has been recently shown in
\cite{alikakosBasicFacts} that each nonconstant solution to the
system (\operatorname{Re}f{eqEq}) satisfies:
\begin{equation}\label{eqGrande}
\int_{B_R}^{}\left\{\frac{1}{2}|\nabla u|^2+ W\left(u\right)
\right\}dx\geq \left\{\begin{array}{ll}
c R^{n-2} & \textrm{if}\ n\geq 3, \\
& \\
c \ln R & \textrm{if}\ n=2,
\end{array}
\right.
\end{equation}
for all $R>1$, and some $c>0$, where $B_R$ stands for the
$n$-dimensional ball of radius $R$, centered at the origin.
On the other side, if additionally $W$ vanishes at least at one
point, it is easy to see that \begin{equation}\label{equpper}
\int_{B_R}^{}\left\{\frac{1}{2}|\nabla u|^2+ W\left(u\right)
\right\}dx\leq CR^{n-1},\ \ R>0,
\end{equation}
for some $C>0$ (see \cite{fuscoPreprint}).
The
system (\operatorname{Re}f{eqEq}) with $W\geq 0$ vanishing at a finite number of
global minima (typically nondegenerate), and coercive at infinity,
is used to model multi-phase transitions (see \cite{fuscoPreprint}
and the references therein). In this case, the system (\operatorname{Re}f{eqEq})
is frequently referred to as the vectorial Allen-Cahn equation. In
\cite{sourdis14}, we showed the following theorem for globally
minimizing solutions (see \cite{fuscoCPAA,sourdis14} for the precise
definition).
\begin{thm}\label{thmMine} Assume that $W\in
C^1(\mathbb{R}^m;\mathbb{R})$, $m\geq 1$, and that there exist
finitely many $N\geq 1$ points $a_i\in \mathbb{R}^m$ such that
\begin{equation}\label{eqpoints}
W(u)>0\ \ \textrm{in}\ \ \mathbb{R}^m\setminus \{a_1,\cdots, a_N \},
\end{equation}
and there exists small $r_0>0$ such that the functions
\begin{equation}\label{eqmonot}
r\mapsto W(a_i+r\nu),\ \ \textrm{where}\ \ \nu \in \mathbb{S}^1, \ \
\textrm{are strictly increasing\ for}\ r\in (0,r_0),\ \
i=1,\cdots,N.
\end{equation}
Moreover, we assume that
\begin{equation}\label{eqinf}\liminf_{|u|\to \infty}
W(u)>0.\end{equation}
If $u\in C^2(\mathbb{R}^2;\mathbb{R}^m)$ is
a bounded, nonconstant, and globally minimizing solution to the
elliptic system (\operatorname{Re}f{eqEq}) with $n=2$, there exist constants
$c_0, R_0>0$ such that
\[
\int_{B_R}^{}W\left(u(x) \right) dx\geq c_0R\ \ \textrm{for}\ \
R\geq R_0.
\]
\end{thm}
In view of (\operatorname{Re}f{equpper}), the above result captures the optimal
growth rate in the case $n=2$. The purpose of this note is to
establish the following Liouville type theorem which holds in any
dimension. Similarly to \cite{sourdis14}, we combine ideas from the
study of vortices in the Ginzburg-Landau model \cite{bethuel} with
variational maximum principles from the study of the vector
Allen-Cahn equation \cite{alikakosPreprint}.
\begin{thm}\label{thmNote}
Let $W$ be as in Theorem \operatorname{Re}f{thmMine}. Suppose that $u\in
C^2(\mathbb{R}^n;\mathbb{R}^m)$, $n\geq 2$, is a bounded and
globally minimizing solution to the elliptic system (\operatorname{Re}f{eqEq})
such that
\[
\int_{\mathbb{R}^n}^{}W\left(u(x) \right) dx<\infty.
\]
Then, we have that
\[
u\equiv a_i\ \ \textrm{for some}\ i\in \{1,\cdots,N \}.
\]
\end{thm}
\begin{proof}
It follows that there exists a constant $C_0>0$ such that
\begin{equation}\label{eqj1}
\int_{B_R}^{}W\left(u(x) \right) dx\leq C_0,\ \ R>0.
\end{equation}
Let
\[
\varepsilon=\frac{1}{R}\ \ \textrm{and}\ \
u_\varepsilon(y)=u\left(\frac{y}{\varepsilon} \right),\ \ y\in B_1.
\]
Then, relation (\operatorname{Re}f{eqj1}) becomes
\begin{equation}\label{eqj2}
\int_{B_1}^{}W\left(u_\varepsilon(y) \right) dy\leq
C_1\varepsilon^n,\ \ \varepsilon>0,
\end{equation}
for some $C_1>0$. Note that, by standard elliptic regularity
estimates \cite{Gilbarg-Trudinger}, we have that
\begin{equation}\label{eqj3}
|u_\varepsilon|+\varepsilon|\nabla u_\varepsilon|\leq C_2\ \
\textrm{in}\ \mathbb{R}^n, \ \varepsilon>0,
\end{equation}
for some $C_2>0$.
Let $d>0$ be any small number. As in \cite{bethuel}, by combining
(\operatorname{Re}f{eqj2}) and (\operatorname{Re}f{eqj3}), we deduce that the set where
$W(u_\varepsilon)$ is above $d>0$ is included in a uniformly bounded
number of balls of radius $\varepsilon$, as $\varepsilon\to 0$.
Certainly, there exists $r_\varepsilon \in
(\frac{1}{4},\frac{3}{4})$ such that
\[
W\left(u_\varepsilon(y) \right)\leq d\ \ \textrm{if}\
|y|=r_\varepsilon.
\]
Since $d>0$ is arbitrary, we are led to $\tilde{r}_\varepsilon \in
(\frac{1}{4},\frac{3}{4})$ such that
\[
\max_{|y|=\tilde{r}_\varepsilon}W\left(u_\varepsilon(y) \right)\to
0\ \ \textrm{as}\ \varepsilon\to 0.
\]
In terms of $u$ and $R$, we have
\[
\max_{|x|=s_R}W\left(u(x) \right)\to 0\ \ \textrm{as}\ R\to \infty,\
\ \textrm{for some}\ s_R \in \left(\frac{1}{4}R,\frac{3}{4}R\right).
\]
In view of (\operatorname{Re}f{eqinf}), the above relation implies that there
exist $i_j\in \{1,\cdots, N \}$ such that
\[
\max_{|x|=s_R}\left|u(x)-a_{i_j} \right|\to 0\ \ \textrm{as}\ R\to
\infty.
\]
By virtue of (\operatorname{Re}f{eqmonot}), as in \cite{sourdis14}, exploiting
the fact that $u$ is a globally minimizing solution, we can apply a
recent variational maximum principle from \cite{alikakosPreprint} to
deduce that
\[
\max_{|x|\leq s_R}\left|u(x)-a_{i_j} \right|\leq
\max_{|x|=s_R}\left|u(x)-a_{i_j} \right|\ \ \textrm{for} \ \ R\gg
1.
\]
The above two
relations imply the existence of an $i_0\in \{1,\cdots,N\}$ such
that
\[
\max_{|x|\leq s_R}\left|u(x)-a_{i_0} \right|\to 0\ \ \textrm{as}\ \
R\to \infty.
\]
Since $s_R\to \infty$ as $R\to \infty$, we conclude that $u\equiv
a_{i_0}$.
\end{proof}
\end{document}
|
\begin{document}
\widetildetle{A better bound on the size of rainbow matchings}
\author{Hongliang Lu\footnote{Partially supported by the National Natural
Science Foundation of China under grant No.11871391 and
Fundamental Research Funds for the Central Universities}\\
School of Mathematics and Statistics\\
Xi'an Jiaotong University\\
Xi'an, Shaanxi 710049, China\\
\\
Yan Wang\\
School of Mathematics\\
Georgia Institute of Technology\\
Atlanta, GA 30332, USA\\
\\
Xingxing Yu\footnote{Partially supported by NSF grant DMS-1600738}\\
School of Mathematics\\
Georgia Institute of Technology\\
Atlanta, GA 30332, USA}
\date{}
\title{A better bound on the size of rainbow matchings}
\date{}
\title{A better bound on the size of rainbow matchings}
\begin{abstract}
Aharoni and Howard conjectured that, for positive integers $n,k,t$
with $n\ge k$ and $n\ge t$, if $F_1,\ldots, F_t\precequbseteqbseteq {[n]\choose k}$ such
that $|F_i|>{n\choose k}-{n-t+1\choose k}$ for $i\in [t]$ then there exist
$e_i\in F_i$ for $i\in [t]$ such that $e_1,\ldots,e_t$ are
pairwise disjoint. Huang, Loh, and Sudakov proved this conjecture for
$t<n/(3k^2)$. In this paper, we show that this conjecture holds for
$t < n/(2k)$ and $n$ sufficiently large.
\end{abstract}
\preceqection{Introduction}
For a positive integer $n$, let $[n]$ denote the set $\{1,\ldots,n\}$.
For a set $S$ with at least $k$ elements, let ${S\choose
k}=\{e\precequbseteqbseteq S\ : \ |e|=k\}$.
Let $k \ge 2$ be an integer.
A \emph{$k$-uniform hypergraph} or \textit{$k$-graph} is a pair
$H=(V,E)$, where $V=V(H)$ is a finite set of {\it vertices} and
$E=E(H)\precequbseteqbseteq {V\choose k}$ is the set of {\it edges}.
We use $e(H)$ to denote the number of edges in $H$.
For any $S \precequbseteqbseteq V(H)$, let $H[S]$ denote the
subgraph of $H$ with $V(H[S])=S$ and $E(H[S])=\{e\in E(H): e \precequbseteqbseteq
S\}$, and let $H-S:= H[V(H)\preceqetminus S]$.
A \emph{matching} in a hypergraph $H$ is a subset of $E(H)$ consisting
of disjoint edges. The maximal size of a matching in a hypergraph $H$
is denoted by $\nu(H)$. A classical problem in extremal set
theory is to determine $\max e(H)$ with $\nu(H)$ fixed.
Erd\H{o}s
\cite{Erdos65} in 1965 made the following conjecture:
For positive integers $k,n,t$ with $n \ge kt$, every $k$-graph $H$ on $n$ vertices with $\nu(H) <
t$ satisfies $e(H)\leq \max \left\{ {n\choose k}-{n-t+1\choose k}, {kt-1\choose k} \right\}.$
This bound is tight for the complete $k$-graph on $kt-1$ vertices and for the $k$-graph on $n$ vertices in which every edge intersects a fixed set of $t-1$ vertices.
There have been recent activities on this conjecture, see \cite{AHS12,AFH12,FLM,Fr13,Fr17,HLS,LM}.
In particular, Frankl \cite{Fr13}
proved that if $n\geq (2t-1)k-(t-1)$ and $\nu(H)<t$ then $e(H)\le
{n\choose k}-{n-t+1\choose k}$, with further improvement by Frankl
and Kupavskii \cite{FK18}.
There are also attempts to extend the above conjecture of Erd\H{o}s to a family
of hypergraphs. Let $\mathcal{F} = \{F_1,\ldots, F_t\}$ be a family
of hypergraphs. A set of pairwise disjoint edges, one from
each $F_i$, is called a \emph{rainbow matching} for $\mathcal{F}$. In this case, we also say that ${\cal F}$ or
$\{ F_1,\ldots, F_t\}$ {\it admits} a rainbow matching. Aharoni and
Howard \cite{AH} made the following conjecture, also see Huang, Loh, and Sudakov \cite{HLS}.
\begin{conjecture}\label{AH-HLS}
Let ${\cal F}=\{F_1,\ldots, F_t\}$ be a family of subsets in ${[n]\choose k}$. If
\[
e(F_i)> \max\left\{{n\choose k}-{n-t+1\choose k},{kt-1\choose k}\right\}
\]
for all $1\leq i\leq t$, then ${\cal F}$ admits a rainbow matching.
\end{conjecture}
Huang, Loh, and Sudakov \cite{HLS} proved that Conjecture \ref{AH-HLS} holds for $n > 3k^2t$.
\begin{theorem}[Huang, Loh, and Sudakov]\label{HLS}
Let $n,k,t$ be three positive integers such that $n > 3k^2t$.
Let ${\cal F}=\{F_1,\ldots, F_t\}$ be a family of subsets of ${[n]\choose k}$. If
\[
e(F_i)> {n\choose k}-{n-t+1\choose k}
\]
for all $1\leq i\leq t$, then ${\cal F}$ has a rainbow matching.
\end{theorem}
Recently, Frankl and Kupavskii \cite{FK20} proved that Conjecture
\ref{AH-HLS} holds when $n\ge 12kt\log(e^2t)$, providing an almost
linear bound. In this paper, we show that Conjecture \ref{AH-HLS} holds when $n > 2kt$ and $n$ is sufficiently large.
\begin{theorem}\label{main}
Let $n,k,t$ be three positive integers such that $n > 2kt$ and $n$ is
sufficiently large.
Let ${\cal F}=\{F_1,\ldots, F_t\}$ be a family of subsets of ${[n]\choose k}$.
If
\[
e(F_i)> {n\choose k}-{n-t+1\choose k}
\]
for all $1\leq i\leq t$, then ${\cal F}$ has a rainbow matching.
\end{theorem}
Note that the lower bound on $e(F_i)$ is best possible. Indeed, For $i\in [t]$ let $F_i$ be the $k$-graph on $[n]$ consisting of all edges intersecting
$[t-1]$. Then for $i\in [t]$, $e(F_i)= {n\choose k}-{n-t+1\choose k}$ and $\nu(F_i)=t-1$. Hence, $\{F_1, \ldots, F_t\}$ does not admit any rainbow matching.
This example naturally corresponds to a special class of
$(k+1)$-graphs ${\cal F}_t(k,n)$. This is defined in Section 2, where
we reduce the problem for finding one such rainbow matching to a problem
about finding ``near'' perfect matchings in a larger class of
$(k+1)$-graphs, denoted by ${\cal F}^t(k,n)$. This will
allow us to apply various techniques used previously to find large matchings in uniform hypergraphs.
We show in Section 3 that Theorem~\ref{main} holds when ${\cal F}^t(k,n)$ is close to ${\cal F}_t(k,n)$, in the sense that most edges of ${\cal F}_t(k,n)$ are also edges of ${\cal F}^t(k,n)$.
To deal with the case ${\cal F}^t(k,n)$ is not close to ${\cal F}_t(k,n)$, we follow the approach in \cite{BMS15} and
\cite{ST15}. First, we find a small absorbing matching $M_1$ in ${\cal F}^t(k,n)$ which is done in Section 4. (However, the existence of this absorbing matching does not require that ${\cal F}^t(k,n)$ be not close to ${\cal F}_t(k,n)$.) Then we take random samples from ${\cal F}^t(k,n)-V(M_1)$ so that they satisfy various properties,
in particular they all have fractional perfect matchings, see Section 5. In Section 6, we use fractional perfect matchings in those random samples to perform a second round of randomization to find a spanning subgraph $H'$ of ${\cal F}^t(k,n)-V(M_1)$. We then
apply a result of Pippenger to find a matching in $H'$ covering all but a small constant fraction of the vertices, and use the matching $M_1$
to find the desired matching in ${\cal F}^t(k,n)$ covering all but fewer than $k$ vertices.
\preceqection{Notation and reduction}
To prove Theorem~\ref{main}, we convert
this rainbow matching problem on $k$-graphs to a matching problem for a special class of $(k+1)$-graphs.
Let $Q,V$ be two disjoint sets. A $(k+1)$-graph $H$ with vertex $Q \cup V$
is called \emph{$(1,k)$-partite} with {\it partition classes} $Q,V$ if, for each edge $e\in E(H)$, $|e\cap Q|=1$ and $|e\cap V|=k$.
A $(1,k)$-partite $(k+1)$-graph $H$ with partition classes $Q,V$ is \emph{balanced} if $|V|=k|Q|$.
We say that $S \precequbseteqbseteq V(H)$ is \textit{balanced} if $|S\cap V|=k|S\cap Q|$.
Let $F_1,\ldots, F_t$ be a family of subsets of ${[n]\choose k}$ and $X:=\{x_1,\ldots,x_t\}$ be a set of $t$ vertices.
We use $\mathcal{F}^t(k,n)$ to denote the $(1,k)$-partite $(k+1)$-graph with partition classes $X, [n]$ and edge set
\[
E(\mathcal{F}^t(k,n))=\bigcup_{i=1}^t \{\{x_i\}\cup e\ :\ e\in F_i\}.
\]
If $F_1=\cdots=F_t=H_k(t,n)$, where $H_k(t,n)$ denotes the $k$-graph with vertex set $[n]$
and edge set ${[n]\choose k}\preceqetminus {[n]-[t]\choose k}$, then
we denote $\mathcal{F}^t(k,n)$ by $\mathcal{F}_t(k,n)$.
\textbf{Observation 1:} $\{F_1,\ldots,F_t\}$ admits a rainbow matching if, and only if, $\mathcal{F}^t(k,n)$ has a matching of size $t$.
Hence, to prove Theorem~\ref{main}, it suffices to show that $\mathcal{F}^t(k,n)$ has a matching of size $t$. For convenience, we further reduce this problem to a near perfect matching problem.
Write $n-kt=km+r$, where $0\leq r\leq k-1$. Let $F_1, \ldots, F_t\precequbseteqbseteq {[n]\choose k}$, and let $F_i={[n]\choose k}$ for $i=t+1, \ldots, t+m$.
Let $Q=\{x_1,\ldots,x_{m+t}\}$ and let $\mathcal{H}^t(k,n)$ be the $(1,k)$-partite $(k+1)$-graph with partition classes $Q, [n]$ and edge set
\begin{align*}
E(\mathcal{H}^t(k,n))=\bigcup_{i=1}^{m+t}\{\{x_i\}\cup e\ :\ e\in F_i\}.
\end{align*}
When $F_1=\cdots=F_t=H_k(t,n)$,
we denote $\mathcal{H}^t(k,n)$ by $\mathcal{H}_t(k,n)$. Note that $\nu(\mathcal{H}_t(k,n))= m+t=(n-r)/k$, i.e., ${\cal H}_t(k,n)$ has a matching covering
all but less than $k$ vertices (and such a matching is said to be near perfect).
\begin{lemma}\label{Rain-PM}
$\mathcal{F}^t(k,n)$ has a matching of size $t$ if, and only if, $\mathcal{H}^t(k,n)$ has a matching of size $m+t=\lfloor n/k\rfloor$.
\end{lemma}
\noindent {\bf Proof.} First, suppose that $\mathcal{F}^t(k,n)$ has a matching $M_{1}$ of size $t$. Since $n-kt= km+r\ge km$, $[n]\preceqetminus V(M_{1})$
contains $m$ pairwise disjoint $k$-sets, say $e_1,\ldots, e_m$. Let $M_{2}=\{e_i\cup \{x_{i+t}\}\ :\ i\in [m]\}$.
Then $M_{1}\cup M_{2}$ is a matching of size $m+t$ in $\mathcal{H}^t(k,n)$.
Now assume that $\mathcal{H}^t(k,n)$ has a matching $M$ of size $m+t$. Note that each edge in $M$ contains exactly one vertex in
$\{x_1, \ldots, x_{m+t}\}$. Thus, the $t$ edges in $M$ containing one of $\{x_1, \ldots, x_t\}$ form a matching in $\mathcal{F}^t(k,n)$ of size $t$.
$\Box $
For the proof of Theorem \ref{main}, we need additional concepts and notation.
Given two hypergraphs $H_1, H_2$ with $V(H_1)=V(H_2)$, let $c(H_1,H_2)$ be the minimum of $|E(H_1)\backslash E(H')|$
taken over all isomorphic copies $H'$ of $H_2$ with $V(H') = V(H_2)$.
For a real number $\varepsilon > 0$,
we say that $H_2$ is \textit{$\varepsilon$-close} to $H_1$ if $V(H_1) = V(H_2)$ and $c(H_1,H_2)\leq \varepsilon|E(H_1)|$.
The following is obvious.
\noindentndent\textbf{Observation 2:} $\mathcal{F}^t(k,n)$ is $\varepsilon$-close to $\mathcal{F}_t(k,n)$ if, and only if,
$\mathcal{H}^t(k,n)$ is $\varepsilon$-close to $\mathcal{H}_t(k,n)$.
As mentioned in Section 1, our proof of Theorem~\ref{main} will be divided into two parts, according to whether or
not $\mathcal{F}^t(k,n)$ is $\varepsilon$-close to $\mathcal{F}_t(n,k)$.
If $\mathcal{F}^t(k,n)$ is close to $\mathcal{F}_t(n,k)$, we will apply greedy argument to construct a matching of size $t$.
If $\mathcal{F}^t(k,n)$ is not close to $\mathcal{F}_t(n,k)$, then, by Observation 2, $\mathcal{H}^t(k,n)$ is not close to $\mathcal{H}_t(n,k)$, and
we will show that $\mathcal{H}^t(k,n)$ has a spanning subgraph with properties that enable us to find a large matching $M_2$ and to use absorbing matching $M_1$ to enlarge $M_2$ to a near perfect matching.
\preceqection{The extremal case: $\mathcal{F}^t(k,n)$ is $\varepsilon$-close to $\mathcal{F}_t(k,n)$}
In this section, we prove Theorem~\ref{main} for the case when $\mathcal{F}^t(k,n)$
is $\varepsilon$-close to the extremal configuration $\mathcal{F}_t(k,n)$, where, given any real $\zeta$ with $0<\zeta<1$, $\varepsilon$ satisfies
$$2^k \preceqqrt{\varepsilon} < \min\{((k+1)24^kk^{2k})^{-1}, \zeta^{k-1}(6 k^2 2^k (k-1)!)^{-1} \}.$$
Note that $\zeta$ will be determined when we consider the non-extremal case where $\mathcal{F}^t(k,n)$ is not
$\varepsilon$-close to $\mathcal{F}_t(k,n)$.
Let $H$ be a $(k+1)$-graph and $v \in V(H)$. We define the neighborhood $N_H(v)$ of $v$ in $H$ to be the set $ \{ S \in {V(H) \choose k} \ :\ S \cup \{v\} \in E(H) \} $.
Let $H$ be a $(k+1)$-graph with the same vertex set as $\mathcal{F}_t(k,n)$.
Given real number $\alpha$ with $0<\alpha < 1$, a vertex $v$ in $H$ is called
\emph{$\alpha$-good} with respect to $\mathcal{F}_t(k,n)$ if
$$\left|N_{\mathcal{F}_t(k,n)}(v)\preceqetminus N_H(v)\right|\le \alpha n^{k} $$
and, otherwise, $v$ is called
\emph{$\alpha$-bad}.
Clearly, if $H$ is $\varepsilon$-close to $\mathcal{F}_t(k,n)$, then the
number of $\alpha$-bad vertices in $H$ is at most $(k+1)\varepsilon n/\alpha$.
\begin{lemma}\label{good-lem}
Let $\zeta, \alpha $ be real numbers and $n,k,t$ be positive integers such that $0<\zeta <1$, $n\ge 24k^3$, $t<(1-\zeta)n/k$,
and $\alpha < \min\{((k+1)24^kk^{2k})^{-1}, \zeta^{k-1}(6 k^2 2^k (k-1)!)^{-1} \}$.
Let $H$ be a $(1,k)$-partite $(k+1)$-graph with $V(H)=V(\mathcal{F}_t(k,n))$. If every vertex of $H$ is $\alpha$-good with respect to $\mathcal{F}_t(k,n)$, then $H$ has a matching of size $t$.
\end{lemma}
\noindent {\bf Proof.}
Let $X := \{x_1, x_2, ..., x_t\}$, $W := [t]$, and $U := [n] \preceqetminus [t]$, such that $X,[n]$ are the partition classes of $H$.
Let $M$ be a maximum matching such that $|e\cap X| = |e\cap W|=1$ for all $e\in E(H)$. Let $X'=X\preceqetminus V(M)$, $W'=W\preceqetminus V(M)$, and $U'=U\preceqetminus V(M)$.
We claim that $|M|\geq n/12k^2$. For,
otherwise, assume $|M| < n/(12k^2)$.
Consider any vertex $x\in X'$.
Since $x$ is $\alpha$-good, we have
\[
\left| \left(W\widetildemes {U\choose k-1}\right)\preceqetminus N_H(x)\right|\leq \alpha n^k.
\]
Note that, since $t<(1-\zeta)n/k$ and $\alpha< \zeta^{k-1}(6k^22^k(k-1)!)^{-1}$,
\[
\left|W'\widetildemes {U'\choose k-1}\right|\geq (|W|-|M|){n-kt\choose k-1}>\frac{n}{12k^2}{\zeta n\choose k-1}
>\frac{n}{12k^2} \frac{(\zeta n/2)^{k-1}}{(k-1)!}
> \alpha n^k.
\]
Thus there exists $f\in N_{H}(x)\cap \left(W'\widetildemes {U'\choose k-1}\right)$. Now $M'=M\cup \{\{x\}\cup f\}$ is a matching of size $|M|+1$ in $H$, and
$|f\cap X|= |f\cap W|=1$ for all $f\in M'$. Hence, $M'$ contradicts the choice of $M$.
Let $S=\{u_1,\ldots,u_{k+1}\}\precequbseteqbseteq V(H)\preceqetminus V(M)$, where $u_1\in X'$, $u_{k+1}\in W'$ and $u_i\in U'$ for $i\in [k]\preceqetminus \{1\}$.
Let $\{e_1,\ldots,e_{k}\}$ be an arbitrary $k$-subset of $M$, and let
$e_i := \{v_{i,1},v_{i,2},\ldots,v_{i,k+1}\}$ with $v_{i,1} \in X$, $v_{i,k+1}\in W$, and $v_{i,j}\in U$ for $i \in [k]$ and $j \in [k] \preceqetminus \{1\}$.
For $j \in [k+1]$, let $f_j := \{u_{j}, v_{1,j+1},
v_{2,j+2},\ldots,$ $v_{k,j+k}\}$ with addition in the subscripts
modulo $k+1$ (except we write $k+1$ instead of $0$). Note that $f_1, \ldots, f_{k+1}$ are pairwise disjoint.
If $f_j \in E(H)$ for all $j \in [k+1]$ then $M':= (M \cup
\{f_1,\ldots,f_{k+1}\})\preceqetminus \{e_1,\ldots,e_{k}\}$ is a matching in $H$
such that $|M'| = |M| + 1 > |M|$ and $|f\cap X|=|f\cap W|=1$ for all $f\in M'$, contradicting the choice of $M$.
Hence, $f_j\not\in E(H)$ for some $j \in [k+1]$.
Note that there are $\binom{|M|}{k}k!$ choices of $(e_1,\ldots, e_{k}) \precequbseteqbseteq M^k$ and that for any two different such choices the corresponding $f_j'$s are distinct.
Hence,
\begin{eqnarray*}
& & |\{e \in E(\mathcal{F}_t(k,n)) \preceqetminus E(H): |e\cap \{u_i: i\in [k+1]\}|=1\}|\\
&\geq & |M|(|M| - 1) \cdots (|M| - k +1) \\
&> & \left(n/(12k^2) - k \right)^{k} \\
&> & \left(n/(24k^2)\right)^{k} \quad \mbox{ (since $n\ge 24k^3$)}\\
&> & (k+1) \alpha n^{k} \quad \mbox{ (since $\alpha< ((k+1)24^{k}k^{2k}))^{-1}$}.
\end{eqnarray*}
This implies that there exists $i \in [k+1]$ such that
$|N_{\mathcal{F}_t(k,n)}(u_i) \preceqetminus N_{H}(u_i)| > \alpha n^{k}$,
contradicting the fact that all $u_i$ are $\alpha$-good.
$\Box $
We can now prove Theorem~\ref{main} when $\mathcal{F}^t(k,n)$ is $\varepsilon$-close to $\mathcal{F}_t(k,n)$.
\begin{lemma}\label{close-lem}
Let $0 < \zeta, \varepsilon < 1$ be real numbers and $k\ge 3$ and $t\ge 0$ be integers, such that
$t< (1-\zeta)(1-k(k+1)\preceqqrt{\varepsilon}) n/k$, $n\ge 48k^3$,
and $2^k \preceqqrt{\varepsilon} < \min\{((k+1)24^kk^{2k})^{-1}, \zeta^{k-1}(6 k^2 2^k (k-1)!)^{-1} \}$.
Let $(F_1, \ldots, F_t)$ be a family of subsets of ${[n]\choose k}$ such that
$e(F_i)> {n\choose k}-{n-t+1\choose k}$ for $i\in [t]$, and let ${\cal F}^t(k,n)$ denote the corresponding $(1,k)$-partite $(k+1)$-graph.
Suppose $\mathcal{F}^t(k,n)$ is $\varepsilon$-close to $\mathcal{F}_t(k,n)$. Then $\mathcal{F}^t(k,n)$ has a matching of size $t$.
\end{lemma}
\noindent {\bf Proof.} We may assume $n\le 3k^2t$ as otherwise the assertion follows from Theorem~\ref{HLS}.
Let $B$ denote the set of $\preceqqrt{\varepsilon}$-bad vertices in $\mathcal{F}^t(k,n)$. Since $\mathcal{F}^t(k,n)$ is
$\varepsilon$-close to $\mathcal{F}_t(k,n)$, $|B|\leq (k+1)\preceqqrt{\varepsilon}n$.
Let $X, [n]$ be the partition classes of ${\cal F}_t(k,n)$, and let $X := \{x_1, x_2, ..., x_t\}$, $W := [t]$, and $U := [n] \preceqetminus [t]$.
Note that each edge of ${\cal F}_t(k,n)$ intersects $W$.
Let $b:=\max\{|B\cap X|, |B\cap W|\}$; so $b \le (k+1)\preceqqrt{\varepsilon}n$.
We choose $X_1,W_1$ such that $B\cap X \precequbseteqbseteq X_1 $, $B\cap W\precequbseteqbseteq W_1$ and $|X_1|=|W_1|=b$.
Let $\mathcal{F}_1=\mathcal{F}^t(k,n)[X_1\cup W_1\cup U]$.
For every $x\in X_1$, we have
\[
|N_{\mathcal{F}_1}(x)|\geq |N_{\mathcal{F}}(x)|-\left({n\choose k}-{n-(t-b)\choose k}\right)
>{n-(t-b)\choose k}-{n-(t-1)\choose k}.
\]
Since $n-(t-b) > n/2 \ge 3k^2(k+1)\preceqqrt{\varepsilon}n > 3k^2 b$, it follows from
Theorem \ref{HLS} that the family $\{N_{\mathcal{F}_1}(x)\ |\ x\in X_1\}$ admits a rainbow matching.
Thus, by Observation 1, $\mathcal{F}_1$ has a matching $M$ of size $b$. Clearly, $M$ covers $B\cap X$.
Let ${\cal F}_2:={\cal F}^t(k,n)[(X\preceqetminus X_1)\cup ([n]\preceqetminus (V(M)\cup B)]$, and let $a:=|B\preceqetminus V(M)|$.
By the choice of $W_1$ and $X_1$, we have $B\cap (W\preceqetminus W_1)=\emptyset$. Note that ${\cal F}_2$ may be viewed as the $(1,k)$-partite $(k+1)$-graph ${\cal F}_2=\mathcal{F}_{t-b}(k, n-kb-a)$, with partiton classes $X\preceqetminus X_1, [n]\preceqetminus (V(M)\cup B)$ from the family
$(F_{i}[(X\preceqetminus X_1)\cup ([n]\preceqetminus (V(M)\cup B)]: i\in X\preceqetminus X_1)$.
Put $n'=n-kb-a$ and $t'=t-b$. We wish to apply Lemma~\ref{good-lem}.
Note that $n'=n-kb-a \ge n - k|B| \ge n-k(k+1)\preceqqrt{\varepsilon}n \ge n/2 \ge 24k^3$.
Moreover, since $b \le (k+1) \preceqqrt{\varepsilon} n \le n/6k^2 \le t/2$, we have $n'/6k^2 \le n/6k^2 \leq t/2 < t-b=t'$.
Also, $t' \le t < (1-\zeta)(n-k(k+1)\preceqqrt{\varepsilon}n)/k \le (1-\zeta)(n - k|B|)/k \le (1-\zeta)(n-kb-a)/k= (1-\zeta)n'/k$.
For every $x\in V(\mathcal{F}_2)$, since $x$ is $\preceqqrt{\varepsilon}$-good with respect to $\mathcal{F}_t(k,n)$,
\begin{align*}
|N_{\mathcal{F}_{t'}(k, n')}(x)\preceqetminus N_{\mathcal{F}_2}(x)|&\leq |N_{\mathcal{F}_t(k,n)}(x)\preceqetminus N_{\mathcal{F}}(x)|\\
&\leq \preceqqrt{\varepsilon} n^k\\
&<2^k\preceqqrt{\varepsilon}(n-kb-a)^k\quad\mbox{(since $kb+a\leq (k+1)^2\preceqqrt{\varepsilon}n<n/2$)}\\
&=2^k\preceqqrt{\varepsilon} (n')^k.
\end{align*}
Thus every vertex $x$ of $\mathcal{F}_2$ is $2^k\preceqqrt{\varepsilon}$-good with respect to $\mathcal{F}_{t'}(k,n')$. By Lemma \ref{good-lem}, $\mathcal{F}_2$ has a matching $M'$ of size $t-b$.
Hence $M\cup M'$ is a matching in $\mathcal{F}$ of size $t$.
$\Box $
\preceqection{Absorbing Lemma}
The purpose of this section is to prove the existence of a small matching $M$ in $\mathcal{H}^t(k,n)$ such that for any small balanced set $S$,
${\cal H}^t(k,n)[V(M)\cup S]$ has a perfect matching. We need to use Chernoff bounds here and in the next section.
Let $Bi(n,p)$ denote a binomial random variable with parameters $n$ and $p$.
The following well-known concentration inequalities, i.e. Chernoff bounds, can be found in Appendix A in \cite{AS08}, or Theorem 2.8, inequalities (2.9) and (2.11) in \cite{JLR}.
\begin{lemma}[Chernoff inequality for small deviation]\label{chernoff1}
If $X=\precequbseteqm_{i=1}^n X_i$, each random variable $X_i$ has Bernoulli distribution with expectation $p_i$, and $\alpha \le 3/2$, then
$$ \mathbb{P}(|X-\mathbb{E}X| \ge \alpha \mathbb{E}X ) \le 2e^{-\frac{\alpha^2}{3}\mathbb{E}X}.$$
In particular, when $X \preceqim Bi(n,p)$ and $\lambda < \frac{3}{2}np$, then
$$ \mathbb{P}(|X-np| \ge \lambda ) \le e^{-\Omega(\lambda^2/(np))}.$$
\end{lemma}
We can now prove an absorbing lemma for $H={\cal H}^t(k,n)$.
\begin{lemma}\label{Absorb-lem}
Let $k\geq 3$ be an integer, $\zeta > 0$ be a real number an $n\ge n_1(k,\zeta)$ sufficiently large.
Let $H$ be a $(1,k)$-partite $(k+1)$-graph with partition classes $\{x_1, \ldots, x_{\lfloor n/k\rfloor} \}, [n]$ such that
$d_H(x_i)>{n\choose k}-{n-t+1\choose k}$ for $i\in [t]$ and $d_H(x_i)={n\choose k}$ for $i=t+1, \ldots, \lfloor n/k\rfloor$.
Suppose $ n/3k^2 \le t \le (1-\zeta)n/k$.
Then for any $c$ with $0<c<\zeta^{2k} (12k^22^k (k!)^k)^{-2}$, there exists a matching $M$ in $H$
such that $|M|\le 2k c n$ and, for any balanced subset $S\precequbseteqbseteq V(H)$
with $|S|\le (k+1)c^{1.5} n/2$, $H[V(M)\cup S]$
has a perfect matching.
\end{lemma}
\noindent {\bf Proof.} For balanced $R\in \binom{V(H)}{k+1}$ and balanced $Q\in \binom{V(H)}{k(k+1)}$,
we say that $Q$ is \emph{$R$-absorbing}
if $\nu(H[Q\cup R])=k+1$ and $Q$ is the vertex set of a matching in $H$. Let $\mathcal{L}(R)$ denote the collection of all $R$-absorbing sets in $H$.
\textbf{Claim 1.~} For each balanced $(k+1)$-set $R\precequbseteqbseteq V(H)$,
the number of $R$-absorbing sets in $H$ is at least $\zeta^k ({n\choose k})^{k+1} (6k^2 2^kk^2!)^{-1}$.
Let $R=\{x,u_1,\ldots,u_k\}$ be fixed with $x\in X$ and $u_i\in [n]$ for $i\in [k]$. Note that the number of edges in $H$ containing
$x$ and intersecting $\{u_1, \ldots, u_k\}$ is at most $k{n\choose k-2}$, and $d_{\mathcal{H}}(x)>{n\choose k}-{n-t+1\choose k}$. So
the number of edges $\{x,v_1,\ldots, v_k\}$ in $H$ such that
$v_i\in [n]$ for $i\in [k]$ and $\{v_1, \ldots, v_k\}\cap \{u_1,\ldots, u_k\}=\emptyset$ is at least
\begin{align*}
d_{\mathcal{H}}(x)>{n\choose k}-{n-t+1\choose k}-{n\choose k-2}\geq \frac{1}{6k^2}{n\choose k},
\end{align*}
since $3k^2t\geq n\geq kt$.
Fix a choice of an edge $\{x,v_1,\ldots, v_k\}$ in $H$ such that
$v_i\in [n]$ for $i\in [k]$ and $\{v_1, \ldots, v_k\}\cap \{u_1,\ldots, u_k\}=\emptyset$, and
let $W_0 = \{v_1,\ldots, v_k\}$.
For each $j\in [k]$ and each pair $u_j,v_j$, we choose a $k$-set $U_j$ such that $U_j$ is disjoint from
$W_{j-1}\cup R$ and both $U_j\cup \{u_j\}$ and $U_j\cup\{v_j\}$ are edges in $H$, and let $W_{j} :=U_{j}\cup W_{j-1}$.
Then if $W_k$ is defined then $W_k$ is an absorbing $k(k+1)$-set for $R$.
Note that in each step $j\in [k]$ there are $k+1 +jk$ vertices in $W_{j-1} \cup R$. Thus, the number of edges in $H$ containing
$u_j$ (respectively, $v_j$) and at least one other vertex in $W_{j-1} \cup R$ is at most $(k+1 +jk){n\choose k-2} \lfloor n/k \rfloor <(k+1)n{n\choose k-2}$.
Note that by definition of $x_{t+1}, x_{t+2}, ..., x_{\lfloor n/k \rfloor}$, there are at least ${n-2\choose k-1}( \lfloor n/k \rfloor -t) \ge {n-2\choose k-1}\zeta n/k$ sets $U_j$ such that both $U_j\cup \{u_j\}$ and $U_j\cup \{v_j\}$ are edges in $\mathcal{H}$ for large $n$.
Hence, for each $j\in[k]$, there are at least ${n-2\choose k-1}\zeta n/k-(k+1)n{n\choose k-2}\geq \zeta n{n-1\choose k-1}/2k$ such choices for $U_j$ (as $n$ is sufficiently large). Thus, in total we obtain $\frac{1}{6k^2}{n\choose k}(\zeta n{n-1\choose k-1}/2k)^k$ absorbing, ordered $k(k+1)$-sets for $R$, with multiplicity at most $(k^2)!$; so
\[
\mathcal{L}(R)\geq \frac{\frac{1}{6k^2}{n\choose k}(\zeta n{n-1\choose k-1}/2k)^k}{(k^2)!}\geq \frac{\zeta^k {n\choose k}^{k+1}}{6k^2 2^k(k^2)!}.
\]
This completes the proof of Claim 1.
Now, let $c$ be fixed constant with $0<c< \zeta^{2k} (12k^22^k (k!)^k)^{-2}$, and
choose a family ${\cal G}$ of balanced $k(k+1)$-sets by selecting each of the ${ \lfloor n/k \rfloor \choose k}{n\choose k^2}$ balanced sets of size $k(k+1)$ with probability
\[
p := \frac{c n}{{\lfloor n/k \rfloor\choose k}{n\choose k^2}}.
\]
It follows from Lemma~\ref{chernoff1} that, with probability $1-o(1)$, the family ${\cal G}$ satisfies the following properties:
\begin{align}\label{absor-eq1}
|{\cal G}|\leq 2c n
\end{align}
and
\begin{align}\label{absor-eq2}
|\mathcal{L}(R)\cap {\cal G}|\geq p |\mathcal{L}(R)|/2\geq \frac{c \zeta^k n}{12k^2 2^k (k!)^{k}} \geq c^{1.5} n
\end{align}
for all balanced $(k+1)$-sets $R$. Furthermore, we can bound the expected number of intersecting pairs of $k(k+1)$-sets from above by
\begin{align*}
{\lfloor n/k\rfloor \choose k}{n\choose k^2}k(k+1)\left({\lfloor n/k \rfloor-1\choose k-1}{n\choose k^2}+{\lfloor n/k\rfloor \choose k}{n-1\choose k^2-1}\right)p^2\leq c^{1.9} n.
\end{align*}
Thus, using Markov's inequality, we derive that with probability at least $1/2$
\begin{align}\label{absor-eq3}
\mbox{${\cal G}$ contains at most $c^{1.9} n$ intersecting pairs of $k(k+1)$-sets. }
\end{align}
Hence, there exists a family ${\cal G}$ satisfying (\ref{absor-eq1}), (\ref{absor-eq2}) and (\ref{absor-eq3}). Delete one $k(k+1)$-set from each intersecting pair in such a family ${\cal G}$. Further removing all non-absorbing $k(k+1)$-sets, we obtain a subfamily ${\cal G}'$ consisting of pairwise disjoint balanced, absorbing $k(k+1)$-sets, which satisfies
\[
|\mathcal{L}(R)\cap {\cal G}'|\geq \frac{1}{2}c^{1.5} n,
\]
for all balanced $(k+1)$-sets $R$.
Since ${\cal G}'$ consists only of absorbing $k(k+1)$-sets, $H[V ({\cal G}')]$ has a perfect matching $M$,
of size at most $2kcn$ by (\ref{absor-eq2}). For a balanced set $S\precequbseteqbseteq V(H)$ of size $|S|\leq (k+1)c^{1.5} n/2$, $S$ can be partitioned into at most $c^{1.5} n/2$ balanced $(k+1)$-sets. For each balanced $(k+1)$-set $R$, since $|\mathcal{L}(R)\cap {\cal G}'|\geq \frac{1}{2}c^{1.5} n$,
we can successively choose a distinct absorbing $k(k+1)$-set for $R$ in ${\cal G}'$.
Hence, $\mathcal{H}[V(M)\cup S]$ has a perfect matching.
$\Box $
\preceqection{Fractional perfect matchings}
When $\mathcal{F}^t(k,n)$ is not $\varepsilon$-close to $\mathcal{F}_t(k,n)$,
we will use fractional perfect matchings in random
subgraphs of ${\cal H}^t(k,n)$.
Let $H$ be a hypergraph. A {\it fractional} matching in $H$ is a function $h: E(H) \to [0,1]$ such that
$\precequbseteqm_{e \ni x} h(e) \le 1$ for all $x \in V(H)$. Let $\nu_f(H):=\max_{h} \precequbseteqm_{e \in E(H)} h(e)$ which is
the maximum size of fractional matching of $H$. A fractional matching in a $k$-uniform hypergraph with $n$ vertices
is {\it perfect} if its size is $n/k$.
First, we need a concept of dense graphs used in the hypergraph container result of Balogh, Morris, and Samotij \cite{BMS15} and independently Sexton and Thomassen \cite{ST15}.
Let $H$ be a hypergraph, $\lambda>0$ be a real number, and ${\cal A}$ be a family of subsets of $V(H)$. We say that $H$ is \textit{$(\mathcal {A}, \lambda)$-dense} if $e(H[A])\ge \lambda e(H)$ for every $A \in \mathcal{A}$.
\begin{lemma}\label{dense}
Let $n, k, t$ be positive integers and $\varepsilon$ be a constant such that $n \le 3k^2t$,
$0<\varepsilon \ll 1$, and $n\geq 40k^2/\varepsilon$. Let $a_0 = \varepsilon/8k, a_1 = \varepsilon/24k^2, a_2=\varepsilon/8k^2$
and $a_3 < \varepsilon/(2^k \cdot k! \cdot 30k)$.
Let $H$ be a $(1,k)$-partite $(k+1)$-graph with vertex partition classes $X,[n]$ with $|X|=t$.
Suppose $d_H(x)\geq {n\choose k}-{n-t+1 \choose k}- a_3 n^k$ for any $x\in X$. If $H$ is not $\varepsilon$-close to $\mathcal{F}_t(k,n)$, then $H$
is $(\mathcal{A}, a_0)$-dense, where $\mathcal{A}=\{A\precequbseteqbseteq V(H) : |A\cap X|\ge (t/n-a_1) n,\ |A\cap [n]|\ge (1-t/n-a_2) n\}$.
\end{lemma}
\noindent {\bf Proof.} We prove this by way of contradiction. Suppose that there exists $A\precequbseteqbseteq V(H)$ such that $|A\cap X|\ge (t/n-a_1) n$,
$|A\cap [n]|\ge (1-t/n-a_2) n$, and $e(H[A])\le a_0 e(H)$. Without loss of generality, we may choose $A$ such that $|A\cap X|= (t/n-a_1) n$ and
$|A\cap[n]|= (1-t/n-a_2) n$. Let $U\precequbseteqbseteq [n]$ such that $A \cap [n] \precequbseteqbseteq U$ and $|U|=n-t$. Let $A_1=A\cap X$, $A_2=X\preceqetminus A$, $B_1=A \cap [n]$, and $B_2=U\preceqetminus A$.
Let $H_0$ denote the isomorphic copy of $H$ by naming vertices such that $X = \{x_1, ..., x_t\}$ and $U = [n] \preceqetminus [t]$.
We derive a contradiction by showing
that $|E(\mathcal{F}_t(k,n))\preceqetminus E(H_0)|< \varepsilon e(\mathcal{F}_t(k,n))$.
Note that, since $n\le 3k^2t$,
$$e(\mathcal{F}_t(k,n)) = t\left( {n \choose k} - {n-t \choose k}\right)
\geq t\left( {n \choose k} - {n-n/3k^2 \choose k}\right) \ge t {n \choose k}/(3k).$$
Moreover,
$$e(\mathcal{F}_t(k,n)) \ge t {n \choose k}/(3k) = \frac{tn}{3k^2}{ n-1 \choose k-1},$$
and since $n > 2k$,
$$e(\mathcal{F}_t(k,n)) \ge t {n \choose k}/(3k) > \frac{tn^k}{2^k \cdot k! \cdot 3k}.$$
Consider $x \in A$.
Let $E_{H_0}(B_1,x)$ denote the set of edges contained entirely in $B_1 \cup \{x\}$ in $H_0$.
The number of edges in $H_0$ containing $x$ that also exist in $\mathcal{F}_t(k,n)$ is the number of edges in $H_0$ containing $x$ and
intersecting $[t]$. Hence,
\begin{eqnarray*}
& & |\{e : x \in e, e \in E(H_0), e \cap [t] \neq \emptyset\}| \\
&\ge & d_{H_0}(x) - |\{e : x \in e, e \in E(H_0-[t]), e \cap B_2 \neq \emptyset\}| - |E_{H_0}(B_1,x)| \\
&\ge & \left({n\choose k}-{n-t+1 \choose k}-a_3 n^k\right) - a_2 n {n-t \choose k-1} - |E_{H_0}(B_1,x)|.
\end{eqnarray*}
Therefore, we have
\begin{eqnarray*}
& & |E(\mathcal{F}_t(k,n))\preceqetminus E(H_0)| \\
&= & \precequbseteqm_{x \in A_1} |\{e : x \in e, e \in E(\mathcal{F}_t(k,n))\preceqetminus E(H_0)\}| + \precequbseteqm_{x \in A_2} |\{e : x \in e, e \in E(\mathcal{F}_t(k,n))\preceqetminus E(H_0)\}| \\
&\le & \precequbseteqm_{x \in A_1} \left( {n \choose k} - {n-t \choose k} - |\{e : x \in e, e \in E(H_0), e \cap [t] \neq \emptyset\}|\right) + |A_2| \left( {n \choose k} - {n-t \choose k}\right) \\
&\le & \precequbseteqm_{x \in A_1} \left[ \left({n \choose k} - {n-t \choose k}\right) - \left({n\choose k} - {n-t+1 \choose k}-a_3 n^k - a_2 n{n-t \choose k-1} - E_{H_0}(B_1,x) \right)\right] \\
& & + a_1 n \cdot e(\mathcal{F}_t(k,n))/t \\
&\le &\precequbseteqm_{x \in A_1} \left[{n-t+1 \choose k} - {n-t \choose k} + a_3 n^k + a_2 n {n-t \choose k-1} + E_{H_0}(B_1,x)\right] + (3k^2 a_1) \cdot e(\mathcal{F}_t(k,n)) \\
&= &t {n-t \choose k-1} + a_3 t n^k + a_2 t n {n-t \choose k-1} + \precequbseteqm_{x \in A_1} E_{H_0}(B_1,x) + (3k^2 a_1) \cdot e(\mathcal{F}_t(k,n)) \\
&\le &(3k^2/n) \cdot e(\mathcal{F}_t(k,n)) + (2^k \cdot k! \cdot 3k a_3) \cdot e(\mathcal{F}_t(k,n)) + (3k^2 a_2) \cdot e(\mathcal{F}_t(k,n)) \\
& & + e(H_0[A]) + (3k^2 a_1) \cdot e(\mathcal{F}_t(k,n)) \\
&< & a_0 e(H_0) + \left(3k^2/n + 2^k \cdot k! \cdot 3k a_3 + 3k^2 a_2 + 3k^2 a_1\right) \cdot e(\mathcal{F}_t(k,n)) \\
&\le & a_0 t {n \choose k} +\left(3k^2/n + 2^k \cdot k! \cdot 3k a_3 + 3k^2 a_2 + 3k^2 a_1\right) \cdot e(\mathcal{F}_t(k,n)) \\
&\le & \left(3k a_0 + 3k^2/n + 2^k \cdot k! \cdot 3k a_3 + 3k^2 a_2 + 3k^2 a_1\right) \cdot e(\mathcal{F}_t(k,n)) \\
&\leq & \varepsilon \cdot e(\mathcal{F}_t(k,n)),
\end{eqnarray*}
a contradiction since $H$ is not $\varepsilon$-close to $\mathcal{F}_t(k,n)$.
$\Box $
We also need a result of Lu, Yu, and Yuan \cite{LYY1}, which is a stability result on matchings in ``stable'' graphs.
For subsets $e=\{u_1, ...,u_k\}, f=\{v_1,...,v_k\} \precequbseteqbseteq [n]$ with $u_i < u_{i+1}$ and $v_i < v_{i+1}$ for $i \in [k-1]$, we write $e \le f$ if $u_i\leq v_i$ for all $i \in [k]$.
A hypergraph $H$ with $V(H) = [n]$ and $E(H) \precequbseteqbseteq {[n] \choose k}$ is said to be \textit{stable} if for $e, f \in {[n] \choose k}$ with $e \le f$, $e \in E(H)$ implies $f \in E(H)$. The following is Lemma 4.2 in \cite{LYY1}.
\begin{lemma}[Lu, Yu and Yuan]\label{stafrankl}
Let $k$ be a positive integer and let $b$ and $\eta$ be constants, such that $0<b<1/(2k)$ and $0<\eta\le (1+18(k-1)!/b)^{-2}$.
Let $n,m$ be positive integers such that $n$ is sufficiently large and
$bn\leq m\leq n/(2k)$.
Let $H$ be a $k$-graph with vertex set $[n]$.
Suppose $H$ is stable and $e(H)>{n\choose k}-{n-m\choose k}-\eta n^{k}$.
If $H$ is not $\preceqqrt{\eta}$-close to $H_k(m,n)$, then $\nu(H) > m$.
\end{lemma}
We now state and prove the main result of this section.
\begin{lemma}\label{Fr-PM}
Let $n, k, t$ be positive integers such that $n\equiv 0\pmod k$ and let $c, \varepsilon$ be constants such that $0 < c\ll \varepsilon \ll 1$.
Suppose that $n$ is sufficiently large and $n/(3k^2) \le t \le n/(2k)$. Let $H$ be a balanced $(1,k)$-partite $(k+1)$-graph with partition classes $X,[n]$, and let $X=\{x_1,\ldots,x_{n/k}\}$ and $X'=\{x_1,\ldots,x_t\}$.
Suppose $d_{H}(x)\geq {n\choose k}-{n-t+1 \choose k}-\preceqqrt{c} n^k$ for $x\in X'$, and $d_H(x)={n\choose k}$ for $x\in X\preceqetminus X'$, and assume that
for any independent set $S$ in $H$, $|S\cap X|\leq (t/n-\varepsilon)n$ or $|S\cap [n]|\leq (1-t/n-\varepsilon)n$.
Then $H$ has a fractional perfect matching.
\end{lemma}
\noindent {\bf Proof.} We use linear programming duality between vertex cover and matchings.
Let $\omega:V(H)\rightarrow [0,1]$ such that $\precequbseteqm_{v \in e} \omega(v) \ge 1$ for all $e \in E(H)$, and, subject to this,
$\omega(H):=\precequbseteqm_{v\in V(H)}\omega(v)$ is minimum. (Thus, $\omega$ is a minimum fractional vertex cover of $H$.)
Without loss of generality, we may assume that $\omega(x_1)\leq \cdots\leq \omega(x_{n/k})$ and $\omega(1)\leq \omega(2)\cdots\leq \omega(n)$.
Let $CL(H)$ be a graph with vertex set $V(H)$ and edge set
\[
E(CL(H))=\left\{e\in {V(H)\choose k+1}\ :\ |e\cap Q|=1\mbox{ and } \precequbseteqm_{x\in e}\omega(x)\geq 1\right\}.
\]
Note that $H$ is a subgraph of $CL(H)$ and $\omega$ is also a vertex cover of $CL(H)$. Thus $\omega$ is also a minimum vertex cover of $CL(H)$.
By Linear Programming Duality Theory, we have $\nu_f(H)=w(H)=w(CL(H))=\nu_f(CL(H))$.
Thus it suffices to show that $CL(H)$ has a fractional perfect matching.
Indeed, we will prove that $\nu(CL(H))=n/k$, i.e., $CL(H)$ has a perfect matching.
By the definition of $E(CL(H))$, we may assume that
\begin{align}\label{STA-Neigh}
N_{CL(H)}(x_1)\precequbseteqbseteq N_{CL(H)}(x_2)\precequbseteqbseteq\cdots\precequbseteqbseteq N_{CL(H)}(x_{n/k}).
\end{align}
Hence, $N_{H}(x_i)={[n]\choose k}$ for $i\in [n/k] \preceqetminus [t]$.
It is also easy to see that $N_{H}(x_i)$ is stable for all $i\in [n/k]$.
Let $\eta$ be a constant satisfying $c^{1/4}\ll \eta\le \min \{ (1+54k^2(k-1)!)^{-1}, \varepsilon(k(k+1))^{-2}\}$. We distinguish two cases.
\preceqmallskip
\textbf{Case 1.~} $N_{H}(x_1)$ is not $\eta$-close to $H_k(t,n)$.
We observe that $e(N_H(x_1)) = d_H(x_1) \geq {n\choose k}-{n-t+1 \choose k}-\preceqqrt{c} n^k
= {n\choose k}-{n-t \choose k} - {n-t \choose k-1} -\preceqqrt{c} n^k$.
By Lemma \ref{stafrankl} with $m=t$ and $b=1/(3k^2)$, $N_{H}(x_1)$ has a matching $M_1$ of size $t$, and
let $M_1=\{e_1,\ldots,e_t\}$.
By (\ref{STA-Neigh}), $M_1\precequbseteqbseteq N_{CL(H)}(x_i)$ for $i\in [n/k]$. Thus
$M_2=\{e_i\cup \{x_{i}\}\ : \ i\in [t]\}$ is a matching in $CL(H)$.
Partition $[n]\preceqetminus V(M_2)$ into $n/k-t$ pairwise disjoint $k$-sets, say $f_1,\ldots,f_{n/k-t}$.
Then by (\ref{STA-Neigh}),
$M_2'=\{f_i\cup \{x_{i+t}\}\ : \ i\in [n/k-t]\}$
is a matching in $CL(H)\preceqetminus V(M_2)$. Hence $M_2\cup M_2'$ is a perfect matching in $CL(\mathcal{H})$.
\preceqmallskip
\textbf{Case 2.~} $N_{H}(x_1)$ is $\eta$-close to $H_k(t,n)$. (Thus, $N_{CL(H)}(x_1)$ is $\eta$-close to $H_k(t,n)$.)
Let $B$ denote the set of $\preceqqrt{\eta}$-bad vertices of $N_{CL(H)}(x_1)$ and let $b=|B|$.
Since $N_{CL(H)}(x_1)$ is $\eta$-close to $H_k(t,n)$, we have $b\leq (k+1)\preceqqrt{\eta}n$. Consider
$H'=CL(H)- (\{x_{t+1},\ldots,x_{n/k}\}\cup \{n-t+1,\ldots,n\})$.
Note that $kb \le k(k+1)\preceqqrt{\eta} n < \varepsilon n$; so $b<\varepsilon n/k$.
Since for any independent set $S$ in $H'$, $|S\cap X|\leq (t/n-\varepsilon)n$ or $|S\cap [n]|\leq (1-t/n-\varepsilon)n$,
we can greedily find pairwise disjoint edges $f_1,\ldots,f_{b}$ in $H'$ such that $x_{t-i+1}\in f_i$ in $H'$. Write $M_{21}=\{f_1,\ldots,f_{b}\}$.
Note that for each vertex $v\in \left([n]\preceqetminus V(M_{21})\right)\preceqetminus B$, we have
\begin{align*}
&|N_{H_k(t-b,n')}(v)\preceqetminus N_{CL(H)-(V(M_{21})\cup B)}(\{v,x_1\})|\\
\leq &|N_{H_k(t,n)}(v)\preceqetminus N_{CL(H)}(\{v,x_1\})|\\
<&\preceqqrt{\eta}n^{k-1}\\
<& \eta^{1/3}(n')^{k-1},
\end{align*}
where $n'=|[n]\preceqetminus V(M_{21})\preceqetminus B|$.
Thus, all vertices of $N_{CL(H)}(x_1)- (V(M_{21})\cup B)$ in $[n]\preceqetminus V(M_{21})$ are $\eta^{1/3}$-good with respect to $H_k(t-b,n')$. Hence by Lemma \ref{good-lem}, $N_{CL(H)}(x_{1})- (V(M_{21})\cup B)$ has a matching $M_{22}'$ of size $t-b$. Write $M_{22}'=\{e_1,\ldots, e_{t-b}\}$. By (\ref{STA-Neigh}),
$M_{22}=\{e_i\cup \{x_{i}\}\ :\ i \in [t-b]\}$ is a matching in $H'$.
Thus, $M_{22}\cup M_{21}$ is a matching of size $t$ in $H'$.
Partition $[n]\preceqetminus V(M_{21}\cup M_{22})$ into $n/k-t$ disjoint $k$-sets, say $g_1,\ldots, g_{n/k-t}$. Let
$M_{23}=\{g_i\cup \{x_{i+t}\}\ :\ i\in [n/k]\preceqetminus [t]\}$. Then $M_{21}\cup M_{22}\cup M_{23}$ is a perfect matching in $CL(H)$.
This competes the proof.
$\Box $
\preceqection{Random rounding}
In this section, we will complete the proof of Theorem~\ref{main}. For convenience, in this section we will not round certain numbers to integers this does not
affect calculations.
First, we need another result of
Lu, Yu, and Yuan \cite{LYY2} on the independence number of
a subgraph of a $k$-graph induced by a random subset of vertices, which is a generalization of Lemma 4.3 in \cite{LYY2} where it was shown for
$(1,3)$-partite graphs. The same proof for Lemma 4.3 in \cite{LYY2} works here as well by using Lemma~\ref{dense} in the place of Lemma 4.1 in \cite{LYY2}.
\begin{lemma}[Lu, Yu, and Yuan]\label{indep}
Let $l, \varepsilon', \alpha_1,\alpha_2$ be positive reals, let $\alpha>0$ with $\alpha \ll \min\{\alpha_1,\alpha_2\}$,
let $k,n$ be positive integers, and let
$H$ be a $(1,k)$-partite $(k+1)$-graph with partition classes $Q,P$ such that $k|Q|=|P|=n$, $e(H)\ge ln^{k+1}$, and $e(H[F])\ge
\varepsilon' e(H)$ for all $F\precequbseteqbseteq
V(H)$ with $|F\cap P|\ge \alpha_1 n$ and $|F\cap Q|\ge \alpha_2 n$.
Let $R\precequbseteqbseteq V(H)$ be obtained by taking each vertex of
$H$ uniformly at random with probability $n^{-0.9}$.
Then, with probability at least $1-n^{O(1)}e^{-\Omega (n^{0.1})}$, every independent set $J$ in $H[R]$
satisfies $|J\cap P|\le (\alpha_1 +\alpha+o(1))n^{0.1}$ or $|J\cap Q|\le (\alpha_2 +\alpha+o(1))n^{0.1}$.
\end{lemma}
Next, we also need the Janson's inequality to provide an exponential upper bound for the lower tail of a sum of dependent zero-one random variable. (See Theorem 8.7.2 in \cite{AS08})
\begin{lemma}[Janson] \label{janson}
Let $\Gamma$ be a finite set and $p_i \in [0,1]$ be a real for $i \in \Gamma$.
Let $\Gamma_{p}$ be a random subset of $\Gamma$ such that the elements are chosen independently with $\mathbb{P}[i \in \Gamma_p] = p_i$ for $i \in \Gamma$.
Let $S$ be a family of subsets of $\Gamma$.
For every $A \in S$, let $I_A = 1$ if $A \precequbseteqbseteq \Gamma_p$ and $0$ otherwise.
Define $X = \precequbseteqm_{A \in S} I_A$,
$\lambda = \mathbb{E}[X]$,
$\Delta = \frac{1}{2}\precequbseteqm_{A \neq B} \precequbseteqm_{A \cap B \neq \emptyset} \mathbb{E}[I_A I_B]$
and $\bar{\Delta} = \lambda + 2\Delta$.
Then, for $0 \le t \le \lambda$, we have
$$\mathbb{P}[X \le \lambda - t] \le \exp(-\frac{t^2}{2\bar{\Delta}}). $$
\end{lemma}
Now, we use Chernoff bound and Janson's inequality to prove a result on several properties of certain random subgraphs.
\begin{lemma}\label{lem1-5}
Let $n, k$ be integers such that $n\ge k\geq 3$,
let $H$ be a $(1,k)$-partite $(k+1)$-graph with partition classes $A,B$
and $k|A| = |B| = n$,
let $A_1,A_2$ be a partition of $A$ with $|A_1|\ge n/(3k^2)$ and $|A_2|\ge n/(3k^2)$, and
let $A_3\precequbseteqbseteq A$ and $A_4\precequbseteqbseteq B$ with $|A_i|=n^{0.99}$ for $i=3,4$.
Take $n^{1.1}$ independent copies of $R$ and denote them by $R^i$, $1\le i\le n^{1.1}$, where $R$ is chosen from $V(H)$ by taking each vertex uniformly at random with probability $n^{-0.9}$ and then deleting $O(n^{0.06})$ vertices uniformly at random so that $|R|\in (k+1) \mathbb{Z}$ and $k|R\cap A|=|R\cap B| $.
For each $S\precequbseteqbseteq V(H)$, let $Y_S:=|\{i: \ S\precequbseteqbseteq R^i\}|$.
Then, with probability at least $1-o(1)$, all of the following statements hold:
\begin{itemize}
\item [$(i)$] $Y_{\{v\}}=(1 \pm n^{-0.01}) n^{0.2}$ for all $v\in V(H)$.
\item [$(ii)$] $Y_{\{u,v\}}\le 2$ for all $\{u, v\} \precequbseteqbseteq V(H)$.
\item [$(iii)$] $Y_e\le 1$ for all $e \in E(H)$.
\item [$(iv)$] For all $i= 1, \dots ,n ^{1.1}$, we have
$|R_i\cap A| =(1/k\pm o(n^{-0.04}))n^{0.1}$ and $|R_i\cap B| =(1\pm o(n^{-0.04}))n^{0.1}$,
\item [$(v)$] Suppose $n/k^3 \le m\le n/k$ and $\rho$ is a constant with $0<\rho <1$ such that
$d_{H}(v)\ge {n\choose
k}-{n-m\choose k}-\rho n^{k}$ for all $v\in A$. Then
for $1 \le i \le n^{1.1}$ and $v\in R_i \cap A$, we have
$$d_{R_i}(v)> {|R_i \cap B|\choose k}-{|R_i \cap B|-mn^{-0.9} \choose k}-3\rho |R_i \cap B|^{k},$$
\item[$(vi)$] $|R_i\cap A_j|= |A_j|n^{-0.9}\pm n^{0.06}$ for $1 \le i \le n^{1.1}$ and $j\in \{1,2,3,4\}$.
\end{itemize}
\end{lemma}
\noindent {\bf Proof.}
For $1 \le i \le n^{1.1}$ and $j\in \{1,2,3,4\}$, $\mathbb{E}[|R_i \cap A|] = n^{0.1}/k$, $\mathbb{E}[|R_i \cap B|] = n^{0.1}$ and $\mathbb{E}[|R_i \cap A_j|] = n^{-0.9}|A_j|$. Recall the assumptions $|A_1|\ge n/(3k^2)$, $|A_2|\ge n/(3k^2)$, and $|A_3|=|A_4|=n^{0.99}$.
By Lemma~\ref{chernoff1}, we have
\begin{itemize}
\item [] $\mathbb{P}\left(\left||R_i \cap A| - n^{0.1}/k\right| \ge n^{0.06} \right) \le e^{-\Omega(n^{0.02})}$,
\item [] $ \mathbb{P}\left(\left||R_i \cap B| - n^{0.1}\right| \ge n^{0.06} \right) \le e^{-\Omega(n^{0.02})}$, and
\item [] $\mathbb{P}\left(\left||R_i \cap A_j| - |A_j|n^{-0.9}\right| \ge n^{0.06} \right) \le e^{-\Omega(n^{0.02})}$.
\end{itemize}
Hence, with probability at least $1-O(n^{1.1})e^{-\Omega(n^{0.02})}$, $(iv)$ and $(vi)$ hold.
For every $v\in V(H)$, $\mathbb{E}[Y_{\{v\}}]=n^{1.1} \cdot n^{-0.9}= n^{0.2}$. By Lemma~\ref{chernoff1},
$$ \mathbb{P}\left(\left||Y_{\{v\}}| - n^{0.2} \right| \ge n^{0.19} \right) \le e^{-\Omega(n^{0.18})}$$
Hence, with probability at least $1-O(n)e^{-\Omega(n^{0.18})}$, $(i)$ holds.
Let $Z_{p,q} = \left|S \in {V(H) \choose p} : Y_S \ge q \right|$. Then
$$\mathbb{E}\left[Z_{p,q}\right] \le {n \choose p} {n^{1.1} \choose q} (n^{-0.9})^{pq} \le n^{p + 1.1q - 0.9pq}. $$
So $\mathbb{E}[Z_{2,3}] \le n^{-0.1}$ and $\mathbb{E}Z_{k,2} \le n^{2.2 - 0.8k} \le n^{-0.2}$ for $k \ge 3$.
Hence by Markov's inequality, $(ii)$ and $(iii)$ hold with probability at least $1-o(1)$.
Finally we show $(v)$.
For all $v\in A$, since
$d_{H}(v)\ge {n\choose
k}-{n-m\choose k}-\rho n^{k}$, we see that, for $1 \le i \le n^{1.1}$ and $v\in R_i \cap A$,
$$\mathbb{E}\left[d_{R_i}(v)\right]> {n\choose k} n^{-0.9k}-{n-m\choose k}n^{-0.9k}-\rho n^{k}n^{-0.9k}
> {n^{0.1} \choose k} -{n^{0.1} - mn^{-0.9} \choose k}-\rho n^{0.1k}.$$
By $(iv)$, with probability at least $1-O(n^{1.1})e^{-\Omega(n^{0.02})}$, for all $i= 1, \dots ,n ^{1.1}$, we have
$|R_i\cap B| =(1+o(n^{-0.04}))n^{0.1}$.
Thus for all $v\in A\cap R_i$,
$$\mathbb{E}\left[d_{R_i}(v)\right] > {|R_i \cap B|\choose k}-{|R_i \cap B|-mn^{-0.9} \choose k}-2\rho |R_i \cap B|^{k}.$$
We wish to apply Lemma~\ref{janson} with $\Gamma = B$, $\Gamma_p = R_i$ and
$S$ be a family of all $k$-set of $B$.
We define
$$\Delta = \frac{1}{2} \precequbseteqm_{b_1, b_2 \precequbseteqbseteq B, b_1 \ne b_2, b_1 \cap b_2 \ne \emptyset} \mathbb{E}[I_{b_1}I_{b_2}] \le \frac{1}{2} |R_i \cap B|^{2k-1} $$
By Lemma~\ref{janson},
\begin{align*}
&\mathbb{P}\left( d_{R_i}(v) \leq {|R_i \cap B|\choose k}-{|R_i \cap B|-mn^{-0.9} \choose k}-3\rho |R_i \cap B|^{k} \right) \\
\leq & \mathbb{P} \left( d_{R_i}(v) \leq \mathbb{E}[d_{R_i}(v)] - \rho |R_i \cap B|^{k} \right) \\
\leq & \exp(- \frac{\rho^2 |R_i \cap B|^{2k}}{2{|R_i \cap B|\choose k} + 2|R_i \cap B|^{2k-1}}) \\
\leq & \exp(-\Omega(n^{0.1})).
\end{align*}
Therefore, with probability at least $1-O(n^{1.1})e^{-\Omega(n^{0.1})}$, $(v)$ holds.
By applying union bound, $(i)$ -- $(v)$ all hold with probability $1-o(1)$.
$\Box $
Now we use random subgraphs and fractional matchings to perform a second round of randomization to find a sparse subgraph
in a hypergraph that is not $\varepsilon$-close to $\mathcal{H}_t(k,n)$.
\begin{lemma}\label{Span-subgraph}
Let $k\ge 3$ be an integer, $0 < \rho \ll \varepsilon \ll 1$ be reals, and $n\in k\mathbb{Z}$ be sufficiently large.
Suppose $n/(3k^2) \le t \le n/(2k)$.
Let $H$ is a $(1,k)$-partite $(k+1)$-graph with partition classes $A,B$ such that $k|A|=|B|=n$.
Let $A_1$ and $A_2$ be a partition of $A$ such that $|A_1|=t$ and $|A_2|=n/k-t$. Suppose that $d_{H}(x)>{n\choose k}-{n-t+1\choose k}-\rho n^k$ for all $x\in A_1$ and $d_{H}(x)={n\choose k}$ for all $x\in A_2$. If $H$ is not $\varepsilon$-close to $\mathcal{H}_t(k,n)$, then there exists a spanning subgraph $H'$ of $H$
such that the following conditions hold:
\begin{itemize}
\item[$(1)$] For all $x\in V(H')$, with at most $n^{0.99}$ exceptions,
$d_{H'}(x)=(1\pm n^{-0.01})n^{0.2}$;
\item[$(2)$] For all $x\in V(H')$, $d_{H'}(x)< 2 n^{0.2}$;
\item[$(3)$] For any two distinct $x,y\in V(H')$, $d_{H'}(\{x,y\})< n^{0.19}$.
\end{itemize}
\end{lemma}
\noindent {\bf Proof.}
Let $A_3\precequbseteqbseteq A$ and $A_4\precequbseteqbseteq B$ with $|A_i|=n^{0.99}$ for $i=3,4$.
Let $R_1,\ldots,R_{n^{1.1}}$ be defined as in Lemma \ref{lem1-5}. By Lemma \ref{lem1-5} $(iv)$, we have, for all $i= 1, \dots ,n ^{1.1}$,
$$|R_i\cap A| =(1/k+o(n^{-0.04}))n^{0.1} \mbox{ and } |R_i\cap B| =(1+o(n^{-0.04}))n^{0.1}.$$
By Lemma \ref{lem1-5} $(vi)$, we have
$$|R_i\cap A_1| =(t/n+o(n^{-0.04}))n^{0.1} \mbox{ and } |R_i\cap A_2| =(1/k-t/n+o(n^{-0.04}))n^{0.1}.$$
By Lemma \ref{lem1-5} $(v)$, we have for $1 \le i \le n^{1.1}$ and $x\in A \cap R^i$,
$$d_{R_i}(x)> {|R_i \cap B|\choose k}-{|R_i \cap B|-(t-1)n^{-0.9} \choose k}-3\rho |R_i \cap B|^{k};$$
By $(iv)$ and $(vi)$ of Lemma \ref{lem1-5},
we may choose $I_i\precequbseteqbseteq R_i\cap (A_3\cup A_4)$ such that $R_i\preceqetminus I_i$ is balanced and $|R_i'|=(1 - o(1))|R_i|$, where $R_i'=R_i\preceqetminus I_i$ for $i=1, \ldots, n^{1.1}.$
Let $H_1=H[A_1 \cup B]$.
Since $H$ is not $\varepsilon$-close to $\mathcal{H}_t(k,n)$, $H_1$ is not $\varepsilon$-close to $\mathcal{F}_t(k,n)$ by Observation 2 in Section 2.
Let $a_0 = \varepsilon/(8k), a_1 = \varepsilon/(24k^2), a_2=\varepsilon/(8k^2)$, and $a_3 < \varepsilon(2^k \cdot k! \cdot 30k)^{-1}$.
By applying Lemma \ref{dense} to $H_1, a_0, a_1, a_2, a_3$, we see that $H_1$ is $(\mathcal{F},a_0)$-dense, where
$$\mathcal{F}=\{U\precequbseteqbseteq V(H) : |U\cap A_1|\ge (t/n-a_1 ) n,\ |U\cap B|\ge (1-t/n-a_2 ) n\}.$$
Now we apply Lemma \ref{indep} to $H_1$ with $l = (3k^3k!)^{-1}$, $\alpha_1 = t/n - a_1$, $\alpha_2 = 1-t/n-a_2$, and $\varepsilon' =a_0$.
Therefore, with probability at least $1-n^{O(1)}e^{-\Omega (n^{0.1})}$, for any independent set $S$ of $R_i'$, $|S\cap R_i'\cap A_1|\leq (t/n - a_1+o(1))n^{0.1}$ or $|S\cap R_i'\cap B|\leq (1-t/n-a_2+o(1))n^{0.1}$.
By definition, for $x \in R_i' \cap A_2$, $d_{R_i'}(x) = {|R_i'| \choose k}$.
By applying Lemma \ref{Fr-PM} to each $H[R_i']$, we see that each $H[R_i']$ contains a fractional perfect matching $\omega_i$.
Let $H^*=\cup_{i=1}^{n^{1.1}}R_i'$. We select a generalized binomial subgraph $H'$ of $H^*$ by letting $V(H')=V(H)$ and
independently choosing edge $e$ from $E(H^*)$,
with probability $\omega_{i_e}(e)$ if $e\precequbseteqbseteq R_{i_e}'$. (By Lemma \ref{lem1-5} $(iii)$, for each $e\in E(H^*)$, $i_e$ is uniquely defined.)
Note that since $w_i$ is a fractional perfect matching of $H[R_i']$ for $1 \le i \le n^{1.1}$, $\precequbseteqm_{e \ni v} w_i(e) \le 1$ for $v \in R_i'.$
By Lemma \ref{lem1-5} $(i)$ and by Lemma~\ref{chernoff1}, $d_{H'}(v)=(1 \pm n^{-0.01}) n^{0.2}$ for any vertex $v\in V(H)-(\cup_{i=1}^{n^{1.1}} I_i) \precequbseteqbseteq V(H)-(A_3 \cup A_4)$
and $d_{H'}(v) \le (1 \pm n^{-0.01}) n^{0.2} < 2n^{0.2}$ for vertex $v \in \cup_{i=1}^{n^{1.1}} I_i$.
By Lemma \ref{lem1-5} (ii) $d_{H'}(\{x,y\})\leq 2 < n^{0.19}$ for any $\{x,y\}\in {V(H) \choose 2}$.
Therefore, $H'$ is the desired hypergraph.
$\Box $
To prove Theorem~\ref{main}, we also need the following result which was attributed to Pippenger \cite{PS}
(see Theorem 4.7.1 in \cite{AS08}). An {\it edge cover} in a hypergraph $H$
is a set of edges whose union is $V(H)$.
\begin{theorem}[Pippenger]\label{nibble}
For every integer $k\ge 2$ and real $r\ge 1$ and $a>0$, there are $\gamma=\gamma(k,r,a)>0$ and $d_0=d_0(k,r,a)$ such that for every $n$ and $D\ge d_0$ the following holds: Every $k$-uniform hypergraph $H=(V,E)$ on a set $V$ of $n$ vertices in which all vertices have positive degrees and which satisfies the following conditions:
\begin{itemize}
\preceqetlength{\itemsep}{0pt}
\preceqetlength{\parsep}{0pt}
\preceqetlength{\parskip}{0pt}
\item[$(1)$] For all vertices $x\in V$ but at most $\gamma n$ of them, $d_H(x)=(1\pm \gamma)D$;
\item[$(2)$] For all $x\in V$, $d_H(x)<r D$;
\item[$(3)$] For any two distinct $x,y\in V$, $d_H(\{x,y\})<\gamma D$;
\end{itemize}
contains an edge cover of at most $(1+a)(n/k)$ edges.
\end{theorem}
\noindentndent\textbf{Proof of Theorem \ref{main}.}
By Theorem \ref{HLS}, we may assume that $2kt < n\leq 3k^2 t$.
Let $0 < \varepsilon \ll 1$ be sufficiently small and $n$ be sufficiently large.
By Observation 1, it suffices to show $\mathcal{F}^t(k,n)$ has a matching of size $t$.
Applying Lemma \ref{close-lem} to $\mathcal{F}^t(k,n)$ with $\zeta = 1/3$,
we may assume that $\mathcal{F}^t(k,n)$ is not $\varepsilon$-close to $\mathcal{F}_t(k,n)$. That is,
$\mathcal{H}^t(k,n)$ is not $\varepsilon$-close to $\mathcal{H}_t(k,n)$ by Observation 2.
Now we apply Lemma \ref{Absorb-lem} to $\mathcal{H}^t(k,n)$ with $\zeta = 1/2$.
Thus there exists some constant $0 < c \ll \varepsilon$ such that $n-kcn \ge 2kt$ and $\mathcal{H}^t(k,n)$ contains an absorbing matching $M_1$ with $m_1:=|M_1|\leq c n$ and for any balanced subset $S$ of vertices with $|S|\leq (k+1)c^{1.5} n$, $\mathcal{H}^t(k,n)[V(M_1)\cup S]$ has a perfect matching. Let $H:=\mathcal{H}^t(k,n)-V(M_1)$ and $n' := n-km_1$.
Next, we see that $H$ is not $(\varepsilon/2)$-close to $\mathcal{H}_{t}(k,n-km_1)$. For, suppose otherwise.
Then
\begin{align*}
&|E(\mathcal{H}_t(k,n)) \preceqetminus E(\mathcal{H}^t(k,n))| \\
&\le |E(\mathcal{H}_{t}(k,n-km_1)) - E(H)| + |e \in E(\mathcal{H}_t(k,n)) : e \cap V(M_1) \neq \emptyset| \\
&\leq (\varepsilon/2) |E(\mathcal{H}_{t}(k,n-km_1))| + (k+1)cn \cdot n^k \\
&\le \varepsilon |E(\mathcal{H}_t(k,n))|.
\end{align*}
This is a contradiction as $\mathcal{H}^t(k,n)$ is not $\varepsilon$-close to $\mathcal{H}_t(k,n)$.
Since $n' \ge n - kcn \ge 2kt$, by Lemma \ref{Span-subgraph} $H$ has a spanning subgraph $H'$ such that
\begin{itemize}
\item[(1)] For all vertices $x\in V(H')$ but at most $n'^{0.99}$ of them,
$d_{H'}(x)=(1\pm n'^{-0.01})n'^{0.2}$;
\item[(2)] For all $x\in V(H')$, $d_{H'}(x)< 2 n'^{0.2}$;
\item[(3)] For any two distinct $x,y\in V(H')$, $d_{H'}(\{x,y\})< n'^{0.19}$.
\end{itemize}
Hence by applying Lemma \ref{nibble} to $H'$ with $0 < a \ll c^{1.5}$,
$H'$ contains an edge cover of at most $(1+a)((n'/k+n')/(k+1))$ edges.
Thus, at most $a(n'/k+n')$ vertices are each covered by more than one edge
in the cover. Hence,
after removing at most $a(n'/k+n')$ edges from the edge cover, we obtain a matching $M_2$ covering all but at most $(k+1)a(n'/k+n') \le 3kan' \le 3kan$ vertices.
Now we may choose a balanced subset $S$ of $V(H)\preceqetminus V(M_2)$ such that $|V(H)\preceqetminus (V(M_2)\cup S)|\leq k$.
Since $|S| \le 3kan \le (k+1)c^{1.5}n$, $\mathcal{H}^t(k,n)[V(M_1)\cup S]$ has a perfect matching, say $M_3$. Thus,
$M_2\cup M_3$ is matching of $H^t(k,n)$ covering all but at most $k$ vertices, and, hence, has size $\lfloor n/k\rfloor$.
Therefore, by Lemma \ref{Rain-PM}, $\mathcal{F}^t(k,n)$ has a matching of size $t$.
$\Box $
\end{document}
|
\begin{document}
\begin{frontmatter}
\title{Handling Covariates in the Design of Clinical Trials}
\runtitle{Covariates in Clinical Trials}
\begin{aug}
\author[a]{\fnms{William F.} \snm{Rosenberger}\corref{}\ead[label=e1]{[email protected]}} \and
\author[b]{\fnms{Oleksandr} \snm{Sverdlov}\ead[label=e2]{[email protected]}}
\runauthor{W. F. Rosenberger and O. Sverdlov}
\affiliation{George Mason University and Bristol-Myers Squibb}
\address[a]{William F. Rosenberger is Professor and Chair,
Department of Statistics, George Mason University,
4400 University Drive, MS 4A7 Fairfax, Virginia
22030-4444, USA, \printead{e1}.}
\address[b]{Oleksandr Sverdlov is Senior Research Biostatistician,
Bristol-Myers Squibb, Route 206 and Province Line Road,
Lawrenceville, New Jersey 08540, USA, \printead{e2}.}
\end{aug}
\begin{abstract}
There has been a split in the statistics community about the need
for taking covariates into account in the design phase of a clinical
trial. There are many advocates of using stratification and
covariate-adaptive randomization to promote balance on certain known
covariates. However, balance does not always promote efficiency or
ensure more patients are assigned to the better treatment. We
describe these procedures, including model-based procedures, for
incorporating covariates into the design of clinical trials, and
give examples where balance, efficiency and ethical considerations
may be in conflict. We advocate a new class of procedures,
covariate-adjusted response-adaptive (CARA) randomization
procedures that attempt to optimize both efficiency
and ethical considerations, while maintaining randomization. We
review all these procedures, present a few new simulation studies,
and conclude with our philosophy.
\end{abstract}
\begin{keyword}
\kwd{Balance}
\kwd{covariate-adaptive randomization}
\kwd{covari\-ate-adjusted response-adaptive randomization}
\kwd{efficiency}
\kwd{ethics}.
\end{keyword}
\pdfkeywords{Balance, covariate-adaptive randomization,
covariate-adjusted response-adaptive randomization, efficiency, ethics}
\end{frontmatter}
\section{Introduction}\label{s1}
Clinical trials are often considered the ``gold standard'' in
convincing the medical community that a therapy is beneficial in
practice. However, not all clinical trials have been universally
convincing. Trials that have inadequate power, or incorrect
assumptions made in planning for power, imbalances on important
baseline covariates directly related to patient outcomes, or
heterogeneity in the patient population, have contributed to a lack
of scientific consensus. Hence, it is generally recognized that the
planning and design stage of the clinical trial is of great
importance. While the implementation of the clinical trial can often
take years, incorrect assumptions and forgotten factors in the
sometimes rushed design phase can cause controversy following a trial.
For example, take the trial of
erythropoietin in maintaining normal hemoglobin concentrations in
patients with metastatic breast cancer (Leyland-Jones, \citeyear{ley2003}). This
massive scientific effort involved 139 clinical sites and 939
patients. The study was terminated early because of an increase in
mortality in the erythropoietin group. The principal investigator\break
explains:
\begin{quote}
\ldots drawing definitive conclusions has been
difficult because the study was not designed to prospectively
collect data on many potential prognostic survival factors that
might have affected the study outcome$\ldots.$ The results of this trial
must be interpreted with caution in light of the potential for an
imbalance of risk factors between treatment groups$\ldots.$ The
randomisation design of the study may not have fully protected
against imbalances because the stratification was only done for one
parameter$,\ldots$ and was not done at each participating centre$\ldots.$ It
is extremely unfortunate that problems in design\ldots\ have complicated
the interpretation of this study. Given the number of design issues
uncovered in the post hoc analysis, the results cannot be considered
conclusive.
\end{quote}
An accompanying commentary calls this article\break
``alarmist,'' thus illustrating the scientific conundrum that
covariates present in clinical trials. There is no agreement in the
statistical community about how to deal with potentially important
baseline covariates in the design phase of the trial. Traditionally,
prestratification has been used on a small number of very important
covariates, followed by stratified analyses. But what if the
investigator feels there are many covariates that are important---too
many, in fact, to feasibly use prestratification?
The very act of randomization tends to mitigate the probability that
important covariates will be distributed differently among treatment
groups. This property is what distinguishes randomized clinical
trials from observational studies. However, this is a large sample
property, and every clinical trialist is aware of randomized trials
that resulted in significant baseline covariate imbalances.
Grizzle (\citeyear{griz1982}) distinguished two factions of the statistical
community, the ``splitters'' and the ``lumpers.'' The splitters
recommend incorporating important covariates into randomization,
thus ensuring balance over these covariates at the design stage. The
lumpers suggest ignoring covariates in the design and use simple
randomization to allocate subjects to different treatment groups,
and adjust for covariates at the analysis stage.
As Nathan Mantel once pointed out (Gail, \citeyear{gai1992}):
\begin{quote}
\ldots After looking at a data set, I might see
that in one group there are an unusually large number of males. I
would point out to the investigators that even though they had
randomized the individuals to treatments, or claimed that they had,
I could still see that there was something unbalanced. And the
response I would get was ``Well, we randomized and therefore we
don't have to bother about it.'' But that isn't true. So, as long
as the imbalance is an important factor you should take it into
account. Even though it is a designed experiment, in working with
humans, you cannot count on just the fact that you randomized.
\end{quote}
Today, many statisticians would argue that the only
legitimate adjusted analyses are for prespecified important
covariates planned for in the analysis according to protocol, and
that these adjustments should be done whether or not the
distributions are imbalanced (e.g., Permutt, \citeyear{per2000}). In addition,
these covariates should be accounted for in the \textit{design} of the
trial, usually by prestratification, if possible.
The three-stage philosophy of prestratifying on important known
covariates, followed by a stratified analysis, and allowing for
randomization to ``take care of'' the other less important (or
unknown) covariates, has become a general standard in clinical
trials. This method breaks down, however, when there are a large
number of important covariates. This has led to the introduction of
\textit{covariate-adaptive randomization} procedures, sometimes
referred to as \textit{minimization} procedures or \textit{dynamic
allocation}.
\footnote{Or sometimes, unfortunately, as
just \textit{adaptive designs}, which could refer to any number of
statistical methods having nothing to do with covariates, including
response-adaptive randomization, sequential monitoring, and
flexible interim decisions.}
Some of these ``covariate-adaptive''
procedures (the term we will use) that have been proposed have been
randomized, and others not.
There is no consensus in either the statistics world or the clinical
trials world as to whether and when these covariate-adaptive
procedures should be used, although they are gaining in popularity
and are now used frequently. Recently clinical trialists using
these procedures have grown concerned that regulatory agencies have
expressed skepticism and caution about the use of these techniques.
In Europe, The Committee on Proprietary Medicinal Products (CPMP)
Points to Consider Document (see Grouin, Day and Lewis, \citeyear{groDayLew2004})
states:
\begin{quote}
Dynamic allocation is strongly
discourag\-ed$\ldots.$ Without adequate and appropriate
supporting/sensitivity analysis, an application is unlikely to be
successful.
\end{quote}
This document has led to much controversy. In a
commentary, Buyse and McEntegart (\citeyear{buyMce2004}) state:
\begin{quote}
In our view, the CPMP's position is unfair,
unfounded, and unwise$\ldots.$ It favors the use of randomization
methods that expose trialists and the medical community to the risk
of accidental bias, when the risk could have been limited through
the use of balancing methods that are especially valuable$\ldots.$ If
there were any controversy over the use of minimization, it would be
expected of an independent agency to weigh all the scientific
arguments, for and against minimization, before castigating the use
of a method that has long been adopted in the clinical community.
\end{quote}
In a letter to the editor, Day, Grouin and Lewis (\citeyear{dayGrLew2005})
respond that
\begin{quote}
\ldots the scientific community is not of one
mind regarding the use of covariate-adaptive randomization
procedures$\ldots.$ Rosenberger and Lachin cautiously state that ``very
little is known about its theoretical properties.'' This is a
substantial point. The direct theoretical link between randomization
and methods of statistical analysis has provided a solid foundation
for reliable conclusions from clinical trial work for many years.
\end{quote}
It is in the context of this controversy that this paper is written.
The intention of this paper is to explore the role of covariates in
the \textit{design} of clinical trials, and to examine the burgeoning
folklore in this area among practicing clinical trialists. Just
because a technique is widely used does not mean that it is
valuable. And just because there is little theoretical evidence
validating a method does not mean it is not valid. The
nonspecificity of the language in these opinion pieces is becoming
troubling: what is meant by the terms ``minimization,'' ``dynamic,''
``adaptive''? Many procedures to mitigate covariate imbalances have
been proposed. Are they all equally effective or equally
inappropriate? We add to the controversy by discussing the often
competing criteria of balance, efficiency and ethical
considerations. We demonstrate by example that clinical trials that
balance on known covariates may not always lead to the most
efficient or the most ethically attractive design, and vice versa.
This paper serves as both a review and a summary of some of our
thoughts on the matter; in particular, we advocate a new class of
procedures called \textit{covariate-adjusted response-adaptive} (CARA)
randomization procedures
(e.g., Hu and Rosenberger, \citeyear{huRos2006}).
The outline of the paper is as follows. In Section~\ref{s2}, we review the
most popular covariate-adaptive randomization procedures. In
Section~\ref{s3}, we describe randomization-based inference and its
relationship to clinical trials employing covariate-adaptive
randomization methods. In Section~\ref{s4}, we discuss what is known from
the literature about the properties of the procedures in Section~\ref{s2}.
In Section~\ref{s5}, we describe the alternative model-based optimal design
approach to the problem and describe properties of these procedures
in Section~\ref{s6}. In Section~\ref{s7}, we discuss the relationship between
balance, efficiency and ethics, and describe philosophical arguments
about whether balance or efficiency is a more important criterion.
We demonstrate by example that balance does not necessarily imply
efficiency and vice versa, and demonstrate that balanced and
efficient designs do not necessarily place more patients on the
better treatment. In Section~\ref{s8}, we describe CARA randomization
procedures and their properties. In Section~\ref{s9}, we report the results
of a simulation study comparing different CARA and
covariate-adaptive randomization procedures for a binary response
trial with covariates.
Finally, we
give a summary of our own opinions in Section~\ref{s10}.
\section{Covariate-Adaptive Randomization}\label{s2}
Following Rosenberger and Lachin (\citeyear{rosLac2002}), a \textit{randomization
sequence} for a two-treatment clinical trial of $n$ patients is a
random vector $\mathbf{T}_n=(T_1,\ldots,T_n)^{\prime}$, where $T_j=1$ if
the $j$th patient is assigned to treatment $1$ and $T_j=-1$ if the
patient is assigned to treatment $2$. A \textit{restricted
randomization procedure} is given by
$\phi_{j+1}=\Pr(T_{j+1}=1|\mathbf{T}_j)$, that is, the probability that the
$(j+1)$th patient is assigned to treatment 1, given the previous
$j$ assignments. When the randomization sequence is dependent on a
patient's covariate vector $\mathbf{Z}$, we have \textit{covariate-adaptive
randomization}. In particular, the randomization procedure can then
be described by
$\phi_{j+1}=\Pr(T_{j+1}=1|\mathbf{T}_j,\mathbf{Z}_1,\ldots,\mathbf{Z}_{j+1})$,
noting that the current patient is randomized based on the history of
previous treatment assignments, the covariate vectors of past
patients \textit{and} the current patient's covariate vector. The goal
of covariate-adaptive randomization is to adaptively balance the
covariate profiles of patients randomized to treatments 1 and 2.
Most techniques for doing so have focused on minimizing the
differences of numbers on treatments $1$ and $2$ across strata,
often marginally. Note that covariate-adaptive randomization
induces a complex covariance structure, given by
$\operatorname{Var}(\mathbf{T}_n|\mathbf{Z}_1=\mathbf{z}_1,\ldots,\mathbf{Z}_n
=\mathbf{z}_n)=\bolds{\Sigma}_{n,\mathbf{z}}$.
For a small set of known discrete covariates, prestratification is
the most effective method for forcing balance with respect to those
covariates across the treatment groups. The technique of
prestratification uses a separate restricted randomization
procedure within each stratum. For notational purposes, if discrete
covariate $Z_i, i=1,\ldots,K$, has $k_i$ levels, then restricted
randomization is used within each of the $\prod_{i=1}^K k_i$ strata.
The first covariate-adaptive randomization procedures were proposed
in the mid-1970s. Taves (\citeyear{tav1974}) proposed a deterministic method to
allocate treatments designed to minimize imbalances on important
covariates, called the \textit{minimization} method.\break Pocock and Simon
(\citeyear{pocsim75}) and Wei (\citeyear{wei1978})
described generalizations of minimization to
randomized clinical trials. We will refer to this class of
covariate-adaptive randomization procedures as \textit{marginal}
procedures, as they balance on covariates marginally, within each of
$\sum_{i=1}^K k_i$ levels of given covariates.
The general marginal procedure can be described as follows for a
two-treatment clinical trial. Let\break $N_{ijl}(n)$ be the number of
patients on treatment $l$ in level $j$ of covariate $Z_i$,
$i=1,\ldots,K, j=1,\ldots,k_i, l=1,2$, after $n$ patients have been
randomized. When patient $n+1$ is ready for randomization, the
patient's baseline covariate vector $(Z_1,\ldots,Z_K)$ is observed as
$(z_1,\ldots,z_K)$. Then $D_i(n)=N_{iz_i1}(n) -\break N_{iz_i2}(n)$ is
computed for each $i=1,\ldots,K$. A weighted sum is then taken as
$D(n)=\sum_{i=1}^K w_i D_i(n)$. The measure $D(n)$ is used to
determine the treatment of patient $n+1$. If $D(n)>0$ ($<0$),
then one decreases (increases) the probability of being assigned to
treatment $1$
accordingly.
Pocock and Simon (\citeyear{pocsim75})
formulated a general rule using Efron's
(\citeyear{efr1971}) biased coin design as:
\begin{eqnarray*}
\phi_{n+1} & = & \cases{
1/2, & if $D(n)=0$, \cr
p, & if $D(n)<0$, \cr
1-p, & if $D(n)>0$.}
\end{eqnarray*}
When $p=1$, we have Taves's (\citeyear{tav1974}) minimization method, which is
nonrandomized. Pocock and Simon (\citeyear{pocsim75}) investigated $p=3/4$.
Wei (\citeyear{wei1978}) proposed a different marginal procedure using urns. At
the beginning of the trial, each of $\sum_{i=1}^K k_i$ urns contain
$\alpha_1$ balls of type 1 and $\alpha_2$ balls of type 2. Let
$U_{ij}$ denote the urn representing level $j$ of covariate $z_i$,
and let $Y_{ijk}(n)$ be the number of balls of type $k$ in urn
$U_{ij}$ after $n$ patients have been randomized. For each urn
compute the imbalance
$D_{ij}(n)=(Y_{ij1}(n)-Y_{ij2}(n))/(Y_{ij1}(n)+Y_{ij2}(n))$. Suppose
patient $n+1$ has covariate vector $(z_1,\ldots,z_K)$. Select the urn
such that $D_{iz_i}(n)$ is maximized. Draw a ball and replace. If
it is a type $k$ ball, assign the patient to treatment $k$, and add
$\alpha_k$ balls of type $k$ with $\beta_k\ge0$ balls of the
opposite type to each of the observed urns. The procedure is
repeated for each new eligible patient entering the trial. Wei
proved that if there is no interaction between the covariates or
between the treatment effect and covariates in a standard linear
model, then marginal balance is sufficient to achieve an unbiased
estimate of the treatment difference. Efron (\citeyear{efr1980}) provided a
covariate-adaptive randomization procedure that balances both
marginally and within strata, but the method applies only to two
covariates.
There has been substantial controversy in the literature as to
whether the introduction of randomization is necessary when
covariate-adaptive procedures are used. Randomization mitigates the
probability of selection bias and accidental bias, and provides a basis
for inference (e.g., Rosenberger and Lachin, \citeyear{rosLac2002}). Taves's
original paper did not advocate randomization, and, in fact, he
still supports the view that randomization is unnecessary, writing
in a letter to the editor (Taves, \citeyear{tav2004}, page~180):
\begin{quote}
I hope that the day is not too far distant
when we look back on the current belief that randomization is
essential to good clinical trial design and realize that it was\ldots\
``credulous idolatry.''
\end{quote}
Other authors have argued for using minimization without
the additional component of randomization. Aickin (\citeyear{aic2001})
argued
that randomization is not needed in covariate-adaptive procedures
because the covariates themselves are random, leading to randomness
in the treatment assignments. He also argued that the usual
selection bias argument for randomization is irrelevant in
double-masked clinical trials with a central randomization unit.
Several authors, such as Zelen (\citeyear{zel1974}),
Nordle and Brandmark (\citeyear{nordBra1977}),
Efron (\citeyear{efr1980}),
Signorini et al. (\citeyear{sigLeuSimBelGeb93}) and
Heritier, Gebski and Pillai (\citeyear{herGebPil2005}),
proposed covariate-adaptive randomization procedures which
achieve balanced allocation both within margins of the chosen
factors and within strata. These methods emphasize the importance of
balancing over interactions between factors when such exist.\break
Raghavarao (\citeyear{rag1980}) proposed an allocation procedure based on distance
functions. When the new patient enters the trial, one computes
$d_k$, the Mahalanobis distance between the covariate profile of the
patient and the average of the patients already assigned to
treatment~$k$, where $k=1,\ldots,K$. Then the patient is assigned to
treatment~$k$ with probability $p_k\propto d_k$.
\section{Randomization-Based Inference}\label{s3}
One of the benefits of randomization is that it provides a basis for
inference (see Chapter 7 of Rosenberger and Lachin, \citeyear{rosLac2002}). Despite
this, assessment of treatment effects in clinical trials is often
conducted using standard likelihood-based methods that ignore the
randomization procedure used. Letting $\mathbf{Y}^{(n)}=(Y_1,\ldots,Y_n)$
be the response vector, $\mathbf{T}_n^{(n)}=(T_1,\ldots,T_n)$ the treatment
assignment vector and
$\mathbf{Z}^{(n)}=(\mathbf{Z}_1,\ldots,\mathbf{Z}_n)$ the
covariate vectors of patients $1,\ldots,n$, the likelihood can simply
be written as\looseness=1
\begin{eqnarray*}
\mathcal{L}_n&=&\mathcal{L}
\bigl(\mathbf{Y}^{(n)},\mathbf{T}^{(n)},\mathbf{Z}^{(n)};\theta\bigr)
\\
&=&\mathcal{L}\bigl(Y_n|\mathbf{Y}^{(n-1)},\mathbf{T}^{(n)},
\mathbf{Z}^n;\theta\bigr)
\\
&&{} \cdot \mathcal{L}\bigl(T_n|\mathbf{Y}^{(n-1)},{\mathbf{T}}^{(n-1)},
\mathbf{Z}^{(n)};\theta\bigr)
\\
&&{} \cdot \mathcal{L}\bigl(\mathbf{Z}_n|\mathbf{Y}^{(n-1)},
\mathbf{T}^{(n-1)},\mathbf{Z}^{(n-1)}\bigr)\mathcal{L}_{n-1}.
\end{eqnarray*}
As $\mathcal{L}(Y_n|\mathbf{Y}^{(n-1)},\mathbf{T}^{(n)},
\mathbf{Z}^n;\theta)=\mathcal{L}(Y_n|T_n, \mathbf{Z}_n;\theta)$,
the treatment assignments do not
depend on $\theta$, and the covariates are considered i.i.d.,
we can reduce this to the recursion
\begin{eqnarray*}
\mathcal{L}_n&\propto&\mathcal{L}(Y_n|T_n, \mathbf{Z}_n;\theta)\mathcal{L}_{n-1}
\\
&=& \prod_{i=1}^n \mathcal{L}(Y_i|T_i,\mathbf{Z}_i;\theta).
\end{eqnarray*}
This is the standard regression equation under a population model;
that is, the randomization is ancillary to the likelihood. Thus,
a proponent of the likelihood principle would ignore the design in
the analysis, and proceed with tests standardly available in SAS.
The alternative approach is to use a randomization test, which is a
simple nonparametric alternative. Under the null hypothesis of no
treatment effect, the responses should be a deterministic sequence
unaffected by the treatment assigned. Therefore, the distribution
of the test statistic under the null hypothesis is computed with
reference to all possible sequences of treatment assignments under
the randomization procedure.
Various authors have struggled with the appropriate way to perform
randomization tests following covariate-adaptive randomization.
Pocock and Simon (\citeyear{pocsim75}) initially suggested that the sequence of
covariate values and responses be treated as deterministic, and the
sequence of treatment assignments be permuted for those specific
covariate values. This is the approach taken by most authors.
Ebbutt et al. (\citeyear{ebbKayMcEng97}) presented an example where results
differed when the randomization test took into consideration the
sequencing of patient arrivals. Senn concluded from this that the
disease was changing in some way through the course of the trial and
thus there was a time trend present (see the discussion of Atkinson,
\citeyear{atk1999}).
\section{What We Know About Covariate- Adaptive Randomization Procedures}\label{s4}
Our knowledge of covariate-adaptive randomization comes from (a) the
original source papers; (b) a vast number of simulation papers; (c)
advocacy or regulatory papers (for or against); and (d) review
papers. Very little theoretical work has been done in this area,
despite the proliferation of papers. The original source papers are
fairly uninformative about theoretical properties of the procedures.
In Pocock and Simon (\citeyear{pocsim75}), for instance, there is a small
discussion, not supported by theory, on the appropriate selection of
biasing probability $p$. There is no discussion about the effect of
the choice of weights for the covariates; no discussion about the
effect on inference; no theoretical justification that the procedure
even works as intended: Do covariate imbalances (loosely defined)
tend to zero? Does marginal balance imply balance within strata or
overall? Wei (\citeyear{wei1978}) devotes less than one page to a description of
his procedure; he does prove that marginal balance implies balance
within strata for a linear model with no interactions. Taves (\citeyear{tav1974})
is a nontechnical paper with only intuitive justification of the
method. Simulation papers have been contradictory.
Klotz (\citeyear{klo1978}) formalized the idea of finding an optimal value of
biasing probability $p$ as a constrained maximization problem.
Consider a trial with $K$ treatments and covariates. When patient
$n+1$ is ready to be randomized, one computes $D_k$, the measure of
overall covariate imbalance if the new patient is assigned to
treatment $k=1,\ldots,K$. The goal is to find the vector of
randomization probabilities $\bolds{\rho}=(\rho_1,\ldots,\rho_K)$ which
maximizes the entropy measure subject to the constraint on the
expected imbalance. Titterington (\citeyear{tit1983}) built upon Klotz's idea and
considered minimization of the quadratic distance between $\bolds{\rho}$
and the vector of uniform probabilities $\bolds{\rho}_0=(1/K,\ldots,1/K)$
subject to the constraints on the expected imbalance.
Aickin (\citeyear{aic2001}) provides perhaps one of the few theoretical analyses
of covariate-adaptive randomization procedures. He gives a very
short proof contradicting some authors' claims that
covariate-adaptive randomization can promote imbalances in
unmeasured covariates. If $X_2$ is an unmeasured covariate, and
covariate-adaptive randomization was used to balance on covariate
$X_1$, then $X_2$ can be decomposed into its \mbox{linear} \mbox{regression} part,
given by $L(X_2|X_1)$, and its linear regression residual
$X_2-L(X_2|X_1)$. If $X_1$ and $X_2$ are correlated positively or
negatively, balancing on $X_1$ will improve the balance of
$L(X_2|X_1)$. Since the residual is not correlated with the
randomization procedure, $X_2-L(X_2|X_1)$ will balance as well as
with restricted or complete randomization. This is a formal
justification of the intuitive argument that Taves (\citeyear{tav1974}) gave in
his original paper, an argument that Aickin (\citeyear{aic2001}) says
is a ``remarkably insightful observation.'' Aickin also uses causal
inference modeling to show that, if the unobserved errors correlated
with the treatment assignments and known covariates are linearly
related to the known covariates, the treatment effect should be
unbiased.
There seems to be a troubling misconception in the literature with
regard to covariate-adaptive randomization. For example, in an
editorial in the \textit{British Medical Journal} (Treasure and MacRae,
\citeyear{treMac1998}) we have the statement:
\begin{quote}
The theoretical validity of the method of
minimisation was shown by Smith$\ldots.$
\end{quote}
The quotation refers to Smith (\citeyear{smi84b}), which actually
derives the asymptotic distribution of the randomization test
following a model-based optimal design approach favored by many
authors. We shall discuss this approach momentarily, but it is
important to point out that \textit{there is no justification},
\textit{theoretical or otherwise},
\textit{of minimization methods in Smith's paper}.
In contrast to the dearth of publications exploring
covariate-adaptive randomization from a theoretical perspective, a
literature search revealed about 30 papers reporting results of
simulation studies. Some of these papers themselves are principally
a review of various other simulation papers. A glance at the recent
Society for Clinical Trials annual meeting abstract guide revealed
about 10 contributed talks reporting additional simulation results
and their use in clinical trials, indicating the continuing
popularity of these designs.
Papers dealing with the comparison of stratified block designs with
covariate-adaptive randomization methods with respect to achieving
balance on covariates include the original paper of Pocock and Simon
(\citeyear{pocsim75}),
Therneau (\citeyear{ther1993}), and review papers by
Kalish and Begg (\citeyear{kalBeg1985}) and
Scott et al. (\citeyear{scoMcpRamCam02}).
The general consensus is that covariate-adaptive
randomization does improve balance for large numbers of covariates.
Inference following covariate-adaptive randomization has been
explored by simulation in Birkett (\citeyear{bir1985}),
using the $t$-test, Kalish and Begg (\citeyear{kalBeg1987})
using randomization tests, and Frane (\citeyear{fra1998}), using
analysis of covariance. Recent papers by
Tu, Shalay and Pater (\citeyear{tuShaPat2000})
and McEntegart (\citeyear{mce2003}) cover a wide-ranging number of questions. Tu
et al. found that minimization method is inferior to
stratification in reducing error rates, and argued that marginal
balance is insufficient in the presence of interactions. McEntegart
concluded that there is little difference in power between
minimization method and stratification. Hammerstrom (\citeyear{ham2003}) performed
some simulations and found that covariate-adaptive randomization
does not significantly improve error rates, but does little harm,
and therefore is useful only for cosmetic purposes.
We conclude this section by interjecting some relevant questions.
Does marginal balance improve power and efficiency, or is it simply
cosmetic? Is covariate-adaptive randomization the proper approach
to this problem?
\section{Model-Based Optimal Design~Approaches}\label{s5}
An alternate approach to balance is to find the optimal design that
minimizes the variance of the treatment effect in the presence of
covariates. This approach is first found in Harville (\citeyear{har1974}), not in
the context of clinical trials, and in Begg and Iglewicz (\citeyear{begIgl1980}).
The resulting designs are deterministic.
Atkinson (\citeyear{atk1982}) adopted the approach and has advocated it in a
series of papers, and in the 1982 paper, introduced randomization
into the solution. In order to keep consistency with the original
paper, we summarize Atkinson's approach for a general case of $K\ge
2$ treatments. Suppose $K$ treatments are to be compared, and
responses follow the classical linear regression model given by
\begin{eqnarray*}
E(Y_i)=\mathbf{x}_i^{\prime} \bolds{\beta},\quad i=1,\ldots,n,
\end{eqnarray*}
where the $Y_i$'s are independent with
$\operatorname{Var}({\mathbf{Y}})=\sigma^2{\mathbf{I}}$ and
$\mathbf{x}_i$ is $(K+q)\times1$ vector which includes treatment indicators and
selected covariates of interest ($q$ is the number of covariates in
the model). Let $\hat{\bolds{\beta}}$ be the least squares estimator of
$\bolds{\beta}$. Then
$\operatorname{Var}(\hat{\bolds{\beta}})=\sigma^2({\mathbf{X}}^{\prime}{\mathbf{X}})^{-1}$,
where ${\mathbf{X}}^{\prime}{\mathbf{X}}$ is the dispersion matrix
from $n$ observations.
For the construction of optimal designs we wish to find the $n$
points of experimentation at which some function is optimized (in
our case we will be finding the optimal sequence of $n$ treatment
assignments). The dispersion matrix evaluated at these $n$ points
is given by $\mathbf{M}(\xi_n)={\mathbf{X}}^{\prime}{\mathbf{X}}/n$, where
$\xi_n$ is the $n$-point design. It is convenient, instead of
thinking of $n$ points, to formulate the problem in terms of a
measure $\xi$ (which in this case is a frequency distribution) over a
design region $\Xi=\{1,\ldots,K\}$.
Atkinson formulated the optimal design problem as a design that
minimizes, in some sense, the variance of
${\mathbf{A}}^{\prime}{\hat{\bolds{\beta}}}$, where $\mathbf{A}$ is a matrix
of contrasts. One possible criterion is Sibson's (\citeyear{sib1974})
$D_A$-optimality that maximizes
\begin{eqnarray}\label{aaaa}
|{\mathbf{A}}^{\prime}{\mathbf{M}}^{-1}(\xi){\mathbf{A}}|^{-1}.
\end{eqnarray}
For any multivariable optimization problem, we compute the
directional derivative of the criterion. In the case of the $D_A$
criterion in (\ref{aaaa}), we can derive the Fr\`{e}chet derivative
as
\[
d_A({\mathbf{x}},\xi)={\mathbf{x}}^{\prime}{\mathbf{M}}^{-1}
(\xi){\mathbf{A}}({\mathbf{A}}^{\prime}{\mathbf{M}}^{-1}(\xi)
{\mathbf{A}})^{-1}{\mathbf{A}}^{\prime}{\mathbf{M}}^{-1}(\xi)
{\mathbf{x}},
\]
for $x\in\Xi$. By the classical Equivalence theorem of
Kiefer and Wolfowitz (\citeyear{kieWol1960}), the optimal design $\xi^*$ that
maximizes the criterion (\ref{aaaa}) then satisfies the following
equations:
\begin{eqnarray*}
\sup_{{\mathbf{x}}\in\Xi} d_A({\mathbf{x}},\xi) \le s \quad \forall
\xi\in\Xi
\end{eqnarray*}
and
\begin{eqnarray*}
\sup_{{\mathbf{x}}\in\Xi} d_A({\mathbf{x}},\xi^*)=s.
\end{eqnarray*}
Such a design is optimal for estimating linear contrasts of
$\bolds{\beta}$. Assume $n$ patients have already been allocated, and
the resulting $n$-point design is given by $\xi_n$. Let the value of
$d_A(x,\xi)$ for allocation of treatment $k$ be $d_A(k,\xi)$.
Atkinson proposed a sequential design which allocates the $(n+1)$th
patient to the treatment $k=1,\ldots,K$ for which $d_A(k,\xi_n)$ is a
maximum, given the patient's covariates. The resulting design is
deterministic.
In order to randomize the allocation, Atkinson suggested biasing a
coin with probabilities
\begin{eqnarray}\label{eeee}
\rho_k=\frac{\psi(d_A(k,\xi_n))}{\sum_{k=1}^K\psi(d_A(k,\xi_n))},
\end{eqnarray}
where $\psi(x)$ is any monotone increasing function, and allocating
to treatment $k$ with the corresponding probability. With two
treatments, $k=1,2$,
we have $s=1$, ${\mathbf{A}}^{\prime}=(-1,1,0,\ldots,0)$, and the
probability of assigning treatment $1$ is given by
\begin{eqnarray}\label{ffff}
\phi_{n+1}=\frac{\psi(d_A(1,\xi_n))}{\psi(d_A(1,\xi_n))
+ \psi(d_A(2,\xi_n))}.
\end{eqnarray}
(We consider only the case of two treatments in this paper.)
Equation (\ref{ffff}) gives a broad class of covariate-adaptive
randomization procedures. The choice of function $\psi$ has not been
explored adequately. Atkinson (\citeyear{atk1982}) suggested using $\psi(x)=x$;
Ball, Smith and Verdinelli (\citeyear{balSmiVer1993})
suggested $\psi(x)=(1+x)^{1/\gamma}$ for a
parameter $\gamma\ge0$, which is a compromise between randomness
and efficiency.
Atkinson (\citeyear{atk1999}, \citeyear{atk2002})
performed careful simulation studies to
compare the performance of several covariate-adaptive randomization
procedures for a linear model with constant variance and trials up
to $n=200$ patients. One criterion of interest was \emph{loss}, the
expected amount of information lost due to treatment and covariate
imbalance. Another criterion was selection bias, measuring
the probability of correctly guessing the next treatment assignment.
Atkinson observed that the deterministic procedure based on the
$D_A$-optimality criterion has the smallest value of loss, and
Atkinson's randomized procedure (\ref{ffff}) with \mbox{$\psi(x)=x$}
increases the loss. He noted that $D_A$-optimal designs are
insensitive to correlation between the covariates, while complete
randomization and minimization method increase the loss when
covariates are correlated.
\section{What We Know About Atkinson's Class of Procedures}\label{s6}
Considerably more theoretical work has been done on the class of
procedures in (\ref{ffff}) than for the covariate-adaptive
randomization procedures in Section~\ref{s2}. Most of the work has been
done in a classic paper by Smith (\citeyear{smi84a}), although he dealt with a
variant on the procedure in (\ref{ffff}). It is instructive to
convert to his notation:
\[
E(Y_n)=\alpha t_n+\sum_{j=1}^q z_{nj}\beta_j,
\]
where $Y_n$ and $t_n$ are the response and treatment assignments of
the $n$th patient, respectively, and $z_{nj}$ represent~$q$
covariates, and may include an intercept. Let $\mathbf{T}_n$ be the
treatment assignment vector and let $\mathbf{Z}_n$ be the matrix of
covariates. Then Atkinson's procedure in (\ref{ffff}) can be
formulated as follows: assign $t_{n+1}=\pm1$ with probabilities
proportional to $(\pm1-\mathbf{z}_{n+1}'(\mathbf{Z}_n\mathbf{Z}_n)^{-1}
\mathbf{Z}_n\mathbf{t}_n)^2$ (Smith, \citeyear{smi84b}, page~543).
Smith (\citeyear{smi84a}) introduced a more general class of
allocation procedures given by
\begin{equation}\label{2-Smith_allocation_general}
\phi_{n+1}=\psi(n^{-1}\mathbf{z}'_{n+1}\mathbf{Q}^{-1}
\mathbf{Z}_n'\mathbf{t}_n),
\end{equation}
where $\psi$ is nonincreasing, twice continuously differentiable
function with bounded second derivative satisfying
$\psi(x)+\psi(-x)=1$, and
$\mathbf{Q}=E(\mathbf{z}_n\mathbf{z}_n')=\lim_{n\rightarrow
\infty}n^{-1}(\mathbf{Z}_n'\mathbf{Z}_n)$.
It is presumed that the $\{\mathbf{z}_n\}$ are independent, identically
distributed random vectors, $\mathbf{Q}$~is nonsingular and all third
moments of $\mathbf{z}_n$ are finite. Note that the procedure
(\ref{2-Smith_allocation_general}) can be implemented only if the
distribution of covariates is known in the beginning of
the trial.
Smith suggested various forms of $\psi$, most leading to a
proportional biased coin raised to some power $\rho$. In general,
$\rho=-2\psi^{\prime}(0)$. Without covariates, Atkinson's procedure
in (\ref{eeee}) leads to
\[
\phi_{n+1}= \frac{n_2^{\rho}}{n_1^{\rho}+n_2^{\rho}},
\]
where
$\rho=2$. Smith found the asymptotic variance of the randomization
test based on the simple treatment effect, conditional on
$\mathbf{Z}_n$. He did not do any further analysis or draw conclusions
except to suggest that $\rho$ should be selected by the investigator
to be as large as possible to balance the competing goals of
balance, accidental bias and selection bias.
\section{Balance, Efficiency or Ethics?}\label{s7}
\begin{figure}
\caption{Multiple objectives of a phase \textup{III}
\label{Multiple objectives}
\end{figure}
Clinical trials have multiple objectives. The principal
considerations are given in the schematic in Figure \ref{Multiple
objectives}. \textit{Balance} across treatment groups is often
considered essential both for important covariates and for treatment
numbers themselves. \textit{Efficiency} is critical for demonstrating
efficacy. \textit{Randomization} mitigates certain biases.
\textit{Ethics} is an essential component in any human experimentation, and
dictates our treatment of patients in the trial. These
considerations are sometimes compatible, and sometimes in conflict.
In this section, we describe the interplay among balance,
efficiency and ethics in the context of randomized clinical trials,
and give some examples where they are in conflict.
In a normal error linear model with constant variance, numerical
balance between treatments on the margins of the covariates is
equivalent to minimizing the variance of the treatment effect. This
is not true for nonlinear models, such as logistic regression or
traditional models for survival analysis (Begg and Kalish, \citeyear{begKal1984};
Kalish and Harrington, \citeyear{kalHar1988}). As we shall discuss further in the
next section, balance does not imply efficiency except in
specialized cases. This leaves open the question, is balance on
covariates important?
We have the conflict recorded in a fascinating interchange among
Atkinson, Stephen Senn and John Whitehead (Atkinson, \citeyear{atk1999}).
Whitehead argues:
\begin{quote}
I think that one criterion is really to
reduce the probability of some large imbalance rather than the
variance of the esti\-mates$\ldots.$ And to make sure that these
unconvincing trials, because of the large imbalance, happen with
very low probability, perhaps is more important$\ldots.$ I would always
be wanting to adjust for these variables. None the less, the message
is simpler if my preferred adjusted analysis is similar to the
simple message of the clinicians.
\end{quote}
Senn gives the counterargument:
\begin{quote}
I think we should avoid pandering to these
foibles of physicians$\ldots.$ I think people worry far too much about
imbalance from the inferrential (sic) point of view$\ldots.$ The way I
usually describe it to physicians is as follows: if we have an
unbalanced trial, you can usually show them that by throwing away
some patients you can reduce it to a perfectly balanced trial. So
you can actually show that within it there is a perfectly balanced
trial. You can then say to them: `now, are you prepared to make an
inference on this balanced subset within the trial?' and they nearly
always say `yes.' And then I~say to them, `well how can a little
bit more information be worse than having just this balance trial
within it?'
\end{quote}
We thus encounter once again deep philosophical differences and the
ingrained culture of clinical trialists. Fortunately, balance and
efficiency are equivalent in homoscedastic linear models. Thus,
stratified randomization and covariate-adaptive randomization
procedures (such as Pocock and Simon's\break method) are valid to the
degree in which they force balance over covariates. Atkinson's
model-based approach is an alternative method that can incorporate
treatment-by-covariate interactions and continuous covariates.
Atkinson's class of procedures for linear models has an advantage of
being based on formal optimality criteria as opposed to ad hoc
measures of imbalance used in covariate-adaptive randomization
procedures. On the other hand, balanced designs may not be most
efficient in the case of nonlinear and heteroscedastic models. We
agree with Senn that cosmetic balance, while psychologically
reassuring, should not be the goal if power or efficiency is lost in
the process of forcing balance.
First, let us illustrate that balanced allocation can be less
efficient and less ethically appealing than unbalanced allocation in
some instances, and that there may exist unbalanced designs which
outperform balanced designs in terms of compound objectives of
efficiency and ethics. Consider a binary response trial of size $n$
comparing two treatments $A$ and $B$, and suppose there is an
important binary covariate $Z$, say gender ($Z=0$ if a patient is
male, and $Z=1$ if female), such that there are $n_0$ males and
$n_1$ females in the trial. Also assume that success probabilities
for treatment $k$ are $p_{k0}$ for males and $p_{k1}$ for females,
where $k=A,B$. Let $q_{kj}=1-p_{kj}$, $j=0,1$. For the time being we
will assume that the true success probabilities are known. One
measure of the treatment effect for binary responses is the log-odds
ratio, which can be expressed as
\begin{equation}\label{2_log-OR}
\log\operatorname{OR}(Z=j)=\log\frac{p_{Aj}/q_{Aj}}{p_{Bj}/q_{Bj}}, \quad j=0,1.
\end{equation}
An experimental design question is to determine allocation proportions
$\pi_{Aj}$ and $\pi_{Bj}$ in stratum $j$ for treatments $A$ and $B$,
respectively, where $j=0$ (male) or $j=1$ (female). Let us consider the
following three allocation rules:
\begin{longlist}[Rule 3:]
\item[\textit{Rule} 1:] Balanced treatment assignments in the two strata,
given by
\[
\pi_{Aj}=\pi_{Bj}=1/2, \quad j=0,1;
\]
\item[\textit{Rule} 2:] Neyman allocation maximizing the power of the stratified
asymptotic test of the log-odds ratio:
\[
T_j=\frac{\log\widehat{\operatorname{OR}}(Z=j)}
{\sqrt{\widehat{\operatorname{var}} (\log\widehat{\operatorname{OR}}(Z=j) )}},
\quad j=0,1.
\]
The allocation proportion is given by
\[
\pi^*_{Aj}=\frac{1/\sqrt{p_{Aj}q_{Aj}}}{1/\sqrt
{p_{Aj}q_{Aj}}+1/\sqrt{p_{Bj}q_{Bj}}},\quad j=0,1;
\]
\item[\textit{Rule} 3:] the analog of
Rosenberger et al.'s (\citeyear{rosStaIvaHarRic01}) optimal
allocation minimizing the expected number of treatment failures in the
trial subject to the fixed variance of the log-odds ratio. This is
given by
\[
\pi^{**}_{Aj}=\frac{1/\sqrt{p_{Aj}q_{Aj}^2}}
{1/\sqrt{p_{Aj}q_{Aj}^2}+1/\sqrt{p_{Bj}q_{Bj}^2}}, \quad j=0,1.
\]
\end{longlist}
Note that unlike Rule 1, Rules 2 and 3 depend on success probabilities
in the two strata, and are unbalanced, in general. Consider a case when
$n_0=n_1=100$ and let $(p_{A0},p_{B0})=(0.95,0.7)$ and
$(p_{A1},\break p_{B1})=(0.7,0.95)$. This represents a case when one of the
treatments is highly successful, there is significant treatment
difference between $A$ and $B$, and there is treatment-by-covariate
interaction (treatment $A$ is more successful for males and is less
successful for females). Then allocation proportions for treatment $A$
in the two strata are $\pi_{A0}=0.68$ and $\pi_{A1}=0.32$ for Rule 2,
and $\pi_{A0}=0.84$ and $\pi_{A1}=0.16$ for Rule 3.
All three rules are very similar in terms of efficiency, as measured
by the asymptotic variances of stratum-specific estimates of the
log-odds ratio. However, Rules 2 and 3 provide extra ethical
savings. For the sample size considered, Rule 3 is expected to have
16 fewer failures than the balanced design. At the same time, Rule
2, whose primary purpose is optimizing efficiency, is expected to
have 8 fewer failures than the balanced allocation. Therefore, in
addition to maximizing efficiency, Rule 2 provides additional
ethical savings, and is certainly far more attractive than balanced
allocation.
So far we have compared different target allocations for ``fixed'' designs,
that is, for a given number of patients in each treatment group and known
model parameters.
In practice, true success probabilities are not available at the trial onset,
which precludes direct implementation of Rules 2 and 3. Since clinical
trials are
sequential in nature, one can use accruing responses to estimate the parameters,
and then cast a randomization procedure which asymptotically achieves
the desired allocation.
To study operating characteristics of response-adaptive
randomization procedures targeting Neyman allocation (Rule 2) and
optimal allocation (Rule 3) we ran a simulation study in R using
10,000~replications (results are available from the second author upon
request). In the
simulations we assumed that two strata (male and female) are equally
likely. For Rules 2 and 3, the doubly adaptive biased coin design
(DBCD) procedure of Hu and Zhang (\citeyear{huZha2004}) was used within each stratum
to sequentially allocate patients to treatment groups. In addition,
balanced allocation was implemented using stratified permuted block
design (PBD) with block size $m=8$. We assumed that responses are
immediate, and compared the procedures with respect to power of the
stratified asymptotic test of the log-odds ratio for testing the
null hypothesis $H_0$: $(p_{A0}=p_{B0})$ and $(p_{A1}=p_{B1})$ versus
$H_A$: not $H_0$ using significance level $\alpha=0.05$, and the
expected number of treatment failures. We considered several
experimental scenarios for success probabilities
$(p_{Aj},p_{Bj})$, $j=0,1$, including the one described in the example above.
To facilitate comparisons, the sample size for each experimental
scenario was chosen such that the stratified block design achieves
approximately 80$\%$ power of the test. In summary,
response-adaptive randomization procedures worked as expected: for
chosen sample sizes they converged to the targeted allocations and
preserved the nominal significance level. Additionally,
response-adaptive randomization procedures had similar average power
to the PBD, but on average they had fewer treatment failures.
Ethical savings of response-adaptive designs were more pronounced
when one of the treatments had high success probability (0.8--0.9)
and treatment differences were large.
We would also like to emphasize that phase III trials are pivotal
studies, and one typically has an idea about the success probabilities
of the treatments from early stage trials. If a particular allocation
is such that it leads to high power of the test, and it is also skewed
toward the better treatment, then it makes sense to implement such a
procedure. The additional ethical savings can be prominent if the
ethical costs associated with trial outcomes are high, such as deaths
of trial participants.
\section{CARA Randomization}\label{s8}
Hu and Rosenberger (\citeyear{huRos2006}) define a covariate-adjusted
response-adaptive (CARA) randomization procedure as one for which
randomization probabilities for a current patient depend on the
history of previous patients' treatment assignments, responses and
covariates, and the covariate vector of the current patient, that is,
\begin{equation}\label{2_CARA}
\quad
\phi_{j}=\Pr(T_{j+1}=1|\mathbf{T}_j,\mathbf{Y}_j,
\mathbf{Z}_1,\ldots,\mathbf{Z}_j,\mathbf{Z}_{j+1}).
\end{equation}
There have been only few papers dealing with CARA
randomization, and it has become an area of active research. CARA
randomization is
an extension of \textit{response-adaptive randomization} which deals with
adjustment for covariates.
Response-adaptive randomization has a rich history in the literature,
and the
interested reader is referred to
Section~1.2 of Hu and Rosenberger (\citeyear{huRos2006}).
Bandyopadhyay and Biswas (\citeyear{banBis2001})
considered a linear regression model
for two treatments and covariates with an additive treatment effect and
constant variance. Suppose large values of response correspond to a
higher efficacy. Then the new patient is randomized to treatment $1$
with probability
\begin{eqnarray}\label{BB}
\phi_{j+1}=\Phi(d_j/T),
\end{eqnarray}
where $d_j$ is the difference of covariate-adjusted treatment means
estimated from the first $j$ patients, $T$ is a scaling constant
and $\Phi$ is the standard normal c.d.f. Although procedure
(\ref{BB}) depends on the full history from $j$ patients, it does
not account for covariates of the $(j+1)$th patient, and it is not
a CARA procedure in the sense of (\ref{2_CARA}). Also, this
procedure depends on the choice of $T$, and small values of $T$ can
lead to severe treatment imbalances which can lead to high power
losses.
Atkinson and Biswas (\citeyear{atkBis05a}, \citeyear{atkBis05b})
improved the allocation rule of
Bandyopadhyay and Biswas (\citeyear{banBis2001})
by proposing CARA procedures that are
based on a weighted $D_A$-optimal criterion combining both efficiency
and ethical considerations. They investigated operating characteristics
of the proposed designs\break through simulation, but they did not derive
asymptotic properties of the estimators and allocation proportions.
Without the asymptotic properties of the estimators, it is difficult to
assess the validity of statistical inferences following CARA designs.
A few papers describe CARA designs for binary response trials. One of
the first papers in this field is by Rosenberger, Vidyashankar and
Agarwal (\citeyear{rosVidAga2001}).
They assumed that responses in treatment group $k=A,B$
follow the logistic regression model
\begin{eqnarray*}
\operatorname{logit}\bigl(\Pr(Y_{k}=1|\mathbf{Z}=\mathbf{z})\bigr)
=\bolds{\theta}_k'\mathbf{z},
\end{eqnarray*}
where $\bolds{\theta}_k$ is a vector of model parameters for treatment~$k$. Let
$\hat{\bolds{\theta}}_{jA}$ and $\hat{\bolds{\theta}}_{jB}$ be the maximum
likelihood estimators of model parameters computed from the data from
$j$ patients. Then the $(j+1)$th patient is randomized to treatment
$A$ with probability
\begin{eqnarray*}
\phi_{j+1}=F\bigl((\hat{\bolds{\theta}}_{jA}
-\hat{\bolds{\theta}}_{jB})'\mathbf{z}_{j+1}\bigr),
\end{eqnarray*}
where $F$ is the standard logistic c.d.f. Basically, each patient is
allocated according to the current value of covariate-adjusted odds
ratio comparing treatments $A$ and $B$. The authors compared their
procedure with complete randomization through simulations assuming
delayed responses. They showed that for larger treatment effects
both procedures have similar power, but at the same time the former
results in a smaller expected proportion of treatment failures.
Bandyopadhyay, Biswas and Bhattacharya (\citeyear{banBisBha2007}) also dealt with binary
responses. They proposed a two-stage design for the logistic
regression model. At the first stage, $2m$ patients are randomized
to treatment $A$ or $B$ in a $1\dvtx1$ ratio and accumulated data are
used to estimate model parameters. At the second stage, each patient
is randomized to treatment $A$ with a probability which depends on
the treatment effect estimated from the first stage and the current
patient's covariate vector.
Theoretical properties of CARA procedures have been developed in a
recent paper by Zhang et al. (\citeyear{zhaHuCheChan2007}).
This paper proposed a general framework for CARA randomization
procedures for a very broad class of models,
including generalized linear models. In the paper the authors proved
strong consistency and asymptotic normality
of both maximum likelihood estimators and allocation proportions. They
also examined the CARA design of
Rosenberger, Vidyashankar and Agarwal (\citeyear{rosVidAga2001})
and provided\break asymptotic
properties of the procedure.
CARA procedures do not lend themselves to analysis via
randomization-based inference. The theoretical validity of
randomization tests is based on conditioning on the outcome data as
a set of sufficient statistics, and then permuting the treatment
assignments. Under the null hypothesis of no treatment difference,
the observed outcome data should be exchangeable, leading to a valid
randomization $p$-value (see Pesarin, \citeyear{pes2001}). However, under the
CARA procedure, the treatment assignments and outcomes form the
sufficient statistics, and conditioning on both would leave nothing.
One could perform a standard permutation test on the resulting data
by introducing a ``sham'' equiprobable randomization, but one would
lose information about treatment efficacy.
Therefore, we rely on likelihood-based methods to conduct inference
following a CARA randomization procedure, and Zhang et al.
(\citeyear{zhaHuCheChan2007})
provide the necessary asymptotic theory. For further
discussion of appropriate inference procedure following general
response-adaptive randomization procedures, refer to Chapter 3 of Hu
and Rosenberger (\citeyear{huRos2006}) and
Baldi Antognini and Giovagnoli (\citeyear{baldiGio2005}, \citeyear{baldiGio2006}).
\section{Comparing Different Randomization Procedures Which Account
for~Covariates}\label{s9}
In the following we used simulation to compare the operating
characteristics of several covariate-adaptive randomization procedures
and CARA procedures for the logistic regression model. We used the
covariate structure considered in Rosenberger, Vidyashankar and Agarwal
(\citeyear{rosVidAga2001}). Assume that responses
for treatment~$k$ satisfy the following logistic regression model:
\begin{equation}\label{7-Logistic_model_Zhang}
\operatorname{logit}\bigl(\Pr(Y_k=1|\mathbf{z})\bigr)
= \alpha_k+\sum_{j=1}^3\beta_{kj}z_j,
\end{equation}
where $\alpha_k$ is the treatment effect, and $\beta_{kj}$ is the
effect due to the $j$th covariate in treatment group $k=A,B$. The
parameter of interest is the covariate-adjusted treatment difference
$\alpha_A-\alpha_B$. The\break components of covariate vector
$\mathbf{z}'=(z_1,z_2,z_3)$,\break which represent \emph{gender}, \emph{age} and
\emph{cholesterol level}, were assumed to be independently distributed
as Bernoulli$(1/2)$, Discrete Uniform$[30,75]$ and\break Normal$(200,20)$. Note
that model~(\ref{7-Logistic_model_Zhang}) allows for\break
treatment-by-covariate interactions, since covariate
effects $\beta_{kj}$'s are not the same across the treatments.
The operating characteristics of designs included measures of
balance, efficiency and ethics. For \emph{balance} we considered the
allocation proportion\break $N_A(n)/n$, and the allocation proportions
within the male category of covariate \emph{gender},
$N_{A0}(n)/N_0(n)$. Also, we examined the Kolmogorov--Smirnov
distance $d_{\mathrm{KS}}(z_2)$ between empirical distributions of covariate
\emph{age} in treatment groups $A$ and $B$. The \emph{efficiency} of
procedures was measured by the average power of the asymptotic test
of the log-odds ratio evaluated at a given $\mathbf{z}_0$. The
\emph{ethical} aspect of a procedure was assessed by the total
number of treatment failures, $F(n)$.
The sample size $n$ was chosen in such a way that complete
randomization yields approximately 80$\%$ or 90$\%$ power of the test
of log-odds ratio under a particular alternative. For each choice of
$n$ we also estimated the significance level of the test under the null
hypotheses. We report the results for three sets of parameter values
given in Table~\ref{Table7-0}. Under the null hypothesis of no
treatment difference (Model 1), $n=200$. When $\alpha_A-\alpha_B=-1$
(Model 2), the choice of $n=200$ yields $80\%$ power for complete
randomization. When $\alpha_A-\alpha_B=-1.25$ (Model 3), we let
$n=160$, which corresponds to $90\%$ power for complete randomization.
\begin{table}[b]
\tabcolsep=0pt
\caption{Parameter values for the logistic
regression model (\protect\ref{7-Logistic_model_Zhang}) used in~simulations\label{Table7-0}}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l d{2.3}d{2.3}d{2.3}d{2.3}d{2.3}d{2.3}@{}}
\hline
& \multicolumn{6}{c@{}}{\textbf{Model}}\\
\cline{2-7}
& \multicolumn{2}{c}{\textbf{1}}
& \multicolumn{2}{c}{\textbf{2}}
& \multicolumn{2}{c@{}}{\textbf{3}}\\
\ccline{2-3,4-5,6-7}
\textbf{Parameters}
& \multicolumn{1}{c}{$\bolds{A}$}
& \multicolumn{1}{c}{$\bolds{B}$}
& \multicolumn{1}{c}{$\bolds{A}$}
& \multicolumn{1}{c}{$\bolds{B}$}
& \multicolumn{1}{c}{$\bolds{A}$}
& \multicolumn{1}{c@{}}{$\bolds{B}$}\\
\hline
$\alpha_k$ & -1.652 & -1.652 & -1.402 & -0.402 & -1.652 & -0.402\\
$\beta_{k1}$ & -0.810 & -0.810 & -0.810 & 0.173 & -0.810 & 0.173 \\
$\beta_{k2}$ & 0.038 & 0.038 & 0.038 & 0.015 & 0.038 & 0.015 \\
$\beta_{k3}$ & 0.001 & 0.001 & 0.001 & 0.004 & 0.001 & 0.004 \\
\hline
\end{tabular*}
\end{table}
The first class of procedures are CARA designs. For their
implementation, we need to sequentially estimate model parameters. In
our simulations we assumed that all responses are immediate after
randomization, although we can add a queuing structure to explore the
effects of delayed response. For CARA procedures, some data must
accumulate so that the logistic model is estimable. We used Pocock and
Simon's method to allocate the first $2m_0$ patients to treatments $A$
and $B$.
Suppose after $n>2m_0$ allocations the m.l.e. of $\bolds{\theta}_k$ has
been computed as $\hat{\bolds{\theta}}_{n,k}$. Then, for a sequential
m.l.e. CARA procedure, the $(n+1)$th patient with covariate
$\mathbf{z}_{n+1}$ is allocated to treatment $A$ with probability
$\phi_{n+1}=\rho(\hat{\bolds{\theta}}_{n,A},\hat{\bolds{\theta}}_{n,B},\mathbf{z}_{n+1})$.
We explored four different choices of $\rho$:
\begin{enumerate}
\item Rosenberger, Vidyashankar and Agarwal's (\citeyear{rosVidAga2001})
target:
\begin{eqnarray*}\label{RVA}
\rho_1=\frac{p_A(\mathbf{z})/q_A(\mathbf{z})}{p_A(\mathbf{z})/
q_A(\mathbf{z})+p_B(\mathbf{z})/q_B(\mathbf{z})}.
\end{eqnarray*}
\item Covariate-adjusted version of Rosenberger et al.'s
(\citeyear{rosStaIvaHarRic01}) allocation:
\begin{eqnarray*}\label{RSIHR}
\rho_2&=& \frac{\sqrt{p_A(\mathbf{z})}}{\sqrt{p_A(\mathbf{z})}+\sqrt{p_B(\mathbf{z})}}.
\end{eqnarray*}
\item Covariate-adjusted version of Neyman allocation:
\begin{eqnarray*}\label{Neyman}
\quad
\rho_3=\frac{\sqrt{p_B(\mathbf{z})q_B(\mathbf{z})}}
{\sqrt{p_B(\mathbf{z})q_B(\mathbf{z})}
+\sqrt{p_A(\mathbf{z})q_A(\mathbf{z})}}.
\end{eqnarray*}
\item Covariate-adjusted version of optimal allocation:
\begin{eqnarray*}\label{5-CARA_optimal}
\quad
\rho_4=\frac{\sqrt{p_B(\mathbf{z})}q_B(\mathbf{z})}
{\sqrt{p_B(\mathbf{z})}q_B(\mathbf{z})
+ \sqrt{p_A(\mathbf{z})}q_A(\mathbf{z})}.
\end{eqnarray*}
\end{enumerate}
Here $p_k(\mathbf{z})=1/(1+\exp(-\bolds{\theta}'_k\mathbf{z}))$
and $q_k(\mathbf{z})=1-p_k(\mathbf{z})$, $k=A,B$. We will
refer to CARA procedures with four described targets as \textit{CARA} 1,
\textit{CARA} 2, \textit{CARA}~3 and \emph{CARA} 4, respectively.
We also considered an analogue of Akinson and Biswas's (\citeyear{atkBis05a})
procedure for the binary response case. It is worthwhile to describe
this approach in more detail. Consider model
(\ref{7-Logistic_model_Zhang}) and let $\bolds{\theta}_k=(\alpha_k,\break \beta
_{1k},\beta_{2k}, \beta_{3k})'$. Suppose that a trial has $n_A$
patients allocated to treatment $A$ and $n_B=n-n_A$ patients allocated
to treatment $B$. Then the information matrix about
$\bolds{\theta}=(\bolds{\theta}_A,\bolds{\theta}_B)$ based on $n$
observations is of the form
\[
\mathbf{M}_n={\rm diag}\{\mathbf{Z}_A'\mathbf{W}_A
\mathbf{Z}_A,\mathbf{Z}_B'\mathbf{W}_B\mathbf{Z}_B\},
\]
where $\mathbf{Z}_k$ is the $n_k\times p$ matrix of covariates for
treatment $k$, $\mathbf{W}_k$ is $n_k\times n_k$ diagonal matrix with
elements $p_kq_k$. Here $p_k=p_k(\mathbf{z}_i,\theta_k)$ denote the success
probability on treatment $k$ given $\mathbf{z}_i$ and $q_k=1-p_k$, $k=A,B$.
Suppose the $(n+1)$th patient enters the trial. Then the directional
derivative of the criterion $\det(\mathbf{M})$ for treatment $k$ given
$\mathbf{z}_{n+1}$ is computed as
\begin{equation}\label{Directional derivative}
\qquad
d(k,\bolds{\theta}_n,\mathbf{z}_{n+1})
= \mathbf{z}'_{n+1}(\mathbf{Z}'_k\mathbf{W}_k\mathbf{Z}_k)^{-1}
\mathbf{z}_{n+1}p_kq_k.
\end{equation}
Note that (\ref{Directional derivative}) depends on $\bolds{\theta}_k$,
which must be estimated using the m.l.e. $\hat{\bolds{\theta}}_{n,k}$.
The $(n+1)$th
patient is randomized to treatment $A$ with probability
\begin{equation}\label{7-Atkinson-Biswas}
\phi_{n+1}=\frac{\hat{f}_A d(A,\hat{\bolds{\theta}}_{n,A},
\mathbf{z}_{n+1})}
{\sum_{k=A}^B\hat{f}_k d(k,\hat{\bolds{\theta}}_{n,k},
\mathbf{z}_{n+1})},
\end{equation}
where $f_k$ is the desired proportion on treatment
$k$. We take $f_k=p_k(\mathbf{z})/q_k(\mathbf{z})$. The CARA
procedure (\ref{7-Atkinson-Biswas}) will be referred to as \emph{CARA 5}.
The second class of allocation rules are covariate-adaptive
randomization procedures. For Pocock and Simon's (P--S)
procedure, each component of $\mathbf{z}_{n+1}$ is discretized
into two levels, and the sum of marginal imbalances within these levels
is computed. The $(n+1)$th patient is allocated with probability $3/4$
to the treatment which would minimize total covariate imbalance. If
imbalances for treatments $A$ and $B$ are equal, then the patient is
assigned to either treatment with probability $1/2$.
For the stratified permuted block design (SPBD), the stratum of
the current patient is determined based on the observed combination of
the patient's covariate profile. Within that stratum allocations are
made using permuted blocks of size $m=10$. It is possible that had some
unfilled last blocks, and thus perfect balance is not achieved.
However, we did not specifically examine this feature of SPBD. We also
report the results for complete randomization (CRD).
The program performing the simulations was written in R. For each
procedure, a trial with $n$ patients was simulated 5000 times. To
facilitate the comparison of the procedures, the $n\times4$ matrix of
covariates $\mathbf{Z}$ was generated once and was held fixed for all
simulations. For CARA procedures, the first $2m_0=80$ patients were
randomized by Pocock and Simon's procedure with biasing probability
$p=3/4$. The response probabilities of patients in treatment group
$k=A,B$ were computed by multiplying the rows of $\mathbf{Z}$ by the
vector of model parameters and calculating the logistic c.d.f.
$F(x)=1/(1+\exp(-x))$ at the computed values. The significance level
of the test was set $\alpha=0.05$, two-sided.
\begin{table*}
\caption{Simulation results for
Model 1 with $\bolds{\theta}_A=\bolds{\theta}_B$ and
$n=200$\label{Table7-1}}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l ccccc@{}}
\hline
\textbf{Procedure} & $\bolds{\frac{N_A(n)}{n}}$ \textbf{(S.D.)}
& $\bolds{\frac{N_{A0}(n)}{N_0(n)}}$ \textbf{(S.D.)}
& $\bolds{d_{\mathrm{KS}}(z_2)}$ \textbf{(S.D.)}
& \textbf{Err. rate}
& $\bolds{F(n)}$ \textbf{(S.D.)} \\
\hline
CRD & 0.50 (0.03) & 0.50 (0.05) & 0.12 (0.04) & 0.05 & 90 (6) \\
SPBD & 0.50 (0.03) & 0.50 (0.04) & 0.12 (0.03) & 0.05 & 90 (6) \\
P--S & 0.50 (0.00) & 0.50 (0.01) & 0.10 (0.03) & 0.05 & 90 (6) \\
CARA 1 & 0.50 (0.03) & 0.50 (0.04) & 0.11 (0.03) & 0.06 & 90 (6) \\
CARA 2 & 0.50 (0.03) & 0.50 (0.04) & 0.12 (0.03) & 0.05 & 90 (6)\\
CARA 3 & 0.50 (0.02) & 0.50 (0.04) & 0.11 (0.03) & 0.06 & 90 (6) \\
CARA 4 & 0.50 (0.02) & 0.50 (0.04) & 0.12 (0.03) & 0.06 & 90 (6) \\
CARA 5 & 0.50 (0.02) & 0.50 (0.04) & 0.12 (0.04) & 0.05 & 90 (6) \\
\hline
\end{tabular*}
\end{table*}
Table \ref{Table7-1} shows the results under the null hypothesis
(Model 1). We see that all rules produce balanced allocations. CARA 1,
CARA 3 and CARA 4 procedures are slightly anticonservative, with a
type I error rate of $0.06$, while the procedures CARA 2 and CARA 5
preserve the nominal significance level of $0.05$. Pocock and Simon's
procedure is the least variable among the eight rules considered; the
other procedures are almost identical in terms of variability of
allocation proportions.
\begin{table*}[b]
\caption{Simulation results for
Model 2 with $\alpha_A-\alpha_B=-1.0$ and $n=200$\label{Table7-2}}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l ccccc@{}}
\hline
\textbf{Procedure}
& $\bolds{\frac{N_A(n)}{n}}$ \textbf{(S.D.)}
& $\bolds{\frac{N_{A0}(n)}{N_0(n)}}$ \textbf{(S.D.)}
& $\bolds{d_{\mathrm{KS}}(z_2)}$ \textbf{(S.D.)}
& \textbf{Power}
& $\bolds{F(n)}$ \textbf{(S.D.)} \\
\hline
CRD & 0.50 (0.04) & 0.49 (0.05) & 0.12 (0.04) & 0.80 & 62 (6) \\
SPBD & 0.50 (0.03) & 0.50 (0.04) & 0.12 (0.03) & 0.81 & 62 (6) \\
P--S & 0.50 (0.01) & 0.50 (0.01) & 0.10 (0.03) & 0.81 & 62 (6) \\
CARA 1 & 0.40 (0.04) & 0.45 (0.04) & 0.12 (0.03) & 0.76 & 56 (6) \\
CARA 2 & 0.48 (0.03) & 0.49 (0.04) & 0.12 (0.03) & 0.81 & 60 (6)\\
CARA 3 & 0.48 (0.03) & 0.49 (0.04) & 0.12 (0.03) & 0.81 & 60 (6) \\
CARA 4 & 0.45 (0.03) & 0.48 (0.04) & 0.12 (0.03) & 0.80 & 58 (6) \\
CARA 5 & 0.47 (0.03) & 0.50 (0.04) & 0.12 (0.04) & 0.81 & 60 (6) \\
\hline
\end{tabular*}
\end{table*}
Tables \ref{Table7-2} and \ref{Table7-3} show the results for Models
2~and~3, respectively. The conclusions are similar in the two cases,
and so we will focus on Model 2. Balanced designs equalize the
treatment assignments very well. As expected, the stratified blocks
and Pocock and Simon's procedure are less variable than complete
randomization. Similar conclusions about balancing properties of the
designs apply to balancing with respect to the continuous covariates.
The average power is $90\%$ for the stratified blocks
and Pocock and Simon's procedure, and $89\%$ for complete randomization.
Let us now examine the performance of CARA procedures. All CARA
procedures are more variable than the stratified blocks and Pocock
and Simon's method, but a little less variable than complete
randomization. In addition, all CARA procedures do a good job in
terms of balancing the distributions of the continuous covariates
[estimated $d_{\mathrm{KS}}(z_2)=0.13$ (S.D.${}=0.04$) versus $0.14$ (S.D.${}=0.04$) for
complete randomization]. CARA 2, CARA 3 and CARA~5 procedures are
closest to the balanced design. The simulated allocation proportions
for treatment $A$ and the corresponding standard deviations are $0.48$
(0.03) for CARA 2, and $0.48$ (0.03) for CARA 3, and $0.47$
(0.03) for CARA~5 procedure. These three CARA procedures have average
power of $81\%$, same as for stratified blocks and Pocock
and Simon's procedure, but at the same time they yield two fewer failures
than the balanced designs. CARA 4 procedure has the power
of $80\%$ (same as for complete randomization), but it has, on average,
four fewer failures than the balanced designs. CARA 1 procedure
is the most skewed: the simulated allocation proportion for treatment
$A$ and the standard deviation is 0.40 (0.04), and it results, on
average, in six fewer treatment failures than in the balanced design
case. On the other hand, it is less powerful than balanced designs (the
average power is $76\%$).
\begin{table*}
\caption{Simulation results for
Model 3 with $\alpha_A-\alpha_B=-1.25$ and $n=160$\label{Table7-3}}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l ccccc@{}}
\hline
\textbf{Procedure}
& $\bolds{\frac{N_A(n)}{n}}$ \textbf{(S.D.)}
& $\bolds{\frac{N_{A0}(n)}{N_0(n)}}$ \textbf{(S.D.)}
& $\bolds{d_{\mathrm{KS}}(z_2)}$ \textbf{(S.D.)}
& \textbf{Power}
& $\bolds{F(n)}$ \textbf{(S.D.)} \\
\hline
CRD & 0.50 (0.04) & 0.49 (0.05) & 0.14 (0.04) & 0.89 & 54 (6)\\
SPBD & 0.50 (0.01) & 0.50 (0.01) & 0.12 (0.03) & 0.89 & 54 (6) \\
P--S & 0.50 (0.01) & 0.50 (0.01) & 0.11 (0.03) & 0.90 & 54 (6) \\
CARA 1 & 0.39 (0.04) & 0.43 (0.04) & 0.13 (0.04) & 0.86 & 50 (6) \\
CARA 2 & 0.47 (0.03) & 0.48 (0.04) & 0.13 (0.04) & 0.90 & 53 (6) \\
CARA 3 & 0.48 (0.03) & 0.48 (0.04) & 0.13 (0.04) & 0.90 & 54 (6) \\
CARA 4 & 0.44 (0.03) & 0.45 (0.04) & 0.13 (0.04) & 0.89 & 51 (6) \\
CARA 5 & 0.47 (0.02) & 0.50 (0.03) & 0.12 (0.03) & 0.91 & 53 (5) \\
\hline
\end{tabular*}
\end{table*}
The overall conclusion is that CARA procedures may be a good
alternative to covariate-adaptive procedures targeting balanced
allocations in the nonlinear response case. Although incorporating
responses in randomization induces additional variability of
allocation proportions, which may potentially reduce power, one can see
from our simulations that such an impact is not dramatic.
For CARA procedures, it is essential that the first allocations to
treatment groups are made by using some covariate-adaptive procedure
or the stratified block design, so that some data accrue and one can
estimate the unknown model parameters with reasonable accuracy. From
numerical experiments we have found that at least $80$ patients must be
randomized to treatment groups before m.l.e.'s can be computed.
Alternatively, one can check after each allocation the convergence of
the iteratively reweighted least squares \mbox{algorithm} for fitting the
logistic model, as Rosenberger,
Vidyashankar and Agarwal (\citeyear{rosVidAga2001}) did. However, due to the slow
convergence of m.l.e.'s, we have found that it is better, first, to
achieve reasonable quality estimators by using a covariate-adaptive
randomization procedure with good balancing properties (such as Pocock
and Simon's method).
From our simulations one can see that there are CARA procedures (such
as CARA 4 procedure) which have the same average power as complete
randomization, but at the same time they result in three to four fewer failures
than the balanced allocations. Such extra ethical savings together with
high power for showing treatment efficacy can be a good reason for
using CARA procedures to design efficient and more ethically attractive
clinical trials.
\section{Discussion}\label{s10}
The design of clinical trials has become a rote exercise, often
driven by regulatory constraints. Boilerplate design sections in
protocols and grant proposals are routinely presented to steering
committees, review committees, and data and safety monitoring
boards. It is not uncommon for the randomization section of a
protocol to state ``double-blinded randomization will be performed''
with no further details. The fact that randomization is rarely if
ever used as a basis for inference means that the particular
randomization sequence is not relevant in the analysis, with the
exception that stratified designs typically lead to stratified
tests. Balance among important baseline covariates is seen to be an
essential cosmetic component of the clinical trial, and many
statisticians recommend adjusting for imbalanced covariates
following the trial, even if such analyses were not planned in the
design phase. While efficiency is usually gauged by a sample size
formula, the role that covariates play in efficiency, and the idea
that imbalances may sometimes lead to better efficiency and more
patients assigned to the superior treatment, are not generally
considered in the design phase of typical clinical trials.
In clinical trials with normally distributed outcomes, where it is assumed
that the variability of the outcomes is similar across treatments,
a balanced design across treatments and covariates will be the most
efficient. In these cases, if there are several important
covariates, stratification can be employed successfully, and if
there are many covariates deemed of sufficient importance,
covariate-adaptive randomization can be used to create balanced, and
therefore efficient, designs.
However, as we have seen, these simple ideas break down when there
are heterogeneous variances, including those found in commonly
performed trials with binary responses or survival responses. The
good news is that there are new randomization techniques that can be
incorporated in the design stage that can lead to more efficient and
more ethically attractive clinical trials. These randomization
techniques are based on the optimal design of experiments and also
tend to place more patients on the better treatment (Zhang et al., \citeyear{zhaHuCheChan2007}).
While more work needs to be done on the properties of
these procedures, we agree with Senn's comments that efficiency is
much more important than cosmetic balance.
The design of clinical trials is as important as the analysis of
clinical trials. Ethical considerations and efficiency should
dictate the randomization procedure used; careful selection of a
good design can save time, money, and in some cases patients' lives.
As Hu and Rosenberger (\citeyear{huRos2006}) point out, modern information
technology has progressed to the point where logistical difficulties
of implementing more complex randomization procedures are no longer
an issue. Careful design involves an understanding of both the
theoretical properties of a design in general, and simulated
properties under a variety of standard to worst-case models. In
some cases, the trade-offs in patient benefits and efficiency are so
modest compared to the relative gravity of the outcome, that
standard balanced designs may be acceptable. However, when outcomes
are grave, and balanced designs may produce severe inefficiency or
too many patients assigned to the inferior treatment, careful design
is essential.
\section*{Acknowledgments}
William F. Rosenberger is supported by
NSF Grant DMS-05-03718. The authors thank the referees for
helpful comments.
\vspace*{-1.5pt}
\end{document}
|
\begin{document}
\title{Deriving Grover's lower bound from simple physical principles}
\author{Ciar{\'a}n~M. Lee}
\email{[email protected]}
\affiliation{University of Oxford, Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, UK.}
\author{John~H. Selby}
\email{[email protected]}
\affiliation{University of Oxford, Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, UK.}
\affiliation{Imperial College London, London SW7 2AZ, UK.}
\begin{abstract}
Grover's algorithm constitutes the optimal quantum solution to the search problem and provides a quadratic speed-up over all possible classical search algorithms. Quantum interference between computational paths has been posited as a key resource behind this computational speed-up. However there is a limit to this interference, at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power? Sorkin has defined a hierarchy of possible interference behaviours---currently under experimental investigation---where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In this work, we consider how Grover's speed-up depends on the order of interference in a theory. Surprisingly, we show that the quadratic lower bound holds regardless of the order of interference. Thus, at least from the point of view of the search problem, post-quantum interference does not imply a computational speed-up over quantum theory.
\end{abstract}
\maketitle
Grover's algorithm \cite{grover1997quantum} provides the optimal quantum solution to the search problem and is one of the most versatile and influential quantum algorithms. The search problem---in its simplest form---asks one to find a single ``marked'' item from an unstructured list of $N$ elements by querying an oracle which can recognise the marked item. The importance of Grover's algorithm stems from the ubiquitous nature of the search problem and its relation to solving \textbf{NP}-complete problems \cite{bennett1997strengths}. Classical computers require $O(N)$ queries to solve this problem, but quantum computers---using Grover's algorithm---only require $O(\sqrt{N})$ queries. Quantum interference between computational paths has been posited \cite{stahlke2014quantum} as a key resource behind this computational ``speed-up''. However, as first noted by Sorkin \cite{sorkin1994quantum,sorkin1995quantum}, there is a limit to this interference---at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power?
Sorkin has defined a hierarchy of possible interference behaviours---currently under experimental investigation \cite{sinha2008testing,park2012three,sinha2010ruling}---where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. To get a greater understanding of the role of interference in computation, we consider how Grover's speed-up depends on the order of interference in a theory.
Restriction to the second level of this hierarchy implies many ``quantum-like'' features, which, at first glance, appear to be unrelated to interference. For example, such interference behaviour restricts correlations \cite{dowker2014histories} to the ``almost quantum correlations'' discussed in \cite{navascues2015almost}, and bounds contextuality in a manner similar to quantum theory \cite{henson2015bounding,niestegge2012conditional}. This, in conjunction with interference being a key resource in the quantum speed-up, suggests that post-quantum interference may allow for a speed-up over quantum computation.
Surprisingly, we show that this is not the case---at least from the point of view of the search problem. We consider this problem within the framework of generalised probabilistic theories, which is suitable for describing arbitrary operationally-defined theories \cite{Pavia1,Pavia2,Hardy2011,lee2015computation,lee2015proofs,barrett2015landscape}. Classical probability theory, quantum theory, Spekken's toy model \cite{janotta2013generalized,spekkens2007evidence}, and the theory of PR boxes \cite{popescu1998causality} all provide examples of theories in this framework. We consider theories satisfying certain natural physical principles which are sufficient for the existence of a well-defined search oracle. Given these physical principles, we prove that a theory at level $h$ in Sorkin's hierarchy requires $\Omega(\sqrt{N/h})$ queries to solve the search problem.
Thus, post-quantum interference does not imply a computational speed-up over quantum theory. Moreover, from the point of view of the search problem, all (finite) orders of interference are asymptotically equivalent.
\section{Generalised probabilistic theories}
A basic requirement of any physical theory is that it should provide a consistent account of experimental data. This idea underlies the framework of generalised probabilistic theories---developed in \cite{Pavia1, Pavia2, Hardy2011, barrett2007information}---which allows for the description of arbitrary theories satisfying this requirement. Informally, a theory in this framework specifies a set of \emph{physical processes} which can be connected together to form experiments. Each process corresponds to a single use of a piece of laboratory apparatus, each having a number of input and output ports, as well as a classical pointer. When the physical apparatus is used in an experiment, the classical pointer comes to rest at one of a number of positions, indicating a specific outcome has occurred.
Each port is associated with a \emph{physical system} of a particular type (labelled $A,B,...$). Intuitively one can consider these physical systems as passing from outputs of one process to inputs of another. Processes can thus be connected together---both in sequence and in parallel---to form \emph{circuits}, where it is required that types match and there are no cycles.
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Closed circuits (i.e. circuits with no disconnected ports) correspond to the probability of obtaining a particular set of outcomes from the experiment represented by that circuit. Processes that yield the same probabilities in all closed circuits are identified, giving rise to equivalence classes of processes. Each element of such an equivalence class has the same input and output ports, and are denoted ${}_AT_B\in{}_A\mathcal{T}_B$, where ${}_A\mathcal{T}_B$ is the set of possible \emph{transformations} from systems $A$ to $B$. Transformations with no input ports are called \emph{states} $S_A \in \mathcal{S}_A$, and no output ports, \emph{effects}, ${}_AE\in{}_A\mathcal{E}$.
Given the probabilistic structure provided by closed circuits, each transformation ${}_AT_B$ can be associated with a real vector such that the set ${}_A\mathcal{T}_B$ is a subset of some real vector space, denoted ${}_AV_B$ \cite{Pavia1}. We assume in this work that all vector spaces are finite dimensional. It can be shown that transformations and effects act linearly on the vector space of states, $V_A$ \cite{Pavia1}. A measurement corresponds to a set of effects $\{e^r\}$ labelled by the position of the classical pointer $r$. The probability of preparing state $s$ and observing outcome $r$ is (suppressing system types for readability) given by: $$e^r(s) = P(r,s).$$
A state is \emph{pure} if it does not arise as a \emph{coarse-graining} of other states \footnote{The process $\{\mathcal{U}_j\}_{j\in{Y}}$, where $j$ indexes the classical pointer, is a coarse-graining of $\{\mathcal{E}_i\}_{i\in{X}}$ if there is a disjoint partition $\{X_j\}_{j\in{Y}}$ of $X$ such that $\mathcal{U}_j=\sum_{i\in{X_j}}\mathcal{E}_i$.}; a pure state is one for which we have maximal information. A state is \emph{mixed} if it is not pure. Similarly, one says a transformation is pure if it does not arise as a coarse-graining of other transformations. It can be shown that reversible transformations preserve pure states \cite{Pavia2}.
We now introduce five physical principles which will be assumed throughout the rest of this work. These can be though of as an abstraction of basic characteristics of the behaviour of information in quantum theory. Note however that these principles are not unique to quantum theory, indeed, real vector space quantum theory, fermionic quantum theory and the classical theory of pure states each satisfy all of these principles.
\begin{principle} \textbf{Causality \cite{Pavia1}:}
There exists a unique deterministic effect ${}_AU$ for every system $A$, such that $\sum_r e^r=U $ for all measurements, $\{e^r\}_r$.
\end{principle}
In quantum theory the unique deterministic effect is provided by the partial trace. Mathematically, causality is equivalent to the statement: ``probabilities of present experiments are independent of future measurement choice'' \cite{Pavia1}, and so this can be interpreted as saying that ``information propagates from present to future''.
The deterministic effect allows one to define a notion of \emph{marginalisation} for multipartite states.
\begin{principle}\textbf{Purification \cite{Pavia1}:}
Given a state $s_{A}$ there exists a system $B$ and a pure state $S_{AB}$ on $AB$ such that $s_{A}$ is the marginalisation of $S_{AB}$: $${}_BU(S_{AB})=s_A.$$ Moreover, the purification $S_{AB}$ is unique up to reversible transformations on the purifying system, $B$ \footnote{Two states $S_{AB}$ and $S'_{AB}$ purifying $s_A$ satisfy $S_{AB}={\mathbb{I}\otimes_BT_B}(S'_{AB})$, with $_BT_B$ a reversible transformation.}.
\end{principle}
For example, in quantum theory any mixed state $\rho=\sum_i p_i \ket{\psi_i}\bra{\psi_i}$ can be written as $\rho= tr_B(\ket{\Psi}\bra{\Psi}_{AB})$ where $\ket{\Psi}_{AB}:=\sum_i\sqrt{p_i}\ket{\psi_i}\ket{i}$. Moreover, any other purification $\ket{\widetilde{\Psi}}_{AB}$ must satisfy $\ket{\Psi}_{AB}=\left(\mathbb{I}_A\otimes U_B\right)\ket{\widetilde{\Psi}}_{AB}$ with $U_B$ a unitary transformation. More generally this can be thought of as saying that information cannot be fundamentally destroyed, only discarded.
\begin{principle}\textbf{Purity Preservation \cite{chiribella2015operational}:}
The composite of pure transformations is pure.
\end{principle}
Pure transformations in quantum theory can be characterised by having Kraus rank $1$. Given two such transformations, their sequential or parallel composition will each also be rank $1$, and so composition preserves purity.
\begin{principle}\textbf{Pure Sharp Effect \cite{chiribella2015operational}:}
For each system $A$ there exists a pure effect that occurs with unit probability on some state.
\end{principle}
Pure states $\{a^i\}_{i=1}^n$ are \emph{perfectly distinguishable} if there exists a measurement, corresponding to effects $\{e^j\}_{j=1}^n$, such that $e^j(a^i)=\delta_{ij}$ for all $i,j$. For example, in quantum theory the computational basis $\{\ket{i}\}$ provide a perfectly distinguishable set, where the corresponding effects are just $\{\bra{j}\}$ such that $\braket{j}{i}=\delta_{ij}$. Such an $n$-tuple of states can reliably encode an $n$-level classical system.
\begin{principle}\textbf{Strong symmetry \cite{barnum2014higher}:}
For any two $n$-tuples of pure and perfectly distinguishable states $\{a^i\}$, and $\{b^i\},$ there exists a reversible transformation $T$ such that $T(a^i)=b^i$ for all $i$.
\end{principle}
An example in quantum theory is the Hadamard transformation reversibly mapping between the bases $\{\ket{0},\ket{1}\}$ and $\{\ket{+},\ket{-}\}$.
These last two principles imply that one can encode classical data in a system, and moreover, that any encoding is equivalent. In other words, information is independent of the encoding medium.
Principles $1$ to $4$ imply the following result (see \cite{chiribella2015operational} for a proof): for any given state $s$, there exists a natural number $n$ and a set of pure and perfectly distinguishable states $\{a^i\}_{i=1}^n$ such that $s=\sum_i p_ia_i$ where $0 \leq p_i \leq 1,\ \forall i$ and $\sum_i p_i=1$.
This result, together with principle $5$, implies the existence of a ``self-dualising'' \cite{muller2012structure,barnum2014higher} inner product $\langle {\cdot,\cdot}\rangle$. That is, to every pure state $s$, there is associated a unique pure effect $e^s$,
satisfying $e^s(s)=1$, such that: $e^s(\cdot)=\langle s, \cdot\rangle $. This inner product is invariant under all reversible transformations; satisfies $0 \leq \langle r, s \rangle \leq 1$ for all states $r, s$; $\langle s, s\rangle=1$ for all pure states $s$; and $\langle s, r \rangle=0$ if $s$ and $r$ are perfectly distinguishable. It also gives rise to the norm $\Vert \cdot \Vert =\sqrt{\langle{\cdot,\cdot}\rangle}$, satisfying $\Vert s \Vert \leq 1$ for all states $s$, with equality for pure states. We will make use of this norm in proving our main result.
\section{Higher-order interference}
Informally, a theory is said to have $n$th order interference if one can generate interference patterns in an $n$-slit experiment which cannot be created in any experiment with only $m$-slits, for all $m<n$.
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More precisely, this means that the interference pattern created on the screen cannot be written as a particular linear combination of the patterns generated when different subsets of slits are blocked. In the two slit experiment, quantum interference corresponds to the fact that the interference pattern cannot be written as the sum of the single slit patterns:
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It was first shown by Sorkin \cite{sorkin1994quantum,sorkin1995quantum} that---at least for ideal experiments \cite{sinha2015superposition}---quantum theory is limited to the $n=2$ case. That is, the interference pattern created in a three---or more---slit experiment \emph{can} be written in terms of the two and one slit interference patterns obtained by blocking some of the slits.
Schematically:
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If a theory does not have $n$th order interference then one can show it will not have $m$th order interference, for any $m>n$ \citep{sorkin1994quantum}. As such, one can classify theories according to their maximal order of interference, $h$. For example quantum theory lies at $h=2$ and classical theory at $h=1$.
Higher order interference was initially formalised by Sorkin in the framework of Quantum Measure Theory \cite{sorkin1994quantum} but has more recently been adapted to the setting of generalised probabilistic theories in \cite{barnum2014higher,lee2015generalised,ududec2011three,lee2015higher}. The most direct translation to this setting describes the order of interference in terms of probability distributions corresponding to the different experimental setups (which slits are open, etc.) \cite{lee2015generalised}. However, given our five principles, it is possible to define physical transformations that correspond to the action of blocking certain subsets of slits. In this case, there is a more convenient (and equivalent, given the five princples) definition in terms of such transformations \cite{barnum2014higher}.
If there are $N$ slits, labelled $1, \dots, N$, these transformations are denoted $P_I$, where $I \subseteq \{1, \dots, N\}:=\mathbf{N}$ corresponds to the subset of slits which are not blocked. In general we expect that $P_I P_J = P_{I\cap J}$, as only those slits belonging to both $I$ and $J$ will not be blocked by either $P_I$ or $P_J$. This intuition suggests that these transformations should correspond to projectors (i.e. idempotent transformations $P_IP_I=P_I$). Given principles $1$ to $5$, it was shown in \cite{barnum2014higher} that this is indeed the case. Given this structure, one can define the maximal order of interference as follows \cite{barnum2014higher}.
\begin{definition}
A theory satisfying principles $1$ to $5$ has maximal order of interference $h$ if, for any $N \geq h$, one has:
\[\mathds{1}_N = \sum_{{\small \begin{array}{c} I\subseteq \mathbf{N} \\ |I|\leq h \end{array}}}\mathcal{C}\left(h,|I|,N\right)P_I\]
where $\mathds{1}_N $ is the identity on a system with $N$ pure and perfectly distinguishable states and
\[\mathcal{C}\left(h,|I|,N\right):=(-1)^{h-|I|}\left(\begin{array}{c}N-|I|-1\\h-|I|\end{array}\right)\]
\end{definition}
The factor $\mathcal{C}\left(h,|I|,N\right)$ in the above definition corrects for the overlaps that occur when different combinations of slits are blocked.
Note that, for the case $h=N$, this reduces to the expected expression of $\mathds{1}_h=P_{\{1,...,h\}}$ i.e. the identity is given by the projector with all slits open. The case of $N=h+1$ corresponds to $\mathcal{C}\left(h,|I|,h+1\right) = (-1)^{h-|I|}$, which is the situation depicted in the previous figures, as well as the one most commonly discussed in the literature \cite{sorkin1994quantum,ududec2011three}.
Rather than work directly with these physical projectors, it is mathematically more convenient to work with (generally) unphysical transformations corresponding to projectors onto the ``coherences'' of a state. For example, in the case of a qutrit, the projector $P_{\{0,1\}}$ projects onto a two dimensional subspace:
\[ P_{\{0,1\}}::\left(\begin{array}{ccc} \rho_{00} & \rho_{01} & \rho_{02} \\ \rho_{10} & \rho_{11} & \rho_{12} \\\rho_{20} & \rho_{21} & \rho_{22} \end{array}\right)\mapsto \left(\begin{array}{ccc} \rho_{00} & \rho_{01} & 0 \\ \rho_{10} & \rho_{11} & 0 \\0 & 0 & 0 \end{array}\right)\]
whilst the coherence-projector $\omega_{\{0,1\}}$ projects only onto the coherences in that two dimensional subspace:
\[ \omega_{\{0,1\}}::\left(\begin{array}{ccc} \rho_{00} & \rho_{01} & \rho_{02} \\ \rho_{10} & \rho_{11} & \rho_{12} \\\rho_{20} & \rho_{21} & \rho_{22} \end{array}\right)\mapsto \left(\begin{array}{ccc} 0 & \rho_{01} & 0 \\ \rho_{10} & 0 & 0 \\0 & 0 & 0 \end{array}\right).\]
That is, $\omega_{\{0,1\}}$ corresponds to the linear combination of projectors: $P_{\{0,1\}}-P_{\{0\}}-P_{\{1\}}$.
There is a coherence-projector $\omega_I$ for each subset of slits $I \subseteq \mathbf{N}$, defined in terms of the physical projectors:
$$\omega_I:=\sum_{\tilde{I}\subseteq I}(-1)^{|I|+|\tilde{I}|}P_{\tilde{I}}.$$ These have the following useful properties, proved in appendix~\ref{AppendixCoherence}.
\begin{lemma} \label{lemma: decompisition of the identity into coherences}
An equivalent definition of the maximal order of interference, $h$, is: $\mathds{1}_N=\sum_{I,|I|=1}^h\omega_I,$ for all $ N \geq h.$
\end{lemma}
The above lemma implies that any state (indeed, any vector in the vector space generated by the states) can be decomposed as $s=\sum_{I,|I|=1}^h s_I,$ where $s_I:=\omega_Is$.
\begin{lemma} \label{lemma: orthogonality of coherences}
``Coherences are orthogonal'': $\mathrm{i)}$ $\omega_I\omega_J=\delta_{IJ}\omega_I$, for all $I,J$ and $\mathrm{ii)}$ $\Norm{s}^2=\sum_I\lVert\omega_Is\rVert^2$
\end{lemma}
\section{Setting up the problem}
In the standard search problem, one is asked to find a specific ``marked'' item from among a large collection of items in some unstructured list. The items are indexed $1, \dots, N$ and one has access to an oracle, which, when asked whether item $i$ is the marked item, denoted $x$, returns the answer ``yes'' or ``no''. Informally, the search problem asks for the minimal number of queries to this oracle required to find $x$ in the worst case.
In the standard bra-ket formalism of quantum theory, this oracle corresponds to a controlled unitary transformation $U$, defined by its action on the (product) computational basis:
$ U |i\rangle |q\rangle = |i\rangle |q\oplus f(i)\rangle,$
where $|i\rangle$ is the index, or control, register, $|q\rangle$ is the target register, $\oplus$ denotes addition modulo 2 and $f:\{1,\dots, N\}\rightarrow\{0,1\}$ satisfies $f(i)=1$ if and only if $i=x$. Inputting $|-\rangle$ into the target register results in a phase being ``kicked-back'' to the control register: $U|i \rangle |-\rangle=(-1)^{f(i)}|i\rangle|-\rangle.$ Discarding the target register reduces the action of the oracle to applying the phase transformation $O_{x}|i \rangle =(-1)^{f(i)}|i\rangle$.
Changing to the density matrix formalism, we see that this phase oracle, whose action on states $\rho$ is now denoted by $\Ora{x}\rho$,
acts as the identity on the diagonal elements of all density matrices whilst adding a `$-$' to the off diagonal elements $\{\rho_{xi},\rho_{ix}\}_{i}.$
Previous work \cite{lee2015generalised} has shown that the conjunction of principles 1, 2, 3 and 5 implies the existence of reversible controlled transformations. These can be used to define oracles in a manner analogous to quantum theory \cite{lee2015generalised}.
Moreover, every controlled transformation gives rise to a ``kicked-back'' reversible phase transformation on the control system \cite{lee2015generalised}. Thus---as in quantum theory---from the point of view of querying the oracle, we can reduce all considerations involving the controlled transformation to those involving the kicked-back phase.
To highlight the role of interference in searching an unstructured list, we describe the action of querying the oracle in terms of the physically motivated set-up of $N$-slit experiments.
Consider first the quantum case. Note that an $N$-slit experiment defines a set of $N$ pure and perfectly distinguishable states $\ket{i}\bra{i}$, each of which can be associated to a distinct element in the $N$ item list. Querying the oracle about item $i$ is equivalent to applying the oracle transformation to state $\ket{i}\bra{i}$. In quantum theory, preparing such a state can be achieved by passing a uniform superposition through the $N$-slit experiment with all but the $i$th slit blocked. The oracle can be implemented by placing a phase shifter behind slit $x$. Querying the oracle in a superposition of states can then be achieved by varying which slits are blocked. This is illustrated schematically below:
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As discussed previously, the physical act of blocking slits is represented by the projectors $P_I$. The action of the quantum oracle can thus be rephrased in terms of these projectors: i) $\Ora{x} P_I=P_I$, if $x \notin I$ or $|I|=1$ and, ii) $\Ora{x}$ can act non-trivially on projectors $P_I$ with $x \in I$ and $|I|>1$, but must satisfy $\Ora{x}P_I=P_I\Ora{x}$, for all $P_I$, which corresponds to the fact that a quantum oracle does not ``create'' or ``destroy'' coherence between states passing through different slits.
By analogy with the quantum case we can define the oracle which encodes the search problem in theories satisfying principles $1$ to $5$ as follows. Note that in this paper we only deal with the case of a single marked item.
\begin{definition} \label{Definition: Search Oracle}
A reversible transformation is a \emph{search oracle}, denoted $\Ora{x}$, if and only if:
$$ \begin{aligned} \mathrm{i)} \ \Ora{x}P_I&=P_I \text{ for} \ \text{all} \ x\notin I \ \text{or} \ |I|=1 \ \text{and,} \\
\mathrm{ii)} \ \Ora{x}P_I&=P_I \Ora{x}\text{, for all }P_I. \end{aligned}$$
\end{definition}
In the above definition, the requirement $\Ora{x}P_I=P_I \Ora{x}$, for all $P_I$, is quite natural. This requirement ensures that one cannot gain any information about item $i$ when querying the oracle using a state with no support on $i$, i.e. a state $s$ such that $P_I s=s$ where $i \notin I$.
In an arbitrary theory, it may not be the case that a transformation satisfying definition~\ref{Definition: Search Oracle} and acting non-trivially on $P_I$, with $x \in I$, exists. This is not an issue as in such theories we cannot even define the search problem, let alone show it can be solved using fewer queries than quantum theory. In this work, we shall assume the existence of a search oracle in any theory we consider.
Given the definition of coherence-projectors $\omega_I$ we can equivalently write definition~\ref{Definition: Search Oracle} as:
$\Ora{x} \omega_I=\omega_I$, for $x \notin I$ or $|I|=1$, and
$\Ora{x}\omega_I=\omega_I \Ora{x}$, for all $I$.
Indeed, in the quantum case, the action of the oracle can be equivalently described as:
$\Ora{x} \omega_I = \omega_I \ \text{ if } x\not\in I \ \text{ or } |I|=1 $, and $\Ora{x}\omega_I= -\omega_I \ \text{ otherwise}$.
We can now formally state the search problem for a single marked item---defined for the quantum case in \cite{nielsen2010quantum,zalka1999grover,boyer1996tight}---as:
\begin{search problem}
Given an $N$ element list with search oracle $\Ora{x}$ and an arbitrary collection of reversible transformations $\{G_i\}$, what is the minimal $k \in \mathbb{N}$ such that $G_k\Ora{x}G_{k-1}\dots G_1\Ora{x}s$ can be found, with probability greater than $1/2$, to be in the state $x$, for arbitrary state $s$, averaged over all possible marked items?
\end{search problem}
\section{Main result}
\begin{theorem} \label{Main Theorem}
In theories satisfying principles $1$ to $5$, with finite maximal order of interference $h$, the number of queries needed to solve the search problem is $\Omega(\sqrt{N/h})$.
\end{theorem}
\begin{proof}[Proof of theorem~\ref{Main Theorem}]
The basic idea is based on the proof of the quantum case presented in \cite{nielsen2010quantum,boyer1996tight,zalka1999grover}.
Let
$$\begin{aligned}
s_k^x&=G_k\Ora{x}G_{k-1}\dots G_1\Ora{x}s, \\
s_k&=G_kG_{k-1}\dots G_1s,
\end{aligned}$$
where $G_i$ is some reversible transformation from the theory, and define
$$D_k=\sum_x \Vert s_k^x - s_k \Vert^2.$$
It will be shown that, for $\langle x,s_k^x \rangle \geq 1/2$, we have $cN \leq D_k \leq 4hk^2$, where $c$ is any constant less than $\left(\sqrt{2}-1\right)^2$, from which the result $k \geq O\left(\sqrt{\frac{N}{h}}\right)$ follows. The lower bound goes through as in the quantum case and is derived in appendix~\ref{Appendix: lower bound for main proof}. The upper bound will now be proved by induction.
We have
$$\begin{aligned}
D_{k+1}&=\sum_x \Vert G_{k+1} \left( \Ora{x} s_k^x -s_k\right)\Vert^2=\sum_x \Vert \Ora{x} s_k^x -s_k\Vert^2 \\
&=\sum_x \Vert \Ora{x}\left( s_k^x -s_k\right) + \left(\Ora{x}-\mathds{1}\right)s_k\Vert^2\\
&\leq \sum_x \Vert s_k^x-s_k\Vert^2 \\
& \quad \quad +2\sum_x \Vert \Ora{x}\left( s_k^x -s_k\right)\Vert \Vert \left(\Ora{x}-\mathds{1}\right)s_k\Vert \\
& \quad \quad \quad \quad \quad +\sum_x \Vert\left(\Ora{x}-\mathds{1}\right)s_k\Vert^2 \\
&\leq D_k +2\sqrt{D_k \sum_x \Vert\left(\Ora{x}-\mathds{1}\right)s_k\Vert^2} +\Vert\left(\Ora{x}-\mathds{1}\right)s_k\Vert^2 \\
&\leq \left(\sqrt{D_k} + \sqrt{\sum_x \Vert (\mathds{1}-\mathcal{O}_x)s_k \Vert^2} \right)^2,
\end{aligned}
$$
which follows from the triangle inequality, the Cauchy-Schwarz inequality, and the fact the norm is invariant under reversible transformations.
The quantity $\sum_x\lVert(\mathds{1}-\mathcal{O}_x)s_k\rVert^2$---which can be thought of as how much some state is ``moved'' in a single query, averaged over all possible marked items $x$---is the only theory dependent quantity that features in this proof. We upper bound it as follows:
$$ \begin{aligned}
&\sum_x \lVert(\mathds{1}-\mathcal{O}_x){s_k}\rVert^2 \\
&=\sum_x\sum_I \lVert(\mathds{1}-\mathcal{O}_x)\omega_I{s_k}\rVert^2\\
&=\sum_x\sum_{{\tiny \begin{array}{c}I \\|I|>1\\ x\in I\end{array}}} \lVert\omega_I(\mathds{1}-\mathcal{O}_x){s_k}\rVert^2\\
&\leq\sum_x\sum_{{\tiny \begin{array}{c}I \\|I|>1\\ x\in I\end{array}}} \left(\lVert\mathds{1}\omega_I{s_k}\rVert+\lVert\mathcal{O}_x\omega_I{s_k}\rVert\right)^2 \\
& \leq\sum_x\sum_{{\tiny \begin{array}{c}I \\|I|>1\\ x\in I\end{array}}} 4\lVert\omega_I{s_k}\rVert^2,
\end{aligned}$$
where the first line follows from lemma~\ref{lemma: decompisition of the identity into coherences}, lemma~\ref{lemma: orthogonality of coherences}, and the definition of the search oracle $\Ora{x}$, and second from the triangle inequality and the fact that the norm is invariant under reversible transformations. We need to know how many times each $\lVert\omega_I{s_k}\rVert^2$ appears when we sum over the marked item $x$. Each given $I=\{i_1, i_2, \dots, i_{|I|}\}$ will appear $|I|$ times as we sum over $x$, one for every time $i_j$ is the marked item. Thus
$$ \begin{aligned}
&\sum_x \lVert(\mathds{1}-\mathcal{O}_x){s_k}\rVert^2 \leq \sum_{{\tiny \begin{array}{c}I \\|I|>1\end{array}}} 4|I|\lVert\omega_I{s_k}\rVert^2\\
&\leq 4 \sum_{I} |I|\lVert\omega_I{s_k}\rVert^2\leq 4h \sum_{I} \lVert\omega_I{s_k}\rVert^2= 4h \lVert{s_k}\rVert^2\leq 4h.
\end{aligned}$$
The second line follows from $\sum_{|I|=1}\Vert \omega_I s_k \Vert^2 \geq 0$, lemma~\ref{lemma: orthogonality of coherences}, $\Vert s_k\Vert \leq 1$, and $|I| \leq h$, for all $I$.
We thus have:
$
\begin{aligned}
D_{k+1}\leq \left(\sqrt{D_k} + \sqrt{4h}\right)^2.
\end{aligned}$
Assuming that $D_k \leq 4hk^2$ gives us $D_{k+1} \leq 4h(k+1)^2,$ from which the result follows via induction.
\end{proof}
\section{Discussion}
In this work, we considered theories satisfying certain natural physical principles which are sufficient for the existence of controlled transformations and a phase kick-back mechanism, necessary features for a well-defined search oracle. Given these physical principles, we proved that a theory with maximal order of interference $h$ requires $\Omega(\sqrt{N/h})$ queries to this oracle to find a single marked item from some $N$-element list. This result challenges our pre-conceived notions about how quantum computers achieve their computational advantage and is somewhat surprising as one might expect more interference to imply more computational power.
Further work will focus on determining sufficient physical principles for there to exist an algorithm that achieves the quadratic lower bound derived here.
Recent work has also investigated Grover's algorithm from the point of view of post-quantum theories \cite{aaronson2016space,bao2015grover}. These works considered modifications of quantum theory which allow for superluminal signalling and cloning of states. In contrast, the generalised probabilistic theory framework employed here allowed us to investigate Grover's lower bound in alternate theories that are physically reasonable and which, for example, do not allow for superluminal signalling \cite{barrett2007information} or cloning \cite{barnum}.
As theories satisfying our five physical principles appear `quantum-like'---at least from the point of view of the search problem---investigating interference behaviour in them may inform current experiments searching for post-quantum interference.
\emph{Acknowledgements---}The authors thank H. Barnum and M. J. Hoban for useful discussions and M. J. Hoban for proof reading a draft of the current paper. The authors also acknowledge encouragement and support from J. J. Barry. This work was supported by the EPSRC through the Controlled Quantum Dynamics Centre for Doctoral Training and the Oxford Department of Computer Science. CML also acknowledges funding from University College, Oxford.
\appendix
\section{Results for coherences} \label{AppendixCoherence}
\subsection{Proof of lemma~\ref{lemma: decompisition of the identity into coherences}}
In a theory with maximal order of interference $h$ one has
\[\mathds{1}_N = \sum_{{\small \begin{array}{c} I\subseteq \mathbf{N} \\ |I|\leq h \end{array}}}\mathcal{C}\left(h,|I|,N\right)P_I.\]
Thus, showing $\mathds{1}_N=\sum_{|I|=1}^h \omega_I$ reduces to showing
\[\sum_{|I|=1}^h \omega_I= \sum_{{\small \begin{array}{c} I\subseteq \mathbf{N} \\ |I|\leq h \end{array}}}\mathcal{C}\left(h,|I|,N\right)P_I.\] As $\omega_I:=\sum_{\tilde{I}\subseteq I}(-1)^{|I|+|\tilde{I}|}P_{\tilde{I}}$, we just have to count the number of $P_I$'s that appear as we sum over $|I|$. For some fixed $I$, this is just
$$ \sum_{\alpha=|I|}^h (-1)^{\alpha-|I|} \left(\begin{array}{c}N-|I|\\\alpha-|I|\end{array}\right).$$ By expanding and rearranging this, one can straightforwardly (if tediously) show that this equals $\mathcal{C}\left(h,|I|,N\right)$, and we are done.
\subsection{Proof of lemma~\ref{lemma: orthogonality of coherences} part $\mathrm{i)}$\label{AppendixCoherence1}}
From the definition of $\omega_I$, it follows that
\[\omega_I\omega_J = (-1)^{|I|+|J|}\sum_{\widetilde{I}\subseteq I}\sum_{\widetilde{J}\subseteq J}(-1)^{|\widetilde{I}|+|\widetilde{J}|}P_{\widetilde{I}}P_{\widetilde{J}}\]
\[=(-1)^{|I|+|J|}\sum_{\widetilde{K}\subseteq I\cap J}\mathcal{D}\left(I,J,\widetilde{K}\right) P_{\widetilde{K}}\]
where $\mathcal{D}\left(I,J,\widetilde{K}\right)$ is the number of distinct pairings of $\widetilde{I}$ and $\widetilde{J}$ such that $\widetilde{I}\cap\widetilde{J}=\widetilde{K}$ and $|\widetilde{I}|+|\widetilde{J}|$ is even, minus the number of distinct pairings where $\widetilde{I}\cap\widetilde{J}=\widetilde{K}$ and $|\widetilde{I}|+|\widetilde{J}|$ is odd. It will now be shown that
\[\mathcal{D}\left(I,J,\widetilde{K}\right)= \left\{ \begin{array}{cc} 0 & \text{ if } I\neq J \\ (-1)^{|I|+|\widetilde{K}|} & \text{ if } I=J \end{array}\right. \]
For the $I\neq J$ case fix some particular $i \in I$ such that $i\not\in J$ and consider some $\widetilde{I}\subseteq{I},\widetilde{J}\subseteq{J}$ such that $\widetilde{I}\cap\widetilde{J}=\widetilde{K}$. If $x \notin \widetilde{I}$ alter $\widetilde{I}$ by adding $i$, otherwise alter $\widetilde{I}$ by removing $x$. This procedure turns each even $|\widetilde{I}|+|\widetilde{J}|$, odd. We have thus shown that for each $\widetilde{I}\subseteq{I}$ and $\widetilde{J}\subseteq{J}$ such that $\widetilde{I}\cap\widetilde{J}=\widetilde{K}$ and $|\widetilde{I}|+|\widetilde{J}|$ is even, there exists an $\widetilde{I}'\subseteq{I}$ such that $\widetilde{I}'\cap\widetilde{J}=\widetilde{K}$ and $|\widetilde{I}'|+|\widetilde{J}|$ is odd, and vice versa. Thus the number of distinct pairings of $\widetilde{I}$ and $\widetilde{J}$ such that $\widetilde{I}\cap\widetilde{J}=\widetilde{K}$ and $|\widetilde{I}|+|\widetilde{J}|$ is even is equal to the number of distinct pairings of $\widetilde{I}$ and $\widetilde{J}$ such that $\widetilde{I}\cap\widetilde{J}=\widetilde{K}$ and $|\widetilde{I}|+|\widetilde{J}|$ is odd, and so $\mathcal{D}\left(I,J,\widetilde{K}\right)=0$ when $I\neq J$.
For the $I=J$ case we can make a similar argument by picking some $i\in I, i\not\in \widetilde{J}$ except for when $\widetilde{J}=J=I$. This case gives an excess $\pm 1$ depending on whether $|J|+|\widetilde{K}|$ is odd or even, implying $\mathcal{D}\left(I,J,\widetilde{K}\right)=(-1)^{|I|+|\widetilde{K}|}$ when $I=J$.
This immediately gives $\omega_I\omega_J=0$ if $I\neq J$ and,
\[\omega_I\omega_I = (-1)^{2|I|}\sum_{\widetilde{K}\subseteq I}(-1)^{|I|+|\widetilde{K}|} P_{\widetilde{K}} = \omega_I\]
if $I=J$.
\subsection{Proof of lemma~\ref{lemma: orthogonality of coherences} part $\mathrm{ii)}$\label{AppendixCoherence2}}
To prove the lemma, we need the fact that the $\omega_I$'s are self-dual $\omega_I^\dagger=\omega_I$, where the $\dagger$ is defined by the the self-dualising inner-product as: $\langle \cdot, \omega_I \cdot\rangle=\langle\omega_I^\dagger\cdot, \cdot\rangle$. Recalling that the $\omega_I$'s correspond to linear combinations of the $P_I$'s, this follows immediately from self-duality of the projectors $P_I$, which is proved in \cite{barnum2014higher} (Recall that principles $1$ to $5$ imply the first two axioms of \cite{barnum2014higher}). We now have
\[\begin{aligned}
\Vert s \Vert^2 &= \langle s, s\rangle = \langle\sum_I\omega_I s, \sum_J\omega_J s\rangle \end{aligned} \]
\[=\sum_{I,J}\langle\omega_I s,\omega_J s\rangle =\sum_{I,J}\langle s,\omega_I^\dagger\omega_J s\rangle\]
\[=\sum_{I,J}\langle s,\omega_I\omega_J s\rangle=\sum_{I,J}\delta_{IJ}\langle s,\omega_I s\rangle\]
where the last equality follows from the orthogonality of the $\omega_I$'s. Finally
\[\begin{aligned}\Vert s\Vert^2=\sum_{I}\langle s,\omega_I{s}\rangle=\sum_{I}\langle\omega_I{s},\omega_I{s}\rangle= \sum_I\lVert\omega_I{s}\rVert^2
\end{aligned}\]
\subsection{Proof of $D_k \geq cN$} \label{Appendix: lower bound for main proof}
We assume that $\langle x,{s}_k^x \rangle \geq1/2$ for all $x$, so a measurement of $s_k^x$ yields a solution to the search problem with probability at least $1/2$. Let $E_k=\sum_x \Vert {s}_k^x - x \Vert^2$ and $F_k=\sum_x \Vert {s}_k - x \Vert^2$. It follows that
$$\begin{aligned} &\text{i) }E_k=\sum_x 2(1- \langle x,{s}_k^x \rangle ) \leq \sum_x 2(1- 1/2) \leq N\text{ and,} \\
&\text{ii) } F_k\geq 2\left(N-\Vert {s}_k \Vert \sqrt{\left\langle \sum_x x, \sum_y y \right\rangle}\right)\geq 2\left(N-\sqrt{N}\right) \end{aligned}$$
where $\text{ii)}$ follows from the Cauchy-Schwarz inequality, $\Vert {s}_k \Vert \leq 1$ and $\langle x,y \rangle = \delta_{xy}$. As explicitly calculated on page $270$ of \cite{nielsen2010quantum}, by using the reverse triangle inequality and the Cauchy-Schwarz inequality, it follows that
$D_k \geq \left( \sqrt{F_k}-\sqrt{E_k}\right)^2$. Combing this with the upper bound on $E_k$ and the lower bound on $F_k$, we have that $D_k \geq cN,$ for sufficiently large $N$, where $c$ is any constant less than $\left(\sqrt{2}-1\right)^2\approx 0.17$.
\end{document}
|
\begin{document}
\title{Minimality, (Weighted) Interpolation in Paley-Wiener Spaces \& Control
Theory}
\author{Fr\'ed\'eric Gaunard}
\begin{abstract}
It is well known from a result by Shapiro-Shields that in the Hardy
spaces, a sequence of reproducing kernels is uniformly minimal if
and only if it is an unconditional basis in its span. This property
which can be reformulated in terms of interpolation and so-called
weak interpolation is not true in Paley-Wiener spaces in general.
Here we show that the Carleson condition on a sequence $\Lambda$
together with minimality in Paley-Wiener spaces $PW_{\tau}^{p}$ of
the associated sequence of reproducing kernels implies the interpolation
property of $\Lambda$ in $PW_{\tau+\epsilon}^{p}$, for every $\epsilon>0$.
With the same technics, using a result of McPhail, we prove a similary
result about minimlity and weighted interpolation in $PW_{\tau+\epsilon}^{p}$..
We apply the results to control theory, establishing that, under some
hypotheses, a certain weak type of controllability in time $\tau>0$
implies exact controllability in time $\tau+\epsilon$, for every
$\epsilon>0$.
\end{abstract}
\subjclass[2000]{30E05, 42A15, 93B05.}
\keywords{Interpolation, Paley-Wiener spaces, minimal sequences, controllability.}
\date{\today}
\maketitle
\section{Introduction}
Let $X$ be a Banach space. A sequence $\left\{ \phi_{n}\right\} _{n\geq1}$
of vectors of $X$ is said to be \emph{minimal }in $X$ if $\phi_{n}\not\in\bigvee_{k\neq n}\phi_{k}:=\overline{\text{span}}^{X}\left(\phi_{k}:\: k\not=n\right)$,
$n\geq1$, and \emph{uniformly minimal} if moreover \begin{equation}
\inf_{n\geq1}\text{dist}\left(\frac{\phi_{n}}{\left\Vert \phi_{n}\right\Vert },\:\bigvee_{k\neq n}\phi_{k}\right)>0.\label{def: unif min.}\end{equation}
It is well known (see e.g. \cite[p. 93]{Ni02a}) that minimality of
$\left\{ \phi_{n}\right\} _{n\geq1}$ in $X$ is equivalent to the
existence of a sequence $\left\{ \psi_{n}\right\} _{n\geq1}\subset X^{\star}$
such that $\left\langle \phi_{n},\psi_{k}\right\rangle =\delta_{nk}$
and the minimality is said uniform if and only if \begin{equation}
\sup_{n\geq1}\:\left\Vert \phi_{n}\right\Vert \cdot\left\Vert \psi_{n}\right\Vert <\infty.\label{eq: unif min equiv}\end{equation}
We consider the case where $X$ is a Banach space of analytic functions
on a domain $\Omega$. Let $\Lambda=\left\{ \lambda_{n}\right\} _{n\geq1}$
be a sequence of complex numbers lying in $\Omega$. We use the terminology
\emph{minimal }also for the sequence $\Lambda$ if there exists a
sequence of functions $(f_{n})_{n\geq1}$ of $X$ such that\[
f_{n}(\lambda_{k})=\delta_{nk},\quad n,k\geq1,\]
and we say that $\Lambda$ is a \emph{weak interpolating sequence}
in $X$, which is denoted by $\Lambda\in\text{Int}_{w}\left(X\right)$,
if there exists a sequence of functions $\left(f_{n}\right)_{n\geq1}$
of $X$ such that\begin{equation}
f_{n}\left(\lambda_{k}\right)=\delta_{nk}\left\Vert k_{\lambda_{n}}\right\Vert _{X^{\star}},\; n\geq1,\text{ and }\sup_{n\geq1}\left\Vert f_{n}\right\Vert <\infty.\label{Int w}\end{equation}
When $X$ is reflexive, this is equivalent to the fact that $\mathcal{K}(\Lambda)$
is uniformly minimal in $X^{\star}$. Such a sequence $\Lambda$ could
also be called a uniformly minimal sequence in $X$ but we prefer
to keep the existing terminology of weak interpolating sequence.
In the case where $X=H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$, $1<p<\infty$,
the Hardy space of the half-plane $\mathbb{C}_{a}^{\pm}$, we can
identify \[
\left(H^{p}\left(\mathbb{C}_{a}^{\pm}\right)\right)^{\star}\simeq H^{q}\left(\mathbb{C}_{a}^{\pm}\right),\qquad\frac{1}{p}+\frac{1}{q}=1,\]
and it is known that the reproducing kernel at $\lambda\in\mathbb{C}_{a}^{\pm}$
is given by $k_{\lambda_{n}}\left(z\right)=\frac{i}{2\pi}\left(z-\overline{\lambda}_{n}\right)^{-1}$.
We have the estimate \[
\left\Vert k_{\lambda_{n}}\right\Vert _{H^{q}\left(\mathbb{C}_{a}^{\pm}\right)}\asymp\left|\text{Im}\left(\lambda_{n}\right)-a\right|^{-\frac{1}{p}}.\]
From factorization in Hardy spaces, it can be deduced that the condition
(\ref{Int w}) is equivalent to the so-called\emph{ Carleson condition}
\begin{equation}
\text{inf}_{n\geq1}\prod_{k\not=n}\left|\frac{\lambda_{n}-\lambda_{k}}{\lambda_{n}-\overline{\lambda}_{k}-2ia}\right|>0,\label{Carleson}\end{equation}
and by the above observations, this is equivalent to $\mathcal{K}(\Lambda)$
being uniformly minimal in $H^{q}\left(\mathbb{C}_{a}^{\pm}\right)$.
In this paper, the spaces $X$ that we consider will only be the Hardy
spaces $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$ or the Paley-Wiener
spaces $PW_{\tau}^{p}$ to be defined later.
We say that $\Lambda$ is an \emph{interpolating sequence} for $X=H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$
or $PW_{\tau}^{p}$, which is denoted by $\Lambda\in\text{Int}\left(X\right)$,
if for each sequence $a=\left(a_{n}\right)_{n\geq1}\in l^{p}$, there
is a fonction $f\in X$ such that \begin{equation}
f\left(\lambda_{n}\right)=a_{n}\left\Vert k_{\lambda_{n}}\right\Vert _{X^{\star}},\quad n\geq1,\label{int def}\end{equation}
and a \emph{complete interpolating sequence} for $X$ ($\Lambda\in\text{Int}_{c}\left(X\right)$)
if the function satisfying (\ref{int def}) is unique. We will give
the explicit formula of $k_{\lambda_{n}}$ for $PW_{\tau}^{p}$ and
an estimate of $\left\Vert k_{\lambda_{n}}\right\Vert _{PW_{\tau}^{q}}$
in the next section.
A famous result by Shapiro and Shields (\cite{SS61}) states that
in $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$, the Carleson condition
(\ref{Carleson}) for a sequence $\Lambda$ is equivalent to the interpolation
property of $\Lambda$. It is also known that $\Lambda\in\text{Int}\left(H^{p}\left(\mathbb{C}_{a}^{\pm}\right)\right)$
if and only if $\mathcal{K}\left(\Lambda\right)$ is an unconditional
basis (or, for $p=2$, a Riesz basis) in its span in $H^{q}\left(\mathbb{C}_{a}^{\pm}\right)$.
We refer to \cite[Section C, Chapter 3]{Ni02b} for definitions and
details. It appears that in the Hardy spaces, the uniformly minimal
sequences are exactly the unconditional sequences.
This property of equivalence between uniform minimality and unconditionnality
is not isolated. It turns out to be true in the Bergmann space (\cite{ScS98}),
in the Fock spaces and in the Paley Wiener spaces for certain values
of $p$ (\cite{ScS00}).
In \cite{AH10}, the authors show that uniform minimality implies
unconditionality in a bigger space for certain backward shift invariant
spaces $K_{I}^{p}:=H^{p}\cap I\overline{H_{0}^{p}}$ (considered here
on the unit circle $\mathbb{T}$) for which the Paley-Wiener spaces
are a particular case. We will use here a different approach to obtain
a stronger result. More precisely, considering the unit disk, the
authors of that paper increase the size of the space $K_{I}^{p}$
in two directions: $K_{J}^{s}$, where $s<p$ and $J$ is an inner
multiple of $I$. In our situation of the Paley-Wiener space $PW_{\tau}^{p}$,
which is isometric to $K_{I_{\mathbb{D}}^{\tau}}^{p}$, $I_{\mathbb{D}}^{\tau}(z)=\exp\left(2\tau(z+1)/(z-1)\right)$
for $z\in\mathbb{D}$, we still increase the size of the space by
taking an inner multiple of $I_{\mathbb{D}}^{\tau}$ but we keep the
same $p$.
We have already mentioned that $\Lambda$ is an interpolating sequence
for the Hardy space $H^{p}$, $1<p<\infty$, if and only if the sequence
$\mathcal{K}(\Lambda)$ is an unconditional basis in its span in $H^{q}$
(see e.g. \cite{Ni02b} or \cite{Se04}). Hence, the result of Shapiro
and Sheilds implies that weak interpolation is equivalent to interpolation
in Hardy spaces. A characterization of complete interpolating sequences
in $PW_{\tau}^{p}$ obtained by Lyubarskii and Seip (\cite{LS97})
(involing in particular Carleson's condition and the Muckenhoupt $(A_{p})$
condition on the generating function of $\Lambda$) implies that Paley-Wiener
spaces do not have this property.
Indeed, as shown in \cite{ScS00}, the sequence $\Lambda=\left\{ \lambda_{n}:\: n\in\mathbb{Z}\right\} $
defined by\[
\lambda_{0}=0,\quad\lambda_{n}=n+\frac{\text{sign}\left(n\right)}{2\max\left(p,q\right)},\; n\in\mathbb{Z}\setminus\left\{ 0\right\} ,\]
is weak interpolating in $PW_{\pi}^{p}$ and complete, but does not
satisfy the conditions of Lyubarskii-Seip's result, and so, $\Lambda$
is not a (complete) interpolating sequence in $PW_{\pi}^{p}$. Nevertheless,
as we will discuss in Subsection \ref{sub:Upper-Uniform-Density},
a density argument (see \cite{Se95}) allows to show that this sequence
is actually an interpolating sequence in a bigger space, i.e. in $PW_{\pi+\epsilon}^{p}$,
for arbitrary $\epsilon>0$. This is a special case of the main result
of this paper.
\begin{thm}
\label{thm: MAIN RESULT}Let $\tau>0$, $1<p<\infty$ and $\Lambda$
be a minimal sequence in $PW_{\tau}^{p}$ such that for every $a\in\mathbb{R}$,
$\Lambda\cap\mathbb{C}_{a}^{\pm}$ satisfies the Carleson condition
(\ref{Carleson}). Then, for every $\epsilon>0$, $\Lambda$ is an
interpolating sequence in $PW_{\tau+\epsilon}^{p}$.
\end{thm}
It should be emphasized that surprisingly, we do not need to require
uniform minimality here. The Carleson condition allows in a way to
compensate this lack of uniformity. As a consequence of this result,
we will see that if $\Lambda\in\text{Int}_{w}\left(PW_{\tau}^{p}\right)$,
then, for every $\epsilon>0$, $\Lambda\in\text{Int}\left(PW_{\tau+\epsilon}^{p}\right)$.
Finally, we recall that a positive measure $\sigma$ on some half-plane
$\mathbb{C}_{a}^{\pm}$ is called a \emph{Carleson measure} in $\mathbb{C}_{a}^{\pm}$
if\begin{equation}
\sup_{Q}\frac{\sigma\left(Q\right)}{h}<\infty,\label{eq:Carleson measure}\end{equation}
where the supremum is taken over all the squares $Q$ of the form\[
Q=\left\{ z=x+iy\in\mathbb{C}_{a}^{\pm}:\: x_{0}<x<x_{0}+h,\;\left|y-a\right|<h\right\} ,\]
for $x_{0}\in\mathbb{R}$ and $h>0$. It is well known from a result
of Carleson (see e.g. \cite[pp. 61 and 278]{Ga81}) that $(1)\Rightarrow(2)\Leftrightarrow(3)$,
where
$(1)$ The sequence $\Lambda=\left\{ \lambda_{n}:\: n\geq1\right\} \subset\mathbb{C}_{a}^{\pm}$
satisfies the Carleson condition (\ref{Carleson});
$(2)$ The measure\[
\sigma_{\Lambda}:=\sum_{n\geq1}\left|\text{Im}\left(\lambda_{n}\right)-a\right|\delta_{\lambda_{n}}\]
is a Carleson measure in $\mathbb{C}_{a}^{\pm}$;
$(3)$ For every $f\in H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$, \[
\int\left|f\right|^{p}d\sigma_{\Lambda}\lesssim\left\Vert f\right\Vert _{p}^{p}.\]
It is also known that $(2)$ or $(3)$ together with the \emph{uniform
pseudo-hyperbolic separation} of $\Lambda$ in $\mathbb{C}_{a}^{\pm}$,
which is \begin{equation}
\inf_{n\neq m}\left|\frac{\lambda_{n}-\lambda_{m}}{\lambda_{n}-\overline{\lambda}_{m}-2ia}\right|>0,\label{unif psh separation}\end{equation}
imply $(1)$. Moreover, if $\Lambda$ lies in a strip of finite width,
$i.e.$ $M:=\sup_{n}\left|\text{Im}\left(\lambda_{n}\right)\right|<\infty$,
the Carleson condition (\ref{Carleson}) in $\mathbb{C}_{\mp2M}^{\pm}$
is equivalent to the \emph{uniform separation condition} \begin{equation}
\inf_{n\neq m}\left|\lambda_{n}-\lambda_{m}\right|>0\label{uniform separation}\end{equation}
which is, in this case, equivalent to the uniform pseudo-hyperbolic
separation since the pseudo-hyperbolic metric defined in $\mathbb{C}_{\mp2M}^{\pm}$
by\[
\rho\left(\lambda,\mu\right)=\left|\frac{\lambda-\mu}{\lambda-\overline{\mu}-2iM}\right|\]
is locally equivalent to the euclidian distance.
This paper is organized as follows. The next section will be devoted
to interpolation in Paley-Wiener spaces. After having recalled some
properties of these spaces, we discuss links between density and interpolation
(in the case of the sequence $\Lambda$ lying in a strip of finite
width) and prove our main result and some consequences.
In the third section, we define and discuss weighted interpolation.
Indeed, after having defined weighted interpolation in Hardy and Paley-Wiener
spaces, we use a result of McPhail (\cite{McP90}) characterizing
the weighted interpolation sequences in $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$
and technics of Theorem \ref{thm: MAIN RESULT} to prove that a minimal
sequence in $PW_{\tau}^{p}$ such that its intersection with every
half-plane satisfies the McPhail condition is a weighted interpolation
sequence in $PW_{\tau+\epsilon}^{p}$, for every $\epsilon>0$.
This theorem will be used in the fourth and last section where we
consider controllability of linear differential systems, establishing
a link between exact and a certain weak type of controllability.
\section{Interpolation in Paley-Wiener Spaces}
We begin by recalling some facts about Paley-Wiener spaces. For $\tau>0$,
the Paley-Wiener space $PW_{\tau}^{p}$ consists of all entire functions
of exponential type at most $\tau$ satisfying\[
\left\Vert f\right\Vert _{p}^{p}=\int_{-\infty}^{+\infty}\left|f(t)\right|^{p}dt<\infty.\]
A result, known as Plancherel-Polya inequality (see e.g. \cite{Le96}
or \cite[p.95]{Se04}) states that if $f\in PW_{\tau}^{p}$, then\begin{equation}
\int_{-\infty}^{+\infty}\left|f(x+iy)\right|^{p}dx\leq e^{p\tau|y|}\left\Vert f\right\Vert _{p}^{p}.\label{eq:plancherel polya}\end{equation}
In particular, it follows that for every $f\in PW_{\tau}^{p}$, $z\mapsto e^{\pm i\tau z}f(z)$
belongs to $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$, with same norm
as $f$. It also follows that translation is an isomorphism from $PW_{\tau}^{p}$
into itself.
Using Cauchy's formula and Cauchy's theorem in an appropriate way,
we see that \[
k_{\lambda}(z)=\frac{\sin\tau\left(z-\overline{\lambda}\right)}{\tau\left(z-\overline{\lambda}\right)}\]
is the reproducing kernel of $PW_{\tau}^{p}$ associated to $\lambda$
and we obtain the norm estimate\[
\left\Vert k_{\lambda}\right\Vert _{PW_{\tau}^{q}}\asymp\left(1+\left|\text{Im}(\lambda)\right|\right)^{-\frac{1}{p}}e^{\tau\left|\text{Im}(\lambda)\right|}.\]
This implies a useful pointwise estimate; recalling that for $\frac{1}{p}+\frac{1}{q}=1$,
$\left(PW_{\tau}^{p}\right)^{\star}\simeq PW_{\tau}^{q}$, we deduce
that there exists a constant $C=C(p)$ such that for every $f\in PW_{\tau}^{p}$,
we have\begin{equation}
\left|f(z)\right|\leq C\left\Vert f\right\Vert _{p}\left(1+\left|\text{Im}\left(z\right)\right|\right)^{-\frac{1}{p}}e^{\tau\left|\text{Im}(z)\right|},\quad z\in\mathbb{C}.\label{pointwise est. PW}\end{equation}
The Paley-Wiener theorem states that\[
L^{2}(0,2\tau)\simeq\mathcal{F}L^{2}(-\tau,\tau)=PW_{\tau}^{2},\]
where $\mathcal{F}$ denotes the Fourier transform\[
\mathcal{F}\phi(z)=\int_{-\tau}^{\tau}\phi(t)e^{-itz}dt.\]
Hence, another approach to interpolation problems in $PW_{\tau}^{2}$
is to consider geometric properties of exponentials in $L^{2}(0,2\tau)$,
which is a famous problem with several applications, see e.g. \cite{HNP81}
or \cite{AI95}.
From the definitions given previously, a sequence $\Lambda$ is interpolating
in $PW_{\tau}^{p}$ if, for every sequence $a=\left(a_{n}\right)_{n\geq1}\in l^{p}$,
it is possible to find a function $f\in PW_{\tau}^{p}$ such that\begin{equation}
f\left(\lambda_{n}\right)\left(1+\left|\text{Im}\left(\lambda_{n}\right)\right|\right)^{\frac{1}{p}}e^{-\tau\left|\text{Im}\left(\lambda_{n}\right)\right|}=a_{n},\quad n\geq1.\label{eq:def int PW}\end{equation}
The condition of weak interpolation for $\Lambda$ in $PW_{\tau}^{p}$
can be reformulated as the existence of a sequence of functions $\left(f_{n}\right)_{n\geq1}\subset PW_{\tau}^{p}$
such that\[
f_{n}\left(\lambda_{k}\right)\left(1+\left|\text{Im}\left(\lambda_{n}\right)\right|\right)^{\frac{1}{p}}e^{-\tau\left|\text{Im}\left(\lambda_{n}\right)\right|}=\delta_{nk},\quad n\geq1,\]
and $\sup_{n\geq1}\left\Vert f_{n}\right\Vert <\infty$.
In particular, if $\Lambda\in\text{Int}_{w}\left(PW_{\tau}^{p}\right)$,
then the Plancherel-Polya inequality (\ref{eq:plancherel polya})
implies that the sequence $\left(e^{\pm i\tau\cdot}f_{n}\right)_{n\geq1}$is
in $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$, with uniform control
of the norm. So, it is easy to see that $\Lambda\cap\mathbb{C}_{a}^{\pm}\in\text{Int}_{w}\left(H^{p}\left(\mathbb{C}_{a\pm\eta}^{\pm}\right)\right)$,
for every $\eta>0$, and hence satisfies the Carleson condition in
the corresponding half-plane, in view of Shapiro-Shields 's theorem.
Moreover, we can affirm that the sequence \[
\Lambda_{a,\eta}:=\Lambda\cap\left\{ z\in\mathbb{C}:\;0<\left|\text{Im}(z)-a\right|<2\eta\right\} \]
is uniformly separated, in view of the discussion in the end of the
previous section. These two observations imply the following result
(for more details, see \cite{Gau11}).
\begin{prop}
\label{w Int implies Carleson}If $\Lambda$ is a weak interpolating
sequence in $PW_{\tau}^{p}$, then, for every $a\in\mathbb{R}$, the
sequence $\Lambda\cap\mathbb{C}_{a}^{\pm}$ satisfies the Carleson
condition (\ref{Carleson}).
\end{prop}
\subsection{\label{sub:Upper-Uniform-Density}Upper Uniform Density and Interpolation}
In this subsection, we assume that the sequence $\Lambda$ satisfies\begin{equation}
M:=\sup_{n\geq1}\left|\text{Im}\left(\lambda_{n}\right)\right|<\infty,\label{lambda incluse dans bande}\end{equation}
which means that $\Lambda$ lies in a strip of finite width parallel
to the real axis. We define the \emph{upper uniform density} $\mathcal{D}_{\Lambda}^{+}$
by\[
\mathcal{D}_{\Lambda}^{+}:=\lim_{r\to\infty}\frac{n_{\Lambda}^{+}\left(r\right)}{r},\]
where \[
n_{\Lambda}^{+}\left(r\right):=\sup_{x\in\mathbb{R}}\left|\text{Re}\left(\Lambda\right)\cap\left[x,x+r\right]\right|,\]
counting multiplicities.
The reader would have remembered that, from previous remarks, Proposition
\ref{w Int implies Carleson} implies that a weak interpolating sequence
in a Paley-Wiener space $PW_{\tau}^{p}$ satisfies the uniform separation
condition.
The next theorem is stated as follows in a paper of Seip (\cite[Theorem 2.2]{Se95})
the proof of which is based on a more general result by Beurling (\cite{Be89}).
\begin{thm}
(\cite{Se95})\label{thm:(densite et interpolation)} Let $\Lambda$
be a sequence satisfying (\ref{lambda incluse dans bande}).
If $\Lambda$ is uniformly separated and $\mathcal{D}_{\Lambda}^{+}<\frac{\tau}{\pi}$
, then $\Lambda\in\text{Int}\left(PW_{\tau}^{p}\right)$. Conversely,
if $\Lambda\in\text{Int}\left(PW_{\tau}^{p}\right)$, then, $\Lambda$
is necessarily uniformly separated and $\mathcal{D}_{\Lambda}^{+}\leq\frac{\tau}{\pi}$.\end{thm}
\begin{cor}
\label{cor:densite n+sg/2p'}The sequence $\Lambda=\left\{ \lambda_{n}:\: n\in\mathbb{Z}\right\} $
defined by \[
\lambda_{0}=0\text{ et }\lambda_{n}=n+\frac{\text{sign}(n)}{2\max\left(p,q\right)},\quad n\neq0,\]
is interpolating in $PW_{\pi+\epsilon}^{p}$, for every $\epsilon>0$.\end{cor}
\begin{proof}
We have already mentioned that this sequence is uniformly minimal.
The uniform separation condition is obvious. Its upper uniform density
is clearly equal to $1$. The Corollary now follows from Theorem \ref{thm:(densite et interpolation)}.
\end{proof}
\subsection{Proof of Main Result}
We recall our main theorem, previously stated in the first section.
\begin{thm*}
(Theorem \ref{thm: MAIN RESULT}) Let $\tau>0$, $1<p<\infty$ and
$\Lambda$ be a minimal sequence in $PW_{\tau}^{p}$ such that for
every $a\in\mathbb{R}$, $\Lambda\cap\mathbb{C}_{a}^{\pm}$ satisfies
the Carleson condition (\ref{Carleson}). Then, for every $\epsilon>0$,
$\Lambda$ is an interpolating sequence in $PW_{\tau+\epsilon}^{p}$.\end{thm*}
\begin{proof}
Using an idea of Beurling, let $\epsilon>0$ be fixed and $\phi_{\epsilon}\in\mathcal{C}^{\infty}$,
with compact support contained in $\left(-\frac{\epsilon}{2},\frac{\epsilon}{2}\right)$,
be such that $H_{\epsilon}:=c\mathcal{F}\phi_{\epsilon}$ satisfies
$H_{\epsilon}(0)=1$. In particular, the Paley-Wiener theorem implies
that $H_{\epsilon}$ is an entire function of exponential type $\epsilon$.
Moreover, since $\phi_{\epsilon}$ belongs to the Schwarz class (and
in particular is $\mathcal{C}^{1}$), we have t\[
\left|H_{\epsilon}\left(x\right)\right|\lesssim\frac{1}{1+|x|},\quad x\in\mathbb{R}.\]
Now, from a a Phragmen-Lindelf principle (see e.g. \cite[p.39]{Le96}),
we can deduce that \begin{equation}
\left|H_{\epsilon}\left(z\right)\right|\lesssim\frac{e^{\epsilon\left|\text{Im}\left(z\right)\right|}}{1+\left|z\right|},\quad z\in\mathbb{C}.\label{majoration Hepsilon}\end{equation}
On the other hand, since $\Lambda$ is minimal in $PW_{\tau}^{p}$,
there exists a sequence of functions $\left(f_{\lambda}\right)_{\lambda\in\Lambda}\subset PW_{\tau}^{p}$
such that $f_{n}(\lambda_{k})=\delta_{nk}$. Let $a=\left(a_{n}\right)_{n\geq1}$
be a finitely supported sequence and $f$ be the solution of the interpolation
problem \[
f(\lambda_{n})=a_{n},\quad n\geq1,\]
defined by\[
f\left(z\right)=\sum_{n\geq1}a_{n}f_{n}\left(z\right)H_{\epsilon}\left(z-\lambda_{n}\right).\]
(Notice that $f$ is a finite sum of functions belonging to $PW_{\tau+\epsilon}^{p}$.)
From (\ref{eq:def int PW}), it suffices to bound the quantity\[
\inf\left\{ \left\Vert f-g\right\Vert _{p}:\; g\in PW_{\tau+\epsilon}^{p},\; g(\lambda)=0,\;\lambda\in\Lambda\right\} \]
by a constant times the following norm of $a$ \[
\left\Vert a\right\Vert :=\left(\sum_{\lambda\in\Lambda}\left|a_{\lambda}\right|^{p}\left(1+\left|\text{Im}\left(\lambda\right)\right|\right)e^{-p\left(\tau+\epsilon\right)\left|\text{Im}\left(\lambda\right)\right|}\right)^{\frac{1}{p}}.\]
We split the above sum in two parts: $f^{+}$ and $f^{-}$ corresponding
respectively to $\Lambda_{0}^{+}:=\Lambda\cap\left(\mathbb{C}^{+}\cup\mathbb{R}\right)$
and $\Lambda\cap\mathbb{C}^{-}$, and estimate each part separately.
Here, we will only estimate the first one, the method is the same
for the second one. Let $\eta>0$ be such that $\left\{ \text{Im}\left(z\right)=-\eta\right\} \cap\Lambda=\emptyset$.
The Plancherel-Polya inequality allows us to estimate the norm of
$f^{+}-g^{+}$, $g^{+}\in PW_{\tau+\epsilon}^{p}$ and $g^{+}|\Lambda=0$,
on the axis $\left\{ \text{Im}\left(z\right)=-\eta\right\} $. We
consider the Blaschke product associated to $\Lambda_{-\eta}^{+}$,
in the half-plane $\mathbb{C}_{-\eta}^{+}$\[
B_{-\eta}\left(z\right)=\prod_{\lambda_{n}\in\Lambda_{-\eta}^{+}}c_{\lambda_{n}}\frac{z-\lambda_{n}}{z-\overline{\lambda}_{n}-2i\eta},\quad z\in\mathbb{C}_{-\eta}^{+},\]
with suitable unimodular coefficients $c_{\lambda_{n}}$. For $\lambda_{n}\in\Lambda_{0}^{+}$,
we consider the function\[
G_{\lambda_{n},\epsilon}:z\mapsto\left(z-\lambda_{n}\right)H_{\epsilon}\left(z-\lambda_{n}\right)f_{n}\left(z\right)e^{i\left(\tau+\epsilon\right)z}\]
which belongs to $H^{\infty}\left(\mathbb{C}_{-\eta}^{+}\right)$
(this follows from (\ref{eq:plancherel polya}), (\ref{pointwise est. PW})
and (\ref{majoration Hepsilon})) and vanishes on $\Lambda_{-\eta}^{+}$
(it actually vanishes on $\Lambda$). We recall that $\Lambda_{-\eta}^{+}$
satisfies the Carleson condition in $\mathbb{C}_{-\eta}^{+}$. Also,
the function $G_{\lambda_{n},\epsilon}^{0}:=G_{\lambda_{n},\epsilon}/B_{-\eta}$
belongs to $H^{\infty}\left(\mathbb{C}_{-\eta}^{+}\right)$. Let $B^{-}$
be the Blaschke product associated to $\Lambda_{-\eta}^{-}$ (in $\mathbb{C}_{-\eta}^{-}$).
Observe that\[
\inf\left\{ \left\Vert f^{+}-g^{+}\right\Vert _{p}:\; g^{+}\in PW_{\tau+\epsilon}^{p},\; g^{+}|\Lambda=0\right\} \]
\[
=\inf\left\{ \left\Vert \sum_{\lambda_{n}\in\Lambda^{+}}a_{n}\frac{G_{\lambda_{n},\epsilon}^{0}}{z-\lambda_{n}}-g_{0}^{+}\right\Vert _{p}:\; g_{0}^{+}\in Y\right\} \]
with\[
g_{0}^{+}=\frac{g^{+}}{B_{-\eta}}e^{i\left(\tau+\epsilon\right)\cdot}\]
and \[
Y:=H_{+}^{p}\cap\overline{B_{-\eta}}\left(K_{I^{\tau+\epsilon}}^{p}\cap I^{\tau+\epsilon}B^{-}H_{-}^{p}\right)\subset L^{p}\left(\mathbb{R}\right).\]
By duality arguments inspired by Shapiro and Shields (see \cite[p. 516]{SS61}
and \cite{Gau11} where we consider the bilinear form $\left(f,g\right):=\int_{\mathbb{R}-i\eta}fg$,
for $f,g\in L^{p}\left(\mathbb{R}-i\eta\right)$), and because\[
Y^{\bot_{\left(\cdot,\cdot\right)}}=H_{+}^{q}+Z\]
where $Z$ is such that, for every $h\in Z$, we have \[
\int_{\mathbb{R}-i\eta}\left(\sum_{\lambda_{n}\in\Lambda^{+}}a_{n}\frac{G_{\lambda_{n},\epsilon}^{0}}{z-\lambda_{n}}\right)hdm=0,\]
it is enough to estimate\[
\sup\left\{ N(h):\; h\in H^{q}\left(\mathbb{C}_{-\eta}^{+}\right),\left\Vert h\right\Vert =1\right\} ,\]
where \begin{eqnarray*}
N\left(h\right) & := & \left|\int_{\mathbb{R}-i\eta}\left(\sum_{\lambda_{n}\in\Lambda_{0}^{+}}a_{n}\frac{G_{\lambda_{n},\epsilon}^{0}}{z-\lambda_{n}}\right)hdm\right|.\\
& \:= & \left|\sum_{\lambda_{n}\in\Lambda_{0}^{+}}a_{n}\int_{\mathbb{R}}\frac{G_{\lambda_{n},\epsilon}^{0}\left(x-i\eta\right)h\left(x-i\eta\right)}{x-\left(\lambda_{n}+i\eta\right)}dx\right|.\end{eqnarray*}
Now, $z\mapsto G_{\lambda_{n},\epsilon}^{0}\left(z-i\eta\right)h\left(z-i\eta\right)$
is a function in $H_{+}^{q}$ and the Cauchy formula gives\[
N(h)=\left|\sum_{\lambda_{n}\in\Lambda_{0}^{+}}a_{n}G_{\lambda_{n},\epsilon}^{0}\left(\lambda_{n}+i\eta-i\eta\right)h\left(\lambda_{n}+i\eta-i\eta\right)\right|.\]
Moreover, since $\Lambda_{0}^{+}$ satisfies the Carleson condition
in $\mathbb{C}_{-\eta}^{+}$ , we have $\left|\frac{B_{-\eta}}{b_{\lambda_{n}}}\left(\lambda_{n}\right)\right|\asymp1$
and since $f_{\lambda_{n}}\left(\lambda_{n}\right)H_{\epsilon}\left(0\right)=1$,
we can estimate \[
\left|G_{\lambda_{n},\epsilon}^{0}\left(\lambda_{n}\right)\right|\asymp\left(\text{Im}\left(\lambda_{n}\right)+\eta\right)e^{-\left(\tau+\epsilon\right)\text{Im}\left(\lambda_{n}\right)},\quad\lambda_{n}\in\Lambda_{0}^{+}.\]
It follows from the triangle inequality and Hlder's inequality that\begin{eqnarray*}
N(h) & \lesssim & \left(\sum_{\lambda_{n}\in\Lambda_{0}^{+}}\left|a_{n}\right|^{p}\left(1+\text{Im}(\lambda_{n})\right)e^{-p\left(\tau+\epsilon\right)\text{Im}(\lambda_{n})}\right)^{\frac{1}{p}}\\
& & \times\left(\sum_{\lambda_{n}\in\Lambda_{0}^{+}}\text{Im}\left(\lambda_{n}+i\eta\right)\left|\tilde{h}\left(\lambda_{n}+i\eta\right)\right|^{q}\right)^{\frac{1}{q}},\end{eqnarray*}
where $\tilde{h}=h\left(\cdot-i\eta\right)\in H_{+}^{q}$ . Now, the
Carleson condition satisfied by $\Lambda_{0}^{+}+i\eta$ in $\mathbb{C}^{+}$
gives \[
\left(\sum_{\lambda_{n}\in\Lambda_{0}^{+}}\text{Im}\left(\lambda_{n}+i\eta\right)\left|\tilde{h}\left(\lambda_{n}+i\eta\right)\right|^{q}\right)^{\frac{1}{q}}\lesssim\left\Vert h\right\Vert =1.\]
See (\ref{eq:Carleson measure}) and properties mentioned thereafter.
Finally, we obtain\[
\inf\left\{ \left\Vert f^{+}-g\right\Vert :\; g\in PW_{\tau+\epsilon}^{p},\; g|\Lambda=0\right\} \]
\[
\qquad\qquad\lesssim\left(\sum_{\lambda_{n}\in\Lambda_{0}^{+}}\left|a_{n}\right|^{p}\left(1+\text{Im}(\lambda_{n})\right)e^{-p\left(\tau+\epsilon\right)\text{Im}(\lambda_{n})}\right)^{\frac{1}{p}}\]
which is the required estimate and ends the proof.
\end{proof}
To conclude this section, we give two immediate corollaries to our
main theorem. First, since, by Proposition \ref{w Int implies Carleson},
a weak interpolating sequence in $PW_{\tau}^{p}$ has to satisfy
the Carleson condition in every half-plane $\mathbb{C}_{a}^{\pm}$,
we can deduce the following result.
\begin{cor}
If $\Lambda\in\text{Int}_{w}\left(PW_{\tau}^{p}\right)$, then, for
every $\epsilon>0$, $\Lambda$ is interpolating in $PW_{\tau+\epsilon}^{p}$.
\end{cor}
We also give a result involving density conditions as a second corollary
to our main result, which does not seem easy to prove directly.
\begin{cor}
Let $\Lambda$ satisfying (\ref{lambda incluse dans bande}) be a
weak interpolating sequence in $PW_{\tau}^{p}$. Then, $\mathcal{D}_{\Lambda}^{+}\leq\frac{\tau}{\pi}$.\end{cor}
\begin{proof}
It follows from Theorem \ref{thm: MAIN RESULT} that $\Lambda$ is
interpolating in $PW_{\tau+\epsilon}^{p}$, for every $\epsilon>0$.
Thus, Theorem \ref{thm:(densite et interpolation)} implies that $\mathcal{D}_{\Lambda}^{+}\leq\frac{\tau+\epsilon}{\pi}$,
for every $\epsilon>0$, thus $\mathcal{D}_{\Lambda}^{+}\leq\frac{\tau}{\pi}$.
\end{proof}
\section{Weighted Interpolation and McPhail's Condition}
The previous technics can be used to show a more general result. We
need to introduce some more definitions. Let $X$ be the Hardy space
$H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$ or the Paley-Wiener space
$PW_{\tau}^{p}$, $\Lambda=\left\{ \lambda_{n}\right\} _{n\geq1}$
a sequence of complex numbers lying in the corresponding domain $\mathbb{C}_{a}^{\pm}$
or $\mathbb{C}$ and $\omega=\left(\omega_{n}\right)_{n\geq1}$ a
sequence of strictly positive numbers. We say that $\Lambda$ is $\omega-$\emph{interpolating}
in $X$ if for every $\left(a_{n}\right)_{n\geq1}\in l^{p}$, there
is $f\in X$ such that\begin{equation}
\omega_{n}f\left(\lambda_{n}\right)=a_{n},\qquad n\geq1.\label{def: weighted interpolation}\end{equation}
The reader has noticed that the previous definition of interpolation
in $X$ is equivalent to $\omega-$interpolation in $X$, with \[
\omega_{n}=\left\Vert k_{\lambda_{n}}\right\Vert _{X^{\star}}^{-1},\qquad n\geq1.\]
Let $\Lambda\subset\mathbb{C}_{a}^{\pm}$. In this section, the sequence
$\Lambda$ is \emph{a priori} not a Carleson sequence. We only assume
the Blaschke condition\begin{equation}
\sum_{n\geq1}\frac{\left|\text{Im}\left(\lambda_{n}\right)-a\right|}{1+\left|\lambda_{n}\right|^{2}}<\infty.\label{eq:Blaschke condition}\end{equation}
We set\[
\vartheta_{n}:=\prod_{k\neq n}\left|\frac{\lambda_{n}-\lambda_{k}}{\lambda_{n}-\overline{\lambda}_{k}-2ia}\right|,\qquad n\geq1.\]
The couple $\left(\Lambda,\omega\right)$ is said to satisfy the \emph{McPhail
condition} $\left(M_{q}\right)$, denoted $\left(\Lambda,\omega\right)\in\left(M_{q}\right)$,
if the measure\begin{equation}
\nu_{\Lambda,\omega}:=\sum_{n\geq1}\frac{\left|\text{Im}\left(\lambda_{n}\right)-a\right|^{q}}{\omega_{n}^{q}\vartheta_{n}^{q}}\delta_{\lambda_{n}}\label{eq:mc phail measure def}\end{equation}
is a Carleson measure in $\mathbb{C}_{a}^{\pm}$. The following theorem
is a special case of McPhail's theorem (\cite{McP90}) and is stated
as follows in \cite{JP06}.
\begin{thm}
\label{thm:(McPhail)}(McPhail)
Let $1<p<\infty$, $\Lambda\subset\mathbb{C}_{a}^{\pm}$ a sequence
satisfying the Blaschke condition (\ref{eq:Blaschke condition}) and
$\omega=\left(\omega_{n}\right)_{n\geq1}$ be a sequence of positive
numbers. The following assertions are equivalents.
$(i)$ $\Lambda$ is $\omega-$interpolating in $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$;
$(ii)$ $\left(\Lambda,\omega\right)$ satisfies the McPhail condition
$\left(M_{q}\right)$, $\frac{1}{p}+\frac{1}{q}=1$.\end{thm}
\begin{rem}
\label{Rem: weighted int PW entraine MP}It follows directly from
McPhail's Theorem and the Plancherel-Polya inequality that
if $\Lambda\subset\mathbb{C}$ is $\omega-$interpolating in $PW_{\tau}^{p}$,
then for every $a\in\mathbb{R}$, we necessarily have\[
\left(\left(\Lambda\cap\mathbb{C}_{a}^{\pm}\right),e^{\pm\tau\left|\text{Im}\left(\lambda\right)\right|}\omega\right)\in\left(M_{q}\right).\]
\end{rem}
The following result is a weighted version of Theorem \ref{thm: MAIN RESULT}.
We will only sketch the proof which is analogous to that of our main
result.
\begin{thm}
\label{thm:WEIGHTED VERSION}Let $\tau>0$, $1<p<\infty$, $\omega=\left(\omega_{n}\right)_{n\geq1}$
a sequence of strictly positive numbers and $\Lambda$ be a minimal
sequence in $PW_{\tau}^{p}$ such that for every $a\in\mathbb{R}$,
\[
\left(\left(\Lambda\cap\mathbb{C}_{a}^{\pm}\right),e^{\pm\tau\left|\text{Im}\left(\lambda\right)\right|}\omega\right)\in\left(M_{q}\right).\]
Then, for every $\epsilon>0$, $\Lambda$ is $\omega-$interpolating
in $PW_{\tau+\epsilon}^{p}$.\end{thm}
\begin{proof}
As in the proof of the main result of this paper, we fix $\epsilon>0$
and we take a finitely supported sequence $\left(a_{n}\right)_{n\geq1}$.
We consider the solution of the weighted interpolation problem (\ref{def: weighted interpolation})
given by\[
f(z)=\sum_{n\geq1}\frac{a_{n}}{\omega_{n}}f_{n}(z)H_{\epsilon}\left(z-\lambda_{n}\right).\]
As previously, it is possible to split the sum in two parts that we
estimate separately. In order to avoid technical details, let us assume
here that $\Lambda$ lies in the half-plane $\mathbb{C}_{1}^{+}$.
As before, we set \[
G_{\lambda_{n},\epsilon}\left(z\right)=e^{i\left(\tau+\epsilon\right)z}\left(z-\lambda_{n}\right)f_{n}(z)H_{\epsilon}\left(z-\lambda_{n}\right)\in H_{+}^{\infty}.\]
If $B$ denotes the Blaschke product associated to $\Lambda$, we
again write\[
G_{\lambda_{n},\epsilon}=BG_{\lambda_{n},\epsilon}^{0}\]
with $G_{\lambda_{n},\epsilon}^{0}$ still in $H_{+}^{\infty}$. By
duality , we need to estimate \[
\sup\left\{ N\left(h\right):\quad h\in H_{+}^{q},\;\left\Vert h\right\Vert _{q}=1\right\} ,\]
where \[
N(h):=\left|\sum_{n\geq1}\frac{a_{n}}{\omega_{n}}\int_{-\infty}^{\infty}\frac{G_{\lambda_{n},\epsilon}^{0}\left(x\right)h\left(x\right)}{x-\lambda_{n}}dx\right|.\]
The Cauchy formula gives then \begin{eqnarray*}
\left|\int_{-\infty}^{\infty}\frac{G_{\lambda_{n},\epsilon}^{0}\left(x\right)h\left(x\right)}{x-\lambda_{n}}dx\right| & = & \left|G_{\lambda_{n},\epsilon}^{0}\left(\lambda_{n}\right)h\left(\lambda_{n}\right)\right|\\
& = & \frac{\left|2\text{Im}\left(\lambda_{n}\right)\right|}{\vartheta_{n}}e^{-\left(\tau+\epsilon\right)\left|\text{Im}\left(\lambda_{n}\right)\right|}\left|h\left(\lambda_{n}\right)\right|,\end{eqnarray*}
and, applying Hlder's inequality, we finally find\[
N(h)\lesssim\left(\sum_{n\geq1}\left|a_{n}\right|^{p}\right)^{\frac{1}{p}}\left(\sum_{n\geq1}\frac{\left|\text{Im}\left(\lambda_{n}\right)\right|^{q}}{\vartheta_{n}^{q}\omega_{n}^{q}}e^{-q\left(\tau+\epsilon\right)\left|\text{Im}\left(\lambda_{n}\right)\right|}\left|h\left(\lambda_{n}\right)\right|^{q}\right)^{\frac{1}{q}}.\]
By assumption, $\nu_{\Lambda,\tilde{\omega}}$, with $\tilde{\omega}=\left(\omega_{n}e^{\pm\tau\left|\text{Im}\left(\lambda_{n}\right)\right|}\right)_{n}$
(we recall that $\nu_{\Lambda,\tilde{\omega}}$ is defined by (\ref{eq:mc phail measure def})
) is a Carleson measure in $\mathbb{C}^{+}$ and so\[
\left(\sum_{n\geq1}\frac{\left|\text{Im}\left(\lambda_{n}\right)\right|^{q}}{\vartheta_{n}^{q}\omega_{n}^{q}e^{q\tau\left|\text{Im}\left(\lambda_{n}\right)\right|}}\left|h\left(\lambda_{n}\right)\right|^{q}\right)^{\frac{1}{q}}\lesssim\left\Vert h\right\Vert _{q}=1.\]
Since \[
e^{-q\epsilon\left|\text{Im}\left(\lambda_{n}\right)\right|}\leq1,\]
we obtain\[
\sup\left\{ N\left(h\right):\quad h\in H_{+}^{q},\;\left\Vert h\right\Vert _{q}=1\right\} \lesssim\left\Vert a\right\Vert _{l^{p}},\]
which permits us to end the proof.\end{proof}
\begin{rem}
As we have seen in the previous section, the weak interpolation of
a sequence $\Lambda$ in $PW_{\tau}^{p}$ implies the interpolation
property on $\Lambda$ in $PW_{\tau+\epsilon}^{p}$, which follows
from the fact that a uniformly minimal sequence $\Lambda$ in the
Hardy space is an interpolating sequence in the same space. We wonder
if we could have an analog result in the weighted case. More precisely,
we say that the sequence $\Lambda\subset\mathbb{C}_{a}^{\pm}$ is
\emph{uniformly} $\omega-$\emph{minimal} in $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$
if there exists a sequence $\left(f_{n}\right)_{n\geq1}$ of functions
of $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$ such that\[
\omega_{n}f_{n}\left(\lambda_{k}\right)=\delta_{nk}\]
and\[
\sup_{n\geq1}\left\Vert f_{n}\right\Vert <\infty.\]
The question is to know whether a uniformly $\omega-$minimal sequence
$\Lambda$ in $H^{p}\left(\mathbb{C}_{a}^{\pm}\right)$ is necessarily
such that the couple $\left(\Lambda,\omega\right)$ satisfies the
McPhail condition $\left(M_{q}\right)$, $\frac{1}{p}+\frac{1}{q}=1$.
\end{rem}
\section{(Weak) Controllability of Linear Differential Systems}
We consider linear differential systems of the form \begin{equation}
\begin{cases}
x'(t) & =Ax(t)+Bu(t),\qquad t\geq0,\\
x(0) & =x_{0},\end{cases}\label{system AB}\end{equation}
where $A$ is the generator of a $c_{0}-$semigroup $\left(S(t)\right)_{t\geq0}$
on a Hilbert space $H$ and $B:\mathbb{C}\to H$ is an operator, called
the \emph{control operator} which we \emph{a priori} do not assume
bounded. We are thus interested in \emph{rank-$1$ control}. We refer
to \cite[Part D]{JP06} and references therein for more details on
this terminology and on the subject. Note that those authors also
consider unbounded control. We will assume that the semigroup $\left(S(t)\right)_{t\geq0}$
is exponentially stable, $i.e.$ there exists $\alpha>0$ such that
we can find $M\geq1$ for which\begin{equation}
\left\Vert S(t)\right\Vert \leq Me^{-\alpha t},\quad t\geq0.\label{def: expo stable}\end{equation}
Controlling the system (\ref{system AB}) means to act on the system
by means of a suitable \emph{input function $u$}. More precisely,
starting from an initial state $x_{0}\in H$, we want the system to
attain in time $\tau>0$ the in advance given final state $x_{1}=x(\tau)$.
Here the solution $x$ of (\ref{system AB}) is given by \begin{equation}
x(t)=S(t)x_{0}+\int_{0}^{t}S\left(t-r\right)Bu(r)dr=:S(t)x_{0}+\mathcal{B}_{t}u.\label{eq:solution AB}\end{equation}
The operator $\mathcal{B}_{t}$ is called \emph{controllability operator}
and we are interested in the study of its range, the so-called space
of reachable states. More precisely, we say that the system (\ref{system AB})
is\emph{ exactly controllable in finite time} $\tau>0$ (respectively
in \emph{infinite time}) if for every $x_{0},x_{1}\in H$, there is
$u\in L^{2}(0,\tau)$ (respectively $u\in L^{2}\left(0,\infty\right))$
such that $x(0)=x_{0}$ and $x(\tau)=x_{1}$ (respectively $\lim_{t\to\infty}x(t)=x_{1}$)
or, equivalently, if $\mathcal{B}_{\tau}$ (respectively $\mathcal{B_{\infty}}:u\mapsto\int_{0}^{\infty}S(t)Bu(t)dt$)
is surjective. It is well known that a bounded compact controllability
operator (and in particular a rank one operator) can never cover the
whole space $H$ (see \cite[p. 215]{Ni02b}).
In all what follows, we will assume that the generator $A$ admits
a Riesz basis of (normalized) eigenvectors $\left(\phi_{n}\right)_{n\geq1}$:\[
A\phi_{n}=-\lambda_{n}\phi_{n},\quad n\geq1,\]
and that the sequence of eigenvalues $\Lambda:=\left\{ \lambda_{n}\right\} _{n\geq1}$
satisfies the Blaschke condition in the right half-plane\[
\sum_{n\geq1}\frac{\text{Re}\left(\lambda_{n}\right)}{1+\left|\lambda_{n}\right|^{2}}<\infty.\]
Note that by the exponential stability, $\Lambda$ indeed lies in
the right half-plane.The Riesz basis property gives the representation\begin{equation}
H=\left\{ x=\sum_{n\geq1}a_{n}\phi_{n}:\;\sum_{n\geq1}\left|a_{n}\right|^{2}<\infty\right\} .\label{representation riesz basis}\end{equation}
We denote by $\left(\psi_{n}\right)_{n\geq1}$ the biorthogonal family
to $\left(\phi_{n}\right)_{n\geq1}$ (which also forms a Riesz basis
of $H$ and satisfies $\left\Vert \psi_{n}\right\Vert \asymp\left\Vert \phi_{n}\right\Vert ^{-1}\asymp1$).
We suppose that $B$ has the following representation\[
Bv=v\left(\sum_{n\geq1}\overline{b}_{n}\phi_{n}\right),\qquad v\in\mathbb{C},\]
with a sequence $\left(b_{n}\right)_{n\geq1}\subset\mathbb{C}$. Observe
that $B$ does not map $\mathbb{C}$ boundedly in $H$, but it does
map boundedly into some extrapolation space in which the sequence
$\left(\phi_{n}\right)_{n\geq1}$ has dense linear span: for example,
we may define \[
H_{B}:=\left\{ x:=\sum_{n\geq1}x_{n}\phi_{n}:\;\left\Vert x\right\Vert _{B}^{2}:=\sum_{n\geq1}\frac{\left|x_{n}\right|^{2}}{n^{2}\left(1+\left|b_{n}\right|^{2}\right)}<\infty\right\} .\]
It appears that (\ref{eq:solution AB}) can be written
\begin{eqnarray*}
x(\tau) & = & S(\tau)x_{0}+\mathcal{B}_{\tau}u\\
& = & S(\tau)x_{0}+\sum_{n\geq1}\left(\overline{b}_{n}\int_{0}^{\tau}u\left(t\right)e^{-\lambda_{n}\left(\tau-t\right)}dt\right)\phi_{n}\\
& = & S(\tau)x_{0}+\sum_{n\geq1}\left(\overline{b}_{n}e^{-\frac{\tau}{2}\lambda_{n}}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\tilde{u}\left(t\right)e^{\lambda_{n}t}dt\right)\phi_{n},\end{eqnarray*}
with $\tilde{u}:=u\left(\cdot+\frac{\tau}{2}\right)\in L^{2}\left(-\frac{\tau}{2},\frac{\tau}{2}\right)$.
We have already introduced the Fourier transform $\mathcal{F}$ and
we have mentioned that $\mathcal{F}L^{2}\left(-\frac{\tau}{2},\frac{\tau}{2}\right)=PW_{\frac{\tau}{2}}^{2}.$
Hence, if\[
f:=\mathcal{F}\tilde{u}=\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\tilde{u}(t)e^{-it\cdot}dt\in PW_{\frac{\tau}{2}}^{2},\]
we have\[
\mathcal{B}_{\tau}u=\sum_{n\geq1}\left(\overline{b}_{n}e^{i\frac{\tau}{2}\left(i\lambda_{n}\right)}f\left(i\lambda_{n}\right)\right)\phi_{n}.\]
With the same method, we obtain\[
\mathcal{B}_{\infty}u=\sum_{n\neq1}\left(\overline{b}_{n}g\left(i\lambda_{n}\right)\right)\phi_{n},\]
where $g:=\mathcal{F}u$ which belongs to $H_{+}^{2}$ from well known
facts about Hardy spaces. Since exact controllability translates to
surjectivity of $\mathcal{B}_{\tau}$ or $\mathcal{B}_{\infty}$,
and by (\ref{representation riesz basis}), we can reformulate exact
controllability in terms of a weighted interpolation problem.
\begin{thm}
\label{thm:ECO INT}The following assertions are equivalent.
$(i)$ The system (\ref{system AB}) is exactly controllable in finite
time $\tau>0$ (respectively in infinite time);
$(ii)$ The sequence $i\Lambda=\left\{ i\lambda_{n}\right\} _{n\geq1}$
is $\omega-$interpolating in $PW_{\frac{\tau}{2}}^{2}$, with\[
\omega_{n}:=e^{-\frac{\tau}{2}\text{\emph{Im}}\left(i\lambda_{n}\right)}\left|b_{n}\right|,\qquad n\geq1\]
(respectively $\left(\left|b_{n}\right|\right)_{n\geq1}-$interpolating
in $H_{+}^{2}$).\end{thm}
\begin{rem}
As a consequence, and in view of Remark \ref{Rem: weighted int PW entraine MP},
an exact controllable system (in finite time $\tau$) has necessarily
to be such that $\left(i\Lambda,\left(\left|b_{n}\right|\right)_{n}\right)$
satisfies $\left(M_{2}\right)$ in $\mathbb{C}^{+}$ and hence the
system has to be controllable in infinite time.
\end{rem}
In \cite[p. 289-290]{Ni02b}, the author introduces a weaker type
of control, called \emph{control for simple oscillations}, requiring
that the control operator maps boundedly some Hilbert space $\mathcal{U}$
into $H$. As already mentioned above, compact (and hence finite rank)
control is impossible with such hypotheses so that we have to deal
here with unbounded control operators $B$. Nevertheless, we keep
the terminology of \cite{Ni02b} in our situation.
The system (\ref{system AB}) is said \emph{controllable for simple
oscillations} \emph{in time} $\tau>0$ if it is possible to find a
sequence $\left(u_{n}\right)_{n\geq1}$ of functions in $L^{2}\left(0,\tau\right)$
such that\begin{equation}
\mathcal{B}_{\tau}u_{n}=\phi_{n},\quad n\geq1.\label{def CSO}\end{equation}
\begin{rem}
Since \[
\left\langle \mathcal{B}_{\tau}u,\:\psi_{n}\right\rangle =\overline{b}_{n}e^{i\frac{\tau}{2}\left(i\lambda_{n}\right)}f\left(i\lambda_{n}\right),\qquad n\geq1,\]
we easily see that (\ref{def CSO}) is equivalent to the minimality
of $i\Lambda$ in $PW_{\tau}^{2}$.
\end{rem}
We can now use Theorems \ref{thm:WEIGHTED VERSION}, \ref{thm:ECO INT}
and the previous remark to establish a link between control for simple
oscillations at time $\tau$ and exact control at time $\tau+\epsilon$.
More precisely, we have the following result.
\begin{thm}
Under the above hypotheses, if the system (\ref{system AB}) is exactly
controllable in infinite time (or equivalently if $\left(i\Lambda,\left(\left|b_{n}\right|\right)_{n}\right)$
satisfies $\left(M_{2}\right)$ in $\mathbb{C}^{+}$) and if it is
controllable for simple oscillations in time $\tau>0$, then the system
is exactly controllable in finite time $\tau+\epsilon$, for every
$\epsilon>0$.\end{thm}
$\\$
\textsc{\small \'Equipe d'Analyse, Institut de Math\'ematiques de Bordeaux,
Universit\'e Bordeaux 1, 351 cours de la Lib\'eration 33405 Talence C\'edex,
France. }{\small \par}
\end{document}
|
\begin{document}
\markboth{X. Wang, B. Jiang \and J. S. Liu}{Generalized R-squared for detecting dependence}
\title{Generalized R-squared for detecting dependence}
\author{X. WANG}
\affil{Department of Statistics, Harvard University, Cambridge, Massachusetts, U.S.A Email{[email protected]}}
\author{B. JIANG}
\affil{Two Sigma Investments, Limited Partnership, New York, New York, U.S.A Email{[email protected]}}
\author{\and J. S. LIU}
\affil{Department of Statistics, Harvard University, Cambridge, Massachusetts, U.S.A Email{[email protected]}}
\maketitle
\begin{abstract}
Detecting dependence between two random variables is a fundamental problem. Although the Pearson correlation is effective for capturing linear dependency, it can be entirely powerless for detecting nonlinear and/or heteroscedastic patterns. We introduce a new measure, G-squared, to test whether two univariate random variables are independent and to measure the strength of their relationship. The G-squared is almost identical to the square of the Pearson correlation coefficient, R-squared, for linear relationships with constant error variance, and has the intuitive meaning of the piecewise R-squared between the variables. It is particularly effective in handling nonlinearity and heteroscedastic errors. We propose two estimators of G-squared and show their consistency. Simulations demonstrate that G-squared estimates are among the most powerful test statistics compared with several state-of-the-art methods.
End{abstract}
\begin{keywords}
Bayes factor; Coefficient of determination; Hypothesis test; Likelihood ratio.
End{keywords}
\section{Introduction}
\label{sec:int}
The Pearson correlation coefficient is widely used to detect and measure the dependence of two random quantities. The square of its least-squares estimate, popularly known as R-squared, is often used to quantify how linearly related two random variables are. However, the shortcomings of the R-squared as a measure of the strength of dependence are also significant, as discussed recently by \citet{Reshef:2011}, which has inspired the development of many new methods for detecting dependence.
The Spearman correlation calculates the Pearson correlation coefficient between rank statistics. Although more robust than the Pearson correlation, this method still cannot capture non-monotone relationships. The alternating conditional expectation method was introduced by \citet{Breiman:1985} to approximate the maximal correlation between $X$ and $Y$, i.e., to find the optimal transformations of the data, $f(X)$ and $g(Y)$, so that their correlation is maximized. The implementation of the method has its limitations because it is unfeasible to search over all possible transformations. Estimating mutual information is another popular approach due to the fact that the mutual information is zero if and only if $X$ and $Y$ are independent. Furthermore, \citet{Kraskov:2004} proposed an efficient method by estimating the entropy of $X$, $Y$ and $(X,Y)$ separately. The method was claimed to be numerically exact for independent cases and to also work for high dimensional variables. An energy distance-based method \citep{Szekely:2007,Szekely:2009} and a kernel-based method \citep{Gretton:2005,Gretton:2012} appeared separately in statistics and machine learning literature to solve the two-sample test problem and have corresponding usage in independence tests. The two methods were recently shown to be equivalent \citep{Sejdinovic:2013}. Methods based on empirical cumulative distribution functions \citep{Hoeffding:1948}, empirical copula \citep{Genest:2004} and empirical characteristic functions \citep{Kankainen:1998, Huskova:2008} have also been proposed for detecting dependence.
Another set of approaches is based on discretizations of the random variables. Known as grid-based methods, they are primarily designed to test independence between univariate random variables. \citet{Reshef:2011} introduced a new statistic, the maximum information coefficient, which focuses on the generality and equitability of a dependence statistic. Y.~Reshef and coauthors (arXiv:1505.02213) proposed two new estimators for this quantity, which are empirically more powerful and easier to compute. \citet{Heller:2016} proposed a grid based method, which utilizes the $\chi^2$ statistic to test independence and is a distribution-free test.
To measure how accurately an independence test can reflect the strength of dependence between two random variables, \citet{Reshef:2011} introduced the idea of equitability, which was more carefully defined and examined in (Y.~Reshef and coauthors, arXiv:1505.02212). Equitability requires that the same value of the statistic implies the same amount of dependence, regardless of the type of relationship. Whether there exists a statistic that can achieve exact equitability is still subject to debate. However, given a collection of functional relationships with varying noise levels, we can compare the empirical equitability of different statistics through simulation studies.
Intuitively, if there is a functional relationship between two random variables $X$ and $Y$, it is natural to estimate their relationship using a nonparametric technique and use the fraction of reduction in the sum of squares as a measure of the strength of the relationship. In this way, one can both detect dependence and provide an equitable statistic. In contrast, it is more challenging for other types of dependence measures, such as energy-based or entropy-based methods, to be equitable. \citet{Doksum:1994} and \citet{Blyth:1994} discussed the correlation curve to measure the strength of the relationship. However, a direct use of nonparametric curve estimation may rely too heavily on the smoothness assumption of the relationship; it also cannot deal with heteroscedastic noises.
The $G^2$ proposed in this paper is derived from a regularized likelihood ratio test for piecewise linear relationships and can be viewed as an integration of continuous and discrete methods. The G-squared statistic is a function of both the conditional mean and conditional variance of one variable given the other. It is thus capable of detecting general functional relationships with heteroscedastic error variances. An estimate of $G^2$ can be derived via the same likelihood ratio approach as the $R^2$ when the true underlying relationship is linear. Thus, it is reasonable that $G^2$ is almost identical to the $R^2$ for linear relationships.
Efficient estimates of $G^2$ can be computed quickly by a dynamic programming method, whereas \citet{Reshef:2011} and \cite{Heller:2016} have to consider grids on two variables simultaneously and hence require longer computational time, as shown by our simulation studies. We will also show that, in terms of both power and equitability, $G^2$ is among the best statistics for independence testing in consideration of a wide range of functional relationships.
\section{Measuring dependence with G-squared}
\label{sec:def}
\subsection{Defining the $G^2$ as a generalization of the $R^2$}\label{subsec:def}
The R-squared measures how well the data fit a linear regression model. Given $Y=\mu+\beta X+e$ with $e\sim N(0,\sigma^2)$, the standard estimate of R-squared can be derived from a likelihood ratio test statistic for testing $\mathcal{H}_0:\beta=0$ against $\mathcal{H}_1:\beta\neq 0$, i.e.,
\begin{eqnarray*}
R^2 = 1-\left(\frac{L(\widehat{\theta})}{L_0(\widehat{\theta}_0)}\right)^{-2/n},
End{eqnarray*}
and $L_0(\widehat{\theta}_0)$ and $L(\widehat{\theta})$ are the maximized likelihoods under $\mathcal{H}_0$ and $\mathcal{H}_1$.
Throughout the paper, we let $X$ and $Y$ be univariate continuous random variables. As a working model, we assume that the relationship between $X$ and $Y$ can be characterized as $Y =f(X)+Epsilon\sigma_X$, $Epsilon \sim N(0,1)$ and $\sigma_X>0$. If $X$ and $Y$ are independent, then $f(X)Equiv\mu$ and $\sigma^2_XEquiv\sigma^2$. Now, let us look at the piecewise linear relationship
\begin{eqnarray*}
f(X)=\mu_h+\beta_h X,\quad \sigma^2_X=\sigma^2_h,\quad c_{h-1}<X\leq c_h,
End{eqnarray*}
where $c_h \ (h=0,\ldots,K)$ are called the breakpoints. While this working model allows for heteroscedasticity, it requires constant variance within each segment between two adjacent breakpoints. Testing whether $X$ and $Y$ are independent is equivalent to testing whether $\mu_h=\mu$ and $\sigma_h^2=\sigma^2$. Given $c_h\ (h=0,\ldots,K)$, the likelihood ratio test statistic can be written as
\begin{eqnarray*}
\textsc{lr} = Exp\left(\frac{n}{2}\log \widehat{\nu}^2 - \sum_{h=1}^K \frac{n_h}{2}\log \widehat{\sigma}^2_h\right),
End{eqnarray*}
where $\widehat{\nu}^2$ is the overall sample variance of $Y$ and $\widehat{\sigma}^2_h$ is the residual variance after regressing $Y$ on $X$ for $X\in (c_{h-1}, c_h]$. Because $R^2$ is a transformation of the likelihood ratio and converges to the square of Pearson correlation coefficient, we perform the same transformation on $\textsc{lr}$. The resulting test statistic converges to a quantity related to the conditional mean and the conditional variance of $Y$ on $X$. It is easy to show that, as $n\to\infty$,
\begin{eqnarray}
1 - \left(\textsc{lr}\right)^{-2/n}\to 1-Exp\left[E\{\log\mathrm{var}(Y\mid X)\} - \log\mathrm{var}(Y)\right].
\label{equ:gyx}
End{eqnarray}
When $h=1$, the relationship degenerates to a simple linear relationship and $1 - \left(\textsc{lr}\right)^{-2/n}$ is exactly $R^2$.
More generally, because a piecewise linear function can approximate any almost-everywhere continuous function, we can employ the same hypothesis testing framework as above to derive (\ref{equ:gyx}) for any such approximation. Thus, for any pair of random variables $(X,Y)$, the following concept is a natural generalization of the R-squared:
\begin{eqnarray*}
G^2_{Y\mid X} &=&1 - Exp\left[E\{\log\mathrm{var}(Y\mid X)\} - \log\mathrm{var}(Y)\right],
End{eqnarray*}
in which we require that $\mathrm{var}(Y)<\infty$. Evidently, $G^2_{Y\mid X}$ lies between zero and one, and is equal to zero if and only if both $E(Y\mid X)$ and $\mathrm{var}(Y\mid X)$ are constant. The definition of $G^2_{Y\mid X}$ is closely related to the R-squared defined by segmented regression \citep{Oosterbaan:2006} discussed in the Supplementary Material. We symmetrize $G^2_{Y\mid X}$ to arrive at the following quantity as the definition of the G-squared:
\begin{eqnarray*}
G^2=\max(G^2_{Y\mid X},\ G^2_{X\mid Y}),
End{eqnarray*}
provided $\mathrm{var}(X)+\mathrm{var}(Y)<\infty$. Thus, $G^2 = 0$ if and only if $E(X\mid Y)$, $E(Y\mid X)$, $\mathrm{var}(Y\mid X)$ and $\mathrm{var}(X\mid Y)$ are all constant, which is not equivalent to independence of $X$ and $Y$. In practice, however, dependent cases with $G^2=0$ are rare.
\subsection{Estimation of $G^2$}
Without loss of generality, we focus on the estimation of $G^2_{Y\mid X}$; $G^2_{X\mid Y}$ can be estimated in the same way by flipping $X$ and $Y$. When $Y = f(X)+ Epsilon\sigma_X$ and $Epsilon\sim N(0,1)$ for an almost-everywhere continuous function $f(\cdot)$, we can use a piecewise linear function to approximate $f(X)$ and estimate $G^2$. However, in practice the number and locations of the breakpoints are unknown. We propose two estimators of $G^2_{Y\mid X}$, the first aiming to find the maximum penalized likelihood ratio among all possible piecewise linear approximations, and the second focusing on a Bayesian average of all approximations.
Suppose we have $n$ sorted independent observations, $(x_i,y_i) \ (i=1,\ldots,n)$, such that $x_1<\cdots <x_n$. For the set of breakpoints, we only need to consider $c_h=x_i$. Each interval $s_h=(c_{h-1}, c_h]$ is called a slice of the observations, so that $c_h\ (h=0,\ldots,K)$ divide the range of $X$ into $K$ non-overlapping slices. Let $n_h$ denote the number of observations in slice $h$, and let $S(X)$ denote a slicing scheme of $X$, that is, $S(x_i)=h$ if $x_i \in s_h$, which is abbreviated as $S$ whenever the meaning is clear. Let $|S|$ be the number of slices in $S$ and let $m_S$ denote the minimum size of all the slices.
To avoid overfitting when maximizing log-likelihood ratios over both unknown parameters and all possible slicing schemes, we restrict the minimum size of each slice as $m_S\geq \ceil{n^{1/2}}$ and maximize the log-likelihood ratio with a penalty on the number of slices. For simplicity, let $m=\ceil{n^{1/2}}$. Thus, we focus on the following penalized log-likelihood ratio
\begin{equation}
nD(Y\mid S,\lambda_0) = 2 \log \textsc{lr}_{S}-\lambda_0(|S|-1)\log n, \label{d-stat}
End{equation}
where $\textsc{lr}_{S}$ is the likelihood ratio for $S$ and $\lambda_0\log n > 0$ is the penalty for incurring one additional slice. From a Bayesian perspective, this is equivalent to assigning the prior distribution for the number of slices to be proportional to $n^{-\lambda_0(|S|-1)/2}$. Suppose each observation $x_i\ (i=1,\ldots,n-1)$ has probability $p_n=n^{-\lambda_0/2}/(1+n^{-\lambda_0/2})$ of being the breakpoint independently. Then the probability of a slicing scheme $S$ is
\[p_n^{|S|-1}(1-p_n)^{n-|S|}\propto\left(\frac{p_n}{1-p_n}\right)^{|S|-1}= n^{-\lambda_0(|S|-1)/2}.\]
When $\lambda_0=3$, the statistic $-nD(Y\mid S,\lambda_0) $ is equivalent to the Bayesian information criterion \citep{Schwarz:1978} up to a constant.
Treating the slicing scheme as a nuisance parameter, we can maximize over all allowable slicing schemes to obtain that
\[D(Y\mid X,\lambda_0) = \max_{m_S\geq m} D(Y\mid S,\lambda_0).\]
Our first estimator of $G^2_{Y\mid X}$, which we call $G_m^2$ with m representing the maximum likelihood ratio, can be defined as
\begin{equation*}
G_m^2(Y\mid X,\lambda_0) = 1- Exp\{-D(Y\mid X,\lambda_0)\}.
End{equation*}
Thus, the overall G-squared can be estimated as
\[
G_m^2(\lambda_0) = \max\{G_m^2(Y\mid X,\lambda_0),\ G_m^2(X\mid Y,\lambda_0)\}.
\]
By definition, $G_m^2(\lambda_0)$ lies between 0 and 1 and $G_m^2(\lambda_0)=R^2$ when the optimal slicing schemes for both directions have only one slice. Later, we will show that when $X$ and $Y$ are a bivariate normal, $G_m^2(\lambda_0)=R^2$ almost surely for large $\lambda_0$.
Another attractive way to estimate $G^2$ is to integrate out the nuisance slicing scheme parameter. A full Bayesian approach would require us to compute the Bayes factor \citep{Kass:1995}, which may be undesirable since we do not wish to impose too strong a modeling assumption. On the other hand, however, the Bayesian formalism may guide us to a desirable integration strategy for the slicing scheme. We thus put the problem into a Bayes framework and compute the Bayes factor for comparing the null and alternative models. The null model is only one model while the alternative is any piecewise linear model, possibly with countably infinite pieces. Let $p_0(y_1,\ldots,y_n)$ be the marginal probability of the data under the null. Let $\omega_{S}$ be the prior probability for slicing scheme $S$ and let $p_{S}(y_1,\ldots,y_n)$ denote the marginal probability of the data under $S$. The Bayes factor can be written as
\begin{eqnarray}
\textsc{bf}&=&\sum_{m_s\geq m}\omega_{S} \times p_{S}(y_1,\ldots,y_n)/p_0(y_1,\ldots,y_n).
\label{equ:bf}
End{eqnarray}
The marginal probabilities are not easy to compute even with proper priors.
\citet{Schwarz:1978} states that if the data distribution is in the exponential family and the parameter is of dimension $k$, the marginal probability of the data can be approximated as
\begin{eqnarray}
p(y_1,\ldots,y_n) \approx \textsc{l}Exp\left\{-k(\log n -\log 2\pi)/2\right\},
\label{equ:bic}
End{eqnarray}
where $\textsc{l}$ is the maximized likelihood. In our setup, the number of parameters $k$ for the null model is two, and for an alternative model with a slicing scheme $S$ is $3|S|$. Plugging expression (\ref{equ:bic}) into both the numerator and the denominator of (\ref{equ:bf}), we obtain
\begin{eqnarray}
\textsc{bf} \approx \sum_{S: \ m_s\geq m} \omega_{S} \textsc{lr}_{S}Exp\left\{-(3|S|-2)(\log n-\log 2\pi)/2\right\}.\label{bf-approx}
End{eqnarray}
If we take $\omega_{S} \propto n^{-\lambda_0(|S|-1)/2}\ (\lambda_0>0)$, which corresponds to the penalty term in (\ref{d-stat}) and is involved in defining $G_m^2$, the approximated Bayes factor can be restated as
\begin{equation}
\textsc{bf}(\lambda_0)=\left\{\sum_{S: \ m_S\geq m} n^{-\frac{\lambda_0(|S|-1)}{2}}\right\} ^{-1}\sum_{S:
\ m_S\geq m} \left(\frac{2\pi}{n}\right)^{\frac{3|S|-2}{2}}Exp\left\{ \frac{n}{2} D(Y\mid S,\lambda_0)\right\}.\label{bf-original}
End{equation}
As we will discuss in Section~\ref{lambda0}, $\textsc{bf}(\lambda_0)$ can serve as a marginal likelihood function for $\lambda_0$ and can be used to find an optimal $\lambda_0$ suitable for a particular data set. This quantity also looks like an average version of $G_m^2$, but with an additional penalty. Since $\textsc{bf}(\lambda_0)$ can take values below 1, its transformation $1- \textsc{bf}(\lambda_0)^{-2/n}$, as in the case where we derived the $R^2$ via the likelihood ratio test, can take negative values, especially when $X$ and $Y$ are independent, and it is therefore not an ideal estimator of $G^2$.
By removing the model size penalty term in (\ref{bf-approx}), we obtain a modified version, which is simply a weighted average of the likelihood ratios and is guaranteed to be greater than or equal to 1:
\begin{eqnarray*}
\textsc{bf}^*(\lambda_0)=\left\{\sum_{S: \ m_S\geq m} n^{-\frac{\lambda_0(|S|-1)}{2}} \right\} ^{-1}\sum_{S: m_S\geq m} Exp\left\{\frac{n}{2}D(Y\mid S,\lambda_0)\right\}.
End{eqnarray*}
We can thus define a quantity similar to our likelihood formulation of R-squared,
\[G_t^2(Y\mid X,\lambda_0)=1-\textsc{bf}^*(\lambda_0)^{-2/n},\]
which we call the total G-squared, and define
\[ G_t^2(\lambda_0)=\max\{G_t^2(Y\mid X,\lambda_0),\ G_t^2(X\mid Y,\lambda_0)\}.\]
We show later that $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$ are both consistent estimators of $G^2$.
\subsection{Theoretical properties of the $G^2$ estimators}
\label{sec:pro}
In order to show that $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$ converge to $G^2$ as the sample size goes to infinity, we introduce the notations: $\mu_X(y) = E(X\mid Y=y)$, $\mu_Y(x) = E(Y\mid X=x)$, $\nu_X^2(y) = \mathrm{var}(X\mid Y=y)$ and $\nu_Y^2(x) = \mathrm{var}(Y\mid X=x)$ as well as the following regularity conditions:
\begin{condition}
The random variables $X$ and $Y$ are bounded continuously with finite variances such that $\nu_Y^2(x)$, $\nu_X^2(y)>b^{-2}>0$ almost everywhere for some constant $b$.
End{condition}
\begin{condition}
The functions $\mu_Y(x)$, $\mu_X(y)$, $\nu_Y^2(x)$ and $\nu_X^2(y)$ have continuous derivatives almost everywhere.
End{condition}
\begin{condition}
There exists a constant $C > 0$ such that
\begin{eqnarray*}
\max\{\left|\mu_X'(y)\right|,\ \left|\nu_X'(y)\right|\} \leq C\nu_X(y),\quad \max\{\left|\mu_Y'(x)\right|,\ \left|\nu_Y'(x)\right|\} \leq C\nu_Y(x)
End{eqnarray*}
almost surely.
End{condition}
With these preparations, we can state our main results.
\begin{theorem}
\label{the:1} Under Conditions 1-3, for all $\lambda_0 > 0$,
\begin{eqnarray*}
G_m^2(Y\mid X,\lambda_0)\rightarrow G^2_{Y\mid X}, \quad G_t^2(Y\mid X,\lambda_0)\rightarrow G^2_{Y\mid X}
End{eqnarray*}
almost surely as $n\to\infty$. Thus, $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$ are consistent estimators of $G^2$.
End{theorem}
A proof of the theorem and numerical studies of the consistency are in the Supplementary Material. It is expected that $G_m^2(\lambda_0)$ should converge to $G^2$ just because of its construction. It is surprising that $G_t^2(\lambda_0)$ also converges to $G^2$. The result, which links $G^2$ estimation with the likelihood ratio and Bayesian formalism, suggests that most of the information up to the second moment has been fully utilized in the two test statistics. The theorem thus supports the use of $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$ for testing whether $X$ and $Y$ are independent. The null distributions of the two statistics depend on the marginal distributions of $X$ and $Y$, which can be generated empirically using permutation. One can also do a quantile-based transformation on $X$ and $Y$ such that their marginal distributions are standard normal; however, the $G^2$ based on the transformed data tends to lose some power.
When $X$ and $Y$ are bivariate normal, the G-squared statistic is almost the same as the R-squared when $\lambda_0$ is large enough.
\begin{theorem}
\label{the:2}
If $X$ and $Y$ follow bivariate normal distribution, then for $n$ large enough
\begin{eqnarray*}
\mathrm{pr} \left\{G_m^2(\lambda_0) = R^2\right\} &>& 1-3n^{-\lambda_0/3+5}.
End{eqnarray*}
So for $\lambda_0>18$ and $n\to\infty$, we have $G_m^2(\lambda_0)=R^2 $ almost surely .
End{theorem}
The lower bound on $\lambda_0$ is not tight and can be relaxed in practice. Empirically, we have observed that $\lambda_0=3$ is large enough for $G_m^2(\lambda_0)$ to be very close to $R^2$ in the bivariate normal setting.
\subsection{Dynamic programming algorithm for computing $G_m^2$ and $G_t^2$}\label{dp}
The brute force calculation of either $G_m^2$ or $G_t^2$ has a computational complexity of $O(2^n)$ and is prohibitive in practice. Fortunately, we have found a dynamic programming scheme for computing both quantities with a time complexity of $O(n^2)$. The algorithms for computing $G_m^2(Y\mid X,\lambda_0)$ and $G_t^2(Y\mid X,\lambda_0)$ are roughly the same except for one operation, i.e., maximization versus summation, and can be summarized by the following steps:
\begin{step}[Data preparation] Arrange the observed pairs $(x_i, y_i)\ (i=1,\ldots,n)$ according to the sorted $x$s from low to high. Then normalize $y_i \ (i=1,\ldots,n)$ such that $\sum_{i=1}^ny_i=0$ and $\sum_{i=1}^n y_i^2=1$.
End{step}
\begin{step}[Main algorithm] Define $m=\ceil{n^{1/2}}$ as the smallest slice size, $\lambda=-\lambda_0\log(n)/2$ and $\alpha=e^\lambda$. Initialize three sequences: $(M_i,\ B_i,\ T_i) \ (i=1,\ldots,n)$ with $M_1=0$ and $B_1=T_1=1$. For $i=m,\ldots, n$, recursively fill in entries of the tables with
\begin{eqnarray*}
M_i = \max_{k\in K_i}\left( \lambda + M_k + l_{k:i}\right),\quad B_i = \sum_{k\in K_i} \alpha B_k, \quad T_i = \sum_{k\in K_i} \alpha T_k L_{k:i},
End{eqnarray*}
where $K_i=\{1\}\cup\{k:k=m+1,\ldots, i-m+1\}$, $l_{k:i} = -(i-k)\log(\widehat{\sigma}_{k:i}^2)/2$ and $L_{k:i}=Exp\{l_{k:i}\}$, with $\widehat{\sigma}_{k:i}^2$ as the residual variance of regressing $y$ on $x$ for observations $(x_{j}, y_{j}) \ (j=k,\ldots,i)$.
End{step}
\begin{step} The final result is
\begin{eqnarray*}
G_m^2 = 1-Exp\left\{ M_n - \lambda \right\},\quad G_t^2 = 1- (T_n/B_n)^{-2/n}.
End{eqnarray*}
End{step}
Here, $M_i\ (i=m,\ldots, n)$ stores the partial maximized likelihood ratio up to the ordered observation $(x_k,y_k) \ (k=1,\ldots, i)$, $B_i \ (i=m,\ldots, n)$ stores the partial normalizing constant, and $T_i \ (i=m,\ldots, n)$ stores the partial sum of the likelihood ratios. When $n$ is extremely large, we can speed up the algorithm by considering fewer slice schemes. For example, we can divide $X$ into chunks of size $m$ by rank and consider only slicing schemes between the chunks. For this method, the computational complexity is $O(n)$. We can compute $G_m^2(X\mid Y,\lambda_0)$ and $G_t^2(X\mid Y,\lambda_0)$ similarly to get $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$. Empirically, the algorithm is faster than many other powerful methods as shown in the Supplementary Material.
\subsection{An empirical Bayes strategy for selecting $\lambda_0$}
\label{lambda0}
\begin{figure}
\centering{\includegraphics[height=2.4in,width=4.8in]{den_0_5.pdf}}
\caption{Sampling distributions of $G_m^2$ and $G_t^2$ under the two models in Section~\ref{lambda0} with $G^2_{Y\mid X}=0.5$ for $\lambda_0$ = 0.5 (dashes), 1.5 (dots), 2.5 (dot-dash) and 3.5 (solid). The density function in each case is estimated by the histogram. The sampling distributions of $G_m^2$ and $G_t^2$ with the empirical Bayes selection of $\lambda_0$ are in gray shadow and overlaid on top of other density functions.}
\label{fig:den_0.5_part}
End{figure}
Although the choice of the penalty parameter $\lambda_0$ is not critical for the general use of $G^2$, we typically use $\lambda_0 = 3$ for $G_m^2$ and $G_t^2$ because $D(Y\mid X, 3)$ is equivalent to the Bayesian information criterion. Fine-tuning $\lambda_0$ can improve the estimation of $G^2$. We thus propose a data-driven strategy for choosing $\lambda_0$ adaptively. $\textsc{bf}(\lambda_0)$ in (\ref{bf-original}) can be viewed as an approximation to $\mathrm{pr}(y_1,\ldots,y_n\mid \lambda_0)$ up to a normalizing constant. We thus can use the maximum likelihood principle to choose the $\lambda_0$ that maximizes $\textsc{bf}(\lambda_0)$. We then use the chosen $\lambda_0$ to compute $G_m^2$ and $G_t^2$ as estimators of $G^2$. In practice, we evaluate $\textsc{bf}(\lambda_0)$ for a set of discrete $\lambda_0$ values, e.g., $\{0.5,\ 1,\ 1.5,\ 2,\ 2.5,\ 3,\ 3.5,\ 4\}$, and pick the one that maximizes $\textsc{bf}(\lambda_0)$. $\textsc{bf}(\lambda_0)$ can be computed efficiently via a dynamic programming algorithm similar to that described in Section~\ref{dp}. As an illustration, we consider the sampling distributions of $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$ with $\lambda_0=0.5,\ 1.5,\ 2.5$ and $3.5$ for the following two examples
\begin{example} $X\sim N(0,1)$, $Y=X+\sigma Epsilon$ and $Epsilon\sim N(0,1)$.
End{example}
\begin{example}
$X\sim N(0,1)$, $Y=\sin(4\pi x)/0.7+\sigma Epsilon $ and $Epsilon\sim N(0,1)$.
End{example}
We simulated $n=225$ data points. For each model, we set $\sigma=1$ so that $G^2_{Y\mid X}=0.5$ and performed 1,000 replications. Figure~\ref{fig:den_0.5_part} shows histograms of $G_m^2(\lambda_0)$ and $G_t^2(\lambda_0)$ with different $\lambda_0$ values. The results demonstrate that, for relationships that can be approximated well by a linear function, a larger $\lambda_0$ is preferred because it penalizes the number of slices more heavily and the resulting sampling distributions are less biased. On the other hand, for complicated relationships such as the trigonometric function, a smaller $\lambda_0$ is preferable because it allows more slices, which can help capture fluctuations in the functional relationship. The figure also shows that the empirical Bayes selection of $\lambda_0$ worked very well, leading to a proper choice of $\lambda_0$ for each simulated data set from both examples and resulting in the most accurate estimates of $G^2$. Additional simulation studies and consistency of the data-driven strategy are in the Supplementary Material.
\section{Simulation Studies}
\label{sec:sim}
\subsection{Power analysis}
\label{subsec:pow}
Now we compare the power of different independence testing methods for various types of relationships. Here, we again fixed $\lambda_0=3$ for both $G_m^2$ and $G_t^2$. Other methods we tested include the alternating conditional expectation \citep{Breiman:1985}, Genest's test \citep{Genest:2004}, Pearson correlation, distance correlation \citep{Szekely:2007}, the method of \citet{Heller:2016}, the characteristic function method Ecf, Hoeffding's test \mathcal{H}oef, the mutual information method \citep{Kraskov:2004} \ and two methods, $\textsc{mic}_e$ and $\textsc{tic}_e$, based on the maximum information criterion \citep{Reshef:2011}. We follow the procedure for computing the powers of different methods as described in previous studies
of (D.~Reshef and coauthors, arXiv:1505.02214) and a 2012 online note by N.~Simon and R.~Tibshirani.
For different functional relationships $f(X)$ and different values of noise levels $\sigma^2$, we simulated $(X,Y)$ with the following model:
\begin{eqnarray*}
X\sim U(0,1),\quad Y=f(X)+Epsilon \sigma,\quad Epsilon \sim N(0,1).
End{eqnarray*}
where $\mathrm{var}\{f(X)\}=1$. Thus $G^2_{Y\mid X}=(1+\sigma^2)^{-1}$ is a monotone function of the signal-to-noise ratio and it is of interest for us to observe how the performances of different methods deteriorate as the signal strength weakens for various functional relationships. We used permutations to generate the null distribution and to set the rejection region for all testing methods in all examples.
Figure~\ref{fig:pow} shows the power comparisons for eight functional relationships. We set the sample size $n=225$ and performed 1,000 replications for each relationship and $G^2_{Y\mid X}$ value. We only plot Pearson correlation, distance correlation, method by \citet{Heller:2016}, $\textsc{tic}_e$, $G_m^2$ and $G_t^2$ for a clear presentation. More simulations are in the Supplementary Material. For any method with tuning parameters, we chose the ones that resulted in the highest average power over all the examples. Due to computational concerns, we chose $K=3$ for the method of \citet{Heller:2016}. It is seen that $G_m^2$ and $G_t^2$ performed robustly, always being among the most powerful methods, with $G_t^2$ slightly more powerful than $G_m^2$ in almost all examples. They outperformed other methods in cases such as the high frequency sine, triangle and piecewise constant functions, where piecewise linear approximation is more appropriate than other approaches. For monotonic examples such as linear and radical relationships, $G_m^2$ and $G_t^2$ had slightly lower powers than Pearson correlation, distance correlation and the method of \citet{Heller:2016}, but were still highly competitive.
We also studied the performances of these methods with different sample sizes, i.e. for $n=$50, 100 and 400, respectively, and found that $G_m^2$ and $G_t^2$ still showed high power regardless of $n$ although their advantages were much less obvious when $n$ is small. More details can be found in the Supplementary Material.
\begin{figure}
\centering{\includegraphics[height=5.12in, width=4.8in]{power.pdf}}
\caption{The powers of $G_m^2$ (black solid), $G_t^2$ (grey solid), Pearson correlation (grey markers), distance correlation (black dashes), method of \citet{Heller:2016} \ (black dots) and $\textsc{tic}_e$ (black markers) for testing independence between $X$ and $Y$ when the underlying true functional relationships are linear, quadratic, cubic, radical, low freq sine, triangle, high freq sine and piecewise constant, respectively. The x-axis is $G^2_{Y \mid X}$, a monotone function of the signal-to-noise ratio, and the y-axis is the power. We chose $n=225$ and performed 1,000 replications for each relationship and $G^2_{Y\mid X}$.}
\label{fig:pow}
End{figure}
\subsection{Equitability}
\label{subsec:equ}
Y.~Reshef and coauthors (arXiv:1505.02212) gave two equivalent theoretical definitions of the equitability of a statistic that measures dependence. Intuitively, equitable statistics can be used to gauge the degree of dependence. They used $\Psi=\mathrm{cor}^2\{Y,f(X)\}$ to define the degree of dependence when the dependence of $Y$ on $X$ can be described by a functional relationship. When $\mathrm{var}(Y\mid X)$ is a constant, $\PsiEquiv G^2_{Y\mid X}$. For a perfectly equitable statistic, its sampling distribution should be almost identical for different relationships with the same $\Psi$.
We repeated the equitability study by \citet{Reshef:2011}. In Fig.~\ref{fig:ei}, we plot the $95\%$ confidence bands of $G_m^2$ and $G_t^2$, compared with alternating conditional expectation, Pearson correlation, distance correlation and $\textsc{mic}_e$, for the following relationships:
\begin{example}
$X\sim U(0,1)$, $Y=X+Epsilon\sigma$ and $Epsilon\sim N(0,1)$.
\label{ex:3}
End{example}
\begin{example}
$X\sim U(0,1)$, $Y=X+Epsilon\sigma$ and $Epsilon\sim N(0,e^{-|X|})$.
\label{ex:4}
End{example}
\begin{example}
$X\sim U(0,1)$, $Y=\frac{X^2}{\surd{2}}+Epsilon\sigma$ and $Epsilon\sim N(0,1)$.
\label{ex:5}
End{example}
\begin{example}
$X\sim U(0,1)$, $Y=\frac{X^2}{\surd{2}}+Epsilon\sigma $ and $Epsilon\sim N(0,e^{-|X|})$.
\label{ex:6}
End{example}
We choose different $\Psi$ values with $n=225$ and 1,000 replications for each case. The plots show that $G_m^2$ and $G_t^2$ increased along with $\Psi$ for all relationships, as they should, and that the confidence bands obtained under different functional relationships have a similar size and location for the same $\Psi$. The confidence bands are also comparably narrow. The $\textsc{mic}_e$ displayed good performances of equitability, though slightly worse than $G_m^2$ and $G_t^2$, while other three statistics did poorly for non-monotone relationships. The alternating conditional expectation tended to have a wider confidence band for Example~\ref{ex:5} and~\ref{ex:6} than the aforementioned three methods, while Pearson correlation and distance correlation had non-overlapping confidence intervals for different relationships when $\Psi$ is moderately large. In other words, Pearson correlation and distance correlation can yield drastically different values for two relationships with the same $\Psi$. This phenomenon is as expected since it is known that these two statistics do not perform well for non-monotone relationships.
\begin{figure}
\centering{\includegraphics[height=3.2in,width=4.8in]{equ_int.pdf}}
\caption{The plots from the top left to the bottom right are the $95\%$ confidence bands for $G_m^2$, $G_t^2$, $\textsc{mic}_e$, alternating conditional expectation, Pearson correlation and distance correlation, respectively. We chose $n=225$ and performed 1,000 replications for each relationship and each $\Psi$ value for the four examples in Section~\ref{subsec:equ}. The fill is the lightest for Example~\ref{ex:3} and darkest for Example~\ref{ex:6}. $\Psi$ is a monotone function of the signal-to-noise ratio when the error variance is constant, and the y-axis shows the values of the corresponding statistic, each estimating its own population mean, which may or may not be $\Psi$.}
\label{fig:ei}
End{figure}
An alternative strategy to study equitability uses a hypothesis testing framework, i.e., to test $\mathcal{H}_0: \Psi = x_0$ against $\mathcal{H}_1:\Psi = x_1 \ (x_1>x_0)$ on a broad set of functional relationships using a statistic. The more powerful a test statistic for this testing problem with all types of relationships, the better its equitability. For each aforementioned method, we performed right-tailed tests with the type-I error fixed at $\alpha=0.05$ and different combinations of $(x_0, x_1)\ (0<x_0<x_1<1)$. Given a fixed sample size, a perfectly equitable statistic should yield the same power for all kinds of relationships so that it is able to reflect the degree of dependency by a single value regardless of the type of relationship. In reality, most statistics can perform well only for a small class of relationships.
In Fig.~\ref{fig:equ}, we use a heat map to demonstrate the average power of a test statistic with different pairs of $(x_0,x_1)\ (0<x_0<x_1<1)$. Each dot in the plot represents the average power of a testing method over a class of functional relationships; the darker the color is, the higher the power. We used the same set of functional relationships as in N.~Reshef and coauthors (arXiv:1505.02214) and carried out the testing for $(x_0,x_1)=(i/50,j/50)\ (i<j=1,\ldots,49)$. We set the sample size as $n=225$ and conducted 1,000 replications for each relationship and each $(x_0, x_1)\ (0<x_0<x_1<1)$. For any method with a tuning parameter, we chose parameters that resulted in the greatest average power. We observed that $G_m^2$, $G_t^2$ and $\textsc{mic}_e$ had the best equitability, followed by alternating conditional expectation and $\textsc{tic}_e$. The average powers for $G_m^2$, $G_t^2$ and $\textsc{mic}_e$ over the entire range of $(x_0,x_1)\ (0<x_0<x_1<1)$ were all $0.6$, although $G_m^2$ and $G_t^2$ were slightly better for larger $x_0$'s. Besides, with our empirical Bayes method for selecting $\lambda_0$, the equitability of $G_m^2$ and $G_t^2$ can be further improved. In comparison, all the remaining methods were not as equitable.
\begin{figure}
\centering{\includegraphics[height=6.4in,width=4.8in]{equ.pdf}}
\caption{Heat map plot for comparing equitability of different methods. From the top left to the bottom right: $G_m^2$, $G_t^2$, alternating conditional expectation, Genest's test, Pearson correlation, distance correlation, the method of \citet{Heller:2016}, characteristic function method, Hoeffding's test, mutual information, $\textsc{mic}_e$ and $\textsc{tic}_e$. The value corresponding to $(x_1, x_0)\ (0<x_0<x_1<1)$ is the power of the method for testing the hypothesis: $\mathcal{H}_0: \Psi = x_0$ against $\mathcal{H}_1:\Psi = x_1$, averaging over a class of functions. The darker a dot, the higher the average power of the corresponding test. We chose sample size $n=225$ and performed 1,000 replications for each relationship and $(x_0, x_1)\ (0<x_0<x_1<1)$.}
\label{fig:equ}
End{figure}
\section{Discussion}
The G-squared can be viewed as a direct generalization of the R-squared. While maintaining the same interpretability as the R-squared, the G-squared is also a powerful and equitable measure of dependence for general relationships. Instead of resorting to curve-fitting methods for estimating the underlying relationship and the G-squared, we employed the more flexible piecewise linear approximations with penalty and dynamic programming algorithms. Although we only consider piecewise linear functions, one can potentially approximate a relationship between two variables with piecewise polynomials or other flexible basis functions, with perhaps additional penalty terms to control the complexity. Furthermore, it is a worthwhile effort to generalize the slicing idea for testing dependence between two multivariate random variables.
\section*{Acknowledgment}
We are grateful to the two referees for helpful comments and suggestions. This research was supported in part by grants from the U.S. National Science Foundation and National Institutes of Health. We thank Ashley Wang for her proofreading of the paper. The views expressed herein are the authors alone and are not necessarily the views of Two Sigma Investments, Limited Partnership or any of its affiliates.
\section*{Supplementary material}
Further material available at \textit{Biometrika} online includes proofs of theorems, software implementation details, discussions on segmented regression and more simulation results.
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|
\begin{document}
\title{\ Covariate Adaptive False Discovery Rate Control with Applications to Omics-Wide Multiple Testing}
\author{Xianyang Zhang and Jun Chen}
{ \mathcalbox{d} }ate{}
\footnotetext[1]{Xianyang Zhang ([email protected]) is Associate
Professor of Statistics at Texas A\&M University. Jun Chen ([email protected]) is Associate Professor of Biostatistics at Mayo Clinic. Zhang acknowledges partial support from
NSF DMS-1830392 and NSF DMS-1811747. Chen acknowledges support from Mayo Clinic Center for Individualized Medicine.} \mathcalaketitle
\onehalfspacing
\mathcalaketitle
\sloppy
\textbf{Abstract} Conventional multiple testing procedures often assume hypotheses for different features are exchangeable. However, in many scientific applications, additional covariate
information regarding the patterns of signals and nulls are available. In this paper, we introduce an FDR control procedure in large-scale inference problem that can incorporate covariate information. We develop a fast algorithm to implement the proposed procedure and prove its asymptotic validity even when the underlying likelihood ratio model is misspecified and the p-values are weakly dependent (e.g., strong mixing). Extensive simulations are conducted to study the finite sample performance of the proposed method and we demonstrate that the new approach improves over the state-of-the-art approaches { by being flexible, robust, powerful and computationally efficient}. We finally apply the method to several omics datasets arising from genomics studies with the aim to identify omics features associated with some clinical and biological phenotypes. We show that the method is overall the most powerful among competing methods, especially when the signal is sparse. The proposed \textbf{C}ovariate \textbf{A}daptive \textbf{M}ultiple \textbf{T}esting procedure is implemented in the R
package \texttt{CAMT}.
\\
\strut \textbf{Keywords:} Covariates, EM-algorithm, False Discovery Rate, Multiple Testing.
{ \mathcalbox{d} }oublespacing
\section{Introduction}
Multiple testing refers to simultaneous testing of more than one
hypothesis. Given a set of hypotheses, multiple testing deals with
deciding which hypotheses to reject while guaranteeing some notion
of control on the number of false rejections. A traditional measure
is the family-wise error rate (FWER), which is the probability of
committing at least one type I error. As the number of trials
increases, FWER still measures the probability of at least one false
discovery, which is overly stringent in many applications. This
absolute control is in contrast to the proportionate control
afforded by the false discovery rate (FDR).
Consider the problem of testing $m$ distinct hypotheses. Suppose a
multiple testing procedure rejects $R$ hypotheses among which $V$
hypotheses are null, i.e., it commits $V$ type I errors. In the
seminal paper by Benjamini and Hochberg, the authors introduced the
concept of FDR defined as
$$\text{FDR}=E\left[\frac{V}{R\vee
1}\right],$$ where $a\vee b=\mathcalax\{a,b\}$ for $a,b\in\mathcalathbb{R}$, and the expectation is with respect to the random quantities $V$ and $R$.
FDR has many advantageous features comparing to other existing error
measures. Control of FDR is less stringent than the control of FWER
especially when a large number of hypothesis tests are performed. FDR is also
adaptive to the underlying signal structure in the data. The
widespread use of FDR is believed to stem from and motivated by the
modern technologies which produce big datasets, with huge
numbers of measurements on a comparatively small number of
experimental units. Another reason for the popularity of FDR is the
existence of a simple and easy-to-use procedure proposed in
Benjamini and Hochberg (1995) (the BH procedure, hereafter) to
control the FDR at a prespecified level.
Although the BH procedure is more powerful than procedures aiming to
control the FWER, it assumes hypotheses for different features
are exchangeable which could result in suboptimal power as demonstrated in recent literature when individual tests
differ in their true effect size, signal-to-noise ratio or prior probability of being false.
In many scientific applications, particularly those from genomics studies,
there are rich covariates that are informative of either the statistical power or the prior null probability.
These covariates can be roughly derived into two classes: statistical covariates and external covariates (Ignatiadi et al., 2016). Statistical covariates are derived from the data itself and could reflect the power or null probability. Generic statistical covariates include the sample variance, total sample size and sample size ratio (for two-group comparison), and the direction of the effects. There are also specific statistical covariates for particular applications. For example, in transcriptomics studies using RNA-Seq, the sum of read counts per gene across all
samples is a statistical covariate informative of power since the low-count genes are subject to more sampling variability. Similarly, the minor allele frequency and the prevalence of the bacterial species can be taken as statistical covariates for genome-wide association studies (GWAS) and microbiome-wide association studies (MWAS), respectively. Moreover, the average methylation level of a CpG site in epigenome-wide association studies (EWAS) can be a statistical covariate informative of the prior null probability due to the fact that differential methylation frequently occurs in highly or lowly methylated region depending on the biological context. Besides these statistical covariates, there are a plethora of covariates that are derived from external sources and are usually informative of the prior null probability. These external covariates include the deleteriousness of the genetic variants for GWAS, the location (island and shore) of CpG methylation variants for EWAS, and pathogenicity of the bacterial species for MWAS. Useful external covariates also include p-values from previous or related studies which suggest that some hypotheses are more likely to be non-null than others. Exploiting such external covariates in multiple testing could lead to improved statistical power as well as
enhanced interpretability of research results.
Accommodating covariates in multiple testing has recently been a very active
research area. We briefly review some contributions that are most
relevant to the current work. The basic idea of many existing works
is to relax the p-value thresholds for hypotheses that are more
likely to be non-null and tighten the thresholds for the other
hypotheses so that the overall FDR level can be controlled. For
example, Genovese et al. (2006) proposed to weight the p-values with
different weights, and then apply the BH procedure to the weighted
p-values. Hu et al. (2010) developed a group BH procedure by
estimating the proportions of null hypotheses for each group
separately, which extends the method in Storey (2002). Li and Barber
(2017) generalized this idea by using the censored p-values (i.e.,
p-values that are greater than a pre-specified threshold) to
adaptively estimate the weights that can be designed to reflect any
structure believed to be present. Ignatiadi et al. (2016) proposed
the independent hypothesis weighting (IHW) for multiple testing with
covariate information. Their idea is to bin the covariates into
several groups and then apply the weighted BH procedure with
piecewise constant weights. Boca and Leek (2018) extended the idea by using a regression approach to estimate weights. Another related method (named AdaPT) was
proposed in Lei and Fithian (2018), which iteratively estimates the
p-value thresholds using partially censored p-values. The above
procedures can be viewed to some extent as different variants of the
weighted BH procedure. Along a separate line, Local FDR (LFDR) based
approaches have been developed to accommodate various forms of
auxiliary information. For example, Cai and Sun (2009) considered
multiple testing of grouped hypotheses using the pooled LFDR
statistic. Sun et al. (2015) developed a LFDR-based procedure to
incorporate spatial information.
Scott et al. (2015) and Tansey et al. (2017) proposed EM-type algorithms to estimate the LFDR by
taking into account covariate and spatial information, respectively.
Although the approaches mentioned above excel in certain aspects, a method that is flexible, robust, powerful and computationally efficient is still lacking. For example, IHW developed in Ignatiadi et al. (2016) cannot handle multiple covariates. AdaPT in Lei and Fithian (2018) is computationally intensive and may suffer from significant power loss when the signal is sparse, and covariate is not very informative. Li and Barber (2017)'s procedure is not Bayes optimal as shown in Lei and Fithian (2018) and thus could lead to suboptimal power as observed in our numerical studies. The FDR regression method proposed in Scott et al. (2015) lacks a rigorous FDR control theory. Table 1 provides a detailed comparison of these methods.
In this paper, in addition to a thorough evaluation of these methods using comprehensive simulations covering different signal structures, we propose a new procedure to incorporate covariate information with generic applicability.
The covariates can be any continuous or categorical variables that are thought to be informative of the statistical properties of the hypothesis tests. The main contributions of our paper are two-fold:
\begin{enumerate}
\item Given a sequence of p-values $\{p_1,{ \mathcalbox{d} }ots,p_m\}$, we introduce a general decision rule of the form
\begin{equation}\label{rej0}
(1-k_i)p^{-k_i}_i\geq \frac{(1-t)\pi_{i}}{t(1-\pi_{i})},\quad
0<k_i<1, \quad 1\leq i\leq m,
\end{equation}
which serves as a surrogate for the optimal decision rule
derived under the two-component mixture model with varying mixing
probabilities and alternative densities. Here $\pi_i$ and $k_i$ are parameters that can be estimated from the covariates and p-values, and $t$ is a cut-off value to be determined by our FDR control method. We develop a new procedure
to estimate $(k_i,\pi_i)$ and find the optimal threshold value for $t$ in (\ref{rej0}).
We show that (i) when $\pi_i$ and $k_i$ are chosen
independently of the p-values, the proposed procedure provides
finite sample FDR control; (ii) our procedure provides
asymptotic FDR control when $\pi_i$ and $k_i$ are chosen to maximize a potentially misspecified likelihood based on the covariates and p-values; (iii) Similar to some recent works (e.g., Ignatiadi et al., 2016; Lei and Fithian, 2017; Li and Barber, 2017), our method allows
the underlying likelihood ratio model to be misspecified. { A distinctive feature is that our asymptotic analysis does not require the p-values to be marginally independent or conditionally independent given the covariates. More specifically, we allow the pairs of p-value and covariate across different hypotheses to be strongly mixing as specified in Assumption \ref{ass-ad3}.}
\item We develop an efficient algorithm to estimate $\pi_i$ and $k_i$.
The developed algorithm is scalable to problems with millions of tests.
Through extensive numerical studies, we show that our procedure is highly competitive to
several existing approaches in the recent literature in terms of finite sample performance.
The proposed procedure is implemented in the R package \texttt{CAMT}.
\end{enumerate}
Our method is related to Lei and Fithian (2018), and it is worth highlighting the differences from their work.
(i) Lei and Fithian (2018) uses partially censored p-values to determine the threshold, which can discard useful information concerning the alternative distribution of p-values (i.e., $f_{1,i}$ in (\ref{m1}) below) since small p-values
that are likely to be generated from the alternative are censored.
In contrast, we use all the p-values to determine the threshold. Our method is seen to exhibit more power as compared to Lei and Fithian (2018) when signal is (moderately) sparse.
Although our method no longer offers theoretical finite sample FDR control, we show empirically that the power gain is not at the cost of FDR control.
(ii) Different from Lei and Fithian (2018) which requires multiple stages for practitioners to make their final decision, our method is a single-stage procedure that only needs to be run one time; Thus the implementation of our method is faster and scalable to modern big datasets.
(iii) Our theoretical analysis is entirely different from those in Lei and Fithian (2018). In particular, we show that our method achieves asymptotic FDR control even when the p-values are dependent.
\begin{comment}
The rest of the article is organized as follows. We consider the
two-component mixture models and study the identifiability in
Section \ref{sec2}. We introduce several types of structural
information that can be incorporated into our framework in Section
\ref{sec-structure}. We introduce two oracle adaptive multiple
testing procedures in Section \ref{sec:fdr} and study their FDR
control properties. Section \ref{sec:plugin} describes a general
EM-type algorithm to estimate the unknown parameters as well as the
alternative density of p-values. Section \ref{sec:asy} presents some
asymptotic results concerning FDR control and asymptotic power.
Sections \ref{sec:num}-\ref{sec:data} are devoted respectively to
the simulation studies and real data analysis. The computational and
technical details are gathered in the appendix.
\end{comment}
\section{Methodology}\label{sec2}
\subsection{Rejection rule}
We consider simultaneous testing of $m$ hypotheses $H_i$ for
$i=1,2,{ \mathcalbox{d} }ots,m$. Let $p_i$ be the p-value associated with the $i$th
hypothesis, and with some abuse of notation, let $H_i$ indicate the underlying truth of the $i$th
hypothesis. In other words, $H_i=0$ if the $i$th hypothesis is true
and $H_i=1$ otherwise. For each hypothesis, we observe a
covariate $x_i$ lying in some space $\mathcalathcal{X}\subseteq
\mathcalathbb{R}^{q}$ with $q\geq 1$. From a Bayesian viewpoint, we can model $H_i$ given $x_i$ as a
Bernoulli random variable with success probability $1-\pi_{0i}$,
where $\pi_{0i}$ denotes the prior probability that the $i$th
hypothesis is under the null when conditioning on $x_i$. One approach to model the p-value distribution is via a two-component mixture model,
\begin{align}
&H_i|x_i \sim ~\text{Bernoulli}(1-\pi_{0i}), \label{m0}\\
&p_i|x_i,H_i \sim ~(1-H_i)f_0+H_if_{1,i}, \label{m1}
\end{align}
where $f_0$ and $f_{1,i}$ are the density functions corresponding to
the null and alternative hypotheses respectively. { In the following discussions, we shall assume that $f_0$ satisfies the following condition: for any $a\in [0,1]$
\begin{align}\label{eq-con-f0}
\int^{a}_{0}f_0(x)dx \leq \int^{1}_{1-a}f_0(x)dx.
\end{align}
This condition relaxes the assumption of uniform distribution on the unit interval. It is fulfilled when $f_0$ is non-decreasing or $f_0$ is symmetric about 0.5 (in which case the equality holds in (\ref{eq-con-f0})). We demonstrate that this relaxation is capable of describing plausible data generating processes that would create a non-uniform null distribution. Let $T$ be a test statistic such that under the null its z-score $Z=(T-\mathcalu_0)/\sigma_0$ is standard normal. In practice, one uses $\hat{\mathcalu}$ and $\hat{\sigma}$ to estimate $\mathcalu_0$ and $\sigma_0$ respectively. Let $\Phi$ be the standard normal CDF. The corresponding one-sided p-value is given by $\Phi((T-\hat{\mathcalu})/\hat{\sigma})$ whose distribution function is $P(\Phi((T-\hat{\mathcalu})/\hat{\sigma})\leq x)=\Phi((\Phi^{-1}(x)\hat{\sigma}+\hat{\mathcalu}-\mathcalu_0)/\sigma_0)$. When $\mathcalu_0\geq \hat{\mathcalu}$ (i.e., we underestimate the mean), one can verify that $f_0$ is a non-decreasing. In the case of $\mathcalu_0=\hat{\mathcalu}$ and $\sigma_0 \neq \hat{\sigma}$, $f_0$ is non-uniformly symmetric about 0.5.
}
Compared to the
classical two-component mixture model, the varying null probability
reflects the relative importance of each hypothesis given the
external covariate information $x_i$ and the varying alternative density $f_{1,i}$
emphasizes the heterogeneity among signals. { In the context without covariate information, it is well known that the optimal rejection is based on the LFDR, see e.g., Efron (2004) and Sun and Cai (2007). The result has been generalized to the setups with group or covariate information, see e.g., Cai and Sun (2009) and Lei and Fithian (2018).} Based on these insights, one can indeed show that the optimal rejection rule that
controls the expected number of false positives while maximizes the
expected number of true positives takes the form of
\begin{equation}\label{rule}
\frac{f_{1,i}(p_i)}{f_0(p_i)}\geq \frac{(1-t)\pi_{0i}}{t(1-\pi_{0i})},
\end{equation}
where $t\in (0,1)$ is a cut-off value. This decision rule is generally unobtainable because
$f_{1,i}$ is unidentifiable without extra assumptions on
its form. Moreover, consistent estimation of the decision rule (\ref{rule}) is difficult, and even with the use of additional approximations, such as splines or piecewise constant functions.
In this work, we do not aim to estimate the optimal rejection rule directly. Instead, we try to find
a rejection rule that can mimic some useful operational characteristics of the optimal rule.
Our idea is to first replace $f_{1,i}/f_0$ by a
surrogate function $h_i$. We emphasize that $h_i$ needs not agree with
the likelihood ratio $f_{1,i}/f_0$ for our method to be valid. In fact, the validity of our method does not rely on the correct specification of model (\ref{m0})-(\ref{m1}).
We require $h_i$ to satisfy (i) $h_i(p)\geq 0$ for $p\in[0,1]$; (ii)
$\int^{1}_{0}h_i(p)dp=1$; (iii) $h$ is decreasing. Requirement (iii) is
imposed to mimic the common likelihood ratio assumption in the
literature, see e.g. Sun and Cai (2007). In this paper, we suggest to use the beta density,
\begin{align}\label{eq-LR}
h_i(p)=(1-k_i)p^{-k_i},\quad 0<k_i<1,
\end{align}
where $k_i$ is a parameter that depends on $x_i$. { Suppose that under the null hypothesis, $p_i$ is uniformly distributed, whereas under the alternative, it follows a beta distribution with parameters $(1-k_i,1)$, then
the true likelihood ratio would take exactly the form given in (\ref{eq-LR}). To demonstrate the approximation of the proposed surrogate likelihood ratio to the actual likelihood ratio for realistic problems, we simulated two binary variables and generated four alternative distributions $f_{1, i}$ depending on the four levels of the two variables (details in the legend of Figure \ref{fig:sim:0}). We used the proposed procedure to find the best $k_i$ and compared the CDF of the empirical distribution (reflecting the actual likelihood ratio) to that of the fitted beta distribution (reflecting the surrogate likelihood ratio). We can see from Figure \ref{fig:sim:0} the approximation was reasonably well and the accuracy increases with the signal density. }
Based on the surrogate likelihood ratio, the corresponding rejection rule is given by
\begin{align}\label{rej1}
h_i(p_i)\geq w_i(t):=\frac{(1-t)\pi_{i}}{t(1-\pi_{i})},
\end{align}
for some weights $\pi_i$ to be determined later. See Section \ref{alg} for more details about the estimation of $k_i$ and $\pi_i.$
\subsection{Adaptive procedure}
We first note that the false discovery proportion (FDP) associated
with the rejection rule (\ref{rej1}) is equal to
\begin{align*}
\text{FDP}(t):=&\frac{\sum^{m}_{i=1}(1-H_i)\mathcalathbf{1}\{h_i(p_i)\geq
w_i(t)\}}{1\vee\sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}}.
\end{align*}
{Then for a cut-off value $t$, we have
\begin{align*}
\text{FDP}(t)=&\frac{\sum^{m}_{i=1}(1-H_i)\mathcalathbf{1}\{p_i\leq h_i^{-1}(w_i(t))\}}{1\vee \sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}}
\\ \approx & \frac{\sum^{m}_{i=1}(1-H_i)P(p_i\leq h_i^{-1}(w_i(t)))}{1\vee \sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}}
\\ \leq & \frac{\sum^{m}_{i=1}(1-H_i)P(1-p_i\leq h_i^{-1}(w_i(t)))}{1\vee \sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}}
\\ \approx & \frac{1+\sum^{m}_{i=1}(1-H_i)\mathcalathbf{1}\{h_i(1-p_i)\geq w_i(t)\}}{1\vee \sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}}
\\ \leq & \frac{1+\sum^{m}_{i=1}\mathcalathbf{1}\{h_i(1-p_i)\geq w_i(t)\}}{1\vee \sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}}:=\text{FDP}_{\text{up}}(t),
\end{align*}
where the approximations are due to the law of large numbers and the inequality follows from Condition (\ref{eq-con-f0}).}\footnote{Rigorous theoretical justifications are provided in Theorem \ref{thm-add} and Theorem \ref{thm}.}
This strategy is partly motivated by the recent distribution-free method proposed in Barber and Cand\`{e}s (2015). We refer any FDR estimator constructed using this strategy as the BC-type estimator.
Both the adaptive procedure in Lei and Fithian (2018) and the proposed method fall into this category.
A natural idea is
to select the largest threshold such that
$\text{FDP}_{\text{up}}(t)$ is less or equal to a prespecified FDR level $\alpha.$
Specifically, we define
\begin{align*}
t^*=\mathcalax\left\{t\in [0,t_{\text{up}}]: \text{FDP}_{\text{up}}(t)=\frac{1+\sum^{m}_{i=1}\mathcalathbf{1}\{h_i(1-p_i)\geq
w_i(t)\}}{1\vee\sum^{m}_{i=1}\mathcalathbf{1}\{h_i(p_i)\geq
w_i(t)\}}\leq
\alpha\right\},
\end{align*}
where $t_{\text{up}}$ satisfies that $w_i(t_{\text{up}})\geq h_i(0.5)$ for all $i,$
and we reject all hypotheses such that $h_i(p_i)\geq w_i(t^*)$. The
following theorem establishes the finite sample control of the above
procedure when $\pi_i$ and $h_i$ are prespecified and thus independent of the p-values. For example, $\pi_i$ and $h_i$ are estimated based on data from an independent but related study.
\begin{theorem}\label{thm-add}
Suppose $h_i$ is strictly decreasing for each $i$ and $f_0$ satisfies Condition (\ref{eq-con-f0}).
If the p-values are independent and the choice of $h_i$ and $\pi_i$ is independent of the p-values, then the adaptive procedure provides finite sample FDR control at level
$\alpha$.
\end{theorem}
\subsection{An algorithm}\label{alg}
The optimal choices of $\pi_i$ and $k_i$ are rarely known in practice, and a generally applicable data-driven method is desirable.
In this subsection, we propose an EM-type algorithm to estimate $\pi_i$ and $k_i$. In particular, we model
both $\pi_i$ and $k_i$ as functions of the covariate $x_i$. As an illustration, we
provide the following example.
\begin{example}\label{example}
{\rm Suppose
\begin{align*}
&p_{i}|x_i,H_i \sim (1-H_i)f_0+H_if_{1,i},
\\&x_{i}|H_i \sim (1-H_i)g_0+H_ig_1,
\end{align*}
where $H_i\sim^{\text{i.i.d}}\text{Bernoulli}(1-\pi_{0})$.
Using the Bayes rule, we have
\begin{align*}
f(p_{i}|x_{i})=&\frac{f(p_{i},x_{i}|H_i=0)\pi_0+f(p_{i},x_{i}|H_i=1)(1-\pi_0)}{f(x_{i}|H_i=0)\pi_{0}+f(x_{i}|H_i=1)(1-\pi_0)}
\\=&\frac{f(p_{i}|x_i,H_i=0)f(x_{i}|H_i=0)\pi_0+f(p_{i}|x_i,H_i=1)f(x_{i}|H_i=1)(1-\pi_0)}{f(x_{i}|H_i=0)\pi_{0}+f(x_{i}|H_i=1)(1-\pi_0)}
\\=&\pi(x_{i})f_{0}(p_{i})+(1-\pi(x_{i}))f_{1,i}(p_{i}),
\end{align*}
where $\pi(x)=g_0(x)\pi_0/\{g_0(x)\pi_{0}+g_1(x)(1-\pi_0)\}=f(H_i=0|x_i=x).$
Therefore, $\pi_i$ is the conditional probability that the $i$th
hypothesis is under the null given the covariate $x_i$. }
\end{example}
{ To motivate our estimation procedure for $\pi_i$ and $k_i$, let us define $\pi_{\theta}(x)=1/(1+e^{-\theta_0-\theta_1'x})$ and $k_{\beta}(x)=1/(1+e^{-\beta_0-\beta_1'x})$ for $x\in\mathcalathbb{R}^q$, where $\theta=(\theta_0,\theta_1)$ and $\beta=(\beta_0,\beta_1)$. Suppose that conditional on $x_i$ and marginalizing over $H_i$,
\begin{align*}
f(p_i|x_i)=&\pi_{\theta}(x_i)f_0(p_i)+(1-\pi_{\theta}(x_i))f_{1,i}(p_i)
=f_0(p_i)\left\{\pi_{\theta}(x_i)+(1-\pi_{\theta}(x_i))\frac{f_{1,i}(p_i)}{f_0(p_i)}\right\}.
\end{align*}
Replacing $f_{1,i}/f_0$ by the surrogate likelihood ratio whose parameters $k_i$ depend on $x_i$, we obtain
$$\tilde{f}(p_i|x_i)=f_0(p_i)\left\{\pi_{\theta}(x_i)+(1-\pi_{\theta}(x_i))(1-k_{\beta}(x_i))p_i^{-k_\beta(x_i)}\right\}.$$
Moving to a log scale and summing up the individual log likelihoods, we see that the null density is a nuisance parameter that does not depend on $\theta$ and $\beta$:
$$\sum^{m}_{i=1}\log \tilde{f}(p_i|x_i)=\sum^{m}_{i=1}\log\left\{\pi_{\theta}(x_i)+(1-\pi_{\theta}(x_i))(1-k_{\beta}(x_i))p_i^{-k_\beta(x_i)}\right\}+C_0,$$
where $C_0=\sum_{i=1}^m \log f_0(p_i)$.} The above discussions thus motivate the following optimization problem for estimating the unknown parameters:
\begin{align}\label{eq-mle}
\mathcalax_{\theta=(\theta_0,\theta_1)'\in { \mathcalathrm{\scriptscriptstyle T} }heta,\beta=(\beta_0,\beta_1)'\in \mathcalathcal{B}}\sum^{m}_{i=1}\log\{\pi_i+(1-\pi_i)(1-k_i)p^{-k_i}\},
\end{align}
where
\begin{align}\label{eq-par}
& \log\left(\frac{\pi_i}{1-\pi_i}\right)=\theta_0+\theta_1'x_i, \quad \log\left(\frac{k_i}{1-k_i}\right)=\beta_0+\beta_1'x_i,
\end{align}
and ${ \mathcalathrm{\scriptscriptstyle T} }heta,\mathcalathcal{B}\subseteq \mathcalathbb{R}^{q+1}$ are some compact parameter spaces. This problem can be solved using the EM-algorithm together with the Newton's method in its M-step.
Let $\hat{\theta}$ and $\hat{\beta}$ be the maximizer from (\ref{eq-mle}). Define
$$\hat{\pi}_i=W(1/(1+e^{-\tilde{x}_i'\hat{\theta}}),\epsilon_1,\epsilon_2):=\begin{cases}
\epsilon_1, & \mathcalbox{if } 1/(1+e^{-\tilde{x}_i'\hat{\theta}})\leq \epsilon_1, \\
1/(1+e^{-\tilde{x}_i'\hat{\theta}}), & \mathcalbox{if } \epsilon_1<1/(1+e^{-\tilde{x}_i'\hat{\theta}})<1-\epsilon_2, \\
1-\epsilon_2, & \mathcalbox{otherwise},
\end{cases}$$
and $\hat{k}_i=1/(1+e^{-\tilde{x}_i'\hat{\beta}})$ with $\tilde{x}_i=(1,x_i')'$, and
$$\hat{w}_i(t)=\frac{(1-t)\hat{\pi}_i}{t(1-\hat{\pi}_i)}.$$
We use winsorization to prevent $\hat{\pi}_i$ from being too close to zero. In numerical studies, we found the choices of $\epsilon_1=0.1$
and $\epsilon_2=10^{-5}$ perform reasonably well. Further denote
\begin{align*}
\hat{t}=\mathcalax\left\{t\in [0,1]:
\frac{1+\sum^{m}_{i=1}\mathcalathbf{1}\{(1-\hat{k}_i)(1-p_i)^{-\hat{k}_i}>
\hat{w}_i(t)\}}{1\vee\sum^{m}_{i=1}\mathcalathbf{1}\{(1-\hat{k}_i)p_i^{-\hat{k}_i}\geq
\hat{w}_i(t)\}}\leq \alpha\right\}.
\end{align*}
Then we reject the $i$th hypothesis if
$$(1-\hat{k}_i)p_i^{-\hat{k}_i}\geq
\hat{w}_i(\hat{t}).$$
\begin{rem}
{\rm
We can replace $x_i\in\mathcalathbb{R}^q$ by $(g_1(x_i),{ \mathcalbox{d} }ots,g_{q_0}(x_i))\in\mathcalathbb{R}^{q_0}$ for some transformations $(g_1,{ \mathcalbox{d} }ots,g_{q_0})$ to allow nonlinearity in the logistic regressions. In numerical studies, we shall consider the spline transformation.
}
\end{rem}
\section{Asymptotic results}\label{sec:asy}
\subsection{FDR control}
In this subsection, we provide asymptotic justification for the proposed procedure. Note that
\begin{align*}
\mathcalathbf{1}\{(1-\hat{k}_i)p^{-\hat{k}_i}\geq
\hat{w}_i(t)\}=\mathcalathbf{1}\{p\leq c(t,\hat{\pi}_{i},\hat{k}_i)\}\text{
for }
c(t,\hat{\pi}_{i},\hat{k}_i)=1\wedge\left\{\frac{t(1-\hat{k}_i)(1-\hat{\pi}_{i})}{(1-t)\hat{\pi}_{i}}\right\}^{1/\hat{k}_i}.
\end{align*}
Define
\begin{align*}
\text{FDR}(t,\Pi,K)=E\left[\frac{\sum_{i=1}^{m}(1-H_i)\mathcalathbf{1}\{p_i\leq
c(t,\pi_{i},k_i)\}}{\sum_{i=1}^{m}\mathcalathbf{1}\{p_i\leq
c(t,\pi_{i},k_i)\}}\right]
\end{align*}
with $\Pi=(\pi_1,{ \mathcalbox{d} }ots,\pi_m)$ and $K=(k_1,{ \mathcalbox{d} }ots,k_m)$. We make the following assumptions to
facilitate our theoretical derivations.
\begin{ass}\label{ass-ad1}
Suppose the parameter spaces ${ \mathcalathrm{\scriptscriptstyle T} }heta$ and $\mathcalathcal{B}$ are both compact.
\end{ass}
\begin{ass}\label{ass-ad2}
Suppose
$$\lim_m\frac{1}{m}\sum^{m}_{i=1}E\log\{\pi_{\theta}(x_i)+(1-\pi_{\theta}(x_i))(1-k_{\beta}(x_i))p_i^{-k_\beta(x_i)}\}$$
converges uniformly over $\theta\in{ \mathcalathrm{\scriptscriptstyle T} }heta$ and $\beta\in\mathcalathcal{B}$ to $R(\theta,\beta)$, which has a unique
maximum at $(\theta^*,\beta^*)$ in ${ \mathcalathrm{\scriptscriptstyle T} }heta\times \mathcalathcal{B}.$
\end{ass}
Let $\mathcalathcal{F}_a^b=\sigma((x_i,p_i),a\leq i\leq b)$ be the Borel $\sigma$-algebra generated by the random variables $(x_i,p_i)$
for $a\leq i \leq b$. Define the $\alpha$-mixing and $\phi$-mixing coefficients respectively as
\begin{align*}
&\alpha(v)=\sup_b\sup_{A\in\mathcalathcal{F}_{-\infty}^b,B\in\mathcalathcal{F}_{b+v}^{+\infty}}|P(AB)-P(A)P(B)|,
\\&\phi(v)=\sup_b\sup_{A\in\mathcalathcal{F}_{-\infty}^b,B\in\mathcalathcal{F}_{b+v}^{+\infty},P(B)>0}|P(A|B)-P(A)|.
\end{align*}
\begin{ass}\label{ass-ad3}
Suppose $(x_i,p_i)$ is $\alpha$-mixing with $\alpha(v)=O(v^{-\xi})$ for $\xi>r/(r-1)$ and $r>1$ (or $\phi$-mixing with $\phi(v)=O(v^{-\xi})$ for $\xi>r/(2r-1)$ and $r\geq 1$).
Further assume $\sup_i E|\log(p_i)|^{r+{ \mathcalbox{d} }elta}<\infty$ and $\mathcalax_i\|x_i\|_{\infty}<C$, where $\|\cdot\|_{\infty}$ denotes the $l_{\infty}$ norm of a vector and $C,{ \mathcalbox{d} }elta>0$.
\end{ass}
Assumption \ref{ass-ad1} is standard. Assumption \ref{ass-ad2} is a typical condition in
the literature of maximum likelihood estimation for misspecified
models, see e.g. White (1982). Assumption \ref{ass-ad3} relaxes the usual independence assumption by allowing $(x_i,p_i)$ to be weakly dependent.
It is needed to establish the uniform strong law of large numbers for the process $R_m(\theta,\beta)$ defined in the proof of Lemma \ref{lem-30} below which establishes the uniform strong consistency for $\hat{\pi}_i$ and $\hat{k}_i$. The boundedness assumption on $x_i$ could be relaxed with a more
delicate analysis to control its tail behavior and study the convergence rate of $\hat{\theta}$ and $\hat{\beta}$.
Denote by $\|\cdot\|$ the $l_{2}$ norm of a vector. An essential condition required in our proof of Lemma \ref{lem-30} is $\|\hat{\theta}-\theta^*\|\mathcalax_{1\leq i\leq n}\|x_i\|=o_{a.s.}(1)$. { If $\|\hat{\theta}-\theta^*\|=O_{a.s.}(n^{-a})$ for some $a>0$, then by the Borel-Cantelli lemma, we require $\mathcalax_{1\leq i\leq n} E\|x_i\|^{k}<\infty$ for some $k$ with $ak>2,$ i.e., $x_i$ should have a sufficiently light polynomial tail.} We remark that Assumption \ref{ass-ad3} can be replaced by more primitive conditions which allow other weak dependence conditions,
see, e.g., P\"{o}tscher and Prucha (1989). Let $\pi_i^*=W(1/(1+e^{-\tilde{x}_i'\theta^*}),\epsilon_1,\epsilon_2)$
and $k_i^*=1/(1+e^{-\tilde{x}_i'\beta^*})$.
\begin{lemma}\label{lem-30}
Under Assumptions \ref{ass-ad1}-\ref{ass-ad3}, we have $$\mathcalax_{1\leq i\leq
m}|\hat{\pi}_i-\pi_i^*|\rightarrow^{a.s.} 0,\quad \mathcalax_{1\leq i\leq
m}|\hat{k}_i-k_i^*|\rightarrow^{a.s.} 0.$$
\end{lemma}
We impose some additional assumptions to study the asymptotic FDR control.
\begin{ass}\label{ass-31}
For two sequences $a_i,b_i \in [\epsilon,1]$ with small enough $\epsilon$ and large enough $m$,
\begin{align*}
&\left|\frac{1}{m}\sum_{i=1}^{m}\left\{P(p_i\leq a_i|x_i)-P(p_i\leq
b_i|x_i)\right\}\right| \leq c_0\mathcalax_{1\leq i\leq m}|a_i-b_i|,
\end{align*}
where $c_0$ depends on $\epsilon$ but is independent of $m, x_i, a_i$ and
$b_i$.
\end{ass}
\begin{ass}\label{ass-32}
Assume that
\begin{align}
&\frac{1}{m}\sum_{i=1}^{m}P(p_i\leq
c(t,\pi_{i}^*,k^*_i))\rightarrow G_0(t),\\
&\frac{1}{m}\sum_{i=1}^{m}P(1-p_i<
c(t,\pi_{i}^*,k^*_i))\rightarrow G_1(t),\\
&\frac{1}{m}\sum_{H_i=0}P(p_i\leq
c(t,\pi_{i}^*,k^*_i))\rightarrow \tilde{G}_1(t),
\end{align}
for any $t\geq t_0$ with $t_0>0$, where $G_0(t)$, $G_1(t)$ and
$\tilde{G}_1(t)$ are all continuous functions of $t$. Note that the probability here is taken with respect to the joint distribution of $(p_i,x_i)$.
\end{ass}
Let $U(t)=G_1(t)/G_0(t)$, where $G_1$ and $G_0$ are defined in
Assumption \ref{ass-32}.
\begin{ass}\label{ass-33}
There exists a $t'>t_0>0$ such that $U(t')<\alpha.$
\end{ass}
Assumption \ref{ass-31} is fulfilled if the conditional density of $p_i$ given $x_i$ is
bounded uniformly across $i$ on $[\epsilon,1]$. This assumption
is not very strong as we still allow the density to be unbounded
around zero. Assumptions \ref{ass-32}-\ref{ass-33} are similar to those in Theorem 4 of Storey et al. (2004).
In particular, Assumption \ref{ass-33} ensures the existence of a cut-off to control the FDR
at level $\alpha.$
We are now in position to state the main result of this section
which shows that the proposed procedure provides asymptotic FDR
control. The proof is deferred to the supplementary material.
\begin{theorem}\label{thm}
Suppose Assumptions \ref{ass-ad1}-\ref{ass-33} hold and $f_0$ satisfies Condition (\ref{eq-con-f0}). Then we have
$$\limsup_{m}\text{FDR}(\hat{t},\hat{\Pi},\hat{K})\leq \alpha,$$
where $\hat{\Pi}=(\hat{\pi}_1,{ \mathcalbox{d} }ots,\hat{\pi}_m)$ and $\hat{K}=(\hat{k}_1,{ \mathcalbox{d} }ots,\hat{k}_m)$.
\end{theorem}
It is worth mentioning that the validity of our method does not rely on the mixture model assumption (\ref{m0})-(\ref{m1}). In this sense, our method is misspecification robust as the classical BH procedure does.
We provide a comparison between our method and some recently proposed approaches in the following table.
\begin{table}[!ht]\label{tab}
\small
\centering
\begin{tabular}{p{2.5cm}p{1cm}p{1.5cm}p{2.2cm}p{1.75cm}p{1.6cm}p{1.6cm}p{1.6cm}}
\hline
Procedure & $\pi_0$ & $f_1$ & FDR control & Dependent p-values & Misspec. robust & Multiple covariates & Computation\\
\hline
Ignatiadis et al. (2016) & Varying & Partially used & Asymptotic control & Unknown & Yes & No & ++++\\
\hline
Li and Barber (2017) & Varying & Not used & Finite sample upper bound & Gaussian copula & Yes & No$^*$ & ++++\\
\hline
Lei and Fithian (2016) & Varying & Varying & Finite sample control & Unknown & Yes & Yes & +\\
\hline
Scott et al. (2015) & Varying & Fixed & No guarantee & Unknown & Unknown & Yes & +++\\
\hline
Boca and Leek (2018) & Varying & Not used & Unknown & Unknown & Yes & Yes & +++\\
\hline
\textbf{The proposed method} & Varying & Varying & Asymptotic control & Asymptotic & Yes &Yes & +++\\
\hline
\end{tabular}
\\ \caption{Comparison of several covariate adaptive FDR control procedures in recent literature. The number of ``+'' represents the speed. *The framework of Li and Barber (2017) allows accommodating multiple covariates, but the provided software did not implement.}
\end{table}
\subsection{Power analysis}
We study the asymptotic power of the oracle procedure. Suppose the mixture model (\ref{m0})-(\ref{m1}) holds with
$\pi_{0i}=\pi_0(x_i)$ and $f_{1,i}(\cdot)=f_{1}(\cdot;x_i)$, where $f_{1}(\cdot;x)$ is a density function for any fixed $x\in\mathcalathcal{X}$.
Denote by $F_1(\cdot;x)$ and $\bar{F}_1(\cdot;x)$ the distribution and survival functions associated with $f_1(\cdot;x)$ respectively.
Suppose the empirical distribution of $x_i$'s converges to the probability law $\mathcalathcal{P}$. Consider the oracle procedure with $\pi_i=\pi_0(x_i)$ and $k_i=k_0(x_i)$. Here $k_0(\cdot)$ minimizes the integrated Kullback-Leibler divergence, i.e.,
\begin{align*}
&k_0=\mathcalathop{\rm argmin~}_{k\in\mathcalathcal{K}}\int\text{D}_{\text{KL}}(f(\cdot;x)||g(;k(x)))\mathcalathcal{P}(dx),
\\&\text{D}_{\text{KL}}(f(\cdot;x)||g(\cdot;k(x)))=\int^{1}_{0} f(p;x)\log\frac{f(p;x)}{g(p;k(x))}dp,
\end{align*}
with $f(p;x)=\pi_0(x)f_0(p)+(1-\pi_0(x))f_{1}(p;x)$ and $g(p;k(x))=\pi_0(x)+(1-\pi_0(x))(1-k(x))p^{-k(x)}$, and $\mathcalathcal{K}=\left\{k(x):\log\left(\frac{k(x)}{1-k(x)}\right)=\beta_0+\beta_1'x,(\beta_0,\beta_1)\in\mathcalathcal{B}\right\}$.
Write $c(t,x)=c(t,\pi_0(x),k_0(x))$. By the law of large numbers, the realized power of the oracle procedure has the
approximation
\begin{align*}
\text{Power}=&\frac{\sum_{i=1}^{m}\mathcalathbf{1}\{i:H_i=1,p\leq c(t,x_i)\}}{\sum_{i=1}^{m}\mathcalathbf{1}\{i:H_i=1\}}
\approx
\frac{\int(1-\pi(x))F_1(c(t_{\text{opt}},x);x)\mathcalathcal{P}(dx)}{\int(1-\pi(x))\mathcalathcal{P}(dx)},
\end{align*}
where $t_{\text{opt}}$ is the largest positive number such that
\begin{align}\label{eq-pow}
\frac{\int \{\pi_0(x)F_0(c(t,x))+(1-\pi_0(x))\bar{F}_1(1-c(t,x);x) \}\mathcalathcal{P}(dx)}{\int \{\pi_0(x)F_0(c(t,x))+(1-\pi_0(x))F_1(c(t,x);x) \}\mathcalathcal{P}(dx)}\leq
\alpha.
\end{align}
We remark that when
\begin{align}\label{eq-pow1}
\frac{\int (1-\pi_0(x))\bar{F}_1(1-c(t_{\text{opt}},x);x)\mathcalathcal{P}(dx)}{\int \{\pi_0(x)F_0(c(t_{\text{opt}},x))+(1-\pi_0(x))F_1(c(t_{\text{opt}},x);x) \}\mathcalathcal{P}(dx)}\approx 0,
\end{align}
the asymptotic power of the proposed procedure is closed to the oracle procedure based on the LFDR given by
\begin{align}\label{lfdr}
\text{LFDR}_i(p_i)=\frac{\pi_{0i}f_0(p_i)}{\pi_{0i}f_0(p_i)+(1-\pi_{0i})f_{1,i}(p_i)}.
\end{align}
\begin{comment}
Note that (\ref{eq-pow}) is equivalent to
$Q_\alpha(t)=\int \{\alpha(1-\pi_0(x))F_1(c(t,x);x)-(1-\pi_0(x))\bar{F}_1(1-c(t,x);x)-(1-\alpha)\pi_0(x)c(t,x)\}\mathcalathcal{P}(dx)\geq 0$.
As $f_1(p;x)$ is decreasing, it is straightforward to verify that the derivative of
$$Q_\alpha(p;x)=\alpha(1-\pi_0(x))F_1(p;x)-(1-\pi_0(x))\bar{F}_1(1-p;x)-(1-\alpha)\pi_0(x)p$$
is a decreasing function of $p.$ Also note that $c(t,x)$ is an increasing function of $t$ with $\frac{\partial c(t,x)}{\partial t}\big|_{t=0}=0.$
To ensure the existence of $t_{\text{ap}}>0$, we should require that
\begin{align*}
\int\{\alpha(1-\pi_0(x))f_1(0;x)-(1-\pi_0(x))f_1(1;x)-(1-\alpha)\pi_0(x)\}\mathcalathcal{P}(dx)>0.
\end{align*}
As a comparison, we consider the optimal procedure with the rejection rule $p_i\leq c_0(x)$ with $c_0(t,x)=f_1^{-1}(\frac{(1-t)\pi_0(x)}{t(1-\pi_0(x))}).$
The optimal procedure has the approximate realized power
\begin{align*}
\text{Power}\approx
\frac{\int(1-\pi(x))F_1(c_0(t_{\text{opt}},x);x)\mathcalathcal{P}(dx)}{\int(1-\pi(x))\mathcalathcal{P}(dx)},
\end{align*}
where
\begin{align*}
t_{\text{opt}}=\mathcalax\left\{t\in[0,1]:\frac{\int\pi_0(x)c_0(t,x)\mathcalathcal{P}(dx)}{\int \{\pi_0(x)c_0(t,x)+(1-\pi_0(x))F_1(c_0(t,x);x)\}\mathcalathcal{P}(dx)}\leq
\alpha\right\}.
\end{align*}
\end{comment}
\section{Simulation studies}
We conduct comprehensive simulations to evaluate the finite-sample performance of the proposed method and compare it to competing methods. For genome-scale multiple testing, the numbers of hypotheses could range from thousands to millions. For demonstration purpose, we start with $m{=}10,000$ hypotheses. To study the impact of signal density and strength, we simulate three levels of signal density (sparse, medium and dense signals) and six levels of signal strength (from very weak to very strong). To demonstrate the power improvement by using external covariates, we simulate covariates of varying informativeness (non-informative, moderately informative and strongly informative). For simplicity, we simulate one covariate $x_i \sim N(0, 1)$ for $i=1,\cdots,m$. Given $x_i$, we let
$$\pi_{0i} = \frac{\exp (\eta_i)}{1 + \exp(\eta_i)}, ~~~ \eta_i = \eta_0 + k_d x_i, $$
where $\eta_0$ and $k_d$ determine the baseline signal density and the informativeness of the covariate, respectively. { For each simulated dataset, we fix the value of $\eta_0$ and $k_d$. } We set $\eta_0 \in \{3.5, 2.5, 1.5\}$, which achieves a signal density around $3\%$, $8\%$, and $18\%$ respectively at the baseline (i.e., no covariate effect), representing sparse, medium and dense signals. We set $k_d \in \{0, 1,1.5\}$, representing a non-informative, moderately informative and strongly informative covariate, respectively. { Thus, we have a total of $3 \times 3 = 9$ parameter settings.}
Based on $\pi_{0i}$, the underlying truth $H_i$ is simulated from
$$H_i \sim \text{Bernoulli}(1 - \pi_{0i}).$$
Finally, we simulate independent z-scores using $$z_i \sim N(k_sH_i, 1), $$ where $k_s$ controls the signal strength (effect size) and we use values equally spaced on $[2, 2.8]$. Z-scores are converted into p-values using the one-sided formula $1 - \Phi(z_i)$. P-values together with $x_{i}$ are used as the input for the proposed method.
In addition to the basic setting (denoted as Setup S0), we investigate other settings to study the robustness of the proposed method. Specifically, we study
\begin{itemize}
\item[Setup S1.] {\it Additional $f_1$ distribution}. Instead of simulating normal z-scores under $f_1$, we simulate z-scores from a non-central gamma distribution with the shape parameter $k{=2}$. The scale/non-centrality parameters of the non-central gamma distribution are chosen to match the variance and mean of the normal distribution under S0.
\item[Setup S2.] {\it Covariate-dependent $\pi_{0i}$ and $f_{1,i}$}. On top of the basic setup S0, we simulate another covariate $x_i' \sim N(0, 1)$ and let $x_i'$ affect $f_{1,i}$. Specifically, we scale $k_s$ by
$\frac{{ \mathcalbox{d} }isplaystyle 2\exp (k_f x_i')}{{ \mathcalbox{d} }isplaystyle 1 + \exp(k_f x_i')}, $
where we set $k_f \in \{0, 0.25, 0.5\}$ for non-informative, moderately informative and strongly informative covariate scenarios, respectively.
\item[Setup S3.] {\it Dependent hypotheses}. We further investigate the effect of dependency among hypotheses by simulating correlated multivariate normal z-scores. Four correlation structures, including two block correlation structures and two AR(1) correlation structures, are investigated. For the block correlation structure, we divide the 10,000 hypotheses into 500 equal-sized blocks. Within each block, we simulate equal positive correlations ($\rho{=}0.5$) (S3.1). We also further divide the block into 2 by 2 sub-blocks, and simulate negative correlations ($\rho{=}-0.5$) between the two sub-blocks (S3.2). For AR(1) structure, we investigate both $\rho{=}0.75^{|i - j|}$ (S3.3) and $\rho{=}(-0.75)^{|i - j|}$ (S3.4).
{
\item[Setup S4.] {\it Heavy-tail covariate}. In this variant, we generate $x_i$ from the t distribution with 5 degrees of freedom.
\item[Setup S5.] {\it Non-theoretical null distribution}. We simulate both increasing and decreasing $f_0$. For an increasing $f_0$ (S5.1), we generate null z-score $z_i | H_0 \sim N(-0.15, 1)$. For a decreasing $f_0$ (S5.2), we generate null z-score $z_i | H_0 \sim N(0.15, 1)$.
}
\end{itemize}
We present the simulation results for the Setup S0-S2 in the main text and the results for the Setup S3-S5 in the supplementary material. { To allow users to conveniently implement our method and reproduce the numerical results reported here, we make our code and data publicly available at https://github.com/jchen1981/CAMT}.
\subsection{Competing methods}
We label our method as CAMT (Covariate Adaptive Multiple Testing) and compare it to the following competing methods:
\begin{itemize}
\item Oracle: Oracle procedure based on LFDR (see e.g., (\ref{lfdr})) with simulated $\pi_{0i}$ and $f_{1, i}$, which theoretically has the optimal performance;
\item BH: Benjamini-Hochberg procedure (Benjamini et al., 1995, \textit{p.adjust} in R 3.4.2);
\item ST: Storey's BH procedure (Storey 2002, \textit{qvalue} package, v2.10.0);
\item { BL: Boca and Leek procedure (Boca and Leek, 2018, \textit{swfdr} package, v1.4.0);}
\item IHW: Independent hypothesis weighting (Ignatiadis et al., 2016, \textit{IHW} package, v1.6.0);
\item { FDRreg: False discovery rate regression (Scott et al., 2015, \textit{FDRreg} package, v0.2, \textit{https://github.com/jgscott/FDRreg}), FDRreg(T) and FDRreg(E) represent FDRreg with the theoretical null and empirical null respectively; }
\item SABHA: Structure adaptive BH procedure (Li and Barber, 2017, $\tau = 0.5, \epsilon = 0.1$ and stepwise constant weights, \textit{https://www.stat.uchicago.edu/$\sim$rina/sabha/All\_q\_est\_functions.R});
\item AdaPT: Adaptive p-value thresholding procedure (Lei and Fithian, 2018, \textit{adaptMT} package, v1.0.0).
\end{itemize}
We evaluate the performance based on FDR control (false discovery proportion) and power (true positive rate) with a target FDR level of 5\%. Results are averaged over 100 simulation runs.
\subsection{Simulation results}
We first study the performance of the proposed method under the basic setup (S0, Figure \ref{fig:sim:1}). All compared methods generally controlled the FDR around/under the nominal level of 0.05 and no serious FDR inflation was observed at any of the parameter setting (Figure \ref{fig:sim:1}A). { However, FDRreg exhibited a slight FDR inflation under some parameter settings and the inflation seemed to increase with the informativeness of the covariate and signal density.} Conservativeness was also observed for some methods in some cases. As expected, the BH procedure, which did not take into account $\pi_0$, was conservative when the signal was dense. IHW procedure was generally more conservative than BH and the conservativeness increased with the informativeness of the covariate. CAMT, the proposed method, was conservative when the signal was sparse and the covariate was less informative. The conservativeness was more evident when the effect size was small but decreased as the effect size became larger. AdaPT was more conservative than CAMT under sparse signal/weak covariate. In terms of power (Figure \ref{fig:sim:1}B), there were several interesting observations. First, as the covariate became more informative, all the covariate adaptive methods became more powerful than ST and BH. The power differences between these methods also increased. Second, { FDRreg was the most powerful across settings. Under a highly informative covariate, it was even slightly above the oracle procedure, which theoretically had an optimal power. The superior power of FDRreg could be partly explained by a less well controlled FDR.} The IHW was more powerful than BL/SABHA when the signal was sparse; but the trend reversed when the signal was dense. Third, AdaPT was very powerful when the signal was dense and the covariate was highly informative. However, the power decreased as the signal became more sparse and the covariate became less informative. In fact, when the signal was sparse and the covariate was not informative or moderately informative, AdaPT had the lowest power. In contrast, the proposed method CAMT was close to the oracle procedure. It was comparable to AdaPT when AdaPT was the most powerful, but was significantly more powerful than AdaPT in its unfavorable scenarios. CAMT had a clear edge when the covariate was informative and signal was sparse. Similar to AdaPT, CAMT had some power loss under sparse signal and non-informative covariate, probably due to the discretization effect from the BC-type estimator.
We conducted more evaluations on type I error control under S0. We investigated the FDR control across different target levels. Figure \ref{fig:sim:2} showed excellent FDR control across target levels for all methods except FDRreg. The actual FDR level of BH and IHW was usually below the target level. CAMT was slightly conservative at a small target level under the scenario of sparse signal and less informative covariate, but it became less conservative at larger target levels. We also simulated a complete null, where no signal was included (Figure \ref{fig:sim:3}). In such case, FDR was reduced to FWER. { Interestingly, FDRreg was as conservative as CAMT and AdaPT under the complete null.}
{ It is interesting to study the performance of the competing methods under a much larger feature size, less signal density, and weaker signal strength, representing the most challenging scenario in real problems. To achieve this end, we simulated $m=100,000$ features with a signal density of $0.5\%$ at the baseline (no covariate effect). Under a moderately informative covariate, we observed a substantial power improvement of CAMT over all other methods including FDRreg while controlling the FDR adequately at different target levels (Figure \ref{fig:sim:3:1}). } We further reduced the feature size to 1,000 (Figure \ref{fig:sim:6:1} in the supplement) to study the robustness of the methods to a much smaller feature size. Although CAMT and AdaPT were still more powerful than the competing methods when the signal was dense and the covariate was informative, a significant power loss was observed in other parameter settings, particularly under sparse signal and a less informative covariate. As we further decreased the feature size to 200, CAMT and AdaPT became universally less powerful than ST across parameter settings (data not shown). Therefore, application of CAMT or AdaPT to datasets with small numbers of features was not recommended unless the signal was dense and the covariate was highly informative.
{ We also simulated datasets, where the z-scores under the alternative were drawn from a non-central gamma distribution (Setup S1). Under such setting, the trend remained almost the same as the basic setup (Figure \ref{fig:sim:4}), but FDRreg had a more marked FDR inflation. When both $\pi_{0i}$ and $f_{1, i}$ depended on the covariate (Setup S2), CAMT became slightly more powerful without affecting the FDR control, especially when the covariate was highly informative (Figure \ref{fig:sim:5}). Meanwhile, the performance of FDRreg was also remarkable with a very small FDR inflation. However, if we increased the effect on $f_{1, i}$ by reducing the standard deviation of the z-score under the alternative, FDRreg was no longer robust and the observed FDP was substantially above the target level when the signal strength was weak, indicating the benefit of modeling covariate-dependent $f_1$ (Figure \ref{fig:sim:6} in the supplement). } CAMT was also robust to different correlation structures (Setup S3.1, S3.2, S3.3, S3.4) and we observed similar performance under these correlation structures (Figures \ref{fig:sim:7}-\ref{fig:sim:10} in the supplement). { The performance of CAMT was also robust to a heavy-tail covariate (Setup S4, Figure \ref{fig:sim:11} in the supplement). } In an unreported numerical study, we
added different levels of perturbation to the covariate by multiplying random small values drawn from Unif(0.95, 1.05), Unif(0.9, 1.1), and Unif(0.8, 1.2), respectively.
We observed that the $\pi_0$ estimates under perturbation are highly correlated with the $\pi_0$ estimates without perturbation, which showed the stability of our method against
data perturbations.
{ We also examined the robustness of CAMT to the deviation from the theoretical null (Setup S5). Specifically, we simulated both decreasing and increasing $f_0$. The new results were presented in Figures \ref{fig:sim:12} and \ref{fig:sim:13} in the supplement. We observed that, for an increasing $f_0$, all the methods other than FDRreg were conservative and had substantial less power than the oracle procedure. FDRreg using a theoretical null was conservative when the covariate was less informative but was anti-conservative under a highly informative covariate. On the other hand, FDRreg using an empirical null had an improved power and controlled the FDR closer to the target level for most settings. However, it did not control the FDR well when the signal was dense and the prior information was strong. When $f_0$ was decreasing, all the methods without using the empirical null failed to control the FDR. FDRreg with an empirical null improved the FDR control substantially for most settings but still could not control the FDR well under the dense-signal and strong-prior setting. Therefore, there is still room for improvement to address the empirical null problem.}
Finally, we compared the computational efficiency of these competing methods (Figure \ref{fig:sim:14}). SABHA (step function) and IHW were computationally the most efficient and they completed the analysis for one million p-values in less than two minutes. CAMT and the new version of FDRreg (v0.2) were also computationally efficient, followed by BL, and they all could complete the computation in minutes for one million p-values under S0. AdaPT was computationally the most intensive and completed the analysis in hours for one million p-values. { We note that all the methods including AdaPT are computationally feasible for a typical omics dataset. }
In summary, CAMT improves over existing covariate adaptive multiple testing procedures, and is a powerful, robust and computationally efficient tool for large-scale multiple testing.
\begin{figure}\label{fig:sim:0}
\end{figure}
\begin{figure}\label{fig:sim:1}
\end{figure}
\begin{figure}\label{fig:sim:2}
\end{figure}
\begin{figure}\label{fig:sim:3}
\end{figure}
\begin{figure}\label{fig:sim:3:1}
\end{figure}
\begin{figure}\label{fig:sim:4}
\end{figure}
\begin{figure}\label{fig:sim:5}
\end{figure}
\begin{figure}\label{fig:sim:14}
\end{figure}
\begin{figure}\label{fig:real:1}
\end{figure}
\begin{figure}\label{fig:real:2}
\end{figure}
\section{Application to omics-wide multiple testing}
To demonstrate the use of the proposed method for real-world applications, we applied CAMT to several omics datasets from transcriptomics, proteomics, epigenomics and metagenomics studies with the aim to identify omics features associated with the phenotype of interest. Since AdaPT is the most start-of-the-art method, we focused our comparison to it. To make a fair comparison, we first run the analyses on the four omics datasets, which were also evaluated by AdaPT (Lei and Fithian, 2018), including Bottomly (Bottomly et al., 2011), Pasilla (Brooks et al., 2011), Airway (Himes et al., 2014) and Yeast Protein dataset (Dephoure et al., 2012). The Bottomly, Pasilla and Airway were three transcriptomics datasets from RNA-seq experiments with a feature size of 13,932, 11,836 and 33,469, respectively. The yeast protein dataset was a proteomics dataset from with a feature size of 2,666. We used the same methods to calculate the p-values for these datasets as described in Lei and Fithian (2018). The distributions of the p-values for these four datasets all exhibited a spike in the low p-value region, indicating that the signal was dense. The logarithm of normalized count (averaged across all samples) was used as the univariate covariate for the three RNA-seq data (Bottomly, Pasilla and Airway). The logarithm of the total number of peptides across all samples was used as the univariate covariate for the yeast protein data. Following AdaPT, we used a spline basis with six equiquantile knots for $\pi_{0i}, f_{1,i}$ (CAMT and AdaPT) and for $\pi_{0i}$ (FDRreg, BL) to account for potential complex nonlinear effects. Since IHW and SABHA could only take univariate covariate, we used the univariate covariate directly. We summarized the results in Figure \ref{fig:real:1}. We were able to reproduce the results in Lei and Fithian (2018). Indeed, AdaPT was more powerful than SABHA, IHW, ST and BH on the four datasets. { FDRreg and BL, which were not compared in Lei and Fithian (2018), also performed well and made more rejections than other methods on the Yeast dataset and the Bottomly dataset, respectively.} The performance of the proposed method, CAMT, was almost identical to AdaPT, which was consistent with the simulation results in the scenario of dense signal and informative covariate (Figure \ref{fig:sim:1}).
We next applied to two additional omics datasets from an epigenome-wide association study (EWAS) of congenital heart disease (CHD) (Wijnands et al., 2017) and a microbiome-wide association study (MWAS) of sex effect (McDonald et al., 2018).
\begin{itemize}
\item \textit{EWAS data}. The aim of the EWAS of CHD was to identify the CpG loci in the human genome that were differentially methylated between healthy ($n=196$) and CHD ($n=84$) children. The methylation levels of 455,741 CpGs were measured by the the Illumina 450K methylation beadchip and was normalized properly before analysis. The p-values were produced by running a linear regression to the methylation outcome for each CpG, adjusting for potential confounders such as age, sex and blood cell mixtures as described in Wijnands et al. (2017). Since widespread hyper-methylation (increased methylation in low-methylation regions) or hypo-methylation (decreased methylation in high-methylation regions) are common in many diseases (Robertson, 2005), we use the mean methylation across samples as the univariate covariate.
\item \textit{MWAS data}. The aim of the MWAS of sex was to identify differentially abundant bacteria in the gut microbiome between males and females, where the abundances of the gut bacteria were determined by sequencing a fingerprint gene in the bacteria 16S rRNA gene. We used the publicly available data from the AmericanGut project (McDonald et al., 2018), where more than the gut microbiome from more than 10,000 subjects were sequenced. We focused our analysis on a relatively homogenous subset consisting of 481 males and 335 males (age between 13-70, normal BMI, from United States). We removed OTUs (clustered sequencing units representing bacteria species) observed in less than 5 subjects, and a total of 2, 492 OTUs were tested using Wilcoxon rank sum test on the normalized abundances. We use the percentage of zeros across samples as the univariate covariate since we expect a much lower power for OTUs with excessive zeros.
\end{itemize}
The results for these two datasets were summarized in Figure \ref{fig:real:2}. For the EWAS data, the signal density was very sparse ($\hat{\pi}_0{= }0.99$, \textit{qvalue} package). CAMT identified far more loci than the other methods at various FDR levels. { The performance was consistent with the simulation results in the scenario of extremely sparse signal and informative covariate, where CAMT was substantially more powerful than the competing methods (Figure \ref{fig:sim:3:1}). }At an FDR of 20\%, we identified 55 differentially methylated CpGs, compared to 19 for AdaPT. These 55 CpG loci were mainly located in CpG islands and the gene promotor regions, which were known for their important role in gene expression regulation (Robertson, 2005). Interestingly, all but one CpG loci had low levels of methylation, indicating the methylation level was indeed informative to help identify differential CpGs. We also did gene set enrichment analysis for the genes where the identified CpGs were located (https://david.ncifcrf.gov/). Based on the GO terms annotated to biological processes (BP\_DIRECT), three GO terms were found to be significant (unadjusted p-value $<$0.05) including one term ``embryonic heart tube development", which was very relevant to the congenital heart disease under study (Wijnands et al., 2017). As a sanity check, we randomized the covariate and re-analyzed the data using CAMT. As expected, CAMT became similar to BH/ST and identified the same eight CpGs at 20\% FDR level.
For the MWAS data, although the difference was not as striking as the EWAS data, CAMT was still overall more powerful than other competing methods except FDRreg. { However, given the fact that FDRreg was not robust under certain scenarios, the interpretation of the increased power should be cautious. } The relationship between the fitted $\pi_{0i}$ and the covariate (number of nonzeros) was very interesting: $\hat{\pi}_{0i}$ first decreased, reached a minimum at around 70 and then increased (Figure \ref{fig:real:3}). When the OTU was rare (e.g., a small number of nonzeros, only a few subjects had it), it was either very individualized or we had limited power to reject it, leading to a large $\pi_{0i}$. In the other extreme where the OTU was very prevalent (e.g., a large number of nonzeros, most of the subjects had it), it was probability not sex-specific either. Therefore, taking into account the sparsity level could increase the power of MWAS. It is also informative to compare CAMT to the traditional filtering-based procedure for MWAS. In practice, we usually apply a prevalence-based filter before performing multiple testing correction, based on the idea that rare OTUs are less likely to be significant and including them will increase the multiple testing burden. A subjective filtering criterion has to be determined beforehand. For this MWAS dataset, if we removed OTUs present in less than 10\% of the subjects, ST and BH recovered 116 and 85 significant OTUs at an FDR of 10\%, compared to 69 and 65 on the original dataset, indicating that filtering did improve the statistical power of traditional FDR control procedures. However, if we removed OTUs present in less than 20\% of the subjects, the numbers of significant OTUs by ST and BH reduced to 71 and 50 respectively. Therefore, filtering could potentially leave out biologically important OTUs. In contrast, CAMT did not require an explicit filtering criterion, and was much more powerful (141 significant OTUs at 10\% FDR) than the filtering-based method.
\begin{figure}\label{fig:real:3}
\end{figure}
\section{Discussions}\label{sec:dis}
There are generally two strategies for estimating the number of false rejections $\sum^{m}_{i=1}(1-H_i)\mathcalathbf{1}\{h_i(p_i)\geq w_i(t)\}$
given the form of the rejection rule $h_i(p_i)\geq w_i(t)$. The first approach (called BH-type estimator) is to replace the number of false rejections by its expectation assuming that $p_i$
follows the uniform distribution on $[0,1]$ under the null, which leads to the quantity $\sum^{m}_{i=1}\pi_{0i}c(t,\pi_{0i},k_i)$ for $c(\cdot)$
defined in Section \ref{sec:asy}. The second approach (called BC-type estimator) estimates the false rejection conservatively by
$\xi+\sum^{m}_{i=1}\mathcalathbf{1}\{h_i(1-p_i)\geq w_i(t)\}$ for a nonnegative constant $\xi$ under the assumption that the null distribution of p-values is symmetric about 0.5.
Both procedures enjoy optimality in some asymptotic sense, see, e.g., Arias-Castro and Chen (2017). The advantage of the BC-type procedure lies on that its estimation of the number of false rejections is asymptotically conservative when the rejection rule converges to a non-random limit (which holds even under a misspcified model, see e.g., White, 1982) and $f_0$ is mirror conservative (see equation (3) of Lei and Fithian, 2018).
This fact allows us to estimate the rejection rule by maximizing a potentially misspecified likelihood as the resulting rejection rule has a non-random limit under suitable conditions. This is not necessarily the case for the BH-type estimator without imposing additional constraint when estimating $\pi_{0i}$ and $k_i$. Specific restriction on the estimators of $\pi_{0i}$ is required for the BH-type estimator to achieve FDR control, see, e.g., equation (3) of Li and Barber (2018).
On the other hand, as the BC-type estimator uses a counting approach to estimate the number of false rejections, it suffers from
the discretization issue (i.e., the BC-type estimator is a step function of $t$ while the BH-type estimator is continuous), which may result in a large variance for the FDR estimate. This is especially the case when the FDR level is small.
For small FDR level, the number of rejections is usually small, and thus both the denominator and numerator of the FDR estimate become small and more variable.
Another issue with the BC-type estimator is the selection of $\xi$. We follow the idea of $\text{knockoff}+$ in Barber and Cand\`{e}s (2015)
by setting $\xi=1$. This choice could make the procedure rather conservative when the signal is very sparse, and the target FDR level is small.
A choice of smaller $\xi$ (e.g. $\xi=0$) often leads to inflated FDR in our unreported simulation studies. To alleviate this issue, one may
consider a mixed strategy by using
\begin{align*}
\mathcalax\left\{\sum^{m}_{i=1}\pi_{0i}c(t,\pi_{0i},k_i),\sum^{m}_{i=1}\mathcalathbf{1}\{h_i(1-p_i)\geq w_i(t)\}\right\}
\end{align*}
as a conservative estimate for the number of false rejections when $t$ is relatively small. Our numerical results in Figure \ref{fig:sim:mix} in the supplementary material show that the resulting method can successfully
reduce the power loss in the case of sparse signals (or small FDR levels) and less informative covariates while maintaining the good power performance in other cases. A serious investigation of this mixed procedure and the BH-type estimator is left for future research.
{ Since our method is not robust to a decreasing $f_0$, some diagnostics are needed before running CAMT. To detect a decreasing $f_0$, the genomic inflation factor (GIF) can be employed (Devin and Roeder, 1999). GIF is defined as the ratio of the median of the observed test statistic to the expected median based on the theoretical null distribution. GIF has been widely used in genome-wide association studies to assess the deviation of the empirical distribution of the null p-values from the theoretical uniform distribution. To accommodate potential dense signals for some genomics studies, we recommend to confine the GIF calculation to p-values between 0.5 and 1. If the GIF is larger, using CAMT may result in excess false positives. In such case, the user should not trust the results and may consider recalculating the p-values by adjusting potential confounding factors, either known or estimated based on some latent variable approach such as surrogate variable analysis (Leek and Storey, 2007), or using the simple genomic control approach based on p-values (Devin and Roeder, 1999). }
\end{document}
|
\begin{document}
\title{Lipschitz extensions of definable $p$-adic functions}
\abstract{In this paper, we prove a definable version of Kirszbraun's theorem in a non-Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function $f:X\times Y\to\mathbb{Q}_p^s$, where $X\subset \mathbb{Q}_p$ and $Y\subset \mathbb{Q}_p^r$, that is $\lambda$-Lipschitz in the first variable, extends to a definable function $\tilde{f}:\mathbb{Q}_p\times Y\to\mathbb{Q}_p^s$ that is $\lambda$-Lipschitz in the first variable.}
\section{Introduction}
In 1934, Kirszbraun proved that every $\lambda$-Lipschitz function $f:S\subset \mathbb{R}^r\to\mathbb{R}^s$ extends to a $\lambda$-Lipschitz function $\tilde{f}:\mathbb{R}^r\to\mathbb{R}^s$ (see \cite{kirszbraun}). In 1983, Bhaskaran proved that a version of Kirszbraun's theorem still holds in a non-Archimedean setting, more precisely, for all spherically complete fields (see \cite{bhaskaran}). Recently, in 2010, Aschenbrenner and Fischer proved a definable version of Kirszbraun's theorem. In particular, they proved that every $\lambda$-Lipschitz function $f:S\subset\mathbb{R}^r\to\mathbb{R}^s$, that is definable in an expansion of the ordered field of real numbers, extends to a $\lambda$-Lipschitz function $\tilde{f}:\mathbb{R}^r\to\mathbb{R}^s$ that is definable in the same structure (see \cite{aschenbrenner}).
The proof of Bhaskaran relies in an essential way on Zorn's Lemma, which makes it far from being applicable to a definable setting. Therefore Aschenbrenner posed the question whether there could be a \emph{definable} version of Kirszbraun's theorem in a non-Archimedean setting. In this paper we partially answer that question and prove a definable version of Kirszbraun's theorem in a non-Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function $f:X\times Y\to\mathbb{Q}_p^s$, where $X\subset \mathbb{Q}_p$ and $Y\subset \mathbb{Q}_p^r$, that is $\lambda$-Lipschitz in the first variable, extends to a definable function $\tilde{f}:\mathbb{Q}_p\times Y\to\mathbb{Q}_p^s$ that is $\lambda$-Lipschitz in the first variable. By \emph{definable}, we mean definable in either a semi-algebraic or a subanalytic structure on $\mathbb{Q}_p$. Working with these languages will allow us to use a cell decomposition result (see Theorem \ref{thm_preparation}) that is essential for the construction of Lipschitz extensions.
In a first approach we use a more easy construction to obtain a $\Lambda$-Lipschitz extension, where $\Lambda$ is possibly larger than $\lambda$. In a second and more involved approach, we show one can take $\Lambda$ equal to $\lambda$. More generally, we prove our results for finite field extensions of $\mathbb{Q}_p$.
\subsection*{Acknowledgments}
The autor would like to thank Raf Cluckers for proposing the idea for this paper, the many fruitful discussions and the constant optimism during the preparation of this paper. The author is also grateful to Matthias Aschenbrenner, for it was his question that formed the inspiration for this research project.
\section{Preliminary definitions and facts}
Let $p$ be a prime number, and $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $K$ be a finite field extension of $\mathbb{Q}_p$. Denote by $\mathrm{ord}:K^\times \to \mathbb{Z}$ the valuation. Denote by $\mathcal{O}_K$ the valuation ring, by $\mathcal{M}_K$ the maximal ideal of $\mathcal{O}_K$ and by $\pi_K$ a fixed generator of $\mathcal{M}_K$. Let $q$ denote the cardinality of the residue field. Finally, let $\overline{\mathrm{ac}}_m: K\to \mathcal{O}_K/(\pi_K^m)$ be the angular component map of depth $m$, sending every nonzero $x$ to $x\pi_K^{-\mathrm{ord}(x)} \mod (\pi_K^m)$ and 0 to 0.
The valuation induces a non-Archimedean norm on $K$ by setting $\abs{x}=q^{-\mathrm{ord}(x)}$ for nonzero $x$, and $\abs{0}=0$. This extends to a non-Archimedean norm on $K^s$ by setting $\abs{(x_1,\ldots,x_s)} = \max_i \{\abs{x_i}\}$. A function $f:K^r\to K^s$ is said to be $\lambda$-Lipschitz, with $\lambda\in\mathbb{R}$, if $\abs{f(x)-f(y)}\leq \lambda \abs{x-y}$ for all $x,y\in K^r$. One calls $\lambda$ the \emph{Lipschitz constant} of $f$.
Say a set $X\subset K^r$ is \emph{definable} if it is definable in either a semi-algebraic or a subanalytic structure on $K$. This means that $X$ is given by a first-order formula, possibly with parameters form $K$, in the semi-algebraic or subanalytic language (see \cite{ccl} for more details). For the convenience of the reader, we recall these languages. The semi-algebraic (or \emph{Macintyre}) language is the language $\mathcal{L}_{\text{Mac}}=(+,-,\cdot,\{P_n\}_{n>0},0,1)$, where the predicates $P_n$ stand for the $n$-th powers in $K$. The subanalytic language is the language $\mathcal{L}_{\text{an}}=\mathcal{L}_{\text{Mac}} \cup (^{-1},\cup_{m>0}K\{x_1,\ldots,x_m\})$, where $^{-1}$ is interpreted as the multiplicative inverse extended by $0^{-1}=0$, and where every function symbol from $K\{x_1,\ldots,x_m\}$ is interpreted as the restricted analytic function $K^m\to K$ given by
\[x\mapsto \begin{cases} f(x) &\text{if }x\in \mathcal{O}_K^m,\\0&\text{otherwise},\end{cases}\]
where $f$ is a formal power series converging on $\mathcal{O}_K^m$. Let $X\subset K^r$ be a definable set, then a function $f:X\to K^s$ is definable if its graph is a definable subset of $K^{r+s}$.
We work with the notion of $p$-adic cells as given in \cite{ch}. We recall the main definitions and properties. For integers $m,n>0$, let $Q_{m,n}$ be the (definable) set
\[Q_{m,n} = \{x\in K^\times \mid \mathrm{ord}(x)\in n\mathbb{Z},\ \overline{\mathrm{ac}}_m(x)=1\}.\]
\begin{definition}
Let $Y$ be a definable set. A \emph{cell} $C\subset K\times Y$ over $Y$ is a (nonempty) set of the form
\begin{equation*}
\resizebox{.85\hsize}{!}{$C = \{(x,y)\in K\times Y\mid y\in Y',\ \abs{\alpha(y)}\mathrel{\square_1}\abs{x-c(y)}\mathrel{\square_2} \abs{\beta(y)},\ x-c(y)\in\xi Q_{m,n}\}$,}
\end{equation*}
where $Y'\subset Y$ is a definable set, $\xi\in K$, $\alpha, \beta: Y'\to K^\times$ and $c:Y'\to K$ are definable functions, $\square_i$ is either $<$ or ``no condition'', and such that $C$ projects surjectively onto $Y'$. We call $c$ and $\xi Q_{m,n}$ the \emph{center} and the \emph{coset} of the cell $C$, respectively. If $\xi = 0$ we call $C$ a \emph{0-cell}, otherwise we call $C$ a \emph{1-cell}. We call $Y'$ the \emph{base} of the cell $C$.
\end{definition}
\begin{definition}
Let $Y$ be a definable set. Let $C\subset K\times Y$ be a 1-cell over $Y$ with center $c$ and coset $\xi Q_{m,n}$. Then, for each $(t,y) \in C$ with $y\in Y$, there exists a unique maximal ball $B$ containing $t$ and satisfying $B\times \{y\}\subset C$, where maximality is under inclusion. If $\mathrm{ord}(t-c(y))=l$, this ball is of the form
\[B = B_{l,c(y),m,\xi} = \{x\in K\mid \mathrm{ord}(x-c(y)) = l,\, \overline{\mathrm{ac}}_m(x-c(y)) = \overline{\mathrm{ac}}_m(\xi)\}.\]
We call the collection of all these maximal balls the \emph{balls of the cell $C$}. For fixed $y_0\in Y$, we call the collection of balls $\{B_{l,c(y_0),m,\xi}\mid B_{l,c(y_0),m,\xi}\times \{y_0\}\subset C\}$ the balls of the cell $C$ \emph{above $y_0$}. If $C\subset K\times Y$ is a $0$-cell, we define the collection of balls of $C$ to be the empty collection.
\end{definition}
Notice that $B_{l,c(y),m,\xi}$ is a ball of diameter $q^{-(l+m)}$, in particular, for every $x_1,x_2\in B_{l,c(y),m,\xi}$ it holds that $\abs{x_1-x_2}\leq q^{-(l+m)}$.
\begin{definition}[Jacobian property]
Let $f:B\to B'$ be a function, where $B,B'\subset K$ are balls. Say that $f$ has the \emph{Jacobian property} if the following conditions hold:
\begin{enumerate}
\item $f$ is a bijection;
\item $f$ is continuously differentiable on $B$, with derivative $\mathrm{d}eriv{f}{x}$;
\item $\mathrm{ord} (\mathrm{d}eriv{f}{x})$ is constant (and finite) on $B$;
\item for all $x,y\in B$ with $x\neq y$, one has:
\[\mathrm{ord}(f(x)-f(y)) = \mathrm{ord}(\mathrm{d}eriv{f}{x}) + \mathrm{ord}(x-y).\]
\end{enumerate}
\end{definition}
\begin{definition}
Let $f:S\subset K\times Y\to K$ be a function. Then we define
\[f\times \mathrm{id}: S\to K\times Y: (x,y)\mapsto (f(x,y),y),\]
and we denote with $S_f$ the image of $f\times\mathrm{id}$.
\end{definition}
\begin{definition}
Let $f:S\subset K\times Y\to K^s$ be a function. Then we define for every $y\in Y$
\[f_y: S_y\to K^s: x\mapsto f(x,y),\]
where $S_y$ denotes the fiber $S_y = \{x\in K\mid (x,y)\in S\}$.
\end{definition}
\begin{definition}
Let $Y$ be a definable set, let $C\subset K\times Y$ be a 1-cell over $Y$, and let $f:C\to K$ be a definable function. Say that $f$ is \emph{compatible} with the cell $C$ if either $C_f$ is a 0-cell over $Y$, or the following holds: $C_f$ is a 1-cell over $Y$ and for each $y\in Y$ and each ball $B$ of $C$ above $y$ and each ball $B'$ of $C_f$ above $y$, the functions $\restr{f_y}{B}$ and $\restr{f_y^{-1}}{B'}$ have the Jacobian property.
If $g:C\to K$ is a second definable function which is compatible with the cell $C$ and if we have $C_f = C_g$ and $\mathrm{ord}(\frac{\partial f(x,y)}{\partial{x}})= \mathrm{ord} (\frac{\partial g(x,y)}{\partial x})$ for every $(x,y)\in C$, then we say that $f$ and $g$ are \emph{equicompatible} with $C$.
If $C'\subset K\times Y$ is a 0-cell over $Y$, any definable function $h:C'\to K$ is said to be compatible with $C'$, and $h$ and $k:C'\to K$ are equicompatible with $C'$ if and only if $h=k$.
\end{definition}
The following theorem is based on Theorem 3.3 of \cite{ch}. This theorem is the result of a constant refinement of the concept of $p$-adic cell decomposition for semi-algebraic and subanalytic structures. Earlier versions are due to Cohen \cite{cohen}, Denef \cite{denef-84,denef}, Cluckers \cite{cluckers}, and relate to the quantifier elimination results from Macintyre \cite{mac} and Denef-van den Dries \cite{denef-vdd}.
\begin{theorem}\label{thm_preparation}
Let $S\subset K\times Y$ and $f:S\to K$ be definable. Then there exists a finite partition of $S$ into cells $C$ over $Y$ such that the restriction $\restr{f}{C}$ is compatible with $C$ for each cell $C$. Moreover, for each cell $C$ there exists a definable function $m:C\to K$, a definable function $e:Y\to K$ and coprime integers $a$ and $b$ with $b>0$, such that for all $(x,y)\in C$
\[m(x,y)^b = e(y)(x-c(y))^a,\]
where $c$ is the center of $C$, and such that if one writes $c'$ for the center of $C_f$, one has that $g = m+c'$ and $f$ are equicompatible with $C$ (we use the conventions that $b=1$ whenever $a=0$, that $a=0$ whenever $C$ is a 0-cell, and that $0^0=1$).
Furthermore, if $C$ and $C_f$ are 1-cells, then for every $y\in Y$ one has that $f_y(B) = g_y(B)$ for every ball $B$ of $C$ above $y$, and the formula
\begin{equation}\label{eq_formula_order}
\mathrm{ord}\left(\frac{\partial f(x,y)}{\partial{x}}\right) = \mathrm{ord}(e(y)^{1/b}q) + (q-1)\mathrm{ord}(x-c(y))
\end{equation}
holds for all $(x,y)\in C$, where $q=a/b$ and where we use the convenient notation $\mathrm{ord}(t^{1/b}) = \mathrm{ord}(t)/b$, for $t\in K$ and $b>0$ a positive integer.
\end{theorem}
\begin{proof}
The existence of a finite partition of $S$ in cells $C$ over $Y$, and for every such a cell $C$ the existence of $g=m+c'$ such that $f$ and $g$ are equicompatible with $C$, follows immediately from Theorem 3.3 in \cite{ch}.
Now assume that $C$ and $C_f$ are 1-cells. It is easy to see that $f_y(B) = g_y(B)$ for every $y\in Y$ and every ball $B$ of $C$ above $y$.
We now prove \eqref{eq_formula_order}. Fix $(x,y)\in C$. Since $f$ and $g$ are equicompatible, we have $\mathrm{ord}(\frac{\partial f(x,y)}{\partial{x}}) = \mathrm{ord}(\frac{\partial g(x,y)}{\partial{x}})$, so we only need to prove that \eqref{eq_formula_order} holds for $f$ replaced by $g$. For this, we first note that
\begin{equation}\label{eq_ord_g}
\mathrm{ord}(g(x,y)-c'(y)) = [\mathrm{ord}(e(y))+a\cdot\mathrm{ord}(x-c(y))]/b.
\end{equation}
It is also immediate that
\begin{equation}\label{eq_g_rechts}
\mathrm{ord}\left(\frac{\partial ([g(x,y)-c'(y)]^b)}{\partial{x}}\right) = \mathrm{ord}(e(y)a(x-c(y))^{a-1}).
\end{equation}
On the other hand, by the chain rule, the left hand side of \eqref{eq_g_rechts} also equals
\begin{equation}\label{eq_g_links}
\mathrm{ord}\left(\frac{\partial ([g(x,y)-c'(y)]^b)}{\partial{x}}\right) =\mathrm{ord}\left(b[g(x,y)-c'(y)]^{b-1}\frac{\partial g(x,y)}{\partial{x}}\right).
\end{equation}
Equating the right hand sides of \eqref{eq_g_rechts} and \eqref{eq_g_links}, and using \eqref{eq_ord_g}, one easily finds the required formula.
\end{proof}
Comparing sizes of balls between which there is a function with the Jacobian property, we obtain the following useful formula.
\begin{lemma}\label{lemma_l'm'ordf'lm}
Let $f:B_{l,c(y),m,\xi}\to B_{l',c'(y),m',\xi'}$ be a function with the Jacobian property. Then $l'+m' = \mathrm{ord}(df/dx)+l+m$.
\end{lemma}
\section{Existence of Lipschitz extensions}
We now proceed towards proving the existence of definable Lipschitz extensions of definable families of functions in one variable. Let us first formulate the main theorem of this paper:
\begin{theorem}\label{thm_main_param_const1}
Let $Y\subset K^r$ and $X\subset K$ be definable sets and let $f:X\times Y\to K^s$ be a definable function that is $\lambda$-Lipschitz in the first variable. Then $f$ extends to a definable function $\tilde{f}:K\times Y\to K^s$ that is $\lambda$-Lipschitz in the first variable, i.e. $\tilde{f}_y$ is $\lambda$-Lipschitz for every $y\in Y$.
\end{theorem}
\begin{remark}\label{remark_lipschitz_constant}
By rescaling, if suffices to proof the theorem for $\lambda=1$. Also, since we use the max-norm on $K^s$, it is enough to prove the theorem for $s=1$.
\end{remark}
Firstly, we present a very general way of \emph{gluing} Lipschitz extensions of a given function to obtain a Lipschitz extension with a larger domain (this is Lemma \ref{lemma_glue}).
Secondly, given a definable function that is $\lambda$-Lipschitz in the first variable, we give a more easy construction to obtain a definable extension that is $\Lambda$-Lipschitz in the first variable, where $\Lambda$ is possibly larger than $\lambda$ (this is Theorem \ref{thm_main_param}).
Thirdly and lastly, using a more involved argument, we show that one can take $\Lambda$ equal to $\lambda$ (this is Theorem \ref{thm_main_param_const1}).
\begin{lemma}[Gluing extensions]\label{lemma_glue}
Let $X\subset K^r$ be a definable set and let $f:X\to K$ be a definable and $\lambda$-Lipschitz function. Let $X = \cup_{i=1}^k X_i$ be a finite covering of $X$ by definable subsets $X_i$. Call $f_i = \restr{f}{X_i}: X_i\to K$.
If every $f_i$ extends to a definable and $\Lambda_i$-Lipschitz map $\tilde{f_i}:K^r\to K$, with $\Lambda_i\geq\lambda$, then $f$ extends to a definable and $\Lambda$-Lipschitz map $\tilde{f}:K^r\to K$, where $\Lambda = \max_i \{\Lambda_i\}$.
\end{lemma}
\begin{proof}
We prove the lemma first for $k=2$. Define $T_1 = \{ x\in K^r\mid \mathrm{d}(x,X_1)\leq \mathrm{d}(x,X_2)\}$, where $\mathrm{d}(x,A)$ denotes the distance from $x$ to the set $A$, i.e. $\mathrm{d}(x,A) = \inf\{\abs{x-a}\mid a\in A\}$. Define $T_2 = K^r\setminus T_1$, and let
\[ \tilde{f}:K^r\to K: x\mapsto \begin{cases} \tilde{f_1}(x)&\text{if }x\in T_1,\\\tilde{f_2}(x)&\text{if }x\in T_2.\end{cases}\]
Clearly $\tilde{f}$ is a definable extension of $f$. We prove that $\tilde{f}$ is $\Lambda$-Lipschitz, where $\Lambda = \max\{\Lambda_1,\Lambda_2\}$. The only nontrivial fact to verify is that for $t_1\in T_1$ and $t_2\in T_2$, we have $\abs{\tilde{f}(t_1)-\tilde{f}(t_2)}\leq \Lambda\abs{t_1-t_2}$.
Since every definable and $\lambda$-Lipschitz function extends uniquely to a definable and $\lambda$-Lipschitz function on the topological closure of its domain, we may assume that $X$, $X_1$ and $X_2$ are topologically closed.
Fix elements $a_i\in X_i$ such that $\abs{t_i-a_i} = \mathrm{d}(t_i,X_i)$, for $i=1,2$.
\begin{comment}
$a_1\in X_1$ and $a_2\in X_2$ according to the following case distinction. If $t_1\in\overline{X_1}$ and $t_2\not\in\overline{X_2}$, choose $a_2\in X_2$ such that $\abs{t_2-a_2}= \mathrm{d}(t_2,X_2)$ and $a_1\in X_1$ such that $\abs{t_1-a_1}<\abs{a_1-a_2}$. If $t_1\not\in \overline{X_1}$ and $t_2\in\overline{X_2}$, choose $a_1\in X_1$ such that $\abs{t_1-a_1} = \mathrm{d}(t_1,X_1)$ and $a_2\in X_2$ such that $\abs{t_2-a_2}<\abs{a_1-a_2}$. If $t_i\not\in \overline{X_i}$ for $i=1,2$, choose $a_i\in X_i$ such that $\abs{t_i-a_i} = \mathrm{d}(t_i,X_i)$.
\end{comment}
It then always holds that
\begin{equation}\label{eq_a2x2<a1a2}
\abs{t_2-a_2}<\abs{a_1-a_2}.
\end{equation}
We can now calculate as follows:
\begin{align*}
\abs{\tilde{f}(t_1)-\tilde{f}(t_2)} &= \abs{\tilde{f_1}(t_1)-f(a_1)+f(a_2)-\tilde{f_2}(t_2)+f(a_1)-f(a_2)}\\
&\leq\max\{\abs{\tilde{f_1}(t_1)-f(a_1)},\abs{f(a_2)-\tilde{f_2}(t_2)},\abs{f(a_1)-f(a_2)}\}\\
&\leq\max\{\Lambda_1\abs{t_1-a_1},\Lambda_2\abs{a_2-t_2},\lambda\abs{a_1-a_2}\}\\
&\leq\max\{\Lambda_1,\Lambda_2\}\max\{\abs{t_1-a_1},\abs{a_2-t_2},\abs{a_1-a_2}\}\\
&\stackrel{\eqref{eq_a2x2<a1a2}}{=}\Lambda\max\{\abs{t_1-a_1},\abs{a_1-a_2}\}\\
&\leq\Lambda\abs{t_1-t_2},
\end{align*}
where we only have to verify the last inequality. For this we prove that
\begin{equation}\label{eq_preparatory}\max\{\abs{t_1-a_1},\abs{a_1-a_2}\}\leq \abs{t_1-t_2},\end{equation}
by considering two cases.
\begin{description}
\item[Case 1: $\abs{t_1-a_1}<\abs{a_1-a_2}$.]
It then holds that
\begin{equation}\label{eq_a1a2=x1a2N}
\abs{a_1-a_2}=\abs{t_1-a_2}.
\end{equation}
So
\begin{equation}\label{eq_lange}
\abs{t_2-a_2}\stackrel{\eqref{eq_a2x2<a1a2}}{<}\abs{a_1-a_2}\stackrel{\eqref{eq_a1a2=x1a2N}}{=}\abs{t_1-a_2},
\end{equation}
hence
\begin{equation}
\abs{a_1-a_2}\stackrel{\eqref{eq_a1a2=x1a2N}}{=}\abs{t_1-a_2}\stackrel{\eqref{eq_lange}}{=}\abs{t_1-t_2}.
\end{equation}
\begin{comment}
Suppose, for the sake of deriving a contradiction, that
\begin{equation}\label{eq_x1x2<a1a2}\abs{t_1-t_2}<\abs{a_1-a_2}.\end{equation}
By the case we are in, it holds that
\begin{equation}\label{eq_a1a2=x1a2}\abs{a_1-a_2}=\abs{t_1-a_2},\end{equation}
and combined with \eqref{eq_x1x2<a1a2} we find
\begin{equation}\label{eq_x1x2<x1a2} \abs{t_1-t_2}<\abs{t_1-a_2}.\end{equation}
Therefore
\[\abs{t_2-a_2} \stackrel{\eqref{eq_x1x2<x1a2}}{=}\abs{t_1-a_2} \stackrel{\eqref{eq_a1a2=x1a2}}{=} \abs{a_1-a_2} \stackrel{\eqref{eq_a2x2<a1a2}}{>} \abs{t_2-a_2},\]
which is a contradiction.
\end{comment}
\item[Case 2: $\abs{a_1-a_2}\leq\abs{t_1-a_1}$.] Suppose that
\begin{equation}\label{eq_x1x2<x1a1} \abs{t_1-t_2}<\abs{t_1-a_1},\end{equation}
then
\begin{equation}\label{eq_x1a1=x2a1=a1a2}\abs{t_1-a_1}\stackrel{\eqref{eq_x1x2<x1a1}}{=}\abs{t_2-a_1} \stackrel{\eqref{eq_a2x2<a1a2}}{=} \abs{a_1-a_2}.\end{equation}
By the choice of $a_1$ and the fact that $t_1\in T_1$, we know $\abs{t_1-a_2}\geq \abs{t_1-a_1}$, so by \eqref{eq_x1a1=x2a1=a1a2} equality holds:
\begin{equation}\label{eq_x1a1=x1a2}\abs{t_1-a_1}=\abs{t_1-a_2}.\end{equation}
Together with \eqref{eq_x1x2<x1a1}, this implies
\begin{equation}\label{eq_x1x2<x1a2twee}\abs{t_1-t_2}<\abs{t_1-a_2}.\end{equation}
So finally,
\begin{equation*}
\abs{a_1-a_2}\stackrel{\eqref{eq_x1a1=x2a1=a1a2}}{=}\abs{t_1-a_1}\stackrel{\eqref{eq_x1a1=x1a2}}{=}\abs{t_1-a_2}\stackrel{\eqref{eq_x1x2<x1a2twee}}{=}\abs{t_2-a_2} \stackrel{\eqref{eq_a2x2<a1a2}}{<}\abs{a_1-a_2},
\end{equation*}
which is a contradiction.
\end{description}
This proves \eqref{eq_preparatory}, and therefore the lemma is proved for $k=2$. An easy induction argument then proves the lemma for general $k$.
\end{proof}
\begin{remark} Lemma \ref{lemma_glue} remains true is one replaces every instance of the word ``Lipschitz'' by ``Lipschitz in the first variable''.
\end{remark}
\begin{theorem}\label{thm_main_param}
Let $Y\subset K^r$ and $X\subset K$ be definable sets and let $f:S=X\times Y\to K^s$ be a definable function that is $\lambda$-Lipschitz in the first variable. Then there exists $\Lambda\geq \lambda$ such that $f$ extends to a definable function $\tilde{f}:K\times Y\to K^s$ that is $\Lambda$-Lipschitz in the first variable, i.e. $\tilde{f}_y$ is $\Lambda$-Lipschitz for every $y\in Y$.
\end{theorem}
\begin{proof}
By Remark \ref{remark_lipschitz_constant}, we may assume that $\lambda=1$ and $s=1$. By Theorem \ref{thm_preparation} and (the remark after) Lemma \ref{lemma_glue}, we may assume that $S$ is a cell over $Y$ with which $f$ is compatible. Furthermore, we may assume that the base of $S$ is $Y$.
If $S_f$ is a $0$-cell over $Y$ with center $c'$, we define
\[\tilde{f}:K\times Y\to K: (x,y)\mapsto c'(y).\]
Clearly, $\tilde{f}$ is a definable extension of $f$ and for all $y\in Y$, $\tilde{f}_y$ is $1$-Lipschitz.
Assume from now on that $S$ and $S_f$ are 1-cells over $Y$, with center $c$ and $c'$, and coset $\xi Q_{m,n}$ and $\xi' Q_{m',n'}$, respectively.
We define $\tilde{f}$ as follows:
\[\tilde{f}:K\times Y\to K: (x,y)\mapsto\begin{cases} f(x,y) & \text{if } (x,y)\in S,\\c'(y)&\text{if }(x,y)\not\in S.\end{cases}\]
Clearly, $\tilde{f}$ is a definable extension of $f$. We prove that $\tilde{f}_y$ is $q^{m'}$-Lipschitz for every $y\in Y$.
Fix $y\in Y$. Let $t_1\in X$ and $t_2\not \in X$. Let $l$ and $l'$ be such that $t_1\in B_{l,c(y),m,\xi}$ and $f(t_1,y)\in B_{l',c'(y),m',\xi'}$. Then
\begin{align} \abs{f(t_1,y)-c'(y)} &= q^{-l'}\notag\\
&= q^{-\mathrm{ord}(\partial f(t_1,y)/\partial x)}q^{m'-m}q^{-\mathrm{ord}(t_1-c(y))}\notag\\
&\leq q^{m'-m}\abs{t_1-c(y)},\label{eq_mm'xc}
\end{align}
where the second equality follows from Lemma \ref{lemma_l'm'ordf'lm} and the last inequality holds because $f$ is $1$-Lipschitz in the first variable, and therefore $\abs{\partial f(t_1,y)/\partial x}\leq 1$.
There are two cases to consider.
\begin{description}
\item[Case 1: $\abs{t_1-c(y)} = \abs{t_2-c(y)}$.] Because $B_{l,c(y),m,\xi}$ is a ball of diameter $q^{-m-l}$, it holds that $\abs{t_1-t_2}> q^{-m-l}$, or put differently:
\begin{equation}\label{eq_l<mx1x2}
q^{-m}\abs{t_1-c(y)}<\abs{t_1-t_2}.
\end{equation}
Therefore
\begin{align*}
\abs{\tilde{f}_y(t_1)-\tilde{f}_y(t_2)} &= \abs{\tilde{f}(t_1,y)-\tilde{f}(t_2,y)}\\
&= \abs{f(t_1,y)-c'(y)}\\
&\stackrel{\eqref{eq_mm'xc}}{\leq} q^{m'-m}\abs{t_1-c(y)}\\
&\stackrel{\eqref{eq_l<mx1x2}}{<} q^{m'}\abs{t_1-t_2}.
\end{align*}
\item[Case 2: $\abs{t_1-c(y)} \neq \abs{t_2-c(y)}$.] From the non-Archimedean property it then follows that
\begin{equation}\label{eq_x1c<x1x2}
\abs{t_1-c(y)}\leq\abs{t_1-t_2},
\end{equation}
so we find
\begin{align*}
\abs{\tilde{f}_y(t_1)-\tilde{f}_y(t_2)} &= \abs{\tilde{f}(t_1,y)-\tilde{f}(t_2,y)}\\
&=\abs{f(t_1,y)-c'(y)} \\
&\stackrel{\eqref{eq_mm'xc}}{\leq} q^{m'-m}\abs{t_1-c(y)}\\
&\stackrel{\eqref{eq_x1c<x1x2}}{\leq}q^{m'-m}\abs{t_1-t_1}.\qedhere
\end{align*}
\end{description}
\end{proof}
\begin{remark}\label{remark_M}
Analyzing the proof of Theorem \ref{thm_main_param}, we find that one can take $\Lambda = \lambda\max_i\{q^{m_i'}\}$, where $\lambda$ is the Lipschitz constant of $f$ (in the first variable), and the $m_i'$ correspond to the 1-cells in the cell decomposition of $S_f$.
\end{remark}
\begin{remark}\label{rem_phi}
We can even improve (i.e. decrease) $\Lambda$ from Remark \ref{remark_M} as follows. In the proof, the worst Lipschitz constant occurs in \textbf{Case 1}. We can get around this case in the following way (as in the beginning of Theorem \ref{thm_main_param}, we assume that $S$ and $S_f$ are $1$-cells over $Y$ with center $c$ and coset $\xi Q_{m,n}$).
For every nonzero $a\in \mathcal{O}_K^\times/(\pi_K^m)$, choose $\xi_m(a)\in \mathcal{O}_K^\times$ to be a class representative of $a$. Since we only need to make a finite number of representative choices, $\xi_m:\mathcal{O}_K^\times/(\pi_K^m)\to K$ is a definable map. Let $\varphi:K\times Y\to K$ be the definable map rescaling the angular component as follows:
\begin{align*}
\varphi:K\times Y&\to K:\\
(x,y)&\mapsto \begin{cases}(x-c(y))\xi_m(\overline{\mathrm{ac}}_m(x-c(y))^{-1}\overline{\mathrm{ac}}_m(\xi))+c(y) & \text{if }x\neq c(y),\\ c(y) & \text{if }x=c(y).\end{cases}
\end{align*}
It is not difficult to see that for every $y\in Y$, $\varphi_y:K\to K$ is $1$-Lipschitz. Now let $\hat{f}$ be the extension described in the proof of Theorem \ref{thm_main_param} in the case that $S$ and $S_f$ are $1$-cells over $Y$ (remark that in Theorem \ref{thm_main_param}, this extension is denoted with $\tilde{f}$). Then $\tilde{f}:K\times Y\to K: (x,y)\mapsto \hat{f}(\varphi(x,y),y)$ is a definable extension of $f$ that is $q^{m'-m}$-Lipschitz in the first variable. One can therefore take $\Lambda = \lambda\max_i\{q^{m_i'-m_i}\}$, where $\lambda$ is the Lipschitz constant of $f$ (in the first variable), and the $m_i$ and $m_i'$ correspond to the 1-cells in the cell decomposition of $S$ and $S_f$, respectively.
\end{remark}
Note that in the proof of Theorem \ref{thm_main_param} we did not use the full generality of Theorem \ref{thm_preparation}. We will now prove Theorem \ref{thm_main_param_const1}, the main theorem of this paper, which uses a more involved extension for which the Lipschitz constant doesn't grow. For this, the full power of Theorem \ref{thm_preparation} is used. Again, the result is formulated in definable families of functions. For clarity, we repeat the formulation of Theorem \ref{thm_main_param_const1}.
\begin{theorem*}
Let $Y\subset K^r$ and $X\subset K$ be definable sets and let $S=X\times Y$. Let $f:S\to K^s$ be a definable function that is $\lambda$-Lipschitz in the first variable. Then $f$ extends to a definable function $\tilde{f}:K\times Y\to K^s$ that is $\lambda$-Lipschitz in the first variable, i.e. $\tilde{f}_y$ is $\lambda$-Lipschitz for every $y\in Y$.
\end{theorem*}
\begin{proof}
By Remark \ref{remark_lipschitz_constant}, we may assume that $\lambda=1$ and $s=1$. By Theorem \ref{thm_preparation} and (the remark after) Lemma \ref{lemma_glue}, we may assume that $S$ is a cell over $Y$ with which $f$ is compatible. Furthermore, we may assume that the base of $S$ is $Y$.
If $S_f$ is a $0$-cell, extend $f$ as in Theorem \ref{thm_main_param}.
Assume from now on that $S$ and $S_f$ are 1-cells over $Y$, with center $c$ and $c'$, and coset $\xi Q_{m,n}$ and $\xi' Q_{m',n'}$, respectively. Let $g$ be as in Theorem \ref{thm_preparation}, in particular $f$ and $g$ are equicompatible with $S$, and $(g(x,y)-c'(y))^b = e(y)(x-c(y))^a$ for every $(x,y)\in S$.
Fix $y\in Y$ and let $B_{l,c(y),m,\xi}$ be a ball of $S$ above $y$. By Theorem \ref{thm_preparation} we can write $f_y(B_{l,c(y),m,\xi})=B_{l',c'(y),m',\xi'}=g_y(B_{l,c(y),m,\xi})$, where $B_{l',c',m',\xi'}$ is a ball of $S_f$ above $y$. Also, we have that $\mathrm{ord}(\partial f/\partial x) = \mathrm{ord}(\partial g/\partial x)$. Let $q=a/b$, then there are three different cases to consider, depending on whether $q=1$, $q<1$ or $q>1$.
\begin{description}
\item[Case 1: $q=1$.] From equation \eqref{eq_formula_order} we have $\mathrm{ord}(\partial f(x,y)/\partial x) = \mathrm{ord}(e(y))$ for all $(x,y)\in S$. So for $x\in B_{l,c(y),m,\xi}$ we have
\begin{align*}
l' &= \mathrm{ord}(e(y)(x-c(y))+c'(y)-c'(y))\\
&=\mathrm{ord}(e(y))+\mathrm{ord}(x-c(y))\\
&=\mathrm{ord}(\partial f(x,y)/\partial x) + l,
\end{align*}
which implies $l'\geq l$, since $f$ is 1-Lipschitz in the first variable. In particular note that in this case $m=m'$, by Lemma \ref{lemma_l'm'ordf'lm}. This allows us to use the same extension as described in Remark \ref{rem_phi}, namely $\tilde{f}:K\times Y\to K: (x,y)\mapsto \hat{f}(\varphi_y(x),y)$, where $\hat{f}$ is as in the proof of Theorem \ref{thm_main_param} in the case that $S$ and $S_f$ are $1$-cells over $Y$, and $\varphi_y$ is as in Remark \ref{rem_phi} (again, remark that in Theorem \ref{thm_main_param} this extension is denoted with $\tilde{f}$). We prove that $\tilde{f}_y$ is 1-Lipschitz. Let $t_1\in\cup_l D_{l,c(y)}\cap X$ and $t_2\not\in\cup_l D_{l,c(y)}\cap X$, where $D_{l,c(y)} = \{x\in K\mid \mathrm{ord}(x-c(y))=l\}$. Then
\begin{align*}
\abs{\tilde{f}_y(t_1)-\tilde{f}_y(t_2)} &= \abs{f(t_1,y)-c'(y)}\\
&\leq \abs{t_1-c(y)}\\
&\leq \abs{t_1-t_2},
\end{align*}
where the first inequality follows from $l'\geq l$ and the second from the non-Archimedean property.
\item[Case 2: $q>1$.] Because $f$ is 1-Lipschitz in the first variable, we have $ \mathrm{ord}(\partial f/\partial x) \geq 0$, and together with \eqref{eq_formula_order} this gives the following lower bound:
\[l\geq -\mathrm{ord}(e(y)^{1/b}q)/(q-1).\]
Recall that $\mathrm{ord}(e(y)^{1/b}q)$ is short for $\mathrm{ord}(e(y))/b+\mathrm{ord}(q)$. On the other hand, as soon as $l\geq (m'-m-\mathrm{ord}(e(y)^{1/b}q))/(q-1)$, we have $l'\geq l$. Indeed, this follows immediately from Lemma \ref{lemma_l'm'ordf'lm} and from \eqref{eq_formula_order}. So up to partitioning $S$ into two cells over $Y$, we may assume that either $l'\geq l$ for all balls of $S$ above $y$, for every $y\in Y$, or that $S$ has at most $N$ balls above $y$, for every $y\in Y$, where $N$ does not depend on $y$. In the former case we can extend $f$ as we did in \textbf{Case 1}. In the latter case we can, after partitioning $Y$ in a finite number of definable sets, assume that there are \emph{exactly} $N$ balls of $S$ above $y$, for every $y\in Y$. Using (the remark after) Lemma \ref{lemma_glue} we may assume that there is exactly one ball of $S$ above $y$, for every $y\in Y$. By definable selection (see \cite{denef-vdd} and \cite{Dries1984-DRIATW}) there is a definable function $h:Y\to K$ such that for each $(x,y)\in S$ with $x\in K$ and $y\in Y$, $(h(y),y)\in S$. We then extend $f$ as follows:
\[\tilde{f}:K\times Y\to K: (x,y)\mapsto\begin{cases}f(x,y) &\text{if }(x,y)\in S,\\ f(h(y),y)&\text{if }(x,y)\not\in S.\end{cases}\]
Fix $y\in Y$, we show that $\tilde{f}_y$ is 1-Lipschitz. Recall that by the argument given above, $S_y$ is a ball in $K$. The only nontrivial case to consider is the following. Let $t_1\in S_y$ and $t_2\not\in S_y$, then
\begin{align*}
\abs{\tilde{f}_y(t_1)-\tilde{f}_y(t_2)} &= \abs{f(t_1,y)-f(h(y),y)}\\
&\leq \abs{t_1-h(y)}\\
&<\abs{t_1-t_2},
\end{align*}
where the last inequality holds because of the non-Archimedean property and the fact that $t_1$ and $h(y)$ are both contained in the ball $S_y$, and $t_2$ is not.
\item[Case 3: $q<1$.] This case is similar to \textbf{Case 2}, where now one finds an upper bound for $l$ instead of a lower bound. The proof is omitted. \qedhere
\end{description}
\end{proof}
\begin{remark}
Note that we proved the main theorem for semi-algebraic and subanalytic structures on $K$. It is, for now, unclear whether the main theorem could also hold in other structures on $K$, such as, for example, $P$-minimal structures, as defined by Haskell and Macpherson in \cite{has-mac-97}. Also, it is unclear whether the extension that we constructed could be used to extend a definable function $f:X\subset K^r\to K^s$ that is $\lambda$-Lipschitz in \emph{all} variables to a definable function $\tilde{f}:K^r\to K^s$ that is $\lambda$-Lipschitz in all variables. For now, there is no evidence towards either a positive or a negative answer to this question.
\end{remark}
\vspace*{10pt}
\textsc{Tristan Kuijpers\\ KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium}\\
\textit{E-mail:} [email protected]
\end{document}
|
\begin{document}
\title[A scheme for time fractional equations]{\protect{On a discrete scheme for time fractional \\
fully nonlinear evolution equations}}
\author[Y. GIGA, Q. LIU, H. MITAKE]
{Yoshikazu Giga, Qing Liu, Hiroyoshi Mitake}
\thanks{
The work of YG was partially supported by Japan Society for the Promotion of Science (JSPS) through grants KAKENHI \#26220702, \#16H03948, \#18H05323, \#17H01091.
The work of QL was partially supported by the JSPS grant KAKENHI \#16K17635 and the grant \#177102 from Central Research Institute of Fukuoka University.
The work of HM was partially supported by the JSPS grant KAKENHI \#16H03948.
}
\address[Y. Giga]{
Graduate School of Mathematical Sciences,
University of Tokyo
3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan}
\email{[email protected]}
\address[Q. Liu]{Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan.
}
\email{[email protected]}
\address[H. Mitake]{
Graduate School of Mathematical Sciences,
University of Tokyo
3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan}
\email{[email protected]}
\keywords{Approximation to solutions; Caputo's time fractional derivatives; Second order fully nonlinear equations; Viscosity solutions.}
\subjclass[2010]{
35R11,
35A35,
35D40.
}
\maketitle
\begin{abstract}
We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions.
The discrete scheme can be viewed as a resolvent-type approximation.
\end{abstract}
\section{Introduction}
In this paper, we are concerned with the second order fully nonlinear PDEs with Caputo's time fractional derivatives:
\begin{numcases}{}
\partial_t^{\alpha}u(x,t)+F(x, t, Du, D^2u)=0 &\qquad\text{for all $x\in\mathbb{R}^n, t>0,$} \langlebel{eq:1}\\
u(x,0)=u_0(x) &\qquad\text{for all $x\in\mathbb{R}^n$}, \langlebel{eq:ini}
\end{numcases}
where $\alphapha\in(0,1)$ is a given constant, $u:\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$ is an unknown function and
$Du$ and $D^2 u$, respectively, denote its spatial gradient and Hessian of $u$.
We \textit{always} assume that $u_0\in BUC(\mathbb{R}^n)$,
which denotes the space of all bounded uniformly continuous functions in $\mathbb{R}^n$. We denote
\textit{Caputo's time fractional derivative} by $\partial_t^{\alpha}u$, i.e.,
\[
\partial_t^{\alpha}u(x,t):=\frac{1}{\Gamma(1-\alpha)}\int_0^t(t-s)^{-\alpha}\partial_{s}u(x,s)\,ds,
\]
where $\Gamma$ is the Gamma function.
We assume that $F$ is a continuous \textit{degenerate elliptic} operator, that is,
\[
F(x, t, p, X_1) \leq F(x, t, p, X_2)
\]
for all $x\in\mathbb{R}^n, t\geq 0, p \in \mathbb{R}^n$ and $X_1, X_2 \in \mathbb{S}^n \text{ with } X_1 \geq X_2$, where $\mathbb{S}^n$ denotes the space of $n \times n$ real symmetric matrices.
Moreover, throughout this work we assume that $F$ is locally bounded in the sense that
\begin{equation}\langlebel{eq:op bound}
M_R:=\sup_{\substack{(x, t)\in \mathbb{R}^n\times [0, \infty)\\ |p|, |X|\leq R}}|F(x, t, p, X)|<\infty\qquad \text{for any $R>0$}.
\end{equation}
Studying differential equations with fractional derivatives is motivated by mathematical models that
describe diffusion phenomena in complex media like fractals, which is sometimes called \textit{anomalous diffusion}
(see \cite{MK} for instance).
It has inspired further research on numerous related topics.
We refer to a non-exhaustive list of references \cite{L, SY, ACV, C, GN, TY, A, N, KY, CKKW} and the references therein.
Among these results, the authors of \cite{ACV, A} mainly study regularity of solutions to a space-time nonlocal equation with
Caputo's time fractional derivative in the framework of viscosity solutions.
More recently, unique existence of a viscosity solution to the initial value problem with Caputo's time fractional derivatives has been established in the thesis of Namba \cite{N-thesis} and independently and concurrently by Topp and Yangari \cite{TY}. The main part of \cite{N-thesis} on this subject has been published in \cite{GN, N}. For example, a comparison principle, Perron's method, and stability results for \eqref{eq:1} in bounded domains with various boundary conditions have been established in \cite{GN, N}. Similar results for whole space has been established in \cite{TY} for nonlocal parabolic equations.
Motivated by these works, in this paper we introduce a discrete scheme for \eqref{eq:1}--\eqref{eq:ini}, which will be explained in detail in the subsection below.
\subsection{The discrete scheme}\langlebel{subsec:scheme}
Our scheme is naturally derived from the definitions of Riemann integral and Caputo's time fractional derivative.
We first observe that
\begin{align*}
\partial_t^{\alpha}u(\cdot, mh)&
=\frac{1}{\Gamma(1-\alpha)}\int_0^{mh}(mh-s)^{-\alpha}\partial_{s}u(x,s)\,ds\\
&=\frac{1}{\Gamma(1-\alpha)}\sum_{k=0}^{m-1} \int_{kh}^{(k+1)h}(mh-s)^{-\alpha}\partial_{s}u(x,s)\,ds
\end{align*}
for $m\in\mathbb{N}$ and $h>0$.
If $u$ is smooth in $\mathbb{R}^n\times [0, \infty)$ and $h$ is small, then we can approximately think that
\[
\int_{kh}^{(k+1)h}(mh-s)^{-\alpha}\partial_{s}u(x,s)\,ds
\approx
\int_{kh}^{(k+1)h}(mh-s)^{-\alpha}\frac{u(x,(k+1)h)-u(x,kh)}{h}\,ds.
\]
Note that $z\Gamma(z)=\Gamma(z+1)$ and
\begin{align*}
\int_{kh}^{(k+1)h}(mh-s)^{-\alpha}\,ds
=&\,
\frac{1}{1-\alpha}\left(((m-k)h)^{1-\alpha}-((m-k-1)h)^{1-\alpha}\right)\\
=&\,
\frac{1}{1-\alpha}f(m-k)h^{1-\alpha},
\end{align*}
where we set
\begin{equation}\langlebel{func:f}
f(r):=r^{1-\alpha}-(r-1)^{1-\alpha}\quad\text{for} \ r\ge1.
\end{equation}
Thus,
\begin{align*}
\partial_t^{\alpha}u(\cdot, mh)\approx
&\,
\frac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^{m-1}
f(m-k)\left(u(x,(k+1)h)-u(x,kh)\right)\\
&\,
=
\frac{1}{\Gamma(2-\alpha)h^\alpha}\left\{
u(x,mh)
-\sum_{k=0}^{m-1} C_{m, k}u(x,kh)\right\},
\end{align*}
where we set
\[
C_{m,0}:=f(m), \quad C_{m,k}:=f(m-k)-f(m-(k-1)) \quad\text{for} \ k=1,\ldots, m-1.
\]
Since $f$ is a non-increasing function, we easily see that
\begin{equation}\langlebel{positive}
C_{m,k}\ge 0 \quad\text{for} \ k=0,\ldots, m-1,
\end{equation}
which implies monotonicity of the scheme
(see Proposition \ref{prop:monotone}).
Inspired by this observation, for any fixed $h>0$,
we below define a family of functions
$\{U^h(\cdot, mh)\}_{m\in\mathbb{N}\cup\{0\}}\subset BUC(\mathbb{R}^n)$ by induction.
Set $U^h(\cdot, 0):=u_0^h$,
where $u_0^h\in BUC(\mathbb{R}^n)$ satisfies
\begin{equation}\langlebel{initial approx}
\sup_{\mathbb{R}^n} \left|u_0^h-u_0\right|\to 0\quad \text{as $h\to 0$.}
\end{equation}
Let $U^h(\cdot, mh)\in C(\mathbb{R}^n)$ for $m\geq 1$ be the viscosity solution of
\begin{equation}\langlebel{eq:m}
\frac{1}{\Gamma(2-\alpha)h^\alpha}\left\{
u(x)
-\sum_{k=0}^{m-1} C_{m,k}U^h(x,kh)\right\}
+F\left(x, t, D u, D^2u\right)=0 \quad \text{in $\mathbb{R}^n$}.
\end{equation}
Let us emphasize here that the equation \eqref{eq:m} is an (degenerate) elliptic problem with the elliptic operator strictly monotone in $u$.
In fact, for any $m\geq 1$ the elliptic equation is of the form
\begin{equation}\langlebel{eq:elliptic}
\langlembda u(x)+F\left(x, t, Du, D^2u\right)=g(x)\quad \text{in $\mathbb{R}^n$,}
\end{equation}
where $\langlembda>0$ and $g\in BUC(\mathbb{R}^n)$. We can obtain such a unique viscosity solution $U^h(\cdot, mh)\in BUC(\mathbb{R}^n)$ to \eqref{eq:m} with $t=mh$ for any $m\in \mathbb{N}$ under appropriate assumptions on $F$.
Define the function $u^{h}:\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$ by
\begin{equation}\langlebel{def:Uh}
u^h(x,t):=U^h(x, mh)\quad\text{for each}\ x\in \mathbb{R}^n, \ t\in[mh,(m+1)h), \ m\in\mathbb{N}\cup\{0\}.
\end{equation}
Our main result of this paper is to show the convergence of $u^h$ to the unique viscosity solution of \eqref{eq:1}--\eqref{eq:ini}.
We remark that our scheme can be regarded as a resolvent-type approximation. Recall the implicit Euler scheme for the differential equation:
\[
u_t + F[u]:=u_t+F\left(x, t, Du,D^2u\right) = 0,
\]
which is given by
\[
u^h(\cdot, mh) - u^h(\cdot, (m-1)h) + hF[u^h(\cdot, mh)]=0\quad (m\in \mathbb{N}).
\]
This is a typical scheme by approximating $u$ by a function $u^h$ piecewise linear in time with time grid length $h$. The resulting equation is a resolvent type
equation for $u^h(\cdot, mh)$ if $u^h(\cdot, (m-1)h)$ is given. It is elliptic if the
original equation is parabolic.
\subsection{Main Results}
We first give an abstract framework on the convergence of $u^h$.
\begin{thm}[Scheme convergence]\langlebel{thm:main}
Assume that \eqref{eq:op bound} and the following two conditions hold.
\begin{enumerate}
\item[{\rm(H1)}] For any $g\in BUC(\mathbb{R}^n)$, there exists a viscosity solution $u\in BUC(\mathbb{R}^n)$ to \eqref{eq:elliptic} for any $t>0$. Moreover, if $u, v\in BUC(\mathbb{R}^n)$ are, respectively, a subsolution and a supersolution of \eqref{eq:elliptic} with any fixed $t>0$, then $u\leq v$ in $\mathbb{R}^n$.
\item[{\rm(H2)}] Let $u\in USC(\mathbb{R}^n\times [0, \infty))$ and $v\in LSC(\mathbb{R}^n\times [0, \infty))$ be, respectively, a sub- and a supersolution of \eqref{eq:1}. Assume $u$ and $v$ are bounded in $\mathbb{R}^n\times [0, T)$ for any $T>0$.
If $u(\cdot, 0)\leq v(\cdot, 0)$ in $\mathbb{R}^n$,
then $u\leq v$ in $\mathbb{R}^n\times [0, \infty)$.
\end{enumerate}
Let $u^h$ be given by \eqref{def:Uh} for any $h>0$, where initial data $u^h_0$ is assumed to fulfill \eqref{initial approx}.
Then, $u^h\to u$ locally uniformly in $\mathbb{R}^n\times [0, \infty)$ as $h\to 0$, where $u$ is the unique viscosity solution to \eqref{eq:1}--\eqref{eq:ini}.
\end{thm}
We obtain the following corollary of Theorem \ref{thm:main} under more explicit sufficient conditions of (H1) and (H2).
\begin{comment}
\begin{cor}\langlebel{cor:periodic}
Assume that $x\mapsto u_0(x), F(x, t, p, X)$ are periodic for all $t>0, p\in\mathbb{R}^n, X\in\mathbb{S}^n$, \eqref{eq:op bound} and
\begin{enumerate}
\item[{\rm(F1)}] There exists a modulus of continuity $\omegaega: [0, \infty)\to [0, \infty)$ such that
\[
F\left(x, t, \mu(x-y),Y\right)-F\left(y, t, \mu(x-y),X\right)\le \omegaega\left(|x-y|(\mu|x-y|+1)\right)
\]
for all $\mu>0$, $x, p\in\mathbb{R}^n$, $t\geq 0$ and $X,Y\in\mathbb{S}^n$ satisfying
\[
\left(
\begin{array}{cc}
X & 0 \\
0 & -Y
\end{array}
\right)
\le
\mu
\left(
\begin{array}{cc}
I & -I \\
-I & I
\end{array}
\right).
\]
\end{enumerate}
Then, \eqref{conv} holds.
\end{cor}
\end{comment}
\begin{cor}\langlebel{cor:non-periodic}
Assume that \eqref{eq:op bound} and the following two conditions hold.
\begin{enumerate}
\item[{\rm(F1)}] There exists a modulus of continuity $\omegaega: [0, \infty)\to [0, \infty)$ such that
\[
F\left(x, t, \mu(x-y),Y\right)-F\left(y, t, \mu(x-y),X\right)\le \omegaega\left(|x-y|(\mu|x-y|+1)\right)
\]
for all $\mu>0$, $x, p\in\mathbb{R}^n$, $t\geq 0$ and $X,Y\in\mathbb{S}^n$ satisfying
\[
\left(
\begin{array}{cc}
X & 0 \\
0 & -Y
\end{array}
\right)
\le
\mu
\left(
\begin{array}{cc}
I & -I \\
-I & I
\end{array}
\right).
\]
\item[{\rm(F2)}] There exists a modulus of continuity $\tilde{\omegaega}: [0, \infty)\to [0, \infty)$ such that
\[
|F(x, t, p, X)-F(x, t, q, Y)|\leq \tilde{\omegaega}(|p-q|+|X-Y|)
\]
for all $x\in \mathbb{R}^n$, $t\geq 0$, $p, q\in \mathbb{R}^n$ and $X, Y\in \mathbb{S}^n$.
\end{enumerate}
Then, the conclusion of Theorem \ref{thm:main} holds.
\end{cor}
\begin{rem}
The assumption (F2) can be removed in the presence of periodic boundary condition, that is, $x\mapsto u_0(x)$ and $x\mapsto F(x, t, p, X)$
are periodic with the same period.
Recall that in a bounded domain or with the periodic boundary condition, (H1) is established in \cite{CIL}
and (H2)
is available in \cite[Theorem 3.1]{GN} \cite[Theorem 3.4]{N} under (F1).
\end{rem}
The comparison result in (H1) under (F1), (F2) in an unbounded domain is due to \cite{JLS}. Existence of solutions in this case can be obtained by Perron's method. In fact, thanks to \eqref{eq:op bound} with $R=0$, we can take $C>0$ large such that $C$ and $-C$ are, respectively, a supersolution and a subsolution of \eqref{eq:elliptic}. We then can prove the existence of solutions by adopting the standard argument in \cite{CIL, G}.
In addition, as shown in \cite{TY}, (H2) is also guaranteed by (F1) and (F2).
Our results above apply to a general class of nonlinear parabolic equations. We refer the reader to \cite[Example 3.6]{CIL} for concrete examples of $F$ that satisfy our assumptions, especially the condition (F1).
Finally, it is worthwhile to mention that the idea for a discrete scheme in this paper can be adopted to handle a more general type of time fractional derivatives as in \cite{C, CKKW}, provided that the comparison theorems can be obtained.
In this paper, we choose Caputo's time fractional derivatives to simplify the presentation.
This paper is organized as follows. In Section \ref{sec:pre}, we give the monotonicity and boundedness of discrete schemes. Section \ref{sec:main} is devoted to the proof of Theorem \ref{thm:main}.
\section{Preparations}\langlebel{sec:pre}
We first recall the definition of viscosity solutions to \eqref{eq:1}.
\begin{defn}[Definition of viscosity solutions]\langlebel{defn vis}
For any $T>0$, a function $u\in USC(\mathbb{R}^n\times[0,T))$ {\rm(}resp., $u\in LSC(\mathbb{R}^n\times[0,T))${\rm)}
is called a viscosity subsolution {\rm(}resp., supersolution{\rm)} of \eqref{eq:1}
if for any $\phi\in C^{2}(\mathbb{R}^n\times [0,T))$
one has
\begin{equation}\langlebel{def:vis}
\partial_t^\alphapha \phi(x_0, t_0)+F(x_0, t_0, D \phi(x_0, t_0), D^2 \phi(x_0, t_0))
\le {\rm(resp.,}\ge{\rm)}\ 0
\end{equation}
whenever $u-\phi$ attains a local maximum {\rm(}resp., minimum{\rm)} at $(x_0, t_0)\in \mathbb{R}^n\times (0,T)$.
We call $u\in C(\mathbb{R}^n\times[0,T))$ a viscosity solution of \eqref{eq:1} if
$u$ is both a viscosity subsolution and a supersolution of \eqref{eq:1}.
\end{defn}
\begin{rem}
Our definition essentially follows \cite[Definition 2.2]{N}. In fact, since
\begin{equation}\langlebel{equiv time der}
\partial_t^{\alpha}\phi(x, t)
=
\frac{1}{\Gamma(1-\alpha)}\left(\frac{\phi(x, t)-\phi(x, 0)}{t^\alpha}+\alpha\int_0^t\frac{\phi(x, t)-\phi(x, s)}{(t-s)^{1+\alpha}}\,ds \right)
\end{equation}
for any $\phi\in C^{1}(\mathbb{R}^n\times [0, \infty))$, our definition is thus the same as \cite[Definition 2.2]{N}. A similar definition of viscosity solutions is cocurrently and independently proposed in \cite[Definition 2.1]{TY} for general space-time nonlocal parabolic problems.
Another possible way to define sub- or supersolutions is to separate the term $\partial_t^{\alpha}\phi(x, t)$ in \eqref{def:vis} into two parts like \eqref{equiv time der} and replace $\phi$ in one or both of the parts by $u$.
See \cite[Definition 2.1]{TY} and \cite[Definition 2.5]{GN}.
Such definitions are proved to be equivalent to Definition \ref{defn vis}. We refer to \cite[Lemma 2.3]{TY} and \cite[Proposition 2.5]{N} for proofs.
Note that the original definition of viscosity solutions in \cite{N-thesis, GN} looks stronger but it turns out that it is the same \cite[Lemma 2.9, Proposition 3.6]{N-thesis}.
\end{rem}
For any $h>0$, define $\partial_{t}^{\alpha,h}: L^\infty_
{loc}(\mathbb{R}^n\times [0, \infty))\to L^\infty_{loc}(\mathbb{R}^n\times [0, \infty))$ to be
\begin{equation}\langlebel{eq:op discrete}
\partial_{t}^{\alpha,h}u(x, t):=\frac{1}{\Gamma(2-\alpha)h^\alpha}\left\{
u(x, mh)
-\sum_{k=0}^{m-1} C_{m,k}u(x,kh)\right\}
\end{equation}
for $(x, t)\in \mathbb{R}^n\times [0, \infty)$, and $m\in \mathbb{N}\cup\{0\}$ satisfying $m=\lfloor t/h\rfloor$,
where $\lfloor s\rfloor$ denotes the greatest integer less than or equal to $s\geq 0$.
A locally bounded function $u: \mathbb{R}^n\times [0, \infty)\to \mathbb{R}$ is said to be a subsolution (resp., supersolution) of
\begin{equation}\langlebel{eq:discrete}
\partial_t^{\alpha, h} u+F\left(x, t, Du, D^2u\right)=0 \quad \text{in $\Omegaega\times (0, \infty)$}
\end{equation}
if for any $m\in \mathbb{N}$, $U=u(\cdot, mh)\in USC(\mathbb{R}^n)$ (resp., $U=u(\cdot, mh)\in LSC(\mathbb{R}^n)$) is a viscosity subsolution (resp, supersolution) of
\[
\frac{1}{\Gamma(2-\alpha)h^\alpha}\left\{
U-\sum_{k=0}^{m-1} C_{m,k}u(\cdot, kh)\right\}
+F\left(x, mh, D U, D^2U\right)=0 \quad \text{in $\mathbb{R}^n$.}
\]
By definition, it is clear that $u^h$ given by \eqref{def:Uh} is a solution of \eqref{eq:discrete}.
\begin{prop}[Monotonicity]\langlebel{prop:monotone}
Fix $h>0$. Assume that {\rm(H1)} holds. Let $U^h(\cdot, t)$, $V^h(\cdot, t)\in BUC(\mathbb{R}^n)$ for all $t\geq 0$ be, respectively, a subsolution and supersolution to \eqref{eq:discrete}.
Then, $U^h(\cdot, mh)\leq V^h(\cdot, mh)$ in $\mathbb{R}^n$ for all $m\in \mathbb{N}$.
\end{prop}
\begin{proof}
Due to the positiveness \eqref{positive} of $C_{m,k}$, one can easily see that the scheme is monotone by iterating the comparison principle in (H1) for elliptic problems.
\end{proof}
We next discuss below the boundedness of the scheme.
\begin{lem}[Barrier]\langlebel{lem:est1}
For any $h>0$, let $V^h(x, t):=(mh)^\alphapha$ for all $x\in \mathbb{R}^n$ and $t\geq 0$ with $m=\lfloor t/h\rfloor$.
Then,
\[
\partial_t^{\alphapha, h} V^h(x, t)\geq {(1-\alphapha)\alphapha\over \Gammama(2-\alphapha)}\qquad \text{for all $x\in \mathbb{R}^n$ and $t\geq h$}.
\]
\end{lem}
\begin{proof}
We have
\begin{equation}\langlebel{eq:barrier1}
\partial_t^{\alphapha,h} V^h(x, t)=\frac{1}{\Gamma(2-\alphapha)}\sum_{k=0}^{m-1}f(m-k)\big((k+1)^\alphapha-k^\alphapha\big)
\end{equation}
for all $x\in \mathbb{R}^n$ and $t\geq h$. Noting that
\begin{align*}
& f(m-k)\ge (1-\alphapha)/(m-k)^\alphapha\ge (1-\alphapha)/m^\alphapha, \\
& (k+1)^\alphapha-k^\alphapha\ge \alphapha/(k+1)^{1-\alphapha}\ge\alphapha/m^{1-\alphapha},
\end{align*}
we can plug these estimates into \eqref{eq:barrier1} to deduce the .
\end{proof}
\begin{lem}[Uniform boundedness]\langlebel{lem:bound}
Assume that \eqref{eq:op bound} and {\rm(H1)} hold.
Let $u^h$ be given by \eqref{def:Uh} for any fixed $h>0$. Then,
\[
|u^h(x, t)|\leq \sup_{\mathbb{R}^n}\left|u_0^h\right|+\frac{\Gamma(2-\alpha)M_0}{(1-\alpha)\alpha}t^\alphapha
\quad\text{for all} \
h>0, x\in \mathbb{R}^n, t\geq 0.
\]
\end{lem}
\begin{proof}
We define
\[
W^h(x, t):=\sup_{\mathbb{R}^n}\left|u_0^h\right|+\frac{\Gamma(2-\alpha)M_0}{(1-\alpha)\alpha}V^h(x, mh)
\]
for any $(x, t)\in \mathbb{R}^n\times [0, \infty)$, where $m=\lfloor t/h\rfloor$ and $V^h$ is given in Lemma \ref{lem:est1}.
In light of Lemma \ref{lem:est1}, we have
\[
\partial_t^{\alpha, h} W^h(x,mh)+F\left(x, t, DW^h(x,mh), D^2W^h(x,mh)\right)
\ge M_0+F(x, t, 0, 0) \ge 0
\]
for all $m\in\mathbb{N}$.
Combining with $U^h(\cdot,0)\le W^h(\cdot,0)$ on $\mathbb{R}^n$, by Proposition \ref{prop:monotone}, we get $U^h(\cdot,mh)\le W^h(\cdot,mh)$ for all $m\in\mathbb{N}$.
Symmetrically, we get $U^h(x, mh)\ge -W^h(\cdot,mh)$ for all $m\in\mathbb{N}\cup\{0\}$,
which implies the conclusion.
\end{proof}
\section{Convergence of discrete schemes}\langlebel{sec:main}
Let $u^h$ be the function defined by \eqref{def:Uh}. By Lemma \ref{lem:bound} and \eqref{initial approx}, we can define the half-relaxed limit of $u^h$ as follows:
\begin{equation}\langlebel{half-relax}
\begin{aligned}
\overline{u}(x,t)&:=
\lim_{\deltata\to 0}\sup\left\{u^h(y,s): |x-y|+|t-s|\le \deltata,\ s\geq 0,\ 0<h\le \deltata\right\},\\
\underline{u}(x,t)&:=
\lim_{\deltata \to 0}\inf\left\{u^h(y,s): |x-y|+|t-s|\le \deltata,\ s\geq 0,\ 0<h\le \deltata\right\}
\end{aligned}
\end{equation}
for all $(x,t)\in\mathbb{R}^n\times[0,\infty)$.
By the definition of Riemann integral and the operator $\partial_{t}^{\alpha,h}$, we have the following.
\begin{lem}\langlebel{lem:limit}
Let $\partial_t^{\alphapha, h}$ be given by \eqref{eq:op discrete}. Then for any $\psi\in C^{1}(\mathbb{R}^n\times[0,\infty))$, we have
\[
\partial_{t}^{\alpha, h}\psi\to \partial_t^{\alpha}\psi
\quad
\text{locally uniformly in} \ \mathbb{R}^n\times(0,\infty) \ \text{as}\ h\to0.
\]
\end{lem}
\begin{prop}[Sub- and supersolution property]\langlebel{prop:half}
Let $\overline{u}$ and $\underline{u}$ be the functions defined by \eqref{half-relax}.
Then $\overline{u}$ and $\underline{u}$ are, respectively, a subsolution and supersolution to \eqref{eq:1}.
\end{prop}
\begin{proof}
We only prove that $\overline{u}$ is a subsolution to \eqref{eq:1} as we can similarly prove that
$\underline{u}$ is a supersolution to \eqref{eq:1}.
Take a test function $\varphi\in C^2(\mathbb{R}^n\times[0,\infty))$ and
$(\hat{x},\hat{t})\in \mathbb{R}^n\times(0,\infty)$ so that $\overline{u}-\varphi$ takes
a strict maximum at $(\hat{x},\hat{t})$ with $(\overline{u}-\varphi)(\hat{x},\hat{t})=0$.
By adding $|x-\hat{x}|^4$ to $\varphi$ (we still denote it by $\varphi$),
we may assume that $\varphi(x,t)\to \infty$ as $|x|$ uniformly for all $t\geq 0$.
We first claim that there exists $(x_{j},t_j)\in\mathbb{R}^n\times(0,\infty)$, $h_j>0$ so that
$(x_j,t_j)\to(\hat{x},\hat{t})$ and $h_j\to0$ as $j\to\infty$,
\begin{align}
&u^{h_j}(\cdot, t_j)-\varphi(\cdot,t_j) \ \text{takes a maximum at} \ x_{j},
\langlebel{claim2}\\
&
\sup_{(x,t)\in\mathbb{R}^n\times(0,\infty)}(u^{h_j}-\varphi)(x,t)
<(u^{h_j}-\varphi)(x_j,t_j)+h_j. \langlebel{claim3}
\end{align}
Indeed, by definition of $\overline{u}$, there exists $(y_j,s_j)\in\mathbb{R}^n\times(0,\infty)$, and $h_j>0$ so that
\begin{align*}
&(y_j,s_j)\to(\hat{x},\hat{t}), \ h_j\to0,
\ \text{and} \
u^{h_j}(y_j,s_j)\to\overline{u}(\hat{x},\hat{t}) \quad\text{as} \ j\to\infty.
\end{align*}
We next take $t_j>0$ such that
\[
\sup_{(x,t)\in\mathbb{R}^n\times(0,\infty)}(u^{h_j}-\varphi)(x,t)
<
\sup_{x\in\mathbb{R}^n}(u^{h_j}-\varphi)(x,t_j)+h_j.
\]
Also, by Lemma \ref{lem:bound} again, there exists $x_j\in\mathbb{R}^n$ so that
\[
\sup_{x\in\mathbb{R}^n}(u^{h_j}-\varphi)(x,t_j)
=\max_{x\in\mathbb{R}^n}(u^{h_j}-\varphi)(x,t_j)
=(u^{h_j}-\varphi)(x_j,t_j).
\]
Then, we can also easily check that
$(x_j,t_j)\to(\hat{x},\hat{t})$ as $j\to\infty$.
Set $N_j:=\lfloor t_j/h_j\rfloor$.
Then we have $u^{h_j}(\cdot ,t_j)=U^{h_j}(\cdot, N_jh_{j})$ in $\mathbb{R}^n$.
Since $U^{h_j}(\cdot, N_jh_{j})$ is a viscosity solution to \eqref{eq:m} with $m=N_j$ and $h=h_j$,
in light of \eqref{claim2},
we obtain
\[
\partial_{t}^{\alpha, h_j}u^{h_j}(x_{j}, t_j)+F\left(x_j, t_j, D\varphi(x_{j},t_{j}),D^2\varphi(x_{j},t_{j})\right)\le 0.
\]
Set
$\sigma_j:=\max_{x\in\mathbb{R}^n}(u^{h_j}-\varphi)(x,t_j)
=u^{h_j}(x_j,t_j)-\varphi(x_j, t_j)$.
In light of \eqref{claim3}, we have
\[
(u^{h_j}-\varphi)(x_j,kh_j)
\le h_j+\sigma_j
\]
for all $k=0,\ldots,N_j-1$.
Hence,
\begin{align*}
& \Gamma(2-\alpha)(h_j)^\alpha\partial_{t}^{\alpha, h_j}u^{h_j}(x_j, t_j)=
u^{h_j}(x_j,N_jh_j)
-\sum_{k=0}^{N_j-1} C_{N_j,k}u^{h_j}(x_j, kh_j)\\
\ge&\,
\varphi(x_j,N_jh_j)+\sigma_j
-\sum_{k=0}^{N_j-1} C_{N_j, k}\left(\varphi(x_j, kh_j)+h_j+\sigma_j\right).
\end{align*}
Noting that
\begin{equation*}\langlebel{eq:sum}
\sum_{k=0}^{N_j-1} C_{N_j, k}=f(1)=1,
\end{equation*}
we obtain
\begin{align*}
\partial_{t}^{\alpha,h_j}u^{h_j}(x_j,N_jh_j)
\ge&\, \frac{1}{\Gamma(2-\alpha)(h_j)^\alpha}\left\{
\varphi(x_j,N_jh_j)-\sum_{k=0}^{N_j-1} C_{N_j, k}\varphi(x_j, kh_j)-h_j\right\}\\
=&\,
\partial_{t}^{\alpha,h_j}\varphi(x_j,N_jh_j)+O(h_j^{1-\alphapha}).
\end{align*}
We therefore obtain
\[
\partial_{t}^{\alpha,,h_j}\varphi(x_{j},N_jh_j)
+F\left(x_j, t_j, D\varphi(x_{j},t_{j}),D^2\varphi(x_{j},t_{j})\right)
\le O(h_j^{1-\alphapha}).
\]
By Lemma \ref{lem:limit} and the continuity of $F$, sending $j\to\infty$ yields
\[
\partial_{t}^{\alpha}\varphi(\hat{x},\hat{t})+
F\left(\hat{x}, \hat{t}, D\varphi(\hat{x},\hat{t}),D^2\varphi(\hat{x},\hat{t})\right)
\le 0.
\qedhere
\]
\end{proof}
\begin{prop}[Initial consistency]\langlebel{prop:ini}
Assume that \eqref{eq:op bound} and {\rm(H1)} hold. Let $\overline{u}$ and $\underline{u}$ be the functions defined by \eqref{half-relax}.
Then $\overline{u}\leq u_0\leq \underline{u}$ in $\mathbb{R}^n$.
\end{prop}
\begin{proof}
Fix any $x_0\in \mathbb{R}^n$. Since $u_0\in BUC(\mathbb{R}^n)$ and \eqref{initial approx} holds, for any $\sigmama>0$ we can find a bounded smooth function $\phi_\sigmama$ such that
$\phi_\sigmama(x_0)\leq u_0(x_0)+\sigmama$ and
$u_0^h(x)\leq \phi_\sigmama(x)$ for all $x\in \mathbb{R}^n$ and all $h>0$ small.
We claim that
\[
\phi^h(x , t)=\phi_\sigmama(x)+{M_{R_\sigmama}\Gammama(2-\alphapha)\over (1-\alphapha)\alphapha}t^\alphapha
\]
is a supersolution of \eqref{eq:discrete} with $h>0$ small, where $M_{R_\sigmama}$ is given in \eqref{eq:op bound} with
\[
R_\sigmama=\sup_{\mathbb{R}^n}\left(|D \phi_\sigmama|+|D^2 \phi_\sigmama|\right).
\]
Indeed, for any $x\in \mathbb{R}^n$, applying Lemma \ref{lem:est1},
we deduce that for all $m\in\mathbb{N}$,
\[
\partial_t^{\alphapha, h}\phi^h(x, mh)\geq M_{R_\sigmama}\geq -F\left(x, mh, D \phi_\sigmama(x), D^2 \phi_\sigmama(x)\right)
\]
for all $x\in \mathbb{R}^n$.
We thus can adopt Proposition \ref{prop:monotone} to obtain that
$u^h(x, Nh)\leq \phi^h(x, Nh)$ for all $x\in \mathbb{R}^n$ and $t\geq 0$ with $N=\lfloor t/h\rfloor$,
which implies that
\[
u^h(x, t)\leq \phi_\sigmama(x)+{M_{R_\sigmama}\Gammama(2-\alphapha)\over (1-\alphapha)\alphapha}t^\alphapha
\]
for all $x\in \mathbb{R}^n$ and $t\geq 0$. We thus have
\[
\overline{u}(x_0, 0)\leq \phi_\sigmama(x_0),
\]
which implies, by letting $\sigmama\to 0$, that
$\overline{u}(x_0, 0)\leq u_0(x_0)$.
The proof for the part on $\underline{u}$ is symmetric and therefore omitted here.
\end{proof}
\begin{proof}[Proof of Theorem {\rm\ref{thm:main}}]
If (H2) holds, then the conclusion of the theorem is a straightforward result of Propositions \ref{prop:half} and \ref{prop:ini}.
\end{proof}
\begin{comment}
\appendix
\section{A comparison principle in $\mathbb{R}^n$}\langlebel{sec:appendix}
We give a comparison principle for \eqref{eq:1} under (F1) and (F2) for completeness.
\begin{thm}\langlebel{thm:comparison unbounded}
Assume that $F$ is a continuous elliptic operator satisfying {\rm(F1)} and {\rm(F2)}.
Then for any $\alphapha\in (0, 1)$, the comparison principle {\rm(H2)} holds.
\end{thm}
\begin{proof}
Assume by contradiction that there exists $(x_0, t_0)\in \mathbb{R}^n\times (0, \infty)$ such that $u(x_0, t_0)-v(x_0, t_0)\geq \theta$ for some $\theta>0$.
For $\mu, \beta>0$ and $T>t_0$, set
\[
\Phi(x, y, t)=u(x, t)-v(x, t)-{\mu\over T-t}-\beta f(x)-\beta f(y).
\]
where $f(x)=(1+|x|^2)^{1/2}$ for $x\in \mathbb{R}^n$. By letting $\mu, \beta$ small, we get
$
\max_{\mathbb{R}^n\times [0, T)} \Phi\geq {\theta/ 2}.
$
Let $(\hat{x}, \hat{t})$ be any maximizer of $\Phi$.
We can easily see that $\hat{t}>0$.
The penalty near space infinity essentially enables us to pursue our argument in the same manner as in the case for a bounded domain (\cite[Theorem 3.4]{N}). Note that
\[
\Phi_{\sigmama}(x, y, t)=u(x, t)-v(y, t)-{|x-y|^2\over \sigmama}-{\mu\over T-t}-\beta f(x)-\beta f(y)
\]
attains a maximum at $(x_{\sigmama}, y_{\sigmama}, t_{\sigmama})\in \mathbb{R}^{2n}\times (0, \infty)$.
Since $(x_\sigmama, y_\sigmama)$ is uniformly bounded in $\sigmama$ due to the penalty terms $\beta f(x)$ and $\beta f(y)$, we see that $(x_{\sigmama}, y_{\sigmama}, t_{\sigmama})$ converges to a maximizer of $\Phi$, denoted again by $(\hat{x}, \hat{x}, \hat{t})$, via a subsequence as $\sigmama\to 0$. A standard argument also yields that $|x_{\sigmama}-y_{\sigmama}|^2/\sigmama\to 0$ as $\sigmama\to 0$.
Applying
an equivalent definition of solutions involving semijets (\cite[Proposition 2.7]{N}), we have $p_{\sigmama}, q_{\sigmama}\in \mathbb{R}^n$, $X_{\sigmama}, Y_{\sigmama}\in \mathbb{S}^n$ satisfying
\begin{equation}\langlebel{eq:ishii1}
p_{\sigmama}={2(x_\sigmama-y_\sigmama)\over \sigmama}+\beta Df(x_\sigmama), \quad q_\sigmama={2(x_\sigmama-y_\sigmama)\over \sigmama}-\beta Df(y_\sigmama),
\end{equation}
\begin{equation}\langlebel{eq:ishii2}
\left(
\begin{array}{cc}
X_{\sigmama}-\beta D^2 f(x_{\sigmama}) & 0 \\
0 & -Y_{\sigmama}-\beta D^2 f(y_{\sigmama})
\end{array}
\right) \le
\frac{2}{\sigmama}
\left(
\begin{array}{cc}
I & -I \\
-I & I
\end{array}
\right)
\end{equation}
such that
\[
\partial_t^\alpha u(x_{\sigmama}, t_{\sigmama})+F(x_{\sigmama}, t_{\sigmama}, p_{\sigmama}, X_{\sigmama})\leq 0,
\quad
\partial_t^\alpha v(y_{\sigmama}, t_{\sigmama})+F(y_{\sigmama}, t_{\sigmama}, q_{\sigmama}, Y_{\sigmama})\geq 0.
\]
Taking the difference of both inequalities above, we have
\begin{equation}\langlebel{vis ineq}
\partial_t^\alpha u(x_{\sigmama}, t_{\sigmama})-\partial_t^\alpha v(y_{\sigmama}, t_{\sigmama})\leq F(y_{\sigmama}, t_{\sigmama}, q_{\sigmama}, Y_{\sigmama})-F(x_{\sigmama}, t_{\sigmama}, p_{\sigmama}, X_{\sigmama}).
\end{equation}
We next use \eqref{eq:ishii1}, \eqref{eq:ishii2} and
(F1), (F2) to estimate the right hand side of \eqref{vis ineq} as below.
\[
F(y_{\sigmama}, t_{\sigmama}, q_{\sigmama}, Y_{\sigmama})-F(x_{\sigmama}, t_{\sigmama}, p_{\sigmama}, X_{\sigmama})
\leq \omegaega\left({2|x_{\sigmama}-y_{\sigmama}|^2\over\sigmama}+|x_{\sigmama}-y_{\sigmama}|\right)+2\tilde{\omegaega}(C\beta)
\]
for some $C>0$ independent of $\sigmama$ and $\beta$.
On the other hand, we can use the same argument as in the proof of \cite[Theorem 3.4]{N} to get
\[
\partial_t^\alpha u(x_{\sigmama}, t_{\sigmama})-\partial_t^\alpha v(y_{\sigmama}, t_{\sigmama})\geq \frac{\theta-u(x_{\sigmama}, 0)+v(y_{\sigmama}, 0)}{t_{\sigmama}^\alphapha\Gammama(1-\alphapha)}\geq \frac{\theta}{t_{\sigmama}^\alphapha\Gammama(1-\alphapha)}
\]
by using \eqref{equiv time der}.
Combining these estimates with \eqref{vis ineq} and passing to the limit as $\sigmama\to 0$,
we get
\[
{\theta\over T^\alphapha \Gammama(1-\alphapha)}\leq {\theta\over \hat{t}^\alphapha \Gammama(1-\alphapha)}\leq 2\tilde{\omegaega}(C\beta).
\]
We reach a contradiction by letting $\beta\to 0$.
\end{proof}
\end{comment}
\begin{comment}
We first adopt (i) to get
\[
|F_\varepsilon(x_{\sigmama, \varepsilon}, t_{\sigmama, \varepsilon}, p_{\sigmama, \varepsilon}, X_{\sigmama, \varepsilon})-F(x_{\sigmama, \varepsilon}, t_{\sigmama, \varepsilon}, p_{\sigmama, \varepsilon}, X_{\sigmama, \varepsilon})|\leq \omegaega(M\varepsilon^{1/2}),
\]
\[
|F^\varepsilon(y_{\sigmama, \varepsilon}, t_{\sigmama, \varepsilon}, q_{\sigmama, \varepsilon}, Y_{\sigmama, \varepsilon})-F(y_{\sigmama, \varepsilon}, t_{\sigmama, \varepsilon}, q_{\sigmama, \varepsilon}, Y_{\sigmama, \varepsilon})|\leq \omegaega(M\varepsilon^{1/2}).
\]
We next use \eqref{eq:ishii1}, \eqref{eq:ishii2} and (ii), (iii) to obtain
\[
\begin{aligned}
&F(y_{\sigmama, \varepsilon}, t_{\sigmama, \varepsilon}, q_{\sigmama, \varepsilon}, Y_{\sigmama, \varepsilon})-F(x_{\sigmama, \varepsilon}, t_{\sigmama, \varepsilon}, p_{\sigmama, \varepsilon}, X_{\sigmama, \varepsilon})\\
&\leq \omegaega\left({2|x_{\sigmama, \varepsilon}-y_{\sigmama, \varepsilon}|^2\over\sigmama}+|x_{\sigmama, \varepsilon}-y_{\sigmama, \varepsilon}|\right)+\tilde{\omegaega}(\beta)+2\omegaega_R(\beta).
\end{aligned}
\]
where $R>0$ depends only on $\varepsilon$. Hence, we have
\end{comment}
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|
\begin{document}
\title{\PSPACE-completeness of Pulling Blocks to Reach a Goal}
\begin{abstract}
We prove \textsc{PSPACE}\xspace-completeness of all but one problem in a large space of pulling-block problems where the goal is for the agent to reach a target destination. The problems are parameterized by whether pulling is optional, the number of blocks which can be pulled simultaneously, whether there are fixed blocks or thin walls, and whether there is gravity. We show \textsc{NP}\xspace-hardness for the remaining problem, \PullkFG[1][?] (optional pulling, strength 1, fixed blocks, with gravity).
\end{abstract}
\section{Introduction}
\label{sec:intro}
In the broad field of \emph{motion planning}, we seek algorithms for actuating
or moving mobile agents (e.g., robots) to achieve certain goals.
In general settings, this problem is PSPACE-complete
\cite{Canny-1988-pspace,Reif-1979-mover},
but much attention has been given to finding simple variants near the threshold
between polynomial time and PSPACE-complete;
see, e.g., \cite{hearn2009games}.
One interesting and well-studied case, arising in warehouse maintenance,
is when a single robot with $O(1)$ degrees of freedom navigates an environment
with obstacles, some of which can be moved by the robot (but which cannot move
on their own).
Research in this direction was initiated in 1988 \cite{Wilfong-1991}.
A series of problems in this space arise from computer puzzle games,
where the robot is the agent controlled by the player,
and the movable obstacles are \emph{blocks}.
The earliest and most famous such puzzle game is \emph{Sokoban},
first released in 1982 \cite{sokoban-wiki}.
Much later, this game was proved PSPACE-complete
\cite{sokoban,hearn2009games}.
In Sokoban, the agent can \emph{push} movable $1 \times 1$ blocks
on a square grid, and the goal is to bring those blocks to target locations.
Later research in \emph{pushing-block puzzles} considered the simpler
goal of simply getting the robot to a target location,
proving various versions NP-hard, NP-complete, or PSPACE-complete
\cite{demainepush,demaine2003pushing,demaine2004pushpush}.
In this paper, we study the \Pull series of motion-planning problems
\cite{Ritt10,PRB16}, where the agent can \emph{pull} (instead of push)
movable $1 \times 1$ blocks on a square grid.
Figure~\ref{fig:example} shows a simple example.
This type of block-pulling mechanic (sometimes together with a block-pushing
mechanic) appears in many real-world video games,
such as Legend of Zelda, Tomb Raider, Portal, and Baba Is You.
\begin{figure}
\caption{A pulling-block problem. The robot is the agent, the flag is the goal square, the light gray blocks can be moved, and the bricks are fixed in place.
\sl \href{https://fontawesome.com/icons/robot?style=solid}
\label{fig:example}
\end{figure}
We study several different variants of \Pull, which can be combined in arbitrary combination:
\begin{enumerate}
\setlength\itemsep{0pt}
\setlength\parskip{0pt}
\item \textbf{Optional/forced pulls:} In \textsc{Pull!}, every agent motion that can also pull blocks must pull as many as possible (as in many video games where the player input is just a direction). In \textsc{Pull?}, the agent can choose whether and how many blocks to pull. Only the latter has been studied in the literature, where it is traditionally called \textsc{Pull}; we use the explicit ``?''\ to indicate optionality and distinguish from \textsc{Pull!}.
\item \textbf{Strength:} In \Pullk[$k$][], the agent can pull an unbroken horizontal or vertical line of up to $k$ pullable blocks at once. In \Pullk[$\ast$][], the agent can pull any number of blocks at once.
\item \textbf{Fixed blocks/walls:} In \PullkF[][], the board may have fixed $1 \times 1$ blocks that cannot be traversed or pulled. In the \PullkW[][], the board may have fixed thin ($1 \times 0$) walls; this is more general because a square of thin walls is equivalent to a fixed block. Thin walls were introduced in \cite{demaine2017push}.
\item \textbf{Gravity:} In \textsc{Pull-G}, all movable blocks fall downward after each agent move. Gravity does not affect the agent's movement.
\end{enumerate}
Table~\ref{tab:results} summarizes our results: for all variants that include fixed blocks or
walls, we prove \textsc{PSPACE}\xspace-completeness for any strength, with optional or forced pulls, and with or
without gravity, with the exception of \PullkFG[1][?] for which we only show \textsc{NP}\xspace-hardness.
\definecolor{header}{rgb}{0.29,0,0.51}
\definecolor{gray}{rgb}{0.85,0.85,0.85}
\def\header#1{\multicolumn{1}{c}{\textcolor{white}{\textbf{#1}}}}
\def\tableref#1{[\S\ref{#1}]}
\begin{table}
\centering
\tabcolsep=0.5\tabcolsep
\begin{tabular}{l c c c c l l}
\rowcolor{header}
\header{Problem} & \header{\hspace{-0.3em}Forced} & \header{Strength\hspace{-0.2em}} & \header{Features} & \header{\hspace{-0.2em}Gravity\hspace{-0.2em}} & \header{Our result} & \header{\hspace{-0.3em}Previous best} \\
\Pull?-$k$F & no & $k \ge 1$ & fixed blocks & no & \textsc{PSPACE}\xspace-complete \tableref{sec:no gravity} & \textsc{NP}\xspace-hard \cite{Ritt10} \\
\rowcolor{gray}
\Pull?-$\ast$F & no & $\infty$ & fixed blocks & no & \textsc{PSPACE}\xspace-complete \tableref{sec:no gravity} & \textsc{NP}\xspace-hard \cite{Ritt10} \\
\Pull!-$k$F & yes & $k \ge 1$ & fixed blocks & no & \textsc{PSPACE}\xspace-complete \tableref{sec:no gravity} & \\
\rowcolor{gray}
\Pull!-$\ast$F & yes & $\infty$ & fixed blocks & no & \textsc{PSPACE}\xspace-complete \tableref{sec:no gravity} & \\
\Pull?-1FG & no & $k = 1$ & fixed blocks & yes & \textsc{NP}\xspace-hard \tableref{sec:Pull1FG NP} & \\
\rowcolor{gray}
\Pull?-1WG & no & $k = 1$ & thin walls & yes & \textsc{PSPACE}\xspace-complete \tableref{sec:optional pull} & \\
\Pull?-$k$FG & no & $k \ge 2$ & fixed blocks & yes & \textsc{PSPACE}\xspace-complete \tableref{sec:optional pull} & \\
\rowcolor{gray}
\Pull?-$\ast$FG & no & $\infty$ & fixed blocks & yes & \textsc{PSPACE}\xspace-complete \tableref{sec:optional pull} & \\
\Pull!-$k$FG & yes & $k \ge 1$ & fixed blocks & yes & \textsc{PSPACE}\xspace-complete \tableref{sec:mandatory gravity} & \\
\rowcolor{gray}
\Pull!-$\ast$FG & yes & $\infty$ & fixed blocks & yes & \textsc{PSPACE}\xspace-complete \tableref{sec:mandatory gravity} & \\
\end{tabular}
\caption[]{Summary of our results.}
\label{tab:results}
\end{table}
The only previously known hardness result for this family of problems is NP-hardness for both \PullkF and \PullkF[$*$] \cite{Ritt10}.
In some cases, our results are stronger than the best known results for the corresponding \textsc{Push} (pushing-block) problem; see \cite{PRB16}.
More complex variants \PullPull (where pulled blocks slide maximally), \PushPull (where blocks can be pushed and pulled), and \textsc{Storage Pull} (where the goal is to place multiple blocks into desired locations) are also known to be PSPACE-complete \cite{demaine2017push,PRB16}.
Our reductions are from Asynchronous Nondeterministic Constraint Logic (NCL)
\cite{hearn2009games, DBLP:conf/cccg/Viglietta13} and
planar 1-player motion planning \cite{demaine2018general, doors}.
In Section~\ref{sec:no gravity}, we reduce from NCL to prove \textsc{PSPACE}\xspace-hardness of all nongravity variants.
In Section~\ref{sec:gravity pspace}, we use the motion-planning-through-gadgets framework \cite{demaine2018general} to prove \textsc{PSPACE}\xspace-completeness of most variants with gravity, including all variants with forced pulling and variants with optional pulling and either thin walls or fixed blocks with $k\ge2$.
These reductions use two particular gadgets for 1-player motion planning,
the newly introduced \emph{nondeterministic locking 2-toggle\xspace}
(a variant of the locking 2-toggle from \cite{demaine2018general})
and the \emph{3-port self-closing door} (one of the self-closing doors from
\cite{doors}).
Although the latter gadget is proved hard in \cite{doors}, for completeness,
we give a more succinct proof in Appendix~\ref{app:self-closing door}.
In Section~\ref{sec:Pull1FG NP}, we prove \textsc{NP}\xspace-hardness for the one remaining case of \PullkFG[1][?],
again reducing from 1-player planar motion planning,
this time with an NP-hard gadget called the crossing NAND gadget \cite{doors}.
\section{Pulling Blocks with Fixed Blocks is \textsc{PSPACE}\xspace-complete}
\label{sec:no gravity}
In this section, we show the PSPACE-completeness of all variants of pulling-block problems we have defined without gravity, namely \PullkF, \PullkW, \PullkF[$k$][!], and \PullkW[$k$][!] for $k \ge 1$, and \PullkF[$\ast$], \PullkW[$\ast$], \PullkF[$*$][!], and \PullkW[$*$][!]. We do this through a reduction from Nondeterministic Constraint Logic \cite{hearn2009games}, which we describe briefly before moving on to the main proof.
\subsection{Asynchronous Nondeterministic Constraint Logic}
\label{ssec:NCL}
\textsc{Nondeterministic Constraint Logic}\xspace (NCL) takes place on \emph{constraint graphs}: a directed graph where each edge has weight 1 or 2. Weight-1 edges are called \emph{red}; weight-2 edges are called \emph{blue}. The ``constraint'' in NCL is that each vertex must maintain in-weight at least 2. A \emph{move} in NCL is a reversal of the direction of one edge, while maintaining compliance with the constraint.
In \emph{asynchronous} NCL, the process of switching the orientation of an edge does not happen instantaneously, but instead it takes a positive amount of time, and it is possible to be in the process of switching several edges simultaneously. When an edge is in the process of being reversed, it is not oriented towards either vertex. Viglietta \cite{DBLP:conf/cccg/Viglietta13} showed that this model is equivalent to the regular (synchronous) model, because there is no benefit to having an edge in the intermediate unoriented state. In this work, we only use the asynchronous NCL model; any mention of NCL should be understood to mean asynchronous NCL.
An instance of \textsc{Nondeterministic Constraint Logic}\xspace consists of a constraint graph $G$ and an edge $e$ of $G$, called the \emph{target edge}.
The output is \textsc{yes}\xspace if there is a sequence of moves on $G$ that reverses the direction of $e$, and \textsc{no}\xspace otherwise. \textsc{Nondeterministic Constraint Logic}\xspace is \textsc{PSPACE}\xspace-complete, even for planar constraint graphs that have only two types of vertices: AND (two red edges, one blue edge) and OR (three blue edges).
We will reduce from the planar, AND/OR, asynchronous version of NCL to show pulling-block problems without gravity \textsc{PSPACE}\xspace-hard.
For more description of NCL, including a proof of \textsc{PSPACE}\xspace-completeness, the reader is referred to \cite{hearn2009games}.
\subsection{NCL Gadgets in Pulling Blocks}
In order to embed an NCL constraint graph into \PullkF, we need three
components, corresponding to NCL edges (which can attach to AND and OR gadgets
in all necessary orientations, and that allows the player to win if the
winning edge is flipped), AND vertices, and OR vertices.
In each of these gadgets, we will show that if the underlying NCL constraint is violated, then the agent will be ``trapped'', meaning that the state is in an \emph{unrecoverable configuration}, a concept used in several previous blocks games \cite{sokoban,hearn2009games}. This occurs when the agent makes a pull move after which no set of moves will lead to a solution, generally because the agent has trapped itself in a way that no pull can be made \textit{at all} (or only a few more pull moves may be made, and all of them lead to a state such that there are no more pull moves).
\textbf{Diode Gadget.} Before describing the three main gadgets, we describe a helper gadget, the \emph{diode}, shown in Figure~\ref{fig:diode}. The diode can be repeatedly traversed in one direction but never the other. It was introduced in \cite{Ritt10}.
\begin{figure}
\caption{Diode gadget, which can be repeatedly traversed from left to right but never from right to left.
In diagrams to follow it will be represented by the diode symbol.}
\label{fig:diode}
\end{figure}
In the next three sections, we describe the three main gadgets in turn.
\input{Gadgets/NCLWire}
\input{Gadgets/NCLor.tex}
\input{Gadgets/NCLand.tex}
\subsection{Proof of \textsc{PSPACE}\xspace-completeness}
We first observe that every pulling-block problem we consider is in \textsc{PSPACE}\xspace.
\begin{lemma}
\label{lem:pull-in-PSPACE}
Every pulling-block problem defined in Section~\ref{sec:intro} is in \textsc{PSPACE}\xspace.
\end{lemma}
\begin{proof}
The entire configuration while playing on instance of a pulling-block problem can be stored in polynomial space (e.g., as a matrix recording whether each cell is empty, a fixed block, a movable block, the agent's location, or the finish tile). There is a simple nondeterministic algorithm which guesses each move and keeps track of the configuration using only polynomial space, accepting if the agent reaches the goal square.
Thus the problem is in \textsc{NP}\xspaceSPACE, so by Savitch's Theorem \cite{savitch1970relationships} it is also in \textsc{PSPACE}\xspace.
\end{proof}
\begin{theorem}
\label{thm:pull-kF-PSPACE-complete}
\PullkF and \PullkF[$k$][!] PSPACE-complete for $k \ge 1$ and $k=*$.
\end{theorem}
\begin{proof}
Lemma~\ref{lem:pull-in-PSPACE} gives us containment in \textsc{PSPACE}\xspace.
For \textsc{PSPACE}\xspace-hardness, we reduce from asynchronous NCL
(as defined in Section~\ref{ssec:NCL}).
Given a planar AND/OR NCL graph, we construct an instance of \PullkF or \PullkF[$k$][!] as follows. First, embed the graph in a grid graph. Scale this grid graph by enough to fit our gadgets; $20\times20$ suffices. At each vertex, place the appropriate AND or OR vertex gadget. Place edge gadgets in the appropriate configuration along each edge, using corner gadgets on turns. Adjust the vertex gadgets to accommodate the alignment of the edge gadgets incident to them. Finally, place the goal tile in the edge gadget corresponding to the target edge so that it is accessible only if the target edge is flipped, and place the agent on any empty tile.
The agent can walk through edge gadgets to visit any NCL edge or vertex, and by
Lemmas~\ref{lem:NCL-vertices-OR} and~\ref{lem:NCL-vertices-AND}, flip edges in accordance with the rules of NCL. Ultimately, it can reach the goal tile if and only if the target edge of the NCL instance can be reversed.
In our construction, the agent never has the opportunity to pull more than 1 block at a time. Thus the reduction works for \PullkF for any $k\geq1$, including $k=*$. In addition, the agent never has to choose not to pull a block when taking a step, so the reduction works for \PullkF[$k$][!] as well as \PullkF.
\end{proof}
\begin{corollary}
\label{cor:pull-W-PSPACE-complete}
\PullkW and \PullkW[$k$][!] are \textsc{PSPACE}\xspace-complete for $k\geq1$ and $k=*$.
\end{corollary}
\begin{proof}
A fixed block can be simulated using four thin walls drawn around a single tile, so our constructions can be built using thin walls instead of fixed blocks. Formally, this is a reduction from \PullkF to \PullkW and a reduction from \PullkF[$k$][!] to \PullkW[$k$][!].
\end{proof}
\section{\texorpdfstring{\PullkFG}{Pull?-kFG} is \textsc{PSPACE}\xspace-complete for \texorpdfstring{$k \ge 2$}{k ≥ 2} and \texorpdfstring{\PullkFG[$k$][!]}{Pull!-kFG} is \textsc{PSPACE}\xspace-complete for \texorpdfstring{$k \ge 1$}{k ≥ 1}}
\label{sec:gravity pspace}
In this section, we show \textsc{PSPACE}\xspace-completeness results for most of the
pulling-block variants with gravity.
In Section~\ref{sec:gadgets}, we introduce and prove results about
\emph{1-player motion planning} from the motion-planning-through-gadgets
framework introduced in \cite{demaine2018computational}, which will be the
basis for the later proofs.
In Section~\ref{sec:optional pull},
we show \textsc{PSPACE}\xspace-completeness for \PullkFG with $k \ge 2$, for
\PullkFG[$\ast$], for \PullkWG with $k \ge 1$, and for \PullkWG[$\ast$].
In Section~\ref{sec:mandatory gravity}, we show \textsc{PSPACE}\xspace-completeness for
\PullkFG[$k$][!] with $k \ge 1$, and for \PullkFG[$\ast$][!].
The one case missing from this collection is \PullkFG[1], which we prove
NP-hard later in Section~\ref{sec:Pull1FG NP}.
\subsection{1-player Motion Planning}
\label{sec:gadgets}
\emph{1-player motion planning} refers to the general problem of planning an agent's motion to complete a path through a series of gadgets whose state and traversability can change when the agent interacts with them. In particular, a \emph{gadget} is a constant-size set of locations, states, and traversals, where each traversal indicates that the agent can move from one location to another while changing the state of the gadget from one state to another. A system of gadgets is constructed by connecting the locations of several gadgets with a graph, which is sometimes restricted to be planar. The decision problem for 1-player motion planning is whether the agent, starting from a specified stating location, can follow edges in the graph and transitions within gadgets to reach some goal location.
Our results use that 1-player planar motion planning is \textsc{PSPACE}\xspace-complete
for the following gadgets:
\begin{enumerate}
\item The \emph{locking 2-toggle}, shown in Figure~\ref{fig:l2t}, is a three-state two-tunnel reversible deterministic gadget. In the \emph{middle state}, both tunnels can be traversed in one direction, switching to one of two \emph{leaf states}. Each leaf state only allows the transition back across that tunnel in the opposite direction, returning the gadget to the middle state. Traversing one tunnel ``locks'' the other side from being used until the prior traversal is reversed.
1-player planar motion planning with locking 2-toggles was shown
\textsc{PSPACE}\xspace-complete in \cite{demaine2018general}.
In Section~\ref{sec:leaf2toggle}, we strengthen the result in \cite{demaine2018general} by showing that 1-player motion planning with locking 2-toggle remains hard even if the initial configuration of the system has all gadgets in leaf (locked) states.
\begin{figure}
\caption{State space of the locking 2-toggle.}
\label{fig:l2t}
\caption{State space of the nondeterministic locking 2-toggle.}
\label{fig:nl2t}
\end{figure}
\item The \emph{nondeterministic locking 2-toggle\xspace}, shown in Figure~\ref{fig:nl2t}, is a four-state gadget where each state has two transitions, each across the same tunnel. The top pair of states each allow a single traversal downward, and allow the agent to choose either of the two bottom states for the gadget. Similarly, the bottom pair of states each allow a single traversal upward to one of the top states. We can imagine this as being similar to the locking 2-toggle if the tunnel to be taken next is guessed ahead of time: the bottom state of the locking 2-toggle is split into two states which together allow the same traversals, but only if the agent picks the correct one ahead of time.
In Section~\ref{sec:nondet2toggle}, we show that 1-player motion planning with the nondeterministic locking 2-toggle\xspace is \textsc{PSPACE}\xspace-complete.
\item The \emph{door gadget} has three directed tunnels called \emph{open}, \emph{close}, and \emph{traverse}. The traverse tunnel is open or closed depending on the state of the gadget and does not change the state. Traversing the open or close tunnel opens or closes the traverse tunnel, respectively.
1-player motion planning with door gadgets was shown
\textsc{PSPACE}\xspace-complete in \cite{nintendoor} and explored more thoroughly
(in particular, proved hard for most planar cases) in \cite{doors}.
\item The \emph{3-port self-closing door}, shown in Figure~\ref{fig:statespace-scd}, is a gadget with a tunnel that becomes closed when the agent traverses it and a location that the agent can visit to reopen the tunnel.
It has an \emph{opening port}, which opens the gadget,
and a \emph{self-closing tunnel}, which is the tunnel that closes when traversed.
In Appendix~\ref{app:self-closing door}, we prove that 1-player planar motion planning with the \emph{3-port self-closing door} is \textsc{PSPACE}\xspace-complete.
A more general result on self-closing doors can be found in \cite{doors}, but we include this more succinct proof for completeness and conciseness.
\begin{figure}
\caption{State space of the 3-port self-closing door, used in the \PullkFG[$k$][!] reduction.}
\label{fig:statespace-scd}
\end{figure}
\end{enumerate}
\subsubsection{Nondeterministic Locking 2-toggle}
\label{sec:nondet2toggle}
\label{sec:leaf2toggle}
In this section, we prove that 1-player motion planning with the nondeterministic locking 2-toggle\xspace is \textsc{PSPACE}\xspace-complete. We also show that 1-player motion planning with the locking 2-toggle remains \textsc{PSPACE}\xspace when the gadgets are restricted to start in leaf states.
We use the construction shown in Figure~\ref{fig:nl2t-to-l2t} to show simultaneously that locking 2-toggles starting in leaf states can simulate a locking 2-toggle starting in a nonleaf state, and nondeterministic locking 2-toggles can simulate a locking 2-toggle. This construction consists of two nondeterministic locking 2-toggles and a 1-toggle. A \emph{1-toggle} is a two-state, two-location, reversible, deterministic gadget where each state admits a single (opposite) transition between the locations and these transitions flip the state. It can be trivially simulated by taking a single tunnel of a locking 2-toggle or nondeterministic locking 2-toggle.
\begin{theorem} \label{thm:nlt}
1-player planar motion planning with the nondeterministic locking 2-toggle\xspace is \textsc{PSPACE}\xspace-complete.
\end{theorem}
\begin{proof}
In the construction shown in Figure~\ref{fig:nl2t-to-l2t}, the agent can enter through either of the top lines; suppose they enter on the left. Other than backtracking, the agent's only path is across the bottom 1-toggle, then up the leftmost tunnel, having chosen the state of the nondeterministic locking 2-toggle\xspace which makes that tunnel traversable.
Now the only place the agent can usefully enter the construction is the leftmost line. The agent can only go down the leftmost tunnel, up the 1-toggle, and out the top right entrance, again making the appropriate nondeterministic choice when traversing the left gadget.
Symmetrically, if (from the unlocked state) the agent enters the top right, they must exit the bottom right, and the next traversal must go from the bottom right to the top right and return the construction to the unlocked state. Thus this construction simulates a locking 2-toggle.
\end{proof}
\begin{figure}
\caption{Constructing a locking 2-toggle from a nondeterministic locking 2-toggle. It is currently in the unlocked state. The nondeterministic locking 2-toggles are in leaf states (top states in Figure~\ref{fig:nl2t}
\label{fig:nl2t-to-l2t}
\end{figure}
If we instead build the above construction with locking 2-toggles in leaf states, then all three of the locking 2-toggles used are in leaf states (the 1-toggle is one tunnel of a locking 2-toggle). A very similar argument as the nondeterministic locking 2-toggle\xspace construction shows this gadget also simulates a locking 2-toggle. Thus, given a 1-player motion planning problem with locking 2-toggles, we can replace all of the locking 2-toggles in nonleaf states with this gadget to obtain an instance where all starting gadgets are in leaf states.
\begin{corollary}
1-player motion planning with the locking 2-toggle where all of the locking 2-toggles start in leaf states is \textsc{PSPACE}\xspace-complete.
\end{corollary}
\later{
\section{3-port Self-Closing Door}
\label{app:self-closing door}
Ani et al.~\cite{doors} proved \textsc{PSPACE}\xspace-completeness of 1-player planar motion planning with many types of self-closing door gadgets and all of their planar variations.
For completeness, we give a proof specific to the 3-port self-closing door gadget in this section.
Our proof is more succinct because it does not consider other variants of the gadget.
The reduction is from 1-player motion planning with the door gadget from \cite{nintendoor}.
\begin{theorem}\label{thm:scd}
1-player planar motion planning with the 3-port self-closing door is \textsc{PSPACE}\xspace-hard.
\end{theorem}
\begin{proof}
We will show that the 3-port self-closing door planarly simulates a crossover, which lets us ignore planarity.
We will then show that the 3-port self-closing door simulates the door gadget. Because 1-player motion planning with the door gadget is \textsc{PSPACE}\xspace-hard \cite{nintendoor}, so is 1-player motion planning with the 3-port self-closing door, and because it simulates a crossover, so is 1-player planar motion planning with the 3-port self-closing door. Along the way, we will construct a self-closing door with multiple door and opening ports as well as a diode.
\paragraph{Diode.}We can simulate a diode (one-way tunnel which is always traversable)
by connecting the opening port to the input of the self-closing tunnel. The agent can always go to the opening port
and then through the self-closing tunnel, but can never go the other way because the self-closing tunnel is directed.
\paragraph{Port Duplicator.} The construction shown in Figure~\ref{fig:scd-port-duplicator} simulates a self-closing door with two equivalent opening ports. If the agent enters from the top, it can
open only one of the upper gadgets, then open the lower gadget, and then must exit the same way it came. Note, this same idea can be used to construct more than two ports, which will be needed later.
\begin{figure}
\caption{3-port self-closing door simulating a version of it that has 2 opening ports. Opening ports are shown in green.
A dotted self-closing tunnel is closed, and a solid self-closing tunnel is open.}
\label{fig:scd-port-duplicator}
\end{figure}
We use these to simulate an intermediate gadget composed of two of self-closing doors each connected to two opening ports in a particular order arrangement, shown in Figure~\ref{fig:planar-scd-crossoverish}. If the agent enters from port 1 or 4,
it will open door E or F, respectively, and then leave. If the agent enters from port 2, it can open doors A, B, and C. If it then traverses door B and opens door E, it will get stuck because both B and D are closed. So the agent cannot open door E and exit.
Instead, it can traverse doors B and A, ending up back at port 2 with no change except that door C is open. Entering
port 2 or 3 always gives the agent an opportunity to open door C, so leaving door C open does not help.
So the only useful path after entering port 2 is to traverse door C. The agent is then forced to go right and can open door F. Then
it is forced to traverse door B. Again if the agent opens door E, it will be stuck, so the agent traverses door A instead and
returns to port 2, leaving door F open.
Similarly, if the agent enters from port 3, the only useful thing it can do is open
door E and return to port 3.
\begin{figure}
\caption{3-port self-closing door simulating the gadget on the right, where each port opens the door of the same color (the top and third-from-top open the top door, and the others open the bottom door).}
\label{fig:planar-scd-crossoverish}
\end{figure}
\paragraph{Crossover.} This intermediate gadget can simulate a directed crossover, shown in Figure~\ref{fig:planar-scd-crossover}. If the agent enters at the top left, it can open the left door on the top gadget, open both doors on the bottom gadget, and then exit the bottom right while closing all three opened doors. If the agent opens both doors on the top gadget it will get stuck. Similarly if the agent enters the bottom left, all it can do is exit the top right.
The directed crossover can simulate an undirected crossover, as in
Figure~\ref{fig:dir-crossover} and shown in \cite{Push100}.
\begin{figure}
\caption{3-port self-closing door simulating a crossover.}
\label{fig:planar-scd-crossover}
\end{figure}
\begin{figure}
\caption{Directed crossover simulating an undirected crossover.}
\label{fig:dir-crossover}
\end{figure}
\paragraph{Door Duplicator.} Now, we use this crossover to simulate a gadget with two self-closing doors controlled by the same opening port, as shown in Figure~\ref{fig:scd-tunnel-duplicator}. This gadget has two states, open and closed. Both doors are either open or closed and going through either door closes both of them.
The construction is similar to the construction for the port duplicator, but goes through a tunnel instead.
\begin{figure}
\caption{3-port self-closing door simulating a gadget with 2 self-closing tunnels.}
\label{fig:scd-tunnel-duplicator}
\end{figure}
\paragraph{Door Gadget.} Finally, we triplicate the opening port by adding a third entrance to the construction in Figure~\ref{fig:scd-port-duplicator} similar to the other two, and use these ports to simulate a door gadget as shown
in Figure~\ref{fig:scd-otc}. Recall the whole three-port two-door gadget has only two states, open and closed. The agent can open both doors from any of the open ports and going across either self-closing door will close both doors. If the agent enters from port $O$, it can open the doors and leave.
If the agent enters from port $T_0$ and the gadget is open, the agent can traverse the door and then reopen it using the third port. The agent then leaves at port $T_1$. If the agent enters from port $C_0$, it can open the gadget and then must traverse the bottom tunnel and leave at port $C_1$, closing the
gadget.
\end{proof}
\begin{figure}
\caption{Simulation of the door gadget in \cite{nintendoor}
\label{fig:scd-otc}
\end{figure}
}
\subsection{\texorpdfstring{\PullkFG}{Pull?-kFG}}
\label{sec:optional pull}
In this section, we show that several versions of pulling-block problems with optional pulling and gravity are \textsc{PSPACE}\xspace-complete by a reduction from 1-player motion planning with nondeterministic locking 2-toggles, shown \textsc{PSPACE}\xspace-hard in Section~\ref{sec:nondet2toggle}.
We begin with a construction of a 1-toggle, and then use those and an intermediate construction to build a nondeterministic 2 toggle.
\paragraph{1-toggle.}
A \emph{1-toggle} is a gadget with a single tunnel, traversable in one direction. When the agent traverses it, the direction that it can be traversed is flipped, meaning that the agent must backtrack and return the way it came in order to be able to traverse it the first way again.
Our 1-toggle construction in \PullkFG for $k\geq2$ is shown in Figure~\ref{fig:1toggle}. In the state shown, it can only be traversed from left to right by pulling both blocks to the left. This traversal flips the direction that the gadget can be traversed---it can now only be traversed from right to left.
\begin{figure}
\caption{1-toggle in \PullkFG[2].}
\label{fig:1toggle}
\end{figure}
\paragraph{Nondeterministic Locking 2-toggle.}
Our construction of a nondeterministic locking 2-toggle\xspace, shown in Figure~\ref{fig:locking2toggle}, uses two 1-toggles plus a connecting section at the top.
\begin{figure}
\caption{Locking 2-toggle in \PullkFG[2].}
\label{fig:locking2toggle}
\caption{Locking 2-toggle in \PullkWG[1].}
\label{fig:locking2toggle-W}
\end{figure}
The configuration shown in Figure~\ref{fig:locking2toggle} is a leaf state. The right tunnel is traversable from to right to bottom right. If the agent traverses that tunnel, it can choose whether to pull the top pair of blocks to the right (because pulling is optional), corresponding to the nondeterministic choice in the nondeterministic locking 2-toggle\xspace. Both 1-toggles will be in the state where they can be traversed from bottom (outside) to top (inside). One of these paths will be blocked by the top pair of blocks and the other will be traversable, depending on whether the agent chose to pull those blocks. Traversing the traversable path then puts the gadget in a leaf state, either the one shown or its reflection.
It is possible for the agent to pull only one block instead of two, but this can only prevent future traversals, so never benefits the agent.
\begin{theorem}
\label{thm:pull-kFG-PSPACE-complete}
\PullkFG is \textsc{PSPACE}\xspace-complete for $k \ge 2$ and $k=*$.
\end{theorem}
\begin{proof}
Lemma~\ref{lem:pull-in-PSPACE} gives containment in \textsc{PSPACE}\xspace.
For hardness, we reduce from 1-player planar motion planning with the nondeterministic locking 2-toggle\xspace, shown \textsc{PSPACE}\xspace-hard in Theorem~\ref{thm:nlt}. We embed any planar network of gadgets in a grid, and replace each nondeterministic locking 2-toggle\xspace with the construction described above in the appropriate state. The resulting pulling-block problem is solvable if and only if the motion planning problem is.
This reduction works for \PullkFG for any $k \ge 2$ including $k=*$, because the player only ever has the opportunity to pull 2 blocks at a time. This proof requires optional pulling because the player must choose whether to pull blocks while traversing a nondeterministic locking 2-toggle\xspace.
\end{proof}
\begin{corollary}
\PullkWG is \textsc{PSPACE}\xspace-complete for $k\ge1$ and $k=*$.
\end{corollary}
\begin{proof}
With thin walls, the tunnels can be separated by a thin wall instead of a fixed block, which means that only one block is required in each of the toggles. This is shown in Figure~\ref{fig:locking2toggle-W}. The rest of the proof follows in the same manner, demonstrating \textsc{PSPACE}\xspace-completeness of \PullkWG for $k \ge 1$.
\end{proof}
\subsection{\texorpdfstring{\PullkFG[$k$][!]}{Pull!-kFG}}
\label{sec:mandatory gravity}
In this section, we show \textsc{PSPACE}\xspace-completeness for pulling-block problems with forced pulling and gravity, using a reduction from 1-player planar motion planning with the 3-port self-closing door, shown \textsc{PSPACE}\xspace-hard in Theorem~\ref{thm:scd}.
\begin{theorem}
\label{thm:pull!-kFG-PSPACE-complete}
\PullkFG[$k$][!] is \textsc{PSPACE}\xspace-complete for $k \ge 1$ and $k=*$.
\end{theorem}
\begin{proof}
Lemma~\ref{lem:pull-in-PSPACE} gives containment in \textsc{PSPACE}\xspace.
We show \textsc{PSPACE}\xspace-hardness by a reduction from 1-player planar motion planning with the 3-port self-closing door. It suffices to construct a 3-port self-closing door in \PullkFG[$k$][!].
First, we construct
a diode, shown in Figure~\ref{fig:pull!-kFG-diode}. The agent cannot enter from the right. If the agent enters from the left,
it must pull the left block to the left to advance. If it pulls the left block left and then exits, they still cannot enter from the right,
so doing so is useless. The agent then advances and is forced to pull the left block back to its original position. The agent then must
pull the right block left to advance, and must actually advance because the way back is blocked. As the agent exits the gadget, it
is forced to pull the right block back to its original position. Therefore, the agent can always cross the gadget from left to right
and never from right to left, simulating a diode.
\begin{figure}
\caption{A diode in \PullkFG[$k$][!].}
\label{fig:pull!-kFG-diode}
\end{figure}
Using this diode, we then construct a 3-port self-closing door, shown in Figure~\ref{fig:pull!-kFG-scd}; the diode icons indicate the diode shown in Figure~\ref{fig:pull!-kFG-diode}. The bottom is exit-only. In the closed
state, the agent should not enter from the top because it would become trapped between a block and the wrong end of a diode. The
agent can enter from the right, pull the block 1 tile right, and leave, opening the gadget. In the open state, the agent can enter
from the top and exit out the bottom, and is forced to pull the block back to its original position, closing the gadget. So this construction
simulates a 3-port self-closing door.
\begin{figure}
\caption{A 3-port self-closing door in \PullkFG[$k$][!].}
\label{fig:pull!-kFG-scd}
\end{figure}
Because the player never has the opportunity to pull multiple blocks, this reduction works for all $k\ge1$ including $k=*$.
\end{proof}
\section{\texorpdfstring{\PullkFG[1]}{Pull?-1FG} is NP-hard}
\label{sec:Pull1FG NP}
In this section, we show \textsc{NP}\xspace-hardness for \PullkFG[1] by reducing from
1-player planar motion planning with the crossing NAND gadget from \cite{doors}.
A \emph{crossing NAND gadget} is a three-state gadget with two crossing
tunnels, where traversing either tunnel permanently closes the other tunnel.
1-player planar motion planning with the crossing NAND gadget is NP-hard
in \cite[Lemma~4.9]{doors} based on the constructions in
\cite{demaine2003pushing,friedman2002pushing} which originally reduce from \textsc{Planar 3-Coloring}\xspace.
\begin{theorem}
\label{thm:pull-1FG-NP-hard}
\PullkFG[1] is \textsc{NP}\xspace-hard.
\end{theorem}
\begin{proof}
We reduce from 1-player planar motion planning with the crossing NAND gadget
\cite[Lemma~4.9]{doors}.
First we first construct a ``single-use'' one-way gadget, shown in
Figure~\ref{fig:single-one-way}.
This gadget can initially can be crossed in one way, but then becomes
impassable in both directions.
\begin{figure}
\caption{Single-use one-way gadget that initially allows traversal from left-to-right and then
prevents traversal in both directions.}
\label{fig:single-one-way}
\end{figure}
Figure~\ref{fig:crossover} shows our construction of the crossing NAND gadget.
Single-use one-way gadgets enforce that the agent must enter
through one of the top paths.
The agent must pull two blocks to enter the gadget;
these blocks end up stacked in the vertical tunnel on top of the block below.
The agent cannot exit via the bottom tunnel underneath its entry tunnel:
the agent can pull one block into the slot on the bottom, and then can pull
one block one square, but that still leaves the third block of the stack
blocking off the exit path.
The agent cannot exit via the other top path, because it is blocked by the
single-use one-way gadget.
The only path remaining is for the agent to cross diagonally by pulling the
single block in the lower layer into the slot, revealing a path to the exit
opposite where the agent entered.
After leaving, both the entry tunnel and exit tunnel are impassable
because the single-use one-way gadgets have become impassable.
If the agent later enters via the other entry tunnel, the agent will be trapped,
because it will not be able to leave via the tunnel that was ``collapsed''
in the initial entry.
\end{proof}
\begin{figure}
\caption{Crossing NAND gadget allowing traversal either from the top-left to
the bottom-right, or from the top-right to the bottom-left. After being
traversed once, the entire gadget becomes impassable in any direction.}
\label{fig:crossover}
\end{figure}
We leave open the question of whether \PullkFG[1] is in \textsc{NP}\xspace or \textsc{PSPACE}\xspace-hard.
\section{Open Problems}
\label{sec:Open Problems}
There are several open problems remaining related to the pulling-block problems considered in this paper.
\begin{enumerate}
\item What is the complexity of \PullkFG[1] (the last remaining problem in Table~\ref{tab:results})? We leave a gap between \textsc{NP}\xspace-hardness and containment in \textsc{PSPACE}\xspace.
\item What is the complexity of pulling-block puzzles without fixed blocks (say, on a rectangular board)? With block pushing, one can generally construct effectively fixed blocks by putting enough blocks together. This technique no longer works in the block-pulling context.
\item Do all of these variants remain \textsc{PSPACE}\xspace-hard when we ask about storage (can the player place blocks covering some set of squares?)\ or reconfiguration (where blocks are distinguishable and must reach a desired configuration) instead of reachability? The storage question for \PullkFG[$k$][?] for $k \geq 1$ and \PullkFG[$*$][?] has been proved PSPACE-hard \cite{PRB16}.
\item What about the studied variants applied to \PushPull (where blocks can be pushed and pulled) and \PullPull (where blocks must be pulled maximally until the robot backs against another block)? Standard versions are proved PSPACE-complete in \cite{demaine2017push,PRB16}, but variations with mandatory pulling, gravity, and/or no fixed blocks all remain open.
\end{enumerate}
\appendix
\latertrue \the\magicAppendix
\end{document}
|
\begin{document}
\title{Circuit Quantum Electrodynamics: A New Look Toward Developing Full-Wave Numerical Models}
\author{Thomas E. Roth~\IEEEmembership{Member,~IEEE},
and Weng C. Chew~\IEEEmembership{Life Fellow,~IEEE}
\thanks{This work was supported by NSF ECCS 169195, a startup fund at Purdue University, and the Distinguished Professorship Grant at Purdue University.
Thomas E. Roth is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. Weng C. Chew is with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA and the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (contact e-mail: [email protected]).
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.}
}
\maketitle
\begin{abstract}
Devices built using circuit quantum electrodynamics architectures are one of the most popular approaches currently being pursued to develop quantum information processing hardware. Although significant progress has been made over the previous two decades, there remain many technical issues limiting the performance of fabricated systems. Addressing these issues is made difficult by the absence of rigorous numerical modeling approaches. This work begins to address this issue by providing a new mathematical description of one of the most commonly used circuit quantum electrodynamics systems, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our new model is better suited to developing numerical solvers than the circuit element descriptions commonly used in the literature. We present details on the quantization of our new model, and derive quantum equations of motion for the coupled field-transmon system. These results can be used in developing full-wave numerical solvers in the future. To make this work more accessible to the engineering community, we assume only a limited amount of training in quantum physics and provide many background details throughout derivations.
\end{abstract}
\begin{IEEEkeywords}
Circuit quantum electrodynamics, computational electromagnetics, quantum mechanics.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}
\label{sec:intro}
\IEEEPARstart{C}{ircuit} quantum electrodynamics (QED) architectures are one of the leading candidates currently being pursued to develop quantum information processing devices \cite{blais2004cavity,blais2007quantum,gu2017microwave}, such as quantum simulators, gate-based quantum computers, and single photon sources \cite{ma2019dissipatively,arute2019quantum,zhou2020tunable,houck2007generating}. These circuit QED devices are most often formed by embedding superconducting Josephson junctions into planar microwave circuitry (also made from superconductors), such as coplanar waveguides \cite{blais2004cavity}. Doing this, planar ``on-chip'' realizations of many cavity QED and quantum optics concepts can be implemented at microwave frequencies. This provides a pathway to leveraging the fundamental light-matter interactions necessary to create and process quantum information in these new architectures \cite{gu2017microwave}. To achieve this, circuit QED systems implement ``artificial atoms'' (also frequently called qubits) through various configurations of Josephson junctions that are then coupled to coplanar waveguide resonators \cite{blais2004cavity,blais2007quantum,gu2017microwave}.
Circuit QED systems have garnered a high degree of interest in large part because of the achievable strength of light-matter coupling and the engineering control that is possible with these systems. The unique aspects of using artificial atoms formed by Josephson junctions allows for them to achieve substantially higher coupling strengths to electromagnetic fields than what is possible with natural atomic systems \cite{devoret2007circuit}. As a result, circuit QED systems have been able to achieve some of the highest levels of light-matter coupling strengths seen in any physical system to date, providing an avenue to explore and harness untapped areas of physics \cite{kockum2019ultrastrong}.
In addition to the strong coupling, circuit QED systems also provide a much higher degree of engineering control than is typically possible with natural atoms or ions. This is because many artificial atoms can be designed to have different desirable features by assembling Josephson junctions in various topologies \cite{gu2017microwave,vion2002manipulating,manucharyan2009fluxonium,koch2007charge}. Further, the operating characteristics of these artificial atoms can be tuned \textit{in situ} by applying various biases, such as voltages, currents, or magnetic fluxes \cite{gu2017microwave}. This allows dynamic reconfiguration of the artificial atom, opening possibilities for device designs that are not feasible using fixed systems like natural atoms or ions \cite{gu2017microwave}. Finally, many aspects of the fabrication processes for these systems are mature due to their overlap with established semiconductor fabrication technologies \cite{gu2017microwave}.
Although there have been many successes with circuit QED systems to date (e.g., achieving ``quantum supremacy'' \cite{arute2019quantum}), a substantial amount of progress is still needed to truly unlock the potential of these systems. One area that could help accelerate the maturation of these technologies is the development of rigorous numerical modeling methods. Current models used in the physics community incorporate numerous approximations to simplify them to the point that they can be solved using semi-analytical approaches to build intuition about the physics \cite{blais2004cavity,blais2007quantum,koch2007charge,gu2017microwave,vool2017introduction,langford2013circuit,girvin2011circuit}. For instance, in almost all circuit QED studies, the electromagnetic aspects of the system are represented as simple combinations of lumped element LC circuits. Although this is appropriate and useful for building intuition, performing the engineering design and optimization of a practical device requires a level of precision that is not possible with this kind of \textit{circuit-based} description.
Instead, models that retain the full three-dimensional vector representation of the electromagnetic aspects of these circuit QED devices are needed. With these \textit{field-based} models, accurate full-wave numerical methods can begin to be formulated. These numerical methods can then be used to enable studies on the engineering optimization of circuit QED devices. Unfortunately, to the authors' best knowledge, this kind of detailed field-based description is not available in the literature.
To address this issue, we present details in this work on the desired field-based framework for circuit QED systems and show how it can be used to derive the more commonly used circuit-based models found in the physics literature. To make the discussion more concrete, we focus on developing a field-based model for the transmon qubit \cite{koch2007charge}, which is one of the most widely used qubits in modern circuit QED systems \cite{ma2019dissipatively,arute2019quantum,zhou2020tunable,houck2007generating}. Similar procedures to those shown in this work can be applied to develop field-based models for other commonly used artificial atoms.
In an effort to make this work accessible to the engineering community, we provide many details in the derivations. We also assume only a limited amount of background knowledge in quantum physics at the level of \cite{chew2016quantum,chew2016quantum2,chew2021qme-made-simple}. Although circuit QED uses superconducting qubits, a minimal knowledge of superconductivity is needed to understand the general physics of these systems. Introductions to superconductivity in the context of circuit QED can be found in \cite{girvin2011circuit,langford2013circuit,vool2017introduction}.
The remainder of this work is organized in the following way. In Section \ref{sec:transmon-background}, we review basic details of circuit QED systems using transmon qubits and introduce the field-based Hamiltonian developed in this work. Following this, we discuss quantization procedures in Section \ref{sec:field-quantization} for the electromagnetic fields that are devised specifically for developing numerical methods. Next, Section \ref{sec:field-to-circuit} presents details on how field-based descriptions can be converted into a transmission line formalism. Using these details, Section \ref{sec:field-transmon-hamiltonian} shows how the field-based Hamiltonian for circuit QED systems is consistent with the circuit-based descriptions found in the literature. With an appropriate Hamiltonian developed, Section \ref{sec:eom} derives the field-based quantum equations of motion that can be used for formulating new numerical modeling strategies. Finally, we present conclusions on this work in Section \ref{sec:conclusion}.
\section{Circuit QED Preliminaries}
\label{sec:transmon-background}
To support the development of the field-based description of circuit QED systems, it will first be necessary to review a few properties of these systems. We begin this by discussing the basic physical properties of the transmon qubit in Section \ref{subsec:transmon-basic-physics}. Following this, we discuss in Section \ref{subsec:transmon-coupling} how the coupling of a transmon qubit to a transmission line is typically handled using a circuit theory description. Finally, in Section \ref{subsec:field-based} we briefly introduce the field-based description of the transmon qubit coupled to a transmission line structure. We will demonstrate the consistency of the field- and circuit-based descriptions of this system in Section \ref{sec:field-transmon-hamiltonian} after developing the necessary tools in the intervening sections.
For readers interested in a more complete description of the transmon qubit, we refer them to the seminal work of \cite{koch2007charge} that presents an in-depth theoretical analysis. More details on the derivation of the typically used Hamiltonians introduced in this section can be found in \cite{vool2017introduction,girvin2011circuit,langford2013circuit,kockum2019quantum}. We focus our discussions on the basic physics of the Hamiltonians to provide an intuitive understanding only.
\subsection{Basic Physical Properties of a Transmon}
\label{subsec:transmon-basic-physics}
Typically, quantum effects are only observable at microscopic levels due to the fragility of individual quantum states. To observe quantum behavior at a macroscopic level (e.g., on the size of circuit components), a strong degree of coherence between the individual microscopic quantum systems must be achieved \cite{girvin2011circuit}. One avenue for this to occur is in superconductors cooled to extremely low temperatures (on the order of 10 mK) \cite{kockum2019quantum}. At these temperatures, electrons in the superconductor tend to become bound to each other as Cooper pairs \cite{vool2017introduction,girvin2011circuit,langford2013circuit}. These Cooper pairs exhibit bosonic properties and become the charge carriers of the superconducting system. Importantly, they have the necessary degree of coherence over large length scales to make observing macroscopic quantum behavior possible. Circuit QED systems interact with these macroscopic quantum states using microwave photons and other circuitry \cite{gu2017microwave}.
One of the earliest qubits used to observe macroscopic quantum behavior in circuit QED systems was the Cooper pair box (CPB) \cite{nakamura1999coherent}, which can be viewed as a predecessor to the transmon. The traditional CPB is formed by a thin insulative gap (on the order of a nm thick) that connects a superconducting ``island'' and a superconducting ``reservoir'' \cite{kockum2019quantum}. The superconductor-insulator-superconductor ``sandwich'' formed between the island and reservoir is known as a Jospehson junction, and has the property that Cooper pairs may tunnel through the junction without requiring an applied voltage \cite{tafuri2019introductory}. For the basic CPB, the island is not directly connected to other circuitry, while the reservoir can be in contact with external circuit components (if desired). Since the superconducting island is isolated from other circuitry, the CPB is very sensitive to the number of Cooper pairs that have tunneled through the Josephson junction. Due to this sensitivity, the CPB is also often referred to as a charge qubit \cite{kockum2019quantum}.
As is common in quantum physics, it is desirable to consider a Hamiltonian mechanics description of the CPB \cite{chew2016quantum,chew2016quantum2,chew2021qme-made-simple}. For an isolated system, this amounts to expressing the total energy of the Josephson junction in terms of \textit{canonical conjugate variables}. These conjugate variables vary with respect to each other in a manner to ensure that the total energy of the system is conserved \cite{chew2021qme-made-simple}. For the CPB system, the canonical conjugate variables are the Cooper pair density difference $n$ and the Josephson phase $\varphi$. Initially considering the classical case, these variables are real-valued deterministic numbers.
For this case, $n$ is the net density of Cooper pairs that have tunneled through the Josephson junction relative to an equilibrium level \cite{vool2017introduction,girvin2011circuit,langford2013circuit}. Due to its relationship to a microscopic theory of superconductivity, $\varphi$ is more challenging to interpret \cite{langford2013circuit}. Briefly, there exists a long-range phase coherence for a collective description of all the Cooper pairs in a superconductor. As a result, the phase of the collective description becomes a meaningful variable to characterize the momentum of all Cooper pairs. This phase can then be related to a current flowing in the superconductor \cite{langford2013circuit}. For a Josephson junction, the phase difference of the two superconductors is important, and is denoted as $\varphi$ \cite{vool2017introduction,girvin2011circuit,langford2013circuit}.
The Hamiltonian of the CPB can be found by considering the total energy of the junction in terms of an effective capacitance and inductance expressed with $n$ and $\varphi$. The capacitance is due to the ``parallel plate'' configuration of the junction. The energy is found by first noting that $2en = Q$, where $Q$ is the total charge ``stored'' in the junction capacitance and $2e$ is the charge of a Cooper pair. Now, considering that the single electron charging energy of a capacitor is $E_C = e^2/2C$, the total capacitive energy of the junction capacitance, $Q^2/2C$, can be written as $4 E_C n^2$ \cite{girvin2011circuit}.
The inductance of the Josephson junction is more complicated to understand because it involves the tunneling physics. However, for the discussion here it only needs to be known that the supercurrent flowing through the junction is $I=I_c \sin{\varphi}$, where $I_c$ is the critical current of the junction. Since $\partial_t \varphi$ can be related to the voltage drop over the junction, this current-phase relationship can be used to derive an effective Josephson inductance that is proportional to $1/\cos{\varphi}$ \cite{girvin2011circuit}. Overall, the energy associated with the inductance is $-E_J\cos{\varphi}$, where $E_J$ is the Josephson energy.
Combining the results for the effective capacitive and inductive energy, the Hamiltonian for the CPB is
\begin{align}
H_T = 4E_C n^2 - E_J \cos\varphi.
\label{eq:classical-JJ-Hamiltonian}
\end{align}
This Hamiltonian can be viewed as being equivalent to a linear capacitor in parallel with a nonlinear inductor. It is this nonlinear inductance that allows Josephson junctions to form qubits. Without the nonlinearity, the energy levels of a quantized form of (\ref{eq:classical-JJ-Hamiltonian}) would be evenly spaced, making it impossible to selectively target a single pair of energy levels to perform qubit operations. If needed, the equations of motion for the CPB can be derived from (\ref{eq:classical-JJ-Hamiltonian}) using Hamilton's equations \cite{chew2016quantum,chew2016quantum2,chew2021qme-made-simple}.
Typically, it is desirable to be able to control the operating point of the CPB system. This can be done using a voltage source capacitively coupled to the superconducting island. This induces a ``background'' Cooper pair density that has tunneled onto the island, denoted by $n_g$ (this is also often referred to as the offset charge) \cite{kockum2019quantum}. The basic circuit diagram of this qubit is shown in Fig. \ref{subfig:basic_cpb}, where $V_g$ is an applied voltage bias capacitively coupled to the CPB through $C_g$. For this system, the Hamiltonian is modified to be
\begin{align}
H_T = 4E_C (n-n_g)^2 - E_J \cos\varphi.
\label{eq:classical-JJ-Hamiltonian2}
\end{align}
\begin{figure}
\caption{Circuit schematics for (a) a traditional CPB and (b) a transmon. A Josephson junction consists of a pure tunneling element (symbolized as a box with an ``X'' through it) in parallel with a small junction capacitance $C_J$.}
\label{fig:cpb_schematics}
\end{figure}
The system can now be quantized by elevating the canonical conjugate variables to be non-commuting quantum operators. In particular, we now characterize the CPB with the quantum operators $\hat{n}$ and $\hat{\varphi}$ that have commutation relation \cite{vool2017introduction}
\begin{align}
[\hat{\varphi},\hat{n}] = i.
\end{align}
Combined with a complex-valued quantum state function, these quantum operators take on a statistical interpretation and share an uncertainty principle relationship \cite{chew2021qme-made-simple}. As a result, measurements of observables associated with these quantum operators (e.g., the number of Cooper pairs that have tunneled through the junction) become random variables with means and variances dictated by the laws of quantum mechanics \cite{chew2021qme-made-simple}. The combination of these properties allows for the non-classical interference between possible states of a system, which is key to quantum information processing.
Now, one advantage of determining the classical Hamiltonian of the CPB system is that the quantum Hamiltonian follows easily from it \cite{chew2021qme-made-simple}. In particular, the quantum Hamiltonian is \cite{kockum2019quantum,koch2007charge}
\begin{align}
\hat{H}_T = 4 E_C (\hat{n}-n_g)^2 - E_J \cos \hat{\varphi}.
\label{eq:isolated_cpb_hamiltonian}
\end{align}
Note that $n_g$ remains a classical variable that describes the offset charge induced by the applied DC voltage.
The different terms in (\ref{eq:isolated_cpb_hamiltonian}) take on similar physical meaning to the classical case of (\ref{eq:classical-JJ-Hamiltonian2}). However, the second term can be expressed in the charge basis as
\begin{align}
-E_J\cos\hat{\varphi} = \frac{E_J}{2} \sum_N \big[ |N\rangle\langle N+1 | + | N+1\rangle\langle N | \big],
\label{eq:tunneling-hamiltonian}
\end{align}
where $|N \rangle$ is an eigenstate of $\hat{n}$ with eigenvalue $N$ \cite{vool2017introduction}. This eigenvalue is a discrete number that counts how many Cooper pairs have tunneled through the junction. Considering this, we see that the form of the effective inductive energy given in (\ref{eq:tunneling-hamiltonian}) clearly shows the tunneling physics.
Unfortunately, the CPB is very sensitive to charge fluctuations (i.e., noise) that appear from a variety of sources in $n_g$ \cite{koch2007charge}. This sensitivity prevents the CPB from being applicable to scalable quantum information processing systems. To address this issue, the transmon qubit was introduced as an ``optimized CPB''. The differences between these qubits are most easily understood in terms of the ratio of $E_J$ to $E_C$ where the devices are designed to operate. In traditional CPBs, $E_J/E_C \ll 1$, due to the small capacitance $C_J$ naturally provided by the Josephson junction. The transmon qubit is designed to operate like a CPB, but with $E_J/E_C \gg 1$.
This is achieved by dramatically decreasing $E_C$ by adding a large shunting capacitance $C_B$ around the CPB, as shown in Fig. \ref{subfig:transmon}. This capacitance is often implemented by forming interdigital capacitors between two superconducting islands that are connected by a Josephson junction, as shown in Fig. \ref{fig:physical_transmon}. Although the physical implementation of the transmon is different from the CPB, the same Hamiltonian of (\ref{eq:isolated_cpb_hamiltonian}) describes its behavior \cite{koch2007charge}. However, because $E_C \ll E_J$ the transmon becomes insensitive to fluctuations in $n_g$, making it useful for scalable quantum information processing systems \cite{koch2007charge}.
\begin{figure}
\caption{Schematic of a transmon qubit coupled to a coplanar waveguide transmission line. Dimensions (in $\mu$m) are based on the transmon qubit used in \cite{houck2007generating}
\label{fig:physical_transmon}
\end{figure}
\subsection{Coupling the Transmon to a Transmission Line Resonator}
\label{subsec:transmon-coupling}
A completely isolated qubit cannot be controlled, and so is of little use. Circuit QED systems address this by coupling a qubit to transmission line structures that can be used to apply microwave drive pulses to control and read out the qubit's state, as well as to interface separated qubits to implement qubit-qubit interactions \cite{blais2004cavity,blais2007quantum,gu2017microwave}. For transmons, capacitive coupling to transmission lines is typical \cite{koch2007charge}. One early strategy using a coplanar waveguide is shown in Fig. \ref{fig:physical_transmon}. More recently, other strategies have emerged to improve the interconnectivity to the transmon \cite{barends2014superconducting}. However, analyzing these newer implementations follows the analysis of the coupling shown in Fig. \ref{fig:physical_transmon}, so only the simpler case will be considered here.
The interaction between the transmon and transmission line can be described quantum mechanically in a number of ways. However, it is usually convenient to express the interaction in terms of $\hat{n}$ and a transmission line voltage operator \cite{koch2007charge}. Often, the transmission line the transmon is coupled to is a resonator. As a result, the voltage operator can be conveniently written in terms of the modes of the resonator \cite{blais2004cavity}. These modes are the one-dimensional sinusoidal functions used to describe the spatial dependence of a resonator's voltage and current in microwave engineering \cite{pozar2009microwave}. This spatial dependence is integrated out of the Hamiltonian to arrive at a circuit-based (i.e., lumped element) description of the interaction.
Following this process, the resulting Hamiltonian that describes a single transmon qubit coupled to a single mode of a transmission line resonator is
\begin{align}
\hat{H} = \hat{H}_T + \hat{H}_R + 2e\beta V_g \hat{V}_r \hat{n},
\label{eq:coupled-transmon}
\end{align}
\begin{align}
\hat{H}_R = \frac{1}{2}\big[ L_r \hat{I}_r^2 + C_r \hat{V}_r^2 \big],
\label{eq:free-resonator}
\end{align}
where $\hat{I}_r$ and $\hat{V}_r$ are the \textit{integrated} resonator voltage and current (i.e., the spatial variation of the mode has been integrated out) \cite{blais2004cavity,koch2007charge}. The Hamiltonian in (\ref{eq:free-resonator}) is the \textit{free resonator Hamiltonian}, i.e., it describes the total energy of the uncoupled resonator mode (and can be viewed as being equivalent to an LC tank circuit). The final term in (\ref{eq:coupled-transmon}) represents the interaction between the transmon and resonator, in terms of the resonator voltage and transmon charge operators.
Other terms in (\ref{eq:coupled-transmon}) and (\ref{eq:free-resonator}) include the magnitude of the resonator voltage at the location of the transmon $V_g$ and $\beta = C_g/(C_g+C_B)$. The latter term represents a voltage divider to capture the portion of the resonator voltage applied to the transmon within a lumped element approximation (note $C_J$ has been absorbed into the much larger capacitance $C_B$ in $\beta$). Further, $L_r = \ell L$ and $C_r = \ell C$, where $\ell$ is the length of the resonator and $L$ and $C$ are the per-unit-length inductance and capacitance of the transmission line. These terms arise from the normalization and subsequent spatial integration of the resonator voltage and current modes. Extending this model to contain multiple resonator modes is in principle quite simple, and will be considered later in this work.
Depending on the numerical model being developed, it may or may not be desirable to keep the interdigital capacitor in the geometric description of the system. When it is desirable to explicitly model the interdigital capacitor, $\beta$ should be omitted from equations since the ``voltage divider'' it represents would already be accounted for. To allow the equations in this work to be used in either situation, we keep $\beta$ in all equations.
Most circuit QED studies work with the transmission line operators expressed in terms of bosonic ladder operators. In particular, each mode of the transmission line is viewed as an independent quantum harmonic oscillator characterized by bosonic ladder operators \cite{chew2016quantum2}. The integrated resonator voltage and current operators in terms of the ladder operators are
\begin{align}
\hat{V}_r = \sqrt{\frac{\hbar \omega_r}{2C_r}} (\hat{a}+\hat{a}^\dagger),
\label{eq:total-v}
\end{align}
\begin{align}
\hat{I}_r = -i\sqrt{\frac{\hbar \omega_r}{2L_r}} (\hat{a}-\hat{a}^\dagger),
\label{eq:total-i}
\end{align}
where $\omega_r$ is the resonant frequency of the mode \cite{blais2004cavity,koch2007charge}. The ladder operators satisfy the bosonic commutation relation,
\begin{align}
[\hat{a},\hat{a}^\dagger] = 1.
\end{align}
Using properties of the ladder operators, the Hamiltonian in (\ref{eq:coupled-transmon}) can be expressed as
\begin{align}
\hat{H} = \hat{H}_T + \hbar\omega_r \hat{a}^\dagger\hat{a} + 2e\beta V^\mathrm{rms}_g \hat{n} (\hat{a}+\hat{a}^\dagger),
\label{eq:coupled-transmon2}
\end{align}
where $V^\mathrm{rms}_g = V_g \sqrt{\hbar\omega_r/2C_r}$ and the zero point energy of the transmission line resonator has been adjusted to remove constant terms.
The isolated transmon Hamiltonian of (\ref{eq:isolated_cpb_hamiltonian}) can be diagonalized exactly \cite{koch2007charge}. Using these eigenstates (denoted as $|j\rangle$), the complete system Hamiltonian of (\ref{eq:coupled-transmon2}) can be rewritten as
\begin{multline}
\hat{H} = \sum_j \hbar\omega_j |j\rangle\langle j | + \hbar\omega_r \hat{a}^\dagger\hat{a} \\ + 2e\beta V_g^\mathrm{rms} \sum_{i,j} \langle i | \hat{n} | j \rangle \, |i\rangle \langle j | (\hat{a}+\hat{a}^\dagger) ,
\label{eq:coupled-transmon3}
\end{multline}
where $\omega_j$ is the eigenvalue associated with the $j$th transmon eigenstate. A more complete description of these eigenstates can be found in \cite{koch2007charge}. For this work, the essential property is that transitions between two transmon eigenstates corresponds to a high probability event of some number of Cooper pairs tunneling through the Josephson junction.
In the transmon operating regime, (\ref{eq:coupled-transmon3}) can be further simplified because it is safe to assume that the $\hat{n}$ operator only couples nearest neighbor transmon eigenstates. Hence, (\ref{eq:coupled-transmon3}) can be rewritten as
\begin{multline}
\hat{H} = \sum_j \hbar\omega_j |j\rangle\langle j | + \hbar\omega_r \hat{a}^\dagger\hat{a} + 2e\beta V_g^\mathrm{rms} \times \\ \sum_{i} \langle i | \hat{n} | i+1 \rangle \, \big(|i\rangle \langle i+1 | + |i+1\rangle \langle i | \big) \big(\hat{a}+\hat{a}^\dagger \big) ,
\label{eq:coupled-transmon4}
\end{multline}
where the fact that $\langle i | \hat{n} | i+1 \rangle = \langle i+1 | \hat{n} | i \rangle $ has also been used \cite{koch2007charge}. This is the circuit-based Hamiltonian that is typically used as a starting point for many circuit QED studies.
\subsection{Field-Based Description of the Transmon System}
\label{subsec:field-based}
With the physics of the transmon understood, we can now briefly introduce the field-based description of a circuit QED system using a transmon. In particular, the postulated field-transmon system Hamiltonian is
\begin{align}
\hat{H} = \hat{H}_T + \hat{H}_F - \iiint \hat{\mathbf{E}} \cdot \partial_t^{-1} \hat{\mathbf{J}}_t d\mathbf{r}.
\label{eq:field-transmon-hamiltonian1}
\end{align}
In (\ref{eq:field-transmon-hamiltonian1}), $\hat{H}_T$ is the transmon Hamiltonian given in (\ref{eq:isolated_cpb_hamiltonian}),
\begin{align}
\hat{H}_F = \frac{1}{2} \iiint \big( \epsilon \hat{\mathbf{E}}^2 + \mu \hat{\mathbf{H}}^2 \big) d\mathbf{r}
\end{align}
is the free field Hamiltonian consisting of the electric and magnetic field operators $\hat{\mathbf{E}}$ and $\hat{\mathbf{H}}$, and the final term is the interaction between $\hat{\mathbf{E} }$ and a transmon current density operator $\hat{\mathbf{J}}_t$. The transmon current density operator is
\begin{align}
\hat{\mathbf{J}}_t = -2e \beta \mathbf{d} \delta(z-z_0) \partial_t\hat{n} ,
\label{eq:transmon-current-operator1}
\end{align}
where $\hat{n}$ is the standard Josephson junction charge operator. We use the awkward notation of $\partial_t^{-1}\hat{\mathbf{J}}_t$ in (\ref{eq:field-transmon-hamiltonian1}) since it will be convenient to use a current density operator when deriving equations of motion in Section \ref{sec:eom}. In (\ref{eq:transmon-current-operator1}), $\mathbf{d}$ is a vector parameterizing the integration path taken to define the voltage of the transmission line at the location of the transmon. Further, the $z$-axis has been identified as the longitudinal direction of the transmission line in the region local to the transmon for notational simplicity.
Before continuing, it is worth commenting on the interpretation of $\hat{\mathbf{J}}_t$. Within the transmon basis, (\ref{eq:transmon-current-operator1}) becomes
\begin{multline}
\hat{\mathbf{J}}_t = -2e\beta\mathbf{d} \delta(z-z_0) \\ \times\sum_j \langle j | \hat{n} | j + 1\rangle \partial_t\big[ |j\rangle\langle j+1| + |j+1\rangle\langle j| \big],
\label{eq:transmon-current-operator3}
\end{multline}
after applying the result that only nearest neighbor states couple for a transmon \cite{koch2007charge}. From this, we see that $\hat{\mathbf{J}}_t$ involves transitions between different transmon eigenstates. These transitions correspond to a high probability event of Cooper pairs tunneling through the Josephson junction. This tunneling produces a current, making the designation of (\ref{eq:transmon-current-operator1}) as a current density reasonable.
Although the physics of (\ref{eq:field-transmon-hamiltonian1}) is fairly intuitive, we will need to develop a number of tools in the following sections to demonstrate its consistency with the circuit-based description of (\ref{eq:coupled-transmon}). This will first require a careful look at the quantization of electromagnetic fields for circuit QED systems in Section \ref{sec:field-quantization}, followed by establishing a correspondence between field and transmission line representations of quantum operators in Section \ref{sec:field-to-circuit}. We will then demonstrate the consistency of (\ref{eq:field-transmon-hamiltonian1}) and (\ref{eq:coupled-transmon}) in Section \ref{sec:field-transmon-hamiltonian}.
\section{Field Quantization for Circuit QED Systems}
\label{sec:field-quantization}
We present two approaches for quantizing the electromagnetic field in systems containing inhomogeneous, lossless, and non-dispersive dielectric and perfectly conducting regions. To simplify the notation, we assume there are no magnetic materials present. The two quantization approaches are relevant for developing different numerical methods. The first approach, discussed in Section \ref{subsec:modes-of-the-universe}, follows a standard mode decomposition quantization process \cite{chew2016quantum2,walls2007quantum,gerry2005introductory,viviescas2003field}. We will refer to this as the \textit{modes-of-the-universe} quantization approach, since the spatial modes used extend across all space \cite{viviescas2003field}. For many numerical modeling approaches, it is convenient to consider a finite-sized \textit{simulation domain} with a number of semi-infinite \textit{port regions} attached to it, as illustrated in Fig. \ref{fig:region-illustration}. This is not compatible with the modes-of-the-universe approach, and so, a different quantization approach will be discussed in Section \ref{subsec:projector-quantization} for this case.
\begin{figure}
\caption{Example of region definitions for a simple circuit QED system composed of a transmon qubit (not to scale) coupled to a coplanar waveguide resonator of length $\ell$. The red regions are fictitious boundaries for the port regions. The dielectric substrate of the system is not shown.}
\label{fig:region-illustration}
\end{figure}
Both quantization methods discussed are performed within the framework of macroscopic QED \cite{scheel2008macroscopic}. The key aspect of this is that a microscopic description of a lossless, non-dispersive dielectric medium is unnecessary. As a result, macroscopic permittivities and permeabilities can be directly used in a quantum description of the electromagnetic fields. Accounting for dispersive or lossy media is more complicated, and is outside the scope of this work \cite{wei2018dissipative,milonni1995field,gruner1996green}.
In some situations, requiring a mode decomposition description can be inconvenient. For instance, this can occur when dealing with certain kinds of nonlinearities. If this is the case, an energy conservation argument can be applied to quantize the electromagnetic field directly in coordinate space \cite{chew2021qme-made-simple}.
\subsection{Modes-of-the-Universe Quantization}
\label{subsec:modes-of-the-universe}
Quantizing the electromagnetic field using a modes-of-the-universe framework is one of the most common quantization approaches. Introductory reviews can be found in \cite{chew2016quantum2,gerry2005introductory,walls2007quantum}. This technique is often simply referred to as a mode decomposition approach. The longer terminology will be used in this work to differentiate it from the quantization approach discussed in Section \ref{subsec:projector-quantization}.
The first step of the modes-of-the-universe approach is to use separation of variables to write the electric field as
\begin{align}
\mathbf{E}(\mathbf{r},t) = \sum_k \sqrt{\frac{\omega_k}{2\epsilon_0}} \big( q_k(t) \mathbf{E}_k(\mathbf{r}) + q_k^*(t) \mathbf{E}^*_k(\mathbf{r}) \big).
\end{align}
For initial simplicity, a discrete summation of modes is assumed. A continuum of modes will be considered as part of the quantization procedure discussed in Section \ref{subsec:projector-quantization}. Inserting this representation into
\begin{align}
\nabla\times\nabla\times\mathbf{E} + \mu\epsilon \partial_t^2 \mathbf{E} = 0
\end{align}
yields two separated equations for each mode, given by
\begin{align}
\partial_t^2 q_k(t) = -\omega_k^2 q_k(t),
\label{eq:mode_time}
\end{align}
\begin{align}
\nabla\times\nabla\times\mathbf{E}_k(\mathbf{r}) - \mu\epsilon \omega_k^2 \mathbf{E}_k(\mathbf{r}) = 0.
\label{eq:field-eig-def}
\end{align}
The complex conjugates of $q_k$ and $\mathbf{E}_k$ also obey (\ref{eq:mode_time}) and (\ref{eq:field-eig-def}), respectively, since $\omega_k$, $\epsilon$, and $\mu$ are all real. Considering (\ref{eq:mode_time}), it is easily seen that the time dependence of these modes will be $\exp(\pm i \omega_k t)$.
To simplify the analysis, it is further required that the field modes are orthonormal such that
\begin{align}
\iiint \epsilon_r(\mathbf{r}) \mathbf{E}^*_{k_1}(\mathbf{r}) \cdot \mathbf{E}_{k_2}(\mathbf{r}) d\mathbf{r} = \delta_{k_1,k_2},
\label{eq:orthonormal-def}
\end{align}
where $\delta_{k_1,k_2}$ is the Kronecker delta function. Later, it will be useful to explicitly consider the normalization of the field modes. Hence, we will often write the modes as
\begin{align}
\mathbf{E}_k = \frac{1}{\sqrt{N_{u,k}}} \mathbf{u}_k(\mathbf{r}),
\end{align}
where
\begin{align}
N_{u,k} = \iiint \epsilon_r(\mathbf{r}) \mathbf{u}_k^*(\mathbf{r}) \cdot \mathbf{u}_k(\mathbf{r}) dV.
\label{eq:original-enorm}
\end{align}
Similarly, the magnetic field can be written as
\begin{align}
\mathbf{H}(\mathbf{r},t) = \sum_k \sqrt{\frac{\omega_k}{2\mu_0}} \big( p_k(t) \mathbf{H}_k(\mathbf{r}) + p_k^*(t) \mathbf{H}^*_k(\mathbf{r}) \big),
\end{align}
where
\begin{align}
\mathbf{H}_k = \frac{1}{\sqrt{N_{v,k}} } \mathbf{v}_k(\mathbf{r})
\end{align}
and the normalization for the magnetic field modes take on a similar form to (\ref{eq:original-enorm}). Although the magnetic field modes have been denoted by seemingly independent variables, i.e., $\mathbf{v}_k$ and $p_k$, they are related to the electric field variables $\mathbf{u}_k$ and $q_k$ according to Maxwell's equations.
It should be noted that these expansions are valid for complex-valued spatial modes. In a closed region (e.g., a cavity), it is often advantageous to use real-valued spatial modes. For this situation, the field expansions become
\begin{align}
\mathbf{E}(\mathbf{r},t) = \sum_k \sqrt{\frac{\omega_k}{2\epsilon_0}} \big( q_k(t) + q_k^*(t) \big) \mathbf{E}_k(\mathbf{r}),
\end{align}
\begin{align}
\mathbf{H}(\mathbf{r},t) = -i\sum_k \sqrt{\frac{\omega_k}{2\mu_0}} \big( p_k(t) - p_k^*(t) \big) \mathbf{H}_k(\mathbf{r}).
\end{align}
It will be useful to use both real- and complex-valued spatial mode functions in this work.
With the mode expansion defined, the Hamiltonian for the electromagnetic field system can be expanded in terms of these modes. The electromagnetic field Hamiltonian is equivalent to the total electromagnetic energy in a system, which is
\begin{align}
H_F = \iiint \frac{1}{2}\big( \epsilon |\mathbf{E}(\mathbf{r},t)|^2 + \mu |\mathbf{H}(\mathbf{r},t)|^2 \big) d\mathbf{r}.
\label{eq:free-field-cH}
\end{align}
Substituting in either the real- or complex-valued mode expansions and performing the spatial integrations, the Hamiltonian simplifies to
\begin{align}
H_F = \sum_k \frac{\omega_k}{2} \big( |q_k(t)|^2 + |p_k(t)|^2 \big).
\end{align}
This can be readily identified as a summation of Hamiltonians for uncoupled harmonic oscillators, or equivalently uncoupled LC resonant circuits \cite{chew2016quantum2}.
Hence, a canonical quantization process can now be performed by elevating the conjugate variables of each harmonic oscillator (i.e., the $q_k$ and $p_k$) to be quantum operators \cite{chew2016quantum2}. These operators obey the canonical commutation relation
\begin{align}
[\hat{q}_{k_1},\hat{p}_{k_2}] = i\hbar \delta_{k_1,k_2} .
\end{align}
These operators may be combined to form bosonic ladder operators for each quantum harmonic oscillator. This gives the annihilation operator as
\begin{align}
\hat{a}_k = \frac{1}{\sqrt{2\hbar}}(\hat{q}_k + i \hat{p}_k)
\label{eq:annihilation}
\end{align}
and the creation operator as
\begin{align}
\hat{a}^\dagger_k = \frac{1}{\sqrt{2\hbar}}(\hat{q}_k - i \hat{p}_k).
\end{align}
These operators satisfy the bosonic commutation relation
\begin{align}
[\hat{a}_{k_1},\hat{a}^\dagger_{k_2}] = \delta_{k_1,k_2} .
\label{eq:boson-commutation}
\end{align}
In terms of ladder operators, the field operators become
\begin{align}
\hat{\mathbf{E}}(\mathbf{r},t) = \sum_k N_{E,k} \big( \hat{a}_k(t)\mathbf{u}_k(\mathbf{r}) + \hat{a}_k^\dagger(t)\mathbf{u}^*_k(\mathbf{r}) \big)
\label{eq:c-q-efield}
\end{align}
\begin{align}
\hat{\mathbf{H}}(\mathbf{r},t) = \sum_k N_{H,k} \big( \hat{a}_k(t)\mathbf{v}_k(\mathbf{r}) + \hat{a}_k^\dagger(t)\mathbf{v}^*_k(\mathbf{r}) \big)
\label{eq:c-q-hfield}
\end{align}
for complex-valued spatial modes. In (\ref{eq:c-q-efield}) and (\ref{eq:c-q-hfield}),
\begin{align}
N_{E,k} = \sqrt{\frac{\hbar \omega_k}{2\epsilon_0 N_{u,k} }}, \,\, N_{H,k} = \sqrt{\frac{\hbar \omega_k}{2\mu_0 N_{v,k} }}.
\label{eq:e-norm}
\end{align}
Similarly, the real-valued spatial mode expansions give
\begin{align}
\hat{\mathbf{E}}(\mathbf{r},t) = \sum_k N_{E,k} \big( \hat{a}_k(t) + \hat{a}_k^\dagger(t) \big) \mathbf{u}_k(\mathbf{r}),
\label{eq:r-q-efield}
\end{align}
\begin{align}
\hat{\mathbf{H}}(\mathbf{r},t) = -i\sum_k N_{H,k} \big( \hat{a}_k(t) - \hat{a}_k^\dagger(t) \big) \mathbf{v}_k(\mathbf{r}).
\label{eq:r-q-hfield}
\end{align}
The quantum field Hamiltonian becomes
\begin{align}
\hat{H}_{F} = \iiint \frac{1}{2} \big( \epsilon \hat{\mathbf{E}}^2 + \mu \hat{\mathbf{H}}^2 \big) d\mathbf{r},
\label{eq:free-field-qH}
\end{align}
which after spatial integration and adjusting the zero point energy can be written in a ``diagonalized'' form in terms of the ladder operators as
\begin{align}
\hat{H}_{F} = \sum_k \hbar \omega_k \hat{a}_k^\dagger \hat{a}_k.
\label{eq:universe-hamiltonian}
\end{align}
As expected, the final Hamiltonian closely matches the isolated circuit portion of the Hamiltonian given in (\ref{eq:coupled-transmon4}).
\subsection{Projector-Based Quantization}
\label{subsec:projector-quantization}
The quantization process in Section \ref{subsec:modes-of-the-universe} is not desirable when port regions like those shown in Fig. \ref{fig:region-illustration} are needed to model a circuit QED system. The issue is that the modes-of-the-universe approach makes no distinction between modes that are associated with ``internal'' dynamics (e.g., that of a transmon coupled to a resonator) and ``external'' modes leaving the device (e.g., modes entering or exiting a device via a port). As a result, a different quantization procedure that allows for the modes in the various regions to be independently worked with is of more interest.
One approach to do this is the Feshbach projector technique as applied to quantum optics \cite{viviescas2003field,viviescas2004quantum}. This approach defines a set of projection operators to isolate the behavior in the various regions of the problem. The eigenvalue problem from the modes-of-the-universe approach is then projected into a set of eigenvalue problems for the various regions being considered. The hermiticity of the projected eigenvalue problems can be maintained by selecting complementary boundary conditions to apply at the interfaces between regions \cite{viviescas2003field}. As a result, a complete set of orthogonal eigenmodes can be found for each region. These various modes can then be quantized and coupled to each other.
For modeling a circuit QED system, a natural decomposition would have one projector cover the simulation domain and another set of complementary projectors cover the infinitely long transmission lines that model ports, as illustrated in Fig. \ref{fig:region-illustration}. For clarity, the region of the simulation domain will be denoted by $\mathcal{Q}$ and the various port regions by $\mathcal{P}_p$. The set of all ports will be denoted by $\mathcal{P}$. The surface at the interface between the simulation domain and port $p$ will be denoted by $\partial \mathcal{Q} \cap \partial\mathcal{P}_p$.
We now present a physically-motivated development of this quantization approach based on an analysis method for open cavities \cite[Ch. 2.9]{haus2012electromagnetic}. This approach also has similarities with using waveports in various computational electromagnetics methods \cite{wang2015higher}. To help guide the discussion, an illustration of the problem setup for this quantization approach is shown in Fig. \ref{fig:projector-quantization-setup}. In Fig. \ref{subfig:projector-quantization-setup1}, the original problem is shown with two reference planes for ports identified. The true fields in all regions of the problem are $\mathbf{E}_T$ and $\mathbf{H}_T$. Now, the regions of the problem are separated by introducing perfect electric conductor (PEC) or perfect magnetic conductor (PMC) boundary conditions in the simulation domain at all port interfaces, as shown in Fig. \ref{subfig:projector-quantization-setup2}. To maintain the hermiticity of the entire problem, complementary conditions are used to close the port region problems \cite{viviescas2003field}. That is, if a PMC condition closes the simulation domain the corresponding port is closed with a PEC condition, as shown in Fig. \ref{subfig:projector-quantization-setup2}.
\begin{figure}
\caption{Illustration of the projector-based quantization setup. In (a) an example two-port problem is shown, while in (b) artificial boundaries and equivalent currents are introduced to separate the regions of the problem.}
\label{fig:projector-quantization-setup}
\end{figure}
The artificial ``closing'' surfaces lead to discontinuities in the electric or magnetic fields at these locations that should not be present from the original problem shown in Fig. \ref{subfig:projector-quantization-setup1}. To produce the correct fields within the simulation domain, equivalent electric or magnetic current densities are introduced at the closing surfaces. For instance, if a PMC condition is applied at a region of the simulation domain (c.f. $\partial\mathcal{Q}\cap\partial\mathcal{P}_1$ in Fig. \ref{subfig:projector-quantization-setup2}), the resulting discontinuity in the magnetic field is compensated with an equivalent electric current density given by $\mathbf{J}_{eq,p} = \hat{n}_p\times \mathbf{H}_T$. Here, $\hat{n}_p$ points into the simulation domain and $\mathbf{H}_T$ should be expanded in terms of the port modes to tie the two problems together \cite{haus2012electromagnetic}. Similarly, to produce the correct fields within the corresponding port region, an equivalent magnetic current density given by $\mathbf{M}_{eq,p} = \hat{n}_p\times \mathbf{E}_T$ must be introduced in the port region (c.f. $\partial\mathcal{Q}\cap\partial\mathcal{P}_1$ in Fig. \ref{subfig:projector-quantization-setup2}). Here, $\mathbf{E}_T$ should be expanded in terms of the simulation domain modes and the unusual sign in the definition of $\mathbf{M}_{eq,p}$ is due to the fixed polarity of the unit normal vector $\hat{n}_p$.
From this physical picture, we see that the interaction between the simulation domain and port regions can be achieved by introducing equivalent current densities. Considering, for now, only closing the simulation domain with PMC conditions, a $\mathbf{J}_{eq,p}$ will need to be introduced at each port in the simulation domain. In Lagrangian/Hamiltonian treatments of electromagnetics, the interaction between a current $\mathbf{J}$ and the field is typically given in terms of the vector potential as $\mathbf{A}\cdot\mathbf{J}$ \cite{chew2016quantum2}. Hence, our resulting Hamiltonian should be
\begin{multline}
H_F = \frac{1}{2}\iiint \big( \epsilon |\mathbf{E}_q|^2 + \mu | \mathbf{H}_q|^2 + \sum_{p \in \mathcal{P}} \big[ \epsilon |\mathbf{E}_p|^2 \\ + \mu | \mathbf{H}_p|^2 \big] - \sum_{p \in \mathcal{P}} 2 \mathbf{A}_q \cdot (\hat{n}_p\times \mathbf{H}_p ) \big) d\mathbf{r},
\label{eq:interacting-system-hamiltonian}
\end{multline}
where a subscript of $q$ ($p$) denotes that the term is associated with the simulation domain (ports). Further, the term $\hat{n}_p \times \mathbf{H}_p$ is an equivalent electric current density (with $\hat{n}_p$ pointing into the simulation domain). The more general case involving both artificial PEC and PMC conditions will be handled in Section \ref{sec:eom}, where it will also be shown that this Hamiltonian produces the correct equations of motion (i.e., Maxwell's equations fed by electric and magnetic current sources).
Inspecting the coupling term in (\ref{eq:interacting-system-hamiltonian}), we see that it has been written from the perspective of treating the port fields as a source to the simulation domain. It is of course possible to also look at the Hamiltonian from the alternative viewpoint that the ports are being fed by a current density. This is done by rearranging the coupling term to be $\mathbf{H}_p\cdot(\mathbf{A}_q\times\hat{n}_p)$, which shows the magnetic field coupling to a term that is proportional to a $\mathbf{M}_{eq,p}$. Although difficult to see at this point, this coupling term will produce the correct form of Maxwell's equations with a $\mathbf{M}_{eq,p}$ acting as a source to the port field equations. This will be shown in Section \ref{sec:eom}.
With the Hamiltonian formulated, the modal expansion of the fields needs to be revisited. By construction of the problem, a complete set of modes can be found in each region to expand the fields in a piecewise manner. Hence, we have that the simulation domain electric field is
\begin{align}
\mathbf{E}_q(\mathbf{r},t) = \sum_k \sqrt{\frac{\omega_k}{2\epsilon_0}} \big( q_k(t) \mathbf{E}_k(\mathbf{r}) + q^*_k(t) \mathbf{E}^*_k(\mathbf{r}) \big)
\label{eq:sim-e-field}
\end{align}
and the port region electric fields are
\begin{multline}
\mathbf{E}_p(\mathbf{r},t) = \sum_\lambda \int_0^\infty d\omega_{\lambda p}\, \sqrt{\frac{\omega_{\lambda p}}{2\epsilon_0}} \big( q_{\lambda p}(\omega_{\lambda p},t) \mathbf{E}_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \\ + q^*_{\lambda p}(\omega_{\lambda p},t) \mathbf{E}^*_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \big).
\label{eq:port-e-field}
\end{multline}
In (\ref{eq:sim-e-field}), the summation over $k$ represents a discrete spectrum of modes with eigenvalue $\omega_k$ for the region $\mathcal{Q}$. In (\ref{eq:port-e-field}), the index $p$ is used to differentiate the different ports in the set $\mathcal{P}$. Each port can support different transverse modes (e.g., transverse electromagnetic or transverse electric), which are differentiated by the discrete index $\lambda$. Due to the semi-infinite length of the port regions, each transverse mode will also support a continuous spectrum. Hence, the integration over the eigenvalue $\omega_{\lambda p}$ can be interpreted as ``continuously summing'' over the one-dimensional continuum of modes for each transverse mode. The overall fields are then the summation of the various mode expansions, i.e., $\mathbf{E} = \mathbf{E}_q + \sum_{p\in\mathcal{P}} \mathbf{E}_p$.
A similar expansion also holds for the magnetic field, i.e., $\mathbf{H} = \mathbf{H}_q + \sum_{p\in\mathcal{P}} \mathbf{H}_p$ where
\begin{align}
\mathbf{H}_q(\mathbf{r},t) = \sum_k \sqrt{\frac{\omega_k}{2\mu_0}} \big( p_k(t) \mathbf{H}_k(\mathbf{r}) + p^*_k(t) \mathbf{H}^*_k(\mathbf{r}) \big),
\end{align}
\begin{multline}
\mathbf{H}_p(\mathbf{r},t) = \sum_\lambda \int_0^\infty d\omega_{\lambda p}\, \sqrt{\frac{\omega_{\lambda p}}{2\mu_0}} \big( p_{\lambda p}(\omega_{\lambda p},t) \mathbf{H}_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \\ + p^*_{\lambda p}(\omega_{\lambda p},t) \mathbf{H}^*_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \big).
\end{multline}
As suggested by the Hamiltonian, an expansion for $\mathbf{A}$ is also necessary. To be consistent with the the expansions of $\mathbf{E}$ and $\mathbf{H}$, the modal expansions for $\mathbf{A}$ are
\begin{align}
\mathbf{A}_q(\mathbf{r},t) = -i\sum_k \sqrt{\frac{1}{2\omega_k\epsilon_0}} \big( p_k(t) \mathbf{E}_k(\mathbf{r}) - p^*_k(t) \mathbf{E}^*_k(\mathbf{r}) \big),
\end{align}
\begin{multline}
\mathbf{A}_p(\mathbf{r},t) = -i\sum_\lambda \int_0^\infty d\omega_{\lambda p}\, \sqrt{\frac{1}{2\omega_{\lambda p}\epsilon_0}} \times \\ \big( p_{\lambda p}(\omega_{\lambda p},t) \mathbf{E}_{\lambda p}(\mathbf{r}, \omega_{\lambda p}) - p^*_{\lambda p}(\omega_{\lambda p},t) \mathbf{E}^*_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \big),
\end{multline}
with $\mathbf{A} = \mathbf{A}_q + \sum_{p\in\mathcal{P}} \mathbf{A}_p$. To simplify the derivation, we use the radiation gauge defined by $\nabla\cdot\epsilon\mathbf{A} = 0, \Phi = 0$ in this work. This gauge is valid when there are either no near-field sources or when only transverse currents are considered, i.e., $\nabla\cdot\mathbf{J}=0$ \cite{jackson1999classical}.
These modal expansions can now be substituted into the Hamiltonian given in (\ref{eq:interacting-system-hamiltonian}). This gives
\begin{align}
H_F = H_\mathcal{Q} + H_\mathcal{P} + H_{\mathcal{QP}},
\label{eq:proj-hamiltonian-sho}
\end{align}
where
\begin{align}
H_\mathcal{Q} = \sum_k \frac{\omega_k}{2}\big( |q_k(t)|^2 + |p_k(t)|^2 \big),
\end{align}
\begin{multline}
H_\mathcal{P} = \sum_{p\in\mathcal{P}} \sum_\lambda \int_0^\infty d\omega_{\lambda p} \, \frac{\omega_{\lambda p} }{2} \big( |q_{\lambda p}(\omega_{\lambda p},t)|^2 \\ + |p_{\lambda p}(\omega_{\lambda p},t)|^2 \big),
\end{multline}
\begin{multline}
H_{\mathcal{QP}} = \sum_{p\in\mathcal{P}} \sum_{\lambda, k} \int_0^\infty d\omega_{\lambda p} \, \big( \mathcal{W}_{k,\lambda p}(\omega_{\lambda p}) p^*_k(t) p_{\lambda p}(\omega_{\lambda p},t) \\ + \mathcal{V}_{k, \lambda p}(\omega_{\lambda p}) p_k(t) p_{\lambda p}(\omega_{\lambda p},t) + \mathrm{H.c.} \big),
\label{eq:proj-hamiltonian-sho2}
\end{multline}
\begin{align}
\mathcal{W}_{k,\lambda p}(\omega_{\lambda p}) = - \mathcal{K}_{k,\lambda p} \int \mathbf{E}^*_k(\mathbf{r}) \cdot \hat{n}_p\!\times\!\nabla\!\times\!\mathbf{E}_{ \lambda p}(\mathbf{r}, \omega_{\lambda p}) dS ,
\label{eq:coupling1}
\end{align}
\begin{align}
\mathcal{V}_{k,\lambda p}(\omega_{\lambda p}) = \mathcal{K}_{k, \lambda p} \int \mathbf{E}_k(\mathbf{r}) \cdot \hat{n}_p\!\times\!\nabla\!\times\!\mathbf{E}_{\lambda p}(\mathbf{r},\omega_{\lambda p}) dS,
\label{eq:coupling2}
\end{align}
\begin{align}
\mathcal{K}_{k,\lambda p} = \frac{c_0^2}{2}\sqrt{\frac{1}{\omega_k\omega_{\lambda p}}}.
\end{align}
In (\ref{eq:coupling1}) and (\ref{eq:coupling2}), the integration surface is $\partial \mathcal{Q} \cap \partial\mathcal{P}_p$ and the magnetic field has been rewritten in terms of the vector potential to more closely match the formulas given in \cite{viviescas2003field}. Inspecting (\ref{eq:proj-hamiltonian-sho}) to (\ref{eq:proj-hamiltonian-sho2}), $H_\mathcal{Q}$ and $H_\mathcal{P}$ represent summations of simple harmonic oscillators for each mode of the fields, while $H_{\mathcal{QP}}$ represents coupling between these oscillators. Further, considering that $\hat{n}_p\!\times\!\nabla\!\times\!\mathbf{E}_{\lambda p}$ is proportional to electric field modes \cite[Ch. 2]{haus2012electromagnetic}, the coupling terms given in (\ref{eq:coupling1}) and (\ref{eq:coupling2}) are proportional to overlap integrals of different spatial mode profiles. These overlap integrals will weight how strongly the harmonic oscillators from the different regions interact, which is physically intuitive from a mode matching perspective.
The form of the Hamiltonian given in (\ref{eq:proj-hamiltonian-sho}) suggests the quantization process \cite{viviescas2003field}. Similar to the modes-of-the-universe case, each mode can be quantized by elevating the harmonic oscillator variables to be quantum operators with equal-time commutation relations
\begin{align}
[\hat{q}_{k_1}(t),\hat{p}_{k_2}(t)] = i\hbar \delta_{k_1,k_2},
\end{align}
\begin{multline}
[\hat{q}_{\lambda_1 p_1}(\omega_{\lambda_1 p_1},t),\hat{p}_{\lambda_2 p_2}(\omega_{\lambda_2 p_2}',t)] = i\hbar \delta_{\lambda_1, \lambda_2} \delta_{p_1,p_2} \\ \times \delta(\omega_{\lambda_1 p_1}-\omega_{\lambda_2 p_2}').
\end{multline}
In addition to these commutation relations, the operators from different regions commute with each other.
Now, bosonic ladder operators can be introduced for each mode similar to (\ref{eq:annihilation}) to (\ref{eq:boson-commutation}). Using these, the total electric field operator is $\hat{\mathbf{E}} = \hat{\mathbf{E}}_q + \sum_{p\in\mathcal{P}}\hat{\mathbf{E}}_p$ where
\begin{align}
\hat{\mathbf{E}}_q(\mathbf{r},t) = \sum_k N_{E,k} \big( \hat{a}_k(t) \mathbf{u}_k(\mathbf{r}) + \hat{a}^\dagger_k(t) \mathbf{u}^*_k(\mathbf{r}) \big),
\label{eq:q-sim-e-field}
\end{align}
\begin{multline}
\hat{\mathbf{E}}_p(\mathbf{r},t) = \sum_\lambda \int_0^\infty d\omega_{\lambda p} \, N_{E,\lambda p} \big( \hat{a}_{\lambda p}(\omega_{\lambda p},t) \mathbf{u}_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \\ + \hat{a}^\dagger_{\lambda p}(\omega_{\lambda p},t) \mathbf{u}^*_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \big).
\label{eq:q-port-e-field}
\end{multline}
A similar expansion holds for the magnetic field operator, i.e., $\hat{\mathbf{H}} = \hat{\mathbf{H}}_q + \sum_{p\in\mathcal{P}}\hat{\mathbf{H}}_p$ where
\begin{align}
\hat{\mathbf{H}}_q(\mathbf{r},t) = \sum_k N_{H,k} \big( \hat{a}_k(t) \mathbf{v}_k(\mathbf{r}) + \hat{a}^\dagger_k(t) \mathbf{v}^*_k(\mathbf{r}) \big),
\label{eq:q-sim-h-field}
\end{align}
\begin{multline}
\hat{\mathbf{H}}_p(\mathbf{r},t) = \sum_\lambda \int_0^\infty d\omega_p \, N_{H,\lambda p} \big( \hat{a}_{\lambda p}(\omega_{\lambda p},t) \mathbf{v}_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \\ + \hat{a}^\dagger_{\lambda p}(\omega_{\lambda p},t) \mathbf{v}^*_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \big) .
\label{eq:q-port-h-field}
\end{multline}
Further, we have that $\hat{\mathbf{A}} = \hat{\mathbf{A}}_q + \sum_{p\in\mathcal{P}}\hat{\mathbf{A}}_p$ where
\begin{align}
\hat{\mathbf{A}}_q(\mathbf{r},t) = -i\sum_k N_{A,k} \big( \hat{a}_k(t) \mathbf{u}_k(\mathbf{r}) - \hat{a}^\dagger_k(t) \mathbf{u}^*_k(\mathbf{r}) \big),
\label{eq:q-sim-a-field}
\end{align}
\begin{multline}
\hat{\mathbf{A}}_p(\mathbf{r},t)\! = \!-i\sum_\lambda \int_0^\infty d\omega_{\lambda p } \, N_{A,\lambda p} \big( \hat{a}_{\lambda p}(\omega_{\lambda p},t) \mathbf{u}_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \\ - \hat{a}^\dagger_{\lambda p}(\omega_{\lambda p},t) \mathbf{u}^*_{\lambda p}(\mathbf{r},\omega_{\lambda p}) \big),
\label{eq:q-port-a-field}
\end{multline}
\begin{align}
N_{A,k(\lambda p)} = \sqrt{\frac{\hbar }{2\epsilon_0 \omega_{k(\lambda p)} N_{E,k(\lambda p)} }}.
\end{align}
In (\ref{eq:q-sim-e-field}) to (\ref{eq:q-port-a-field}), the modal representations of $\mathbf{u}_k$, $\mathbf{u}_p$, $\mathbf{v}_k$, and $\mathbf{v}_p$ have been used to allow the modal normalizations to be explicitly included in the field operators. This is useful when establishing a correspondence between Hamiltonians written in terms of field and transmission line operators.
The corresponding Hamiltonian is
\begin{multline}
\hat{H}_F = \frac{1}{2}\iiint \big( \epsilon \hat{\mathbf{E}}^2_q + \mu \hat{\mathbf{H}}^2_q + \sum_{p \in \mathcal{P}} \big[ \epsilon \hat{\mathbf{E}}^2_p + \mu \hat{\mathbf{H}}^2_p \big] \\ - \sum_{p \in \mathcal{P}} 2 \hat{\mathbf{A}}_q \cdot ( \hat{n}_p \times \hat{\mathbf{H}}_p ) \big) d\mathbf{r}.
\label{eq:q-interacting-system-hamiltonian}
\end{multline}
This can be expressed using bosonic ladder operators as
\begin{align}
\hat{H}_F = \hat{H}_\mathcal{Q} + \hat{H}_\mathcal{P} + \hat{H}_{\mathcal{Q}\mathcal{P}},
\label{eq:q-interacting-system-hamiltonian2}
\end{align}
where
\begin{align}
\hat{H}_\mathcal{Q} = \sum_k \hbar \omega_k \hat{a}^\dagger_k (t) \hat{a}_k(t)
\end{align}
\begin{align}
\hat{H}_\mathcal{P} = \sum_{p\in \mathcal{P}} \sum_\lambda \int_0^\infty d\omega_{\lambda p} \, \hbar \omega_{\lambda p} \hat{a}^\dagger_{\lambda p}(\omega_{\lambda p},t) \hat{a}_{\lambda p}(\omega_{\lambda p},t)
\end{align}
\begin{multline}
\hat{H}_{\mathcal{Q}\mathcal{P}} = \sum_{p\in \mathcal{P}} \sum_{\lambda, k} \int_0^\infty d\omega_{\lambda p} \big( \mathcal{W}_{k,\lambda p}(\omega_{\lambda p}) \hat{a}^\dagger_k(t) \hat{a}_{\lambda p}(\omega_{\lambda p},t) \\ + \mathcal{V}_{k,\lambda p}(\omega_{\lambda p}) \hat{a}_k(t) \hat{a}_{\lambda p}(\omega_{\lambda p},t) + \mathrm{H.c.} \big).
\end{multline}
The Hamiltonian given in (\ref{eq:q-interacting-system-hamiltonian2}) can be recognized as the system-and-bath Hamiltonian that is commonly used in quantum optics \cite{viviescas2003field,gardiner2004quantum}. This is a satisfying result, since this Hamiltonian is often used to study the input-output relationship of optical cavities \cite{walls2007quantum}.
\section{Correspondence Between Field and Transmission Line Hamiltonians}
\label{sec:field-to-circuit}
With an appropriate quantization procedure now in place, we can continue the process of developing a field-based description of circuit QED systems. To assist in this, it will be useful to determine a correspondence between the field-based Hamiltonian of (\ref{eq:free-field-qH}) or (\ref{eq:q-interacting-system-hamiltonian}) and a Hamiltonian consisting of transmission line voltages and currents. To simplify this process, the classical case from the modes-of-the-universe approach; i.e., (\ref{eq:free-field-cH}), is considered first.
Since our goal is to reduce our expressions to a form like the circuit-based description given in (\ref{eq:coupled-transmon}), we need to introduce some assumptions implicit in (\ref{eq:coupled-transmon}) for the manipulations in this section and Section \ref{sec:field-transmon-hamiltonian} to work. However, we emphasize that these approximations are only needed to show consistency with the approximate expressions used in the literature. They are not needed in the construction of our general field-based equations provided up to this point.
Now, the first assumption is that our transmission line geometry and operating frequencies are such that only quasi-TEM modes are excited. For simplicity, these modes will be treated as pure TEM modes for the purposes of defining transmission line parameters such as voltages, currents, and per-unit-length impedances. Second, we will assume in this section that we are only considering a finite length transmission line with a constant cross-section and that fringing effects at the end of the transmission line can be accounted for separately (e.g., by shifting mode frequencies). For notational simplicity, the longitudinal direction of the transmission line will be aligned with the $z$-axis. Considering these simplifications, we can arrive at an ``exact'' correspondence between field-based and transmission line-based descriptions in this section.
To begin, we need to revisit the expansion of the electric and magnetic fields in terms of modes. Since we are dealing with TEM waves, we can decompose the electric field as
\begin{multline}
\mathbf{E}(\mathbf{r},t) = \sum_{k,l} \sqrt{\frac{\omega_{k,l}}{2\epsilon_0 N_{E_{T,k}} N_{E_{L,l}} }} \\ \times \big( q_{k,l}(t) \mathbf{u}_{k,l}(\mathbf{r}) + q^*_{k,l}(t) \mathbf{u}^*_{k,l}(\mathbf{r}) \big)
\label{eq:emode}
\end{multline}
where the mode functions $\mathbf{u}_{k,l}$ are split into a transverse vector function and a longitudinal scalar function as
\begin{align}
\mathbf{u}_{k,l}(\mathbf{r}) = \mathbf{u}_{T,k}(x,y) u_{L,l}(z).
\end{align}
These mode functions are orthogonal in the sense that
\begin{align}
\iint \epsilon_r\mathbf{u}_{T,k_1}^* \cdot \mathbf{u}_{T,k_2} dxdy = \delta_{k_1,k_2} N_{E_{T,k_1}},
\label{eq:norm1}
\end{align}
\begin{align}
\int u_{L,l_1}^*(z) u_{L,l_2}(z) dz = \delta_{l_1,l_2} N_{E_{L,l_1}}.
\label{eq:norm2}
\end{align}
Similarly, for the magnetic field we have that
\begin{multline}
\mathbf{H}(\mathbf{r},t) = \sum_{k,l} \sqrt{\frac{\omega_{k,l}}{2\mu_0 N_{H_{T,k}} N_{H_{L,l}} }} \\ \times \big( p_{k,l}(t) \mathbf{v}_{k,l}(\mathbf{r}) + p^*_{k,l}(t) \mathbf{v}^*_{k,l}(\mathbf{r}) \big)
\end{multline}
where the mode functions $\mathbf{v}_{k,l}$ are
\begin{align}
\mathbf{v}_{k,l}(\mathbf{r}) = \mathbf{v}_{T,k}(x,y) v_{L,l}(z).
\end{align}
These mode functions are orthogonal in a similar sense to (\ref{eq:norm1}) and (\ref{eq:norm2}).
The conversion between fields and transmission line quantities can be performed by adopting definitions for the transmission line parameters so that the power and energy densities expressed in terms of field and circuit parameters agree \cite{pozar2009microwave}. Since we consider lossless lines here, only the per-unit-length capacitance and inductance of the line are needed. For a particular transmission line mode, these are denoted as $C_k$ and $L_k$, respectively. Within the notation of this work, the definitions for these become $C_k = \epsilon_0 N_{E_{T,k}}$ and $L_k = \mu_0 N_{H_{T,k}}$ \cite{pozar2009microwave}.
With these definitions, the field-Hamiltonian of (\ref{eq:free-field-cH}) can now be converted into a transmission line form. The electric field term will be considered first. Substituting in the modal expansion, this term becomes
\begin{multline}
\epsilon |\mathbf{E}(\mathbf{r},t)|^2 = \epsilon \! \sum_{k_1,l_1,k_2,l_2}\!\!\sqrt{\frac{\omega_{k_1,l_1}\omega_{k_2,l_2}}{\epsilon^2_0 N_{E_{T,k_1}} N_{E_{L,l_1}} N_{E_{T,k_2}} N_{E_{L,l_2}} } } \\ \times \big( q^*_{k_2,l_2}(t) q_{k_1,l_1}(t) \mathbf{u}^*_{k_2,l_2}(\mathbf{r}) \cdot \mathbf{u}_{k_1,l_1}(\mathbf{r}) \\ + \frac{1}{2} q_{k_2,l_2}(t)q_{k_1,l_1}(t) \mathbf{u}_{k_2,l_2}(\mathbf{r}) \cdot \mathbf{u}_{k_1,l_1}(\mathbf{r}) \\ + \frac{1}{2} q^*_{k_2,l_2}(t)q^*_{k_1,l_1}(t) \mathbf{u}^*_{k_2,l_2}(\mathbf{r}) \cdot \mathbf{u}^*_{k_1,l_1}(\mathbf{r}) \big) .
\label{eq:e1}
\end{multline}
Recalling that the Hamiltonian involves the volume integral of these terms, we can simplify our expression by noting that the terms proportional to $\mathbf{u}_{k_2,l_2}\cdot\mathbf{u}_{k_1,l_1}$ and $(\mathbf{u}_{k_2,l_2}\cdot\mathbf{u}_{k_1,l_1})^*$ will average to zero unless $k_1=k_2$ and $l_1=l_2$. We then also have that for harmonic oscillators, terms of the form $q_{k,l}^2$ and $(q_{k,l}^*)^2$ vanish \cite{haken1976quantum}. Hence, we have that
\begin{multline}
\!\!\iiint \epsilon |\mathbf{E}(\mathbf{r},t)|^2 d\mathbf{r} = \\ \!\!\! \sum_{k_1,l_1,k_2,l_2} \sqrt{\frac{\omega_{k_1,l_1}\omega_{k_2,l_2}}{\epsilon^2_0 N_{E_{T,k_1}} N_{E_{L,l_1}} N_{E_{T,k_2}} N_{E_{L,l_2}} } } \\ \times q^*_{k_2,l_2}(t) q_{k_1,l_1}(t) \iiint \epsilon \, \mathbf{u}^*_{k_2,l_2}(\mathbf{r}) \cdot \mathbf{u}_{k_1,l_1}(\mathbf{r}).
\end{multline}
We can expand the $\mathbf{u}_{k,l}$ functions and use the orthogonality relationships given in (\ref{eq:norm1}) and (\ref{eq:norm2}) to get
\begin{multline}
\iiint \epsilon |\mathbf{E}(\mathbf{r},t)|^2d\mathbf{r} \\ = \int \sum_{k,l} C_k \frac{\omega_{k,l}}{C_k N_{E_{L,l}}} |q_{k,l}(t)|^2 |u_{L,l}(z)|^2 dz,
\label{eq:int1}
\end{multline}
where we have also noted that $C_k = \epsilon_0 N_{E_{T,k}}$. We don't cancel the $C_k$ terms in (\ref{eq:int1}) because it helps suggest the correct form of voltage mode to convert between the electric field and transmission line voltage descriptions of the system.
In particular, we can define a voltage mode to be
\begin{align}
V_{k,l}(z,t) = \sqrt{\frac{\omega_{k,l}}{ C_k N_{E_{L,l}}}} \big(q_{k,l}(t) u_{L,l}(z) + q_{k,l}^*(t) u_{L,l}^*(z) \big).
\end{align}
Following a similar set of steps to those shown in (\ref{eq:e1}) to (\ref{eq:int1}), we can see that
\begin{align}
\iiint \epsilon |\mathbf{E}(\mathbf{r},t)|^2d\mathbf{r} = \int \sum_{k,l} C_k |V_{k,l}(z,t)|^2 dz.
\label{eq:eint}
\end{align}
Similarly, We can define a current mode as
\begin{align}
I_{k,l}(z,t) = \sqrt{\frac{\omega_{k,l}}{ L_k N_{H_{L,l}}}} \big(p_{k,l}(t) v_{L,l}(z) + p_{k,l}^*(t) v_{L,l}^*(z) \big)
\end{align}
to see that
\begin{align}
\iiint \mu |\mathbf{H}(\mathbf{r},t)|^2d\mathbf{r} = \int \sum_{k,l} L_k |I_{k,l}(z,t)|^2 dz.
\label{eq:hint}
\end{align}
Combining the results in (\ref{eq:eint}) and (\ref{eq:hint}), the transmission line Hamiltonian can be written as
\begin{align}
H_{TR} = \int \frac{1}{2} \sum_{k,l} \big( C_k |V_{k,l}(z,t)|^2 + L_k|I_{k,l}(z,t)|^2 \big)dz,
\end{align}
which is equivalent to the field-based Hamiltonian. This equality is of course only valid when considering a portion of a system with a constant transmission line cross-section, as mentioned at the beginning of this section. However, this is exactly the part of a system used in writing a circuit-based Hamiltonian like (\ref{eq:coupled-transmon}). Hence, we now have the tools to relate a field-based Hamiltonian to the simpler circuit descriptions often used in the literature.
Moving now to the quantum case, it is important to note that all the operations used in the classical case still apply. Hence, the process easily generalizes to the quantum case, giving
\begin{align}
\hat{H}_{F} = \hat{H}_{TR} = \int \frac{1}{2} \sum_{k,l} \big( C_k \hat{V}_{k,l}^2 + L_k \hat{I}_{k,l}^2 \big) dz,
\end{align}
where
\begin{align}
\hat{V}_{k,l}(z,t) = N_{V_{k,l}} \big( \hat{a}_{k,l}(t) u_{L,l}(z) + \hat{a}_{k,l}^\dagger(t)u^*_{L,l}(z) \big),
\label{eq:c-q-voltage}
\end{align}
\begin{align}
\hat{I}_{k,l}(z,t) = N_{I_{k,l}} \big( \hat{a}_{k,l}(t)v_{L,l}(z) + \hat{a}_{k,l}^\dagger(t)v^*_{L,l}(z) \big) ,
\label{eq:c-q-current}
\end{align}
\begin{align}
N_{V_{k,l}} = \sqrt{ \frac{\hbar \omega_{k,l}}{2 C_k N_{E_{L,l}}} }, \,\, N_{I_{k,l}} = \sqrt{ \frac{\hbar \omega_{k,l}}{2 L_k N_{H_{L,l}}} }.
\label{eq:sim-domain-mode-norm}
\end{align}
These definitions for $\hat{V}_{k,l}$ and $\hat{I}_{k,l}$ are not immediately seen to be consistent with those given in (\ref{eq:total-v}) and (\ref{eq:total-i}). The reason is that complex-valued spatial modes have been used in (\ref{eq:c-q-voltage}) and (\ref{eq:c-q-current}), while real-valued modes were used in (\ref{eq:total-v}) and (\ref{eq:total-i}). Further, the spatial variation of the voltage and current modes have been completely integrated out for (\ref{eq:total-v}) and (\ref{eq:total-i}).
To see that the expressions derived in this section are consistent with the literature, the steps outlined in this section can be repeated for the real-valued spatial mode expansions like those given in (\ref{eq:r-q-efield}) and (\ref{eq:r-q-hfield}). For this case, the voltage and current operators become
\begin{align}
\hat{V}_{k,l}(z,t) = N_{V_{k,l}} \big( \hat{a}_{k,l}(t) + \hat{a}_{k,l}^\dagger(t) \big) u_{L,l}(z)
\label{eq:r-q-voltage}
\end{align}
\begin{align}
\hat{I}_{k,l}(z,t) = -i N_{I_{k,l}} \big( \hat{a}_{k,l}(t) - \hat{a}_{k,l}^\dagger(t) \big) v_{L,l}(z).
\label{eq:r-q-current}
\end{align}
These can be seen to be consistent with (\ref{eq:total-v}) and (\ref{eq:total-i}) by recalling that $C_r = C\ell$ and $L_r = L\ell$, where $\ell$ is the length of the resonator \cite{blais2004cavity}. Restricting (\ref{eq:r-q-voltage}) and (\ref{eq:r-q-current}) to be a resonator mode, it can be seen that the longitudinal normalizations $N_{E_{L,k}}$ and $N_{H_{L,k}}$ would become $\ell$. The additional factors that occur after integrating out the remaining spatial variation in (\ref{eq:r-q-voltage}) and (\ref{eq:r-q-current}) are grouped with other terms in the overall Hamiltonian to be consistent with \cite{blais2004cavity}.
With the basic process developed, the expressions from the projector-based quantization approach, e.g., (\ref{eq:q-interacting-system-hamiltonian}), can now be converted into a transmission line form as well. This will not be performed here for brevity. However, it should be noted that the interacting part of the Hamiltonian given in (\ref{eq:q-interacting-system-hamiltonian}) cannot be written in a simpler transmission line form. This is because the transverse integrations cannot be concisely written when allowing for complex-valued mode functions. If real-valued mode functions are used, the spatial integrals can be shown to be proportional to overlap integrals of the electric field modes for the different regions of the problem, which is an intuitively satisfying result.
\section{Hamiltonian for the Field-Transmon System}
\label{sec:field-transmon-hamiltonian}
With the ability to convert between field and transmission line representations, it is now possible to show the consistency of the postulated field-based Hamiltonian of (\ref{eq:field-transmon-hamiltonian1}) with the more common circuit-based description given in (\ref{eq:coupled-transmon}). To do this, we must match the assumptions implicit in (\ref{eq:coupled-transmon}), which correspond to only considering the unperturbed TEM modes of a transmission line resonator that interact with the transmon qubit. This approximately corresponds to only considering the portion of Fig. \ref{fig:region-illustration} marked with length $\ell$. To simplify the expressions, only fields from the simulation domain will be considered as these are what interact with the transmon. The subscript of $q$ will be dropped from these fields in this section. Further, all of the simulation domain mode functions will be assumed to be real-valued. Since the simulation domain can typically be made a closed system, this choice does not amount to a loss of generality \cite{chew2016quantum2}. The complex-valued mode function case can be handled easily, but leads to unnecessarily long expressions that are omitted for brevity.
Considering these points, the postulated field-transmon system Hamiltonian of (\ref{eq:field-transmon-hamiltonian1}) is reproduced here in full for ease of reference as
\begin{multline}
\hat{H} = 4E_C (\hat{n}-n_g)^2 - E_J \cos \hat{\varphi} \\ + \iiint \frac{1}{2} \big( \epsilon \hat{\mathbf{E}}^2 + \mu \hat{\mathbf{H}}^2 - 2 \hat{\mathbf{E}} \cdot \partial_t^{-1} \hat{\mathbf{J}}_t \big)d\mathbf{r},
\label{eq:field-transmon-hamiltonian}
\end{multline}
where the transmon current density operator was
\begin{align}
\hat{\mathbf{J}}_t = -2e \beta \mathbf{d} \delta(z-z_0) \partial_t\hat{n} .
\label{eq:transmon-current-operator}
\end{align}
The field-based Hamiltonian of (\ref{eq:field-transmon-hamiltonian}) can be shown to be equivalent to circuit-based Hamiltonians by evaluating the spatial integration in (\ref{eq:field-transmon-hamiltonian}). The first two terms can be simply evaluated following the steps in Section \ref{sec:field-to-circuit}, yielding
\begin{multline}
\iiint \frac{1}{2} \big( \epsilon \hat{\mathbf{E}}^2 + \mu \hat{\mathbf{H}}^2 \big) d\mathbf{r} \\ = \frac{1}{2}\sum_{k,l} \big( C_k N_{E_{L,l}} \hat{V}^2_{I_{k,l}} + L_k N_{H_{L,l}} \hat{I}^2_{I_{k,l}} \big),
\label{eq:f-to-tr1}
\end{multline}
where the integrated voltage and current operators are
\begin{align}
\hat{V}_{I_{k,l}} = N_{V_{k,l}} \big( \hat{a}_{k,l}(t) + \hat{a}_{k,l}^\dagger(t) \big)
\label{eq:q-total-voltage-mode}
\end{align}
\begin{align}
\hat{I}_{I_{k,l}} = -i N_{I_{k,l}} \big( \hat{a}_{k,l}(t) - \hat{a}_{k,l}^\dagger(t) \big).
\label{eq:q-total-current-mode}
\end{align}
The remaining term to consider is the coupling term. To carry out the spatial integration, the expression is expanded in terms of the modes for $\hat{\mathbf{E}}$ given in (\ref{eq:emode}). This gives
\begin{multline}
-\iiint \hat{\mathbf{E}}(\mathbf{r},t) \cdot \partial_t^{-1} \hat{\mathbf{J}}_t(\mathbf{r},t) d\mathbf{r} =
\sum_{k,l} N_{E_{k,l}} \big( \hat{a}_{k,l}(t) + \hat{a}^\dagger_{k,l}(t) \big) \\ \times 2e \beta \hat{n}(t) \iiint \mathbf{u}_{k,l}(\mathbf{r}) \cdot \mathbf{d}(x,y) \delta(z-z_0) d\mathbf{r},
\end{multline}
where
\begin{align}
N_{E_{k,l}} = \sqrt{ \frac{\hbar \omega_{k,l}}{2 \epsilon_0 N_{E_{T,k}} N_{E_{L,l}}} }.
\label{eq:enorm}
\end{align}
Now, the spatial integral along the $z$-axis can be evaluated easily and the remaining transverse integral can be identified as the definition of a voltage, i.e.,
\begin{align}
V_{k,l}(z_0) = \int_{a}^{b} \mathbf{u}_{k,l}(x,y,z_0) \cdot d \mathbf{d}(x,y),
\end{align}
where $a$ and $b$ are the initial and final points of the integration path defined by $\mathbf{d}$ \cite{pozar2009microwave}. Hence, we have that
\begin{multline}
-\iiint \hat{\mathbf{E}}(\mathbf{r},t) \cdot \partial_t^{-1} \hat{\mathbf{J}}_t(\mathbf{r},t) d\mathbf{r} \\ =
2e \beta \hat{n}(t) \sum_{k,l} N_{V_{k,l}} \big( \hat{a}_{k,l}(t) + \hat{a}^\dagger_{k,l}(t) \big) V_{k,l}(z_0) ,
\label{eq:int3}
\end{multline}
where we have rewritten $N_{E_{k,l}}$ as $N_{V_{k,l}}$ by noting the relationship between the transverse normalization in (\ref{eq:enorm}) and the per-unit-length modal capacitance in (\ref{eq:sim-domain-mode-norm}). Finally, we can use (\ref{eq:q-total-voltage-mode}) to get
\begin{multline}
-\iiint \hat{\mathbf{E}}(\mathbf{r},t) \!\cdot\! \partial_t^{-1} \hat{\mathbf{J}}_t(\mathbf{r},t) d\mathbf{r} \\ = \sum_{k,l} 2e \beta V_{k,l}(z_0) \hat{V}_{I_{k,l}}(t) \hat{n}(t).
\label{eq:coupling-derivation}
\end{multline}
Putting the results of (\ref{eq:f-to-tr1}) and (\ref{eq:coupling-derivation}) together, the field-transmon system Hamiltonian of (\ref{eq:field-transmon-hamiltonian}), can now be written as
\begin{multline}
\hat{H} = 4E_C (\hat{n}-n_g)^2 - E_J \cos \hat{\varphi} + \frac{1}{2}\sum_{k,l} \big( C_k N_{E_{L,l}} \hat{V}^2_{I_{k,l}} \\ + L_k N_{H_{L,l}} \hat{I}^2_{I_{k,l}} \big) + \sum_{k,l} 2e \beta V_{k,l}(z_0) \hat{V}_{I_{k,l}} \hat{n}.
\end{multline}
Restricting this Hamiltonian to only consider a single mode of a resonator coupled to the transmon recovers (\ref{eq:coupled-transmon}). Hence, the postulated field-transmon system Hamiltonian can be seen to be consistent with the circuit-based descriptions of circuit QED systems typically used in the literature.
\section{Equations of Motion}
\label{sec:eom}
Now that an appropriate Hamiltonian has been found for the field-transmon system, the quantum equations of motion can be derived using Hamilton's equations \cite{chew2016quantum}. We will consider the full system composed of the transmon qubit, the simulation domain fields, and the port region fields. The Hamiltonian is also generalized by allowing for the termination of the port regions in the simulation domain with either PEC or PMC conditions.
With this understood, the complete Hamiltonian becomes
\begin{align}
\hat{H} = \hat{H}_T + \hat{H}_F + \hat{H}_I ,
\label{eq:quantum-hamiltonian}
\end{align}
where
\begin{align}
\hat{H}_T = 4E_C (\hat{n}-n_g)^2 - E_J \cos \hat{\varphi},
\label{eq:transmon-hamiltonian}
\end{align}
\begin{align}
\hat{H}_F = \frac{1}{2}\iiint \big( \epsilon \hat{\mathbf{E}}^2_q + \mu \hat{\mathbf{H}}^2_q + \sum_{p \in \mathcal{P}} \big[ \epsilon \hat{\mathbf{E}}^2_p + \mu \hat{\mathbf{H}}^2_p \big] \big) d\mathbf{r} ,
\label{eq:field-hamiltonian}
\end{align}
\begin{multline}
\hat{H}_I = - \iiint \big( \hat{\mathbf{E}}_q \cdot \partial_t^{-1}\hat{\mathbf{J}}_t + \sum_{p \in \mathcal{P}_M} \hat{\mathbf{A}}_q \cdot (\hat{n}_p\times \hat{\mathbf{H}}_p) \\ + \sum_{p \in \mathcal{P}_E} \hat{\mathbf{F}}_q \cdot (\hat{\mathbf{E}}_p \times \hat{n}_p) \big) d\mathbf{r}.
\label{eq:interaction-hamiltonian}
\end{multline}
In (\ref{eq:interaction-hamiltonian}), $\mathcal{P}_E$ ($\mathcal{P}_M$) denotes the set of ports terminated in a PEC (PMC) \textit{in the simulation domain}. The union of these sets is all of the ports $\mathcal{P}$. Note that $\hat{n}_p$ is the unit normal vector to the port surface, and it points into the simulation domain. The terms in (\ref{eq:interaction-hamiltonian}) quantify the interactions between the different parts of the total system. The first term is the coupling of the simulation domain field and transmon system, while the next two terms represent coupling between the simulation domain and port region fields.
In (\ref{eq:interaction-hamiltonian}), the electric vector potential, $\hat{\mathbf{F}}_q$, has been introduced into the Hamiltonian. This is necessary because of the set of ports $\mathcal{P}_E$ that introduce equivalent magnetic current densities as sources to the simulation domain. The simplest way to account for the presence of magnetic sources is to introduce another set of auxiliary potentials, as is commonly done in classical electromagnetics \cite{jin2011theory}.
To derive equations of motion, the Hamiltonian needs to be expressed in terms of canonical conjugate operators \cite{chew2016quantum}. The transmon operators $\hat{n}$ and $\hat{\varphi}$ are already in this form. However, the electric and magnetic fields are not canonical conjugate operators in a Hamiltonian mechanics formalism \cite{chew2016quantum}. Instead, the electromagnetic field portions of the Hamiltonian need to be rewritten in terms of the electric and magnetic vector potentials and their conjugate momenta.
To support this, the electric and magnetic fields are first decomposed into the set of fields produced by electric or magnetic sources. Under this decomposition, the fields produced by electric (magnetic) sources are completely specified by the magnetic (electric) vector potential \cite{jin2011theory}. For this to hold, a radiation gauge is being used for both the magnetic and electric vector potentials. Considering this, the field portions of the Hamiltonian are first rewritten as
\begin{multline}
\hat{H}_F = \frac{1}{2} \iiint \big( \epsilon\hat{\mathbf{E}}_{qe}^2 + \mu\hat{\mathbf{H}}_{qe}^2 + \epsilon\hat{\mathbf{E}}_{qm}^2 + \mu\hat{\mathbf{H}}_{qm}^2 \\ + \sum_{p \in \mathcal{P}} \big[ \epsilon \hat{\mathbf{E}}^2_{pe} + \mu \hat{\mathbf{H}}^2_{pe} + \epsilon \hat{\mathbf{E}}^2_{pm} + \mu \hat{\mathbf{H}}^2_{pm}\big] \big) d\mathbf{r} ,
\end{multline}
\begin{multline}
\hat{H}_I = -\iiint \big( \hat{\mathbf{E}}_{qe} \cdot \partial_t^{-1} \hat{\mathbf{J}}_t + \sum_{p \in \mathcal{P}_M} \hat{\mathbf{A}}_{qe} \cdot (\hat{n}_p\times \hat{\mathbf{H}}_{pm}) \\ + \sum_{p \in \mathcal{P}_E} \hat{\mathbf{F}}_{qm} \cdot (\hat{\mathbf{E}}_{pe} \times \hat{n}_{p}) \big) d\mathbf{r},
\end{multline}
where a subscript $e$ ($m$) denotes that this quantity is due to electric (magnetic) sources. The structure of the coupling terms between the fields in different regions reflects the difference in boundary conditions and corresponding equivalent source densities at the interfaces between regions.
With the Hamiltonian decomposed into portions due to electric and magnetic sources, it can now be written in terms of the electric and magnetic vector potentials and their conjugate momenta. This gives
\begin{multline}
\hat{H}_F = \frac{1}{2} \iiint \big( \epsilon^{-1}\hat{\mathbf{\Pi}}_{qe}^2 + \mu^{-1}(\nabla\times\hat{\mathbf{A}}_{qe})^2 + \mu^{-1}\hat{\mathbf{\Pi}}_{qm}^2 \\ + \epsilon^{-1}(\nabla\times\hat{\mathbf{F}}_{qm})^2 + \sum_{p \in \mathcal{P}} \big[ \epsilon^{-1}\hat{\mathbf{\Pi}}_{pe}^2 + \mu^{-1}(\nabla\times\hat{\mathbf{A}}_{pe})^2 \\ + \mu^{-1}\hat{\mathbf{\Pi}}_{pm}^2 + \epsilon^{-1}(\nabla\times\hat{\mathbf{F}}_{pm})^2\big] \big) d\mathbf{r},
\end{multline}
\begin{multline}
\hat{H}_I = \iiint \big( \epsilon^{-1}\hat{\mathbf{\Pi}}_{qe} \cdot \partial_t^{-1} \hat{\mathbf{J}}_t - \! \sum_{p \in \mathcal{P}_M} \!\!\mu^{-1} \hat{\mathbf{A}}_{qe} \!\cdot\! (\hat{n}_p \! \times \! \hat{\mathbf{\Pi}}_{pm}) \\ - \sum_{p \in \mathcal{P}_E} \epsilon^{-1} \hat{\mathbf{F}}_{qm} \cdot (\hat{\mathbf{\Pi}}_{pe} \times \hat{n}_{p}) \big) d\mathbf{r},
\end{multline}
where $\hat{\mathbf{\Pi}}_{qe} = \epsilon\partial_t \hat{\mathbf{A}}_{qe}$ is the conjugate momentum for the vector potential in the simulation domain. Similarly, $\hat{\mathbf{\Pi}}_{pe} = \epsilon\partial_t \hat{\mathbf{A}}_{pe}$ is the conjugate momentum for the vector potential in the port regions. The conjugate momenta for the electric vector potentials are $\hat{\mathbf{\Pi}}_{qm} = \mu \partial_t \hat{\mathbf{F}}_{qm}$ and $\hat{\mathbf{\Pi}}_{pm} = \mu \partial_t \hat{\mathbf{F}}_{pm}$ for the simulation domain and port regions, respectively.
With the Hamiltonian now written completely in terms of canonical conjugate operators, equations of motion can be derived using Hamilton's equations \cite{chew2016quantum}. Equations of motion will first be derived for the transmon operators. For these operators, Hamilton's equations are
\begin{align}
\frac{\partial \hat{\varphi}}{\partial t} = \frac{\partial \hat{H}}{\partial \hat{n}} , \,\, \frac{\partial \hat{n}}{\partial t} = -\frac{\partial \hat{H}}{\partial \hat{\varphi}} .
\end{align}
Evaluating the necessary derivatives gives
\begin{multline}
\frac{\partial \hat{\varphi}}{\partial t} = 8 E_C (\hat{n}-n_g) \\ + 2e\beta \iiint \hat{\mathbf{E}}_{qe} (\mathbf{r},t) \cdot \mathbf{d} \delta(z-z_0) d\mathbf{r},
\end{multline}
\begin{align}
\frac{\partial \hat{n}}{\partial t} = - E_J \sin\hat{\varphi},
\end{align}
where the field-transmon interaction term has been rewritten in terms of the electric field operator.
Equations of motion for the potentials requires taking functional derivatives of $\hat{H}$ with respect to the conjugate operators \cite{chew2016quantum}. These can be easily performed, and will be done in stages for the different sets of potentials. Beginning with the simulation domain magnetic vector potential, we have
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{A}}_{qe}} = \mu^{-1} \nabla\times\nabla\times\hat{\mathbf{A}}_{qe} + \sum_{p \in \mathcal{P}_M}\mu^{-1} \hat{n}_p\times \hat{\mathbf{\Pi}}_{pm},
\end{align}
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{\Pi}}_{qe}} = \epsilon^{-1} \hat{\mathbf{\Pi}}_{qe} + \epsilon^{-1} \partial_t^{-1} \hat{\mathbf{J}}_t .
\end{align}
Hamilton's equations of motion for the magnetic vector potential system are \cite{chew2016quantum}
\begin{align}
\frac{\partial \hat{\mathbf{A}}_{qe}}{\partial t} = \frac{\delta\hat{H}}{\delta \hat{\mathbf{\Pi}}_{qe}}, \,\, \frac{\partial \hat{\mathbf{\Pi}}_{qe}}{\partial t} = -\frac{\delta\hat{H}}{\delta \hat{\mathbf{A}}_{qe}}.
\label{eq:ham1}
\end{align}
These can be combined to give
\begin{align}
\nabla\times\nabla\times\hat{\mathbf{A}}_{qe} + \mu\epsilon \partial_t^2 \hat{\mathbf{A}}_{qe} = \mu \hat{\mathbf{J}}_t - \sum_{p \in \mathcal{P}_M} \hat{n}_p \times \hat{\mathbf{\Pi}}_{pm}.
\label{eq:sim-A-wave}
\end{align}
This inhomogeneous wave equation can be seen to take the expected form for the magnetic vector potential by noting that $-\hat{n}_p\times\hat{\mathbf{\Pi}}_{pm} = \mu \hat{n}_p\times\hat{\mathbf{H}}_{pm}$ is an equivalent electric current density times the permeability.
A similar process can be done for the magnetic vector potential in the port regions. For a particular port, we have
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{A}}_{pe}} = \mu^{-1} \nabla\times\nabla\times\hat{\mathbf{A}}_{pe}
\end{align}
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{\Pi}}_{pe}} = \epsilon^{-1} \hat{\mathbf{\Pi}}_{pe} + \sum_{p' \in \mathcal{P}_E} \delta_{p,p'}\epsilon^{-1} \hat{n}_{p'} \times\hat{\mathbf{F}}_{qm}.
\end{align}
The Kronecker delta function is used to only include a source term if the particular port $p \in \mathcal{P}_E$. Similar functional derivatives can be evaluated for each of the individual port regions. Hamilton's equations can then be used to derive a wave equation. This yields
\begin{align}
\nabla\times\nabla\times\hat{\mathbf{A}}_{pe} + \mu\epsilon \partial^2_t \hat{\mathbf{A}}_{pe} = \!\!\sum_{p' \in \mathcal{P}_E} \delta_{p,p'} \mu \hat{n}_{p'} \! \times\!\partial_t\hat{\mathbf{F}}_{qm}.
\label{eq:port-A-wave}
\end{align}
Note that due to the fixed orientation of $\hat{n}_p$ pointing into the simulation domain, $\hat{n}_p \times\partial_t \hat{\mathbf{F}}_{qm}$ is an equivalent electric current density with a positive sign.
To finish the derivation, equations of motion for the electric vector potential need to be established. As expected, these follow a very similar process to that for the magnetic vector potential. Beginning with the equations for the simulation domain, the necessary functional derivatives are
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{F}}_{qm}} = \nabla\times\epsilon^{-1}\nabla\times\hat{\mathbf{F}}_{qm} - \sum_{p \in \mathcal{P}_E}\epsilon^{-1} \hat{n}_p\times \hat{\mathbf{\Pi}}_{pm}
\end{align}
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{\Pi}}_{qm}} = \mu^{-1} \hat{\mathbf{\Pi}}_{qm}.
\end{align}
Hamilton's equations of motion for the electric vector potential system are
\begin{align}
\frac{\partial \hat{\mathbf{F}}_{qm}}{\partial t} = \frac{\delta\hat{H}}{\delta \hat{\mathbf{\Pi}}_{qm}}, \,\, \frac{\partial \hat{\mathbf{\Pi}}_{qm}}{\partial t} = -\frac{\delta\hat{H}}{\delta \hat{\mathbf{F}}_{qm}}.
\label{eq:ham3}
\end{align}
These can be combined to give
\begin{align}
\epsilon\nabla\times\epsilon^{-1}\nabla\times\hat{\mathbf{F}}_{qm} + \mu\epsilon \partial_t^2 \hat{\mathbf{F}}_{qm} = \sum_{p \in \mathcal{P}_M} \hat{n}_p \times \hat{\mathbf{\Pi}}_{pm}.
\label{eq:sim-F-wave}
\end{align}
By recalling that $\hat{n}_p \times\hat{\mathbf{\Pi}}_{pm} = \epsilon \hat{\mathbf{E}}_{pm} \times\hat{n}_p$, it is seen that the source term for this inhomogeneous wave equation has the form of an equivalent magnetic current density times the permittivity. Hence, this is the expected wave equation for an electric vector potential in the radiation gauge.
The final set of equations are for the electric vector potential in the port regions. The functional derivatives are
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{F}}_{pm}} = \nabla\times\epsilon^{-1}\nabla\times\hat{\mathbf{F}}_{pm}
\end{align}
\begin{align}
\frac{\delta\hat{H}}{\delta \hat{\mathbf{\Pi}}_{pm}} = \mu^{-1} \hat{\mathbf{\Pi}}_{pm} - \sum_{p' \in \mathcal{P}_M} \delta_{p,p'}\mu^{-1} \hat{n}_{p'} \times\hat{\mathbf{A}}_{qe}.
\end{align}
Using Hamilton's equations, the results of these functional derivatives can be combined to give
\begin{multline}
\epsilon\nabla\times\epsilon^{-1}\nabla\times\hat{\mathbf{F}}_{pm} + \mu\epsilon \partial^2_t \hat{\mathbf{F}}_{pm} \\ = -\sum_{p' \in \mathcal{P}_M} \delta_{p,p'} \epsilon \hat{n}_{p'} \times\partial_t\hat{\mathbf{A}}_{qe}.
\label{eq:port-F-wave}
\end{multline}
Similar to (\ref{eq:port-A-wave}), the fixed polarity of $\hat{n}_p$ means that $-\hat{n}_p\times\partial_t\hat{\mathbf{A}}_{qe}$ is equal to an equivalent magnetic current density with a positive sign.
Noting that (\ref{eq:sim-A-wave}), (\ref{eq:port-A-wave}), (\ref{eq:sim-F-wave}), and (\ref{eq:port-F-wave}) are the expected wave equations in each region for the radiation gauge, it can be concluded that the equations of motion for the electromagnetic fields are simply the quantum Maxwell's equations for each region with the necessary sources added \cite{chew2016quantum2}. For the simulation domain, this gives
\begin{multline}
\nabla\times\hat{\mathbf{H}}_q(\mathbf{r},t) - \partial_t \hat{\mathbf{D}}_q(\mathbf{r},t) \\ = \hat{\mathbf{J}}_t(\mathbf{r},t) + \sum_{p \in \mathcal{P}_M} \hat{\mathbf{J}}_{p}(\mathbf{r},t)
\end{multline}
\begin{align}
\nabla\times\hat{\mathbf{E}}_q(\mathbf{r},t) + \partial_t \hat{\mathbf{B}}_q(\mathbf{r},t) = - \sum_{p \in \mathcal{P}_E} \hat{\mathbf{M}}_{p}(\mathbf{r},t)
\end{align}
\begin{align}
\nabla \cdot \hat{\mathbf{D}}_q(\mathbf{r},t) = 0
\end{align}
\begin{align}
\nabla \cdot \hat{\mathbf{B}}_q(\mathbf{r},t) = 0
\end{align}
where the port current densities are
\begin{align}
\hat{\mathbf{J}}_p(\mathbf{r},t) = \hat{n}_p \times \hat{\mathbf{H}}_p(\mathbf{r},t), \,\, \,\,\,\,\, p \in \mathcal{P}_M,
\end{align}
\begin{align}
\hat{\mathbf{M}}_p(\mathbf{r},t) = -\hat{n}_p \times \hat{\mathbf{E}}_p(\mathbf{r},t), \,\, \,\,\,\,\, p \in \mathcal{P}_E.
\end{align}
Similarly, the quantum Maxwell's equations for a single port region are
\begin{align}
\nabla\times\hat{\mathbf{H}}_p(\mathbf{r},t) - \partial_t \hat{\mathbf{D}}_p(\mathbf{r},t) = \hat{\mathbf{J}}_{q}(\mathbf{r},t)
\end{align}
\begin{align}
\nabla\times\hat{\mathbf{E}}_p(\mathbf{r},t) + \partial_t \hat{\mathbf{B}}_p(\mathbf{r},t) = - \hat{\mathbf{M}}_{q}(\mathbf{r},t)
\end{align}
\begin{align}
\nabla \cdot \hat{\mathbf{D}}_p(\mathbf{r},t) = 0
\end{align}
\begin{align}
\nabla \cdot \hat{\mathbf{B}}_p(\mathbf{r},t) = 0
\end{align}
where the port current densities are
\begin{align}
\hat{\mathbf{J}}_q(\mathbf{r},t) = -\hat{n}_p \times \hat{\mathbf{H}}_q(\mathbf{r},t), \,\, \,\,\,\,\, p \in \mathcal{P}_E,
\end{align}
\begin{align}
\hat{\mathbf{M}}_q(\mathbf{r},t) = \hat{n}_p \times \hat{\mathbf{E}}_q(\mathbf{r},t), \,\, \,\,\,\,\, p \in \mathcal{P}_M.
\end{align}
Note that due to the radiation gauge used in this work, the port current densities are all solenoidal and as a result there are no quantum charge densities associated with these currents.
Maxwell's equations can be used to form wave equations for $\hat{\mathbf{E}}$. In the simulation domain, this gives
\begin{multline}
\nabla\times\nabla\times\hat{\mathbf{E}}_q(\mathbf{r},t) + \mu\epsilon\partial_t^2\hat{\mathbf{E}}_q(\mathbf{r},t) = -\mu \partial_t \hat{\mathbf{J}}_t(\mathbf{r},t) \\ -\mu \sum_{p\in\mathcal{P}_M} \partial_t \hat{\mathbf{J}}_p(\mathbf{r},t) - \sum_{p\in\mathcal{P}_E} \nabla\times \hat{\mathbf{M}}_p(\mathbf{r},t),
\label{eq:q-Sim-wave}
\end{multline}
while in a particular port region we have
\begin{multline}
\nabla\times\nabla\times\hat{\mathbf{E}}_p(\mathbf{r},t) + \mu\epsilon\partial_t^2\hat{\mathbf{E}}_p(\mathbf{r},t) \\ = -\mu \partial_t \hat{\mathbf{J}}_q(\mathbf{r},t) - \nabla\times \hat{\mathbf{M}}_q(\mathbf{r},t).
\label{eq:q-Port-wave}
\end{multline}
In general, only one set of sources will be present in (\ref{eq:q-Port-wave}) depending on whether $p\in \mathcal{P}_E$ or $p\in\mathcal{P}_M$.
With appropriate wave equations developed, different modeling strategies can be devised to solve them. For instance, a quantum finite-difference time-domain solver could be used \cite{na2020quantum2}. Alternatively, eigenmodes of the electromagnetic system can be found numerically and used in a quantum information preserving numerical framework \cite{na2020quantum}. In certain cases, the current densities can be treated as impressed sources so that a dyadic Green's function approach can be used to propagate the quantum information \cite{chew2016quantum2}. We will demonstrate this process for a circuit QED system in our future work.
Although full-wave numerical models are of primary interest for practical applications, the development of simpler problems that can be solved using analytical methods are also important. The solutions to these test problems can help validate different full-wave modeling techniques and serve as a useful pedagogical tool for learning how this new formalism can be applied to real-world problems.
\section{Conclusion}
\label{sec:conclusion}
In this work, we have provided a new look at how circuit QED systems using transmon qubits can be described mathematically. Expressed in terms of three-dimensional vector fields, this new approach is well-suited to developing numerical models that can leverage the latest developments in computational electromagnetics research. We have also demonstrated how our new model is consistent with the simpler circuit-based descriptions often used in the literature. Using our new model, we derived the quantum equations of motion applicable to the coupled field-transmon system. Developing solution strategies for this kind of coupled quantum system is an area of active research interest. Numerical methods in this area have the potential to greatly benefit the overall field of circuit QED, and correspondingly, the development of new kinds of quantum information processing hardware.
\ifCLASSOPTIONcaptionsoff
\fi
\end{document}
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\begin{document}
\def\spacingset#1{\renewcommand{\baselinestretch}
{#1}\small\normalsize} \spacingset{1}
\if11
{
\title{\bf Transformed-linear prediction for extremes}
\author{Jeongjin Lee\thanks{
Jeongjin Lee and Daniel Cooley were partially supported by US National Science Foundation Grant DMS-1811657.}
\hspace{.2cm}\\
Department of Statistics, Colorado State University\\
and \\
Daniel Cooley \\
Department of Statistics, Colorado State University}
\maketitle
} \fi
\if01
{
\begin{center}
{\LARGE\bf Transformed-linear prediction for extremes}
\end{center}
} \fi
\begin{abstract}
We consider the problem of performing prediction when observed values are at their highest levels.
We construct an inner product space of nonnegative random variables from transformed-linear combinations of independent regularly varying random variables.
Under a reasonable modeling assumption, the matrix of inner products corresponds to the tail pairwise dependence matrix, which summarizes tail dependence.
The projection theorem yields the optimal transformed-linear predictor, which has the same form as the best linear unbiased predictor in non-extreme prediction.
We also construct prediction intervals based on the geometry of regular variation.
We show that these intervals have good coverage in a simulation study as well as in two applications: prediction of high pollution levels, and prediction of large financial losses.
\end{abstract}
\noindent
{\it Keywords: }Multivariate Regular Variation, Projection Theorem, Tail Pairwise Dependence Matrix, Air Pollution, Financial Risk
\spacingset{1.9}
\section{Introduction}
\label{sec:intro}
Prediction of unobserved quantities is a common objective of statistical analyses.
Figure \ref{fig: washingtonDC} shows the one-hour maximum measurements of the air pollutant nitrogen dioxide (NO$_2$) in parts per billion for four monitoring stations in the Washington DC area on January 23, 2020.
Given these measurements, it is natural to ask what the predicted level would be at a nearby unmonitored location such as Alexandria VA, which is marked ``Alx" in Figure \ref{fig: washingtonDC} and which had NO$_2$ monitoring prior to 2015. What makes this particular day interesting is that measurements are at very high levels; each measurement exceeds its station's empirical 0.98 quantile for the year, and the Arlington station (Arl) is recording its highest measurement for the year.
We propose a linear prediction method which is designed specifically for when observed values are at extreme levels and which is based on a framework from extreme value analysis.
If the joint distribution of all variates were known, the conditional distribution would provide complete information about the variate of interest given the observed values.
The air pollution data's distribution is not known, is clearly non-Gaussian, and there is no clear choice for a candidate joint distribution.
Further, extreme value analysis would caution against using a model that had been fit to the entire data set to describe joint tail behavior.
Linear methods, such as kriging in spatial statistics, offer a straightforward predictor by simply applying weights to each of the observations.
Linear prediction methods do not require specification of the joint distribution and instead provide the best (in terms of mean square prediction error, MSPE) linear unbiased prediction (BLUP) weights given only the covariance structure between the observed and unobserved measurements.
Uncertainty is often summarized by MSPE and prediction intervals are commonly based on Gaussian assumptions.
However, covariance could be a poor descriptor of tail dependence, and Gaussian assumptions may be poorly suited to describe uncertainty in the tail.
\begin{figure}
\caption{Maximum NO$_2$ measurements for January 23, 2020. All observations are above the empirical .98 quantile for each location.}
\label{fig: washingtonDC}
\end{figure}
In this work, we propose a extremal prediction method which is similar in spirit to familiar linear prediction.
We will analyze only data which are extreme.
To provide a framework for modeling dependence in the upper tail, we rely on regular variation on the positive orthant.
Modeling in the positive orthant allows our method to focus only on the upper tail, which is assumed to be the direction of interest; in this example we are interested in predicting when pollution levels are high.
On the way to developing our prediction method, we will construct a vector space of non-negative regularly-varying random vectors arising from transformed-linear operations.
We summarize pairwise tail dependencies in a matrix which has properties analogous to a covariance matrix.
Our transformed-linear predictor has a similar form to the BLUP in non-extreme linear prediction.
Rather than being based on the elliptical geometry underlying standard linear prediction, uncertainty quantification is based on on the polar geometry of regular variation.
We will show that our method has good coverage when applied to the Washington air pollution data and also when applied to a higher dimensional financial data set.
\section{Background}
\label{sec:background}
\subsection{Regular variation on the positive orthant}
Informally, a multivariate regularly varying random variable has a distribution which is jointly heavy tailed.
Regular variation is closely tied to classical extreme value analysis \citep[][Appendix B]{deHaan2007}, and \cite{resnick2007} gives a comprehensive treatment.
Let $\bm{X}$ be a $p$-dimensional random vector that takes values in $\mathbb{R}_{+}^{p}=[0,\bm{\infty})^p$.
$\bm{X}$ is regularly varying (denoted $RV_+^p(\alpha)$) if there exists a function $b(s) \rightarrow \infty$ as $s \rightarrow \infty$ and a non-degenerate limit measure $\nu_{\bm X}$ for sets in $[0,\infty)^{p} \setminus \{\bm{0}\}$ such that
\begin{equation}
\label{eq: regVar1}
s\operatorname{P}(b(s)^{-1}\bm{X}\in \cdot)\xrightarrow{v} \nu_{\bm{X}}(\cdot)
\end{equation}
as $s\rightarrow\infty$, where $\xrightarrow{v}$ indicates vague convergence
in the space of non-negative Radon measures on $[0,\bm{\infty}]^{p}\setminus\{\bm{0}\}$.
The normalizing function is of the form $b(s)=U(s) s^{1/\alpha}$
where $U(s)$ is a slowly varying function, and $\alpha$ is termed the tail index.
For any set $C \subset [0,\bm{\infty}]^{p}\setminus\{\bm{0}\}$ and $k > 0$, the measure has the scaling property $\nu_{\bm{X}}(kC)=k^{-\alpha}\nu_{\bm{X}}(C)$.
This scaling property implies regular variation can be more easily understood in a polar geometry.
Given any norm, $r>0$, and Borel set $B\subset \Theta_{+}^{p-1}=\{\bm{x}\in \mathbb{R}_{+}^{p}:||\bm{x}||=1\}$, the set $C(r,B)=\{\bm{x}\in \mathbb{R}_{+}^{p}:||\bm{x}||>r, \bm{x}/||\bm{x}|| \in B\}$ has measure $\nu_{\bm{X}}{(C(r,B))}=r^{-\alpha}H_{\bm{X}}(B)$, where $H_{\bm X}$ is a measure on $\Theta_{+}^{p-1}$.
The angular measure $H_{\bm X}$ fully describes tail dependence in the limit; however, modeling $H_{\bm X}$ even in moderate dimensions is difficult.
The measure's intensity function in terms of polar coordinates is
\begin{equation}
\label{eq:nu}
\nu_{\bm{X}}(\mathrm{d} r\times \mathrm{d}\bm w)=\alpha r^{-\alpha-1}\mathrm{d} r\mathrm{d} H_{\bm{X}}(\bm w).
\end{equation}
\subsection{Transformed linear operations}
In order to perform linear-like operations for vectors in the positive orthant, \cite{cooley2019decompositions} defined transformed linear operations.
Consider $\bm x \in \mathbb{R}_{+}^{p}=[0,\infty)^{p}$, let $t$ be a monotone bijection mapping from $\mathbb{R}$ to $\mathbb{R}_{+}$, with $t^{-1}$ its inverse.
For $\bm y \in \mathbb{R}^{p}$, $t(\bm y)$ applies the transform componentwise.
For $\bm{x}_1$ and $\bm{x}_2 \in \mathbb{R}_{+}^{p}=[0,\infty)^{p}$, define vector addition as $\bm{x}_1 \oplus \bm{x}_2 = t\{t^{-1}(\bm{x}_1)+t^{-1}(\bm{x}_2)\}$ and define scalar multiplication as $a\circ \bm{x}_1=t\{at^{-1}(\bm{x}_1)\}$ for $a\in \mathbb{R}$.
It is straightforward to show that $\mathbb{R}_{+}^{p}$ with these transformed-linear operations is a vector space as it is isomorphic to $\mathbb{R}^{p}$ with standard operations.
To apply transformed linear operations to non-negative regularly-varying random vectors, \cite{cooley2019decompositions} consider the softplus function $t(y)=\log\{1+\exp(y)\}$,
whose important property is $\lim\limits_{y\to\infty} t(y)/y=\lim\limits_{x\to\infty} t^{-1}(x)/x=1$.
Because $t$ negligibly affects large values, regular variation in the upper tail is preserved when $t$ is used to define transformed-linear operations on regularly-varying random vectors.
More precisely,
assume $\bm X_i$ is regularly varying as in (\ref{eq: regVar1}) with limit measure $\nu_{\bm X_i}(\cdot)$, $i=1,2$.
Further assume that the marginals meet the lower tail condition
$s\operatorname{P}\{X_{i,j} \le \exp(-kb(s))\}\to 0, \mbox{ as } s\rightarrow\infty$,
$j=1,\cdots,p$, for all $k > 0$.
This lower tail condition is specific to $t$ and is required to guarantee that $\operatorname{P}(X_{i,j} < x) \rightarrow 0$ as $x \rightarrow 0$ fast enough so that when $a < 0$, $a \circ \bm X_i$ does not affect the upper tail; it is met by common regularly varying distributions like the Fr\'echet and Pareto.
Applying transformed linear operations, if $\bm X_1, \bm X_2$ are independent,
\begin{eqnarray}
\label{eq:transPlus}
s\operatorname{P}(b(s)^{-1}(\bm X_1 \oplus \bm X_2)\in \cdot) &\xrightarrow{\nu}& \nu_{\bm X_1}(\cdot)+\nu_{\bm X_2}(\cdot)\mbox{; and}\\
\label{eq:transMult}
s\operatorname{P}(b(s)^{-1}(a\circ \bm X_1)\in \cdot) &\xrightarrow{v}&
\left \{
\begin{array}{l}
a^{\alpha}\nu_{\bm X_1}(\cdot) \mbox{ if } a>0\mbox{, and}\\
0 \mbox{ if } a \le 0.
\end{array}
\right.
\end{eqnarray}
Other transforms with the same limiting properties and with appropriately adjusted lower tail condition could be used in place of $t$.
\cite{cooley2019decompositions} go on to construct $\bm X \in RV_+^p(\alpha)$ via transformed linear combinations of independent regularly varying random variables.
Let $A = (\bm a_1, \ldots, \bm a_q)$, where $\bm a_j \in \mathbb{R}^p$ and hence $A \in \mathbb{R}^{p \times q}$.
Let
\begin{equation}
\label{eq:linConst}
\bm X = A \circ \bm Z = t(A t^{-1}(\bm Z)),
\end{equation}
where
$\bm Z = (Z_1, \ldots Z_q)^\top$ is a vector of independent regularly varying random variables where $s\operatorname{P}(b(s)^{-1} Z_{j} > z) \to z^{-\alpha}$ and $Z_j$ meets the aforementioned lower tail condition.
$\bm X$ is regularly varying
with angular measure
\begin{equation}
\label{eq: discreteAngMsr}
H_{\bm X}(\cdot) = \sum_{j = 1}^q \| \bm a^{(0)}_{j} \|^\alpha \delta_{ \bm a^{(0)}_{j} / \| \bm a^{(0)}_{j} \|}(\cdot),
\end{equation}
where $\delta$ is the Dirac mass function.
The zero operation $a^{(0)} := \max(a, 0)$ will be important throughout, and is understood to be componentwise when applied to vectors or matrices.
As $q \rightarrow \infty$ the class of angular measures resulting from this construction method is dense in the class of possible angular measures \citep{cooley2019decompositions}, and one only needs to consider nonnegative matrices $A$ to construct the dense class.
\subsection{Tail Pairwise Dependence Matrix}
For a general $\bm X \in RV_+^p(\alpha)$, if $p$ is even moderately large, it is challenging to describe the angular measure $H_{\bm X}$.
Rather than fully characterize $H_{\bm X}$, we will summarize tail dependence via the tail pairwise dependence matrix (TPDM), a matrix of pairwise summary measures.
Let $\alpha = 2$ and let $\bm X \in RV_+^p(2)$ have angular measure $H_{\bm X}$.
Let $\Sigma_{\bm X} =\{ \sigma_{\bm X_{ij}}\}_{i,j=1,\cdots,p}$ be the $p \times p$ matrix where
\begin{equation}
\label{eq: TPDM}
\sigma_{\bm X_{ik}}=\int_{\Theta_{+}^{p-1}} {w_{i}w_{k}} \mathrm{d} H_{\bm{X}}(w),
\end{equation}
and $\Theta_{+}^{p-1}=\{\bm{x}\in \mathbb{R}_{+}^{p}:\|\bm{x}\|_{2}=1\}$.
Each element $\sigma_{\bm X_{ij}}$ is an extremal dependence measure of \cite{larsson2012},
however we do not require $H$ to be a probability measure.
As (\ref{eq: TPDM}) resembles a second moment, it is not surprising that it has some properties similar to a covariance matrix.
Most importantly, $\Sigma_{\bm X}$ can be shown to be positive semi-definite \citep{cooley2019decompositions}.
Also, the diagonal elements ${\sigma_{\bm X}}_{ii}$ reflect the relative magnitudes of the respective elements $X_i$, as (\ref{eq:nu}) implies
$\lim_{s \rightarrow\infty}s\operatorname{P}(b(s)^{-1} X_i >c)
=\int_{\Theta_{+}^{p-1}}\int_{c/w_i}^{\infty}{2r^{-3}\mathrm{d} r\mathrm{d} H_{\bm{X}}(w)=c^{-2}\int_{\Theta_{+}^{p-1}}w_{i}^{2}\mathrm{d} H_{\bm{X}}(w)=c^{-2}\sigma_{X_{ii}}}.
$
Letting $x = cU(s)s^{1/2}$, there is a corresponding slowly varying function $L$ such that the relation can be rewritten as
\begin{equation}
\label{eq:tailRatio}
\lim_{x \rightarrow \infty} \frac{\operatorname{P}(X_i > x)}{x^{-2} L(x)} = {\sigma_{\bm X}}_{ii}.
\end{equation}
So the `magnitude' of the elements of $\bm X$ described by the diagonal elements of the TPDM is in terms of suitably-normalized tail probabilities rather than variance.
The presence of the slowly varying function $L(x)$ in the denominator means it is ambiguous to discuss the `scale' of a regularly varying random variable, as scale information is in both the normalizing sequence and the angular measure (and consequently, TPDM).
Because the notion of `scale' is inherent in principal component analysis, \cite{cooley2019decompositions} further assumed that $\bm X$ was Pareto-tailed, making $L(x)$ a constant that was pushed into the angular measure $H_{\bm X}$ and subsequently into $\Sigma_{\bm X}$.
Here, we will not require a Pareto tail, and the random variables we will construct in Section \ref{sec:innerProductSpace}
will have a natural normalizing function.
\cite{cooley2019decompositions} choose $\alpha = 2$ because the TPDM has a convenient form for random vectors defined as in (\ref{eq:linConst}).
With the angular measure in (\ref{eq: discreteAngMsr}), $\sigma_{\bm X_{ik}} = \sum_{j = 1}^q a_{ij}^{(0)}a_{kj}^{(0)}$ and $\Sigma_{\bm A \circ \bm Z} = A^{(0)} {A^{(0)}}^\top.$
\cite{kiriliouk2022} recently generalized the TPDM for any $\alpha > 0$ by allowing the integrand to depend on $\alpha$.
For the inner product space we introduce in Section \ref{sec:innerProductSpace}, we will continue to assume $\alpha = 2$.
Additionally, for any $\bm X \in RV_+^p(2)$, $\Sigma_{\bm X}$ is completely positive; that is, there exists $q_* < \infty$ and nonnegative $p \times q_*$ matrix $A$ such that $\Sigma_{\bm X} = A A^T$ \citep[][Proposition 5]{cooley2019decompositions}.
The value of $q_*$ is not known, and $A$ is not unique.
This property implies that given any TPDM, one can find a nonnegative matrix $A$ such that $A \circ \bm Z$, and
in Section \ref{sec:UQ}, we will use this completely positive decomposition to create prediction intervals.
\section{Inner product space and prediction}
\label{sec:innerProductSpace}
\subsection{Inner product space $\mathcal{V}^q$}
We consider a space of regularly varying random variables constructed from transformed-linear combinations.
We assume $\alpha = 2$ to obtain an inner product space.
Let $\bm Z = (Z_{1}, \ldots Z_{q})^\top$ be a vector of independent $Z_{j} \in RV_{+}^{1}(2)$ meeting lower tail condition,
$s\operatorname{P}(Z_{j}\le \exp(-kb(s)))\rightarrow 0$ as $s\rightarrow \infty$ for all $k>0$.
Define $L(z)$ such that $\lim_{z \rightarrow \infty} \frac{\operatorname{P}(Z_j > z)}{z^{-2} L(z)} = 1$ for all $j = 1, \ldots, q$.
For $\bm a \in \mathbb{R}^q$, consider the subspace of $RV_+^1(2)$
\begin{equation}
\label{eq:V}
\mathcal{V}^q = \big\{ X ; X =
\bm a^\top \circ \bm Z = a_{1} \circ Z_{1} \oplus \cdots \oplus a_{q} \circ Z_{q} \}.
\end{equation}
If $X_1 = \bm a_1^\top \circ \bm Z$ and $X_2 = \bm a_2^\top \circ \bm Z$, then $X_1 \oplus X_2 = (\bm a_1 + \bm a_2)^\top \circ \bm Z$.
Also, $c \circ X_1 = c \bm a_1^\top \circ \bm Z$ for $c \in \mathbb{R}$.
$\mathcal{V}^q$ is isomorphic to $\mathbb{R}^q$ as any $X \in \mathcal{V}^q$ is uniquely identifiable by its vector of coefficients $\bm a$.
Like $\mathbb{R}^{q},$ $\mathcal{V}^{q}$ is complete and thus is a Hilbert space \citep{lee2022phd}.
$\mathcal{V}^q$ differs from the vector space in \cite{cooley2019decompositions} which was non-stochastic.
We define the inner product of $X_{1} = \bm a_1^\top \circ \bm Z$ and $X_2=\bm a_2^\top \circ \bm Z$ as
\begin{equation*}
\langle X_{1}, X_{2} \rangle := \bm a_1^\top \bm a_2 = \sum_{i=1}^{q}a_{1i}a_{2i}.
\end{equation*}
We say $X_1, X_2 \in \mathcal{V}^q$ are orthogonal if $\langle X_1, X_2 \rangle = 0$.
The norm is defined as $\| X \|_{\mathcal{V}^{q}} = \sqrt{ \langle X, X \rangle}$, whose subscript $\mathcal{V}^{q}$ distinguishes this norm based on the random variable's coefficients from the usual Euclidean norm.
The norm
defines a metric
$
d(X_1, X_2) = \| X_1 \ominus X_2 \|_{\mathcal{V}^{q}}=\sqrt{\sum_{i=1}^{q}(a_{1i}-a_{2i})^2},
$
which we will further interpret in Section \ref{sec:Vplus}.
Considering vectors $\bm X = (X_1, \ldots, X_p)^\top$ where $X_i = \bm a_i^\top \circ \bm Z \in \mathcal{V}^q$ for $i = 1, \ldots, p$,
$\bm X \in RV_+^p(2)$ and is of the form $A \circ \bm Z$ in (\ref{eq:linConst}).
we denote the matrix of inner products
\begin{equation}
\label{eq:ipMatrix}
\Gamma_{\bm X} = \langle X_i, X_j \rangle_{i,j = 1, \ldots p} = A A^\top.
\end{equation}
We will relate $\Gamma_{\bm X}$ for $X_i$ in $\mathcal{V}^q$ to the TPDM $\Sigma_{\bm X}$ for general $\bm X \in RV_+^p(2)$ in Section \ref{sec:Vplus}.
\subsection{Transformed-linear prediction}
\label{sec:transLinearPred}
As $\mathcal{V}^q$ is isomorphic to Hilbert space $\mathbb{R}^q$, the best transformed-linear predictor follows similarly.
Assume $X_i = \bm a_i^\top \circ \bm Z \in \mathcal{V}^q$ for $i = 1, \ldots, p+1$. Let $\bm X_p = (X_1, \ldots, X_p)^\top$.
We aim to find $\bm b \in \mathbb{R}^p$ such that $d(\bm b^\top \circ \bm X_p, X_{p+1})$ is minimized.
Writing in matrix form
\begin{equation*}
\begin{bmatrix}
\bm{X}_{p}\\
X_{p+1}
\end{bmatrix}
=
\begin{bmatrix}
A_p\\
\bm{a}_{p+1}^\top
\end{bmatrix}
\circ \bm{Z},
\end{equation*}
where $A_{p}= (\bm a_1^\top, \ldots, \bm a_p^\top)^\top$.
The matrix of inner products of $(\bm{X}_{p}^\top, X_{p+1})^\top$ is
\begin{equation}\label{eq:GammaMatrices}
\Gamma_{(\bm{X}_{p}^\top, X_{p+1})^\top}
=
\begin{bmatrix}
A_{p}A_{p}^\top & A_{p}\bm{a}_{p+1}\\
\bm{a}_{p+1}^\top A_{p}^\top & \bm{a}_{p+1}^\top\bm{a}_{p+1}
\end{bmatrix}
:=
\begin{bmatrix}
\Gamma_{11} & \Gamma_{12}\\
\Gamma_{21} & \Gamma_{22}
\end{bmatrix}.
\end{equation}
Minimizing $d(\bm b^\top \circ \bm X_p, X_{p+1})$ is equivalent to minimizing $\|A_{p}^\top\bm{b}-\bm{a}_{p+1}\|_{2}^{2}$.
Taking derivatives with respect to $\bm b$ and setting equal to zero, the minimizer $\hat {\bm b}$ solves
$(A_{p}A_{p}^\top)\hat{\bm b}=A_{p}\bm{a}_{p+1}$.
If $A_{p}A_{p}^\top$ is invertible, then the solution $\hat{\bm{b}}$ is,
\begin{equation}\label{eq: bHat}
\hat{\bm{b}}=(A_{p}A_{p}^\top)^{-1}A_{p} \bm{a}_{p+1}=\Gamma_{11}^{-1}\Gamma_{12}.
\end{equation}
An equivalent way to think of the best transformed-linear prediction is through the projection theorem.
$\hat{X}_{p+1}$ is
such that $X_{p+1}\ominus\hat{X}_{p+1}$ is orthogonal to the plane spanned by $X_1,\ldots, X_p$.
The orthogonality condition can be stated as $\langle X_{p+1}\ominus\hat{X}_{p+1},X_i\rangle=0$, for $i=1,\ldots,p$. By linearity of inner products, this can equivalently be expressed as
\begin{align}\label{eq:4}
\begin{bmatrix}
<X_{p+1},X_{i}>\\
\end{bmatrix}_{i=1}^{p}
&=
\begin{bmatrix}
<X_{i},X_{j}>
\end{bmatrix}_{i,j=1}^{p}
\begin{bmatrix}
b_{i}
\end{bmatrix}_{i=1}^{p}
=
\begin{bmatrix}
\sum_{k=1}^{q}a_{ik}a_{jk}
\end{bmatrix}_{i,j=1}^{p}
\begin{bmatrix}
b_{i}
\end{bmatrix}_{i=1}^{p}.
\end{align}
By (\ref{eq:GammaMatrices}), $\hat {\bm b}$ satisfies $A_{p}\bm{a}_{p+1}=A_{p}A_{p}^\top\hat{\bm b}$ as above.
\section{Modeling and Subset $\mathcal{V}_+^q$}
\label{sec:Vplus}
At this point we can employ transformed linear operations to construct regularly-varying random vectors $\bm X = A \circ \bm Z$ that take values in the positive orthant, and elements are in the vector space $\mathcal{V}^q$.
While it is essential that the elements of the coefficient vectors $\bm a$ are allowed to be negative for $\mathcal{V}^q$ to be a vector space, these negative elements can feel largely academic as they do not influence tail behavior.
The magnitude (as in \eqref{eq:tailRatio}) of $X \in \mathcal{V}^q$ can be understood in terms of the generating $Z_j$'s.
Using the fact $\operatorname{P}(Z_1 + Z_2 > z) \sim \operatorname{P}(Z_1 > z) + \operatorname{P}(Z_2 > z)$ as $z \rightarrow \infty$ if $Z_1, Z_2$ are independent \citep[cf.][Lemma 3.1]{jessen2006},
we call
$$
TR(X) := \lim_{z \rightarrow \infty} \frac{\operatorname{P}(X > z)}{\operatorname{P}(Z_1 > z)} = \sum_{j = 1}^q {(a_j^{(0)})}^2
$$
the tail ratio of $X$ and only the positive elements of $\bm a$ contribute.
The random variables $X = \bm a^\top \circ \bm Z$ and $X_+ = \bm a^{(0)^\top} \circ \bm Z$ have the same tail ratio.
Furthermore, if $\bm X = A \circ \bm Z$, both it and $\bm X_+ = A^{(0)} \circ \bm Z$ have the same angular measure: $H_{\bm X} = H_{\bm X_+} = \sum_{j = 1}^q \| a^{(0)}_{j} \|^2 \delta_{ a^{(0)}_{j} / \| a^{(0)}_{j} \|}(\cdot)$.
$\bm X$ and $\bm X_+$ are indistinguishable in terms of their tail behavior.
In terms of modeling, it seems reasonable to restrict our attention to the subset $\mathcal{V}^q_+ = \big\{ X ; X = \bm a^\top \circ \bm Z = a_{1} \circ Z_{1} \oplus \cdots \oplus a_{q} \circ Z_{q} \},$ where $a_j \in [0, \infty)$, and $\bm Z = (Z_{1}, \ldots Z_{q})^\top$ as in (\ref{eq:V}).
Considering inference for a random vector $\bm X \in RV_+^p$, we assume that $\bm X = A \circ \bm Z$ for some unknown $p \times q$ nonnegative matrix $A$.
Recall such constructions are dense in $RV_+^p$.
Furthermore, we will assume that $p$ is large enough that estimating $H_{\bm X}$ is intractable, so we instead summarize dependence via the TPDM, which is estimable from $\bm X$'s pairwise tail behavior.
Since $X_i \in \mathcal{V}^q_+$, $\Sigma_{\bm X} = \Gamma_{\bm X} = A A^\top$, and we are able to apply the results from Section \ref{sec:innerProductSpace}.
Furthermore, the underlying dimension $q$ is latent and not needed for inference.
Assuming the elements of $\bm X$ are in $\mathcal{V}^q_+$ is not only reasonable, but the results of Section 3 are probably useful only if this assumption is made.
Consider the simple example where
\begin{equation}
\label{eq:reviewerExample}
\bm X =
\left(
\begin{array}{c}
X_1\\
X_2
\end{array}
\right)
=
\left(
\begin{array}{c c}
1 & -10\\
1 & -1
\end{array}
\right)
\circ
\left(
\begin{array}{c}
Z_1\\
Z_2
\end{array}
\right)
= A \circ \bm Z,
\end{equation}
and $Z_i$ is iid with $\operatorname{P}(Z_i \leq z) = 1 - z^{-2}$ for $z \geq 1$.
The left panel of Figure \ref{fig:reviewerExample} shows realizations $\bm x_t, t = 1, \ldots, 20,000$ from (\ref{eq:reviewerExample}).
Here, the angular measure is given by $H_{\bm X}(\cdot) = 2 \delta_{(1/\sqrt{2}, 1/\sqrt{2})}(\cdot)$, and $X_1$ and $X_2$ exhibit perfect tail dependence in the limit as shown in Figure \ref{fig:reviewerExample}.
However, $\| X_1 \ominus X_2 \|_{\mathcal{V}^q} = 9$, and the non-zero distance between these random variables is hard to reconcile with their perfect tail dependence.
This distance arises from the negative elements in $A$ whose influence is not evident in realizations of $\bm X$, but which can be seen in the preimage $\bm Y = t^{-1}(\bm X)$
shown in the right panel of Figure \ref{fig:reviewerExample}.
Furthermore applying \eqref{eq: bHat}, $\hat X_1 = 5.5 \circ X_2$, with the weight 5.5 is the best `average' of the two possible ways that the preimages can be large.
Thus both the norm and the predictor arising from $\mathcal{V}^q$ seem more applicable to the latent preimage space rather than the one observed.
However, given only the data in the left panel of Figure \ref{fig:reviewerExample}, the information in the negative coefficients of $A$ would not be visible.
The TPDM of (\ref{eq:reviewerExample}) is $\Sigma_{\bm X} = {\tiny \begin{pmatrix}1 & 1\\ 1 & 1\end{pmatrix}}$.
If we assume $X_i \in \mathcal{V}^q_+$ and use the TPDM as the inner product matrix in (\ref{eq: bHat}), $\hat X_1 = 1 \circ X_2$.
Further, note that if $\bm X = A^{(0)}\circ \bm Z$, then $\| X_1 \ominus X_2 \|_{\mathcal{V}^q} = 0$.
\begin{figure}
\caption{Left panel: realizations $\bm x_t$ from the model in (\ref{eq:reviewerExample}
\label{fig:reviewerExample}
\end{figure}
Applying transformed-linear prediction in practice, we propose assuming that the elements of $(\bm X_p^\top, X_{p+1})^\top$ are in $\mathcal{V}^q_+$, and using the (estimated) TPDM for prediction: $\hat X_{p+1} = \hat {\bm b}^\top \circ \bm X_p$ where $\hat {\bm b} = \Sigma_{11}^{-1} \Sigma_{12}$.
Although $X_{p+1}$ is assumed to be in $\mathcal{V}^q_+$, the predictor $\hat X_{p+1}$ may not be in this subset as $\hat {\bm b}$ may have negative elements.
We do not see this as a detriment as the tranformed-linear operations guarantee $\hat X_{p+1} > 0$ almost surely and the coefficients defining $\hat X_{p+1}$ in $\mathcal{V}^q$ are latent.
The tail ratio allows us to better discuss the meaning of the metric $d(X_1, X_2) = \| X_1 \ominus X_2 \|_{\mathcal{V}^{q}}$.
$TR(X_1 \ominus X_2)$ does not equal $TR(X_2 \ominus X_1)$, except under the unusual circumstance where $\sum_{j = 1}^q \left( (a_{1j} - a_{2j})^{(0)} \right)^2 = \sum_{j = 1}^q \left( (a_{2j} - a_{1j})^{(0)} \right)^2$.
Consider $TR\left( \max ( X_1 \ominus X_2, X_2 \ominus X_1) \right) =
\label{eq:trMax}
\lim_{z \rightarrow \infty}
\left (
\frac{
{P(X_1 \ominus X_2 > z)}+
{P(X_2 \ominus X_1 > z)}-
{P(X_1 \ominus X_2 > z, X_2 \ominus X_1 > z)} }
{P(Z > z)}
\right ).
$
Let $Q = \{j \in \{1, \ldots, q\} \mid (a_{1j} - a_{2j})t^{-1}(Z_j)>0 \}$ be a set of indices where the sign of $(a_{1j}-a_{2j})$ is aligned with the sign of $t^{-1}(Z_j)\in RV_1(2)$ and $Q^{\complement}=\{1\,\ldots,q\}\setminus{Q}$ be its complement set, then the numerator's third term can be rewritten as
\begin{eqnarray*}
\lim_{z \rightarrow \infty}
\frac{P(X_1 \ominus X_2 > z, X_2 \ominus X_1 > z)}{P(Z>z)} &=&
\lim_{z \rightarrow \infty}
\frac{P\left(\bigoplus\limits_{j \in Q} (a_{1j} - a_{2j})\circ Z_j > z, \bigoplus\limits_{j \in Q^{c}} (a_{2j} - a_{1j})\circ Z_j > z\right)}{P(Z > z)}.
\end{eqnarray*}
Since $Q \cap Q^{\complement} = \emptyset$, the independence of the $Z_j$'s implies that this limit is zero and
\begin{equation}
\label{eq:tailRatioMetric}
TR\left( \max ( X_1 \ominus X_2, X_2 \ominus X_1) \right) = \sum_{j = 1}^q (a_{1j} - a_{2j})^2 = d^2(X_1, X_2).
\end{equation}
In section \ref{sec:innerProductSpace}, the metric for $\mathcal{V}^q$ was defined in terms of the random variables' defining coefficients, but the previous definition is unsatisfying as these coefficients are not visible given realizations of the random variables.
The relationship in (\ref{eq:tailRatioMetric}) explains the metric in terms of a tail ratio, which can be estimated from realizations.
Further, $TR\left( \max ( (X_{p+1} \ominus \hat X_{p+1}), (\hat X_{p+1} \ominus X_{p+1})) \right)$ can be viewed as the risk function which $\hat {\bm b}$ minimizes.
\section{Prediction Error}
\label{sec:UQ}
\subsection{Analogue to Mean Square Prediction Error}
\label{sec:MSPE}
In the non-extreme setting, linear prediction minimizes MSPE.
As MSPE corresponds to the conditional variance under a Gaussian assumption, it is used to generate Gaussian-based prediction intervals.
Similarly, our transformed linear predictor $\hat {\bm b}$ minimizes
\begin{equation}\label{eq:7}
\begin{split}
||\hat{X}_{p+1} \ominus X_{p+1}||^{2}_{\mathcal{V}^{q}}
&=(\hat {\bm{b}}^\top A_{p}-\bm{a}_{p+1}^\top)(\hat {\bm{b}}^\top A_{p}-\bm{a}_{p+1})^\top\\
&=\Sigma_{22}-\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12}:=K.\\
\end{split}
\end{equation}
Unlike MSPE, $K$ is not understood via expectation, but instead via tail probabilities as
$
K = TR \left( \max ( (X_{p+1} \ominus \hat X_{p+1}), (\hat X_{p+1} \ominus X_{p+1}) ) \right).
$
However, despite its similarity to MSPE, $K$ seems not very useful for constructing prediction intervals.
To illustrate, we simulate $n = 20,000$ four dimensional vectors $\bm X$ and obtain $\hat X_4$ predicted on $(X_1, X_2, X_3)^\top$.
$\bm X$ is generated from a $4 \times 10$ matrix $A$ applied to a vector $\bm Z$ comprised of 10 independent $RV_+(2)$ random variables; the elements of $A$ are drawn from a uniform distribution, then normalized to have rows with norm 1.
Using the known TPDM to obtain $K = 0.224$ and known tail behavior of the $Z_j$'s, we calculate $\operatorname{P} \left( D \leq 2.99 \right) \approx 0.95$ where $D = \max ( (X_{p+1} \ominus \hat X_{p+1}), (\hat X_{p+1} \ominus X_{p+1}) )$.
We observe 0.952 of the simulated $D$ values are in fact below this bound.
However, Figure \ref{fig:mspe} shows that knowledge of $K$ is not useful for constructing prediction intervals.
Unlike the Gaussian case where the variance of the conditional distribution does not depend on the predicted value $\hat X_{p+1}$,
in the polar geometry of regular variation, the magnitude of the error is related to the size of the predicted value.
In the next sections we use the polar geometry of regular variation to construct meaningful prediction intervals when $\hat X_{p+1}$ is large.
\begin{figure}
\caption{The plot of $D=\max(\hat{X}
\label{fig:mspe}
\end{figure}
\subsection{Prediction inner product matrix and completely positive decomposition}
\label{sec:predTPDM}
To quantify prediction error, we first aim to describe the tail dependence between the predictor $\hat{X}_{p+1}$ and predictand $X_{p+1}$.
The vector $(\hat{X}_{p+1}, X_{p+1})^\top \in RV_+^2(2)$, and this vector's tail dependence is characterized by $H_{(\hat{X}_{p+1}, X_{p+1})^\top}$.
While this angular measure is not readily available, the $2 \times 2$ `prediction' inner product matrix
\begin{equation}\label{eq:predTPDM}
\begin{split}
\Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}
&=
\begin{bmatrix}
\hat{\bm{b}}^\top{A}_p\\
\bm{a}_{p+1}^\top
\end{bmatrix}
\begin{bmatrix}
{A}_{p}^\top\hat{\bm{b}} &
\bm{a}_{p+1}
\end{bmatrix}
=
\begin{bmatrix}
\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12} & \Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12}\\
\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12} & \Sigma_{22}
\end{bmatrix}
\end{split},
\end{equation}
can be obtained from the partitioned TPDM,
as we have assumed $X_1, \ldots, X_{p+1} \in \mathcal{V}^q_+$.
We then use complete positivity to find an angular measure constrained by knowledge of $\Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$.
Although the entries of $\hat{\bm{b}}^\top{A}_p$ are not guaranteed to be nonnegative, the Cholesky decomposition of the $2 \times 2$ prediction inner product matrix yields positive entries and thus $\Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$ is completely positive.
Given a $q_* \geq 2$, there exist procedures \citep{groetzner2020}
to obtain nonnegative $2 \times q_*$ matrices $B$ such that $B B^\top = \Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$, and we can then use (\ref{eq: discreteAngMsr}) to construct an angular measure consisting of $q_*$ discrete point masses.
Since the completely positive decomposition is not unique,
there would seem to be incentive to set $q_*$ large, thereby distributing the total mass of the angular measure $H_{B \circ \bm Z}$ into many point masses.
On the other hand, as $q_*$ grows, the procedures for obtaining $B$ require more computation.
We take a practical approach.
We choose $q_*$ to be of moderate size, but apply the procedure repeatedly, obtaining nonnegative $B^{(k)}, k = 1, \ldots, n_{decomp},$ such that $B^{(k)} {B^{(k)}}^\top = \Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$ for all $k$.
We then set $\hat H_{(\hat{X}_{p+1}, X_{p+1})^\top} = n_{decomp}^{-1} \sum_{k = 1}^{n_{decomp}} H_{B^{(k)} \circ \bm Z}$, and $n_{decomp}^{-1} \sum_{k = 1}^{n_{decomp}} B^{(k)} {B^{(k)}}^\top = \Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$ as desired.
$\hat H_{(\hat{X}_{p+1}, X_{p+1})^\top}$ consists of $n_{decomp}\times q_*$ point masses.
We use a simulation study to illustrate.
We again begin by generating a matrix $A$ whose elements are drawn from a uniform distribution; however this time the dimension of $A$ is $7 \times 400$ and the true angular measure consists of 400 point masses.
We draw 60,000 random realizations of $\bm X = A \circ \bm Z$, and use the first 40,000 as a training set.
The largest 1\% of this training set is used to estimate the seven-dimensional TPDM, from which we obtain $\hat {\bm b}$ and additionally $\hat \Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$.
We then use the completely positive decomposition to obtain $2 \times 9$ matrices $B^{(k)}, k = 1, \ldots, 51$, resulting in an estimated angular measure $\hat H_{(\hat{X}_{p+1}, X_{p+1})^\top}$ consisting of 459 point masses.
We obtain a 95\% `joint polar region' by drawing bounds at $\bm w_{0.025}$ and $\bm w_{0.975}$, the 0.025
and 0.975 empirical quantiles of the univariate distribution of angles provided by the normalized estimated angular measure.
The left panel of Figure \ref{fig:simStudy} shows the scatterplot of the 20,000 remaining test points $\hat{X}_{p+1}$ and $X_{p+1}$ and the 95\% joint region.
Thresholding at the 0.95 quantile of $\| (\hat{X}_{p+1}, X_{p+1}) \|_{2}$,
we find that 96.3\% of the large values in the test set fall within the joint region.
To informally assess the variability of these quantiles,
we perform the completely positive decomposition under different scenarios on the same data set.
To speak in terms of angles, let $\theta(\bm w) = \arctan(w_2/w_1)$.
For the scenario above, our bounds were $(\theta(\bm w_{0.025}), \theta(\bm w_{0.975}))$ = (0.30, 1.40).
A second completely positive decomposition where $q_* = 6$ and consisting of 510 point masses yielded bounds of (0.29 1.40), and a third decomposition where $q_* = 7$ and consisting of 560 point masses yielded bounds of (0.33, 1.41).
It seems that constraining $\hat H_{(\hat{X}_{p+1}, X_{p+1})^\top}$ by $\hat \Gamma_{(\hat{X}_{p+1}, X_{p+1})^\top}$ and requiring it to consist of a large enough number of point masses result in bounds with low variability.
If a continuous angular measure is desired, we propose performing a kernel density estimate of the angular masses obtained from the completely positive decomposition.
We use the adjusted boundary bias approach of \cite{marron1994transformations} for the kernel density estimation since the support of $H_{(\hat X_{p+1}, X_{p+1})^\top}$ is bounded.
The bounds obtained by automatically choosing the bandwidth and applying to the three decompositions above are
(0.28, 1.42), (0.32 1.43), and (0.28 1.41).
\subsection{Prediction intervals for $X_{p+1}$ given large $\hat{X}_{p+1}$}
\label{sec:condtlInterval}
The region obtained in the previous section describes the joint behavior of $\hat{X}_{p+1}$ and $X_{p+1}$, but the quantity of interest is the conditional behavior of $X_{p+1}$ given a specific large value $\hat{X}_{p+1} = x$.
In $(p+1)$-dimensional space, \cite{cooley2012approximating} fit a parametric model for angular density $h_{(\bm X_p^\top, X_{p+1})^\top}$, and use the limiting intensity function of regular variation to get an approximate density of $X_{p+1}$ given large $\bm X_p = \bm x_p$.
Following their approach with $\alpha = 2$ and the $L_2$ norm, and letting $\bm x = (\bm x_p^\top, x_{p+1})^\top$, transforming (\ref{eq:nu}) from polar to Cartesian coordinates has Jacobian $|J|=\|\bm{x}\|^{-(p+1)}x_{p+1}$ \citep{song1997} and yields a limiting measure of $\nu_{(\bm X_p^\top, X_{p+1})^\top}(\bm x)\mathrm{d}\bm x = 2 \|\bm x \|^{-(p+4)} x_{p+1} h( \bm x \|\bm x\|^{-1}) \mathrm{d} \bm x$.
The approximate conditional density is $f_{X_{p+1}|\bm X_p}(x_{p+1} | \bm x_p) \approx c^{-1} \nu_{(\bm X_p^\top, X_{p+1})^\top}( \bm x_p, x_{p+1} )$, where $c = \int_0^\infty \nu_{(\bm X_p^\top, X_{p+1})^\top}(\bm x) \mathrm{d} x_{p+1}.$
\cite{cooley2012approximating} applied their method in moderate dimension ($p = 4$); applying the approach for larger $p$ would require a high dimensional angular measure model.
We adapt the method of \cite{cooley2012approximating} to model the relationship between $X_{p+1}$ and $\hat X_{p+1}$.
Regardless of $p$, we only need to describe this bivariate relationship.
In two dimensions, the problem simplifies.
To find the bound of the prediction interval for a given value $\hat x_{p+1}$, we wish to find $d$
such that
$$
\rho
= \int_0^d f_{X_{p+1}|\hat X_{p+1}}(x_{p+1}|\hat x_{p+1})\mathrm{d} x_{p+1}
=\frac
{\int_0^d 2 \|\bm x\|^{-5} x_{p+1} h(\bm x \|\bm x\|^{-1}) \mathrm{d} x_{p+1}}
{\int_0^\infty 2 \|\bm x\|^{-5} x_{p+1} h(\bm x \|\bm x\|^{-1}) \mathrm{d} x_{p+1}},
$$
where $\bm x = (\hat x_{p+1}, x_{p+1})^\top$, and where $\rho$ sets the prediction level; below we set $\rho$ to 0.025 and 0.975 to yield a 95\% prediction interval of $x_{p+1}$ given $\hat x_{p+1}$
Letting $\theta$
be such that $x_{p+1} = \hat x_{p+1} \tan \theta$, simple substitution and cancellation of $\hat x_{p+1}$ yield the equivalent problem
$$
\rho
=\frac
{\int_0^{\theta^*} 2 \tan\theta (1 + \tan^2 \theta)^{-5/2} h(\cos \theta, \sin \theta) \sec^2\theta \mathrm{d} \theta}
{\int_0^\infty 2 \tan\theta (1 + \tan^2 \theta)^{-5/2} h(\cos \theta, \sin \theta) \sec^2\theta \mathrm{d} \theta}.
$$
With $\rho$ specified, $\theta^*$ can be solved independently of the value of $\hat x_{p+1}$, and given this value, the bound is $d = \hat x_{p+1} \tan (\theta^*)$.
We use the kernel density estimated in Section \ref{sec:predTPDM} in place of $h$ and numerically integrate to solve for $\theta^*$.
The center panel of Figure \ref{fig:simStudy} illustrates the conditional density for a particular realization from the aforementioned simulation study where $\hat x_{p+1}$ = 33.17 and with actual value $x_{p+1}$ = 48.15 denoted by the star.
The right panel shows a scatterplot of the largest 5\% (by $\hat x_{p+1}$) of the test set from the aforementioned simulation along with the upper and lower bounds from the conditional density approximation.
Scatterplots of realizations from regularly-varying random vectors can be difficult to interpret, because weak dependence implies that large points occur near the axes.
The fact that the points occur in the interior implies that there is a strong relationship between $\hat X_{p+1}$ and $X_{p+1}$, and clearly the width of the prediction interval needs to increase with $\hat x_{p+1}$.
The coverage rate of these intervals is 0.947.
\begin{figure}
\caption{(Left) The estimated joint 95\% joint prediction region based on the approximated angular measure $\hat H_{(\hat{X}
\label{fig:simStudy}
\end{figure}
\section{Applications}
\label{sec:applications}
\subsection{Nitrogen dioxide air pollution.}
NO$_2$ is one of six air pollutants for which the US Environmental Protection Agency (EPA) has national air quality standards.
We analyze daily EPA NO$_2$ data\footnote{https://www.epa.gov/outdoor-air-quality-data/download-daily-data} from five locations in the Washington DC metropolitan area (see Figure \ref{fig: washingtonDC}).
The first four stations (McMillan 11-001-0043, River Terrace 11-001-0041, Takoma 11-001-0025, Arlington 51-013-0020) have long data records spanning 1995-2020.
Alexandria does not have observations after 2016.
We will perform prediction at Alexandria given data at the other four locations.
Observations in Alexandria actually come from two different stations: 51-510-0009 which has measurements from January1995 to August 2012 and 51-510-00210 from August 2012 to April 2016.
Exploratory analysis did not indicate any detectable change point in the Alexandria data either with respect to the marginal distribution or with dependence with other stations, so we treat this data as coming from a single station.
There are 5163 days between 1995 and 2016 where all five locations have measurements.
Because NO$_2$ levels have decreased over the study period, we detrend at each location using a moving average mean and standard deviation with window of 901 days to center and scale.
Our inner product space assumes each $X_i \in RV_+^1(\alpha = 2)$, and the detrended NO$_2$ data must be transformed to meet this assumption.
In fact, it is unclear whether the NO$_2$ data are even heavy tailed.
Nevertheless, we believe the regular variation framework is useful for describing the tail dependence for this data after marginal transformation.
Characterizing dependence after marginal transformation is justified by Sklar's theorem (\citet{sklar1959}, see also \citet[Proposition 5.15]{resnick1987}), and such transformations are regularly used in multivariate extremes studies.
After viewing standard diagnostic plots, we fit a generalized Pareto distribution above each location's 0.95 quantile and obtain the marginal estimated cdf's $\hat F_i$ which are the empirical cdf below the 0.95 quantile and the fitted generalized Pareto above.
Letting $X_i^{(orig)}$ denote the random variable for detrended NO$_2$ at location $i$, we define $X_i = 1/\sqrt{(1-\hat{F_i}(X_i^{(orig)}))}-\delta$ obtaining a `shifted' Pareto distribution for $i = 1, \ldots, 5$.
Each $X_i \in RV_+(\alpha = 2)$ and the shift $\delta = 0.9352$ is such that $\operatorname{E}[t^{-1}(X_i)]$ = 0.
This shift makes the preimages of the transformed data centered which we found reduced bias in the estimation of the TPDM.
We assume $\bm X = (X_1, \ldots, X_5)^\top \in RV_+^5(\alpha = 2)$.
Further, we let $\bm X_t$ denote the random vector of observations on day $t$, which we assume to be iid copies of $\bm X$.
This is a simplifying assumption as there is temporal dependence in the NO$_2$ data, but it seems less informative that the spatial dependence exhibited by each day's observations.
We first predict during the period prior to 2015 in order that we can use the observed data at Alexandria to assess performance.
Indices are randomly drawn to divide the data set into training and test sets consisting of 3442 and 1721 observations respectively, and both sets cover the entire observational period.
Using the training set, the five-dimensional TPDM $\hat \Sigma_{\bm X}$ is estimated as follows.
Let $\bm x_t$ denote the observed measurements on day $t$.
For each $i \neq j$ in $1, \ldots, 5$, let $r_{t,ij} = \| (x_{t,i}, x_{t,j}) \|_2$ and $(w_{t,i},w_{t,j})=(x_{t,i},x_{t,j})/r_{t,ij}$.
We let $\hat{\sigma}_{ij}=2 n_{exc}^{-1}\sum_{t=1}^{n}{w_{t,i}w_{t,j}\mathbb{I}(r_{t,ij}>r^*_{ij})}$, where $n_{exc} = \sum_{t=1}^{n} \mathbb{I}(r_{t,ij}>r^*_{ij})$.
We choose $r^*_{ij}$ to correspond to the 0.95 quantile.
The constant 2 arises from knowledge that the tail ratio of each $X_i$ is one due to the marginal transformation.
This pairwise estimation of the TPDM differs from the method in \cite{cooley2019decompositions} who used the entire vector norm as the radial component.
\cite{mhatre2021transformedlinear} show that the TPDM is equivalent whether it is defined in terms of the angular measure of the entire vector or the angular measure corresponding to the two-dimensional marginals.
From $\hat \Sigma_{\bm X}$, we obtain $\hat X_{t,5} = \hat {\bm b}^\top \circ \bm X_{t,4}$, where $\hat {\bm b} = (-0.047, 0.177, 0.192, 0.482)^\top$.
We note that the largest weighted component is Arlington, which is closest to Alexandria.
Interestingly, McMillan has a slightly negative weight.
We calculate $\hat X_{t,5}$ for all $t$, but only consider those for which $\hat X_{t,5}$ exceeds the 0.95 quantile.
The left panel of Figure \ref{fig:no2} shows the scatterplot of the values $x_{t,5}$ versus $\hat x_{t,5}$.
By taking the inverse of the marginal transformation, multiplying by the moving average standard deviation and adding the moving average mean, the predicted value can be put on the scale of the original data.
The center panel of Figure \ref{fig:no2} shows the scatterplot on the original scale.
We use the method described in Section \ref{sec:predTPDM} to approximate $H_{(\hat X_{p+1}, X_{p+1})}$ and use the method described in Section \ref{sec:condtlInterval} to create 95\% prediction intervals for each large predicted value $\hat x_{t,5}$.
We chose the matrix $B$ arising from the completely positive decomposition to be $2 \times 9$.
Prediction intervals on the Pareto scale are shown in the left panel of Figure \ref{fig:no2} and the coverage rate of these intervals is 0.965.
The intervals can similarly be back-transformed to be on the original scale as shown in the center panel of Figure \ref{fig:no2}.
The lack of monotonicity in these intervals with respect to the predicted value is due to the trend in the data over the observation period.
For comparison to standard linear prediction, we find the BLUP based on the estimated covariance matrix from the entire data set, and create Gaussian-based 95\% confidence intervals from the estimated MSPE.
When done on the original data, we obtain a coverage rate of 0.88, and when done on square-root transformed data to account for the skewness, we obtain a coverage rate of 0.78.
We also compare our prediction method to the extremes-based method of \cite{cooley2012approximating}, which approximated the conditional distribution of the large values of a regularly varying variate via a parametric model for the angular measure.
The method of \cite{cooley2012approximating} can be done due to this application's relatively low dimension.
As done in \cite{cooley2012approximating}, the pairwise beta model \citep{cooley2010pairwise} is fit by maximum likelihood to the preprocessed training data set.
The 95\% prediction intervals are based on the approximated conditional density of $X_5$ given $x_1, \ldots, x_4$, and the achieved coverage rate for the test set is 0.965.
Because the fitted angular measure model would seemingly contain more information than the estimated TPDM, we were surprised that the widths of the prediction intervals were very similar for the two methods.
The average ratio of \cite{cooley2012approximating} average interval width to our TPDM-based approach was 1.04.
We then apply our prediction method to five dates in 2019 and 2020 (including January 23, 2020 in Figure \ref{fig: washingtonDC}) when observed values at the four recording stations were large and no observation was taken at Alexandria.
Here, we use the entire period from 1995-2016 to estimate the TPDM, and we obtain a slightly different estimate $\hat {\bm b} = (0.026, 0.153, 0.118, 0.461)^\top$.
The right panel of Figure \ref{fig:no2} shows the point estimate and 95\% prediction intervals from our transformed-linear approach (after back transformation to original scale).
The trend at Arlington was used for the unobserved trend at Alexandria.
For comparison, covariance matrix-based BLUP's and MSPE-based 95\% prediction intervals for these dates are shown with a dashed line.
\begin{figure}
\caption{(Left) Scatterplot of $\hat{X}
\label{fig:no2}
\end{figure}
\subsection{Industry portfolios.}
We apply the transformed-linear prediction method to a higher dimensional financial data set.
The data set obtained from the Kenneth French Data Library\footnote{https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data\_library.html} contains the value-averaged daily returns of 30 industry portfolios.
We analyze data for 1950-2020, consisting of $n=17911$ observations.
Since our interest is in extreme losses, we negate the returns, and set negative returns to zero so that data is in the positive orthant.
Although these data appear to be heavy-tailed, it still requires marginal transformation so that $\alpha = 2$ can be assumed.
Let $\bm{X}^{(orig)}$ denote the random vector of the value-averaged daily returns.
For simplicity we use the empirical CDF to perform the marginal transformation $X_i =1/\sqrt{(1-\hat{F_i}(X_i^{(orig)}))}-\delta$, which is applied to each industry's data so that $X_i$ follows the same shifted Pareto distribution as before.
We again assume $\bm X_t$, the random vector denoting the observations on day $t$, are iid copies of $\bm X$.
A training set consisting of two-thirds of the data ($n_{train}=11940)$ is randomly selected and used to estimate the TPDM and obtain the vector $\hat {\bm b}.$
The test set consists of the remaining one-third of the data ($n_{test}=5970)$ to assess coverage rates.
Following similar steps in the previous application, the $30\times 30$ TPDM $\Sigma_{\bm{X}}$ is estimated first in the training set. We focus on performing the linear prediction for extreme losses of coal, beer, and paper.
The three largest coefficients in $\hat {\bm b}_{coal}$ are $(0.42, 0.36, 0.20)$ and correspond to fabricated products and machinery, steel, and oil respectively.
The three largest coefficients $\hat {\bm b}_{beer}$ are $(0.52, 0.24, 0.12)$ and correspond to food products, retail, and consumer goods (household).
The three largest coefficients for $\hat {\bm b}_{paper}$ are $(0.21, 0.11, 0.08)$ and correspond to chemicals, consumer goods (household), and construction materials.
The assessed coverage rates of our transformed linear $95\%$ prediction intervals for coal, beer, and paper are $97.9\%$, $96.3\%$, and $98\%$, respectively.
For the purpose of comparison, we also assessed coverage rates of the MSPE-based $95\%$ prediction intervals.
Because the data are strongly non-Gaussian, we use the empirical CDF to transform the marginals to be standard normal before estimating the covariance matrix.
The coverage rates of MSPE-based 95\% prediction intervals are $79.3\%$, $66.6\%$, and $51.2\%$ for coal, beer, and paper respectively.
\section{Summary and Discussion}
\label{sec:summary}
We have proposed a method for performing linear prediction when observations are large.
To do so, we constructed an inner product space of nonnegative random variables arising from transformed linear combinations of independent regularly varying random variables.
The elements of the TPDM correspond to these inner products if one is willing to assume that these random variables in $\mathcal{V}^q_+$.
The projection theorem yields the optimal transformed linear predictor.
Our method for obtaining prediction intervals shows very good performance both in a simulation study and in two applications.
The method is simple and is based only on the TPDM which is estimable in high dimensions.
We restrict to nonnegative regularly varying random variables to focus on the upper tail.
Relaxing this restriction could allow one to use standard linear operations.
Even when the data can be negative, we believe there is value in focusing in one direction.
In the financial application, tail dependence for extreme losses is different than for gains, and this information is lost when dependence is summarized with a single number as in the TPDM.
The random vectors $\bm X = A \circ \bm Z$ comprised of elements of our vector space have a simple angular measure consisting of $q$ point masses where $q$ is the number of columns of $A$.
Previous models with angular measures consisting of discrete point masses have been criticized as being overly simple.
A difference here is that we do not have to specify $q$ to use this framework to perform prediction, or more generally, we do not have to really believe that our data arise from such a simple model.
Rather, if we are comfortable with the information contained in the TPDM, then we can use its information to easily obtain a point prediction and sensible prediction intervals that reflect the information contained.
In many applications, dependence cannot be measured between the observed values and the value to be predicted.
In kriging for example, a spatial process model is first fit so that covariance between any two locations is quantified.
One can imagine modeling the extremal pairwise dependence as a function of distance before applying the methods here to perform prediction for extreme levels.
\end{document}
|
\begin{document}
\tilde{t}le{Average diagonal entropy in non-equilibrium isolated quantum systems}
\author{Olivier Giraud}
\affiliation{LPTMS, CNRS, Univ.~Paris-Sud, Universit\'e Paris-Saclay, 91405 Orsay, France}
\author{Ignacio Garc\'ia-Mata} \affiliation{Instituto de Investigaciones F\'isicas de Mar del Plata (CONICET-UNMdP), B7602AYL Mar del Plata, Argentina} \affiliation{Consejo Nacional de Investigaciones Cient\'ificas y Tecnol\'ogicas (CONICET), C1425FQB C.A.B.A, Argentina}
\begin{abstract}
The diagonal entropy was introduced as a good entropy candidate especially for isolated quantum systems out of equilibrium.
Here we present an analytical calculation of the average diagonal entropy for systems undergoing unitary evolution and an external perturbation in the form of a cyclic quench. We compare our analytical findings with numerical simulations of various many-body quantum systems.
Our calculations elucidate various heuristic relations proposed recently in the literature.
\end{abstract}
\title{Average diagonal entropy in non-equilibrium isolated quantum systems}
The precision for manipulating quantum systems attained to date
has led to the next big question: how do basic thermodynamics principles operate at very small scales.
Motivated by this question
an ever growing effort has surged attempting to describe quantum thermodynamics of small isolated quantum systems, and their approach to equilibrium and subsequent thermalization. The incredible advance in experimental techniques allowing to follow the time evolution of closed quantum systems
\cite{Paredes2004,*Kinoshita,*Hofferbert} has been the main boost of these endeavors. The relevance of this subject is
(at least) twofold. On the one hand, quantum technologies (e.g.~for quantum information
\cite{BlattRMP} and quantum simulation \cite{Gerri,*Blatt2012,*Serwane,*Korenblit}) tend to be based on systems with negligible interaction with the environment. On the other hand, a complete microscopic thermodynamical description of these advances in nonequilibrium statistical mechanics of such quantum systems has remained elusive, in part
due to the lack of a suitable definition of entropy.
Although the von Neumann entropy $S_{\rm vN}=-\mathrm{tr} \rho \ln \rho$ (with $k_{\rm B}=1$) is a natural tool to measure the entropy of a quantum state $\rho$, it cannot be used to describe the approach to equilibrium of isolated quantum systems, since they undergo unitary dynamics. As an alternative the diagonal entropy (DE)
\begin{equation}
\label{defDE}
S_{\rm D}=-\sum_n \rho_{nn}\ln\rho_{nn}
\end{equation}
was proposed \cite{Polkov2011}, where $\rho_{nn}$ are the diagonal elements of the density matrix in the energy eigenbasis.
The DE possesses most of the expected features of a thermodynamic entropy, such as additivity, or increase when a system at equilibrium undergoes an external perturbation~\cite{Polkov2011, Ikeda2015}. The DE appears to be a fundamental quantity, which can describe the behavior of a very broad class of out-of-equilibrium systems. Clarifying the universality or the specificities of its properties is therefore an important goal.
Two universal properties of DE have been proposed recently. In the case where a system is perturbed by an external operation during a time $\tau$, one can study the DE as a function of the duration $\tau$ of the perturbation. In \cite{Ikeda2015} a conjecture was made introducing bounds on the difference between the time averaged DE, $\overline{S_{\rm D}(\tau)}$, and the DE of the time averaged state $\overline{\rho(\tau)}$ (denoted as $S_{_{\overline{\rho(\tau)}}}$), namely
\begin{equation}
\label{ikeda}
0 \le \Delta S\le 1- \gamma, \ \text{ where }\ \Delta S \stackrel{\rm def}{=}S_{_{\overline{\rho(\tau)}}}-\overline{S_{\rm D}(\tau)}
\end{equation}
and $\gamma=0.5772\ldots$ is Euler's constant. A second property was numerically uncovered in \cite{GarciaMata2015}, where a universal relation was found between $\Delta S$ and a quantity
measuring localization of the initial state.
The aim of this Letter is to provide analytical support to both of these observations and to determine to which extent they are universal. To this end, we derive an analytical expression for $\Delta S$ as an expansion in terms of average generalized participation ratios (PR), which characterize the localization properties of the vector of transition probabilities between the initial and the perturbed eigenstates. Using perturbation theory, we analyze the behavior of the first terms of this expansion in the two extreme regimes of localized and delocalized states.
To support our analytical findings we performed numerical simulations using two representative physical models displaying chaotic and integrable regimes.
We show that our truncated expansion of $\Delta S$ is accurate independently of the physical model and of the localization
properties of the eigenfunctions.
The importance of our results is twofold.
They provide a precise analytical value for the time average of $S_{\rm D}$ and $\Delta S$, improving \cite{Ikeda2015}, and giving a tighter bound.
Moreover, they relate the DE to localization properties of eigenstates of the perturbed system by an explicit and accurate expression. This should lead to a better understanding of the deep connections between Anderson-type transitions
and equilibration characterized by the DE.
It is noteworthy that the DE is uniquely related to the energy distribution \cite{Polkov2011} and it is thus (in principle) a measurable quantity. Therefore, it could be relevant in the experimental description of equilibration and thermalization processes which have gained so much attention due to a flurry of recent breakthroughs (see \cite{Jensen85,Deutsch91, Srednicki1994,Calabrese2006,Rigol2008,Rigol2009,Linden2009,GogolinMullerEisert2011} to name but a few).
For simplicity we consider a cyclic external operation, where a system, described by a Hamiltonian $H=H(\lambda)$ depending on a fixed parameter $\lambda$, undergoes a sudden quench $H\rightarrow H'=H(\lambda+\delta\lambda)$ at time $t=0$ and is then reverted to the Hamiltonian $H$ at time $\tau$. If $H$ is time-independent, the DE $S_D(t)$ of the state $\rho(t)$ is constant for $t>\tau$. It can thus be studied as a function of the perturbation duration $\tau$, and denoted $S_D(\tau)$. For large enough $\tau$, the system evolving under Hamiltonian $H'$ will have the time to equilibrate, so that $S_D(\tau)$ goes to a constant value that can be estimated by considering the average $\overline{S_{\rm D}(\tau)}$.
Let us label by $\ket{n}$ the basis of normalized eigenvectors of $H$ (with eigenvalues $E_n$), and by $\ket{m}$ the basis of eigenvectors of $H'$ (with eigenvalues $E'_m$). We may consider finite systems of size $N$, or truncate our matrices to
a Hilbert space of dimension $N$. We assume that at $t=0$ the system is in an eigenstate $\ket{n_0}\bra{n_0}$ of $H$ with energy $E_{n_0}$. Let $U=e^{-i H'\tau}$ ( $\hbar\stackrel{\rm def}{=} 1$) be the evolution operator from time 0 to $\tau$. At time $\tau$ the state is $\rho(\tau)=U\ket{n_0}\bra{n_0}U^{\dagger}$. Its DE can be expressed as
\begin{equation}
\label{dent}
S_D(\tau)=-\sum_n h_n\ln h_n,{\bf ???}uad h_n=|\bra{n}U\ket{n_0}|^2,
\end{equation}
where $h_n$ is the probability to observe the system in an eigenstate $\ket{n}$ at time $\tau$ (we have the normalization $\sum_n h_n=1$).
Our aim is to calculate the quantity
\begin{equation}
\label{DeltaS}
\Delta S=-\sum_n \overline{h_n}\ln\overline{h_n}+\sum_n \overline{h_n\ln h_n}.
\end{equation}
Since $h_n\in [0,1]$ has finite support, the knowledge of all its moments uniquely defines its distribution $P(h_n)$, from which it is possible to calculate $\Delta S$. Using
\begin{equation}
\label{Unn0}
\bra{n}U\ket{n_0}=\sum_me^{-iE'_m \tau}\langle n\ketbra{m}n_0{\rm a}ngle,
\end{equation}
the probabilities $h_n$ can be expressed as
\begin{equation}
\label{hnbis}
h_n=\!\!\sum_{m_1,m_2}\!e^{i(E'_{m_1}-E'_{m_2}) \tau}\langle n\ketbra{m_2}n_0{\rm a}ngle\langle n_0\ketbra{m_1}n{\rm a}ngle.
\end{equation}
Moments of $h_n$ are obtained by calculating the averages $\overline{h_n^k}$. From Eq.~\eqref{hnbis}, these averages involve averages of quantities $\exp \left[i\left(\sum_{i=1}^k E'_{m_{2i-1}}-\sum_{i=1}^{k} E'_{m_{2i}}\right)\tau\right]$. We make, as in \cite{Ikeda2015}, the assumption that these quantities average to 1 if the sets $\{m_{2i-1},1\leq i\leq k\}$ and $\{m_{2i},1\leq i\leq k\}$ are permutations of each other, and to 0 otherwise. From Eq.~\eqref{hnbis}, keeping only terms with $m_1=m_2$ we have the first moment
\begin{equation}
\label{hnbar}
\overline{h_n}=\sum_m c_{mn},{\bf ???}uad c_{mn}=|\langle m|n_0{\rm a}ngle|^2|\langle m|n{\rm a}ngle|^2.
\end{equation}
The $\overline{h_n}$ are the average probabilities of transition from $\ket{n_0}$ to $\ket{n}$. For the second moment $\overline{h_n^2}$, keeping only terms for which the sets $\{m_1,m_3\}$ and $\{m_2,m_4\}$ are the same, we get
\begin{equation}
\label{hn2}
\overline{h_n^2}=2\left(\sum_m c_{mn}\right)^2-\sum_m c_{mn}^2.
\end{equation}
For integer $q$, we introduce the PR
\begin{equation}
\label{xiqn}
\xi_{q,n}\equiv\frac{\sum_m c_{mn}^q}{(\sum_m c_{mn})^q},
\end{equation}
which characterize the localization properties of the vectors $(c_{mn})_{m}$. We can then express Eq.~\eqref{hn2} as $\overline{h_n^2}/\overline{h_n}^2=2-\xi_{2,n}$. Higher-order moments can be expressed in the same way. For instance we have $\overline{h_n^3}/\overline{h_n}^3=6-9\xi_{2,n}+4\xi_{3,n}$ and $\overline{h_n^4}/\overline{h_n}^4=24-72\xi_{2,n}+18(\xi_{2,n})^2+64\xi_{3,n}-33\xi_{4,n}$. Keeping only the first two terms in these expressions we get the general formula (see Supplemental material for a detailed proof)
\begin{equation}
\label{hnkordre2}
\overline{h_n^k}=\overline{h_n}^k\, k!\,\left(1-\frac{k(k-1)}{4}\xi_{2,n}\right).
\end{equation}
As $\xi_{2,n}$ does not depend on $k$, the moment generating function of $h_n$ can then be resummed as
\begin{equation}
\label{sumM}
M_n(t)=\sum_{k=0}^{\infty}\frac{\overline{h_n^k}}{k!}t^k=\frac{1}{1-\overline{h_n}t}-\frac{(\overline{h_n}t)^2}{2(1-\overline{h_n}t)^3}\xi_{2,n}.
\end{equation}
The probability distribution for $h_n$ is then obtained by inverse Laplace transform of $M_n(t)$, which gives
\begin{equation}
\label{phnext}
P(h_n)=\frac{1}{\,\overline{h_n}\, }e^{-\tfrac{h_n}{\,\overline{h_n}\,}}\left(1-\left[\frac{h_n^2}{4\overline{h_n}^2}-\frac{h_n}{\,\overline{h_n}\, }+\frac12\right]\xi_{2,n}\right).
\end{equation}
Using this distribution, the calculation of the averages in Eq.~\eqref{DeltaS} is then straightforward and direct integration yields
\begin{equation}
\label{conjecturenext}
\Delta S= 1-\gamma-\frac14\overline{\xi}_{2},{\bf ???}uad \overline{\xi}_{2}\equiv\sum_n\overline{h_n}\,\xi_{2,n}.
\end{equation}
Recalling that $\sum_n\overline{h_n}=1$, the quantity $\overline{\xi}_{2}$ appears as the average over PRs \eqref{xiqn} weighted by the mean transition probability $\overline{h_n}$ to go from $\ket{n_0}$ to $\ket{n}$ during the quench. This first resul, Eq.~\eqref{conjecturenext}, has two consequences. First, it makes a more precise statement than the conjecture $\Delta S\leq 1-\gamma$ of \eqref{ikeda} by showing that for finite dimension $N$, as the value of $\overline{\xi}_2$ is finite, the inequality $\Delta S<1-\gamma$ is strictly fullfilled. Moreover, while the equality $\Delta S = 1-\gamma$ was proved in \cite{Ikeda2015} under some assumptions on the localization properties and in the limit $N\to\infty$, our result provides a value for the first correction to the difference between $\Delta S$ and its upper bound in the finite $N$ case.
The second consequence of Eq.~(\ref{conjecturenext}) is that it relates the DE to the average PRs of the overlaps $c_{mn}$, substantiating the observations of \cite{GarciaMata2015}.
In Eq.~\eqref{hnkordre2} we only kept the first two terms in the expression of the moments $\overline{h_n^k}/\overline{h_n}^k$. It is in fact possible to systematically carry out the calculation by keeping successive terms in the expression of the moments, as we show in the Supplemental material. This yields an expansion of the form
\begin{equation}
\label{bigsum}
\Delta S= 1-\gamma+\sum_{\mu}\frac{a_{\mu}\overline{\xi}_{\mu}}{|\mu|(|\mu|-1)},\quad \overline{\xi}_{\mu}\equiv\sum_n\overline{h_n}\,\xi_{\mu_1,n}\xi_{\mu_2,n}\ldots,
\end{equation}
where $a_{\mu}$ are rational numbers and the sum over $\mu$ runs over all finite integer sequences $\mu=(\mu_1,\mu_2,\ldots)$ such that $\mu_i\geq 2$, and $|\mu|=\sum_i\mu_i$. For instance, keeping the few next lowest-order terms in the expression of moments, \eqref{conjecturenext} can be corrected to
\begin{eqnarray}
\label{conjecture3}
\Delta S&=& 1-\gamma-\frac14\overline{\xi}_2+\left(\frac19\overline{\xi}_3+\frac{1}{16}\overline{\xi}_{22}\right)\nonumber\\
&&-\left(\frac{11}{96} \overline{\xi}_4+\frac{1}{6} \overline{\xi}_{32}+\frac{1}{16} \overline{\xi}_{222}\right).
\end{eqnarray}
This expression is the main result of this work. It provides a connection between the average DE and the structure of the eigenfunctions -- localization on the perturbed basis -- through the generalized PR \eqref{xiqn}. It is all the more accurate that the higher-order terms are negligible (which typically is the case in the delocalized regime, as we discuss below). To distinguish these different orders it will be useful to introduce the quantities $O_1=1-\gamma-\overline{\xi}_2/4$, $O_2=O_1+(\overline{\xi}_3/9+\overline{\xi}_{22}/16)$ and $O_3=O_2-(11\overline{\xi}_4/96+ \overline{\xi}_{32}/6+ \overline{\xi}_{222}/16)$.
\begin{figure}
\caption{$\Delta S$ (solid black line) as a function of the coupling strength $\lambda$ for the DM with $j=20$, $N_t=250$, $\tau=10^7$, $\Delta\tau=250$, and quench strength $\delta \lambda=0.1$. The initial state is $\ket{n_0}
\label{fig:Dickeorder}
\end{figure}
To test the consistency of our analytical results we study two different models.
Both of them undergo a localization-delocalization transition when varying one parameter.
The first model is the Dicke model (DM) \cite{Dicke54} describing the dipole interaction of a single mode of a bosonic field, of frequency $\omega$, with $n_s$ two-level particles, with level splitting $\omega_0$. The corresponding Hamiltonian is
\begin{equation}
H(\lambda)=\omega_0 J_z +\omega a^\dagger a+\frac{\lambda}{\sqrt{2 j}}(a^\dagger + a)(J_{-}+J_{+}).
\end{equation}
The collective angular momentum operators $J_z$, $J_\pm$ correspond to a pseudospin $j=n_s/2$, and $a^\dagger$ ($a$) are
creation (annihilation) operators of the field.
The DM undergoes a superradiant quantum phase transition in the thermodynamic limit ($n_s\to \infty$) at $\lambda_c=\sqrt{\omega_0\omega}/2$ \cite{dicke}. For finite $n_s$, there is a transition from Poissonian to Wigner-Dyson level spacing statistics at $\lambda\approx \lambda_c$. This marks a transition from quasi-integrability at small $\lambda$ to quantum chaos at large $\lambda$, as is verified using a semiclassical model in \cite{EmaryBrandes2003}. In our calculations we consider $\omega=\omega_0=1$ ($\lambda_c=0.5$) and the quench is implemented by changing $\lambda\to \lambda+\delta\lambda$.
\begin{figure}
\caption{$\Delta S$ (solid black line) for the SW model with $p=0.06$, $\delta W=0.3$, $N=2^9$, $\tau=10^6$, $\Delta \tau=2500$ (top) and $\Delta \tau=3500$ (bottom), for one realization of disorder and of the shortcut links. Initial states are $\ket{n_0}
\label{fig:SWorder}
\end{figure}
The second model is a quantum smallworld (SW) system with disorder \cite{PhysRevB.62.14780, Giraud2005}. This is a one-dimensional tight-binding Anderson model having $N=2^{n_r}$ sites, with nearest-neighbor interaction and periodic boundary conditions, to which $p\,N$ shortcut links between sites are added, connecting $p\,N$ random pairs of vertices. The Hamiltonian which describes the quantum version of this system is
\begin{equation}
H=\sum_{i}\varepsilon_i\op{i}{i}+\sum_{\langle i,j{\rm a}ngle}V \op{i}{j}
+\sum_{k=1}^{\lfloor pN \rfloor} V(\op{i_k}{j_k}+\op{j_k}{i_k}),
\end{equation}
where $\varepsilon_i$ are Gaussian random variables with zero mean and width $W$, and $V=1$. The second sum runs over nearest-neighbors, while the third term describes the shortcut links of smallworld type, connecting random pairs $(i_k,j_k)$. When $p=0$ the model coincides with the usual one-dimensional Anderson model where all states are known to be localized with localization length $l\sim1/W^2$ for small $W$ \cite{Kramer1993}. The presence of long-range interacting pairs for $p>0$ induces a delocalization transition from localized states for large $W$ to delocalized states at small $W$ \cite{Giraud2005, NOUSunpub}.
The quench in this case is implemented by keeping the shortcut links $(i_k,j_k)$ fixed and changing $W\to W-\delta W$ .
To compute $\Delta S$ we fully diagonalize $H$ and $H'$ to calculate the $h_n$ and perform the average \eqref{DeltaS} over a window $[\tau,\tau+\Delta \tau]$ for a very large $\tau$. Examples of $S_{_{\overline{\rho(\tau)}}}$ and $\overline{S_{\rm D}(\tau)}$ are plotted in the inset of Fig.~\ref{fig:Dickeorder}. In the case of the DM, we take into account the parity symmetry, and we truncate the phonon basis to a finite size $N_t$ (we only consider energies well inside the converged part of the spectrum).
In Fig.~\ref{fig:Dickeorder} we show $\Delta S$ for the DM as a function of the coupling strength $\lambda$ for two different initial states.
At large $\lambda$ (delocalized regime), order $O_1$ already approximates $\Delta S$ rather well. At small $\lambda$ (localized regime), $O_1$ and $O_2$ tend to a constant value corresponding to the fully localized case $\overline{\xi}_\mu\to 1$, while $O_3$ matches $\Delta S$ very well. The higher the energy, the smaller the localized plateau is, as illustrated in Fig.~\ref{fig:Dickeorder}; this can be understood by the fact that the quench induces more transitions at higher energies (see also \cite{GarciaMata2015}). In Fig.~\ref{fig:SWorder} we present similar results for the SW model: in the delocalized (small-$W$) regime, order $O_1$ gives again a good approximation of $\Delta S$, while in the localized regime $O_3$ gives a very good approximation for $\Delta S$. Note that Fig.~\ref{fig:SWorder} corresponds to a single disorder and shortcut link realization (we have checked that the results are equivalent for any realization with sufficiently large $\Delta \tau$).
In order to understand these features, we consider two limiting situations. When equilibrium is reached `ideally', the eigenvectors $\ket{m}$ of $H'$ are uniformly spread (i.~e.~delocalized) in the $\ket{n}$ basis, thus each overlap is $|\langle m\ket{n}|^2\sim 1/N$, so that $c_{mn}\sim 1/N^2$ for all $m,n$. For the PR this implies $\xi_{q,n}\sim N^{1-q}$, and thus $\overline{\xi}_{\mu}\sim N^{\textrm{lg}(\mu)-|\mu|}$, with lg$(\mu)$ the number of terms in the sequence $\mu$. In the opposite case where eigenvectors $\ket{m}$ and $\ket{n}$ coincide, all $\overline{\xi}_{\mu}$ are equal to 1. These distinct features reflect in the behavior of the moments $\overline{h_n^k}$ (for instance, the sum of mean return probabilities $\sum_{n_0}\overline{h_{n_0}}$ was proposed as a tool to measure the degree of equilibration of an isolated quantum system \cite{Luck2015}).
In these two extreme regimes $\Delta S$ has very distinct behaviors. In the delocalized case where $\overline{\xi}_{\mu}\sim N^{\textrm{lg}(\mu)-|\mu|}$, we have for instance $\overline{\xi}_2\sim 1/N$, while the two last brackets in Eq.~\eqref{conjecture3} correspond to terms scaling as $1/N^2$ and $1/N^3$, respectively. One can thus truncate the sum \eqref{bigsum} to any order by keeping terms with the same power in $N$. Order 0 is simply given by the constant $1-\gamma$, which coincides with the result of \cite{Ikeda2015}. Keeping the term in $1/N$ yields our first main result Eq.~\eqref{conjecturenext}, and as already mentioned explains the conjecture of \cite{Ikeda2015}.
In the localized case, on the other hand, the expansion \eqref{bigsum} is no longer valid, as each term $\overline{\xi}_{\mu}$ is of order 1. However, the truncation \eqref{conjecture3} happens to yield a very good approximation for $\Delta S$, as shown on the physical models in Figs.~\ref{fig:Dickeorder} and \ref{fig:SWorder}, where at small $\lambda$ or large $W$ the numerically computed $\Delta S$ is almost indistinguishable from the expression for $O_3$. This can be understood via the following perturbation-theory approach. Let us consider the simplest case of a localized model, where $H'=H+\epsilon V$ and $V$ is a symmetric matrix with elements of order 1. Standard perturbation theory for $\epsilon\ll 1$ yields
\begin{equation}
\label{overlap_perturb}
\langle n\ket{m}=\delta_{nm}\left(1-\frac{1}{2}\sum_{k\neq m}v_{km}^2\right)+\epsilon(1-\delta_{nm}) v_{nm},
\end{equation}
with $v_{nm}=\epsilon V_{nm}/(E_m-E_n)$ and $\delta_{nm}$ is the Kronecker symbol. Inserting this expression into \eqref{Unn0}, we get, at lowest order in $\epsilon$,
\begin{equation}
\label{hnperturb}
h_n=\left\{
\begin{array}{cc}4\sin^2[(E'_n-E'_{n_0})\tau/2]v_{nn_0}^2\quad&n\neq n_0\\
&\\
1-2\sum_{k\neq n_0}v_{n_0k}^2\quad&n=n_0\,.
\end{array}
\right.
\end{equation}
Upon averaging over $\tau$ in \eqref{DeltaS}, the term $n=n_0$ does not contribute (as it does not depend on $\tau$), while terms $n\neq n_0$ yield a contribution involving the average $\overline{z\log z}-\overline{z}\log(\overline{z})=\frac12(1-\log 2)$ (where $z\stackrel{\rm def}{=}\sin^2x$).
At order $\epsilon^2$ the average \eqref{DeltaS} is thus given by
\begin{equation}
\label{deltaSperturb1}
\Delta S\simeq2(1-\log 2)\sum_{n\neq n_0}v_{nn_0}^2.
\end{equation}
One can similarly calculate a perturbation expansion for the $\overline{\xi}_{\mu}$, by injecting \eqref{overlap_perturb} into $c_{mn}$ given by \eqref{hnbar} and calculating the PR defined by \eqref{xiqn}. At order $\epsilon^2$ one gets
\begin{eqnarray}
\label{xiperturb}
\overline{\xi}_2&\simeq&1-\sum_{n\neq n_0}v_{nn_0}^2,\quad
\overline{\xi}_3\simeq\overline{\xi}_{22}\simeq1-\frac32\sum_{n\neq n_0}v_{nn_0}^2,\nonumber\\
\overline{\xi}_4&\simeq&\overline{\xi}_{32}\simeq\overline{\xi}_{222}\simeq1-\frac74\sum_{n\neq n_0}v_{nn_0}^2.
\end{eqnarray}
Using \eqref{xiperturb} to calculate the successive orders of $\Delta S$ given in \eqref{conjecture3} we obtain
\begin{eqnarray}
\label{deltaSperturb2}
O_1\simeq&\frac34-\gamma+\frac14\sum_{n\neq n_0}v_{nn_0}^2,\\
\label{deltaSperturb3}
O_2\simeq&\frac{133}{144}-\gamma-\frac{1}{96}\sum_{n\neq n_0}v_{nn_0}^2,\\
\label{deltaSperturb4}
O_3\simeq&\frac{167}{288}-\gamma+\frac{227}{384}\sum_{n\neq n_0}v_{nn_0}^2.
\end{eqnarray}
As the constant in front of $\epsilon^2$ in \eqref{deltaSperturb3} is $1/96$, at this order $\Delta S$ is essentially equal to $133/144-\gamma\simeq 0.346395$, as can be seen in the figures in the localized region. At the next order $O_3$ on the other hand, the constant $167/288-\gamma\simeq 0.00264545$ almost vanishes, while $227/384\simeq 0.591146$ is numerically very close to $2(1-\log 2)\simeq 0.613706$. Thus the expression \eqref{deltaSperturb4} almost coincides with \eqref{deltaSperturb1}, which explains why the truncation at $O_3$ works so well. Truncation at order 4 (which can be obtained from the general result given explicitly in the Supplemental material) would yield $\Delta S\simeq 1.2184 - 1.68839 \sum_{n\neq n_0}v_{nn_0}^2$, so that this approximation is worse than order 3 (and the same happens at higher orders). $O_3$ thus appears as the optimal truncation in the localized regime. As was noted above, this truncation also works quite well in the delocalized regime.
One sees from the numerical results that Eq.~\eqref{conjecture3} is in fact a good approximation for $\Delta S$ over the whole range of parameters.
To summarize,
there is a quest for a quantum entropy with all the required properties.
The DE is a good candidate and here we give an analytical expression for its time average as a function of eigenvector properties only, to a very good approximation and independently of the dynamical properties, in particular of the localization-delocalization or chaotic-integrable characteristics.
Our results establish the validity of the conjecture of Ref.~\cite{Ikeda2015} and extend its accuracy.
These results are
relevant in the relation between many-body localization transition and the study of equilibration in non-equilibrium isolated quantum systems.
\acknowledgments
IGM and OG received partial funding from a binational project from CONICET (grant no. 1158/14) and CNRS (grant no PICS06303).
IGM also received a funding from Universit\'e Toulouse III Paul Sabatier as an invited professor.
IGM thanks D. A. Wisniacki for fruitful discussions.
\begin{thebibliography}{27}
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\onecolumngrid
\appendix
\begin{center}
{\large\textbf{Supplemental material for ``Average diagonal entropy in non-equilibrium isolated quantum systems
''}}
\end{center}
In this Supplemental material we provide a detailed proof of the general relation Eq.~(14) of the main text. We recall that $\Delta S$ is defined as
\begin{equation}
\label{DeltaS2}
\Delta S=-\sum_n \overline{h_n}\ln\overline{h_n}+\sum_n \overline{h_n\ln h_n},
\end{equation}
with transition probabilities given by
\begin{equation}
\label{hnbis2}
h_n=\!\!\sum_{m_1,m_2}\!e^{i(E'_{m_1}-E'_{m_2}) \tau}\langle n\ketbra{m_2}n_0{\rm a}ngle\langle n_0\ketbra{m_1}n{\rm a}ngle.
\end{equation}
To make notations lighter we will drop the index $n$ until the very last equation. The demonstration goes as follows. First we obtain an expression for the moments $\overline{h^k}$ as sums over integer partitions (section \ref{subs1}).
Then, through the inverse Laplace transform, we can get the distribution $P(h)$ to calculate \equa{DeltaS} (section \ref{subs2}).
To achieve this we first have to reexpress the moments in a way where resummation and simplification becomes feasible (section \ref{subs3}), and finally obtain the full expression for $\Delta S$ (section \ref{subs4}).
\subsection{Moments as a sum over integer partitions}
\label{subs1}
The first step is to calculate the moments $\overline{h^k}$ averaged over $\tau$, assuming that the quantities
$e^{ \left[i\left(\sum_{i=1}^k E'_{m_{2i-1}}-\sum_{i=1}^{k} E'_{m_{2i}}\right)\tau\right]}$ average to 1 if the sets $\{m_{2i-1},1\leq i\leq k\}$ and $\{m_{2i},1\leq i\leq k\}$ are permutations of each other, and to 0 otherwise. In the main paper we gave the first averages
\begin{eqnarray}
\label{moment2}
\overline{h^2}/\overline{h}^2&=&2-\xi_2\\
\label{moment3}
\overline{h^3}/\overline{h}^3&=&6-9\xi_2+4\xi_3\\
\label{moment4}
\overline{h^4}/\overline{h}^4&=&24-72\xi_2+18(\xi_2)^2+64\xi_3-33\xi_4
\end{eqnarray}
in terms of the participation ratios (PR) of the vectors $(c_{m})_{m}$ for integer $q$,
\begin{equation}
\label{xiqn2}
\xi_{q}\equiv\frac{\sum_m c_{m}^q}{(\sum_m c_{m})^q}, {\bf ???}uad c_{m}=|\langle m|n_0{\rm a}ngle|^2|\langle m|n{\rm a}ngle|^2, {\bf ???}uad \sum_mc_m=\overline{h}.
\end{equation}
It is possible to obtain these expressions in a systematic way by introducing integer partitions. It is usual to denote by $\lambda\vdash k$ a partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ of $k$, with $\lambda_1\geq\lambda_2\geq\ldots$ and $\sum_i\lambda_i=k$ (it can be padded on the right by an arbitrary number of zeros). The products of $\xi_q$ appearing in the expressions \eqref{moment2}--\eqref{moment4} for $\overline{h^k}$ correspond to all possible integer partitions of the integer $k$. For instance, noticing that $\xi_1=1$, the terms $\xi_4$, $\xi_3\xi_1$, $\xi_2^2$, $\xi_2\xi_1^2$ and $\xi_1^4$ contributing to Eq.~\eqref{moment4} correspond to the partitions $4=3+1=2+2=2+1+1=1+1+1+1$. Following standard textbooks \cite{Macdo}, one can define several families of symmetric polynomials of the variables $c_m$, $1\leq m\leq N$. We set
\begin{equation}
\label{mlambda}
m_{\lambda}=\sum_\sigma c_1^{\sigma(\lambda_1)}c_2^{\sigma(\lambda_2)}\ldots c_N^{\sigma(\lambda_N)},
\end{equation}
where the sum runs over all permutations of $(\lambda_1,\lambda_2,\ldots,\lambda_N)$ (see p.~18 of \cite{Macdo}); if $N$ is smaller than the number of nonzero $\lambda_i$'s then by convention $m_{\lambda}=0$. We also define the polynomials
\begin{equation}
\label{plambda}
p_{\lambda}=(\sum_m c_m^{\lambda_1})(\sum_m c_m^{\lambda_2})\ldots
\end{equation}
(see p.~24 of \cite{Macdo}). The $p_{\lambda}$ are simply related to the PR defined by Eq.~\eqref{xiqn2} by
\begin{equation}
\label{pxi}
p_{\lambda}=\overline{h}^k\xi_{\lambda_1}\xi_{\lambda_2}\cdots
\end{equation}
For $\lambda\vdash k$, the $p_{\lambda}$ and $m_{\lambda}$ are related by the linear relation
\begin{equation}
\label{pLm}
p_{\lambda}=\sum_{\mu\vdash k}L_{\lambda\mu}m_{\mu}
\end{equation}
(p.~103 of \cite{Macdo}), with $L_{\lambda\mu}$ an invertible lower-triangular matrix of integers indexed by partitions $\lambda,\mu$, of $k$.
Taking the $k\,$th power of Eq.~\eqref{hnbis2} and keeping only terms where the energies exactly compensate, we get the general expression of the $k\,$th moment as
\begin{equation}
\label{hk1}
\overline{h^k}=\sum_{m_1,\ldots,m_k}P(m_1,\ldots,m_k)c_{m_1}c_{m_2}\ldots c_{m_k}
\end{equation}
with $P(m_1,\ldots,m_k)$ the number of permutations of the $m_i$. We then group together all terms with the same `pattern' of indices, e.g., for $k=3$, terms with $m_1=m_2=m_3$ or $m_1=m_2\neq m_3$ or $m_1\neq m_2\neq m_3$, which corresponds to the different integer partitions 3=2+1=1+1+1. The expression \eqref{hk1} simply reduces to
\begin{equation}
\label{hk2}
\overline{h^k}=\sum_{\lambda\vdash k}P_{\lambda}^2m_{\lambda}
\end{equation}
with $P_{\lambda}$ the multinomial coefficient associated with $(\lambda_1,\lambda_2,\ldots)$ and $m_{\lambda}$ the symmetric polynomial \eqref{mlambda}. In order to relate the mean moments to the PR, we want to express $\overline{h^k}$ by means of the $p_{\lambda}$ rather than the $m_{\lambda}$; inverting the relation \eqref{pLm} we get
\begin{equation}
\label{hk3}
\overline{h^k}=\sum_{\lambda\vdash k}\sum_{\mu\vdash k}P_{\lambda}^2(L^{-1})_{\lambda\mu}p_{\mu}.
\end{equation}
Setting $Z_{\mu}=\sum_{\lambda\vdash k}P_{\lambda}^2(L^{-1})_{\lambda\mu}$ we have the final compact expression
\begin{equation}
\overline{h^k}=\sum_{\mu\vdash k}Z_{\mu}p_{\mu}.
\label{finalhk}
\end{equation}
The $L_{\lambda\mu}$, and thus the $Z_{\mu}$, can be calculated very easily using mathematical software, while $p_{\mu}$ is obtained from Eq.~\eqref{pxi}. For instance, one gets for $k=3=2+1=1+1+1$ (where partitions are ordered in that reverse lexicographic order) the matrix $L=\{\{1,0,0\},\{1,1,0\},\{1,3,6\}\}$, and multinomial coefficients $P_{3}=1$, $P_{21}=3$ and $P_{111}=6$, so that $Z_{\mu}$ is the vector $(4, -9, 6)$, which allows to recover indeed Eq.~\eqref{moment3}.
\subsection{From moments to entropy difference}
\label{subs2}
The knowledge of the moments \eqref{finalhk} allows to reconstruct the probability distribution $P(h)$. Indeed, let
\begin{equation}
\label{defM}
M(t)=\int_{0}^{\infty}dh P(h) e^{t h}
\end{equation}
be the moment generating function of $P(h)$. Then $M(t)$ can be expressed as a series
\begin{equation}
\label{sumM2}
M(t)=\sum_{k=0}^{\infty}\frac{\overline{h^k}}{k!}t^k.
\end{equation}
It is then possible to get $P(h)$ from inverse Laplace transform of $M(t)$, namely
\begin{equation}
\label{inverselaplace}
P(h)=\frac{1}{2i\pi}\int_{c-\textrm{i}\infty}^{c+\textrm{i}\infty}dt e^{t h}M(-t)
\end{equation}
with $c$ a real number such that the contour in \eqref{inverselaplace} goes to the right of all poles (the $M(-t)$ comes from the fact that \eqref{defM} is not exactly a Laplace transform as there is a factor $e^{t h}$ rather than $e^{-t h}$). The entropy difference $\Delta S$ associated with $h$ is then simply given by
\begin{equation}
\label{entropydiff}
-\overline{h}\ln\overline{h}+\overline{h\ln h}=-\overline{h}\ln\overline{h}+\int_{0}^{\infty}dh P(h) h\ln h.
\end{equation}
In order to perform the sum over $k$ in \eqref{sumM2} we first have to rewrite the moments $\overline{h^k}$ under a form where the dependence on $k$ is more transparent.
\subsection{Reexpressing the moments}
\label{subs3}
The aim of this section is to show that $Z_{\mu}$ appearing in Eq.~\eqref{finalhk} can be put under the form
\begin{equation}
\label{zmutilde}
Z_{\mu}=k!\,\binom{k}{s}\tilde{Z}_{\mu'}
\end{equation}
where $s=\sum_{\mu_i\geq 2}\mu_i$ and $\mu'$ is the partition of $s$ obtained by removing the 1's from $\mu$, with $\tilde{Z}_{\mu'}$ rational numbers. For instance, for all partitions of the form $\mu=(211\ldots1)$ we have $\mu'=(2)$ and $s=2$. From its explicit definition $Z_{\mu}=\sum_{\lambda\vdash k}P_{\lambda}^2(L^{-1})_{\lambda\mu}$ it is easy to calculate $Z_{2}=-1$, $Z_{21}=-6$, $Z_{211}=-12$, $Z_{2111}=-20$ and so on, so that Eq.~\eqref{zmutilde} holds with $\tilde{Z}_{2}=-\frac12$. Similarly we can compute $\tilde{Z}_{3}=\frac23$, $\tilde{Z}_{4}=-\frac{11}{8}$, $\tilde{Z}_{22}=\frac34$. In the special case of partitions of the form $\mu=(111\ldots1)$, Eq.~\eqref{zmutilde} can still be satisfied by setting $\mu'=(0)$ and $\tilde{Z}_{0}=1$.
In order to show Eq.~\eqref{zmutilde}, we start by noting that the $L_{\lambda\mu}$ can be interpreted as the number of ways of adding together the $\lambda_i$ in order to obtain the $\mu_i$ (see \cite{Macdo} p.~103). For example, for $k=3$ the partition $\lambda=(111)$ can yield the partition $\mu=(21)$ by adding two 1's. There are three different ways of doing so, hence the entry $L_{111,21}=3$ given in subsection \ref{subs1}.
Let $\lambda\vdash k$ be a fixed partition of $k$ such that all $\lambda_i\geq 2$. From their definition, the quantities $Z_{\mu}$ verify
\begin{equation}
\label{sum1}
\sum_{\mu\vdash k}L_{\mu\lambda}Z_{\mu}= P_{\lambda}^2.
\end{equation}
It can be rewritten as a double sum over the number $r$ of 1's in the partition $\mu$ and over partitions $\mu'$ of $k-r$ not containing any 1. We thus have
\begin{equation}
\label{sumrmup}
\sum_{r=0}^{k}\sum_{\genfrac{}{}{0pt}{}{\mu'\vdash k-r}{\mu'_i\geq 2}}L_{\mu\lambda}\tilde{Z}_{\mu}k!\,\binom{k}{r}= P_{\lambda}^2,
\end{equation}
where we have introduced the notation $\tilde{Z}_{\mu}=Z_{\mu}/(k!\,\binom{k}{r})$, and $\mu=(\mu'1\ldots 1)$ with $r$ numbers 1. The goal is to show that $\tilde{Z}_{\mu}$ in fact only depends on $\mu'$ and not on $r$.
If we adjoin a 1 to the partition $\lambda$, we get the partition $(\lambda 1)$, which is a partition of $k+1$. For this partition, Eq.~\eqref{sumrmup} yields
\begin{equation}
\label{sumkr}
\sum_{r=0}^{k+1}\sum_{\genfrac{}{}{0pt}{}{\nu'\vdash k+1-r}{\nu'_i\geq 2}}L_{\nu(\lambda 1)}\tilde{Z}_{\nu}(k+1)!\,\binom{k+1}{r}= P_{(\lambda 1)}^2=(k+1)^2P_{\lambda}^2.
\end{equation}
Since $L_{\nu(\lambda 1)}$ can be interpreted as the number of ways of adding together the $\nu_i$ in order to obtain the elements of $(\lambda 1)$, which are the $\lambda_i$ and the additional term 1, necessarily this additional 1 has to come from a 1 appearing among the $\nu_i$. In particular this implies that the term $r=0$ in the sum \eqref{sumkr} must vanish, and that $\nu$ is of the form $(\mu 1)$. There are $r$ possible ways of choosing this additional 1 (the total number of 1's in $\nu$), and then the number of ways to group the remaining $\nu_i=\mu_i$ to get the $\lambda_i$ is precisely $L_{\mu\lambda}$. Thus $L_{\nu(\lambda 1)}=L_{(\mu 1)(\lambda 1)}=r L_{\mu\lambda}$, and $\nu'=\mu'$. Shifting the sum in \eqref{sumkr} yields
\begin{equation}
\label{sumkr2}
\sum_{r=0}^{k}\sum_{\genfrac{}{}{0pt}{}{\mu'\vdash k-r}{\mu'_i\geq 2}}(r+1)L_{\mu\lambda}\tilde{Z}_{(\mu 1)}(k+1)!\,\binom{k+1}{r+1}=(k+1)^2P_{\lambda}^2,
\end{equation}
which after simplification reduces to
\begin{equation}
\label{sumkr3}
\sum_{r=0}^{k}\sum_{\genfrac{}{}{0pt}{}{\mu'\vdash k-r}{\mu'_i\geq 2}} L_{\mu\lambda}\tilde{Z}_{(\mu 1)}k!\,\binom{k}{r}=P_{\lambda}^2.
\end{equation}
Comparing Eq.~\eqref{sumkr3} with Eq.~\eqref{sumrmup} one gets that $\tilde{Z}_{(\mu 1)}=\tilde{Z}_{\mu}$. Proceeding in the same way recursively one can show that all $\tilde{Z}_{(\mu 1\ldots 1)}$ are equal: we denote them $\tilde{Z}_{\mu'}$, which proves Eq.~\eqref{zmutilde}.
\subsection{Resummation of contributions}
\label{subs4}
We are now in a position to calculate the distribution $P(h)$. Using the above result, Eq.~\eqref{finalhk} can be rewritten
\begin{equation}
\overline{h^k}=\sum_{\mu\vdash k}k!\,\binom{k}{s}\tilde{Z}_{\mu'}p_{\mu'}p_1^{k-s},
\label{finalhk2}
\end{equation}
using the definition \eqref{plambda} of $p_{\lambda}$ and the notation $\mu=(\mu'11\ldots 1)$, and with $s=\sum_i\mu'_i$. Any given sequence of numbers $\mu'=(\mu'_1\mu'_2\ldots)$ with $\mu'_i\geq 2$ will contribute to each moment $\overline{h^k}$ with $k\geq s$ through the partition $(\mu'11\ldots 1)$ of $k$ with $(k-s)$ 1's. The contribution of $\mu'$ to $M(t)$ will be (using $p_1=\overline{h}$)
\begin{equation}
\sum_{k=s}^{\infty}t^k\binom{k}{s}\tilde{Z}_{\mu'}p_{\mu'}p_1^{k-s}=\frac{t^s\tilde{Z}_{\mu'}p_{\mu'}}{(1-\overline{h}t)^{s+1}}.\label{mtgeneral}
\end{equation}
The inverse Laplace transform \eqref{inverselaplace} then yields the contribution of $\mu'$ to the probability distribution $P(h)$. There is a single pole of order $s+1$ at $-1/\overline{h}$, whose residue is given by
\begin{equation}
\label{bigsum2}
\frac{1}{s!}\frac{\tilde{Z}_{\mu'}p_{\mu'}}{\overline{h}^{s+1}}\lim_{t\to -1/\overline{h}}\frac{\partial^s}{\partial t^s}\left(t^se^{t h}\right)=
\sum_{r=0}^{s}\binom{s}{r}^2\frac{r!}{s!}\left(-\frac{h}{\overline{h}}\right)^{s-r}
\frac{e^{-h/\overline{h}}}{\overline{h}^{s+1}}\tilde{Z}_{\mu'}p_{\mu'}.
\end{equation}
The contribution to the entropy difference \eqref{entropydiff} is then obtained by evaluating integrals of the form
\begin{equation}
\int_{0}^{\infty}\!dh\left(-\frac{h}{\overline{h}}\right)^{a}\frac{e^{-h/\overline{h}}}{\overline{h}}h\ln h =(-1)^a (a+1)!\,\, \overline{h} \left(\ln\overline{h}+\frac{|S_{a+2}|}{(a+1)!}-\gamma\right),
\end{equation}
where $S_{a}$ are Stirling numbers of the first kind and $\gamma=0.5772\ldots\,$ is Euler's constant. Performing the summation over $r$ in \eqref{bigsum2}, this term reduces to $\overline{h}\left(\ln\overline{h}+1-\gamma\right)$ if $s=0$, and to
\begin{equation}
\label{bigsum3}
\frac{\tilde{Z}_{\mu'}p_{\mu'}}{\overline{h}^{s-1}}\sum_{r=0}^{s}\binom{s}{r}(-1)^r(r+1)\left(\ln\overline{h}-\gamma+\frac{|S_{r+2}|}{(r+1)!}\right)=\frac{\tilde{Z}_{\mu'}p_{\mu'}}{s(s-1)\overline{h}^{s-1}}
\end{equation}
if $s\geq 2$. The last expression has been obtained by using the identities for $s\geq 2$ (see e.g.~\cite{spivey})
\begin{equation}
\sum_{r=0}^{s}\binom{s}{r}(-1)^r(r+1)=0
\end{equation}
and
\begin{equation}
\sum_{r=0}^{s}\binom{s}{r}(-1)^r\frac{|S_{r+2}|}{r!}=\frac{1}{s(s-1)}.
\end{equation}
Summing up contributions from all possible $\mu'$ one finally has
\begin{equation}
\label{bigsum4}
-\overline{h}\ln\overline{h}+\overline{h\ln h}=\overline{h}\left(1-\gamma\right)+\sum_{\mu'}\frac{\tilde{Z}_{\mu'}\overline{h}}{s(s-1)}\frac{p_{\mu'}}{\overline{h}^{s}}
\end{equation}
where $s=\sum_{i}\mu'_i$ and the sum runs over all sequences $\mu'=(\mu'_1\mu'_2\ldots)$ with $\mu'_i\geq 2$. By increasing order these partitions are $(2),(3),(4),(22),(5),(32),\ldots$. Using Eq.~\eqref{pxi} the term $p_{\mu'}/\overline{h}^{s}=\xi_{\mu'_1}\xi_{\mu'_2}\ldots$ is just a product of generalized participation ratios. Reintroducing the dependence in $n$ for $h\equiv h_n$, we get (recall that $\sum_n\overline{h_n}=1$) the final expression
\begin{equation}
\label{finalexpr}
-\sum_n \overline{h_n}\ln\overline{h_n}+\sum_n \overline{h_n\ln h_n}=1-\gamma+\sum_{\mu'}\frac{\tilde{Z}_{\mu'}}{s(s-1)}\overline{\xi}_{\mu'}
\end{equation}
with $s=\sum_{i}\mu'_i$ and $\overline{\xi}_{\mu}\equiv\sum_n\overline{h_n}\,\xi_{\mu_1,n}\xi_{\mu_2,n}\ldots$.
\end{document}
|
\begin{document}
\begin{titlepage}
\begin{center}
\Large{\textbf{Experimental demonstration of scalable quantum key distribution over a thousand kilometers}} \\
\large{A.\,Aliev, V.\,Statiev, I.\,Zarubin, N.\,Kirsanov, D.\,Strizhak, A.\,Bezruchenko, A.\,Osicheva, A.\,Smirnov, M.\,Yarovikov, A.\,Kodukhov, V.\,Pastushenko, M.\,Pflitsch, V.\,Vinokur}
\Large{\it{Terra Quantum AG}}
\end{center}
\end{titlepage}
\section*{Abstract}
\noindent
Secure communication over long distances is one of the major problems of modern informatics. Classical transmissions are recognized to be vulnerable to quantum computer attacks.
Remarkably, the same quantum mechanics that engenders quantum computers offer guaranteed protection against these attacks via a quantum key distribution (QKD) protocol.
Yet, long-distance transmission is problematic since the signal decay in optical channels occurs at distances of about a hundred kilometers.
We resolve this problem by creating a QKD protocol, further referred to as the Terra Quantum QKD protocol (TQ-QKD protocol), using
semiclassical pulses containing enough photons for random bit encoding and exploiting erbium amplifiers to retranslate photon pulses and, at the same time, ensuring that at this intensity only a few photons could go outside the channel even at distances about hundred meters.
As a result, an eavesdropper will not be able to efficiently utilize the lost part of the signal.
A central TQ-QKD protocol’s component is the end-to-end control over losses in the transmission channel which, in principle, could allow an eavesdropper to obtain the transmitted information.
However, our control precision is such that if the degree of the leak falls below the control border, then the leaking states are quantum since they contain only a few photons.
Therefore, available to an eavesdropper parts of the bit encoding states representing `0' and `1' are nearly indistinguishable.
Our work presents the experimental realization of the TQ-QKD protocol ensuring secure communication over 1032 kilometers.
Moreover, further refining the quality of the scheme's components will greatly expand the attainable transmission distances.
This paves the way for creating a secure global QKD network in the upcoming years.
\section{Introduction}
The TQ-QKD protocol\,\cite{new_theory} resolves the problem of secure long-distance communications.
A threat of breaking current standardized public key algorithms (e.g. the RSA, ECC, and DSA) was brought by the emergence of Shor's algorithm.
Although Shor's algorithm can only be executed on massive quantum computers that as yet do not exist, this threat must not be ignored.
Fortunately, the same quantum physics brings in a possibility for QKD\cite{qkd:1, qkd:2, qkd:3, qkd:4, qkd:5, qkd:6, qkd:7}, a secure communication method that implements a cryptographic protocol involving components of quantum mechanics.
It enables communicating participants to generate a shared random secret key known only to them, which then can be used to encrypt and decrypt messages.
A unique property of quantum key distribution providing its security relies on the foundations of quantum mechanics: the ability of the communicating users to detect the presence of any third party trying to gain knowledge of the key.
This results from the fundamental quantum mechanics principle, the fact that measuring disturbs the measured quantum states.
A third party trying to eavesdrop on the key inevitably creates detectable anomalies.
By careful analysis of transmitting quantum states, a communication system detects eavesdropping and immediately takes measures to fully secure the transmission.
Most of the existing QKD applications are curtailed by channel losses that result in the exponential decay of the signal with the distance as dictated by the fundamental Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound\,\cite{plob}.
In this framework up-to-date record secret key generation rates\,\cite{compare_qkd_mdi, compare_qkd_bb84} do not exceed several bits per second at distances of 400\,km.
To overcome the PLOB bound, one can introduce trusted nodes\cite{trusted_nodes:1, trusted_nodes:2, trusted_nodes:3} where local secret keys are produced for each QKD link between nodes and stored in the nodes.
This implies re-coding of the quantum information into a fully classical one at these trusted nodes and completely eliminates quantum protection of an overall protocol.
The way to preserve quantumness is the use of quantum repeaters.
Ideally, quantum repeaters\cite{quantum_repeaters:1, quantum_repeaters:2, quantum_repeaters:3, quantum_repeaters:4, quantum_repeaters:5, quantum_repeaters:6, quantum_repeaters:7, quantum_repeaters:8, quantum_repeaters:9, quantum_repeaters:10, quantum_repeaters:11, quantum_repeaters:12, quantum_repeaters:13, quantum_repeaters:14, quantum_repeaters:15, quantum_repeaters:16} would have been expected to decontaminate and forward quantum signals without directly measuring or cloning them.
However, such idealized quantum repeaters remain unavailable for existing technologies.
The only available secure method to beat the PLOB bound is the Twin-Field QKD\,\cite{lucamarini, compare_qkd_tf, compare_qkd_tf_1002} (working only at relatively short distances) which sends quantum states from both Alice and Bob to the intermediate point.
This method, however, is not scalable and dramatically suffers from channel losses as well.
Figure\,\ref{comparing} summarizes some of the previous realizations of long-distance QKD including the current record distance\,\cite{compare_qkd_tf_1002} and underlines the superiority of our work in terms of speed and distance compared to other works. Our work presents the realization of the TQ-QKD protocol eliminating the PLOB bound by using quantum thermodynamic restrictions and quantum mechanics-based loss control in the optical channel.
The implementation of the secure long-distance transmission line is based on using the Erbium-Doped Fiber Amplifiers (EDFA) of our own Terra Quantum construction arranged every 50\,km which have enabled the practical realization of the optical fiber channel transmission line over 1032\,km.
\begin{figure}
\caption{\textbf{Comparing of our results with the earlier long-distance QKD realizations\,\cite{compare_qkd_mdi, compare_qkd_bb84, compare_qkd_tf, compare_qkd_tf_1002}
\label{comparing}
\end{figure}
\section{Terra Quantum QKD transmission line}\label{QKD_section}
\begin{figure}
\caption{\textbf{A setup of the QKD transmission line}
\label{QKDscheme}
\end{figure}
The Terra Quantum QKD transmission line (TQ-QKDTL) realizing the secure information transmission between the legitimate users, Alice and Bob, comprises Terra Quantum-made EDFAs (TQ-EDFA) with the amplifying coefficient 10\,dB. A setup of the implemented TQ-QKDTL is shown in Fig.\ref{QKDscheme}. At Alice's laboratory, the signals that are to be sent via the fiber optical channel are formed from the laser pulses by the amplitude modulator (AM) IxBlue MXAN-LN-10. Before arriving at AM, generated pulses pass through the phase modulator (PM), which shifts the phase of each pulse randomly. Phase randomization\,\cite{phase_rand, phase_rand2, phase_rand3, phase_rand4} decreases the effectiveness of the eavesdropper's attacks without affecting the probability distributions of Bob's measurement results. The phase shift of a particular pulse is controlled by the Terra Quantum-made random signal generator (RSG) using the avalanche breakdown described in Refs.\,\cite{qrng:1,qrng:2}.
Exploiting the quantum interference effect, the AM modulates the intensity of the optical signal.
The phase difference between the interfering components is set by the voltage coming to the radio frequency (RF) port.
Further, an additional direct current (DC) port is used for the bias point shifting.
The high-frequency electric signal is created by the field-programmable gate array (FPGA) where the original information bits created by Alice via using the TQ-QRNG are coming, and the electric amplifier working in the 10\,GHz (IxBlue DR-VE-10-MO). The random bits are generated using the TQ-QRNG that has been certified by the Federal Institute of Metrology METAS (Test Report No 116-05151).
Our TQ-QRNG uses the random time interval between the registrations of the photons generated by a weak LED light source via a single-photon detector as an entropy source. The hashed time intervals constitute the final random number sequences.
The FPGA board sets in the constant offset voltage for the bias control getting the information about the present power level coming from the AM via the monitoring detector. The light irradiation is generated by the TLX1 Thorlabs laser and has the 1530.33\,nm wavelength and the 10\,kHz frequency of the emission bandwidth. This wavelength is chosen since it corresponds to the peak of the EDFA amplification spectrum, see Fig.\,\ref{amplif_spectrum}.
At Bob's spot, Alice's optical signals are first amplified by the 20dB TQ-EDFA, then go through the narrow-band TQ-made optical filter with the 8.5\,GHz bandwidth, see Supplementary Information (SI), section\,\ref{narrow_filter_section} Since this filter is the fiber Bragg grating (FBG), its bandwidth is temperature sensitive. To stabilize the temperature of the narrow bandwidth filter, the Terra Quantum team developed a specific thermostat. Being filtered from the natural amplifier's noise referred to as an amplified spontaneous emission (ASE), Alice's signal arrives at Bob's detector (FPD610FC). Then the voltage coming from the detector is digitized by the oscilloscope, Tektronix DPO4104, and is treated by Bob's computer in an automatic regime.
\section{The protocol}\label{protocol_section}
The structure of the TQ-QKD protocol is shown in Fig.\,\ref{protocol_block_scheme}. Before starting the secret communication, legitimate users must make sure that there is no eavesdropper intercepting the transmission line.
To that end, ideally, they have to execute the loss control procedure using the optical time domain reflectometry (OTDR) technique.
Corresponding devices commonly used for testing the integrity of fiber lines not containing amplifiers are described below in Section\,\ref{REFL_section}
We develop a special OTDR technique suitable for our TQ-QKD protocol which we describe in detail in a forthcoming publication.
The loss control procedure is repeated with a certain frequency enabling the determination of the reference line tomogram, the magnitude of which drifts slowly with time.
The actions of `bits sending' and `test pulse sending' (constituting the transmittometry) occur between each pair of consecutive acts of reflectometry.
The protocol is illustrated in Fig.\,\ref{protocol_block_scheme}.
The reflectometry and transmittometry facilitate continuous loss control in parallel with the bit distribution.
\begin{figure}
\caption{\textbf{The general TQ-QKD protocol structure.}
\label{protocol_block_scheme}
\end{figure}
At the beginning of the key generation process, Alice encodes the logical bits into phase-randomized coherent states. The pulses corresponding to the different bits have different average photon numbers and random phases. At his side, Bob carries out the measurements of the energies of the received states and exercises the subsequent classical post-selection of the results. The measurements can be formalized using the projective operators
\begin{equation}
\hat{E}_0=\sum \limits_{k=\Theta_3}^{\Theta_1} \ket{k} \bra{k},\quad\hat{E}_1 = \sum \limits_{k=\Theta_2}^{ \Theta_4} \ket{k} \bra{k},\quad\hat{E}_\text{fail} = \hat{\mathds{1}} - \hat{E}_0 - \hat{E}_1,
\label{Eint}
\end{equation}
where $\hat{E}_0$ and $\hat{E}_1$ correspond to the outcomes which Bob interprets as `0' and `1', respectively; the operator $\hat{E}_\text{fail}$ corresponds to the failed outcome (the corresponding bit positions will be removed at the postselection stage); $\ket{k}$ stands for a Fock state containing $k$ photons; $\hat{\mathds{1}}$ is the unity operator, and $\Theta_{1-4}$ are the postselection parameters that Bob chooses in correspondence to the amount of the leak $r_\text{E}$ created by an eavesdropper.
The exemplary probability distributions of the measured photon number corresponding to bits `0' and `1' are presented in Fig.\,\ref{postselection}.
Next, Bob sifts the measurement results through four $\Theta_{1-4}$ borders and obtains the raw key.
\begin{figure}
\caption{\textbf{Post-selection.}
\label{postselection}
\end{figure}
\subsection{Error correction}\label{error_correction}
To correct the errors that are bound to occur in the measured bit sequence, we employ the combination of two well-known correction codes, the repetition code and the low-density parity-check (LDPC) code.
The repetition code proves to be effective under the bit error rates (BERs) up to 30\%. This means that about one-third of all bits after the postselective measurement are inverted.
For the $n$ bit-long code, the fraction $R_{\text{rep}}$ of the corrected shared sequence that becomes known to Eve is
\begin{equation}
R_{\text{rep}} = \frac{n-1}{n}.
\end{equation}
The LDPC code is faster but is applicable only when the BER is below 11\%. We use the LDPC code with the code rate $R_\text{LDPC} = 1/2$ which means that the half of information becomes declassified.
In our case, the BERs constitute 35--40\%, and we resolve the issue by combining two methods.
Our sequence consists of 1944 blocks of the length $n$. Firstly, Bob uses Alice's repeat code syndromes to find properties of the bits `0' and `1' in each block. Here we divulge the $R_\text{rep}$ part of the sifted sequence. Then these parts of the signal go as an input of the LDPC code where we use Alice's LDPC syndrome to finally correct the shared sequence. At this step, we disclose the $R_\text{LDPC}$ part of the remaining secret information.
The fraction
\begin{equation}
R_{\text{LDPC\&rep}} = R_\text{rep} + R_\text{LDPC}(1 - R_\text{rep}) = \frac{n-1}{n} + \frac{1}{2}\cdot\frac{1}{n} = 1 - \frac{1}{2n}
\end{equation}
of the shared sequence becomes known to Eve.
The value of $n$ is chosen in accordance with the error value, the bigger the error, the bigger $n$ is needed for the successful correction.
Choosing a large $n$, one can use the repetition code only, however, in this case, a large fraction of the information gets public, and therefore the time for the key accumulation becomes large as well.
Therefore, one chooses the optimal $n$ allowing one to use the optimal combination of codes for every concrete error value enabling legitimate users to achieve the maximum key generation rate.
\subsection{{Privacy amplification}}\label{privacy_amplification}
One of the possible eavesdropper scenarios actions is the attack via the diverting of the part of the signal.
The amount of information that Eve gets during Alice's state transmission through the optical channel, $I(\text{A}, \text{E})$ (described in the section\,\ref{Eves_info_section}), does depend on the precision of the control over the loss coefficient in the line.
To estimate the magnitude to which the final key is to be compressed, we take into account all the possible leaks during the protocol implementation.
To begin, before starting the error correction, Alice divulges the $r_{\text{estim}} = 0.01$ part of her shared sequence in order to estimate the BER.
Then, during the correction itself, another share $R_{\text{LDPC\&rep}}$ of the sifted sequence gets declassified.
Finally, during the signal transmission, an eavesdropper could divert $I(\text{A}, \text{E})$ of the information amount.
As a result, all the losses related to the error corrections are $r_{\text{corr}} = r_{\text{estim}} + R_{\text{LDPC\&rep}}$, while the total losses are $r = r_{\text{corr}} + I(\text{A}, \text{E})$.
The distributed sequence is to be compressed to such an extent that the length of the final secret key became $L_\checkmark \cdot(1 - r)>0$, where $L_\checkmark = 1944 \cdot n$ is the length of the sifted sequence.
In case $r \geq 1$, the distributed sequence is not considered safe and is not added to the general key.
\subsection{Possible attacks and ways of protection}\label{eavesdropper}
Table\,\ref{atacks} presents the possible attacks, their description, and the corresponding ways of protection.
\begin{table}[h!]
\setlength{\tabcolsep}{4pt}
\raggedright
\centering
\begin{tabularx}{\textwidth}{|s|b|b|}
\hline
\textbf{The attack} & \textbf{Brief description} & \textbf{The way of protection}
\\ \hline\hline
Measurement of the natural losses (see subsection\,\ref{atack_demon}).
&
Rayleigh scattering can cause a leak of the signal. Then an eavesdropper can eavesdrop even without intercepting the line.
&
We work with moderate signal power. Corresponding losses constitute a few photons per 100 meters\,\cite{new_theory}, which does not allow measuring them using any realistic photodetectors.
\\ \hline
Measuring losses at the splices (see subsection\,\ref{attack_on_welding}).
&
An eavesdropper can collect the signal lighting up out of the cable at the fusion spliced joints without creating additional losses.
& We make careful specially optimized splicing procedures. Besides, we demonstrate that not all the losses in the channel go outside the cable.
\\ \hline
Creating a local leak (see subsection\,\ref{atack_attenuation}).
&
An eavesdropper can achieve diverting the part of the signal by local bending of the fiber channel.
&
Lock-in control, see subsection\,\ref{lock_in}
\\ \hline
Creation of the local leak and the increase of the amplifying coefficient in the magistral line amplifier (see subsection\,\ref{atack_amplifier}).
&
An eavesdropper can divert a part of the transmitting signal and compensate for the visible losses by increasing the amplifier power.
&
We design the TQ amplifier settled in an active fiber channel in a way that it always performs in a regime of maximal pumping power and population inversion.
\\ \hline
\end{tabularx}
\caption{Possible attacks against the protocol and protection methods.}
\label{atacks}
\end{table}
\subsubsection{Attacks with the measurements of the natural losses}\label{atack_demon}
During the signal transmission along the fiber optical channels, part of the signal scatters out due to Rayleigh scattering.
Then the eavesdropper can arrange an anomalously large distributed detector next to one of the magistral amplifiers and watch the complete signal even without the mechanical intervention into a transmitting channel. This problem has been described in\,\cite{new_theory}. To avoid this problem we work at the signal intensity that is much less than some minimal critical value at which the eavesdropping is still possible.
\subsubsection{Measuring losses at the splices}\label{attack_on_welding}
During the transmission, a portion of the signal may leak outside the channel at the splice joints.
The eavesdropper, thus, gets an opportunity of gathering this locally leaking radiation, avoiding, thereby, the creation of any additional losses through mechanical interventions.
However, as shown in the SI Section\,\ref{loss_section}, not all the losses go outside the transmission line. A significant portion of the losses remains confined within the cable and is inaccessible to potential eavesdroppers.
Our estimates suggest that the loss at a splice joint amounts to a mere 0.1\%, which is acceptable for transmissions spanning over 1000\,km.
Notably, one cannot avoid splice joints near amplifiers, but we implement additional protection at these points.
\subsubsection{The attack with creation of the local leak}\label{atack_attenuation}
Using a beamsplitter an eavesdropper can divert and use the part of the signal.
Our transmission channel design uses stringent physical loss control over the transmission line which makes it impossible for eavesdroppers to introduce a beamsplitter into the line remaining unnoticed.
The eavesdropper may try to arrange the optical fiber bending and collect all the outcoming from the channel signal. Yet we can detect even the fast eavesdropper's intervention using the regular transmittometry via detecting the sharp integral change of the intensity in the whole line, see Subsection\,\ref{lock_in}
\subsubsection{The attack against the amplifier}\label{atack_amplifier}
Diverting the part of the signal using the beamsplitter, an eavesdropper decreases the integral intensity in the line. Therefore, to remain unnoticed the eavesdropper should recover the observable losses using an amplifier.
The eavesdropper cannot introduce its own amplifier since the transmission line is permanently subject to loss control and any mechanical intervention will be immediately detected. The retuning amplifying coefficient of the magistral amplifier is also impossible since it works in the regime of the maximal pumping power. Moreover, because of the amplifier security control mechanism any mechanical action immediately increases the losses.
\subsection{Eavesdropper's information estimation}\label{Eves_info_section}
In the context of the proposed loss control approach, an eavesdropper is not able to conduct conventional attacks including the replacement of the optical line with the ideal (lossless) quantum channel, since such attacks dramatically change the line reflectogram and are to be easily detected.
Thus, Eve has to introduce local losses at some point in the line.
The cascade of $M_{1(2)}$ 50km-long fiber pieces and amplifiers before (after) Eve's intrusion point can be effectively represented as a pair of loss and amplification.
We introduce the notations $G_{1(2)}=GM_{1(2)}\left(1-T\right)+1$ and $T_{1(2)}=1/G_{1(2)}$ for the amplification factor (the ratio of the output photon number to the input photon number in the amplification channel) and transmission probability of the effective amplification and loss channels, respectively.
The detailed analysis of the signal's density matrix evolution is provided in Ref.\,\cite{new_theory}.
Here, we consider only Eve's quantum state $\hat{\rho}_{\text{E}}^{(a)}$ on the condition that the sent bit is $a$ and Bob obtains the conclusive measurement result.
Let $r_\text{E}$ be the minimal detectable artificial leakage and $\gamma_a$ be the amplitude of the bit-encoding pulse.
Eve's density matrix is
\begin{multline}
\hat{\rho}_{\text{E}}^{(a)}
=
\frac{1}{p(\checkmark|a)}
\int\!\!d^2\alpha\,\,
P\left(\alpha;\sqrt{T_1}\gamma_a,G_1\right)
\int\!\!d^2\beta\,\,
P\left(\beta;
\sqrt{(1-r_{\text{E}})T_2}\alpha,G_2\right)
\langle\beta|\left(\hat{E}_0+\hat{E}_1\right)|\beta \rangle
\\\times
\left[
\frac{1}{2\pi}\int\limits_{0}^{2\pi}\!\!d\varphi\,\,
\left|e^{i\varphi}\sqrt{r_{\text{E}}}|\alpha|\right\rangle \left\langle e^{i\varphi} \sqrt{r_{\text{E}}}|\alpha| \right|_\text{E}\right],
\label{eve_matr_ints}
\end{multline}
where
\begin{equation}
P(\alpha,\gamma,G) = \frac{1}{\pi (G-1)} \exp\left(-\frac{|\alpha- \sqrt{G} \gamma|^2}{G-1}\right)
\label{P}.
\end{equation}
The normalizing factor $p(\checkmark|a)$ is the probability of a conclusive measurement result at Bob's end provided that the sent bit is $a$ and can be determined from the condition $\text{tr}\big[\hat{\rho}_{\text{E}}^{(a)}\big]=1$.
The sum $p_\checkmark=p\left(\checkmark|0\right)+p\left(\checkmark|1\right)$ determines the average probability of a conclusive result.
The average maximum amount of information that Eve can extract from the states Eq.\,(\ref{eve_matr_ints}) is upper-bounded by the Holevo quantity\,\cite{Holevo} $\chi$ which, in our case, can be expressed as
\begin{equation}
I(\text{A},\text{E})\leq\chi
=
S\left(\frac{p\left(\checkmark|0\right)}{2p_\checkmark}\hat{\rho}_{\text{E}}^{(0)}+\frac{p\left(\checkmark|1\right)}{2p_\checkmark}\hat{\rho}_{\text{E}}^{(1)}\right)
-
\frac{p\left(\checkmark|0\right)}{2p_\checkmark}S\left(\hat{\rho}_{\text{E}}^{(0)}\right)
-
\frac{p\left(\checkmark|1\right)}{2p_\checkmark}S\left(\hat{\rho}_{\text{E}}^{(1)}\right),
\label{eve_holevo}
\end{equation}
where $S(\hat{\rho})=-\text{tr}\left[ \hat{\rho} \log_2 \hat{\rho} \right]$ is von Neumann entropy.
The value $I(\text{A},\text{E})$ includes information that Eve extracts from the measurement of the intercepted photons and does not comprise information leaked during the error correction procedure.
\subsection{Secret key generation rate}\label{generation_speed}
Alice generates a random bit sequence using the TQ-QRNG with the speed of 5\,Mb/sec and loads generated string into her PC circuits.
The memory of Alice's circuit transfers the bit sequence to FPGA forming the electric pulses of the required length alternating with the sinusoidal signal proving the loss control.
This pulse mixture is transferred to an amplitude modulator transforming it into the optical signal that is sent with the rate of 7\,Kb/sec to the optical magistral line of the 1032\,km length.
Bob takes readings of the signal in portions which, at present, is the major effect restricting the key distribution rate stemming from the use of an oscilloscope. Bob accepts the signal every 3 seconds using his oscilloscope and launches, in parallel, the treatment of the sinusoidal signal component and the identification of the `0' and `1' states.
This is a postselective measurement after which approximately 60\% of the original bit sequence remains.
Bob accumulates these arriving bit sequences in his device memory until their amount becomes sufficient for exercising the error correction procedure. As soon as their number exceeds the minimum necessary for the error corrections, Bob informs Alice via the classical channel, and she sends him her error syndromes. Importantly, this additional communication occurs during the receiving of the signal by oscilloscope and thus does not slow down the work of the protocol, ensuring, at the same time, its reliability and stability.
In the course of the practical protocol implementation, we succeeded in transmitting the raw bit sequence with the rate\,800\,bps. The next step is the compression of the key over the quantity equal to the number of bits declassified during the protocol's work.
The compression coefficient is determined by the specific method of transmission of Alice's error syndromes to Bob and by the amount of information stolen by the eavesdropper. The length of the resulting key is
\begin{equation}
L_\text{f}=p_\checkmark L\cdot\big(S(\text{A})-f S(\text{A}|\text{B})-I(\text{A},\text{E})\big).
\end{equation}
Here, $S(\text{A})$ is Alice's system entropy provided that the postselection is carried out;
$S(\text{A}|\text{B})$ is Alice's system entropy obtained under the condition that the results of Bob's measurements are known and that the inconclusive outcomes are discarded.
The quantity $f$ is determined by the error correction code, and thus
$f S(\text{A}|\text{B})=r_{\text{corr}}$ is the fraction of the sequence leaked to the eavesdropper during the error correction procedure, and $p_\checkmark$ is the probability of a conclusive outcome at Bob's side, i.e., the probability that the bit is going to be taken into account, and $p_\checkmark L=L_\checkmark$ is the length of the sifted bit sequence.
Then the compression of the distributed sequence gives the rate of the final absolutely secret random key as 34\,bps.
\subsection{The encountered problems}
Table\,\ref{problems} summarizes the difficulties and problems that our realized protocol has overcome.
\begin{table}[!h]
\centering
\setlength{\tabcolsep}{4pt}
\raggedright
\begin{tabularx}{\textwidth}{|X|X|X|}
\hline
\textbf{The problem} & \textbf{Brief description} & \textbf{Solution}
\\ \hline\hline
An exponential decay of the signal in an optical fiber (Subsection\,\ref{Low_intensity_defence}).
&
The signal exponential decay in the optical channel results in the long-distance communication problem.
&
The use of the EDFA-like amplifiers was modified to fit our protocol.
\\ \hline
Detecting low-power information-carrying signal\, (Subsection\,\ref{problem:low_intensity}).
&
Our QKD protocol design implies low-power key distribution.
&
The TQ-made amplifier having an amplifying coefficient of 20\,dB and a low signal-to-noise ratio is developed.
\\ \hline
The high level of the ASE from the EDFA (Subsection\,\ref{problem:high_ASE}).
&
The EDFA randomly generates wide-spectrum optical irradiation; long optical lines contain many EDFAs which˙results in a high noise level.
&
The TQ-made narrow-band, 8.5 GHz, thermostabilized filter is developed.
\\ \hline
Laser irradiation generation in a long optical line without optical isolators (Subsection\,\ref{problem:generation}).
&
In long optical lines having active fiber segments in TQ-EDFAs, the unbalanced loss and amplification levels may cause laser irradiation due to multiple reflections at connectors and Rayleigh scattering.
&
The developed optical line has high control over the loss and amplification levels upon adding every amplifier, having enabled the right choice of the amplification coefficient corresponding to losses.
\\ \hline
Floating of the offset voltage at the amplitude modulator (Subsection\,\ref{problem:AM_drift}).
&
The working point of the voltage offset, i.e., the working point of the amplitude modulator may shift with time.
&
The part of the optical irradiation is diverted and controlled using an additional detector at Alice's side. Then the voltage is tuned according to the detector's indications.
\\ \hline
\end{tabularx}
\caption{The encountered problems and the ways of their solution.}
\label{problems}
\end{table}
\subsubsection{Exponential signal decay in the optical fiber}\label{Low_intensity_defence}
Upon spreading along the optical fiber, the optical signal exponentially decays with the distance. Therefore, for long-distance transmission, the TQ-EDFAs are used. The TQ-EDFA ensures the loss compensation at a distance of about 50\,km. Since this amplifier is not an ideal quantum repeater, it generates additional noise. Importantly, the TQ-EDFA design is adjusted to our protocol, see the details in SI, Subsection\,\ref{amplifier_section}
\subsubsection{Detecting the low power level}\label{problem:low_intensity}
Since, as has been mentioned above, a high power level excludes the possibility of the secret information, see Subsection\,\ref{atack_demon}, the information transmission is executed at a moderate intensity of the optical signal. Hence this signal is to be amplified before sending it to the high-sensitivity detector. To that end, our team developed a specific preamplifier with an amplifying coefficient of 20\,dB. An additional constructive feature of the TQ amplifier that differs it from the magistral amplifiers is the presence of optical isolators in its constructive design.
\subsubsection{The high ASE level}\label{problem:high_ASE}
A long-distance optical transmission line exploits magistral amplifiers EDFA to compensate for the exponential optical losses. Since EDFAs are not the ideal quantum repeaters, they generate ASE leading to the noise interfering with the detected information-carrying signal. We have developed a narrow-band 8.5\,GHz filter that has enabled to decrease the ASE power arriving at Bob's detector. An important technological component of the new TQ narrow band filter is a thermal stabilizer that has enabled the $\pm 0.01 \text{ K}$ of thermal stability precision, also developed by our team.
\subsubsection{Laser irradiation generated in the long-distance line not containing optical isolators}\label{problem:generation}
A long-distance optical channel exploits a significant number of EDFAs and an amplification effect is hosted by the erbium-doped fiber segments; these segments are referred to as active medium regions. Since in the optical line, several signal reflections occur the whole line is to be viewed as an active medium containing weak resonators. Therefore, the transfer of the ASE critical power is accompanied by the generation of the irradiation analogous to laser irradiation. We eliminate this parasitic generation by choosing the regime of the information signal transmission ensuring being in the ASE amplification peak, see Fig.\,\ref{amplif_spectrum} and using the TQ-EDFAs-designed to ensure the amplification coefficient of 10\,dB providing thus losses compensation over the 50\,km distance.
\subsubsection{Floating of the offset voltage at the amplitude modulator}\label{problem:AM_drift}
The possibility of the offset voltage floating resulting in the shift of the working point (often referred to as a bias point) of the amplitude modulator, leads also to the change of the average intensity of the optical signal. To compensate for the bias point shift, our TQ team has developed an algorithm that uses the data from the monitoring detector, shown in Fig.\,\ref{QKDscheme}, and returns the changing average intensity of the optical signal to its original magnitude.
\section{Line control}\label{control_section}
The important element of the protocol is the control over the losses in the communication line enabling the legitimate users to momentarily estimate the information stolen by eavesdroppers.
\subsection{Transmittometry}\label{lock_in}
The control over the throughput losses is executed as follows. The sinusoidal-modulated signal with the frequency of 25\, MHz is sent along the optical line. The pulse length of every control sending is 1\,ms. At the line output after the TQ-manufactured preamplifier and optical filters, the signal is received by the FPD610 detector.
This approach is analogous to the lock-in method\,\cite{lock_in} where the signal taken from the detector is received by an oscilloscope and transferred to the user's computer. Further treatment is the search, at the modulation frequency, for the amplitude of the signal Fourier transforms at the reference frequency. The ratio of the obtained and reference amplitudes enables us to determine the magnitude of the losses. The reference loss magnitude is renewed at every reflectometry session. Figure\,\ref{lockin_stabil} shows the loss coefficient time dependence.
To model the eavesdropper presence, one introduces the losses into the line (in the implemented measurements the losses are about 3\%) and observes the loss coefficient time dependence. The moment of switching on the losses is marked by the arrow, see Fig.\,\ref{lockin_loss}.
\begin{figure}
\caption{The time dependence of the loss coefficient of the transmission line. Over a long time, the drifting of the transparency of the line turns out to be significant.}
\label{lockin_stabil}
\caption{Measurements detecting the intervention into the transmission line. The associated introduced losses are about 3\%.}
\label{lockin_loss}
\end{figure}
\subsubsection{Temperature dependence}\label{temp_dep_section}
The optical loss control in the communication line enables not only determining the magnitude of the signal diverted by an eavesdropper but also the natural changes in the optical fiber transparency. One of the reasons for the transparency coefficient change is the temperature variations. Using the OTDR, we measure the losses magnitude change at the coil containing the 50\,km of the fiber line upon the coil heating, see Fig.\,\ref{OTDR_loss_check}. The obtained reflectogram allows us to determine the loss magnitude of the whole segment of the optical fiber line from the slope of the log-log plot, see Fig.\,\ref{OTDR_loss_check}.
\begin{figure}
\caption{The time dependencies of the losses in the optical fiber and the fiber temperature. The temperature change is shown by the blue curve, the corresponding loss coefficient behavior is presented by the magenta one.}
\label{OTDR_loss_check}
\end{figure}
\subsection{Reflectometry}\label{REFL_section}
The OTDR procedure allows for detecting local interceptions in the communication line. The procedure consists of sending a short powerful optical pulse into the line and measuring the backscattered signal. The local losses usually occur in optical connections, splices, or bends, which are seen as the main points where the signal leaks out of the line. Although splices only dissipate some of the loss outward, see SI, Section\,2, and our line is designed to eliminate optical connections in order to prevent excessive leakage, there are still unavoidable splices and fiber interferences that must be observed. Rayleigh scattering, which is almost impossible for an eavesdropper to exploit, helps to detect faults along the line. If the eavesdropper diverts the part of the signal, the intensity of the backscattered radiation drops in the segment containing the intercept point.
This allows local interception to be detected.
Figure\,\ref{QKDscheme_OTDR} presents the setup of the complete TQ-QKD protocol realization. The protocol contains periodic measurements of the local losses. If local losses are not detected, the change in the loss coefficient is assigned to natural reasons. One of them may be the dependence of the fiber transparency upon temperature, see Subsection\,\ref{temp_dep_section} The losses coefficient is also subject to the influence of other slow fluctuation parameters producing a so-called flicker noise, which is a type of electronic noise with a $1/f$ power spectral density, so it is often referred to as $1/f$ noise\,\cite{flicker_noise1}, \cite{flicker_noise2}. The measurements of the line stability are presented in Fig.\,\ref{lockin_stabil}.
Furthermore, to accelerate the secret key distribution, we will use FPGA instead of the oscilloscope at Bob's side.
To remind here, while an ideal TQ-QKD protocol should contain a periodical reflectometry we postpone the detailed description of its experimental realization to the forthcoming publication.
\begin{figure}
\caption{\textbf{A setup of the TQ-QKD protocol realization set up including reflectometry}
\label{QKDscheme_OTDR}
\end{figure}
We present a concise description of the current state of our investigation in this direction.
Using the available FPD610-FC detector and TLX1 laser, according to scheme\,\ref{QKDscheme_OTDR}, we collected the reflectogram of the whole line. Figure\,\ref{real_refl} presents the results of the measurements.
Since the sensitivity of the FPD610-FC detector is by order of magnitude lower than the necessary one, and the TLX1 laser power is also over the order of magnitude lower than the one that is normally used for the pulse formation, we can only approximately detect the main features of the reflectogram.
The collected data enable us to conclude that the power of the detected signal due to the Rayleigh scattering does not decay upon going through amplifiers along the whole transmission line. This follows from the fact that the peaks presented in Fig.\,\ref{real_refl} do not lose their height.
This result allows us to further develop the OTDR technique for our system.
We use the oscilloscope to collect the data from the OTDR detector. The length of the probing pulse is 100\,ns.
\begin{figure}
\caption{The line reflectogram shows 19 peaks corresponding to the amplifier's positions as well as the peak at zero corresponding to the connector at the beginning of the transmission line necessary to connect the reflectometer. The data are reliable despite the insufficiently sensitive detector and insufficiently powerful laser.}
\label{real_refl}
\end{figure}
\section{The BER and key randomness analysis}\label{bitber_section}
To provide an understanding of how the proposed line protection by the TQ-QKD protocol is executed, we describe a picture of the signal-receiving procedure for Bob and the interception scheme for Eve.
We set that our loss control indicates that 1\% of a signal is intercepted by Eve. Since the electromagnetic signal is quantized and carried by discrete particles, photons, the photon number in the pulses corresponding to `0' and `1' states intercepted by Eve is 100 times less than Bob's corresponding photon numbers. Notably, for the quantum coherent states that we use for encoding `0' and `1' bits there are inevitable quantum fluctuations in the measured pulse intensities. Therefore, the less the number of photons $N$ that Eve manages to intercept, the bigger the relative fluctuations which scale as $1/\sqrt{N}$ are. To visualize this effect,
we plot the experimentally obtained distributions of the `0' and `1' states as functions of voltage corresponding to the intensity of an incoming signal, using the statistics obtained for a large number of the secret quantum keys,
at Bob's side, Fig.\,\ref{distribution_Bob}.
For Eve we measure the statistics of the number of photons received during the attack using our equipment, Fig.\,\ref{distribution_Eve}.
The measurement is made for the signal coming to Bob, but attenuated by a factor of 100.
In order to distinguish the signal coming to Eve, the amplification factor was increased.
In our protocol, Alice encodes the bit `0' via the state containing 11360 photons and the bit `1' via the state containing 15100 photons.
One sees that while at Bob's side the `0' and `1' states are still distinguishable, despite tough postselection, at Eve's side two corresponding distribution functions completely overlap making the determining of the bit state impossible. Bob knows only the total signal distribution, where he has to put in the postselection parameters (boundaries) to ensure efficient discrimination between the states. The distance between the distribution peaks depends on the difference between the intensity of the signals. We have also executed the tests verifying the randomness of the distributed key; an example of the auto-correlation function for the final key is shown below in Fig.\,\ref{fig:autocorr}. To check to which extent our random sequence corresponds to the binomial distribution, we compared the corresponding values of the average and the standard deviation.
In our case, where we use the selection of the 705 trials for the sequences of 2111 bits, the theoretical values of the average and the standard deviation for the final key realization are, correspondingly, $mean_{theor} =1055.50$ and $std_{theor} = 527.75$.
Accordingly, the experimental values are, correspondingly, $mean_{exp} = 1055.48$ and $std_{exp} = 531.74$.
\begin{figure}
\caption{Bob}
\label{distribution_Bob}
\caption{Eve}
\label{distribution_Eve}
\caption{{\textbf{The distribution of `0' and `1' bits at Bob's end of the line and for the eavesdropper.}
\label{distribution_1}
\end{figure}
\subsection{The intensity BER dependence}
Upon decreasing the states' distinguishability (the overlap of the distributions in Fig.\,\ref{distribution_Bob}), the value of the error increases. Figure\,\ref{ber_intensity} demonstrates that the error drops approximately linearly with the increase of the intensity differences in `0' and `1' states normalized by the average signal intensity at the line entrance.
\begin{figure}
\caption{\textbf{The BER dependence on the relative difference between the states intensities}
\label{ber_intensity}
\end{figure}
\subsection{Autocorrelation function}
To reveal the systematic dependencies between the bits in sequences at different time intervals we have checked its autocorrelation function. The absence of the peaks testifies to the statistical independence of the bits sequences and their approaching to complete randomness, see Fig.\,\ref{fig:autocorr}.
\begin{figure}
\caption{\textbf{Autocorrelation function of the final key}
\label{fig:autocorr}
\end{figure}
\section{Amplifiers noise}\label{noise_section}
During the quantum states transmission over the optical channel, the basic information signal gets distorted by noise which complicates the recognition of the states and reveals the eavesdropper. To solve this problem one has to understand the nature of the emerging noises since this understanding helps to reveal and identify the information component of the signal.
One of the sources of the noise is magistral amplifiers which amplify not only signal mode with the carrier wavelength but also the modes with other wavelengths (the amplification spectrum of the TQ-EDFA is discussed below in the SI Subsection\,\ref{amplifier_section}, see, in particular, Fig.\,\ref{amplif_spectrum}). Even in the absence of the signal from the laser, the amplifiers generate spontaneous irradiation (ASE) which constitutes a good part of the natural amplifier noise. We present the results of the measurements of the total ASE power depending on the number of amplifiers. The theory is given in\,\cite{new_theory}.
According to the theory, in the absence of a signal at the entrance to the communication line
\begin{equation}
n_\text{mode} = 2(G(M(1-T)+T)-1),
\end{equation}
where $n_\text{mode}$ is the average photon number in a single mode,
$G$ is the amplifying coefficient, $M$ is the number of the amplifiers, $T$ is the intensity dumping per 50\,km of the optical fiber. The factor of 2 appears due to two possible photon polarizations.
The number of the modes $N$ detected after the filter is given by $N = \alpha \Delta \nu \tau$, where $\alpha$ is the coefficient taking into account different intensities of the modes with the different wavelengths, $\Delta\nu$ is the filter spectrum bandwidth, and $\tau$ is the measurement time. To find $\alpha$, one has to assign a certain weight to each mode, determined by the bandwidth of the filter, see Fig.\,\ref{wide_filter_spec}. Here we use the Wavelength Division Multiplexing (WDM) filters with wide bandwidth and sharp wavelength borders. The obtained mode number for the time $\tau=2.5$\,ns is $N=200$.
Then the number of photons coming from the ASE to the detector during time $\tau$, the time of the bit sending, is given by
\begin{equation}
n = Nn_\text{mode}.
\end{equation}
Measurements of the average power at the line exit as a function of the amplifiers' number are presented in Fig.\ref{24}.
\begin{figure}
\caption{Transparency of the wide bandwidth filter as a function of the wavelength.}
\label{wide_filter_spec}
\caption{An average number of the ASE photons coming through the filter of the 68\,GHz bandwidth during 2.5\,ns.}
\label{24}
\end{figure}
\section*{Conclusion and discussion}
In conclusion, we have presented our first-ever practical implementation of the novel TQ-QKD protocol developed in\,\cite{new_theory}. We implemented the quantum key distribution protocol based on our theory for a record distance of 1032 km, see Fig.\ref{comparing}. This required an additional development of certain new devices. Importantly, we silhouetted the ways of further improvement of these devices which will enable us to achieve larger transmission distances leading eventually to developing a global QKD network. This, in its turn, will enable fully secure and high-rate information distribution over the globe.
While preparing our results for presentation, we have found out that the very recent state-of-the-art TF-QKD experiment described in Ref.\,\cite{compare_qkd_tf_1002} achieves a transmission distance of 1002 km utilizing ultra-low-loss fiber. This, in particular, implies that the fiber is produced through a custom manufacturing process. Installing such a custom-produced line is a labor-consuming and financially challenging subject compared to using the standard optical infrastructure. Moreover, even with the employment of this ultra-low-loss fiber and the integration of the most advanced noise suppression methodologies, the achieved key distribution rate registers a mere 0.0034 bits per second. This suggests that the 1000 km mark is the limit, at which the rate is already impractically low.
In contrast, we efficiently achieve 1032 km with the distribution rate 34\,bps which is larger by four orders of magnitude. Notably, this is still far from exhausting our full potential while employing the costly viable standard of the Single-Mode fiber line.
Thus, our protocol can be deployed on the existing fiber infrastructure.
\section*{Suplementary Information}\label{supplem_section}
\setcounter{section}{0}
\renewcommand{NOTE\arabic{section}}{NOTE\arabic{section}}
\input{supplementary.tex}
\end{document}
|
\mathbf{b}egin{document}
\mathbf{b}oldsymbol{\mu}aketitle
\mathbf{b}egin{abstract}
Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent ( OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other is a simple regularizer defined on high dimensional variables. We show that the iteration complexity of ORBCD has the same order as OGD or SGD. For strongly convex functions, by reducing the variance of stochastic gradients, we show that ORBCD can converge at a geometric rate in expectation, matching the convergence rate of SGD with variance reduction and RBCD.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{abstract}
\section{Introduction}
In recent years, considerable efforts in machine learning have been devoted to solving the following composite objective minimization problem:
\mathbf{b}egin{align}\label{eq:compositeobj}
\mathbf{b}oldsymbol{\mu}in_{\mathbf{b}oldsymbol{\mu}athbf{x}}~f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) = \frac{1}{I}\sum_{i=1}^{I}f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) + \sum_{j=1}^{J}g_j(\mathbf{b}oldsymbol{\mu}athbf{x}_j)~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $\mathbf{b}oldsymbol{\mu}athbf{x}\in\R^{n\times 1}$ and $\mathbf{b}oldsymbol{\mu}athbf{x}_j$ is a block coordinate of $\mathbf{b}oldsymbol{\mu}athbf{x}$. $f(\mathbf{b}oldsymbol{\mu}athbf{x})$ is the average of some smooth functions, and $g(\mathbf{b}oldsymbol{\mu}athbf{x})$ is a \mathbf{b}oldsymbol{\mu}athbf{e}mph{simple} function which may be non-smooth. In particular, $g(\mathbf{b}oldsymbol{\mu}athbf{x})$ is block separable and blocks are non-overlapping. A variety of machine learning and statistics problems can be cast into the problem~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj}. In regularized risk minimization problems~\mathbf{c}ite{hastie09:statlearn}, $f$ is the average of losses of a large number of samples and $g$ is a simple regularizer on high dimensional features to induce structural sparsity~\mathbf{c}ite{bach11:sparse}. While $f$ is separable among samples, $g$ is separable among features.
For example, in lasso~\mathbf{c}ite{tibs96:lasso}, $f_i$ is a square loss or logistic loss function and $g(\mathbf{b}oldsymbol{\mu}athbf{x}) = \lambda \| \mathbf{b}oldsymbol{\mu}athbf{x} \|_1$ where $\lambda$ is the tuning parameter. In group lasso~\mathbf{c}ite{yuan07:glasso}, $g_j(\mathbf{b}oldsymbol{\mu}athbf{x}_j) = \lambda\| \mathbf{b}oldsymbol{\mu}athbf{x}_j \|_2$, which enforces group sparsity among variables. To induce both group sparsity and sparsity, sparse group lasso~\mathbf{c}ite{friedman:sglasso} uses composite regularizers $g_j(\mathbf{b}oldsymbol{\mu}athbf{x}_j) = \lambda_1\| \mathbf{b}oldsymbol{\mu}athbf{x}_j \|_2 + \lambda_2 \|\mathbf{b}oldsymbol{\mu}athbf{x}_j\|_1$ where $\lambda_1$ and $\lambda_2$ are the tuning parameters.
Due to the simplicity, gradient descent (GD) type methods have been widely used to solve problem~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj}. If $g_j$ is nonsmooth but simple enough for \mathbf{b}oldsymbol{\mu}athbf{e}mph{proximal mapping}, it is better to just use the gradient of $f_i$ but keep $g_j$ untouched in GD. This variant of GD is often called proximal splitting~\mathbf{c}ite{comb09:prox} or proximal gradient descent (PGD)~\mathbf{c}ite{tseng08:apgm,beck09:pgm} or forward/backward splitting method (FOBOS)~\mathbf{c}ite{duchi09}. Without loss of generality, we simply use GD to represent GD and its variants in the rest of this paper. Let $m$ be the number of samples and $n$ be dimension of features. $m$ samples are divided into $I$ blocks (mini-batch), and $n$ features are divided into $J$ non-overlapping blocks.
If both $m$ and $n$ are large, solving~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj} using batch methods like gradient descent (GD) type methods is computationally expensive.
To address the computational bottleneck, two types of low cost-per-iteration methods, online/stochastic gradient descent (OGD/SGD)~\mathbf{c}ite{Robi51:SP,Judi09:SP,celu06,Zinkevich03,haak06:logregret,Duchi10_comid,duchi09,xiao10} and randomized block coordinate descent (RBCD)~\mathbf{c}ite{nesterov10:rbcd,bkbg11:pbcd,rita13:pbcd,rita12:rbcd}, have been rigorously studied in both theory and applications.
Instead of computing gradients of all samples in GD at each iteration, OGD/SGD only computes the gradient of one block samples, and thus the cost-per-iteration is just
one $I$-th of GD. For large scale problems, it has been shown that OGD/SGD is faster than GD~\mathbf{c}ite{tari13:pdsvm,shsisr07:pegasos,shte09:sgd}. OGD and SGD have been generalized to handle composite objective functions~\mathbf{c}ite{nest07:composite,comb09:prox,tseng08:apgm,beck09:pgm,Duchi10_comid,duchi09,xiao10}. OGD and SGD use a decreasing step size and converge at a slower rate than GD. In stochastic optimization, the slow convergence speed is caused by the variance of stochastic gradients due to random samples, and
considerable efforts have thus been devoted to reducing the variance to accelerate SGD~\mathbf{c}ite{bach12:sgdlinear,bach13:sgdaverage,xiao14:psgdvd,zhang13:sgdvd,jin13:sgdmix,jin13:sgdlinear}.
Stochastic average gradient (SVG)~\mathbf{c}ite{bach12:sgdlinear} is the first SGD algorithm achieving the linear convergence rate for stronly convex functions, catching up with the convergence speed of GD~\mathbf{c}ite{nesterov04:convex}. However, SVG needs to store all gradients, which becomes an issue for large scale datasets. It is also difficult to understand the intuition behind the proof of SVG. To address the issue of storage and better explain the faster convergence,~\mathbf{c}ite{zhang13:sgdvd} proposed an explicit variance reduction scheme into SGD. The two scheme SGD is refered as stochastic variance reduction gradient (SVRG). SVRG computes the full gradient periodically and progressively mitigates the variance of stochastic gradient by removing the difference between the full gradient and stochastic gradient. For smooth and strongly convex functions, SVRG converges at a geometric rate in expectation. Compared to SVG, SVRG is free from the storage of full gradients and has a much simpler proof. The similar idea was also proposed independently by~\mathbf{c}ite{jin13:sgdmix}. The results of SVRG is then improved in~\mathbf{c}ite{kori13:ssgd}. In~\mathbf{c}ite{xiao14:psgdvd}, SVRG is generalized to solve composite minimization problem by incorporating the variance reduction technique into proximal gradient method.
On the other hand, RBCD~\mathbf{c}ite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd,shte09:sgd,chang08:bcdsvm,hsieh08:dcdsvm,osher09:cdcs} has become increasingly popular due to high dimensional problem with structural regularizers. RBCD randomly chooses a block coordinate to update at each iteration. The iteration complexity of RBCD was established in~\mathbf{c}ite{nesterov10:rbcd}, improved and generalized to composite minimization problem by~\mathbf{c}ite{rita12:rbcd,luxiao13:rbcd}. RBCD can choose a constant step size and converge at the same rate as GD, although the constant is usually $J$ times worse~\mathbf{c}ite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd}. Compared to GD, the cost-per-iteration of RBCD is much cheaper.
Block coordinate descent (BCD) methods have also been studied under a deterministic cyclic order~\mathbf{c}ite{sate13:cbcd,tseng01:ds,luo02:cbcd}. Although the convergence of cyclic BCD has been established~\mathbf{c}ite{tseng01:ds,luo02:cbcd}, the iteration of complexity is still unknown except for special cases~\mathbf{c}ite{sate13:cbcd}.
While OGD/SGD is well suitable for problems with a large number of samples, RBCD is suitable for high dimension problems with non-overlapping composite regularizers. For large scale high dimensional problems with non-overlapping composite regularizers, it is not economic enough to use one of them. Either method alone may not suitable for problems when data is distributed across space and time or partially available at the moment~\mathbf{c}ite{nesterov10:rbcd}. In addition, SVRG is not suitable for problems when the computation of full gradient at one time is expensive. In this paper,
we propose a new method named online randomized block coordinate descent (ORBCD) which combines the well-known OGD/SGD and RBCD together. ORBCD first randomly picks up one block samples and one block coordinates, then performs the block coordinate gradient descent on the randomly chosen samples at each iteration. Essentially, ORBCD performs RBCD in the online and stochastic setting.
If $f_i$ is a linear function, the cost-per-iteration of ORBCD is $O(1)$ and thus is far smaller than $O(n)$ in OGD/SGD and $O(m)$ in RBCD.
We show that the iteration complexity for ORBCD has the same order as OGD/SGD.
In the stochastic setting, ORBCD is still suffered from the variance of stochastic gradient. To accelerate the convergence speed of ORBCD, we adopt the varaince reduction technique~\mathbf{c}ite{zhang13:sgdvd} to alleviate the effect of randomness.
As expected, the linear convergence rate for ORBCD with variance reduction (ORBCDVD) is established for strongly convex functions for stochastic optimization. Moreover, ORBCDVD does not necessarily require to compute the full gradient at once which is necessary in SVRG and prox-SVRG. Instead, a block coordinate of full gradient is computed at each iteration and then stored for the next retrieval in ORBCDVD.
The rest of the paper is organized as follows. In Section~\mathbf{r}ef{sec:relate}, we review the SGD and RBCD. ORBCD and ORBCD with variance reduction are proposed in Section~\mathbf{r}ef{sec:orbcd}. The convergence results are given in Section~\mathbf{r}ef{sec:theory}. The paper is concluded in Section~\mathbf{r}ef{sec:conclusion}.
\section{Related Work}\label{sec:relate}
In this section, we briefly review the two types of low cost-per-iteration gradient descent (GD) methods, i.e., OGD/SGD and RBCD. Applying GD on~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj}, we have the following iterate:
\mathbf{b}egin{align}\label{eq:fobos}
\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}}~\langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
In some cases, e.g. $g(\mathbf{b}oldsymbol{\mu}athbf{x})$ is $\mathbf{b}oldsymbol{\mu}athbf{e}ll_1$ norm,~\mathbf{b}oldsymbol{\mu}yref{eq:fobos} can have a closed-form solution.
\subsection{Online and Stochastic Gradient Descent}
In~\mathbf{b}oldsymbol{\mu}yref{eq:fobos}, it requires to compute the full gradient of $m$ samples at each iteration, which could be computationally expensive if $m$ is too large. Instead, OGD/SGD simply computes the gradient of one block samples.
In the online setting, at time $t+1$, OGD first presents a solution $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}$ by solving
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}}~\langle \nabla f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $f_t$ is given and assumed to be convex. Then a function $f_{t+1}$ is revealed which incurs the loss $f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t)$.
The performance of OGD is measured by the regret bound, which is the discrepancy between the cumulative loss over $T$ rounds and the best decision in hindsight,
\mathbf{b}egin{align}
R(T) = \sum_{t=1}^{T} { [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t)] - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^*)+g(\mathbf{b}oldsymbol{\mu}athbf{x}^*)]}~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $\mathbf{b}oldsymbol{\mu}athbf{x}^*$ is the best result in hindsight. The regret bound of OGD is $O(\sqrt{T})$ when using decreasing step size $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = O(\frac{1}{\sqrt{t}})$. For strongly convex functions, the regret bound of OGD is $O(\log T)$ when using the step size $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = O(\frac{1}{t})$. Since $f_t$ can be any convex function, OGD considers the worst case and thus the mentioned regret bounds are optimal.
In the stochastic setting, SGD first randomly picks up $i_t$-th block samples and then computes the gradient of the selected samples as follows:
\mathbf{b}egin{align}\label{eq:sgd}
\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}}~\langle \nabla f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
$\mathbf{b}oldsymbol{\mu}athbf{x}^t$ depends on the observed realization of the random variable $\mathbf{b}oldsymbol{\mu}athbf{x}i = \{ i_1, \mathbf{c}dots, i_{t-1}\}$ or generally $\{ \mathbf{b}oldsymbol{\mu}athbf{x}^1, \mathbf{c}dots, \mathbf{b}oldsymbol{\mu}athbf{x}^{t-1} \}$. Due to the effect of variance of stochastic gradient, SGD has to choose decreasing step size, i.e., $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = O(\frac{1}{\sqrt{t}})$, leading to slow convergence speed. For general convex functions, SGD converges at a rate of $O(\frac{1}{\sqrt{t}})$. For strongly convex functions, SGD converges at a rate of $O(\frac{1}{t})$. In contrast, GD converges linearly if functions are strongly convex.
To accelerate the SGD by reducing the variance of stochastic gradient, stochastic variance reduced gradient (SVRG) was proposed by~\mathbf{c}ite{zhang13:sgdvd}.~\mathbf{c}ite{xiao14:psgdvd} extends SVRG to composite functions~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj}, called prox-SVRG. SVRGs have two stages, i.e., outer stage and inner stage. The outer stage maintains an estimate $\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}$ of the optimal point $x^*$ and computes the full gradient of $\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}$
\mathbf{b}egin{align}
\tilde{\mathbf{b}oldsymbol{\mu}u} &= \frac{1}{n} \sum_{i=1}^{n} \nabla f_i(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) = \nabla f(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}})~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
After the inner stage is completed, the outer stage updates $\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}$. At the inner stage, SVRG first randomly picks $i_t$-th sample, then modifies stochastis gradient by subtracting the difference between the full gradient and stochastic gradient at $\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}$,
\mathbf{b}egin{align}
\mathbf{v}_{t} &= \nabla f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \nabla f_{i_t}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
It can be shown that the expectation of $\mathbf{v}_{t}$ given $\mathbf{b}oldsymbol{\mu}athbf{x}^{t-1}$ is the full gradient at $\mathbf{b}oldsymbol{\mu}athbf{x}^t$, i.e., $\mathbf{b}oldsymbol{\mu}athbb{E}\mathbf{v}_{t} = \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t)$. Although $\mathbf{v}_t$ is also a stochastic gradient, the variance of stochastic gradient progressively decreases. Replacing $\nabla f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t)$ by $\mathbf{v}_t$ in SGD step~\mathbf{b}oldsymbol{\mu}yref{eq:sgd},
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} & = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}}~\langle \mathbf{v}_{t}, \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
By reduding the variance of stochastic gradient, $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ can converge to $\mathbf{b}oldsymbol{\mu}athbf{x}^*$ at the same rate as GD, which has been proved in~\mathbf{c}ite{zhang13:sgdvd,xiao14:psgdvd}.
For strongly convex functions, prox-SVRG~\mathbf{c}ite{xiao14:psgdvd} can converge linearly in expectation if $\mathbf{b}oldsymbol{\mu}athbf{e}ta > 4L$ and $m$ satisfy the following condition:
\mathbf{b}egin{align}\label{eq:svrg_rho}
\mathbf{r}ho = \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta^2}{\gamma(\mathbf{b}oldsymbol{\mu}athbf{e}ta-4L)m} + \frac{4L(m+1)}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-4L)m} < 1~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $L$ is the constant of Lipschitz continuous gradient. Note the step size is $1/\mathbf{b}oldsymbol{\mu}athbf{e}ta$ here.
\subsection{Randomized Block Coordinate Descent}
Assume $\mathbf{b}oldsymbol{\mu}athbf{x}_{j} (1\leq j \leq J)$ are non-overlapping blocks. At iteration $t$, RBCD~\mathbf{c}ite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd} randomly picks $j_t$-th coordinate and solves the following problem:
\mathbf{b}egin{align}\label{eq:rbcd}
\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}}~\langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Therefore, $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}, \mathbf{b}oldsymbol{\mu}athbf{x}_{k\neq j_t}^t)$. $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ depends on the observed realization of the random variable
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}i = \{ j_1, \mathbf{c}dots, j_{t-1}\}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Setting the step size $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = L_{j_t}$ where $L_{j_t}$ is the Lipshitz constant of $j_t$-th coordinate of the gradient $\nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t)$, the iteration complexity of RBCD is
$O(\frac{1}{t})$. For strongly convex function, RBCD has a linear convergence rate. Therefore, RBCD converges at the same rate as GD, although the constant is $J$ times larger~\mathbf{c}ite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd}.
\iffalse
Here we briefly review and simplify the proof of the iteration complexity of RBCD in~\mathbf{c}ite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd}, which paves the way for the proof of ORBCD.
\mathbf{b}egin{thm}
RBCD has the following iteration complexity:
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{T-1}}f(\mathbf{b}oldsymbol{\mu}athbf{x}^T) - f(\mathbf{b}oldsymbol{\mu}athbf{x}) & \leq \frac{J \left[ \mathbf{b}oldsymbol{\mu}athbb{E} [f(\mathbf{b}oldsymbol{\mu}athbf{x}^1)] - f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \frac{L}{2} \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^1 \|_2^2 \mathbf{r}ight ]}{T} ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{thm}
Denoting $g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) \in \mathbf{b}oldsymbol{\mu}athbf{p}artial g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1})$, the optimality condition of~\mathbf{b}oldsymbol{\mu}yref{eq:rbcd} is
\mathbf{b}egin{align}
\langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) + \mathbf{b}oldsymbol{\mu}athbf{e}ta_t (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \leq 0~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Rearranging the terms yields
\mathbf{b}egin{align}
& \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \leq - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t \langle \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 ) \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 ) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Using the smoothness of $f$ and $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}, \mathbf{b}oldsymbol{\mu}athbf{x}_{k\neq j_t}^t)$, we have
\mathbf{b}egin{align}
& f(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})- [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) ] \nonumber \\
& \leq \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \mathbf{r}angle + \frac{L_{j_t}}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) - [g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) ] \nonumber \\
& = \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + \frac{L_{j_t}}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle - [g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) ]\nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle - [g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) ] \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) - \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle - [g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) ]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the last inequality is obtained by setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = L_{j_t}$.
Conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^t$, take expectation over $j_t$ gives
\mathbf{b}egin{align}\label{eq1}
& \mathbf{b}oldsymbol{\mu}athbb{E}_{j_t}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})|\mathbf{b}oldsymbol{\mu}athbf{x}^t] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) +g(\mathbf{b}oldsymbol{\mu}athbf{x}^t)] \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{j_t}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) - \frac{1}{J} \sum_{j=1}^J \langle \nabla_{j_t} f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle - \frac{1}{J}[g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - g(\mathbf{b}oldsymbol{\mu}athbf{x}) ]\nonumber \\
& = \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{j_t}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) - \frac{1}{J} \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle - \frac{1}{J}[g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{j_t}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) - \frac{1}{J} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - (f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})) ]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Taking expectation over $\mathbf{b}oldsymbol{\mu}athbf{x}i_t$, we have
\mathbf{b}egin{align}
& \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_t}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})] -\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t)] \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_t}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) - \frac{1}{J} \{ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) +g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) ] - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})] \} ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Setting $\mathbf{b}oldsymbol{\mu}athbf{x} = \mathbf{b}oldsymbol{\mu}athbf{x}^t$ gives
\mathbf{b}egin{align}
& \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t}}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})] - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t})] \leq - \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_t}[\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Thus, $\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t})]$ decreases monotonically.
Let $\mathbf{b}oldsymbol{\mu}athbf{x} = \mathbf{b}oldsymbol{\mu}athbf{x}^*$, which is an optimal solution. Rearranging the temrs of~\mathbf{b}oldsymbol{\mu}yref{eq1} yields
\mathbf{b}egin{align}
&\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t}}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t)] - [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] \nonumber \\
&\leq J \left[ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}} [f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t}) ] - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t}}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})] + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t}}[\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2]) \mathbf{r}ight]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Let $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = L = \mathbf{b}oldsymbol{\mu}ax_j {L_{j}} $. Summing over $t$, we have
\mathbf{b}egin{align}
& \sum_{t=1}^T\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t})] - [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{*})] \nonumber \\
& \leq J \left\{ f(\mathbf{b}oldsymbol{\mu}athbf{x}^1) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{1}) -\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{T}} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^{T+1})+ g(\mathbf{b}oldsymbol{\mu}athbf{x}^{T+1})] + \frac{L}{2}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^1 \|_2^2 \mathbf{r}ight \} \nonumber \\
& \leq J \left\{ [f(\mathbf{b}oldsymbol{\mu}athbf{x}^1) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{1})] - [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) +g(\mathbf{b}oldsymbol{\mu}athbf{x}^*)] + \frac{L}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^1 \|_2^2 \mathbf{r}ight \} ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Using the monotonicity of $\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{t-1}}f(\mathbf{b}oldsymbol{\mu}athbf{x}^t)$ and dividing $T$ on both sides complete the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
\fi
\section{Online Randomized Block Coordinate Descent}\label{sec:orbcd}
In this section, our goal is to combine OGD/SGD and RBCD together to solve problem~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj}. We call the algorithm online randomized block coordinate descent (ORBCD), which computes one block coordinate of the gradient of one block of samples at each iteration. ORBCD essentially performs RBCD in online and stochastic setting.
Let $\{ \mathbf{b}oldsymbol{\mu}athbf{x}_1, \mathbf{c}dots, \mathbf{b}oldsymbol{\mu}athbf{x}_J \}, \mathbf{b}oldsymbol{\mu}athbf{x}_j\in \R^{n_j\times 1}$ be J non-overlapping blocks of $\mathbf{b}oldsymbol{\mu}athbf{x}$.
Let $U_j \in \R^{n\times n_j}$ be $n_j$ columns of an $n\times n$ permutation matrix $\mathbf{b}oldsymbol{\mu}athbf{U}$, corresponding to $j$ block coordinates in $\mathbf{b}oldsymbol{\mu}athbf{x}$. For any partition of $\mathbf{b}oldsymbol{\mu}athbf{x}$ and $\mathbf{b}oldsymbol{\mu}athbf{U}$,
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x} = \sum_{j=1}^{J}U_j\mathbf{b}oldsymbol{\mu}athbf{x}_j~, \mathbf{b}oldsymbol{\mu}athbf{x}_j = U_j^T\mathbf{b}oldsymbol{\mu}athbf{x}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The $j$-th coordinate of gradient of $f$ can be denoted as
\mathbf{b}egin{align}
\nabla_j f(\mathbf{b}oldsymbol{\mu}athbf{x}) = U_j^T \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x})~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Throughout the paper, we assume that the minimum of problem~\mathbf{b}oldsymbol{\mu}yref{eq:compositeobj} is attained. In addition, ORBCD needs the following assumption :
\mathbf{v}space{-3mm}
\mathbf{b}egin{asm}\label{asm:orbcd1}
$f_t$ or $f_i$ has block-wise Lipschitz continuous gradient with constant $L_j$, e.g.,
\mathbf{b}egin{align}
\| \nabla_j f_t(\mathbf{b}oldsymbol{\mu}athbf{x} + U_j h_j ) - \nabla_j f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) \|_2 \leq L_j \| h_j \|_2 \leq L \| h_j \|_2~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $L = \mathbf{b}oldsymbol{\mu}ax_j L_j$.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{asm}
\mathbf{b}egin{asm}\label{asm:orbcd2}
1. $\| \nabla f_t (\mathbf{b}oldsymbol{\mu}athbf{x}^t) \|_2 \leq R_f $, or $\| \nabla f (\mathbf{b}oldsymbol{\mu}athbf{x}^t) \|_2 \leq R_f $;
2. $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ is assumed in a bounded set ${\mathbf{c}al X}$, i.e., $\sup_{\mathbf{b}oldsymbol{\mu}athbf{x},\mathbf{b}oldsymbol{\mu}athbf{y} \in {\mathbf{c}al X}} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{y} \|_2 = D$.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{asm}
While the Assumption~\mathbf{r}ef{asm:orbcd1} is used in RBCD, the Assumption~\mathbf{r}ef{asm:orbcd2} is used in OGD/SGD. We may assume the sum of two functions is strongly convex.
\mathbf{b}egin{asm}\label{asm:orbcd3}
$f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})$ or $f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})$ is $\gamma$-strongly convex, e.g.,
we have
\mathbf{b}egin{align}\label{eq:stronggcov}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) \geq f_t(\mathbf{b}oldsymbol{\mu}athbf{y}) + g(\mathbf{b}oldsymbol{\mu}athbf{y}) + \langle \nabla f_t(\mathbf{b}oldsymbol{\mu}athbf{y}) + g'(\mathbf{b}oldsymbol{\mu}athbf{y}), \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t} \mathbf{r}angle + \frac{\gamma}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{y} \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $\gamma > 0$ and $g'(\mathbf{b}oldsymbol{\mu}athbf{y})$ denotes the subgradient of $g$ at $\mathbf{b}oldsymbol{\mu}athbf{y}$.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{asm}
\subsection{ORBCD for Online Learning}
In online setting, ORBCD considers the worst case and runs at rounds.
At time $t$, given any function $f_t$ which may be agnostic, ORBCD randomly chooses $j_t$-th block coordinate and presents the solution by solving the following problem:
\mathbf{b}egin{align}\label{eq:orbcdo}
\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} &= \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}}~\langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 \nonumber \\
& = \text{Prox}_{g_{j_t}}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} -\frac{1}{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}\nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) )~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $\text{Prox}$ denotes the proximal mapping. If $f_t$ is a linear function, e.g., $f_t = l_t\mathbf{b}oldsymbol{\mu}athbf{x}^t$, then $\nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) = l_{j_t}$, so solving~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdo} is $J$ times cheaper than OGD.
Thus, $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = ( \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}, \mathbf{b}oldsymbol{\mu}athbf{x}_{k\neq j_t}^t)$, or
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{b}oldsymbol{\mu}athbf{x}^t + U_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Then, ORBCD receives a loss function $f_{t+1}(\mathbf{b}oldsymbol{\mu}athbf{x})$ which incurs the loss $f_{t+1}(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})$. The algorithm is summarized in Algorithm~\mathbf{r}ef{alg:orbcd_online}.
$\mathbf{b}oldsymbol{\mu}athbf{x}^t$ is independent of $j_t$ but depends on the sequence of observed realization of the random variable
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}i = \{ j_1, \mathbf{c}dots, j_{t-1} \}.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Let $\mathbf{b}oldsymbol{\mu}athbf{x}^*$ be the best solution in hindsight. The regret bound of ORBCD is defined as
\mathbf{b}egin{align}
R(T) = \sum_{t=1}^T\left \{ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}[ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) ] - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^*) +g(\mathbf{b}oldsymbol{\mu}athbf{x}^*)] \mathbf{r}ight \}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
By setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \sqrt{t} + L$ where $L=\mathbf{b}oldsymbol{\mu}ax_jL_j$, the regret bound of ORBCD is $O(\sqrt{T})$. For strongly convex functions, the regret bound of ORBCD is $O(\log T)$ by setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \frac{\gamma t}{J} + L$.
\mathbf{b}egin{algorithm*}[tb]
\mathbf{c}aption{Online Randomized Block Coordinate Descent for Online Learning}
\label{alg:orbcd_online}
\mathbf{b}egin{algorithmic}[1]
\STATE {\mathbf{b}fseries Initialization:} $\mathbf{b}oldsymbol{\mu}athbf{x}^1 = \mathbf{b}oldsymbol{\mu}athbf{0}$
\FOR{$t=1 \text{ to } T$}
\STATE randomly pick up $j_t$ block coordinates
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \in {\mathbf{c}al X}_j}~\langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2$~
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{b}oldsymbol{\mu}athbf{x}^t + U_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t)$
\STATE receives the function $f_{t+1}(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})$ and incurs the loss $f_{t+1}(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})$
\mathbf{b}oldsymbol{\mu}athbf{E}NDFOR
\mathbf{b}oldsymbol{\mu}athbf{e}nd{algorithmic}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{algorithm*}
\subsection{ORBCD for Stochastic Optimization}
In the stochastic setting, ORBCD first randomly picks up $i_t$-th block sample and then randomly chooses $j_t$-th block coordinate. The algorithm has the following iterate:
\mathbf{b}egin{align}\label{eq:orbcds}
\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} & = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}}~\langle \nabla_{j_t} f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 \nonumber \\
& = \text{Prox}_{g_{j_t}}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} -\nabla_{j_t} f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t) )~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
For high dimensional problem with non-overlapping composite regularizers, solving~\mathbf{b}oldsymbol{\mu}yref{eq:orbcds} is computationally cheaper than solving~\mathbf{b}oldsymbol{\mu}yref{eq:sgd} in SGD.
The algorithm of ORBCD in both settings is summarized in Algorithm~\mathbf{r}ef{alg:orbcd_stochastic}.
$\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}$ depends on $(i_t, j_t)$, but $j_{t}$ and $i_{t}$ are independent.
$\mathbf{b}oldsymbol{\mu}athbf{x}^t$ is independent of $(i_t, j_t)$ but depends on the observed realization of the random variables
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}i = \{ ( i_1, j_1), \mathbf{c}dots, (i_{t-1}, j_{t-1}) \}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The online-stochastic conversion rule~\mathbf{c}ite{Duchi10_comid,duchi09,xiao10} still holds here. The iteration complexity of ORBCD can be obtained by dividing the regret bounds in the online setting by $T$. Setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \sqrt{t} + L$ where $L=\mathbf{b}oldsymbol{\mu}ax_jL_j$, the iteration complexity of ORBCD is
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^t) + g(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^t) ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq O(\frac{1}{\sqrt{T}})~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
For strongly convex functions, setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \frac{\gamma t}{J} + L$,
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^t) + g(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^t) ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq O(\frac{\log T}{T})~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The iteration complexity of ORBCD match that of SGD. Simiarlar as SGD, the convergence speed of ORBCD is also slowed down by the variance of stochastic gradient.
\mathbf{b}egin{algorithm*}[tb]
\mathbf{c}aption{Online Randomized Block Coordinate Descent for Stochastic Optimization}
\label{alg:orbcd_stochastic}
\mathbf{b}egin{algorithmic}[1]
\STATE {\mathbf{b}fseries Initialization:} $\mathbf{b}oldsymbol{\mu}athbf{x}^1 = \mathbf{b}oldsymbol{\mu}athbf{0}$
\FOR{$t=1 \text{ to } T$}
\STATE randomly pick up $i_t$ block samples and $j_t$ block coordinates
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \in {\mathbf{c}al X}_j}~\langle \nabla_{j_t} f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2$~
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{b}oldsymbol{\mu}athbf{x}^t + U_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t)$
\mathbf{b}oldsymbol{\mu}athbf{E}NDFOR
\mathbf{b}oldsymbol{\mu}athbf{e}nd{algorithmic}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{algorithm*}
\mathbf{b}egin{algorithm*}[tb]
\mathbf{c}aption{Online Randomized Block Coordinate Descent with Variance Reduction}
\label{alg:orbcdvd}
\mathbf{b}egin{algorithmic}[1]
\STATE {\mathbf{b}fseries Initialization:} $\mathbf{b}oldsymbol{\mu}athbf{x}^1 = \mathbf{b}oldsymbol{\mu}athbf{0}$
\FOR{$t=2 \text{ to } T$}
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}_0 = \tilde{\mathbf{b}oldsymbol{\mu}athbf{x}} = \mathbf{b}oldsymbol{\mu}athbf{x}^t$.
\FOR{$k = 0\textbf{ to } m-1$}
\STATE randomly pick up $i_k$ block samples
\STATE randomly pick up $j_k$ block coordinates
\STATE $\mathbf{v}_{j_k}^{i_k} = \nabla_{j_k} f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^{k}) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u}_{j_k}$ where $\tilde{\mathbf{b}oldsymbol{\mu}u}_{j_k} = \nabla_{j_k} f(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}})$
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} = \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} }~\langle \mathbf{v}_{j_k}^{i_k}, \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle + g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_k}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k}\|_2^2$~
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} = \mathbf{b}oldsymbol{\mu}athbf{x}^k + U_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_j}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k)$
\mathbf{b}oldsymbol{\mu}athbf{E}NDFOR
\STATE $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = \mathbf{b}oldsymbol{\mu}athbf{x}^m$ or $\frac{1}{m}\sum_{k=1}^{m}\mathbf{b}oldsymbol{\mu}athbf{x}^k$
\mathbf{b}oldsymbol{\mu}athbf{E}NDFOR
\mathbf{b}oldsymbol{\mu}athbf{e}nd{algorithmic}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{algorithm*}
\subsection{ORBCD with variance reduction}
In the stochastic setting, we apply the variance reduction technique~\mathbf{c}ite{xiao14:psgdvd,zhang13:sgdvd} to accelerate the rate of convergence of ORBCD, abbreviated as ORBCDVD. As SVRG and prox-SVRG, ORBCDVD consists of two stages. At time $t+1$, the outer stage maintains an estimate $\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}} = \mathbf{b}oldsymbol{\mu}athbf{x}^t$ of the optimal $\mathbf{b}oldsymbol{\mu}athbf{x}^*$ and updates $\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}$ every $m+1$ iterations.
The inner stage takes $m$ iterations which is indexed by $k = 0,\mathbf{c}dots, m-1$. At the $k$-th iteration, ORBCDVD randomly picks $i_k$-th sample and $j_k$-th coordinate and compute
\mathbf{b}egin{align}
\mathbf{v}_{j_k}^{i_k} &= \nabla_{j_k} f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u}_{j_k}~, \label{eq:orbcdvd_vij}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where
\mathbf{b}egin{align}\label{eq:orbcdvd_mu}
\tilde{\mathbf{b}oldsymbol{\mu}u}_{j_k} = \frac{1}{n} \sum_{i=1}^{n} \nabla_{j_k} f_i(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) = \nabla_{j_k} f(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}})~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
$\mathbf{v}_{j_t}^{i_t}$ depends on $(i_t, j_t)$, and $i_t$ and $j_t$ are independent. Conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^k$, taking expectation over $i_k, j_k$ gives
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}\mathbf{v}_{j_k}^{i_k} &= \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k} \mathbf{b}oldsymbol{\mu}athbb{E}_{j_k}[\nabla_{j_k} f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u}_{j_k}] \nonumber \\
&= \frac{1}{J}\mathbf{b}oldsymbol{\mu}athbb{E}_{i_k} [\nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u} ] \nonumber \\
& = \frac{1}{J}\nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^k)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Although $\mathbf{v}_{j_k}^{i_k}$ is stochastic gradient, the variance $\mathbf{b}oldsymbol{\mu}athbb{E} \| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2$ decreases progressively and is smaller than $\mathbf{b}oldsymbol{\mu}athbb{E} \| \nabla f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) \|_2^2$.
Using the variance reduced gradient $\mathbf{v}_{j_k}^{i_k}$, ORBCD then performs RBCD as follows:
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} &= \mathbf{a}rgmin_{\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}}~ \langle \mathbf{v}_{j_k}^{i_k}, \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle + g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k \|_2^2 \label{eq:orbcdvd_xj}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
After $m$ iterations, the outer stage updates $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}$ which is either $\mathbf{b}oldsymbol{\mu}athbf{x}^m$ or $\frac{1}{m}\sum_{k=1}^{m}\mathbf{b}oldsymbol{\mu}athbf{x}^k$. The algorithm is summarized in Algorithm~\mathbf{r}ef{alg:orbcdvd}. At the outer stage,
ORBCDVD does not necessarily require to compute the full gradient at once. If the computation of full gradient requires substantial computational
efforts, SVRG has to stop and complete the full gradient step before making progress. In contrast, $\tilde{\mathbf{b}oldsymbol{\mu}u}$ can be partially computed at each iteration and then stored for the next retrieval in ORBCDVD.
Assume $\mathbf{b}oldsymbol{\mu}athbf{e}ta > 2L$ and $m$ satisfy the following condition:
\mathbf{b}egin{align}\label{eq:orbcdvd_rho}
\mathbf{r}ho = \frac{L(m+1)}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m} + \frac{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)J}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m} - \frac{1}{m}+ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta (\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)J}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m\gamma} < 1~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Then $h(\mathbf{b}oldsymbol{\mu}athbf{x})$ converges linearly in expectation, i.e.,
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - (f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] \leq O(\mathbf{r}ho^t)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta = 4L$ in~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_rho} yields
\mathbf{b}egin{align}
\mathbf{r}ho = \frac{m+1}{2m} + \frac{3J}{2m} - \frac{1}{m}+ \frac{6JL}{m\gamma} \leq \frac{1}{2} + \frac{3 J}{2m}(1+\frac{4 L}{\gamma})~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Setting $m = 18JL/\gamma$, then
\mathbf{b}egin{align}
\mathbf{r}ho \leq \frac{1}{2} + \frac{1}{12}(\frac{\gamma}{L}+4) \mathbf{a}pprox \frac{11}{12}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where we assume $\gamma/L \mathbf{a}pprox 1$ for simplicity.
\section{The Rate of Convergence}\label{sec:theory}
The following lemma is a key building block of the proof of the convergence of ORBCD in both online and stochastic setting.
\mathbf{b}egin{lem}
Let the Assumption~\mathbf{r}ef{asm:orbcd1} and \mathbf{r}ef{asm:orbcd2} hold.
Let $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ be the sequences generated by ORBCD. $j_t$ is sampled randomly and uniformly from $\{1,\mathbf{c}dots, J \}$. We have
\mathbf{b}egin{align}\label{eq:orbcd_key_lem}
& \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{R_f^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $L = \mathbf{b}oldsymbol{\mu}ax_j L_j$.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{lem}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
The optimality condition is
\mathbf{b}egin{align}
\langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + \mathbf{b}oldsymbol{\mu}athbf{e}ta_t (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \leq 0~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Rearranging the terms yields
\mathbf{b}egin{align}
& \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \leq - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t \langle \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 ) \nonumber \\
& = \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 ) ~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the last equality uses $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}, \mathbf{b}oldsymbol{\mu}athbf{x}_{k\neq {j_t}}^t)$.
By the smoothness of $f_t$, we have
\mathbf{b}egin{align}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \leq f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + \langle \nabla_j f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_j^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_j^t \mathbf{r}angle + \frac{L_j}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_j^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_j^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Since $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}^t = U_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t})$,
\iffalse
the convexity of $g$ gives
\mathbf{b}egin{align}
& g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) \leq \langle g'(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}), \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \mathbf{r}angle \leq \langle g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) + \sum_{\mathbf{b}oldsymbol{\mu}athbb{I}_{j_t} \mathbf{c}ap \mathbf{b}oldsymbol{\mu}athbb{I}_k \neq \mathbf{b}oldsymbol{\mu}athbf{e}mptyset} g'_{k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{k}^{t+1}), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \mathbf{r}angle \nonumber \\
& \leq \langle g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \mathbf{r}angle + \frac{1}{2\mathbf{a}lpha}\| \sum_{\mathbf{b}oldsymbol{\mu}athbb{I}_{j_t} \mathbf{c}ap \mathbf{b}oldsymbol{\mu}athbb{I}_k \neq \mathbf{b}oldsymbol{\mu}athbf{e}mptyset} g'_{k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{k}^{t+1}) \|_2^2 + \frac{\mathbf{a}lpha}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 \nonumber \\
& \leq \langle g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \mathbf{r}angle + \frac{(J-1)R_g^2}{2\mathbf{a}lpha} + \frac{\mathbf{a}lpha}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}egin{align}
g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) \leq \langle g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \mathbf{r}angle~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}egin{align}
\| \mathbf{b}oldsymbol{\mu}athbf{x}_{k}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{k}^t \|_2^2 = \| \mathbf{b}oldsymbol{\mu}athbf{x}_{\mathbf{b}oldsymbol{\mu}athbb{I}_{j_t} \mathbf{c}ap \mathbf{b}oldsymbol{\mu}athbb{I}_k \neq \mathbf{b}oldsymbol{\mu}athbf{e}mptyset}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{\mathbf{b}oldsymbol{\mu}athbb{I}_{j_t} \mathbf{c}ap \mathbf{b}oldsymbol{\mu}athbb{I}_k \neq \mathbf{b}oldsymbol{\mu}athbf{e}mptyset}^t \|_2^2 \leq \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}egin{align}
\langle \sum_{\mathbf{b}oldsymbol{\mu}athbb{I}_{j_t} \mathbf{c}ap \mathbf{b}oldsymbol{\mu}athbb{I}_k \neq \mathbf{b}oldsymbol{\mu}athbf{e}mptyset} g'_{k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{k}^{t+1}), \mathbf{b}oldsymbol{\mu}athbf{x}_{k}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{k}^t \mathbf{r}angle \leq \frac{(J-1)R_g^2}{2\mathbf{a}lpha} + \frac{\mathbf{a}lpha}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\fi
we have
\mathbf{b}egin{align}
& f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t)] \nonumber \\
& \leq \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \mathbf{r}angle + \frac{L_{j_t}}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) + g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}) \nonumber \\
& \leq \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle + \frac{L_{j_t}}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t}), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Rearranging the terms yields
\mathbf{b}egin{align}\label{eq:lem1}
\langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g_{j_t}'(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle &\leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 \nonumber \\
&+ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t)- [ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) ]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The convexity of $f_t$ gives
\mathbf{b}egin{align}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \leq \langle \nabla f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}^t - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \mathbf{r}angle = \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} \mathbf{r}angle \leq \frac{1}{2\mathbf{a}lpha} \| \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) \|_2^2 + \frac{\mathbf{a}lpha}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the equality uses $\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} = (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1}, \mathbf{b}oldsymbol{\mu}athbf{x}_{k\neq {j_t}}^t)$. Plugging into~\mathbf{b}oldsymbol{\mu}yref{eq:lem1}, we have
\mathbf{b}egin{align}
& \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'_{j_t}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 + \langle \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t \|_2^2 + \frac{\mathbf{a}lpha}{2}\| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^t - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t+1} \|_2^2 + \frac{1}{2\mathbf{a}lpha} \| \nabla_{j_t} f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) \|_2^2 ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Let $ L = \mathbf{b}oldsymbol{\mu}ax_{j} L_j$. Setting $\mathbf{a}lpha = \mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L$ where $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t > L$ completes the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
This lemma is also a key building block in the proof of iteration complexity of GD, OGD/SGD and RBCD. In GD, by setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = L$, the iteration complexity of GD can be established. In RBCD, by simply setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = L_{j_t}$, the iteration complexity of RBCD can be established.
\subsection{Online Optimization}
Note $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ depends on the sequence of observed realization of the random variable
$\mathbf{b}oldsymbol{\mu}athbf{x}i = \{ j_1, \mathbf{c}dots, j_{t-1} \}$.
The following theorem establishes the regret bound of ORBCD.
\mathbf{b}egin{thm}
Let $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \sqrt{t} + L$ in the ORBCD and the Assumption~\mathbf{r}ef{asm:orbcd1} and \mathbf{r}ef{asm:orbcd2} hold. $j_t$ is sampled randomly and uniformly from $\{1,\mathbf{c}dots, J \}$. The regret bound $R(T)$ of ORBCD is
\mathbf{b}egin{align}
R(T) \leq J ( \frac{\sqrt{T} + L}{2}D^2 + \sqrt{T} R^2 + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) )~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{thm}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
In~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_key_lem}, conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^t$, take expectation over $j_t$, we have
\mathbf{b}egin{align}\label{eq:a}
\frac{1}{J} \langle \nabla f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle &\leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1})
~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Using the convexity, we have
\mathbf{b}egin{align}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq \langle \nabla f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Together with~\mathbf{b}oldsymbol{\mu}yref{eq:a}, we have
\mathbf{b}egin{align}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] &\leq J \left \{ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \mathbf{r}ight \}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Taking expectation over $\mathbf{b}oldsymbol{\mu}athbf{x}i$ on both sides, we have
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} \left [ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] \mathbf{r}ight ] &\leq J \left \{ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) \mathbf{r}ight .\nonumber \\
& + \left. \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \mathbf{r}ight \}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Summing over $t$ and setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \sqrt{t} + L$, we obtain the regret bound
\mathbf{b}egin{align}\label{eq:orbcd_rgt0}
& R(T) = \sum_{t=1}^T\left \{ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) ] - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \mathbf{r}ight \} \nonumber \\
&\leq J \left \{ - \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_{T}}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{T+1} \|_2^2 + \sum_{t=1}^{T}(\mathbf{b}oldsymbol{\mu}athbf{e}ta_{t} - \mathbf{b}oldsymbol{\mu}athbf{e}ta_{t-1}) \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t} \|_2^2 + \sum_{t=1}^{T}\frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{T+1}) \mathbf{r}ight \} \nonumber \\
& \leq J \left \{ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_T}{2} D^2 + \sum_{t=1}^{T}\frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \mathbf{r}ight \} \nonumber \\
& \leq J \left \{ \frac{\sqrt{T} + L}{2} D^2 + \sum_{t=1}^{T}\frac{R^2}{2\sqrt{t} } + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \mathbf{r}ight \} \nonumber \\
& \leq J ( \frac{\sqrt{T} + L}{2} D^2+ \sqrt{T} R^2 + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) )~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
which completes the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
If one of the functions is strongly convex, ORBCD can achieve a $\log(T)$ regret bound, which is established in the following theorem.
\mathbf{b}egin{thm}\label{thm:orbcd_rgt_strong}
Let the Assumption~\mathbf{r}ef{asm:orbcd1}-\mathbf{r}ef{asm:orbcd3} hold and $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \frac{\gamma t}{J} + L$ in ORBCD. $j_t$ is sampled randomly and uniformly from $\{1,\mathbf{c}dots, J \}$. The regret bound $R(T)$ of ORBCD is
\mathbf{b}egin{align}
R(T) \leq J^2R^2 \log(T) + J(g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{thm}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
Using the strong convexity of $f_t + g$ in~\mathbf{b}oldsymbol{\mu}yref{eq:stronggcov}, we have
\mathbf{b}egin{align}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq \langle \nabla f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle - \frac{\gamma}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Together with~\mathbf{b}oldsymbol{\mu}yref{eq:a}, we have
\mathbf{b}egin{align}
f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] &\leq \frac{J\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - \gamma }{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \frac{J\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) \nonumber \\
& + \frac{JR^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + J [ g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) ] ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Taking expectation over $\mathbf{b}oldsymbol{\mu}athbf{x}i$ on both sides, we have
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} \left [ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] \mathbf{r}ight ] &\leq \frac{J\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - \gamma}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \frac{J\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2}\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}[\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2]) \nonumber \\
& + \frac{JR^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + J [ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) ]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Summing over $t$ and setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \frac{\gamma t}{J} + L$, we obtain the regret bound
\mathbf{b}egin{align}
& R(T) = \sum_{t=1}^T\left \{ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f_t(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) ] - [f_t(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \mathbf{r}ight \} \nonumber \\
&\leq - \frac{J\mathbf{b}oldsymbol{\mu}athbf{e}ta_{T}}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{T+1} \|_2^2 + \sum_{t=1}^{T}\frac{J\mathbf{b}oldsymbol{\mu}athbf{e}ta_{t} -\gamma - J\mathbf{b}oldsymbol{\mu}athbf{e}ta_{t-1}}{2}\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t} \|_2^2 + \sum_{t=1}^{T}\frac{JR^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + J ( g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{T+1}) ) \nonumber \\
& \leq \sum_{t=1}^{T}\frac{J^2R^2}{2\gamma t } + J(g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ) \nonumber \\
& \leq J^2R^2 \log(T) + J(g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ) ~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
which completes the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
In general, ORBCD can achieve the same order of regret bound as OGD and other first-order online optimization methods, although the constant could be $J$ times larger.
\iffalse
If setting $f_t = f$, ORBCD turns to batch optimization or randomized overlapping block coordinate descent (ROLBCD). By dividing the regret bound by $T$ and denoting $\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T = \frac{1}{T}\sum_{t=1}^{T} \mathbf{b}oldsymbol{\mu}athbf{x}^t$, we obtain the iteration complexity of ROLBCD, i.e.,
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i_{T-1}^j}[ f(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T) + g(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T) ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] = \frac{R(T)}{T} ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The iteration complexity of ROLBCD is $O(\frac{1}{\sqrt{T}})$, which is worse than RBCD.
\fi
\subsection{Stochastic Optimization}
In the stochastic setting, ORBCD first randomly chooses the $i_t$-th block sample and the $j_t$-th block coordinate.
$j_{t}$ and $i_{t}$ are independent. $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ depends on the observed realization of the random variables
$\mathbf{b}oldsymbol{\mu}athbf{x}i = \{ ( i_1, j_1), \mathbf{c}dots, (i_{t-1}, j_{t-1}) \}$.
The following theorem establishes the iteration complexity of ORBCD for general convex functions.
\mathbf{b}egin{thm}\label{thm:orbcd_stc_ic}
Let $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \sqrt{t} + L$ and $\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T = \frac{1}{T} \sum_{t=1}^{T}\mathbf{b}oldsymbol{\mu}athbf{x}^t $ in the ORBCD. $i_t, j_t$ are sampled randomly and uniformly from $\{1,\mathbf{c}dots, I \}$ and $\{1,\mathbf{c}dots, J \}$ respectively. The iteration complexity of ORBCD is
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^t) + g(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^t) ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq \frac{J ( \frac{\sqrt{T} + L}{2} D^2+ \sqrt{T} R^2 + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) )}{T}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{thm}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
In the stochastic setting, let $f_t$ be $f_{i_t}$ in~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_key_lem}, we have
\mathbf{b}egin{align}\label{eq:orbcd_key_stoc}
\langle \nabla_{j_t} f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g_{j_t}'(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Note $i_t, j_t$ are independent of $\mathbf{b}oldsymbol{\mu}athbf{x}^t$. Conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^t$, taking expectation over $i_t$ and $j_t$, the RHS is
\mathbf{b}egin{align}
& \mathbf{b}oldsymbol{\mu}athbb{E}\langle \nabla_{j_t} f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g_{j_t}'(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle = \mathbf{b}oldsymbol{\mu}athbb{E}_{i_t} [ \mathbf{b}oldsymbol{\mu}athbb{E}_{j_t} [ \langle \nabla_{j_t} f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g_{j_t}'(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_t} \mathbf{r}angle ] ] \nonumber \\
& = \frac{1}{J} \mathbf{b}oldsymbol{\mu}athbb{E}_{i_t} [ \langle \nabla f_{i_t}(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle + \langle g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle ] \nonumber \\
& = \frac{1}{J}\langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Plugging back into~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_key_stoc}, we have
\mathbf{b}egin{align}\label{eq:orbcd_stc_0}
& \frac{1}{J}\langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t) , \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle \nonumber \\
&\leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E} g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Using the convexity of $f + g$, we have
\mathbf{b}egin{align}
f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Together with~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_stc_0}, we have
\mathbf{b}egin{align}
f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] &\leq J \left \{ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \mathbf{r}ight \}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Taking expectation over $\mathbf{b}oldsymbol{\mu}athbf{x}i$ on both sides, we have
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} \left [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) \mathbf{r}ight ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) ] &\leq J \left \{ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta_t}{2} ( \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}[\| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} \|_2^2]) \mathbf{r}ight .\nonumber \\
& + \left. \frac{R^2}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta_t - L)} + \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}g(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \mathbf{r}ight \}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Summing over $t$ and setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \sqrt{t} + L$, following similar derivation in~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_rgt0}, we have
\mathbf{b}egin{align}
\sum_{t=1}^T\left \{ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \mathbf{r}ight \} \leq J ( \frac{\sqrt{T} + L}{2} D^2+ \sqrt{T} R^2 + g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) )~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Dividing both sides by $T$, using the Jensen's inequality and denoting $\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T = \frac{1}{T}\sum_{t=1}^{T}\mathbf{b}oldsymbol{\mu}athbf{x}^t$ complete the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
For strongly convex functions, we have the following results.
\mathbf{b}egin{thm}
For strongly convex function, setting $\mathbf{b}oldsymbol{\mu}athbf{e}ta_t = \frac{\gamma t}{J} + L$ in the ORBCD. $i_t, j_t$ are sampled randomly and uniformly from $\{1,\mathbf{c}dots, I \}$ and $\{1,\mathbf{c}dots, J \}$ respectively. Let $\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T = \frac{1}{T} \sum_{t=1}^{T}\mathbf{b}oldsymbol{\mu}athbf{x}^t $. The iteration complexity of ORBCD is
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T) + g(\mathbf{b}ar{\mathbf{b}oldsymbol{\mu}athbf{x}}^T) ] - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) +g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq \frac{J^2R^2 \log(T) + J(g(\mathbf{b}oldsymbol{\mu}athbf{x}^1) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ) }{T}~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{thm}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
If $f+g$ is strongly convex, we have
\mathbf{b}egin{align}
f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - [f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})] \leq \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g'(\mathbf{b}oldsymbol{\mu}athbf{x}^t), \mathbf{b}oldsymbol{\mu}athbf{x}^{t} - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle - \frac{\gamma}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Plugging back into~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_stc_0}, following similar derivation in Theorem~\mathbf{r}ef{thm:orbcd_rgt_strong} and Theorem~\mathbf{r}ef{thm:orbcd_stc_ic} complete the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
\subsection{ORBCD with Variance Reduction}
According to the Theorem 2.1.5 in~\mathbf{c}ite{nesterov04:convex}, the block-wise Lipschitz gradient in Assumption~\mathbf{r}ef{asm:orbcd1} can also be rewritten as follows:
\mathbf{b}egin{align}
& f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) \leq f_i(\mathbf{b}oldsymbol{\mu}athbf{y}) + \langle \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{y}), \mathbf{b}oldsymbol{\mu}athbf{x}_j - \mathbf{b}oldsymbol{\mu}athbf{y}_j\mathbf{r}angle + \frac{L}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_j - \mathbf{b}oldsymbol{\mu}athbf{y}_j\|_2^2~, \label{eq:blk_lip1} \\
&\| \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{y}) \|_2^2 \leq L \langle \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{y}), \mathbf{b}oldsymbol{\mu}athbf{x}_j - \mathbf{b}oldsymbol{\mu}athbf{y}_j\mathbf{r}angle~.\label{eq:blk_lip2}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Let $\mathbf{b}oldsymbol{\mu}athbf{x}^*$ be an optimal solution. Define an upper bound of $f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) -(f(\mathbf{b}oldsymbol{\mu}athbf{x}^*)+g(\mathbf{b}oldsymbol{\mu}athbf{x}^*))$ as
\mathbf{b}egin{align}\label{eq:def_h}
h(\mathbf{b}oldsymbol{\mu}athbf{x},\mathbf{b}oldsymbol{\mu}athbf{x}^*)= \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}), \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^*\mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
If $f(\mathbf{b}oldsymbol{\mu}athbf{x}) + g(\mathbf{b}oldsymbol{\mu}athbf{x})$ is strongly convex, we have
\mathbf{b}egin{align}\label{eq:orbcd_strong_h}
h(\mathbf{b}oldsymbol{\mu}athbf{x},\mathbf{b}oldsymbol{\mu}athbf{x}^*) \geq f(\mathbf{b}oldsymbol{\mu}athbf{x}) - f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + g(\mathbf{b}oldsymbol{\mu}athbf{x}) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \geq \frac{\gamma}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^* \|_2^2 ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}egin{lem}\label{lem:orbcdvd_lem1}
Let $\mathbf{b}oldsymbol{\mu}athbf{x}^*$ be an optimal solution and the Assumption~\mathbf{r}ef{asm:orbcd1}, we have
\mathbf{b}egin{align}
\frac{1}{I} \sum_{i=1}^{I} \| \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 \leq L h(\mathbf{b}oldsymbol{\mu}athbf{x},\mathbf{b}oldsymbol{\mu}athbf{x}^*)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $h$ is defined in~\mathbf{b}oldsymbol{\mu}yref{eq:def_h}.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{lem}
\mathbf{b}oldsymbol{\mu}athbf{p}roof Since the Assumption~\mathbf{r}ef{asm:orbcd1} hold, we have using
\mathbf{b}egin{align}\label{eq:orbcd_bd_h}
&\frac{1}{I} \sum_{i=1}^{I} \| \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 = \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^J \| \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 \nonumber \\
&\leq \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^J L \langle \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^*), \mathbf{b}oldsymbol{\mu}athbf{x}_j - \mathbf{b}oldsymbol{\mu}athbf{x}_j^*\mathbf{r}angle \nonumber \\
& = L [ \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}), \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^*\mathbf{r}angle + \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^*), \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle]~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the inequality uses~\mathbf{b}oldsymbol{\mu}yref{eq:blk_lip2}. For an optimal solution $\mathbf{b}oldsymbol{\mu}athbf{x}^*$, $g'(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) = 0$ where $g'(\mathbf{b}oldsymbol{\mu}athbf{x}^*)$ is the subgradient of $g$ at $\mathbf{b}oldsymbol{\mu}athbf{x}^*$. The second term in~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_bd_h} can be rewritten as
\mathbf{b}egin{align}
& \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^*), \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle = - \langle g'(\mathbf{b}oldsymbol{\mu}athbf{x}^*), \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x} \mathbf{r}angle = g(\mathbf{b}oldsymbol{\mu}athbf{x}) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Plugging into~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_bd_h} and using~\mathbf{b}oldsymbol{\mu}yref{eq:def_h} complete the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
\mathbf{b}egin{lem}\label{lem:orbcdvd_lem2}
Let $\mathbf{v}_{j_k}^{i_k} $ and $\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1}$ be generated by~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_vij}-\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_xj}. Conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^k$, we have
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E} \| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2 \leq \frac{2L}{J} [h(\mathbf{b}oldsymbol{\mu}athbf{x}^{k},\mathbf{b}oldsymbol{\mu}athbf{x}^*) + h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*)]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{lem}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
Conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^k$, we have
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}[ \nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u} ] = \frac{1}{I} \sum_{i=1}^{I} [ \nabla f_{i}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_{i}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u} ] = \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^k)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Note $\mathbf{b}oldsymbol{\mu}athbf{x}^k$ is independent of $i_k, j_k$. $i_k$ and $j_k$ are independent. Conditioned on $\mathbf{b}oldsymbol{\mu}athbf{x}^k$, taking expectation over $i_k, j_k$ and using~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_vij} give
\mathbf{b}egin{align}
&\mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2 = \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k} [ \mathbf{b}oldsymbol{\mu}athbb{E}_{j_k}\| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2] \nonumber \\
&= \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}[ \mathbf{b}oldsymbol{\mu}athbb{E}_{j_k}\| \nabla_{j_k} f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u}_{j_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2] \nonumber \\
&= \frac{1}{J}\mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}\| \nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) + \tilde{\mathbf{b}oldsymbol{\mu}u} - \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2 \nonumber \\
& \leq \frac{1}{J} \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}\| \nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) \|_2^2 \nonumber \\
& \leq \frac{2}{J} \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}\| \nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 + \frac{2}{J} \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}\| \nabla f_{i_k}(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) - \nabla f_{i_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 \nonumber \\
& = \frac{2}{IJ} \sum_{i=1}^{I}\| \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 + \frac{2}{IJ}\sum_{i=1}^{I} \| \nabla f_i(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}) - \nabla f_i(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \|_2^2 \nonumber \\
& \leq \frac{2L}{J} [ h(\mathbf{b}oldsymbol{\mu}athbf{x}^{k}, \mathbf{b}oldsymbol{\mu}athbf{x}^*) + h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}, \mathbf{b}oldsymbol{\mu}athbf{x}^*)]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The first inequality uses the fact $\mathbf{b}oldsymbol{\mu}athbb{E} \| \mathbf{b}oldsymbol{\mu}athbf{z}eta - \mathbf{b}oldsymbol{\mu}athbb{E}\mathbf{b}oldsymbol{\mu}athbf{z}eta \|_2^2 \leq \mathbf{b}oldsymbol{\mu}athbb{E} \| \mathbf{b}oldsymbol{\mu}athbf{z}eta \|_2^2$ given a random variable $\mathbf{b}oldsymbol{\mu}athbf{z}eta$, the second inequality uses $\| \mathbf{a} + \mathbf{b} \|_2^2 \leq 2 \| \mathbf{a} \|_2^2 + 2\|\mathbf{b}\|_2^2$, and the last inequality uses Lemma~\mathbf{r}ef{lem:orbcdvd_lem1}.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
\mathbf{b}egin{lem}\label{lem:orbcdvd_lem3}
Under Assumption~\mathbf{r}ef{asm:orbcd1}, $f(\mathbf{b}oldsymbol{\mu}athbf{x}) = \frac{1}{I} \sum_{i=1}^{I}f_i(\mathbf{b}oldsymbol{\mu}athbf{x})$ has block-wise Lipschitz continuous gradient with constant $L$, i.e.,
\mathbf{b}egin{align}
\| \nabla_j f(\mathbf{b}oldsymbol{\mu}athbf{x} + U_j h_j ) - \nabla_j f(\mathbf{b}oldsymbol{\mu}athbf{x}) \|_2 \leq L \| h_j \|_2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{lem}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
Using the fact that $f(\mathbf{b}oldsymbol{\mu}athbf{x}) = \frac{1}{I} \sum_{i=1}^{I}f_i(\mathbf{b}oldsymbol{\mu}athbf{x})$, we have
\mathbf{b}egin{align}
&\| \nabla_j f(\mathbf{b}oldsymbol{\mu}athbf{x} + U_j h_j ) - \nabla_j f(\mathbf{b}oldsymbol{\mu}athbf{x}) \|_2 = \| \frac{1}{I} \sum_{i=1}^{I} [\nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x} + U_j h_j ) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) ] \|_2 \nonumber \\
& \leq \frac{1}{I} \sum_{i=1}^{I} \| \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x} + U_j h_j ) - \nabla_j f_i(\mathbf{b}oldsymbol{\mu}athbf{x}) \|_2 \nonumber \\
& \leq L \| h_j \|_2~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the first inequality uses the Jensen's inequality and the second inequality uses the Assumption~\mathbf{r}ef{asm:orbcd1}.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
Now, we are ready to establish the linear convergence rate of ORBCD with variance reduction for strongly convex functions.
\mathbf{b}egin{thm}\label{thm:orbcdvd}
Let $\mathbf{b}oldsymbol{\mu}athbf{x}^t$ be generated by ORBCD with variance reduction~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_mu}-\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_xj}. $j_k$ is sampled randomly and uniformly from $\{1,\mathbf{c}dots, J \}$. Assume $\mathbf{b}oldsymbol{\mu}athbf{e}ta > 2L$ and $m$ satisfy the following condition:
\mathbf{b}egin{align}
\mathbf{r}ho = \frac{L(m+1)}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m} + \frac{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)J}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m} - \frac{1}{m}+ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta (\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)J}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m\gamma} < 1~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Then ORBCDVD converges linearly in expectation, i.e.,
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^t) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^t) - (f(\mathbf{b}oldsymbol{\mu}athbf{x}^*)+g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] \leq \mathbf{r}ho^t [ \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} h(\mathbf{b}oldsymbol{\mu}athbf{x}^1, \mathbf{b}oldsymbol{\mu}athbf{x}^*)]~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where $h$ is defined in~\mathbf{b}oldsymbol{\mu}yref{eq:def_h}.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{thm}
\mathbf{b}oldsymbol{\mu}athbf{p}roof
The optimality condition of~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_xj} is
\mathbf{b}egin{align}
\langle \mathbf{v}_{j_k}^{i_k} + \mathbf{b}oldsymbol{\mu}athbf{e}ta (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k) + g'_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1}), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle \leq 0~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Rearranging the terms yields
\mathbf{b}egin{align}
& \langle \mathbf{v}_{j_k}^{i_k} + g'_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1}) , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle \leq - \mathbf{b}oldsymbol{\mu}athbf{e}ta \langle \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle \nonumber \\
& \leq \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k \|_2^2 ) \nonumber \\
& = \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k \|_2^2 ) ~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the last equality uses $\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} = (\mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1}, \mathbf{b}oldsymbol{\mu}athbf{x}_{k\neq {j_k}}^t)$.
Using the convecxity of $g_j$ and the fact that $g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1}) = g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})$, we have
\mathbf{b}egin{align}
& \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle + g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}) \leq \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1}) \nonumber \\
& + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x} - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k \|_2^2 ) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
According to Lemma~\mathbf{r}ef{lem:orbcdvd_lem3} and using~\mathbf{b}oldsymbol{\mu}yref{eq:blk_lip1}, we have
\mathbf{b}egin{align}
\langle \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} \mathbf{r}angle \leq f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1}) + \frac{L}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Letting $\mathbf{b}oldsymbol{\mu}athbf{x} = \mathbf{b}oldsymbol{\mu}athbf{x}^*$ and using the smoothness of $f$, we have
\mathbf{b}egin{align}
& \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k} \mathbf{r}angle + g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \leq \langle \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k), \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} \mathbf{r}angle + f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - [f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})+g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})] \nonumber \\
& + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^k \|_2^2 ) + \frac{L}{2} \| \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k+1} \|_2^2\nonumber \\
& \leq \frac{1}{2(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} \| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) \|_2^2 + f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - [f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})+g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})] + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 ) ~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Taking expectation over $i_k, j_k$ on both sides and using Lemma~\mathbf{r}ef{lem:orbcdvd_lem2}, we have
\mathbf{b}egin{align}\label{eq:orbcdvd_expbd}
& \mathbf{b}oldsymbol{\mu}athbb{E} [ \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^* \mathbf{r}angle + g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^*)] \nonumber \\
&\leq \frac{L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} [h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) + h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*)] + f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \mathbf{b}oldsymbol{\mu}athbb{E}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})+g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})] \nonumber \\
& + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 )~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
The left hand side can be rewritten as
\mathbf{b}egin{align}
& \mathbf{b}oldsymbol{\mu}athbb{E} [\langle \mathbf{v}_{j_k}^{i_k} , \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^{k} - \mathbf{b}oldsymbol{\mu}athbf{x}_{j_k}^* \mathbf{r}angle + g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - g_{j_k}(\mathbf{b}oldsymbol{\mu}athbf{x}^*)] = \frac{1}{J} [ \mathbf{b}oldsymbol{\mu}athbb{E}_{i_k}\langle \mathbf{v}^{i_k} , \mathbf{b}oldsymbol{\mu}athbf{x}^k - \mathbf{b}oldsymbol{\mu}athbf{x}^* \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) -g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] \nonumber \\
& = \frac{1}{J} [ \langle \nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) , \mathbf{b}oldsymbol{\mu}athbf{x}^k - \mathbf{b}oldsymbol{\mu}athbf{x}^* \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) -g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] = \frac{1}{J} h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Plugging into~\mathbf{b}oldsymbol{\mu}yref{eq:orbcdvd_expbd} gives
\mathbf{b}egin{align}
\frac{1}{J} [ h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] & \leq \frac{L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} [h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) + h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*)] + f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \mathbf{b}oldsymbol{\mu}athbb{E}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})+g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})] \nonumber \\
&+ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 ) \nonumber \\
& \leq \frac{L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} [h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) + h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*)] + f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \mathbf{b}oldsymbol{\mu}athbb{E}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})+g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})] \nonumber \\
&+ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 ) ~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Rearranging the terms yields
\mathbf{b}egin{align}
\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta - 2L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) &\leq \frac{L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)}[ h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*) ] + f(\mathbf{b}oldsymbol{\mu}athbf{x}^k) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^k) - \mathbf{b}oldsymbol{\mu}athbb{E}[f(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})+g(\mathbf{b}oldsymbol{\mu}athbf{x}^{k+1})] \nonumber \\
& + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^k \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^{k+1} \|_2^2 )~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
At time $t+1$, we have $\mathbf{b}oldsymbol{\mu}athbf{x}_0 = \tilde{\mathbf{b}oldsymbol{\mu}athbf{x}} = \mathbf{b}oldsymbol{\mu}athbf{x}^t$. Summing over $k = 0,\mathbf{c}dots, m$ and taking expectation with respect to the history of random variable $\mathbf{b}oldsymbol{\mu}athbf{x}i$, we have
\mathbf{b}egin{align}
\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta - 2L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} \sum_{k=0}^{m} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}_k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) &\leq \frac{L(m+1)}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}[ f(\mathbf{b}oldsymbol{\mu}athbf{x}_0) + g(\mathbf{b}oldsymbol{\mu}athbf{x}_0) ] - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} [ f(\mathbf{b}oldsymbol{\mu}athbf{x}_{m+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}_{m+1})] \nonumber \\
&+ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} ( \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}_0 \|_2^2 - \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}_{m+1} \|_2^2 ) \nonumber \\
&\leq \frac{Lm}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}_0,\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}_0 \|_2^2 \nonumber ~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where the last inequality uses
\mathbf{b}egin{align}
f(\mathbf{b}oldsymbol{\mu}athbf{x}_0) + g(\mathbf{b}oldsymbol{\mu}athbf{x}_0) - [ f(\mathbf{b}oldsymbol{\mu}athbf{x}_{m+1}) + g(\mathbf{b}oldsymbol{\mu}athbf{x}_{m+1})] & \leq f(\mathbf{b}oldsymbol{\mu}athbf{x}_0) + g(\mathbf{b}oldsymbol{\mu}athbf{x}_0) - [ f(\mathbf{b}oldsymbol{\mu}athbf{x}^*) + g(\mathbf{b}oldsymbol{\mu}athbf{x}^*)] \nonumber \\
& \leq \langle\nabla f(\mathbf{b}oldsymbol{\mu}athbf{x}_0), \mathbf{b}oldsymbol{\mu}athbf{x}_0 - \mathbf{b}oldsymbol{\mu}athbf{x}^* \mathbf{r}angle + g(\mathbf{b}oldsymbol{\mu}athbf{x}_0) - g(\mathbf{b}oldsymbol{\mu}athbf{x}^*) \nonumber \\
& = h(\mathbf{b}oldsymbol{\mu}athbf{x}_0,\mathbf{b}oldsymbol{\mu}athbf{x}^*)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Rearranging the terms gives
\mathbf{b}egin{align}
\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta -2L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} \sum_{k=1}^{m} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}^k,\mathbf{b}oldsymbol{\mu}athbf{x}^*) \leq \frac{L(m+1)}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\tilde{\mathbf{b}oldsymbol{\mu}athbf{x}},\mathbf{b}oldsymbol{\mu}athbf{x}^*) + (1- \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta -2 L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} ) \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}_0,\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}_0 \|_2^2~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Pick $x^{t+1}$ so that $h(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1}) \leq h(\mathbf{b}oldsymbol{\mu}athbf{x}_k), 1\leq k \leq m$, we have
\mathbf{b}egin{align}\label{eq:orbcdvd_lineareq0}
\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta - 2L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} m \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} h(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1},\mathbf{b}oldsymbol{\mu}athbf{x}^*) \leq [ \frac{L(m+1)}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} + 1- \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta -2 L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} ] \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}^t,\mathbf{b}oldsymbol{\mu}athbf{x}^*) + \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{2} \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}\| \mathbf{b}oldsymbol{\mu}athbf{x}^* - \mathbf{b}oldsymbol{\mu}athbf{x}^t \|_2^2~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where ther right hand side uses $\mathbf{b}oldsymbol{\mu}athbf{x}^t = \mathbf{b}oldsymbol{\mu}athbf{x}_0 = \tilde{\mathbf{b}oldsymbol{\mu}athbf{x}}$. Using~\mathbf{b}oldsymbol{\mu}yref{eq:orbcd_strong_h}, we have
\mathbf{b}egin{align}
\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta - 2L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} m \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} h(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1} ,\mathbf{b}oldsymbol{\mu}athbf{x}^*) \leq [ \frac{L(m+1)}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} + 1- \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta -2 L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} +\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta}{\gamma} ] \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}^t,\mathbf{b}oldsymbol{\mu}athbf{x}^*)~.
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
Dividing both sides by $\frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta - 2L}{J(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)} m$, we have
\mathbf{b}egin{align}
\mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i} h(\mathbf{b}oldsymbol{\mu}athbf{x}^{t+1},\mathbf{b}oldsymbol{\mu}athbf{x}^*) \leq \mathbf{r}ho \mathbf{b}oldsymbol{\mu}athbb{E}_{\mathbf{b}oldsymbol{\mu}athbf{x}i}h(\mathbf{b}oldsymbol{\mu}athbf{x}^t,\mathbf{b}oldsymbol{\mu}athbf{x}^*)~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
where
\mathbf{b}egin{align}
\mathbf{r}ho = \frac{L(m+1)}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m} + \frac{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)J}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m} - \frac{1}{m}+ \frac{\mathbf{b}oldsymbol{\mu}athbf{e}ta (\mathbf{b}oldsymbol{\mu}athbf{e}ta-L)J}{(\mathbf{b}oldsymbol{\mu}athbf{e}ta-2L)m\gamma} < 1~,
\mathbf{b}oldsymbol{\mu}athbf{e}nd{align}
which completes the proof.
\mathbf{b}oldsymbol{\mu}athbf{q}ed
\section{Conclusions}\label{sec:conclusion}
We proposed online randomized block coordinate descent (ORBCD) which combines online/stochastic gradient descent and randomized block coordinate descent. ORBCD is well suitable for large scale high dimensional problems with non-overlapping composite regularizers. We established the rate of convergence for ORBCD, which has the same order as OGD/SGD. For stochastic optimization with strongly convex functions, ORBCD can converge at a geometric rate in expectation by reducing the variance of stochastic gradient.
\section*{Acknowledgment}
H.W. and A.B. acknowledge the support of NSF via IIS-0953274, IIS-1029711, IIS- 0916750, IIS-0812183, NASA grant NNX12AQ39A, and the technical support from the University of Minnesota Supercomputing Institute. A.B. acknowledges support from IBM and Yahoo. H.W. acknowledges the support of DDF (2013-2014) from the University of Minnesota. H.W. also thanks Renqiang Min and Mehrdad Mahdavi for mentioning the papers about variance reduction when the author was in the NEC Research Lab, America.
\mathbf{b}ibliographystyle{plain}
\mathbf{b}ibliography{long,bcd,admm,onlinelearn,sparse,map}
\mathbf{b}oldsymbol{\mu}athbf{e}nd{document}
|
\betaegin{document}
\pagestyle{plain}
\title{
A spinning construction for virtual 1-knots and 2-knots,
and the fiberwise and welded equivalence of virtual 1-knots
}
\alphauthor{Louis H. Kauffman, Eiji Ogasa, and Jonathan Schneider}
\date{}
\betaegin{abstract}
Spun-knots (respectively, spinning tori) in $S^4$
made from classical 1-knots compose an important class of
spherical 2-knots (respectively, embedded tori) contained in $S^4$.
Virtual 1-knots are generalizations of classical 1-knots.
We generalize these constructions to the virtual 1-knot case
by using what we call, in this paper, the spinning construction of a submanifold.
The construction proceeds as follows:
It has been known that
there is a consistent way to make an embedded circle $C$ contained in \\
(a closed oriented surface $F$)$\times$(a closed interval $[0,1]$) from any virtual 1-knot $K$.
Embed $F$ in $S^4$ by an embedding map $f$.
Let $F$ also denote $f(F).$
We can regard the tubular neighborhood of $F$ in $S^4$
as $F\times D^2$.
Let $[0,1]$ be a radius of $D^2$.
We can regard $F\times D^2$
as the result of rotating
$F\times [0,1]$ around $F\times \{0\}$.
Assume $C\cap(F\times\{0\})=\phi$.
Rotate $C$ together
when we rotate $F\times [0,1]$ around $F\times \{0\}$.
Thus we obtain an embedded torus $Q\subset S^4$.
We prove the following:
The embedding type $Q$ in $S^4$ depends only on $K$, and does not depend on $f$.
Furthermore,
the submanifolds, $Q$ and the embedded torus made from $K,$ defined by Satoh's method,
of $S^4$ are isotopic.
We generalize this construction in the virtual 1-knot case, and
we also succeed to make a consistent
construction of one-dimensional-higher tubes from
any virtual 2-dimensional knot.
Note that Satoh's method says nothing about the virtual 2-knot case.
Rourke's interpretation of Satoh's method is that
one puts `fiber-circles' on each point of each virtual 1-knot diagram.
If there is no virtual branch point in a virtual 2-knot diagram,
our way gives such fiber-circles to each point of the virtual 2-knot diagram.
Furthermore we prove the following:
If a virtual 2-knot diagram $\alphalpha$ has a virtual branch point,
$\alphalpha$ cannot be covered by such fiber-circles.
We obtain a new equivalence relation,
the $\mathcal E$-equivalence relation
of the set of virtual 2-knot diagrams,
that is much connected with
the welded equivalence relation and our spinning construction.
We prove that
there are virtual 2-knot diagrams, $J$ and $K$,
that are virtually nonequivalent
but are $\mathcal E$-equivalent.
Although Rourke claimed that
two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are
fiberwise equivalent if and only if
$\alphalpha$ and $\betaeta$ are welded equivalent,
we state that this claim is wrong.
We prove that
two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are
fiberwise equivalent if and only if
$\alphalpha$ and $\betaeta$ are rotational welded equivalent (the definiton of rotational welded equivalence is given in the body of the paper).
\epsilonnd{abstract}
\maketitle
\Z[\pi/\pi^{(n)}]ewpage
\tableofcontents
\section{Introduction}\ellongleftarrowbel{jobun}
\subsection{
Spinning tori
}\ellongleftarrowbel{i1}\hskip20mm\\%
Spun-knots (respectively, spinning tori) in $S^4$
made from classical 1-knots compose an important class of
spherical 2-knots (respectively, embedded tori) contained in $S^4$.
See \cite{Zeeman} for the definition of spun-knots.
We review the construction of them below.\\
Let $\mathbb{R}^4=\{(x,y,z,w)|x,y,z,w\iotan\mathbb{R}\}$.
Regard
$\mathbb{R}^4$ as the result of rotating
$H=\{(x,,y,z,w)|x\geqq0,w=0\}$
around
$A=\{(x,,y,z,w)|x=0,w=0\}$
as the axis.
Take a 1-knot $K$ in $H$
so that $K\cap A$ is an arc (respectively, the empty set).
Rotate $K-{\rm Int}(K\cap A)$
around $A$ together
when we rotate $H$ around $A$.
The resultant submanifold of $\mathbb{R}^4$
is the spun knot (respectively, the spinning tori) of $K$.
We can easily also regard them as submanifolds of $S^4$.
We can define a spun link if $K$ is a link although we discuss the knot case mainly in this paper.
Our discussion can be easily generalized to the link case.
One of our themes in this paper is to generalize the spun knots of classical knots to the virtual knot case.
We begin by explaining why virtual knots are important.
\betaigbreak
\subsection{
History of relations between virtual knots and QFT,
and a reason why virtual knots are important
}\ellongleftarrowbel{i2}\hskip20mm\\%
Virtual 1-links are defined in \cite{Kauffman1,Kauffman, Kauffmani} as generalizations
of classical 1-links. One motivation for virtual 1-links is as follows.
Jones \cite{Jones} defined the Jones polynomial for classical 1-links in $S^3$.
The following had been well-known before the Jones polynomial was found:
The Alexander polynomial for classical 1-links in $S^3$
is defined in terms of the topology of the complement of the link and can be generalized to
give invariants of closed oriented 3-manifolds and of links within the 3-manifold.\\
Jones \cite[page 360, \S10]{Jones} tried to define a 3-manifold invariant associated with the Jones polynomial,
and succeeded in some cases.
Of course, when the Jones polynomial was found, the following question was regarded as a very natural one:
\smallbreak
{\betaf Question J.} Can we generalize the definition of the Jones polynomial for classical 1-links in $S^3$ to that in any 3-manifold?
\smallbreak
Note the result may not be a polynomial but a function of $t$.
\\
Witten \cite{W} wrote a quantum field theoretic path integral for any 1-link $L$ in any compact oriented 3-manifold $M$.
His path integral included the Jones polynomial for 1-links in $S^3$,
its generalizations and new (at the time) invariants of 3-manifolds.
This was a breakthrough for the philosophy of physics in that
one of the most natural geometrically intrinsic interpretations of a mathematical object
was done by using a path integral, and had not been done by any other way.
\smallbreak
\Z[\pi/\pi^{(n)}]oindent
{\betaf Note.}
Here, `geometrically intrinsic interpretation' means the point of view that would define a link invariant in terms of the embedding of the link in the ambient 3-dimensional manifolds
just as one can do naturally and easily in the case of the Alexander polynomial of 1-knots.
Jones \cite{Jones} defined the Jones polynomial by using representations of braid groups to an operator algebra (the Temperley-Lieb algebra). Representations, braid groups, operator algebras are mathematically explicit objects so some people may feel that that is enough to consider the meaning of the Jones polynomial.
\betaigbreak
If $M=S^3$, we can say at the physics level that the Witten path integral represents the Jones polynomial
for 1-links in $S^3$.
Reshetikhin, Turaev, Lickorish and others \cite{KM, Lickorish, Lickorishl, RT} etc.
generalized the result in \cite[page 360, \S10]{Jones} and
created rigorous definitions for invariants of 3-manifolds that parallel Witten's ideas, without using the functional integral.
They succeeded to define new invariants of closed oriented 3-manifolds and
invariants of links embedded in 3-manifolds
that we today call quantum invariants.
(Note, here, we distinguish the above invariants of links embedded in 3-manifolds
with the Jones polynomial for them as below.)
In both Witten's version and the Reshetikhin-Turaev versions the invariants of 3-manifolds are obtained by representing
the 3-manifold as surgery on a framed link and summing over invariants corresponding to appropriate representations decorating the surgery link. The same technique
applies when one includes an extra link component that is not part of the surgery data.
In this way, one obtains quantum invariants of links in 3-manifolds.
Another technique, formalized by Crane \cite{Crane} and by Kohno \cite{Kohno} uses a Heegard decomposition of the 3-manifold and algebraic structure of the conformal field theory
for the surface of the Heegard decompositon. These methods produce invariants for 3-manifolds and, in principle, invariants for links in 3-manifolds, but are much more indirect than the original physical idea of Witten that would integrate directly over the many possible evaluations of the Wilson loop for the knot or link in the 3-sphere, or the original combinatorial
skein techniques that produce the invariant of a link from its diagrammatic combinatorics.
See \cite{Kauffmanp}. \\
The Witten path integral is written also in the case where $L\Z[\pi/\pi^{(n)}]eq\phi$ and $M\Z[\pi/\pi^{(n)}]eq S^3$.
It corresponded to
Question J , which had been considered before the Witten path integral appeared.
\betaigbreak
In \cite{KauffmanJ} Kauffman found a definition of the Jones polynomial as a state summation over combinatorial states of the link diagram and found a diagrammatic interpretation of the Temperley-Lieb algebra that put the original definition of Jones in a wider context of generalized partition functions and statistical mecnanics on graphs and knot and link diagrams.
In \cite{Kauffman1,Kauffman, Kauffmani} Kauffman generalized the Jones polynomial
in the case where $M$ is (a closed oriented surface)$\times[-1,1]$.
In fact,
\cite{Kauffman1,Kauffman, Kauffmani} defined virtual 1-links
as
another way of describing
1-links in (a closed oriented surface)$\times[-1,1]$:
the set of virtual 1-links
is the same as
that of 1-links in (a closed oriented surface)$\times[-1,1],$ taken up to handle stabilization.
See Theorem \ref{vk}. We make the point here that the virtual knot theory is a context for links in the fundamental 3-manifolds of the form $F \times I$ where $F$ is a closed
surface. The state summation approach to the Jones polynomial generalizes to invariants of links in such thickened surfaces. This provides a significant and direct arena for
examining such structures without the functional integral. It also provides challenges for corresponding approaches that use the functional integral methods. It remains a serious
challenge to produce ways to work with the functional integrals that avoid difficulties in analysis.
\\
Path integrals represent the superposition principle dramatically.
This is a marvelous idea of Feynman.
The Witten path integral also represents a geometric idea of the Jones polynomial and quantum invariants physics-philosophically very well.
Witten found a Lagrangian via the Chern-Simons 3-form and Wilson line with a tremendous insight, and he calculated the path integral of the Lagrangian rigorously at physics level,
and showed that the result of the calculation is the Jones polynomial for links in $S^3$, and the quantum invariants of any closed oriented 3-manifold with or without embedded circles. It is a wonderful work of Witten.
However recall the following facts:
The Witten path integral for any 1-link in any closed 3-manifold has not been calculated
in mathematical level nor in physics level in any way that can be regarded as direct. This means that Question J is open in the general case.
That is,
nobody has succeeded to generalize the Jones polynomial in a direct way, and mathematically rigorously to the case where $M$ is not $S^3$, (respectively, $B^3$, $\mathbb{R}^3$), nor
(a closed oriented surface $F$)$\times[-1,1]$. (Note the last manifold is not closed.
Note that the discussion in the $S^3$ case is the same as that in the $B^3$ (respectively, $\mathbb{R}^3$) case. Virtual knot theory can also discuss the case where $F$ is compact and non-closed, but then we need to fix the embedding type of $F$ in $F\times[-1,1]$.)\\
Recall the following fact: Even if we make a (seemingly) meaningful Lagrangian,
the path integral associated with the Lagrangian cannot always be calculated.
An example is
the Witten path integral associated with
the general case of Question J.
Another one is the following. Today they do not know how to calculate the path integral
if we replace Chern-Simons-3-form on 3-manifolds with
Cern-Simons-$(2p+1)$-form on $(2p+1)$-manifolds,
where $p$ is any integer$\geq2$,
in the Witten path integral.
Indeed nowadays they only calculate path integrals only when they can calculate them.
If the path integral of the Lagrangian is not calculated explicitly, neither mathematicians nor physicists
regard the theory of the Lagrangian as a meaningful one.
Furthermore, even if we calculate path integrals, the result of the calculation is sometimes what we do not expect.
See an example of \cite{LeeYang} explained in
\cite[the last part of section 5.1]{Ryder}.\\
The heuristics of the Witten path integral have not been fully mined. See \cite{KauffmanPath} for a survey of the results of some of these heuristics in relation to the Jones
polynomial and Vassiliev invariants. It is possible that good heuristics will emerge for understanding invariants of links in 3-manifolds. But at the present time it is worth examining the
cases we do understand for working with generalizations of the Jones polynomial for links in thickened surfaces.
We had begun considering Question J
before the Witten path integral
appeared in this discussion.
Question J is also natural and important
even if we do not consider path integrals.
\\
\Z[\pi/\pi^{(n)}]oindent
{\betaf Note.}
(1)
We can observe some historical correspondences.
Feynman discovered path integrals
by using an analogy with (quantum) statistical mechanics,
and he interpreted quantum theory by using path integrals.
Operator algebras, path integrals, (quantum) statistical mechanics are closely related.
The Jones polynomial is discovered by using operator algebras (\cite{Jones}), next
is interpreted via (quantum) statistical mechanics (\cite{KauffmanJ}), then by using path integrals (\cite{W}).
Operator algebras, path integrals, and (quantum) statistical mechanics are related again with topology in
the background.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(2)
The Jones polynomial of 1-links in (a closed oriented surface)$\times$(the interval)
is discovered in \cite{Kauffman1,Kauffman, Kauffmani},
by using the analogy with state sums in (quantum) statistical mechanics in \cite{KauffmanJ}.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(3) \cite{KauffmanSaleur} found a relation between the Alexander-Conway polynomial between
1-dimensional classical knots and quantum field theory. The relation gives a different aspect from the Homflypt polynomial and the Witten path integral.
\cite{Ogasapath} found a relation between the degree of the Alexander polynomial of high dimensional knots and the Witten index of a supersymmetric quantum system.
It is also an outstanding open question whether we can define an analog to the Jones polynomial for high dimensional knots.
\betaigbreak
Virtual 1-links have many other important properties than the above one.
See \cite{Kauffman1,Kauffman, Kauffmani}.
Thus it is very natural to consider whether any property of classical 1-knots is possessed by virtual 1-knots, as below.
\betaigbreak
\subsection{
Main results
}\ellongleftarrowbel{i3}\hskip20mm\\%
We generalize the construction of spun-knots (respectively, spinning tori) of classical 1-knots
to the virtual 1-knot case as follows.
Recall that, in \cite{Kauffman1, Kauffman, Kauffmani} there is given a consistent way
to make an embedded circle $C$ contained in
(a closed oriented surface $F$)$\times$(a closed interval $[0,1]$) from any virtual 1-knot $K$ diagram
(see Theorem \ref{vk}).
Note the following.
When we construct spun knots (spinning tori),
we regard $\mathbb{R}^4$ itself as the total space of the normal bundle of $A$ in $\mathbb{R}^4$.
Recall that $A$ is defined in \S\ref{i1}.
Embed $F$ in $\mathbb{R}^4\subset S^4$ by an embedding map $f$.
Let $F$ stand for $f(F).$.
Note that the tubular neighborhood of $F$ in $S^4$ is diffeomorphic to $F\times D^2$.
Let $[0,1]$ be a radius of $D^2$.
We can regard $F\times D^2$
as the result of rotating
$F\times [0,1]$ around $F\times \{0\}$.
Assume $C\cap(F\times\{0\})=\phi$.
Rotate $C$ together
when we rotate $F\times [0,1]$ around $F\times \{0\}$.
Thus we obtain an embedded torus $Q\subset S^4$. \\
We prove the following (Theorems \ref{honto} and \ref{mainkore}):
The embedding type $Q$ in $S^4$ depends only on $K$, and does not depend on $f$.
Furthermore the submanifolds,
$Q$ and the embedded torus made from $K$ defined by Satoh in \cite{Satoh},
of $S^4$ are isotopic.\\
This construction of $Q$ is an example of what we call the spinning construction of submanifolds
in Definition \ref{spinningsubmanifold}.
This paper does not discuss the case where $C\cap(F\times\{0\})\Z[\pi/\pi^{(n)}]eq\phi$.
\\
There are classical 1-knots, virtual 1-knots, and classical 2-knots
so it is natural to consider virtual 2-knots. We define virtual 2-knots in Definition \ref{JV}.
It is very natural to consider whether
any property of `classical 1-, and 2-knots and virtual 1-knots'
is possessed by virtual 2-knots.
It is natural to ask whether we can define one-dimensional-higher tubes
for virtual 2-knots
(Question \ref{North Carolina})
since
we succeed in the virtual 1-knot case
as explained above.
Note that Satoh's mehtod in \cite{Satoh} does not treat the virtual 2-knot case. \\
In the virtual 1-knot case,
in \cite{Rourke},
Rourke interpreted Satoh's method as follows:
Let $\alphalpha$ be any virtual 1-knot diagram.
Put `fiber-circles' on each point of $\alphalpha$ and obtain a one-dimensional-higher tube.
(We review this construction in Theorem \ref{Montana} and Definition \ref{Nebraska}).
If we try to generalize Rourke's way to the virtual 2-knot case,
we encounter the following situation.\\
Let $\alphalpha$ be any virtual 1-knot diagram. There are two cases:
\smallbreak\Z[\pi/\pi^{(n)}]oindent(1)
The case where $\alphalpha$ has no virtual branch point.
(We define virtual branch point in Definitions \ref{oyster} and \ref{JV}.)
\smallbreak\Z[\pi/\pi^{(n)}]oindent(2)
The case where $\alphalpha$ has a virtual branch point.
\smallbreak
In the case (1) , we can make a tube by Rourke's method.
See \cite[section 3.7.1]{J}, Note \ref{kaiga}, and Definition \ref{suiri}.
In the case (2), however, Schneider \cite{J} found
it difficult to define a tube near any virtual branch point. \\
Thus we consider the following two problems. \\
Can we put fiber-circles over each point of any virtual 2-knot
in a consistent way as described above,
and make a one-dimensional-higher tube
(Question \ref{North Dakota})? \\
Is there a one-dimensional-higher tube construction which is defined for all virtual 2-knots, and which agrees with the method in the case (1) written above
when there are no virtual branch points
(Question \ref{South Dakota})? \\
In Theorem \ref{vv} we give an affirmative answer to Question \ref{South Dakota}.
Our solution is a generalization of our method in the virtual 1-knot case
used in \S\S\ref{E}-\ref{Proof}.
We also use
a spinning construction of submanifolds
explained in Definition \ref{spinningsubmanifold}. \\
In Theorem \ref{Rmuri}
we give a negative answer to Question \ref{North Dakota}. \\
\\
We obtain a new equivalence relation,
the $\mathcal E$-equivalence relation of
the set of virtual
1- and 2-knot diagrams (Definition \ref{zoo}).
It is done by using the above spinning construction.
The $\mathcal E$-equivalence relation is closley connected with
the welded equivalence relation and our spinning construction.
Welded 1-links are defined in \cite{Rourke} associated with virtual 1-links.
Welded 1-links are related to tubes very much as we discuss in this paper.
We introduce welded 2-knots in Definition \ref{JW}. \\
We prove that there are virtual 2-knot diagrams, $J$ and $K$,
that are virtually nonequivalent
but are $\mathcal E$-equivalent
(Theorem \ref{Maine}).
Welded 1-,and 2-knots are recipients of the tube construction or the above spinning construction.
The above spinning construction is related to the fiberwise equivalence explained below.
We will explain their connection in this paper and this is a theme of this research.\\
Although Rourke claimed in \cite[Theorem 4.1]{Rourke} that
two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are fiberwise equivalent if and only if
$\alphalpha$ and $\betaeta$ are welded equivalent,
we state that this claim is wrong.
(See
\cite{Rourke} and Definitions \ref{Nevada}
of this paper for the definition of the fiberwise equivalence,
and
\cite{Rourke, Satoh} for that of the welded equivalence.)
The reason for the failure of Rourke's claim is given in
Theorems \ref{smooth} and \ref{fwrw}, and Claim \ref{panda}.
We prove in Theorems \ref{smooth} and \ref{fwrw} that {\iotat virtual 1-knot diagrams, $\alphalpha$ and $\betaeta$,
are fiberwise equivalent if and only if they are rotational welded equivalent.} The reader can recall that in virtual 1-knot theory there are Reidemeister-type moves for virtual
crossings. Rotational equivalence for virtual knots is obtained by making the virtual curl (analog of the first Reidemeister move) forbidden. Rotational equivalence for welded knots
also forbids the virtual curl move in the context of the rules for welded knots.
(See \cite{Kauffman, Kauffmanrw, J} for rotational welded equivalence.)
Our result is proved by using the property of virtual 2-knots found in Theorem \ref{Rmuri}.
Virtual 2-knots themselves are important, and furthermore they are also important for research in virtual 1-knots.
Our main results are
Theorems \ref{honto}, \ref{mainkore},
\ref{vv}, \ref{Rmuri},
\ref{Maine},
\ref{smooth},
and \ref{Montgomery}. \\
\betaigbreak
\section{$\mathcal K(K)$ for a virtual 1-knot $K$}\ellongleftarrowbel{K}
\Z[\pi/\pi^{(n)}]oindent
We work in the smooth category unless we indicate otherwise.
In a part of \S\ref{New Mexico}
we will use the PL category in order to prove our results in the smooth category.
See Note \ref{haruwa}.
We review some facts on virtual 1-knots in this section before we state two of our main results,
Theorems \ref{honto} and \ref{mainkore},
in the following section. \\
\betaegin{figure}
\iotancludegraphics[width=140mm]{v.pdf}
\vskip-50mm
\caption{{\betaf A virtual crossing point and a surgery by a 1-handle}\ellongleftarrowbel{Alabama}}
\epsilonnd{figure}
Let $\alphalpha$ be a virtual 1-knot diagram.
In this paper we use Greek lowercase letters for virtual diagrams and
Roman capital letters for virtual knots.
See \cite{Kauffman1, Kauffman, Kauffmani}
for the definition and properties of virtual 1-knot diagrams and
those of virtual 1-knots.
For $\alphalpha$ there are a nonnegative integer $g$ and
an embedded circle contained in $\Sigma_g\times[0,1]$ as follows,
where $\Sigma_g$ is a closed oriented surface with genus $g$.
Take $\alphalpha$ in $\mathbb{R}^2$. (Recall that we can make the infinity point $\{*\}$
and $\mathbb{R}^2$ into $S^2$.)
Carry out a surgery on $\mathbb{R}^2$
by using a 3-dimensional 1-handle near a virtual crossing point as shown in
Figure \ref{Alabama}
and obtain $T^2-\{*\}$.
Note that the virtual 1-knot $K$ is oriented and that the arrows in
Figure \ref{Alabama}
denote the orientation.
Segments are changed as shown in the right figure of Figure \ref{Alabama}.
Do this procedure near all virtual crossing points.
Suppose that $\alphalpha$ has $g$ copies of virtual crossing point ($g\iotan\mathbb{N}\cup\{0\}$).
Here, $\mathbb{N}$ denotes the set of natural numbers.
Note that $a$ is a natural number if and only if $a$ is a positive integer.
What we obtain is $\Sigma_g-\{*\}$. We call it $\Sigma_g^\betaullet$.
(In \S\ref{Proof}, for a closed oriented surface $F$,
we define $F^\circ$ to be $F-$(an open 2-disc).
So, here, we use $^\betaullet$ not $^\circ$.)
Thus we obtain an immersed circle in $\Sigma_g^\betaullet$ from $\alphalpha$.
Call it $\mathcal I(\alphalpha)$.
Note that it is an immersion in ordinary sense (that is, it has no `virtual crossing point').
Regard $\Sigma_g$ as an abstract manifold.
Make $\Sigma_g^\betaullet\times[0,1]$.
There is a naturally embedded circle $\mathcal L(\alphalpha)$ contained in $\Sigma_g^\betaullet\times[0,1]$
whose projection by the projection $\Sigma_g^\betaullet\times[0,1]\to\Sigma_g^\betaullet\times\{0\}$ is
$\mathcal I(\alphalpha)$.
Suppose that $\mathcal L(\alphalpha)\cap(\Sigma_g^\betaullet\times\{0\})=\phi$.
Let ${\mathcal{K}}(\alphalpha)$ be
an embedded circle in $\Sigma_g\times[0,1]$ which we obtain naturally from $\mathcal L(\alphalpha)$.
$\Sigma_g$ is called a {\iotat representing surface}.
$\Sigma_g^\betaullet=\Sigma_g-\{*\}$
is also sometimes called a {\iotat representing surface}.
(The closure of ) any neighborhood of the immersed circle
in $\Sigma_g$ is also
called a {\iotat representing surface}.\\
\betaegin{thm}\ellongleftarrowbel{vk}
{\rm (\cite{Kauffman1, Kauffman, Kauffmani}.)}
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams.
$\alphalpha$ and $\betaeta$ represent the same virtual 1-knot
if and only if
${\mathcal{K}}(\alphalpha)$ is obtained from ${\mathcal{K}}(\betaeta)$ by
a sequence of the following operations.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
$(1)$ A surgery on the surface by a 3-dimensional 1-handle, where
\hskip2mm$($The attached part of the handle$)\cap($the projection of the embedded circle$)=\phi$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
$(2)$ A surgery on the surface by a 3-dimensional 2-handle, where
\hskip2mm$($The attached part of the handle$)\cap($the projection of the embedded circle$)=\phi$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
$(3)$ An orientation preserving diffeomorphism map of the surface.
\epsilonnd{thm}
Hence the following definition makes sense.
Let $K$ be a virtual 1-knot.
Let $\alphalpha$ be a virtual 1-knot diagram of $K$.
Define $\mathcal K(K)$ to be $\mathcal K(\alphalpha)$.
\betaigbreak
\section{$\mathcal E(K)$ for a virtual 1-knot $K$}\ellongleftarrowbel{E}
\Z[\pi/\pi^{(n)}]oindent
We generalize spun knots and spinning tori,
and introduce a new class of submanifolds. \\
Let $n$ be a positive integer.
Two submanifolds $J$ and $K$ $\subset S^n$ are {\iotat $($ambient$)$ isotopic}
if there is a smooth orientation preserving family of diffeomorphisms $\epsilonta_t$ of $S^n$, $0\elleqq t\elleqq1$, with $\epsilonta_0$ the identity and $\epsilonta_1(J)=K$. \\
\betaegin{defn}\ellongleftarrowbel{spinningsubmanifold}
Let $F$ be a codimension two submanifold contained in a manifold $X$.
Suppose that the tubular neighborhood $N(F)$ of $F$ in $X$
is the product bundle. That is, we can regard $N(F)$ as $F\times D^2$.
See Figure \ref{tube}.
We can regard the closed 2-disc $D^2$ as the result of
rotating a radius $[0,1]$ around the center $\{o\}$ as the axis.
We can regard $N(F)$ as the result of
rotating $F\times[0,1]$ around $F=F\times\{0\}$
as the axis.
Suppose that a submanifold $P$ contained in $X$ is embedded in $F\times[0,1]$.
Let $P'$ be a submanifold $P\cap(F\times\{0\})$ of $F\times\{0\}$.
When we rotate $F\times[0,1]$ around $F$ and make $F\times D^2$,
rotate $P$ together, and call the resultant submanifold $Q$.
This submanifold $Q$ contained in $X$ is called
the {\iotat spinning submanifold} made from
$P$ by the rotation in $F\times D^2$ under the condition that
$P\cap(F\times\{0\})$ is the submanifold $P'$.
This way of construction of $Q$ is called a {\iotat spinning consruction} of submanifolds.
If $P$ is a subset not a submanifold, we can define $Q$ as well.
\epsilonnd{defn}
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=120mm]{tube.pdf}
\vskip-10mm
\caption{{\betaf The tubular neighborhood which is a product $D^2$-bundle}\ellongleftarrowbel{tube}}
\betaigbreak
\epsilonnd{figure}
Spun knots and spinning tori are spinning submanifolds.
\cite[Proof of Claim in page 3114]{Ogasa98SL} and
\cite[Lemma 5.3]{Ogasa98n} used spinning construction.
By the uniqueness of the tubular neighborhood, we have the following.
\betaegin{cla}\ellongleftarrowbel{atarimae1}
Let $\check f$ $($respectively, $\check g)$
$:F\times D^2\hookrightarrow X$ be an embedding map.
We can regard $\check f(\Sigma_g\times D^2)$
$($respectively, $\check g(\Sigma_g\times D^2))$
as the tubular neighborhood of
$\check f(\Sigma_g\times\{o\})$ $($respectively, $\check g(\Sigma_g\times\{o\}))$.
Let $\check f|_{\Sigma_g\times\{o\}}$
be isotopic to $\check g|_{\Sigma_g\times\{o\}}$.
Then submanifolds,
$\check f(\Sigma_g\times\{o\})$ and $\check g(\Sigma_g\times\{o\})$,
of $X$
are isotopic.
\epsilonnd{cla}\betaigbreak
Let $\alphalpha$ be a virtual 1-knot diagram.
Take $\Sigma_g\times[0,1]$ and $\mathcal K(\alphalpha)$ as in \S\ref{K},
that is, $K(\alphalpha)$ is a 1-knot in $\Sigma_g\times[0,1]$, where $\Sigma_g$ representing $\alphalpha$.
Assume $\mathcal K(\alphalpha)\cap(\Sigma_g\times\{0\})=\phi$.
Suppose $\mathcal K(\alphalpha)\cap(\Sigma_g\times\{0\})=\phi$.
Make $\Sigma_g\times D^2$, where we regard $[0,1]$ as a radius of $D^2$.
Let $\check f:\Sigma_g\times D^2\hookrightarrow S^4$ be an embedding map.
Let $\mathcal E_{\check f}(\alphalpha)$ be
the spinning submanifold made from $\mathcal K(\alphalpha)$
by the rotation in $\check f(\Sigma_g\times D^2)$.
Note $\mathcal E_{\check f}(\alphalpha)\subset S^4$.
Let $f$ be $\check f|_{\Sigma_g\times\{o\}}$.
By Claim \ref{atarimae1} it makes sense that
we call $\mathcal E_{\check f}(\alphalpha)$,
$\mathcal E_f(\alphalpha)$. \\
Suppose that $\alphalpha$ represents a virtual 1-knot $K$.
Theorem \ref{honto} is one of our main results. \\
\betaegin{thm}\ellongleftarrowbel{honto}
For an arbitrary virtual 1-knot $K$,
the submanifold type $\mathcal E_f(\alphalpha)$
of $S^4$ does not depend on the choice of a set $(\alphalpha, f)$.
\epsilonnd{thm} \betaigbreak
By Theorem \ref{honto} we can define $\mathcal E(K)$ for any virtual 1-knot $K$. \\
Let $\mathcal S(\alphalpha)$ be an embedded $S^1\times S^1$ contained in $S^4$
for a virtual 1-knot diagram $\alphalpha$, defined by Satoh in \cite{Satoh}.
It was proved
there that if $\alphalpha$ and $\betaeta$ represent the same virtual 1-knot,
the submanifolds,
$\mathcal S(\alphalpha)$ and $\mathcal S(\betaeta)$,
of $S^4$ are isotopic.
So we can define $\mathcal S(K)$ for any virtual 1-knot $K$. \\
We will prove the following in \S\ref{Proof}.
Theorem \ref{mainkore} is one of our main results.
\betaegin{thm}\ellongleftarrowbel{mainkore}
Let $K$ be a virtual 1-knot. Then
the submanifolds,
$\mathcal E(K)$ and $\mathcal S(K)$,
of $S^4$ are isotopic.
\epsilonnd{thm}
\betaegin{note}\ellongleftarrowbel{yuenchi}
If $K$ in Theorem \ref{mainkore} is a classical knot,
$\mathcal E(K)$ and $\mathcal S(K)$ is the spinning torus of $K$.
\epsilonnd{note}
\betaigbreak\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
\cite[section 10.2]{B1},
\cite[section 3.1.1]{B2} and
\cite{Dylan} proved only a special case of Theorem \ref{mainkore},
which is only Theorem \ref{mainmaenotame} of this paper.
We prove the general case.
Our result is stronger than the result in
\cite[section 10.2]{B1},
\cite[section 3.1.1]{B2} and
\cite{Dylan}.
\betaigbreak
\section{Proof of Theorems \ref{honto} and \ref{mainkore}}\ellongleftarrowbel{Proof}
\Z[\pi/\pi^{(n)}]oindent
We first prove a special case.
\betaegin{thm}\ellongleftarrowbel{mainmaenotame}
Take a virtual 1-knot diagram $\alphalpha$ in \S\ref{K}.
Let $\check\iotaota:\Sigma_g\times D^2\to S^4$ be
an embedding map whose image of
$\Sigma_g^\betaullet$
by $\check\iotaota$
is $\Sigma_g^\betaullet$
in \S\ref{K}.
Let $\iotaota$ be $\check\iotaota\vert_{\Sigma_g}$.
Then the submanifolds,
$\mathcal E_{\iotaota}(\alphalpha)$ and $\mathcal S(\alphalpha)$,
of $S^4$ are isotopic.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{mainmaenotame}.}
Let $\mathbb{R}^4=\{(x,y,u,v)|x,y,u,v\iotan\mathbb{R}\}$,
$\mathbb{R}^2_b=\{(x,y)|x,y\iotan\mathbb{R}\}$,
and $\mathbb{R}^2_F=\{(u,v)|u,v\iotan\mathbb{R}\}$.
Note
$\mathbb{R}^4=\mathbb{R}^2_b\times\mathbb{R}^2_F$.
Regard $\mathbb{R}^3$ in \S2 as \\
$\mathbb{R}^2_b\times\{(u,v)| u\iotan\mathbb{R}, v=0\}$.
Take the tubular neighborhood of $\Sigma_g^\betaullet$ in $\mathbb{R}^3$.
It is diffeomorphic to $\Sigma_g^\betaullet\times[-1,1]$.
We can suppose that
$\Sigma_g^\betaullet, \Sigma_g^\betaullet\times[0,1]
\subset\mathbb{R}^2_b\times\{(u,v)| u\geqq0, v=0\}$. \\
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
Let $\Sigma_g\subset S^4$.
Suppose that $\{*\}\iotan S^4$ is included in $\Sigma_g$.
Then \Z[\pi/\pi^{(n)}]ewline
$S^4-\Sigma_g=(S^4-\{*\})-(\Sigma_g-\{*\})=R^4-\Sigma_g^\betaullet$. \\
Take the tubular neighborhood $N(\Sigma_g^\betaullet)$ of $\Sigma_g^\betaullet$ in $\mathbb{R}^4$.
Note that
$N(\Sigma_g^\betaullet)$ is diffeomorphic to $\Sigma_g^\betaullet\times D^2$.
We can regard $N(\Sigma_g^\betaullet)$
as the result of
rotating $\Sigma_g^\betaullet\times[0,1]$
around $\Sigma_g^\betaullet$ as the axis
(diffeomorphically not isometrically).
Suppose that $\mathcal L(\alphalpha)\cap(\Sigma_g^\betaullet\times\{0\})=\phi$.
Make the spinning submanifold
$\mathcal E_\iotaota(\alphalpha)$
from $\mathcal L(\alphalpha)$.
\\
We can suppose that
each fiber $D^2$ of
$N(\Sigma_g^\betaullet)$
is parallel to $\{(x,y)|x=0, y=0\}\times\mathbb{R}^2_F$
by using an isotopy of an embedding map of the tubular neighborhood. \\
We can suppose that $\mathcal I(\alphalpha)$ intersects each fiber $D^2$ transversely.
{\iotat Reason.}
Note a 1-handle drawn in the right-side of Figure \ref{Alabama}.
If $\mathcal I(\alphalpha)$ near the 1-handle is put like (Ac) in Figure \ref{Alaska},
$\mathcal I(\alphalpha)$ does not intersect each fiber $D^2$ transversely.
However we can do the following operation.
By using an isotopy of a part of $\mathcal I$
we change
the part of $\mathcal I(\alphalpha)$
from (Ac) to (Ob) in Figure \ref{Alaska}.
After this operation, $\mathcal I(\alphalpha)$ intersects each fiber $D^2$ transversely. \\
\betaegin{note}\ellongleftarrowbel{kabuto}
We will explain a property of (Ac), in Note \ref{kuwagata}.
It is important. We will use it in Alternative proof of Claim \ref{shichi} of \S\ref{v2}.
\epsilonnd{note} \betaigbreak
Note that, even if a part of $\mathcal I(\alphalpha)$ is (Ac),
we can make a spinning submanifold $\mathcal E_\iotaota(\alphalpha)$.
However, if there is not (Ac), we have an advantage as below. \\
\betaegin{figure}
\iotancludegraphics[width=130mm]{hand.pdf}
\vskip-50mm
\caption{{\betaf (Ac) and (Ob).
}\ellongleftarrowbel{Alaska}}
\epsilonnd{figure}
\betaegin{figure}
\iotancludegraphics[width=150mm]{obtuse.pdf}
\vskip-40mm
\caption{{\betaf Rotation around a part near (Ob).
The reason why (Ob) is useful for us.
}\ellongleftarrowbel{Arizona}}
\epsilonnd{figure}
\betaegin{figure}
\vskip-7mm
\iotancludegraphics[width=150mm]{acute.pdf}
\vskip-47mm
\caption{{\betaf Rotation around a part near (Ac).
The reason why (Ac) is not useful for us.
This property of (Ac) is used in in Alternative proof of Claim \ref{shichi} of \S\ref{v2}.
}\ellongleftarrowbel{Arkansas}}
\betaigbreak
\epsilonnd{figure}
If there is not (Ac) in $\mathcal I(\alphalpha)$, we have the following. \\
Take a point $q\iotan\mathbb{R}^2_b\times\{(u,v)| u=0, v=0\}$.
Note $\alphalpha\subset\mathbb{R}^2_b\times\{(u,v)| u=0, v=0\}$.
By the above construction of $\mathcal E_\iotaota(\alphalpha)$, we have the following. \\
\smallbreak\Z[\pi/\pi^{(n)}]oindent (i)
If $q\cap \alphalpha=\phi$,
$(\{q\}\times\mathbb{R}^2_F)\cap\mathcal E_\iotaota(\alphalpha)=\phi.$
\smallbreak\Z[\pi/\pi^{(n)}]oindent (ii)
If $q$ is a normal point of $\alphalpha$,
$(\{q\}\times\mathbb{R}^2_F)\cap\mathcal E_\iotaota(\alphalpha)$ is a single circle in $\{q\}\times\mathbb{R}^2$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent (iii)
If $q$ is a real crossing point of $\alphalpha$,
$(\{q\}\times\mathbb{R}^2_F)\cap\mathcal E_\iotaota(\alphalpha)$
is two circles in $\{q\}\times\mathbb{R}^2$
such that one of the two is in the inside of the other.
The inner (respectively, outer) circle corresponds
to the lower (respectively, upper) point of the singular point.
\smallbreak\Z[\pi/\pi^{(n)}]oindent (iv)
If $q$ is a virtual crossing point of $\alphalpha$,
$(\{q\}\times\mathbb{R}^2_F)\cap\mathcal E_\iotaota(\alphalpha)$
is two circles in $\{q\}\times\mathbb{R}^2$
such that each of the two is in the outside of the other each other.
It is Rourke's description of $\mathcal S(\alphalpha)$
in Theorem \ref{Montana} which is cited below from \cite{Rourke}.
(However \cite{Rourke} does not write a proof.
So \cite{J} wrote a proof.)\\
\betaegin{thm}\ellongleftarrowbel{Montana}
{\betaf (\cite{Rourke}.)}
Let $\alphalpha$ be a virtual 1-knot diagram.
Take an embedding map
$\varphi: S^1_b\times S^1_f\hookrightarrow \mathbb{R}^2_b\times\mathbb{R}^2_f$ with the following properties.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(1)$
Let $\pi:\mathbb{R}^2_b\times\mathbb{R}^2_f\to\mathbb{R}^2_b$ be the natural projection.
$\pi\circ\varphi(S^1_b\times S^1_f)\subset\mathbb{R}^2_b$ defines
$\alphalpha$ without the notations of virtual crossings.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(2)$
For points in $\mathbb{R}^2_b$, we have the following:
\betaegin{enumerate}
\iotatem
If $q\Z[\pi/\pi^{(n)}]otin\alphalpha$,
we have $\pi^{-1}(q)=\phi.$
\smallbreak
\iotatem
If $q$ is a normal point of $\alphalpha$,
we have that $\pi^{-1}(q)$ is a circle in $\{q\}\times\mathbb{R}^2$.
\smallbreak
\iotatem
If $q$ is a real crossing point of $\alphalpha$,
we have that $\pi^{-1}(q)$ is two circles in $\{q\}\times\mathbb{R}^2$
such that one of the two is in the inside of the other.
The inner $($respectively, outer$)$ circle corresponds
to the lower $($respectively, upper$)$ point of the singular point.
\smallbreak
\iotatem
If $q$ is a virtual crossing point of $\alphalpha$,
we have that $\pi^{-1}(q)$ is two circles in $\{q\}\times\mathbb{R}^2$
such that each of the two is in the outside of the other each other.
\epsilonnd{enumerate}
\smallbreak
Then the submanifolds,
$\mathcal S(\alphalpha)$ and $\varphi(S^1_b\times S^1_f)$,
of $S^4$ are isotopic
\epsilonnd{thm}
This completes the proof of Theorem \ref{mainmaenotame}.
\qed\\
\betaegin{defn}\ellongleftarrowbel{Nebraska}
In Theorem \ref{Montana},
each circle
$f(S^1_b\times S^1_f)\cap$(each fiber $\mathbb{R}^2_f$)
is called a {\iotat fiber-circle}.
We say that $f(S^1_b\times S^1_f)$ admits {\iotat Rourke's fibration}.
\epsilonnd{defn}\betaigbreak
\betaegin{note}\ellongleftarrowbel{kuwagata}
As we preannounced in Note \ref{kabuto}, we state a comment on (Ac).
If the projection on a surface includes (Ac),
$\mathcal E(\alphalpha)$ does not admit Rourke's fibration.
The reason is explained in Figures \ref{Arizona} and \ref{Arkansas}.
We will use this property,
which is raised by the difference between (Ac) and (Ob),
in Alternative proof of Claim \ref{shichi} of \S\ref{v2}.
\epsilonnd{note}\betaigbreak
We next prove the general case. \\
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorems \ref{honto} and \ref{mainkore}.}
We prove Theorem \ref{oh} below.
The key idea of the proof is Claim \ref{wow}.
Let $\Sigma$ be a closed oriented surface.
Let $G_1$ and $G_2$ be submanifolds of $S^4$
which are orientation preserving diffeomorphic to $\Sigma$.
It is known that
there is a case that
the submanifolds, $G_1$ and $G_2$, of $S^4$ are non-isotopic.
Let $\Sigma^\circ$ denote $\Sigma-(\text{an open 2-disc})$.
Let $$G^\circ_i\\=G_i-(\text{an open 2-disc})$$ be a submanifold of $S^4$
which are orientation preserving diffeomorphic to $\Sigma^\circ$ $(i=1,2)$. \\
\betaegin{cla}\ellongleftarrowbel{wow}
The submanifolds, $G^\circ_1$ and $G^\circ_2$, of $S^4$ are isotopic.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{wow}.}
$\Sigma^\circ$ has a handle decomposition
which consists of one 0-handle, 1-handles, and no 2-handle.
\qed\\
Let $i\iotan\{1,2\}$.
We can regard the tubular neighborhood of $G_i$ in $S^4$ as $G_i\times D^2$.
Embed $S^1$ in $G_i\times [0,1]$, where we regard $[0,1]$ as a radius of $D^2$,
and call the image $J_i$. Assume that $J_i\cap(G_i\times\{0\})=\phi$.
Suppose that there is
a bundle map $\check\sigma:G_1\times D^2\to G_2\times D^2$
such that $\check\sigma$ covers
an orientation preserving diffeomorphism map $\sigma:G_1\to G_2$ and
such that $\check\sigma(J_1)=J_2$. \\
Define a submanifold $E_i$ contained in $S^4$
to be the spinning submanifold made from $J_i$
by the rotation in $G_i\times D^2$.
\betaegin{thm}\ellongleftarrowbel{oh}
The submanifolds, $E_1$ and $E_2$, of $S^4$ are isotopic.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{oh}.}
We can suppose that
$J_i\subset G^\circ_i\times [0,1]$.
By the existence of $\sigma$,
there is a bundle map $\check\tau:G^\circ_1\times D^2\to G^\circ_2\times D^2$
such that $\check\tau$ covers an orientation preserving diffeomorphism map
$\tau: G^\circ_1\to G^\circ_2$ and
such that $\check\tau(J_1)=J_2$. \\
Note the following:
Let $f:\Sigma^\circ\to S^4$ be an embedding map.
We can regard $\tau$ as a diffeomorphism map
$\Sigma^\circ\to \Sigma^\circ$.
By Claim \ref{wow} ,
the submanifolds,
$f(\Sigma^\circ)$ and $f(\tau(\Sigma^\circ))$,
of $S^4$ are isotopic.
Therefore the submanifolds, $E_1$ and $E_2$, of $S^4$ are isotopic,
\qed\\
Theorems \ref{vk} and \ref{oh} imply Theorems \ref{honto} and \ref{mainkore} \qed\\
We can extend all discussions in \S\S\ref{K}-\ref{Proof} and the following \S\ref{rt} to the virtual 1-link case easily.
When we define $\mathcal E(\alphalpha)$ in \S\ref{E},
we assume
$\mathcal L(\alphalpha)\cap\Sigma_g^\betaullet=\phi$.
Suppose that
$\mathcal L(\alphalpha)\cap(\Sigma_g^\betaullet)$ is an arc instead.
Then we obtain a spherical 2-knot in $\mathbb{R}^4$
as the spinning submanifold.
The class of such spherical 2-knots is also a generalization of 2-dimensional spun-knots of 1-knots,
and is also worth studying.
As we state in \S\ref{jobun}, we do not discuss this class in this paper.
\betaigbreak
\section{Immersed solid tori} \ellongleftarrowbel{rt}
\betaegin{figure}
\iotancludegraphics[width=150mm]{3.pdf}
\vskip-40mm
\caption{{\betaf $\zeta^{-1}($each closed 2-disc$)$}\ellongleftarrowbel{Colorado}}
\epsilonnd{figure}
\Z[\pi/\pi^{(n)}]oindent By the definition of $\mathcal S(\hskip2mm)$ in \cite{Satoh}, we have (i)$\mathbb{R}ightarrow$(ii).
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(i) An embedded torus $Y$ contained in $S^4$ is isotopic
to $\mathcal S(\alphalpha)$ for a virtual 1-knot diagram $\alphalpha$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(ii) There is an immersion map $\zeta:S^1\times D^2\ellooparrowright S^4$
with the following properties: \Z[\pi/\pi^{(n)}]ewline
$\zeta(S^1\times\partial D^2)$ is $Y$.
The singular point set of $\zeta$ consists of double points and is a disjoint union of closed 2-discs,
and $\zeta^{-1}($each closed 2-disc$)$ is as shown in Figure \ref{Colorado}.
\smallbreak
By using the construction of $\mathcal E(\alphalpha)$,
we can also describe the immersed solid torus \\ in (ii)
as follows:
By using the projection `$\mathcal L(\alphalpha)\to \mathcal I(\alphalpha)$' in \S\ref{K},
we can make an immersed annulus in
$\Sigma_g\times[0,1]$ naturally.
Note that (the immersed annulus)$\cap\Sigma_g\times\{0\}\Z[\pi/\pi^{(n)}]eq\phi$.
Make a subset from this immersed annulus
by a spinning construction around $\Sigma_g$, defined in Definition \ref{spinningsubmanifold}.
Then the result is an immersed solid torus in (ii).
\smallbreak
We prove the converse of the above claim,
that is, the following.
\betaegin{thm}\ellongleftarrowbel{grt}
{\rm(ii)}$\mathbb{R}ightarrow${\rm(i)}.
\epsilonnd{thm}
We prove this theorem as an application of our results in \S\ref{Proof}
although it may be also proved in another way.
\\
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{grt}.}
Let $q\iotan\partial D^2$.
Let $C$ be $\zeta(S^1\times\{q\})$.
In the following paragraphs,
for $Y$, we will make an embedded oriented surface $F$ contained in $S^4$
so that we will put $C$ in the tubular neighborhood $N(F)$ of $F$ in $S^4$.
We will make $C\cap F=\phi$.
We will make $Y$ so that it will be the spinning submanifold of $C$ around $F$.
Let $\{o\}$ be the center of $D^2$.
We will let $F$ include $\zeta(S^1\times\{o\})$. \\
Let $\timesi:S^1\times D^2\times I\ellooparrowright S^4$ be an immersion map, where $I=[-1,1]$,
to satisfy that
$\timesi\vert_{S^1\times D^2\times \{0\}}=\zeta$ and that
\Z[\pi/\pi^{(n)}]ewline\hskip3cm$\timesi(\{x\}\times \{o\}\times I) \quad\betaot\quad \timesi(\{x\}\times D^2\times \{0\})$ \Z[\pi/\pi^{(n)}]ewline
for each $x$ if we give appropriate metrics to $S^4$ and $S^1\times D^2\times I$.
Then we can suppose the following:
\smallbreak\Z[\pi/\pi^{(n)}]oindent(1)
$P=\timesi(S^1\times \{o\}\times I)$ is a boundary-connected-sum of $n$ copies of the annulus ($n\iotan\mathbb{N}$).
\quad See Figure \ref{Connecticut}
for an example of $P$.
\betaegin{figure}
\betaegin{center}
\iotancludegraphics[width=70mm]{4.pdf}
\epsilonnd{center}
\vskip-20mm
\caption{{\betaf An example of $P$}\ellongleftarrowbel{Connecticut}}
\epsilonnd{figure}
\smallbreak\Z[\pi/\pi^{(n)}]oindent(2)
$Q=\timesi(S^1\times D^2\times I)$ is a boundary-connected-sum of $m$-copies of $S^1\times B^3$
($m\iotan\mathbb{N}$).
\smallbreak\Z[\pi/\pi^{(n)}]oindent(3)
$\partial P\subset\partial Q$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(4)
$Q$ is the tubular neighborhood of $P$ in $S^4$.
$Q$ is diffeomorphic to $P\times D^2$. \\
By the Mayer-Vietoris sequence,
$H_1(S^4, Q;\mathbb{Z})\cong
H_1(S^4- {\rm Int} Q, \partial Q;\mathbb{Z})\cong0$.
Hence there is an embedded oriented compact surface-with-boundary
$G$ contained in $S^4-{\rm Int} Q$ such that
$\partial G=\partial P$ and that
$G\cup P$ is an oriented closed surface $F$.
({\iotat Reason}: Consider a simplicial decomposition of $S^4-{\rm Int} Q$.)
We can regard $Y$ as the spinning submanifold made from $\zeta(S^1\times\{q\})$ around $F$.
Hence we can regard $\zeta(S^1\times\{q\})$ as
$\mathcal K(\betaeta)$ for a virtual 1-knot diagram $\betaeta$
in a fashion which is explained in \S\S\ref{E}-\ref{Proof},
and can regard $Y$ as $\mathcal E(\betaeta)$.\\
This completes the proof of Theorem \ref{grt}. \qed
\betaigbreak
\section{The virtual 2-knot case}\ellongleftarrowbel{v2}
\Z[\pi/\pi^{(n)}]oindent
\Z[\pi/\pi^{(n)}]oindent
Virtual 2-knot theory is defined analogously to Virtual 1-knot theory, using
generic surfaces in 3-space as knot diagrams and using Roseman moves for knot
equivalence, and allowing the double-point arcs to have classical or virtual crossing
data.
See \cite{Takeda, J}.
Virtual 2-knot diagrams (respectively, virtual 2-knots) in \cite{J} and this paper
are the same as
virtual surface-knots (respectively, virtual surface-knot diagrams) in \cite{Takeda}.
\betaegin{defn}\ellongleftarrowbel{oyster}
Let $F$ be a closed surface.
A smooth map $f : F\to\mathbb{R}^3$ is considered {\iotat quasi-generic}
if it fails to be one-to-one only at transverse crossings of orders 2 and 3
as shown in Figures \ref{sashimid} and \ref{sashimit},
\betaegin{figure}
\betaegin{center}
\iotancludegraphics[width=40mm]{sashimid.pdf}
\epsilonnd{center}
\caption{{\betaf Transversal double points
}\ellongleftarrowbel{sashimid}}
\epsilonnd{figure}
\betaegin{figure}
\iotancludegraphics[width=70mm]{sashimit.pdf}
\caption{{\betaf Transversal triple points}\ellongleftarrowbel{sashimit}}
\epsilonnd{figure}
and it fails to be regular only at isolated {\iotat branch points}
where, locally, the image of a disk looks like the cone over a loop, with no other parts of the surface touching the vertex.
See Figure \ref{gbra}.
Branch points include the cone over any closed, regular, transversely self-intersecting
curve.
In particular, the cone over a figure-$\iotanfty$ curve is called
a {\iotat Whitney branch point}. See Figure \ref{sashimiv}.\\
\betaegin{figure}
\iotancludegraphics[width=70mm]{gbra.pdf}
\caption{{\betaf A general branch point}\ellongleftarrowbel{gbra}}
\epsilonnd{figure}
\betaegin{figure}
\iotancludegraphics[width=70mm]{sashimiv.pdf}
\caption{{\betaf Whitney-umbrella branch point
}\ellongleftarrowbel{sashimiv}}
\epsilonnd{figure}
A quasi-generic map $f$ is {\iotat generic} if the only branch points are Whitney
branch points.
The three features of a generic map---
Whitney branch points, double-point arcs,
and triple points---
have slice-histories corresponding respectively to the Reidemeister
$I$-, $II$-, and $III$- moves in 1-knot theory.
\epsilonnd{defn}
\betaegin{figure}\betaigbreak \iotancludegraphics[width=150mm]{s24from89.pdf}
\vskip-40mm
\caption{{\betaf The singular point sets of virtual 2-knots}\ellongleftarrowbel{JV1}} \betaigbreak \epsilonnd{figure}
\betaegin{defn}\ellongleftarrowbel{sashimi}
A {\iotat virtual 2-knot diagram} consists of a generic map $F$
together with classical and virtual crossing data along its double-point arcs.
Crossing data is representedgraphically as broken and unbroken surfaces:
See the left two figures of Figure \ref{JV1}.
Branch points can be classical or virtual:
See the middle three figures,
the figures which are not the above ones nor the following ones,
in Figure \ref{JV1}.
At triple points, three crossings meet. Triple points of the following types are
allowed: See the right three figures in Figure \ref{JV1}.
All other combinations of crossing data are forbidden. Note that the three allowed
triple points have slice-histories corresponding to the Reidemeister $III$-moves
in Virtual 1-knot theory.
A virtual 2-knot diagram may be reduced to its bare combinatorial structure,
forgetting all but the information that is invariant under isotopies of $\mathbb{R}^3$ and $F$.
In this regard, we do not distinguish diagrams that are related by isotopies of
$\mathbb{R}^3$ and $F$.
\epsilonnd{defn}\betaigbreak
\betaegin{defn}\ellongleftarrowbel{JV}{\betaf (\cite[section 3.5]{J}.)}
A virtual 2-knot diagram may be transformed by Roseman moves. There are
seven types of local moves, shown here without crossing data.
When a virtual 2-knot diagram undergoes a Roseman move, its crossing data
carried continuously by the move. Two diagrams related by a series of Roseman
moves are called {\iotat virtually equivalent}.
The equivalence classes are {\iotat Virtual 2-knot types}, or sometimes simply
{\iotat Virtual 2-knots}.
\epsilonnd{defn}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
The readers need not be familiar with Roseman moves in order to read this paper.\\
It is natural to ask whether we can define one-dimensional-higher tubes from virtual 2-knots
since we succeed in the virtual 1-knot case
as written in \S\S\ref{E}-\ref{Proof}. \\
The following facts let it be more natural:
The one-dimensional-higher tube $\mathcal E(K)$ made from a virtual 1-knot $K$
is the spun-knot of $K$ if $K$ is a classical knot (see \cite{Satoh}).
\cite{Zeeman} defined spun-knots not only for classical 1-knot but also for classical 2-knots. \\
\betaegin{que}\ellongleftarrowbel{North Carolina}
Can we define one-dimensional-higher tubes for virtual 2-knots
in a consistent way? Suppose that these tubes are diffeomorphic to
$F\times S^1$ if the virtual 2-knot is defined by $F$.
\epsilonnd{que}\betaigbreak
Note that Satoh's method in \cite{Satoh} did not say anything about the virtual 2-knot case.
In the virtual 1-knot case, in \cite{Rourke},
Rourke interpreted Satoh's method as we reviewed
in Theorem \ref{Montana} and Definition \ref{Nebraska}. \\
\betaegin{note}\ellongleftarrowbel{kaiga}
In the virtual 2-dimensional knot case
we also
use the terms `fiber-circle' and `Rourke-fibration'
in Definition \ref{Nebraska}.
\epsilonnd{note}
\betaegin{defn}\ellongleftarrowbel{suiri}
Let $M$ be a 3-dimensional compact submanifold of $\mathbb{R}^5$.
Regard $\mathbb{R}^5$ as $\mathbb{R}^3\times\mathbb{R}^2$.
We say that the submanifold $M$ admits
{\iotat Rourke fibration}, or
that $M$ is embedded {\iotat fibrewise}
if
$M\cap(p\times\mathbb{R}^2)$ is a collection of circles
for any point $p\iotan\mathbb{R}^3$.
We call the circles in $M\cap(p\times\mathbb{R}^2)$, {\iotat fiber circles}.
\epsilonnd{defn}
If we try to generalize Rourke's way to the virtual 2-knot case,
we will do the following:
Let $\alphalpha$ be a virtual 2-knot diagram.
Let $\mu=0,1,2,3$.
We give $\mu$-copies of circle to any $\mu$-tuple point in $\alphalpha$,
and construct the tube.
Of course we determine the position of fiber-circles in each fiber plane
by the property of the $\mu$-tuple point (See \cite[section 3.7.1]{J} for detail).
See Figure \ref{doremi}. \\
\betaegin{figure}
\iotancludegraphics[width=110mm]{doremi.pdf}
\vskip-20mm
\caption{{\betaf
The nest of circles in fibers.
}\ellongleftarrowbel{doremi}}
\epsilonnd{figure}
However we encounter the following situation.
Let $\alphalpha$ be any virtual 1-knot diagram.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(1)
The case where $\alphalpha$ has no virtual branch point.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(2)
The case where $\alphalpha$ has a virtual branch point.
\smallbreak
In the (1) case, we can make a tube by Rourke's way.
See \cite[section 3.7.1]{J}.
In the (2) case, however, \cite{J} found
it difficult to define a tube near any virtual branch point.
Thus it is natural to ask the following two questions. \\
\betaegin{que}\ellongleftarrowbel{North Dakota}
Can we put fiber-circles over each point of any virtual 2-knot
in a consistent way as written above,
and make one-dimensional-higher tube?
\epsilonnd{que}\betaigbreak
\betaegin{que}\ellongleftarrowbel{South Dakota}
Is there a one-dimensional-higher tube construction which is defined for all virtual 2-knots, and which agrees with the way in the (1) case written above
when there are no virtual branch points?
\epsilonnd{que}\betaigbreak
We generalize our method in \S\S\ref{E}-\ref{Proof}
and
give an affirmative answer to Question \ref{South Dakota},
and hence to Question \ref{North Carolina}.
See Theorem \ref{vv}.
We also use a spinning construction of submanifolds
explained in Definition \ref{spinningsubmanifold}.
Theorem \ref{Rmuri} gives a negative answer
to Question \ref{North Dakota}.
\betaigbreak
We make the virtual 2-knot version of
representing surfaces, which are defined above Theorem \ref{vk}.
\betaegin{defn}\ellongleftarrowbel{Jbase} {\betaf (\cite[section 3.5]{J}.)}
The development of an invariant for virtual 2-knot theory closely parallels that for virtual 1-knot theory. The idea is to think of a virtual 2-knot diagram as a classical 2-knot diagram ``drawn" on a closed 3-manifold. We then define an equivalence relation on these objects that extends classical move-equivalence and allows the 3-manifold to vary. Take as input a virtual 2-knot diagram $\alphalpha$.
Let $N(\alphalpha)$ be a neighborhood of the diagram,
which is a regular neighborhood except at virtual branch points,
in the following sense:
$N(\alphalpha)$ is formed by thickening $\alphalpha$ everywhere except at virtual branch points;
as you approach virtual branch points, let the thickening gradually diminish to zero,
so that near the virtual branch point $N(\alphalpha)$ looks like the cone over a thickened figure-$\iotanfty$. \\
Along each virtual crossing curve of $\alphalpha$, double the square-shaped junction of $N(\alphalpha)$ to create overlapping ``slabs".
Call this 3-manifold-with-boundary $B(\alphalpha)$.
It has a purely classical knot diagram in it. (To be precise, $B(\alphalpha)$ is not technically a 3-manifold-with-boundary at virtual branch points, since the ``slab" is pinched to zero thickness at these points.)
See Figures \ref{Jbase2} and \ref{Jbase3}. \\
\betaegin{figure}
\iotancludegraphics[width=130mm]{s26from91.pdf}
\vskip-30mm
\caption{{\betaf
The left upper figure is
a part of $N(\alphalpha)$ near a double point curve.
The right upper figure is that
near a triple point.
The left lower figure is that
near a virtual branch point.
The right lower figure is that
near a classical branch point.
}\ellongleftarrowbel{Jbase2}}
\epsilonnd{figure}
Now embed $B(\alphalpha)$ into any compact oriented 3-manifold (not necessarily connected).
The resultant compact 3-manifold
is called a {\iotat representing 3-manifold} $M$
associated with a virtual 2-knot diagram $\alphalpha$.
$M$ contains a classical 2-knot diagram $\mathcal I(\alphalpha)$.
\\
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=130mm]{s27from92.pdf}
\vskip-20mm
\caption{{\betaf
Make $B(\alphalpha)$ from $N(\alphalpha)$.
}\ellongleftarrowbel{Jbase3}}
\betaigbreak
\epsilonnd{figure}
\epsilonnd{defn}
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
Virtual 1-knot has two kinds of equivalent definitions:
one is defined by using diagrams with virtual points in $\mathbb{R}^2$.
The other is done by using representing surfaces.
See Theorem \ref{vk} and \S\ref{K} of this paper, and \cite{Kauffman1, Kauffman, Kauffmani}.
It is very natural to ask the following question.
\betaegin{que}\ellongleftarrowbel{imayatteru}
Do we have the virtual 2-knot version of Theorem \ref{vk} by using representing 3-manifolds in Definition \ref{Jbase}?
\epsilonnd{que}
This question is open. \cite{J} gave a partial answer.
We do not discuss it in this paper.
\vskip9mm
Note that $\mathcal I(\alphalpha)$ is an immersed surface in an ordinary sense.
That is, it does not include a virtual point.
Note that we cannot embed a
representing 3-manifold
in $\mathbb{R}^4$ in general.
We show an example.
Take the Boy surface in $\mathbb{R}^3$
(see \cite{Boy, OgasaBoy}).
We can regard it as
a virtual 2-knot diagram as follows:
Suppose that the only one immersed crossing curve
is a virtual one. That is, it consists of
one virtual triple point and other virtual double points.
Then no
representing 3-manifolds for this virtual 2-knot
can be embedded in $\mathbb{R}^4$. It is
proved by using obstruction classes of
the normal bundle of $\mathbb{R} P^2$ in $\mathbb{R}^4$. \\
However, by \cite{Hirsh}, we have the following.
\betaegin{thm}\ellongleftarrowbel{Hirsh}
Any $M$ in Definition \ref{Jbase} can be embedded in $\mathbb{R}^5$.
\epsilonnd{thm}
We will define the virtual 2-knot version of
$\mathcal E(\alphalpha)$ in Definition \ref{wakeru} after some preliminaries.
Let $X$ be a 3-dimensional closed oriented abstract manifold.
Let $G_1$ and $G_2$ be submanifolds of $S^5$
which are diffeomorphic to $X$.
Recall the following fact
\betaegin{cla}\ellongleftarrowbel{kantan}
There is a case that
the submanifolds, $G_1$ and $G_2$, of $S^5$ are non-isotopic.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{kantan}.}
Take a spherical 3-knot $K$ whose Alexander polynomial is nontrivial.
Let $G_2$ be the knot-sum of $G_1$ and $K$.
See e.g \cite{Rolfsen} for the Alexander polynomial of 3-knots and the knot-sum.
\qed
\\
While $G_1$ and $G_2$ may be non-isotopic
submanifolds, which are diffeomorphic to $X$ above,
of $S^5$,
it is the case that they are isotopic
after removing an open three-ball from each of them.
Let $X^\circ_i$ denote $X-\text{(an open 3-ball)}$.
Let $G^\circ_i=G_i-\text{(an open 3-ball)}$ be a submanifold of $S^5$ ($i=1,2$).
\betaegin{cla}\ellongleftarrowbel{wowwow}
The submanifolds, $G^\circ_1$ and $G^\circ_2$, of $S^5$ are isotopic.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
Claim \ref{wowwow} is the virtual 2-knot version of Claim \ref{wow}.
\\
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{wowwow}.}
$X^\circ$ has a handle decomposition
which consists of one 0-handle, 1-handles, 2-handles and no 3-handle.
The dimensions of the cores of these handles are 0, 1, or 2.
Hence the dimensions $\elleqq2$. (Here, it is important the dimension $\Z[\pi/\pi^{(n)}]eq$ 3.) The dimension of $S^5$ is 5.
Since $2<\Bbbkrac{3(2+1)}{2}$, Claim \ref{wowwow} holds by
\cite{Haefliger3}.
\qed
\betaegin{cla}\ellongleftarrowbel{jimeisoku}
Let $M$ be a compact 3-manifold. By Theorem $\ref{Hirsh}$, $M$ is embedded in $\mathbb{R}^5$.
The normal bundle $\Z[\pi/\pi^{(n)}]u$ of $M$ embedded in $\mathbb{R}^5$ is the trivial bundle for any embedding of $M$ in $\mathbb{R}^5$.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{jimeisoku}.}
If $M$ is closed, $M$ bounds a Seifert hypersurface $V$ in $S^5$ (See \cite[Theorem 2 page 49]{Kirby}).
Take the normal bundle $\alphalpha$ of $V$ in $\mathbb{R}^5$.
Then $\Z[\pi/\pi^{(n)}]u$ is a sum of vector bundles $\alphalpha|_M$ and an orientable $\mathbb{R}$-bundle over $M$.
Hence Claim \ref{jimeisoku} holds in this case. \\
In the case where $M$ is nonclosed,
take the double $DM$ of $M$ as abstract manifolds.
Then $DM$ can be embedded in $\mathbb{R}^5$.
By the previous paragraph, the normal bundle of this embedded $DM$ is trivial.
Then the restriction of this normal bundle to $M\subset DM$ is trivial.
By this fact and Claim \ref{wowwow}, Claim \ref{jimeisoku} holds in this case.
This completes the proof of Claim \ref{jimeisoku}.
\qed
\\
We introduce the virtual 2-knot version of
$\mathcal E(\alphalpha)$.
\betaegin{defn}\ellongleftarrowbel{wakeru}
Take an abstract manifold $M$ in Definition \ref{Jbase},
where $\mathcal I(\alphalpha)$ is still contained in $M$.
Make $M\times[0,1]$.
We can obtain an embedded surface
$\mathcal J(\alphalpha)$
contained in $M\times[0,1]$
such that the projection of $\mathcal J(\alphalpha)$
by the projection $M\times[0,1]\to M$
is $\mathcal I(\alphalpha)$.
We suppose $\mathcal J(\alphalpha)\cap(M\times\{0\})=\phi$.
Take any embedding of $M$ in $\mathbb{R}^5$.
Define a submanifold $\mathcal E(\alphalpha)$ contained in $S^5$
to be the spinning submanifold made from $\mathcal J(\alphalpha)$ around $M$.
(Recall Claim \ref{jimeisoku}.)
\epsilonnd{defn}
We prove the virtual 2-knot version of
Theorem \ref{honto},
which is Theorem \ref{vv}.
Theorem \ref{vv} is one of our main results.
It gives an affirmative answer to Question
\ref{North Carolina}.
\betaegin{thm}\ellongleftarrowbel{vv}
Let $\alphalpha$ and $\alphalpha'$ be virtual 2-knot diagrams
which represent the same virtual 2-knot.
Make $\mathcal E(\alphalpha)$ and $\mathcal E(\alphalpha')$
by using a
representing 3-manifold $M$ $($respectively, $M')$ associated with
$\alphalpha$ $($respectively, $\alphalpha').$
Then submanifolds, $\mathcal E(\alphalpha)$ and $\mathcal E(\alphalpha')$, of $\mathbb{R}^5$
are isotopic
even if $M$ is not diffeomorphic to $M'$.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Theorem \ref{vv}.}
It suffices to prove the following two cases:
\smallbreak
(i) $\alphalpha$ is obtained from $\alphalpha'$ by one of classical moves.
\smallbreak
(ii) $\alphalpha$ is obtained from $\alphalpha'$ by one of virtual moves.
\smallbreak
In the case (ii), there is a diffeomorphism map $f:M\to M'$
such that $f(\alphalpha)$ is isotopic to $\alphalpha'$ in $M'$.
Note that $\alphalpha\subset M$ and that $\alphalpha'\subset M'$.\\
In the case (i). Take a closed 3-ball $B$ where the classical move is carried out.
Note that $M\cup B$ (respectively, $M'\cup B$) is a representing 3manifold of
$\alphalpha$ (respectively $\alphalpha'$).
Note that there is a diffeomorphism map $f:M\cup B\to M'\cup B$
such that $f(\alphalpha)$ is isotopic to $\alphalpha'$ in $M'\cup B$.
Note that $\alphalpha\subset M\cup B$ and that $\alphalpha'\subset M'\cup B$.
In both cases, by the following Theorem \ref{bdy},
Theorem \ref{vv} holds.
\qed\\
We prove the following Theorem \ref{ohoh},
which is the virtual 2-knot version of Theorem \ref{oh}.
The key idea of the proof is Claim \ref{wowwow}
(recall Note below Claim \ref{wowwow}.)
Let $i=1,2$.
Take $G_i$ defined in Claim \ref{wowwow}.
We can regard the tubular neighborhood of $G_i$ in $S^5$ as $G_i\times D^2$.
Embed a closed oriented surface
in $G_i\times [0,1]$,
where we regard $[0,1]$ as a radius of $D^2$,
and call the image $J_i$.
Assume that $J_i\cap(G_i\times\{0\})=\phi.$
Suppose that there is a bundle map
$\check\sigma:G_1\times D^2\to G_2\times D^2$
such that $\check\sigma$ covers an orientation preserving diffeomorphism map $\sigma:G_1\to G_2$
and such that $\check\sigma(J_1)=J_2$.
Define a submanifold $E_i$ contained in $S^5$
to be the spinning submanifold made from $J_i$
by the rotation in $G_i\times D^2$.
\betaegin{thm}\ellongleftarrowbel{ohoh}
The submanifolds, $E_1$ and $E_2$, of $S^5$ are isotopic.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{ohoh}.}
We can suppose that $J_i\subset G^\circ_i\times [0,1]$.
By the existence of $\sigma$,
there is a bundle map $\check\tau:G^\circ_1\times D^2\to G^\circ_2\times D^2$
such that $\check\tau$ covers a diffeomorphism map
$\tau:G^\circ_1\to G^\circ_2$
and such that $\check\tau(J_1)=J_2$.
Note the following:
Let $f:M^\circ\to S^5$ be an embedding map.
We can regard $\tau$ as a diffeomorphism map
$M^\circ\to M^\circ$.
By Claim \ref{wowwow},
the submanifolds, $f(M^\circ)$ and $f(\tau(M^\circ))$, of $S^5$ are isotopic.
Therefore
the submanifolds, $E_1$ and $E_2$, of $S^5$ are isotopic.
\qed
\betaegin{thm}\ellongleftarrowbel{bdy}
Replace
the condition that $M$ is a closed compact oriented 3-manifold
with
the condition that $M$ is a non-closed compact oriented 3-manifold.
Then Theorem \ref{ohoh} also holds.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{bdy}.}
The proof of Theorem \ref{bdy}
is done in a similar fashion to that of Theorem \ref{ohoh}.
The proof of Theorem \ref{bdy} is easier than that of Theorem \ref{ohoh}.
\qed\\
We now have completed the proof of Theorem \ref{vv}, and
answered Question
\ref{North Carolina}.
We next answer Question \ref{South Dakota}.
We define a consistent way
to put a
representing 3-manifold in $\mathbb{R}^5$.
\betaegin{defn}\ellongleftarrowbel{hatena}
Let $\alphalpha$ be a virtual 2-knot diagram
contained in $\mathbb{R}^3$.
Regard $\mathbb{R}^3$ as
$\mathbb{R}^3\times\{0\}\times\{0\}$
$\subset\mathbb{R}^5=\mathbb{R}^3\times\mathbb{R}\times\mathbb{R}$.
Put `a
representing 3-manifold
for $\alphalpha$' in $\mathbb{R}^5$ as follows.
Take the neighborhood $T$ of $\alphalpha$ as
defined in Definition \ref{Jbase}.
Take a neighborhood of each of classical and virtual branch points
such that the neighborhood is diffeomorphic to the closed 3-ball
and
such that $\alphalpha\cap$(the neighborhood) is as drawn in Figure \ref{cvbr}. \\
\betaegin{figure}
\iotancludegraphics[width=40mm]{cvbr.pdf}
\caption{{\betaf
The intersection of
a classical or virtual branch point
and
its neighborhood explained in Definition \ref{hatena}
}\ellongleftarrowbel{cvbr}}
\epsilonnd{figure}
Let
$T'
=T-$Int(the neighborhoods of real branch points and those of virtual branch ones).
Along any virtual crossing line
we double $T'$
as done in Definition \ref{Jbase}.
Note that
this operation can be done in
$\mathbb{R}^5$
although it cannot be done in $\mathbb{R}^4$ in general.
Thus we obtain a compact oriented
3-dimensional submanifold $X\subset\mathbb{R}^5$ from $T'$. \\
For a real branch point, we attach
`the closed 3-ball which is a neighborhood of the real branch point'
to $X$.
Note that
near any virtual branch point,
the operation can be done in
$\mathbb{R}^3\times\mathbb{R}\times\{0\}$.
For a virtual branch point, we attach
`the closed 3-ball which is a neighborhood of the virtual branch point', as drawn in
Figures \ref{Delaware}-\ref{Hawaii},
to $X$.
Note that
in Figures \ref{Delaware}-\ref{Hawaii}
we draw $\mathbb{R}^4=\mathbb{R}^3\times\mathbb{R}\times\{0\}$.
Note that the virtual branch point vanishes in this closed 3-ball. \\
The resultant compact oriented 3-manifold is
a
representing 3-manifold with $\mathcal I(\alphalpha)$, which is defined in Definition \ref{Jbase}.
We call it $M_\iotaota$.
Recall that $\mathcal I(\alphalpha)$ has no virtual point and, in particular,
that $\mathcal I(\alphalpha)$ has no virtual branch point.
\\
Figure \ref{Delaware} draws a part of a
representing 3-manifold $M$ near a virtual branch point.
Figure \ref{Florida} adds a part of
$\mathcal I(\alphalpha)$
to Figures \ref{Delaware}.
Figure \ref{Georgia}
draws Figures \ref{Delaware}
by seeing from a different direction.
Figure \ref{Hawaii}
draws Figures \ref{Florida}
by seeing from a different direction.
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=140mm]{another2.pdf}
\caption{{\betaf
A part of a
representing 3-manifold near a virtual branch point. We do not draw
a virtual branch point here.
In \ref {Florida} we do it.
}\ellongleftarrowbel{Delaware}}
\betaigbreak
\epsilonnd{figure}
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=145mm]{another3.pdf}
\vskip-30mm
\caption{{\betaf
A part of a
representing 3-manifold near a virtual branch point. We draw
a virtual branch point here.
}\ellongleftarrowbel{Florida}}
\betaigbreak
\epsilonnd{figure}
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=140mm]{mpvbp2.pdf}
\vskip-40mm
\caption{{\betaf
A part of a
representing 3-manifold near a virtual branch point. We do not draw
a virtual branch point here.
In Figure \ref{Hawaii} we will do it.
}\ellongleftarrowbel{Georgia}}
\epsilonnd{figure}
\betaegin{figure}
\iotancludegraphics[width=140mm]{mpvbp3.pdf}
\vskip-30mm
\caption{{\betaf
A part of a
representing 3-manifold near a virtual branch point. We draw
a virtual branch point here.
We explain the most lower figure
in more detail in Figure \ref{hosoku}.
}\ellongleftarrowbel{Hawaii}}
\betaigbreak
\epsilonnd{figure}
\betaegin{figure}
\iotancludegraphics[width=110mm]{hachix.pdf}
\vskip-10mm
\caption{{\betaf
The explanation of the most lower figure of
Figure \ref{Hawaii}.
The neighborhood of the red curve of that figure is
obtained by curving the middle figure of the above figures.
}\ellongleftarrowbel{hosoku}}
\epsilonnd{figure}
\epsilonnd{defn}
We prove that we have an affirmative answer to
Question \ref{South Dakota}.
\betaegin{thm}\ellongleftarrowbel{konnyaku}
Let $\alphalpha$ be a virtual 2-knot diagram.
Make $\mathcal E(\alphalpha)$ by using $M_\iotaota$,
and call it $\mathcal E_\iotaota(\alphalpha)$.
If $\alphalpha$ has no virtual branch point, then
$\mathcal E_\iotaota(\alphalpha)$ admits Rourke fibration.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{konnyaku}.}
Let $p\iotan\alphalpha$.
Regard $\mathbb{R}^5$ in Definition \ref{hatena}
as $\mathbb{R}^3\times\mathbb{R}\times\mathbb{R}$.
By the construction of $\mathcal E_\iotaota(\alphalpha)$,
$\mathcal E_\iotaota(\alphalpha)\cap(p\times\mathbb{R}\times\mathbb{R})$
is the empty set or a collection of circles
such that this correspondence satisfies Rourke's description.
Hence Theorem \ref{konnyaku} holds.
\qed
\betaegin{note}\ellongleftarrowbel{vrei}
It is trivial that if we use another embedding of another $M$,
$\mathcal E(\alphalpha)$ associated with the embedding
may not admit Rourke fibration.
Such an example exists.
Let $\timesi$ be the trivial 2-knot diagram.
It is trivial that $\timesi$ admits Rourke fibration.
Let $\zeta$ be a virtual 2-knot diagram of the trivial 2-knot.
Assume that the singular point set of $\zeta$ consists of two virtual branch points
and one virtual segment. \\
A {\iotat virtual segment} is the segment with the following properties.
It is a segment included in a virtual 2-knot diagram.
One of the boundary is a virtual branch point.
The points in the interior of the segment are virtual double points.
It is drawn in Figure \ref{Maryland}.
It is drawn in Figure \ref{sashimiv} if the branch point there is a virtual branch point.
See \cite{J}. \\
$\zeta$ does not admit Rourke fibration by Theorem \ref{Rmuri}.
\epsilonnd{note}
Note the following claim.
\betaegin{cla}\ellongleftarrowbel{shichi}
Take $\mathcal E_\iotaota(\alphalpha)$ in
Theorem $\ref{konnyaku}$.
If $\alphalpha$ includes a virtual branch point,
$\mathcal E_\iotaota(\alphalpha)$ does not admit Rourke's fibration.
That is,
$\mathcal E_\iotaota(\alphalpha)$ is not embedded fiberwise.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
It is trivial that if we use another embedding of another $M$,
$\mathcal E(\alphalpha)$ associated with the embedding
may admit Rourke fibration.
Such an example exists.
It is the one in Note \ref{vrei}.
\\
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{shichi}.}
By Theorem \ref{Rmuri}. \qed \\
We give an alternative proof of Claim \ref{shichi} after Proof of Theorem \ref{Rmuri}.
\\
Theorem \ref{Rmuri} is an answer to Question \ref{North Dakota},
and is one of our main results.
\betaegin{thm}\ellongleftarrowbel{Rmuri}
The answer to Question $\ref{North Dakota}$
is negative.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{Rmuri}.}
We prove by `reductio ad absurdum'.
We suppose the following assumption, and will arrive at a contradiction.
\smallbreak
\Z[\pi/\pi^{(n)}]oindent
{\betaf Assumption.} The neighborhood of a virtual branch point can be covered by the fiber-circles.
\smallbreak
Note the fiber over the virtual segment as shown in Figure \ref{Maryland}.
Give a Euclidean metric to $\mathbb{R}^5$.
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=130mm]{0x1y2.pdf}
\vskip-60mm
\caption{{\betaf
The assumption of `reductio ad absurdum'}\ellongleftarrowbel{Maryland}}
\epsilonnd{figure}
By Assumption,
the circles,
$A$ and $B$ in Figure \ref{Maryland},
meet at the circle
$C$ when $\varepsilon\to0$.
Let $s$ be the area of $C$.
When $\varepsilon\to0$, $A\to C$ and $B\to C$.
Hence, we have the following.
\betaegin{equation}\ellongleftarrowbel{Massachusetts}
{\text{When $\varepsilon\to0$, (the area of $B)\to s$.}}
\epsilonnd{equation}
Note that $s$ is a fixed positive real number.
\betaegin{figure}
\iotancludegraphics[width=120mm]{x2.pdf}
\vskip-30mm
\caption{{\betaf A one parameter families}\ellongleftarrowbel{Missouri}}
\epsilonnd{figure}
Take a one-parameter-family for each point $p\iotan C.$ See Figure \ref{Missouri}.
Suppose that $a$ and $b$ go to $p$ when $\varepsilon\to0$.
Let $\delta(a,b)$
the distance along the trace of the one-parameter-family
between $a$ and $b$.
\betaegin{equation}\ellongleftarrowbel{Michigan}
{\text{When $\varepsilon\to0$, $\delta(a,b)\to0.$}}
\epsilonnd{equation}
\Z[\pi/\pi^{(n)}]oindent
In the fiber $\mathbb{R}^2$ which includes $A$ and $B$,
take any point $x\iotan A$.
Suppose that $x$ goes to $y\iotan B$ by the one-parameter-family.
In this fiber $\mathbb{R}^2$
take a disc of radius $2\delta(x,y)$ whose center is $x\iotan A$.
Call the sum of the discs, $N(A)$. See Figure \ref{Mississippi}.
When $\varepsilon\to0$, \\
(the area of $N(A))\to 0$.
By (\ref{Michigan}),
$B\subset N(A)$.
Note that in this fiber $\mathbb{R}^2$,
$B$ (respectively, $A$) is not included in the inside of $A$ (respectively, $B$).
Therefore, by Jordan curve theorem,
the inside of $B$ is also included in $N(A)$.
Hence we have the following.
\betaegin{equation}\ellongleftarrowbel{Minnesota}
{\text{When $\varepsilon\to0$, (the area of $B)\to 0$.}}
\epsilonnd{equation}
\betaegin{figure}
\iotancludegraphics[width=130mm]{z0x1y2.pdf}
\vskip-70mm
\caption{{\betaf $N(A)$}\ellongleftarrowbel{Mississippi}}
\epsilonnd{figure}
By (\ref{Massachusetts}) and (\ref{Minnesota}),
we arrived at a contradiction.
This completes the proof of Theorem \ref{Rmuri}.
\qed\\
We give a direct proof of why $\mathcal E_\iotaota(\alphalpha)$ does not admit Rourke fibration, without using Theorem \ref{Rmuri}.
Note that it is not an alternative one of Theorem \ref{Rmuri}.
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent{\betaf Alternative proof of Claim \ref{shichi}.}
If $p$ is a virtual branch point,
$p$ is in the boundary of a virtual segment in $\mathbb{R}^3$.
Take $\mathcal I(\alphalpha)$ immersed in $M$.
Let $\kappa:\mathcal I(\alphalpha)\to\alphalpha$ be the natural map defined in
Definition \ref{JV}.
We have the following.
$\kappa^{-1}$(the virtual segment)
is a union of two segments, $\Psi$ and $\Phi$.
A point of $\partial\Psi$ and that of $\partial\Phi$ meet at a point
as drawn in Figure \ref{Idaho}.
$\kappa$(this point) is the virtual branch point. \\
The two segments make an angle.
See Figures \ref{Delaware}-\ref{Hawaii}.
The angle is acute.
Even if we take an arbitrary
representing 3-manifold of the virtual 2-knot diagram $\alphalpha$,
the angle is acute not obtuse.
Furthermore the angle is put as drawn there.
The reason for this is that there is always an acute angle as drawn in
Figure \ref{Idaho} whichever representing 3-manifolds we take. \\
As we preannounced in Notes \ref{kabuto} and \ref{kuwagata},
we use Figures \ref{Arizona} and \ref{Arkansas}.
In particular, see the most lower figure of Figure \ref{Arkansas}.
Therefore
$\mathcal E_\iotaota(\alphalpha)\cap(p\times\mathbb{R}_u\times\mathbb{R}_v)$
is a bouquet,
not the empty set or a collection of circles.
\Z[\pi/\pi^{(n)}]ewpage{\color{white}a}
\vskip-30mm
\betaegin{figure}[H]
\iotancludegraphics[width=170mm]{neta.pdf}
\vskip-90mm
\caption{{\betaf
$\mathcal E_\iotaota(\alphalpha)$ made by the spinning construction can be embed in $\mathbb{R}^5$
but $\mathcal E_\iotaota(\alphalpha)$ does not admit Rourke fibration.
The reason why we cannot make the fiber over any virtual branch point a collection of circles is drawn. }\ellongleftarrowbel{Idaho}}
\epsilonnd{figure}
Therefore Claim \ref{shichi} holds.
\qed
\betaigbreak
\section{The $\mathcal E$-equivalence}\ellongleftarrowbel{vw}
We introduce a new equivalence relation
of the set of 1-(respectively, 2-)dimensional virtual knots.
\betaegin{defn}\ellongleftarrowbel{zoo}
Let $K$ and $J$ be 1-(respectively, 2-)dimensional virtual knots.
If the submanifolds, $\mathcal E(K)$ and $\mathcal E(J)$,
of $\mathbb{R}^4$ (respectively, $\mathbb{R}^5$) are isotopic,
$K$ and $J$ are said to be
{\iotat $\mathcal E$-equivalent}.
See Theorem \ref{vv}, and the line right below Theorem \ref{honto} for $\mathcal E(\quad)$.
\epsilonnd{defn}
\betaegin{thm}\ellongleftarrowbel{milk}
{\rm {\betaf (By \cite[Theorem 2.2]{Rourke} and \cite[Proposition 3.3]{Satoh}.)}}
If two virtual 1-knots are welded equivalent,
then they are $\mathcal E$-equivalent.
Hence there are two virtual 1-knots, $J$ and $K$,
such that $J$ is not virtually equivalent to $K$
but such that $\mathcal E(J)$ is isotopic
to $\mathcal E(K)$.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{milk}.}
By \cite[Theorem 2.2]{Rourke}, there are two virtual 1-knots, $J$ and $K$,
such that
$J$ is not virtually equivalent to $K$
but such that
$J$ is welded equivalent to $K$.
By \cite[Proposition 3.3]{Satoh},
$J$ and $K$ are $\mathcal E$-equivalent.
\qed\\
Thus it is natural to ask whether
we have the virtual 2-knot version of Theorem \ref{milk}.
In other words, are there virtual 2-knots, $J$ and $K$,
which are $\mathcal E$-equivalent but which are not virtually equivalent?
We answer this question below.
\\
Let $\alphalpha$ be a 1-dimensional virtual knot diagram defined in $\mathbb{R}^2$.
Regard $\mathbb{R}^3$ as the result of rotating $\mathbb{R}^2_{\geqq0}=\mathbb{R}^1\times\{t\vert t\geqq 0\}$ around $\mathbb{R}^1\times\{t\vert t=0\}$ as the axis.
Take $\alphalpha$ in
$\mathbb{R}^1\times\{t\vert t> 0\}$.
When we rotate $\mathbb{R}^2_{\geqq0}$, rotate $\alphalpha$ together.
Then we obtain a 2-dimensional virtual knot diagram in $\mathbb{R}^3$ naturally,
and call it $\mathcal O(\alphalpha)$.
Note that
$\mathcal O(\alphalpha)$ is a virtual 2-knot diagram made from $T^2$. \\
If 1-dimensional virtual knot diagrams, $\alphalpha$ and $\betaeta$, are virtually equivalent,
it is trivial that
2-dimensional virtual knot diagrams, $\mathcal O(\alphalpha)$ and $\mathcal O(\betaeta)$
are virtually equivalent (see Definition \ref{JV}).
Hence it makes sense that we define an 2-dimensional virtual knot $\mathcal O(K)$ for a 1-dimensional virtual knot $K$. \\
Let $X$ be a classical surface knot contained in $\mathbb{R}^4=\mathbb{R}^3\times\{t\iotan\mathbb{R}\}$. Take $X$ in $\mathbb{R}^3\times\{t>0\}$. Regard $\mathbb{R}^5$ as the result of rotating $\mathbb{R}^3\times\{t\geqq0\}$ around $\mathbb{R}^3\times\{t=0\}$ as the axis. Then we rotate $X$ together.
Call the resultant 3-dimensional submanifold of $\mathbb{R}^5$, ${\mathcal O}(X)$.
Note the following: If $X$ is diffeomorphic to
a closed
surface $\Sigma_g$,
then
${\mathcal O}(X)$ is diffeomorphic to
$\Sigma_g\times S^1$. \\
\betaegin{pr}\ellongleftarrowbel{kakan}
Let $K$ be a virtual 1-knot.
Then
the submanifolds, $\mathcal E({\mathcal O}(K))$ and ${\mathcal O}(\mathcal E(K))$, of $\mathbb{R}^5$
are isotopic.
\epsilonnd{pr}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Proposition \ref{kakan}.}
By the definitions. \qed
\\
The {\iotat standardly embedded torus} or {\iotat standard torus} is
a submanifold of $\mathbb{R}^4$,
diffeomorphic to the torus,
and put in the standard position.
Let $\Sigma_g$ be an oriented closed surface.
The {\iotat standardly embedded surface
$($diffeomorphic to $\Sigma_g)$}
or
{\iotat standard surface $($diffeomorphic to $\Sigma_g)$}
is defined as well.
Note that we can regard classical 1- (respectively, 2-) knots as virtual knots. \\
Let $R$ be the virtual reef knot
whose diagram is drawn
in \cite[Figure 3, section three]{Rourke}.
We cite the diagram in Figure \ref{vr}.
As written there, $R$ is a nontrivial virtual 1-knot,
is welded equivalent to the trivial 1-knot,
and has the group $\mathbb{Z}$. \\
\betaegin{figure}
\iotancludegraphics[width=100mm]{vr.pdf}
\vskip-40mm
\caption{{\betaf Virtual reef knot}\ellongleftarrowbel{vr}}
\epsilonnd{figure}
\betaegin{cla}\ellongleftarrowbel{theta}
The virtual 2-knot $\mathcal O(R)$
is not virtually equivalent to
the standard torus.
\epsilonnd{cla}
\betaegin{note}\ellongleftarrowbel{tau}
The submanifold, $\mathcal E(R)$ and the standard torus,
of $\mathbb{R}^4$ are isotopic
because the virtual 1-knot $R$ is welded equivalent to the unknot.
\epsilonnd{note}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{theta}.}
The proof is done
in a similar way in \cite{Takeda} and a generalized fashion of the manner in
\cite[section three]{Rourke}:
The fundamental group of the virtual reef knot $R$ is $\mathbb{Z}$.
However the fundamental group of the mirror image of $R$ is
non-trivial.
The fundamental group of the mirror image of $R$ is the lower fundamental group of $\mathcal O(R)$ and
its non-triviality demonstrates the non-triviality of
$\mathcal O(R)$
as a virtual 2-knot.
\qed\\
Theorem \ref{Maine} is one of our main results. \\
\betaegin{thm}\ellongleftarrowbel{Maine}
There is a virtual 2-knot $K$
with the following conditions.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(1)$
The virtual 2-knot $K$
is not virtually equivalent to
the standard surface.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(2)$
The virtual 2-knot $K$
is $\mathcal E$-equivalent to
the standard surface.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{Maine}.}
Let $K$ be the virtual 2-knot $\mathcal O(R)$ in Claim \ref{theta}.
Claim \ref{theta} implies Theorem \ref{Maine}.(1).
Let $T$ denote the standard torus.
By Note \ref{tau}, $\mathcal E(R)=T$.
Proposition \ref{kakan} implies
$\mathcal E(K)
=\mathcal E(\mathcal O(R))
=\mathcal O(\mathcal E(R))
=\mathcal O(T)$,
where $=$ denotes the ambient isotopy
of submanifolds.
$\mathcal O(T)$ and $\mathcal E(T)$ are standardly embedded $T^3$ in $\mathbb{R}^5$ by the definition of them.
Hence we have Theorem \ref{Maine}.(2).
Therefore Theorem \ref{Maine} holds. \qed\\
We ask questions.
\betaegin{que}\ellongleftarrowbel{France}
(1) Do we have the following?
Let $\Sigma_g$ be
a closed oriented genus $g$ surface.
Let $Q$ (respectively, $Q'$) be
a virtual surface-knot
made from $\Sigma_g$.
If $Q$ and $Q'$ have the group $\mathbb{Z}$,
then the submanifolds, $\mathcal E(Q)$ and $\mathcal E(Q')$,
of $\mathbb{R}^5$ are isotopic.
\smallbreak
\Z[\pi/\pi^{(n)}]oindent
(2)
Is a virtual 1- (respectively, 2-) knot $K$ welded equivalent to the trivial 1-knot
if $K$ has the group $\mathbb{Z}$?
\epsilonnd{que}
\betaigbreak
\section{The fibrewise equivalence}\ellongleftarrowbel{New Mexico}
\subsection{
The fibrewise equivalence
is equal to the rotational welded equivalence,
and is different from the welded equivalence of virtual 1-knots
}\ellongleftarrowbel{sub1}\hskip20mm\\%
We research relations among
the fiberwise equivalence of virtual 1-knots,
the welded equivalence of them,
and
the rotational welded equivalence of them.
We mentioned it in the last few paragraphs of \S\ref{i3}.
See \cite{Rourke, Satoh} for the definition of the welded equivalence,
and \cite{Kauffman, Kauffmanrw, J} for that of the rotational welded equivalence,
as we also mentioned them in the last few paragraphs of \S\ref{i3}.\\
We first introduce the definition of the fiberwise equivalence of virtual 1-knots.
For our purpose (to prove Theorems \ref{smooth} and \ref{Montgomery}),
we will modify the definition a few times as below.
\betaegin{figure}
\iotancludegraphics[width=100mm]{xaa.pdf}
\vskip-40mm
\caption{{\betaf Fiberwise isotopy}\ellongleftarrowbel{Fib}}
\epsilonnd{figure}
\betaegin{defn}\ellongleftarrowbel{Nevada}
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams.
We say that
$\alphalpha$ and $\betaeta$ are {\iotat fiberwise equivalent}
if
Rourke's description of $\mathcal S(\alphalpha)$
and
that of $\mathcal S(\betaeta)$
are `fiberwise isotopic'.
In other words, this means that
$\alphalpha$ and $\betaeta$ satisfy the following conditions.
There is an
embedding map
$$g:S^1_b\times[0,1]\times S^1_f\hookrightarrow\mathbb{R}^2_b\times[0,1]\times\mathbb{R}^2_f$$
with the following properties.
See Figure \ref{Fib}.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(1)
For any fixed $t\iotan[0,1]$, $g(S_b^1\times\{t\}\times S^1_f)\subset\mathbb{R}^2_b\times\{t\}\times\mathbb{R}^2_f$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(2)
For any fixed $p\iotan S^1_b$ and any fixed $t\iotan[0,1]$,
$g(\{p\}\times\{t\}\times S^1_f)$ is contained in
the same fiber $\{q\}\times\mathbb{R}^2_f$ for a point
$q\iotan\mathbb{R}^2_b\times[0,1]$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent (3)
Let $\pi:\mathbb{R}^2_b\times[0,1]\times\mathbb{R}^2_f\to\mathbb{R}^2_b\times[0,1]$.
$(\pi\circ g)(S_b^1\times\{0\}\times S^1_f)$
(respectively, \\$(\pi\circ g)(S_b^1\times\{1\}\times S^1_f)$)
$\subset \mathbb{R}^2_b\times\{0\}$ (respectively, $\mathbb{R}^2_b\times\{1\}$)
is the diagram $\alphalpha$ (respectively, $\betaeta$) without information whether
each crossing point is a classical one or a virtual one.
This information is given by the fiber-circles over each crossing point as
in Theorem \ref{Montana} and Definition \ref{Nebraska}.
$\pi\circ g$ meets $R^2_b\times\{0,1\}$ transversely.
\betaigbreak
In knot theory we usually use an `ambient' isotopy in order to define the equivalence relation of knots as below.
We impose the following condition (4).
(See \cite[sections 1.1 and 1.2]{BZ} for an explanation on this fact
in the 1-dimensional classical knot case.)
\smallbreak\Z[\pi/\pi^{(n)}]oindent (4)
Let $g_t$ denote
$$g|_{S^1_b\times\{t\}\times S^1_f}: S^1_b\times\{t\}\times S^1_f\hookrightarrow\mathbb{R}^2_b\times\{t\}\times\mathbb{R}^2_f$$
for $0\elleqq t\elleqq1$.
There is an an isotopy
$$H_t:\mathbb{R}^2_b\times\{t\}\times\mathbb{R}^2_f\to\mathbb{R}^2_b\times\{t\}\times\mathbb{R}^2_f (0\elleqq t\elleqq1)$$
such that
$H_0$ is the identity map
and such that
$g_t=H_t\circ g_0$ for any $t\iotan[0,1]$.
We call $g$ a {\iotat special isotopy} between $\alphalpha$ and $\betaeta$.
\epsilonnd{defn}
\betaigbreak
\betaegin{defn}\ellongleftarrowbel{anko}
Take $g$ in Definition \ref{Nevada}.
If $g'$ is obtained by moving $g$ by an ambient isotopy map
$G_t$, where $0\elleq t\elleq1$, $g_0=g$, and $g_1=g'$,
keeping the conditions $(1)$-$(4)$ of Definition \ref{Nevada},
then we say that $g'$ is {\iotat level preserving, fiberwise isotopic} or {\iotat special isotopic} to $g$,
or
that we {\iotat perturb $g$ in the special way} to obtain $g'$.
We write $g\sim g'$.
$G_t$ is called a
{\iotat level preserving, fiberwise isotopy}
or
{\iotat special isotopy} between $g$ and $g'$.
\epsilonnd{defn}
\betaigbreak
\betaegin{note}\ellongleftarrowbel{kakanzu}
The following holds.
Let $\rho:S^1_b\times[0,1]\times S^1_f\to S^1_b\times[0,1]$ be the natural projection.
Then there is a (not necessarily smooth) continuous map \\
$\underline{g}:S^1_b\times[0,1]\to \mathbb{R}^2\times[0,1]$
such that $\pi\circ g=\underline{g}\circ\rho$.
That is, there is the following commutative diagram.
$$
\betaegin{matrix}
S^1_b\times[0,1]\times S^1_f&\stackrel{g}\to&\mathbb{R}^2_b\times[0,1]\times \mathbb{R}^2_f\\
\downarrow_\rho&\circlearrowright &\downarrow_\pi \\
S^1_b\times[0,1]&\stackrel{\underline{g}}\to&\mathbb{R}^2_b\times[0,1]
\epsilonnd{matrix}
$$
\epsilonnd{note}
\betaegin{defn}\ellongleftarrowbel{neba}
Under the above condition, we say that $\underline{g}$ is {\iotat covered} by $g$.
\epsilonnd{defn}
\betaigbreak
The following theorem is one of our main results.
\betaegin{thm}\ellongleftarrowbel{smooth}
Two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are smooth fiberwise equivalent if and only if
$\alphalpha$ and $\betaeta$ are smooth rotational welded equivalent.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.} See Note \ref{xmikan}. \betaigbreak
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Theorem \ref{smooth}.}
The `if' part is easy.
We prove the `only if' part.
\\
\Z[\pi/\pi^{(n)}]oindent{\betaf Strategy.}
See (I) and (II) below.
We want to prove (I)$\Leftrightarrow$(II).
It is easy to prove (II)$\mathbb{R}ightarrow$(I).
We will prove (I) $\mathbb{R}ightarrow$ (II) as follows.
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent(I) Smooth virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are smooth fiberwise equivalent.
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent(II) Smooth virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are smooth rotational welded equivalent.
\betaigbreak
See (1) below.
In Claim \ref{xbeef}
we will prove (I)$\mathbb{R}ightarrow$(1).
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent(1) There is a PL virtual 1-knot diagram $\alphalpha'$ (respectively, $\betaeta'$)
which is piecewise smooth isotopic to $\alphalpha$ (respectively, $\betaeta$)
such that $\alphalpha'$ and $\betaeta'$ are PL fiberwise equivalent.
\betaigbreak
See (2) below.
In Theorem \ref{fwrw}, we will prove (1)$\mathbb{R}ightarrow$(2).
It will be proved in the text
which starts from Proposition \ref{polygon},
and ends in Claim \ref{takusan}.
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent(2) $\alphalpha'$ and $\betaeta'$ are PL rotational welded equivalent.
\betaigbreak
In Lemma \ref{PLtosmooth}, we will prove (2)
$\mathbb{R}ightarrow$ (II). Thus we will finish the proof of (I) $\mathbb{R}ightarrow$ (II).
\betaigbreak
\betaigbreak
Assume that smooth virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are smooth fiberwise equivalent.
We do not know whether or not there are two
special isotopies $g$ and $g'$ between $\alphalpha$ and $\betaeta$
with the following properties.
$g$ and $g'$ are not smooth
special isotopic
but piecewise smooth special isotopic
Although we do not answer this question,
we accomplish the proof of (I)$\Leftrightarrow$(II).
\betaigbreak
Take $g$
in Definition \ref{Nevada}.
We do not know
whether there is a smooth $g'$ with $g'\sim g$
with the following properties:
There is a finite simplicial structure on
Im $\pi\circ g'$
which restricts to
a finite simplicial structure on the singular subset
of Im $\pi\circ g'$.
One reason is as follows. Im $\pi\circ g$ may be the projection of a wild embedding
for a smooth $g$ even if $g$ is not a wild embedding map.
Although we do not answer this question, we accomplish the proof of (I)$\Leftrightarrow$(II).
\betaegin{defn}\ellongleftarrowbel{PLNevada}
Consider the conditions of Definition \ref{Nevada} in the PL category.
The equivalence relation is called {\iotat PL fiberwise equivalence}.
\epsilonnd{defn}
\betaegin{cla}\ellongleftarrowbel{xbeef}
If virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are smooth fiberwise equivalence,
then $\alphalpha$ and $\betaeta$ are PL fiberwise equivalence.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{xbeef}.}
It is enough to prove that
the map $g$ in Definition \ref{Nevada} is approximated
by a fiberwise level-preserving PL embedding map. We prove it below.
Regard $S^1_b$ as $[0,1]/\sim$, where $0\sim1$.
Regard $S^1_f$ as $[0,1]/\sim$, where $0\sim1$.
Hence we can regard
$S^1_b\times[0,1]\times S^1_f$
as the one made from $[0,1]\times[0,1]\times[0,1]$ by these equivalence relations.
Let $n$ be any positive integer.
Take points
$(\Bbbkrac{i}{2n},\Bbbkrac{j}{2n},\Bbbkrac{k}{2n})\iotan[0,1]\times[0,1]\times[0,1]$,
where $i$ (respectively, $j$, $k$) is any integer with the condition
$0\elleqq$$i$ (respectively, $j$, $k$) $\elleqq 2n$.
Let $l$ be any integer with the condition $0\elleqq 2l$(respecctively, $2l+2)\elleqq 2n$.
Take any cube $C$ whose vertices are
$(\Bbbkrac{\alphalpha}{2n},\Bbbkrac{\betaeta}{2n},\Bbbkrac{\gamma}{2n})$,
where $\alphalpha$ (respectively, $\betaeta$, $\gamma$) is any integer in $\{2l,2l+2\}$.
Take a simplicial division on $S^1_b\times[0,1]\times S^1_f$ as follows.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(1)
0-simplices are all $(\Bbbkrac{i}{2n},\Bbbkrac{j}{2n},\Bbbkrac{k}{2n})\iotan[0,1]\times[0,1]\times[0,1]$ as above.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(2)
1-simplices are defined as follows.
Take any cube $C$.
Note that each of six sites includes nine 0-simlices,
that the sum of six sites includes 26 0-simlices,
and
that $C$ includes 27 0-simlices.
Take the 0-simplex $P$ in $C$ which is not included in any site.
Take any segment whose boundary is $P$ and one of the other 28 0-simplices.
Take 16 segments in each site of $C$ as drawn in Figure \ref{menkirikata}.
1-simplices are these two kinds of segment.
\betaegin{figure}
\iotancludegraphics[width=120mm]{menkirikata.pdf}
\vskip-30mm
\caption{{\betaf 16 1-simplices on a site of $C$}\ellongleftarrowbel{menkirikata}}
\epsilonnd{figure}
\smallbreak\Z[\pi/\pi^{(n)}]oindent(3)
The set of 1-simplices defines 2-simplices naturally.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(4)
The set of 2-simplices defines 3-simplices naturally.
\betaigbreak
Let $n$ be sufficiently large.
Take the image of all 0-simplices by $g$ in $\mathbb{R}^2_b\times[0,1]\times\mathbb{R}^2_f$.
They determine a fiberwise level-preserving PL embedding map of $S^1_b\times[0,1]\times S^1_f$ naturally.
{\iotat Reason.} Im $g$ is a smooth regular submanifold.
Hence it has a tubular neighborhood.
This completes the proof of Claim \ref{xbeef}. \qed
\betaegin{note}\ellongleftarrowbel{bango}
If $C=$(Im $g$)$\cap$(a fiber $\mathbb{R}^2_f$) is PL homeomorphic to a circle,
then $C$ is a polygon. However the number of the vertices of $C$ depends on fibers.
\epsilonnd{note}
\betaegin{note}\ellongleftarrowbel{haruwa}
From here to the end of the proof of Theorem \ref{fwrw}, we work in the PL category
unless we indicate otherwise.
After that, we will go back to the smooth category.
When we move a map by isotopy, we take a PL subdivision if necessary.
\epsilonnd{note}
Claim \ref{xbeef} implies the following.
\betaegin{pr}\ellongleftarrowbel{polygon}
$g$ in Definition \ref{PLNevada} satisfies the condition that a finite simplicial structure on Im $\pi\circ g$
which restricts to a finite simplicial structure on the singular subset of Im $\pi\circ g$.
\epsilonnd{pr}
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=150mm]{zudelta.pdf}
\vskip-40mm
\caption{{\betaf The $\Delta^1$-move.}\ellongleftarrowbel{zudelta}}
\epsilonnd{figure}
We call the operation drawn in Figure \ref{zudelta},
the $\Delta^1$-move of virtual 1-knot diagrams.
Note that we do not draw the other part of this diagram. The other part may intersect the part drawn in Figure \ref{zudelta}.
By Proposition \ref{polygon}, $\alphalpha, \betaeta$ in Definition \ref{PLNevada}
have the following properties:
\betaegin{cla}\ellongleftarrowbel{Delta}
$\alphalpha$ $($respectively, $\betaeta$$)$ is obtained from $\betaeta$ $($respectively, $\alphalpha$$)$ by
a finite step of $\Delta^1$-moves.
\epsilonnd{cla}
\betaegin{defn}\ellongleftarrowbel{PLyugentsuki}
Add the following condition to Definition \ref{PLNevada} without changing the other parts.
(Note we work in the PL category.)
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(\ref{PLyugentsuki}.1)$ In each fiber $\mathbb{R}^2_f$, there are a finite number of circles.
$($That is, $<\iotanfty.)$
\epsilonnd{defn}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
See Note \ref{xudon}.
Recall Note \ref{bango}.
\\
Indeed, the following holds.
\betaegin{thm}\ellongleftarrowbel{herasu}
Definitions $\ref{PLNevada}$ and $\ref{PLyugentsuki}$ are equivalent.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{herasu}.}
It is trivial that if $g$ satisfies Definition \ref{PLyugentsuki}, then $g$ satisfies Definition \ref{PLNevada}.
We prove that if $g$ satisfies Definition \ref{PLNevada},
then we can perturb $g$ in the special way so that $g$ satisfies Definition \ref{PLyugentsuki}.
Suppose that $g$ satisfies Definition \ref{PLNevada}.
Let $q\iotan\mathbb{R}^2_b$ and $t\iotan[0,1]$.
Since Im$g$ is a compact PL regular submanifold,
Im$g\cap(\{q\}\times\{t\}\times\mathbb{R}_f)$ is
a disjoint union of a finite number of circles and a finite number of annuli.
Note that the union of them is a regular submanifold of $\{q\}\times\{t\}\times\mathbb{R}^2_f$.
Take a tubular neighborhood $N$ of each annulus
in $\mathbb{R}^2_b\times[0,1]\times\mathbb{R}^2_f$, to be small enough.
Stretch each annulus into the direction perpendicular to $\{x\}\times\{t\}\times\mathbb{R}^2_f$.
Then we can obtain a new $g$ which satisfies Definition \ref{PLyugentsuki}.
The idea of how we stretch is drawn in Figure \ref{hipparu}
Note that
Figures \ref{hipparu} draws `figures in PL category' although the figures are smoothened.
\qed
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=145mm]{hipparu.pdf}
\vskip-37mm
\caption{{\betaf
The idea of how we stretch $g(S^1_b\times[0,1]\times S^1_f)\cap N$
}\ellongleftarrowbel{hipparu}}
\epsilonnd{figure}
\betaigbreak
A point
$p\iotan$Im$\pi\circ g$
$=(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)$
$=(\underline{g}\circ\rho)(S^1_b\times[0,1]\times S^1_f)$
$=\underline{g}(S^1_b\times[0,1])$
is called a {\iotat multiple point} or {\iotat $n$-tuple point}
if $\underline{g}^{-1}(p)\iotan S^1_b\times[0,1]$
consists of $n$ points ($n\geqq2$).
(Note that in Definition \ref{PLyugentsuki}, $n<\iotanfty$.)
A point $p\iotan$Im$\pi\circ g$
is called a {\iotat single point}
if $\underline{g}^{-1}(p)$ consists of a single points.
The {\iotat singular point set} of $p\iotan$Im$\pi\circ g$ consists of branch points and multiple points. \\
Note the following facts.
Take $g$ in Definition \ref{PLyugentsuki}, and $\underline{g}$ which is covered by $g$.
Recall that `cover' is defined in Definition \ref{neba}.
Suppose that $\underline{g}$ is a generic map.
Note Im $\underline{g}$.
We can define whether each double point is classical or virtual
by using the information of the fiber-circles over each point as in Theorem \ref{Montana},
Definition \ref{Nebraska}, Note \ref{kaiga}, and Definition \ref{suiri}.
There is a case where a classical (respectively, virtual) double point appears.
The information of fiber-circles over each branch point determines that the branch point is classical.
{\iotat Reason.} By Theorem \ref{Rmuri},
there are no virtual branch point. \\
Note each triple point.
There are three circles in the fiber over each triple point.
There are four cases how three circles are put in the fiber.
See Notes \ref{umeboshi} and \ref{faso}, Definnition \ref{JW}, and Figure \ref{JW1}.
There is a case where each of the four occurs.
\betaegin{note}\ellongleftarrowbel{umeboshi}
$(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)$ in $\mathbb{R}^2_b\times[0,1]$
is a welded 2-knot with a fixed boundary in general,
and
is not a virtual 2-knot with a fixed boundary in general.
See \cite[sections 3.5-3.7]{J} for their definitions
and their difference.
In the welded 2-knot case we also use the terms, `fiber-circle' and `Rourke-fibration'.
See Note \ref{faso}.
\epsilonnd{note}
Here we cite the definition of welded 2-knots from \cite{J}.
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=140mm]{l83from93.pdf}
\vskip-30mm
\caption{{\betaf The singular point sets of welded 2-knots}
\ellongleftarrowbel{JW1}}
\betaigbreak
\epsilonnd{figure}
Recall that a 2-knot diagram is (the image of) a generic map of a surface in 3-space, with
classical and virtual crossing data along the double-point arcs.
Also recall that 2-knot diagrams may be transformed by Roseman moves, which preserve the
crossing data locally.
\betaegin{defn}\ellongleftarrowbel{JW}{\betaf (\cite[section 3.6]{J}.)}
If all triple points of a 2-knot diagram are of the four types shown in Figure \ref{JW1},
the diagram is called a Welded 2-knot diagram. If a pair of Welded 2-knot
diagrams are related by a series of Roseman moves, with only Welded diagrams
appearing throughout the process, then the diagrams are Welded equivalent and
belong to the same Welded 2-knot type.
\epsilonnd{defn}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
The above definition makes sense both in the smooth and the PL category.
The readers need not be familiar with Roseman moves in order to read this paper.
\betaegin{note}\ellongleftarrowbel{faso}
When we consider circles in fiber $\mathbb{R}^2$ as in Note \ref{umeboshi},
there is a new type drawn in Figure \ref{rashi},
which is not in Figure \ref{doremi}.
\epsilonnd{note}
\betaegin{figure}
\iotancludegraphics[width=100mm]{rashi.pdf}
\vskip-20mm
\caption{{\betaf A new type of a nest of circles.}\ellongleftarrowbel{rashi}}
\epsilonnd{figure}
\betaegin{note}\ellongleftarrowbel{oshii}
By Proposition \ref{polygon}, $\underline{g}$ in Definition \ref{PLyugentsuki}
satisfies the conditions (I)-(III) below, but $\underline{g}$ is not generic.
\smallbreak\Z[\pi/\pi^{(n)}]oindent (I)
$\underline{g}:S^1_b\times[0,1]\to\mathbb{R}^2_b\times[0,1]$
is a continuous map such that
$\underline{g}(S^1_b\times\{t\})\subset\mathbb{R}^2_b\times\{t\}$ for any $t\iotan[0,1]$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent (II)
Let $t\iotan[0,1]$.
There are closed intervals,
$I_1,..., I_\mu$ ($\mu\iotan\mathbb{N}$),
such that
$S^1_b\times\{t\}\\=I_1\cup...\cup I_\mu$
and such that
$\underline{g}|_{I_i}$ is a PL
embedding for each $i$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent (III)
There are closed 2-discs,
$D^2_1,..., D^2_\Z[\pi/\pi^{(n)}]u$ ($\Z[\pi/\pi^{(n)}]u\iotan\mathbb{N}$),
such that
$S^1_b\times[0,1]=D^2_1\cup...\cup D^2_\Z[\pi/\pi^{(n)}]u$
and such that
$\underline{g}|_{D^2_i}$ is a PL
embedding for each $i$.
\epsilonnd{note}
\betaegin{defn}\ellongleftarrowbel{wasabi}
If a map
$\underline{g}:S^1_b\times[0,1]\to\mathbb{R}^2_b\times[0,1]$ satisfies the conditions (I)-(III) in Note \ref{oshii},
then $\underline{g}$ is said to be {\iotat level preserving}.
If $\underline{g}'$ is obtained by moving $\underline{g}$ by a homotopy
$\underline{G_t}$, where $0\elleq t\elleq1$, $\underline{G_0}=g$ and $\underline{G_1}=g'$,
keeping the conditions (I)-(III) of in Note \ref{oshii},
then we say that $\underline{g}'$ is {\iotat level preserving homotopic} to $\underline{g}$
or
that we {\iotat perturb $\underline{g}$ in the special way} and obtain $\underline{g}'$.
We write $\underline{g}\sim\underline{g}'$.
$\underline{G_t}$ is called a {\iotat level preserving homotopy} or a {\iotat special homotopy}.
Let $g: S^1_b\times[0,1]\times S^1_f\to\mathbb{R}^2_b\times[0,1]\times \mathbb{R}^2_f$
be a map in Definition \ref{PLyugentsuki}.
Take a special homotopy
$\underline{G_t}$ of $\underline{g}$, and
a special isotopy
$G_t$ of $g$
where $0\elleqq t\elleqq1.$
If $\underline{G_t}$
is covered by $G_t$
for any element $t$ in $\{t|0\elleqq t\elleqq1\}$,
then we say that
$\underline{G_t}$ is {\iotat covered} by $G_t$.
\epsilonnd{defn}
\betaegin{defn}\ellongleftarrowbel{gene}
Add the following condition to
Definition \ref{PLyugentsuki} without changing the other parts.
\Z[\pi/\pi^{(n)}]oindent$(\ref{gene}.1)$
We can perturb $g$ in Definition \ref{PLyugentsuki} in the special way
so that $g$ covers a PL level preserving, generic map $S^1_b\times[0,1]\to\mathbb{R}^2_b\times[0,1]$.
\epsilonnd{defn}
We prove the following theorem.
\betaegin{thm}\ellongleftarrowbel{Wyoming}
Definition $\ref{gene}$ is equivalent to Definition $\ref{PLyugentsuki}$
$($and, by Theorem $\ref{herasu},$ is equivalent to Definition $\ref{PLNevada}.)$
\epsilonnd{thm}
\betaegin{note}\ellongleftarrowbel{shio}
Even if we perturb $\underline{g}: S^1_b\times[0,1]\to\mathbb{R}^2_b\times[0,1]$, which is covered by $g$,
in the special way
by a special homotopy $\underline{G_t}$,
$\underline{G_t}$ is not covered by a special isotopy $G_t$ of $g$
in general.
We must make
$\underline{G_t}$
under the condition that
$\underline{G_t}$ is covered by $G_t$.
\epsilonnd{note}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Theorem \ref{Wyoming}.}
It is trivial that
if $g$ satisfies Definition \ref{gene}, then $g$ satisfies Definition \ref{PLyugentsuki}.
We prove the following.
\betaegin{cla}\ellongleftarrowbel{kamen}
If $g$ satisfies Definition \ref{PLyugentsuki},
we can perturb $g$ so that $g$ satisfies Definition \ref{gene}.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note}.
Recall that $\pi\circ g$ does not cover a generic map $\underline{g}$ in general.
\\
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Claim \ref{kamen}.}
\Z[\pi/\pi^{(n)}]oindent
{\betaf The first step.}
Recall that by Definition \ref{PLyugentsuki}, for each $t\iotan[0,1]$, (Im$(\pi\circ g)$)$\cap(\mathbb{R}^2\times\{t\})$ is an immersed circle. We prove the following.
\betaegin{cla}\ellongleftarrowbel{Oregon}
We can perturb $g$
in the special way
so that the singular point set of \\
$(${\rm Im}$(\pi\circ g))\cap(\mathbb{R}^2\times\{t\})$
is a finite number of points
except for a finite number of levels $t\iotan[0,1]$.
In other words, we can do so that for only a finite number of levels $t\iotan[0,1]$,
the singular point set of $($Im$(\pi\circ g))\cap(\mathbb{R}^2\times\{t\})$ includes a finite number of segments.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Claim \ref{Oregon}.}
Let $I$ denote the interior of a 1-simplex which is in a simplicial complex structure of
the singular point set of (Im$(\pi\circ g)$)$\cap(\mathbb{R}^2\times\{\gamma\})$
for a real number $\gamma\iotan[0,1]$.
Assume that $I$ consists of multiple PL points.
Suppose that there are real numbers
$\alphalpha,\betaeta\iotan[0,1]$ with the following properties: \\
$\alphalpha<\gamma<\betaeta$.
For $\alphalpha<t<\betaeta$,
$h_t:R^2_b\times\{\gamma\}\to\mathbb{R}^2_b\times(\alphalpha,\betaeta)$ is an isotopy
($t$ runs in $(\alphalpha,\betaeta)$)
such that
$h_t(({\rm Im}(\pi\circ g))\cap(\mathbb{R}^2\times\{\gamma\}))=$
$({\rm Im}(\pi\circ g))\cap(\mathbb{R}^2\times\{t \})$
for all $t\iotan(\alphalpha, \betaeta)$,
and such that
$h_t$ preserves
the information of fiber-circles over
two immersed circles
which are put in the both sides of $=$.
(Here,
the information of fiber-circles means what we define in
Theorem \ref{Montana}, Definition \ref{Nebraska}, Note \ref{kaiga}, and Definition \ref{suiri}.)
Note that $(\underline{g})^{-1}(I)$
is a disjoint union of $n$ open segments $I_1,...,I_n$ in $S^1_b\times[0,1]$.
Note that $\underset{\alphalpha<t<\betaeta}{\cup} h_t(I)$
consists of $n$-tuple points,
is an open set, and
is a discrete submanifold of $\mathbb{R}^2_b\times[0,1]$.
We can perturb $g$ in the special way
so that $\underset{\alphalpha<t<\betaeta}{\cup} h_t(I)$
separates $n$ copies of
$\underset{\alphalpha<t<\betaeta}{\cup} h_t(I)$
and
so that we keep the boundary of the closure of
$\underset{\alphalpha<t<\betaeta}{\cup} h_t(I)$
since
there does not appear a new singularity of the immersed annulus.
Figure \ref{step1} is an example. \\
\betaegin{note}\ellongleftarrowbel{ultra}
Figures \ref{step1}-\ref{dia1} draw `figures in PL category' although the figures are smoothened.
When we move a map by isotopy, we take a PL subdivision if necessary.
\epsilonnd{note}
\betaegin{figure}
\vskip-30mm
\iotancludegraphics[width=170mm]{step1.pdf}
\vskip-50mm
\caption{{\betaf An example of fiberwise isotopy. }\ellongleftarrowbel{step1}}
\epsilonnd{figure}
Note that
the boundary of the closure of each $I$ may have a singular point set.
The repetition of this procedure and
Proposition \ref{polygon}
imply
Claim \ref{Oregon}. \qed
\betaegin{note}\ellongleftarrowbel{honmaya}
Note each point in the resultant part which is made
from $\underset{\alphalpha<t<\betaeta}{\cup} h_t(I)$ by the separation.
By the definition of $I$, it is a single point.
\epsilonnd{note}
\Z[\pi/\pi^{(n)}]oindent
{\betaf The second step.} We prove the following.
\betaegin{cla}\ellongleftarrowbel{Pennsylvania}
Suppose that $g$ satisfies
the condition of Claim {\rm \ref{Oregon}}.
We can perturb $g$ in the special way
so that $\pi\circ g$ covers a level preserving transverse immersion $\underline{g}$
except for a finite number of points contained in $S^1_b\times[0,1]$
with the following property:
Let $P$ be an exceptional point.
Then $\underline{g}^{-1}(\underline{g}(P))$ may be more than one point.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Claim \ref{Pennsylvania}.}
Since $g$ satisfies the condition of Claim \ref{Oregon},
the singular point set of Im$(\pi\circ g)$ is a 1-dimensional finite simplicial complex.
Recall Proposition \ref{polygon}
and Note \ref{honmaya}.
Take the interior $I$ of a 1-simplex in the singular point set of the simplicial complex structure
with the following property:
\smallbreak\Z[\pi/\pi^{(n)}]oindent(1)
$\underline{g}^{-1}(I)$ is disjoint $n$ open segments $I_1,...,I_n$ in $S^1_b\times[0,1]$ ($n\iotan\mathbb{N}$).
$\underline{g}\vert_{I_i}$ is an embedding map.
\smallbreak\Z[\pi/\pi^{(n)}]oindent(2)
There is an open neighborhood $U$ of $I$ in $\mathbb{R}^2_b\times[0,1]$
with the following property:
There are open discs
$D^2_i$ embedded in $S^1_b\times[0,1]$
each of which is a tubular neighborhood of $I_i$ in $S^1_b\times[0,1]$
for each $i$.
$D^2_i\cap D^2_j=\phi$ for each distinct $i,j$.
$\underline{g}|_{D^2_i}$ is an embedding map.
$U\cap\underline{g}(S^1_b\times[0,1])$ is
$\underline{g}(D^2_1)\cup...\cup\underline{g}(D^2_n)$.
$\underline{g}(D^2_i)\cap\underline{g}(D^2_j)=I$ for each distinct $i,j$.
\smallbreak
We perturb $\underline{g}|_{(D^2_1\cup...\cup D^2_n)\cap(\underline{g}^{-1}(U))}$
in the special way
below
but we must remember Note \ref{shio}.
\smallbreak
Let $V$ be an open neighborhood of $U$ in $\mathbb{R}^2_b\times[0,1]$.
Hence $\overline{U}\subset V$.
Let $\wp:\mathbb{R}^2_b\times[0,1]\times\mathbb{R}^2_f\to\mathbb{R}^2_f.$
Combine this map $\wp$ and the diagram in Note \ref{kakanzu}:
$$
\betaegin{matrix}
S^1_b\times[0,1]\times S^1_f&\stackrel{g}\to&\mathbb{R}^2_b\times[0,1]\times \mathbb{R}^2_f
&\stackrel{\wp}\to\mathbb{R}^2_f\\
\downarrow_\rho&\circlearrowright &\downarrow_\pi&& \\
S^1_b\times[0,1]&\stackrel{\underline{g}}\to&\mathbb{R}^2_b\times[0,1]&&
\epsilonnd{matrix}
$$
\betaegin{cla}\ellongleftarrowbel{shoga}
We can perturb $g$ in the special way,
keeping out $V$ $($not $U)$,
with the following properties:
The image $\wp(g(\rho^{-1}(D^2_i)))$
is a circle $C_i$.
We have $C_i\cap C_j=\phi$ for each distinct $i, j$.
The map $\wp\vert_{g(\rho^{-1}(D^2_i))}$ is the projection.
\epsilonnd{cla}
\betaigbreak\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{shoga}.}
Take a point $\sigma\iotan I$.
Let $\underline{g}^{-1}(\sigma)=\{\sigma_1,...,\sigma_n\}$ and $\sigma_i\iotan I_i$.
Then the image of $\wp(g(\rho^{-1}(\sigma_i)))$ is a circle $C'_i$,
and we have $C'_i\cap C'_j=\phi$ for each distinct $i, j$.
We can take $g$ so that the circle $C_i$ which we want is this circle $C'_i$ for each $i$.
Then Claim \ref{shoga} holds. \qed
\betaigbreak\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
The reason why we prepare $V$ is
as follows:
Before the perturbation, the map $\wp\vert_{g(\rho^{-1}(\partial D^2_i))}$ is not a projection.
Note $\partial D^2_i\subset\overline U$.
However
$\wp\vert_{g(\rho^{-1}(\partial D^2_i))}$ is the projection
after the perturbation.
\\
We next make $\underline{g}|_{(D^2_1\cup...\cup D^2_n)\cap(\underline{g}^{-1}(U))}$
a level preserving transverse immersion
since we can perturb $g$ in the special way,
keeping out $U$, with the following properties:
For any point $q\iotan D^2_i$
and any point $r$ in the circle $\rho^{-1}(q)$,
$\wp(g(r))\iotan\mathbb{R}^2_f$ is fixed while we perturb $g$.
Claim \ref{shoga} ensures that while we perturb $g$ in this way,
we keep a property that $g$ is an embedding map.
Figure \ref{Ohio} is an example of this
procedure. The repetition of this procedure implies Claim \ref{Pennsylvania}. \qed\\
\betaegin{figure}
\iotancludegraphics[width=150mm]{tsume.pdf}
\vskip-35mm
\caption{{\betaf A special
isotopy of $g$. The intersection of four sheets in the upper figure
is perturbed and is changed into the one in the lower figure.}\ellongleftarrowbel{Ohio}}
\epsilonnd{figure}
\Z[\pi/\pi^{(n)}]oindent
{\betaf The third step.} We prove the following.
\betaegin{cla}\ellongleftarrowbel{Rhode Island}
Suppose that
$g$ satisfies the condition of Claim {\rm\ref{Pennsylvania}}.
We can perturb $g$
in the special way
so that $\pi\circ g$ covers a level preserving transverse immersion $\underline{g}$
except for a finite number of points contained in $S^1_b\times[0,1]$
with the following property:
Let $P$ be any exceptional point.
The set
$\underline{g}^{-1}(\underline{g}(P))$ consists of only one point.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Claim \ref{Rhode Island}.}
Assume that
$f^{-1}(f(P))$ consists of $m$ points
$P_1,...,P_m$ ($m\iotan\mathbb{N}$) in $S^1_b\times[0,1]$.
Take an open neighborhood $U$ of $f(P)$ in $\mathbb{R}^2_b\times[0,1]$ with the following properties:
There are open discs
$D^2_i$ in $S^1_b\times[0,1]$
which is a tubular neighborhood of $P_i$ in $S^1_b\times[0,1]$
for each $i$.
$D^2_i\cap D^2_j=\phi$ for each distinct $i,j$.
$U\cap\underline{g}(S^1_b\times[0,1])$ is
$\underline{g}(D^2_1)\cup...\cup\underline{g}(D^2_m)$.
$\underline{g}(D^2_1))\cap...\cap\underline{g}(D^2_m)=f(P)$.
For a pair $(i,j)$, we may have
$\underline{g}(D^2_i))\cap\underline{g}(D^2_j)\underset{\Z[\pi/\pi^{(n)}]eq}{\supset}f(P)$.
\smallbreak
We perturb $\underline{g}|_{(D^2_1\cup...\cup D^2_n)\cap(\underline{g}^{-1}(U))}$
in the special way
below
but we must remember Note \ref{shio}.
Take an open neighborhood $V$ of $U$ in $\mathbb{R}^2_b\times[0,1]$
such that $\overline{U}\subset V$.
We can perturb $g$ in the special way,
keeping out $V$ (not $U$),
with the following properties:
$\wp(g(\rho^{-1(}D^2_i)))$ is a circle $C_i$.
$\wp\vert_{g(\rho^{-1(}D^2_i))}$ is the projection. \\
We can make $\underline{g}|_{(D^2_1\cup...\cup D^2_n)\cap(\underline{g}^{-1}(U))}$
a level preserving transverse immersion except for a finite number of points
since we can perturb $g$ in the special way,
keeping out $U$,
with the following properties:
For any point $q\iotan D^2_i$
and any point $r$ in the circle $\rho^{-1}(q)$,
$\wp(g(r))\iotan\mathbb{R}^2_f$ is fixed while we perturb $g$.
(Note that while we perturb $g$ in this way,
we keep a property that $g$ is a
embedding map.)
The repetition of this procedure
and Note \ref{honmaya} imply
Claim \ref{Rhode Island}. \qed\\
\Z[\pi/\pi^{(n)}]oindent{\betaf The fourth step.}
Take $\pi\circ g$ and $\underline{g}$ in Claim \ref{Rhode Island}.
Let $P$ be any exceptional point. Recall that $P\iotan S_b^1\times[0,1]$.
Let $N(P)$ be the tubular neighborhood of $P$ in $S_b^1\times[0,1]$.
Take the tubular neighborhood $B$ of $\underline{g}(P)$ in $\mathbb{R}^2_b\times[0,1]$.
We can suppose that
$\underline{g}(N(P))\subset B$
and that
$\underline{g}(\partial N(P))\subset \partial B$.
The image $\underline{g}(N(P))$ makes $\underline{g}(P)$
a branch point
(recall Definition \ref{oyster}).
Here we ignore the information
of fiber circles over $P$.
The information of Rourke fiber makes $\underline{g}(\partial N(P))\subset \partial B$,
a virtual 1-knot diagram $\omega$ in $\partial B-$(a point).
Note that $\partial B-$(a point) is the 2-space and
that the point is not included in $\omega$.
Recall virtual segments defined in Note \ref{vrei}.
A {\iotat classical segment} is the segment with the following properties.
It is a segment included in a virtual 2-knot diagram.
One of the boundary is a classical branch point.
The points in the interior of the segment are classical double points.
An example is drawn in Figure \ref{sashimiv} if the branch point there is a classical branch point. \\
\betaegin{cla}\ellongleftarrowbel{mochi}
We can assume that all branch points of Im $\underline{g}$ are classical Whitney branch points.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{mochi}.}
Since $\underline{g}(P)$ is a branch point,
$n$ virtual segments and $m$ classical segments
meet at
$\underline{g}(P)$,
where $\{n,m\}\subset\mathbb{N}\cup\{0\}$ and $n+m\geqq2$.
We can prove that there is no virtual segment
in the same fashion as the one in the proof of Theorem \ref{Rmuri}.
(Note that in \S\ref{v2} we proved Theorem \ref{Rmuri} in the smooth category
but we can prove the PL version of Theorem \ref{Rmuri} in the same way.)
Therefore
more than one classical segment meet at $\underline{g}(P)$.
Hence $\omega$ is a classical diagram and
determines a classical 1-knot.
\\
In order to complete the proof of Claim \ref{mochi}, we will prove Claim \ref{koma}.
In order to prove Claim \ref{koma}, we prove the following Claim \ref{tantei}.
\betaegin{defn}\ellongleftarrowbel{prod}
Let $u,v\iotan[0,1]$. Let $u\elleq t\elleq v$.
The map $\underline{g}|_{S^1_b\times[u,v]}$ is called a {\iotat product map}
if there is an isotopy $\iotaota_t$
of $\mathbb{R}^2$ from the identity map
such that $\iotaota_t:\mathbb{R}^2\times\{u\}\to\mathbb{R}^2\times\{t\}$
carries $({\rm Im}\underline{g})\cap(\mathbb{R}^2\times\{u\})$ to $({\rm Im}\underline{g})\cap(\mathbb{R}^2\times\{t\})$.
Let $B^3$ be an embedded closed 3-ball in $\mathbb{R}^2_b\times[0,1]$.
The map $\underline{g}|_{S^1_b\times[u,v]}$ is called a {\iotat product map out $B^3$}
if there is an isotopy $\iotaota_t$
of $\mathbb{R}^2$ from the identity map
such that $\iotaota_t:\mathbb{R}^2\times\{u\}\to\mathbb{R}^2\times\{t\}$
carries $({\rm Im}\underline{g}-B^3)\cap(\mathbb{R}^2\times\{u\})$
to $({\rm Im}\underline{g}-B^3)\cap(\mathbb{R}^2\times\{t\})$.
\epsilonnd{defn}
We have the following.
\betaegin{figure}\iotancludegraphics[width=140mm]{todome.pdf}
\vskip-40mm
\caption{{\betaf
A branch point moved by a special
isotopy of $g$ }}\ellongleftarrowbel{todome}
\epsilonnd{figure}
\betaegin{cla}\ellongleftarrowbel{tantei}
By using a special
isotopy of $g$,
any branch point is moved
as drawn in Figure $\ref{todome}$:
Let $\alphalpha_u$ $($respectively, $\alphalpha_v)$ be an immersed circle determined by
$\underline{g}(S^1_b\times\{u\})\subset\mathbb{R}^2_b\times\{u\}$
$($respectively, $\underline{g}(S^1_b\times\{v\})\subset\mathbb{R}^2_b\times\{v\})$
with the information of Rourke fiber determined by
$g(S^1_b\times\{u\}\times S^1_f)\subset\mathbb{R}^2_b\times\{u\}\times\mathbb{R}^2_f$
$($respectively, \\$g(S^1_b\times\{v\}\times S^1_f)\subset\mathbb{R}^2_b\times\{v\}\times\mathbb{R}^2_f).$
The map $\underline{g}|_{S^1_b\times[u,v]}$ is a product map out $B$.
Hence $\alphalpha_u=\alphalpha_v\#\omega$,
where $\#$ denotes the connected sum of immersed circles into $\mathbb{R}^2$
and $=$ means that there is an orientation preserving diffeomorphism
of $\mathbb{R}^2$ which carries the left hand side to the right side one.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{tantei}.}
For each $t$, $(\mathbb{R}^2_b\times\{t\})\cap$Im $\underline{g}$ is connected.
Hence
the branch point is not a local maximal (respectively, minimal) point of
the restriction of the height function $\mathbb{R}^2_b\times[0,1]\to[0,1]$ to Im $\underline{g}$.
\\
Claim \ref{sukoshi} follows from Claim \ref{tako}.
\betaegin{cla}\ellongleftarrowbel{sukoshi}
Let $X$ be the closed interval contained in $\mathcal S$. Assume that $X$ does not have self-intersection.
Then, by using a special
isotopy of $g$,
we can move {\rm Int}$X$ as drawn in Figure \ref{igaini}
with the following properties:
We move {\rm Int}$X$ by an isotopy of embedding of {\rm Int}$X$,
keeping the position of $\partial X$ in $\mathbb{R}^2_b\times[0,1]$.
We keep the position of $\overline{\mathcal S-X}$ in $\mathbb{R}^2_b\times[0,1]$.
We keep the condition $X\cap(\mathcal S-X)=\phi.$
\epsilonnd{cla}
\betaegin{figure}
\iotancludegraphics[width=140mm]{igai.pdf}
\vskip-20mm
\caption{{\betaf Changing $X$.}\ellongleftarrowbel{igaini}}
\epsilonnd{figure}
In Figure \ref{beefsteak} there is an example of Claim \ref{sukoshi}.
Threre is drawn
how $X$ changes by a special
isotopy of $g$
in the case of the upper two figures in
the right column of Figure \ref{igaini}.
Note that Int$X$ consists of double points.
Each point of $\partial X$ is a branch, double or triple one.
Let $B$ be an open disc contained in $S^1_b\times[0,1]\times S^1_f$. By Definition \ref{PLNevada}.(1), $\pi\circ g(B)$ is not parallel to $\mathbb{R}^2_b\times\{0\}$.
Note that if $\pi\circ g(B)$ is parallel to $\mathbb{R}^2_b\times\{0\}$,
the phenomenon in the right column of Figure \ref{beefsteak} does not occur. \\
\betaegin{figure}
\iotancludegraphics[width=120mm]{beef.pdf}
\caption{{\betaf
While the middle part of two sheets approaches by
a special
isotopy of $g$,
the intersection $X$ in Lemma \ref{sukoshi} changes.
}\ellongleftarrowbel{beefsteak}}
\epsilonnd{figure}
\betaigbreak
\betaegin{cla}\ellongleftarrowbel{tako}
Let $B^3$ be a closed $($respectively, open$)$ 3-ball embedded in $R^2_b\times[0,1]$.
Take any orientation preserving isotopy of
diffeomorphism of $B^3$ fixing $\partial B^3$ from the identity map.
We can give a coordinate $(x,y,t)$ to $p\iotan B^3\subset\mathbb{R}^2_b\times[0,1]$.
Suppose that this isotopy carries $p$ to a point whose coordinate is $(x,y,t')$,
where $t'\Z[\pi/\pi^{(n)}]eq t$ or $t'=t$ holds.
Use this isotopy and make a homotopy of $\underline{g}.$
Suppose that this homotopy is a special
homotopy of $\underline{g}$.
Then this homotopy of $\underline{g}$ can be covered by a
special
isotopy of $g$.
\epsilonnd{cla}
By Claim \ref{sukoshi},
we can let the interior of all classical segments exist below (respectively, over) the branch point
with respect to the height
as drawn in Figure \ref{todome}.
By the first, second and third steps,
the singular point set of Im $\underline{g}$
is a finite simplicial complex.
Hence we have the following.
\betaegin{cla}\ellongleftarrowbel{koma}
There are only finite number of $t\iotan[0,1]$ with the following properties:
There is no real number $\varepsilon$ such that
the map $\underline{g}|_{S^1_b\times[t-\varepsilon, t+\varepsilon]}$
is a product map out $B$.
\epsilonnd{cla}
By Claims \ref{tako} and \ref{koma}, we have Claim \ref{tantei}.
\qed\\
\betaegin{cla}\ellongleftarrowbel{kinako}
$\omega$ defines the trivial knot.
Hence we obtain $\alphalpha_v$ from $\alphalpha_u$
by using only classical Reidemeister moves.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{kinako}.}
By the map $g|_{S^1_b\times[u,v]\times S^1_f}$,
$\alphalpha_u$ and $\alphalpha_v$ are fiberwise equivalent.
Therefore
the submanifolds, $\mathcal S(\alphalpha_u)$ and $\mathcal S(\alphalpha_v)$, of $\mathbb{R}^4$
are isotopic.
Hence
$\pi_1(\mathbb{R}^4-\mathcal S(\alphalpha_u))\\\cong
\pi_1(\mathbb{R}^4-\mathcal S(\alphalpha_v))$.
Hence
the group of $\alphalpha_u$ and that of $\alphalpha_v$ are isomorphic.
Since $\alphalpha_u\\=\alphalpha_v\#\omega$,
the group of $\alphalpha_u$ is
the free product of that of $\alphalpha_v$ and that of $\omega$.
Hence
the group of $\omega$ is $\mathbb{Z}$.
Since $\omega$ defines a classical 1-knot,
$\omega$ defines the trivial 1-knot.
Since $\omega$ is a classical 1-knot diagram and represents the trivial 1-knot,
$\omega$ is changed into the trivial 1-knot diagram by using only classical Reidemeister moves.
Hence Claim \ref{kinako} holds.
\qed
\\
It is easy to prove that
if two virtual 1-knot diagrams are obtained each other
by using only classical Reidemeister moves,
they are fiberwise equivalent.
Therefore,
we change $\underline{g}|_{S^1_b\times[u,v]}$ in $B$
so that we let
the map $\underline{g}|_{S^1_b\times[u,v]}$ be a level preserving,
generic map.
Hence the following holds: If $\underline{g}(S^1_b\times[u,v])$ includes a branch point,
it is the classical Whitney branch point.
These classical Whitney branch points appear
when we carry out classical Reidemeister $I$ move
while we change $\alphalpha_u$ into $\alphalpha_v$.
After repeating this procedure,
all branch points of Im $\underline{g}$ are classical Whitney branch points.
This completes the proof of Claim \ref{mochi}. \qed\\
This completes the proof of Claim \ref{kamen}. \qed\\
This completes the proof of Theorem \ref{Wyoming}. \qed \\
Claim \ref{kinako} implies the following Proposition \ref{amakara}.
\betaegin{defn}\ellongleftarrowbel{shuza}
Virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are said to be
{\iotat strongly fiberwise equivalent}
if $\alphalpha$ and $\betaeta$ satisfy the conditions
which are made by replacing the phrase
`level preserving generic map' with
`level preserving transverse immersion'
without changing other parts in Definition \ref{gene}.
\epsilonnd{defn}
\betaegin{pr}\ellongleftarrowbel{amakara}
If virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are fiberwise equivalent,
there is a sequence of virtual 1-knot diagrams,
$\alphalpha=\alphalpha_1,\betaeta_1,\alphalpha_2,\betaeta_2,...,
\alphalpha_{k-1},\betaeta_{k-1},\alphalpha_k,\betaeta_k=\betaeta$
$(k\iotan\mathbb{N})$,
such that
$\alphalpha_i$ and $\betaeta_i$
are strongly fiberwise equivalent
$(1\elleqq i\elleqq k)$
and such that
$\betaeta_i$ and $\alphalpha_{i+1}$ $(1\elleqq i\elleqq k-1)$
are classical move equivalent $($and therefore rotational welded equivalent$)$.
\epsilonnd{pr}
We prove the following theorem.
\betaegin{thm}\ellongleftarrowbel{ike}
If $g$ satisfies Definition $\ref{shuza}$,
then the following hold.
Let $\mathcal S$ be the singular point set of
$(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)$ in Definition $\ref{shuza}.$
Note that $\mathcal S$ is a finite 1-dimensional simplicial complex.
\smallbreak\Z[\pi/\pi^{(n)}]oindent{\rm (i)}
$\mathcal S\cap(\mathbb{R}^2_b\times\{0$ $($respectively, $1)\})$ is
a set of virtual and classical crossing points of $\alphalpha$ $($respectively, $\betaeta),$
and therefore is a set of double points.
It consists of 0-simplices.
Only one 1-simplex
is attached to each of these 0-simplices.
These 1-simplices
meet \\
$\mathbb{R}^2_b\times\{0$ $($respectively, $1)\}$ transversely.
\smallbreak\Z[\pi/\pi^{(n)}]oindent{\rm (ii)}
Triple points are 0-simplices.
$($Recall
Notes $\ref{umeboshi}$ and $\ref{faso}$, Definnition $\ref{JW}.)$
\smallbreak\Z[\pi/\pi^{(n)}]oindent{\rm (iii)}
The restriction of
`the height function $\mathfrak h:\mathbb{R}^2_b\times[0,1]\to[0,1]$'
to the interior of any 1-simplex
in $\mathcal S$
has no critical point.
$($Hence we have the following:
For each $\zeta\iotan[0,1]$, $\mathcal S\cap(\mathbb{R}^2_b\times\{\zeta\})$ is a finite number of points.
No 1-simplex is parallel to $\mathbb{R}^2_b\times\{0\}.)$
\smallbreak\Z[\pi/\pi^{(n)}]oindent{\rm (iv)}
Let $\zeta\iotan(0,1).$
$\mathcal S\cap(\mathbb{R}^2_b\times\{\zeta\})$ includes no or only one 0- simplex.
\smallbreak\Z[\pi/\pi^{(n)}]oindent{\rm (v)}
In $\mathbb{R}^2_b\times(0,1)$, 0-simplices
appear only in the two
cases of Figure $\ref{zero}$.
\betaegin{figure}
\vskip-30mm
\iotancludegraphics[width=135mm]{zero.pdf}
\vskip-30mm
\caption{{\betaf 0-simplices in $\mathcal S$ }}
\ellongleftarrowbel{zero}
\epsilonnd{figure}
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Theorem \ref{ike}.}
Theorem \ref{ike}.(i)
follows from Definition \ref{shuza}.(5) for any simplicial complex structure.
Theorem \ref{ike}.(ii)
holds for any simplicial complex structure
by the definition of simplicial complex structure.
\\
Proof of Theorem \ref{ike}.(iii).
Suppose that there is an $X$
as is an example explained in Claim \ref{sukoshi} and Figure \ref{beefsteak}.
Repeating this procedure
we can take a simplicial complex structure
such that
(any 1-simplex)$\cap(\mathbb{R}^2_b\times\{t\})$
for any $t\iotan[0,1]$ is a finite number of points.
Therefore
the restriction of $\mathfrak h$ to the interior of any 1-simplex of this simplicial complex structure
has a finite number of critical points.
Make a new simplicial complex structure
so that the critical points are new 0-simplicies
so that we keep the condition of Theorems \ref{ike}.(i) and (ii).
Suppose that
there is a 0-simplex
$e^0$ to which only two 1-simplices
$e^1_1$ and $e^1_2$, attach,
and that
$e^0$ is not a critical point of the restriction of $\mathfrak h$ to
(Int$e^1_1)\cup e^0\cup$(Int$e^1_2$)
=Int($e^1_1\cup e^0\cup e^1_2$).
Make a new simplicial complex structure
such that
$e^1_1\cup e^0\cup e^1_2$
is changed into a new 1-simplex
without changing other simplicial complex structure.
This completes the proof of Theorem \ref{ike}.(iii).
\\
Theorem \ref{ike}.(iv)
holds because, by Claims \ref{koma} and \ref{tako},
we can change the height of any 0-simplex
so that we keep the condition of Theorems \ref{ike}.(i)-(iii).
\\
Proof of Theorem \ref{ike}.(v).
There are only two
cases:
(P) Only two 1-simplices attach to a 0-simplex.
(Q) Only six 1-simplices attach to a 0-simplex.
Note that, by Theorem \ref{ike}.(iii), each 1-simplex is attached to two different 0-simplices.
In the case (P),
by Theorem \ref{ike}.(iii),
the 0-simplex
exists as drawn in Type P of Figure \ref{zero}.
In the case (Q),
as drawn in Figure \ref{beefsteak} associated with Claim \ref{sukoshi},
we can move 1-simplex
so that we have the condition as drawn in Type Q of Figure \ref{zero}, and
so that we keep the condition of Theorems \ref{ike}.(i)-(iv).
See Figure \ref{tsuika2} for an example of this move.
\betaegin{figure}
\iotancludegraphics[width=150mm]{tsuika2.pdf}
\vskip-40mm
\caption{{\betaf
The singularity in the upper figure is made into the one in the lower which consists of one Type Q and three Type P.}}
\ellongleftarrowbel{tsuika2} \epsilonnd{figure}
This completes
the proof of Theorem \ref{ike}.(v).\\
This completes the proof of Theorem \ref{ike}.
\qed
\\
We have the following theorem.
\betaegin{thm}\ellongleftarrowbel{fwrw}
Two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are PL fiberwise equivalent if and only if
$\alphalpha$ and $\betaeta$ are PL rotational welded equivalent.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent
{\betaf Proof of Theorem \ref{fwrw}.}
The `if' part is easy.
We prove the `only if' part.
By Proposition \ref{amakara}, it suffices to prove Claim \ref{takusan}
\betaegin{cla}\ellongleftarrowbel{takusan}
Two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are
PL strongly fiberwise equivalent only if
$\alphalpha$ and $\betaeta$ are PL rotational welded equivalent.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{takusan}.}
Let $C_\zeta=\underline{g}(S^1_b\times\{\zeta\})=$
$\underline{g}(S^1_b\times[0,1])
\cap(\mathbb{R}^2_b\times\{\zeta\})\\=$
$(\pi\circ g(S^1_b\times[0,1]\times S^1_f))
\cap(\mathbb{R}^2_b\times\{\zeta\})$.
By Theorem \ref{ike},
$C_\zeta$ is
an immersed circle in $\mathbb{R}^2_b\times\{\zeta\}$ and
its singular point set is a finite number of points.
$C_\zeta$ changes from $\alphalpha$ to $\betaeta$ step by step
as $\zeta$ runs from 0 to 1.
If, for a $\zeta_p$, $C_{\zeta_p}$ includes a 0-simplex
of the simplicial complex structure
in Theorem \ref{ike}.
A classical or virtual Reidemeister move is done there.
We do any of them only there.
If $\zeta_q<\zeta<\zeta_r$,
$C_\zeta$ includes no 0-simplex.
Then $C_\zeta$ is not changed while $\zeta$ runs from $\zeta_q$ to $\zeta_r$.
We investigate how $C_\zeta$ changes in detail.
Near a 0-simplex
in $\mathbb{R}^2_b\times[0,1]$,
Im $\underline{g}$ is drawn as in Figure \ref{tsuika}
since $\underline{g}$ is a transverse immersion.
Here, note that we can move $\mathcal S$ by using
a special
isotopy of $g$. \\
\betaegin{figure}
\iotancludegraphics[width=110mm]{tsuika.pdf}
\vskip-10mm
\caption{{\betaf How sheets intersect near Types P and Q.}}
\ellongleftarrowbel{tsuika} \epsilonnd{figure}
Therefore
we have only the following two facts on $\mathcal S$ and local moves on the knot diagrams.
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent (i)
Let $\sigma, \tau\iotan[0,1]$.
Suppose that $\mathcal S\cap(\mathbb{R}^2_b\times(\sigma,\tau))$ includes no 0-simplex.
It holds that $\underline{g}|_{S^1_b\times[\sigma,\tau]}$ is a product map.
Then $(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)\cap(\mathbb{R}^2_b\times\{\sigma\})$ can be obtained from $(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)\cap(\mathbb{R}^2_b\times\{\tau\})$ by an isotopy of $\mathbb{R}^2_b$.
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent (ii)
Let $\timesi\iotan[0,1].$
Suppose that $\mathcal S\cap(\mathbb{R}^2_b\times\{\timesi\})$ includes only one 0-simplex.
Suppose that
$\mathcal S\cap(\mathbb{R}^2_b\times(\timesi, \timesi+\varepsilon])$
(respectively,
$\mathcal S\cap(\mathbb{R}^2_b\times[\timesi-\varepsilon, \timesi))$)
includes no 0-simplex.
Let
$D=(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)\cap(\mathbb{R}^2_b\times\{\timesi-\varepsilon'\})$
and
$U\\
=(\pi\circ g)(S^1_b\times[0,1]\times S^1_f)\cap(\mathbb{R}^2_b\times\{\timesi+\varepsilon'\})$.
If the 0-simplex
is put in Type P or Q,
then $U$ is obtained from $D$ by
one welded move other than a virtual Reidemeister $I$ move.
(Note.
Type P causes classical and virtual Reidemeister $II$ moves.
Type Q causes classical and virtual Reidemeister $III$ moves.
Four types of triple points correspond to four types of classical and virtual Reidemeister $III$ moves.)
Therefore
$\alphalpha$ is changed into $\betaeta$ by welded moves other than the virtual Reidemeister $I$ move.
Hence $\alphalpha$ is rotational welded equivalent to $\betaeta$.
This completes the proof of Claim \ref{takusan}. \qed\\
This completes the proof of Theorem \ref{fwrw}.
\qed\\
We will complete the proof of Theorem \ref{smooth} and go back to the smooth category.
We said that the `if' part of Theorem \ref{smooth} is easy.
We will prove the `only if' part of Theorem \ref{smooth} by using the following lemma.
\betaegin{lem}\ellongleftarrowbel{PLtosmooth}
Let $\alphalpha$ and $\betaeta$ be smooth virtual 1-knot diagrams.
Let $\alphalpha'$ $($respectively, $\betaeta')$ be a PL virtual 1-knot diagram
which is piecewise smooth, planar, ambient isotopic to $\alphalpha$ $($respectively, $\betaeta)$.
Then we have the following.
$\alphalpha$ and $\betaeta$ are smooth rotational welded equivalent if and only if
$\alphalpha'$ and $\betaeta'$ are PL rotational welded equivalent.
\epsilonnd{lem}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Lemma \ref{PLtosmooth}.}
Let $\timesi$ and $\zeta$ be smooth virtual 1-knot diagrams.
If $\timesi$ and $\zeta$ are PL, planar, ambient isotopic to a PL virtual 1-knot diagram $\gamma$,
then $\timesi$ is smooth, planar, ambient isotopic to $\zeta$.
{\iotat Reason.} Smoothen the corner of $\gamma$.
Each of PL rotational welded Reidemeister moves is regarded as smooth rotational welded Reidemeister move.
This completes the proof of Lemma \ref{PLtosmooth}.
\qed\\
Assume that two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are smooth fiberwise equivalent.
By Claim \ref{xbeef}, they are PL fiberwise equivalent.
By Theorem \ref{fwrw}, they are PL rotational welded equivalent.
By Lemma \ref{PLtosmooth}, they are smooth rotational welded equivalent.
Therefore the `only if' part of Theorem \ref{smooth} is true.
This completes the proof of Theorem \ref{smooth}. \qed\\
We are now back to the smooth category.
\\
\Z[\pi/\pi^{(n)}]oindent {\betaf Note.}
Figure \ref{dia1} explains Figure \ref{tsuika2} in more detail.
\betaegin{figure} \vskip-10mm \iotancludegraphics[width=120mm]{dia1.pdf}
\caption{{\betaf
This pair of the left figure and the right one is an example of a pair of the figures of Figure \ref{tsuika2}. The left (respectively, right) figure is an example of a sequence of diagrams associated with the upper (respectively, lower) figure of Figure \ref{tsuika2}. The left sequence is perturbed and is made into the right sequence.These diagrams are drawn without information of virtual multiple points and classical ones.
}}\ellongleftarrowbel{dia1} \epsilonnd{figure}
\betaegin{note}\ellongleftarrowbel{xudon}
In \cite{Rourke}, the fiberwise equivalence is defined by the following definition.
We call the equivalence relation the {\iotat $f$-fiberwise equivalence} in this paper.
Note that we work in the smooth category.
\betaegin{defn}\ellongleftarrowbel{yugentsuki}
Add the following condition to Definition \ref{Nevada} without changing the other parts.
We call the equivalence relation the {\iotat $f$-fiberwise equivalence}.
Note that we work in the smooth category.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(\ref{yugentsuki}.1)$ In each fiber $\mathbb{R}^2_f$, there are a finite number of circles.
$($That is, $<\iotanfty.)$
\epsilonnd{defn}
We said that it is easy to prove the following (i). It is also easy to prove the \\Bbbkollowing (ii).
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(i) If virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are rotational welded equivalent,
then $\alphalpha$ and $\betaeta$ are fiberwise equivalent.
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(ii) If virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are rotational welded equivalent,
then $\alphalpha$ and $\betaeta$ are $f$-fiberwise equivalent.
\betaigbreak
Theorem \ref{smooth} and the above (ii) imply the following (iii).
\smallbreak\Z[\pi/\pi^{(n)}]oindent
(iii) If virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are fiberwise equivalent,
then $\alphalpha$ and $\betaeta$ are $f$-fiberwise equivalent.
(Note that we work in the smooth category.)
\\
The converse of (iii) is trivial. Hence we have the following:
Virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are fiberwise equivalent
if and only if $\alphalpha$ and $\betaeta$ are $f$-fiberwise equivalent
\epsilonnd{note}
\betaigbreak
\betaegin{note}\ellongleftarrowbel{xmikan}
Although Rourke claimed in \cite[Theorem 4.1]{Rourke} that
two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are
fiberwise equivalent if and only if $\alphalpha$ and $\betaeta$ are welded equivalent
in the PL (respectively, smooth) category,
we state that this claim is wrong, as we mentioned it
in the last few paragraphs of \S\ref{i3}.
The reason for the wrongness is Theorems \ref{smooth} and \ref{fwrw} and Claim \ref{panda}.
\epsilonnd{note}
We introduce a new equivalence relation of the set of virtual 1-knot diagrams.
\betaegin{defn}\ellongleftarrowbel{parity}
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams.
We say that
$\alphalpha$ and $\betaeta$ are {\iotat virtually parity equivalent}
if $\alphalpha$ and $\betaeta$ have the same parity of virtual crossing points.
\epsilonnd{defn}
We prove several results associated with the virtual parity.
\betaegin{cla}\ellongleftarrowbel{hirumeshi}
If two virtual 1-knot diagrams
$\alphalpha$ and $\betaeta$ are
rotational welded equivalent,
then $\alphalpha$ and $\betaeta$ are virtually parity equivalent.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{hirumeshi}.}
We can obtain $\alphalpha$ from $\betaeta$
by some welded-moves
other than virtual Reidemeister $I$ move.
\qed
\betaegin{cla}\ellongleftarrowbel{panda}
The welded equivalence does not imply the rotational welded equivalence.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Note.}
By their definitions, the rotational welded equivalence implies the welded equivalence.
\\
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{panda}.}
Call the virtual 1-knot diagram in Figure \ref{New Hampshire},
the {\iotat virtual figure $\iotanfty$ knot diagram}.
\betaegin{figure}
\iotancludegraphics[width=80mm]{x5.pdf}
\vskip-30mm
\caption{{\betaf The virtual figure $\iotanfty$ knot diagram}\ellongleftarrowbel{New Hampshire}}
\epsilonnd{figure}
The virtual figure $\iotanfty$ knot diagram
and the trivial 1-knot diagram
are welded equivalent by the definition
but are not rotational welded equivalent
by Claim \ref{hirumeshi}.
This completes the proof of Claim \ref{panda}.
\qed\\
\vskip10mm
\subsection{Related topics}\ellongleftarrowbel{sub2}\hskip10mm\\%
Theorem \ref{Montgomery} is one of our main results.
\betaegin{thm}\ellongleftarrowbel{Montgomery}
If two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$ are fiberwise equivalent,
then $\alphalpha$ and $\betaeta$ are virtually parity equivalent.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{Montgomery}.}
Theorem \ref{smooth}
and Claim \ref{hirumeshi} imply Theorem \ref{Montgomery}. \qed\\
It is known that
the usual trefoil knot diagram is not welded equivalent
to the trivial knot diagram
(see
\cite{Kauffman, Kauffmanrw, Rourke, Satoh, J}).
Hence these two diagrams are not also rotational welded equivalent.
Hence we have the following.
\betaegin{cla}\ellongleftarrowbel{ice}
The converse of Theorem $\ref{Montgomery}$ is not true in general.
\epsilonnd{cla}
We have the following.
\betaegin{cla}\ellongleftarrowbel{cream}
The number of virtual crossing points of virtual 1-knot diagrams is not
an invariant of the fiberwise equivalence
$($respectively, the rotational welded equivalence$)$
in general.
\epsilonnd{cla}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Claim \ref{cream}.}
The two virtual knot diagrams in Figrue \ref{oldWestVirginia}
are fiberwise equivalent (respectively, rotational welded equivalent). \qed\\
We introduce a `weaker' equivalence relation than the fiberwise equivalence defined by
Definition \ref{Nevada}.
We want to replace `level preserving embedding of $S^1_b\times[0,1]$' in
Definition \ref{Nevada}
with an oriented compact surface
which is not necessarily
`level preserving embedding of $S^1_b\times[0,1]$',
and loose a few conditions there.
We prove
in Theorem \ref{parityhozon}
that
this equivalence relation is equivalent to
the virtual parity equivalence relation.
\betaegin{figure}
\iotancludegraphics[width=70mm]{oldWestVirginia.pdf}
\caption{{\betaf
Two virtual knot diagrams which are rotational welded equivalent
(respectively, fiberwise equivalent).
}}\ellongleftarrowbel{oldWestVirginia}
\epsilonnd{figure}
\betaegin{defn}\ellongleftarrowbel{sigma}
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams.
We say that
$\alphalpha$ and $\betaeta$ are {\iotat weakly fiberwise equivalent}
if $\alphalpha$ and $\betaeta$ satisfy
the following conditions.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(1)$
There is
a compact generic oriented surface $F$ with boundary whose boudary is a disjoint union of two circles, which is contained in $\mathbb{R}^2_b\times[0,1]$, and
$F$ is covered by Rourke's fibration.
Note that thus there is a submanifold of $\mathbb{R}^2_b\times[0,1]\times\mathbb{R}^2_f$ which is diffeomorphic
to $F\times S^1$.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(2)$
The in-out information of fiber circles gives $\alphalpha$ and $\betaeta$
the information whether each multiple (respectively, branch) point is virtual or classical
as in Theorem \ref{Montana}.
\smallbreak\Z[\pi/\pi^{(n)}]oindent$(3)$
$\partial F$ is $\alphalpha$ and $\betaeta$.
$F$ meets $\mathbb{R}^2_b\times\{0\}$ $($respectively, $\mathbb{R}^2_b\times\{1\})$
at $\alphalpha$
$($respectively, $\betaeta)$
transversely.
\smallbreak
If $F$ above is an annulus,
we say that
$\alphalpha$ and $\betaeta$ are {\iotat fiberwise cobordant}.
\epsilonnd{defn}
\betaegin{thm}\ellongleftarrowbel{parityhozon}
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams.
$\alphalpha$ and $\betaeta$ are weakly fiberwise equivalent
if and only if
$\alphalpha$ and $\betaeta$ are virtually parity equivalent.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{parityhozon}.}
The `only if' part:
We use `reductio ad absurdum'.
We suppose an assumption:
$\alphalpha$ and $\betaeta$ are not virtually parity equivalent.
Take a generic surface which connects $\alphalpha$ and $\betaeta$
as in Definition \ref{sigma}.
Then this generic surface must have at least one
virtual branch point because
the union of
$\alphalpha$ and $\betaeta$
has
an odd number of
virtual crossing point.
By Theorem \ref{Rmuri}, this generic surface never exists.
(See Figure \ref{Texas}.)
We arrived at a contradiction.
Hence the above assumption is false and the `only if' part is true.
\\
\betaegin{figure}
\iotancludegraphics[width=80mm]{x5yx.pdf}
\caption{{\betaf Let $\alphalpha$ be the virtual figure $\iotanfty$ knot diagram and $\betaeta$,
the trivial 1-knot diagram.
For example, $(\pi\circ g)(S_b^1\times[0,1]\times S^1_f)$ cannot be realized as drawn above,
by Theorem \ref{Rmuri}.
}}\ellongleftarrowbel{Texas}
\epsilonnd{figure}
The `if' part:
It suffices to prove that
$\alphalpha\alphamalg(-\betaeta)$ in $\mathbb{R}^2$,
where $\alphamalg$ denote the disjoint union of the diagrams,
is weakly fiberwise equivalent to the trivial 1-knot diagram.
We can attach bands
as drawn in Figure \ref{bands}
so that the orientations of virtual knot diagrams and those of the bands are compatible.
Thus $\alphalpha\alphamalg(-\betaeta)$ is
weakly fiberwise equivalent to
the disjoint union of
nonnegative even integer of copies of the virtual figure $\iotanfty$ knot and a classical link diagram.
We can attach a band to two copies of the virtual figure $\iotanfty$ knot diagram and
combine them as drawn in Figure \ref{WestVirginia},
so that the orientation of the band and those of the knot diagrams
are compatible,
and call the resultant diagram $\zeta$.
Thus $\alphalpha\alphamalg(-\betaeta)$in $\mathbb{R}^2$ is
weakly fiberwise equivalent to
the disjoint union of a finite number of copies of $\zeta$.
It is easy to prove that $\zeta$ is
rotational welded equivalent to the trivial knot.
Suppose that we obtain the $\mu$-component trivial 1-link diagram after that.
Attach $\mu-1$ copies of 2-disc to $(\mu-1)$ components of this trivial 1-link diagram.
Hence $\alphalpha\alphamalg(-\betaeta)$ is weakly fiberwise equivalent to the trivial 1-knot diagram.
Hence $\alphalpha$ and $\betaeta$ are weakly fiberwise equivalent.
This completes the proof of Theorem \ref{parityhozon}. \qed
\betaegin{figure}\betaigbreak
\iotancludegraphics[width=140mm]{bands.pdf}
\vskip-70mm
\caption{{\betaf Attaching bands.}\ellongleftarrowbel{bands}}
\epsilonnd{figure}
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=72mm]{WestVirginia.pdf}
\smallbreak
\caption{{\betaf
A combination of two copies of the virtual figure $\iotanfty$ knot diagram
}}\ellongleftarrowbel{WestVirginia}
\betaigbreak
\epsilonnd{figure}
\betaegin{defn}\ellongleftarrowbel{nono}
We define
the `{\iotat nonorientably weakly fiberwise equivalence}'.
In Definition \ref{sigma}
replace `oriented surface' with `non-orientable surface',
and
replace `weakly fiberwise equivalent' with
`nonorientably weakly fiberwise equivalent'.
\epsilonnd{defn}
\betaegin{thm}\ellongleftarrowbel{yuru}
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams.
$\alphalpha$ and $\betaeta$ are nonorientably weakly fiberwise equivalent
if and only if
$\alphalpha$ and $\betaeta$ are virtually parity equivalent.
\epsilonnd{thm}
\Z[\pi/\pi^{(n)}]oindent{\betaf Proof of Theorem \ref{yuru}.}
The `if' part:
By Theorem \ref{parityhozon},
$\alphalpha$ and $\betaeta$ are weakly fiberwise equivalent.
It is trivial that
if $\alphalpha$ and $\betaeta$ are weakly fiberwise equivalent,
then $\alphalpha$ and $\betaeta$ are nonorientably weakly fiberwise equivalent.
{\iotat Reason.}
Take a generic oriented surface
for $\alphalpha$ and $\betaeta$
as in Definition \ref{sigma}.
Take an immersed Klein bottle in $\mathbb{R}^2_b\times[0,1]$.
Connect
the generic oriented surface
and
the immersed Klein bottle
by using an embedded 3-dimensional 1-handle in $\mathbb{R}^3$
such that
the intersection of the 1-handle and
the oriented surface (respectively, immersed Klein bottle)
is only the attaching part of the 1-handle.
The resultant generic nonorientable surface implies that
$\alphalpha$ and $\betaeta$ are nonorientably weakly fiberwise equivalent.
The proof of the `only if' part is
the same as that of the `only if' part of Theorem \ref{parityhozon}
if we replace the words `Definition \ref{sigma}'
with `Definition \ref{nono}',
and remove the sentence `(See Figure \ref{Texas}.)'.
\qed
\betaigbreak
Define {\iotat Whitney degree} of any virtual 1-knot diagram $\alphalpha$ to be
Whitney degree which is defined by $\alphalpha$
when we regard $\alphalpha$ as an immersed oriented circle in $\mathbb{R}^2$.
Two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$
are said to be {\iotat Whitney parity equivalent}
if the parity of Whitney degree of $\alphalpha$ is the same as that of $\betaeta$.
Two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$
are said to be {\iotat classically parity equivalent}
if the parity of the classical crossing points of $\alphalpha$ is the same as that of $\betaeta$.
The following holds.
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams which are rotational welded equivalent (respectively, fiberwise equivalent). (Note Theorem \ref{smooth}.)
Then
$\alphalpha$ and $\betaeta$
are classically parity equivalent
if and only if
$\alphalpha$ and $\betaeta$
are Whitney parity equivalent. \\
\Z[\pi/\pi^{(n)}]oindent{\iotat Reason.}
$\alphalpha$ and $\betaeta$
are Whitney parity equivalent
if and only if
the number of the classical Reidemeister $I$ moves is even
in a sequence of rotational welded moves which $\alphalpha$ is changed into $\betaeta$.
Note that we cannot use the virtual Reidemeister $I$ move by definition.
\betaigbreak
Some readers may ask the following question:
Suppose that
two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$
do not have any classical crossing point
and that Whitney degrees are different.
Then is it valid that
$\alphalpha$ and $\betaeta$ are not rotational welded equivalent?
The answer is negative.
We show a counter example in Figure \ref{Whitneydegree}.(i) (respectively, \ref{Whitneydegree}.(ii)).
The proof that each pair is rotational welded equivalent is left to the reader.
\\
Two virtual 1-knot diagrams $\alphalpha$ and $\betaeta$
are said to be {\iotat mixed parity equivalent}
if the parity of
the sum of the classical and virtual crossing points of $\alphalpha$ is
the same as
that of $\betaeta$.
The following holds.
Let $\alphalpha$ and $\betaeta$ be virtual 1-knot diagrams which are welded equivalent. Then
$\alphalpha$ and $\betaeta$
are mixed parity equivalent
if and only if
$\alphalpha$ and $\betaeta$
are Whitney parity equivalent. \\
\Z[\pi/\pi^{(n)}]oindent{\iotat Reason.}
$\alphalpha$ and $\betaeta$
are Whitney parity equivalent
if and only if
the sum of the number of
the classical and virtual Reidemeister $I$ moves
is even
in a sequence of welded moves which $\alphalpha$ is changed into $\betaeta$.
\betaegin{figure}
\betaigbreak
\iotancludegraphics[width=120mm]{Whitneydegree.pdf}
\smallbreak
\caption{{\betaf
Two pairs of virtual knot diagrams
}}\ellongleftarrowbel{Whitneydegree}
\betaigbreak
\epsilonnd{figure}
\betaigbreak
\section{Virtual high dimensional knots}\ellongleftarrowbel{vhigh}
\Z[\pi/\pi^{(n)}]oindent
See \cite{KauffmanOgasa, KauffmanOgasaII, KauffmanOgasaB, LevineOrr, Ogasa}
for codimension two high dimensional knots.
See \cite{Haefliger, Haefliger2, Levine} for high codimensional high dimensional knots.
It is natural to attempt to define virtual high dimensional knots and
their one-dimensional-higher tubes.
We could define $n$-dimensional virtual knots by using virtual $n$-knot diagrams in $\mathbb{R}^{n+1}$.
We would make any virtual $n$-knot
into a submanifold
of
(a closed oriented $n$-manifold $M$) $\times[0,1]$
as we do in the virtual 1- and 2-dimensional cases.
We want to make a bijection between
the set of such submanifolds and that of virtual $n$-knots.
The 1-dimensional case is done (see Theorem \ref{vk} and Definition \ref{Jbase}).
We should define a one-dimensional-higher tube
as the spinning submanifold made from $K$ around $M$.
Satoh's method makes no sense in the dimension greater than one.
Rourke's way also makes non-sense by Theorem \ref{Rmuri}.
Furthermore we must note
that
the $n$-dimensional case ($n\iotan\mathbb{N}-\{1,2\}$) of Theorems \ref{oh} and \ref{ohoh}
does not necessarily hold
in smooth category (respectively, PL category)
because
it is not trivial to produce an analogue of their proof
by the following fact of \cite{Hudson}:
There is an integer $p\geqq3$ and
are two smooth (respectively, PL) $a$-dimensional submanifolds, $X$ and $Y$, of $S^{a+p}$ which are diffeoomorphic (respectively, PL homeomorphic) each other
but which are non-isotopic
as smooth submanifolds (respectively, PL submanifolds).
To complete these topics in this section is left to the readers as problems.
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{\Bbbkootnotesize \Z[\pi/\pi^{(n)}]oindent{\betaf Acknowledgment.} Kauffman's work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University
(contract no.14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).}
\betaigbreak
\Z[\pi/\pi^{(n)}]oindent
Louis H. Kauffman:
Department of Mathematics, Statistics and Computer Science \\ 851 South Morgan Street University of Illinois at Chicago
Chicago, Illinois 60607-7045, and\\ Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia\quad [email protected]
\betaigbreak\Z[\pi/\pi^{(n)}]oindent
Eiji Ogasa: Computer Science, Meijigakuin University, Yokohama, Kanagawa, 244-8539, Japan
\quad [email protected] \quad
[email protected]
\betaigbreak\Z[\pi/\pi^{(n)}]oindent
Jonathan Schneider:
Department of Mathematics, College of DuPage,
425 Fawell Boulevard, Glen Ellyn, Illinois, 60137, USA \quad
[email protected]
\epsilonnd{document}
|
\begin{document}
\title{\sc\bf\large\MakeUppercase{Dirichlet approximation of equilibrium distributions in Cannings models with mutation}}
\author{\sc Han~L.~Gan, Adrian R\"ollin, and Nathan Ross}
\deltaate{\it Washington University in St.\ Louis, National University of Singapore, and University of Melbourne}
\maketitle
\begin{abstract}
Consider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general.
We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite population stationary distribution for the Wright-Fisher genealogy with general mutation structure and the Cannings exchangeable genealogy with parent independent mutation structure. In the first case, the bound is small if the population is large and the mutations do not depend too much on parent type; ``too much" is naturally quantified by our bound. In the second case, the bound is small if the population is large and the chance of three-mergers in the Cannings genealogy is small relative to the chance of two-mergers; this is the same condition to ensure convergence of the genealogy to Kingman's coalescent. These results follow from a new development of Stein's method for the Dirichlet distribution based on Barbour's generator approach and a probabilistic description of the semigroup of the Wright-Fisher diffusion due to Griffiths, and Li and Tavar\'e.
\epsnd{abstract}
\section{Introduction}
We consider a neutral Cannings model with mutation in a haploid population of constant size~$N$ with~$K$ alleles. In each generation every individual has a random number of offspring such that the total number of offspring is~$N$. Different generations have i.i.d.\ offspring count vectors with distribution given by an exchangeable vector~$\tV:=(V_1, \ldots, V_N)$ not identically equal to~$(1, \ldots, 1)$;~$V_i$ is the number of offspring of individual~$1\leq i \leq N$. The random genealogy induced from this description is referred to as the \epsmph{Cannings model} \citep*{Cannings1974}; particular instances are the \epsmph{Wright-Fisher model}, where~$\mathscr{L}(\tV)$ is multinomial with~$N$ trials and probabilities~$(1/N, \ldots, 1/N)$, and the \epsmph{Moran model}, where~$\mathscr{L}(\tV)$ is described by choosing a uniform pair of indices~$(I,J)$ and setting~$V_I=2$,~$V_J=0$ and~$V_i=1$ for~$i\not=I, J$. On top of the random genealogy given by~$\mathscr{L}(\tV)$, we put a mutation structure as follows. Given a child's parent type is~$i$, the child is of type~$j$ with probability~$p_{i j}$, where~$\sum_{j} p_{i j} = 1$. The type of each child in a given generation is chosen independently conditional on the genealogy of that generation and the parent's type. It is easy to see that this rule induces a time-homogeneous Markov chain~$(\tX(0), \tX(1), \ldots)$ with state space~$\{\tx\in\mathrm{I}Z_{\geq0}^{K-1}: \sum_{i=1}^{K-1} x_i\leq N\},$ where for~$i=1,\ldots, K-1$,~$X_i(n)$ is the number of individuals in the population having allele~$i$ at time~$n$; note that the count for allele~$K$ is given by~$N-\sum_{i=1}^{K-1} X_i(n)$.
Since~$\tX(n)$ is a Markov chain on a finite state space, it has a stationary distribution. But it is typically not possible to write down an expression for such a stationary distribution --- an important exception is the Wright-Fisher model with parent independent mutation (PIM), meaning~$p_{i j}$ does not depend on~$i$ for~$j\not=i$. In general, if the population size~$N\to\infty$, then under some weak conditions \citep*{Mohle2000} (discussed in more detail below in Remark~\ref{rem1}) the Cannings genealogy viewed backwards in time converges to Kingman's coalescent \citep*{Kingman1982, Kingman1982b, Kingman1982a} and the mutation structure on top of the coalescent has a nice Poisson process description. But even in this limit the stationary distribution (of now proportions of the~$K$ alleles)
is notoriously difficult to handle outside of the PIM case; see \citep*{Griffiths1994} \citep*{Bhaskar2012} for work on sampling under the stationary distributions and \citep*{Ethier1992} for a probabilistic construction. Even if a formula in the limit were available, it is in any case important in population-mutation models to understand the difference between finite~$N$ likelihoods and those in the~$N\to\infty$ limit \citep*{Bhaskar2014} \citep*{Fu2006} \citep*{Lessard2007, Lessard2010} \citep*{Mohle2004}.
Our approach to understanding these finite population stationary distributions is to determine when they are close to the Dirichlet distribution, which arises as the stationary limit in the PIM case (in this case the process converges in a suitable sense to the Wright-Fisher diffusion).
In the next section we present our main results. First, we give two approximation theorems providing upper bounds on the distributional distance between the Dirichlet distribution and, for the first result, the finite population stationary distribution for the Wright-Fisher genealogy with general mutation structure and, for the second result, the Cannings exchangeable genealogy with parent independent mutation structure. Second, we discuss a new development of Stein's method for the Dirichlet distribution from which the first two results follow.
\section{Main results}
Before stating our main results, we need some notation and definitions, as well as a short discussion regarding Lipschitz functions defined on open convex sets and their extension to the boundary.
Denote by~$\Dir(\ta)$ the Dirichlet distribution with parameters~$\ta=(a_1,\deltaots,a_K)$, where~$a_1>0, \ldots, a_K>0$, supported on the~$(K-1)$-dimensional open simplex, which we parameterize as
\be{
\Delta_{K}=\left\{\tx=(x_1,\ldots, x_{K-1}): x_1> 0, \ldots, x_{K-1} > 0, \sum_{i=1}^{K-1} x_i < 1\right\}\subset \mathrm{I}R^{K-1}.
}
Denote by~$\-\Delta_K$ the closure of~$\Delta_K$.
On~$\Delta_{K}$,~$\Dir(\ta)$ has density
\ben{\label{1}
\psi_\ta(x_1,\deltaots,x_{K-1}) = \frac{\Gamma(s)}{\prod_{i=1}^K \Gamma(a_i)} \prod_{i=1}^K x_i^{a_i-1},
}
where~$s=\sum_{i=1}^K a_i$, and where we set~$x_K=1-\sum_{i=1}^{K-1} x_i$, as we shall often do in this paper whenever considering vectors taking values in~$\Delta_{K}$.
\deltaef\mathrm{BC}{\mathrm{BC}}
\deltaefC_{\mathrm{L}}{C_{\mathrm{L}}}
\deltaefC_{\mathrm{b}}{C_{\mathrm{b}}}
Let~$U$ be an open subset of~$\mathrm{I}R^n$. For~$m\geq 1$, we denote by~$\mathrm{BC}^{m,1}(U)$ the set of bounded functions~$g:U\to\mathrm{I}R$ that have~$m$ bounded and continuous partial derivatives and whose~$m$-th partial derivatives are Lipschitz continuous. In line with this notation, we denote by~$\mathrm{BC}^{0,1}(U)$ the set of bounded functions that are Lipschitz continuous.
We denote by~$\norm{g}_\infty$ the supremum norm of~$g$, and, if the~$k$-th partial derivatives of~$g$ exist, we let
\be{
\abs{g}_k=\sup_{1\leq i_1,\deltaots,i_k\leq n} \bbnorm{\frac{\partial^k g}{\partial x_{i_1}\cdots\partial x_{i_k}}}_\infty
}
and
\be{
\abs{g}_{k,1} =
\sup_{1\leq i_1,\deltaots,i_k\leq n}\sup_{\tx,\ty\in U}
\bbbabs{
\frac{\partial^k\bklr{g(\tx)-g(\ty)}}{\partial x_{i_1}\cdots\partial x_{i_k}}}\frac{1}{\norm{\tx-\ty}_1}.
}
Note that we use the~$L_1$-norm in our definition of the Lipschitz constant instead of the usual~$L_2$-norm. This is purely a matter of convenience, since the~$L_1$-norm shows up naturally in our proofs.
If~$g\in \mathrm{BC}^{m,1}(U)$ and~$U$ is convex, then all partial derivatives up to order~$m-1$ are Lipschitz continuous, too, and for any~$0\leq k \leq m-1$,
\ben{\label{2}
\abs{g}_{k,1} = \abs{g}_{k+1}.
}
As a result, if~$U$ is an open convex set, then any function~$g\in \mathrm{BC}^{m,1}(U)$ and all its partial derivatives up to order~$m$ can be extended continuously to a function~$\-g$ defined on the closure~$\-U$ in a unique way, and we have~$\norm{\-g}_\infty=\norm{g}_\infty$,~$\abs{\-g}_k=\abs{g}_k$ for~$1\leq k\leq m$ and~$\abs{\-g}_{m,1}=\abs{g}_{m,1}$. We can therefore identify the set of functions~$\mathrm{BC}^{m,1}(U)$ with set of extended functions~$\mathrm{BC}^{m,1}(\-U)$.
\subsection{Wright-Fisher model with general mutation structure}
Our first result is a bound on the approximation of the stationary distribution
of the Wright-Fisher model with general mutation structure by
a Dirichlet distribution.
\begin{theorem}\label{THM1}
Let the~$(K-1)$-dimensional vector\/~$\tX$ be distributed as a stationary distribution of the Wright-Fisher model for a population of~$N$ haploid individuals with~$K$ types and mutation
structure~$p_{i j}$,~$1\leq i,j\leq K$; set~$\textnormal{\textbf{W}}=\tX/N$. Let\/~$\ta$ be a~$K$-vector of positive numbers, set~$s=\sum_i a_i$, and let\/~$\tZ\sim \Dir(\ta)$. Then, for any~$h\in \mathrm{BC}^{2,1}(\-\Delta_K)$,
\be{
\left|\mathrm{I}E h(\textnormal{\textbf{W}})-\mathrm{I}E h(\tZ) \right| \leq \frac{\abs{h}_1}{s} A_1+\frac{\abs{h}_2}{2(s+1)} A_2 + \frac{\abs{h}_{2,1}}{18(s+2)} A_3,
}
where
\bg{
A_1 = 2N(K+1)\tau,\quad
A_2= N K^2 \mu^2 + 2K\mu, \quad
A_3= 8 N K^3 \mu^3 +\frac{16\sqrt{2}K^3}{N^{1/2}},
}
with
\ben{
\tau = \sum_{i=1}^K\sum_{\substack{j=1\\j\neq i}}^K\,\bbabs{\,p_{ij}-\frac{a_j}{2N}},\qquad \mu = \sum_{i=1}^K\sum_{\substack{j=1\\j\neq i}}^K p_{ij}.
}
Moreover, there is a constant~$C=C(\ta)$ such that
\be{
\sup_{A\in \mathcal{C}_{K-1}}\babs{\mathrm{I}P[\textnormal{\textbf{W}}\in A] - \mathrm{I}P[\tZ\in A]}\leq C\bklr{A_1+A_2+A_3}^{\theta/(3+\theta)},
}
where~$\mathcal{C}_{K-1}$ is the family of convex sets on\/~$\mathrm{I}R^{K-1}$ and where
$\theta=\theta(\ta)>0$ is given at~\epsqref{10}.
\epsnd{theorem}
\begin{remark}
To interpret the bounds of the theorem, if
$p_{i j}=\frac{a_j}{2N}+\varepsilon_{ij}$ for~$i\not=j$, and we assume~$\abs{\varepsilon_{ij}}\leq \varepsilon$, then
\ba{
\tau \leq K(K-1)\varepsilon
\qquad
\mu \leq \frac{(K-1)s}{2N}+K(K-1)\varepsilon,
}
so that for fixed~$K$ and~$\ta$ (though note that, for smooth functions, the reliance on these parameters is explicit),
\be{
A_1 = \mathrm{O}(N\varepsilon), \qquad A_2 = \mathrm{O}(N^{-1}+N \varepsilon^2), \qquad A_3 = \mathrm{O}(N^{-1/2}+N \varepsilon^3).
}
In particular, in the PIM case, where~$\varepsilon=0$, our bound on smooth functions is of order~$N^{-1/2}$, and for the convex set metric of order~$N^{-1/8}$ if~$\min\{a_1,\deltaots,a_K\}\geq 1$ and otherwise the order of the bound is some negative power of~$N$ having
a more complicated relationship to~$\ta$, but which is easily read from~\epsqref{10}.
In the special case where~$K=2$,~$\varepsilon=0$ and~$h$ has six bounded derivatives, \citep*{Ethier1977} derived a bound analogous to that of Theorem~\ref{THM1}, but of order
$N^{-1}$.
In the general case our bound quantifies the effect of non-PIM: if
$N\varepsilon \to 0$ as~$N\to\infty$ then the stationary distribution converges to the Dirichlet distribution.
\epsnd{remark}
\subsection{Cannings model with parent-independent mutation structure}
Our next result is for the general Cannings exchangeable non-degenerate genealogy. The bounds
are in terms of the moments of the offspring vector~$\tV$; hence, let
\ban{\label{3}
\alpha:=\mathrm{I}E\klg{V_1 (V_1-1)},
\quad
\beta:=\mathrm{I}E\klg{ V_1 (V_1-1)(V_1-2)},
\quad
\gamma:=\mathrm{I}E \klg{V_1 (V_1-1)V_2(V_2-1)}
}
(and note that these quantities depend on~$N$).
\begin{theorem}\label{THM2}
Let the $(K-1)$-dimensional vector~$\tX$ be a stationary distribution of the Cannings model for a population of size~$N\geq 4$ with non-degenerate exchangeable genealogy~$\mathscr{L}(\tV)$. Assume we have
parent independent mutation structure; that is,~$p_{ij}=\pi_j$,~$1\leq i\not= j \leq K$,~ for some~$\pi_1,\deltaots,\pi_K>0$,
and $p_{ii}=1-\sum_{j\not=i} \pi_j$.
Let~$\alpha$,~$\beta$, and~$\gamma$ be as defined at~\epsqref{3},
and for~$i=1,\ldots, K$, set~$a_i=\frac{2(N-1)\pi_i}{\alpha}$ and~$s=\sum_i a_i$. Let
$\textnormal{\textbf{W}}=\tX/N$, and let~$\tZ\sim \Dir(\ta)$. Then, for any~$h\in\mathrm{BC}^{2,1}(\-\Delta_K)$,
\be{
\left|\mathrm{I}E h(\textnormal{\textbf{W}})-\mathrm{I}E h(\tZ) \right| \leq \frac{\abs{h}_2}{2(s+1)} A_2 + \frac{\abs{h}_{2,1}}{18(s+2)} A_3,
}
where, with~$\epsta=N\alpha^{-1}\sum_{j=1}^K\pi_j=\frac{sN}{2(N-1)}$,
\ba{
A_2&= \bbklr{\frac{\alpha}{N}}^2\epsta^2 K^2 + \frac{\alpha}{N}\bbklr{\epsta^2 (K^2+1) + 2\epsta K^2} + \frac{3\epsta K}{N} ,\\
A_3 &= 2K^3\left(1+\epsta\sqrt{\frac{\alpha}{N}}+\sqrt{\frac{\epsta}{N}} \right)\left(
\epsta\bbbklr{\frac{\alpha}{N}}^{3/4}
+ \left( \frac{12\beta}{\alpha N}+ \frac{ 24 \gamma}{\alpha N}\right)^{1/4}
+\frac{1}{N^{1/2}}\left(3\epsta^2\frac{\alpha}{N} + \frac{\epsta}{N} \right)^{1/4}
\right)^2.
}
Moreover, there is a constant~$C=C(\ta)$ such that
\ben{
\sup_{A\in \mathcal{C}_{K-1}}\babs{\mathrm{I}P[\textnormal{\textbf{W}}\in A] - \mathrm{I}P[\tZ\in A]}\leq C\bklr{A_2+A_3}^{\theta/(3+\theta)},
}
where~$\mathcal{C}_{K-1}$ is the family of convex sets on\/~$\mathrm{I}R^{K-1}$ and where
$\theta=\theta(\ta)>0$ is given at~\epsqref{10}.
\epsnd{theorem}
\begin{remark}\label{rem1}
To interpret the bound of the theorem, we note that the bound goes to zero if, as~$N\to\infty$, $\epsta\leq s$ remains bounded and all three of
\ban{\label{4}
&\frac{\alpha}{N}, \qquad \frac{\beta}{\alpha N}, \qquad
\frac{\gamma}{\alpha N},
}
tend to zero. And for the convergence to be to non-degenerate, we must have
\ben{\label{5}
a_i = \frac{2(N-1)\pi_i}{\alpha}\to \tilde{a}_i,
}
for some limiting positive~$\tilde{a}_i$,~$i=1,\ldots, K$, which also implies that
$\epsta=sN/(2(N-1))$ converges to a positive constant.
As briefly mentioned above, under appropriate assumptions, the exchangeable genealogy alone (that is, without mutation structure)
converges to Kingman's coalescent as~$N\to\infty$. This convergence occurs if and only if
\citep*{Mohle2000} \citep*{Mohle2001, Mohle2003}
\ben{\label{6}
\frac{\beta}{ \alpha N}\to 0.
}
In this case and also assuming a limiting scaling of the mutation
probabilities given by~\epsqref{5},
the finite population stationary distribution converges to the
stationary distribution of a Wright-Fisher diffusion, that is,~$\Dir(\ta)$.
At first glance it appears that demanding the terms of~\epsqref{4} tend to zero
is a stronger requirement for convergence than the sufficient~\epsqref{6},
but \citep*[Lemma~5.5]{Mohle2003}, \citep*[Display~(16)]{Mohle2000}
show that~\epsqref{6} also implies
\be{
\frac{\alpha}{N}\to 0\, \qquad \text{ and } \qquad \frac{\gamma}{\alpha N}\to 0.
}
So, in fact, our bound goes to zero assuming only~\epsqref{6} and thus quantifies the convergence of the stationary distribution in terms of natural quantities. Assuming~$\epsta$ remains bounded, we obtain
\be{
A_2 = \mathrm{O}\bbbklr{K^2\frac{\alpha}{N}+\frac{K}{N}},\qquad A_3 = \mathrm{O}\bbbklr{K^3\bbklr{\frac{\alpha}{N}}^{3/2} + K^3\bbklr{\frac{\beta}{\alpha N}}^{1/2}+K^3\bbklr{\frac{\gamma}{\alpha N}}^{1/2}+\frac{K^3}{N}}.
}
\epsnd{remark}
\begin{remark}
For the stationary distribution of types in an exchangeable Cannings genealogy with general mutation structure,
a bound with features similar to those of Theorems~\ref{THM1} and~\ref{THM2} should be possible using our methods. However,
the formulation and proof of such a result would be rather messy, and so, for the sake of exposition and clarity, we
present two separate theorems to handle more specific situations.
\epsnd{remark}
\subsection{Stein's method of exchangeable pairs for the Dirichlet distribution}
Theorems~\ref{THM1} and~\ref{THM2} follow from a new development of Stein's method for the Dirichlet distribution.
Stein's method \citep*{Stein1972, Stein1986} is a powerful tool for providing bounds on the approximation of
a probability distribution of interest by a well understood target distribution; see \citep*{Chen2011} and \citep*{Ross2011}
for recent introductions, and \citep*{Chatterjee2014} for a recent literature survey. We show a general exchangeable pairs Dirichlet approximation theorem very much in the spirit of exchangeable pairs approximation results for other distributions; e.g., normal \citep*[Theorem~1.1]{Rinott1997}; multivariate normal \citep*[Theorem~2.3]{Chatterjee2008} \citep*[Theorem~2.1]{Reinert2009};
exponential \citep*[Theorem~1.1]{Chatterjee2011} \citep*[Theorem~1.1]{Fulman2013}; beta \citep*[Theorem~4.4]{Dobler2015}; limits in Curie-Weiss models \citep*[Theorem~1.1]{Chatterjee2011a}. In what follows, sums range from~$1$ to~$K$ unless otherwise stated.
\begin{theorem}\label{THM3}
Let~$\ta=(a_1,\ldots, a_K)$ be a vector of positive numbers and set~$s=\sum_{i=1}^K a_i$.
Let~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ be an exchangeable pair of~$(K-1)$ dimensional random vectors with non-negative entries with sum no greater than one.
Also let~$\Lambda$ be an invertible matrix and~$\tR$ be a random vector such that
\ben{
\mathrm{I}E [\textnormal{\textbf{W}}'-\textnormal{\textbf{W}}| \textnormal{\textbf{W}}]=\Lambda (\ta-s \textnormal{\textbf{W}}) + \tR. \label{7}
}
Then, for any~$h\in\mathrm{BC}^{2,1}(\-\Delta_K)$,
\ben{\label{8}
\abs{\mathrm{I}E h(\textnormal{\textbf{W}})-\mathrm{I}E h(\tZ)}\leq \frac{\abs{h}_1}{s} A_1 + \frac{\abs{h}_2}{2(s+1)} A_2 +\frac{\abs{h}_{2,1}}{6(s+2)} A_3,
}
where
\ba{
A_1&:=\sum_{m,i} \babs{(\Lambda^{-1})_{i,m}}\mathrm{I}E \abs{R_m },\\
A_2&:= \sum_{m,i,j} \babs{(\Lambda^{-1})_{i,m}} \mathrm{I}E \left| \Lambda_{m,i} W_i(\deltaelta_{i j}-W_j)-\frac{1}{2}\mathrm{I}E[ (W_m'-W_m)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|, \\
A_3&:=\sum_{m,i,j,k} \babs{(\Lambda^{-1})_{i,m}} \mathrm{I}E \left|(W_m'-W_m)(W_j'-W_j)(W_k'-W_k) \right|.
}
Moreover, there exists a constant~$C=C(\ta)$ such that
\ben{\label{9}
\sup_{A\in \mathcal{C}_{K-1}}\babs{\mathrm{I}P[\textnormal{\textbf{W}}\in A] - \mathrm{I}P[\tZ\in A]}\leq C\bklr{A_1+A_2+A_3}^{\theta/(3+\theta)},
}
where~$\mathcal{C}_{K-1}$ is the family of convex sets on\/~$\mathrm{I}R^{K-1}$ and
\ben{\label{10}
\theta = \frac{\theta_\wedge}{\theta_\wedge+\theta_\circ},\qquad \theta_\wedge = 1\wedge\min\{a_1,\deltaots,a_K\},\qquad \theta_\circ=\sum_{i=1}^K\bklr{1-1\wedge a_i}.
}
Additionally, if~$\Lambda$ is a multiple of the identity matrix, then
the result still holds assuming only that~$\mathscr{L}(\textnormal{\textbf{W}})=\mathscr{L}(\textnormal{\textbf{W}}')$, in which
case the factor~$\frac{\abs{h}_{2,1}}{6(s+2)}$ in \epsqref{8} can be improved
to~$\frac{\abs{h}_{2,1}}{18(s+2)}$
\epsnd{theorem}
The layout of the remainder of the paper is as follows. We finish the introduction by applying
Theorem~\ref{THM3} in an easy example, the multi-colored P\'olya urn.
In Section~\ref{sec1} we develop Stein's method for the Dirichlet distribution and
prove Theorem~\ref{THM3}.
In Section~\ref{sec2} we prove Theorem~\ref{THM1}, the bounds for the Wright-Fisher model and
in Section~\ref{sec3} we prove Theorem~\ref{THM2}, the bounds for the PIM Cannings model.
\paragraph{A simple example: Multi-colored P\'olya Urn.}
In order to illustrate how Theorem~\ref{THM3} is applied, we use it to bound the error in approximating the counts in the classical P\'olya urn by a Dirichlet distribution. The result is new to us, but a bound in the
Wasserstein distance could be obtained
from analogous bounds for the beta distribution \citep*{Goldstein2013} \citep*{Dobler2015} using
the iterative urn approach of \citep*{Pekoz2014a}.
An urn initially contains~$a_i>0$ ``balls" of color~$i$ for~$i =1,2,\ldots, K$ with a total number of balls~$s= \sum_{i=1}^K a_i$. At each time step, draw a ball uniformly at random from the urn and replace it along with another ball of the same color.
Let~$\tX(n) = (X_1(n), X_2(n), \ldots , X_{K-1}(n))$, where~$X_i(n)$ is the number of times color~$i$ was drawn up to and including the~$n$th draw. It is well known (see, e.g., \citep*{Mahmoud2009}) that as~$n \to \infty$,
\be{
\textnormal{\textbf{W}}(n):= \frac{\tX(n)}{n} \stackrel{d}{\longrightarrow} \Dir(\ta),
}
and we provide a bound on the approximation of the distribution of~$\textnormal{\textbf{W}}(n)$ by the Dirichlet limit.
\begin{theorem}\label{THM4}
Let~$\ta=(a_1, \ldots, a_K)$ be a vector of positive numbers,~$s=\sum_{i=1}^K{a_i}$,~$\tZ\sim\Dir(\ta)$, and~$\textnormal{\textbf{W}}(n)$ be the P\'olya urn proportions as defined above. Then, for any~$h\in\mathrm{BC}^{2,1}(\-\Delta_K)$,
\be{
\abs{\mathrm{I}E h(\textnormal{\textbf{W}}(n)) -\mathrm{I}E h(\tZ)}\leq \frac{s}{n(s+1)} \abs{h}_2 + \frac{(K-1)(3K-5) (n+s-1)}{18n^2(s+2)} \abs{h}_{2,1}.
}
Moreover, there exists a constant~$C=C(\ta)$ such that
\be{
\sup_{A\in \mathcal{C}_{K-1}}\abs{\mathrm{I}P[\textnormal{\textbf{W}}\in A] - \mathrm{I}P[\tZ\in A]}\leq Cn^{-\theta/(3+\theta)},
}
where $\theta=\theta(\ta)>0$ is defined at~\epsqref{10}.
\epsnd{theorem}
We use Theorem~\ref{THM3} to prove the result. To define the exchangeable pair,
note that we can set~$\tX(n) = \sum_{j=1}^n \textnormal{\textbf{Y}}(j)$ where, for~$\te_i$ equal to the~$i$th unit vector,~$\textnormal{\textbf{Y}}(j)=\te_i$ if color~$i$ is drawn on the~$j$th draw.
It is easy to check that
\be{
\mathrm{I}P\bkle{\textnormal{\textbf{Y}}(j) = \te_i \big\vert \tX(j-1)} = \frac{X_i(j-1) + a_i}{j-1+s}.
}
We define the exchangeable pair~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ (dropping the~$n$ to ease notation) by resampling the last draw~$\textnormal{\textbf{Y}}(n)$; that is,
\be{
\textnormal{\textbf{W}}' = \textnormal{\textbf{W}} - \frac{\textnormal{\textbf{Y}}(n)}{n} + \frac{\textnormal{\textbf{Y}}'(n)}{n},
}
where conditional on~$\tX(n-1)$,~$\textnormal{\textbf{Y}}'(n)$ and~$\textnormal{\textbf{Y}}(n)$ are i.i.d.\ Before computing the terms appearing in the bound of Theorem~\ref{THM3}, we record a lemma.
\begin{lemma}\label{lem1}
Recalling the notation and definitions above, and let~$\delta_{ij}$ denote the Kronecker delta function,
\ba{
\mathrm{I}E(\textnormal{\textbf{Y}}'(n) | \textnormal{\textbf{W}}) &= \frac{1}{n+s-1}\left[ \ta + (n-1)\textnormal{\textbf{W}}\right],\\
\mathrm{I}E(Y_i'(n)Y_j(n) | \textnormal{\textbf{W}}) &= \frac{1}{n+s-1} \mathrm{I}E \left[ a_i W_j + nW_iW_j - W_i \delta_{ij}\right].
}
\epsnd{lemma}
\begin{proof}
First note that
\be{
\mathrm{I}E[\textnormal{\textbf{Y}}'(n)| (\textnormal{\textbf{Y}}(1), \ldots, \textnormal{\textbf{Y}}(n))]=\frac{1}{n+s-1} \left[ \ta + n\textnormal{\textbf{W}} - \textnormal{\textbf{Y}}(n) \right].
}
The first equality now follows by taking expectation conditional on~$\textnormal{\textbf{W}}$ and noting that exchangeability implies~$\mathrm{I}E [\textnormal{\textbf{Y}}(n)| \textnormal{\textbf{W}}]=\textnormal{\textbf{W}}$.
For the second identity, use the previous display to find
\ba{
\mathrm{I}E[ Y_i'(n) Y_j(n) | Y_j(1), \ldots Y_j(n)] = \frac{Y_j(n)}{n+s-1} \left[ a_i +n W_i - Y_i(n) \right],
}
and taking expectation conditional on~$\textnormal{\textbf{W}}$, noting~$\mathrm{I}E \textnormal{\textbf{Y}}(n)=\textnormal{\textbf{W}}$ and~$Y_i(n) Y_j(n)=\deltaelta_{i j} Y_i(n)$,
yields
\be{
\mathrm{I}E(Y_i'(n)Y_j(n)| \textnormal{\textbf{W}})= \frac{1}{n+s-1} \mathrm{I}E \left[ a_i W_j + nW_iW_j - W_i \delta_{ij}\right]. \qedhere
}
\epsnd{proof}
\begin{proof}[Proof of Theorem~\ref{THM4}]
We apply Theorem~\ref{THM3} with the exchangeable pair defined above. We show below
that for~$i,j,k\in\{1,\ldots, K\}$,
\ban{
&\mathrm{I}E[\textnormal{\textbf{W}}'-\textnormal{\textbf{W}}|\textnormal{\textbf{W}}]= \frac{1}{n(n+s-1)}\left(\ta - s \textnormal{\textbf{W}}\right), \label{12}\\
&\mathrm{I}E[(W_i' - W_i)(W_j' - W_j)| \textnormal{\textbf{W}}] =\frac{\delta_{ij}(a_i + (2n+s)W_i) - a_i W_j - a_j W_i - 2nW_iW_j}{n^2(n+s-1)}, \label{13}\\
&\mathrm{I}E \left| (W_i'-W_i)(W_j'-W_j)(W_k'-W_k) \right| \leq n^{-3}(1-\mathrm{I}[\text{$i,j,k$ distinct}]), \label{14}
}
so we can apply the Theorem~\ref{THM3} with~$\Lambda=\frac{1}{n (n+s-1)} \times \mathrm{I}d$.
In this case, using~\epsqref{13},
\ba{
A_2&= n(n+s-1)\sum_{i,j=1}^{K-1} \mathrm{I}E \left| \frac{1}{n(n+s-1)} W_i(\delta_{ij}-W_j) - \frac{1}{2} \mathrm{I}E\left[(W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}\right]\right|\\
&=\frac{1}{2n} \sum_{i,j=1}^{K-1} \mathrm{I}E \left| \delta_{ij}(a_i + sW_i) - a_iW_j - a_jW_i\right|\leq \frac{2s}{n}.
}
Now using~\epsqref{14}, we have
\be{
A_3\leq \frac{(K-1)(3K-5) (n+s-1)}{n^2}.
}
Finally the form of~\epsqref{12} makes it clear that~$R=0$ and so~$A_1=0$. Putting together the last two displays yields the result.
All that is left is to show~\epsqref{12},~\epsqref{13}, and~\epsqref{14}. Lemma~\ref{lem1} implies
\be{
\mathrm{I}E[\textnormal{\textbf{W}}' - \textnormal{\textbf{W}} | \textnormal{\textbf{W}}]= \frac{1}{n} \mathrm{I}E [ \textnormal{\textbf{Y}}'(n) - \textnormal{\textbf{Y}}(n) | \textnormal{\textbf{W}} ]=\frac{1}{n(n+s-1)}\left(\ta - s \textnormal{\textbf{W}}\right),
}
which is~\epsqref{12}. For~\epsqref{13}, use Lemma~\ref{lem1} and that~$Y_i(n) Y_j(n)=\deltaelta_{i j} Y_i(n)$ and~$Y_i'(n) Y_j'(n)=\deltaelta_{i j} Y_i'(n)$ to find
\ba{
&\mathrm{I}E[(W_i' - W_i)(W_j' - W_j)| \textnormal{\textbf{W}}] = \frac{1}{n^2} \mathrm{I}E[ Y_i'(n)Y_j'(n) + Y_i(n)Y_j(n) - Y_i'(n)Y_j(n) - Y_i(n)Y_j'(n) | \textnormal{\textbf{W}}]\\
&= \frac{1}{n^2} \left[\frac{\delta_{ij}}{n+s-1}(a_i + (n-1)W_i) + \delta_{ij}W_i - \frac{1}{n+s-1} \left(a_i W_j + a_j W_i + 2nW_iW_j - 2 W_i \delta_{ij}\right) \right]\\
&= \frac{1}{n^2(n+s-1)} \left[ \delta_{ij}(a_i + (2n+s)W_i) - a_i W_j - a_j W_i - 2nW_iW_j\right].
}
Finally~\epsqref{14} follows noting that~$\abs{W_i'-W_i}\leq 1/n$ and that at most two of the~$(W_i'-W_i)$ can be non-zero.
The bound on the convex set metric is immediate from the bounds on~$A_2$ and~$A_3$ and \epsqref{9}.
\epsnd{proof}
\section{Stein's~method~for~the~Dirichlet~distribution}\label{sec1}
\subsection{Stein operator}
In order to apply Stein's method we need a characterizing operator for the Dirichlet distribution, which is provided below.
Let~$\deltaelta_{ij}$ denote the Kronecker delta function, and for a function~$f$, let~$f_j$ be the partial derivative of~$f$ with respect to the~$j$th component,~$f_{i j}$ the 2nd partial derivative, and so on.
\begin{lemma}
Let~$a_1, \ldots, a_K$ be positive numbers and~$s=\sum_{i=1}^K a_i$. The random vector~$\textnormal{\textbf{W}}\in\Delta_{K}$ has distribution~$\Dir (a_1,\ldots,a_K)$ if and only if for all~$f\in\mathrm{BC}^{2,1}(\Delta_K)$
\be{
\mathrm{I}E\left[ \sum_{i,j=1}^{K-1} W_i(\deltaelta_{i j}-W_j)f_{i j}(\textnormal{\textbf{W}})
+\sum_{i=1}^{K-1}(a_i-s W_i)f_{i}(\textnormal{\textbf{W}})\right]=0.
}
\epsnd{lemma}
The forward implication of the lemma is straightforward
and the backwards follows by taking expectations against polynomials~$f$ to yield formulas for mixed moments of~$\textnormal{\textbf{W}}$. Also note that
\ben{\label{15}
\mathcal{A}f(\tx) := \frac{1}{2}\left[\sum_{i,j=1}^{K-1} x_i(\deltaelta_{i j}-x_j) f_{i j}(\tx)
+\sum_{i=1}^{K-1}(a_i-s x_i)f_i(\tx)\right]
}
is the generator of the Wright-Fisher diffusion which has the Dirichlet as its unique stationary distribution; see \citep*{Wright1949}, \citep*{Ethier1976}, \citep*{Shiga1981}.
\subsection{Bounds on the solution to the Stein equation}
To apply Stein's method, we proceed as follows. Let~$\tZ\sim\Dir(\ta)$, and let~$h:\-\Delta_K\to\mathrm{I}R$ be some measurable test function. If~$h$ is bounded, then clearly~$\mathrm{I}E \abs{h(\tZ)} < \infty$. Assume we have a function~$f:=f_h$ that solves
\ben{\label{16}
\sum_{i,j=1}^{K-1} x_i(\deltaelta_{i j}-x_j)f_{i j}(\tx)
+\sum_{i=1}^{K-1}(a_i-s x_i) f_i (\tx)=h(\tx)-\mathrm{I}E h(\tZ) =: \tilde{h}(\tx),
}
and note that replacing~$\tx$ by~$\textnormal{\textbf{W}}$ in this equation and taking expectation gives an expression for~$\mathrm{I}E h(\textnormal{\textbf{W}})-\mathrm{I}E h(\tZ)$
in terms of just~$\textnormal{\textbf{W}}$ and~$f$.
Since this operator is twice the generator of the Wright-Fisher diffusion given by~\epsqref{15}, we use the generator approach of \citep*{Barbour1990} \citep*{Gotze1991},
and observe we may set~$f$ to be
\ben{\label{17}
f(\tx) = - \frac{1}{2}\int_0^\infty \mathrm{I}E \left[ h(\tZ_\txt) - h(\tZ) \right] dt = - \frac{1}{2}\int_0^\infty \mathrm{I}E\tilde{h}(\tZ_\txt) dt,
\qquad\tx\in\Delta_K, }
where~$(\tZ_\txt)_{t\geq0}$ is the Wright-Fisher diffusion, defined by the generator~$\mathcal{A}$ with~$\tZ_\tx(0) = \tx$ (the factor
of~$1/2$ in the expression appears since~\epsqref{15} is twice the generator of~$\tZ$).
Using a probabilistic description of the Wright-Fisher semigroup due to
\citep*{Griffiths1983} and \citep*{Tavare1984} we show that the above integral is well defined, and we obtain the following bounds on the solution~\epsqref{17} to the Stein equation~\epsqref{16}.
\begin{theorem}\label{THM5}
If~$h:\-\Delta_K\to\mathrm{I}R$ is continuous, then~$f$ defined by \epsqref{17} is twice partially differentiable and solves \epsqref{16} for all~$\tx\in\Delta_K$, and we have the bound
\ben{\label{18}
\norm{f}_\infty\leq \frac{(s+1)}{s}\norm{\tilde h}_\infty.
}
If~$h\in\mathrm{BC}^{m,1}(\-\Delta_K)$ for some~$m\geq 0$, then~$f\in\mathrm{BC}^{m,1}(\-\Delta_K)$, and we have the bounds
\ben{\label{19}
\abs{f}_k\leq \frac{\abs{h}_k}{k(s+k-1)},\quad 1\leq k\leq m,\qquad\text{and}
\qquad
\abs{f}_{m,1}\leq \frac{\abs{h}_{m,1}}{m(s+m-1)}.
}
If~$m\geq 2$, then
equation \epsqref{16} holds for all~$\tx\in\-\Delta_K$.
\epsnd{theorem}
\begin{remark}
The Dirichlet distribution is a multivariate generalization of the Beta distribution for which Stein's method
has recently been developed \citep*{Dobler2012, Dobler2015} \citep*{Goldstein2013} where bounds are derived
for the~$K=2$ case of the Stein equation used here. Direct comparisons are difficult in general since
typically different derivatives of the test function appear. However one easily comparable bound is \citep*[Proposition~4.2(b)]{Dobler2015} that~$|f|_1 \leq |h|_1/s$, which is the same as our bound in this case.
In general, the bounds appearing in these other works are quite complicated, involving different expressions for different regions of the parameter space, whereas
our bounds are very clean and have a simple relationship to the parameters. Furthermore our bounds apply in the multivariate setting.
\epsnd{remark}
\begin{proof}[Proof of Theorem~\ref{THM5}] Throughout the proof we make the simplifying assumption that~$\mathrm{I}E h(\tZ)=0$ so that~$\tilde h = h$.
Following the generator approach of \citep*{Barbour1990}, \citep*{Gotze1991}; see also \citep*[Appendix~B]{Gorham2016}; let~$\tx\in\-\Delta_K$,
let~$(\tZ_\txt)_{t\geq0}$ be the Wright-Fisher diffusion defined by the generator~$\mathcal{A}$ with~$\tZ_\tx(0) = \tx$,
and let~$f$ be as defined in~\epsqref{17}.
\noindent{\bf Construction of semigroup.}
The key to our bounds is a construction of the marginal variable~$\tZ_\txt$ from \citep*{Griffiths1983} \citep*{Tavare1984}; see also the introduction of~\citep*{Barbour2000}.
Let~$L_t$ be a pure death process on~$\{0, 1, \ldots\}\cup\{\infty\}$ started at~$\infty$ with death rates
\ben{\label{20}
q_{i, i-1}= \frac{1}{2} i(i-1+s).
}
Denote by~$\mathrm{MN}_K(n; p_1,\deltaots,p_K)$ the~$K$-dimensional multinomial distribution with~$n$ trials and probabilities~$p_1,\deltaots,p_K$; by slight misuse of notation, we write~$\mathrm{MN}_K(L_t;\tx,x_K)$ to be
short for~$\mathrm{MN}_K(L_t;x_1,\deltaots,x_{K-1},x_K)$. Conditional on~$L_t$, let~$\tN\sim\mathrm{MN}_K(L_t;\tx,x_K)$, where~$x_K=1-\sum_{i=1}^{K-1}x_i$.
Then,
\be{
\mathscr{L}\bklr{\tZ_\txt\big\vert L_t,\tN} \sim \Dir(\ta+\tN).
}
\noindent{\bf Existence of solution to Stein equation on~$\boldsymbol{\Delta_K}$ and bound (\ref{18}).}
For~$n\geq 1$, let~$Y_n$ be the time the process~$L_t$ spends in state~$n$ and note that~$Y_n$ is exponentially distributed with rate~$n(n-1+s)/2$. Since
\be{
\sum_{n\geq 1} \mathrm{I}E Y_n = \sum_{n\geq 1}\frac{2}{n(n+s-1)} \leq \frac{2(s+1)}{s},
}
the random variable~$T = \inf\{t>0\,:\,L_t=0\} = \sum_{n\geq 1}Y_n$ is finite almost surely and has finite expectation. Observing that~$\mathrm{I}E\bklr{{ h}(\tZ_\txt)\big| L_t=0}=0$ since, given~$L_t=0$, we have~$\tZ_\txt \sim \Dir(\ta)$, it follows that
\besn{\label{457}
\int_0^\infty \babs{\mathrm{I}E\bklr{ h(\tZ_\txt) }} dt &\leq \int_0^\infty \norm{h}_\infty \mathrm{I}P(L_t>0) dt \\
&\leq \norm{h}_\infty \int_0^\infty \mathrm{I}P(T>t) dt =\norm{ h}_\infty\mathrm{I}E T < \infty.
}
Thus,
$f$ in \epsqref{17} is well-defined.
To show that $f$ is in the domain of~$\cal{A}$ and satisfies $\mathcal{A}f= h$ under the assumption that~$h\in\mathrm{BC}(\-\Delta_K)$, the Banach space of bounded and continuous functions equipped with sup-norm, we follow the argument of \citep*[Pages~301-2]{Barbour1990} also used in \citep*[Appendix~B]{Gorham2016}. First,
\citep*[Theorem~1]{Ethier1976} implies that
the semigroup $(T_t)_{t\geq0}$ defined by $T_t g(\tx)=\mathrm{I}E g(\tZ_\txt)$ for $g\in\mathrm{BC}(\-\Delta_K)$ is strongly continuous. Note also that $\mathrm{BC}^{m,1}(\-\Delta_K)\subset\mathrm{BC}(\-\Delta_K)$ for all $m\geq 0$.
We can therefore apply
\citep*[Proposition~1.5(a), Page~9]{Ethier1986}, which implies that $f\s u(\tx):=- \frac{1}{2}\int_0^u \mathrm{I}E{h}(\tZ_\txt) dt$ is in the domain of~$\mathcal{A}$ and satisfies
\be{
\mathcal{A} f\s u (\tx)=h(\tx)-\mathrm{I}E h(\tZ_\tx(u)).
}
Furthermore, \citep*[Corollary~1.6, Page~10]{Ethier1986} implies that $\mathcal{A}$ is a closed operator,
so it is enough to show that as $u\to\infty$,
\ben{\label{458}
\norm{f\s u-f}_\infty\to0 \,\,\, \mbox{ and } \,\,\, \norm{\mathcal{A}f\s u-h}_{\infty}\to 0.
}
By definitions,~\epsqref{458} is implied by
\be{
\sup_{\tx\in\-\Delta_K}\int_u^\infty \babs{\mathrm{I}E\bklr{ h(\tZ_\txt) }} dt\to 0 \,\,\, \mbox{ and } \,\,\, \sup_{\tx\in\-\Delta_K}\mathrm{I}E h(\tZ_\tx(u))\to 0,
}
as $u\to\infty$. But the first limit follows from~\epsqref{457} and the second is because
\be{
\sup_{\tx\in\-\Delta_K}\mathrm{I}E h(\tZ_\tx(u))\leq \norm{h}_\infty \sup_{\tx\in\-\Delta_K} \mathop{d_{\mathrm{TV}}}\bklr{\mathscr{L}(\tZ_\tx(u)), \Dir(\ta)} \leq \norm{h}_\infty \mathrm{I}P(L_u>0)\to 0.
}
The boundedness of the solution follows essentially from the computations above, but we give a slightly different argument
in detail, since a similar but more complicated one is used later. Compute
\bes{
-2f(\tx)
& =\int_0^\infty \mathrm{I}E h(\tZ_\txt) dt
=\mathrm{I}E \int_0^\infty \mathrm{I}E \bklr{h(\tZ_\txt)\big|L_t} dt \\
& =\mathrm{I}E \int_0^\infty \sum_{n\geq1} \mathrm{I}E\bklr{h(\tZ_\txt)\big| L_t=n}\mathrm{I}[L_t=n] dt\\
&=\mathrm{I}E \sum_{n\geq1} \int_0^\infty \mathrm{I}E\bklr{h(\tZ_\tx(1))\big| L_1=n}\mathrm{I}[L_t=n] dt \\
& =\mathrm{I}E \sum_{n\geq1} \mathrm{I}E\bklr{h(\tZ_\tx(1))\big| L_1=n}Y_n
=\sum_{n\geq 1} \mathrm{I}E\bklr{ h(\tZ_\tx(1))\big| L_1=n}\mathrm{I}E Y_n,
}
where we have used dominated convergence multiple times to interchange expectation, integration and summation, along with the fact that~$\mathrm{I}E \bklr{h(\tZ_\txt)\big|L_t=n}$ only depends on~$n$ and not on~$t$ and can therefore be replaced by~$\mathrm{I}E \bklr{h(\tZ_\tx(1))\big|L_1=n}$ (or with~$t$ being replaced by any other fixed positive time).
This leads to
\be{
\abs{f(\tx)}\leq \frac{1}{2}\norm{ h}_\infty \sum_{n\geq 1} \mathrm{I}E Y_n
= \norm{ h}_\infty \sum_{n\geq 1}\frac{1}{n(n-1+s)}\leq \frac{(s+1)}{s}\norm{ h}_\infty,
}
which is \epsqref{18}.
\noindent{\bf Preliminaries for partial derivatives.}
To show the existence and bounds for the partial derivatives,
we need some couplings. Let $\te_i$ denote the unit vector with a one in the $i$th coordinate (and zeros in all others with dimension from context).
Fix~$m\geq 0$ and~$1\leq i_1,\deltaots,i_{m+1}\leq K-1$. Let~$\tx = (x_1, x_2, \ldots, x_{K-1})\in\Delta_K$. Choose~$\varepsilon_1,\deltaots,\varepsilon_{m+1}>0$ arbitrarily, but small enough that~$x_K:=1-\sum_{j=1}^{K-1} x_j>\sum_{j=1}^{m+1}\varepsilon_j$, or equivalently, that~$\tx+\sum_{j=1}^{m+1}\varepsilon_j\te_{i_j}\in\Delta_K$. Then, proceed with the following steps.
\begin{enumerate}[$(i)$]
\item Let~$L_t$ be the pure death process as described above.
\item Given~$L_t$, let~$\textnormal{\textbf{M}} :=(\tB, \tN)\sim \mathrm{MN}_{m+1+K}(L_t;\eps_1, \deltaots, \eps_{m+1}, \tx , x_K-\sum_{j=1}^{m+1}\varepsilon_j)$, where~$\tB=(B_1,\deltaots,B_{m+1})$ and~$\tN=(N_1,\deltaots,N_K)$.
\item Given~$L_t$ and~$\textnormal{\textbf{M}}$, let
\ben{\label{21}
( \delta_1, \deltaots, \delta_{m+1}, \textnormal{\textbf{D}}_\tx) \sim \Dir(\tB, \ta+\tN).
}
\item Set~$\boldsymbol{\eps}_j = \eps_j \te_{i_j}$ for~$1\leq j\leq m+1$.
As described immediately below, basic facts
about the multinomial and Dirichlet distributions imply that
\ben{\label{22}
\textnormal{\textbf{D}}_\tx \epsqreflaw \tZ_\txt
}
and that
\ben{\label{23}
\textnormal{\textbf{D}}_{\tx} + \sum_{j\in A} \delta_{j}\te_{i_j} \epsqreflaw \tZ_{\tx+\sum_{j\in A} \boldsymbol{\eps}_{j}}(t)
}
for any subset~$A\subset\{1,\deltaots,m+1\}$.
\epsnd{enumerate}
To see why~\epsqref{22} and~\epsqref{23} are true, use the
following standard facts.
\begin{itemize}
\item Let~$(\xi_1, \deltaots, \xi_{p-1}) \sim \Dir(y_1,\ldots,y_p)$. If~$A=\{i_1,\ldots,i_j\}\subset\{1,\ldots, p-1\}$
is any subset of indices, then
\be{
\bklr{\xi_{i_1},\ldots, \xi_{i_j} } \sim \Dir\bbklr{y_{i_1},\ldots,y_{i_j}, y_p+\sum_{k\not\in A} y_k}.
}
Furthermore, letting~$\boldsymbol{\xi}^{(k)}\in \mathrm{I}R^{p-2}$ denote~$(\xi_1, \deltaots, \xi_{p-1})$ with the~$k$th coordinate
removed,
we have for~$i<k$ (and a similar statement for $i>k$),
\be{
\boldsymbol{\xi}^{(k)}+\te_i \xi_k
\sim \Dir\bklr{y_{1},\ldots,y_{i-1}, y_{i}+y_{k},y_{i+1} \ldots, y_{k-1}, y_{k+1}, \ldots, y_{p} }.
}
\item Let~$(\zeta_1,\ldots,\zeta_p)\sim \mathrm{MN}_p(b; y_1,\ldots,y_p)$. If~$A=\{i_1,\ldots,i_j\}\subset\{1,\ldots, p\}$
is any
subset of indices, then
\be{
\bbklr{\zeta_{i_1},\ldots, \zeta_{i_j}, \sum_{k\not\in A} \zeta_k} \sim \mathrm{MN}_{j+1}\bbklr{b; y_{i_1},\ldots,y_{i_j}, \sum_{k\not\in A} y_k}.
}
Furthermore, letting~$\boldsymbol{\zeta}^{(k)}\in \mathrm{I}R^{p-1}$ denote~$(\zeta_1, \deltaots, \zeta_{p})$ with the~$k$th coordinate
removed,
we have for~$i<k$ (and a similar statement for $i>k$),
\be{
\boldsymbol{\zeta}^{(k)}+\te_i \zeta_k
\sim \mathrm{MN}_{p-1}\bklr{b; y_{1},\ldots,y_{i-1}, y_{i}+y_{k},y_{i+1} \ldots, y_{k-1}, y_{k+1}, \ldots, y_{p} }.
}
\epsnd{itemize}
The first item above follows from the usual decomposition of the components of the Dirichlet distribution in terms of ratios of gamma variables, and the second is straightforward from the
probabilistic description of the multinomial distribution.
\noindent {\bf Existence of partial derivatives and bounds (\ref{19}).} To ease notation, for vectors $\tx,\ty$ and a function $g$, define $\Delta_{\ty}g(\tx)=g(\tx+\ty)-g(\ty)$ (context should clarify when we mean the simplex or the difference operator). Assume now that $h\in\mathrm{BC}^{m,1}(\-\Delta_K)$ for some $m\geq 0$, let~$1\leq i_1,\deltaots,i_{m+1}\leq K-1$, and recall the coupling and associated notation defined above. For any~$1\leq k\leq m+1$, we have
\bes{
\frac{\babs{\Delta_{\boldsymbol{\eps}_{1}}\cdots\Delta_{\boldsymbol{\eps}_{k}}f(\tx)}}{{\varepsilon_1\cdots\varepsilon_k}}
& = \frac{1}{2{\varepsilon_1\cdots\varepsilon_k}}\bbbabs{\int_{0}^\infty \mathrm{I}E \bklr{\Delta_{\boldsymbol{\eps}_{i}}\cdots\Delta_{\boldsymbol{\eps}_{k}}\bklr{{ h}(\tZ_{\cdot}(t))}(\tx)}dt} \\
& = \frac{1}{2{\varepsilon_1\cdots\varepsilon_k}}\bbbabs{\int_{0}^\infty \mathrm{I}E \bklr{\Delta_{\boldsymbol{\eps}_{i}}\cdots\Delta_{\boldsymbol{\eps}_{k}}{{ h}(\textnormal{\textbf{D}}_{\tx})}}dt} \\
& \leq \frac{\abs{h}_{k-1,1}}{2{\varepsilon_1\cdots\varepsilon_k}}\int_{0}^\infty {\mathrm{I}E \bklr{\deltaelta_1\cdots\deltaelta_m}}dt,
}
where in the last step we have applied Lemma~\ref{lem2}. Now using formulas for Dirichlet and multinomial moments, we have
\be{
\mathrm{I}E \bklr{\deltaelta_1\cdots\deltaelta_m\big\vert L_t,M} = \frac{B_1\cdots B_m}{(L_t+s)(L_t+s+1)\cdots(L_t+s+m-1)}
}
and
\be{
\mathrm{I}E \bklr{B_1\cdots B_m\big\vert L_t} = \varepsilon_1\cdots\varepsilon_k L_t(L_t-1)\cdots(L_t-m+1).
}
Thus,
\bes{
\frac{1}{\varepsilon_1\cdots\varepsilon_k}\int_0^\infty \mathrm{I}E\bklr{\deltaelta_1\cdots\deltaelta_k} dt
& = \sum_{n\geq 1}\frac{n(n-1)\cdots(n-k+1)}{(n+s)(n+s+1)\cdots(n+s+k-1)}\mathrm{I}E Y_n \\
& = \sum_{n\geq 1}\frac{2(n-1)\cdots(n-k+1)}{(n+s-1)(n+s)\cdots(n+s+k-1)} = \frac{2}{k(s+k-1)}.
}
Hence, it follows that
\be{\label{24}
\frac{\babs{\Delta_{\boldsymbol{\eps}_{1}}\cdots\Delta_{\boldsymbol{\eps}_{k}}f(\tx)}}{{\varepsilon_1\cdots\varepsilon_k}} \leq \frac{\abs{h}_{k-1,1}}{k(s+k-1)} =: M_k
}
Since~$\tx$ and~$\varepsilon_1,\cdots,\varepsilon_k$ are arbitrary,
\epsqref{31} in Lemma~\ref{lem5} is satisfied, and we conclude that~$f\in\mathrm{BC}^{m,1}(\-\Delta_K)$ and that
for $k=1,\ldots, m+1$,
$\abs{f}_{k-1,1} \leq \frac{\abs{h}_{k-1,1}}{k(s+k-1)}$, which is \epsqref{19}.
\noindent {\bf Extension to~$\boldsymbol{\-\Delta_K}$.} Assume now~$m\geq 2$. Since~$f\in\mathrm{BC}^{m,1}(\-\Delta_K)$, we have in particular that the~$f_i$ and the~$f_{ij}$ can be extended continuously and uniquely to the boundary of~$\-\Delta_K$. Since the left hand side of \epsqref{16} only consists of finite sums and continuous transformations of the~$f_i$ and~$f_{ij}$ and is equal to the right hand side of \epsqref{16} on~$\Delta_K$, it follows
that \epsqref{16} also holds on the boundary of~$\-\Delta_K$.
\epsnd{proof}
\subsection{Proof of Theorem~\ref{THM3}}
\begin{proof}[Proof of Theorem~\ref{THM3}] Since $h\in\mathrm{BC}^{2,1}(\-\Delta_K)$, Theorem~\ref{THM5} implies that there is a function $f\in\mathrm{BC}^{2,1}(\-\Delta_K)$ solving \epsqref{16}. Exchangeability implies
\bes{
0&=\tsfrac{1}{2}\mathrm{I}E[ (\textnormal{\textbf{W}}'-\textnormal{\textbf{W}})^t \Lambda^{-t} (\nabla f(\textnormal{\textbf{W}}')+\nabla f(\textnormal{\textbf{W}}))] \\
&=\mathrm{I}E[ (\textnormal{\textbf{W}}'-\textnormal{\textbf{W}})^t \Lambda^{-t}\nabla f(\textnormal{\textbf{W}}))] +\tsfrac{1}{2}\mathrm{I}E[ (\textnormal{\textbf{W}}'-\textnormal{\textbf{W}}) ^t\Lambda^{-t} (\nabla f(\textnormal{\textbf{W}}')-\nabla f(\textnormal{\textbf{W}}))],
}
and applying the linearity condition~\epsqref{7} yields
\bes{
&\mathrm{I}E [(\ta-s\textnormal{\textbf{W}})^t \nabla f(\textnormal{\textbf{W}})] \\
&\qquad= -\tsfrac{1}{2}\mathrm{I}E[ (\textnormal{\textbf{W}}'-\textnormal{\textbf{W}})^t \Lambda^{-t} (\nabla f(\textnormal{\textbf{W}}')-\nabla f(\textnormal{\textbf{W}}))]-\mathrm{I}E [\tR^t\Lambda^{-t}\nabla f(\textnormal{\textbf{W}})].
}
By the fundamental theorem of calculus,
\bes{
f_i(\textnormal{\textbf{w}}') & = f_i(\textnormal{\textbf{w}}) + \int_{0}^1\sum_{j=1}^{K-1}(w'_j-w_j)f_{ij}(\textnormal{\textbf{w}}+(\textnormal{\textbf{w}}'-\textnormal{\textbf{w}})t)dt \\
& = f_i(\textnormal{\textbf{w}}) + \sum_{j=1}^{K-1}(w'_j-w_j)f_{ij}(\textnormal{\textbf{w}}) + \sum_{j=1}^{K-1}\int_{0}^1(w'_j-w_j)\bklr{f_{ij}(\textnormal{\textbf{w}}+(\textnormal{\textbf{w}}'-\textnormal{\textbf{w}})t)-f_{ij}(\textnormal{\textbf{w}})}dt.
}
Since~$f_{ij}$ is Lipschitz continuous,
\be{
\babs{f_{ij}(\textnormal{\textbf{w}}+(\textnormal{\textbf{w}}'-\textnormal{\textbf{w}})t)-f_{ij}(\textnormal{\textbf{w}})}
\leq \abs{f}_{2,1}t\sum_{k=1}^{K-1}\abs{w'_k-w_k};
}
hence, there are~$\tilde{Q}_{ijk} = \tilde{Q}_{ijk}(\textnormal{\textbf{w}},\textnormal{\textbf{w}}',f)$ such that~$\abs{\tilde{Q}_{ijk}}\leq \abs{f}_{2,1}$ and
\bes{
&(\textnormal{\textbf{w}}'-\textnormal{\textbf{w}})^t \Lambda^{-t} (\nabla f(\textnormal{\textbf{w}}')-\nabla f(\textnormal{\textbf{w}})) \\
&\qquad= \sum_{m,i,j} (\Lambda^{-1})_{i,m} (w_m'-w_m)(w_j'-w_j) f_{i j}(\textnormal{\textbf{w}}) \\
&\qquad\qquad+\frac{1}{2}\sum_{m,i,j,k} (\Lambda^{-1})_{i,m} (w_m'-w_m)(w_j'-w_j)(w_k'-w_k)\tilde{Q}_{ijk}.
}
Combining the previous three displays, we have
\ban{
&\mathrm{I}E\left[ \sum_{i,j=1}^{K-1} W_i(\deltaelta_{i j}-W_j) f_{i j}(\textnormal{\textbf{W}})+\sum_{i=1}^{K-1}(a_i-s W_i) f_i(\textnormal{\textbf{W}})\right] \notag \\
&\quad=\mathrm{I}E\left[\sum_{i,j=1}^{K-1} \left(W_i(\deltaelta_{i j}-W_j) -\frac{1}{2}\sum_{m=1}^{K-1}(\Lambda^{-1})_{i,m} (W_m'-W_m)(W_j'-W_j) \right)f_{i j}(\textnormal{\textbf{W}})\right]\notag \\
&\qquad\quad-\frac{1}{2}\sum_{m,i,j,k}(\Lambda^{-1})_{i,m} \mathrm{I}E\left[(W_m'-W_m)(W_j'-W_j)(W_k'-W_k)\tilde{Q}_{ijk}\right] \notag \\
&\qquad\quad\qquad -\mathrm{I}E \left[\sum_{i,j} R_j(\Lambda^{-1})_{i,j}f_i(\textnormal{\textbf{W}})\right].\notag
}
We can further simplify the first summand above to
\bes{
\mathrm{I}E\left[\sum_{i,j,m} (\Lambda^{-1})_{i,m} \left( \Lambda_{m,i} W_i(\deltaelta_{i j}-W_j) -\frac{1}{2} (W_m'-W_m)(W_j'-W_j) \right)f_{i j}(\textnormal{\textbf{W}})\right],
}
and now the theorem follows from judicious use of the triangle inequality and the bound \epsqref{19} from Theorem~\ref{THM5}.
If~$\Lambda$ is a multiple of the identity matrix
then, following ideas of \citep*{Rollin2008},
the proof is nearly identical but started from
\bes{
f(\textnormal{\textbf{W}})-f(\textnormal{\textbf{W}}')&=\sum_{i=1}^{K-1} (W_i'-W_i) f_i(\textnormal{\textbf{W}})+\frac{1}{2}\sum_{i,j=1}^{K-1} (W_i'-W_i)(W_j'-W_j) f_{i j}(\textnormal{\textbf{W}}) \\
&\qquad+\frac{1}{6} \sum_{i,j,k=1}^{K-1}(W_i'-W_i)(W_j'-W_j) (W_k'-W_k) \tilde S_{ijk},
}
where~$\tilde S_{ijk}=\tilde S_{ijk}(\textnormal{\textbf{W}},\textnormal{\textbf{W}}',f)$ satisfies $\abs{\tilde S_{ijk}}\leq \abs{f}_{2,1}$.
From here, the proof follows as above by taking expectation, noting that~$\mathrm{I}E[f(\textnormal{\textbf{W}})-f(\textnormal{\textbf{W}}')]=0$ (since~$\mathscr{L}(\textnormal{\textbf{W}})=\mathscr{L}(\textnormal{\textbf{W}}')$)
and that the expectation of the first term on the right hand side above can be simplified using the linearity condition~\epsqref{7}.
The bound on the convex set distance \epsqref{9} directly follows from \epsqref{8} and Lemma~\ref{lem9}.
\epsnd{proof}
\subsection{Auxiliary results}
In what follows, we define, as usual,~$\Delta_\ty g(\tx)=g(\tx +\ty)-g(\tx)$ and denote by~$\te_i$ the~$i$th unit vector in~$\mathrm{I}R^n$ .
\begin{lemma} \label{lem2}
Let~$U\subset\mathrm{I}R^n$ be a convex open set, and let~$g\in\mathrm{BC}^{m,1}(U)$ for some~$m\geq 0$. Let~$\tx\in U$, let~$1\leq k\leq m+1$, and let~$\ty\s 1, \ldots, \ty\s k\in\mathrm{I}R^n$ be such that~$\tx + \sum_{i=1}^j \ty\s{j} \in U$ for all~$1\leq j\leq k$.
Then, if~$k\leq m$,
\be{
\left|\left(\prod_{i=1}^k \Delta_{\ty\s i}\right) g(\tx)\right|\leq \abs{g}_{k} \prod_{i=1}^k \norm{\ty\s i}_1,
}
and if~$k=m+1$, the same estimate holds with~$\abs{g}_{k}$ replaced by~$\abs{g}_{m,1}$ on the right hand side.
\epsnd{lemma}
\begin{proof} Assume~$k\leq m$.
Applying the easy identity
\be{
g(\tx +\ty)-g(\tx)=\int_0^1 \sum_{i=1}^n \frac{\partial g}{\partial x_i} (\tx + t\ty) \ty_i dt
}
repeatedly~$k$ times yields
\ben{\label{25}
\left(\prod_{i=1}^k \Delta_{\ty\s i}\right) g(\tx)=\int_{[0,1]^k} \sum_{i_1, \ldots, i_k=1}^{n}\frac{\partial^k g}{\prod_{ j =1}^k \partial x_{i_j }} \left(\tx + \sum_{j=1}^k \ty\s{j} t_j\right)\prod_{j=1}^k y\s{j}_{i_j} d\textbf{t}.
}
Thus
\be{
\left|\left(\prod_{i=1}^k \Delta_{\ty\s i} \right) g(\tx)\right| \leq \abs{g}_k \sum_{i_1, \ldots, i_k=1}^{n} \prod_{j=1}^k \abs{ y\s{j}_{i_j}} = \abs{g}_k \prod_{i=1}^k \norm{\ty\s i}_1.
}
For~$k=m+1$, use~\epsqref{25} for $k=m$ to find
\bes{
\left(\prod_{i=1}^{m+1} \Delta_{\ty\s i}\right) g(\tx)&=\int_{[0,1]^m} \sum_{i_1, \ldots, i_m=1}^{n}\frac{\partial^m \Delta_{\ty\s{m+1}}g}{\prod_{ j =1}^m \partial x_{i_j }} \left(\tx + \sum_{j=1}^m \ty\s{j} t_j\right)\prod_{j=1}^k y\s{j}_{i_j} d\textbf{t} \\
&=\int_{[0,1]^m} \sum_{i_1, \ldots, i_m=1}^{n}\Delta_{\ty\s{m+1}}\frac{\partial^m g}{\prod_{ j =1}^m \partial x_{i_j }} \left(\tx + \sum_{j=1}^m \ty\s{j} t_j\right)\prod_{j=1}^k y\s{j}_{i_j} d\textbf{t}.
}
Since the $m+1$ partials are Lipschitz, we find
\be{
\bbbbabs{\Delta_{\ty\s{m+1}}\frac{\partial^m g}{\prod_{ j =1}^m \partial x_{i_j }}\left(\tx + \sum_{j=1}^m \ty\s{j} t_j\right)}
\leq \abs{g}_{m,1} \norm{\ty\s{m+1}}_1,
}
and the result now easily follows by combining this with the previous display.
\epsnd{proof}
\begin{lemma}\label{lem3} Let~$U\subset\mathrm{I}R^n$ be an open convex set, and let~$g:U\to\mathrm{I}R$ be a function. Then,~$g$ is~$M$-Lipschitz continuous with respect to the~$L_1$-norm, if and only if it is
coordinate-wise~$M$-Lipschitz continuous; that is,
\be{
\sup_{x\in U}\sup_u \frac{\abs{\Delta_u g(x)}}{\abs{u}}\leq M.
}
\epsnd{lemma}
\begin{proof} It is clear that if~$g$ is~$M$-Lipschitz continuous, then it is in particular~$M$-Lipschitz continuous in each coordinate. The reverse direction is easily proved using convexity and a telescoping sum argument along the coordinates.
\epsnd{proof}
We were not able to locate the next two lemmas in the literature; hence, we give self-contained proofs.
There is strong resemblance with the theory of \epsmph{bounded~$k$-th variation}, see for example \citep*[Theorem~11]{Russell1973}, but we were not able to find a result that would directly apply to our situation; we also refer to recent survey textbooks \citep*{Mukhopadhyay2012} and \citep*{Appell2014}.
In what follows, we assume that~$u$ and~$v$ appearing in terms like~$\Delta_u f(z)$,~$\Delta_u\Delta_v f(z)$,~$\Delta_{u\varepsilon_i} g(\tx)$ and~$\Delta_{u\varepsilon_i}\Delta_{v\varepsilon_j} g(\tx)$ are such that~$z+u$,~$z+u+v$,~$\tx+u\te_i$ and~$\tx+u\te_i+v\te_j$ are within the domains of the functions being evaluated.
\begin{lemma}\label{lem4} Let~$f:(a,b)\to\mathrm{I}R$ be a function. If
\ben{\label{26}
M_1:=\sup_{z}\sup_{u}\frac{\abs{\Delta_uf(z)}}{\abs{u}}<\infty, \qquad
M_2:=\sup_{z}\sup_{ u,v}\frac{\abs{\Delta_u\Delta_v f(z)}}{\abs{uv}}<\infty,
}
then~$f$ is differentiable and~$f'$ is~$M_2$-Lipschitz-continuous.
\epsnd{lemma}
\begin{proof}
Since, by the first condition of~\epsqref{26}, $f$ is $M_1$-Lipschitz, Rademacher's theorem implies that there is a dense set $E\subset (a,b)$ on which $f$ has a derivative $f'$. On $E$, the second condition of~\epsqref{26} implies that $f'$ is $M_2$-Lipschitz, and so by Kirszbraun's theorem,~$f'$ can be extended to an $M_2$-Lipschitz function $\tilde f'$ on $(a,b)$. We show that for $x\not\in E$, $\tilde f'(x)$ is in fact the derivative of $f$ at~$x$. Fix $\varepsilon>0$. Let $x'\in E$ such that $\abs{x'-x}<\varepsilon/(3M_2)$, such that $\abs{h^{-1}\Delta_hf(x')-f'(x')}\leq \varepsilon/3$, and such that $\abs{\tilde f'(x')-\tilde f'(x)}< \varepsilon/3$. Then for any $0<h\leq \varepsilon$,
\ba{
\bbbabs{\frac{\Delta_h f(x)}{h}- \tilde f'(x)}
&\leq \bbbabs{\frac{\Delta_h f(x)}{h}-\frac{\Delta_h f(x')}{h}}
+ \bbbabs{\frac{\Delta f(x')}{h}-\tilde f'(x')}
+ \babs{\tilde f'(x')-\tilde f'(x)} \\
&= \bbbabs{\frac{\Delta_{x-x'}\Delta_h f(x')}{h}}
+ \bbbabs{\frac{\Delta f(x')}{h}- f'(x')}
+ \babs{\tilde f'(x')-\tilde f'(x)} \\
&\leq \frac{\varepsilon}{3} + \frac{\varepsilon}{3}+ \frac{\varepsilon}{3}=\varepsilon.
}
Hence, $\lim_{h\to0}h^{-1}\Delta_h f(x) = \tilde f'(x)$, as desired.
\epsnd{proof}
\begin{lemma}\label{lem5} Let~$U\subset\mathrm{I}R^n$ be an open convex set, let~$g:U\to\mathrm{I}R$ be a bounded function, and let~$m\geq 0$. If, for each~$1\leq k\leq m+1$, there is a constant~$M_k<\infty$, such that, for each set of indices~$1\leq i_1,\deltaots,i_k\leq n$,
\ben{\label{31}
\sup_{x\in U }\sup_{u_1,\deltaots,u_k}\frac{\abs{\Delta_{u_1\te_{i_1}}\cdots\Delta_{u_k\te_{i_k}} g(x)}}{\abs{u_1\cdots u_k}} \leq M_k,
}
then~$g\in \mathrm{BC}^{m,1}(U)$
and
\ben{\label{32}
\abs{g}_{k,1}\leq M_{k+1},\qquad 0\leq k\leq m.
}
\epsnd{lemma}
\begin{proof} If~$m=0$, the result is immediate since \epsqref{31} is just the coordinate-wise~$M_1$-Lipschitz condition, which implies that~$g$ is Lipschitz, and \epsqref{32} follows from Lemma~\ref{lem3}. Now, assume~$m\geq 1$. Fix a set of~$m$ indices~$1\leq i_1,\deltaots,i_m\leq n$. We proceed by induction and start with~$k=1$. Let~$\tx= (x_1,\deltaots,x_n)\in U~$, let~$a<x_i<b$ such that
\be{
t(x_1,\deltaots,x_{i_1-1},a,x_{i_1-1},\deltaots,x_n)
+ (1-t)(x_1,\deltaots,x_{i_1-1},b,x_{i_1-1},\deltaots,x_n) \in U, \qquad 0\leq t\leq1,
}
and, with~$\tx_z = (x_1,\deltaots,x_{i_1-1},z,x_{i_1-1},\deltaots,x_n)$, let~$f(z) = g(\tx_z)$. By the assumptions on~$g$, we have
\bg{
\sup_{z}\sup_{u}\frac{\abs{\Delta_{u}f(z)}}{\abs{u}}
= \sup_{z}\sup_{u_1}\frac{\abs{\Delta_{u_1\eps_{i_1}}g(\tx_z)}}{\abs{u_1}}\leq M_1<\infty,\\
\sup_{z}\sup_{u,v}\frac{\abs{\Delta_{u}\Delta_{v}f(z)}}{\abs{uv}}
=\sup_{z}\sup_{u_1,u_2}\frac{\abs{\Delta_{u_1\eps_{i_1}}\Delta_{u_2\eps_{i_1}}g(\tx_z)}}{\abs{u_1u_2}}\leq M_2<\infty
}
(note that in the second expression, the second difference is also in the direction~$\eps_{i_1}$)
so that the conditions \epsqref{26} are satisfied. Applying Lemma~\ref{lem4}, we conclude that~$f'(z)=\frac{\partial}{\partial x_{i_1}} g(\tx_z)$ exists and that it is~$M_2$-Lipschitz continuous in direction~$i_1$, but the same argument there together with \epsqref{31} yields~$M_2$-Lipschitz continuity in any other direction, so that by Lemma~\ref{lem3},~$\frac{\partial}{\partial x_{i_1}} g(\tx)$ is~$M_2$-Lipschitz. Since~$\tx$ was arbitrary,~$\frac{\partial}{\partial x_{i_1}} g(\tx)$ exists in all of~$U$ and is~$M_2$-Lipschitz, which concludes the base case.
Assume now that~$1<k<m$ and that~$\frac{\partial^{k-1}}{\partial x_{i_1}\cdots\partial x_{i_{k-1}}} g(\tx)$ exists in all of~$U$. Let~$x\in U$, let~$a$,~$b$ and~$\tx_z$ be as before, and let~$f(z) = \frac{\partial^{k-1}}{\partial x_{i_1}\cdots\partial x_{i_{k-1}}} g(\tx_z)$.
From the assumptions on~$g$ and since~$\frac{\partial^{k-1}}{\partial x_{i_1}\cdots\partial x_{i_{k-1}}} g(\tx)$ exists, we have
\bg{
\sup_{z}\sup_{u}\frac{\abs{\Delta_{u}f(z)}}{\abs{u}}
= \sup_{z}\sup_{u_{k}}\bbbabs{\lim_{u_1\to0}\cdots\lim_{u_{k-1}\to0}\frac{\Delta_{u_1\te_{i_1}}\cdots\Delta_{u_{k}\te_{i_{k}}}g(\tx_z)}{u_1\deltaots u_{k}}} \leq M_{k} < \infty \\
\sup_{z}\sup_{u,v}\frac{\abs{\Delta_{u}\Delta_v f(z)}}{\abs{uv}}
= \sup_{z}\sup_{u_{k},u_{k+1}}\bbbabs{\lim_{u_1\to0}\cdots\lim_{u_{k-1}\to0}\frac{\Delta_{u_1\te_{i_1}}\cdots\Delta_{u_{k}\te_{i_{k}}}\Delta_{u_{k+1}\te_{i_{k}}}g(\tx_z)}{u_1\deltaots u_{k}u_{k+1}}} \leq M_{k+1} < \infty
}
so that the conditions \epsqref{26} are satisfied. Applying Lemma~\ref{lem4}, we conclude that~$f'(z)=\frac{\partial^{k}}{\partial x_{i_1}\cdots\partial x_{i_{k}}} g(\tx_z)$ exists
and that it is~$M_{k+1}$-Lipschitz continuous in direction~$i_k$, but the same argument there together with \epsqref{31} yields~$M_{k+1}$-Lipschitz continuity in any other direction, so that by Lemma~\ref{lem3},~$\frac{\partial^{k}}{\partial x_{i_1}\cdots\partial x_{i_{k}}} g(\tx_z)$ is~$M_{k+1}$-Lipschitz. Since~$\tx$ was arbitrary,~$\frac{\partial^{k}}{\partial x_{i_1}\cdots\partial x_{i_{k}}} g(\tx_z)$ exists in all of~$U$ and is~$M_{k+1}$-Lipschitz, which concludes the induction step.
\epsnd{proof}
The following is a specialisation of \citep*[Lemma~2.1]{Bentkus2003} to the convex set metric; we need some notation first. Let~$A\subset \mathrm{I}R^K$ be convex, let~$d(\tx,A) = \inf_{\ty\in A}|\tx-\ty|$, and define the sets
\ben{\label{33}
A^\varepsilon = \{\tx\in \mathrm{I}R^K\,:\,d(\tx,A)\leq \varepsilon\},\qquad A^{-\varepsilon} = \{\tx\in A\,:\, B(\tx;\varepsilon)\subset A\},
}
where~$B(\tx;\varepsilon)$ is the closed ball of radius~$\varepsilon$ around~$\tx$.
\deltaef\mathcal{C}_{K-1}{\mathcal{C}_{K-1}}
\begin{lemma}[{\citep*[Lemma~2.1]{Bentkus2003}}] \label{lem6} Let~$\mathcal{C}_K$ be the family of convex sets of\/~$\mathrm{I}R^K$, and for fixed~$\varepsilon>0$, let~$\{\varphi_{\varepsilon,A}; \, A\in\mathcal{C}_K\}$ be a family of functions satisfying
\ben{\label{34}
0\leq \varphi_{\varepsilon,A} \leq 1,
\qquad
\text{$\varphi_{\varepsilon,A}(\tx) = 1$ for~$\,\tx\in A$},
\qquad
\text{$\varphi_{\varepsilon,A}(\tx) = 0$ for~$\,\tx\not\in A^{\varepsilon}$.}
}
Then, for any two random vectors~$\tX$ and~$\textnormal{\textbf{Y}}$,
\bes{
&\sup_{A\in\mathcal{C}_K}\babs{\mathrm{I}P[\tX\in A]-\mathrm{I}P[\textnormal{\textbf{Y}}\in A]} \\
&\quad \leq \sup_{A\in\mathcal{C}_K}\babs{\mathrm{I}E\varphi_{\varepsilon,A}(\tX)-\mathrm{I}E\varphi_{\varepsilon,A}(\textnormal{\textbf{Y}})}
+ \sup_{A\in\mathcal{C}_K}\max\bklg{\mathrm{I}P[\textnormal{\textbf{Y}}\in A\setminus A^{-\varepsilon}],\mathrm{I}P[\textnormal{\textbf{Y}}\in A^\varepsilon\setminus A]}
}
\epsnd{lemma}
\deltaef\mathcal{S}{\mathcal{S}}
\begin{lemma}[Smoothing operator] \label{lem7}Let~$f:\mathrm{I}R^n\to \mathrm{I}R$ be a bounded and Lebesgue measurable function. For~$\varepsilon>0$, define the smoothing operator~$\mathcal{S}_\varepsilon$ as
\be{
(\mathcal{S}_\varepsilon f)(\tx) = \frac{1}{(2\varepsilon)^n}\int\limits_{x_1-\varepsilon}^{x_1+\varepsilon}\cdots\int\limits_{x_n-\varepsilon}^{x_n+\varepsilon}f(z)\,dz_n\cdots dz_1.
}
Then, for any $m\geq 1$, we have that $\mathcal{S}_\varepsilon^m f\in\mathrm{BC}^{m-1,1}(\mathrm{I}R^n)$, and for fixed~$\tx\in\mathrm{I}R^n$,~$(\mathcal{S}_\varepsilon^ m f)(\tx)$ does not depend on~$f(\ty)$,~$\ty\in\mathrm{I}R^n\setminus B(\tx;mn^{1/2}\varepsilon)$. Moreover, we have the bounds
\ben{\label{35}
\norm{\mathcal{S}^m_\varepsilon f}_{\infty}\leq\norm{f}_\infty,
\qquad
\abs{\mathcal{S}^m_\varepsilon f}_{k-1,1}\leq \frac{\norm{f}_\infty}{\varepsilon^k}, \quad 1\leq k\leq m.
}
\epsnd{lemma}
\begin{proof} The claim that $f(\tx)$ does not depend on $f(\ty)$,~$\ty\in\mathrm{I}R^n\setminus B(\tx;mn^{1/2}\varepsilon)$, is a straightforward consequence of the definition, as is the bound
\ben{\label{36}
\norm{\mathcal{S}_\varepsilon f}_\infty \leq \norm{f}_\infty.
}
Now, it is easy to see that for $u>0$ and $1\leq i\leq n$,
\bes{
\abs{\Delta_{u\te_{i}}\mathcal{S}_\varepsilon f (\tx)}
\leq \begin{cases}
\deltaisplaystyle2\norm{f}_\infty& \text{if $u>2\varepsilon$,}\\[2ex]
\deltaisplaystyle\frac{u\norm{f}_\infty}{\varepsilon}& \text{if $u\leq 2\varepsilon$,}\\
\epsnd{cases}
}
so that $\abs{\Delta_{u\te_{i}}\mathcal{S}_\varepsilon f (\tx)}\leq u\norm{f}_{\infty}/\varepsilon$ for all $x$ and all $u$, which implies that
\ben{\label{37}
\bbbnorm{\frac{\Delta_{u\te_{i}}\mathcal{S}_\varepsilon f}{u}}_\infty
\leq \frac{\norm{f}_\infty}{\varepsilon}.
}
Fix $1\leq k\leq m$, $u_1,\deltaots,u_k>0$ and $1\leq i_1,\deltaots,i_k\leq n$. Noting that~$\Delta_{u\te_i}\mathcal{S}_\varepsilon g = \mathcal{S}_\varepsilon \Delta_{u\te_i}g$, we can write
\be{
\Delta_{u_1\te_{i_1}}\cdots\Delta_{u_k\te_{i_k}}\mathcal{S}_\varepsilon^m = (\Delta_{u_1\te_{i_1}}\mathcal{S}_\varepsilon)\cdots(\Delta_{u_k\te_{i_k}}\mathcal{S}_\varepsilon) \mathcal{S}_\varepsilon^{l-m}.
}
Applying \epsqref{37} repeatedly~$k$ times and if $k<m$ applying in addition \epsqref{36}, we obtain \epsqref{31} with $M_k = \norm{f}/\varepsilon^k$, so that the claim follows from Lemma~\ref{lem5}.
\epsnd{proof}
\begin{lemma}\label{lem8} Let~$\varepsilon>0$, and let~$A\subset \mathrm{I}R^n$ be convex. There exists a function $\varphi=\varphi_{\varepsilon,A}\in\mathrm{BC}^{2,1}(\mathrm{I}R^n)$ satisfying~\epsqref{34} with
\ben{\label{38}
\abs{\varphi}_1\leq \frac{9n^{1/2}}{\varepsilon},
\qquad
\abs{\varphi}_2\leq \frac{81n}{\varepsilon^2},
\qquad
\abs{\varphi}_{2,1}\leq \frac{729n^{3/2}}{\varepsilon^3}.
}
\epsnd{lemma}
\begin{proof} Let~$\deltaelta = \frac{\varepsilon}{9\sqrt{n}}$. Define
\be{
\varphi(\tx) = \mathcal{S}_\deltaelta^3 I_{A^{\varepsilon/3}}(\tx);
}
the claim then follows from Lemma~\ref{lem7}.
\epsnd{proof}
\begin{lemma}\label{lem9} Let~$\mathcal{C}_{K-1}$ be the class of convex sets on~$\mathrm{I}R^{K-1}$. Let~$\tZ\sim \Dir(a_1,\deltaots,a_K)$ and assume
\ben{\label{39}
\abs{\mathrm{I}E h(\textnormal{\textbf{W}}) - \mathrm{I}E h(\tZ)} \leq c_0\abs{h}_0 + c_1\abs{h}_1 + c_2\abs{h}_2 + c_3\abs{h}_{2,1},
}
for any $h\in\mathrm{BC}^{2,1}(\-\Delta_K)$.
Then there is a constant~$C>0$ depending only on~$a_1,\deltaots,a_K$ such that
\ben{\label{40}
\sup_{A\in \mathcal{C}_{K-1}}\abs{\mathrm{I}P[\textnormal{\textbf{W}}\in A]-\mathrm{I}P[\tZ\in A]} \leq c_0+C(c_1+c_2+c_3)^{\theta/(3+\theta)},
}
where
\ben{
\theta = \frac{\theta_\wedge}{\theta_\wedge+\theta_\circ},\qquad \theta_\wedge = 1\wedge\min\{a_1,\deltaots,a_K\},\qquad \theta_\circ=\sum_{i=1}^K\bklr{1-1\wedge a_i}.
}
\epsnd{lemma}
\begin{proof} Since both $\textnormal{\textbf{W}}$ and $\tZ$ take values in $\-\Delta_K$, we may assume without loss of generality that~$A\subset \-\Delta_K$.
Fix~$\varepsilon>0$; from Lemma~\ref{lem6} we have
\bes{
& \sup_{A\in \mathcal{C}_{K-1}}\abs{\mathrm{I}P[\textnormal{\textbf{W}}\in A]-\mathrm{I}P[\tZ\in A]} \\
& \qquad\leq \sup_{A\in \mathcal{C}_{K-1}}\abs{\mathrm{I}E \varphi_{\varepsilon,A}(\textnormal{\textbf{W}}) - \mathrm{I}E \varphi_{\varepsilon,A}(\tZ)} + \sup_{A\in \mathcal{C}_{K-1}} \mathrm{I}P[\tZ\in A^\varepsilon\setminus A]\vee \mathrm{I}P[\tZ\in A\setminus A^{-\varepsilon}] \\
& =: R_1 + R_2,
}
where the $\varphi_{\varepsilon,A}$ are chosen as in Lemma~\ref{lem8}. Using \epsqref{39} and \epsqref{38}
\be{
R_1\leq c_0 + \frac{9(K-1)^{1/2}c_1}{\varepsilon} + \frac{81(K-1)c_2}{\varepsilon^2} + \frac{729(K-1)^{3/2}c_3}{\varepsilon^3}.
}
In order to bound~$R_2$ we proceed as follows. Let~$\deltaelta\geq\varepsilon$ (to be chosen later), and consider~$\-\Delta^{-\deltaelta}_{K-1}$, the~$\deltaelta$-shrinkage of~$\-\Delta_{K-1}$ as defined in~\epsqref{33}.
For given convex~$A\subset \-\Delta_{K-1}$, let~$A_\circ = A \cap \-\Delta^{-\deltaelta}_{K-1}$ (which is again convex) and note that
\ba{
\mathrm{I}P[\tZ\in A^\varepsilon\setminus A] & \leq \mathrm{I}P[\tZ\in \-\Delta_{K-1}\setminus\-\Delta_{K-1}^{-\deltaelta}] + \mathrm{I}P[\tZ\in A^{\varepsilon}_\circ\setminus A_\circ], \\
\mathrm{I}P[\tZ\in A\setminus A^{-\varepsilon}] & \leq \mathrm{I}P[\tZ\in \-\Delta_{K-1}\setminus\-\Delta_{K-1}^{-\deltaelta}] + \mathrm{I}P[\tZ\in A_\circ\setminus A^{-\varepsilon}_\circ],
}
so that
\be{
\mathrm{I}P[\tZ\in A^\varepsilon\setminus A]\vee \mathrm{I}P[\tZ\in A\setminus A^{-\varepsilon}]\leq
\mathrm{I}P[\tZ\in \-\Delta_{K-1}\setminus\-\Delta_{K-1}^{-\deltaelta}] + \mathrm{I}P[\tZ\in A_\circ^\varepsilon\setminus A_\circ]\vee \mathrm{I}P[\tZ\in A_\circ\setminus A_\circ^{-\varepsilon}].
}
Using a union bound and the fact that the marginals of~$\tZ$ have beta distributions,
\bes{
\mathrm{I}P\bkle{\tZ\in \-\Delta_{K-1}\setminus\-\Delta^{-\deltaelta}_{K-1}}
&\leq \sum_{i=1}^K\bklr{\mathrm{I}P[Z_i\leq \deltaelta] + \mathrm{I}P[Z_i\geq 1-\deltaelta]} \\
& \leq \sum_{i=1}^K\frac{\Gamma(s)}{\Gamma(a_i)\Gamma(s-a_i)}\bbklr{\frac{\deltaelta^{a_i}}{a_i}+\frac{\deltaelta^{s-a_i}}{s-a_i}}
\leq C\deltaelta^{\theta_\wedge}.
}
Now, the density~$\psi_\ta$ of the Dirichlet distribution (see~\epsqref{1}), restricted to~$\-\Delta^{-\deltaelta}_{K-1}$, is bounded by
\ben{\label{41}
\bbnorm{\psi_\ta\big\vert_{\-\Delta^{-\deltaelta}_{K-1}}} \leq \frac{\Gamma(s)}{\prod_{i=1}^K \Gamma(a_i)} \deltaelta^{-\theta_\circ}.
}
From Steiner's formula for convex bodies, which describes the volume of~$\varepsilon$-enlargements of convex bodies (see e.g.\ \citep*[Theorem~46]{Morvan2008}), the Hausdorff-continuity of the corresponding coefficients in Steiner's formula (so called \epsmph{Quermassintegrale}; see \citep*[Theorem~50]{Morvan2008}) and compactness of~$\-\Delta_{K-1}$, and the bound \epsqref{41}, we conclude that there is a constant~$S_K$ that only depends on the dimension~$K$ such that, for convex~$A\subset \-\Delta^{-\deltaelta}_{K-1}$,~$\mathop{\mathrm{Vol}}(A^\varepsilon\setminus A)\leq \varepsilon S_K$ and~$\mathop{\mathrm{Vol}}(A\setminus A^{-\varepsilon})\leq \varepsilon S_K$, so that if~$\deltaelta>\varepsilon$,
\be{
\mathrm{I}P[\tZ\in A_\circ^\varepsilon\setminus A_\circ]\vee \mathrm{I}P[\tZ\in A_\circ\setminus A_\circ^{-\varepsilon}]\leq \frac{\Gamma(s)}{\prod_{i=1}^K \Gamma(a_i)}(\deltaelta-\varepsilon)^{-\theta_\circ}\cdot \varepsilon S_K
}
(note that an upper bound on~$S_K$ could be obtained in principle by evaluating the coefficients in the Steiner formula for the convex set~$\-\Delta_{K-1}$).
Note that in the previous display if~$\theta_\circ=0$ then the inequality still
holds without the factor of~$(\deltaelta-\varepsilon)$ even if~$\deltaelta=\varepsilon$.
Thus we have
\be{
\mathrm{I}P[\tZ\in A^\varepsilon\setminus A]\vee \mathrm{I}P[\tZ\in A\setminus A^{-\varepsilon}]\leq C \bkle{\deltaelta^{\theta_\wedge} + \varepsilon\bklr{\mathrm{I}[\theta_\circ>0](\deltaelta-\varepsilon)^{-\theta_\circ}
+\mathrm{I}[\theta_\circ=0]}}.
}
Choosing~$\deltaelta = \varepsilon^{1/(\theta_\wedge+\theta_\circ)}$,
we have that~$\deltaelta\geq\varepsilon$,
and~$\deltaelta=\varepsilon$ only if~$\theta_\circ=0$, so that
\be{
\sup_{A\in \mathcal{C}_{K-1}}\abs{\mathrm{I}P[\textnormal{\textbf{W}}\in A]-\mathrm{I}P[\tZ\in A]} \leq C\bbklr{c_0 + \frac{c_1}{\varepsilon} + \frac{c_2}{\varepsilon^2} + \frac{c_3}{\varepsilon^3} + \varepsilon^\theta},
}
for some constant~$C=C(\ta)$.
Without loss of generality we may assume that~$C\geq 1$ in \epsqref{40} so that if~$c_1+c_2+c_3 \geq 1$, then~\epsqref{40} is trivially true. If~$c_1+c_2+c_3 < 1$, choose~$\varepsilon = (c_1+c_2+c_3)^{1/\klr{3+\theta}}<1$ and bound both~$1/\varepsilon$ and~$1/\varepsilon^2$ by~$1/\varepsilon^3$; this again yields \epsqref{40}.
\epsnd{proof}
\section{Proof of Theorem~\ref{THM1}: Wright-Fisher model}\label{sec2}
Recall the description in the introduction of the Wright-Fisher model with neutral mutation in a haploid population of constant size~$N$.
The process is driven by offspring vector having distribution~$\mathrm{MN}(N;1/N, \ldots, 1/N)$, and the
mutation structure is general with~$K$ types. The process is a time-homogeneous Markov chain~$\tX(0), \tX(1), \ldots$, where~$\tX(n)$ is a~$(K-1)$ dimensional vector that represents the counts of the first~$K-1$ alleles in the population, so~$\tX(n)/N \in \-\Delta_K$.
Since~$(\tX(n))_{n\geq0}$ is a Markov chain on a finite state space, it has a stationary distribution, and we apply Theorem~\ref{THM3} to prove the
bound on the approximation of this stationary distribution by the Dirichlet distribution given by Theorem~\ref{THM1}.
To define a stationary pair~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$, let~$\tX$ be distributed as a stationary distribution of the chain of Theorem~\ref{THM1}
and let~$\tX'$ be a step in the chain from~$\tX$.
Set~$\textnormal{\textbf{W}}=\tX/N$ and~$\textnormal{\textbf{W}}'=\tX'/N$.
It is not difficult to see that
the distribution of~$\tX'$ given~$\tX$ is the first~$K-1$ coordinates of a multinomial with~$N$ trials with success probabilities given by the vector
$\tq(\tX)$, where
\ben{\label{42}
q_j(\tX)
=\sum_{k=1}^Kp_{ k j}\frac{X_ k}{N} = \sum_{k=1}^{K-1} p_{ k j}\frac{X_ k}{N} + p_{Kj}\bbbklr{1-\sum_{k=1}^{K-1}\frac{X_k}{N}}.
}
Hence
\be{
\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]
= p_{jj}W_j + \sum_{\substack{k=1\\k\neq j}}^{K-1}p_{ k j} W_k + p_{Kj} - \sum_{k=1}^{K-1}p_{Kj} W_k,
}
so that
\bes{
\mathrm{I}E[W_j'-W_j|\textnormal{\textbf{W}}]
& = -(1-p_{jj})W_j + \sum_{\substack{k=1\\k\neq j}}^{K-1}p_{ k j} W_k + p_{Kj} - \sum_{k=1}^{K-1}p_{Kj} W_k \\
& =\frac{1}{2N}(a_j -sW_j) + R_j(\textnormal{\textbf{W}}),
}
where
\besn{\label{43}
R_j(\textnormal{\textbf{W}})
& = -\frac{a_j}{2N}+ \bbklr{\frac{s}{2N}-(1-p_{jj})}W_j+\sum_{\substack{k=1\\k\neq j}}^{K-1}p_{ k j} W_k + p_{Kj} - \sum_{k=1}^{K-1}p_{Kj} W_k \\
& = \bbklr{p_{Kj}-\frac{a_j}{2N}}(1-W_j)+\bbbklr{\,\sum_{\substack{k=1\\k\neq j}}^K\bbklr{\frac{a_k}{2N}-p_{jk}}}W_j+\sum_{\substack{k=1\\k\neq j}}^{K-1}(p_{ k j}-p_{Kj}) W_k.
}
Thus we are in the setting of Theorem~\ref{THM3} with~$\ta$ as above,~$\Lambda=(2N)^{-1}\times \mathrm{I}d$, and~$\tR$ given by~\epsqref{43}.
Applying the theorem is a relatively straightforward but somewhat tedious calculation involving conditioning and multinomial moment formulas. We need the quantities
\bg{
T_j=p_{K j}+\sum_{ \substack{k=1\\k \neq j}}^{K-1} (p_{ k j}- p_{K j}) W_{ k },\\
\sigma_j= p_{Kj} + \sum_{\substack{k=1\\k\neq j}}^K p_{jk},\qquad \tau_j = p_{Kj} + \sum_{\substack{k=1\\k\neq j}}^K\abs{p_{kj}-p_{Kj}}, \qquad 1\leq j\leq K-1.
}
Note that we can write
\ben{\label{44}
q_j:=q_j(\tX)=W_j(1-\sigma_j)+T_j.
}
We also record the following multinomial moment formula lemma. Let~$(n)_{ k \deltaownarrow}=n(n-1)\cdots(n- k +1)$ denote the falling factorial.
\begin{lemma}\label{lem10}
For~$(\tX, \tX')$ defined above,~$i,j,k\in\{1,\ldots, K-1\}$ all distinct and non-negative integers~$ k _i, k _j, k _k$,
\be{
\mathrm{I}E\left[ \left(X_i'\right)_{ k _i\deltaownarrow}\left(X_j'\right)_{ k _j\deltaownarrow}\left(X_k'\right)_{ k _k\deltaownarrow} \big| \tX \right]
=\left( N \right)_{( k _i+ k _j+ k _k) \deltaownarrow} q_i(\tX)^{ k _i} q_j(\tX)^{ k _j} q_k(\tX)^{ k _k}.
}
\epsnd{lemma}
\begin{lemma}\label{lem11}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above,
\bes{
\mathrm{I}E\left[(W_j'-W_j)^2\big| \textnormal{\textbf{W}} \right]&=W_j^2\left[-\frac{1}{N}+\frac{2\sigma_j}{N} +\sigma_j^2\left(1-\frac{1}{N}\right)\right] \\
&\qquad + W_j\left[\frac{1}{N} - \frac{2T_j}{N}-\frac{\sigma_j}{N}-2T_j\sigma_j\left(1-\frac{1}{N}\right)\right]
+T_j\left[T_j+\frac{1-T_j}{N}\right].
}
\epsnd{lemma}
\begin{proof}
We first expand
\be{
\mathrm{I}E\left[(W_j'-W_j)^2\big| \textnormal{\textbf{W}} \right]=\frac{1}{N^2}\mathrm{I}E\left[ X_j'\left(X_j'-1\right) \big| \textnormal{\textbf{W}}\right]
-\left(2W_j-\frac{1}{N}\right) \mathrm{I}E[W_j'|\textnormal{\textbf{W}}] + W_j^2.
}
Using Lemma~\ref{lem10} and the expression for~$\tq$ given at~\epsqref{44} we find
\be{
\frac{1}{N^2}\mathrm{I}E\left[ X_j'\left(X_j'-1\right) \big| \textnormal{\textbf{W}}\right]=\frac{N-1}{N} \left(W_j \left(1-\sigma_j\right)+T_j\right)^2,
}
and
\be{
\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]=W_j \left(1-\sigma_j\right)+T_j.
}
Combining these last three displays and simplifying yields the result.
\epsnd{proof}
\begin{lemma}\label{lem12}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above, and~$i\not=j$,
\bes{
\mathrm{I}E\left[(W_i'-W_i)(W_j'-W_j) \big| \textnormal{\textbf{W}} \right]&=W_iW_j\left[-\frac{1}{N}+\frac{\sigma_i+\sigma_j}{N}+\sigma_i\sigma_j\left(1-\frac{1}{N}\right)\right] +T_iT_j\left(1-\frac{1}{N}\right) \\
&\qquad
-W_iT_j\left (\sigma_i+\frac{1-\sigma_i}{N}\right) -W_jT_i\left (\sigma_j+\frac{1-\sigma_j}{N}\right).
}
\epsnd{lemma}
\begin{proof}
We first expand
\be{
\mathrm{I}E\left[(W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}\right]=\mathrm{I}E [W_i'W_j' | \textnormal{\textbf{W}}] -W_i\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]-W_j\mathrm{I}E[W_i'|\textnormal{\textbf{W}}]+W_i W_j.
}
Using Lemma~\ref{lem10} and the expression for~$\tq$ given at~\epsqref{44} we find
\be{
\mathrm{I}E [W_i'W_j' | \textnormal{\textbf{W}}]=\frac{N-1}{N} \left(W_i \left(1-\sigma_i\right)+T_i\right) \left(W_j \left(1-\sigma_j\right)+T_j\right),
}
and
\be{
W_i\mathrm{I}E[W_j'|\textnormal{\textbf{W}}]=W_i\left(W_j \left(1-\sigma_j\right)+T_j\right).
}
Combining these last three displays and simplifying yields the result.
\epsnd{proof}
\begin{lemma}\label{lem13}
For~$(\textnormal{\textbf{W}},\textnormal{\textbf{W}}')$ defined above, and~$\lambda=(2N)^{-1}~$,
\ba{
\sum_{i,j=1}^{K-1} \mathrm{I}E &\left| W_i(\deltaelta_{i j}-W_j)-\frac{1}{2 \lambda}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\leq N\sum_{i,j=1}^{K-1}(\sigma_i+\tau_i) \left(\sigma_j+\tau_j+\frac{2}{N}\right).
}
\epsnd{lemma}
\begin{proof}
The lemma follows in a straightforward way from Lemmas~\ref{lem11} and~\ref{lem12}, the triangle inequality, that~$0\leq W_i\leq 1$,
and~$\abs{T_j}\leq \tau_j$.
\epsnd{proof}
\begin{lemma}\label{lem14}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ defined above and~$\lambda=(2N)^{-1}~$,
\ba{
\frac{1}{\lambda}\sum_{i,j,k=1}^{K-1}&\mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)}\\
&\leq \frac{2}{N^{1/2}} \left(\sum_{i=1}^{K-1} \left[\sqrt2+\sqrt N (\tau_i + \sigma_i) \right]\right)^{2} \left(\sum_{i=1}^{K-1} \left[1+\sqrt N (\tau_i + \sigma_i)\right]\right).
}
\epsnd{lemma}
\begin{proof}
Conditional on~$\tX$,~$\tX'$ is distributed as the first~$(K-1)$ entries of a multinomial distribution with~$N$ trials
and success probabilities given by the vector at~\epsqref{44}:
\be{
q_ k :=q_ k (\tX)=\left(W_ k \left(1-\sigma_ k \right)+T_ k \right).
}
Decompose
\ba{
X_i' - X_i&=X_i' -\mathrm{I}E[X_i'| X_i] + \mathrm{I}E[X_i'| X_i] - X_i \\
&= [X_i' - (X_i(1-\sigma_i) + NT_i)] + [NT_i - \sigma_iX_i]\\
&=: E_i + G_i.
}
Using H\"older's inequality followed by Minkowski's inequality, we find
\ban{
\sum_{i,j,k}^{K-1} &\mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)} = \frac{1}{N^3}\sum_{i,j,k}^{K-1}\mathrm{I}E \abs{(E_i + G_i)(E_j + G_j)(E_k + G_k)} \notag \\
&\leq \frac{1}{N^3}\sum_{i,j,k}^{K-1}\left[\mathrm{I}E (E_i+G_i)^4 \mathrm{I}E (E_j+G_j)^4 \right]^{1/4}\left[\mathrm{I}E (E_k+G_k)^2\right]^{1/2} \notag \\
&\leq \frac{1}{N^3}\left(\sum_{i=1}^{K-1}\left[(\mathrm{I}E E_i^4)^{1/4}+(\mathrm{I}E G_i^4)^{1/4} \right]\right)^2\sum_{k=1}^{K-1}\left[(\mathrm{I}E E_k^2)^{1/2}+(\mathrm{I}E G_i^2)^{1/2}\right]. \label{45}
}
Now noting that for~$Y\sim\mathop{\mathrm{Bi}}n(n,p)$,
\bes{
\mathrm{I}E(Y-np)^4 &= 3(np(1-p))^2 + np(1-p)(1-6p(1-p)) \leq 3(np(1-p))^2 + np(1-p),
}
which, along with the variance formula for the binomial distribution, yields
\ba{
\mathrm{I}E E_i^4 &= 3(X_iq_i(1-q_i))^2 + X_iq_i(1-q_i) \leq 4N^2, \\
\mathrm{I}E E_i^2 &= X_i q_i (1-q_i) \leq N.
}
Plugging these bounds along with~$\abs{G_i} = \abs{NT_i - \sigma_i X_i} \leq N\tau_i + N\sigma_i$ into~\epsqref{45} yields the result.
\epsnd{proof}
\begin{proof}[Proof of Theorem~\ref{THM1}] We apply Theorem~\ref{THM3} with~$\Lambda=(2N)^{-1}\times \mathrm{I}d$. Using the bounds in Lemmas~\ref{lem13} for~$A_2$ and~\ref{lem14} for~$A_1$ along with a straightforward bound on~$\abs{R_j}$ ($R_j$ given at~\epsqref{43}) for~$A_1$, we obtain
\ba{
A_1 &\leq 2N\sum_{j=1}^{K-1} \left[ | p_{K j} - \frac{a_j}{2N}| +\sum_{\substack{k=1\\k\not=j}}^{K}\abs{p_{j k}-\frac{a_k}{2N}}+\sum_{\substack{k=1\\k\not=j}}^{K-1} \abs{p_{k j}- p_{Kj}}\right],\\
A_2&\leq N\sum_{i,j=1}^{K-1}(\sigma_i+\tau_i) \bbklr{\sigma_j+\tau_j+\frac{2}{N}}, \\
A_3&\leq \frac{2}{N^{1/2}}\left(\,\sum_{i=1}^{K-1} \bklr{\sqrt{2}+\sqrt{N}(\sigma_i+ \tau_i)}\right)^{2}\left(\,\sum_{i=1}^{K-1} \bklr{1+\sqrt{N}(\sigma_i+ \tau_i)}\right),
}
The final bound in Theorem~\ref{THM1} is now obtained through straightforward manipulations and applying some standard analytic inequalities, in particular
that~$\abs{x+y}^p\leq 2^{p-1}(\abs{x}^p+\abs{y}^p)$ for~$p\geq1$, and that
\be{
\sum_{i=1}^{K-1}(\sigma_i+\tau_i)\leq 2\sum_{i=1}^{K-1}p_{K i}+
\sum_{i=1}^{K-1}\sum_{\substack{ j\neq i}}^{K}p_{i j}+\sum_{i=1}^{K-1}\sum_{\substack{ j\neq i}}^{K-1}p_{ij}
+(K-2)\sum_{i=1}^{K-1} p_{K i}
\leq K \sum_{i=1}^K\sum_{\substack{j=1\\j\neq i}}^K p_{ij}
=K\mu.
}
\epsnd{proof}
\section{Proof of Theorem~\ref{THM2}: Cannings model}\label{sec3}
Recall the description in the introduction of the Cannings exchangeable
model with neutral PIM mutation in a haploid population of constant size~$N$.
The process is driven by a generic exchangeable offspring vector
$\tV$ with mutation structure such that~$p_{ij} = \pi_j$ for~$1\leq i \neq j\leq K$ and~$p_{ii}=1-\sum_{j\not=i} \pi_j$. To distinguish from the~$\pi_i$, we write~$p_i:=1-p_{ii}$ for the chance that an individual with parent of type~$i$ is not of type~$i$. As in the previous section, we apply Theorem~\ref{THM3}, and to define a stationary pair~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$, let~$\tX$ be distributed as a stationary distribution of the chain
and let~$\tX'$ be a step in the chain from~$\tX$.
Set~$\textnormal{\textbf{W}}=\tX/N$ and~$\textnormal{\textbf{W}}'=\tX'/N$.
We first compute~$\mathrm{I}E[X_i'-X_i|\tX]$.
The first thing to note is that we can decompose the number of individuals of
type~$i$ in the~$\tX'$-generation into those that have parent of type~$i$ and those that do not. In particular, if we denote by~$M_i$ the number of offspring in~$\tV$ that originate from a parent of type~$i$ in the~$\tX$-generation and write~$\textnormal{\textbf{M}}=(M_1,\ldots, M_K)$, then
\ben{
\mathscr{L}(X_i' | \textnormal{\textbf{M}})=\mathscr{L}(Y_1(\textnormal{\textbf{M}})+Y_2(\textnormal{\textbf{M}})), \label{46}
}
where~$Y_1(\textnormal{\textbf{M}})\sim\mathop{\mathrm{Bi}}n(N-M_i, \pi_i)$,~$Y_2(\textnormal{\textbf{M}})\sim \mathop{\mathrm{Bi}}n(M_i,1- p_i)$,
and these two variables are independent given~$\textnormal{\textbf{M}}$. From here we easily have
\be{
\mathrm{I}E(X_i'|\tX,\textnormal{\textbf{M}}) = \pi_i(N-M_i) + (1-p_i) M_i.
}
Now noting the exchangeability of~$\tV$ implies~$\mathrm{I}E V_j = 1$, and hence~$\mathrm{I}E (M_i|\tX) = X_i$, take the expectation with respect to~$\textnormal{\textbf{M}}$ to find
\bes{
\mathrm{I}E(X_i'|\tX) &= \pi_i(N-X_i) + (1-p_i)X_i\\
&= \pi_i N + (1-\sigma)X_i,
}
where~$\sigma = \sum_{i=1}^K \pi_i$. If we now set~$\textnormal{\textbf{W}} = \tX/N$ and~$\textnormal{\textbf{W}}' = \tX' / N$ we find that
\be{
\mathrm{I}E[\textnormal{\textbf{W}}' - \textnormal{\textbf{W}} | \textnormal{\textbf{W}}] = \boldsymbol{\pi}-\sigma\textnormal{\textbf{W}}.
}
Recalling our definition of~$\alpha$ from~\epsqref{3} and letting
\be{
\ta=\frac{2(N-1)}{\alpha} \boldsymbol{\pi},
}
we are in the setting of Theorem~\ref{THM3} with
$\Lambda=\frac{\alpha}{2(N-1)}\times \mathrm{I}d$ and~$\tR=0$.
As in Section~\ref{sec2}, applying the theorem is a relatively straightforward but
tedious calculation involving conditioning and computing various moment formulas.
For the latter we record the following lemma.
\begin{lemma}\label{47}
If\/~$\tV$ is a Cannings exchangeable offspring vector, if
$\alpha$,~$\beta$, and~$\gamma$ are the moments defined at~\epsqref{3}, and
$\deltaelta:=\mathrm{I}E \klg{V_1(V_1-1)(V_1-2)(V_1-3)}$,
then
\ban{
\mathrm{I}E V_1^2 &= 1+\alpha, \label{48}\\
\mathrm{I}E V_1V_2 &=1 - \alpha\frac{1}{N-1},\label{49}\\
\mathrm{I}E V_1^3 &= 1+3\alpha+\beta, \label{50}\\
\mathrm{I}E V_1 V_2 V_3&=1-\alpha\frac{3}{N-1}+\beta\frac{2}{(N-1)(N-2)}, \label{51}\\
\mathrm{I}E V_1^2 V_2&= 1+\alpha \frac{N-3}{N-1}-\beta\frac{1}{N-1},\label{52} \\
\mathrm{I}E V_1^2 V_2^2 &= 1+ \alpha \frac{2N-5}{N-1} -\beta \frac{2}{N-1} + \gamma,\label{53}\\
\mathrm{I}E V_1^4&= 1+7\alpha + 6 \beta + \deltaelta \label{54}, \\
\begin{split}\label{55}
\mathrm{I}E V_1V_2V_3V_4&=1-\alpha\frac{6}{N-1}+\beta\frac{8}{(N-1)(N-2)} \\
&\qquad+\gamma\frac{3}{(N-2)(N-3)}-\deltaelta\frac{3}{(N-1)(N-2)(N-3)},
\epsnd{split}\\
\mathrm{I}E V_1^2 V_2 V_3&= 1+\alpha\frac{N-6}{N-1}-\beta \frac{2N-8}{(N-1)(N-2)}-\gamma\frac{1}{N-2}+\deltaelta\frac{1}{(N-1)(N-2)}, \label{56} \\
\mathrm{I}E V_1^3 V_2 &= 1+\alpha \frac{3N-7}{N-1} +\beta \frac{N-6}{N-1}-\deltaelta \frac{1}{N-1}. \label{57}
}
\epsnd{lemma}
\begin{proof}
Since~$\sum_{i=1}^N V_1 =N$ and the~$V_i$'s are exchangeable, we have that~$\mathrm{I}E V_1=1$.
Thus~$\alpha=\mathrm{I}E V_1 (V_1-1)=\mathrm{I}E V_1^2 -1$ which is~\epsqref{48}. Note that similarly,
\be{
N=\mathrm{I}E V_1 (V_1+\cdots+V_N) = \mathrm{I}E V_1^2 + (N-1) \mathrm{I}E V_1 V_2 = \alpha+1 +(N-1)\mathrm{I}E V_1V_2,
}
and rearranging gives~\epsqref{49}.
For~\epsqref{50}, we have that
\be{
\mathrm{I}E V_1^3= \mathrm{I}E V_1(V_1-1)(V_1-2)+3\mathrm{I}E V_1^2 -2\mathrm{I}E V_1=\beta+3(\alpha+1)-2,
}
and further,
\ba{
N^2&=\mathrm{I}E V_1(V_1+\cdots + V_N)^2 = \mathrm{I}E V_1^3 + 3(N-1)\mathrm{I}E V_1^2 V_2 + (N-1)(N-2) \mathrm{I}E V_1 V_2 V_3,\\
(\alpha+1)N&=\mathrm{I}E V_1^2(V_1+\cdots+V_N)=\mathrm{I}E V_1^3 + (N-1) \mathrm{I}E V_1^2 V_2.
}
Solving these two equations yields the expressions for~\epsqref{51} and~\epsqref{52}.
Moving forward similarly, we have
\ba{
\mathrm{I}E V_1^2V_2^2&= \mathrm{I}E V_1(V_1-1)V_2(V_2-1)+2 \mathrm{I}E V_1^2V_2-\mathrm{I}E V_1 V_2, \\
\mathrm{I}E V_1^4&=\mathrm{I}E V_1(V_1-1)(V_1-2)(V_1-3)+6\mathrm{I}E V_1 (V_1-1)(V_1-2)+7\mathrm{I}E V_1(V_1-1)+\mathrm{I}E V_1,
}
and using previous expressions gives~\epsqref{53} and~\epsqref{54}. Along the same lines, we have
\ba{
\mathrm{I}E(V_1V_2V_3(V_1 + \cdots + V_N)) &= N \mathrm{I}E(V_1V_2V_3) = 3\mathrm{I}E(V_1^2V_2V_3) + (N-3) \mathrm{I}E(V_1V_2V_3V_4)\\
\mathrm{I}E(V_1^2V_2(V_1 + \cdots + V_N)) &= N\mathrm{I}E(V_1^2V_2) = \mathrm{I}E(V_1^3V_2) + \mathrm{I}E(V_1^2V_2^2) + (N-2)\mathrm{I}E(V_1^2V_2V_3)\\
\mathrm{I}E(V_1^3(V_1 + \cdots + V_N)) &= N \mathrm{I}E(V_1^3) = \mathrm{I}E(V_1^4) + (N-1)\mathrm{I}E(V_1^3V_2).
}
Plugging in values for known quantities in these three equations and solving yields~\epsqref{55},~\epsqref{56}, and~\epsqref{57}.
\epsnd{proof}
We first work on the~$A_2$ term from Theorem~\ref{THM3} which only requires
two moments.
\begin{lemma}\label{58}
For~$\tV,\tX, \textnormal{\textbf{M}}$ defined above,~$\alpha$ defined at~\epsqref{3}, and~$1\leq i\not=j\leq (K-1),$
\bes{
\mathrm{I}E(M_i|\tX) &= X_i,\\
\mathrm{I}E(M_i^2|\tX) &= X_i^2\left(1-\frac{\alpha}{N-1}\right) + X_i\frac{\alpha N}{N-1},\\
\mathrm{I}E(M_iM_j|\tX)& = X_iX_j\left(1 - \frac{\alpha}{N-1}\right).
}
\epsnd{lemma}
\begin{proof}
Using exchangeability, without loss of generality,
\bes{
\mathrm{I}E(M_i|\tX) &= \mathrm{I}E[V_1 + \cdots + V_{X_i}|\tX] = X_i, \\
\mathrm{I}E(M_i^2|\tX) &= \mathrm{I}E[( V_1 + \cdots + V_{X_i})^2|\tX] = X_i \mathrm{I}E(V_1^2) + X_i(X_i-1)\mathrm{I}E(V_1V_2),\\
\mathrm{I}E(M_iM_j|\tX) &= \mathrm{I}E[ (V_1 + \cdots + V_{X_i})(V_{X_i+1} + \cdots + V_{X_i+X_j})|\tX]= X_i X_j \mathrm{I}E(V_1V_2),
}
The lemma now follows by using the formulas
for the moments of the~$V_i$ in Lemma~\ref{47}.
\epsnd{proof}
\begin{lemma}\label{59}
For~$1\leq i\leq (K-1)$,~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}'), \pi_i, p_i, \sigma$ defined above, and~$\alpha$
defined at~\epsqref{3},
\bes{
\mathrm{I}E[ (W_i' - W_i)^2 | \textnormal{\textbf{W}}] &= W_i^2\left[\frac{-\alpha}{N-1} - \alpha \left(\frac{\sigma^2 - 2\sigma}{N-1}\right) + \sigma^2\right]\\
&\ \ \ + W_i\left[ \frac{\alpha}{N-1} - \alpha \left(\frac{2\sigma -\sigma^2}{N-1}\right) + \frac{p_i(1-p_i) - \pi_i (1-\pi_i)}{N} - 2\pi_i\sigma\right]\\
&\ \ \ + \pi_i(1-\pi_i)/N + \pi_i^2.
}
\epsnd{lemma}
\begin{proof}
Using the decomposition of~\epsqref{46},
\bes{
\mathrm{I}E[ (X_i' - X_i)^2 | \tX, \textnormal{\textbf{M}}] &= (N-M_i)\pi_i(1-\pi_i) + (N-M_i)^2\pi_i^2 + M_i(1-p_i)p_i + M_i^2(1-p_i)^2\\
&\ \ \ + 2(N-M_i)\pi_iM_i(1-p_i)- 2X_i(\pi_i N + (1-\sigma)M_i) + X_i^2\\
&= M_i^2(1-\sigma)^2\\
&\ \ \ + M_i[p_i(1-p_i) -\pi_i(1-\pi_i) - 2N\pi_i^2 + 2N \pi_i(1-p_i)-2X_i(1-\sigma)]\\
&\ \ \ + N\pi_i(1-\pi_i) + N^2 \pi_i^2 - 2N \pi_iX_i + X_i^2. \\
}
Now taking expectation with respect to~$M_i$ using Lemma~\ref{58},
\bes{
\mathrm{I}E[ (X_i' - X_i)^2 | \tX]
&= \left[X_i^2\left(1-\frac{\alpha}{N-1}\right) + X_i\frac{\alpha N}{N-1}\right](1-\sigma)^2\\
&\qquad + X_i[ p_i(1-p_i) -\pi_i(1-\pi_i) - 2N\pi_i^2 + 2N\pi_i(1-p_i)-2X_i(1-\sigma)]\\
&\qquad + N\pi_i(1-\pi_i) + N^2 \pi_i^2 - 2N \pi_iX_i + X_i^2 \\
&= X_i^2\left[1 + \left(1-\frac{\alpha}{N-1}\right)(1-\sigma)^2 - 2(1-\sigma)\right]\\
&\ \ \ + X_i\left[ \frac{\alpha N}{N-1}(1-\sigma)^2- 2N\pi_ip_i + p_i(1-p_i) - \pi_i(1-\pi_i) -2N\pi_i^2 \right]\\
&\ \ \ + N\pi_i(1-\pi_i) + N^2 \pi_i^2.
}
Dividing this last expression by~$N^2$ and rearranging gives the lemma.
\epsnd{proof}
\begin{lemma}\label{60}
For~$1\leq i\not=j \leq (K-1)$,~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}'), \pi_i, \pi_j, p_i, p_j,\sigma$ defined above, and~$\alpha$
defined at~\epsqref{3},
\bes{
\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j) | \textnormal{\textbf{W}} ] &= W_iW_j\left[ -\frac{\alpha}{N-1} + \frac{\alpha\sigma(2-\sigma)}{N-1} + \sigma^2\right]\\
&\qquad+\left(W_i\pi_j+W_j\pi_i\right)\left[ -\frac{1}{N} -\frac{N-1}{N}\sigma \right]+\frac{N-1}{N}\pi_i\pi_j.
}
\epsnd{lemma}
\begin{proof}
Given~$\textnormal{\textbf{M}}$, we can write~$(X_i', X_j', N-X_i'-X_j')$
as the sum of three independent multinomial random
variables corresponding to the counts of types~$i$,~$j$ and neither~$i$ or~$j$
in~$\tX'$ coming from
individuals in the previous~$\tX$-generation having types~$i$,~$j$, and neither~$i$ or~$j$.
Then the parameters of these multinomials are
$M_i, (1-p_i, \pi_j, 1-p_i-\pi_j)$;
$M_j, (\pi_i,1-p_j, \pi_j, 1-p_j-\pi_i)$; and
$N-M_i-M_j, (\pi_i, \pi_j, 1-\pi_i-\pi_j)$.
From this description and multinomial moment formulas (e.g., Lemma~\ref{lem10}),
it's straightforward to find that
\bes{
\mathrm{I}E[X_i'X_j'| \tX, \textnormal{\textbf{M}}] &= M_i(M_i-1)(1-p_i)\pi_j + M_j(M_j-1)\pi_i(1-p_j) \\
&\qquad + (N-M_i-M_j)(N-M_i-M_j-1)\pi_i\pi_j\\
&\qquad+M_i(1-p_i)M_j(1-p_j) + M_i(1-p_i)(N-M_i-M_j) \pi_j \\
&\qquad+ M_j\pi_iM_i\pi_j+ M_j\pi_i(N-M_i-M_j)\pi_j \\
&\qquad+ (N-M_i-M_j)\pi_iM_i\pi_j + (N-M_i-M_j)\pi_i M_j (1-p_j)\\
&=M_i M_j (1-\sigma)^2+(M_i \pi_j+M_j \pi_i)(1-\sigma)(N-1)+N(N-1)\pi_i \pi_j.
}
Also note that
\be{
\mathrm{I}E[X_i'X_j | \tX, \textnormal{\textbf{M}}] = X_j[(N-M_i)\pi_i + (1-p_i)M_i],
}
so that these last two displays and Lemma~\ref{58}
imply
\bes{
\mathrm{I}E[(X_i'-X_i)(X_j'-X_j)|\tX] &= X_iX_j\left[ (1-\sigma)^2\left(1-\frac{\alpha}{N-1}\right) - 2(1-\sigma) + 1\right]\\
&\qquad+ ( X_i \pi_j+X_j\pi_i)[-(N-1) \sigma -1]+ N(N-1) \pi_i\pi_j.
}
Dividing this last expression by~$N^2$ and rearranging gives the lemma.
\epsnd{proof}
The next lemma summarizes the bound on the~$A_2$ term of Theorem~\ref{THM3}
for this example.
\begin{lemma}\label{lem15}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ and~$\sigma$ defined above and~$\alpha$ defined at~\epsqref{3},
if~$\lambda = \alpha/(2(N-1))$, then
\bes{
\frac{1}{\lambda} \sum_{i,j=1}^{K-1} \mathrm{I}E& \left|\lambda W_i(\deltaelta_{i j}-W_j)-\frac{1}{2}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\\
&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right].
}
\epsnd{lemma}
\begin{proof}
Using Lemmas~\ref{59} and~\ref{60},
\bes{
\frac{1}{\lambda} &\sum_{i,j=1}^{K-1} \mathrm{I}E \left|\lambda W_i(\deltaelta_{i j}-W_j)-\frac{1}{2}\mathrm{I}E[ (W_i'-W_i)(W_j'-W_j)|\textnormal{\textbf{W}}] \right|\\
&\leq \frac{N-1}{\alpha}\sum_{i=1}^{K-1} \left(\alpha \left(\frac{\sigma^2 + 2\sigma}{N-1}\right) + \sigma^2 + \frac{p_i(1-p_i) + \pi_i (1-\pi_i)}{N} + 2\pi_i\sigma+ \pi_i^2\right) \\
&\ \ \ + \frac{N-1}{\alpha} \sum_{i \neq j}^{K-1} \left( \frac{\alpha}{N-1}(2\sigma + \sigma^2) + \frac{\pi_i + \pi_j}{N} + \sigma^2 +\sigma\pi_i + \sigma\pi_j + \pi_i \pi_j\right)\\
&\leq (K-1)(\sigma^2 + 2\sigma) + \frac{N-1}{\alpha}[(K-1)\sigma^2 + (K-1)\sigma/N + 3\sigma^2 ]\\
&\quad + (K-1)(K-2)(\sigma^2 + 2\sigma) + \frac{N-1}{\alpha}\left[\frac{ 2(K-2)\sigma}{N} + \sigma^2((K-1)(K-2)+2(K-2)+1) \right],\\
&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right].\qedhere
}
\epsnd{proof}
To compute the~$A_3$ term of Theorem~\ref{THM3}, we
need higher moment information.
\begin{lemma}\label{61}
For~$\tV,\tX, \textnormal{\textbf{M}}$ defined above
and~$1\leq i\leq (K-1),$
\bes{
\mathrm{I}E(M_i^3|\tX)
&= X_i^3 [\mathrm{I}E(V_1V_2V_3)] + X_i^2[ 3\mathrm{I}E(V_1^2V_2) - 3\mathrm{I}E(V_1V_2V_3)]\\
& \qquad + X_i[ \mathrm{I}E(V_1^3) - 3\mathrm{I}E(V_1^2V_2) + 2\mathrm{I}E(V_1V_2V_3)]\\
\mathrm{I}E(M_i^4|\tX)
&= X_i^4 [\mathrm{I}E(V_1V_2V_3V_4)] + X_i^3[ 6\mathrm{I}E(V_1^2V_2V_3) - 6\mathrm{I}E(V_1V_2V_3V_4)]\\
&\qquad + X_i^2 [ 4\mathrm{I}E(V_1^3V_2) + 3\mathrm{I}E(V_1^2V_2^2) - 18\mathrm{I}E(V_1^2V_2V_3) + 11\mathrm{I}E(V_1V_2V_3V_4)]\\
&\qquad + X_i[ \mathrm{I}E(V_1^4) - 4\mathrm{I}E(V_1^3V_2) - 3\mathrm{I}E(V_1^2V_2^2) + 12\mathrm{I}E(V_1^2V_2V_3) - 6 \mathrm{I}E(V_1V_2V_3V_4)].
}
\epsnd{lemma}
\begin{proof}
Similar to the proof of Lemma~\ref{47}, exchangeability implies
\bes{
\mathrm{I}E(M_i^3|\tX) &= \mathrm{I}E[(V_1 + \cdots + V_{X_i})^3|\tX],\\
&= X_i \mathrm{I}E(V_1^3) + 3X_i(X_i-1)\mathrm{I}E(V_1^2V_2) + X_i(X_i-1)(X_i-2) \mathrm{I}E(V_1V_2V_3),\\
\mathrm{I}E(M_i^4|\tX) &= \mathrm{I}E[(V_1 + \cdots + V_{X_i})^4|\tX],\\
&=X_i \mathrm{I}E(V_1)^4 + 4X_i(X_i-1)\mathrm{I}E(V_1^3V_2) + 3X_i(X_i-1)\mathrm{I}E(V_1^2V_2^2),\\
&\qquad + 6X_i(X_i-1)(X_i-2) \mathrm{I}E(V_1^2V_2V_3) + X_i(X_i-1)(X_i-2)(X_i-3) \mathrm{I}E(V_1V_2V_3V_4).
}
The lemma now follows by rearranging these equations.
\epsnd{proof}
\begin{lemma}\label{lem16}
For~$(\textnormal{\textbf{W}}, \textnormal{\textbf{W}}')$ and~$\sigma$ defined above and~$\alpha,\beta,\gamma, \deltaelta$ defined at~\epsqref{3}
and~$N>1$,
if~$\lambda = \alpha/(2(N-1))$, then
\ba{
&\frac{1}{\lambda}\sum_{i,j,k=1}^{K-1}\mathrm{I}E |(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)|\\
&\quad \leq 2(K-1)^3\left(\left( \frac{3\sigma^2}{N\alpha} + \frac{\sigma}{N^2\alpha} \right)^{1/4} + \left( \frac{\rho}{N^3\alpha}\right)^{1/4} + \left(\frac{N\sigma^4}{\alpha}\right)^{1/4} \right)^2 \left(\sqrt{\frac\sigma\alpha}+ 1 + \sqrt{\frac{N\sigma^2}{\alpha}}\right).
}
where
\be{
\rho :=
\frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}.
}
\epsnd{lemma}
\begin{proof}
Decompose
\ba{
X_i' - X_i &= [X_i' - (M_i(1-p_i) + (N-M_i)\pi_i)] + [(M_i-X_i)(1-\sigma)] + [N\pi_i - X_i \sigma]\\
&=:E_i + F_i + G_i.
}
Using H\"older's inequality followed by Minkowski's inequality, we find
\ban{
\sum_{i,j,k}^{K-1}& \mathrm{I}E \abs{(W_i'-W_i)(W_j'-W_j)(W_k'-W_k)} \notag\\
&\qquad= \frac{1}{N^3}\sum_{i,j,k}^{K-1}\mathrm{I}E \abs{(E_i +F_i+ G_i)(E_j +F_j+ G_j)(E_k +F_k+ G_k)}\notag \\
&\qquad \leq \frac{1}{N^3}\sum_{i,j,k}^{K-1}\left[\mathrm{I}E (E_i+F_i+G_i)^4 \mathrm{I}E (E_j+F_j+G_j)^4 \right]^{1/4}\left[\mathrm{I}E (E_k+F_k+G_k)^2\right]^{1/2}\notag\\
\begin{split}\label{62}
& \qquad \leq \frac{1}{N^3}\left(\sum_{i=1}^{K-1}\left[(\mathrm{I}E E_i^4)^{1/4}+(\mathrm{I}E F_i^4)^{1/4}+(\mathrm{I}E G_i^4)^{1/4} \right]\right)^2 \\
&\qquad\qquad\qquad\qquad \times \sum_{k=1}^{K-1}\left[(\mathrm{I}E E_k^2)^{1/2}+(\mathrm{I}E F_k^2)^{1/2}+(\mathrm{I}E G_k^2)^{1/2}\right].
\epsnd{split}
}
Recall the decomposition~\epsqref{46} of~$\mathscr{L}(X_i'|\textnormal{\textbf{M}})=\mathscr{L}(Y_1(\textnormal{\textbf{M}})+Y_2(\textnormal{\textbf{M}}))$ as a sum of
conditionally (on~$\textnormal{\textbf{M}}$)
independent binomials
and note that if~$Y\sim\mathop{\mathrm{Bi}}n(n,p)$ then
\bes{
\mathrm{I}E(Y-np)^4 &= 3(np(1-p))^2 + np(1-p)(1-6p(1-p)) \leq 3(np(1-p))^2 + np(1-p),
}
so that
\ba{
\mathrm{I}E[E_i^4|\textnormal{\textbf{M}}]&= \mathrm{I}E[(Y_1(\textnormal{\textbf{M}}) - \mathrm{I}E [Y_1(\textnormal{\textbf{M}})|\textnormal{\textbf{M}}] + Y_2(\textnormal{\textbf{M}})- \mathrm{I}E [Y_2(\textnormal{\textbf{M}})|\textnormal{\textbf{M}}])^4| \textnormal{\textbf{M}}] \\
&\leq 3 (M_ip_i(1-p_i))^2 + M_ip_i(1-p_i) + 6 M_i(1-p_i)p_i(N-M_i)\pi_i(1-\pi_i)\\
& \ \ \ + 3((N-M_i)\pi_i(1-\pi_i))^2 + (N-M_i)\pi_i(1-\pi_i)\\
&\leq 3(N(p_i(1-p_i) + \pi_i(1-\pi_i)))^2 + N(p_i(1-p_i) + \pi_i(1-\pi_i))\\
&\leq 3(N \sigma)^2 + N\sigma.
}
Using a similar argument for the second moment, we thus have for all~$1\leq i \leq K-1$,
\ben{\label{63}
\mathrm{I}E E_i^2\leq N \sigma, \hspace{1cm} \mathrm{I}E E_i^4\leq 3(N\sigma)^2 + N\sigma.
}
Now note that~$|G_i| \leq (N-X_i) \pi_i + X_i(\sigma-\pi_i) \leq N\sigma$, so that for all~$1\leq i \leq K-1$,
\ben{\label{64}
\mathrm{I}E G_i^2\leq (N\sigma)^2, \hspace{1cm} \mathrm{I}E G_i^4\leq (N\sigma)^4.
}
For the~$F_i=M_i-X_i$ moments, first note that Lemma~\ref{58} implies
\ben{\label{65}
\mathrm{I}E[F_i^2|\tX]=\mathrm{I}E[(M_i - X_i)^2|\tX]
=\frac{\alpha X_i (N-X_i)}{N-1} \leq \frac{\alpha N^2}{N-1}.
}
Furthermore, using Lemmas~\ref{47},~\ref{58}, and~\ref{61},
\ba{
&\mathrm{I}E[(M_i - X_i)^4| \tX] = \mathrm{I}E(M_i^4 | \tX) - 4X_i \mathrm{I}E(M_i^3|\tX) + 6X_i^2 \mathrm{I}E(M_i^2|\tX) - 4X_i^3 \mathrm{I}E(M_i|X_i) + X_i^4\\
&\quad= X_i^4\left\{ \frac{3\gamma}{(N-2)(N-3)} + \frac{(-3)\deltaelta}{(N-1)(N-2)(N-3)}\right\}\\
&\qquad + X_i^3\left\{ \frac{-6 N\gamma}{(N-2)(N-3)} + \frac{6N \deltaelta}{(N-1)(N-2)(N-3)}\right\}\\
&\qquad + X_i^2 \left\{ \frac{-\alpha}{N-1} + \frac{(-2N+4)\beta}{(N-1)(N-2)} + \frac{(3N^2 + 3N -3)\gamma}{(N-2)(N-3)} + \frac{(-4N^2 + 2N - 3)\deltaelta}{(N-1)(N-2)(N-3)} \right\}\\
&\qquad + X_i \left\{ \frac{(-5N + 6)\alpha}{N-1} + \frac{(2N^2 -4N)\beta}{(N-1)(N-2)}+ \frac{(-3N^2+3N)\gamma}{(N-2)(N-3)} + \frac{(N^3 - 2N^2 + 3N)\deltaelta}{(N-1)(N-2)(N-3)}\right\}.
}
Now using that~$0\leq X_i \leq N$ (and assuming~$N > 1$),
we have
\ba{
&\alpha X_i (-X_i -5N+6) \leq 0, & &\beta X_i((-2N+4)X_i+(2N^2-4N))\leq \beta N^2(N-2)/2, \\
& 3\gamma X_i^3(X_i-2N)\leq 0, & &3\gamma X_i((N^2 + N -1)X_i-N^2+N)\leq 3\gamma N^4, \\
& 3 \deltaelta X_i^3(-X_i+2N)\leq 3\deltaelta N^4, & & \deltaelta[X_i^2(-4N^2 + 2N - 3)+X_i(N^3 - 2N^2 + 3N)]\leq \deltaelta(N^4+3N^2).
}
Combining these inequalities with the previous display, we have
\ben{\label{66}
\mathrm{I}E F_i^4=\mathrm{I}E[(M_i - X_i)^4] \leq \frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}= \rho.
}
Now using the inequalities~\epsqref{63},~\epsqref{64},~\epsqref{65}, and~\epsqref{66} in~\epsqref{62} yields
the lemma.
\epsnd{proof}
\begin{proof}[Proof of Theorem~\ref{THM2}] We apply Theorem~\ref{THM3} with~$\Lambda = \frac{\alpha}{2(N-1)}\times \mathrm{I}d$.
From Lemmas~\ref{lem15} for~$A_2$
and~\ref{lem16} for~$A_3$ we obtain
\ba{
A_2&\leq \sigma^2\left[ (K-1)^2 + \frac{N-1}{\alpha}\left( K^2 +1\right)\right] + \sigma \left[ 2(K-1)^2 + \frac{3K-5}{\alpha} \right],\\
A_3 &\leq 2(K-1)^3\left(\left( \frac{3\sigma^2}{N\alpha} + \frac{\sigma}{N^2\alpha} \right)^{1/4} + \left( \frac{\rho}{N^3\alpha}\right)^{1/4} + \left(\frac{N\sigma^4}{\alpha}\right)^{1/4} \right)^2 \left(\sqrt{\frac\sigma\alpha}+ 1 + \sqrt{\frac{N\sigma^2}{\alpha}}\right),
}
where
\be{
\rho :=
\frac{ N^2\beta}{2(N-1)}+ \frac{(3 N^4) \gamma }{(N-2)(N-3)}+ \frac{(4N^4+3N^2)\deltaelta}{(N-1)(N-2)(N-3)}.
}
The final bound in Theorem~\ref{THM2} is now obtained through straightforward manipulations and applying some standard analytic inequalities,
in particular,~$\sigma=\epsta(\alpha/N)$ and~$\deltaelta\leq (N-3)\beta$.
\epsnd{proof}
\section*{Acknowledgments}
We thank the anonymous referee for helpful comments and for pointing out an omission in an earlier version of the manuscript (proof of existence of partial derivatives of the solution to the Stein equation).
NR received support from ARC grant DP150101459; AR received support from NUS Research Grant R-155-000-124-112. This work was done
partially while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore in 2015. The visit was supported by the Institute. HG would also like to thank the School of Mathematics at the University of Melbourne for their hospitality while some of this work was done.
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{{\mathfrak{m}}athfrak{b}}egin{document}
{\mathfrak{m}}aketitle
{{\mathfrak{m}}athfrak{b}}egin{abstract}
We study the birational geometry of irregular varieties and the singularities of Theta divisors of PPAV's in positive characteristic by applying recent generic vanishing results of Hacon and Patakfalvi. In particular, we prove that irreducible Theta divisors of principally polarized abelian varieties are strongly F-regular, which extends an old result of Ein and Lazarsfeld to fields of positive characteristic. In order to prove this, we formulate a positive characteristic analogue of another result of Ein and Lazarsfeld, to the effect that the Albanese image of a smooth projective variety of maximal Albanese dimension with vanishing holomorphic Euler characteristic is fibered by abelian subvarieties.
{\epsilon}nd{abstract}
\tableofcontents
\section{Introduction}
The purpose of this paper is to apply recent generic vanishing results in positive characteristic due to Hacon and Patakfalvi ~{{\mathfrak{m}}athfrak{c}}ite{hp13} to the study of the birational geometry of irregular varieties and the singularities of Theta divisors of principally polarized abelian varieties. Over fields of characteristic zero, seminal work of Ein and Lazarsfeld ~{{\mathfrak{m}}athfrak{c}}ite{el97} applied generic vanishing techniques over the complex numbers to settle a number of questions concerning the geometry of irregular varieties. One of their main results states that irreducible Theta divisors on principally polarized abelian varieties have mild singularities:
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 1]{el97})
Let $A$ be an abelian variety and let $\Theta\subset A$ be a principal polarization (i.e. an ample divisor such that $h^0(A,{\mathcal{O}}_A(\Theta))=1$). If $\Theta$ is irreducible, then it is normal and has rational singularities. \label{el97-main-theorem}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
The conclusion of the theorem is captured by the adjoint ideal of $\Theta$: given any log resolution ${\mathfrak{m}}u:A' {\rightarrow} A$ of the pair $(A,\Theta)$ and writing ${\mathfrak{m}}u^{{{\mathfrak{m}}athfrak{a}}st}\Theta=\Theta'+F$ with $\Theta'$ smooth and $F$ ${\mathfrak{m}}u$-exceptional, one may define ${{\mathfrak{m}}athfrak{a}}dj(A,\Theta)={\mathfrak{m}}u_{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{O}}_{A'}(K_{A'/A}-F)$. Standard arguments show that ${{\mathfrak{m}}athfrak{a}}dj(A,\Theta)={\mathcal{O}}_A$ is equivalent to $\Theta$ being normal and having rational singularities (see section 9.3.E in ~{{\mathfrak{m}}athfrak{c}}ite{laz04}). Bearing this in mind, Ein and Lazarsfeld's argument breaks into the following steps: let $X{\rightarrow} \Theta$ be a resolution of singularities.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item If $\Theta$ is irreducible, then $X$ is of general type. This relies on a classical argument due to Ueno (see ~{{\mathfrak{m}}athfrak{c}}ite{mor00}, Theorem 3.7), characterizing the Itaka fibration and the Kodaira dimension of an irreducible subvariety of an abelian variety.
\item If $X$ is of general type, then ${{\mathfrak{m}}athfrak{c}}hi({\omega}_X)>0$. More concretely, if $X$ is a smooth projective variety of maximal Albanese dimension and ${{\mathfrak{m}}athfrak{c}}hi(X,\omega_X)=0$, then the image of the Albanese map is fibred by tori. In particular, this shows that if $X$ is birational onto its image under the Albanese map, then $X$ is not of general type.
\item Generic vanishing theorems and Nadel vanishing yield ${{\mathfrak{m}}athfrak{a}}dj(\Theta)={\mathcal{O}}_A {\Lambda}ongleftrightarrow {{\mathfrak{m}}athfrak{c}}hi(X,\omega_X)>0$.
{\epsilon}nd{enumerate} {\mathfrak{m}}edskip
Therefore if $\Theta$ is irreducible, its adjoint ideal must be trivial, and by the characterization described above, it must be normal an have rational singularities. Work of Abramovich ~{{\mathfrak{m}}athfrak{c}}ite{abr95} shows that the statement in (i) remains valid in positive characteristic once an appropriate notion of Kodaira dimension is defined for possibly singular varieties. In this paper we provide positive characteristic analogues of items (ii) and (iii). {\mathfrak{m}}edskip
The only known results in this direction are due to Hacon ~{{\mathfrak{m}}athfrak{c}}ite{hac11}, where he proved that for principally polarized abelian varieties $(A,\Theta)$ over algebraically closed fields of positive characteristic, the pair $(A,\Theta)$ is a limit of strongly F-regular pairs. More precisely:
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 1.1]{hac11})
Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field of characteristic $p>0$. If $D\in |m\Theta|$, then $\left(A,\frac{1-{\epsilon}psilon}{m}D\right)$ is strongly F-regular for any rational number $0<{\epsilon}psilon<1$
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We summarize briefly our main results. The arguments employed in the proofs bear a strong resemblance to their characteristic zero analogues, albeit plenty of technicalities arise. Not only is resolution of singularities unavailable in general, but also generic vanishing for canonical sheaves is known to fail in positive characteristic (c.f. ~{{\mathfrak{m}}athfrak{c}}ite{hk12}). Nevertheless, recent work of Hacon and Patakfalvi ~{{\mathfrak{m}}athfrak{c}}ite{hp13} provides strong generic vanishing statements for objects arising from Cartier modules (see section 2.4 for precise statements): given a coherent Cartier module ${\mathcal{O}}mega_0\in {{\mathfrak{m}}athbb C}oh(A)$, the traces of the Frobenius iterates yield an inverse system $${{\mathfrak{m}}athfrak{c}}dots {\rightarrow} F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0 {\rightarrow} F_{{{\mathfrak{m}}athfrak{a}}st}^{e-1}{{\mathcal{O}}mega}_0 {\rightarrow} {{\mathfrak{m}}athfrak{c}}dots$$ and denoting by ${{\mathcal{O}}mega}=\varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0$ its inverse limit, there exists a closed subset $Z\subset \hat{A}$ such that $H^i(A,{{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})=0$ for every $i>0$ and every ${{\mathfrak{m}}athfrak{a}}lpha\in Z$ such that $p^e{{\mathfrak{m}}athfrak{a}}lpha\notin Z$ for all $e>>0$ (c.f. Corollary 3.3.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). This grounds on the following Theorem, which is the main result in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}.
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 3.1.1 and Lemma 3.1.2]{hp13})
Let $k$ be an algebraically closed field of characteristic $p>0$ and
$A$ be an abelian variety over $k$. Let $\{{\mathcal{O}}mega_e\}$ be Cartier module on $A$. If for any sufficiently
ample line bundle $L$ on $\hat{A}$ and any $e\gg 0$,
$H^i(A,{\mathcal{O}}mega_e\otimes \hat{L}^\vee)=0$ for all $i>0$, then the
complex\footnote{Here $S_{A,\hat{A}}$ denotes the Fourier-Mukai functor with kernel given by the Poincar{\'e} bundle of $A\times \hat{A}$} $${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}} RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$$ is a
quasi-coherent sheaf in degree 0, i.e.,
${\Lambda}ambda={\mathfrak{m}}athcal{H}^0({\Lambda}ambda)$.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
This result is generalized further in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, where a notion of M-regularity in positive characteristic is also introduced. Concretely, one has the following:
{{\mathfrak{m}}athfrak{b}}egin{thm}[c.f. Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}]
Let $A$ be an abelian variety and
$\{{\mathcal{O}}mega_e\}$ be a GV-inverse system of coherent sheaves on $A$
such that $\{{\mathcal{O}}mega_e\}$ is M-regular, in the sense that ${\mathfrak{m}}athcal{H}^0({\Lambda}ambda)$ is torsion-free. Then for any scheme-theoretic point $P\in A$, if ${\textrm{dim }} P\geqslant i$, then $P$ is not in the support of $$Im (R^i\hat{S}({\mathcal{O}}mega) {\rightarrow} R^i\hat{S}({\mathcal{O}}mega_e))$$ for any $e$.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
Our main technical result is a partial converse to the previous theorem: the presence of torsion in ${\mathcal{H}}^0({\Lambda})$ induces the following non-vanishing statement:
{{\mathfrak{m}}athfrak{b}}egin{thm}[c.f. Theorem 3.1]
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a
$g$-dimensional abelian variety satisfying the Mittag-Leffler
condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let
${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$.
Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. If ${\mathcal{H}}^0({\Lambda})$ is has a torsion point $P$ of dimension $g-k$, then the maps $$\varprojlim \left(R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {\rightarrow} R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)$$ are non-zero for every $e>>0$.
{\epsilon}nd{thm}
Using this, we can derive a fibration statement similar to that of Ein and Lazarsfeld:
{{\mathfrak{m}}athfrak{b}}egin{thm}[c.f. Theorem 4.2]
Let $X$ be a smooth projective variety of maximal Albanese dimension and denote by $a:X{\rightarrow} A$ the Albanese map. Let $g={\textrm{dim }} A$. Consider the inverse system $\{{{\mathcal{O}}mega}_e:=F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X\}_e$ and denote ${{\mathcal{O}}mega}=\varprojlim {{\mathcal{O}}mega}_e$. Define ${\Lambda}_e=RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$ and assume that the sheaf ${\mathcal{H}}^0({\Lambda})=\varinjlim {\mathcal{H}}^0({\Lambda}_e)$ has torsion. Then the image of the Albanese map is fibered by abelian subvarieties of $\hat{A}$.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
An identical argument to the one we employ to prove the previous theorem also yields the following result \label{Main-theorem} describing the singularities of Theta divisors in positive characteristic.
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $A$ be an ordinary abelian variety over an algebraically closed field of positive characteristic and let $\Theta$ be an irreducible Theta divisor. Then $\Theta$ is strongly F-regular. \label{Main-theorem}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
Over fields of positive characteristic, work of Smith and Hara (c.f. {{\mathfrak{m}}athfrak{c}}ite{smi97}, {{\mathfrak{m}}athfrak{c}}ite{har98}) shows that F-rationality is the positive characteristic counterpart to rational singularities, and the former is implied by strong F-regularity, so in this sense Theorem \ref{Main-theorem} is stronger than one might expect. {\mathfrak{m}}edskip
This paper is structured as follows. We start by recording all the background results we need in section 2: in 2.1 we recall the main definitions and some useful properties of the Fourier-Mukai transform in the context of abelian varieties, in 2.2 we record the relevant definitions of F-singularities; in 2.3 we record results of Pink and Roessler characterizing subvarieties of abelian varieties in positive characteristic; in 2.4 we outline a few useful facts concerning inverse systems that will ease the exposition of the proofs and in 2.5 we collect the generic vanishing statements in positive characteristic that will be needed in the sequel. Sections 3 and 4 constitute the technical core of the paper: section 3 contains the proof of the non-vanishing statement in the presence of torsion of ${\mathcal{H}}^0({\Lambda})$ and in section 4 we generalize Ein and Lazarsfeld's fibration statement. Finally in section 5 we present the proof of Theorem \ref{Main-theorem} on the singularities of Theta divisors. We start with the case of simple abelian varieties in section 5.1, which is a simple computation following easily from the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that does not require the arguments from sections 3 and 4. The general case is more involved and is presented section 5.2.
\section{Preliminaries}
\subsection{Derived categories and Fourier-Mukai transforms}
Let $A$ be a g-dimensional abelian variety, denote by $\hat{A}={{\mathfrak{m}}athbb P}ic^0(A)$ its dual and let ${\mathbb{F}}F\in {{\mathfrak{m}}athbb C}oh(A)$. Let ${{\mathfrak{m}}athbb P}P\in {{\mathfrak{m}}athbb P}ic(A\times \hat{A})$ be the Poincar{\'e} bundle and denote and consider the usual Fourier-Mukai functors: $$RS_{A,\hat{A}}^{{{\mathfrak{m}}athbb P}P}:D(A) {\rightarrow} D(\hat{A}), {{\mathfrak{m}}athfrak{q}}quad RS_{A,\hat{A}}^{{{\mathfrak{m}}athbb P}P}({{\mathfrak{m}}athfrak{b}}ullet)=Rp_{\hat{A}{{\mathfrak{m}}athfrak{a}}st}(p_A^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athfrak{b}}ullet)\otimes {{\mathfrak{m}}athbb P}P)$$ even though we will most often omit ${{\mathfrak{m}}athbb P}P$ from the notation and simply write $RS_{A,\hat{A}}({{\mathfrak{m}}athfrak{b}}ullet)$. {\mathfrak{m}}edskip
We start by stating Mukai's inversion theorem in the derived category of quasi-coherent sheaves:
{{\mathfrak{m}}athfrak{b}}egin{thm}[~{{\mathfrak{m}}athfrak{c}}ite{muk81}]
If $[-g]$ denotes the rightwise shift by $g$ places and $-1_A$ is the inverse on $A$, the following equalities hold on $D_{qc}(A)$ and $D_{qc}(\hat{A})$ $$RS_{\hat{A},A} {{\mathfrak{m}}athfrak{c}}irc RS_{A,\hat{A}}= (-1_A)^{{{\mathfrak{m}}athfrak{a}}st}[-g], {{\mathfrak{m}}athfrak{q}}quad RS_{A,\hat{A}} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A}= (-1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st}[-g]$$
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We will also be using the following two results:
{{\mathfrak{m}}athfrak{b}}egin{lemma}[~{{\mathfrak{m}}athfrak{c}}ite{muk81}, Proposition 3.8]
The Fourier-Mukai transform commutes with the dualizing functor in $D_{qc}(\hat{A})$ up to inversions and shifts, namely $$D_A {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A} \simeq \left((-1_{A})^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A} {{\mathfrak{m}}athfrak{c}}irc D_{\hat{A}}\right)[g]$$
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f, ~{{\mathfrak{m}}athfrak{c}}ite[Lemma 3.4]{muk81})
Let ${{\mathfrak{m}}athfrak{p}}hi:A{\rightarrow} B$ be an isogeny between abelian varieties and denote by $\hat{{{\mathfrak{m}}athfrak{p}}hi}:\hat{B} {\rightarrow} \hat{A}$ the dual isogeny. Then the following equalities hold on $D_{qc}(B)$ and $D_{qc}(A)$ respectively. $${{\mathfrak{m}}athfrak{p}}hi^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{B},B} \simeq RS_{\hat{A},A} {{\mathfrak{m}}athfrak{c}}irc \hat{{{\mathfrak{m}}athfrak{p}}hi}_{{{\mathfrak{m}}athfrak{a}}st}, {{\mathfrak{m}}athfrak{q}}quad {{\mathfrak{m}}athfrak{p}}hi_{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athfrak{c}}irc RS_{\hat{A},A} \simeq RS_{\hat{B},B} {{\mathfrak{m}}athfrak{c}}irc \hat{{{\mathfrak{m}}athfrak{p}}hi}^{{{\mathfrak{m}}athfrak{a}}st}$$ In particular, this holds for the (e-th iterate) Frobenius map $F^e$ and its dual isogeny, namely the Verschiebung map $V^e=\hat{F}^e$. \label{Mukai-vs-isogenies}
{\epsilon}nd{lemma}
We will also be using the following simple remark.
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Exercise 5.12]{huy06})
Let ${{\mathfrak{m}}athfrak{p}}i:B{\rightarrow} A$ be a morphism between abelian varieties and let ${{\mathfrak{m}}athbb P}P$ be a locally free sheaf on $A\times \hat{A}$. Denote ${{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{p}}i}=({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athbb P}P)$. Then $$S_{{{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{p}}i}} \simeq S_{{{\mathfrak{m}}athbb P}P} {{\mathfrak{m}}athfrak{c}}irc {{\mathfrak{m}}athfrak{p}}i_{{{\mathfrak{m}}athfrak{a}}st}$$ \label{Mukai-vs-push-forward}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
We next record the notions of homotopy limits and colimits in the derived category. Given a direct system of objects ${{\mathfrak{m}}athbb C}C_i\in D(A)$ $${{\mathfrak{m}}athbb C}C_1 {\rightarrow} {{\mathfrak{m}}athbb C}C_2 {\rightarrow} \ldots$$ its homotopy colimit ${\textrm{hocolim}_{\rightarrow}} {{\mathfrak{m}}athbb C}C_i$ is defined by the triangle $$\oplus {{\mathfrak{m}}athbb C}C_i {\longrightarrow} \oplus {{\mathfrak{m}}athbb C}C_i {\longrightarrow} {\textrm{hocolim}_{\rightarrow}} {{\mathfrak{m}}athbb C}C_i \stackrel{[+1]}{{\longrightarrow}}$$ where the first map is the homomorphism given by $id-shift$ where $shift:\oplus {{\mathfrak{m}}athbb C}C_i {\rightarrow} \oplus {{\mathfrak{m}}athbb C}C_i$ is given on ${{\mathfrak{m}}athbb C}C_i$ by the composition ${{\mathfrak{m}}athbb C}C_i {\rightarrow} {{\mathfrak{m}}athbb C}C_{i+1} \hookrightarrow \oplus {{\mathfrak{m}}athbb C}C_j$. {\mathfrak{m}}edskip
Given an inverse system of objects ${{\mathfrak{m}}athbb C}C_i \in D_{qc}(X)$ $${{\mathfrak{m}}athbb C}C_1 \longleftarrow {{\mathfrak{m}}athbb C}C_2 \longleftarrow {{\mathfrak{m}}athfrak{c}}dots$$ its homotopy limit ${\textrm{holim}_{\leftarrow}} {{\mathfrak{m}}athbb C}C_i$ is given by the triangle $${\textrm{holim}_{\leftarrow}} {{\mathfrak{m}}athbb C}C_i {\longrightarrow} {{\mathfrak{m}}athfrak{p}}rod {{\mathfrak{m}}athbb C}C_i {\longrightarrow} {{\mathfrak{m}}athfrak{p}}rod {{\mathfrak{m}}athbb C}C_i \stackrel{+1}{{\longrightarrow}}$$ where the map between products is ${{\mathfrak{m}}athfrak{p}}rod(id-shift)$ and where by product we mean product of chain complexes as opposed to the product inside $D_{qc}(X)$. {\mathfrak{m}}edskip
Note that if ${{\mathfrak{m}}athbb C}C_i$ are coherent sheaves, then ${\textrm{hocolim}_{\rightarrow}} {{\mathfrak{m}}athbb C}C_i = \varinjlim {{\mathfrak{m}}athbb C}C_i$. {\mathfrak{m}}edskip
If $X$ is an n-dimensional variety over a field $k$ and ${\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$ denotes its dualizing complex, so that ${\mathcal{H}}^{-{\textrm{dim }} X}({\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}) \simeq {\omega}_X$, we define the dualizing functor $D_X$ on $D_{qc}(X)$ as $D_X({\mathbb{F}}F)= R{\mathcal{H}} om({\mathbb{F}}F,{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet})$. In this context, Grothendieck duality reads as follows:
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $f:X{\rightarrow} Y$ be a proper morphism of quasi-projective varieties over a field $k$. Then for any complex ${\mathbb{F}}F\in D_{qc}(X)$ we have an isomorphism $$Rf_{{{\mathfrak{m}}athfrak{a}}st}D_X({\mathbb{F}}F) \simeq D_Y Rf_{{{\mathfrak{m}}athfrak{a}}st}({\mathbb{F}}F)$$ Assuming that $X$ and $Y$ are smooth, then we equivalently have that for any ${\mathbb{F}}F \in D_{qc}(X)$ and ${\mathcal{E}}\in D_{qc}(Y)$, if ${\omega}_f={\omega}_X\otimes f^{{{\mathfrak{m}}athfrak{a}}st}{\omega}_Y$ denotes the relative dualizing sheaf, there is a functorial isomorphism $$Rf_{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om({\mathbb{F}}F,Lf^{{{\mathfrak{m}}athfrak{a}}st}({\mathcal{E}}) \otimes {\omega}_f[{\textrm{dim }} X-{\textrm{dim }} Y]) \simeq R{\mathcal{H}} om (Rf_{{{\mathfrak{m}}athfrak{a}}st}{\mathbb{F}}F,{\mathcal{E}})$$ \label{grothendieck-duality}
{\epsilon}nd{thm}
\subsection{F-singularities and linear subvarieties of abelian subvarieties}
In this section we recall the basic notions from the theory of F-singularities following {{\mathfrak{m}}athfrak{c}}ite{sch12} and ~{{\mathfrak{m}}athfrak{c}}ite{bst12}. Let $X$ be a separated scheme of finite type over an F-finite perfect field of characteristic $p>0$. A variety is a connected reduced equidimensional scheme over $k$. We denote the canonical sheaf of $X$ by ${\omega}_X={\mathcal{H}}^{-{\textrm{dim }} X}({\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet})$, where ${\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}={\epsilon}ta^{{{\mathfrak{m}}athfrak{a}}st}k$ is the dualizing complex of $X$ and ${\epsilon}ta:X {\rightarrow} k$ is the structural map. If $X$ is normal, a \textit{canonical divisor} on $X$ is any divisor $K_X$ such that ${\omega}_X \simeq {\mathcal{O}}_X(K_X)$. {\mathfrak{m}}edskip
By a pair $(X,{\Delta})$ we mean the combined information of a normal integral scheme $X$ and an effective ${{\mathfrak{m}}athbb Q}$-divisor ${\Delta}$. Denote by $F^e:X {\rightarrow} X$ the e-th iterated absolute Frobenius, where the source has structure map ${\epsilon}ta{{\mathfrak{m}}athfrak{c}}irc F^e:X{\rightarrow} k$. Since $(F^e)^!{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} = (F^e)^!{\epsilon}ta^!k={\epsilon}ta^! (F^e)^!k={\epsilon}ta^!k={\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$. In general for a finite morphism $f:X{\rightarrow} Y$, a coherent sheaf ${\mathbb{F}}F$ on $X$ and a quasi-coherent sheaf ${\mathcal{G}}$ on $Y$, we have the duality ${\mathcal{H}} om(f_{{{\mathfrak{m}}athfrak{a}}st}{\mathbb{F}}F,{\mathcal{G}}) \simeq f_{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{H}} om({\mathbb{F}}F, f^!{\mathcal{G}})$, so the identity ${\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} {\rightarrow} {\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} \simeq (F^e)^!{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$ yields a trace map $F_{{{\mathfrak{m}}athfrak{a}}st}^e{\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet} {\rightarrow} {\omega}_X^{{{\mathfrak{m}}athfrak{b}}ullet}$ and taking cohomology we obtain ${{\mathfrak{m}}athbb P}hi^e:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\omega}_X {\rightarrow} {\omega}_X$. Given a variety $X$, the \textit{parameter test submodule} $\tau({\omega}_X)$ of $X$ is the unique smallest ${\mathcal{O}}_X$-submodule $M\subseteq {\omega}_X$, non-zero on any component of $X$, such that ${{\mathfrak{m}}athbb P}hi^1(F_{{{\mathfrak{m}}athfrak{a}}st}M) \subseteq M$. {\mathfrak{m}}edskip
Assume that $(X,{\Delta}elta)$ is a pair such that $K_X+{\Delta}$ is ${{\mathfrak{m}}athbb Q}$-Cartier with index not divisible by $p$. Choose $e>0$ such that $(p^e-1)(K_X+{\Delta})$ is Cartier and define the line bundle ${\mathfrak{m}}athcal{L}_{e,{\Delta}}={\mathcal{O}}_X((1-p^e)(K_X+{\Delta}))$. By ~{{\mathfrak{m}}athfrak{c}}ite{sch09}, there is a canonically determined map ${{\mathfrak{m}}athfrak{p}}hi_{e,{\Delta}elta}:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\mathfrak{m}}athcal{L}_{e,{\Delta}} {\rightarrow} {\mathcal{O}}_X$. We define the \textit{test ideal} $\tau(X,{\Delta})$ of the pair $(X,{\Delta})$ to be the smallest non-zero ideal $J\subseteq {\mathcal{O}}_X$ such that $${{\mathfrak{m}}athfrak{p}}hi_{e,{\Delta}}(F_{{{\mathfrak{m}}athfrak{a}}st}^e(J{{\mathfrak{m}}athfrak{c}}dot {\mathfrak{m}}athcal{L}_{e,{\Delta}})) \subseteq J.$$ Similarly one defines the \textit{non-F-pure ideal} $\sigma(X,{\Delta})$ of $(X,{\Delta})$ to be the the largest such ideal $J\subseteq {\mathcal{O}}_X$. {\mathfrak{m}}edskip
Ever since Hochster and Huneke introduced test ideals and tight closure theory in ~{{\mathfrak{m}}athfrak{c}}ite{hh90}, deep connections have been established between the classes of singularities defined in terms of Frobenius splittings and those arising within the minimal model program. For instance, a normal domain $(R,{\mathfrak{m}})$ of characteristic $p>0$ is said to be F-pure if the inclusion induced by the Frobenius $R\hookrightarrow F_{{{\mathfrak{m}}athfrak{a}}st}^eR{\epsilon}quiv R^{1/p^e}$ splits for every $e$. Similarly, a pair $(R,{\Delta})$ is said to be F-pure if the inclusion $R\hookrightarrow R^{1/p^e} \hookrightarrow R\left(\lceil(p^e-1){\Delta}\rceil\right)^{1/p^e}$ splits for every $e$ and it was shown in ~{{\mathfrak{m}}athfrak{c}}ite{hw02} that F-pure pairs are the analogues of log canonical pairs in characteristic zero, in the sense that if $(X,{\Delta})$ is a log canonical pair, then its reduction mod $p$ $(X_p,{\Delta}_p)$ is F-pure for all $p>>0$. {\mathfrak{m}}edskip
In this paper we shall be concerned with the two classes of F-singularities that we define next. We will be recording the original definition in terms of Frobenius splittings and we will then state their description in terms of test ideals that will be used in the sequel.
{{\mathfrak{m}}athfrak{b}}egin{defi}
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item A pair $(X={{\mathfrak{m}}athbb S}pec R,{\Delta})$ is \textit{strongly F-regular} if for every non-zero element $c\in R$, there exists $e$ such that the map $R\hookrightarrow R^{1/p^e} \hookrightarrow R((p^e-1){\Delta})^{1/p^e}$ that sends $1{\mathfrak{m}}apsto c^{1/p^e} {\mathfrak{m}}apsto c^{1/p^e}$ splits as an $R$-module homomorphism.
\item A reduced connected variety $X$ is F-rational if it is Cohen-Macaulay and there is no non-zero submodule $M\subsetneq {\omega}_R$ such that the Grothendieck trace map ${{\mathfrak{m}}athbb P}hi_X:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\omega}_X {\rightarrow} {\omega}_X$ satisfies ${{\mathfrak{m}}athbb P}hi(F_{{{\mathfrak{m}}athfrak{a}}st}^e M)\subseteq M$.
{\epsilon}nd{enumerate}
{\epsilon}nd{defi} {\mathfrak{m}}edskip
Strongly F-regular pairs are the analog of log terminal pairs in characteristic zero (c.f. ~{{\mathfrak{m}}athfrak{c}}ite{hw02}) and F-rational varieties are the analogue of varieties with rational singularities (c.f. ~{{\mathfrak{m}}athfrak{c}}ite{smi97}). The notion of strong F-regularity is also captured by the test ideal, as the following well-known result shows.
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 2.4]{hw02})
A pair $(X,{\Delta})$ is strongly F-regular if, and only if, $\tau(X,{\Delta})={\mathcal{O}}_X$.
{\epsilon}nd{lemma}
Assume that $(X,{\Delta})$ is a pair, where $X$ is a normal proper variety over an algebraically closed field of characteristic $p>0$ and ${\Delta}\geq 0$ is a ${{\mathfrak{m}}athbb Q}$-divisor such that $K_X+{\Delta}$ is ${{\mathfrak{m}}athbb Q}$-Cartier with index not divisible by $p$. The map ${{\mathfrak{m}}athfrak{p}}hi_{{\Delta}elta}^e:F_{{{\mathfrak{m}}athfrak{a}}st}^e{\mathfrak{m}}athcal{L}_{e,{\Delta}} {\rightarrow} {\mathcal{O}}_X$ defined in ~{{\mathfrak{m}}athfrak{c}}ite{sch09} restricts to surjective maps $$F_{{{\mathfrak{m}}athfrak{a}}st}^e(\sigma(X,{\Delta}) \otimes {\mathfrak{m}}athcal{L}_{e,{\Delta}}) {\longrightarrow} \sigma(X,D), {{\mathfrak{m}}athfrak{q}}quad F_{{{\mathfrak{m}}athfrak{a}}st}^e(\tau(X,{\Delta}) \otimes {\mathfrak{m}}athcal{L}_{e,{\Delta}}) {\longrightarrow} \tau(X,D).$$
The power of vanishing theorems in characteristic zero relies on the fact that they allow us to lift global sections of adjoint bundles. The full space of global sections is not so well behaved in positive characteristic, so one instead focuses on a subspace of it that is stable under the Frobenius action. {\mathfrak{m}}edskip
If $M$ is any Cartier divisor, one thus defines the subspace $S^0(X, \tau(X,{\Delta}) \otimes {\mathcal{O}}_X(M))$ as
{{\mathfrak{m}}athfrak{b}}egin{multline*} S^0(X, \tau(X,{\Delta}) \otimes {\mathcal{O}}_X(M)) \\ := {{\mathfrak{m}}athfrak{b}}igcap_{n\geq0} Im\left( H^0(X,F_{{{\mathfrak{m}}athfrak{a}}st}^{ne}\tau(X,{\Delta}) \otimes {\mathfrak{m}}athcal{L}_{ne,{\Delta}}(p^{ne}M)) {\longrightarrow} H^0(X,\tau(X,{\Delta}) \otimes {\mathcal{O}}_X(M)) \right) \\ \subseteq H^0(X,{\mathcal{O}}_X(M)) {\epsilon}nd{multline*}
Among the many applications of these subspaces, for instance, they can be used to prove global generation statements: concretely, suppose that $X$ is a $d$-dimensional variety of characteristic $p>0$ and that ${\Delta}$ is a ${{\mathfrak{m}}athbb Q}$-divisor such that $K_X+{\Delta}$ is ${{\mathfrak{m}}athbb Q}$-Cartier with index not divisible by $p$. It was shown in ~{{\mathfrak{m}}athfrak{c}}ite{sch09} that if $L$ and $M$ are Cartier divisors such that $L-K_X-{\Delta}$ is ample and $M$ is ample and globally generated, then the sheaf $\tau(X,{\Delta})\otimes {\mathcal{O}}_X(L+nM)$ is globally generated for all $n\geq d$ by $S^0(X, \tau(X,{\Delta}) \otimes {\mathcal{O}}_X(L+nM))$. {\mathfrak{m}}edskip
\subsection{The Frobenius morphism on Abelian varieties}
Throughout this paper, $A$ will denote an abelian variety of dimension $g$ over a field $k$ and $\hat{A}={{\mathfrak{m}}athbb P}ic^0(A)$ will denote the dual abelian variety
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 2.13]{hp13})
For a g-dimensional abelian variety $A$ over a field $k$, the following conditions are equivalent.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item There are $p^g$ p-torsion points.
\item The Frobenius action $H^g(A,{\mathcal{O}}_A) {\rightarrow} H^g(A,{\mathcal{O}}_A)$ is bijective
\item The Frobenius action $H^i(A,{\mathcal{O}}_A) {\rightarrow} H^i(A,{\mathcal{O}}_A)$ is bijective for all $0\leq i \leq g$
\item $S^0(A,{\omega}_A) = H^0(A, {\omega}_A)$
{\epsilon}nd{enumerate}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
If any of these equivalent conditions is satisfied we say that $A$ is \textit{ordinary}. Given an isogeny $\varphi:A {\rightarrow} B$ between abelian varieties of dimension $g$, $A$ is ordinary if and only if $B$ is ordinary. Given a surjective morphism $\varphi:A {\rightarrow} B$ of abelian varieties, if $A$ is ordinary then so is $B$ (see Lemmas 2.14 and 2.14 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). {\mathfrak{m}}edskip
We finally record a characterization of linear subvarieties of abelian varieties following ~{{\mathfrak{m}}athfrak{c}}ite{pr03}. Let $A$ be an abelian variety endowed with an isogeny $\varphi:A {\rightarrow} A$. We say that $A$ is pure of positive weight if there exist integers $r,s>0$ such that $\varphi^s=F_{p^r}$ for some model of $A$ over ${\mathbb{F}}_{p^r}$. If $A$ is defined over a finite field, we say $A$ is \textit{supersingular} if and only if it is pure of positive weight for the isogeny given by multiplication by $p$; in general, we say that $A$ is supersingular if it is isogenous to a supersingular variety defined over a finite field. We say that $A$ \textit{has no supersingular factors} is there exist no non-trivial homomorphism to an abelian variety which is pure of positive weight for the isogeny given by multiplication by $p$. One sees that $A$ has no supersingular factors if there does not exist a non-trivial homomorphism to a supersingular abelian variety. In particular, if $A$ is an ordinary abelian variety, it follows from the observations in the previous paragraph that $A$ has no supersingular factors (see Lemma 2.16 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). {\mathfrak{m}}edskip
The following result of Pink and Roessler characterizing linear subvarieties of abelian varieties will be crucial in our proof:
{{\mathfrak{m}}athfrak{b}}egin{thm}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 2.2]{pr03})
Let $A$ be an abelian variety over a field $K$ of characteristic $p>0$ and let $X\subset A$ be a reduced closed subscheme $p(X) \subset X$, where $p$ denotes the isogeny given by multiplication by $p$. If $A$ has no supersingular factors, then all the maximal dimensional irreducible components of $X$ are completely linear (namely, torsion translates of subabelian varieties).
{\epsilon}nd{thm} {\mathfrak{m}}edskip
\subsection{Generalities on inverse systems and spectral sequences}
\textbf{Mittag-Leffler inverse systems}. We start by recording a few results that will be used in the sequel, most of which are taken directly from ~{{\mathfrak{m}}athfrak{c}}ite{har78}. Recall that a sheaf is countably quasi-coherent if it is quasi-coherent and locally countably generated. Also recall that an inverse system of coherent sheaves $\{{\mathcal{O}}mega_e\}$ is said to satisfy the Mittag-Leffler condition if for any $e\geq 0$, the image of ${\mathcal{O}}mega_{e'}{\rightarrow} {\mathcal{O}}mega_{e}$ stabilizes for $e'$ sufficiently large. The inverse limit functor is always left exact in the sense that if $$\xymatrix{0 {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{F}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d] & {\mathfrak{m}}athcal{G}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d] & {\mathfrak{m}}athcal{H}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{F}_{e-1} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{G}_{e-1} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathfrak{m}}athcal{H}_{e-1} {{\mathfrak{m}}athfrak{a}}r[r] & 0}$$ is an exact sequence of inverse systems, then $$0 {\rightarrow} \varprojlim{\mathfrak{m}}athcal{F}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{G}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{H}_e$$ is exact in the category of quasi-coherent sheaves. By a theorem of Roos (c.f. Proposition I.4.1 in ~{{\mathfrak{m}}athfrak{c}}ite{har78}), the right derived functors $R^i\varprojlim$ are $0$ for $i\geqslant 2$. Hence, we have a long exact sequence $$0 {\rightarrow} \varprojlim{\mathfrak{m}}athcal{F}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{G}_e {\rightarrow} \varprojlim{\mathfrak{m}}athcal{H}_e {\rightarrow} R^1\varprojlim{\mathfrak{m}}athcal{F}_e {\rightarrow} R^1\varprojlim{\mathfrak{m}}athcal{G}_e {\rightarrow} R^1\varprojlim{\mathfrak{m}}athcal{H}_e {\rightarrow} 0.$$
We start by recording a characterization of the Mittag-Leffler condition in terms of the first right-derived inverse limit.
{{\mathfrak{m}}athfrak{b}}egin{lemma}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition I.4.9]{har78})
Let $\{{{\mathcal{O}}mega}_e\}_e$ be an inverse system of countably quasi-coherent sheaves on a scheme $X$ of finite type. Then the following conditions are equivalent:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item $\{{{\mathcal{O}}mega}_e\}_e$ is satisfies the Mittag-Leffler condition.
\item $R^1\varprojlim {{\mathcal{O}}mega}_e=0$
\item $R^1\varprojlim {{\mathcal{O}}mega}_e$ is countably quasi-coherent.
{\epsilon}nd{enumerate}
\label{ML-charact}
{\epsilon}nd{lemma}{\mathfrak{m}}edskip
The following is basic result about the cohomology of an inverse system of sheaves:
{{\mathfrak{m}}athfrak{b}}egin{prop}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem I.4.5]{har78})
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a variety $X$. Let $T$ be a functor on $D(X)$ which commutes with arbitrary direct products. Suppose that $\{{\mathcal{O}}mega_e\}$ satisfies the Mittag-Leffler condition. Then for each $i$, there is an exact sequence $$0{\rightarrow} R^1\varprojlim R^{i-1}T({\mathcal{O}}mega_e){\rightarrow} R^iT(\varprojlim {\mathcal{O}}mega_e){\rightarrow} \varprojlim R^iT({\mathcal{O}}mega_e) {\rightarrow} 0.$$ In particular, if for some $i$, $\{R^{i-1}T({\mathcal{O}}mega_e)\}$ satisfies the Mittag-Leffler condition, then $R^iT(\varprojlim {\mathcal{O}}mega_e){{\mathfrak{m}}athfrak{c}}ong \varprojlim R^iT({\mathcal{O}}mega_e)$ (by Lemma \ref{ML-charact}). \label{inverse-limits-commute-functor}
{\epsilon}nd{prop} {\mathfrak{m}}edskip
We will be applying this theorem to the push-forward $f_{{{\mathfrak{m}}athfrak{a}}st}$ under a proper morphism of schemes. We finally record a standard statement about the commutation of inverse limits and tensor products. Recall that a sheaf is countably quasi-coherent if it is quasi-coherent and locally countably generated. Then one has the following:
{{\mathfrak{m}}athfrak{b}}egin{lemma}(see ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 4.10]{har78})
Let $\{{\mathbb{F}}F_e\}_e$ be an inverse system of countably quasi-coherent sheaves on a scheme $X$ of finite type and let $E$ be a flat ${\mathcal{O}}_X$-module. Consider the natural map $${{\mathfrak{m}}athfrak{a}}lpha:(\varprojlim {\mathbb{F}}F_e)\otimes E \rightarrow \varprojlim ({\mathbb{F}}F_e \otimes E)$$ If $\varprojlim {\mathbb{F}}F_e$ is countably quasi-coherent then ${{\mathfrak{m}}athfrak{a}}lpha$ is injective and if furthermore $R^1\varprojlim {{\mathcal{O}}mega}_e$ is countably quasi-coherent, then ${{\mathfrak{m}}athfrak{a}}lpha$ is surjective. In particular, if $\{{\mathbb{F}}F_e\}$ is an inverse system of coherent sheaves satisfying the Mittag-Leffler condition on a scheme $X$ with generic point $w$, then there is an isomorphism $$\left(\varprojlim_e {\mathbb{F}}F_e \right) \otimes k(w) {\longrightarrow} \left(\varprojlim_e {\mathbb{F}}F_e \otimes k(w) \right)$$
\label{inverse-limit-tensor-product-commute}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
\textbf{Inverse systems of convergent spectral sequences}. The following observation is taken from ~{{\mathfrak{m}}athfrak{c}}ite{car08}. Let $\{E(n)\}$ be an inverse system of spectral sequences with morphisms of spectral sequences $E(n) {\rightarrow} E(n-1)$ and consider the tri-graded abelian groups $E_{p,q}^r=\varprojlim_n E_{p,q}^r(n)$, with differentials given by the inverse limits of the differentials in the $E(n)$. Concretely, if $d^r(n)$ is the r-th differential in $E(n)$ and $x(n)\in E_{p,q}^r$, then the r-th differential $d^r$ in the limit sequence $E_{p,q}^r$ is given by $d^r(x(n))=d^r(n)(x(n))$. The resulting object is a spectral sequence provided that $H(E_{p,q}^r,d^r)=E_{p,q}^{r+1}$, which is in turn equivalent to showing that $$H(\varprojlim_n E_{p,q}^r(n),d^r)=\varprojlim_n H(E_{p,q}^r(n),d^r)$$ and this is precisely the statement of Proposition \ref{inverse-limits-commute-functor} above (for the functor of global sections). {\mathfrak{m}}edskip
Note that, in particular, if the terms $E_{p,q}^r$ are all finite-dimensional vector spaces, the hypotheses of Proposition \ref{inverse-limits-commute-functor} hold and all the $\varprojlim^{(1)}$ terms are 0. Besides, given that the inverse limit of the spectral sequences is again a spectral sequence and provided that every spectral sequence in the inverse system is bounded and convergent, one observes that the limit spectral sequence is also convergent: for fixed $p,q$, there is a fixed $N$ such that $E_{p,q}^{\infty}(n)=E_{p,q}^N(n)$.{\mathfrak{m}}edskip
\textbf{Morphisms between spectral sequences}. We recall the definition of a spectral sequence from EGA III [$0_{III}$, 1.1]. Let ${\mathfrak{m}}athscr{C}$ be an abelian category. A \textbf{(biregular) spectral sequence} $E$ on ${\mathfrak{m}}athscr{C}$ consists of the following ingredients:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item A family of objects $\{E^{p,q}_{r}\}$ in ${\mathfrak{m}}athscr{C}$,
where $p,q,r\in {\mathfrak{m}}athbb{Z}$ and $r\geqslant 2$, such that for any
fixed pair $(p,q)$, $E^{p,q}_r$ stabilizes when $r$ is sufficiently
large. We denote the stable objects by $E^{p,q}_\infty$.
\item A family of morphisms $d^{p,q}_r:E^{p,q}_r{\rightarrow}
E^{p+r,q-r+1}_r$ satisfying $$d^{p+r,q-r+1}_r{{\mathfrak{m}}athfrak{c}}irc d^{p,q}_r=0.$$
\item A family of isomorphisms
${{\mathfrak{m}}athfrak{a}}lpha^{p,q}_r:{\kappa}er(d^{p,q}_r)/Im(d^{p-r,q+r-1}_r)\stackrel{\rightarrow}{\sim}
E^{p,q}_{r+1}$.
\item A family of objects $\{E^n\}$ in ${\mathfrak{m}}athscr{C}$. For every
$E_n$, there is a bounded decreasing filtration $\{F^pE^n\}$ in the
sense that there is some $p$ such that $F^pE^n=E^n$ and there is
some $p$ such that $F^pE^n=0$.
\item A family of isomorphisms
${{\mathfrak{m}}athfrak{b}}eta^{p,q}:E^{p,q}_\infty\stackrel{\rightarrow}{\sim}F^pE^{p+q}/F^{p+1}E^{p+q}$.
{\epsilon}nd{enumerate}
We say that the spectral sequence $\{E^{p,q}_r\}$ converges to $\{E^n\}$
and write $$E^{p,q}_2{{\mathfrak{m}}athbb R}ightarrow E^{p+q}.$$
A morphism ${{\mathfrak{m}}athfrak{p}}hi:E{\rightarrow} H$ between two spectral sequences on
${\mathfrak{m}}athscr{C}$ is a family of morphisms ${{\mathfrak{m}}athfrak{p}}hi^{p,q}_r:E^{p,q}_r{\rightarrow}
H^{p,q}_r$ and ${{\mathfrak{m}}athfrak{p}}hi^n:E^n{\rightarrow} H^n$ such that ${{\mathfrak{m}}athfrak{p}}hi$ is compatible
with $d$, ${{\mathfrak{m}}athfrak{a}}lpha$, the filtration and ${{\mathfrak{m}}athfrak{b}}eta$. The following result is useful in order to obtain information about the limiting map ${{\mathfrak{m}}athfrak{p}}hi^n$ from the maps ${{\mathfrak{m}}athfrak{p}}hi_2^{p,q}$.
{{\mathfrak{m}}athfrak{b}}egin{lemma}[see Lemma 2.15 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}]
Let $$\xymatrix{ E_2^{i,j} {{\mathfrak{m}}athfrak{a}}r@2{->}[r] {{\mathfrak{m}}athfrak{a}}r^{{{\mathfrak{m}}athfrak{p}}hi_2^{i,j}}[d] & E^{i+j} {{\mathfrak{m}}athfrak{a}}r^{{{\mathfrak{m}}athfrak{p}}hi^{i+j}}[d] \\
H_2^{i,j} {{\mathfrak{m}}athfrak{a}}r@2{->}[r] & H^{i+j}}$$ be two spectral sequences with
commutative maps. Let $l$ and $a$ be integers. Suppose that
$E_2^{i,l-i}=0$ for $i<a$, $H_2^{i,l-i}=0$ for $i>a$ and
${{\mathfrak{m}}athfrak{p}}hi_2^{a,l-a}=0$. Then ${{\mathfrak{m}}athfrak{p}}hi^l=0$. \label{spec-seq-zero-map}
{\epsilon}nd{lemma} {\mathfrak{m}}edskip
\subsection{Generic vanishing in positive characteristic}
A smooth projective variety $X$ over an algebraically closed field is said to have maximal Albanese dimension if it admits a generically finite morphism to an abelian variety $X{\rightarrow} A$. Over fields of characteristic zero, the main tool that is employed when studying properties of varieties of maximal Albanese dimension is the generic vanishing theorem of Green and Lazarsfeld (~{{\mathfrak{m}}athfrak{c}}ite{gl87}, ~{{\mathfrak{m}}athfrak{c}}ite{gl91}). Even though it is shown in ~{{\mathfrak{m}}athfrak{c}}ite{hk12} that the obvious generalization of this result to fields of positive characteristic if false, recent work of Hacon and Patakfalvi ~{{\mathfrak{m}}athfrak{c}}ite{hp13} provides a weaker generic vanishing statement in positive characteristic which albeit necessarily weaker, is strong enough prove positive characteristic versions of Kawamata's celebrated characterization of abelian varieties. In this subsection we collect the results of ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that we shall be using throughout. {\mathfrak{m}}edskip
The following is the main theorem in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}:
{{\mathfrak{m}}athfrak{b}}egin{thm}( c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Theorem 3.1, Lemma 3.2]{hp13})
Let $A$ be an abelian variety defined over an algebraically closed field of positive characteristic and let ${\mathcal{O}}mega_{e+1} {\rightarrow} {{\mathcal{O}}mega}_e$ be an inverse system of coherent sheaves on $A$.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item If for any sufficiently ample line bundle $L\in {{\mathfrak{m}}athbb P}ic(\hat{A})$ and for any $e>>0$ we have $H^i(A,{{\mathcal{O}}mega}_e \otimes RS_{\hat{A},A}(L)^{\vee})=0$ for every $i>0$, then the complex ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}} RS_{A,\hat{A}}(D_A{{\mathcal{O}}mega}_e)$ (which in general is concentrated in degrees $[-g,\ldots,0]$), is actually a quasi-coherent sheaf concentrated in degree 0, namely ${\Lambda}ambda={\mathcal{H}}^0({\Lambda}ambda)$. Besides ${{\mathcal{O}}mega}=\varprojlim {{\mathcal{O}}mega}_e = \left((-1_A)^{{{\mathfrak{m}}athfrak{a}}st}D_A RS_{\hat{A},A}({\Lambda}ambda)\right)[g]$.
\item The condition in (i) is satisfied for coherent Cartier modules: if $F_{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_0 {\rightarrow} {{\mathcal{O}}mega}_0$ is a coherent Cartier module and we denote ${{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0$, then for any ample line bundle $L\in {{\mathfrak{m}}athbb P}ic(\hat{A})$ we have $$H^i(A,{{\mathcal{O}}mega}_e \otimes RS_{\hat{A},A}(L)^{\vee} \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0, {{\mathfrak{m}}athfrak{q}}uad \forall e>>0, {{\mathfrak{m}}athfrak{q}}uad \forall i>0, {{\mathfrak{m}}athfrak{q}}uad \forall {{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$$
{\epsilon}nd{enumerate} \label{generic-vanishing-char-p}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
From the above result and the cohomology and base change theorem one derives the following corollary:
{{\mathfrak{m}}athfrak{b}}egin{corol}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Corollaries 3.5 and 3.6]{hp13})
With the same notations as above we have the following:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item For every ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ we have ${\Lambda}ambda \otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq \varinjlim H^0(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$, and for every integer $e\geq0$, ${\mathcal{H}}^0({\Lambda}ambda_e) \otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq H^0(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$.
\item There exists a proper closed subset $Z\subset \hat{A}$ such that if $i>0$ and $p^ey\notin Z$ for all $e>>0$, then $\varinjlim H^i(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}=0$. Furthermore, if $W^i=\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}, {{\mathfrak{m}}athfrak{q}}uad \varprojlim H^i(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee} \neq 0\}$, then $$W^i \subset Z'=\overline{{{\mathfrak{m}}athfrak{b}}igcup_{e\geq 0} \left([p_{\hat{A}}^e]^{-1}(Z)\right)_{red}}$$ where $pZ' \subset Z'$ If besides $\hat{A}$ has no supersingular factors, then the top dimensional components of $Z'$ are a finite union of torsion translates of subtori of $A$.
{\epsilon}nd{enumerate} \label{GV-corollary1}
{\epsilon}nd{corol} {\mathfrak{m}}edskip
We quote two more results from ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that will provide a simple proof of a special case of our main theorem:
{{\mathfrak{m}}athfrak{b}}egin{prop}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Proposition 3.17]{hp13})
Let $A$ be an ordinary abelian variety and consider the same notations as above. Then each maximal dimensional irreducible component of the set $Z$ of points ${{\mathfrak{m}}athfrak{a}}lpha \in \hat{A}$ such that the image of the natural map $${\mathcal{H}}^0({\Lambda}ambda_0) \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\mathcal{H}}^0({\Lambda}ambda) \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \simeq {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha}$$ is non-zero, is a torsion translate of an abelian subvariety of $\hat{A}$ and ${\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \neq 0$ if and only if ${{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}^e \in Z$. \label{GV-corollary2}
{\epsilon}nd{prop} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{prop}(c.f. ~{{\mathfrak{m}}athfrak{c}}ite[Lemma 3.9, Corollary 3.10]{hp13})
Let ${{\mathcal{O}}mega}_0$ be a coherent sheaf on an abelian variety $A$ and assume that $F_{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_0 {\rightarrow} {{\mathcal{O}}mega}_0$ is surjective. Then ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega} = {{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}_0$, so that ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}$ is a closed subvariety. Let $\hat{B} \subset \hat{A}$ be an abelian subvariety such that $$V^0({{\mathcal{O}}mega}_0)=\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: h^0({{\mathcal{O}}mega}_0 \otimes {{\mathfrak{m}}athbb P}P_{{{\mathfrak{m}}athfrak{a}}lpha}) \neq 0\}$$ is contained in finitely many translates of $\hat{B}$. Then $t_x^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega} \simeq {{\mathcal{O}}mega}$ for every $x\in {\omega}idehat{\hat{A}/\hat{B}}$. In particular, ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}$ is fibered by the projection $A{\rightarrow} B$, namely ${{\mathfrak{m}}athbb S}upp {{\mathcal{O}}mega}$ is a union of fibers of $A{\rightarrow} B$. \label{GV-corollary3}
{\epsilon}nd{prop} {\mathfrak{m}}edskip
Note that, in particular, Proposition \ref{GV-corollary3} applies to the subvariety $$Z=\left\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: {{\mathfrak{m}}athfrak{q}}uad Im \left({\mathcal{H}}^0({\Lambda}ambda_0)\otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \right) \neq 0\right\}$$ from Proposition \ref{GV-corollary2}. {\mathfrak{m}}edskip
Finally, grounding on the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}, part of Pareschi and Popa's generic vanishing theory was extended to positive characteristic in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}. The main result in that paper is the following:
{{\mathfrak{m}}athfrak{b}}egin{thm} Let $A$ be an abelian variety. Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on $A$ satisfying the Mittag-Leffler condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let ${\Lambda}ambda_e=R\hat{S}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. The following are equivalent:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item[(1)] For any ample line bundle $L$ on $\hat{A}$, $H^i(A,{\mathcal{O}}mega\otimes \hat{L}^\vee)=0$ for any $i>0$.
\item[(1')] For any fixed positive integer $e$ and any $i>0$, the homomorphism $$H^i(A, {\mathcal{O}}mega\otimes \hat{L}^\vee) {\rightarrow} H^i(A,{\mathcal{O}}mega_e\otimes \hat{L}^\vee)$$ is 0 for any sufficiently ample line bundle $L$.
\item[(2)] ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$.
{\epsilon}nd{enumerate}
These conditions imply the following:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item[(3)] For any scheme-theoretical point $P\in A$, if ${\textrm{dim }} P>g-i$, then $P$ is not in the support of the image of $$\varprojlim R^i\hat{S}({\mathcal{O}}mega_e) {\rightarrow} R^i\hat{S}({\mathcal{O}}mega_e)$$ for any $e$.
{\epsilon}nd{enumerate}
If $\{R^i\hat{S}({\mathcal{O}}mega_e)\}$ satisfies the Mittag-Leffler condition for any $i\geq 0$, then (3) also implies (1), (1') and (2) and, moreover, the support of the image of the map in (3) is a closed subset.
{\epsilon}nd{thm} {\mathfrak{m}}edskip
We also record the following variant of the implication $(2){{\mathfrak{m}}athbb R}ightarrow (3)$ in the previous theorem.
{{\mathfrak{m}}athfrak{b}}egin{prop}
Let ${{\mathfrak{m}}athfrak{p}}i:A {{\mathfrak{m}}athfrak{p}}roj W$ be a projection between abelian varieties with generic fiber dimension $f$ and with ${\textrm{dim }} W=k$. Let $\{{{\mathcal{O}}mega}_e\}_e$ be a Cartier module on $A$ and let $S_{A,\hat{W}}$ be the Fourier-Mukai functor with kernel $\left({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{W}}\right)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. Denote ${\Lambda}_e = RS_{A,\hat{W}} (D_A ({{\mathcal{O}}mega}_e))$. If $P\in \hat{W}$ is a scheme-theoretic point with ${\textrm{dim }}(P)>k+f-{\epsilon}ll$, then $P$ is not in the support of the image of the map $$\varprojlim_e R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) {\longrightarrow} R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$$ Moreover, if the inverse system $\{R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the Mittag-Leffler condition, then the support of the image of the above map is closed and its codimension is $\geq {\epsilon}ll-f$. \label{GV-k}
{\epsilon}nd{prop}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The proof is identical (modulo shifts) to that of Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, but we include it for the sake of completeness. {\mathfrak{m}}edskip
We need to show that if $P\in \hat{W}$ is a scheme-theoretic point with ${\textrm{dim }}(P)>k+f-{\epsilon}ll$, then $$\left(R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim {{\mathcal{O}}mega}_e)\right)_P \stackrel{0}{{\longrightarrow}} \left(R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_P$$
Note in the first place that for any ${\epsilon}ll$, we have the following isomorphisms
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p({\Lambda}_e,{\mathfrak{m}}athcal{O}_{\hat{W}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-k}(D_{\hat{W}}({\Lambda}_e)) \simeq
{\mathfrak{m}}athcal{H}^{p-k}(D_{\hat{W}}(RS_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e)))) \nonumber \\
&\stackrel{[{{\mathfrak{m}}athfrak{a}}st]}{\simeq}& {\mathfrak{m}}athcal{H}^{p}(
R\tilde{S}_{A,\hat{W}}(D_A(D_A({{\mathcal{O}}mega}_e)))) \simeq
{\mathfrak{m}}athcal{H}^{p}(R\tilde{S}_{A,\hat{W}}({{\mathcal{O}}mega}_e)) \label{comp1} {\epsilon}nd{eqnarray}
and
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p({\Lambda},{\mathfrak{m}}athcal{O}_{\hat{W}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-g}(D_{\hat{W}}({\Lambda})) \simeq
{\mathfrak{m}}athcal{H}^{p-g}(D_{\hat{W}}({\textrm{hocolim}_{\rightarrow}}_e RS_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e)))) \nonumber \\
&\simeq& {\mathfrak{m}}athcal{H}^{p-g}({\textrm{holim}_{\leftarrow}}_e D_{\hat{W}}(RS_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e)))) \nonumber \\ &\simeq&
{\mathfrak{m}}athcal{H}^p({\textrm{holim}_{\leftarrow}}_e (-1_{\hat{W}})^* RS_{A,\hat{W}}(\tilde{{{\mathcal{O}}mega}}_e)) \nonumber \\
&\simeq&
(-1_{\hat{W}})^*{\mathcal{H}}^p\left({\textrm{holim}_{\leftarrow}}_e RS_{A,\hat{W}}(\tilde{{{\mathcal{O}}mega}}_e)\right) \label{comp2} {\epsilon}nd{eqnarray}
where in $[{{\mathfrak{m}}athfrak{a}}st]$ we used Lemma 2.2 in ~{{\mathfrak{m}}athfrak{c}}ite{pp11}. Here, if $S$ is the Fourier-Mukai functor with kernel ${{\mathfrak{m}}athbb P}P$, $\tilde{S}$ denotes the Fourier-Mukai functor with kernel ${{\mathfrak{m}}athbb P}P^{\vee}$; the codimension computation is unaffected by this, so we omit the tildes in the remainder of the proof. {\mathfrak{m}}edskip
From the following factorization of the map $R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim {{\mathcal{O}}mega}_e) {\rightarrow} R^{{\epsilon}ll}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$
$$R^{{\epsilon}ll}S_{A,\hat{W}}(\varprojlim_e {{\mathcal{O}}mega}_e) {\longrightarrow} {\mathcal{H}}^{{\epsilon}ll}\left( {\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) {\longrightarrow} {\mathcal{H}}^{{\epsilon}ll}\left( RS_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right)$$
we see that it suffices to show that the map $${\mathcal{E}} xt^{{\epsilon}ll}({\Lambda},{\mathcal{O}}_{\hat{W}})_P {\longrightarrow} {\mathcal{E}} xt^{{\epsilon}ll}({\Lambda}_e,{\mathcal{O}}_{\hat{W}})_P$$ is zero for ${\textrm{dim }}(P)>k+f-{\epsilon}ll$. In order to see this, we may proceed as in the proof of Theorem 4.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}, computing the above map via the commutative diagram $$\xymatrix{ {\mathcal{E}} xt^i({\mathcal{H}}^j({\Lambda}),{\mathcal{O}}_{\hat{W}})_P {{\mathfrak{m}}athfrak{a}}r[d]_{{{\mathfrak{m}}athfrak{p}}hi^{i,j}} {{\mathfrak{m}}athfrak{a}}r@{=>}[r] & {\mathcal{E}} xt^{{\epsilon}ll}({\Lambda},{\mathcal{O}}_{\hat{W}})_P {{\mathfrak{m}}athfrak{a}}r[d]^{{{\mathfrak{m}}athfrak{p}}hi^{{\epsilon}ll}} \\ {\mathcal{E}} xt^i({\mathcal{H}}^j({\Lambda}_e),{\mathcal{O}}_{\hat{W}})_P {{\mathfrak{m}}athfrak{a}}r@{=>}[r] & {\mathcal{E}} xt^{{\epsilon}ll}({\Lambda}_e,{\mathcal{O}}_{\hat{W}})_P}$$
with $i-j={\epsilon}ll$. We seek to apply Lemma \ref{spec-seq-zero-map} with $a={\epsilon}ll-1-f$. If $i>a$, then $i \geq {\epsilon}ll-f > [k+f-{\textrm{dim }}(P)] -f = k-{\textrm{dim }}(P)$ and hence ${\mathcal{E}} xt^i({\mathcal{H}}^j({\Lambda}_e),{\mathcal{O}}_{\hat{W}})_P=0$ and if $i\leq a$, then $j = i-{\epsilon}ll \leq ({\epsilon}ll -1 -f) -{\epsilon}ll = -1-f$, so that ${\mathcal{H}}^j({\Lambda})=0$ (c.f. proof of Theorem 3.1.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}). Lemma \ref{spec-seq-zero-map} then implies that ${{\mathfrak{m}}athfrak{p}}hi^{{\epsilon}ll}=0$ as claimed.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
\section{Main technical result}
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a $g$-dimensional abelian variety satisfying the Mittag-Leffler condition and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let ${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. It was shown in Theorem 4.2 of ~{{\mathfrak{m}}athfrak{c}}ite{wz14} that if ${\mathcal{H}}^0({\Lambda})$ is torsion-free, then the maps $$\left(\varprojlim R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)\right)_P {\longrightarrow} R^kS_{A,\hat{A}}({{\mathcal{O}}mega}_e)_P$$ are zero for any point $P\in \hat{A}$ such that ${\textrm{dim }}(P) \geq g-k$. We next show a partial converse to this statement. {\mathfrak{m}}edskip
In the sequel, we will say that ${\mathcal{H}}^0({\Lambda})$ has torsion if it is not torsion-free. More concretely, we will say that ${\mathcal{H}}^0({\Lambda})$ has torsion at a point $P$ if there exists a section $s \in {\mathcal{O}}_{\hat{A}}$ such that the multiplication map ${\mathcal{H}}^0({\Lambda})_P \stackrel{\times s_P}{{\longrightarrow}} {\mathcal{H}}^0({\Lambda})_P$ is not injective. {\mathfrak{m}}edskip
Before stating our main result we need to introduce some notation. Consider the following commutative diagram {{\mathfrak{m}}athfrak{b}}egin{equation} \xymatrix{0 {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathbb{F}}F={\Lambda}/{\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & \tilde{{\Lambda}}_e^t {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & \tilde{{\Lambda}}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & {\mathbb{F}}F_e {{\mathfrak{m}}athfrak{a}}r[u] {{\mathfrak{m}}athfrak{a}}r[r] & 0} \label{main-thm-notation} {\epsilon}nd{equation} where ${\Lambda}^t$ denotes the torsion subsheaf of ${\Lambda}$, $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$, ${\mathbb{F}}F_e=\textrm{Im}(\tilde{{\Lambda}}_e {\rightarrow} {\mathbb{F}}F)$ and $\tilde{{\Lambda}}_e^t={\kappa}er\left( \tilde{{\Lambda}}_e {\rightarrow} {\mathbb{F}}F\right)$. It is easy to see that the second row is exact and that $\tilde{{\Lambda}}_e^t$ is the torsion subsheaf of $\tilde{{\Lambda}}_e$. It is also clear by construction that ${\Lambda} = \varinjlim_e \tilde{{\Lambda}}_e$ (c.f. Theorem \ref{generic-vanishing-char-p}). {\mathfrak{m}}edskip
Since the direct limit is exact, by its universal property there is also a commutative diagram $$\xymatrix{0 {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & {\Lambda} {{\mathfrak{m}}athfrak{a}}r[r] & {\mathbb{F}}F={\Lambda}/{\Lambda}^t {{\mathfrak{m}}athfrak{a}}r[r] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & \varinjlim_e \tilde{{\Lambda}}_e^t {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & \varinjlim_e \tilde{{\Lambda}}_e {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u]_{\simeq} & \varinjlim_e {\mathbb{F}}F_e {{\mathfrak{m}}athfrak{a}}r[u] {{\mathfrak{m}}athfrak{a}}r[r] & 0}$$ Note that ${\Lambda}^t \simeq \varinjlim_e \tilde{{\Lambda}}_e^t$. Indeed, an element ${\epsilon}ta \in {\Lambda}^t \hookrightarrow {\Lambda}$ lifts to a class $[\tilde{{\epsilon}ta}_e] \in \varinjlim \tilde{{\Lambda}}_e = {\Lambda}$ and this class maps to 0 under the composition $\varinjlim \tilde{{\Lambda}}_e {\rightarrow} {\mathbb{F}}F$, so it lies in $\varinjlim \tilde{{\Lambda}}_e^t$ (by exactness of the direct limit). By the 5-lemma, we have that the right vertical map is also an isomorphism. {\mathfrak{m}}edskip
We are now ready to state our main technical result: {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $\{{\mathcal{O}}mega_e\}$ be an inverse system of coherent sheaves on a $g$-dimensional abelian variety satisfying the Mittag-Leffler condition. Let ${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$. Denote as above $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$ and define $\tilde{{\mathcal{O}}mega}_e = RS_{\hat{A},A} (D_{\hat{A}} (\tilde{{\Lambda}}_e))$. Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. If ${\mathcal{H}}^0({\Lambda})$ has a torsion point $P$ of maximal dimension $g-k$, then $$\varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0.$$ Equivalently, the maps $$\varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) {\rightarrow} R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)$$ are non-zero for every $e\gg0$. \label{torsion-non-zero-map}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{pf}
We start by performing a sequence of reductions. {\mathfrak{m}}edskip
\textbf{Reduction 1}: With the notation introduced in diagram (\ref{main-thm-notation}), in order to show that $$\varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0$$ it is sufficient to show that $$\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] \neq 0.$$
Indeed, consider the long exact sequence for ${\mathcal{E}} xt$ induced by the short exact sequence $$0 {\longrightarrow} \tilde{{\Lambda}}_e^t {\longrightarrow} \tilde{{\Lambda}}_e {\longrightarrow} {\mathbb{F}}F_e {\longrightarrow} 0$$ namely $${{\mathfrak{m}}athfrak{c}}dots {\longrightarrow} {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) {\longrightarrow} {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) {\longrightarrow} {\mathcal{E}} xt^{k+1}({\mathbb{F}}F_e, {\mathcal{O}}_{\hat{A}}) {\longrightarrow} {{\mathfrak{m}}athfrak{c}}dots$$ By Lemma 6.3 in ~{{\mathfrak{m}}athfrak{c}}ite{pp08b} it follows that ${\mathcal{E}} xt^{k+1}({\mathbb{F}}F_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) = 0$ for all $e$, so since $\otimes k(P)$ is right-exact, we obtain a surjection $${\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\longrightarrow} {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ and hence\footnote{Note that the system $\left\{ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P)\right\}_e$ satisfies the ML-condition.} a surjection $$\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] {\longrightarrow} \varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right].$$
Therefore, if $\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e^t, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] \neq0$, then $$\varprojlim_e \left[ {\mathcal{E}} xt^k(\tilde{{\Lambda}}_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) \right] \stackrel{[{{\mathfrak{m}}athfrak{a}}st]}{\simeq} \varprojlim \left(R^kS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0$$ as claimed, where the isomorphism $[{{\mathfrak{m}}athfrak{a}}st]$ follows from the following computation
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} {\mathcal{E}} xt^p(\tilde{{\Lambda}}_e,{\mathfrak{m}}athcal{O}_{\hat{A}}) &\simeq&
{\mathfrak{m}}athcal{H}^{p-g} \left( D_{\hat{A}}(\tilde{{\Lambda}}_e) \right) \simeq
{\mathfrak{m}}athcal{H}^{p-g} ( (-1_{\hat{A}})^{{{\mathfrak{m}}athfrak{a}}st} RS_{A,\hat{A}} \overbrace{RS_{\hat{A},A} (D_{\hat{A}}(\tilde{{\Lambda}}_e))}^{:=\tilde{{{\mathcal{O}}mega}}_e} [g]) \nonumber \\
&\simeq& {\mathfrak{m}}athcal{H}^p \left( (-1_{\hat{A}})^* RS_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e) \right). {\epsilon}nd{eqnarray}
\textbf{Reduction 2}: Denoting $i:Z:=\overline{\{P\}}\hookrightarrow \hat{A}$, we next reduce to showing that $$\varprojlim_e \left[ {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}^{t}_e,{\mathcal{O}}_Z) \otimes k(P) \right] \neq0$$
In order to see this, it suffices to show that for every $e$ there is an isomorphism {{\mathfrak{m}}athfrak{b}}egin{equation} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P) \simeq {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t, {\mathcal{O}}_Z) \otimes k(P). \label{key-iso} {\epsilon}nd{equation}
But observe that {{\mathfrak{m}}athfrak{b}}egin{multline*} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}})_{|Z} \otimes k(P) \simeq L^0i^{{{\mathfrak{m}}athfrak{a}}st} {\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P) \stackrel{[1]}{\simeq} {\mathcal{H}}^k\left( Li^{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \right) \otimes k(P) \\ \stackrel{[2]}{\simeq} {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}^t_e, {\mathcal{O}}_Z) \otimes k(P) {\epsilon}nd{multline*}
where:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(i)]
\item The isomorphism in [1] follows from Grothendieck's spectral sequence\footnote{C.f. equation (3.10) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}. Also note that since tensoring by $k(P)$ is exact on $D(Z)$, there is a spectral sequence as written.} $$E_2^{p,q} = L^pi^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^q(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\Lambda}ongrightarrow L^{p+q}i^{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ Note in the first place that ${\mathcal{E}} xt^q(\tilde{{\Lambda}}^t_e,{\mathcal{O}}_{\hat{A}})=0$ near $P$ for all $q<k$. Indeed, since $\tilde{{\Lambda}}_e^t$ is supported on $Z$\footnote{Note that if $Z$ is an irreducible component of ${{\mathfrak{m}}athbb S}upp {\Lambda}$, then it is also an irreducible component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for every $e>>0$. Let $Z$ be one such component, denote by ${\mathcal{I}}_Z$ its ideal sheaf and take a section ${\epsilon}ta=\{{\epsilon}ta_e\}\in {\Lambda}$ supported on $Z$. It is then clear that $Z\subset {{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for every $e>>0$. Now, if $f\in {\mathcal{I}}_Z$ is in $\operatorname{ann}({\epsilon}ta)$, it is clear that $f\in \operatorname{ann}({\epsilon}ta_e)$ for $e>>0$, so that $Z={{\mathfrak{m}}athbb S}upp {\epsilon}ta_e$ for all $e>>0$, as claimed.} near $P$, which has codimension $k$, our claim follows from the fact that ${\mathcal{E}} xt^q({{\mathfrak{m}}athfrak{b}}ullet,{\mathcal{O}}_{\hat{A}})=0$ for all $q<{{\mathfrak{m}}athfrak{c}}odim{{\mathfrak{m}}athbb S}upp({{\mathfrak{m}}athfrak{b}}ullet)$ (c.f. Lemma 6.1 in ~{{\mathfrak{m}}athfrak{c}}ite{pp08b}). {\mathfrak{m}}edskip
The differentials coming out of $E_2^{0,k} = L^0i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}})$ are hence trivial and the differential targeting $E_2^{0,k}$ is $$d_2^{-2,k+1}: L^{-2}i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^{k+1}(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P) {\longrightarrow} L^0i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^k(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}}) \otimes k(P)$$ which is also trivial since there is an open neighborhood $U$ of $P$ over which $$\left[L^{-2}i^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^{k+1}(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}})\right]_{|U} \simeq L^{-2} i_{U}^{{{\mathfrak{m}}athfrak{a}}st} {\mathcal{E}} xt^{k+1}(\tilde{{\Lambda}}^t_e, {\mathcal{O}}_{\hat{A}})_{U} = 0$$ where $i_U: Z {{\mathfrak{m}}athfrak{c}}ap U \hookrightarrow U$ and where the last vanishing follows from the coherence of $\tilde{{\Lambda}}^t_e$ and the fact that $P$ has codimension $k$.
\item The isomorphism in [2] follows from the $Lf^{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om({\mathbb{F}}F^{{{\mathfrak{m}}athfrak{b}}ullet},{\mathfrak{m}}athcal{G}^{{{\mathfrak{m}}athfrak{b}}ullet}) \simeq R{\mathcal{H}} om(Lf^{{{\mathfrak{m}}athfrak{a}}st}{\mathbb{F}}F^{{{\mathfrak{m}}athfrak{b}}ullet}, Lf^{{{\mathfrak{m}}athfrak{a}}st}{\mathfrak{m}}athcal{G}^{{{\mathfrak{m}}athfrak{b}}ullet})$ (c.f. equation (3.17) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}).
{\epsilon}nd{enumerate}
Finally, in order to show that $$\varprojlim_e \left( {\mathcal{E}} xt^k(Li^{{{\mathfrak{m}}athfrak{a}}st} \tilde{{\Lambda}}_e^{t},{\mathcal{O}}_Z)\otimes k(P) \right)$$ is non-zero, we will use the isomorphism $$\varprojlim_e \left( {\mathcal{E}} xt^k_{{\mathcal{O}}_Z} (Li^{{{\mathfrak{m}}athfrak{a}}st} \tilde{{\Lambda}}_e^{t},{\mathcal{O}}_Z)\otimes k(P) \right) = \varprojlim_e {\mathcal{E}} xt^k_{k(P)} \left( Li^{{{\mathfrak{m}}athfrak{a}}st} \tilde{{\Lambda}}_e^{t} \otimes k(P), k(P) \right) \neq 0$$
where we used that $k(P)\simeq {\mathcal{O}}_P$ and Proposition III.6.8 in ~{{\mathfrak{m}}athfrak{c}}ite{har77}. Denote by $i:Z=\overline{\{P\}} \hookrightarrow \hat{A}$ the inclusion. By Grothendieck duality (c.f. discussion in section 2.3 from ~{{\mathfrak{m}}athfrak{c}}ite{bst12}), since all the higher direct images of a closed immersion are zero, we have a functorial isomorphism {{\mathfrak{m}}athfrak{b}}egin{equation} R{\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e \otimes k(P) \right], k(P)[-k] \right) \simeq Ri_{{{\mathfrak{m}}athfrak{a}}st} R{\mathcal{H}} om_{Z} \left( Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e \otimes k(P), Li^! k(P) \right) \label{GD} {\epsilon}nd{equation}
Taking k-th cohomology, we obtain {{\mathfrak{m}}athfrak{b}}egin{eqnarray}{\mathcal{H}} om_{k(P)} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P) \right], k(P) \right) &\simeq& {\mathcal{H}}^k\left(Ri_{{{\mathfrak{m}}athfrak{a}}st}R{\mathcal{H}} om_{k(P)}(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P), Li^! k(P))\right) \nonumber \\ &\simeq& i_{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{E}} xt^k_{k(P)}(Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P), Li^! k(P)). \label{GD2} {\epsilon}nd{eqnarray}
Note that the inverse limit of the left hand side is $$\varprojlim {\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}\tilde{{\Lambda}}_e^t \otimes k(P) \right], k(P) \right) = {\mathcal{H}} om_{\hat{A}} \left( i_{{{\mathfrak{m}}athfrak{a}}st}\left[ Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} \otimes k(P) \right], k(P) \right).$$
We claim that the latter sheaf is non-zero. Indeed, note that {{\mathfrak{m}}athfrak{b}}egin{equation} H om_{\hat{A}} \left( i^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} \otimes k(P) , k(P) \right) \neq0 \label{non-zero-hom} {\epsilon}nd{equation}
since it is simply the $k(P)$-dual of the non-zero $k(P)$-vector space ${\Lambda}_{|Z} \otimes k(P)$. The natural map $Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda} {\rightarrow} L^0 i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda}$ induces $$R^0i_{{{\mathfrak{m}}athfrak{a}}st} \left[ Li^{{{\mathfrak{m}}athfrak{a}}st}({\Lambda}) \otimes k(P) \right] \stackrel{\simeq}{{\longrightarrow}} R^0i_{{{\mathfrak{m}}athfrak{a}}st} \left[ i^{{{\mathfrak{m}}athfrak{a}}st}({\Lambda}) \otimes k(P) \right] \stackrel{\neq0}{{\longrightarrow}} k(P)$$ where the last map is just a non-zero morphism from (\ref{non-zero-hom}) with the source sheaf extended by zero. Since ${\Lambda}$ is locally free in a neighborhood of $P$ and closed immersions have no higher direct images, the fact that the first map is an isomorphism follows from the degeneration of the spectral sequence (c.f. equation (3.10) in ~{{\mathfrak{m}}athfrak{c}}ite{huy06}) $$R^s i_{{{\mathfrak{m}}athfrak{a}}st} \left( L^t i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right) \simeq R^s i_{{{\mathfrak{m}}athfrak{a}}st} \left( {\mathcal{H}}^t \left( L i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right)\right) \stackrel{s+t=p}{{\Lambda}ongrightarrow} R^pi_{{{\mathfrak{m}}athfrak{a}}st} \left( Li^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P) \right)$$ since $L^t i^{{{\mathfrak{m}}athfrak{a}}st} {\Lambda} \otimes k(P)=0$ for all $t<0$. Hence, the inverse limit on the right hand side of (\ref{GD2}) is also non-zero, and in particular that $$\varprojlim {\mathcal{E}} xt^k_Z(Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda}_e \otimes k(P), Li^! k(P)) \neq0$$
But recall that $Z$ is a torsion translate of an abelian subvariety of $\hat{A}$, so ${\omega}_Z\simeq {\mathcal{O}}_Z \simeq {\mathcal{O}}_P \simeq k(P)$, so $Li^! k(P)\simeq k(P)$ and we may hence conclude that $$\varprojlim {\mathcal{E}} xt^k_Z(Li^{{{\mathfrak{m}}athfrak{a}}st}{\Lambda}_e \otimes k(P), k(P)) \neq0$$ as claimed.
{\epsilon}nd{pf} {{\mathfrak{m}}athfrak{b}}igskip
{{\mathfrak{m}}athfrak{b}}egin{rmk}
Theorem \ref{torsion-non-zero-map} has shown that $$\varprojlim \left(R^k S_{A,\hat{A}}(\tilde{{{\mathcal{O}}mega}}_e)\otimes k(P)\right) \neq 0.$$ where recall, we defined $\tilde{{\mathcal{O}}mega}_e = RS_{\hat{A},A} (D_{\hat{A}} (\tilde{{\Lambda}}_e))$ where $\tilde{{\Lambda}}_e=\textrm{Im}({\Lambda}_e {\rightarrow} {\Lambda})$. In what follows we will drop the tildes in order to ease de notation: all we need is a projective system of coherent sheaves satisfying the generic vanishing property and inducing a non-zero limit as stated in the theorem.
{\epsilon}nd{rmk} {\mathfrak{m}}edskip
If $A$ has no super-singular factors, we know by Proposition 3.3.5 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} that for every $e\geq0$, the top dimensional components of the set of points $P\in \hat{A}$ such that the map ${\mathcal{H}}^0({\Lambda}_e)_P{\rightarrow} {\mathcal{H}}^0({\Lambda})_P$ is non-zero is a torsion translate of an abelian subvariety of $\hat{A}$. {\mathfrak{m}}edskip
Let $P\in \hat{A}$ be a torsion point of maximal dimension (namely, ${\textrm{dim }}(P)$ is maximal such that ${\mathcal{H}}^0({\Lambda})_P$ has torsion) and consider $W=\overline{\{P\}}$. In particular, $W$ is a component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda})$, and we already argued earlier that $W$ must also be an irreducible component of ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}_e)$ for $e>>0$, so $W$ is also a top dimensional component of the support of the image of the map ${\mathcal{H}}^0({\Lambda}_e){\rightarrow} {\mathcal{H}}^0({\Lambda})$. We thus conclude that if $P\in \hat{A}$ is a torsion point of maximal dimension, then $W=\overline{\{P\}}$ is a torsion translate of an abelian subvariety of $\hat{A}$. {\mathfrak{m}}edskip
In this context, Theorem \ref{torsion-non-zero-map} yields the following.
{{\mathfrak{m}}athfrak{b}}egin{corol}
Let $\{{\mathcal{O}}mega_e\}$ be a Mittag-Leffler inverse system of coherent sheaves on a
$g$-dimensional abelian variety $A$ with no supersingular factors, and let ${\mathcal{O}}mega=\varprojlim {\mathcal{O}}mega_e$. Let
${\Lambda}ambda_e=RS_{A,\hat{A}}(D_A({\mathcal{O}}mega_e))$ and ${\Lambda}ambda={\textrm{hocolim}_{\rightarrow}}{\Lambda}ambda_e$.
Suppose that $\{{{\mathcal{O}}mega}_e\}$ is a GV-inverse system, in the sense that ${\mathfrak{m}}athcal{H}^i({\Lambda}ambda)=0$ for any $i\neq 0$. If $P$ is a torsion point of ${\mathcal{H}}^0({\Lambda})$ of maximal dimension (so that $W=\overline{\{P\}}$ is a torsion translate of an abelian subvariety of $\hat{A}$), then there are non-zero maps $$\varprojlim \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {\rightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)$$ for $e\gg0$, where the Fourier-Mukai kernel of $S_{A,\hat{W}}$ is given by ${{\mathfrak{m}}athbb P}P^{A\times \hat{W}}=(id\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$, ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$ being the normalized Poincar{\'e} bundle of $A\times \hat{A}$. \label{torsion-non-zero-map2}
{\epsilon}nd{corol}
{{\mathfrak{m}}athfrak{b}}egin{pf}
By Theorem \ref{torsion-non-zero-map} we have a non-zero map $$\varprojlim \left(R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {\rightarrow} R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)$$ for some $e>0$. Consider the base-change maps $$R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P) {\rightarrow} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}), {{\mathfrak{m}}athfrak{q}}uad R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) {\rightarrow} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}})$$ Note that the second map is an isomorphism by flat base change: indeed, denoting by $\iota:\hat{W} \hookrightarrow \hat{A}$ the inclusion, we have {{\mathfrak{m}}athfrak{b}}egin{eqnarray*} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) &\stackrel{def}{=}& R^{g-k}p_{\hat{W}{{\mathfrak{m}}athfrak{a}}st}\left(p_A^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}} \right)\otimes k(P) \\ &\stackrel{FBC}{\simeq}& R^{g-k}p_{\hat{W}{{\mathfrak{m}}athfrak{a}}st}'\left(p_A^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}\otimes k(P) \right) \\ &\stackrel{def}{\simeq}& R^{g-k}p_{\hat{W}{{\mathfrak{m}}athfrak{a}}st}'\left(p_A^{{{\mathfrak{m}}athfrak{a}}st}{{\mathcal{O}}mega}_e \otimes (id\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}\otimes k(P) \right) \\ &\stackrel{[{{\mathfrak{m}}athfrak{a}}st]}{\simeq}& H^{g-k}(A,{{\mathcal{O}}mega}_e \otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}) {\epsilon}nd{eqnarray*} where in $[{{\mathfrak{m}}athfrak{a}}st]$ we used Proposition III.8.5 in ~{{\mathfrak{m}}athfrak{c}}ite{har77} and where $p_{\hat{W}}'$ is the base change of the projection, as illustrated in the diagram $$\xymatrix{A \times_{\hat{W}} {{\mathfrak{m}}athbb S}pec k(P) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{p_{\hat{W}}'} & A \times \hat{W} {{\mathfrak{m}}athfrak{a}}r[d]^{p_{\hat{W}}} \\ {{\mathfrak{m}}athbb S}pec k(P) {{\mathfrak{m}}athfrak{a}}r[r]^{\textrm{flat}} & \hat{W}}$$
However note that we have {{\mathfrak{m}}athfrak{b}}egin{equation} H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}) \simeq H^{g-k}(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}) \label{obvious-iso} {\epsilon}nd{equation} so we can write both base change maps in the following diagram $$\xymatrix{\varprojlim\left(R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {{\mathfrak{m}}athfrak{a}}r[r]^-{\neq 0} {{\mathfrak{m}}athfrak{a}}r[d] & R^{g-k}S_{A,\hat{A}}({{\mathcal{O}}mega}_e)\otimes k(P) {{\mathfrak{m}}athfrak{a}}r[d]^-{[{{\mathfrak{m}}athfrak{a}}st]} \\ \varprojlim H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[d]^{\simeq} \\ \varprojlim H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}\right) {{\mathfrak{m}}athfrak{a}}r[r] & H^{g-k}\left(A,{{\mathcal{O}}mega}_e\otimes {{\mathfrak{m}}athbb P}P^{A\times \hat{W}}_{|A\times \{P\}}\right) \\ \varprojlim\left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P)\right) {{\mathfrak{m}}athfrak{a}}r[u]^{\simeq} {{\mathfrak{m}}athfrak{a}}r[r] & R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(P) {{\mathfrak{m}}athfrak{a}}r[u]_{\simeq}}$$ where the top horizontal map is non-zero for $e\gg0$ by Theorem \ref{torsion-non-zero-map}, the middle isomorphisms are the ones in (\ref{obvious-iso}), and where the isomorphisms at the bottom follow from flat base change as described above. {\mathfrak{m}}edskip
We seek to show that the bottom horizontal map is non-zero for some $e$. Nevertheless, note that if it this were not the case, then the top horizontal map could not possibly be non-zero, since in any case the base change maps $[{{\mathfrak{m}}athfrak{a}}st]$ are injective by the proof of Proposition III.12.5 in ~{{\mathfrak{m}}athfrak{c}}ite{har77}\footnote{In a nutshell, let $f:X{\rightarrow} Y={{\mathfrak{m}}athbb S}pec A$ be a projective morphism and let ${\mathbb{F}}F$ be a coherent sheaf on $X$. For any $A$-module $M$, define $T^i(M):=H^i(X,{\mathbb{F}}F\otimes_A M)$, which is a covariant additive functor from $A$-modules to $A$-modules which is exact in the middle (by Proposition III.12.1 in ~{{\mathfrak{m}}athfrak{c}}ite{har77}). Writing $$A^r{\rightarrow} A^s {\rightarrow} M {\rightarrow} 0$$ we have a diagram $$\xymatrix{T^i(A)\otimes A^r {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & T^i(A)\otimes A^s {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]_{\simeq} & T^i(A) \otimes M {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[d]^{\varphi} & 0 \\ T^i(A^r) {{\mathfrak{m}}athfrak{a}}r[r] & T^i(A^s) {{\mathfrak{m}}athfrak{a}}r[r] & T^i(M)}$$ where $\varphi:T^i(A)\otimes M {\rightarrow} T^i(M)$ is the base change map and where the two first vertical arrows are isomorphisms. A straight-forward diagram chase then shows that $\varphi$ is injective.}.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{rmk}
We will be using two different Fourier-Mukai kernels on $A\times \hat{W}$. If $\iota:\hat{W}\hookrightarrow \hat{A}$ denotes the inclusion and ${{\mathfrak{m}}athfrak{p}}i:A{{\mathfrak{m}}athfrak{p}}roj W$ is the dual projection, we have a diagram $$\xymatrix{A\times \hat{W} {{\mathfrak{m}}athfrak{a}}r[r]^{id_A\times \iota} {{\mathfrak{m}}athfrak{a}}r[d]_{{{\mathfrak{m}}athfrak{p}}i \times id_{\hat{W}}} & A \times \hat{A} \\ W \times \hat{W} & }$$ If ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$ and ${{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$ denote the normalized Poincar{\'e} bundles on $A\times \hat{A}$ and $W\times \hat{W}$ respectively, on $A\times \hat{W}$ we may consider the locally-free sheaves $(id_A\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}$ and $({{\mathfrak{m}}athfrak{p}}i\times id_{\hat{W}})^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. {\mathfrak{m}}edskip
In Corollary \ref{torsion-non-zero-map2} we proved a non-vanishing statement for the derived Fourier-Mukai transform with respect to the former kernel and in what follows we need an analogous statement for the transform with respect to the latter. Nevertheless, note that we are simply looking at fibers over points $w\in \hat{W}\subset \hat{A}$ (concretely over the generic point of $\hat{W}$), and over these points both sheaves are isomorphic. Indeed, $w\in \hat{W}\subset \hat{A}$ determines ${{\mathfrak{m}}athbb P}P^{W\times \hat{W}}_{|W\times \{w\}}\in {{\mathfrak{m}}athbb P}ic(W)$ and ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{w\}}\in {{\mathfrak{m}}athbb P}ic(A)$, with ${{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{w\}}\simeq {{\mathfrak{m}}athfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athbb P}P^{W\times \hat{W}}_{|W\times \{w\}}$, and therefore $$\overbrace{\left[(id_A\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{A\times \hat{A}}\right]_{|A\times \{w\}}}^{\simeq {{\mathfrak{m}}athbb P}P^{A\times \hat{A}}_{|A\times \{w\}}} \simeq \overbrace{\left[({{\mathfrak{m}}athfrak{p}}i\times id_{\hat{W}})^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}\right]_{|A\times \{w\}}}^{\simeq {{\mathfrak{m}}athfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st} {{\mathfrak{m}}athbb P}P^{W\times \hat{W}}_{|W\times \{w\}}}$$ \label{different-FM-kernels}
{\epsilon}nd{rmk} {\mathfrak{m}}edskip
\section{Fibering of the Albanese image}
In ~{{\mathfrak{m}}athfrak{c}}ite{el97}, Ein and Lazarsfeld showed the following statement:
{{\mathfrak{m}}athfrak{b}}egin{thm}[see ~{{\mathfrak{m}}athfrak{c}}ite{el97}, Theorem 3]
$X$ is a smooth projective variety of maximal Albanese dimension over a field of characteristic zero and ${{\mathfrak{m}}athfrak{c}}hi({\omega}_X)=0$, then the image of the Albanese map is fibered by subtori of $A$. \label{EL-fibered-by-tori}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{skpf} We sketch the proof given in ~{{\mathfrak{m}}athfrak{c}}ite{pp08} (Theorem E). {\mathfrak{m}}edskip
If ${{\mathfrak{m}}athfrak{c}}hi({\omega}_X)=0$, it follows that $V^0({\omega}_X)\subset \hat{A}$ is a proper subvariety (c.f. Lemma 1.12(b) in ~{{\mathfrak{m}}athfrak{c}}ite{par11}). Choose an irreducible component $W\subset V^0({\omega}_X)$ of codimension $p>0$, which is a torsion translate of an abelian subvariety of $\hat{A}$ that we also denote by $W$. Let ${{\mathfrak{m}}athfrak{p}}i:A{{\mathfrak{m}}athfrak{p}}roj \hat{W}$ denote the dual projection and consider the diagram $$\xymatrix{X {{\mathfrak{m}}athfrak{a}}r[r]^-a & Y:=a(X) {{\mathfrak{m}}athfrak{a}}r[d]_-{h={{\mathfrak{m}}athfrak{p}}i_{|Y}} {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[r] & A {{\mathfrak{m}}athfrak{a}}r@{->>}[dl]^{{{\mathfrak{m}}athfrak{p}}i} \\ & \hat{W}}$$
Since the fibers of the projection $A{{\mathfrak{m}}athfrak{p}}roj \hat{W}$ are abelian subvarieties of dimension $p$, the conclusion of the theorem will follow provided that $f\geq p$, where $f$ denotes the dimension of the generic fiber of $h$. In order to see this, recall the following standard facts:
{{\mathfrak{m}}athfrak{b}}egin{enumerate}[(a)]
\item $a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X$ is a GV-sheaf on $Y=a(X)$ and $V^0({\omega}_X)=V^0(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)$.
\item If $W$ is an irreducible component of $V^0(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)$ of codimension $p$, then it is also a component of $V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)$.
\item $a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X$ is a $GV_{-f}$-sheaf with respect the the Fourier-Mukai functor with kernel $({{\mathfrak{m}}athfrak{p}}i \times 1_W)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$, so in particular ${{\mathfrak{m}}athfrak{c}}odim_W V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X) \geq p-f$ for every $p\geq0$.
{\epsilon}nd{enumerate} {\mathfrak{m}}edskip
By (b) we have $W \subseteq V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X) \subseteq W$ so that ${{\mathfrak{m}}athfrak{c}}odim_W V^p(a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X)=0$, and finally (c) yields $f\geq p$, which is what we sought to show.
{\epsilon}nd{skpf} {\mathfrak{m}}edskip
Our goal in this section is to prove a positive characteristic analogue of Theorem \ref{EL-fibered-by-tori}. {\mathfrak{m}}edskip
Let $X$ be a smooth projective variety of maximal Albanese dimension and denote by $a:X{\rightarrow} A$ the Albanese map. Then $a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X$ is a Cartier module and we may consider the inverse system $\{{{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^e S^0a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X\}_e$. Define ${\Lambda}_e=RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$ and set ${\Lambda}={\textrm{hocolim}_{\rightarrow}} {\Lambda}_e$. By Corollary 3.3.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} we have that $H^i({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for every $i>0$ and very general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. Thus, defining as above ${{\mathfrak{m}}athfrak{c}}hi({{\mathcal{O}}mega}):={{\mathfrak{m}}athfrak{c}}hi({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ for very general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$, we see that ${{\mathfrak{m}}athfrak{c}}hi({{\mathcal{O}}mega})=h^0({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ and it seems that in trying to extend Theorem \ref{EL-fibered-by-tori} to positive characteristic, one should assume that $h^0({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$. {\mathfrak{m}}edskip
This leaves us in a setting which is similar to the one we encountered in the proof of Theorem \ref{Main-Theorem}. If $rk({\Lambda})=h^0({{\mathcal{O}}mega}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$, then in particular ${\Lambda}ambda$ must be a torsion sheaf. In light of this observation, we show the following:
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $X$ be a smooth projective variety of maximal Albanese dimension and let $a:X{\rightarrow} A$ be a generically finite map to an abelian variety $A$ with no supersingular factors. Let $g={\textrm{dim }} A$. Consider the inverse system $\{{{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0a_{{{\mathfrak{m}}athfrak{a}}st}{\omega}_X\}_e$ and denote ${{\mathcal{O}}mega}=\varprojlim {{\mathcal{O}}mega}_e$. Define ${\Lambda}_e=RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$ and assume that the sheaf ${\mathcal{H}}^0({\Lambda})=\varinjlim {\mathcal{H}}^0({\Lambda}_e)$ has torsion. Then the image of the Albanese map is fibered by linear subvarieties of $\hat{A}$. \label{alb-fibered-by-tori}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{pf}
Let $w\in \hat{A}$ be a torsion point of ${\mathcal{H}}^0({\Lambda})$ of maximal dimension $k$; by the remark preceding Corollary \ref{torsion-non-zero-map2}, we have that $\hat{W}:=\overline{\{w\}}\subset \hat{A}$ is a torsion translate of an abelian subvariety of $\hat{A}$, which we still denote by $\hat{W}$. Denote by ${{\mathfrak{m}}athfrak{p}}i:A{\rightarrow} W$ the projection dual to the inclusion $\hat{W}\hookrightarrow \hat{A}$. $${{\mathfrak{m}}athfrak{b}}egin{diagram} \node{Y:=a(X)} {{\mathfrak{m}}athfrak{a}}rrow{e,t,J}{} {{\mathfrak{m}}athfrak{a}}rrow{s,l}{h={{\mathfrak{m}}athfrak{p}}i_{|Y}} \node{A} {{\mathfrak{m}}athfrak{a}}rrow{sw,r}{{{\mathfrak{m}}athfrak{p}}i} \\ \node{W} {\epsilon}nd{diagram}$$
By Corollary \ref{torsion-non-zero-map2} we know that the map $$\varprojlim \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(w)\right) {\rightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\otimes k(w)$$ is non-zero for every $e\gg 0$, where the Fourier-Mukai kernel of $S_{A,\hat{W}}$ is given by $(id\times \iota)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P$, ${{\mathfrak{m}}athbb P}P$ being the normalized Poincar{\'e} bundle of $A\times \hat{A}$. {\mathfrak{m}}edskip
Recall that, in general, even though the system $\{{{\mathcal{O}}mega}_e\}$ satisfies the Mittag-Leffler condition, the inverse system $\{R^tS({{\mathcal{O}}mega}_e)\}_e$ may fail to do so (c.f. Example 3.2 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}). We handle the Mittag-Leffler case first, however, since the proof is neater and the subsequent generalization does not rely on new ideas. {\mathfrak{m}}edskip
\textbf{Case in which $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the ML-condition}. The proof in this case goes along the lines of that of Theorem E in ~{{\mathfrak{m}}athfrak{c}}ite{pp08}. Note that if $w$ is the generic point of $\hat{W}\hookrightarrow \hat{A}$, we have the following:
{{\mathfrak{m}}athfrak{b}}egin{eqnarray*} w &\stackrel{[1]}{\in}& \left\{w\in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \right\} \\ &\stackrel{[2]}{=}& \left\{ w \in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \right\} \\ &\subseteq& \left\{ w \in \hat{W}: {{\mathfrak{m}}athfrak{q}}uad \left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \stackrel{\neq0}{{\longrightarrow}} \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \right\} \\ &\subseteq& \hat{W} {\epsilon}nd{eqnarray*}
where $w$ lies in the first set by Corollary \ref{torsion-non-zero-map2} and where the equality [2] follows from Lemma \ref{inverse-limit-tensor-product-commute}, since we are under the assumption that the system $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$ satisfies the Mittag-Leffler condition. {\mathfrak{m}}edskip
This implies that the codimension (in $\hat{W}$) of the support of image of the map $$\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) {\longrightarrow} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)$$ is zero (this support is closed - under the Mittag-Leffler assumption - by Proposition 4.3 in ~{{\mathfrak{m}}athfrak{c}}ite{wz14}). At the same time, by Proposition \ref{GV-k} we know that this codimension must be $\geq g-k-f$, where $f$ is the dimension of a general fiber of $h$, so in particular $f\geq g-k$ and this concludes the proof under the Mittag-Leffler assumption (indeed, the fibers of the projection $A{{\mathfrak{m}}athfrak{p}}roj W$ are abelian subvarieties of dimension $g-k$). {\mathfrak{m}}edskip
\textbf{General case}. We finally observe that, in our setting, we can actually do without the Mittag-Leffler assumption on $\{R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\}_e$. We only used this assumption in order to guarantee the closedness of the support of the image of the map $\left(\varprojlim_e R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w \stackrel{\neq0}{{\longrightarrow}} \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right)_w$ and in order to ensure that the inverse limit commutes with $\otimes k(w)$. {\mathfrak{m}}edskip
Note in the first place that we don't need the support of the image of the above map to be closed for the previous argument to work. Proposition \ref{GV-k} shows that in order for $w$ to belong to the support, we need ${{\mathfrak{m}}athfrak{c}}odim \overline{\{w\}}\geq g-k-f$, and this suffices in order to conclude that $f\geq g-k$. {\mathfrak{m}}edskip
With regards to the commutation of the inverse limit and $\otimes k(w)$, note in the first place that there is always an inclusion $\supseteq$ induced by the natural map $$\varprojlim_e \left( R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) {\longrightarrow} \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right)$$ Moreover, in our setting, the opposite inclusion $\subseteq$ follows from the flatness of $k(w)$ as an ${\mathcal{O}}_{\hat{W}}$-module and the fact that the projection formula and its consequences still hold in the category of quasi-coherent sheaves under some perfection assumptions (c.f. Lemma 71 in ~{{\mathfrak{m}}athfrak{c}}ite{mur06}). We state this below as a lemma, and the proof is hence complete.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{lemma}
With the same notations as above, if $$\varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)$$ then $$\varprojlim_e \left( R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)$$ In particular, equality [2] in the above chain still holds.
{\epsilon}nd{lemma}
{{\mathfrak{m}}athfrak{b}}egin{pf}
As in the proof of Proposition \ref{GV-k}, let $\tilde{{\Lambda}}_e = R\tilde{S}_{A,\hat{W}}(D_A({{\mathcal{O}}mega}_e))$, where $\tilde{S}_{A,\hat{W}}$ denotes the Fourier-Mukai transform with kernel ${\mathfrak{m}}athcal{P}^{\vee}$, with ${\mathfrak{m}}athcal{P}=\left({{\mathfrak{m}}athfrak{p}}i \times 1_{\hat{W}}\right)^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athbb P}P^{W\times \hat{W}}$. Note in the first place that we have the following isomorphisms of ${\mathcal{O}}_{\hat{W}}$-modules: {{\mathfrak{m}}athfrak{b}}egin{eqnarray*} \varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)\right) &\stackrel{[1]}{\simeq}& \varprojlim_e \left({\mathcal{E}} xt^{g-k}_{{\mathcal{O}}_{\hat{W}}}(\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \otimes k(w)\right) \\ &\simeq& \varprojlim_e \left({\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \right) \otimes k(w)\right) \\ &\stackrel{[2]}{\simeq}& \varprojlim_e {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \otimes k(w) \right) \\ &\stackrel{[3]}{\simeq}& {\mathcal{H}}^{g-k}\left(\varprojlim_e \left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,{\mathcal{O}}_{\hat{W}}) \otimes k(w) \right)\right) \\ &\stackrel{[4]}{\simeq}& {\mathcal{H}}^{g-k}\left(\varprojlim_e R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}}_e,k(w)) \right) \\ &\stackrel{[5]}{\simeq}& {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}},k(w)) \right) \\ &\stackrel{[6]}{\simeq}& {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}},{\mathcal{O}}_{\hat{W}}) \otimes k(w) \right) \\ &\stackrel{[7]}{\simeq}& {\mathcal{H}}^{g-k}\left(R{\mathcal{H}} om_{{\mathcal{O}}_{\hat{W}}} (\tilde{{\Lambda}},{\mathcal{O}}_{\hat{W}})\right) \otimes k(w) \\ &\stackrel{[8]}{\simeq}& {\mathcal{H}}^{g-k}\left({\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) {\epsilon}nd{eqnarray*}
where [1] and [8] follow from the computations (\ref{comp1}) and (\ref{comp2}) in the proof of Proposition \ref{GV-k}, [2] and [7] follow from the flatness of $\otimes k(w)$ as an ${\mathcal{O}}_{\hat{W}}$-module, [3] follows from Proposition \ref{inverse-limits-commute-functor}, since the system $\left\{{\mathcal{E}} xt^{p-1}({\Lambda}_e,{\mathcal{O}}_{\hat{W}})\otimes k(w)\right\}_e$ satisfies the ML-condition, and [4] and [6] follow from the isomorphism\footnote{This isomorphism holds for complexes of sheaves of modules ${\mathfrak{m}}athcal{F},{\mathfrak{m}}athcal{G},{\mathfrak{m}}athcal{H}$ provided that either ${\mathfrak{m}}athcal{F}$ or ${\mathfrak{m}}athcal{H}$ are perfect (c.f. Lemma 71 in ~{{\mathfrak{m}}athfrak{c}}ite{mur06}). Note that $k(w)$ is a perfect complex, being a coherent sheaf.} $$R{\mathcal{H}} om({\mathbb{F}}F,{\mathfrak{m}}athcal{G}) \otimes {\mathfrak{m}}athcal{H} \simeq R{\mathcal{H}} om({\mathbb{F}}F,{\mathfrak{m}}athcal{G} \otimes {\mathfrak{m}}athcal{H}).$$
Hence, by assumption we have a non-zero map $${\mathcal{H}}^{g-k}\left({\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) \stackrel{\neq0}{{\longrightarrow}} R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w)$$ and the conclusion of the lemma then follows from the following commutative diagram $$\xymatrix{\varprojlim_e \left(R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \right) \otimes k(w) {{\mathfrak{m}}athfrak{a}}r[r] & R^{g-k}S_{A,\hat{W}}({{\mathcal{O}}mega}_e) \otimes k(w) \\ {\mathcal{H}}^{g-k}\left({\textrm{holim}_{\leftarrow}} RS_{A,\hat{W}}({{\mathcal{O}}mega}_e)\right) \otimes k(w) {{\mathfrak{m}}athfrak{a}}r[u] {{\mathfrak{m}}athfrak{a}}r[ur]_{\neq0}}$$
{\epsilon}nd{pf} {\mathfrak{m}}edskip
In particular, within the context of principally polarized abelian varieties, the same argument yields the following statement:
{{\mathfrak{m}}athfrak{b}}egin{corol}
Let $(A,\Theta)$ is a principally polarized abelian variety with no supersingular factors defined over an algebraically closed field of characteristic $p>0$. Assume further that $\Theta$ is irreducible. Consider the inverse system $\{{{\mathcal{O}}mega}_e:=F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau_{\Theta})\}_e$ on $A$ and set ${\Lambda}={\textrm{hocolim}_{\rightarrow}}_e RS_{A,\hat{A}}D_A({{\mathcal{O}}mega}_e)$. Then ${\Lambda}$ is a torsion-free quasi-coherent sheaf concentrated in degree 0. \label{theta-divisors-not-ruled}
{\epsilon}nd{corol}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The fact that ${\Lambda}={\mathcal{H}}^0({\Lambda})$ is a quasi-coherent sheaf concentrated in degree zero follows from Theorem \ref{generic-vanishing-char-p}(i), since ${\omega}_{\Theta}$ is a Cartier module. {\mathfrak{m}}edskip
Assume for a contradiction that ${\mathcal{H}}^0({\Lambda})$ is not torsion-free and fix an irreducible component $\hat{W}:=\overline{\{w\}}\hookrightarrow \hat{A}$ of maximal dimension of the closure of the set of torsion points of ${\mathcal{H}}^0({\Lambda})$. Denote by ${{\mathfrak{m}}athfrak{p}}i:A{{\mathfrak{m}}athfrak{p}}roj W$ the dual projection. $$\xymatrix{\Theta {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[r] {{\mathfrak{m}}athfrak{a}}r[d]_{h} & A {{\mathfrak{m}}athfrak{a}}r@{->>}[dl]_{{{\mathfrak{m}}athfrak{p}}i} \\ W & }$$
We may then argue as in the proof of Theorem \ref{alb-fibered-by-tori} to conclude that $\Theta$ is fibered by abelian subvarieties of $A$, but this is not possible given that $\Theta$ is irreducible (and hence of general type) in light of Abramovich's work (c.f. Section 2.3 or ~{{\mathfrak{m}}athfrak{c}}ite{abr95}).
{\epsilon}nd{pf} {\mathfrak{m}}edskip
\section{Singularities of Theta divisors}
We now focus on the singularities of Theta divisors and embark on the proof of Theorem \ref{Main-theorem}. As a warm-up, we focus on simple abelian varieties to start with, namely those which do not contain smaller dimensional abelian varieties.
\subsection{Case of simple abelian varieties}
The crux of the argument resides in the construction of sections of ${\mathcal{O}}_A(\Theta)$ which vanish along the test ideal $\tau(\Theta)$ and, in the case of simple abelian varieties, it is a direct consequence of the results in ~{{\mathfrak{m}}athfrak{c}}ite{hp13}. The proof of the general case will follow the same pattern, albeit further work will be required to prove that the required sections exist. {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $(A,\Theta)$ be a PPAV over an algebraically closed field $K$ of characteristic $p>0$ such that $A$ is simple and ordinary. Then $\Theta$ is strongly F-regular. (In particular, $\Theta$ is F-rational, and by Lemma 2.34 in ~{{\mathfrak{m}}athfrak{c}}ite{bst12}, it is normal and Cohen-Macaulay).
\label{Main-theorem-simple}
{\epsilon}nd{thm} {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{pf}
On $\hat{A}$ consider the inverse system ${{\mathcal{O}}mega}_e=F_{{{\mathfrak{m}}athfrak{a}}st}^e{{\mathcal{O}}mega}_0$, where ${{\mathcal{O}}mega}_0=\omega_{\Theta}\otimes \tau(\Theta)$. This yields a direct system ${\Lambda}ambda_e= R\hat{S}D_A{{\mathcal{O}}mega}_e$ equipped with natural maps ${\mathcal{H}}^0({\Lambda}ambda_e) {\rightarrow} {\mathcal{H}}^0({\Lambda}ambda)={\Lambda}ambda={\textrm{hocolim}_{\rightarrow}} {\Lambda}ambda_e$. By \ref{generic-vanishing-char-p}, we know that ${\Lambda}ambda$ is quasi-coherent sheaf in degree 0. {\mathfrak{m}}edskip
Consider the set $$Z=\left\{{{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}: {{\mathfrak{m}}athfrak{q}}uad Im \left({\mathcal{H}}^0({\Lambda}ambda_0)\otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} {\longrightarrow} {\Lambda}ambda \otimes {\mathcal{O}}_{\hat{A},{{\mathfrak{m}}athfrak{a}}lpha} \right) \neq 0\right\}$$ By Proposition \ref{GV-corollary2} we know that $Z$ is a finite union of torsion translates of subtori. Since $\hat{A}$ is simple by assumption, this implies that either $Z=\hat{A}$ or $Z$ is a finite set. {\mathfrak{m}}edskip
Assume for a contradiction that $Z$ is finite. By Proposition \ref{GV-corollary3} we have $t_x^{{{\mathfrak{m}}athfrak{a}}st}{\mathcal{O}}mega={\mathcal{O}}mega$ for every $x\in {\omega}idehat{\hat{A}/Z}={\omega}idehat{\hat{A}}$, so that ${{\mathfrak{m}}athbb S}upp {\mathcal{O}}mega=A$. Since the maps in the inverse system $F_{{{\mathfrak{m}}athfrak{a}}st}^e{\mathcal{O}}mega_0=F_{{{\mathfrak{m}}athfrak{a}}st}^e(\omega_{\Theta}\otimes \tau(\Theta))$ are surjective, we know by Proposition \ref{GV-corollary3} that ${{\mathfrak{m}}athbb S}upp {\mathcal{O}}mega={{\mathfrak{m}}athbb S}upp {\mathcal{O}}mega_0={{\mathfrak{m}}athbb S}upp {\omega}_{\Theta}\otimes \tau(\Theta)=A$, which is absurd. {\mathfrak{m}}edskip
We must thus have $Z=\hat{A}$, so that ${{\mathfrak{m}}athbb S}upp {\mathcal{H}}^0({\Lambda}ambda_0)=\hat{A}$, and hence cohomology and base change yields $H^0(A,{\omega}_{\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$ for all ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. {\mathfrak{m}}edskip
Consider the following commutative diagram: $$\xymatrix{ H^0(A,P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta} \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \\ H^0(A,K\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & }$$
where $K$ is defined so that the diagram commutes. In the top row we have $H^1(A,P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for ${{\mathfrak{m}}athfrak{a}}lpha\neq 0$ since $P_{{{\mathfrak{m}}athfrak{a}}lpha}$ is topologically trivial. The polarization induced by $\Theta$ is principal, so $h^0({\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=1$. Since by the above discussion $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}\otimes \tau(\Theta))\neq 0$, it follows that $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$ and hence both the right inclusion and the top right restriction are equalities. {\mathfrak{m}}edskip
The ideal sheaf $\tilde{\tau}$ on $A$ is defined as follows: fix an open subset $U={{\mathfrak{m}}athbb S}pec R\subseteq A$ and assume that $\Theta$ is given by an ideal sheaf $I=I(\Theta)$. Let $J=\tau_{\Theta}(U)$ be the test ideal of $\Theta$ and let $\tilde{J}\subset R$ be an ideal such that $J=\tilde{J}/I$. Omitting the twist by $P_{{{\mathfrak{m}}athfrak{a}}lpha}$, the diagram above locally boils down to
$$\xymatrix{ && R/\tilde{J} {{\mathfrak{m}}athfrak{a}}r[r]^-{\simeq} & (R/I)/(\tilde{J}/I) \simeq R/\tilde{J} & \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & I {{\mathfrak{m}}athfrak{a}}r[r] & R {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & R/I {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & 0 \\ 0 {{\mathfrak{m}}athfrak{a}}r[r] & I {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & \tilde{J} {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & J {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & 0 }$$
so $\tilde{\tau}(U)=\tilde{J}$. Now taking cohomology, a section $s\in H^0(J)$ embeds as ${{\mathfrak{m}}athfrak{b}}ar{s}\in H^0(R/I)$ and maps to zero in $H^0(R/\tilde{J})$ by exactness. By exactness of the second row, ${{\mathfrak{m}}athfrak{b}}ar{s}$ lifts to $\tilde{{{\mathfrak{m}}athfrak{b}}ar{s}}\in H^0(R)$, which still projects to zero in $H^0(R/\tilde{J})$ by commutativity of the top square, so $\tilde{{{\mathfrak{m}}athfrak{b}}ar{s}}$ must lift to a non-zero section of $H^0(\tilde{J})$. {\mathfrak{m}}edskip
Finally, since $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}\otimes \tau(\Theta))\neq 0$ for every ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ and these sections lift to sections of $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ vanishing along $\tilde{\tau}$, we conclude that $h^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}\otimes \tilde{\tau})=1$. Hence if $\tilde{\tau}$ were not trivial, we would have $Zeros(\tilde{\tau})\subset \Theta+{{\mathfrak{m}}athfrak{a}}lpha_P$ for every ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ (where ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ is the point corresponding to $P_{{{\mathfrak{m}}athfrak{a}}lpha}\in {{\mathfrak{m}}athbb P}ic^0(A)$), which is absurd since these translates of $\Theta$ don't have any points in common. We thus conclude that $\tilde{\tau}={\mathcal{O}}_A$, and hence $\tau(\Theta)={\mathcal{O}}_{\Theta}$ so that $\Theta$ is strongly F-regular.
{\epsilon}nd{pf} {\mathfrak{m}}edskip
In the proof of Theorem \ref{Main-theorem-simple} we used the simplicity of $A$ in order to show that $H^0(A,{\omega}_{\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$ for all ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. The same argument we employed above will work in the general case provided that we can show the existence of sections in $H^0(A,{\omega}_{\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$, and it turns out that this is somewhat more involved.
\subsection{General case}
We finally study singularities of Theta divisors in the general setting. As we mentioned earlier, the argument will be analogous to the one employed to prove the theorem in the case of simple abelian varieties, although additional work will be required to prove that there exist sections in $H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$. {\mathfrak{m}}edskip
The main ingredient in Ein and Lazarsfeld's proof over fields of characteristic zero was that given a smooth projective variety $X$ of maximal Albanese dimension such that ${{\mathfrak{m}}athfrak{c}}hi(X,{\omega}_X)=0$, the image of its Albanese morphism is fibered by tori (c.f. Theorem 3 in ~{{\mathfrak{m}}athfrak{c}}ite{el97}). Our proof will rely on Corollary \ref{theta-divisors-not-ruled}, where we proved that if the sheaf ${\mathcal{H}}^0({\Lambda})$ associated to the inverse system $\{F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau_{\Theta})\}$ was not torsion-free, then $\Theta$ would be fibered by tori, which is impossible since $\Theta$ is irreducible. {\mathfrak{m}}edskip
In a nutshell, and as in the case of simple abelian varieties, $\Theta$ will be strongly F-regular provided that there exist non-trivial sections in $H^0(\Theta,{\mathcal{O}}_A(\Theta)\otimes {\mathcal{O}}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ and we will show that if that was not the case, then the sheaf ${\mathcal{H}}^0({\Lambda})$ would have torsion, a contradiction. {\mathfrak{m}}edskip
{{\mathfrak{m}}athfrak{b}}egin{thm}
Let $(A,\Theta)$ be an ordinary principally polarized abelian variety over an algebraically closed field $k$ of characteristic $p>0$. If $\Theta$ is irreducible, then $\Theta$ is strongly F-regular. \label{Main-Theorem}
{\epsilon}nd{thm}
{{\mathfrak{m}}athfrak{b}}egin{pf}
The proof goes along the lines of Theorem \ref{Main-theorem-simple}: consider again the commutative diagram: $$\xymatrix{ H^0(A,P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta} \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \\ H^0(A,K\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau}) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & H^0(\Theta,{\mathcal{O}}_A(\Theta)_{|\Theta}\otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) {{\mathfrak{m}}athfrak{a}}r@{^{(}->}[u] & }$$
In the proof of Theorem \ref{Main-theorem-simple} we used the simplicity of $A$ to conclude easily that $$H^0(\Theta,\overbrace{{\mathcal{O}}_A(\Theta)\otimes {\mathcal{O}}_{\Theta}}^{{\omega}_{\Theta}} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$$ It then followed from the commutative diagram above that $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau})\neq 0$ and this in turn forced $\tilde{\tau}$ to be trivial, whence $\tau(\Theta)={\mathcal{O}}_{\Theta}$. {\mathfrak{m}}edskip
We shall now use the previous results in order to conclude that $H^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})\neq 0$. In a nutshell, assuming for a contradiction that $\tau(\Theta)$ is not trivial we will show that $0\neq S^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}) \subseteq H^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})$ for general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$. The diagram above then yields $H^0(A,{\mathcal{O}}_A(\Theta)\otimes P_{{{\mathfrak{m}}athfrak{a}}lpha} \otimes \tilde{\tau})\neq 0$ and since by assumption $\tau(\Theta)\ne {\mathcal{O}}_{\Theta}$, we conclude that $\textrm{Zeroes}(\tilde{\tau})\subset \Theta+{{\mathfrak{m}}athfrak{a}}lpha_P$ for general ${{\mathfrak{m}}athfrak{a}}lpha_P\in A$ (as before, ${{\mathfrak{m}}athfrak{a}}lpha_P$ denotes the point in $A$ corresponding to $P_{{{\mathfrak{m}}athfrak{a}}lpha}\in {{\mathfrak{m}}athbb P}ic^0A$), but this is not possible since general translates of $\Theta$ do not have points in common. Therefore we must have $\tau(\Theta)={\mathcal{O}}_{\Theta}$, and hence $\Theta$ is strongly F-regular. {\mathfrak{m}}edskip
We now argue by contradiction: let ${{\mathcal{O}}mega}=\varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^eS^0{\omega}_{\Theta}= \varprojlim F_{{{\mathfrak{m}}athfrak{a}}st}^e({\omega}_{\Theta}\otimes \tau(\Theta))$. By Corollary 3.2.1 in ~{{\mathfrak{m}}athfrak{c}}ite{hp13} and its proof, for all closed ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$ we have that $${\mathcal{H}}^0({\Lambda})\otimes k({{\mathfrak{m}}athfrak{a}}lpha) \simeq H^0(\Theta,\varprojlim {{\mathcal{O}}mega}_e \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha}^{\vee})^{\vee}$$ so assuming for a contradiction that $S^0(\Theta,{\omega}_{\Theta} \otimes \tau(\Theta) \otimes P_{{{\mathfrak{m}}athfrak{a}}lpha})=0$ for general ${{\mathfrak{m}}athfrak{a}}lpha\in \hat{A}$, it follows that $rk ({\Lambda}ambda)=0$, and hence that ${\mathcal{H}}^0({\Lambda}ambda)$ has torsion. Nevertheless, this is not possible by Corollary \ref{theta-divisors-not-ruled}, since we are assuming that $\Theta$ is irreducible, and this concludes the proof.
{\epsilon}nd{pf}
\small
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\begin{document}
\title{Cavity QED with Multiple Hyperfine Levels}
\author{K.~M.~Birnbaum}
\altaffiliation[Permanent address: ]{Jet Propulsion Laboratory, California Institute of Technology, Mail Stop 161-135, 4800 Oak Grove Drive, Pasadena, CA 91109, U.S.A.}
\affiliation{Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,
Pasadena, CA 91125, U.S.A.}
\author{A.~S.~Parkins}
\altaffiliation[Permanent address: ]{Department of Physics, University of Auckland,
Private Bag 92019, Auckland, New Zealand}
\affiliation{Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,
Pasadena, CA 91125, U.S.A.}
\author{H. J. Kimble}
\affiliation{Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,
Pasadena, CA 91125, U.S.A.}
\date{June 8, 2006}
\begin{abstract}
We calculate the weak-driving transmission of a linearly polarized cavity mode strongly coupled to the D2 transition of a single Cesium atom. Results are relevant to future experiments with microtoroid cavities, where the single-photon Rabi frequency $g$ exceeds the excited-state hyperfine splittings, and photonic bandgap resonators, where $g$ is greater than both the excited- and ground-state splitting.
\end{abstract}
\pacs{42.50.Pq, 42.50.-p, 32.10.Fn}
\maketitle
\section{Introduction}
The Jaynes-Cummings model of cavity QED treats an atom as a two-level system. This is appropriate for a realistic atom when that atom has a cycling transition, typically reached by optical pumping with circularly polarized light \cite{hood98,hood00}. However, new types of optical resonators such as microtoroids \cite{toroid} and photonic band gap cavities \cite{pbg} do not support circularly polarized modes. Though these structures with extremely low critical atom and photon numbers show great promise for strong coupling, a more detailed model of the atom must be employed when calculating the properties of these atom-cavity systems \cite{kmb,iqec}. A linearly polarized mode may couple multiple Zeeman states of the atom. Additionally, for these very small resonators, the single photon Rabi frequency ($2g$) can be comparable to or larger than the hyperfine splitting of the atom, so that multiple hyperfine levels must be considered when calculating the excitations of the system. We will consider a linearly polarized single-mode resonator coupled to the D2 ($6S_{1/2} \to 6P_{3/2}$) transition of a single Cesium atom. However, this may also give some intuition for other multilevel scatterers, such as molecules and excitons \cite{exciton1, exciton2}.
\section{Coupling to Multiple Excited Levels}
\label{sec:toroid}
In order to describe the interaction of the atom with various light fields, it is useful to define the atomic dipole transition operators
\begin{multline}
\label{dipole}
D_{q}(F,F')= \\
\sum_{m_{F}=-F}^{F}|F,m_{F}\rangle \langle F,m_{F}| \mu_{q} |F',m_{F}+q \rangle \langle F',m_{F}+q|
\end{multline}
where $q=\{-1,0,1\}$ and $\mu_{q}$ is the dipole operator for $\{\sigma_{-},\pi, \sigma_{+}\}$-polarization, normalized such that for a cycling transition $\langle \mu \rangle =1$. We will approximate all atom-field interactions to be dipole interactions.
First, let us consider the case when $g$ is comparable to the hyperfine splitting of the excited states, but still small compared to the ground-state splitting. This limit is appropriate for the proposed parameters of microtoroid resonators \cite{toroid} and small Fabry-Perot cavities~\cite{hood01}. If the cavity is tuned near the $F=4 \to F'$ transitions, then the Hamiltonian for the atom cavity system can be written using the rotating wave approximation as
\begin{eqnarray}
\label{H_0}
H_{0} &=& \omega_c a^{\dag} a + \sum_{F'=2}^{5}\omega_{F'}|F'\rangle \langle F'| \nonumber \\
&+& g\Big(\sum_{F'=3}^{5} a^{\dag}D_{0}(4,F') + D_{0}^{\dag}(4,F')a\Big),
\end{eqnarray}
where $\omega_{F'}$ is the frequency of the $F=4 \to F'$ transition, $\omega_c$ is the frequency of the cavity, and $a$ is the annihilation operator for the cavity mode. The operator $|F'\rangle \langle F'|$ projects onto the manifold of excited states with hyperfine number $F'$, and may be written more explicitly as $|F'\rangle \langle F'| = \sum_{m_F'} |F',m_F'\rangle \langle F',m_F'|$. We use units such that $\hbar=1$ and energy has the same dimensions as frequency. Note that we are treating the cavity as a single-mode resonator with linear polarization. Fabry-Perot cavities have two modes with orthogonal polarizations, so this model is only appropriate if there is a birefringent splitting which makes one of the modes greatly detuned (compared to $g$) from the atomic resonance.
In the weak-driving limit of an atom-cavity system in the regime of strong coupling, we expect that high transmission will occur when the probe light is resonant with a transition from a ground state of the system to a state in the $N=1$ lowest excitation manifold. Furthermore, we expect a higher transmission when resonantly exciting an eigenstate which is ``cavity-like,'' i.e., an eigenstate which has larger weight in the field excitation rather than the atomic dipole.
\begin{figure}\label{toroid_eigen}
\end{figure}
In Fig.~\ref{toroid_eigen}, we plot the eigenfrequencies $\{\epsilon_k^{(1)}\}$ of $H_0$ determined by the equation $H_0|\psi_k^{(N)}\rangle = \epsilon_k^{(N)} |\psi_k^{(N)}\rangle$. Here $N$ is the excitation manifold, where $\epsilon_k^{(N+1)}-\epsilon_k^{(N)} \sim \omega_c$ and $\epsilon_k^{(0)}=0$. Also displayed is $\langle \psi_k^{(1)} | a^{\dag} a |\psi_k^{(1)}\rangle$ for each eigenstate $|\psi_k^{(1)}\rangle$ corresponding to each eigenfrequency $\epsilon_k^{(1)}$, which is a measure of how ``cavity-like'' that state is. This should give some indication of what cavity and probe detunings yield high transmission.
In order to study the system properties more precisely, we can find the Hamiltonian of the driven system, write the Liouvillian that describes the time-evolution including damping, and calculate the steady state of the system. We will assume that the cavity resonance is tuned near the $F=4\to F'$ atomic transitions. We expect that, absent any repumping fields, atomic decays to the $F=3$ ground state will leave the atom uncoupled to the resonator. To avoid this, we will assume that a classical (coherent-state) driving field tuned near the $F=3\to F'$ transitions is applied to the atom in addition to the probe field which drives the cavity. In the rotating wave approximation, the Hamiltonian of this driven atom-cavity system in the frame rotating with the cavity probe is
\begin{eqnarray}
\label{H_1}
H_1 &=&\sum_{F'=2}^{5}\Delta_{F'}|F'\rangle \langle F'| + \Delta_r |F=3\rangle\langle F=3| + \Delta_c a^{\dag} a \nonumber \\
&+& g\sum_{F'=2}^{5} \Big(a^{\dag}D_{0}(4,F') + D_{0}^{\dag}(4,F')a\Big) \nonumber \\
&+& \Omega_r \sum_{F'=2}^{5} \Big( D_{0}(3,F') + D_{0}^{\dag}(3,F') \Big)\nonumber \\
&+&\varepsilon a^{\dag} +\varepsilon^{*} a,
\end{eqnarray}
where $\Delta_{F'} = \omega_{4\to F'} - \omega_p$, $\Delta_r = \omega_r - \omega_{GSS} - \omega_p$, and $\Delta_c = \omega_c-\omega_p$. Here $\omega_p$ is the probe frequency, $\omega_r$ is the repump frequency, and $\omega_{GSS}\approx 9.2$~GHz is the ground-state splitting of Cs. The cavity is driven at a rate $\varepsilon$ so that in the absence of an atom the intracavity photon number would be $N_{no~atom} = |\varepsilon|^2 /(\kappa^2 + \Delta_c^2)$, and the atom is driven by the repump field with Rabi frequency $2\Omega_r$. Here, we have assumed that there is no off-resonant coupling of the cavity mode to the $F=3$ ground states, nor is there off-resonant coupling of the repump light to the $F=4$ states. We expect that corrections due to those terms will be small when $g,\Omega_r \ll \omega_{GSS}$.
\begin{figure}\label{toroidplot}
\end{figure}
The time evolution of the density matrix $\rho$ of the atom-cavity system is given by the master equation,
\begin{eqnarray}
\label{master_1}
\dot{\rho} = -i[H_1,\rho] +
\kappa \mathcal{D}[a]\rho +
\gamma \sum_{q,F}\mathcal{D} \Big[\sum_{F'}D_{q}(F,F')\Big]\rho ,
\end{eqnarray}
where $\kappa$ and $\gamma$ are the cavity field and atomic dipole decay rates, respectively,
and the zero-temperature decay superoperator $\mathcal{D}$ acts on the density matrix such that $\mathcal{D}[c]\rho \equiv 2c\rho c^{\dag} - c^{\dag}c\rho -\rho c^{\dag}c$ for any operator $c$. Note that in the atomic spontaneous emission term (proportional to $\gamma$), we have assumed that all $F'\to F=4$ transitions of the same polarization couple to a common reservoir of vacuum electromagnetic field modes and similarly for all $F'\to F=3$ transitions (but the reservoirs for $F'\to F=4$ and $F'\to F=3$ transitions are independent). This assumption arises from the fact that level shifts due to the atom-cavity coupling will be comparable to the atomic excited state hyperfine splittings (but small compared to the ground state splitting)
and, therefore, there exists the possibility for coherence, or quantum interference effects between
transitions of the same polarization from different $F'$ states to a single, common ground-state level
\cite{cardimona82,cardimona83,kmb}.
Such a possibility is described in the common-reservoir master equation (\ref{master_1}) by
generalized atomic damping terms which couple such transitions.
Note that the choice of independent reservoirs for transitions to the different hyperfine ground states is consistent with our assumption that there is no off-resonant coupling between transitions from different hyperfine ground-state manifolds.
From the steady-state solution to Eq.~\ref{master_1}, $\dot{\rho}_{ss}=0$, we can compute steady-state expectation values of an operator $c$ by evaluating $\mathrm{Tr}(\rho_{ss}c)$. We define the normalized cavity transmission $T = \mathrm{Tr}(\rho_{ss}a^{\dag}a) \kappa^2/|\varepsilon|^2$, where $T=1$ for an empty cavity on resonance. $T$ is plotted in Fig.~\ref{toroidplot} versus cavity and probe detunings. Notice the similarity to Fig.~\ref{toroid_eigen}, which demonstrates that the qualitative features of the transmission are indeed determined by the eigenvalues and eigenstates of the Hamiltonian.
\begin{figure}\label{toroid_slice}
\end{figure}
Fig.~\ref{toroid_slice} shows $T$ as a function of probe detuning for fixed cavity frequency along with atomic ground-state populations $\langle F=4,m_F|\rho_{ss}|F=4,m_F\rangle$. The large swings in the relative populations of various Zeeman states demonstrate the importance of optical pumping in understanding the steady-state behavior of the transmission. The rapid variation of the populations that occurs near the transmission peaks can be understood by noting in Fig.~\ref{toroid_eigen} that each transmission peak is associated with multiple eigenstates with similar eigenvalues. These eigenstates have different amplitudes of the Zeeman states and therefore lead to different optical pumping effects. It should be noted that the width of the transmission peaks are therefore not simply determined by $\kappa$ and $\gamma$ but also by the separation of the various eigenvalues contributing to each peak, making the peaks wider than would be naively expected.
\begin{figure}\label{toroid_4_5}
\end{figure}
Fig.~\ref{toroid_4_5} demonstrates the importance of incorporating multiple hyperfine levels into the model of the atom when calculating the cavity transmission for the large values of $g$ expected in upcoming experiments \cite{toroid}. The solid red curve denotes the transmission $T$ from Fig.~\ref{toroidplot} for a cavity fixed to be resonant with the $F=4\to F'=5'$ transition. The dashed black curve indicates the transmission calculated using a model of the atom which includes all Zeeman states of the $F=4$ and $F'=5'$ manifolds, but no other hyperfine levels. The substantial differences between the curves indicates that although the other hyperfine transitions are not resonant, the large coupling $g$ causes these transitions to have a significant effect on the atom-cavity system.
\section{Coupling to the Entire D2 Transition}
\label{sec:pbg}
\begin{figure}\label{pbg_eigen}
\end{figure}
\begin{figure}\label{pbg_levels}
\end{figure}
Now we will turn to the regime where $g$ is larger than both the ground- and excited-state hyperfine splittings. This case is applicable for the expected parameters of cavity QED with photonic band gap cavities \cite{pbg}. In this regime, the cavity mode couples to both ground-state hyperfine manifolds, and the Hamiltonian of the atom-cavity system in the absence of a driving field can be written
\begin{eqnarray}
\label{H_2}
H_2 &=&\sum_{F'}\omega_{F'}|F'\rangle \langle F'| - \omega_{GSS} |F=3\rangle\langle F=3| + \omega_c a^{\dag} a \nonumber \\
&+& g\sum_{F,F'}\Big( a^{\dag}D_{0}(F,F') + D_{0}^{\dag}(F,F')a \Big)
\end{eqnarray}
As we did earlier for $H_0$, we find the eigenvalues and eigenvectors of this Hamiltonian determined by the condition $H_2|\phi_k^{(N)}\rangle = \eta_k^{(N)} |\phi_k^{(N)}\rangle$. In Fig.~\ref{pbg_eigen}, we plot the the frequencies $\eta_k^{(1)}$ of the lowest lying excitations, as well as how ``cavity-like'' the corresponding eigenmodes are, $\langle\phi_k^{(1)}| a^{\dag}a |\phi_k^{(1)}\rangle$. The eigenvalues in the first excitation manifold separate into five bands. The lowest and second-highest of these bands have eigenstates which are superpositions of $F=3$ atomic ground states with one photon in the cavity and $F'=\{2',3',4'\}$ atomic excited states with zero photons; the highest and second-lowest bands have eigenstates which are superpositions of $F=4$ states with one photon and $F'=\{3',4',5'\}$ states with zero photons.
The central band is occupied by eigenstates the composition of which is dominated by atomic excited states. In particular, these eigenstates have a greatly suppressed coupling to the cavity mode as a result of quantum interference between transition amplitudes from atomic excited states with the same $m_F$ number but different values of $F'$. Similarly, with the assumption of a common reservoir for atomic spontaneous
emission from the various hyperfine states (see below),
these eigenstates also exhibit strongly suppressed spontaneous emission via $\pi$-polarized dipole transitions. It should be noted that coupling to the D1 transition ($6S_{1/2} \to 6P_{1/2}$) does not result in a similar set of eigenstates with suppressed coupling; the absence of $F'=2',5'$ states precludes the possibility of the
required destructive quantum interference between $\pi$-polarized transitions.
\begin{figure}\label{pbgplot}
\end{figure}
We expect high transmission when a probe is tuned to be resonant with a transition from a ground state of the atom-cavity system to one of the states in the first excitation manifold. The eigenvalues of the ground states are $\eta^{(0)}=0$ for states with the atom in the $F=4$ manifold and $\eta^{(0)}=-\omega_{GSS}$ for states with the atom in $F=3$. In Fig.~\ref{pbg_levels}, we plot the difference frequencies for transitions between ground and first excited states, $\eta_k^{(1)}-\eta_j^{(0)}$, where $k,j$ are restricted to single-quantum transitions that can be excited by the cavity probe.
Notice that although the eigenvalues of the Hamiltonian do not cross, the differences of eigenvalues between the ground and first excitation manifolds do have crossings. These crossings correspond to a dual resonance condition, in which a transition from one hyperfine ground state to an excited state is resonant with a transition from the other hyperfine ground state to a different excited state. As we will show in a moment, this can lead to some distinctive features in the probe transmission spectrum.
\begin{figure}\label{pbg_slice_m13}
\end{figure}
We will now calculate the steady state of the driven, damped system. We will consider the cavity to be driven by a single coherent-state field at the frequency $\omega_p$. Since the cavity mode can couple to all of the atomic ground states, a repump field is not needed. The Hamiltonian of the driven atom-cavity system under the rotating wave approximation, in the frame rotating with the probe, is
\begin{eqnarray}
\label{H_3}
H_3 &=&\sum_{F'}\Delta_{F'}|F'\rangle \langle F'| - \omega_{GSS} |F=3\rangle\langle F=3| + \Delta_c a^{\dag} a \nonumber \\
&+& g\sum_{F,F'}\Big( a^{\dag}D_{0}(F,F') + D_{0}^{\dag}(F,F')a \Big)\nonumber \\
&+&\varepsilon a^{\dag} +\varepsilon^{*} a ,
\end{eqnarray}
and the master equation for the evolution of the density matrix is
\begin{eqnarray}
\label{master_2}
\dot{\rho} = -i[H_3,\rho] +
\kappa \mathcal{D}[a]\rho +
\gamma \sum_{q}\mathcal{D}\Big[\sum_{F,F'}D_{q}(F,F')\Big]\rho .
\end{eqnarray}
Note that in this limit, in which level shifts produced by the atom-field coupling may yield transitions
of similar frequencies to and from {\it different} hyperfine ground states (i.e., $F=3$ and $F=4$),
we assume that all atomic decays of a given polarization are into a common reservoir (without regard for the initial $F'$ and final $F$) \cite{cardimona83}.
From the master equation (\ref{master_2}),
we find the steady-state density matrix $\rho_{ss}$ and the steady-state normalized transmission $T$, plotted versus probe and cavity detunings in Fig.~\ref{pbgplot}.
\begin{figure}\label{pbg_slice_p20}
\end{figure}
These transmission spectra reflect the structure of the eigenvalues plotted in Fig.~\ref{pbg_levels}, although since $\kappa$ is not substantially smaller than $g$ for the parameter set considered, the correspondence is perhaps not as pronounced as for the previous section.
While the transmission spectra are dominated by a pair of broad peaks with widths of the order of $\kappa$, of particular interest are sharp features at $\omega_p \approx \omega_{4\to5'}-0.3$~GHz and $\omega_p \approx \omega_{4\to5'}+8.9$~GHz. These transmission features are particularly strong at the cavity detunings where transition frequencies of $H_2$ cross, i.e., where the dual resonance condition is satisfied, which for the parameters $(g,\kappa,\gamma)=(17,4.4,0.0026)$GHz occurs at $\omega_c \approx \omega_{4\to5'}+20$~GHz and $\omega_c \approx \omega_{4\to5'}-13$~GHz. The steady-state transmission at these cavity detunings is plotted in Figs.~\ref{pbg_slice_m13}(a) and \ref{pbg_slice_p20}(a) versus probe detuning.
Also plotted, in Figs.~\ref{pbg_slice_m13}(b) and \ref{pbg_slice_p20}(b), are the total populations in the $F=3$ and $F=4$ ground-state manifolds, which illustrate that
the sharp peaks in cavity transmission are associated with significant optical pumping effects.
For the case illustrated in Fig.~\ref{pbg_slice_p20},
the transitions which satisfy the dual resonance condition are between the $F=3$ ground-state manifold and a manifold of excited eigenstates which have a significant photon component, and between the $F=4$ ground-state manifold and the central band of atom-like eigenstates. Weak dissipative channels (primarily atomic spontaneous emission) can transfer population between the two transitions in a manner that depends sensitively on the probe field detuning and the atomic state compositions of the excited eigenstates.
Pronounced optical pumping effects between the different $m_F$ levels also occur as the probe field is tuned to the various atom-like eigenstates as
a result of the suppression of $\pi$-polarized spontaneous emission from each of
these states.
In Fig.~\ref{pbg_slice}(a), we plot the normalized steady-state transmission versus probe detuning with the cavity frequency fixed between the frequencies of the $F=3\to F'$ and $F=4\to F'$ transitions. Two small narrow peaks associated with the dual resonance effect are still apparent, and the atomic populations, plotted in Fig.~\ref{pbg_slice}(b), now show very strong and abrupt pumping into the $F=3$ or $F=4$ manifolds around these peaks.
\begin{figure}\label{pbg_slice}
\end{figure}
Future experiments with single atoms coupled to photonic band gap cavities should be able to study these sharp features. They should be relatively easy to measure because although they are narrow in probe frequency (which is easily controlled), they are robust against changes in cavity frequency (which is harder to control experimentally) of the order of $\kappa$.
\section{Conclusion}
We have presented results of the calculation of the weak-field steady-state transmission of a single-mode linearly polarized optical resonator coupled to the D2 transition of a single Cesium atom.
Our results are for a regime of single-photon dipole coupling strength not previously
considered, but of relevance to planned experiments with microtoroid and photonic bandgap cavities, as well as with other recently-implemented atom-chip microcavity systems \cite{treutlein06,barclay06}.
They necessarily take into account the entire atomic hyperfine
structure and comparison with simpler models highlights the importance of doing so.
In addition to features expected from a strongly coupled atom-cavity system, they also
reveal interesting and significant quantum interference phenomena associated with
the coupling of different atomic transitions to the same mode or modes of
the electromagnetic field.
\begin{acknowledgments}
This research is supported by the National Science Foundation, by the Caltech MURI Center for Quantum Networks, and by the Advanced Research and Development Activity (ARDA). ASP acknowledges support from the Marsden Fund of the Royal Society of New Zealand.
\end{acknowledgments}
\end{document}
|
\begin{document}
\begin{abstract} \noindent
We study the stable pairs theory of local curves in 3-folds
with descendent insertions.
The rationality of the partition function
of descendent invariants is established
for the full local curve geometry (equivariant
with respect
to the scaling 2-torus) including relative
conditions and odd degree insertions for higher genus curves.
The capped 1-leg descendent vertex (equivariant
with respect to the 3-torus)
is also proven to be rational.
The results are obtained by combining geometric constraints
with a detailed analysis of the poles of the descendent
vertex.
\end{abstract}
\title{ extbf{Descendents on local curves: Rationality}
\setcounter{tocdepth}{1}
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
\subsection{Descendents}\label{dess}
Let $X$ be a nonsingular 3-fold, and let
$$\beta \in H_2(X,\mathbb{Z})$$ be a nonzero class. We will study here the
moduli space of stable pairs
$$[\OO_X \stackrel{s}{\rightarrow} F] \in P_n(X,\beta)$$
where $F$ is a pure sheaf supported on a Cohen-Macaulay subcurve of $X$,
$s$ is a morphism with 0-dimensional cokernel, and
$$\chi(F)=n, \ \ \ [F]=\beta.$$
The space $P_n(X,\beta)$
carries a virtual fundamental class obtained from the
deformation theory of complexes in
the derived category \cite{pt}. A review can be found in Section \ref{ooo}.
Since $P_n(X,\beta)$ is a fine moduli space, there exists a universal sheaf
$$\FF \rightarrow X\times P_{n}(X,\beta),$$
see Section 2.3 of \cite{pt}.
For a stable pair $[\OO_X\to F]\in P_{n}(X,\beta)$, the restriction of
$\FF$
to the fiber
$$X \times [\OO_X \to F] \subset
X\times P_{n}(X,\beta)
$$
is canonically isomorphic to $F$.
Let
$$\pi_X\colon X\times P_{n}(X,\beta)\to X,$$
$$\pi_P\colon X\times P_{n}(X,\beta)
\to P_{n}(X,\beta)$$
be the projections onto the first and second factors.
Since $X$ is nonsingular
and
$\FF$ is $\pi_P$-flat, $\FF$ has a finite resolution
by locally free sheaves.
Hence, the Chern character of the universal sheaf $\FF$ on
$X \times P_n(X,\beta)$ is well-defined.
By definition, the operation
$$
\pi_{P*}\big(\pi_X^*(\gamma)\cdot \text{ch}_{2+i}(\FF)
\cap(\pi_P^*(\ \cdot\ )\big)\colon
H_*(P_{n}(X,\beta))\to H_*(P_{n}(X,\beta))
$$
is the action of the descendent $\tau_i(\gamma)$, where
$\gamma \in H^*(X,\Z)$.
For nonzero $\beta\in H_2(X,\Z)$ and arbitrary $\gamma_i\in H^*(X,\Z)$,
define the stable pairs invariant with descendent insertions by
\begin{eqnarray*}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_j)
\right\rangle_{\!n,\beta}^{\!X}& = &
\int_{[P_{n}(X,\beta)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_j) \\
& = &
\int_{P_n(X,\beta)} \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\Big( [P_{n}(X,\beta)]^{vir}\Big).
\end{eqnarray*}
The partition function is
$$
\ZZ^X_{\beta}\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)
=\sum_{n}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right\rangle_{\!n,\beta}^{\!X}q^n.
$$
Since $P_n(X,\beta)$ is empty for sufficiently negative
$n$,
$\ZZ^X_{\beta}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)$
is a Laurent series in $q$. The following conjecture was made in
\cite{pt2}.
\begin{conj}
\label{111}
The partition function
$\ZZ_{\beta}^X\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)$ is the
Laurent expansion of a rational function in $q$.
\end{conj}
If only primary field insertions $\tau_0(\gamma)$ appear,
Conjecture \ref{111} is known for
toric $X$ by \cite{moop, mpt} and for Calabi-Yau $X$ by
\cite{bridge,toda} together with \cite{joy}.
In the presence of descendents $\tau_{i>0}(\gamma)$,
very few results have been obtained.
The central result of the present paper is the proof of
Conjecture 1 in case $X$ is the total space of
an rank 2 bundle over a curve, a {\em local curve}.
In fact, the rationality
of the stable pairs descendent theory of relative local curves
is proven.
\subsection{Local curves} \label{lc1}
Let $N$ be a split
rank 2 bundle on a nonsingular projective curve $C$ of genus $g$,
\begin{equation}\label{ffg}
N=L_1\oplus L_2.
\end{equation}
The splitting determines a scaling action of a 2-dimensional torus
$$T=\C^* \times \C^*$$ on $N$.
The {\em level} of the splitting is the pair of integers
$(k_1,k_2)$ where,
$$k_i= {\text {deg}}(L_i).$$
Of course, the scaling action and the level
depend upon the choice of splitting \eqref{ffg}.
Let $s_1,s_2 \in H^*_\mathbf{T}(\bullet)$ be the first Chern classes
of the standard representations of the first and second
$\C^*$-factors of $T$ respectively.
We define
\begin{equation}\label{lwww}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_j)
\right\rangle_{\!n,d}^{\!N} =
\int_{[P_{n}(N,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_j) \ \ \ \in \mathbb{Q}(s_1,s_2)\ .
\end{equation}
Here,
the curve class is $d$ times the zero section $C \subset N$ and
$$\gamma_j \in H^*(C,\mathbb{Z})\ .$$
The right side of \eqref{lwww} is defined by $T$-equivariant
residues as in \cite{BryanP,lcdt}. Let
$$
\ZZ_{d}^{N}\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T
=\sum_{n}
\left\langle \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right\rangle_{\!n,d}^{\! N}q^n.
$$
\begin{thm}
\label{onnn}
$\ZZ_{d}^{N}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$ is the
Laurent expansion in $q$ of a rational function in $\mathbb{Q}(q,s_1,s_2)$.
\end{thm}
The rationality of Theorem \ref{onnn} holds even when
$\gamma_j \in H^1(C,\mathbb{Z})$.
Theorem \ref{onnn} is proven via
the stable pairs theory of relative
local curves and the 1-leg descendent vertex.
The proof provides a method to
compute $\ZZ_{d}^{N}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$.
\subsection{Relative local curves} \label{lc2}
\label{relgeom}
The fiber of $N$ over a point $p\in C$ determines a $T$-invariant
divisor
$$N_p \subset N$$
isomorphic to $\com^2$ with the standard $T$-action.
For $r>0$, we will consider the local theory of $N$
relative to the divisor
$$S= \bigcup_{i=1}^r N_{p_i} \subset N$$
determined by the fibers over $p_1,\ldots,p_r\in C$.
Let $P_n(N/S,d)$ denote the relative moduli space of stable pairs, see
\cite{pt} for a discussion.
For each $p_i$, let $\eta^i$
be a partition of $d$ weighted
by the equivariant Chow ring,
$$A_T^*(N_{p_i},{\mathbb Q})\stackrel{\sim}{=} {\mathbb Q}[s_1,s_2],$$
of the fiber $N_{p_i}$.
By Nakajima's construction,
a weighted partition $\eta^i$ determines a $T$-equivariant class
$$\CC_{\eta^i} \in A_T^*(\text{Hilb}(N_{p_i},d), \mathbb{Q})$$
in the
Chow ring of the Hilbert scheme of points.
In the theory of stable pairs, the weighted partition $\eta^i$
specifies relative
conditions via the boundary map
$$\epsilon_i: P_n(N/S,d)\rightarrow \text{Hilb}(N_{p_i},d).$$
An element $\eta\in {\mathcal P}(d)$ of the set of
partitions of $d$ may be
viewed as a
weighted partition with all weights set to the identity class
$$1\in H^*_T(N_{p_i},{\mathbb Q})\ .$$
The Nakajima basis of $A_T^*(\text{Hilb}(N_{p_i},d), \mathbb{Q})$ consists of
identity weighted partitions indexed by ${\mathcal P}(d)$.
The $T$-equivariant intersection pairing in the Nakajima basis is
$$g_{\mu\nu}=\int_{\text{Hilb}(N_{p_i},d)} \CC_\mu \cup \CC_\nu =
\frac{1}{(s_1s_2)^{\ell(\mu)}}
\frac{(-1)^{d-\ell(\mu)}}
{{\mathfrak{z}}(\mu)}\ {\delta_{\mu,\nu}},$$
where
$${\mathfrak z}(\mu) = \prod_{i=1}^{\ell(\mu)} \mu_i \cdot
|\text{Aut}(\mu)|.$$
Let $g^{\mu\nu}$ be the inverse matrix.
The notation $\eta([0])$ will be used to set all
weights to $[0]\in A^*_T(N_{p_i},{\mathbb Q} )$.
Since
$$[0]= s_1s_2 \in A^*_T(N_{p_i}, {\mathbb Q} ),$$
the weight choice has only a mild effect.
Following the
notation of \cite{BryanP,lcdt},
the relative stable pairs partition function with
descendents,
\begin{equation*}
{\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T
=\sum _{n\in \Z }q^{n}
\int _{[P_{n} (N/S,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_{j})\
\prod_{i=1}^r \epsilon_i^*(\CC_{\eta^i}),
\end{equation*}
is well-defined for local curves.
\begin{thm}
\label{tnnn}
$\ZZ_{d,\eta^1,\dots,\eta^r}
^{N/S}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$ is the
Laurent expansion in $q$ of a rational function in $\mathbb{Q}(q,s_1,s_2)$.
\end{thm}
Theorem \ref{tnnn} implies Theorem \ref{onnn} by the degeneration formula.
The
proof of Theorem \ref{tnnn} uses the TQFT formalism exploited in
\cite{BryanP,lcdt} together with an analysis of
the capped 1-leg descendent vertex.
\subsection{Capped 1-leg descendent vertex} \label{legger}
The capped 1-leg geometry concerns
the trivial bundle,
$$N = \mathcal{O}_{\PP^1} \oplus \mathcal{O}_{\PP^1} \rightarrow \PP^1\ ,$$
relative to the fiber
$$N_\infty \subset N$$
over $\infty \in \PP^1$.
Capped geometries have been studied (without descendents)
in \cite{moop}.
The total space $N$ naturally carries an action of a
3-dimensional torus $$\mathbf{T} = T \times \com^*\ .$$
Here, $T$ acts as before by scaling the
factors of $N$ and preserving the relative divisor $N_\infty$.
The $\com^*$-action
on the base $\PP^1$ which fixes the points $0, \infty\in \PP^1$
lifts to an additional $\com^*$-action on $N$ fixing
$N_\infty$.
The equivariant cohomology
ring $H_{\mathbf{T}}^*(\bullet)$ is generated by
the Chern classes $s_1$, $s_2$, and $s_3$
of the standard representation of the three $\com^*$-factors.
We define
\begin{equation}\label{pppw}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}
=\sum _{n\in \Z }q^{n}
\int _{[P_{n} (N/N_\infty,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}(\gamma_{j})\
\cup \epsilon_\infty^*(\mathsf{C}_{\eta}),
\end{equation}
by $\mathbf{T}$-equivariant residues.\footnote{The $T$-equivariant
series associated to the cap will be denoted
$${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^T \ ,
$$
for $\gamma_j\in H^*(\Pp,\mathbb{Z})$.}
Here, $\gamma_j \in H^*_{\mathbf{T}}(\PP^1,\mathbb{Z})$.
By definition, the partition function \eqref{pppw} is
a Laurent series in $q$ with coefficients in the field
$\mathbb{Q}(s_1,s_2,s_3)$.
\begin{thm}
\label{cnnn}
$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{thm}
Theorem \ref{cnnn} is the main contribution of the paper.
The result relies upon
a delicate cancellation of
poles in the vertex formula of \cite{pt2}
for stable pairs invariants.
Theorem \ref{tnnn} is derived as a consequence.
\subsection{Stationary theory}
In \cite{parttwo}, we prove reduction rules for
stationary descendents in the
$T$-equivariant local theory of curves. Let
$\mathsf{p}\in H^2(C,\mathbb{Z})$ be the class of a point on a
nonsingular curve $C$. The stationary descendents are
$\tau_i(\mathsf{p})$.
For the degree $d$ local theory of $C$, we
find universal formulas expressing the descendents
$\tau_{i>d}(\mathsf{p})$ in terms of the descendents
$\tau_{i\leq d}(\mathsf{p})$.
The reduction rules provide an alternative (and more effective)
approach
to
the rationality of Theorem \ref{tnnn} in the
stationary case.
The exact calculation in \cite{parttwo} of the basic stationary
descendent series
$$\mathsf{Z}^{\mathsf{cap}}_{d,(d)}( \tau_d(\mathsf{p}))^T =
\frac{q^d}{d!}\left(\frac{s_1+s_2}{s_1s_2}\right)
\frac{1}{2}\sum_{i=1}^d \frac{ 1+(-q)^{i}}{1-(-q)^i} \ $$
plays a special role.
The coefficient of $q^d$,
$$ \left\langle \tau_d, (d) \right\rangle_{\text{Hilb}(\com^2,d)}=
\frac{1}{2\cdot (d-1)!} \left(\frac{s_1+s_2}{s_1s_2}\right),$$
is the classical $T$-equivariant pairing on the
Hilbert scheme of $d$ points in $\C^2$.
The $T$-equivariant stationary descendent theory is simpler
than the full descendent theories studied here. We do not know an
alternative approach to the rationality of the
full $T$-equivariant descendent theory
of local curves. Even the rationality of the $\mathbf{T}$-equivariant
stationary theory of the cap does not appear to
be accessible via \cite{parttwo}.
The methods of \cite{parttwo} also prove a functional equation for
the partition function for stationary descendents which is
a special case of the following conjecture we make here.
\begin{conj}
\label{33345}
Let $\ZZ_{d,\eta^1,\dots,\eta^r}
^{N/S}\big( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\big)^T$ be the Laurent expansion in $q$ of
$F(q,s_1,s_2) \in \mathbb{Q}(q,s_1,s_2)$. Then, $F$
satisfies the functional equation
\[
F(q^{-1},s_2,s_2) = (-1)^{\Delta+|\eta|-\ell(\eta)
+
\sum_{j=1}^k
i_j}q^{-\Delta}F(q,s_1,s_2),
\]
where the constants are defined by
$$\Delta = \int_{\beta}c_1(T_N),\ \ \
|\eta|=\sum_{i=1}^r |\eta^i|,\ \ \ \text{and} \ \ \
\ell(\eta)=\sum_{i=1}^r \ell(\eta^i)
\ .$$
\end{conj}
Here, $T_N$ is the tangent bundle of the 3-fold $N$, and
$\beta$ is the curve class given by $d$ times the $0$-section.
We believe the straightforward generalization of Conjecture \ref{33345}
to all descendent partition functions for the stable
pairs theories of relative 3-folds (equivariant and non-equivariant) holds.
If there are no descendents, the functional equation is known
to hold in the toric case \cite{moop}.
The strongest evidence with descendents
is the stationary result of Theorem 2
of \cite{parttwo}.
\subsection{Denominators}
The descendent partition
functions for the stable pairs theory of local
curves have very restricted
denominators when considered as rational functions in $q$
with coefficients in $\Q(s_1,s_2)$ for Theorems \ref{onnn}-\ref{tnnn}
and rational functions in $q$
with coefficients in $\Q(s_1,s_2,s_3)$ for Theorem \ref{cnnn}.
\begin{conj}
\label{222}
The denominators of the degree $d$ descendent partition functions
$\ZZ$ of Theorems \ref{onnn}, \ref{tnnn}, and \ref{cnnn}
are products of factors of the form $q^k$ and
$$1-(-q)^r$$
for $1\leq r \leq d$.
\end{conj}
In other words, the poles in $-q$ are conjectured to occur
only at 0 and
$r^{th}$ roots for $r$ at most $d$ (and have no
dependence on the variables $s_i$).
Conjecture \ref{222} is proven in Theorem \ref{2222} of
Section \ref{ennd}
for descendents
of even cohomology.
The denominator restriction yields new
results about the 3-point functions of the Hilbert scheme of
points of $\mathbb{C}^2$ stated as a Corollary to Theorem \ref{2222}.
\subsection{Descendent theory of toric 3-folds}
Calculation of the descendent theory of stable pairs on nonsingular
toric 3-folds requires
knowledge of the capped 3-leg descendent vertex.{\footnote{The
capped 2-leg descendent vertex is, of course, a specialization
of the 3-leg vertex.}}
The rationality of the capped 3-leg descendent vertex is proven
in \cite{part3} via a geometric reduction to the 1-leg case
of Theorem \ref{cnnn}.
As a result, Conjecture \ref{111} is
established for all nonsingular toric 3-folds. The rationality
of the descendent theory of
several log Calabi-Yau geometries
is also proven in \cite{part3}.
\subsection{Plan of the paper}
After a brief review of the theory of
stable pairs in Section \ref{ooo}, the
vertex formalism of \cite{pt2} is summarized in
Section \ref{ttt}.
The proof of Theorem \ref{cnnn} is presented in Section \ref{333}
for descendents of the nonrelative $\mathbf{T}$-fixed point
$0\in \PP^1$ modulo the
pole cancellation property established
in Section \ref{polecan}.
Depth and the rubber calculus for stable pairs of local curves
are discussed in Sections \ref{depp} and \ref{rubc}.
The full statement of Theorem \ref{cnnn}
is obtained in Section
\ref{444}. In fact,
the rationality of the $\mathbf{T}$-equivariant descendent
theories of all twisted caps and tubes is established
in Section \ref{444}.
Theorems \ref{onnn} and \ref{tnnn} are proven as a
consequence of Theorem \ref{cnnn} in Section \ref{555} using the
methods of \cite{BryanP,vir,lcdt}. Denominators
are studied in Section \ref{ennd}.
\subsection{Other directions}
Whether parallel results can be obtained for the
local Gromov-Witten theory of curves \cite{BryanP} is
an interesting question. Although conjectured to be equivalent,
the descendent theory of stable pairs on $3$-folds appears more accessible
than descendents in Gromov-Witten theory. The direct vertex analysis undertaken here for
Theorem \ref{cnnn} must be replaced
in Gromov-Witten theory with a deeper
understanding of Hodge integrals \cite{FP}.
Another advantage of stable pairs, at least for Calabi-Yau
geometries, is the possibility of using
motivic integrals with respect to Beh\-rend's $\chi$-function \cite{Beh},
see \cite{pt3} for an early use. Recently, D. Maulik and R. P. Thomas
have been pursuing $\chi$-functions in the log Calabi-Yau setting.
Applications to the rationality of descendent series in
Fano geometries might be possible.
A principal motivation of studying descendents for stable
pairs is the perspective of \cite{mptop}.
Descendents constrain relative invariants. With the
degeneration formula, the possibility emerges of
studying stable pairs on arbitrary (non-toric) 3-folds.
\section{Stable pairs on $3$-folds}
\label{ooo}
\subsection{Definitions}
Let $X$ be a nonsingular quasi-projective $3$-fold over $\mathbb{C}$
with
polarization $L$.
Let $\beta\in H_2(X,\mathbb{Z})$ be a nonzero class.
The moduli space $P_n(X,\beta)$ parameterizes \emph{stable pairs}
\begin{equation}\label{vqq2}
\OO_X \stackrel{s}{\rightarrow} F
\end{equation}
where $F$ is a sheaf with Hilbert polynomial
$$ \chi(F\otimes L^k) = k\int_\beta c_1(L) + n$$
and $s\in H^0(X,F)$ is a section.
The two stability conditions are:
\begin{enumerate}
\item[(i)]
the sheaf $F$ is {pure} with proper support,
\item[(ii)] the section $\OO_X \stackrel{s}{\rightarrow} F$ has 0-dimensional
cokernel.
\end{enumerate}
By definition, {\em purity} (i) means
every nonzero
subsheaf of $F$ has support of dimension 1 \cite{HLShaves}. In particular,
purity implies the (scheme-theoretic)
support $C_F$ of $F$ is a Cohen-Macaulay curve.
A quasi-projective moduli space of stable pairs
can be constructed by a standard GIT analysis of Quot scheme
quotients \cite{LPPairs1}.
For convenience, we will often refer to the stable pair
\eqref{vqq2} on $X$ simply by $(F,s)$.
\subsection{Virtual class}
A central result of \cite{pt} is the construction of a
virtual class on $P_n(X,\beta)$.
The standard approach to the deformation theory of pairs
fails to yield an appropriate 2-term deformation theory
for $P_n(X,\beta)$.
Instead, $P_n(X,\beta)$ is viewed in \cite{pt}
as a moduli space
of complexes in the derived category.
Let $D^b(X)$ be the bounded derived category of coherent
sheaves on $X$.
Let
$${I}\udot = \left\{ \OO_X \rightarrow F \right\}\in
D^b(X)$$
be the complex determined by a stable pair.
The tangent-obstruction theory obtained by deforming ${I}\udot$
in $D^b(X)$ while fixing its determinant is 2-term and governed by the
groups{\footnote{The subscript 0 denotes traceless $\Ext$.}}
$$\Ext^1({I}\udot, {I}\udot)_0, \ \
\Ext^2({I}\udot, {I}\udot)_0.$$
The virtual class
$$[P_n(X,\beta)]^{vir} \in A_{\text{dim}^{vir}}
\left(P_n(X,\beta),\mathbb{Z}\right)$$
is then obtained by standard methods \cite{BehFan,LiTian}.
The virtual dimension is
$$\text{dim}^{vir} = \int_\beta c_1(T_X).$$
Apart from the derived category deformation theory,
the construction of the virtual class of $P_n(X,\beta)$
is parallel to virtual class construction in DT theory \cite{Thomas}.
\subsection{Characterization}
Consider the kernel/cokernel exact sequence associated to a stable
pair $(F,s)$,
\beq \label{IOFQ}
0\to\I_{C_F}\to\OO_X\Rt{s}F\to Q\to0.
\eeq
The kernel is the ideal sheaf of the Cohen-Macaulay support
curve $C_F$ by Lemma 1.6 of \cite{pt}. The cokernel
$Q$ has dimension 0 support by stability.
The {\em reduced} support scheme, $\text{Support}^{red}(Q)$, is
called the {\em zero locus} of the pair.
The zero locus lies on $C_F$.
Let $C\subset X$ be a fixed Cohen-Macaulay curve.
Stable pairs with support $C$ and bounded zero locus are characterized
as follows.
Let $$\m\subset\OO_C$$
be the ideal in $\OO_C$ of a 0-dimensional subscheme.
Since $$\hom(\m^r/\m^{r+1},\OO_C)=0$$ by
the purity of $\OO_C$, we obtain an inclusion $$\hom(\m^r,\OO_C)\subset
\hom(\m^{r+1},\OO_C).$$
The inclusion $\m^r\into\OO_C$ induces a canonical section
$$\OO_C\into\hom(\m^r,\OO_C).$$
\begin{prop} \label{descl}
A stable pair $(F,s)$ with support $C$ satisfying
$$\text{\em Support}^{red}(Q) \subset \text{\em Support}(\OO_C/\m)$$
is equivalent to a subsheaf
of $\hom(\m^r,\OO_C)/\OO_C,\ r\gg0.$
\end{prop}
Alternatively, we may work with coherent subsheaves of the quasi-coherent sheaf
\begin{equation}\label{infhom}
\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_C)/\OO_C
\end{equation}
Under the equivalence of Proposition \ref{descl}, the
subsheaf of \eqref{infhom} corresponds to $Q$, giving a subsheaf $F$ of
$\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_C)$ containing the canonical subsheaf $\OO_C$
and the sequence
$$0\to\OO_C\stackrel{s}{\rightarrow} F \rightarrow Q \rightarrow 0.$$
Proposition \ref{descl} is proven in \cite{pt}.
\section{$\mathbf{T}$-fixed points with one leg} \label{ttt}
\subsection{Affine chart}
Let $N$ be the 3-fold total space of
$$\OO_{\PP^1} \oplus \OO_{\PP^1} \rightarrow \PP^1 \ $$
carrying the action of the 3-dimensional torus $\mathbf{T}$ as in Section
\ref{legger}.
Let
\begin{equation}\label{vqaa}
[\OO_N \stackrel{s}{\rightarrow} F] \in P_n(N,d)^\mathbf{T}
\end{equation}
be a $\mathbf{T}$-fixed stable pair. The curve class is
$d[\PP^1]$.
Let $U\subset N$ be the $\mathbf{T}$-invariant affine chart
associated to the
$\mathbf{T}$-fixed point of $N$ lying over $0\in \PP^1$.
The restriction of the stable pair \eqref{vqaa} to
the chart $U$,
\begin{equation}\label{vvvt}
\OO_{U} \stackrel{s_U}{\rightarrow} F_U\ ,
\end{equation}
determines an invariant section $s_U$ of an
equivariant sheaf $F_U$.
Let $x_1,x_2,x_3$ be coordinates on the affine chart $U$
in which the $\mathbf{T}$-action takes the diagonal form,
$$(t_1,t_2,t_3) \cdot x_i = t_i x_i.$$
By convention, $x_1$ and $x_2$ are coordinates on the fibers of
$N$ and $x_3$ is a coordinate on the base $\PP^1$.
We will characterize the restricted data $(F_U,s_U)$
in the coordinates $x_i$ closely following the presentation
of \cite{pt2}.
\subsection{Monomial ideals and partitions}
Let $x_1,x_2$ be coordinates on the plane
$\C^2$.
A subscheme $S\subset \C^2$ invariant under the
action of the diagonal torus,
$$(t_1,t_2)\cdot x_i = t_ix_i$$
must be defined by a monomial ideal
$\I_S \subset \C[x_1,x_2]$.
If
$$\dim_\C \C[x_1,x_2]/\I_S < \infty$$
then $\I_S$ determines a finite partition $\mu_S$
by considering lattice points corresponding
to monomials of $\C[x_1,x_2]$
{\em not} contained in $\I_S$.
Conversely, each partition $\mu$ determines a monomial ideal
$$\mu[x_1,x_2]\subset \C[x_1,x_2].$$
Similarly, the subschemes $S\subset \C^3$ invariant under
the diagonal $\mathbf{T}$-action are in bijective correspondence with
$3$-dimensional partitions.
\subsection{Cohen-Macaulay support}
The first step in the characterization of the restricted
data
\eqref{vvvt}
is to determine the scheme-theoretic support
$C_U$ of $F_U$. If nonempty, $C_U$ is a
$\mathbf{T}$-invariant, Cohen-Macaulay subscheme of pure dimension 1.
The $\mathbf{T}$-fixed subscheme $C_U \subset \C^3$ is defined by a
monomial ideal $$\I_C \subset \C[x_1,x_2,x_3].$$
associated to the 3-dimensional partition $\pi$.
The localisation
$$(\I_C)_{x_3} \subset \C[x_1,x_2,x_3]_{x_3},$$
is $T$-fixed and corresponds to a 2-dimensional partition
$\mu$.
Alternatively, the 2-dimensional partitions $\mu$ can be defined
as the infinite limit of
the $x_3$-constant cross-sections of $\pi$.
In order for $C_U$ to have dimension 1,
$\mu$ can not be empty.
There exists
a unique {\em minimal} $\mathbf{T}$-fixed subscheme
$$C_{\mu}\subset \C^3$$
with outgoing partition $\mu$.
The $3$-dimensional partition corresponding to $C_\mu$ is
the infinite cylinder on
the $x_3$-axis determined by the $2$-dimensional
partitions $\mu$.
Let
\begin{eqnarray*}
\I_{\mu}= \mu[x_1,x_2] \cdot \C[x_1,x_2,x_3],& \ \ &
C_{\mu}= \OO_{\C^3}/\I_{\mu}\ .
\end{eqnarray*}
\label{cmmm}
\subsection{Module $M_3$}
The kernel/cokernel sequence associated to the
$\mathbf{T}$-fixed restricted data \eqref{vvvt} takes the form
\begin{equation}\label{cvrw}
0 \rightarrow \I_{C_\mu} \rightarrow \OO_{U} \stackrel{s}
{\rightarrow} F_U \rightarrow Q_U \rightarrow 0\
\end{equation}
for an outgoing partition $\mu$.
Since the
support of the quotient $Q_U$ in \eqref{cvrw} is 0-dimensional
by stability and $\mathbf{T}$-fixed,
$Q_U$
must be supported at the origin.
By Proposition \ref{descl}, the pair $(F_U,s_U)$
corresponds to a $\mathbf{T}$-invariant subsheaf of
$$\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_{C_\mu})/\OO_{C_\mu}
,$$
where
$\m$ is the
ideal sheaf of the origin in $C_\mu\subset\C^3$.
Let
$$M_3 = (\OO_{C_{\mu}})_{x_3}$$
be the $\C[x_1,x_2,x_3]$-module obtained by localisation. Explicitly
$$
M_3=\C[x_3,x_3^{-1}]\otimes\frac{\C[x_1,x_2]}{\mu[x_1,x_2]}\,.
$$
By elementary algebraic arguments,
\begin{eqnarray*}
\lim\limits_{\mathbf{T}o}\hom(\m^r,\OO_{C_\mu}) &
\cong & M_3\ .
\end{eqnarray*}
The $\mathbf{T}$-equivariant
$\C[x_1,x_2,x_3]$-module $M_3$ has a canonical $\mathbf{T}$-invariant
element 1.
By Proposition \ref{descl}, the $\mathbf{T}$-fixed pair $(F_U,s_U)$
corresponds to a finitely generated $\mathbf{T}$-invariant
$\C[x_1,x_2,x_3]$-submodule
\beq\label{datum}
Q_U\subset M_3/\langle 1 \rangle.
\eeq
Conversely, {\em every} finitely generated{\footnote{Here, finitely generated
is equivalent to finite dimensional or Artinian.}}
$\mathbf{T}$-invariant $\C[x_1,x_2,x_3]$-sub\-module
$$Q \subset M_3/\langle 1 \rangle$$
occurs as the restriction to $U$ of
a $\mathbf{T}$-fixed stable pair on $N$.
\subsection{The 1-leg stable pairs vertex}
\label{vc}
Let $R$ be the coordinate ring,
$$
R = \C[x_1,x_2,x_3] \cong \Gamma(U).
$$
Following
the conventions of Section \ref{legger}, the $\mathbf{T}$-action on $R$ is
\begin{equation*}
(t_1,t_2,t_3)\cdot x_i = t_i x_i \,.
\end{equation*}
Since the tangent spaces are dual to the coordinate functions,
the tangent weight of $\mathbf{T}$ along the third axis is $-s_3$.
Let $Q_U \subset M/\langle 1 \rangle$ be a $\mathbf{T}$-invariant
submodule viewed as a stable pair on $U$.
Let ${\mathbb{I}}_U\udot$ denote the universal complex on
$[Q_U] \times U$.
Consider a $\mathbf{T}$-equivariant free
resolution{\footnote{Here, ${\mathbb{I}}_U\udot$ is viewed to
live in degrees 0 and -1.}}
of ${\mathbb{I}}_U\udot$,
\begin{equation}
\label{resol}
\{ \F_{s} \rightarrow \dots \rightarrow \F_{-1}\}
\cong
{\mathbb{I}}_U\udot \ \in D^b([{Q}_U] \times U).
\end{equation}
Each term in \eqref{resol}
can be taken to have the form
$$
\F_i = \bigoplus_j
R(d_{ij})\,, \quad d_{ij} \in \Z^3.$$
The Poincar\'e polynomial
$$
P_U = \sum_{i,j} (-1)^{i+1} \ t^{d_{ij}}
\
\in \Z[t_1^\pm,t_2^\pm,t_3^\pm]$$
does not depend on the choice of the resolution
\eqref{resol}.
We denote the $\mathbf{T}$-character of $F_U$ by $\FFF_U$.
By the
sequence
$$0 \rightarrow \OO_{C_U} \rightarrow F_U \rightarrow Q_U
\rightarrow 0,$$
we have a complete understanding of
the representation $\FFF_U$.
The $\mathbf{T}$-eigenspaces of $F_U$ correspond to the
$\mathbf{T}$-eigenspaces of $\OO_{C_U}$ and
$Q_U$.
The result determines
$$\FFF_U \in \Z(t_1,t_2,t_3).$$
The rational dependence on the $t_i$ is elementary.
From
the resolution \eqref{resol}, we see
that the Poincar\'e polynomial
$P_U$ is related to the $\mathbf{T}$-character
of $F_U$ as follows:
\begin{equation}
\FFF_U =
\frac{1+P_U}{(1-t_1)(1-t_2)(1-t_3)} \label{PQ}
\,.
\end{equation}
The virtual represention
$\chi({\mathbb{I}}_U\udot,{\mathbb{I}}_U\udot)$ is given by the
following alternating sum
\begin{align*}
\chi({\mathbb{I}}_U\udot,{\mathbb{I}}_U\udot) &= \sum_{i,j,k,l} (-1)^{i+k}
\Hom_R(R(d_{ij}), R(d_{kl}))
\\
&= \sum_{i,j,k,l} (-1)^{i+k}
R(d_{kl}-d_{ij})\,.
\end{align*}
Therefore, the $\mathbf{T}$-character is
$$
\tr_{\chi({\mathbb{I}}_U,{\mathbb{I}}_U)} =
\frac{P_U \,\overline{P}_U}
{(1-t_1)(1-t_2)(1-t_3)} \,.
$$
The bar operation
$$\gamma \in \Z(\!(t_1,t_2,t_3)\!) \mapsto
\Z(\!(t_1^{-1},t_2^{-1},
t_3^{-1})\!)$$
is
$t_i \mapsto t_i^{-1}$
on the variables.
We find the $\mathbf{T}$-character of
the $U$ summand of virtual tangent
space $\mathcal{T}_{\left[{I}\udot\right]}$
of the moduli space of stable pairs of the 1-leg cap is
$$ \tr_{R-\chi({\mathbb{I}}\udot_U,{\mathbb{I}}_U\udot)} =
\frac{1-P_U \, \overline{P}_U}
{(1-t_1)(1-t_2)(1-t_3)} \, ,
$$
see \cite{pt2}.
Using \eqref{PQ}, we may express the answer in terms of
$\FFF_U$,
\begin{equation}\label{vertexchar}
\tr_{R-\chi({\mathbb{I}}\udot_U,{\mathbb{I}}_U\udot)}
= \FFF_{U} -
\frac{\overline{\FFF}_U}{t_1t_2t_3} + \FFF_{U}
\overline{\FFF}_U \frac{(1-t_1)(1-t_2)(1-t_3)}{t_1 t_2 t_3} \,.
\end{equation}
On the right side of
\eqref{vertexchar}, the rational functions
should be expanded
in ascending powers in the $t_i$.
The stable pairs vertex is obtained from \eqref{vertexchar}
after a redistribution of edge terms following \cite{pt2}.
Let
$$
\FFF_{\mu} = \sum_{(k_1,k_2) \in \mu} t_1^{k_1}
t_2^{k_2}\
$$
correspond to the outgoing partition $\mu$.
Define
$$
\GGG_{\mu} = - \FFF_{\mu} -
\frac{\overline{\FFF}_{\mu}}{t_1 t_2} + \FFF_{\mu}
\overline{\FFF}_{\mu} \frac{(1-t_1)(1-t_2)}{t_1 t_2} \,.
$$
Define the vertex character $\mathsf{V}_U$ by the following
modification,
\begin{equation}\label{gx34}
\mathsf{V}_U = \tr_{R-\chi({\mathbb{I}}\udot_U,{\mathbb{I}}_U\udot)}
+
\frac{\GGG_{\mu}(t_{1},t_{2})}{1-t_3}\, .
\end{equation}
The character $\mathsf{V}_U$ depends
{\em only on the local data ${Q}_U$}.
By the results of \cite{pt2}, $\mathsf{V}_U$ is a
Laurent polynomial in $t_1$, $t_2$, and $t_3$.
\subsection{Descendents}
Let $[0]\in H^*_\mathbf{T}(\PP^1,\Z)$ be the class of the
$\mathbf{T}$-fixed point
$0\in \PP^1$. Consider the $\mathbf{T}$-equivariant
descendent (with value in the $\mathbf{T}$-equivariant cohomology
of a point),
\begin{multline}\label{vpzz}
\left\langle \tau_{i_1}([0]) \cdots \tau_{i_k}([0]) \right
\rangle_{n,d}^N =\\
\int_{P_n(N,d)} \prod_{j=1}^k \tau_{i_j}([0])
\Big( [P_{n}(N,d)]^{vir}\Big)\in \Q(s_1,s_2,s_3)\ ,
\end{multline}
following the notation of Section \ref{dess}.
In order to calculate \eqref{vpzz} by $\mathbf{T}$-localization, we
must determine the action of
the operators $\tau_{i}([0])$ on the $\mathbf{T}$-equivariant
cohomology of the $\mathbf{T}$-fixed loci.
The calculation of \cite{pt2} yields a formula for the
descendent weight,
\begin{multline}
\mathsf{w}_{{i_1},\cdots, {i_m}}
(Q_U) =\\
e(-\mathsf{V}_{U})
\cdot \prod_{j=1}^m
\text{ch}_{2+i_j}\big(\FFF_{U}\cdot
(1-t_1)(1-t_2)(1-t_3)\big) \ .
\end{multline}
The {\em descendent vertex} $\bW_\mu^{\mathsf{Vert}}(\tau_{i_1}([0]) \cdots
\tau_{i_m}([0]))$
is obtained from the descendent weight,
\begin{multline}\label{vvped}
\bW_\mu^{\mathsf{Vert}} (\tau_{i_1}([0]) \cdots \tau_{i_k}([0]))
= \\
\left(\frac{1}{s_1s_2}\right)^k
\sum_{Q_U}
\mathsf{w}_{{i_1}, \cdots, {i_k}}
(Q_U)\ q^{\ell({Q}_U)+|\mu|}\ \in
\Q(s_1,s_2,s_3)(\!(q)\!)\ .
\end{multline}
\label{heyle}
Here, $\ell(Q_U)$ is the length of $Q_U$.
\subsection{Edge weights}
The edge weight in the cap geometry is
$$\bW^{(0,0)}_\mu = e(\GGG_\mu)\ \in \Q(s_1,s_2).$$
In fact, $\bW^{(0,0)}_\mu$ is simply the inverse product
of the tangent weights of the Hilbert scheme of points of $\C^2$
at the $T$-fixed point corresponding to the partition $\mu$.
\section{Capped 1-leg descendents: stationary} \label{333}
\subsection{Overview}
Consider the capped geometry of Section \ref{legger}. As before, let
$0\in \PP^1$ be the $\mathbf{T}$-fixed point away
from the relative divisor over $\infty \in \PP^1$, and
let
$$[0] \in H^*_{\mathbf{T}}(\PP^1, \mathbb{Z})$$
be the associated class.
The $\mathbf{T}$-weight on the tangent space to $\PP^1$ at 0
is $-s_3$.
We study here the stationary{\footnote{Stationary
refers to descendents of point classes.}} series
\begin{equation}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}\ .
\label{gbn}
\end{equation}
Our main result is a special case of Theorem \ref{cnnn}.
\begin{prop}
\label{cttt}
$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
\subsection{Dependence on $s_3$}
The function \eqref{gbn} is the generating series of the
integrals
\begin{equation} \label{krt}
\left\langle \prod_{j=1}^k \tau_{i_j}([0])
\right\rangle_{\!n,\eta}^{\mathsf{cap},{\mathbf{T}}}
=
\int _{[P_{n} (N/N_\infty,d)]^{vir}}
\prod_{j=1}^k \tau_{i_j}([0])\
\cup \epsilon_\infty^*(C_{\eta})\ ,
\end{equation}
following the notation of Section \ref{legger}.
Let $\ell(\eta)$ denote the length of the partition
$\eta$ of $d$, and let \begin{equation}\label{ktgg}
\delta=\sum_{j=1}^k i_j + d-\ell(\eta)\ .
\end{equation}
The dimension of
$[P_{n} (N/N_\infty,d)]^{vir}$ after applying
the integrand of \eqref{krt} is $2d-\delta$.
\begin{lem} The
integral $\left\langle \prod_{j=1}^k \tau_{i_j}([0])
\right\rangle_{\!n,\eta}^{\mathsf{cap},\mathbf{T}}$ \label{rq2}
is a {\em polynomial} in $s_3$ of degree
$\delta$
with coefficients in the subring
$${\mathbb Q}[s_1,s_2]_{(s_1s_2)}\subset {\mathbb Q}(s_1,s_2).$$
\end{lem}
\begin{proof}
Let $N=\mathcal{O}_{\Pp} \oplus \mathcal{O}_{\Pp}$.
Let
${\mathbb{F}} \rightarrow {\mathcal N}$
denote the universal sheaf over the universal total space
$${\mathcal N} \rightarrow P_n(N/N_\infty,d).$$
Since $N=\PP^1 \times \com^2$, there is a proper morphism
$${\mathcal N} \rightarrow P_n(N/N_\infty,d) \times \com^2.$$
The locations and multiplicities of the supports of the universal
sheaf determine a
morphism of Hilbert-Chow type,
$$\iota:P_n(N/N_\infty,d) \rightarrow \text{Sym}^{d}(\com^2).$$
A $\mathbf{T}$-equivariant, proper
morphism,
$$\widehat{\iota}: \text{Sym}^{d}(\com^2) \rightarrow \oplus_{1}^{d}
(\com^2),$$
is obtained via the higher moments,
\begin{multline*}
\widehat{\iota}\Big( \ \{(x_i,y_i)\} \ \Big) = \\
\Big(\sum_i x_i, \sum_i y_i\Big) \oplus
\Big(\sum_i x^2_i, \sum_i y^2_i\Big) \oplus \cdots \oplus
\Big(\sum_i x^d_i, \sum_i y^d_i\Big).
\end{multline*}
Let $\rho=\widehat{\iota}\circ \iota$.
Since $\rho$ is a $\mathbf{T}$-equivariant, proper morphism,
there is a $\mathbf{T}$-equivariant push-forward
$$\rho_*: A^{\mathbf{T}}_*(P_n(N/N_\infty,d), {\mathbb Q}) \rightarrow
A^{\mathbf{T}}_*( \oplus_1^{d}(\com^2) , {\mathbb Q}).$$
Descendent invariants
are defined via the $\mathbf{T}$-equivariant residue of
$$\left(\prod_{j=1}^k \tau_{i_j}([0]) \cup \epsilon^*_{\infty}(C_\eta)
\right) \ \cap
[P_n(N/S,d)]^{vir} \ \in A^{\mathbf{T}}_*(P_n(N/N_\infty,d), {\mathbb Q}).$$
We may instead calculate the $\mathbf{T}$-equivariant residue of
\begin{equation}\label{ress}
\rho_*\left( \left(
\prod_{j=1}^k \tau_{i_j}([0]) \cup \epsilon^*_{\infty}(C_\eta)\right) \ \cap
[P_n(N/N_\infty,d)]^{vir}\right)
\end{equation} in
$A^{\mathbf{T}}_*(
\oplus_1^{d}(\com^2) , {\mathbb Q})$.
The codimension of the class \eqref{ress} in
$\oplus_1^{d}(\com^2)$
is $\delta$.
Since the third factor of $\mathbf{T}$ acts trivially on
$\oplus_1^{d}(\com^2)$, the class \eqref{ress} may be written as
\begin{equation}\label{htty3}
\gamma_0 s_3^0 + \gamma_1 s_3^1 + \ldots + \gamma_{\delta} s_3^{\delta}
\end{equation}
where $\gamma_i \in A^{T}_{2d-\delta+i}(
\oplus_1^{d}(\com^2) , {\mathbb Q})$.
Since the space $\oplus_1^{d}
(\com^2)$ has a unique $T$-fixed point
with tangent weights,
$$-s_1,-s_2,-2s_1,-2s_2, \ldots, -ds_1, -ds_2,$$
we conclude the localization of $\gamma_i$
has only monomial poles in the variables $t_1$ and $t_2$.
\end{proof}
As a consequence of Lemma \ref{rq2}, we may write
\begin{equation} \label{krtt}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
=\sum_{r=0}^\delta s_3^r \cdot \Gamma_r(q,s_1,s_2)
\end{equation}
where $\Gamma_r \in \mathbb{Q}(s_1,s_2)((q))$.
\subsection{Localization: rubber contribution} \label{rubcon}
The $\mathbf{T}$-equivariant
localization formula for the series
${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
has three parts:
\begin{enumerate}
\item[(i)] vertex terms over $0\in \PP^1$,
\item[(ii)] edge terms,
\item[(iii)] rubber integrals over $\infty \in \PP^1$.
\end{enumerate}
The vertex and edge terms have been explained already in Section \ref{ttt}.
We discuss the rubber integrals here.
The stable pairs theory of {\em rubber}{\footnote{We
follow the terminology and conventions of the
parallel rubber discussion for the local Donaldson-Thomas
theory of curves treated in \cite{lcdt}.}} naturally arises at the
boundary of $P_n(N/N_\infty,d)$.
Let $R$ be a rank 2 bundle of level $(0,0)$ over $\Pp$. Let
$$R_0, R_\infty\subset R$$
denote the fibers over $0, \infty\in \Pp$.
The 1-dimensional torus $\C^*$ acts on $R$ via the symmetries of
$\Pp$.
Let $P_n(R/R_0\cup R_\infty,d)$ be the relative moduli space
of stable pairs, and let
$$P_n(R/R_0 \cup R_\infty,d)^\circ \subset P_n(R/R_0\cup R_\infty,d)$$
denote the open set with finite stabilizers for the $\C^*$-action
and {\em no} destabilization over $\infty\in \Pp$.
The rubber moduli space,
$${P_n(R/R_0\cup R_\infty,d)}^\sim
= P_n(R/R_0 \cup R_\infty,d)^\circ/\C^*,$$
denoted by a superscripted tilde,
is determined by the (stack) quotient. The moduli space is
empty unless $n>d$.
The rubber theory of $R$ is defined by integration against the
rubber virtual class,
$$[{P_n(R/R_0\cup R_\infty,d)}^\sim ]^{vir}.$$
All of the above rubber constructions are $T$-equivariant for the
scaling action on the fibers of $R$ with weights $s_1$ and $s_2$.
The rubber moduli space $P_n(R/R_0\cup R_\infty, d)^\sim$ carries
a cotangent line at the dynamical point $0 \in \Pp$. Let
$$\psi_0 \in A^1_T({P_n(R/R_0\cup R_\infty,d)}^\sim, {\mathbb Q})$$
denote the associated cotangent line class.
Let $$\mathsf{P}_\mu \in A^{2d}_T(\text{Hilb}(\C^2,d),\mathbb{Z})$$
be the class corresponding to the $T$-fixed point determined
by the monomial ideal $\mu[x_1,x_2]\subset \C[x_1,x_2]$.
In the localization formula for the cap, special
rubber integrals with relative conditions $\mathsf{P}_\mu$ over $0$ and $\CC_\eta$
(in the Nakajima basis) over $\infty$ arise. Let
\begin{equation*}
\mathsf{S}^\mu_\eta =
\sum_{n\geq d} q^{n}
\left\langle \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}\ \in \Q(s_1,s_2,s_3)((q)) \ .
\end{equation*}
The bracket on the right is the rubber
integral defined by $T$-equivariant
residues. If $n=d$, the rubber moduli space in undefined ---
the bracket is then taken to be the $T$-equivariant intersection pairing
between the classes $\mathsf{P}_\mu$ and $\CC_\eta$ in
$\text{Hilb}(\C^2,d)$.
The $s_3$ dependence of the rubber integral
$$ \left\langle \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}\ \in \Q(s_1,s_2,s_3)$$
enter {\em only} through the term $s_3-\psi_0$.
On the $T$-fixed loci of the moduli space $P_n(R/R_0\cup R_\infty, d)^\sim$, the
cotangent line class $\psi_0$ is either equal to
a weight of $\text{Tan}_\mu$ (if $0$ lies on a twistor component)
or is nilpotent (if $0$ lies on a non-twistor component).
We conclude the following result.
\begin{lem} The evaluation of $\mathsf{S}_\eta^\mu$ at \label{plle}
$$s_3= n_1 s_1 + n_2 s_2,\ \ \ \ n_1,n_2\in \Q$$
is well-defined if $(n_1,n_2) \neq (0,0)$
and $n_1 s_1 + n_2 s_2$ is not a weight of $\text{\em Tan}_\mu$.
\end{lem}
The weights of $\text{Tan}_\mu$ are either proportional to
$s_1$ or $s_2$ or of the form
$$n_1s_1+n_2s_2 ,\ \ \ \ n_1,n_2\neq 0$$
where $n_1$ is the {\em opposite} sign of $n_2$.
\subsection{Localization: full formula}
The localization formula \cite{GraberP} for the capped 1-leg descendent
vertex is the following:
\begin{equation}\label{fred}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}} = \sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
{\bW_\mu^{(0,0)}} \cdot \mathsf{S}^{\mu}_{\eta}\ .
\end{equation}
The form is the same as the Donaldson-Thomas localization formulas
used in \cite{moop,lcdt}.
\subsection{Proof of Proposition \ref{cttt}} \label{ggtt2}
We will consider the evaluations of ${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
( \prod_{j=1}^k \tau_{i_j}([0]))^{\mathbf{T}}$ at the values
\begin{equation}\label{gthh4}
s_3 = \frac{1}{a}(s_1+s_2)
\end{equation}
for all integers $a>0$.
By Theorem \ref{canpole}, the main cancellation of poles result
of Section \ref{polecan}, the evaluation \eqref{gthh4} of
$\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right)$
is well-defined and yields a Laurent {\em polynomial} in $q$
with coefficients in $\Q(s_1,s_2)$.
The edge term $\bW_\mu^{(0,0)}$ has no $s_3$ dependence (and
$q$ dependence given by $q^{-d}$).
The evaluation \eqref{gthh4} of
$\mathsf{S}^\mu_\eta$ is well-defined by Lemma \ref{plle} and
is the Laurent series
associated to a rational function in $\Q(q,s_1,s_2)$
by Lemma \ref{hyy3} below.
We have proven the evalution of ${\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
at \eqref{gthh4} for all integers $a>0$ is well-defined and yields
a rational function in $\Q(q,s_1,s_2)$.
By \eqref{krtt} and the invertibility of the
Vandermonde matrix, we see
$$\Gamma_r(q,s_1,s_2) \in \Q(q,s_1,s_2)$$
for all $0 \leq r \leq \delta$.
\qed
\subsection{Evaluation of $\mathsf{S}_\eta^\mu$}
\label{dfv}
The following result is well-known from the study of the
quantum differential equation of the Hilbert scheme of
points \cite{hilb1,hilb2}. We include the proof
for the reader's convenience.
\begin{lem} For all integers $a\neq 0$, the evaluation
$$\mathsf{S}_\eta^\mu
|_{s_3=\frac{1}{a}(s_1+s_2)}$$
yields the Laurent series associated to a rational
function in $\Q(q,s_1,s_2)$. \label{hyy3}
\end{lem}
\begin{proof}
Let $\com^*$ act on $\PP^1$ with tangent weights $-s_3$ and $s_3$
at $0,\infty \in \PP^1$ respectively.
Lift the $\com^*$-action to $\mathcal{O}_{\Pp}(-a)$
with fiber weights{\footnote{Remember, weights on the
coordinate functions are the opposite of the weights on the fibers.}} $as_3$ and $0$ over $0,\infty\in \PP^1$.
Lift $\com^*$ to $\mathcal{O}_{\Pp}$ with fiber
weights $0$ and $0$ over $0,\infty\in \PP^1$.
The $(-a,0)$-tube is the geometry of total
space of
\begin{equation} \label{gttr}
\mathcal{O}_{\Pp}(-a) \oplus \mathcal{O}_{\Pp} \rightarrow \Pp
\end{equation}
relative to the fibers over both $0,\infty \in \Pp$.
The 2-dimensional torus $T$ acts on the $(-a,0)$-tube
as before by scaling the line summands. For
$$\mathbf{T}=T \times \com^* ,$$
we obtain a $\mathbf{T}$-action on the $(-a,0)$-tube.
Define the generating series of $\mathbf{T}$-equivariant integrals
\begin{equation} \label{hllw}
{\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty}
= \sum_{n} q^{n}
\Big\langle \CC_{\eta^0} \ \Big| \ 1 \ \Big|\ \CC_{\eta^\infty}
\Big\rangle_{n,d}^{(-a,0)}\ \in \Q(s_1,s_2,s_3)((q))\
\end{equation}
where the superscript $(-a,0)$ refers to the geometry \eqref{gttr}.
The series ${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty} $
has no insertions. Hence, the results of
\cite{moop,mpt} show
${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty}$
is actually the Laurent series associated to a rational function
in $\Q(q,s_1,s_2,s_3)$.
The $\mathbf{T}$-equivariant localization formula yields
$${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty} =
\sum_{|\mu|=d} \mathsf{S}_{\eta^0}^\mu\Big|_{s_1=s_1-as_3,s_2,s_3=-s_3} \cdot \bW^{(-a,0)}_\mu \cdot \mathsf{S}_{\eta^\infty}^\mu
\ .$$
The formula for the edge term $\bW^{(-a,0)}_\mu$ can be found in Section 4.6
of \cite{pt2}.
Next, we consider the evaluation of the three terms of the
localization formula at
\begin{equation}\label{jjttf}
s_3= \frac{1}{a}({s_1+s_2})\ .
\end{equation}
After evaluation, the first term becomes
\begin{equation} \label{oldd}
\mathsf{S}_{\eta^0}^\mu\Big|_{s_1=-s_2,s_2,s_3=-s_3}
\end{equation}
which only has $q^d$ terms by holomorphic symplectic vanishing \cite{mpt,lcdt}.
The evaluation of $\bW^{(-a,0)}_\mu$ at \eqref{jjttf} is easily
seen to be well-defined and nonzero by inspection of the formulas in
Section 4.6 of \cite{pt2}.
The $q$ dependence of $\bW^{(-a,0)}_\mu$ is monomial.
The evaluation of the third term
$ \mathsf{S}_{\eta^\infty}^\mu\ $
at \eqref{jjttf} is well-defined by Lemma \ref{plle}.
We conclude the evaluation of ${\mathsf Z}^{(-a,0),\mathbf{T}}_{d,\eta^0,\eta^\infty} $
at \eqref{jjttf} is a well-defined rational function in
$\Q(q,s_1,s_2)$.
By the invertibility of \eqref{oldd} and the edge terms,
$\mathsf{S}_{\eta^\infty}^\mu$ must also be a rational function in
$\Q(q,s_1,s_2)$ after the evaluation \eqref{jjttf}.
\end{proof}
\subsection{Twisted cap}
The twisted $(a_1,a_2)$-cap is the geometry of the total space of
\begin{equation} \label{gttrr}
\mathcal{O}_{\Pp}(a_1) \oplus \mathcal{O}_{\Pp}(a_2) \rightarrow \Pp
\end{equation}
relative to the fiber over $\infty \in \Pp$.
We lift the $\com^*$-action on $\Pp$ to $\mathcal{O}_{\Pp}(a_i)$
with fiber weights $0$ and $-a_is_3$ over $0,\infty\in \PP^1$.
The 2-dimensional torus $T$ acts on the $(a_1,a_2)$-cap
by scaling the line summands, so
we obtain a $\mathbf{T}$-action on the $(a_1,a_2)$-cap.
Define the generating series of $\mathbf{T}$-equivariant integrals
\begin{multline*}
{\mathsf Z}^{(a_1,a_2)}_{d,\eta} \left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
= \\
\sum_{n} q^{n}
\left\langle \prod_{j=1}^k \tau_{i_j}([0]) \ \Bigg|\ \CC_{\eta}
\right\rangle_{n,d}^{(a_1,a_2)}\ \in \Q(s_1,s_2,s_3)((q))\
\end{multline*}
where the superscript $(a_1,a_2)$ refers to the geometry \eqref{gttrr}.
\begin{prop}
\label{ctttt}
$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
\begin{proof}
The twisted $(a_1,a_2)$-cap admits a $\mathbf{T}$-equivariant
degeneration to a standard $(0,0)$-cap and an $(a_1,a_2)$-tube by
bubbling off $0\in \Pp$.
The insertions $\tau_{i_j}([0])$ are sent $\mathbf{T}$-equivariantly
to the non-relative point of the $(0,0)$-cap.
The rationality of $
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}$
then follows from Proposition \ref{cttt}, the
$\mathbf{T}$-equivariant
rationality
results for the $(a_1,a_2)$-tube without insertions \cite{mpt,lcdt}, and
the degeneration formula.
\end{proof}
\section{Cancellation of poles} \label{polecan}
\subsection{Overview}
Our goal here is to prove the following result.
\begin{thm}\label{canpole}
For all integers $a>0$, the evaluation
\begin{equation*}
\bW^{\mathbf{V}er}_\mu\left(\prod_{j=1}^k\tau_{i_j}([0])\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)}
\end{equation*}
is well-defined and yields a
Laurent polynomial in $q$ with coefficients in $\Q(s_1,s_2)$.
\end{thm}
We regard the partition $\mu$, the
descendent factor $\prod_{j=1}^k\tau_{i_j}([0])$, and
the integer $a$ as fixed throughout Section \ref{polecan}.
Recall $\bW^{\mathbf{V}er}_\mu\left(\prod_{j=1}^k\tau_{i_j}([0])\right)$ is
defined as an infinite sum over the fixed loci $Q_U$,
\begin{equation}\label{infinite sum}
\bW^{\mathbf{V}er}_\mu\left(\prod_{j=1}^k\tau_{i_j}([0])\right) = \left(\frac{1}{s_1s_2}\right)^k\sum_{Q_U}\mathsf{w}_{\tau_{i_1},\ldots,\tau_{i_k}}(Q_U) q^{l(Q_U)+|\mu|}.
\end{equation}
The $Q_U$ are determined by $\FFF_U$, the weight of the corresponding box configuration. Although $\FFF_U$ is just a Laurent series in $t_1,t_2,t_3$, the
product
$(1-t_3)\FFF_U$
is a Laurent polynomial.
Our approach to proving Theorem~\ref{canpole}
is to break (\ref{infinite sum}) into finite sums
based on the Laurent polynomial
$$(1-t_3)\FFF_U|_{t_3=(t_1t_2)^{\frac{1}{a}}}\ .$$
For any Laurent polynomial $f\in\Z[t_1,t_2,(t_1t_2)^{-\frac{1}{a}}]$, define
\begin{equation*}
\mathcal{S}_f = \left\{Q_U \ \bigg| \ (1-t_3)\FFF_U|_{t_3=(t_1t_2)^{\frac{1}{a}}} = f
\right\} \ .
\end{equation*}
Theorem~\ref{canpole} follows from the following result
regarding the subsums of (\ref{infinite sum})
corresponding to the sets $\mathcal{S}_f$.
\begin{prop}\label{vanishing}
Let $f\in\Z[t_1,t_2,(t_1t_2)^{-\frac{1}{a}}]$ be a Laurent polynomial. The evaluation
\begin{equation*}
\left(\sum_{Q_U\in\mathcal{S}_f}\mathsf{w}_{i_1,\ldots,i_k}(Q_U)\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)}
\end{equation*}
is well-defined. Moreover, the evaluation vanishes for all but finitely many choices of $f$.
\end{prop}
\subsection{Notation and Preliminaries}
We introduce here the notation and conventions required
to analyze the sums appearing in Proposition~\ref{vanishing}.
First, we view the partition $\mu$ as a subset of $\Z_{\ge 0}^2$.
The lattice points,
for which we use the coordinates $(i,j)\in\mu$,
correspond to the lower left corners of
the boxes of $\mu$.
We also write
$$(\delta; j) = (i,j)$$ for $\delta = i-j$.
The points $(\delta; j)\in\mu$ for fixed $\delta$ lie on a single diagonal.
The diagonals will play an important role. Let
$\mu_\delta = \{j \mid (\delta; j)\in\mu\}$, and
define
\begin{equation*}
\Sym_\mu = \prod_{\delta\in\Z}\Sym(\mu_\delta),
\end{equation*}
where $\Sym(S)$ is the group of permutations of a set $S$. Thus,
$\Sym_\mu$ may be viewed
as the group of permutations of $\mu$ which move points only
inside their diagonals.
Let $$\sgn:\Sym_\mu\to\{\pm 1\}$$ be the sign of the permutation of $\mu$.
Recall the Laurent polynomials $(1-t_3)\FFF_U$ are of the form
\begin{equation*}
(1-t_3)\FFF_U = \sum_{(i,j)\in\mu}t_1^it_2^jt_3^{-h_U(i,j)},
\end{equation*}
where $h_U(i,j)$ is the depth of the box arrangement below $(i,j)$.
Because of our reparametrization of the partition $\mu$ and the evaluation $t_3=(t_1t_2)^{\frac{1}{a}}$, the
following change of variables will be convenient:
\begin{equation*}
v_1=t_1, \quad v_2=t_1t_2, \quad v_3=t_1t_2t_3^{-a}
\end{equation*}
and $u_i=e(v_i)$, so
\begin{equation*}
u_1 = s_1, \quad u_2=s_1+s_2, \quad u_3=s_1+s_2-as_3.
\end{equation*}
The evaluations under consideration are then simply $v_3=1$ and $u_3=0$.
From now on we will assume $\mathcal{S}_f$ to be nonempty, so
$$f = (1-t_3)\FFF_U|_{t_3=(t_1t_2)^{\frac{1}{a}}}$$
for some $Q_U$ and thus $f$ can be written in the form
\begin{equation*}
f = \sum_{(\delta; j)\in\mu}v_1^{\delta}v_2^{e_\delta(j)}
\end{equation*}
for some exponents $e_\delta(j)$. These exponents are made unique by requiring that $e_\delta(j)$ is a weakly decreasing function of $j$, for each $\delta$. We generally regard $f$ as fixed and thus do not indicate the $f$-dependence in $e_\delta(j)$.
We now classify all $Q_U\in\mathcal{S}_f$. Given any $\sigma=(\sigma_\delta)\in\Sym_\mu$, we define a function $h_\sigma: \mu \to \Z$ by
\begin{equation*}
h_\sigma(\delta; j) = a\cdot(j - e_\delta(\sigma_\delta^{-1}(j))).
\end{equation*}
When $h_\sigma$ defines a valid box arrangement, we say
$\sigma$ is {\it admissible}. Admissibility
is equivalent to the following conditions on $\sigma$:
\begin{align*}
\sigma_0(j)&\ne 0 \text{ if } e_0(j)>0 \\
\sigma_{\delta+1}(j)&\ne\sigma_{\delta}(k) \text{ if } e_{\delta+1}(j)>e_{\delta}(k) \\
\sigma_{\delta}(j)&\ne\sigma_{\delta+1}(k)+1 \text{ if } e_{\delta}(j)>e_{\delta+1}(k)+1.
\end{align*}
For admissible $\sigma$, let $Q_\sigma$ denote the corresponding $\mathbf{T}$-fixed locus.
Unraveling the definitions, we compute
\begin{align*}
(1-t_3)\FFF_\sigma &= \sum_{(i,j) \in \mu}t_1^it_2^jt_3^{-h_\sigma(i-j,j)} \\
&= \sum_{(\delta; j)\in \mu}v_1^\delta v_2^j v_2^{-\frac{1}{a}h_\sigma(\delta; j)}v_3^{\frac{1}{a}h_\sigma(\delta; j)} \\
&= \sum_{(\delta; j)\in \mu}v_1^\delta v_2^{e_\delta(\sigma_\delta^{-1}(j))} v_3^{j - e_\delta(\sigma_\delta^{-1}(j))} \\
&= \sum_{(\delta; j)\in \mu}v_1^\delta v_2^{e_\delta(j)} v_3^{\sigma_\delta(j) - e_\delta(j)}.
\end{align*}
We conclude $(1-t_3)\FFF_\sigma|_{v_3=1} = f$ and $Q_\sigma\in\mathcal{S}_f$. In fact,
a direct examination shows
every $Q_{U'}\in\mathcal{S}_f$ can be obtained as $Q_\sigma$
for some admissible
$\sigma\in \Sym_\mu$.
If we let $\Sym_\mu^0$ be the subgroup of $\Sym_\mu$ consisting of elements $\tau$ such that $e_\delta(\tau_\delta(j))=e_\delta(j)$, then $Q_\sigma = Q_{\sigma'}$ if and only if $\sigma^{-1}\sigma'\in \Sym_\mu^0$.
We thus can replace the sum over $Q_U\in\mathcal{S}_f$ with a sum over admissible $\sigma\in\Sym_\mu$:
\begin{multline}\label{kk449}
\left(\sum_{Q_U\in\mathcal{S}_f}\mathsf{w}_{i_1,\ldots,i_k}(Q_U)\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)} = \\
\frac{1}{|\Sym_\mu^0|}\left(\sum_{\sigma\in\Sym_\mu\text{ admissible}}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)\right)\bigg|_{s_3=\frac{1}{a}(s_1+s_2)}.
\end{multline}
We will show the evaluation is well-defined
by choosing $\kappa_0$ such that each
term $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$ in the above sum has order of vanishing along $u_3=0$ at least $-\kappa_0$, and then showing
\begin{equation}\label{differentiated}
\sum_{\sigma\in\Sym_\mu\text{ admissible}}\left(\frac{\partial}{\partial u_3}\right)^\kappa(u_3^{\kappa_0}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma))\bigg|_{u_3=0} = 0
\end{equation}
for $0\le\kappa<\kappa_0$.
The second part of Proposition \ref{vanishing},
the vanishing of the evaluation \eqref{kk449}
for all but finitely many $f$, is
then equivalent to proving that (\ref{differentiated}) holds for $\kappa=\kappa_0$ (for all but finitely many $f$).
In order to prove these vanishing results, we will need to analyze the dependence of the terms $\left(\frac{\partial}{\partial u_3}\right)^\kappa(u_3^{\kappa_0}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma))\bigg|_{u_3=0}$ on the permutation $\sigma\in\Sym_\mu$.
For each $\kappa$, we will find
the corresponding term is equal to a polynomial in the values $\sigma_\delta(j)$ of relatively low degree which
vanishes at all inadmissible permutations $\sigma$.
Let $\Q[\sigma]$ and $\Q(\sigma)$ denote the
ring of polynomials and the field of rational functions respectively
in the variables $\sigma_\delta(j)$. For a polynomial $P\in\Q[\sigma]$, let $\deg(P)$ be the (total) degree of $P$. For rational functions $\frac{P}{Q}\in\Q(\sigma)$, we set
$$\deg\left(\frac{P}{Q}\right) = \deg(P)-\deg(Q).$$
We observe that if $P\in\Q[\sigma]$ has degree $\deg(P)<\sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1)$, then
\begin{equation*}
\sum_{\sigma\in\Sym_\mu}\sgn(\sigma)P(\sigma) = 0,
\end{equation*}
since a nonzero alternating polynomial with respect to $\Sym_\mu$ would have to have greater degree.
\subsection{Proof of Proposition~\ref{vanishing}}
We need to study the $\sigma$-dependence of
\begin{equation*}
\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma) = e(-\mathsf{V}_\sigma)\prod_{j=1}^k\ch_{2+i_j}(\FFF_\sigma\cdot(1-t_1)(1-t_2)(1-t_3)).
\end{equation*}
We begin by explicitly writing $\mathsf{V}_\sigma$ in terms of $\sigma$ and the numbers $e_\delta(j)$. Recall
\begin{equation*}
\mathsf{V}_\sigma = \frac{\FFF'_\sigma -\FFF'_0}{1-t_3}
+ \frac{\overline{\FFF'_\sigma} -\overline{\FFF'_0}}{t_1t_2(1-t_3)}
- \frac{\FFF'_\sigma\overline{\FFF'_\sigma} - \FFF'_0\overline{\FFF'_0}}{1-t_3}(1-t_1^{-1})(1-t_2^{-1}),
\end{equation*}
where $\FFF'_\sigma = (1-t_3)\FFF_\sigma$ and $$\FFF'_0 =
\sum_{(i,j)\in\mu}t_1^it_2^j.$$
In particular, $\mathsf{V}_\sigma|_{v_3=1}$ does not
depend on $\sigma$. Hence,
the order of vanishing of $e(-\mathsf{V}_\sigma)$ along $u_3=0$ is an
integer $-\kappa_0$ independent of $\sigma$.
Since the descendent factor is a polynomial in $u_1,u_2,u_3$,
the order of vanishing of $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$ along $u_3=0$
is at least $-\kappa_0$. If $\kappa_0\le 0$,
then the evaluation is well-defined on each $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$
and thus on their sum. If $\kappa_0<0$, then
the evaluation in fact yields zero.
So we may assume $\kappa_0\ge 0$.
We now rewrite $\mathsf{V}_\sigma$ in terms of $v_1,v_2,v_3$. We find
$\mathsf{V}_\sigma$ equals
\footnotesize
\begin{align*}
&\ \ \ \sum_{(\delta; j)\in\mu}\frac{v_1^\delta v_2^{e_\delta(j)}v_3^{\sigma_\delta(j) - e_\delta(j)}-v_1^\delta v_2^j}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}\\
& + \sum_{(\delta; j)\in\mu}\frac{v_1^{-\delta} v_2^{-e_\delta(j)-1}v_3^{-\sigma_\delta(j) + e_\delta(j)}-v_1^{-\delta} v_2^{-j-1}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\
&- \sum_{(\delta_1; j_1),(\delta_2; j_2)\in\mu}\frac{v_1^{\delta_1-\delta_2} v_2^{e_{\delta_1}(j_1)-e_{\delta_2}(j_2)}v_3^{\sigma_{\delta_1}(j_1)-\sigma_{\delta_2}(j_2)-e_{\delta_1}(j_1)+e_{\delta_2}(j_2)}-v_1^{\delta_1-\delta_2} v_2^{j_1-j_2}}{(1-(\frac{v_2}{v_3})^{\frac{1}{a}}) \cdot(1-v_1^{-1})^{-1}(1-v_1v_2^{-1})^{-1}}.
\end{align*}
\normalsize
Let $C > 2\max(e_\delta(j))$ be a large positive integer. We break up each of the three above sums above using $C$. Then, $\mathsf{V}_\sigma$ equals
\footnotesize
\begin{align*}
&\ \ \
\sum_{(\delta; j)\in\mu}\frac{v_1^\delta v_2^{e_\delta(j)}v_3^{\sigma_\delta(j) - e_\delta(j)}-v_1^\delta v_2^{-C}v_3^{\sigma_\delta(j)+C}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}
\\ &+ \sum_{(\delta; j)\in\mu}\frac{v_1^\delta v_2^{-C}v_3^{j+C}-v_1^\delta v_2^j}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\
&+ \sum_{(\delta; j)\in\mu}\frac{v_1^{-\delta} v_2^{-e_\delta(j)-1}v_3^{-\sigma_\delta(j) + e_\delta(j)}-v_1^{-\delta} v_2^{-C-1}v_3^{-\sigma_\delta(j)+C}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\ &
+ \sum_{(\delta; j)\in\mu}\frac{v_1^{-\delta} v_2^{-C-1}v_3^{-j+C}-v_1^{-\delta} v_2^{-j-1}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}} \\
&- \sum_{(\delta_1; j_1),(\delta_2; j_2)\in\mu} \Bigg(
\frac{v_1^{\delta_1-\delta_2} v_2^{e_{\delta_1}(j_1)-e_{\delta_2}(j_2)}v_3^{\sigma_{\delta_1}(j_1)-\sigma_{\delta_2}(j_2)-e_{\delta_1}(j_1)+e_{\delta_2}(j_2)}}
{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}
\\&
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{v_1^{\delta_1-\delta_2} v_2^{-C}v_3^{\sigma_{\delta_1}(j_1)-\sigma_{\delta_2}(j_2)+C}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}\Bigg)
\cdot(1-v_1^{-1})(1-v_1v_2^{-1}) \\
&- \sum_{(\delta_1; j_1),(\delta_2; j_2)\in\mu}\frac{v_1^{\delta_1-\delta_2} v_2^{-C}v_3^{j_1-j_2+C}-v_1^{\delta_1-\delta_2} v_2^{j_1-j_2}}{1-(\frac{v_2}{v_3})^{\frac{1}{a}}}\cdot(1-v_1^{-1})(1-v_1v_2^{-1}).
\end{align*}
\normalsize
We now expand out the above sums into monomials:
all of the resulting terms will be of the form
$$\pm v_1^x v_2^y v_3^{z(\sigma)},$$
where $x$ and $y$ have no dependence on the permutation $\sigma = (\sigma_\delta)$ and $z\in\Q[\sigma]$ is a linear function of the values $\sigma_\delta(j)$.
After separating out the monomials with $x=y=0$, we write
\begin{equation*}
\mathsf{V}_\sigma = \sum_{(c,0,0,z)\in S}c v_3^{z(\sigma)}+\sum_{\substack{(c,x,y,z)\in S \\ (x,y)\ne(0,0)}}c v_1^x v_2^y v_3^{z(\sigma)},
\end{equation*}
where $S$ is a finite set containing the data of the monomials which
appear
(with
coefficients $c \in \mathbb{Z}$).
Then
\begin{equation*}
e\left(-\sum_{(c,0,0,z)\in S}c v_3^{z(\sigma)}\right) = \phi(\sigma)u_3^{-\kappa_0},
\end{equation*}
for a rational function $\phi=\phi_f\in\Q(\sigma)$
which will be explicitly described below.
We analyze first
the descendent factors in $\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma)$.
The descendent terms can be expressed in the form
\begin{multline*}
\prod_{j=1}^k\ch_{2+i_j}\left(\FFF_\sigma\cdot(1-t_1)(1-t_2)(1-t_3)\right)
=\\
\prod_{j=1}^k\sum_{(c',x,y,z)\in S'_j}c'(xu_1+yu_2+z(\sigma)u_3)^{2+i_j},
\end{multline*}
where the $S'_j$ are more fixed finite sets containing the data of the
terms which appear. As before, $z\in\Q[\sigma]$ is linear.
We then find
\begin{multline*}
u_3^{\kappa_0}\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma) = \phi(\sigma)\prod_{\substack{(c,x,y,z)\in S \\
(x,y)\ne(0,0)}}(xu_1+yu_2+z(\sigma)u_3)^{-c}
\\
\cdot \prod_{j=1}^k\sum_{(c',x,y,z)\in S'_j}c'(xu_1+yu_2+z(\sigma)u_3)^{2+i_j}.
\end{multline*}
Differentiating the above
product $\kappa$ times with respect to $u_3$ and then setting $u_3$ equal to $0$ is easily done. We obtain
\begin{equation*}
\left(\frac{\partial}{\partial u_3}\right)^\kappa(u_3^{\kappa_0}
\mathsf{w}_{i_1,\ldots,i_k}(Q_\sigma))
|_{u_3=0}
= \sum_{i\in\mathcal{I}}\phi(\sigma)Z_i(\sigma)R_i(u_1,u_2),
\end{equation*}
where $\mathcal{I}$ is an indexing set, $Z_i\in\Q[\sigma]$ has degree at most $\kappa$, and $R_i(u_1,u_2)\in\Q(u_1,u_2)$ does not depend on $\sigma$.
Proposition~\ref{vanishing} will follow from the claim that
\begin{equation}\label{cancel}
\sum_{\sigma\in\Sym_\mu\text{ admissible}}\phi(\sigma)Z(\sigma) = 0
\end{equation}
for any polynomial $Z$ of degree $\kappa < \kappa_0$
(or degree $\kappa=
\kappa_0$ for all but finitely many $f$). The
vanishing property \eqref{cancel}
is purely a property of the rational function $\phi\in\Q(\sigma)$.
We will now study $\phi$ in more detail.
The goal is to find a polynomial $\psi\in\Q[\sigma]$ of
sufficiently low degree satisfying
$$\phi(\sigma) = \sgn(\sigma)\psi(\sigma)$$
for every admissible $\sigma\in\Sym_\mu$ and
satisfying
$\psi(\sigma) = 0$ for every inadmissible $\sigma\in\Sym_\mu$.
From the formula for $\mathsf{V}_\sigma$,
we can describe $\phi\in\Q(\sigma)$ explicitly as a product of linear factors:
\footnotesize
\begin{align*}
\phi(\sigma) =
&\left(\prod_{\substack{(0; j)\in\mu \\ e_0(j)>0}}\sigma_0(j)\right)
\left(\prod_{\substack{(0; j)\in\mu \\ j>0}}j\right)^{-1}
\left(\prod_{\substack{(0; j)\in\mu \\ e_0(j)<-1}}(-\sigma_0(j)-1)\right)
\\ &
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)>e_\delta(j_2)}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))\right)^{-1}
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(j_1-j_2)\right)
\\ &
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)>e_\delta(j_2)+1}}(\sigma_\delta(j_1)-\sigma_\delta(j_2)-1)\right)^{-1}
\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2+1}}(j_1-j_2-1)\right) \\
&\left(\prod_{\substack{(\delta+1; j_1),(\delta; j_2)\in\mu \\ e_{\delta+1}(j_1)>e_\delta(j_2)}}(\sigma_{\delta+1}(j_1)-\sigma_{\delta}(j_2))\right)
\left(\prod_{\substack{(\delta+1; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(j_1-j_2)\right)^{-1} \\
&\left(\prod_{\substack{(\delta; j_1),(\delta+1; j_2)\in\mu \\ e_{\delta}(j_1)>e_{\delta+1}(j_2)+1}}(\sigma_{\delta}(j_1)-\sigma_{\delta+1}(j_2)-1)\right)
\left(\prod_{\substack{(\delta; j_1),(\delta+1; j_2)\in\mu \\ j_1>j_2+1}}(j_1-j_2-1)\right)^{-1}.
\end{align*}
\normalsize
The degree of $\phi$ is easily computed to be $-\kappa_0$, since there are the same number of constant factors appearing on the numerator and denominator in the above expression.
\begin{lem} \label{frrg} We have
\begin{equation*}
\frac{\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(j_1-j_2)}{\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)>e_\delta(j_2)}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))} = \pm\sgn(\sigma)\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ e_\delta(j_1)=e_\delta(j_2) \\ j_1>j_2}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))
\end{equation*}
for every $\sigma\in\Sym_\mu$.
\end{lem}
\begin{proof}
The formula is obtained by cancelling equal terms on the left side.
\end{proof}
Suppose that $\{\delta \mid \mu_\delta \ne \emptyset\} = \{\delta \mid a\le\delta\le b\}$. By using the identity of Lemma \ref{frrg}
and grouping terms appropriately, we find
$$\phi(\sigma) = \sgn(\sigma)\phi_0(\sigma)$$
for $\phi_0\in\Q(\sigma)$ given by
\begin{equation}
\phi_0 = XPQ\frac{\prod_{a\le \delta\le b-1}R_\delta}{\prod_{a+1 \le \delta \le b-1}S_\delta},
\end{equation}
where
\begin{equation*}
P = \prod_{\substack{j\in\mu_0 \\ e_{0}(j)<-1}}(-\sigma_{0}(j)-1),\ \ \
Q = \prod_{\substack{j\in\mu_0 \\ e_{0}(j)>0}}\sigma_{0}(j),
\end{equation*}
\footnotesize
\begin{equation*}
R_\delta = \left(\prod_{\substack{j_1\in\mu_{\delta+1}, j_2\in\mu_{\delta} \\ e_{\delta+1}(j_1)>e_\delta(j_2)}}(\sigma_{\delta+1}(j_1)-\sigma_{\delta}(j_2))\right)\left(\prod_{\substack{j_1\in\mu_{\delta}, j_2\in\mu_{\delta+1} \\ e_{\delta}(j_1)>e_{\delta+1}(j_2)+1}}(\sigma_{\delta}(j_1)-\sigma_{\delta+1}(j_2)-1)\right),
\end{equation*}
\normalsize
\begin{equation*}
S_\delta = \prod_{\substack{j_1,j_2\in\mu_{\delta} \\ e_{\delta}(j_1)>e_\delta(j_2)+1}}(\sigma_{\delta}(j_1)-\sigma_{\delta}(j_2)-1),
\end{equation*}
and $X\in\Q[\sigma]$ is a polynomial. The total degree of the
rational function $\phi_0$ is
\begin{equation*}
\deg(\phi)+\deg\left(\prod_{\substack{(\delta; j_1),(\delta; j_2)\in\mu \\ j_1>j_2}}(\sigma_\delta(j_1)-\sigma_\delta(j_2))\right) = -\kappa_0 + \sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1).
\end{equation*}
We now require an algebraic result in order to
convert $\phi_0$ into a polynomial.
Let $m,n\geq 0$ be integers, and
let $$A=\Q[x_1,\ldots,x_n,y_1,\ldots,y_m].$$
Let $P$ be the collection of $n!m!$
points $$(x_1,\ldots,x_n,y_1,\ldots,y_m)\in \Q^{n+m}$$ satisfying
$\{x_1,\ldots,x_n\} = \{1,\ldots,n\}$ and
$\{y_1,\ldots,y_m\} = \{1,\ldots,m\}$.
Let $a_1\le a_2\le\cdots\le a_n$ be integers with $0\le a_i < i$, and set
\begin{equation*}
F = \prod_{1\le j\le a_i}(x_j-x_i+1) \in A.
\end{equation*}
The following Proposition will be proven in Section \ref{division}.
\begin{prop}\label{division}
If $G\in A$ vanishes when evaluated at every point of $P$ at which $F$ vanishes,
then there exists $H\in A$ with $$\deg(H) \le \deg(G)-\deg(F)$$ satisying
$G=FH$ for every point of $P$.
\end{prop}
If $S_{\delta+1}(\sigma)=0$ for a given $\sigma\in\Sym_\mu$ (which is then
necessarily inadmissible), then
\begin{equation}
\label{y34}
R_\delta(\sigma)=R_{\delta+1}(\sigma)=0.
\end{equation}
By reindexing the permutation sets $\mu_\delta$ and $\mu_{\delta+1}$ as necessary, we can apply Proposition~\ref{division} with $G = R_\delta$ and $F = S_\delta$, since $S_{\delta}$ is of the appropriate form.\footnote{By definition,
$e_\delta(j)$ is a weakly decreasing function of $j$. We use the {\emph{opposite}} ordering on the variables $\sigma_\delta(j)$ to write $S_\delta$ in the desired form. Explicitly, if $$\mu_\delta = \{A, A+1, \ldots, B\},$$ then we take $x_i = \sigma_\delta(B-i+1)-A+1$.} Thus for
$a+1\le\delta\le b-1$, there exist polynomials $T_\delta\in\Q[\sigma]$
with $\deg(T_\delta)\le \deg(R_\delta) - \deg(S_\delta)$
satisfying
$$T_\delta(\sigma) = \frac{R_\delta(\sigma)}{S_\delta(\sigma)}$$
for all $\sigma$ for which
which $S_\delta(\sigma)\neq 0$. Then
\begin{equation*}
\psi = XPQR_a\prod_{a+1 \le \delta \le b-1}T_\delta \in \Q[\sigma]
\end{equation*}
has degree at most equal to that of $\phi_0$ and satisfies
$$\sgn(\sigma)\psi(\sigma) = \sgn(\sigma)\phi_0(\sigma)=\phi(\sigma)$$
for any admissible $\sigma$.
For a polynomial $\theta\in\Q[\sigma]$, let $V(\theta)$
denote the set of $\sigma\in\Sym_\mu$ such that $\theta(\sigma)=0$.
We see
\begin{align*}
V(\psi) &\supseteq V(Q)\cup V(R_a) \cup \left(\bigcup_{a+1 \le \delta \le b-1}V(T_\delta)\right) \\
&\supseteq V(Q)\cup V(R_a) \cup \left(\bigcup_{a+1 \le \delta \le b-1}(V(R_\delta)-V(S_\delta))\right) \\
&\supseteq V(Q)\cup \left(\bigcup_{a \le \delta \le b-1}V(R_\delta)\right) \\
&= \{\sigma\in\Sym_\mu \mid \sigma\text{ is not admissible}\}.
\end{align*}
The third inclusion is by repeated application of \eqref{y34}.
We conclude $\psi$ vanishes when evaluated at any inadmissible $\sigma$.
We are finally able to evaluate the sum \eqref{cancel}. We have
\begin{equation*}
\sum_{\sigma\in\Sym_\mu\text{ admissible}}\phi(\sigma)Z(\sigma) = \sum_{\sigma\in\Sym_\mu}\sgn(\sigma)\psi(\sigma)Z(\sigma).
\end{equation*}
If $\deg(Z)<\kappa_0$, then $\deg(\psi Z)<\sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1)$, and thus
\begin{equation*}
\sum_{\sigma\in\Sym_\mu}\sgn(\sigma)\psi(\sigma)Z(\sigma) = 0.
\end{equation*}
We have proven the evaluation of Proposition \ref{vanishing}
is well-defined.
The second part of Proposition~\ref{vanishing} asserts the
vanishing of the evaluation for all but finitely many $f$. We will use a combination of two
ideas to prove the assertion.
First, if $\mathcal{S}(f) = \emptyset$, then the evaluation is trivially zero.
Second, we replace
the polynomial $\psi$ above with another polynomial
$\psi'$ which assumes the same values but has lower degree. Then
$$\deg(\psi')<\deg(\phi_0)
= -\kappa_0 + \sum_{\delta}\frac{1}{2}|\mu_\delta|(|\mu_\delta|-1).$$
So for $\deg(Z)\leq\kappa_0$,
\begin{equation*}
\sum_{\sigma\in\Sym_\mu}\sgn(\sigma)\psi'(\sigma)Z(\sigma) = 0.
\end{equation*}
As we have seen, a choice of $f$ such that $\mathcal{S}(f)\ne\emptyset$ uniquely determines
constants $e_\delta(j)$ weakly decreasing in $j$. We use linear inequalities in
the constants $e_\delta(j)$
to describe four cases in which either $\mathcal{S}(f) = \emptyset$ or $\psi$ can be
replaced by $\psi'$ as above. In the end, we will check that only finitely many
possibilities avoid all four cases. The finiteness will come from giving upper
and lower bounds for the $e_\delta(j)$.
For the lower bound, since
$e_\delta(j)$ is weakly decreasing in $j$, we introduce the notation
\[
m_\delta = \max(\mu_\delta)
\]
and focus on the values $e_\delta(m_\delta)$.
\noindent {\bf Case I.}
Let $J = \max\{j\mid (\delta; j)\in\mu\text{ for some }\delta\}$ and suppose
$e_\delta(j) > J$ for some $(\delta; j)\in\mu$.
Then for any $\sigma\in\Sym_\mu$,
\[
h_\sigma(\delta; \sigma_\delta(j)) = a\cdot(\sigma_\delta(j)-e_\delta(j)) < 0,
\]
so $\sigma$ is not admissible. Thus $\mathcal{S}(f)=\emptyset$.
\noindent {\bf Case II.}
Consider the sequence
$$e_0(0)\ge e_0(1)\ge \cdots \ge e_0(m_0).$$
Suppose there exists $i\in\{0,\ldots,m_0\}$ for which
the conditions
\begin{enumerate}
\item[$\bullet$]
$e_0(i)<-1$
\item[$\bullet$]
$i=0$ or $e_0(i)<e_0(i-1)-1$
\end{enumerate}
hold.
Then, for
admissible $\sigma\in\Sym_\mu$, the factor $\sigma_0$ must map
$\{i,\ldots,m_0\}$ to itself, as the box configuration function
$$h_\sigma(\delta; j) = a(j-e_\delta(\sigma_\delta^{-1}(j)))$$
must be weakly increasing in $j$.
The factor $P$ of $\psi$ is a multiple of
\begin{equation*}
\prod_{j=i}^{m_0}(-\sigma_0(j)-1).
\end{equation*}
Since $\frac{\psi}{P}$ vanishes at all inadmissible $\sigma$, we can take
\begin{equation*}
\psi' = \frac{\prod_{j=i}^{m_0}(-j-1)}{\prod_{j=i}^{m_0}(-\sigma_0(j)-1)}\psi,
\end{equation*}
and then $\psi'(\sigma)=\psi(\sigma)$ at all $\sigma\in\Sym_\mu$.
We have $\deg(\psi')<\deg(\psi)$, as desired.
\noindent {\bf Case III.}
Suppose $\delta \ge 0$ and $e_{\delta+1}(m_{\delta+1}) +1 < e_{\delta}(m_{\delta})$.
Then, either $m_{\delta+1} = m_{\delta} - 1$ or $m_{\delta+1} = m_{\delta}$. We consider the two options separately.
\noindent({\bf{i}})
If $m_{\delta+1} = m_{\delta} - 1$, then for any $\sigma\in\Sym_\mu$, we can take
$$i = \sigma_\delta^{-1}(\sigma_{\delta+1}(m_{\delta+1})+1)\ .$$ Then,
$
\sigma_\delta(i) = \sigma_{\delta+1}(m_{\delta+1})+1$
and
$e_\delta(i) \ge e_\delta(m_\delta) > e_{\delta+1}(m_{\delta+1})+1$,
so $\sigma$ is not admissible. Thus $\mathcal{S}(f)=\emptyset$.
\noindent({\bf{ii}})
If $m_{\delta+1} = m_{\delta}$, then we have
$e_{\delta+1}(m_{\delta}) +1 < e_{\delta}(m_{\delta}) \le e_\delta(j)$
for $0\le j \le m_\delta$, so $R_\delta$ is a multiple of
\begin{equation}\label{jj45}
\prod_{j=0}^{m_\delta}(\sigma_\delta(j)-\sigma_{\delta+1}(m_\delta)-1).
\end{equation}
The product \eqref{jj45} vanishes unless $\sigma_{\delta+1}(m_\delta)=m_\delta$.
Hence
\begin{equation*}
(-m_\delta-1)\prod_{j=1}^{m_\delta}(j-\sigma_{\delta+1}(m_\delta)-1)
\end{equation*}
equals \eqref{jj45}
for all $\sigma\in\Sym_\mu$ and is of lower degree, so we may replace $\psi$ with $\psi'$ of lower degree.
\pagebreak
\noindent {\bf Case IV.}
Suppose $\delta < 0$ and $e_{\delta}(m_\delta) < e_{\delta+1}(m_{\delta+1})$.
The situation is parallel to Case III. As before, either $\mathcal{S}(f)=\emptyset$ or we can replace a divisor of $R_\delta$ with a polynomial of lower degree.
To complete the proof of Proposition \ref{vanishing}, we
must check there are only
finitely many $f$
which avoid Cases I-IV.
If $f$ does not fall into Case I, then $e_\delta(j)\le J$ for all $(\delta; j)\in\mu$. If $f$ does not fall into Case II, then $e_0(j)\ge -j-1$ for each $j$, and in particular $e_0(m_0) \ge -m_0-1$. If $f$ also does not fall
into either of the other two cases,
we can extend the inequality to obtain
$$e_\delta(m_\delta) \ge -m_0 - 1 - \max\{\delta \mid \mu_\delta\ne\emptyset\}$$
for all $\delta$.
Since $e_\delta(j)$ is a weakly decreasing function of $j$, the
bounds imply bounds for all of the $e_\delta(j)$. Since the $e_\delta(j)$ belong to $\frac{1}{a}\Z$, we conclude
there are only a finite number of possibilities for each if $f$ does not fall into any of the Cases I-IV. \qed
\subsection{Proof of Proposition~\ref{division}}
Let $R=\Q[x_1,\ldots,x_n]$, and let
$$e_1, e_2,\ldots,e_n\in R$$ be the elementary
symmetric polynomials with $c_1, c_2, \ldots, c_n \in \Z$ their evaluations at $x_i=i$.
Let $$I = (e_1-c_1,\ldots,e_n-c_n) \subset R$$ denote the ideal of polynomials
vanishing on every permutation of $(1,\ldots,n)$.
For a polynomial $f\in R$, let $f_0$ denote the homogeneous part
of $f$ of highest degree.
For an ideal $J\subset R$, let $J_0$ denote the homogeneous
ideal generated by the top-degree parts,
$$J_0 = \langle \ f_0 \ | \ f \in J \ \rangle \ .$$
Using the regularity of $e_1,\ldots, e_n$,
we easily see $I_0 = (e_1,\ldots,e_n)$.
We define $R'=\Q[y_1,\ldots,y_m]$ and ideals $I',I'_0\subset R'$
as above with respect to the permutations of $(1,\ldots,m)$.
We have
$$A = R\otimes_\Q R'= \Q[x_1,\ldots, x_n,y_1,\ldots,y_m]\ .$$
For notational convenience, we let
$$I,I_0,I',I'_0 \subset A$$
denote the extensions of the respective ideals of $R$ and $R'$ in $A$.
The ideal of $A$ vanishing on the set $P\subset \Q^{n+m}$
of Proposition \ref{division} is precisely $I+I'$.
The basic equality
$$(I+I')_0 = I_0 + I_0'$$
holds.
Let $\widehat{P} = \{p\in P \mid F(p)\ne 0\}$. Let $H\in A$ be a polynomial with
the prescribed values
$$H(p) = \frac{G(p)}{F(p)}$$ for
$p\in \widehat{P}$, of minimum possible degree $d = \deg(H)$.
We must show $d\le \deg(G)-\deg(F)$.
For contradiction, assume $d > \deg(G)-\deg(F)$.
Then, the polynomial $G - FH$ vanishes at every $p\in P$ and has top degree part $F_0H_0$.
Since $F_0\in R$, we verify the following equality
\begin{equation*}
H_0\in\{f\in A \mid F_0f \in (I+I')_0\} = \{r\in R \mid F_0r \in I_0\} + I'_0 \ \ \subset A.
\end{equation*}
We claim the above ideal is equal to
\begin{equation*}
\{f\in A \mid Ff \in I+I'\}_0 = \{r\in R \mid Fr \in I\}_0 + I'_0 \ \ \subset A,
\end{equation*}
Assuming the equality,
there exists $H'\in A$ with top degree part $H_0$ and $FH' \in I+I'$ vanishing at
every $p\in P$.
But then $H_0-H'$ has degree less than that of $H_0$ and still interpolates the desired values, so we have a contradiction.
To complete the proof of Proposition \ref{division},
we must show
\begin{equation*}
\{r\in R \mid F_0r \in I_0\} + I'_0 = \{r\in R \mid Fr \in I\}_0 + I'_0,
\end{equation*}
or equivalently
\begin{equation} \label{befff}
\{r\in R \mid F_0r \in I_0\} = \{r\in R \mid Fr \in I\}_0.
\end{equation}
The left hand side contains the right hand side.
The equality \eqref{befff} is thus a
consequence of the following Lemma which implies
the two sides have equal (and finite) codimension in $R$.
\begin{lem}\label{ranks}
Let $n\geq 0$ be an integer, and
let $a_1\le a_2\le\cdots\le a_n$ be integers satisfying $0\le a_i < i$. Let
\begin{equation*}
F = \prod_{1\le j\le a_i}(x_j-x_i+1) \ \ \ \ {\text and} \ \ \ \
F_0 = \prod_{1\le j\le a_i}(x_j-x_i).
\end{equation*}
Then, we have
\begin{eqnarray*}
\rk_\Q(m_F: R/I \to R/I) & =& \rk_\Q(m_{F_0}:R/I_0 \to R/I_0) \\
& = & \prod_{i=1}^n(i-a_i),
\end{eqnarray*}
where $m_F$ and $m_{F_0}$ denote multiplication operators by $F$ and $F_0$ respectively.
\end{lem}
\begin{proof}
We first show $\rk_\Q(m_F) = \prod_{i=1}^n(i-a_i)$.
Since $R/I$ is the coordinate ring of the set of $n!$ permutations of $(1,\ldots,n)$,
the rank is simply the number of permutations at which $F$ does not vanish.
We must count the number of permutations $\sigma\in \Sym_n$
satisfying $$\sigma(i)-1\ne\sigma(j)$$
for $1\le j\le a_i$.
We view the permutation $\sigma$ (extended by $\sigma(0)=0$)
as a directed path on vertices labeled $0,1,\ldots,n$ with an edge from $i$ to $j$
if $\sigma(i)-1=\sigma(j)$.
We are then counting permutations which
do not have an edge from $i$ to $j$ if $1\le j\le a_i$.
We count the number of ways of building such a path by first choosing an edge leading out of $n$, then an edge leading out of $n-1$, and so on. The edge leading out of $n$ can go to $0$ or to any $j$ with $a_n < j < n$; there are $n-a_n$ choices. After placing the edges leading out of $n,n-1,\ldots,k+1$, the digraph will be a disjoint union of $k+1$ paths. One of these paths will end at $k$ and $a_k$ of the other paths will end at $1,\ldots,a_k$, so the choices for the edge leading out of $k$ are to go to the start of one of the $k-a_k$ other paths. Thus,
the number of such permutations is indeed the product $(n-a_n)\cdots(1-a_1)$.
Proving $\rk_\Q(m_{F_0}) = \prod_{i=1}^n(i-a_i)$ will require more work.
Let $$J = \{f\in R \mid F_0f\in I_0\},$$
so multiplication by $F_0$ induces an isomorphism between $R/J$ and
$\text{Image}(m_{F_0})\subset R/I_0$.
We will show
\begin{equation}\label{uu23}
\rk_\Q(R/J) = \prod_{i=1}^n(i-a_i)\ .
\end{equation}
In fact, we claim $R/J$ is a 0-dimensional complete intersection of
multidegree $(1-a_1,\ldots,n-a_n)$. The
dimension \eqref{uu23}
will then follow from Bezout's Theorem.
For $1\le k \le n$, let
\begin{equation*}
f_k = \sum_{i=k}^n x_i\prod_{j=a_k+1}^{k-1}(x_j-x_i).
\end{equation*}
We claim $J = (f_1,\ldots,f_n)$. Note $f_k$ has degree $k-a_k$ as desired.
We will prove this claim by induction on the sequence $(a_i)_{i=1}^n$. The base case is $a_i=0$ for all $i$
where
$$F=1,\ \ \ J = I_0,\ \ \ \text{and}\ \ \
f_k = \sum_{i=k}^nx_i\prod_{j=1}^{k-1}(x_j-x_i)\ .$$
We must show $(f_1,\ldots,f_n) = (e_1,\ldots,e_n)$.
First, suppose $f_1=f_2=\cdots=f_n=0$
at some point $$(t_1,\ldots,t_n)\in\overline{\Q}^n.$$
From $f_n=0$, we find either $t_n=0$ or $t_n=t_i$ for some $i<n$.
Since $f_{n-1}=0$, either $t_{n-1}=0$ or $t_{n-1}=t_i$ for some $i<n-1$.
Continuing, we conclude for every $k$, either $t_k=0$ or $t_k=t_i$ for some $i<k$.
Thus, $t_k=0$ for all $k$.
Therefore $R/(f_1,\ldots,f_n)$ is a complete intersection and has $\Q$-rank
$$(\deg f_1)\cdots(\deg f_n) = n! = \rk_\Q(R/(e_1,\ldots,e_n))\ .$$
By the rank computation, we
need only show
\begin{equation}\label{httyy}
(f_1,\ldots,f_n) \subseteq (e_1,\ldots,e_n)
\end{equation}
to complete the base case of the induction.
But the inclusion \eqref{httyy} is easily seen. For every $k$, we have
\begin{align*}
f_k &= \sum_{i=1}^n x_i\prod_{j=1}^{k-1}(x_j-x_i) \\
&= \sum_{i=1}^n \sum_{e=1}^n c_ex_i^e \\
&=\sum_{e=1}^n c_e \left(\sum_{i=1}^n x_i^e\right),
\end{align*}
where $c_e\in R$. The power sum
$\sum_{i=1}^n x_i^e$ is symmetric and can be written as a polynomial in the elementary symmetric functions $e_1,\ldots,e_n$.
The base case is now established.
We now consider two sets of indices $a_1,\ldots, a_n$ and $a'_1,\ldots,a'_n$
for which such that $a'_i=a_i$ except when $i=l$ and
\begin{equation}
a'_l=a_l+1.
\label{grtt}
\end{equation}
We moreover require either $l = n$ or $a_{l+1}=a_l+1$. We assume
inductively our claim holds for $a_1,\ldots, a_n$ and show
the claim for $a'_1,\ldots,a'_n$.
Every $(a'_i)_{1\le i \le n}$ which is not identically zero can be reached by
taking $l = \min\{l \mid a'_l=a'_n\}$, so the inductive step will imply the Lemma.
Let $J$,$J'$ be the corresponding ideals and let $f_1,\ldots,f_n$ and $f'_1\ldots,f'_n$ be the claimed generators.
We are assuming $J = (f_1,\ldots,f_n)$ and want to prove $J' = (f'_1\ldots,f'_n)$.
From the definition of $J$ and $J'$, we easily see
$$J' = \{g\in R \mid (x_{a_l+1}-x_l)g\in J\}.$$
Also note $f'_k = f_k$ for $k\ne l$.
If $l=n$, then
$$f'_l = \frac{f_l}{x_{a_l+1}-x_l}$$
and otherwise
$$f'_l = \frac{f_l-f_{l+1}}{x_{a_l+1}-x_l}$$
by condition \eqref{grtt}.
Let $\overline{R} = R/(x_{a_l+1}-x_l)$. For an element $r\in R$,
let $\overline{r}$ denote the projection in $\overline{R}$. Consider
the $\overline{R}$-module homomorphism
$$\psi: \overline{R}^n\to \overline{R}$$
defined by $\psi(\overline{r}_1,\ldots,\overline{r}_n) = \overline{f}_1\overline{r}_1+\cdots+\overline{f}_n\overline{r}_n$. Let $s_i^{(j)}$ for $1\le i\le n$ and $1\le j\le m$ be such that the $m$ elements $(\overline{s}_1^{(j)},\ldots, \overline{s}_n^{(j)})\in\overline{R}^n$ generate the kernel of $\psi$. Clearly,
$J'$ is the ideal generated by $J$ and the $m$ elements
$$\frac{1}{x_{a_l+1}-x_l}\sum_{i=1}^nf_is^{(j)}_i \ .$$
In other words, we must find all the relations between the elements
$$\overline{f}_1,\ldots,\overline{f}_n.$$
Now $\overline{f}_l = \overline{f}_{l+1}$ if $l\ne n$, or $\overline{f}_l=0$ if $l = n$, so we need only consider relations between the $n-1$ elements with $\overline{f}_l$ removed.
These $n-1$ elements in $\overline{R}$ form a complete intersection,
so the relations are generated by the trivial ones
$\overline{f}_i\overline{f}_j-\overline{f}_j\overline{f}_i = 0$.
We have proven that
$J'$ is the ideal generated by $J = (f_1,\ldots,f_n)$, either $\frac{f_l}{x_{a_l+1}-x_l}$ if $l=n$ or $\frac{f_l-f_{l+1}}{x_{a_l+1}-x_l}$ otherwise, and the elements $$\frac{f_if_j-f_jf_i}{x_{a_l+1}-x_l} = 0.$$
Thus $J' = (f'_1\ldots,f'_n)$, as desired.
\end{proof}
\section{Descendent depth}\label{depp}
\subsection{$T$-Depth}
Let $N$ be a split rank 2 bundle on
a nonsingular projective curve $C$ of genus $g$.
Let $S\subset N$ be the relative divisor associated
to the points $p_1,\ldots, p_r\in C$.
We consider the $T$-equivariant stable pairs theory of $N/S$
with respect to the scaling action.
The $T$-{\em depth} $m$ theory of $N/S$ consists of all
$T$-equivariant series
\begin{equation}
\label{hkkq}
{\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}
\left( \prod_{{j'}=1}^{k'}
\tau_{i'_{j'}}(\mathsf{1}) \ \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T
\end{equation}
where $k' \leq m$.
As before, $\mathsf{p}\in H^2(C,\mathbb{Z})$ is the class of a point.
The $T$-depth $m$ theory has at most $m$ descendents
of $1$ and arbitrarily many descendents of $\mathsf{p}$ in the integrand.
The $T$-depth $m$ theory of $N/S$ is {\em rational} if all
$T$-depth $m$ series \eqref{hkkq} are Laurent expansions in $q$
of rational functions in $\Q(q,s_1,s_2)$.
The $T$-depth 0 theory concerns only descendents of $\mathsf{p}$.
By taking the specialization $s_3=0$ of Proposition \ref{cttt},
$$
\ZZ_{d,\eta}
^{\mathsf{cap}}\left( \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T=
\ZZ_{d,\eta}
^{\mathsf{cap}}\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}\Big|_{s_3=0}\ , $$
we see
the depth 0 theory of the cap is rational.
\begin{lem} The $T$-depth 0 theory of $N/S$ over a curve $C$ is rational.
\label{ht99}
\end{lem}
\begin{proof} By the degeneration formula, all the descendents
$\tau_{i_j}(\mathsf{p})$ can be degenerated on to a $(0,0)$-cap.
The $T$-depth 0 theory of the cap is rational.
The pairs
theory of local curves without any insertions is rational by \cite{mpt,lcdt}.
Hence, the result follows by the degeneration formula.
\end{proof}
\subsection{Degeneration}
We have already used the degeneration formula in simple cases
in Proposition \ref{ctttt}
and Lemma \ref{ht99} above. We review here the
full $T$-equivariant formula for descendents of
$\mathsf{1},\mathsf{p}\in H^*(C,\mathbb{Z})$.
Let $C$ degenerate to a union $C_1\cup C_2$ of nonsingular
projective curves $C_i$
meeting at a node $p'$.
Let $N$ degenerate to split bundles
$$N_1 \rightarrow C_1, \ \ \ \ N_2 \rightarrow C_2 \ .$$
The levels of $N_i$ must sum to the level of $N$.
The relative points $p_i$, distributed to nonsingular points
of $C_1\cup C_2$, specify relative points $S_i\subset C_i$ away from
$p'$. Let $S_i^+= S_i \cup \{ p' \}$.
In order to apply the degeneration formula to the series
\eqref{hkkq}, we must also specify the distribution of
the point classes occuring in the descendents $\tau_{i_j}(\mathsf{p})$.
The disjoint union $$J_1\cup J_2 = \{1,\ldots, k\}$$
specifies the descendents $\tau_{i_j}(\mathsf{p})$
distribute to $C_i$ for $j\in J_i$.
The degeneration formula for \eqref{hkkq} is
\begin{multline*}
\sum_{J'_1\cup J'_2=\{1,\ldots, k'\}}
{\mathsf Z}^{N_1/S^+_1}_{d,\eta^1,\dots,\eta^{|S_1|}, \mu}
\left( \prod_{{j'}\in J'_1}
\tau_{i'_{j'}}(\mathsf{1}) \ \prod_{j\in J_1} \tau_{i_j}(\mathsf{p})
\right)^T\ \frac{g^{\mu\widehat{\mu}}}{q^d} \\ \cdot
{\mathsf Z}^{N_2/S^+_2}_{d,\eta^{|S_1|+1},\dots,\eta^{|S_2|}, \widehat{\mu}}
\left( \prod_{{j'}\in J'_2}
\tau_{i'_{j'}}(\mathsf{1}) \ \prod_{j\in J_2} \tau_{i_j}(\mathsf{p})
\right)^T
\end{multline*}
A crucial point in the derivation of the degeneration formula
is the pre-deformability condition (ii) of Section 3.7 of \cite{pt}.
The condition insures the existence of finite resolutions of
the universal sheaf $\mathbb{F}$ in the relative geometry (needed
for the definition of the descendents) and guarantees the
splitting of the descendents under pull-back via the gluing
maps of the relative geometry. The foundational treatment for
stable pairs is essentially the same as for ideal sheaves \cite{liwu}.
\subsection{Induction I}
To obtain the rationality of the $T$-depth $m$ theory of
$N/S$ over a curve $C$, further knowledge of the descendent
theory of twisted caps is required.
\begin{lem} The rationality of the
$T$-depth $m$ theories of all twisted caps implies
the rationality of the
$T$-depth $m$ theory
of $N/S$ over a curve $C$. \label{rtt5}
\end{lem}
\begin{proof}
We start by proving rationality for the $T$-depth $m$ theories of
all $(0,0)$ geometries,
\begin{equation} \label{hxxz}
\mathcal{O}_{\C} \oplus \mathcal{O}_{\C} \rightarrow \Pp\ ,
\end{equation}
relative to $p_1,\ldots, p_r \in \Pp$.
If $r=1$, the geometry is the cap and rationality of the
$T$-depth $m$ theory is given.
Assume rationality holds for $r$. We will show rationality holds for
$r+1$.
Let $p(d)$ be the number of partitions of size $d>0$.
Consider the $\infty \times p(d)$ matrix $M_d$, indexed by
monomials
$$L= \prod_{i\geq 0} \tau_i (\mathsf{p})^{n_i} $$
in the descendents of $\mathsf{p}$ and partitions $\mu$ of $d$,
with
coefficient
$
{\mathsf Z}^{\mathsf{cap}}_{d,\mu}
\left( L
\right)^T$
in position $(L,\mu)$.
The lowest Euler characteristic for a degree $d$
stable pair on the cap is $d$.
The leading $q^d$
coefficients of $M_d$ are well-known to be of maximal
rank.{\footnote{ The leading $q^d$ coefficients
are obtained from the Chern characters
of the tautological rank $d$ bundle
on $\text{Hilb}(N_\infty,d)$.
The Chern characters generate the ring
$H^*_T(\text{Hilb}(N_\infty,d),\mathbb{Q})$ after
localization as can easily
be seen in the $T$-fixed point basis.
A more refined result is
discussed in Section \ref{ennd}.}}
Hence, the full matrix $M_d$ is also of maximal rank.
Consider the level $(0,0)$ geometry
over $\Pp$ relative to $r+1$ points in $T$-depth $m$,
\begin{equation}
\label{yone}
{\mathsf Z}^{(0,0)}_{d,\eta^1, \ldots, \eta^r,\mu}
\left( \prod_{j'=1}^{k'}
\tau_{i'_{j'}}(1) \ \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T\ .
\end{equation}
We will determine the series \eqref{yone} from the
$T$-depth $m$ series relative to $r$ points,
\begin{equation}
\label{yall}
{\mathsf Z}^{(0,0)}_{d,\eta^1, \ldots, \eta^r}
\left( L \ \prod_{j'=1}^{k'}
\tau_{i'_{j'}}(1) \ \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T
\end{equation}
defined by all monomials $L$ in the descendents of $\mathsf{p}$.
Consider the $T$-equivariant degeneration of
the $(0,0)$ geometry relative to $r$ points obtained
by bubbling off a single $(0,0)$-cap.
All the descendents of $\mathsf{p}$
remain on the original $(0,0)$ geometry in the degeneration except for those
in $L$ which distribute to the cap.
By induction on $m$, we need only analyze the terms of the degeneration formula
in which the descendents of the identity distribute away from the cap.
Then,
since $M_d$ has full rank,
the invariants \eqref{yone} are determined
by the invariants \eqref{yall}.
We have proven the rationality of the $T$-depth $m$ theory of the $(0,0)$-cap
implies the rationality of the $T$-depth $m$ theories of all
$(0,0)$ relative geometries over $\Pp$. By degenerations of
higher genus curves $C$ to rational curves with relative points,
the rationality of the $(0,0)$ relative geometries over
curves $C$ of arbitrary genus is established.
Finally, consider a relative geometry $N/S$ over $C$ of
level $(a_1,a_2)$. We can degenerate $N/S$ to the
union of a $(0,0)$ relative geometry over $C$ and
a twisted $(a_1,a_2)$-cap. Since the rationality of the
$T$-depth $m$ theory of
the twisted cap is given, we conclude the rationality of
$N/S$ over $C$.
\end{proof}
The proof of Lemma \ref{rtt5} yields a slightly refined
result which will be half of our induction argument
relating the descendent theory of the $(0,0)$-cap and
the $(0,0)$-tube.
\begin{lem}\label{nndd}
The rationality of the $T$-depth $m$ theory of the
$(0,0)$-cap implies the rationality of the $T$-depth $m$ theory of
the $(0,0)$-tube.
\end{lem}
\subsection{$\mathbf{T}$-depth}
The $\mathbf{T}$-{\em depth} $m$ theory of the $(a_1,a_2)$-cap
consists of all the
$\mathbf{T}$-equivariant series
\begin{equation}
\label{hkkqq}
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left(
\prod_{j=1}^k \tau_{i_j}([0]) \ \prod_{{j'}=1}^{k'}
\tau_{i'_{j'}}([\infty])
\right)^{\mathbf{T}}
\end{equation}
where $k' \leq m$.
Here, $0\in \Pp$ is the non-relative $\mathbf{T}$-fixed point and
$\infty\in \Pp$ is the relative point.
The $\mathbf{T}$-depth $m$ theory of the $(a_1,a_2)$-cap
is {\em rational} if all
$\mathbf{T}$-equivariant
depth $m$ series \eqref{hkkq} are Laurent expansions in $q$
of rational functions in $\Q(q,s_1,s_2,s_3)$.
\begin{lem} The rationality of the $\mathbf{T}$-depth $m$
theory of the $(a_1,a_2)$-cap implies the
rationality of the $T$-depth $m$ theory of the $(a_1,a_2)$-cap.
\end{lem}
\begin{proof}
The identity class
$1\in H^*_T(\Pp,\mathbb{Z})$ has a well-known
expression in terms of the $\mathbf{T}$-fixed point classes
$$1 = -\frac{[0]}{s_3} + \frac{[\infty]}{s_3}\ .$$
We can calculate at most $m$ descendents of $1$ in the $T$-equivariant
theory via at most
$m$ descendents of $[\infty]$ in the $\mathbf{T}$-equivariant
theory (followed the specialization $s_3=0$).
\end{proof}
\section{Rubber calculus} \label{rubc}
\subsection{Overview}
We collect here results concerning
the rubber calculus which will be needed to complete the
proof of Theorem \ref{cnnn}. Our discussion of the
rubber calculus follows the treatment given in Section 4.8-4.9 of
\cite{lcdt}.
\subsection{Universal 3-fold $\mathcal{R}$}
Consider the moduli space of stable pairs on rubber
$P_n(R/R_0\cup R_\infty)^\sim$ discussed in Section \ref{rubcon}.
Let
$$\pi:\mathcal{R} \rightarrow {P_n(R/R_0\cup R_\infty,d)}^\sim$$
denote the universal 3-fold.
The space $\mathcal{R}$ can be viewed as a moduli space of stable pairs on
rubber {\em together} with a point $r$ of the 3-fold rubber.
The point $r$ is {\em not} permitted to lie on the relative divisors $R_0$ and
$R_\infty$. The stability condition is given by finiteness of the
associated automorphism group.
The virtual class of ${\mathcal R}$ is obtained via $\pi$-flat pull-back,
$$[{\mathcal R}]^{vir} = \pi^* \Big( [{P_n(R/R_0\cup R_\infty,d)}^\sim]^{vir}\Big).$$
As before, let
$$\mathbb{F} \rightarrow {\mathcal R}$$
denote the universal sheaf on ${\mathcal R}$.
The target point $r$ together with $R_0$ and $R_\infty$
specifies 3 distinct points of the destabilized $\Pp$ over which the rubber is fibered.
By viewing the target point as $1\in \Pp$,
we obtain a rigidification map to the tube,
$$\phi: \mathcal{R} \rightarrow P_n(N/N_0\cup N_\infty,d),$$
where $N=\mathcal{O}_\Pp \oplus \mathcal{O}_\Pp$ is the trivial bundle over $\Pp$.
By a comparison of deformation theories,
\begin{equation}\label{zek}
[{\mathcal R}]^{vir} = \phi^* \Big( [{P_n(N/N_0\cup N_\infty,d)}]^{vir}\Big).
\end{equation}
\subsection{Rubber descendents} \label{papap}
Rubber calculus transfers $T$-equivariant rubber descendent
integrals to $T$-equivariant descendent integrals
for the $(0,0)$-tube geometry via the maps $\pi$ and $\phi$.
Consider the rubber descendent
\begin{equation} \label{dref}
\Big\langle \mu \ \Big| \ \psi_0^\ell \ \tau_{c}\cdot \prod_{j=1}^k \tau_{i_j}
\ \Big|\ \nu \Big
\rangle_{n,d}^{\sim}\ .
\end{equation}
As before, $\psi_0$ is the cotangent line at the dynamical point
$0\in \Pp$. The action of the rubber descendent $\tau_{i}$ is defined
via the universal sheaf $\mathbb{F}$ by the operation
$$
\pi_{*}\big( \text{ch}_{2+i}(\FF)
\cap(\pi^*(\ \cdot\ )\big)\colon
H_*(P_{n}(N/N_0\cup N_\infty,d))\to H_*(P_{n}(N/N_0\cup N_\infty,d))\ .
$$
By the push-pull formula, the integral \eqref{dref} equals
\begin{equation}\label{zex}
\Big\langle \mu \ \Big|\ \text{ch}_{2+c}(\mathbb{F}) \ \pi^*\left(\psi_0^\ell
\cdot \prod_{j=1}^k \tau_{i_j}
\right) \ \Big|\ \nu \Big
\rangle_{n,d}^{{\mathcal R}\sim}.
\end{equation}
Next,
we compare the cotangent lines $\pi^*(\psi_0)$ and $\phi^*(\psi_0)$ on
${\mathcal R}$. A standard argument yields
$$\pi^*(\psi_0)=\phi^*(\psi_0) - \phi^*(D_0),$$
where
$$D_0 \subset I_n(N/N_0\cup N_\infty,d)$$
is the virtual boundary divisor
for which the rubber over $\infty$ carries Euler characteristic $n$.
We will apply the cotangent line comparisons to \eqref{zex}.
The basic vanishing
\begin{equation}\label{gbb6}
\psi_0|_{D_0}=0
\end{equation}
holds.
Consider the Hilbert scheme of points ${\text{Hilb}}(R_0,d)$ of the
relative divisor.
The boundary condition $\mu$ corresponds to a Nakajima basis element
of $A^*_T({\text{Hilb}}(R_0,d))$.
Let
${\mathbb{F}}_0$ be the universal quotient sheaf on
$$\text{Hilb}(R_0,d) \times R_0,$$ and define the descendent
\begin{equation}\label{mrr}
\tau_c=\pi_*\Big( {\text{ch}}_{2+c}({\mathbb{F}}_0)\Big)
\in A^c_T({\text{Hilb}}(R_0,d))
\end{equation}
where $\pi$ is the projection
$$\pi: \text{Hilb}(R_0,d) \times R_0 \rightarrow
\text{Hilb}(R_0,d)\ .
$$
The cotangent line comparisons, equation \eqref{zex},
and the vanishing \eqref{gbb6} together yield
the following result,
\begin{multline}\label{dx}
\Big\langle \mu \ \Big| \ \psi_0^\ell \ \tau_{c}\cdot \prod_{j=1}^k \tau_{i_j}
\ \Big|\ \nu \Big\rangle_{n,d}^{\sim} =
\\ \Big\langle \mu \ \Big|\
\psi_0^\ell \ \tau_{c}(\mathsf{p}) \cdot \prod_{j=1}^k \tau_{i_j}
\ \Big|\ \nu \Big
\rangle_{n,d}^{\mathsf{tube},T} \\
- \Big\langle \tau_c\cdot \mu \ \Big|
\ \psi_0^{\ell-1} \prod_{j=1}^k \tau_{i_j} \ \Big|\ \nu
\Big\rangle_{n,d}^{\sim} \ .
\end{multline}
Equation \eqref{dx} will be the main required property of the rubber
calculus.
\section{Capped 1-leg descendents: full} \label{444}
\subsection{Overview} We complete the proof of Theorem \ref{cnnn}
using the interplay between the $\mathbf{T}$-equivariant localization of the cap
and the theory of rubber integrals. A similar strategy was used in \cite{vir}
to prove the Virasoro constraints for target curves.
As a consequence, we will also obtain a special case of Theorem
\ref{tnnn}.
Let $N$ be a split rank 2 bundle on
a nonsingular projective curve $C$ of genus $g$.
Let $S\subset N$ be the relative divisor associated
to the points $p_1,\ldots, p_r\in C$.
We consider the $T$-equivariant stable pairs theory of $N/S$
with respect to the scaling action.
\begin{prop}
If
$\gamma_j \in H^{2*}(C,\mathbb{Z})$ are {\em even} cohomology classes, then
\label{pnnn}
$$\ZZ_{d,\eta^1,\dots,\eta^r}
^{N/S}\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{{T}}$$ is the
Laurent expansion in $q$ of a rational function in $\mathbb{Q}(q,s_1,s_2)$.
\end{prop}
Proposition \ref{pnnn} is the restriction of Theorem \ref{tnnn}
to even cohomology. The proof is given in Section \ref{jj367}.
The proof of Theorem \ref{tnnn} will be completed with the
inclusion of descendents of odd cohomology in Section \ref{555}.
\subsection{Induction II}
The first half of our induction argument was established
in Lemma \ref{nndd}.
The second half relates the $(0,0)$-tube back to the
$(0,0)$-cap with an increase in depth.
\begin{lem} The rationality of \label{p45}
the ${T}$-depth $m$
theory of the $(0,0)$-tube implies the
rationality of $\mathbf{T}$-depth $m+1$ theory of the $(0,0)$-cap.
\end{lem}
\begin{proof}
The result follows from the $\mathbf{T}$-equivariant
localization
formula for the $(0,0)$-cap and the rubber calculus of Section
\ref{papap}.
To illustrate the method, consider first the $m=0$
case of Lemma \ref{p45}.
The localization formula for
$\mathbf{T}$-depth 1 series for the $(0,0)$-cap is the following:
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \tau_{i'_1}([\infty])
\right)^{\mathbf{T}} =
\\
\sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
\bW_\mu^{(0,0)} \cdot
\left( \mathsf{S}^{\tau_{i'_1}\cdot\mu}_\eta+
\mathsf{S}^{\mu}_{\eta}(\tau_{i'_1}) \right) \ ,
\end{multline*}
where the rubber terms on the right are
\begin{eqnarray*}
\mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_1}\cdot \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim} ,
\\
\mathsf{S}^\mu_\eta(\tau_{i'_1}) & = &
\sum_{n\geq d} q^{n}
\left\langle \mathsf{P}_\mu \ \left| \ \frac{ s_3 \tau_{i'_1}}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}. \\
\end{eqnarray*}
In the first rubber term, $\tau_{i'_1}$ acts
on the boundary condition $P_\mu$ via
\eqref{mrr}. The term arises from the
distribution of the Chern character
of the descendent
$\tau_{i'_1}([\infty])$
away from the rubber.
The second rubber term simplifies
via the
topological recursion relation for $\psi_0$ after
writing
\begin{equation}\label{nhhk}
\frac{s_3}{s_3-\psi_0} = 1 + \frac{\psi_0}{s_3-\psi_0}\
\end{equation}
and the rubber calculus relation \eqref{dx}. We find
\begin{eqnarray*}
\mathsf{S}^\mu_\eta(\tau_{i'_1})
& = &
\sum_{|\widehat{\eta}|=d} \mathsf{S}^\mu_{\widehat{\eta}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta
.
\end{eqnarray*}
The leading $1$ on the right side of \eqref{nhhk} corresponds to
the degenerate leading term of $\mathsf{S}^\mu_{\widehat{\eta}}$.
The topological recursion applied to the $\psi_0$ prefactor
of the second term produces the rest of $\mathsf{S}^\mu_{\widehat{\eta}}$.
The superscript $\mathsf{tube}$ refers here to
the $(0,0)$-tube.
The rubber calculus produces the correction
$-\mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta$.
After reassembling the localization formula, we find
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \tau_{i'_1}([\infty])
\right)^{\mathbf{T}} =
\\
\sum_{|\widehat{\eta}|=d}
{\mathsf Z}^{\mathsf{cap}}_{d,\widehat{\eta}}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
\end{multline*}
which implies the $m=0$ case of Lemma \ref{p45}.
The above method of expressing the $\mathbf{T}$-depth
$m+1$ theory of the $(0,0)$-cap in terms of
the $\mathbf{T}$-depth
$0$ theory of the $(0,0)$-cap and the
$T$-depth $m$ theory of the $(0,0)$-tube is valid for
all $m$.
Consider the $m=1$ case.
The localization formula for
$\mathbf{T}$-depth 2 series for the $(0,0)$-cap is the following:
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \tau_{i'_1}([\infty])
\tau_{i_2}([\infty])
\right)^{\mathbf{T}} =
\\
\sum_{|\mu|=d}
\bW_\mu^{\mathsf{Vert}} \left(\prod_{j=1}^k \tau_{i_j}([0]) \right) \cdot
\bW_\mu^{(0,0)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \\
\cdot \left( \mathsf{S}^{\tau_{i'_1}\tau_{i'_2}\cdot\mu}_\eta+
\mathsf{S}^{\tau_{i'_1}\cdot \mu}_{\eta}(\tau_{i'_2})
+\mathsf{S}^{\tau_{i'_2}\cdot \mu}_{\eta}(\tau_{i'_1})
+\mathsf{S}^{\mu}_{\eta}(\tau_{i'_1}\tau_{i'_2})
\right) \ ,
\end{multline*}
where the rubber terms on the right are
\begin{eqnarray*}
\mathsf{S}^{\tau_{i'_1}\tau_{i'_2}\cdot \mu}_\eta & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_1}\tau_{i'_2}\cdot \mathsf{P}_\mu \ \left| \ \frac{1}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim} ,
\\
\mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta(\tau_{i'_2}) & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_1}\cdot \mathsf{P}_\mu \ \left| \ \frac{ s_3 \tau_{i'_2}}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}, \\
\\
\mathsf{S}^{\tau_{i'_2}\cdot \mu}_\eta(\tau_{i'_1}) & = &
\sum_{n\geq d} q^{n}
\left\langle \tau_{i'_2}\cdot \mathsf{P}_\mu \ \left| \ \frac{ s_3 \tau_{i'_1}}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}, \\
\\
\mathsf{S}^{\mu}_\eta(\tau_{i'_1}\tau_{i'_2}) & = &
\sum_{n\geq d} q^{n}
\left\langle \mathsf{P}_\mu \ \left| \ \frac{ s_3^2 \tau_{i'_1}\tau_{i'_2}
}{s_3-\psi_0} \ \right|\ \CC_\eta
\right\rangle_{n,d}^{
\sim}. \\
\end{eqnarray*}
Using \eqref{nhhk} and the rubber calculus relation \eqref{dx}, we find
\begin{eqnarray*}
\mathsf{S}^\mu_\eta(\tau_{i'_1}\tau_{i'_2})
& = &\ \
\sum_{|\widehat{\eta}|=d} \mathsf{S}^\mu_{\widehat{\eta}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty])\cdot \tau_{i'_2}(1) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_1}\cdot \mu}_\eta(\tau_{i'_2})\\
& & + \sum_{|\widehat{\eta}|=d} \mathsf{S}^\mu_{\widehat{\eta}}(\tau_{i'_2})
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d}
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
.
\end{eqnarray*}
As we have seen before,
\begin{eqnarray*}
\mathsf{S}^\mu_{\widehat{\eta}}(\tau_{i'_2})
& = &
\sum_{|\widehat{\mu}|=d} \mathsf{S}^\mu_{\widehat{\mu}}
\cdot \frac{g^{\widehat{\mu}\widehat{\mu}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\mu},\widehat{\eta}}
\left( \tau_{i_2'}([\infty]) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_2}\cdot \mu}_{\widehat{\eta}}, \\
\mathsf{S}^{\tau_{i'_2}\cdot \mu}_\eta(\tau_{i'_1})
& = &
\sum_{|\widehat{\eta}|=d} \mathsf{S}^{\tau_{i'_2}\cdot\mu}_{\widehat{\eta}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty]) \right)^T
\ -\ \mathsf{S}^{\tau_{i'_1}\tau_{i'_2}\cdot \mu}_\eta
.
\end{eqnarray*}
After adding everything together, we have for $m=1$
the relation:
\begin{multline*}
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}([0]) \cdot \prod_{j'=1}^2
\tau_{i'_{j'}}([\infty])
\right)^{\mathbf{T}} =
\\
+s_3 \sum_{|\widehat{\eta}|=d}
{\mathsf Z}^{\mathsf{cap}}_{d,\widehat{\eta}}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},\eta}
\left( \tau_{i_1'}([\infty])\cdot \tau_{i_2'}(1) \right)^T \\
+\sum_{|\widehat{\mu}|,|\widehat{\eta}|=d}
{\mathsf Z}^{\mathsf{cap}}_{d,\widehat{\mu}}
\left( \prod_{j=1}^k \tau_{i_j}([0])
\right)^{\mathbf{T}}
\cdot \frac{g^{\widehat{\mu}\widehat{\mu}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\mu},\widehat{\eta}}
\left(
\tau_{i'_{2}}([\infty]) \right)^T
\\
\cdot \frac{g^{\widehat{\eta}\widehat{\eta}}}{q^d} \cdot
{\mathsf Z}^{\mathsf{tube}}_{d,\widehat{\eta},{\eta}}
\left(
\tau_{i'_{1}}([\infty]) \right)^T
\ . \ \ \ \ \ \
\end{multline*}
We leave the derivation of the parallel formula for general $m$
(via elementary bookkeeping) to
the reader.
\end{proof}
An identical argument yields the twisted version of
Lemma \ref{p45} for the $(a_1,a_2)$-cap.
\begin{lem} The rationality of \label{p456}
the ${T}$-depth $m$
theory of the $(0,0)$-tube implies the
rationality of the $\mathbf{T}$-depth $m+1$ theory of the $(a_1,a_2)$-cap.
\end{lem}
\subsection{Proof of Theorem \ref{cnnn}}
Lemmas \ref{nndd} and \ref{p45}
together provide an induction which results in the
rationality of the $\mathbf{T}$-depth $m$ theory of
the $(0,0)$-cap for all $m$.
Since the classes of the $\mathbf{T}$-fixed points $0,\infty \in \Pp$
generate $H_{\mathbf{T}}^*(\Pp, \mathbb{Z})$
after localization, all partition functions
$$
{\mathsf Z}^{\mathsf{cap}}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}},\ \ \ \ \gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$$
are Laurent series
in $q$ of rational functions in $\mathbb{Q}(q,s_1,s_2,s_3)$.
\qed
\subsection{Proof of Proposition \ref{pnnn}} \label{jj367}
Using Lemma \ref{p456}, we obtain the extension of
Theorem \ref{cnnn} to twisted $(a_1,a_2)$-caps.
\begin{prop}
\label{qnnn}
For $\gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$, the descendent series
$$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$$
of the $(a_1,a_2)$-cap
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
By taking the $s_3=0$ specialization of Proposition
\ref{qnnn}, we obtain the rationality of the $T$-depth $m$
theory of the $(a_1,a_2)$-cap for all $m$.
Proposition \ref{pnnn} then follows from Lemma \ref{rtt5}. \qed
\subsection{$\mathbf{T}$-equivariant tubes}
The $(a_1,a_2)$-tube is the total space of
$$\mathcal{O}_{\Pp}(a_1) \oplus \mathcal{O}_{\Pp}(a_2) \rightarrow \Pp$$
relative to the fibers over both $0,\infty \in \Pp$.
We lift the $\com^*$-action on $\Pp$ to $\mathcal{O}_{\Pp}(a_i)$
with fiber weights $0$ and $a_is_3$ over $0,\infty\in \PP^1$.
The 2-dimensional torus $T$ acts on the $(a_1,a_2)$-tube
by scaling the line summands, so
we obtain a $\mathbf{T}$-action on the $(a_1,a_2)$-tube.
\begin{prop}
\label{qqnnn}
For $\gamma_{j}\in H^*_{\mathbf{T}}(\Pp,\mathbb{Z})$, the
descendent series
$$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta_1\eta_2}
\left( \prod_{j=1}^k \tau_{i_j}(\gamma_{j})
\right)^{\mathbf{T}}$$
of the $(a_1,a_2)$-tube
is the
Laurent expansion in $q$ of a rational function in
$\mathbb{Q}(q,s_1,s_2,s_3)$.
\end{prop}
\begin{proof}
Consider the descendent series
\begin{equation}\label{pw4}
{\mathsf Z}^{(a_1,a_2)}_{d,\eta_2}
\left( L \prod_{j'=1}^{k'} \tau_{i'_{j'}} (\mathsf{1})\ \prod_{j=1}^k \tau_{i_j}([\infty])
\right)^{\mathbf{T}}
\end{equation}
of the $(a_1,a_2)$-cap where $L$ is a monomial in the descendents of
$[0]$.
The $(a_1,a_2)$-cap admits a $\mathbf{T}$-equivariant
degeneration to a standard $(0,0)$-cap and an $(a_1,a_2)$-tube by
bubbling off $0\in \Pp$.
The insertions $\tau_{i_j}([0])$ of $L$ are sent $\mathbf{T}$-equivariantly
to the non-relative point of the $(0,0)$-cap.
Since \eqref{pw4} is rational by Proposition \ref{qnnn} and the
matix $M_d$ of Lemma \ref{rtt5} is full rank,
the rationality of
$$
{\mathsf Z}^{(a_1,a_2)}_{d,\eta_1\eta_2}
\left( \prod_{j'=1}^{k'} \tau_{i'_{j'}} (\mathsf{1})\ \prod_{j=1}^k \tau_{i_j}([\infty])
\right)^{\mathbf{T}}$$
follows by induction on $k'$ from the degeneration formula.
The classes $\mathsf{1}$ and $[\infty]$ generate $H_{\mathbf{T}}^*(\Pp,\mathbb{Z})$
after localization.
\end{proof}
\section{Descendents of odd cohomology} \label{555}
\subsection{Reduction to $(0,0)$}
Let $N/S$ be the relative
geometry of
a split rank 2 bundle on a nonsingular projective curve $C$ of genus $g$.
Let
$$\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g \in H^1(C,\mathbb{Z})$$
be a standard symplectic basis of the odd cohomology of $C$.
Proposition \ref{pnnn} establishes Theorem \ref{tnnn}
in case only the descendents of the even classes
$\mathsf{1},\mathsf{p}\in H^*(C,\mathbb{Z})$ are present.
The descendents of $\alpha_i$ and $\beta_j$
will now be considered.
The relative geometry $N/S$ may be $T$-equivariantly degenerated to
$$\mathcal{O}_C \oplus \mathcal{O}_C \rightarrow C$$
and an $(a_1,a_2)$-cap. The relative points and
the
descendents $\tau_k(\alpha_i)$ and $\tau_k(\beta_j)$ in the
integrand remain on $C$. Since the rationality of the
$T$-equivariant descendent theory of the $(a_1,a_2)$-cap
has been proven, we may restrict our study of the descendents
of odd cohomology to the $(0,0)$ relative geometry over $C$.
\subsection{Proof of Theorem \ref{tnnn}}
The full descendent theories of $(0,0)$ relative geometries of
curves $C$
are uniquely determined by the even descendent theories of $(0,0)$ relative geometries
by the following four properties:
\begin{enumerate}
\item[(i)] Algebraicity of the virtual class,
\item[(ii)] Degeneration formulas for the relative theory in the
presence of odd cohomology,
\item[(iii)] Monodromy invariance of the relative theory,
\item[(iv)] Elliptic vanishing relations.
\end{enumerate}
The properties (i)-(iv) were used in \cite{vir} to determine the
full relative Gromov-Witten descendents of target curves in terms
of the descendents of even classes.
The results of Section 5 of \cite{vir} are entirely formal and
apply verbatim to the descendent theory of $(0,0)$ relative geometries of
curves. Moreover, the rationality of the even theory implies
the rationality of the full descendent theory. \qed
\section{Denominators} \label{ennd}
\subsection{Summary} We prove the
denominator claims of Conjecture \ref{222}
when only descendents of $\mathsf{1}$ and $\mathsf{p}$
are present.
\begin{thm}
\label{2222}
If only descendents of even cohomology are considered,
the denominators of the degree $d$ descendent partition functions
$\ZZ$ of Theorems \ref{onnn}, \ref{tnnn}, and \ref{cnnn}
are products of factors of the form $q^k$ and
$$1-(-q)^r$$
for $1\leq r \leq d$.
\end{thm}
Theorem \ref{2222} is proven by carefully tracing
the denominators through the proofs of Theorems \ref{onnn}-\ref{cnnn}.
When the descendents of odd cohomology are included,
the strategy of Section 5 of \cite{vir} requires
matrix inversions{\footnote{Specifically,
the matrix associated to Lemma 5.6 of \cite{vir}
has an inverse with denominators we cannot
at present constrain.}} for which we can not control the
denominators.
Theorem \ref{2222} is new even when {\em no} descendents
are present. For the trivial bundle
$$N = \OO_{\mathbb{P}^1} \oplus \OO_{\mathbb{P}^1} \rightarrow
\mathbb{P}^1\ ,$$
the $T$-equivariant partition $\mathsf{Z}^{N/S}_{d,\eta^1,\eta^2,\eta^3}$
of Theorem \ref{tnnn}
is (up to $q$ shifts) equal to the 3-point
function $$\langle \eta^1, \eta^2,\eta^3 \rangle$$
in the quantum cohomology of the Hilbert scheme of points
of $\mathbb{C}^2$, see \cite{hilb1,lcdt}.
\noindent{\bf Corollary.} {\em The 3-point functions in the $T$-equivariant
quantum cohomology of $\text{Hilb}(\mathbb{C}^2,d)$
have possible poles in -q only at the $r^{th}$ roots of
unity for $r$ at most $d$.}
\begin{proof}
By Theorem \ref{2222}, we see the possible poles in $-q$
of the 3-point
functions are at $0$ and the $r^{th}$ roots of
unity for $r$ at most $d$.
By definition, the 3-point functions have
no poles at 0.
\end{proof}
\subsection{Denominators for Proposition \ref{cttt}}
We follow here the notation used in the proof of Proposition
\ref{cttt} in Section \ref{333}.
The matrix $\mathsf{S}_\eta^\mu$ is a fundamental solution of a
linear differential equation with singularities only at
0 and $r^{th}$ roots of unity for $r$ at most $d$, see
\cite{hilb2}. Hence, the poles in $-q$ of the evaluation
$$\mathsf{S}_\eta^\mu
|_{s_3=\frac{1}{a}(s_1+s_2)}$$
can occur only at $0$ and $r^{th}$ roots of unity for $r$ at most $d$.
The denominator claim of Theorem \ref{2222} for
Proposition \ref{cttt} then follows directly from the proof
in Section \ref{ggtt2}.
While only the rationality of Theorem \ref{canpole} is needed
in the proof of Proposition \ref{cttt}, the much stronger Laurent
{polynomiality}
of Theorem \ref{canpole} is used here.
\subsection{Denominators for $T$-equivariant stationary theory}
Consider the denominators of
\begin{equation*}
{\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}
\left( \prod_{j=1}^k \tau_{i_j}(\mathsf{p})
\right)^T \ .
\end{equation*}
The denominator result for the $T$-equivariant stationary theory of
the
$(0,0)$-cap is obtained from the denominator result for
Proposition \ref{cttt} by the specialization $s_3=0$.
By degenerating all the descendents $\tau_{i_j}(\mathsf{p})$
on to a $(0,0)$-cap, we need only study the denominators
of $T$-equivariant
partition functions ${\mathsf Z}^{N/S}_{d,\eta^1,\dots,\eta^r}$
with no descendent insertions.
The denominator result for
the $T$-equivariant $(a,b)$-tube with no descendents
is again a consequence
of the study of the fundamental solution in \cite{hilb2}.
By repeated degenerations (using the $(a,b)$-tube for the
twists in $N$), we need only study the
denominators
of $T$-equivariant
partition functions ${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$
with 3 relative insertions.
\subsection{Relative/descendent correspondence}
Relative conditions in the theory of local curves
were exchanged for descendents
in the proof of Lemma \ref{rtt5}.
For the denominator result for
${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$, we require a
more efficient correspondence.
\begin{prop}\label{gbb5}
Let $d>0$ be an integer.
The square matrix with coefficients
\begin{equation}\label{gredd}
\mathsf{Z}^{\mathsf{cap}}_{d,\lambda}\left( \tau_{\mu_1-1}([0])
\cdots \tau_{\mu_{\ell(\mu)}-1}([0]) \right)^T
\end{equation}
as $\lambda$ and $\mu$ vary among
partitions of $d$
\begin{enumerate}
\item[(i)] is triangular with respect to the partial ordering
by length,
\item[(ii)] has
diagonal entries given
by monomials in $q$,
\item[(iii)] and is of
maximal rank.
\end{enumerate}
\end{prop}
\begin{proof}
The Proposition follows from the
results of Section 4.6 of \cite{lcdt} applied to
the theory of stable pairs.
Our relative conditions $\lambda$ are defined
with identity weights in the $T$-equivariant cohomology
of $\mathbb{C}^2$. For the proof, we weight all the parts of
$\lambda$ with he $T$-equivariant class of the origin in
$\mathbb{C}^2$. Then, by compactness and dimension
constraints, the triangularity of the matrix is
immediate for partitions of different lengths.
On the diagonal, the expected dimension of
the integrals are 0. Using the compactification
\begin{equation}\label{jttm}
\mathbb{C}^2 \times \mathbb{P}^1 \subset \mathbb{P}^2 \times
\mathbb{P}^1
\end{equation}
as in Section 4.6 of \cite{lcdt}, we
obtain the triangularity of equal length partitions.
Consider the Hilbert scheme of points ${\text{Hilb}}(\com^2,d)$ of the
plane.
Let
${\mathbb{F}}$ be the universal quotient sheaf on
$$\text{Hilb}(\com^2,d) \times \com^2,$$ and define the
descendent{\footnote{The Chern character of $\mathbb{F}$
is properly supported over ${\text{Hilb}}(\com^2,d)$.} }
\begin{equation*}
\tau_k=\pi_*\Big( {\text{ch}}_{2+k}({\mathbb{F}})\Big)
\in A^{k}({\text{Hilb}}(\com^2,d),\mathbb{Q})
\end{equation*}
as before \eqref{mrr}.
Using the compactification \eqref{jttm}, we
reduce the calculation of the diagonal entries to
the pairing
\begin{equation}\label{appp}
s_1s_2\ \Big\langle \tau_{c-1} \ \Big| \ (c) \Big\rangle _
{{\text{Hilb}}(\com^2,d)}
= \frac{1}{c!}\
\end{equation}
which appears in \cite{parttwo}.
We conclude the diagonal entries do not vanish.
The diagonal entries are monomial in $q$ by the usual
vanishing obtained by the holomorphic symplectic form on $\mathbb{C}^2$.
\end{proof}
The denominator result holds for the
nonvanishing entries of the correspondence
matrix \eqref{gredd}. Since the matrix is triangular
with monomials in $q$ on the diagonal, the denominator
result holds for the {\em inverse} matrix.
We can now establish the denominator result for
the $T$-equivariant 3-point function
${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$.
We start with the descendent series
\begin{equation}\label{jj87}
{\mathsf{Z}}^{0,0}_{d,\eta^3}\left(
\tau_{\mu_1-1}(\mathsf{p})
\cdots \tau_{\mu_{\ell(\mu)}-1}(\mathsf{p})\cdot \tau_{\widehat{\mu}_1-1}(\mathsf{p})
\cdots \tau_{\widehat{\mu}_{\ell(\widehat{\mu})}-1}(\mathsf{p})\right)
\end{equation}
for partitions $\mu$ and $\widehat{\mu}$ of $d$.
The denominator result holds for all
series \eqref{jj87}. By bubbling all the
descendents $\tau_{\mu_i-1}(\mathsf{p})$ off of the point $0\in \mathbb{P}^1$
and
bubbling all the
descendents $\tau_{\widehat{\mu}_i-1}(\mathsf{p})$ off of the point $1\in \mathbb{P}^1$, we conclude the
denominator result for ${\mathsf Z}^{(0,0)}_{d,\eta^1,\eta^2,\eta^3}$
from the denominator result for the
inverse of the correspondence matrix \eqref{gredd}.
\subsection{Denominators for Theorems \ref{tnnn}-\ref{cnnn}}
The denominator result for Theorem \ref{cnnn} is obtained by following
the proof given in Sections \ref{depp}-\ref{444}. An important point is
to replace the matrix $M_d$ appearing in the proof of
Lemma \ref{rtt5} with the correspondence matrix \eqref{gredd}. The required
matrix inversion then keeps the denominator form.
The rest of the proof of Theorem \ref{cnnn} respects the
denominators.
Proposition \ref{pnnn} is the statement of Theorem \ref{tnnn} for descendents
of even cohomology. Again, the proof respects the denominators.
The proof of Theorem \ref{2222} is complete.\qed
\noindent
Department of Mathematics\\
Princeton University\\
[email protected]
\noindent
Department of Mathematics\\
Princeton University\\
[email protected]
\end{document}
|
\begin{document}
\title[Large-time asymptotics]{Large-time asymptotics for degenerate \\
cross-diffusion population models \\
with volume filling}
\author[X. Chen]{Xiuqing Chen}
\address{School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082,
Guang\-dong Province, China}
\email{[email protected]}
\author[A. J\"ungel]{Ansgar J\"ungel}
\address{Institute of Analysis and Scientific Computing, Technische Universit\"at Wien,
Wiedner Hauptstra\ss e 8--10, 1040 Wien, Austria}
\email{[email protected]}
\author[X. Lin]{Xi Lin}
\address{Department of Mathematics and Physics, Guangzhou Maritime University, Guangzhou 510765,
Guang\-dong Province, China}
\email{[email protected]}
\author[L. Liu]{Ling Liu}
\address{School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082,
Guang\-dong Province, China}
\email{[email protected]}
\date{\today}
\thanks{The first, third, and fourth authors acknowledge support from the National Natural
Science Foundation of China (NSFC), grant 11971072.
The second author acknowledges partial support from the Austrian Science Fund (FWF), grants P33010 and F65. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, ERC Advanced Grant no.~101018153.}
\begin{abstract}
The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including ``iterated'' degenerated functions.
\end{abstract}
\keywords{Degenerate parabolic equations, cross-diffusion systems, entropy method, large-time asymptotics.}
\subjclass[2000]{35K51, 35K59, 35K65, 35Q92, 92D25.}
\maketitle
\section{Introduction}
The aim of this note is to extend the large-time asymptotics result of \cite{ZaJu17} on multi-species cross-diffusion systems with volume-filling effects to the degenerate case. Such systems describe, for instance, the spatial segregation of population species \cite{SKT79}, chemotactic cell migration in tissues \cite{Pai09}, motility of biological cells \cite{SLH09}, or ion transport in fluid mixtures \cite{BSW12}. The main difficulties of the cross-diffusion systems are the lack of positive semidefiniteness of the diffusion matrix and the nonstandard degeneracies. The first issue was overcome by applying the boundedness-by-entropy method \cite{Jue15}, which exploits the underlying entropy (or formal gradient-flow) structure. This allows for both a global existence analysis and the proof of lower and upper bounds, without the use of a maximum principle. The second issue was handled by extending the Aubin--Lions compactness lemma \cite{ZaJu17}. However, the large-time asymptotics in \cite{ZaJu17} only holds if the problem is not degenerate. In the present note, we remove this restriction.
The evolution of the volume fraction $u_i(x,t)$ of the $i$th species is given by
\begin{align}\label{1.eq}
& \pa_t u_i = \operatorname{div}\sum_{j=1}^n A_{ij}(u)\na u_j\quad\mbox{in }\Omega,\ t>0,\ i=1,\ldots,n \\
& \sum_{j=1}^n A_{ij}(u)\na u_j\cdot\nu = 0\quad\mbox{on }\pa\Omega, \quad
u_i(\cdot,0) = u_i^0\quad\mbox{in }\Omega, \label{1.bic}
\end{align}
where $u_0=1-\sum_{i=1}^n u_i$ is the solvent volume fraction or the proportion of unoccupied space (depending on the application), $\Omega\subset{\mathbb R}^d$ ($d\ge 1$) is a bounded domain with Lipschitz boundary, $\nu$ is the exterior unit normal vector to $\pa\Omega$, and the diffusion coefficients are given by
\begin{equation}\label{1.A}
A_{ij}(u) = D_ip_i(u)q(u_0)\delta_{ij} + D_iu_ip_i(u)q'(u_0)
+ D_iu_i q(u_0)\frac{\pa p_i}{\pa u_j}(u),
\end{equation}
where $i,j=1,\ldots,n$, $u=(u_1,\ldots,u_n)$ is the solution vector, $D_i>0$ are the diffusivities, $\delta_{ij}$ denotes the Kronecker symbol, and $p_i$ and $q$ are smooth functions. In particular, the bounds $0\le u_i\le 1$ should hold for all $i=0,\ldots,n$. The boundary condition in \eqref{1.bic} means that the physical or biological system is isolated.
We note that equations \eqref{1.eq} and \eqref{1.A} can be written as
\begin{align}\label{1.eq2}
\pa_t u_i = D_i\operatorname{div}\bigg(u_ip_i(u)q(u_0)\na\log\frac{u_ip_i(u)}{q(u_0)}\bigg)
= D_i\operatorname{div}\bigg(q(u_0)^2\na{\frac{u_ip_i(u)}{q(u_0)}}
\bigg).
\end{align}
In some applications, drift or reaction terms need to be added; see, e.g., \cite{BDPS10,GeJu18} for systems with drift terms and \cite{DJT20} for reaction rates.
Equations \eqref{1.eq} and \eqref{1.A} can be formally derived from a random-walk lattice model in the diffusion limit \cite[Appendix A]{ZaJu17}. The functions $p_i$ and $q$ are related to the transition rates of the lattice model with $p_i$ measuring the occupancy and $q$ measuring the non-occupancy. This class of systems contains the population model of Shigesada, Kawasaki, and Teramoto \cite{SKT79} (if $p_i$ is a linear function and $q=1$) and Nernst--Planck-type equations accounting for finite ion sizes (if $p_i=1$ and $q(u_0)=u_0$; see \cite{GeJu18}).
In this note, we consider the degenerate case $q'(0)=0$ and assume that there exists a smooth function $\chi$ such that $p_i=\exp(\pa\chi/\pa u_i)$ to guarantee an entropy structure via the entropy density
\begin{equation}\label{1.h}
h(u) = \sum_{i=1}^n (u_i(\log u_i-1)+1) + \int_1^{u_0}\log q(s)ds + \chi(u),
\end{equation}
where $u\in\mathcal{D}:=\{u\in(0,1)^n:\sum_{i=1}^n u_i<1\}$.
There exist other approaches to model volume filling. The finite particle size may be taken into account by adding cross-diffusion terms of the type $u_i\na\sum_{j=1}^n b_{ij}u_j$ to the standard Nernst--Planck flux \cite{Hsi19} or by using the Bikerman-type flux $J_i=-D_i(\na u_i-u_i\na\log u_0)$ in the mass conservation equation $\pa_t u_i+\operatorname{div} J_i=0$ \cite{Bik42}.
The global existence of bounded weak solutions to \eqref{1.eq}--\eqref{1.A} was shown in \cite[Theorem 1]{ZaJu17} assuming $D_i=1$ for $i=1,\ldots,n$ and the following conditions:
\begin{itemize}
\item[\bf (H1)] Domain: $\Omega\subset{\mathbb R}^d$ ($d\ge 1$) is a bounded convex domain with Lipschitz boundary, $T>0$. Set $\mathcal{D}=\{u\in(0,1)^n:\sum_{i=1}^n u_i<1\}$ and $\Omega_T=\Omega\times(0,T)$.
\item[\bf (H2)] Initial datum: $u^0(x)\in\mathcal{D}$ for a.e.\ $x\in\Omega$ and $h(u^0)\in L^1(\Omega)$.
\item[\bf (H3)] Functions $p_i$: $p_i=\exp(\pa\chi/\pa u_i)$, where $\chi\in C^3(\overline{\mathcal{D}})$ is convex.
\item[\bf (H4)] Function $q$: $q\in C^3([0,1])$ satisfies $q(0)=0$, $q(1)=1$, $q'(0)\ge 0$ and $q(s)>0$, $q'(s)>0$ for all $0<s\le 1$.
\end{itemize}
The convexity of $\Omega$ in Hypothesis (H1) is used for the convex Sobolev inequality; see Lemma \ref{lem.csi} below. For generalized Nernst--Planck systems with $p_i=\mbox{const.}$, we may choose $\chi(u)=\sum_{i=1}^n u_i$, which satisfies Hypothesis (H3). Moreover, if $p_i(u)=P_i(u_i)$ for some functions $P_i:[0,1]\to[0,\infty)$, condition $p_i=\exp(\pa\chi/\pa u_i)$ is satisfied with $\chi(u)=\sum_{i=1}^n\chi_i(u_i)$ and $\chi_i(s)=\int_0^s\log P_i(\tau)d\tau$. The functions $q(s)=s^\alpha$ with $\alpha\ge 1$ satisfy Hypothesis (H4).
We claim that the existence result also holds for arbitrary $D_i>0$. Indeed, it is sufficient to define $\widetilde\chi(u)=\chi(u)+\sum_{j=1}^n u_j\log D_j$, since $\exp(\pa\widetilde\chi/\pa u_i)=D_i\exp(\pa\chi/\pa u_i)=D_ip_i$, and we can apply Theorem 1 in \cite{ZaJu17} with $\widetilde\chi$. We observe that the condition $q'(s)/q(s)\ge c_1>0$ in \cite{ZaJu17} is not needed for the existence analysis.
The weak solution $u=(u_1,\ldots,u_n)$ to \eqref{1.eq}--\eqref{1.A} satisfies $u(x,t)\in\mathcal{D}$ for a.e.\ $(x,t)\in\Omega_T$, mass conservation, the regularity
\begin{align*}
& \sqrt{q(u_0)},\ \sqrt{q(u_0)u_i}\in L^2(0,T;H^1(\Omega)), \quad
\sqrt{q(u_0)}\na u_i\in L^2(\Omega_T), \\
& \pa_t u_i\in L^2(0,T;H^1(\Omega)') \quad\mbox{for }i=,1\ldots,n,
\end{align*}
and the weak formulation
\begin{align*}
\int_0^T\langle\pa_t u_i,\phi_i\rangle dt
= -\int_0^T\int_\Omega D_i \sqrt{q(u_0)}\big[\na\big(u_ip_i(u)\sqrt{q(u_0)}\big)
- 3u_ip_i(u)\na\sqrt{q(u_0)}\big]\cdot\na\phi_i dxdt
\end{align*}
for all $\phi_i\in L^2(0,T;H^1(\Omega))$, $i=1,\ldots,n$, where $\langle\cdot,\cdot\rangle$ denotes the duality product of $H^1(\Omega)'$ and $H^1(\Omega)$. Moreover, the initial datum in \eqref{1.bic} is satisfied in the sense of $H^1(\Omega)'$ and the entropy inequality
\begin{align}\label{1.ei}
\int_\Omega h(u(t))dx
+ c_0\int_s^t\int_\Omega\bigg(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2
+ |\na\sqrt{q(u_0)}|^2\bigg)dxdr \le \int_\Omega h(u(s))dx,
\end{align}
holds for $0\le s<t$, $t>0$ for some $c_0>0$ depending on $D_i$, $p_i$, and $q$, recalling definition \eqref{1.h} of $h(u)$. The $L^\infty(\Omega_T)$ bound for $u_i$ and the $L^2(\Omega_T)$ for $\sqrt{q(u_0)}\na u_i$ imply that $\na(u_ip_i(u)\sqrt{q(u_0)})\in L^2(\Omega_T)$, so that the weak formulation is well defined.
Our main result is the convergence of the solutions to \eqref{1.eq}--\eqref{1.A} towards the constant steady state
$$
u_i^\infty = \frac{1}{|\Omega|}\int_\Omega u_i^0dx
\quad\mbox{for }i=1,\ldots,n,
\quad u_0^\infty= 1 - \sum_{i=1}^n u_i^\infty
$$
for large times under the following additional hypothesis:
\begin{itemize}
\item[\bf (H5)] $q$ is convex, $q/q'$ is concave, and there exist $\beta\in[0,1]$, $c_1>0$ such that
$$
\lim_{s\to 0} \frac{s^\beta q'(s)}{q(s)}=c_1>0.
$$
\end{itemize}
Examples of functions satisfying Hypothesis (H5) are $q(s)=s^\alpha$ with $\alpha\ge 1$.
The convergence (with exponential decay rate) was proved in \cite{ZaJu17} for the nondegenerate case $q'(0)>0$ only. In the degenerate situation $q'(0)=0$, the numerical results of \cite{GeJu18} indicate that exponential rates cannot be expected. Therefore, we show the convergence without rate.
\begin{theorem}[Large-time asymptotics]\label{thm.time}
Let Hypotheses (H1)--(H5) hold and let $u=(u_1,$ $\ldots,u_n)$ be a weak solution to \eqref{1.eq}--\eqref{1.A} satisfying the entropy inequality \eqref{1.ei}. Then $u_i(t)\to u_i^\infty$ strongly in $L^p(\Omega)$ as $t\to\infty$ for all $i=1,\ldots,n$ and $1\le p<\infty$.
\end{theorem}
The idea of the proof is to exploit, as in \cite{ZaJu17}, the relative entropy density (or Bregman distance)
\begin{equation}\label{1.hstar}
h^*(u|u^\infty) = h(u) - h(u^\infty) - h'(u^\infty)\cdot(u-u^\infty),
\end{equation}
where $u=(u_1,\ldots,u_n)$ is the weak solution to \eqref{1.eq}--\eqref{1.A}. The entropy inequality implies that
$$
\frac{dh^*}{dt}(u|u^\infty)
+ \frac{c_0}{2}\int_\Omega\sum_{i=1}^n|\na\sqrt{q(u_0)u_i}|^2 dx \le 0.
$$
Unfortunately, the entropy production integral cannot be estimated in terms of the relative entropy directly by applying a logarithmic Sobolev inequality to $u_i$. We overcome this issue by using two ideas.
First, we apply the logarithmic Sobolev inequality to $\sqrt{q(u_0)u_i}$,
$$
\int_\Omega q(u_0)u_i\log\frac{q(u_0)u_i}{|\Omega|^{-1}\int_\Omega q(u_0)u_idx}dx
\le C\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dx.
$$
The idea is to relate the integrand of the left-hand side to the relative entropy part
$h_1^*(u|u^\infty)=\sum_{i=1}^n(u_i\log(u_i/u_i^\infty)-u_i+u_i^\infty)dx$. For this, we define
$$
f_1(u) = \sum_{i=1}^n\bigg(q(u_0)u_i\log\frac{q(u_0)u_i}{|\Omega|^{-1}\int_\Omega q(u_0)u_idx}
- q(u_0)u_i + \frac{1}{|\Omega|}\int_\Omega q(u_0)u_idx\bigg).
$$
Since $\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dxdt<\infty$, we also have
$\int_0^\infty\int_\Omega f_1(u)dxdt<\infty$, and there exists a subsequence $t_k\to\infty$
such that $f_1(u(t_k))\to 0$. The key result is the limit (see Lemma \ref{lem.key})
$$
\lim_{t_k\to\infty}\bigg(\frac{f_1(u(t_k))}{|\Omega|^{-1}\int_\Omega q(u_0(t_k))dx}
- h_1^*(u(t_k)|u^\infty)\bigg) = 0.
$$
This result shows that $h_1^*(u(t_k)|u^\infty)\to 0$ as $t_k\to\infty$.
Second, instead of the part $h_2^*(u|u^\infty)=\int_{u_0^\infty}^{u_0}\log(q(s)/q(u_0^\infty))ds$ of the relative entropy density, we analyze the function
$$
f_2(u_0) = \int_{\bar{q}}^{q(u_0)}\log\frac{q(s)}{q(\bar{q})}ds,
$$
where $\bar{q}:=|\Omega|^{-1}\int_\Omega q(u_0)dx$, which can be seen as an ``iterated'' version of $h_2^*(u|u^\infty)$, since it involves $q\circ q$ instead of $q$. Then an application of the convex Sobolev inequality yields a bound for the integral over $|\na\sqrt{q(u_0)}|^2$ without the need of condition $q'(0)>0$; see Remark \ref{rem.f2} for details. It follows from $\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)}|^2 dxdt<\infty$ that $\int_0^\infty\int_\Omega f_2(u)dxdt<\infty$, and there exists a subsequence $t_k\to\infty$ such that $f_2(u(t_k))\to 0$.
The convergences $f_1(u(t_k))\to 0$ and $f_2(u(t_k))\to 0$ as well as the monotonicity of the entropy imply that $h^*(u(t_k)|u^\infty) \to 0$ pointwise. The monotonicity of $t\mapsto \int_\Omega h^*(u(t)|u^\infty)dx$ then implies the convergence for all sequences $t\to\infty$ and finally $u_i(t)\to u_i^\infty$ strongly in $L^2(\Omega)$.
To conclude the introduction, we mention some results on the large-time asymptotics for diffusion systems. Exponential equilibration rates in $L^p(\Omega)$ norms were shown for reaction-diffusion systems in \cite{DeFe06,DFM08}, for electro-reaction-diffusion systems in \cite{GlHu97}, and for Maxwell--Stefan systems for chemically reacting fluids in \cite{DJT20,JuSt13}. The convergence to equilibrium was proved for Shigesada--Kawasaki--Teramoto cross-diffusion systems without rate in \cite{Shi06}, for instance. All these results concern nondegenerate diffusion equations. The work \cite{BDPS10} is concerned with the large-time asymptotics for systems like \eqref{1.eq} with $D_i=p_i=1$ and $q(u_0)=u_0$ without rate.
The asymptotics for solutions to Poisson--Nernst--Planck-type equations with quadratic nonlinearity was investigated in \cite{Zin16} using Wasserstein techniques. Decay rates for degenerate diffusion systems without cross-diffusion terms were derived in \cite{CJMTU01}.
An extension of our results to cross-diffusion systems with drift or reactions seems delicate; see Remark \ref{rem.drift} for drift terms and \cite{DJT20} for cross-diffusion systems with reversible reactions.
\section{Proof of Theorem \ref{thm.time}}
We first recall the convex Sobolev inequality; see \cite[Lemma 11]{ZaJu17}.
\begin{lemma}[Convex Sobolev inequality]\label{lem.csi}
Let $\Omega\subset{\mathbb R}^d$ ($d\ge 1$) be a convex domain and let $g\in C^4({\mathbb R})$ be convex such that $1/g''$ is concave. Then there exists $C_S>0$ such that for all $v\in L^1(\Omega)$ such that $g(v)$, $g''(v)|\na u|^2\in L^1(\Omega)$,
$$
\frac{1}{|\Omega|}\int_\Omega g(v)dx - g\bigg(\frac{1}{|\Omega|}\int_\Omega vdx\bigg)
\le \frac{C_S}{|\Omega|}\int_\Omega g''(v)|\na v|^2 dx.
$$
\end{lemma}
The logarithmic Sobolev inequality is obtained for the choice $g(v)=v(\log v-1)+1$:
\begin{equation}\label{3.lsi}
\int_\Omega v\log\frac{v}{|\Omega|^{-1}\int_\Omega vdx}dx
\le 4C_S\int_\Omega|\na\sqrt{v}|^2 dx
\end{equation}
and for functions $\sqrt{v}\in H^1(\Omega)$.
Since $h(u^\infty)$ is independent of time (because of mass conservation), the entropy inequality \eqref{1.ei} implies the relative entropy inequality
\begin{align}\label{3.ei}
\int_\Omega h^*(u(t)|u^\infty)dx &+ c_0\int_s^t\int_\Omega
\bigg(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2 + |\na\sqrt{q(u_0)}|^2\bigg)dxdr \\
&\le \int_\Omega h^*(u(s)|u^\infty)dx \nonumber
\end{align}
for $0\le s<t$ and $t>0$, where $h^*(u|u^\infty)$ is defined in \eqref{1.hstar}. As mentioned in the introduction, we cannot apply the logarithmic Sobolev inequality \eqref{3.lsi} with $v=u_i$ since $q(u_0)=0$ for $u_0=0$. Instead we apply this inequality to $v=q(u_0)u_i$.
We split the relative entropy density $h^*$ into three parts, $h^*=h_1^* + h_2^* + h_3^*$, where
\begin{align*}
h_1^*(u|u^\infty) &= \sum_{i=1}^n\bigg(u_i\log\frac{u_i}{u_i^\infty}
- u_i + u_i^\infty\bigg), \\
h_2^*(u|u^\infty) &= \int_{u_0^\infty}^{u_0}\log\frac{q(s)}{q(u_0^\infty)}ds, \\
h_3^*(u|u^\infty) &= \chi(u) - \chi(u^\infty) - \sum_{i=1}^n(u_i-u_i^\infty)\log p_i(u^\infty),
\end{align*}
where $\chi$ is introduced in Hypothesis (H3).
\begin{lemma}
The functions $h_i^*(\cdot|u^\infty)$, $i=1,2,3$, are nonnegative and bounded on $\overline\mathcal{D}$.
\end{lemma}
\begin{proof}
The function $h_1^*$ is bounded since $u_i\mapsto u_i\log u_i$ is bounded for $0\le u_i\le 1$, and $h_3^*$ is bounded thanks to Hypothesis (H3) on $p_i$. Integrating by parts in $h_2^*(u|u^\infty)$ and observing that $u_0\log q(u_0)\le 0$, we find that
\begin{align}\label{3.aux}
h_2^*(u|u^\infty) = u_0\log\frac{q(u_0)}{q(u_0^\infty)} - \int_{u_0^\infty}^{u_0}
s\frac{q'(s)}{q(s)}ds \le -\log q(u_0^\infty) + \int_0^1 s\frac{q'(s)}{q(s)}ds.
\end{align}
By Hypothesis (H5), $\lim_{s\to 0} sq'(s)/q(s) = \lim_{s\to 0}s^{1-\beta}\cdot s^\beta q'(s)/q(s)$ is finite (here, we need $\beta\le 1$). Therefore, $s\mapsto sq'(s)/q(s)$ is bounded on $[0,\delta]$ for some $\delta>0$. On the other hand, $s\mapsto sq'(s)/q(s)$ is also bounded on $[\delta,1]$ since this function is continuous and $q(s)>0$ for $s>0$ is nondecreasing. This shows that $\int_0^1 (sq'(s)/q(s))ds$ is bounded, proving the claim.
\end{proof}
\subsection{Study of some auxiliary functions}
The study of the large-time behavior is based on the analysis of the two functions
\begin{align}\label{3.f}
f_1(u) = \sum_{i=1}^n\bigg(q(u_0)u_i\log\frac{q(u_0)u_i}{\bar{q}_i} - q(u_0)u_i
+ \bar{q}_i\bigg), \quad
f_2(u_0) = \int_{\bar{q}}^{q(u_0)}\log\frac{q(s)}{q(\bar{q})}ds,
\end{align}
for $u\in\overline{\mathcal{D}}$, where
\begin{equation}\label{3.qbar}
\bar{q} = \frac{1}{|\Omega|}\int_\Omega q(u_0)dx, \quad
\bar{q}_i = \frac{1}{|\Omega|}\int_\Omega q(u_0)u_idx.
\end{equation}
\begin{lemma}\label{lem.f}
The function $f_1$ is nonnegative, and the function $f_2$ is nonnegative and bounded on $\overline\mathcal{D}$.
\end{lemma}
\begin{proof}
Set $z=q(u_0)u_i/\bar{q}_i$ and let $u\in\overline{\mathcal{D}}$. Then
$$
f_1(u) = \sum_{i=1}^n\bar{q}_i(z\log z - z+1)\ge 0,
$$
proving the first claim. To show the nonnegativity of $f_2$, we distinguish two cases. If $q(u_0(x,t))\ge\bar{q}$ at some $(x,t)\in\Omega_T$, then $\log(q(s)/q(\bar{q}))\ge 0$ for any $\bar{q}\le s\le q(u_0(x,t))$ and consequently $f_2(u(x,t))\ge 0$. If $q(u_0(x,t))<\bar{q}$, we have
$\log(q(s)/q(\bar{q})) < 0$ for $q(u_0(x,t))\le s\le\bar{q}$ and
$f_2(u_0(x,t)) = \int_{q(u_0(x,t))}^{\bar{q}}\log(q(\bar{q})/q(s))ds \ge 0$.
It remains to show that $f_2$ is bounded. Since $q$ is convex, Jensen's inequality shows that $\bar{q}\ge q(|\Omega|^{-1}\int_\Omega u_0dx)=q(u_0^\infty)$.
Then, using integration by parts and arguing as in \eqref{3.aux},
\begin{align*}
f_2(u_0) &= q(u_0)\log\frac{q(q(u_0))}{q(\bar{q})}
- \int_{\bar{q}}^{q(u_0)} s\frac{q'(s)}{q(s)}ds
\le -q(u_0)\log q(\bar{q}) + \int_0^1 s\frac{q'(s)}{q(s)}ds \\
&\le -\log q(q(u_0^\infty)) + \int_0^1 s\frac{q'(s)}{q(s)}ds.
\end{align*}
We already showed above that the last integral is bounded. This finishes the proof.
\end{proof}
\subsection{Convergence of $f_1$ and $f_2$}
\begin{lemma}\label{lem.convf}
It holds for a.e.\ $x\in\Omega$, $s\in(0,1]$ that
$$
\lim_{N\to\infty}f_1(u(x,s+N)) = 0, \quad \lim_{N\to\infty}f_2(u_0(x,s+N)) = 0.
$$
\end{lemma}
\begin{proof}
The idea is to exploit the boundedness of the entropy production integrated over $t\in(0,\infty)$. First, we consider $f_1$. We know from \eqref{3.ei} for $s=0$ and $t\to\infty$ that
\begin{equation}\label{3.infty}
c_0\int_0^\infty\int_\Omega\bigg(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2
+ |\na\sqrt{q(u_0)}|^2\bigg)dxdt \le \int_\Omega h^*(u^0|u^\infty)dx.
\end{equation}
Thus, in view of $q(u_0)u_i\ge 0$ and
\begin{align*}
|\na\sqrt{q(u_0)u_i}|^2 &= q(u_0)|\na\sqrt{u_i}|^2
+ 2\sqrt{q(u_0)u_i}\na\sqrt{q(u_0)}\cdot\na\sqrt{u_i} + u_i|\na\sqrt{q(u_0)}|^2 \\
&\le 2q(u_0)|\na\sqrt{u_i}|^2 + 2|\na\sqrt{q(u_0)}|^2,
\end{align*}
it follows for a constant $C>0$ being independent of time that
\begin{equation*}
\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dxdt \le C.
\end{equation*}
Furthermore, by the logarithmic Sobolev inequality \eqref{3.lsi}, applied to $v=q(u_0)u_i$,
$$
\int_0^\infty\int_\Omega q(u_0)u_i\log\frac{q(u_0)u_i}{\bar{q}_i}dx
\le C\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)u_i}|^2 dx \le C,
$$
recalling definition \eqref{3.qbar} of $\bar{q}_i$. Taking into account definition \eqref{3.f} of $f_1$, we see that
$$
\int_0^\infty\int_\Omega f_1(u(x,t))dxds
= \sum_{N=0}^{\infty}\int_0^1\int_\Omega f_1(u(x,s+N))dx ds < \infty.
$$
Therefore, the sequence $N\mapsto \int_0^1\int_\Omega f_1(u(\cdot,s+N))dx ds$ converges to zero,
$$
\lim_{N\to\infty}f_1(u(x,s+N)) = 0\quad\mbox{for a.e. }x\in\Omega,\ s\in(0,1].
$$
Next, we prove the limit for $f_2$. For any fixed $t>0$, we introduce the nonnegative function
$$
f(s;t) = \int_{\bar{q}(t)}^s\log\frac{q(\sigma)}{q(\bar{q}(t))}d\sigma, \quad 0<s\le 1.
$$
By Lemma \ref{lem.f}, $x\mapsto f(q(u_0(x,t));t) = f_2(u(x,t))$ is integrable in $\Omega$
for any fixed $t>0$. Moreover, $f(\cdot,t)$ is twice differentiable in $(0,1)$:
$$
\frac{df}{ds}(s;t) = \log\frac{q(s)}{q(\bar{q}(t))}, \quad
\frac{d^2 f}{ds^2}(s;t) = \frac{q'(s)}{q(s)} > 0.
$$
We infer from the positivity of $d^2f/ds^2$ that $f(\cdot,t)$ is convex. By Hypothesis (H5), $(d^2f/ds^2)^{-1} = q/q'$ is concave. Thus, the assumptions of the convex Sobolev inequality (Lemma \ref{lem.csi}) are satisfied for $f(q(u_0(x,t));t)$:
\begin{align*}
\frac{1}{|\Omega|}\int_\Omega & f(q(u_0(x,t));t)dx
- f\bigg(\frac{1}{|\Omega|}\int_\Omega q(u_0(x,t))dx;t\bigg) \\
&\le C(\Omega)\int_\Omega \frac{q'(q(u_0(x,t)))}{q(q(u_0(x,t)))}|\na q(u_0)|^2 dx.
\end{align*}
Hence, since $f(\bar{q}(t);t)=0$ by definition and recalling that $f(q(u_0(x,t));t) = f_2(u_0(x,t))$, the previous inequality becomes
\begin{equation}\label{2.f2est}
\int_\Omega f_2(u_0)dx
\le C(\Omega)\int_\Omega\frac{q(u_0)q'(q(u_0))}{q(q(u_0))}\frac{|\na q(u_0)|^2}{q(u_0)}dx
\le C\int_\Omega|\na\sqrt{q(u_0)}|^2 dx,
\end{equation}
where we used Hypothesis (H5) to infer that
$$
\frac{s q'(s)}{q(s)} = s^{1-\beta}\frac{s^\beta q'(s)}{q(s)}\quad\mbox{with }
s = q(u_0)
$$
is bounded in $[0,1]$. By \eqref{3.infty}, the integrated entropy dissipation is finite:
$$
\int_0^\infty\int_\Omega f_2(u_0)dxdt \le C\int_0^\infty\int_\Omega|\na\sqrt{q(u_0)}|^2
dxdt \le C.
$$
Therefore, arguing as for the function $f_1$, we obtain
$\lim_{N\to\infty}f_2(u_0(x,s+N)) = 0$ for a.e.\ $x\in\Omega$, $s\in(0,1]$,
which finishes the proof.
\end{proof}
\begin{remark}\label{rem.f2}\rm
In the nondegenerate case $q'(0)>0$, it was shown in \cite[Section 5]{ZaJu17} that $t\mapsto
h_2^*(u(t)|u^\infty)$ converges to zero exponentially fast. Indeed, applying the convex Sobolev inequality similarly as in the previous proof,
\begin{equation}\label{2.h2est}
\int_\Omega h_2^*(u|u^\infty)dx \le C\int_\Omega\frac{q'(u_0)}{q(u_0)}|\na u_0|^2 dx
= 4C\int_\Omega\frac{|\na\sqrt{q(u_0)}|^2}{q'(u_0)} dx,
\end{equation}
and we conclude from the entropy inequality \eqref{1.ei} and Gronwall's lemma. Since we allow for $q'(0)=0$, this argument cannot be used here. We solve this issue by considering the ``iterated'' function $f_2$ involving $q\circ q$ and assuming that $s\mapsto sq'(s)/q(s)$ is bounded; see \eqref{2.f2est}. The iterated use of $q$ gives the term $|\na\sqrt{q(u_0)}|^2$ in \eqref{2.f2est} without requiring the nondegeneracy condition $q'(0)>0$.
\qed\end{remark}
A consequence of the limit for $f_2$ is the following result.
\begin{lemma}\label{lem.conv1}
If $\lim_{N\to\infty}f_2(u_0(x,s+N))=0$ for some $x\in\Omega$, $s\in(0,1]$ then
$$
\lim_{N\to\infty}\frac{q(u_0(x,s+N))}{\bar{q}(s+N)} = 1.
$$
\end{lemma}
\begin{proof}
We write $u_i^N:=u_i(x,s+N)$ and $\bar{q}^N=\bar{q}(s+N)$
to simplify the notation. We recall from Lemma \ref{lem.f} that $f_2$ is nonnegative and change the variable $\sigma=s/\bar{q}^N$ in the integral:
\begin{align*}
f_2(u_0^N) &= \int_{\bar{q}^N}^{q(u_0^N)}\log\frac{q(s)}{q(\bar{q}^N)}ds
= \bar{q}^N\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma \\
&\ge q(u_0^\infty)\int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma,
\end{align*}
where we used Jensen's inequality to find that $\bar{q}^N\ge q(|\Omega|^{-1}\int_\Omega u_0^N dx) = q(u_0^\infty)$. Moreover, since $\bar{q}^N\le 1$,
$$
q(u_0^\infty)\int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
\le f_2(u_0^N) \le \int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma.
$$
This shows that $\lim_{N\to\infty}f_2(u_0^N)=0$ if and only if
\begin{equation}\label{3.con}
\lim_{N\to\infty}\int_1^{q(u_0^N)/\bar{q}^N}
\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma = 0.
\end{equation}
Set $A:=\{(x,s)\in\Omega\times(0,1]:\lim_{N\to\infty} f_2(u_0(x,s+N))=0\}$. We want to show that $\lim_{N\to\infty} q(u_0^N)/\bar{q}^N=1$ for $(x,s)\in A$. If not, there exist $(x_0,s_0)\in A$ and $\eps_0>0$ such that either
$$
\frac{q(u_0^N)}{\bar{q}^N} > 1+\eps_0 \quad\mbox{or}\quad
\frac{q(u_0^N)}{\bar{q}^N} < 1-\eps_0\quad\mbox{for all }N\in{\mathbb N}.
$$
In the former case, we have $q(\bar{q}^N\sigma)\ge q(\bar{q}^N(1+\eps_0/2))$
for $\sigma\ge 1+\eps_0/2$, since $q$ is increasing, and therefore,
\begin{equation}\label{3.aux2}
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
\ge \int_{1+\eps_0/2}^{1+\eps_0}\log\frac{q(\bar{q}^N(1+\eps_0/2))}{q(\bar{q}^N)}d\sigma.
\end{equation}
Using the convexity of $q$, a Taylor expansion shows that
$q(\bar{q}^N + \bar{q}^N\eps_0/2) \ge q(\bar{q}^N) + q'(\bar{q}^N)\bar{q}^N\eps_0/2$.
Then the integrand of the previous integral can be estimated according to
$$
\log\bigg(\frac{q(\bar{q}^N(1+\eps_0/2))}{q(\bar{q}^N)}\bigg)
\ge \log\bigg(1 + \frac{q'(\bar{q}^N)}{q(\bar{q}^N)}\bar{q}^N\frac{\eps_0}{2}\bigg)
\ge \log\bigg(1 + c_0q(u_0^\infty)^{1-\beta}\frac{\eps_0}{2}\bigg),
$$
where we used Hypothesis (H5) and $\bar{q}^N\ge q(u_0^\infty)$ in the last step, and $c_0>0$ is some constant. As the right-hand side is independent of $\sigma$, we infer from \eqref{3.aux2} that
$$
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
\ge \frac{\eps_0}{2}\log\bigg(1 + c_0q(u_0^\infty)^{1-\beta}\frac{\eps_0}{2}\bigg).
$$
In the latter case $q(u_0^N)/\bar{q}^N<1-\eps_0$, we estimate as
\begin{align*}
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma
&= \int_{q(u_0^N)/\bar{q}^N}^1\log\frac{q(\bar{q}^N)}{q(\bar{q}^N\sigma)}d\sigma \\
&\ge \int_{1-\eps_0}^{1-\eps_0/2}\log\frac{q(\bar{q}^N)}{q(\bar{q}^N(1-\eps_0/2)}d\sigma.
\end{align*}
We apply again a Taylor expansion, similarly as in the first case,
$$
q(\bar{q}^N) = q\bigg(\bar{q}^N\bigg(1-\frac{\eps_0}{2}\bigg)
+ \frac{\eps_0}{2}\bar{q}^N\bigg)
\ge q\bigg(\bar{q}^N\bigg(1-\frac{\eps_0}{2}\bigg)\bigg)
+ q'\bigg(\bar{q}^N\bigg(1-\frac{\eps_0}{2}\bigg)\bigg)\frac{\eps_0}{2}\bar{q}^N,
$$
which leads to
$$
\log\frac{q(\bar{q}^N)}{q(\bar{q}^N(1-\eps_0/2)}
\ge \log\bigg(1 + \frac{q'(\bar{q}^N(1-\eps_0/2))}{q(\bar{q}^N(1-\eps_0/2))}
\frac{\eps_0}{2}\bar{q}^N\bigg)
\ge \log\bigg(1 + c_0q(u_0^\infty)^{1-\beta}\frac{\eps_0}{2}\bigg).
$$
Thus, in both cases,
$$
\int_1^{q(u_0^N)/\bar{q}^N}\log\frac{q(\bar{q}^N\sigma)}{q(\bar{q}^N)}d\sigma > 0
\quad\mbox{uniformly in }N\in{\mathbb N},
$$
which contradicts \eqref{3.con} and consequently $\lim_{N\to\infty}f_2(u_0^N)=0$.
\end{proof}
\subsection{Key lemma}
We show that $f_1(u(\cdot,s+N))/\bar{q}(s+N)$ and $h_1^*(u(\cdot,s+N)|u^\infty)$ are close for sufficiently large $N\in{\mathbb N}$. The following lemma is the key of the proof.
\begin{lemma}\label{lem.key}
For a.e.\ $x\in\Omega$, $s\in(0,1]$, it holds that
$$
\lim_{N\to\infty}\bigg(\frac{f_1(u(x,s+N))}{\bar{q}(s+N)}
- h_1^*(u(x,s+N)|u^\infty)\bigg) = 0.
$$
\end{lemma}
\begin{proof}
We set $u^N:=u(\cdot,s+N)$, $\bar{q}^N=\bar{q}(s+N)$, and $\bar{q}_i^N=|\Omega|^{-1}\int_\Omega q(u_0^N)u_i^Ndx$. Inserting definition \eqref{3.f} of $f_1$, the lemma is proved if we can show that for any $i=1,\ldots,n$,
\begin{align}\label{3.aux3}
0 &= \lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N}
- \frac{q(u_0^N)}{\bar{q}^N}u_i^N + \frac{\bar{q}_i^N}{\bar{q}^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}
+ u_i^N - u_i^\infty\bigg) \\
&= \lim_{N\to\infty}\bigg\{\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N} - u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg)
- \bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N - u_i^N\bigg) \nonumber \\
&\phantom{xx}+ \bigg(\frac{\bar{q}_i^N}{\bar{q}^N} - u_i^\infty\bigg)\bigg\}. \nonumber
\end{align}
Fix $i\in\{1,\ldots,n\}$. We know from Lemmas \ref{lem.convf} and \ref{lem.conv1} that
$\lim_{N\to\infty} q(u_0^N)/\bar{q}^N=1$ a.e. Together with the boundedness of $u_i^N$,
this shows that
$$
\lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N - u_i^N\bigg) = 0
$$
as well as
\begin{align*}
\lim_{N\to\infty}&\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg) \\
&= \lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{(q(u_0^N)/\bar{q}^N)u_i^N}{\bar{q}_i^N/\bar{q}^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg) \\
&= \lim_{N\to\infty}\bigg\{\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{q(u_0^N)}{\bar{q}^N}
+ \bigg(\frac{q(u_0^N)}{\bar{q}^N}-1\bigg)u_i^N\log\frac{u_i^N}{u_i^\infty} \\
&\phantom{xx}{}
- \frac{q(u_0^N)}{\bar{q}^N}u_i^N\log\frac{\bar{q}_i^N/\bar{q}^N}{u_i^\infty}\bigg\}
= - \lim_{N\to\infty}\frac{q(u_0^N)}{\bar{q}^N}u_i^N
\log\frac{\bar{q}_i^N/\bar{q}^N}{u_i^\infty}.
\end{align*}
To show that the limit on the right-hand side equals zero, we observe that, because of mass conservation and dominated convergence,
\begin{align*}
&\lim_{N\to\infty}\bigg(\frac{\bar{q}_i^N}{\bar{q}^N} - u_i^\infty\bigg)
= \lim_{N\to\infty}\bigg(\frac{1}{|\Omega|}\int_\Omega\frac{q(u_0^N)}{\bar{q}^N}u_i^N dx
- u_i^\infty\bigg) \\
&\phantom{x}
= \lim_{N\to\infty}\bigg(\frac{1}{|\Omega|}\int_\Omega\frac{q(u_0^N)}{\bar{q}^N}u_i^N dx
- \frac{1}{|\Omega|}\int_\Omega u_i^0 dx\bigg)
= \lim_{N\to\infty}\frac{1}{|\Omega|}\int_\Omega\bigg(\frac{q(u_0^N)}{\bar{q}^N}-1\bigg)
u_i^N dx = 0,
\end{align*}
and this is equivalent to $\lim_{N\to\infty}\log((\bar{q}_i^N/\bar{q}^N)/u_i^\infty)=0$. We conclude that
$$
\lim_{N\to\infty}\bigg(\frac{q(u_0^N)}{\bar{q}^N}u_i^N\log\frac{q(u_0^N)u_i^N}{\bar{q}_i^N}
- u_i^N\log\frac{u_i^N}{u_i^\infty}\bigg) = 0.
$$
Putting together the previous limits, we have proved \eqref{3.aux3}.
\end{proof}
\subsection{Convergence of $h^*$}
We conclude from Lemmas \ref{lem.convf} and \ref{lem.key} that
$\lim_{N\to\infty}h_1^*(u^N|u^\infty) = 0$. We claim that also $h_2^*$ and $h_3^*$ converge to zero as $N\to\infty$. Since $u_i^N$ and $u_i^\infty$ are bounded in $[0,1]$, we have the estimate
\cite[Lemma 16]{HJT22}
$$
\frac12\sum_{i=1}^n(u_i^N-u_i^\infty)^2
\le \sum_{i=1}^n\bigg(u_i^N\log\frac{u_i^N}{u_i^\infty} - (u_i^N-u_i^\infty)\bigg)
= h_1^*(u^N|u^\infty)\to 0,
$$
showing that $u_i^N\to u_i^\infty$ a.e.\ in $\Omega\times(0,1]$ as $N\to\infty$ for $i=1,\ldots,n$. We deduce from the continuity of $\chi$ that also $\lim_{N\to\infty}h_3^*(u^N|u^\infty)=0$.
For the limit of $h_2^*$, we observe that $u_0^N=1-\sum_{i=1}^n u_i^N\to u_0^\infty$ a.e. Hence, for any fixed $(x,s)\in\Omega\times(0,1]$, there exists $N_0\in{\mathbb N}$ such that
$1/2\le u_0(x,s+N)/u_0^\infty\le 3/2$ for $N>N_0$. Next, we write $h_2^*$ as
$$
h_2^*(u^N|u^\infty) = \int_{u_0^\infty}^{u_0^N}\log\frac{q(s)}{q(u_0^\infty)}ds
= u_0^\infty\int_1^{u_0^N/u_0^\infty}\log\frac{q(u_0^\infty\sigma)}{q(u_0^\infty)}d\sigma.
$$
Since the integrand is a function in $L^1(1/2,3/2)$, it follows from the absolute continuity of the integral that $\lim_{N\to\infty}h_2^*(u^N|u^\infty)=0$ a.e.\ in $\Omega\times(0,1]$. By definition of $h^*$, we have proved that $\lim_{N\to\infty}h^*(u^N|u^\infty)=0$.
\subsection{Convergence in $L^p(\Omega)$}
We deduce from the relative entropy inequality \eqref{3.ei} that $t\mapsto \int_\Omega h^*(u(t)|u^\infty)dx$ is bounded and nonincreasing. Then it follows from the limit $\lim_{N\to\infty}h^*(u^N|u^\infty)=0$ that in fact we have the convergence for all sequences $t\to\infty$,
$\lim_{t\to \infty}\int_\Omega h^*(u(t)|u^\infty)dx=0$ and in particular, since $h_2^*\ge 0$ and $h_3^*\ge 0$,
$$
\lim_{t\to \infty}\int_\Omega h_1^*(u(t)|u^\infty)dx=0.
$$
Using \cite[Lemma 16]{HJT22} again, we have
$$
\lim_{N\to\infty}\frac12\sum_{i=1}^n\int_\Omega(u_i(t)-u_i^\infty)^2 dx
\le \lim_{N\to\infty}\int_\Omega h_1^*(u(t)|u^\infty)dx=0.
$$
The convergence in $L^p(\Omega)$ for any $p<\infty$ then follows from the uniform bound
for $(u_i(t))_{t>0}$, finishing the proof.
\begin{remark}[Drift terms]\label{rem.drift}\rm
Equations \eqref{1.eq2} with drift terms read as
$$
\pa_t u_i = D_i\operatorname{div}\bigg\{u_ip_i(u)q(u_0)\na\bigg(\log\frac{u_ip_i(u)}{q(u_0)}
+ \Phi_i\bigg)\bigg\}, \quad i=1,\ldots,n,
$$
where $\Phi_i=\Phi_i(x)$ are given (electric or environmental) potentials. Adding the associated energy to the entropy density \eqref{1.h},
$$
h_2(u) = \sum_{i=1}^n(u_i(\log u_i-1)+1) + \int_1^{u_0}\log q(s)ds + \chi(u)
+ \sum_{i=1}^n u_i\Phi_i,
$$
we can compute (formally) the entropy inequality, giving
$$
\frac{d}{dt}\int_\Omega h_2(u)dx + \int_\Omega\sum_{i=1}^n D_iu_ip_i(u)q(u_0)
\bigg|\na\bigg(\log\frac{u_ip_i(u)}{q(u_0)} + \Phi_i\bigg)\bigg|^2 dx = 0.
$$
It was shown in \cite[Section 3.2]{ZaJu17} that the entropy production term with $\Phi_i=0$ can be bounded from below by $p_i(u)(q(u_0)\sum_{i=1}^n|\na\sqrt{u_i}|^2+|\na\sqrt{q(u_0)}|^2)$. Such an estimate seems to be impossible in the presence of $\na\Phi_i$. Indeed, the entropy inequality shows that
\begin{align*}
4\int_0^\infty&\int_\Omega q(u_0)^2e^{-\Phi_i}
\bigg|\na\bigg(\frac{u_ip_i(u)e^{\Phi_i}}{q(u_0)}\bigg)^{1/2}
\bigg|^2 dx \\
&= \int_0^\infty\int_\Omega u_ip_i(u)q(u_0)\bigg|\na\bigg(\log\frac{u_ip_i(u)}{q(u_0)}
+ \Phi_i\bigg)\bigg|^2< \infty.
\end{align*}
Thus, in the special case $q(0)>0$ and if $\Phi_i$ is bounded from above,
we conclude the existence of a subsequence $t_k\to \infty$ such that $\na(u_ip_i(u)e^{\Phi_i}/q(u_0))^{1/2}(t_k)\to 0$ strongly in $L^2(\Omega)$ as $k\to\infty$, and one may proceed similarly as in \cite[Section 5]{BFS14}. However, the condition $q(0)=0$ is needed to model correctly the transition rate of nonoccupied cells in the lattice model \cite{BDPS10,ZaJu17}.
\qed\end{remark}
\end{document}
|
\begin{document}
\author{M. I. Katsnelson\cite{mik}, V. V. Dobrovitski, and
B. N. Harmon}
\address{Ames Laboratory, Iowa State University, Ames, Iowa, 50011}
\title{Propagation of local decohering action in distributed quantum
systems}
\date{today}
\maketitle
\draft
\begin{abstract}
We study propagation of the decohering influence caused by a local
measurement performed on a distributed quantum system. As an
example, the gas of bosons forming a Bose-Einstein condensate is
considered. We demonstrate that the local decohering perturbation
exerted on the measured region propagates over the system in the
form of a decoherence wave, whose dynamics is governed by
elementary excitations of the system. We argue that the
post-measurement evolution of the system (determined by
elementary excitations) is of importance for transfer of
decoherence, while the initial collapse of the wave function
has negligible impact on the regions which are not directly
affected by the measurement.
\end{abstract}
\pacs{03.65.Bz, 05.30.Jp, 03.75.Fi}
\section*{Introduction}
The theory of quantum measurement begun
in the 1920s still remains an active topic of interest (see, e.g.
Ref.\ \onlinecite{meas1} and references therein). According to
von Neumann's theory of measurement \cite{neumann}, unitary
evolution of a system prepared initially in a pure quantum state
is interrupted by an instant decohering action of the measuring
apparatus, so that the density matrix describing an ensemble of
such systems changes radically (it ceases to be a projection
operator) and entropy rises. This view has been shown to describe
rather accurately the consequences of an act of measurement, but
the dynamics of the measurement process itself is lacking.
The contemporary theory of quantum measurements, which provides much
deeper analysis of the measurement process, is based on the
concept of decoherence \cite{meas}. To be measured, the system
has to interact with its environment, which consists of a large
number of degrees of freedom. The Hilbert space of the system
becomes divided into subspaces corresponding to the same
eigenvalue of the system-environment interaction Hamiltonian. As
a result of this interaction, coherence between different
subspaces is quickly lost, and after the measurement the system
appears in a mixed state. The concept of decoherence turned out
to be successful in many areas of fundamental physics, such as
the study of macroscopic quantum effects \cite{leg} and
consistent histories interpretation of quantum mechanics
\cite{omnes}, so that investigation of this process and related
effects is of considerable importance.
At present, decoherence and its consequences for point-like
quantum systems have been studied in detail (for review, see
Ref.\ \onlinecite{zurnew}), but distributed quantum systems
have received
significantly less attention. Mostly, linear systems have been
investigated, where separation into noninteracting modes is
possible, and each mode is considered as an independent
oscillator \cite{zurfield}. However, this approach is difficult to
apply to sufficiently
nonlinear systems (e.g., spin systems, or the Bose-Einstein
condensate as described by the Gross-Pitaevskii equation) possessing
localized soliton-like excitations. For systems where localized
excitations prevail, dealing explicitely with
real-space coordinates could be a more suitable strategy.
A real-space description of decoherence in distributed systems is a
very general and complicated issue. In this paper, we consider
only one aspect of the problem, namely, how {\it local\/}
properties of different regions in a distributed quantum system
are affected by a {\it local\/} measurement, that acts only on
some part of the system. Indeed, different regions in the system
are not isolated from each other, and correlations between them
exist (or can build up). Therefore, in spite of the fact that a
local measurement initially affects only one region, other regions
can ``acquire knowledge'' that some part of the system has been
measured. In this paper we explicitely show that the decohering
influence of the local measurement propagates through the system
in the form of a decoherence wave. Dynamics of the decoherence
wave is governed by elementary excitations, while the effect of
entanglement is very small for macroscopically large systems.
The consideration presented here can be applied to other similar
situations, so that a decoherence wave propagating with a
characteristic velocity of excitations is likely to be quite
common. This phenomenon, being a notable part of any real
measurement, is of fundamental interest. Moreover, propagation of
decoherence can be also of importance for the design of quantum
computers. Such a computer is a system of interacting quantum
entities, representing quantum bits (qubits). Fault-tolerant
quantum computations involve measurements performed on some
qubits and it is important to know how such measurement may
affect other qubits \cite{comput}. Moreover, decoherence is
introduced by a dissipative environment of qubits, so that
analysis of decoherence propagation may lead to strategies to
minimize influences detrimental to performance of the computer.
In this paper we consider a Bose-Einstein condensate of an ideal or
weakly non-ideal gas of bosons, which constitutes a good example
of a distributed system in a pure quantum state. It can be
implemented in reality as a gas of trapped atoms cooled down to
very low temperatures \cite{bose}. Suppose we measure the number
of particles in some region of space. If two such measurements are
done {\it simultaneously\/} at two different parts of the trap we
obtain the trivial result corresponding to the ground-state
wavefunction of the condensate. But if the second measurement is
carried out after some delay then the result is different and
provides information about the propagation of the perturbation
induced by the first measurement.
The situation considered here is related to the problem of broken
gauge symmetry and existence of a relative phase of two
interfering condensates \cite{phase}, which has been extensively
discussed recently. If we have a condensate with a definite
number of particles, its phase is spread uniformly between 0 and
$2\pi$, while a definite phase requires a non-conservation of the
number of particles in the condensate. It has been shown that a
well-defined phase (evidenced experimentally by appearance of the
interference fringes) builds up in the course of the measurement
(atoms detection), due to increasing uncertainty in the number of
particles in each of the interfering condensates: each detected
atom may well belong to either of them. For the circumstances
considered in this paper, we have a similar situation: the local
phase of the condensate is the same in every region. Identity of
the phase throughout the condensate is due to uncertainty in the
local number of the particles inside each region. However, when
the number of particles in some region is determined by a local
measurement, the phase coherence in the condensate as a whole is
partially destroyed, what leads to observable consequences,
propagation of the decoherence wave in the system. Note that
decoherence wave is the same both for the condensate with
definite number of particles (with uncertain global phase, the
case of non-interacting bosons) and for the condensate with
definite global phase (but with uncertain number of condensed
particles, the case of weakly interacting bosons): the results
for the latter case transform exactly to the results for the
former as interaction goes to zero.
We describe the dynamics of the condensate in a linear approximation,
i.e. we use the approximation of noninteracting quasiparticles to
study a weakly non-ideal Bose-gas. In so doing, we loose the
ability to investigate some interesting nonlinear effects, but
we gain in clarity of presentation: it is reasonable to
start from a simplified (and not totally unrealistic) case to
emphasize the main idea.
We do not specify the way of measuring the local density of
condensate, and the dynamics of the measurement process is not
considered here. Analysis of a specific experimental scheme is
a distinct problem, requiring separate study, while here we
focus on the post-measurement evolution of the condensate. In
principle, the local density of the Bose-condensate can be measured by
placing some probe into the trap, which interacts with the
condensate so that an entangled state is formed
\begin{equation}
|X\rangle = \sum C_n |n\rangle \otimes |\alpha_n\rangle
\end{equation}
where $|n\rangle$ is the state of condensate with the number of
particles $n$ in the measured region, and $|\alpha_n\rangle$ is
the state of the probe. If the probe interacts with a large
number of environmental degrees of freedom, so that
$|\alpha_n\rangle$ are the eigenstates corresponding to different
eigenvalues of the probe-environment interaction Hamiltonian,
then the coherence between different probe states is being lost,
and the condensate's state also becomes an incoherent mixture of
different states $|n\rangle$. If the probe (and, correspondingly,
the condensate) decoheres quickly enough (as is usually the case) we
can consider the measurement as instantaneous and safely use von
Neumann's theory to describe the condensate's state immediately
after the measurement.
Although the situation considered above is in many
respects too idealized to apply rigorously to a real experiment, it
is detailed enough to capture the essential processes of
concern in this paper.
\section*{Propagation of decoherence in Bose-Einstein condensate}
To study quantitatively the effect of decoherence propagation,
let us consider first an ideal Bose-gas confined by external fields and
described by the Hamiltonian
\begin{equation}
H = \sum_{\mu} E_{\mu} \alpha^{\dag}_{\mu} \alpha_{\mu},
\end{equation}
where $\alpha^{\dag}_{\mu}$ and $\alpha_{\mu}$ are the
boson creation and annihilation operators. $E_{\mu}$
are the one-particle energies, and we denote the
corresponding one-particle wavefunctions as
$\varphi_{\mu}({\bf r})$, where $\mu =0$ stands for
the ground state having minimal energy $E_0=0$.
Then, the
ground-state eigenfunction of the system of $M$ bosons
can be written as
\begin{equation}
\label{ground}
|\Psi\rangle =\frac{1}{\sqrt{M!}}
\left(\alpha_{0}^{\dag}\right)^{M} |0\rangle,
\end{equation}
where $|0\rangle$ is the vacuum state. For simplicity, we can consider
the trap as being divided into a large number $N_c$ of small
cells each having the
volume $V_0$ (it can be considered as the volume directly affected
by the measuring apparatus), satisfying the relation $V_0\ll V$,
where $V$ is the total volume of the trap. Then, the
coordinate ${\bf r}$ is understood as a discrete quantity
(the number of a cell). This is similar to a general practice
in solid-state theory, where $V_0$ is analogous to the volume
of an elementary cell of the crystal \cite{ziman}.
Note that in so doing, the
number of one-particle states taken into account becomes equal to
$N_c$, which is finite, though very large.
This corresponds to the fact that the number of states inside
the first Brillouin zone equals to the number of lattice cells.
At the instant $t=0$ we perform measurement
of the number of bosons in the cell ${\bf r}=0$. This observable
is represented by the operator $N=a^{\dag}(0) a(0)$,
where
\begin{equation}
\label{sum}
a({\bf r}) = \sum\limits_{\mu} \varphi_{\mu}({\bf r}) \alpha_{\mu}.
\end{equation}
is the boson field operator. Eigenvalues of the operator $N$
are $n=0,1,2...$ and, suppose, the measurement
has given us one of them. According to von
Neumann's theory, it corresponds to the action
of the operator $W_n$ on the system, where
\begin{equation}
\label{w}
W_n=\delta _{n,N}=\int\limits_{0}^{2\pi} \frac{d\phi}{2\pi}
\exp{\left[i\phi (n-N)\right]}
\end{equation}
is a projector onto the state with the number
of particles $n$ in the measured region. The operator $W_n$
has the value equal to unity on this state and it has
zero value on all others states.
Further development of the system is to be described by the density
matrix of the system $U(t)$, since the measurement interrupts unitary
evolution and casts the system into mixed quantum state.
According to the standard theory of measurement \cite{neumann,meas},
the density matrix at the time $t$ is
\begin{equation}
\label{evolution}
U(t)=\sum\limits_{n=0}^{\infty} \exp{(-iHt)} W_n U_{\text{in}}
W_n^{\dag} \exp{(iHt)} ,
\end{equation}
where $U_{\text{in}}=|\Psi\rangle \langle\Psi |$
is the density matrix before the measurement.
To trace propagation of decoherence in the system, we
study evolution of the one-particle density matrix
\begin{equation}
\label{ro}
\rho({\bf r}, {\bf r}', t)=\mathop{\rm Tr}\left[ U(t) a^{\dag}({\bf r}')
a({\bf r}) \right].
\end{equation}
This quantity describes local properties of the Bose-Einstein condensate;
in particular, the average number of particles resulting
from the second
measurement, which is performed at the point ${\bf r}$ at
the instant $t$, is given by the value $\rho({\bf r}, {\bf r}, t)$.
To simplify calculations, we use
the fact that the total number of particles is large,
$M\gg 1$, so that operators $\alpha_0$ and $\alpha^{\dag}_0$ acting
on the state $|\Psi\rangle$ can be replaced by the number $\sqrt{M}$
with relative accuracy $1/\sqrt{M}$; this is a standard approximation
in the theory of Bose-Einstein condensation \cite{agd}. Therefore,
Eq. (\ref{sum}) can be rewritten as
\begin{equation}
\label{sum1}
a({\bf r})=\sqrt{n_B({\bf r})}+\bar a({\bf r}),
\qquad \bar a({\bf r)} = \sum\limits_{\mu \neq 0}
\varphi_{\mu}({\bf r}) \alpha_{\mu}
\end{equation}
where $n_B ({\bf r}) = M\varphi_0^2 ({\bf r})$ is the average number
of condensate particles contained in the volume $V_0$ at the cell
${\bf r}$. The expression for the one-particle density matrix
can be written as
\begin{eqnarray}
\label{rho1}
\rho({\bf r}, {\bf r}', t) &=& \sum\limits_{n=0}^{\infty }
\rho_n({\bf r},{\bf r}', t) ,\\
\nonumber
\rho_n({\bf r},{\bf r}', t) &=&
\langle\Psi |W_n^{\dag} a^{\dag}({\bf r}',t)
a({\bf r},t)W_n |\Psi\rangle,
\end{eqnarray}
where $a({\bf r}, t) = \exp{(iHt)} a({\bf r})\exp{(-iHt)}$. The
operator product in Eq.\ (\ref{rho1}) is to be ordered normally,
i.e. it is to be rewritten in such a way that
all $a^{\dag}$ stand to the left of all $a$ in each term
of the Taylor series expansion. In so doing, we take into account that
\begin{equation}
\label{commut}
\left[ a({\bf r}, t), \bar a^{\dag}(0)\right] = \sum\limits_{\mu\neq 0}
\varphi_{\mu}({\bf r}) \varphi^*_{\mu}(0)
{\rm e}^{-iE_{\mu}t} \equiv g({\bf r,}t).
\end{equation}
Note that for a system containing a large number of particles $M\gg 1$,
the function $g({\bf r}, t)$ can be replaced by the Green's function
\begin{equation}
G({\bf r},t) = \sum\limits_{\mu} \varphi_{\mu}({\bf r}) \varphi^*_{\mu}(0)
{\rm e}^{-iE_{\mu}t}
\end{equation}
with accuracy of order of $1/M$,
since $G({\bf r},t) = g({\bf r},t)+\varphi_0({\bf r})\varphi_0^*(0)$.
Performing the calculations, we obtain
\begin{eqnarray}
\label{answer}
\rho_n({\bf r},{\bf r}', t) &=& p_n \left[\sqrt{n_B({\bf r})}
- G({\bf r}, t) \sqrt{n_0} \right] \\
\nonumber
&&\times \left[ \sqrt{n_B({\bf r}')}
- G^*({\bf r}', t) \sqrt{n_0} \right] \\
\nonumber
&&+p_{n-1} n_0 G({\bf r}, t) G^*({\bf r}',t),
\end{eqnarray}
where $n_0=n_B(0)$, and $p_n = \mathop{\rm e}^{-n_0} n_0^n/(n!)$
is the Poisson distribution function.
Summation over $n$ can be performed
explicitly, yielding
\begin{eqnarray}
\label{answ1}
\rho({\bf r},{\bf r}',t) &=& \sqrt{n_B({\bf r}) n_B({\bf r}')}
- G^*({\bf r}',t) \sqrt{n_B({\bf r}) n_0}\\
\nonumber
&& - G({\bf r},t) \sqrt{n_B({\bf r}') n_0}
+ 2 n_0 G^*({\bf r}',t) G({\bf r},t).
\end{eqnarray}
This result shows that the measurement made at
the point ${\bf r=}0$
produces a decohering perturbation which propagates
over the trap in the form of a decoherence wave, and
this propagation is
governed by the Green's function $G({\bf r},t)$. It
can be explicitely demonstrated by considering
an example of the gas
consisting of free Bose-particles of mass $m$. The
corresponding Green's function at the distances $r\gg V_0^{1/3}$
and times $t\gg mV_0^{2/3}/\hbar$ is \cite{feynman}
\begin{equation}
\label{feyn1}
G({\bf r},t) = V_0 \left( \frac{m}{2\pi i\hbar t}\right)
^{3/2}\exp{\left( \frac{im{\bf r}^2}{2\pi \hbar t}\right)}.
\end{equation}
Local density of the
condensate after the measurement is given by the value
\begin{eqnarray}
\label{dens}
\rho({\bf r}, {\bf r}, t) &=& n_B + 2 n_B V_0^2
\left(\frac{m}{2\pi\hbar t}\right)^3 \\
\nonumber
&& - 2 n_B V_0 \left( \frac{m}{2\pi\hbar t}\right)^{3/2}
\cos{\left(\frac{m {\bf r}^2}{2\pi\hbar t}\right)},
\end{eqnarray}
where $n_B=M/V$ is density of the condensate before the
measurement, which is independent on position $\bf r$.
This is an observable effect, which, in principle, can
be detected experimentally.
The entropy of the system, being initially zero,
after the measurement is
\begin{equation}
S= -\mathop{\rm Tr} \left[ U(t)\ln{U(t)}\right] =
-\sum\limits_{n=0}^{\infty} p_n \ln{p_n} > 0,
\end{equation}
which is a clear indication of the decohering
effect of measurement. The increase of entropy of
condensate as a whole happens only at the instant of measurement
and further evolution, being unitary, keeps it constant (decoherence
only propagates in the system from one region to another). Note that
local entropy (in contrast to the one-particle density matrix, where
the decoherence propagation is clearly seen) can be hardly used
to track the decoherence wave. The value of the local entropy
is nonzero even in the initial pure state, while the total
entropy of the system is zero. It happens because of ``negative
entropy'' stored in the form of correlations between
different parts of the condensate (for more detailed discussion
see Ref.\ \cite{zurinfo}).
The results obtained can be qualitatively interpreted as follows.
The measurement performed at ${\bf r}=0$ leads to localization
of some number of particles
within the cell ${\bf r}=0$. The localized particles
acquire rather large momenta, of order
$\hbar/V_0^{1/3}$; the average number of such particles is
$n_0=n_B(0)$. Immediately after being localized, these
particles start to propagate over the trap, and their
propagation is governed by the Green's function (\ref{feyn1}).
Because of indistinguishability of particles in the trap,
we can not say that these are ``the same'' particles which
were measured at ${\bf r}=0$, so that the effect we consider
is not a physical motion of some separate particles in the trap,
but is the propagation of the decohering influence of
the measurement through the system.
An interesting feature of the decoherence propagation
can be illustrated by the gas of bosons trapped
in a parabolic external potential, so that each particle is
represented by an isotropic harmonic oscillator of
eigenfrequency $\Omega$. In this case, provided that
$r\gg V_0^{1/3}$ and $V_0\ll (\hbar /\Omega)^{3/2}\sim V$,
the Green's function has the form \cite{feynman}
\begin{equation}
\label{feynman}
G({\bf r},t) =V_0 \left(\frac{\Omega}{2\pi i\hbar
\sin{\Omega t}} \right)^{3/2} \exp{\left(\frac{i\Omega {\bf r}^2}
{2\pi\hbar}\cot{\Omega t}\right)}
\end{equation}
where the particles are assumed to have unitary mass. This function
is periodic in time with the period $2\pi/\Omega$.
Therefore, the decoherence propagation is also periodic
in time with the same period. In the general case of Bose-gas
trapped in a finite volume, the decoherence propagation becomes
a quasiperiodic process, according to Eq.\ (\ref{commut}).
And, last but not least, decoherence propagation is a wave process,
possessing both amplitude and phase. Existence of coherent waves
in the system without quantum coherence is not
unusual, the same property is shared, e.g., by the sound wave
propagating in the classical fluid. Therefore, in principle, an
interference of two decoherence waves is possible.
Above, we have considered the system of noninteracting bosons.
Now, let us investigate the case of weakly interacting particles,
i.e. a weakly non-ideal Bose-gas contained in a trap of
large volume $V$. We assume no external potential acting on
the particles, so that the one-particle states
are simple plane waves
\begin{equation}
\varphi_{\bf k}({\bf r}) = \sqrt{\frac{V_0}{V}} \exp{(i{\bf kr})},
\end{equation}
where the normalization reflects the fact that the
trap is divided into cells of volume $V_0\ll V$.
This system is described by the Hamiltonian
\begin{eqnarray}
\label{nonideal}
H&=&\sum\limits_{\bf k} E_{\bf k} \alpha_{\bf k}^{\dag} \alpha_{\bf k}\\
\nonumber
&&+ \frac 1{2V} \sum\limits_{{\bf k}_1+{\bf k}_2={\bf k}'_1+{\bf k}'_2}
v ({\bf k}_1-{\bf k}'_1) \alpha_{{\bf k}'_1}^{\dag}
\alpha_{{\bf k}'_2}^{\dag} \alpha_{{\bf k}_2} \alpha_{{\bf k}_1}
\end{eqnarray}
where $v({\bf k})$ is the Fourier transform of the interaction
potential (which is assumed to be repulsive). Since the interaction
is small, new Bose operators can be introduced according to
Bogoliubov transformation
\begin{eqnarray}
\label{bogol}
\alpha_{\bf k} &=& \xi_{\bf k} \cosh{\chi_{\bf k}} +
\xi_{-{\bf k}}^{\dag} \sinh{\chi_{\bf k}}, \\
\nonumber
\alpha_{-{\bf k}}^{\dag} &=& \xi_{\bf k} \sinh{\chi_{\bf k}} +
\xi_{-{\bf k}}^{\dag} \cosh{\chi_{\bf k}},
\end{eqnarray}
with the parameters $\chi_{\bf k}$ defined as
\begin{equation}
\label{bogol1}
\tanh{2\chi_{\bf k}} = -\frac{v({\bf k})n_B}{E_{\bf k} +
v({\bf k})n_B},
\end{equation}
where $n_B$ is the average number of particles belonging to Bose-Einstein
condensate contained in the volume $V_0$. Provided that the interaction
is small (or the gas density $M/V$ is small), almost all
particles belong to the condensate, so we can take
$n_B= M V_0/V$ with relative accuracy of order of $\sqrt{v(0) M/V}$
\cite{agd}.
By using the Bogoliubov transformation, we pass to the ideal gas
of new excitations with the dispersion law
\begin{equation}
\label{bogol2}
\omega_{\bf k}=\sqrt{E_{\bf k}^2 + 2 E_{\bf k} v({\bf k}) n_B}.
\end{equation}
Again, we consider dynamical behavior of the one-particle density matrix.
The calculation procedure remains essentially the same as for
the ideal Bose-gas. In so doing, we obtain the result:
\begin{eqnarray}
\label{answnew}
\rho_n ({\bf r},{\bf r}',t) &=& \frac{n_B}{(n!)^2}
\frac{\partial^{2n}}{\partial z^n \partial z^{\prime n}}
\Bigl\{ [1+(z-1) G({\bf r},t)] \\
\nonumber
&&\times [1+(z'-1) G^*({\bf r}',t)] \\
\nonumber
&&\times\exp{[n_B X(z,z')]}\Bigr\}_{z=z'=0}
\end{eqnarray}
where the following notations were used,
\begin{eqnarray}
\label{answlast}
X(z,z') &=& B(zz'-1) + (1-B)(z+z'-2) \\
&& + A\left[ (z-1)^2+(z'-1)^2\right], \\
\nonumber
A &=& \frac{V_0}{2V} \sum\limits_{\bf k} \frac{v({\bf k})n_B}
{\omega_{\bf k}}, \\
\nonumber
B &=& \frac{V_0}{2V} \sum\limits_{\bf k} \left[1+\frac{E_{\bf k}
+v({\bf k}) n_B} {\omega_{\bf k}}\right],
\end{eqnarray}
and $G({\bf r},t)$ is the Green's function of the weakly interacting
Bose-gas:
\begin{eqnarray}
\label{green}
G({\bf r},t) &=& \sum\limits_{\bf k} \exp{(i{\bf kr})}\\
\nonumber
&&\times \left\{ \cos{\omega_{\bf k}t}
-i\frac{E_{\bf k} + v({\bf k}) n_B}{\omega_{\bf k}}
\sin{\omega_{\bf k}t} \right\}.
\end{eqnarray}
Again, we see that the decoherence wave propagating in the
system follows the dynamics of the
Green's function (\ref{green}). Dynamic behavior of $G({\bf r},t)$
at large times $t$ and large distances $r$
can be analyzed by the
method of stationary phase \cite{witham}. According to
this method, the value
of the function $G(r,t)$ at the point $\bf r$ at the
instant $t$ is determined by those excitations which have
a group velocity ${\bf u}({\bf k})\equiv d\omega_{\bf k}/ d{\bf k}$
obeying the requirement ${\bf u}({\bf k}) = {\bf r}/t$.
The excitations with large wavevectors ${\bf k}$ are
subject to considerable damping
\cite{agd}, so that at large distances only the undamped long-wavelength
excitations determine the dynamics of the Green's function.
These excitations represent
sound propagating in the Bose-gas with the velocity $c=\sqrt{n_B v(0)/m}$,
so the decoherence wave
in a system of weakly interacting bosons propagates with the
sound velocity $c$.
This result can be interpreted in the same way as the
decoherence wave in an ideal Bose-gas. The measurement
affects the particles situated at ${\bf r}=0$. Due to
the interparticle interaction, the decohering perturbation
is transferred to other regions of the system.
The decoherence transfer is governed by the
undamped excitations present in the system, i.e. by the
long-wavelength excitations traveling with
the sound velocity $c$.
\section*{Discussion}
Summarizing, we have studied the decohering
influence of a local measurement performed on a distributed
quantum system. We show that the decohering perturbation exerted
on the measured region propagates over the system by forming a
{\it decoherence wave\/}, whose dynamics is determined by the
Green's function of the system. This result, although
not totally unexpected, is not as trivial as it might seem, since
decoherence is a rather peculiar effect, and the decohering impact
of a measurement can be quite different from other physical
influences (see, e.g. the discussion in Ref.\ \cite{scully}).
The usual scenario for few-particle systems is
based on the Einstein-Podolsky-Rosen (EPR) situation \cite{epr}
of strong entanglement, when, e.g. two
particles with spins $1/2$ form a singlet state
\begin{equation}
|\psi\rangle = \frac 1{\sqrt{2}}\left(\,|\!\uparrow\downarrow\rangle
- |\!\downarrow\uparrow\rangle\, \right).
\end{equation}
If the first spin has been measured, and as a result of this
measurement has been cast in the state $|\!\uparrow\rangle$ (here
again we use von Neumann's theory of instant measurement), then
the transfer of decoherence is instant: the second spin
immediately occurs in the state $|\!\downarrow\rangle$. In
distributed systems this effect is also present: the wave function of
the system collapses immediately after the measurement. But the
impact of the collapse upon the one-particle density matrix
(and even $s$-particle density matrix, for $s\ll M$) is
practically unobservable for the system of macroscopic size
(where $M\gg 1$): the change in the density matrix element
$\rho({\bf r},{\bf r}',t)$ immediately after the measurement is
of order of $n_0/M$ (provided, of course, that ${\bf r},{\bf r}
'\neq 0$), and the same is true for the $k$-particle density
matrix
$\rho({\bf r}_1,\dots{\bf r}_k; {\bf r}'_1,\dots{\bf r}'_k)$
if $k\ll M$. This result is rather obvious: localization of the
number $n_0$ of particles in some cell can not affect noticeably
other cells if the total number of particles is macroscopically
large. Therefore, the post-measurement evolution of the system,
which is governed by the Green's function, becomes important since it
provides much more noticeable changes in the density
matrix elements: in Eq.\ (\ref{dens}) the term corresponding to
the decoherence wave does not go to zero as $M\to\infty$.
Obviously, it happens because in the EPR-like situation the
entanglement is very ``stiff'', so that each state of one
particle determines completely the state of the other. But in the
many-particle system there is no one-to-one correspondence,
since the total number of degrees of freedom is much larger than
the number of degrees of freedom fixed during the measurement.
This difference is the reason for the different dynamics of
decoherence propagation.
Finally, we remark that another aspect of decoherence in
distributed systems has been studied within the context of
decoherent quantum histories \cite{gellmann,brun}. Although the
effects studied there, as well as systems considered and methods
used, are different from those investigated here, it is
interesting to note that local properties of distributed quantum
systems are often ``intrinsically'' decoherent \cite{brun} if a
coarse enough description is used. For the effects considered
here, sufficient coarse graining leads to averaging of the
oscillating Green's function over the spatial scale of several
oscillations, so that the details of the decoherence wave becomes
negligible. Therefore, the intrinsic structure of the decoherence
wave can be distinguished only at fine scales, where coherence of
the Green's function holds.
This work was partially carried out at the Ames Laboratory, which
is operated for the U.\ S.\ Department of Energy by Iowa State
University under Contract No.\ W-7405-82 and was supported by
the Director for Energy Research, Office of Basic Energy Sciences
of the U.\ S.\ Department of Energy.
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\bibitem[*]{mik} Permanent address: Institute of Metal Physics,
Ekaterinburg 620219\, Russia.
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(Princeton, Princeton University Press, 1983); M. B. Mensky,
{\it Continuous Quantum Measurements and Path Integrals\/}
(Bristol, IOP Publishing, 1993).
\bibitem{neumann} J. von Neumann, {\it Mathematical
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\end{document}
|
\begin{document}
\title{Arithmetic formulas for the Fourier coefficients of Hauptmoduln of level 2, 3, and 5}
\begin{abstract}
We give arithmetic formulas for the coefficients of Hauptmoduln of higher levels as analogues of Kaneko's formula for the elliptic modular $j$-invariant. We also obtain their asymptotic formulas by employing Murty-Sampath's method.
\end{abstract}
\section{Introduction}
For the elliptic modular function $j(\tau)$, let $\textbf{t}_m(d)$ be the modular trace function (the precise definition will be given later) and $c_n$ ($n \geq 1$) the $n$th Fourier coefficient of $j(\tau)$, that is, $j(\tau) = q^{-1} + 744 + \sum_{n=1}^{\infty} c_n q^n$. Zagier \cite{Zag02} studied the traces of singular moduli and showed that the generating function of $\textbf{t}_m(d)$ is a meromorphic modular form of weight $3/2$ on the right group for each $m$. Multiplying it by theta function and observing the modular forms of weight 2, Kaneko \cite{Kan95} gave the following arithmetic formula for $c_n$ experimentally, and showed it.
\begin{eqnarray*}
c_n &=& \frac{1}{n} \biggl\{ \sum_{r \in \mathbb{Z}} \textbf{t}_1(n - r^2) + \sum_{\substack{r \geq 1,\ odd}} ((-1)^n \textbf{t}_1(4n - r^2) - \textbf{t}_1(16n - r^2)) \biggr\} \\
&=& \frac{1}{2n} \sum_{r \in \mathbb{Z}} \textbf{t}_2(4n - r^2) .
\end{eqnarray*}
On the other hand, by using the circle method, Petersson \cite{Pet32} and later Rademacher \cite{Rad} independently derived the asymptotic formula for $c_n$:
\begin{eqnarray*}
c_n \sim \frac{e^{4 \pi \sqrt{n}}}{\sqrt{2} n^{3/4}}\ as\ n \to \infty.
\end{eqnarray*}
The circle method is introduced by Hardy and Ramanujan \cite{HR18} to prove the asymptotic formula for the partition function
\begin{eqnarray*}
p(n) \sim \frac{e^{\pi \sqrt{2n/3}}}{4\sqrt{3}n}\ as\ n \to \infty,
\end{eqnarray*}
where $p(n)$ is defined by $\sum_{n=0}^{\infty} p(n)q^n = \prod_{n=1}^{\infty}(1-q^n)^{-1}$. In 2013, Bruinier and Ono \cite{BO13} considered certain traces of singular moduli for weak Maass forms and derived the algebraic formula for $p(n)$. Combining this formula with Laplace's method, Dewar and Murty \cite{DM13, DM132} proved the asymptotic formulas for $p(n)$ and $c_n$ without the circle method. More recently, Murty and Sampath \cite{Sam15} derived the asymptotic formula for $c_n$ from Kaneko's arithmetic formula with Laplace's method.\\
In this article, we generalize these formulas to Hauptmoduln (defined in section 2) for the congruence subgroups $\Gamma_0(p)$ and $\Gamma_0^*(p)$ (the extension of $\Gamma_0(p)$ by the Atkin-Lehner involution) with $p = 2, 3,$ and $5$. \\
Let $j_p(\tau)$ and $j_p^*(\tau)$ be the corresponding Hauptmoduln for $\Gamma_0(p)$ and $\Gamma_0^*(p)$, respectively. Ohta \cite{Ohta09} gave the arithmetic formulas for the Fourier coefficients of $j_2(\tau)$ and $j_2^*(\tau)$, and a part of those of $j_3(\tau)$. She also treated the cases of $j_4(\tau)$ and $j_4^*(\tau)$. Let $c_n^{(p)}$ and $c_n^{(p*)}$ be the $n$th Fourier coefficients of $j_p(\tau)$ and $j_p^*(\tau)$, respectively. We express these coefficients in terms of the modular trace functions $\textbf{t}_m^{(p*)}(d)$.
\begin{thm}
\label{main1}
For any $n \geq 1$, we have
\begin{eqnarray*}
c_n^{(2)} &=& \frac{1}{2n} \times \left\{ \begin{array}{ll}
- \sum_{r \equiv 0 (2)} \textbf{t}_2^{(2*)}(4n - r^2) + 24\sigma_1^{(2)}(n) & (n \equiv 0 \bmod 2), \\
\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(2*)}(4n - r^2) + 24\sigma_1(n) & (n \not\equiv 0 \bmod 2), \\ \end{array} \right. \\
c_n^{(3)} &=& \frac{1}{2n} \times \left\{ \begin{array}{ll}
- \sum_{r \equiv 0 (3)} \textbf{t}_2^{(3*)}(4n - r^2) + 36\sigma_1^{(3)}(n) & (n \equiv 0 \bmod 3), \\
\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(4n - r^2) + 36\sigma_1(n) & (n \not\equiv 0 \bmod 3), \\ \end{array} \right. \\
c_n^{(5)} &=& \frac{1}{2n} \times \left\{ \begin{array}{ll}
-\sum_{r \equiv 0 (5)} \textbf{t}_2^{(5*)}(4n - r^2) + 18\sigma_1^{(5)}(n) & (n \equiv 0 \bmod 5), \\
\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(5*)}(4n - r^2) +18\sigma_1(n) & (n \not\equiv 0 \bmod 5), \\ \end{array} \right. \\
c_n^{(p*)} &=& c_n^{(p)} - p c_{pn}^{(p)} \ \ (p = 2, 3, 5)\\
\end{eqnarray*}
where $\sigma_1(n) = \sum_{d | n}d$, and $\sigma_1^{(p)}(n) = \sum_{\substack{d | n \\ p \nmid d}}d$.
\end{thm}
\begin{rmk}
These formulas are different from those in Ohta \cite{Ohta09}. In \cite{Ohta09}, the definition of $\textbf{t}_m^{(p)}(d)$ was mixed with that of $\textbf{t}_m^{(p*)}(d)$, and used the values of $\textbf{t}_m^{(p)}(d)$ instead of $\textbf{t}_m^{(p*)}(d)$.
\end{rmk}
Combining these formulas with Laplace's method as in \cite{Sam15}, we obtain the asymptotic formulas of $c_n^{(p)}$.
\begin{thm}
\label{main2}
We have
\begin{eqnarray*}
c_n^{(2)} &\sim& \frac{e^{2\pi \sqrt{n}}}{2n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0 \bmod 2), \\ 1 & (n \equiv 1 \bmod 2), \\
\end{array} \right.\\
c_n^{(3)} &\sim& \frac{e^{4\pi \sqrt{n}/3}}{\sqrt{6}n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0, 2 \bmod 3), \\ 2 & (n \equiv 1\ \ \ \bmod 3), \\
\end{array} \right.\\
c_n^{(5)} &\sim& \frac{e^{4\pi \sqrt{n}/5}}{\sqrt{10}n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0 \bmod 5), \\ (3 + \sqrt{5})/2 & (n \equiv 1 \bmod 5), \\ -1 + \sqrt{5} & (n \equiv 2 \bmod 5), \\ -1 - \sqrt{5} & (n \equiv 3 \bmod 5), \\ (3 - \sqrt{5})/2 & (n \equiv 4 \bmod 5)\\
\end{array} \right.
\end{eqnarray*}\\
as n $\to \infty$.
\end{thm}
\section{Preliminaries}
In this section, we shall define the Hauptmoduln and the modular trace functions.
\begin{dfn}
Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb{R})/{\pm I}$ containing $(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix})$. If the genus of $\Gamma$ is equal to 0, there is a unique modular function f of weight 0 satisfying the following conditions. We call this f the Hauptmodul with respect to $\Gamma$.\\
$(1)$ f is holomorphic in the upper half plane $\mathfrak{H}$,\\
$(2)$ f has a Fourier expansion of the form $f(\tau) = q^{-1} + \sum_{n = 1}^{\infty} H_n q^n \ (q := e^{2\pi i \tau})$,\\
$(3)$ f is holomorphic at cusps of $\Gamma$ except i$\infty$.
\end{dfn}
For $\Gamma_0(p) := \{(\begin{smallmatrix}a & b \\c & d \end{smallmatrix}) \in \mathrm{PSL}_2(\mathbb{Z})\ |\ c \equiv 0 \pmod{p} \}$ and $\Gamma_0^*(p) := \Gamma_0(p) \cup \Gamma_0(p)(\begin{smallmatrix}0 & -1/\sqrt{p} \\ \sqrt{p} & 0 \end{smallmatrix})$ ($p$ = 2, 3, 5), the corresponding Hauptmoduln $j_p(\tau)$ and $j_p^*(\tau)$ can be described by means of the Dedekind $\eta$-function $\eta(\tau):= q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$;
\begin{eqnarray*}
j_2(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(2\tau)} \biggr)^{24} + 24 = \frac{1}{q} + 276q - 2048q^2 + 11202q^3 + \cdots,\\
j_2^*(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(2\tau)} \biggr)^{24} + 24 + 2^{12} \biggl( \frac{\eta(2\tau)}{\eta(\tau)} \biggr)^{24} = \frac{1}{q} + 4372q + 96256q^2 + 1240002q^3 + \cdots,\\
j_3(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(3\tau)} \biggr)^{12} + 12 = \frac{1}{q} + 54q - 76q^2 - 243q^3 + \cdots,\\
j_3^*(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(3\tau)} \biggr)^{12} + 12 + 3^6 \biggl( \frac{\eta(3\tau)}{\eta(\tau)} \biggr)^{12} = \frac{1}{q} + 783q + 8672q^2 + 65367q^3 + \cdots,\\
j_5(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(5\tau)} \biggr)^6 + 6 = \frac{1}{q} + 9q + 10q^2 - 30q^3 + \cdots,\\
j_5^*(\tau) &=& \biggl( \frac{\eta(\tau)}{\eta(5\tau)} \biggr)^6 + 6 + 5^3 \biggl( \frac{\eta(5\tau)}{\eta(\tau)} \biggr)^6 = \frac{1}{q} + 134q + 760q^2 + 3345q^3 + \cdots.
\end{eqnarray*}
For $p$ = 2, 3, and 5, let $d$ be a positive integer such that $-d$ is congruent to a square modulo 4$p$, and $\mathcal{Q}_{d,p}$ the set of positive definite binary quadratic forms $Q(X,Y) = [a,b,c] = a X^2 + b X Y + c Y^2\ (a,b,c \in \mathbb{Z})$ of discriminant $-d$ with $a \equiv 0$ (mod $p$). Moreover, we fix an integer $\beta$ (mod $2p$) with $\beta^2 \equiv -d$ (mod $4p$) and denote by $\mathcal{Q}_{d,p,\beta}$ the set of quadratic forms $[a,b,c] \in \mathcal{Q}_{d,p}$ such that $b \equiv \beta$ (mod $2p$). For every positive integer $m$, let $\varphi_m(j_p^*)$ be a unique polynomial of $j_p^*$ satisfying $\varphi_m(j_p^*(\tau)) = q^{-m} + O(q)$. We define two modular trace functions:
\begin{eqnarray*}
\textbf{t}_m^{(p)}(d) &:=& \sum_{Q \in \mathcal{Q}_{d,p,\beta} / \Gamma_0(p)} \frac{1}{|\Gamma_0(p)_Q|} \varphi_m(j_p^*(\alpha_Q)),\\
\textbf{t}_m^{(p*)}(d) &:=& \sum_{Q \in \mathcal{Q}_{d,p} / \Gamma_0^*(p)} \frac{1}{|\Gamma_0^*(p)_Q|} \varphi_m(j_p^*(\alpha_Q)),
\end{eqnarray*}
where $\alpha_Q$ is the root of $Q(X,1) = 0$ in $\mathfrak{H}$. The definition of $\textbf{t}_m^{(p)}(d)$ is independent of $\beta$. In addition, we set $\textbf{t}_2^{(2*)}(0) := 5, \, \textbf{t}_2^{(3*)}(0) = \textbf{t}_2^{(5*)}(0) := 3, \, \textbf{t}_2^{(p*)}(-1) := -1, \, \textbf{t}_2^{(p*)}(-4) := -2, \, \textbf{t}_2^{(p*)}(d) := 0$ for $d < -4$ or $-d \not\equiv$ square (mod $4p$) ($p$ = 2, 3, 5). For the relation between two modular trace functions, see $\cite{Kim08}$.
\begin{rmk}
For $p = 1$, we put $j_1^*(\tau) := j(\tau) - 744 = \{(\eta(\tau)/\eta(2\tau))^8 + 2^8 (\eta(2\tau)/\eta(\tau))^{16}\}^3-744$ and $\textbf{t}_m(d) := \textbf{t}_m^{(1*)}(d)$.
\end{rmk}
\section{Proof of Theorem \ref{main1}}
We give a proof only for the case $p = 3$; the other cases are proved in the same way.
\begin{dfn}
For every positive integer $t$, we define the operator $U_t$ by
\begin{eqnarray*}
\biggl(\sum a_n q^n\biggr) \biggr|U_t := \sum a_{tn} q^n.
\end{eqnarray*}
\end{dfn}
Then $U_t$ sends a modular form to a modular form of the same weight but raises the level in general. To prove Theorem \ref{main1}, we need the following theorem, which is a special case $f = \varphi_m(j_p^*(\tau))$ of Theorem 1.1 in \cite{BF06}.
\begin{thm}
\label{BF}
The function
\begin{eqnarray*}
g_m^{(p*)}(\tau) := \sum_{d > 0} \textbf{t}_m^{(p*)}(d) q^d + (\sigma_1(m) + p \sigma_1(m/p)) - \sum_{k |m}k q^{-k^2}
\end{eqnarray*}
$($where $\sigma_1(x) = 0$ if $x \not\in \mathbb{Z})$ is a meromorphic modular form of weight $3/2$, holomorphic outside the cusps, with respect to $\Gamma_0(4p)$, that is,
\begin{eqnarray*}
g_m^{(p*)}(\tau) \in M_{3/2}^{mer}(\Gamma_0(4p)).
\end{eqnarray*}
Here $M_k^{mer}(\Gamma)$ denotes the space of meromorphic modular forms of weight $k$ with respect to $\Gamma$.
\end{thm}
We prove Theorem\ref{main1}. For the modular form $f(\tau) = \sum a_n q^n$, we define the functions $\tilde{f}_0, \ \tilde{f}_1 \ and\ \tilde{f}_2$ by
\begin{eqnarray*}
\tilde{f}_0(\tau) &:=& \frac{1}{3} \left\{f(\tau) + f(\tau + \frac{1}{3}) + f(\tau + \frac{2}{3})\right\},\\
\tilde{f}_1(\tau) &:=& \frac{1}{3} \left\{f(\tau) + \zeta^{-1}f(\tau + \frac{1}{3}) + \zeta f(\tau + \frac{2}{3})\right\},\\
\tilde{f}_2(\tau) &:=& \frac{1}{3} \left\{f(\tau) + \zeta f(\tau + \frac{1}{3}) + \zeta^{-1}f(\tau + \frac{2}{3})\right\}
\end{eqnarray*}
where $\zeta = e^{2\pi i/3}$. For each $k \pmod{3}$, then $\tilde{f}_k$ has a Fourier expansion of the form $\tilde{f}_k(\tau) = \sum_{n \equiv k (3)}a_n q^n$, and it is also a modular form of the same weight. By Theorem \ref{BF}, we have
\begin{eqnarray*}
g_2^{(3*)}(\tau) = \sum_{d = -4}^{\infty} \textbf{t}_2^{(3*)}(d) q^d \in M_{3/2}^{mer}(\Gamma_0(12)).
\end{eqnarray*}
Now consider the modular form $g_2^{(3*)}(\tau) \cdot \theta_0(\tau)$ where $\theta_0(\tau) := \sum_{n \in \mathbb{Z}} q^{n^2} \in M_{1/2}(\Gamma_0(4))$. This form is of weight 2 and we have
\begin{eqnarray*}
g_2^{(3*)}(\tau) \cdot \theta_0(\tau) = \biggl(\sum_{d = -4}^{\infty}\textbf{t}_2^{(3*)}(d) q^d \biggr) \cdot \biggl(\sum_{r \in \mathbb{Z}}q^{r^2} \biggr) = \sum_{n = -4}^{\infty} \biggl(\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma_0(12)).
\end{eqnarray*}
Similarly, the product $g_2^{(3*)}(\tau) \cdot \theta_0(9\tau)$ is also a modular form of weight 2 and its Fourier expansion is
\begin{eqnarray*}
g_2^{(3*)}(\tau) \cdot \theta_0(9\tau) = \sum_{n = -4}^{\infty} \biggl(\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(n - (3r)^2) \biggr) q^n = \sum_{n = -4}^{\infty} \biggl(\sum_{r \equiv 0 (3)} \textbf{t}_2^{(3*)}(n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma_0(36)).
\end{eqnarray*}
We put
\begin{eqnarray*}
F(\tau) &:=& \biggl(g_2^{(3*)}(\tau) \cdot \theta_0(\tau)\biggr) \bigg|U_4 = \sum_{n = -1}^{\infty} \biggl(\sum_{r \in \mathbb{Z}} \textbf{t}_2^{(3*)}(4n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma(12)), \\
G(\tau) &:=& \biggl(g_2^{(3*)}(\tau) \cdot \theta_0(9\tau)\biggr) \bigg|U_4 = \sum_{n = -1}^{\infty} \biggl(\sum_{r \equiv 0 (3)} \textbf{t}_2^{(3*)}(4n - r^2) \biggr) q^n \in M_2^{mer}(\Gamma(36)).
\end{eqnarray*}
Then $F(\tau)$ and $G(\tau)$ are meromorphic modular forms of weight 2. Moreover, for
\begin{eqnarray*}
j'_3(\tau) &=& \sum_{n = -1}^{\infty} n c_n^{(3)} q^n \in M_2^{mer}(\Gamma_0(3)),\\
E_2^{(3)}(\tau) &:=& \frac{1}{2}(3E_2(3\tau) - E_2(\tau)) = 1 + 12\sum_{n = 1}^{\infty} \sigma_1^{(3)}(n) q^n \in M_2(\Gamma_0(3)),
\end{eqnarray*}
(where the prime denotes $(2\pi i)^{-1} d/d\tau$ and $E_2(\tau) := 1 - 24\sum_{n = 1}^{\infty} \sigma_1(n) q^n$ is the Eisenstein series of weight 2), we put
\begin{eqnarray*}
H(\tau) := j'_3(\tau) - \frac{3}{2} E_2^{(3)}(\tau) = -\frac{1}{q} - \frac{3}{2} + \sum_{n = 1}^{\infty} (n c_n^{(3)} - 18\sigma_1^{(3)}(n)) q^n \in M_2^{mer}(\Gamma_0(3)).
\end{eqnarray*}
Then, the theorem in the case of $p = 3$ is equivalent to the following identities of modular forms:
\begin{eqnarray*}
2\tilde{H}_0(\tau) = -\tilde{G}_0(\tau), \ \ 2\tilde{H}_1(\tau) = \tilde{F}_1(\tau), \ \ 2\tilde{H}_2(\tau) = \tilde{F}_2(\tau).
\end{eqnarray*}
Since these modular forms are of weight 2 on $\Gamma(36)$, we see that, by the Riemann-Roch theorem, it is enough to check the coincidence of Fourier coefficients on both sides of the equalities up to $q^{3960}$. We checked this by using Mathematica and Pari-GP.\\
Similarly, we can show the equation $j_3^*(\tau) = j_3(\tau) - 3(j_3|U_3)(\tau)$, and we obtain $c_n^{(3*)} = c_n^{(3)} - 3c_{3n}^{(3)}$.
\section{Proof of Theorem \ref{main2}}
In this section, we give an overview of a proof. Since we can prove any case in the same way as \cite{Sam15}, we give a proof only for the case $p =$ 3. First, we prepare for a proof.
\begin{dfn}
The binary quadratic forms
\begin{eqnarray*}
\left\{ \begin{array}{ll}
[3, 0, d/12] & (-d \equiv 0 \pmod{12}), \\
\lbrack 3, 1, (d+1)/12 \rbrack \ ,\ \lbrack 3, -1, (d+1)/12 \rbrack & (-d \equiv 1 \pmod{12}), \\
\lbrack 3, 2, (d+4)/12 \rbrack \ ,\ \lbrack 3, -2, (d+4)/12 \rbrack & (-d \equiv 4 \pmod{12}), \\
\lbrack 3, 3, (d+9)/12 \rbrack & (-d \equiv 9 \pmod{12}) \\
\end{array} \right.
\end{eqnarray*}
are forms with discriminant $-d$ and are called the principal form of discriminant $-d$.
\end{dfn}
\begin{lem}
The following conditions are equivalent for a form $Q \in \mathcal{Q}_{d, 3}$:\\
$(1)$ There are $x, y \in \mathbb{Z}$ such that $Q(x, y) = 3$.\\
$(2)$ $Q$ is $\Gamma_0^*(3)$-equivalent to $[3, B, C]$ for some $B, C \in \mathbb{Z}$.\\
$(3)$ $Q$ is $\Gamma_0^*(3)$-equivalent to a principal form of discriminant $-d$.
\end{lem}
This lemma can be proved in the same way as Lemma 2.2 in \cite{Sam15}. The key theorem for the proof of Theorem\ref{main2} is the following.
\begin{thm}
$($Laplace's method$)$. Suppose that $h(t)$ is a real-valued $C^2$-function defined on the interval $(a, b)$ $($with $a, b \in \mathbb{R}$$)$. If we further suppose that $h$ has a unique maximum at $t = c$ with $a < c < b$ so that $h'(c) = 0$ and $h''(c) < 0$, then, we have
\begin{eqnarray*}
\int_a^b e^{\lambda h(t)} dt \sim e^{\lambda h(c)} \biggl(\frac{-2\pi}{\lambda h''(c)} \biggr)^{1/2}
\end{eqnarray*}
as $\lambda \to \infty$.
\end{thm}
We prove Theorem \ref{main2}. By definition,
\begin{eqnarray*}
\textbf{t}_2^{(3*)}(d) &:=& \sum_{Q \in \mathcal{Q}_{d,3} / \Gamma_0^*(3)} \frac{1}{|\Gamma_0^*(3)_Q|} \varphi_2(j_3^*(\alpha_Q)).
\end{eqnarray*}
If $Q = [a, b, c]$ is the element of $\mathcal{Q}_{d, 3}$, we have
\begin{eqnarray*}
e^{2\pi i \alpha_Q} = \exp \biggl(2\pi i \biggl(\frac{-b + i \sqrt{d}}{2a} \biggr) \biggr) = \exp \biggl(-\frac{\pi i b}{a} \biggr) \exp \biggl(-\frac{\pi \sqrt{d}}{a} \biggr)
\end{eqnarray*}
and consequently;
\begin{eqnarray*}
\varphi_2(j_3^*(\alpha_Q)) &=& q^{-2} + O(q)\\
&=& \exp \biggl(\frac{2\pi i b}{a} \biggr) \exp \biggl(\frac{2\pi \sqrt{d}}{a} \biggr) + O\biggl(\exp \biggl(-\frac{\pi \sqrt{d}}{a} \biggr)\biggr).
\end{eqnarray*}
By this calculation, the contribution to $\textbf{t}_2^{(3*)}(d)$ comes only from classes of forms with $a = 3$. By Lemma 4.2, any such form is equivalent to a principal form, so that we have
\begin{eqnarray*}
\textbf{t}_2^{(3*)}(d) = O\biggl(\exp \biggl(-\frac{\pi \sqrt{d}}{3}\biggr)\biggr) + \exp \biggl(\frac{2\pi \sqrt{d}}{3} \biggr) \times \left\{ \begin{array}{ll} 1 & (d \equiv 0, 3\ \bmod 12), \\
-1 & (d \equiv 8, 11 \bmod 12). \\
\end{array} \right.
\end{eqnarray*}
Combining this formula with Theorem\ref{main1}, we obtain
\begin{eqnarray*}
c_n^{(3)} \sim \frac{1}{2n} \times \left\{ \begin{array}{ll}
-\sum_{\substack{r \equiv 0 (3) \\ 4n \geq r^2}} \exp \bigl(2\pi \sqrt{4n - r^2}/3 \bigr) & (n \equiv 0 \bmod 3), \\
\sum_{\substack{r \equiv 1, 2 (3) \\ 4n \geq r^2}} \exp \bigl(2\pi \sqrt{4n - r^2}/3 \bigr) & (n \equiv 1 \bmod 3), \\
-\sum_{\substack{r \equiv 0 (3) \\ 4n \geq r^2}} \exp \bigl(2\pi \sqrt{4n - r^2}/3 \bigr) & (n \equiv 2 \bmod 3). \\
\end{array} \right.
\end{eqnarray*}
For each $k = 0, 1, 2$, we consider the sum
\begin{eqnarray*}
S_n^{(k)} := \frac{3}{2\sqrt{n}} \sum_{\substack{r \equiv k (3) \\ 4n \geq r^2}} e^{\frac{4}{3}\pi \sqrt{n} \sqrt{1 - \frac{r^2}{4n}}}
= \frac{3}{2\sqrt{n}} \sum_{\substack{l \in \mathbb{Z} \\ 4n \geq (3l + k)^2}} e^{\frac{4}{3}\pi \sqrt{n} \sqrt{1 - \frac{(3l + k)^2}{4n}}},
\end{eqnarray*}
and view this sum as a Riemann sum for the function $t \mapsto e^{4\pi \sqrt{n} \sqrt{1 - t^2}/3}$ : $(-1, 1) \to \mathbb{R}$. We can show that $S_n^{(k)}$ is asymptotic to the corresponding Riemann integral $J_n$ where
\begin{eqnarray*}
J_n := \int_{-1}^1 e^{4\pi \sqrt{n} \sqrt{1 - t^2}/3} dt.
\end{eqnarray*}
(For further detail, see \cite{Sam15}). Moreover, applying Laplace's method to the case $\lambda = \sqrt{n}$ and $h(t) = 4\pi \sqrt{1 - t^2}/3$ on $(-1, 1)$, we have
\begin{eqnarray*}
J_n \sim e^{\sqrt{n} \cdot 4 \pi/3} \cdot \biggl(\frac{-2\pi}{-4\pi \sqrt{n}/3} \biggr)^{1/2} = \frac{\sqrt{3}}{\sqrt{2} n^{1/4}} e^{4\pi \sqrt{n}/3}.
\end{eqnarray*}
Putting these asymptotic formulas together, we obtain
\begin{eqnarray*}
c_n^{(3)} &\sim& \frac{1}{3 \sqrt{n}} \times \left\{ \begin{array}{ll}
-S_n^{(0)} & (n \equiv 0 \bmod 3), \\
S_n^{(1)} + S_n^{(2)} & (n \equiv 1 \bmod 3), \\
-S_n^{(0)} & (n \equiv 2 \bmod 3), \\
\end{array} \right. \\
&\sim& \frac{e^{4\pi \sqrt{n}/3}}{\sqrt{6} n^{3/4}} \times \left\{ \begin{array}{ll}
-1 & (n \equiv 0 \bmod 3), \\
2 & (n \equiv 1 \bmod3), \\
-1 & (n \equiv 2 \bmod3) \\
\end{array} \right.
\end{eqnarray*}
as $n \to \infty$.
\section{Tables of $\textbf{t}_m^{(p*)}(d)$ and $\textbf{t}_m^{(p)}(d)$ \ $(-4 \leq d \leq 50)$}
\begin{table}[h]
\begin{minipage}{9cm}
\begin{tabular}[t]{|c|r|r|r|r|} \hline
$d$ & $\textbf{t}_1^{(2*)}(d)$ & $\textbf{t}_2^{(2*)}(d)$ & $\textbf{t}_1^{(2)}(d)$ & $\textbf{t}_2^{(2)}(d)$\\ \hline \hline
$-$4 & 0 & $-$2 & 0 & $-$4 \\ \hline
$-$1 & $-$1 & $-$1 & $-$1 & $-$1 \\ \hline
0 & 1 & 5 & 2 & 10 \\ \hline
4 & $-$26 & 518 & $-$52 & 1036 \\ \hline
7 & $-$23 & $-$8215 & $-$23 & $-$8215 \\ \hline
8 & 76 & 7180 & 152 &14360 \\ \hline
12 & $-$248 & 52760 & $-$496 & 105520 \\ \hline
15 & $-$1 & $-$385025 & $-$1 & $-$385025 \\ \hline
16 & 518 & 287710 & 1036 & 575420 \\ \hline
20 & $-$1128 &1263640 & $-$2256 & 2527280 \\ \hline
23 & $-$94 & $-$6987870 & $-$94 & $-$6987870 \\ \hline
24 & 2200 & 4831256 & 4400 & 9662512 \\ \hline
28 & $-$4096 & 16572370 & $-$8192 & 33144740 \\ \hline
31 & 93 & $-$78987171 & 93 & $-$78987171 \\ \hline
32 & 7180 & 52263100 & 14360 & 104526200 \\ \hline
36 & $-$12418 & 153553438 & $-$24836 & 307106876 \\ \hline
39 & $-$236 & $-$663068908 & $-$236 & $-$663068908 \\ \hline
40 & 20632 & 425670680 & 41264 & 851341360 \\ \hline
44 & $-$33512 & 1122593352 & $-$67024 & 2245186704 \\ \hline
47 & 235 & $-$4515675925 & 235 & $-$4515675925 \\ \hline
48 & 53256 & 2835914280 & 106512 & 5671828560 \\ \hline
\end{tabular}
\end{minipage}
\begin{minipage}{9cm}
\begin{tabular}[t]{|c|r|r|r|r|} \hline
$d$ & $\textbf{t}_1^{(3*)}(d)$ & $\textbf{t}_2^{(3*)}(d)$ & $\textbf{t}_1^{(3)}(d)$ & $\textbf{t}_2^{(3)}(d)$\\ \hline \hline
$-$4 & 0 & $-$2 & 0 & $-$2 \\ \hline
$-$1 & $-$1 & $-$1 & $-$1 & $-$1 \\ \hline
0 & 1 & 3 & 2 & 6 \\ \hline
3 & $-$7 & 33 & $-$14 & 66 \\ \hline
8 & $-$34 & $-$410 & $-$34 & $-$410 \\ \hline
11 & 22 & $-$1082 & 22 &$-$1082 \\ \hline
12 & 26 & 1428 & 52 & 2856 \\ \hline
15 & $-$69 & 3195 & $-$138 & 6390 \\ \hline
20 & $-$116 & $-$11892 & $-$116 & $-$11892 \\ \hline
23 & 115 & $-$22797 & 115 & $-$22797 \\ \hline
24 & 174 & 28710 & 348 & 57420 \\ \hline
27 & $-$241 & 53223 & $-$482 & 106446 \\ \hline
32 & $-$410 & $-$140222 & $-$410 & $-$140222 \\ \hline
35 & 492 & $-$240500 & 492 & $-$240500 \\ \hline
36 & 492 & 287244 & 984 & 574488 \\ \hline
39 & $-$705 & 477567 & $-$1410 & 955134 \\ \hline
44 & $-$1060 & $-$1081096 & $-$1060 & $-$1081096 \\ \hline
47 & 1272 & $-$1718792 & 1272 & $-$1718792 \\ \hline
48 & 1442 & 2004918 & 2884 & 4009836 \\ \hline
\end{tabular}
\end{minipage}
\end{table}
\begin{table}[h]
\begin{tabular}{|c|r|r|r|r|} \hline
$d$ & $\textbf{t}_1^{(5*)}(d)$ & $\textbf{t}_2^{(5*)}(d)$ & $\textbf{t}_1^{(5)}(d)$ & $\textbf{t}_2^{(5)}(d)$\\ \hline \hline
$-$4 & 0 & $-$2 & 0 & $-$2 \\ \hline
$-$1 & $-$1 & $-$1 & $-$1 & $-$1 \\ \hline
0 & 1 & 3 & 2 & 6 \\ \hline
4 & $-$8 & $-$6 & $-$8 & $-$6 \\ \hline
11 & $-$12 & $-$124 & $-$12 & $-$124 \\ \hline
15 & $-$19 & 93 & $-$38 & 186 \\ \hline
16 & $-$6 & $-$270 & $-$6 & $-$270 \\ \hline
19 & 20 & 132 & 20 & 132 \\ \hline
20 & 6 & 268 & 12 & 536 \\ \hline
24 & $-$44 & 216 & $-$44 & 216 \\ \hline
31 & $-$39 & $-$1863 & $-$39 & $-$1863 \\ \hline
35 & $-$44 & 1668 & $-$88 & 3336 \\ \hline
36 & 20 & $-$3054 & 20 & $-$3054 \\ \hline
39 & 53 & 1653 & 53 & 1653 \\ \hline
40 & 56 & 2868 & 112 & 5736 \\ \hline
44 & $-$136 & 2416 & $-$136 & 2416 \\ \hline
\end{tabular}
\end{table}
\end{ack}
\noindent
T. Matsusaka: Graduate School of Mathematics, Kyushu University, Motooka 744, Nishi-ku Fukuoka 819-0395, Japan\\
e-mail: [email protected]\\
\noindent
R. Osanai:\\
e-mail: [email protected]
\end{document}
|
\begin{document}
\title{A Tube-based MPC Scheme for Interaction Control of Underwater Vehicle Manipulator Systems\\
\thanks{This work was supported by the H2020 ERC Grant BUCOPHSYS, the EU H2020 Co4Robots project, the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR) and the Knut och Alice Wallenberg Foundation (KAW).}
}
\author{\IEEEauthorblockN{Alexandros Nikou, Christos K. Verginis and Dimos V. Dimarogonas}
\IEEEauthorblockA{Department of Automatic Control \\
School of Electrical Engineering and Computer Science\\
KTH Royal Institute of Technology, Stockholm, Sweden \\
{\tt \{anikou,cverginis,dimos\}@kth.se}}
}
\maketitle
\begin{abstract}
Over the last years, the development of Autonomous Underwater Vehicles (AUV) with attached robotic manipulators, the so-called Underwater Vehicle Manipulator System (UVMS), has gained significant research attention, due to the ability of interaction with underwater environments. In such applications, force/torque controllers which guarantee that the end-effector of the UVMS applies desired forces/torques towards the environment, should be designed in a way that state and input constraints are taken into consideration. Furthermore, due to their complicated structure, unmodeled dynamics as well as external disturbances may arise. Motivated by this, we proposed a robust Model Predicted Control Methodology (NMPC) methodology which can handle the aforementioned constraints in an efficient way and it guarantees that the end-effector is exerting the desired forces/torques towards the environment. Simulation results verify the validity of the proposed framework.
\end{abstract}
\section{Introduction}
Most of the underwater manipulation tasks, such as maintenance of ships, underwater weld inspection, surveying oil/gas searching, require the manipulator mounted on the vehicle to be in contact with the underwater object or environment (see \cite{antonelli, cieslak2015autonomous}). The aforementioned tasks are usually complex due to highly nonlinear dynamics, the presence of uncertainties, external disturbances as well as state and control input (actuation) constraints. Thus, these constraints should be taken into account in the force control design process in an efficient way.
Motivated by the aforementioned, this paper considers the modeling of a general UVMS in compliant contact with a planar surface, and the development of a constrained Nonlinear Model Predictive Control (NMPC) scheme for force/torque control. NMPC for manipulation of nominal system dynamics has been proposed in \cite{alex_med} for stabilization of ground vehicles with attached manipulators to pre-defined positions. In this work, we propose a novel robust tube-based NMPC force control approach that efficiently deals with state and input constraints and achieves a desired exerted force from the UVMS to the environment. In particular, the controller consists of two terms: a nominal control input, which is computed on-line and is the outcome of a Finite Horizon Optimal Control Problem (FHOCP) that is repeatedly solved at every sampling time, for its nominal system dynamics; and an additive state feedback law which is computed off-line and guarantees that the real trajectory of the closed-loop system will belong to a hyper-tube centered along the nominal trajectory. The volume of the hyper-tube depends on the upper bound of the disturbances, the bounds of the Jacobian matrix as well as Lipschitz constants of the UVMS dynamics. Under the assumption that the FHOCP is feasible at time $t = 0$, we guarantee the boundedness of the closed-loop system states.
The rest of this manuscript is structured as follows: Section \ref{sec:notation_preliminaries} provides the notation that will be used as well as necessary background knowledge; in Section \ref{sec:problem_formulation}, the problem treated in this paper is formally defined; Section \ref{sec:main_results} contains the main results of the paper; Section \ref{sec:simulation_results} is devoted to numerical simulations; and in Section \ref{sec:conclusions}, conclusions and future research directions are discussed.
\section{Notation and Preliminaries} \label{sec:notation_preliminaries}
Define by $\mathbb{N}$ and $\mathbb{R}$ the sets of positive integers and real numbers, respectively. Given the set $\mathcal{S}$, define by $S^n \coloneqq S \times \dots \times S$, its $n$-fold Cartesian product. Given vector $z \in \mathbb{R}^{n}$ define by $$\|z\|_{2} \coloneqq \sqrt{z^\top z}, \ \ \|z\|_{P} \coloneqq \sqrt{z^\top P z},$$ its Euclidean and weighted norm, with $P \ge 0$. Given vectors $z_1$, $z_2 \in \mathbb{R}^3$, $\mathcal{S}: \mathbb{R}^3 \to \mathfrak{so}(3)$ stands for the skew-symmetric matrix defined according to $\mathcal{S}(z_1) z_2 = z_1 \times z_2$ where $$\mathfrak{so}(3) \coloneqq \left\{\mathcal{S} \in \mathbb{R}^{3\times 3} : z^\top \mathcal{S}(\cdot) z = 0, \forall z \in \mathbb{R}^{3} \right\}.$$ $\lambda_{\scriptscriptstyle \min}(P)$ stands for the minimum absolute value of the real part of the eigenvalues of $P \in \mathbb{R}^{n \times n}$; $0_{m \times n} \in \mathbb{R}^{m \times n}$ and $I_n \in \mathbb{R}^{n \times n}$ stand for the $m \times n$ matrix with all entries zeros and the identity matrix, respectively. Given coordination frames $\Sigma_i$, $\Sigma_j$, denote by $R^j_i$ the transformation from $\Sigma_i$ to $\Sigma_j$. Given~sets~$\mathcal{S}_1$, $\mathcal{S}_2$~$\subseteq \mathbb{R}^n$, $\mathcal{S} \subseteq \mathbb{R}^{m}$~and~matrix $B \in \mathbb{R}^{n \times m}$,~the \emph{Minkowski addition}, the~\emph{Pontryagin~difference} and the \emph{matrix-set multiplication} are respectively defined by:
\begin{align*}
\mathcal{S}_1 \oplus \mathcal{S}_2 & \coloneqq \{s_1 + s_2 : s_1 \in \mathcal{S}_1, s_2 \in \mathcal{S}_2\}, \\
\mathcal{S}_1 \ominus \mathcal{S}_2 & \coloneqq \{s_1 : s_1+s_2 \in \mathcal{S}_1, \forall s_2 \in \mathcal{S}_2\}, \\
B \circ \mathcal{S} & \coloneqq \{b: b = Bs, s \in \mathcal{S} \}.
\end{align*}
\begin{lemma} \cite{alex_IJRNC_2018} \label{lemma:basic_ineq}
For any constant $\rho > 0$, vectors $z_1$, $z_2 \in \mathbb{R}^n$ and matrix $P \in \mathbb{R}^{n \times n}$, $P > 0$ it holds that $$z_1 P z_2 \le \tfrac{1}{4 \rho} z_1^\top P z_1 + \rho z_2^\top P z_2.$$
\end{lemma}
\begin{definition} \label{def:RPI_set} \cite{alex_IJRNC_2018}
Consider a dynamical system $\dot{\chi} = f(\chi,u,d)$ where: $\chi \in \mathcal{X}$, $u \in \mathcal{U}$, $d \in \mathcal{D}$ with initial condition $\chi(0) \in \mathcal{X}$. A set $\mathcal{X}' \subseteq \mathcal{X}$ is a \emph{Robust Control Invariant (RCI) set} for the system, if there exists a feedback control law $u \coloneqq \kappa(\chi) \in \mathcal{U}$, such that for all $\chi(0) \in \mathcal{X}'$ and for all $d \in \mathcal{D}$ it holds that $\chi(t) \in \mathcal{X}'$ for all $t \ge 0$, along every solution $\chi(t)$.
\end{definition}
\section{Problem Formulation} \label{sec:problem_formulation}
\subsection{Kinematic Model}
Consider a UVMS which is composed of an AUV and a $n$ Degree Of Freedom (DoF) manipulator mounted on the base of the vehicle. The AUV can be considered as a six DoF rigid body with position and orientation vector $\eta \coloneqq [x, y, z~\vline~\phi, \theta, \psi]^\top \in \mathbb{R}^6$, where the components of the vectors have been named according to SNAME \cite{SNAME} as surge, sway, heave, roll, pitch and yaw respectively. The joint angular position state vector of the manipulator is defined by $q \coloneqq [ q_1,\dots,q_n]^\top \in \mathbb{R}^n$. Define by $\dot{q} \coloneqq [\dot{q}_1,\dots,\dot{q}_n]^\top \in \mathbb{R}^n$ the corresponding joint velocities.
In order to describe the motion of the combined system, the earth-fixed inertial frame $\Sigma_I$, the body-fixed frame $\Sigma_B$ and the end-effector fixed frame $\Sigma_E$ are introduced (see Fig. \ref{fig:uvms_frames}). Moreover, without loss of generality, the reference frame $\Sigma_0$ is chosen to be located at the manipulator's base, and the frames $\Sigma_1, \ldots, \Sigma_n$ are located to the $1$-st$,\ldots,n$-th link of the manipulator, respectively, under the Denavit-Hartenberg convention \cite{sciavicco2012modelling}. The translational and rotational kinematic equations for the AUV system (see \cite{antonelli}) are given by:
\begin{subequations} \label{eq:kin}
\begin{align}
\dot{\eta}
& =
\begin{bmatrix}
\dot{\eta}_1 \\
\dot{\eta}_2 \\
\end{bmatrix}
= \mathfrak{J}(\eta_2)
\begin{bmatrix}
\nu_1 \\
\nu_2 \\
\end{bmatrix}, \\
\mathfrak{J}(\eta_2) & \coloneqq
\begin{bmatrix}
\mathfrak{J}_1(\eta_2) & 0_{3 \times 3} \\
0_{3 \times 3} & \mathfrak{J}_{2}(\eta_2) \\
\end{bmatrix}, \\
\mathfrak{J}_1(\eta_2) & \coloneqq
\begin{bmatrix}
c_{\theta} c_{\psi} & s_{\phi} s_{\theta} c_{\psi}-s_{\psi} c_{\phi} &
s_{\theta} c_{\phi} c_{\psi}+s_{\phi} s_{\psi} \\
s_{\psi} c_{\theta} & s_{\phi} s_{\theta} s_{\psi}+c_{\phi} c_{\psi} &
s_{\theta} s_{\psi} c_{\phi}-s_{\phi} c_{\psi} \\
-s_{\theta} & s_{\phi} c_{\theta} & c_{\phi} c_{\theta} \\
\end{bmatrix}, \\
\mathfrak{J}_2(\eta_2) & \coloneqq
\begin{bmatrix}
1 & \tfrac{s_{\phi} s_{\theta}}{c_{\theta}} & \tfrac{c_{\phi} s_{\theta}}{c_{\theta}} \\
0 & c_{\phi} & -s_{\phi} \\
0 & \tfrac{s_{\phi}}{c_{\theta}} & \tfrac{c_{\phi}}{c_{\theta}} \\
\end{bmatrix},
\end{align}
\end{subequations}
where $\eta_{1} \coloneqq \left[ x, y, z \right]^{\tau} \in \mathbb{R}^3$, $\eta_{2} \coloneqq \left[\phi, \theta, \psi \right]^\top \in \mathbb{R}^3 $ denote the position vector and the orientation vector of the frame $\Sigma_B$ relative to the frame $\Sigma_I$, respectively; $ \nu_{1}$, $\nu_{2} \in \mathbb{R}^3 $ denote the linear and the angular velocity of the frame $\Sigma_B$ with respect to $\Sigma_I$ respectively; $\mathfrak{J}(\eta_2) \in \mathbb{R}^{6 \times 6}$ stands for the Jacobian matrix transforming the velocities from $\Sigma_B$ to $\Sigma_I$; $\mathfrak{J}_{1}(\eta_2)$, $\mathfrak{J}_2(\eta_2) \in \mathbb{R}^{3 \times 3}$ are the corresponding parts of the Jacobian related to position and orientation, respectively; The notation $s_{\varsigma}$ and $c_{\varsigma}$ stand for the trigonometric functions $\sin(\varsigma)$ and $\cos(\varsigma)$ of an angle $\varsigma \in \mathbb{R}$, respectively.
\begin{figure}
\caption{An AUV Equipped with a n DoF manipulator}
\label{fig:uvms_frames}
\end{figure}
\noindent Denote by $$\mathfrak{q} \coloneqq \left[\eta_1^\top, \eta_2^\top, q^\top \right]^\top\in\mathbb{R}^{6+n},$$ the pose configuration vector of the UVMS. Let $\mathfrak{p}$, $\mathfrak{o} \in \mathbb{R}^3$ be the position and orientation vectors of the end-effector with reference to the frame $\Sigma_I$, respectively. The vectors $\mathfrak{p}$, $\mathfrak{o}$ depend on the pose $\mathfrak{q}$ and they can be obtained by the the following homogeneous transformation:
\begin{equation}
\mathfrak{T}(\mathfrak{q}) \coloneqq
\begin{bmatrix}
R_E^{I}(\mathfrak{q}) & \mathfrak{p}(\mathfrak{q}) \\
0_{1 \times 3} & 1 \\
\end{bmatrix}
=
T_B^I T_0^B T_1^0 \cdots T_n^{n-1} T_E^n, \label{eq:forw_kinematics}
\end{equation}
where: $T^j_i$ is the homogeneous transformation matrix describing the position and orientation of frame $\Sigma_i$ with reference to the frame $\Sigma_j$ with $i$, $j \in \{1,\dots, n, I, 0, B, E\}$. The end-effector linear velocity $\dot{\mathfrak{p}} \in \mathbb{R}^3$ and the time derivative or Euler angles $\dot{\mathfrak{o}} \in \mathbb{R}^3$ are related to the body-fixed velocities $\nu_1$, $\nu_2$ and $\dot{q}$ with the following \emph{kinematics model:}
\begin{equation} \label{eq:kinematics}
\dot{\chi}
=
J(\mathfrak{q}) \zeta,
\end{equation}
where $$\chi \coloneqq [\mathfrak{p}^\top, \mathfrak{o}^\top]^\top \in \mathbb{R}^{6}, \ \ \zeta \coloneqq \left[ \nu_1^\top, \nu_2^\top, \dot{q}^\top \right]^\top \in \mathbb{R}^{{6+n}},$$ is the body-fixed system velocity vector. The Jacobian transformations matrices $$J(\mathfrak{q}) \in \mathbb{R}^{6 \times (6+n)}, \ \ J_{\rm pos}(\mathfrak{q}) \in \mathbb{R}^{3 \times (6 +n)}, \ \ J_{\rm{or}}(\mathfrak{q}) \in \mathbb{R}^{3 \times (6 +n)},$$ are respectively defined by:
\begin{align*}
J(\mathfrak{q}) &\coloneqq \begin{bmatrix}
J_{\rm pos}(\mathfrak{q}) \\
J_{\rm or}(\mathfrak{q}) \\
\end{bmatrix}, \\
J_{\rm pos}(\mathfrak{q}) &\coloneqq
\left[\mathfrak{J}_1(\eta_2)~\vline~- \mathfrak{J}_1(\eta_2) \mathcal{S}(p_{ee})~\vline~R_0^{I} J_{e,1} \right], \\
J_{\rm or}(\mathfrak{q}) &\coloneqq
\left[0_{3 \times 3}~\vline~\mathfrak{J}_2(\mathfrak{o}) R_B^E~\vline~\mathfrak{J}_2(\mathfrak{o}) R_0^{E} J_{e,2} \right].
\end{align*}
In the latter, the vector $p_{ee} \in \mathbb{R}^{3}$ is the local position of the end-effector with reference to the frame $\Sigma_B$; the matrices $J_{e,1}$, $J_{e,2} \in \mathbb{R}^{3 \times n}$ represent the manipulator Jacobian matrices with respect to the frame $\Sigma_0$; and $\mathcal{S}(\cdot)$ the skew-symmetric matrix as given in Section \ref{sec:notation_preliminaries}. For the aforementioned transformations we refer to \cite{sciavicco2012modelling}.
\subsection{Dynamic Model}
When the end-effector of the robotic system is in contact with the environment, the force at the tip of the manipulator acts on the whole system according to the following uncertain nonlinear dynamics:
\begin{align} \label{eq:dynamics}
\dot{\zeta} = f(\chi, \zeta)+ \mathfrak{u} + d(\mathfrak{q}, \zeta, t),
\end{align}
where:
\begin{align}
& \hspace{-4mm} f(\chi, \zeta) \coloneqq \notag \\
& \hspace{-4mm} - M(\mathfrak{q})^{-1} \Big\{ C(\zeta, \mathfrak{q}) \zeta + D(\zeta, \mathfrak{q}) \zeta +g(\mathfrak{q}) + J^{\top}(\mathfrak{q}) \mathfrak{F}(\chi) \Big\}, \hspace{-4mm} \label{eq:func_h}
\end{align}
where $M(\mathfrak{q}) \in \mathbb{R}^{({6+n}) \times ({6+n})}$ is the inertia matrix for which it holds that: $z^\top M(\mathfrak{q}) z > 0$, $\forall z \in \mathbb{R}^{6+n}$; $C(\zeta, \mathfrak{q}) \in \mathbb{R}^{({6+n}) \times ({6+n})}$ is the matrix of Coriolis and centripetal terms; $D(\zeta, \mathfrak{q}) \in \mathbb{R}^{({6+n}) \times ({6+n})}$ is the matrix of dissipative effects; $d(\mathfrak{q}, \zeta, t) \in \mathbb{R}^{6+n}$ is a vector that models the external disturbances, uncertainties and unmodeled dynamics of the system; $g(\mathfrak{q}) \in \mathbb{R}^{({6+n})}$ is the vector of gravity and buoyancy effects; $\mathfrak{u} \in \mathbb{R}^{6+n}$ denotes the vector of the propulsion forces and moments acting on the vehicle in the frame $\Sigma_{B}$ as well as the joint torques; $\mathfrak{F}(\chi) \in \mathbb{R}^{6}$ is the vector of interaction forces and torques exerted by the end-effector towards the environment expressed in $\Sigma_I$.
In this paper, an interaction between the end-effector and a frictionless, elastically compliant surface is assumed. Then, according to \cite{siciliano_force_control}, the vector of interaction forces and torques that is exerted by the end-effector can be written as:
\begin{equation} \label{eq:force_F}
\mathfrak{F}(\chi) \coloneqq K (\chi - \chi_{\scriptscriptstyle \rm eq}),
\end{equation}
where $K \in \mathbb{R}^{6 \times 6}$, $K > 0$ stands for the stiffness matrix, which represents elastic coefficient of the environment, and $\chi_{\scriptscriptstyle \rm eq} \in \mathbb{R}^{6}$ is the given constant vector of the equilibrium position/orientation of the undeformed environment.
We also consider that the UVMS is in the presence of state and input constraints given by $\mathfrak{q} \in \mathcal{Q}$, $\zeta \in \mathcal{Z}$, $u \in \mathcal{U}$,
where $\mathcal{Q} \subseteq \mathbb{R}^{6+n}$, $\mathcal{Z} \subseteq \mathbb{R}^{6+n}$ and $\mathcal{U} \subseteq \mathbb{R}^{6+n}$ are \emph{connected sets containing the origin}. For certain technical reasons that will be presented thereafter, the constraints imposed to the configuration states $\mathfrak{q}$ are given by:
\begin{align} \label{eq:set_Q}
\mathcal{Q} \coloneqq \Big\{ \mathfrak{q} \in \mathbb{R}^{6+n} : & \ \lambda_{\scriptscriptstyle \min}\left[ \tfrac{J^{+}(\mathfrak{q})+J^{+}(\mathfrak{q})^\top}{2} \right] \ge \underline{J}, \notag \\
& \ \ \|J(\mathfrak{q})\|_2 \le \overline{J}, \ \|\dot{J}(\mathfrak{q})\|_{2} \le \widetilde{J} \Big\},
\end{align}
where $J^{+}(\mathfrak{q}) \coloneqq J(\mathfrak{q}) J(\mathfrak{q})^\top$ and $\underline{J}$, $\overline{J}$, $\widetilde{J} > 0$. According to \eqref{eq:forw_kinematics}, the constraints $\mathfrak{q} \in \mathcal{Q}$ impose also constraints on the vector $\chi \in \mathcal{X} \subseteq \mathbb{R}^{6}$, where the set $\mathcal{X}$ can be computed by the transformation $\mathfrak{T}(\mathfrak{q})$, as given in \eqref{eq:forw_kinematics}. Note also that the function $f$ given in \eqref{eq:func_h} is continuously differentiable in the set $\mathcal{Q} \times \mathcal{X} \times \mathcal{Z}$. Furthermore, assume bounded disturbances $d \in \mathcal{D}$ where:
$\mathcal{D}$ $\coloneqq \big\{d \in \mathbb{R}^{6+n}:$ $\|d(\mathfrak{q}, \zeta, t)\|_{2} \le \widetilde{d}$, $\forall (\mathfrak{q}$, $\zeta) \in \mathcal{Q}$ $\times \mathcal{Z}\big\}$, where $\widetilde{d} > 0$.
For the kinematics/dynamics \eqref{eq:kinematics},\eqref{eq:dynamics}, define the corresponding \emph{nominal kinematics/dynamics} by:
\begin{subequations}
\begin{align}
\dot{\overline{\chi}} & = J(\overline{\mathfrak{q}}) \overline{\zeta}, \label{eq:nom_kinematics} \\
\dot{\overline{\zeta}} & = f(\overline{\chi}, \overline{\zeta})+ \overline{u}, \label{eq:nom_dynamics}
\end{align}
\end{subequations}
where $d(\cdot) \equiv 0$, $\overline{\mathfrak{q}} \in \mathcal{Q}$, $\overline{\chi} \in \mathcal{X}$, $\overline{\zeta} \in \mathcal{Z}$ and $\overline{u} \in \mathcal{U}$. Define the stack vector $\overline{\xi} \coloneqq [\overline{\chi}, \overline{\zeta}]^\top \in \mathbb{R}^{12+n}$ and consider the linear nominal system $\dot{\overline{\xi}} = A \overline{\xi} + B \overline{u}, \ \ A \in \mathbb{R}^{(12+n) \times (12+n)}, \ \ B \in \mathbb{R}^{(12+n) \times (6+n)}$,
which is the outcome of the Jacobian linearization of the nominal dynamics \eqref{eq:nom_kinematics},\eqref{eq:nom_dynamics} around the equilibrium point $\xi = 0$. Due to the dimension of the control input ($6+n > 6$), the stabilization of the state $\overline{\chi}$ to the desired state $\chi_{\scriptscriptstyle \rm des}$ can be achieved. Therefore, the linear system is stabilizable.
\subsection{Problem Statement}
\begin{problem} \label{problem}
Consider a UVMS composed of an AUV and an attached manipulator with $n$ DoF, which is in contact with a surface of a compliant environment. The UVMS is governed by the kinematics and dynamics models given in \eqref{eq:kinematics} and \eqref{eq:dynamics}, respectively. The system is in the presence of state and input constraints as well as bounded disturbances which are respectively given by:
\begin{align} \label{eq:constr}
\mathfrak{q} \in \mathcal{Q}, \ \chi \in \mathcal{X}, \ \zeta \in \mathcal{Z}, \ \mathfrak{u} \in \mathcal{U}, \ d \in \mathcal{D}.
\end{align}
Given a vector $\mathfrak{F}_{\scriptscriptstyle \rm des} \in \mathbb{R}^{6}$ that satisfies \eqref{eq:force_F} and stands for the desired force/torque vector that the end-effector is required to exert towards a surface of the environment, design a \emph{feedback control law} $\mathfrak{u} \coloneqq \kappa(\chi, \zeta)$ such that $\lim\limits_{t \to \infty} \|\mathfrak{F}(\chi(t))-\mathfrak{F}_{\scriptscriptstyle \rm des}\|_{2} \to 0$, while all the constraints given in \eqref{eq:constr} are satisfied.
\end{problem}
\section{Main Results} \label{sec:main_results}
In this section, we propose a novel feedback control law that solves Problem \ref{problem} in a systematic way. Due to the fact that it is required to design a feedback control law that guarantees the minimization of the term $\|\mathfrak{F}(t)-\mathfrak{F}_{\scriptscriptstyle \rm des}\|_{2}$, as $t \to \infty$, under state and input constraints given by \eqref{eq:constr}, we utilize a Nonlinear Model Predictive Control (NMPC) framework \cite{michalska_1993, frank_1998_quasi_infinite, mayne_2000_nmpc}. Furthermore, since the UVMS is under the presence of disturbances/uncertainties $d \in \mathcal{D}$, we provide a robust analysis, the so-called tube-based robust NMPC approach \cite{yu_2013_tube, alex_IJRNC_2018}. In particular, first, the error states and the corresponding transformed constraints sets are defined in Section \ref{sec:error_constr}. Then, the proposed feedback control law consists of two parts: an on-line control law which is the outcome of a solution to a Finite Horizon Optimal Control Problem (FHOCP) for the nominal system dynamics (see Section \ref{sec:optimal_contol}); and a state feedback law which is designed off-line and guarantees that the real system trajectories always lie within a hyper-tube centered along the nominal trajectories (see \ref{sec:state_feedback_law}).
\subsection{Errors and Constraints} \label{sec:error_constr}
According to \eqref{eq:force_F}, for the error between the actual $\mathfrak{F}$ and the desired $\mathfrak{F}_{\scriptscriptstyle \rm des}$ forces/torques exerted from the end-effector to the surface it holds that: $\mathfrak{F}-\mathfrak{F}_{\scriptscriptstyle \rm des}$ $= K (\chi - \chi_{\scriptscriptstyle \rm eq})$ $-K (\chi_{\scriptscriptstyle \rm des}$ $- \chi_{\scriptscriptstyle \rm eq})$ $= K (\chi-\chi_{\scriptscriptstyle \rm des})$,
where $\chi_{\scriptscriptstyle \rm des} \coloneqq K^{-1} \mathfrak{F}_{\scriptscriptstyle \rm des} + \chi_{\scriptscriptstyle \rm eq} \in \mathbb{R}^{6}$. The latter implies that if we design a feedback control law $u = \kappa(\chi, \zeta)$ which guarantees that $\lim\limits_{t \to \infty} \|\chi(t)-\chi_{\scriptscriptstyle \rm des}\|_{2} \to 0$, while all the constraints given in \eqref{eq:constr} are satisfied, Problem \ref{problem} will have been solved.
Define the error state $e \coloneqq \chi-\chi_{\scriptscriptstyle \rm des} \in \mathbb{R}^{6}$. Then, the \emph{uncertain error kinematics/dynamics} are given by:
\begin{subequations}
\begin{align}
\dot{e} & = J(\mathfrak{q}) \zeta, \label{eq:unsrt_error_kin} \\
\dot{\zeta} & = f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)+ \mathfrak{u} + d(\mathfrak{q}, \zeta, t), \label{eq:unsrt_error_dyn}
\end{align}
\end{subequations}
and the corresponding \emph{nominal error kinematics/dynamics} by:
\begin{subequations}
\begin{align}
\dot{\overline{e}} & = J(\overline{\mathfrak{q}}) \overline{\zeta}, \label{eq:nom_error_kin} \\
\dot{\overline{\zeta}} & = f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \overline{\zeta})+ \overline{u}, \label{eq:nom_error_dyn}
\end{align}
\end{subequations}
In order to translate the constraints for the state $\chi \in \mathcal{X}$ to constraints that are dictated regarding the error $e$, the constraints set $\mathcal{E} \coloneqq \{e \in \mathbb{R}^{6}: e \in \mathcal{X} \oplus (-\chi_{\scriptscriptstyle \rm des}) \}$ is introduced.
\subsection{Feedback Control Design} \label{sec:state_feedback_law}
\noindent Consider the feedback law:
\begin{equation} \label{eq:control_law_u}
\mathfrak{u} \coloneqq \overline{u}(\overline{e}, \overline{\zeta}) + \kappa(e, \zeta, \overline{e}, \overline{\zeta}),
\end{equation}
which consists of a nominal control law $\overline{u}(\overline{e}, \overline{\zeta}) \in \mathcal{U}$ and a state feedback law $\kappa(\cdot)$. The control action $\overline{u}(\overline{e}, \overline{\zeta})$ will be the outcome of a FHOCP for the nominal kinematics/dynamics \eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn} which is solved on-line at each sampling time. The state feedback law $\kappa(\cdot)$ is used to guarantee that the real trajectories $e(t)$, $\zeta(t)$, which are the solution to \eqref{eq:unsrt_error_kin},\eqref{eq:unsrt_error_dyn}, always remain within a bounded hyper-tube centered along the nominal trajectories $\overline{e}(t)$,~$\overline{\zeta}(t)$ which are~the~solution~ to~\eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn}.
Define by $\mathfrak{e} \coloneqq e - \overline{e} \in \mathbb{R}^{6}$ and $\mathfrak{z} \coloneqq \zeta - \overline{\zeta} \in \mathbb{R}^{6+n}$ the deviation between the real states of the uncertain system \eqref{eq:unsrt_error_kin},\eqref{eq:unsrt_error_dyn} and the states of the nominal system \eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn}, respectively, with $\mathfrak{e}(0) = \mathfrak{z}(0) = 0$. It will be proved hereafter that the trajectories $\mathfrak{e}(t)$, $\mathfrak{z}(t)$ remain invariant in compact sets. The dynamics of the states $\mathfrak{e}$, $\mathfrak{z}$ are written as:
\begin{subequations}
\begin{align}
\dot{\mathfrak{e}} & = \mathfrak{b}(\chi, \overline{\chi}, \zeta) + J(\overline{\mathfrak{q}}) \mathfrak{z}, \label{eq:frak_e} \\
\dot{\mathfrak{z}} & = \mathfrak{l}(e, \overline{e}, \zeta, \overline{\zeta})+(\mathfrak{u}-\overline{u}) + d(\mathfrak{q}, \zeta, t), \label{eq:frak_z}
\end{align}
\end{subequations}
where the functions $\mathfrak{b}$, $\mathfrak{l}$ are defined by: $\mathfrak{b}(\chi, \overline{\chi}, \zeta) \coloneqq \mathfrak{c}(\chi, \zeta)-\mathfrak{c}(\overline{\chi}, \zeta)$, $\mathfrak{l}(e, \overline{e}, \zeta, \overline{\zeta}) \coloneqq f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)-f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \overline{\zeta})$, with $\mathfrak{c}(\chi, \zeta) \coloneqq J(\mathfrak{q}) \zeta$. Since the aforementioned functions are continuously differentiable, the following hold:
\begin{align*}
\|\mathfrak{b}(\cdot)\|_2 & = \|\mathfrak{c}(\chi, \zeta)-\mathfrak{c}(\overline{\chi}, \zeta)\|_2 \le L_{\scriptscriptstyle \mathfrak{c}} \|\chi-\overline{\chi}\|_2 = L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2, \\
\|\mathfrak{l}(\cdot)\|_{2} & \le \|f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)- f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \zeta)\|_{2} \notag \\
&\hspace{12mm}+\|f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \zeta)- f(\overline{e}+\chi_{\scriptscriptstyle \rm des}, \overline{\zeta})\|_2 \notag \\
& \le L_1\|e-\overline{e}\|_{2} + L_2 \|\zeta-\overline{\zeta}\|_{2} \le L \left( \|\mathfrak{e}\|_{2} + \|\mathfrak{z}\|_{2} \right).
\end{align*}
The constant $L_{\scriptscriptstyle \mathfrak{c}}$ stands for the Lipschitz constant of function $\mathfrak{c}$ with respect to the variable $\chi$; $L_1$, $L_2$ stand for the Lipschitz constants of function $h$ with respect to the variables $\chi$ and $\zeta$, respectively, and $L \coloneqq \max\{L_1, L_2\}$.
\begin{lemma} \label{lemma:tube}
The state feedback law designed by:
\begin{equation} \label{eq:kappa_law}
\kappa(e, \overline{e}, \zeta, \overline{\zeta}) \coloneqq - k (e-\overline{e})-k \sigma J(\overline{q})^\top (\zeta -\overline{\zeta}),
\end{equation}
where $k$, $\sigma > 0$ are chosen such that the following hold:
\begin{subequations}
\begin{align}
\underline{\sigma} & > 0,\ \ \sigma \coloneqq \frac{L_{\scriptscriptstyle \mathfrak{c}}+\underline{\sigma}}{\underline{J}},
\ \ \rho > \tfrac{\Lambda_1}{4 \underline{\sigma}}, \ \ k > \rho \Lambda_1 + \Lambda_2, \label{eq:sigma_under_sigma} \\
\Lambda_1 & \coloneqq \left[L + \overline{J}+ \sigma \left( L_{\scriptscriptstyle \mathfrak{c}} + \widetilde{J}\right) \right], \Lambda_2 \coloneqq \left(L + \sigma \overline{J}^2\right), \label{eq:Lambda_1}
\end{align}
\end{subequations}
renders the sets:
\begin{subequations}
\begin{align}
\Omega_1 & \coloneqq \left\{\mathfrak{e} \in \mathbb{R}^{6} : \|\mathfrak{e}\|_2 \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}} \right\}, \label{eq:omega_1} \\
\Omega_2 & \coloneqq \left\{\mathfrak{z} \in \mathbb{R}^{6+n} : \|\mathfrak{z}\|_2 \le \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}} \right\}, \label{eq:omega_2}
\end{align}
\end{subequations}
RCI sets for the error dynamics \eqref{eq:frak_e}, \eqref{eq:frak_z}, according to Definition \ref{def:RPI_set}. The constants $\alpha_1$, $\alpha_2 > 0$ are defined by:
\begin{equation}
\alpha_1 \coloneqq \underline{\sigma}- \tfrac{\Lambda_1}{4 \rho}, \ \alpha_2 \coloneqq k-\rho \Lambda_1-\Lambda_2. \label{eq:a_1_a_2}
\end{equation}
\end{lemma}
\noindent \textbf{Proof :} A backstepping control methodology will be used \cite{krstic1995nonlinear}. The state $\mathfrak{z}$ in \eqref{eq:frak_e} can be seen as virtual input to be designed such that the Lyapunov function $\mathfrak{L}_1(\mathfrak{e}) \coloneqq \frac{1}{2} \|\mathfrak{e}\|^2_2$ for the system \eqref{eq:frak_e} is always decreasing. The time derivative of $\mathfrak{L}_1$ along the trajectories of system \eqref{eq:frak_e} is given by:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{e}) & = \mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{z} + \mathfrak{e}^\top \mathfrak{b}(\cdot) \le \mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{z} + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2. \label{eq:lyap1}
\end{align}
Design the virtual control input as $\mathfrak{z} \equiv - \sigma J(\overline{\mathfrak{q}})^\top \mathfrak{e}$, with $\underline{J}$, $\sigma$ as given in \eqref{eq:set_Q}, \eqref{eq:sigma_under_sigma}, respectively. Then, by employing \eqref{eq:set_Q}, \eqref{eq:lyap1} becomes:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{e}) & \le - \sigma \mathfrak{e}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{e} + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2 \notag \\
& \le - \sigma \lambda_{\min} \left[\tfrac{J^{+}(\overline{\mathfrak{q}})+J^{+}(\overline{\mathfrak{q}})^\top}{2}\right] \|\mathfrak{e}\|_{2}^2 + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2 \notag \\
& \le - \sigma \underline{J} \|\mathfrak{e}\|_{2}^2 + L_{\scriptscriptstyle \mathfrak{c}} \|\mathfrak{e}\|_2^2 = - \underline{\sigma} \|\mathfrak{e}\|_{2}^{2}. \label{eq:lyap11}
\end{align}
Define the backstepping auxiliary error state $\mathfrak{r} \coloneqq \mathfrak{z}+\sigma J(\overline{\mathfrak{q}})^\top \mathfrak{e} \in \mathbb{R}^{6+n}$ and the the stack vector $\mathfrak{y} \coloneqq [\mathfrak{e}^\top, \mathfrak{r}^\top]^\top \in \mathbb{R}^{12+n}$. Consider the Lyapunov function $\mathfrak{L}(\mathfrak{y}) = \tfrac{1}{2}\|\mathfrak{y}\|^2$. Its time derivative along the trajectories of the system \eqref{eq:frak_e},\eqref{eq:frak_z} is given by:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{y}) & = \mathfrak{e}^\top \dot{\mathfrak{e}}+\mathfrak{r}^\top \Big[\dot{\mathfrak{z}}+\sigma J(\overline{\mathfrak{q}})^\top \dot{\mathfrak{e}}+\sigma \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e}\Big] \notag \\
&\hspace{-5mm} = \left[\mathfrak{e} + \sigma J(\overline{\mathfrak{q}}) \mathfrak{r} \right]^\top \dot{\mathfrak{e}} + \mathfrak{r}^\top \dot{\mathfrak{z}} +\sigma \mathfrak{r}^\top \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e} = - \sigma \mathfrak{e}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{e} \notag \\
&\hspace{2mm} + \mathfrak{e}^\top \mathfrak{b}(\cdot) +\sigma \mathfrak{r}^\top J(\overline{\mathfrak{q}})^\top \mathfrak{b}(\cdot) + \mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{r} +\sigma \mathfrak{r}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{r} \notag \\
&\hspace{2mm} +\sigma \mathfrak{r}^\top \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e} + \mathfrak{r}^\top \mathfrak{l}(\cdot) + \mathfrak{r}^\top (\mathfrak{u}-\overline{u}) + \mathfrak{r}^\top d(\cdot). \label{eq:lyap2}
\end{align}
By invoking \eqref{eq:lyap11} as well as the following:
\begin{align*}
\sigma \mathfrak{r}^\top J(\overline{\mathfrak{q}})^\top \mathfrak{b}(\cdot) & \le \sigma \|\mathfrak{r}\|_{2} \|J(\overline{\mathfrak{q}})\|_{2} \|\mathfrak{b}(\cdot)\|_{2} \le \sigma L_{\scriptscriptstyle \mathfrak{c}} \overline{J} \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2}, \\
\mathfrak{e}^\top J(\overline{\mathfrak{q}}) \mathfrak{r} & \le \|\mathfrak{e}\|_{2} \|J(\overline{\mathfrak{q}})\|_2 \|\mathfrak{r}\|_{2} \le \overline{J} \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2}, \\
\sigma \mathfrak{r}^\top J^{+}(\overline{\mathfrak{q}}) \mathfrak{r} & \le \sigma \|\mathfrak{r}\|_{2}^{2} \|J^{+}(\overline{\mathfrak{q}})\|_{2} \le \sigma \|\mathfrak{r}\|_{2}^{2} \|J(\overline{\mathfrak{q}})\|_{2} \big\|J^{\top}(\overline{\mathfrak{q}}) \big\|_{2} \\
& \le \sigma \overline{J}^2 \|\mathfrak{r}\|_{2}^{2}, \\
\sigma \mathfrak{r}^\top \dot{J}(\overline{\mathfrak{q}})^\top \mathfrak{e} & \le \sigma \|\mathfrak{e}\|_{2} \|\dot{J}(\overline{\mathfrak{q}})\|_{2} \|\mathfrak{r}\|_{2} \le \sigma \widetilde{J} \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2} \\
\mathfrak{r}^\top \mathfrak{l}(\cdot) & \le L \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2} + L \|\mathfrak{r}\|_{2}^{2}, \\
\mathfrak{r}^\top d(\cdot) & \le \|\mathfrak{r}\|_{2} \|d(\cdot)\|_{2} \le \|\mathfrak{y}\|_{2} \widetilde{d},
\end{align*}
\eqref{eq:lyap2} becomes:
\begin{align}
\dot{\mathfrak{L}}(\mathfrak{y}) & \le - \underline{\sigma} \|\mathfrak{e}\|_{2}^{2} + \Lambda_1 \|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2} \notag \\
&\hspace{16mm} + \Lambda_2 \|\mathfrak{r}\|^{2}_{2} + \mathfrak{r}^\top (\mathfrak{u}-\overline{u}) + \|\mathfrak{y}\|_{2} \widetilde{d}. \label{eq:lyap3}
\end{align}
with $\Lambda_1$, $\Lambda_2$ given in \eqref{eq:Lambda_1}. By using Lemma \ref{lemma:basic_ineq} for $n = P = 1$, we get $\|\mathfrak{e}\|_{2} \|\mathfrak{r}\|_{2}$ $\le \tfrac{1}{4 \rho} \|\mathfrak{e}\|_{2}^{2}$ $+ \rho \|\mathfrak{r}\|_{2}^{2}$,
with $\rho$ designed so that \eqref{eq:sigma_under_sigma} holds. Combining the latter with \eqref{eq:lyap3} it yields:
\begin{align*}
\dot{\mathfrak{L}}(\mathfrak{y}) & \le - \left(\underline{\sigma}- \tfrac{\Lambda_1}{4 \rho} \right) \|\mathfrak{e}\|_{2}^{2} + \big(\rho \Lambda_1 + \Lambda_2 \big) \|\mathfrak{r}\|^{2}_{2} \notag \\
&\hspace{30mm} + \mathfrak{r}^\top (\mathfrak{u}-\overline{u}) + \|\mathfrak{y}\|_{2} \widetilde{d}.
\end{align*}
By designing $\mathfrak{u} - \overline{u} = -k \mathfrak{r} = -k \mathfrak{e}-k \sigma J(\overline{\mathfrak{q}})^{\top} \mathfrak{z}$, which is compatible with \eqref{eq:control_law_u} and the same as in \eqref{eq:kappa_law}, we have:
\begin{align*}
\dot{\mathfrak{L}}(\mathfrak{y}) & \le - \left(\underline{\sigma}- \tfrac{\Lambda_1}{4 \rho} \right) \|\mathfrak{e}\|_{2}^{2} - \big(k -\rho \Lambda_1 - \Lambda_2 \big) \|\mathfrak{r}\|^{2}_{2} + \|\mathfrak{y}\|_{2} \widetilde{d} \\
& \le - \min\{\alpha_1, \alpha_2\}\|\mathfrak{y}\|_2^2 + \|\mathfrak{y}\|_2 \widetilde{d} \\
& = -\|\mathfrak{y}\|_2 \big[ \min\{\alpha_1, \alpha_2\}\|\mathfrak{y}\|_2 - \widetilde{d} \big],
\end{align*}
as $\alpha_1$ and $\alpha_2$ given in \eqref{eq:a_1_a_2}. Thus, $\dot{\mathfrak{L}}(\mathfrak{y}) < 0$, when $\|\mathfrak{y}\|_2 > \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}$. Taking the latter into consideration and the fact that $\mathfrak{y}(0)$, we have that $\|\mathfrak{y}(t)\| \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}$, $\forall t \ge 0$. Moreover, the following inequalities hold:
\begin{align*}
\|\mathfrak{e}\|_2 &\le \| \mathfrak{y} \|_2 \Rightarrow \|\mathfrak{e}(t)\|_2 \le \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}, \forall t \ge 0, \\
\Big| \|\mathfrak{e}\|_2- \big\|J^\top \mathfrak{z} \big\|_2 \Big| &\le \big\|\mathfrak{e}+J^\top \mathfrak{z} \big\|_2 = \|\mathfrak{z}\|_2 \le \|\mathfrak{y}\|_2 \\
\Rightarrow \|\mathfrak{z}(t)\|_2 &\le \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}}, \forall t \ge 0. \hspace{34mm} \square
\end{align*}
\begin{remark}
According to Lemma \ref{lemma:tube}, the volume of the tube which is centered along the nominal trajectories $\overline{e}(t)$, $\overline{\zeta}(t)$, that are solution of system \eqref{eq:nom_error_kin},\eqref{eq:nom_error_dyn}, depends on the parameters $\widetilde{d}$, $\overline{J}$, $\underline{J}$, $\widetilde{J}$, $L$ and $L_{\scriptscriptstyle \mathfrak c}$. By tuning the parameters $\rho$ and $k$ from \eqref{eq:sigma_under_sigma} appropriately, the volume of the tube can be adjusted.
\end{remark}
\subsection{On-line Optimal Control} \label{sec:optimal_contol}
Consider a sequence of sampling times $\{t_k\}$, $k \in \mathbb{N}$, with a constant sampling period $0 < h < T$, where $T$ is a prediction horizon such that $t_{k+1} \coloneqq t_{k} + h$, $\forall k \in \mathbb{N}$. At each sampling time $t_k$, a FHOCP is solved as follows:
\begin{subequations}
\begin{align}
&\hspace{-7mm}\min\limits_{\overline{u}(\cdot)} \left\{ \|\overline{\xi}(t_k+T)\|^2_{\scriptscriptstyle P} \hspace{-1mm} + \hspace{-2mm}\int_{t_k}^{t_k+T} \hspace{-1mm}\Big[ \|\overline{\xi}(\mathfrak{s})\|^2_{\scriptscriptstyle Q} +\|\overline{u}(\mathfrak{s})\|^2_{\scriptscriptstyle R} \Big] d\mathfrak{s} \right\} \hspace{0mm} \label{eq:mpc_cost_function} \hspace{-7mm}\\
&\hspace{-6mm}\text{subject to:} \notag \\
&\hspace{-3mm} \dot{\overline{\xi}}(\mathfrak{s}) = g(\overline{\xi}(\mathfrak{s}), \overline{u}(\mathfrak{s})), \ \ \overline{\xi}(t_k) = \xi(t_k), \label{eq:diff_mpc} \\
&\hspace{-3mm} \overline{\xi}(\mathfrak{s}) \in \overline{\mathcal{E}} \times \overline{\mathcal{Z}}, \ \ \overline{u}(\mathfrak{s}) \in \overline{\mathcal{U}}, \ \ \forall \mathfrak{s} \in [t_k,t_k+T], \label{eq:mpc_constrained_set} \\
&\hspace{-3mm} \overline{\xi}(t_k+T)\in \mathcal{F}, \label{eq:mpc_terminal_set}
\end{align}
\end{subequations}
where $\xi \hspace{-1mm}\coloneqq\hspace{-1mm}[e^\top,\zeta^\top]^\top \hspace{-2mm}\in \mathbb{R}^{12+n}$, $g(\xi,u)\hspace{-1mm}$ $\coloneqq\hspace{-1mm}\begin{bmatrix} J(\mathfrak{q}) \zeta \\ f(e+\chi_{\scriptscriptstyle \rm des}, \zeta)+u \end{bmatrix}$; $Q$, $P \in \mathbb{R}^{(12+n) \times (12+n)}$ and $R \in \mathbb{R}^{(6+n) \times (6+n)}$ are positive definite gain matrices to be appropriately tuned. We will explain hereafter the sets $\overline{\mathcal{E}}$, $\overline{\mathcal{V}}$, $\overline{\mathcal{U}}$ and $\mathcal{F}$.
In order to guarantee that while the FHOCP \eqref{eq:mpc_cost_function}-\eqref{eq:mpc_terminal_set} is solved for the nominal dynamics \eqref{eq:nom_error_kin}-\eqref{eq:nom_error_dyn}, the real states $e$, $\zeta$ and control input $\mathfrak{u}$ satisfy the corresponding state $\mathcal{E}$, $\mathcal{Z}$ and input constraints $\mathcal{U}$, respectively, the following modification is performed: $\overline{\mathcal{E}} \coloneqq \mathcal{E} \ominus \Omega_1, \ \ \overline{\mathcal{Z}} \coloneqq \mathcal{Z} \ominus \Omega_2, \ \ \overline{\mathcal{U}} \coloneqq \mathcal{U} \ominus \left[ \Lambda \circ \overline{\Omega} \right]$,
with $\Lambda \coloneqq {\rm diag} \{-k I_6, -k \sigma \overline{J} I_{6+n}\} \in \mathbb{R}^{(12+n) \times (12+n)}$, $\overline{\Omega} \coloneqq \Omega_1 \times \Omega_2$, the operators $\ominus$, $\circ$ as defined in Section \ref{sec:notation_preliminaries}, and $\Omega_1$, $\Omega_2$ as given in \eqref{eq:omega_1}, \eqref{eq:omega_2}, respectively. Intuitively, the sets $\mathcal{E}$, $\mathcal{Z}$ and $\mathcal{U}$ are tightened accordingly, in order to guarantee that while the nominal states $\overline{e}$, $\overline{\zeta}$ and the nominal control input $\overline{u}$ are calculated, the corresponding real states $e$, $\zeta$ and real control input $\mathfrak{u}$ satisfy the state and input constraints $\mathcal{E}$, $\mathcal{Z}$ and $\mathcal{U}$, respectively. This constitutes a standard constraints set modification technique adopted in tube-based NMPC frameworks (for more details see \cite{yu_2013_tube}). Define the \emph{terminal set} by:
\begin{align} \label{eq:terminal_set_F}
\mathcal{F} \coloneqq \big\{\overline{\xi} \in \overline{\mathcal{E}} \times \overline{\mathcal{Z}} : \|\overline{\xi}\|_{\scriptscriptstyle P} \le \epsilon \big\}, \ \ \epsilon > 0,
\end{align}
which is used to enforce the stability of the system \cite{frank_1998_quasi_infinite}. In particular, due to the fact that the linearized nominal dynamics $\dot{\overline{\xi}} = A \overline{\xi} + B \overline{u}$ are stabilizable, it can be proven that (see \cite[Lemma 1, p. 4]{frank_1998_quasi_infinite}) there exists a \emph{local controller} $u_{\scriptscriptstyle \rm loc} \coloneqq \mathfrak{K} \overline{\xi} \in \overline{\mathcal{U}}$, $\mathfrak{K} \in \mathbb{R}^{(6+n) \times (6+n)}$, $\mathfrak{K} > 0$ which guarantees that: $\tfrac{d}{dt}\left(\|\overline{\xi}\|^2_{\scriptscriptstyle P}\right) \le -\|\overline{\xi}\|^2_{\scriptscriptstyle \widetilde{Q}}$, $\forall \overline{\xi} \in \mathcal{F}$, with $\widetilde{Q} \coloneqq Q+\mathfrak{K}^\top R$.
\begin{theorem} \label{theorem_main}
Suppose also that the FHOCP \eqref{eq:mpc_cost_function}-\eqref{eq:mpc_terminal_set} is feasible at time $t = 0$. Then, the feedback control law \eqref{eq:control_law_u} applied to the system \eqref{eq:unsrt_error_kin}-\eqref{eq:unsrt_error_dyn} guarantees that there exists a time $\mathfrak{t}$ such that $\forall t \ge \mathfrak{t}$ it holds that:
\begin{subequations}
\begin{align}
\hspace{0mm} \|\chi(t)-\chi_{\scriptstyle \rm des}\|_{\scriptscriptstyle 2} & \le \tfrac{\epsilon}{\sqrt{\lambda_{\scriptscriptstyle \min}(P)}} + \tfrac{\widetilde{d}}{\min\{\alpha_1, \alpha_2\}}, \label{eq:theom_ineq_1} \\
\hspace{0mm} \|\zeta(t)\|_{\scriptscriptstyle 2} & \le \tfrac{\epsilon}{\sqrt{\lambda_{\scriptscriptstyle \min}(P)}} + \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}}. \label{eq:theom_ineq_2}
\end{align}
\end{subequations}
\end{theorem}
\begin{proof}
The proof of the theorem consists of two parts:
\noindent \textbf{Feasibility Analysis}: It can be shown that recursive feasibility is established and it implies subsequent feasibility. The proof of this part is similar to the feasibility proof of \cite[Theorem 2, Sec. 4, p. 12]{alex_IJRNC_2018}, and it is omitted here due to space constraints.
\noindent \textbf{Convergence Analysis}: Recall that $e = \chi-\chi_{\scriptscriptstyle \rm des}$, $\mathfrak{e} = e-\overline{e}$ and $\mathfrak{z} = \zeta-\overline{\zeta}$. Then, we get $\|\chi(t)-\chi_{\scriptstyle \rm des}\|_{\scriptscriptstyle 2}$ $\le \|\overline{e}(t)\|_{\scriptscriptstyle 2} + \|\mathfrak{e}(t)\|_{\scriptscriptstyle 2}$, $\|\zeta(t)\|_{\scriptscriptstyle 2}$ $\le \|\overline{\zeta}(t)\|_{\scriptscriptstyle 2} + \|\mathfrak{z}(t)\|_{\scriptscriptstyle 2}$, which, by using the fact that $\|\overline{e}\|$, $\|\overline{\zeta}\| \le \|\overline{\xi}\|_{2}$ as well as the bounds from \eqref{eq:omega_1}, \eqref{eq:omega_2} the latter inequalities become:
\begin{subequations}
\begin{align}
\|\chi(t)-\chi_{\scriptstyle \rm des}\|_{\scriptscriptstyle 2} & \le \|\overline{\xi}(t)\|_{\scriptscriptstyle 2} + \tfrac{ \widetilde{d}}{\min\{\alpha_1, \alpha_2\}}, \label{eq:conv_1}\\
\|\zeta(t)\|_{\scriptscriptstyle 2} & \le \|\overline{\xi}(t)\|_{\scriptscriptstyle 2} + \tfrac{2 \widetilde{d}}{\overline{J} \min\{\alpha_1, \alpha_2\}}, \forall t \ge 0. \label{eq:conv_2}
\end{align}
\end{subequations}
The nominal state $\overline{\xi}$ is controlled by the nominal control action $\overline{u} \in \overline{\mathcal{U}}$ which is the outcome of the solution to the FHOCP \eqref{eq:mpc_cost_function}-\eqref{eq:mpc_terminal_set} for the nominal dynamics \eqref{eq:nom_error_kin}-\eqref{eq:nom_error_dyn}. Hence, by invoking previous NMPC stability results found in \cite{frank_1998_quasi_infinite}, the state $\overline{\xi}(t)$ is driven to terminal set $\mathcal{F}$, given in \eqref{eq:terminal_set_F}, in finite time, and it remains there for all times. Thus, there exist a finite time $\mathfrak{t}$ such that $\overline{\xi}(t) \in \mathcal{F}$, $\forall t \ge \mathfrak{t}$. From \eqref{eq:terminal_set_F}, the latter implies that: $\|\overline{\xi}(t)\|_{\scriptscriptstyle P} \le \epsilon, \forall t \ge \mathfrak{t} \Rightarrow \|\overline{\xi}(t)\|_{\scriptscriptstyle 2} \le \tfrac{\epsilon}{\sqrt{\lambda_{\scriptscriptstyle \min}(P)}}, \forall t \ge \mathfrak{t}.$ The latter implication combined by \eqref{eq:conv_1}-\eqref{eq:conv_2} leads to the conclusion of the proof.
\end{proof}
\begin{figure}
\caption{ The GIRONA-UVMS composed of Girona500 AUV and ARM 5E Micro manipulator \cite{cieslak2015autonomous}
\label{fig:girona}
\end{figure}
\section{Simulation Results} \label{sec:simulation_results}
For a simulation scenario, consider the Girona 500 AUV depicted in Fig. \ref{fig:girona} equipped with an ARM 5E Micro manipulator from \cite{cieslak2015autonomous}. The manipulator consists of $n = 4$ revolute joints with limits: $-0.52 \le q_1 \le 1.46$, $\-0.1471 \le q_2 \le 1.3114$, $-1.297 \le q_3 \le 0.73$ and $-3.14 \le q_4 \le 3.14$. The end-effector is in ready-to-grasp mode with initial state: $\chi(0) = [\mathfrak{p}(0)^\top, \mathfrak{o}(0)^\top]^\top =[-1.0, 1.3, -1.0, 0.0, -\tfrac{\pi}{8}, \tfrac{\pi}{12}]^\top$. The stiffness matrix is $K = I_{6}$ with $\chi_{\scriptscriptstyle \rm eq} = 0$ which results to $\mathfrak{F}_{\scriptscriptstyle \rm des} = \chi_{\scriptscriptstyle \rm des} = [\mathfrak{p}_{\scriptscriptstyle \rm des}^\top, \mathfrak{o}_{\scriptscriptstyle \rm des}^\top ]^\top = [0, 0, 0, \tfrac{\pi}{3}, \tfrac{\pi}{10}, 0]^\top$.
According to \eqref{eq:forw_kinematics}, the transformation matrices which lead to the forward kinematics are given by:
\begin{align*}
T_B^I & =
\begin{bmatrix}
\mathfrak{J}_1(\eta_2) & \eta_1 \\
0_{1 \times 3} & 1 \\
\end{bmatrix}, \ \ T_0^B =
\begin{bmatrix}
I_{3 \times 3} & \left[ 0.53, 0, 0.36 \right]^\top \\
0_{1 \times 3} & 1 \\
\end{bmatrix},
\end{align*}
and $T_{i}^{i-1}$, $i =1,\dots,4$ are given by the Denavit-Hantenberg parameters which can be calculated from Table \ref{table:DH_parameters}. By imposing the constraints $-\pi \le \phi$, $\psi \le \pi$ and $-\tfrac{\pi}{2}+\epsilon \le \theta \le \tfrac{\pi}{2} - \epsilon$, $\epsilon = 0.1$, according to \eqref{eq:set_Q} we get $\underline{J} = 0.5095$ and $L_{\scriptscriptstyle \mathfrak{c}} = 2 \sqrt{2}$. For simplified calculations, we apply the methodology of this paper by considering disturbance in the following disturbed kinematic model: $\dot{\chi} = J(\mathfrak{q}) \zeta + w(\mathfrak{q}, t)$,
with $w(\cdot) = 0.2 \sin(t) I_6$ $\Rightarrow \|w(\cdot)\|_{2} \le 0.2 = \widetilde{w}$, in which the vector $\zeta$ stands for the virtual control input to be designed such that $\lim_{t \to \infty} \|\chi(t) - \chi_{\scriptscriptstyle \rm des}\| \to 0$. The input constraints are set to $\|\nu_1\|_2 \le 2$, $\|\nu_2\|_2 \le 2$ and $\|\dot{q}\|_2 \le 2$. Then, by using \eqref{eq:frak_e} and \eqref{eq:lyap1} and designing the control gain $\sigma = 3.084$, the resulting RCI is $\Omega = \left\{\mathfrak{e} \in \mathbb{R}^{6} : \|\mathfrak{e}\|_{2} \le \tfrac{\widetilde{w}}{\sigma \underline{J} + L_{\scriptscriptstyle \mathfrak{c}}} = 0.3 \right\}$.
\begin{table}[t!]
\begin{center}
\begin{tabular}{|C{0.4cm}||C{1.2cm}||C{1.2cm}||C{1.2cm}||C{1.2cm}|}
\hline
& $d_i (m)$ & $q_i$ & $a_i (m)$ & $\alpha_i (\text{rad})$ \\
\hline \hline
$1$ & $0$ & $q_1$ & $0.1$ & $-\frac{\pi}{2}$ \\
\hline
$2$ & $0$ & $q_2$ & $0.26$ & $0$ \\
\hline
$3$ & $0$ & $q_3$ & $0.09$ & $\frac{\pi}{2}$ \\
\hline
$4$ & $0.29$ & $q_4$ & $0$ & $0$ \\
\hline \hline
E & \multicolumn{4}{c|}{$\text{Rot}(y,-\frac{\pi}{2})$} \\
\hline
\end{tabular}
\end{center}
\caption{Denavit-Hantenberg Parameters of the ARM 5E Micro}
\label{table:DH_parameters}
\end{table}
The simulation time is $6 \sec$. The optimization horizon and the sampling time are set to $T = 0.7 \sec$ and $h = 0.1 \sec$, respectively. The NMPC gains are set to $Q = P = 0.5 I_{6}$ and $R = 0.5 I_{10}$. Fig. \ref{fig:error} shows the evolution of the real and the nominal position errors of the end-effector. the corresponding real and nominal orientation errors are depicted in Fig. \ref{fig:error2}. Finally, the control inputs are presented in Fig. \ref{fig:inputs}. It can be observed that the desired task is performed while all the state/input constraints are satisfied.
\section{Conclusions and Future Research} \label{sec:conclusions}
This paper addresses the problem of force/torque control of UVMS under state/input constraints as well as external uncertainties/disturbances. In particular, we have proposed a tube-based robust NMPC framework that incorporates the aforementioned constraints in a novel way. Future efforts will be devoted towards extending the current framework under multi-UVMS which interact with each other through a common object in order to perform a collaborative manipulation task.
\begin{figure}
\caption{The evolution of the real position errors of the end-effector $\mathfrak{p}
\label{fig:error}
\end{figure}
\begin{figure}
\caption{The evolution of the real orientation errors $\mathfrak{o}
\label{fig:error2}
\end{figure}
\begin{figure}
\caption{The virtual control input signals $\|\nu_1(t)\|_{2}
\label{fig:inputs}
\end{figure}
\end{document}
|
\begin{document}
\title{Large Sumsets from Medium-Sized Subsets}
\author{B\'ela Bollob\'as \and Imre Leader \and Marius Tiba}
\address{Department of Pure Mathematics and Mathematical Statistics,
Wilberforce Road,
Cambridge, CB3 0WA, UK, and Department of Mathematical Sciences,
University of Memphis, Memphis, TN 38152, USA}\email{[email protected]}
\address{Department of Pure Mathematics and Mathematical Statistics,
Wilberforce Road, Cambridge, CB3 0WA, UK}\email{[email protected]}
\address{IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil}\email{[email protected]}
\thanks{The first author was partially supported by NSF grant DMS-1855745}
\begin{abstract}
The classical Cauchy--Davenport inequality gives a lower bound for the size of
the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to
prove a considerable strengthening of this inequality, where we take only a small number of points from each of
the two subsets when forming the sum. One of our results is that
there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with
$|A|=|B|=n\le p/3$ then there are subsets $A'\subset A$ and
$B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$. In fact, we show that one
may take any sizes one likes: as long as $c_1$ and $c_2$ satisfy $c_1c_2 \ge cn$ then we may choose
$|A'|=c_1$ and $|B'|=c_2$. We prove related
results for general abelian groups.
\end{abstract}
\maketitle
\section{Introduction}
The Cauchy--Davenport theorem~\cite{Cauchy, Dav1, Dav2} states that if $p$ is a prime and $A$ and $B$ are non-empty
subsets of ${\mathbb Z}_p$ with $|A|+|B|\le p+1$ then $|A+B|\ge |A|+|B|-1$. Intervals show that this bound is best possible.
Over the years, this classical result
was followed by a host of important contributions about sums of subsets of groups, including other abelian groups such as
${\mathbb Z}$ itself. For these
contributions, see, among others,
Mann~\cite{Mann1, Mann2}, Kneser~\cite{Knes},
Erd\H{o}s and Heilbronn~\cite{ErdHeil}, Freiman~\cite{Frei-59, Frei-62, Frei-book, Frei-87}, Pl\"unnecke~\cite{Plu-70},
Ruzsa~\cite{Ruz-89}, Dias da Silva and Hamidoune~\cite{DiHa}, Alon, Nathanson and Ruzsa~\cite{AlNaRu},
Shao~\cite{shao}, Stanchescu~\cite{Stan}, Breuillard, Green and Tao~\cite{BGT-doubling}, as well as the books of
Nathanson~\cite{Nathbook}, Tao and Vu~\cite{taobook} and Grynkiewicz~\cite{grynkiewicz-2}.
Recently, a new direction of research was started in ~\cite{BLT}: can we get similar bounds for the size of the sum
if $A+B$ is replaced by $A+B'$, where $B'$ is a {\em small} subset of $B$? Among other results, it was proved that if
$A$ and $B$ are finite non-empty subsets of ${\mathbb Z}$ with $|A| \ge |B|$ then $B'$ can be taken to be really small:
there are {\em three} elements $b_1, b_2, b_3 \in B$ such that $|A+\{b_1,b_2,b_3\}| \ge |A|+|B|-1$. For
${\mathbb Z}_p$, it was shown that if
$A, B \subset {\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then $B$ has a subset $B'$ with at most $c$ elements
such that $|A+B'|\ge 2n-1$, where $c$ is an absolute constant. (Here again $p$ is prime, and for the rest of this paper $p$ will always denote a prime.)
Our aim in this paper is to prove that actually one can replace {\em both} $A$ and $B$ by appropriate small subsets.
In the result just mentioned, the product of the sizes of our two subsets in the sum, namely $A$ and $B'$, is $cn$,
and trivially we cannot ever get a sum of size linear in $n$ without the product of the sizes of the two sets being
linear in $n$. But, remarkably, one can indeed always choose subsets $A'$ of $A$ and $B'$ of $B$, of any desired
sizes, as long as the product of these sizes is linear in $n$.
The result mentioned in the Abstract has both sizes being a constant times $\sqrt{n}$, and the sets $A$ and $B$
themselves have size bounded away from $p/2$. The general form of our result is as follows.
\begin{theorem}\label{all_thm1}
For all $\alpha, \beta>0$ there exists $c>0$ such that the following holds.
Let $A$ and $B$ be non-empty subsets of $\mathbb{Z}_p$ with
$\alpha |B| \leq |A| \leq \alpha^{-1} |B|$ and $|A|+|B| \leq (1-\beta)p$.
Then, for any integers $1 \leq c_1 \leq |A|$ and $1 \leq c_2 \leq |B|$ such that $c_1c_2 \geq c\max(|A|,|B|)$,
there exist subsets $A' \subset A$ and $B' \subset B$, of sizes $c_1$ and $c_2$ such that
$|A'+B'| \geq |A|+|B|-1$.
\end{theorem}
It is rather surprising that this holds with no other restrictions on the values of $c_1$ and $c_2$, whatever the
form of the sets $A$ and $B$. We remark that the constant $c$ does have to depend on $\beta$: if our two sets
are allowed to have sizes whose sum approaches $p$ then taking random subsets shows that $c$ has to grow. Also, it is
easy to see that $c$ depends on $\alpha$. For example, if $B$ is far larger than $A$ then trivially taking $c$ points
from $B$ to be summed with all of $A$ will yield a sumset that is too small.
Our main tool is a similar result that is valid in general abelian groups. The reader will note that it is `worse'
than the above result in that there is an error term, but it is also far `better' because the lower bound comes from
the sum of two actual sets (the sets $A^*$ and $B^*$ below) rather than merely from the lower bound that comes from the
sum of their sizes (or, more precisely, the general Kneser lower bound that generalises the Cauchy--Davenport theorem).
\begin{theorem}\label{implication_shao1-prov} For all $K$ and $\varepsilon >0$ there exists $c$ such that the
following holds.
Let $A$ and $B$ be finite non-empty subsets of an abelian group, and let $1 \leq c_1 \leq |A|$ and
$1 \leq c_2 \leq |B|$ be integers satisfying $c_1c_2 \geq c \max(|A|,|B|)$. Then there are subsets $A^* \subset A$
and $B^* \subset B$, with $|A^*|\ge (1-\varepsilon)|A|$ and
$|B^*|\ge (1-\varepsilon)|B|$, such that
if we select points $a_1, \hdots, a_{c_1}$ and $b_1,\hdots,b_{c_2}$ uniformly at random from $A^*$ and
$B^*$ then, writing $A'$ for $\{a_1, \hdots, a_{c_1}\}$ and $B'$ for $\{b_1, \hdots, b_{c_2}\}$, we have
$${\mathbb E}|A'+ B'|
\geq \min\big( (1-\varepsilon)|A^*+ B^*|,\ K |A^*|,\ K |B^*| \big).$$
\end{theorem}
We remark that the terms $K |A^*|$ and $K |B^*|$ are only present to deal with unimportant cases: the key term is
$(1-\varepsilon)|A^*+ B^*|$. Thus the result is informally somehow saying that, in terms of sumsets, $A^*$ and $B^*$ may really be approximated by very small subsets of themselves (and indeed most subsets will do).
Interestingly, while this theorem is for general abelian groups, Theorem~\ref{all_thm1} is only about $\mathbb{Z}_p$. The passage
between these does require quite a lot of work.
The plan of the paper is as follows. In Section 2 we give various prerequisites that we shall need. Then in Section
3 we prove Theorem~\ref{implication_shao1-prov}, and also provide the consequence of it in $\mathbb{Z}_p$ that we use in
the proof of Theorem~\ref{all_thm1}.
In Section 4 we prove Theorem~\ref{all_thm1},
and the last section, Section 5, contains open problems.
Our notation is standard.
Sometimes we write `$x \mod d$' as shorthand for the infinite arithmetic progression
$\{ y \in \mathbb{Z}: y \equiv x \mod d \}$, and refer to it as a {\em fibre} mod $d$.
When $S$ is a subset of $\mathbb{Z}$ we often write $S^x$ for the intersection of this fibre with
$S$ -- when the value of $d$ is clear. (We sometimes write $S^x$ as $S^x_d$ when we want to stress the value of $d$.)
Thus $S^x=S\cap \pi^{-1}(x)$, where $\pi = \pi_d$ denotes
the natural projection from $\mathbb{Z}$ to $\mathbb{Z}_d$.
When we write a probability or an expectation over a finite set, we always assume that the elements
of the set are being sampled uniformly. Thus, for example,
for a finite set $X$ we denote the expectation and probability when we sample uniformly over
all $x \in X$ by ${\mathbb E}_{x\in X} \text{ and } \Prob_{x\in X}$.
We also often sample uniformly over all $c$-sets of a given set $X$. In most of those cases,
we could instead sample $c$ elements uniformly and independently, but the notation would tend to get
unwieldy, and this is why we use the sampling over all $c$-sets instead.
Before we turn to the next section, let us draw attention to the superficial similarity of
our problems to a beautiful result of Ellenberg~\cite{Ell}. Given a prime $p$ and a positive
integer
$d$, let $f(p^d)$ be the smallest integer such that for any sets $S, T \subset {\mathbb Z}_p^d$
there are subsets $S' \subset S$ and $T'\subset T$ satisfying $(S+T')\cup (S'+T)=S+T$
and $|S'|+|T'| \le f(p^d)$. Ellenberg proved that $f(p^d) \le (cp)^d$, where $c<1$ is an absolute
constant.
\section{Prerequisites}
In this section we collect together the various prerequisites that we will need. Each of these may be treated
as a `black box': knowledge of their proofs will not be required.
The first of the three theorems we shall need is due to Shao~\cite{shao}, and concerns
restricted sums. Let $A$ and $B$ be subsets of an abelian group, and let
$\Gamma \subset A \times B$. The {\em $\Gamma$-restricted sum} of $A$ and $B$ is
$A+_{\Gamma}B=\{a+b: \ a\in A,\ b\in B, (a,b) \in \Gamma \}$. Here is the result of Shao.
\begin{theorem}\label{thm_shao}
For all $\varepsilon, K>0$ there exists $\delta>0$ such that the following holds. Let $G$ be an
abelian group and let $N \in \mathbb{N}$. Let $A,B \subset G$
be two subsets with $|A|, |B| \geq N$. Let $\Gamma \subset A \times B$ be a subset with
$|\Gamma|\geq (1-\delta)|A||B|$. If $|A+_{\Gamma}B| \leq KN$, then there
exist $A_0 \subset A$ and $B_0 \subset B$ such that
\[
|A_0| \geq (1-\varepsilon)|A|
\text{ and } |B_0|\geq (1-\varepsilon)|B| \text{ and } |A_0+B_0| \leq |A+_{\Gamma}B|+\varepsilon N.
\]
\end{theorem}
The second theorem is an easy corollary of a theorem of Grynkiewicz~\cite{grynkiewicz-2}.
\begin{theorem}\label{Freiman_Zp}
Given $\beta, \gamma >0$ there is an $\varepsilon>0$ such that the following holds.
Let $A$ and $B$ be subsets of $\mathbb{Z}_p$. Suppose that
$2\leq \min(|A|,|B|)\text{ and }|A|+|B| \leq (1-\beta)p$
and $|A+B| \leq |A|+|B|-1 + \varepsilon \min(|A|,|B|)$.
Then there are arithmetic progressions $P$ and $Q$ with the same common difference that contain $A$ and $B$ and satisfy $|P \Delta A|\leq \gamma \min(|A|,|B|) \text{ and } |Q \Delta B|\leq \gamma \min(|A|,|B|)$.
\end{theorem}
The last theorem we need is a somewhat technical result from \cite{BLT}. It gives a strengthening of the result from \cite{BLT}
mentioned above about sums in $\mathbb{Z}_p$, when the sets $A$ and $B$ `relate nicely' to intervals.
\begin{theorem}\label{CD_technical}
For all $\beta>0$ there exists $\gamma>0$ such that for every $\alpha>0$ there is a value of $c$ for which the
following
holds. Let $A$ and $B$ be subsets of $\mathbb{Z}_p$ and let $I= [p_l, p_r]$ and $J= [q_l, q_r]$
be intervals in $\mathbb{Z}_p$ satisfying $\alpha |J| \leq |I| \leq \alpha^{-1} |J|$,\ $|I|+|J|\leq (1-\beta)p$,
$\max(|A\Delta I|, |B\Delta J|) \leq \gamma \min (|I|, |J|)$\ and\ $\{q_l, q_r\} \subset B \subset J$.
Then there is a family $\mathcal{F}\subset B^{(c)}$,\ depending only on $I$, $J$ and $B$ (but not on $A$),\ such that
\[
\E_{B'\in \mathcal{F}}|A+B'|\geq |A|+|J|-1 \geq |A|+|B|-1.
\]
\end{theorem}
\section{Proof of Theorem 2}
In this section we prove our main result on general abelian groups, Theorem~\ref{implication_shao1-prov}.
We start by giving a brief overview of the proof. Although this paper is self-contained, we mention that the
reader who is familiar with \cite{BLT} will see that this proof is similar in spirit to the proof of Theorem 10 in
that paper.
We will repeatedly apply Theorem~\ref{thm_shao} in order to construct a decreasing sequence
of $s + 1$ pairs of sets $(A, B) = (A_0, B_0), (A_1, B_1), \dots , (A_s, B_s)$,
satisfying $A_i \subset A_{i-1}$ and $B_i
\subset B_{i-1}$ and $|A_i| >
(1 -
\varepsilon / s)^i |A|$ and $|B_i| > (1 - \varepsilon / s)^i |B|$.
Having constructed $A_i$ and $B_i$ we divide the elements of $A_i+B_i$ into the
set $P_i$ of `popular' ones (those hit at least $\alpha \min(|A_i|,|B_i|)$ times) and
the set $U_i$ of unpopular ones. And we let $\Gamma$ be the pairs summing to
popular elements.
We first deal with the situation when $|\Gamma|$ is much smaller than $|A_i| |B_i|$ or
$|P_i|$ is much larger than $K \min(|A_i|,|B_i|)$. In both cases a simple
computational check shows that the pair $(A_i, B_i)$ has the desired
properties. If we are not in this situation then we can apply Thm~\ref{thm_shao} to $A_i, B_i$ and $\Gamma$ to construct the sets
$A_{i+1}$, $B_{i+1}$. These satisfy $|A_{i+1} +
B_{i+1}| < |P_i| + (\varepsilon / s) \min(|A_{i+1}|, |B_{i+1}|)$.
We then deal with the case when the process continues for at least $s$
steps. In this case we have $|A_{i+1} + B_{i+1}| - (\varepsilon / s)
\min(|A_{i+1}|, |B_ {i+1}|) < |P_i| < |A_i+B_i|$, where the second inequality is
clear and the first is by Theorem~\ref{thm_shao}.
As $|P_0| < 10 K \min (|A_0|, |B_0|)$ and $|A_s + B_s| > |B_s| > |B_0|/2$, we
deduce that there exists $i$ such that $|P_i|$ is about
then easy to check that the pair of sets $(A_i, B_i)$ has the desired
property. Indeed, if $c$ is large enough (depending on $\alpha$),
then $|A'+B'|$ is about $|P_i|$ as each point in $P_i$ is hit with high probability. So
we conclude that $|A'+B'|$ is about $|A_i+B_i|$.
We now turn to the proof itself.
\begin{proof}[Proof of Theorem \ref{implication_shao1-prov}]
Fix $\varepsilon>0$ and $K>0$, where we assume that $\varepsilon$ is sufficiently small and
$K$ is sufficiently large.
Pick $s=\lfloor \frac{50K}{\varepsilon} \rfloor$. Let
$\delta$ be given by Theorem~\ref{thm_shao} with parameters $\frac{\varepsilon}{s}$ and $10K$.
Also pick $\alpha = \delta/16 K$. Finally, pick $c\geq \max(2^{10}K/ \delta, 2^{10} |\log(\epsilon)|/ \alpha)$. We may assume
that $|A| \geq |B|$.
We shall examine a process in which we repeatedly apply Theorem \ref{thm_shao} in order to construct a
decreasing sequence of $s+1$ pairs of sets $(A,B)=(A_0,B_0),(A_1,B_1), \hdots ,(A_s,B_s),$
satisfying $A_i \subset A_{i-1} \text{ and } B_i \subset B_{i-1}$ and
$|A_i|\geq (1-\varepsilon/s)^i |A|$ and $|B_i|\geq (1-\varepsilon/s)^i |B| $.
Fix $i<s$ and assume that the pair of sets $(A_i,B_i)$ has already been
constructed. We shall either stop the process at step $i$ or construct the pair of
sets $(A_{i+1},B_{i+1})$.
Let $A_i+B_i=C_i^+\sqcup C_i^-$ be the partition into `popular' and `unpopular' elements given by
$ C_i^+=\{c \in A_i+B_i: \ |(c-A_i)\cap B_i| \geq \alpha |B_i|\}$, and
$C_i^-=\{c \in A_i+B_i: \ |(c-A_i)\cap B_i| < \alpha |B_i|\}$.
Also, let the partition $A_i \times B_i=\Gamma_i\sqcup \Gamma_i^c$ be
given by $\Gamma_i=\{(a,b) \in A_i \times B_i: \ a+b \in C_i^+ \} \subset A_i \times B_i$, and $\Gamma_i^c=\{(a,b) \in A_i
\times B_i : \ a+b \in C_i^- \} \subset A_i \times B_i$,
so that $A_i+_{\Gamma_i}B_i=C_i^+$ and $A_i+_{\Gamma^c_i}B_i=C_i^-$.
Finally, for
each $x \in A_i+B_i$ set
$$
A_i^x=(x-B_i)\cap A_i \text{ and } B_i^x=(x-A_i)\cap B_i \text{ such that } A_i^x=x-B_i^x,$$
$$
r_i(x)=|A_i^x|=|B_i^x|=|\{(a,b)\in A_i \times B_i: \
x=a+b\}|,
$$
so that $ \sum_x r_i(x)=|A_i||B_i|.$
We stop this process `early', at step $i$, if $$|\Gamma_i|<(1-\delta)|A_i||B_i| \text{ or } |A_i+_{\Gamma_i}B_i|> 10K \min(|A_i|,|B_i|).$$
Otherwise, we apply Theorem~\ref{thm_shao} with parameters
$\varepsilon/s, 10K$ to the pair of sets $(A_i,B_i)$. Thus we produce a pair of sets $(A_{i+1},B_{i+1})$, satisfying
$A_{i+1} \subset A_i$, $B_{i+1} \subset B_i$, \
$|A_{i+1}| \geq (1-\varepsilon/s)|A_i|$, \ $|B_{i+1}|\geq (1-\varepsilon/s)|B_i|$
and $|A_{i+1}+B_{i+1}| \leq |A_i+_{\Gamma_i}B_i|+\frac{\varepsilon}{s} \min(|A_i|,|B_i|)$.
We shall analyse separately the cases in which the process continues until the end and in
which the process stops before that.
Before we begin, we need one easy estimate. Suppose the process continues until step $j$. If we
choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{equation}\label{unification1}
\begin{split}
{\mathbb E}|A_j'+B_j'|
&= \sum_x \Prob(x \in A_j'+B_j')\\
&\geq \sum_{x} \sum_{\substack{ X\subset A_j^x, Y\subset B_j^x \\ X=x-Y \\ |X|=|Y|> \frac{n_Br_j(x)}{2|B_j|} }}
\mathbb{P} (B_j' \cap B_j^x = Y \text{ and } A_j' \cap X \neq \emptyset)\\
&= \sum_{x} \sum_{\substack{ X\subset A_j^x, Y\subset B_j^x \\ X=x-Y \\ |X|=|Y|> \frac{n_Br_j(x)}{2|B_j|} }}
\mathbb{P} (B_j' \cap B_j^x= Y ) \mathbb{P}(A_j' \cap X \neq \emptyset)\\
&= \sum_{x} \Prob(|B_j'\cap B_j^x| > \frac{n_Br_j(x)}{2|B_j|}) \min_{\substack{X \subset A_j^x \\ |X| >
\frac{n_Br_j(x)}{2|B_j|}}}\Prob(|A_j' \cap X| > 0)\\
&\geq \sum_{x} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))).
\end{split}
\end{equation}
Here, the last inequality follows from Chernoff's inequality (see for example Corollary 1.9 in \cite{taobook}) and the fact that $|X| > \frac{n_Br_j(x)}{2|B_j|} $ is equivalent to $|X| \geq \max(1, \frac{n_Br_j(x)}{2|B_j|})$.
{\bf Claim A.}
{\em
Suppose the process
stops early, say at step $j<s$. Then the pair of sets $(A_j,B_j)$ has the desired properties.
}
\begin{proof}
\noindent
\textbf{Case 1:} Consider first the
case when $|C^+_j|=|A_j+_{\Gamma_j}B_j|> 10K \min(|A_j|,|B_j|).$ For $x\in C_j^+$,
by construction we have that $r_j(x)=|(x-A_j)\cap B_j|\geq \alpha
|B_j|.$ If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|
&\geq& \sum_{x} (1-\exp(-\frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))) \\
&\geq& \sum_{x}(1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2
\geq \sum_{x \in C^+_j} (1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2\\
&\geq& \sum_{x \in C^+_j}(1- \exp(-2^{-4} \alpha c))^2
\geq |C_j^+|/2
\geq 5K \min(|A_j|,|B_j|).
\end{eqnarray*}
Here the first inequality follows from \eqref{unification1}; the second from the hypothesis $n_An_B \geq c|A|\geq c|A_j|$
which, in particular, gives $n_B \geq c|B_j|$; the fourth from the construction as $r_j(x) \geq \alpha |B_j|$
for $x \in C_j^+$; the fifth from the hypothesis $c \geq 2^{10}/ \alpha$; and the last inequality follows from
the assumption on the size of $C^+_j$.
\noindent
\textbf{Case 2:} Consider now the case when $|\Gamma_j|<(1-\delta)|A_j||B_j|.$ By construction,
$\sum_{x\in C_j^-}r_j(x) \geq \delta|A_j||B_j|.$ Moreover, for
$x\in C_j^-$ we have $r_j(x)=|(x-A_j)\cap B_j|\leq \alpha
|B_j|.$
If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have the following sequence of inequalities. To make the formulae less cluttered, we define $D_j^-=\{x: n_B r_j(x)\le 2|B_j|\}$ and $D_j^+=\{x: n_B r_j(x)> 2|B_j|\}$.
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|\hspace{-8pt}
&\geq& \hspace{-8pt} \sum_{x} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|}))) \\
&\geq& \hspace{-12pt} \sum_{x \in D_j^-} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} ))
(1-\exp(- \frac{n_A}{4|A_j|})) + \sum_{x\in D_j^+} 2^{-1}(1-\exp(- \frac{n_An_Br_j(x)}{8|A_j||B_j|}))\\
&\geq& \hspace{-12pt} \sum_{x\in D_j^-} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} ))
(1-\exp(- \frac{n_A}{4|A_j|})) + \sum_{x\in D_j^+} 2^{-1}(1-\exp(- \frac{8Kr_j(x)}{\delta|B_j|}))\\
&\geq& \hspace{-19pt} \sum_{x \in C_j^-\cap D_j^-} \frac{n_Br_j(x)}{32|B_j|} \frac{n_A}{8|A_j|}
+ \sum_{x \in C_j^- \cap D_j^+} \frac{Kr_j(x)}{\delta |B_j|}
\geq \sum_{x \in C_j^-} \frac{K r_j(x)}{\delta |B_j|} \geq K|A_j|.
\end{eqnarray*}
Here, the first inequality follows from \eqref{unification1}, the second inequality follows by splitting
into two cases according to how $\frac{n_B r(x)}{2|B_j|}$ compares to 1, the third inequality follows
from the hypothesis $n_An_B \geq c|A| \geq c|A_j|$ and $c \geq 2^{10}K/\delta$, the fourth inequality
follows from the two facts that $1-\exp(-t) \geq t/2$ for $0 \leq t \leq 1/2 $ and $\frac{8Kr_j(x)}{\delta |B_j|}
\leq \frac{8K \alpha}{\delta} <1/2 $ for $x \in C_j^-$, the fifth inequality follows again from the
hypothesis $n_An_B \geq c |A| \geq c |A_j| $ and $c\geq 2^{10} K/ \delta $, and the last inequality
follows from the original assumption that $\sum_{x \in C_j^-} r_j(x) \geq \delta |A_j||B_j|$.
We conclude the pair of sets $(|A_j|,|B_j|)$ has the desired properties, so Claim A is proved.
\end{proof}
We now turn to the case when the process does not stop early.
{\bf Claim B.} {\em
Suppose that the process continues until the terminal step $s$.
Then there is an index $j\leq s$ such that the pair of sets $(A_j,B_j)$ has the desired
properties.}
\begin{proof}
Note that $|A_{1}+B_{1}| \leq |A_0+_{\Gamma_0}B_0|+\frac{\varepsilon}{s} \min(|A_0|,|B_0|) \leq 11K \min(|A_0|,|B_0|).$
Moreover, $11K \min(|A|,|B|)= 11K \min(|A_0|,|B_0|)\geq |A_{1}+B_{1}| \geq \hdots \geq |A_{s}+B_{s}| \geq 0.$ Therefore
$|A_{j+1}+B_{j+1}|\geq |A_{j}+B_{j}|-\frac{11K}{s-1}\min(|A|,|B|)$ for some index $j$,
$1 \leq j \leq s$. We shall
show that the pair $(A_j,B_j)$ has the desired properties.
Indeed, by construction,
\[
|A_{j}+_{\Gamma_j}B_{j}|+\frac{\varepsilon}{s}
\min(|A|,|B|) \geq |A_{j}+_{\Gamma_j}B_{j}|+\frac{\varepsilon}{s}
\min(|A_j|,|B_j|) \geq |A_{j+1}+B_{j+1}|.
\]
It follows that
\[
|A_j+_{\Gamma_j}B_j|\geq |A_{j}+B_{j}|- \big(\frac{11K}{s-1}+\frac{\varepsilon}{s} \big)\min(|A|,|B|)
\geq \big(1- \frac{20K}{s} \big)|A_{j}+B_{j}|.
\]
If we choose elements $a_1, \hdots, a_{n_A}$ and $b_1, \hdots, b_{n_B}$ uniformly at random
from $A_j$ and $B_j$, and we write $A'_j=\{a_1, \hdots, a_{n_A}\}$ and $B'_j=\{b_1, \hdots, b_{n_B}\}$, then we have
\begin{eqnarray*}
{\mathbb E}|A_j'+B_j'|
&\geq& \sum_{x} (1-\exp(- \frac{n_Br_j(x)}{16|B_j|} )) (1-\exp(- \max(\frac{n_A}{4|A_j|},\frac{n_An_Br_j(x)}{8|A_j||B_j|})))\\
&\geq& \sum_{x}(1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2
\geq \sum_{x \in C^+_j} (1-\exp(- \frac{cr_j(x)}{16|B_j|}))^2\\
&\geq& \sum_{x \in C^+_j}(1- \exp(-2^{-4} \alpha c))^2
\geq (1- \exp(-2^{-4} \alpha c))^2 |A_j+_{\Gamma_j}B_j|\\
&\geq& (1- \exp(-2^{-4} \alpha c))^2 (1-\frac{20 K}{s}) |A_j+B_j|
\geq (1-\epsilon) |A_j+B_j|.
\end{eqnarray*}
Here the first inequality follows from \eqref{unification1}; the second from the hypothesis
$n_An_B \geq c|A|\geq c|A_j|$ which, in particular, gives $n_B \geq c|B_j|$; the fourth inequality
holds by the construction, as $r_j(x) \geq \alpha |B_j|$ for $x \in C_j^+$; the fifth inequality follows from the
hypothesis $c \geq 2^{10}/ \alpha |\log(\epsilon)|$; and the last inequality follows from the assumption on the size of $C^+_j$.
Thus the pair $(|A_j|,|B_j|)$ has the desired properties, so Claim B is proved.
\end{proof}
This concludes the proof of Theorem~\ref{implication_shao1-prov}: whether the process stops early
or does not, the pair of sets $(A_j,B_j)$ has the desired
properties.
\end{proof}
When we come to proving Theorem~\ref{all_thm1}, we shall need the following consequence of this.
\begin{theorem}\label{all_thm2}
For all $\beta, \gamma>0$ there exists $\epsilon>0$ such that for all $\alpha>0$ there is a value of $c$
for which the following holds. Let $A$ and $B
$ be subsets of $\mathbb{Z}_p$ and let $1 \leq c_1 \leq |A|$ and $1 \leq c_2 \leq |B|$ be integers such
that $c_1c_2 \geq c\max(|A|,|B|)$. Suppose that
\begin{equation}\label{all_eq2}
2 \leq \min(|A|,|B|), \alpha |B| \leq |A| \leq \alpha^{-1} |B| \text{ and } |A|+|B| \leq (1-\beta)p
\end{equation}
and
\begin{equation}\label{all_eq3}
\E_{A' \in A^{(c_1)},\ B' \in B^{(c_2)}} |A'+B'| \leq |A|+|B|-1 +\eps \min(|A|,|B|).
\end{equation}
Then there exist arithmetic progressions $P$ and $Q$ with the same common difference and
\begin{equation*}
\max(|A\Delta P|, |B \Delta Q|) \leq \gamma \min(|A|,|B|).
\end{equation*}
\end{theorem}
\begin{proof}
We start by fixing some parameters. Fix $2^{-10}>\beta>\gamma>\alpha>0$. Let $\varepsilon_1$ be the output of Theorem~\ref{Freiman_Zp} with
input $\beta,2^{-11}\gamma$. Let
$\varepsilon=\min(2^{-2}\varepsilon_1,2^{-6}\gamma)$. Let $\mu=2^{-12}\min(\alpha \varepsilon, \alpha \gamma)$.
Finally, let $c$ be the output of Theorem~\ref{implication_shao1-prov} with input $(4/\alpha,\mu)$.
By Theorem~\ref{implication_shao1-prov}, there are subsets $A^*\subset A$ and $B^*\subset B$ with
\begin{equation}\label{eq22.1*}
|A^*|\geq \lceil (1-\mu)|A|\rceil \text{ and } |B^*|\geq \lceil (1-\mu)|B|\rceil
\end{equation}
such that
\begin{equation}\label{eq22.2*}
\E_{A' \in A^{(c_1)}, \ B'\in B^{(c_2)}}| A'+B'| \geq \min \bigg((1-\mu)|A^*+B^*|\text{, } \frac{4}{\alpha}|A|\text{, }
\frac{4}{\alpha}|B|\bigg).
\end{equation}
By \eqref{all_eq2} we have
\begin{equation}\label{eq22.25*}
\min\bigg(\frac{4}{\alpha}|A|,\frac{4}{\alpha}|B|\bigg) \geq 4\max(|A|,|B|) >
|A|+|B|-1+\varepsilon\min(|A|,|B|),
\end{equation}
and by \eqref{all_eq2} and \eqref{eq22.1*} we find that
\begin{equation}\label{eq22.27*}
2\leq \min\big(|A^*|,|B^*|\big), \ \ \ \frac{\alpha}{2}
|B^*| \leq |A^*| \leq \frac{2}{\alpha}|B^*| \ \ \ {\rm and} \ \ \ |A^*|+|B^*| \leq (1-\beta)p.
\end{equation}
Hence \eqref{all_eq3}, \eqref{eq22.2*} and \eqref{eq22.25*} imply
\begin{equation*}
|A|+|B|-1+\varepsilon\min(|A|,|B|) \geq (1-\mu)|A^*+B^*|.
\end{equation*}
Combining this with \eqref{eq22.1*} and \eqref{eq22.27*}, we find that
\begin{equation}\label{eq22.3*}
\begin{split}
|A^*+B^*| &\leq (1-\mu)^{-1}(|A|+|B|-1+\varepsilon\min(|A|,|B|)) \\
&\leq (1-\mu)^{-2}|A^*|+(1-\mu)^{-2}|B^*|-1+2\varepsilon \min(|A^*|, |B^*|)\\
&\leq |A^*|+|B^*|-1+8\mu\max(|A^*|,|B^*|)+ 2\varepsilon\min(|A^*|,|B^*|)\\
&\leq |A^*|+|B^*|-1+4\varepsilon\min(|A^*|,|B^*|).
\end{split}
\end{equation}
Recalling Theorem~\ref{Freiman_Zp} and \eqref{eq22.27*}, we see that there
exist arithmetic progressions $P$ and $Q$ with
the same common difference such that
\begin{equation*}
|A^*\Delta P| \leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|) \text{ and } |B^*\Delta Q| \leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|)
\end{equation*}
The conclusion now follows from this and \eqref{eq22.1*}:
\begin{eqnarray*}
|A\Delta P|&\leq& |A^*\Delta P|+|A\Delta A^*| \leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|)+\mu|A|\\
&\leq& \frac{\gamma}{2^{11}}\min(|A|,|B|)+\mu|A|
\leq \frac{\gamma}{2^{10}}\min(|A|,|B|).
\end{eqnarray*}
and
\begin{eqnarray*}
|B\Delta Q|&\leq& |B^*\Delta Q|+|B\Delta B^*|
\leq \frac{\gamma}{2^{11}}\min(|A^*|,|B^*|)+\mu|B|\\
&\leq& \frac{\gamma}{2^{11}}\min(|A|,|B|)+\mu|B|
\leq \frac{\gamma}{2^{10}}\min(|A|,|B|).
\end{eqnarray*}
This completes the proof of Theorem~\ref{all_thm2}.
\end{proof}
\section{Proof of Theorem 1}
We start with a sketch of how the proof of Theorem~\ref{all_thm1} will proceed.
By Theorem~\ref{all_thm2}, we may assume that $A$ and $B$ are close to
intervals $I$ and $J$. Some delicate analysis around the endpoints of $I$ and
$J$, where we may slightly alter these intervals, will allow us to reduce to
the case where $A$ and $B$ are actually contained in $I$ and $J$. Thus there is
no `wraparound', and so we may as well be working in ${\mathbb Z}$ instead of ${\mathbb Z}_p$.
Assume for simplicity that $I = J$ has size $n$, that $A$ and $B$ are
contained in $I$ and $J$ and have size at least say $(1- 1/1000) n$, and that
$c_1 = c_2= c \sqrt{n}$ for some large constant $c$.
Fix $d$ to be about $\sqrt n$, and as usual write $X^y$ for $X \cap (y \mod d)$. Assume $A^0$ is the
largest of the fibres $A^y$. Then clearly $2|A^0| > |B^y|$ for all $y$. By
Theorem~\ref{CD_technical}, if $B'^y$ is a bounded set of $r$ random
points in $B^y$, chosen according to some distribution independent of $A$, we
have that $|A^0 + B'^y| \geq |A^0| + |B^y|-1$.
If we let $B'= \cup_y B'^y$, we have $|A^0 + B'| \geq |A^0| |\pi(B)| +|B| -
| \pi(B)|$ and $A^0 + B' \subset \pi^{-1} \pi(B)$.
Now pick a random fibre $y \mod d$ and let $A' = A^0 \cup A^y$.
We want to show that
for each fibre $z \mod d$ we have ${\mathbb E} |(A'^y+B')^z| \geq n/d$.
Indeed, once we have shown this, then combining the two inequalities and summing
over all $z$ not in $\pi(B)$ gives
${\mathbb E} |A'+B'| \geq |A^0| |\pi(B)| +|B| - |\pi(B)| + (d- |\pi(B)|) n/d \geq n/d
| \pi(B)| +|B| - |\pi(B)| + (d- |\pi(B)|) n/d \geq n+|B| - d$.
So fix sets $A'$ and $B'$ which satisfy this bound. To finish from
here we just note that by adding $d$ extra points in $A'$ and $d$ extra points in
$B'$ we can guarantee $|A'+B'| \geq |A|+|B|-1$. We also note $|A'| = |B'| =
O(\sqrt n)$.
To show ${\mathbb E} |(A'^y+B')^z| \geq n/d$, we proceed as follows. Note that the
proportion of fibres $y \mod d$ such that $A^y$ and $B^y$ have size at least
$9n/10d$ is at least $9/10$. Therefore, for a fixed fibre $z \mod d$, with
probability at least $8/10$ both $A^y$ and $B^{z-y}$ have size at least $9n/10d$.
Conditioned on this event, it follows that $2|A^y| \geq |B^{z-y}|$. Using
Theorem~\ref{CD_technical} again, we obtain $|A^y+B'^{z-y}|
\geq |A^y|+|B^{z-y}|-1 \geq 18n/10d - 1$.
Hence
\[
{\mathbb E} |(A'^y+B')^z| \geq (8/10) (18n/10d - 1) \geq n/d.
\]
We now start to work towards the proof of Theorem~\ref{all_thm1}. We collect together in advance
some results that we shall need. The first of these results will be applied when we already know that our
sets are close to intervals.
\begin{theorem}\label{all_thm3}
There exists $\gamma>0$ such that for all $\alpha>0$ there exists $c$ for which the following holds. Let $X$ and $Y$ be
subsets of two intervals $I$ and $J$ of $\mathbb{Z}_p$, and let $1 \leq c_1 \leq |X|$ and $1 \leq c_2 \leq |Y|$ be
integers such that $c_1c_2 \geq c\max(|X|,|Y|)$. Suppose that $\alpha |J| \leq |I| \leq \alpha^{-1}|J|$, \ $|I| +|J| \leq p$
and $\max(|I\setminus X|, |J \setminus Y| ) \leq \gamma \min(|I|, |J|)$. Then there exist
$X' \in X^{(c_1)}$ and $Y' \in Y^{(c_2)}$ such that $|X'+Y'| \geq |X|+|Y|-1$.
\end{theorem}
\begin{proof}
Since $|I| +|J| \leq p$, we may assume that the ambient space is $\mathbb{Z}$ rather than $\mathbb{Z}_p$.
Let $\gamma'$ and $k$ be the outputs of Theorem~\ref{CD_technical} with input $\alpha/2$ (since we are now in
$\mathbb{Z}$, there is no $\beta$, or more formally we are applying Theorem~\ref{CD_technical} inside $\mathbb{Z}_q$ for
some much larger $q$ with say $\beta=1/2$).
By increasing $\gamma'$ if
necessary we may assume that
$k \geq 100 /\gamma'$. Set $\gamma=\gamma'/100$ and
let $t=\lceil\log_{2/3}(1-(1+\alpha \gamma'/100)^{-1}) \rceil$, and put $c=2^5t(k+1)$. We may
assume by symmetry that
$c_1 \geq c_2$. Let $d= \lfloor c_2(k+1)^{-1}\rfloor$. Note that the hypothesis forces $c_2 \geq c \geq 2(k+1)$, which
ensures that $d$ is positive. \\
The definition of $d$ and the inequality $k \ge 100/\gamma'$ imply
\begin{equation}\label{03.005}
\min(|I|,|J|) \geq \min(|X|,|Y|) \geq c_2 \geq (k+1)d \geq 100d/ \gamma'.
\end{equation}
Given a set $Z$, recall that we write $Z^x_d$ for $Z \cap (x \text{ mod } d)$, the points of $Z$ in a fibre. Since $I$ and $J$ are intervals, for every $x \in \mathbb{Z}_d$ we have
\begin{equation*}\label{03.025}
|I|/d+1 \geq |I^x_d| \geq |I|/d-1 \text{ and } |J|/d+1 \geq |J^x_d| \geq |J|/d-1.
\end{equation*}
Combining the last two inequalities, for every $x \in \mathbb{Z}_d$ we have
\begin{equation}\label{03.027}
|I^x_d| \geq |I|/d- (\gamma'/100 d) \min(|I|,|J|) \text{ and } |J^x_d| \geq |J|/d-(\gamma'/100 d) \min(|I|,|J|)
\end{equation}
and
\begin{equation}\label{03.028}
|I^x_d| \leq |I|/d+ (\gamma'/100 d) \min(|I|,|J|) \text{ and } |J^x_d| \leq |J|/d+(\gamma'/100 d) \min(|I|
,|J|).
\end{equation}
\noindent
We may assume (by taking a translate of $X$, if necessary) that
$|X^0_{d}| = \max_{x \in \mathbb{Z}_{d}}|X^x_{d}|$. Then
\begin{equation*}\label{03.01}
|X^0_{d}| \geq |X|/d \geq |I|/d- (\gamma/ d) \min(|I|,|J|) \geq |I|/d- (\gamma'/3 d) \min(|I|,|J|).
\end{equation*}
Now define the sets $E_X, E_Y \subset \mathbb{Z}_d$ by
\[
E_X=\{x \in \mathbb{Z}_d \text{ : } |X^x_{d}| \geq (|I|/d)- (\gamma'/3 d) \min(|I|,|J|) \}
\]
and
\[
E_Y= \{x \in \mathbb{Z}_d \text{ : } |Y^x_{d}| \geq (|J|/d)- (\gamma'/3 d) \min(|I|,|J|) \}.
\]
For all $x\in E_X$ and $y \in E_Y$, noting that $X^x_d \subset I^x_d$ and $Y^y_d \subset J^y_d$, we see by \eqref{03.027} and
\eqref{03.028} that
\[
\max(|X^x_d \Delta I^x_d|, |Y^x_d \Delta J^y_d|) \leq (1/3+1/100) (\gamma'/d) \min(|I|,|J|) \leq \gamma'
\min(|I^x_d|, |J^y_d|).
\]
Since $(\alpha/2)|J^y_d| \leq |I^x_d| \leq (2/\alpha)|J^y_d|$, by Theorem~\ref{CD_technical}, there is a family $\mathcal{F}^y_d$ of subsets of $Y^y_d$ of size $k$, depending only on $Y^y_d$ (not on $X^x_d$), such that
\begin{equation}\label{03_06}
\E_{Z\in \mathcal{F}^y_d} |X^x_d+Z| \geq |X^x_d|+|Y^y_d|-1.
\end{equation}
Now construct a family $ \mathcal{F} = \{ \cup_{y \in E_Y} F^y_d \text{ : } F^y_d \in \mathcal{F}^y_d \}$,
and note that each set $F \in \mathcal{F}$ satisfies $|F| \leq |E_Y| k \leq dk \leq c_2-d$.
Define sets $ E_x' \subset E_X$ and $E_Y' \subset E_Y$ by
\[
E_X'=\{x \in \mathbb{Z}_d \text{ : } |X^x_{d}| \geq (|I|/d)- (\gamma'/10 d) \min(|I|,|J|) \}
\]
and
\[
E_Y'= \{x \in \mathbb{Z}_d \text{ : } |Y^x_{d}| \geq (|J|/d)- (\gamma'/10 d) \min(|I|,|J|) \}.
\]
By Markov's inequality,
\begin{equation*}
\mathbb{P}(E_X') \geq 1-\frac{\gamma}{(1/10-1/100)\gamma'}
\geq \frac{2}{3} \ \ \text{and} \ \ \mathbb{P}(E_Y') \geq 1- \frac{\gamma}{(
1/10-1/100)\gamma'} \geq \frac{2}{3}.
\end{equation*}
Simple calculations using \eqref{03.027} and \eqref{03.028} now show that for all $x' \in
E_X'\subset E_X, y' \in E_Y'\subset E_Y, x \in \mathbb{Z}_d$ and $y \in \mathbb{Z}_d \setminus E_y $ we have
\begin{eqnarray*}
|X^{x'}_d|+|Y^{y'}_d| &\geq& |I|/d +|J|/d -(\gamma'/5 d) \min(|I|,|J|)\\
&\geq& [|I|/d + (\gamma'/100 d) \min(|I|,|J|)] + [|J|/d- (\gamma'/3d) \min(|I|,|J|)]\\
&+& (1/3-1/5-1/100 )(\gamma'/d) \min(|I|,|J|)\\
&\geq& |X^x_d|+|Y^y_d| +(\gamma'/10 d) \min(|I|,|J|)\\
&\geq& |X^x_d|+|Y^y_d| +(\alpha \gamma'/10 d) \max(|I|,|J|)
\geq (1+\alpha \gamma'/100) (|X^x_d|+|Y^y_d|).
\end{eqnarray*}
\noindent
Now consider the family
$ \mathcal{G} = \{ X^0_d\cup_{i=1}^t X^{x_i}_d \text{ : } x_i \in E_X'\}$.
By \eqref{03.028}, every set $G \in \mathcal{G}$ satisfies
\begin{eqnarray*}
|G| &\leq& (t+1)(|I|/d+ (\gamma'/100 d) \min(|I|,|J|)) \leq 8td^{-1}|X|\\
&\leq& 2^{4}t(k+1)c_2^{-1}|X| \leq 2^{-1}cc_2^{-1}|X| \leq 2^{-1}c_1\leq c_1-d.
\end{eqnarray*}
The last ingredient needed to complete the proof of Theorem~\ref{all_thm3} is the following lemma.
\begin{lemma}\label{F-and-G}
The families ${\mathcal F}$ and ${\mathcal G}$ are such that
$$\E_{X' \in \mathcal{G},\ Y'\in \mathcal{F} } |X'+Y'| \geq |X|+|Y|-d.$$
\end{lemma}
\begin{proof}
By the linearity of expectation, it is enough to show that for all $z \in \mathbb{Z}_d$ we have
$$\E_{X' \in \mathcal{G}, \ Y'\in \mathcal{F} } |(X'+Y')^z_d| \geq |X^z_d|+|Y^z_d|-1.$$
First assume that $z \in E_Y$. Then, using \eqref{03_06}, we get
\begin{equation*}
\E_{X' \in \mathcal{G},\ Y'\in \mathcal{F} } |(X'+Y')^z_d| \geq \E_{Z\in \mathcal{F}^z_d } |(X^0_d+Z)^z_d|
= \E_{Z\in \mathcal{F}^z_d } |X^0_d+Z|
\geq |X^0_d|+|Y^z_d|-1 \geq |X^z_d|+|Y^z_d|-1.
\end{equation*}
Now assume instead that $z \not \in E_Y$. Then, using \eqref{03_06}, ${\mathbb P}(E_X')\ge 2/3$, and our bound on $|X^{x'}_d|+|Y^{y'}_d|$, we obtain
\begin{eqnarray*}
&&\E_{X' \in \mathcal{G}, \ Y'\in \mathcal{F} } |(X'+Y')^z_d| \geq
\E_{\substack{x_1, \hdots, x_t \in E_X' \\ Y'\in \mathcal{F} }} |((\cup_iX^{x_i}_d)+Y' )^z_d|\\
&& \hspace{30pt}\geq \E_{\substack{x_1, \hdots, x_t \in E_X' \\ Y'\in \mathcal{F} }}
\bigg( |((\cup_iX^{x_i}_d)+Y' )^z_d| \text{ : } \exists i \text{ such that } z-x_i \in E_Y' \bigg) \\
&& \hspace{130pt} \times\Prob_{x_1, \hdots, x_t \in E_X'} \bigg(\exists i \text{ such that } z-x_i \in E_Y' \bigg)\\
&&\hspace{30pt}\geq \E_{\substack{x \in E_X' \cap (z-E_Y') \\ Y'\in \mathcal{F} }} |(X^{x}_d+Y' )^z_d| (1-(2/3)^t)
\geq \E_{\substack{x \in E_X' \cap (z-E_Y') \\ Z\in \mathcal{F}^{z-x}_d }} |X^{x}_d+Z | (1-(2/3)^t)\\
&&\hspace{30pt} \geq (|X^x_d|+|Y^{z-d}_d|-1) (1-(2/3)^t)
\geq \bigg((1+\alpha \gamma'/100) (|X^0_d|+|Y^z_d|)-1\bigg)(1-(2/3)^t)\\
&&\hspace{30pt}\geq (1+100^{-1}\alpha \gamma') (|X^0_d|+|Y^z_d|)(1-(2/3)^t)-1
\geq |X^0_d|+|Y^z_d|-1 \geq |X^z_d|+|Y^z_d|-1,
\end{eqnarray*}
proving Lemma~\ref{F-and-G}.
\end{proof}
To complete the proof of Theorem~\ref{all_thm3}, note that if
$X' \in X^{(c_1-d)}$ and $Y' \in Y^{(c_2-d)}$ satisfy
$|X'+Y'| \geq |X|+|Y|-d$, then
there exist sets $X'' \in X^{(c_1)}$ and $Y'' \in Y^{(c_2)}$ such that
$|X''+Y''| \geq |X|+|Y|-1.$
\end{proof}
Having proved Theorem~\ref{all_thm3}, we turn to `improving the setup by changing the ends of the
intervals', the step we mentioned in our sketch of the proof at the start of the section.
\begin{lemma}\label{all_lem4}
For all $\beta$ and $\gamma$ with $2^{-20}> \beta >2^{20} \gamma >0$ the following holds. Let $A$ and $B$ be
subsets of $\mathbb{Z}_p$ and
let $I$ and $J$ be intervals of $\mathbb{Z}_p$ such that
$|A|+|B| \leq (1-\beta)p$ and $\max(|A\Delta I|, |B\Delta J|) \leq \gamma \min(|A|, |B|)$.
Then in $\mathbb{Z}_p$ there are two sets of three consecutive intervals, $I_1, I_2, I_3$ and $J_1, J_2, J_3$, with
\[
\lfloor (\beta/8) p \rfloor \le \min(|I_1|, |I_3|,|J_1|,|J_3|)\le \max(|I_1|, |I_3|,|J_1|,|J_3|) \leq (\beta/4) p
\]
such that, setting $A_i=A\cap I_i$ and $B_i=B \cap J_i$, we have $|A_1|=|B_1|$, $|A_3|=|B_3|$, and
\begin{equation}\label{all_eq7}
\max(|A\Delta I_2|, |B\Delta J_2|) \leq 2^{10}\gamma \min(|A|, |B|).
\end{equation}
Moreover, for $i \in \{1,3\}$ and for any two arithmetic progressions $P$ and $Q$ we have
\begin{equation*}
\max(|A_i \Delta P|, |B_i \Delta Q |) \geq |A_i|/2^{10}= |B_i|/2^{10}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $I_2'$ and $J_2'$ be maximal intervals in $\mathbb{Z}_p$ satisfying \eqref{all_eq7}: note that such intervals do exist by
hypothesis. By our bounds on $|A|+|B|$, $|A\Delta I|$ and $|B\Delta J|$, we have
\begin{equation*}\label{eqp_001}
|I_2'|+|J_2'| \leq |A|+|B|+ 2 \gamma \min(|A|,|B|) \leq (1-\beta +2\gamma) p \leq (1- (\beta/2)) p.
\end{equation*}
Thus we can construct intervals $I_1', I_3'$ and $J_1', J_3'$ such
that $I_1',I_2',I_3'$ and $J_1', J_2', J_3'$ are two families of three consecutive
intervals of $\mathbb{Z}_p$ with
\begin{equation}\label{eqp_002}
|I_1'|=|I_3'|=|J_1'|=|J_3'|= \lfloor (\beta/8) p \rfloor.
\end{equation}
Note that, by the maximality of $I_2'$ and $J_2'$, for any intervals $I_1''', I_3'''$ and $J_1''', J_3'''$
such that $I_1''', I_2', I_3'''$ and $J_1''', J_2', J_3'''$ are familes of consecutive
intervals, if we set $A_i'''=A \cap I_i'''$ and $B_i'''= B \cap J_i'''$, then we have
\begin{equation}\label{eqp_003}
|I_i'''| > 2 |A_i'''| \text{ and } |J_i'''| > 2 |B_i'''|.
\end{equation}
Let $A_i'=A \cap I_i'$ and $B_i'=B\cap J_i'$, and assume that
$|A_1'| \geq |B_1'| \text{ and } |A_3'| \geq |B_3'|$.
(The three other cases are analogous.)
Note that by \eqref{all_eq7} we have
\begin{equation}\label{eqp_0030}
\max(|A_1'|, |A_3'|, |I_2' \setminus A_2'|) \leq \gamma \min(|A|, |B|) \leq 2\gamma \min( |A_2'|, |B_2'|) .
\end{equation}
Now consider subintervals $I_1''$ and $I_3''$ at the ends of interval $I_2'$ that are minimal subject
to the following two properties:
setting $A_i''=A \cap I_i''$, we have
\begin{equation}\label{eqp_004}
|A_1''| \geq 14|A_1'| \text{ and } |A_3''|\geq 14|A_3'|
\end{equation}
and there exist intervals $P_1''$ and $P_3''$ (contained inside $I_1''$ and $I_3''$ respectively) such that
\begin{equation}\label{eqp_005}
|P_1'' \Delta A_1''| \leq |A_1''|/14 \text{ and } |P_3'' \Delta A_3''|\leq |A_3''|/14.
\end{equation}
(Here we insist that if $A_1'=\emptyset$ or $A_3'=\emptyset$ then $I_1''=P_1''=A_1''=
\emptyset$ or $I_3''=P_3''=A_3''=\emptyset$, respectively.)
Note that such intervals $I_1''$ and $I_3''$ do exist, as the intervals $I_1''=P_1''=I_3''=P_3''=I_2'$ have the desired properties by \eqref{eqp_0030}.
We now show that
\begin{equation}\label{eqp_006}
|I_1''|+|I_3''| \leq 2^6\gamma \min(|A_2'|, |B_2'|)<|I_2'|.
\end{equation}
The right-hand inequality is immediate, since $A_2' \subset I_2'$ and $\gamma < 2^{-6}$. For the left-hand
inequality, suppose for a contradiction that say $ |I_1''| > 2^5 \gamma \min(|A_2'|,|B_2'|)$. Consider first the case when $2 \gamma \min(|A_2'|,|B_2'|) <1
$. By \eqref{eqp_0030} we have $A_1'=A_3'=\emptyset$, and hence we obtain $I_1''=I_3''=\emptyset$, which is a contradiction.
Consider now the case when $2 \gamma \min(|A_2'|,|B_2'|) \geq 1$. In this case we consider the proper subinterval $I_1'''$ at the
end of $I_1''$ with $|I_1'''|=\lfloor 2^5\gamma \min(|A_2'|,|B_2'|) \rfloor
\geq 30 \gamma \min(|A_2'|,|B_2'|)$. Let $A_1'''=I_1''' \cap A$. By \eqref{eqp_0030}, we
have $| I_1''' \setminus A_1'''| \leq |I_2' \setminus A_2'| \leq 2 \gamma \min(|A_2'|,|B_2'|)$. In particular,
we have $|A_1'''| \geq 28 \gamma \min(|A_2'|, |B_2'|) $, which implies $|I_1''' \setminus A_1'''| \leq |A_1'''|/14$. But by \eqref{eqp_0030},
we also have $|A_1'| \leq 2\gamma \min(|A_2'|, |B_2'|)$, which implies $|A_1'''| \geq 14|A_1'|$.
Therefore $I_1'''$ is an interval strictly smaller than $I_1''$ with the desired properties, giving a contradiction. This proves
inequality \eqref{eqp_006}.
By \eqref{eqp_006}, the intervals $I_1''$ and $I_3''$ induce a partition $I_2'=I_1''\sqcup I_2 \sqcup I_3''$ into consecutive
intervals. Moreover, by \eqref{eqp_006} and \eqref{all_eq7} we get
\begin{equation}\label{eqp_007}
|A\Delta I_2| \leq |A\Delta I_2'| + |I_1''|+|I_3''| \leq 2^7 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
Note also that by \eqref{all_eq7} we have
\begin{equation}\label{eqp_0031}
|J_2'\setminus B_2'| \leq \gamma \min(|A|,|B|) \leq 2 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
Now consider subintervals $J_1''$ and $J_3''$ at the ends of $J_2'$ such that with $B_i''=B\cap J_i''$ we
have
\begin{equation}\label{eqp_0032}
|B_1'|+|B_1''|=|A_1'|+|A_1''| \text{ and } |B_3'|+|B_3''|=|A_3'|+|A_3''|
\end{equation}
Note that there are such intervals, since for $i \in \{1,3\}$ both $|B_i'| \leq |A_i'| \leq |A_i'|+|A_i''|$
and $|B_i'|+|B_2'| \geq |B_2'| \geq |A_i'|+|I_i''| \geq |A_i'|+|A_i''|$ hold. We now show that
\begin{equation}\label{eqp_0040}
|J_1''|+|J_3''| \leq 2^8\gamma \min(|A_2'|, |B_2'|)<|J_2'|.
\end{equation}
Assume for a contradiction that $|J_1''| \geq 2^7 \gamma \min(|A_2'|, |B_2'|)$. On the one hand, by \eqref{eqp_0031}
we deduce $|B_1''| \geq |J_1''|- |J_2' \setminus B_2'| \geq 126 \gamma \min(|A_2'|,|B_2'|)$. On the other hand,
by \eqref{eqp_0030} we have $|A_1'| \leq 2 \gamma
\min(|A_2'|, |B_2'|)$ and by \eqref{eqp_006} we have $|A_1''| \leq |I_1''| \leq 2^6 \gamma \min(|A_2'|, |B_2'|)$.
Thus we
obtain $|B_1''| > |A_1'|+|A_1''|$, which gives the desired contradiction. This proves inequality \eqref{eqp_0040}.
By \eqref{eqp_0040}, the intervals $J_1''$ and $J_3''$ induce a partition $J_2'=J_1''\sqcup J_2 \sqcup J_3''$ into
consecutive intervals. Moreover, by \eqref{eqp_0040} and \eqref{all_eq7} we have
\begin{equation}\label{eqp_008}
|B\Delta J_2| \leq |B\Delta J_2'| + |J_1''|+|J_3''| \leq 2^9 \gamma \min(|A_2'|,|B_2'|).
\end{equation}
For $i \in \{1,3\} $, set $I_i=I_i' \sqcup I_i''$, $J_i= I_i'\sqcup I_i''$, and note that $I_1, I_2, I_3$ and $J_1, J_2, J_3$ are consecutive intervals. Moreover, by \eqref{eqp_002} we have
\begin{equation}\label{eqp_009}
\min(|I_1|, |I_3|, |J_1|, |J_3|) \geq \lfloor (\beta/8) p \rfloor.
\end{equation}
In addition, by \eqref{eqp_002}, \eqref{eqp_006} and \eqref{eqp_0031}, we also have
\begin{equation}\label{eqp_0010}
\max(|I_1|, |I_3|, |J_1|, |J_3|) \leq (\beta/4) p .
\end{equation}
For $i \in \{1,2,3\}$ let $A_i=A\cap I_i$ and $B_i = B \cap J_i$. By \eqref{eqp_0032} we have
\begin{equation}\label{eqp_010}
|A_1|=|B_1| \text{ and } |A_3|=|B_3|.
\end{equation}
It remains to show that for any arithmetic progressions $P_1$ and $P_3$ we have
\begin{equation}\label{eqp_011}
|A_1 \Delta P_1| \geq 2^{-10}|A_1| \text{ and } |A_3 \Delta P_3| \geq 2^{-10}|A_3|.
\end{equation}
Assume for a contradiction that
\begin{equation}\label{eqp_012}
|A_1 \Delta P_1| < 2^{-10}|A_1|,
\end{equation}
which in particular means that
\begin{equation}\label{eqp_0125}
A_1\neq \emptyset \text{ i.e. } A_1' \neq \emptyset.
\end{equation}
Note that \eqref{eqp_004}, \eqref{eqp_005} and \eqref{eqp_012} imply
\begin{eqnarray*}
|P_1'' \Delta P_1| &&\leq |A_1 \Delta P_1| + |P_1'' \Delta A_1|
\leq |A_1 \Delta P| + |A_1'| + |P_1'' \Delta A_1''| \\
&&\leq |A_1|/2^{10} + |A_1''|/14+|A_1''|/14 \leq (1/2^9+1/7) |A_1''| \le |P_1''|/4.
\end{eqnarray*}
Recall that, when $A_1' \neq \emptyset$, $P_1''$ is a subinterval of $I_1''$ with $p/8 \geq |P_1''| \geq 8$
by \eqref{eqp_004}, \eqref{eqp_005} and \eqref{eqp_006}. It follows that $P_1$ is an interval intersecting the
interval $I_1''$ of size $p/4 \geq |P_1| \geq 4$. By \eqref{eqp_0010} we also
deduce $|P_1|+|I_1| \leq p/2$, which implies
that $P_1 \cap I_1$ is an interval. By replacing $P_1$ with $P_1\cap I_1$, we may assume that $P_1$ is a
subinterval of $I_1$.
We distinguish two cases: recalling that $I_1''$ and $I_3''$ were chosen to be minimal subject
to \eqref{eqp_004} and \eqref{eqp_005}, we ask whether the lower bound on the size of $A_1''$ in \eqref{eqp_004} is attained or not.
\textbf{Case A. $|A_1''|=14|A_1'|$.}\\
In this case, by \eqref{eqp_003}, we have
\begin{equation*}\label{eqp_015}
|(P_1 \cap I_1')\Delta A_1'| \geq |A_1'|.
\end{equation*}
So we obtain
\begin{equation}\label{eqp_016}
|A_1\Delta P_1| \geq |(P_1 \cap I_1')\Delta A_1'| \geq |A_1'| = |A_1|/15.
\end{equation}
\textbf{Case B. $|A_1''|>14|A_1'|\geq 14$.}\\
In this case, by the minimality of $I_1''$, if we let $x$ be the last point inside $A_1''$, then by \eqref{eqp_005} we deduce
\begin{equation*}\label{eqp_018}
|(P_1 \cap I_1'') \Delta (A_1'' \setminus \{x\})| > |A_1'' \setminus \{x\}| /14 .
\end{equation*}
This is equivalent to
\begin{equation*}\label{eqp_019}
|(P_1 \cap I_1'') \Delta (A_1'' \setminus \{x\})| \geq \min(2, |A_1'' \setminus \{x\}| /14) ,
\end{equation*}
and hence
\begin{equation*}\label{eqp_020}
|(P_1 \cap I_1'') \Delta A_1''| \geq \min(1, (|A_1''|-1)/14-1) \geq |A_1''|/28.
\end{equation*}
Thus we obtain
\begin{equation}\label{eqp_021}
|A_1\Delta P_1| \geq |(P_1 \cap I_1'')\Delta A_1''| \geq |A_1''|/28 \geq |A_1|/56.
\end{equation}
Inequalities \eqref{eqp_016} and \eqref{eqp_021} imply that, in either case,
$|A_1 \Delta P_1| \geq |A_1|/56$ , contradicting \eqref{eqp_012} and so proving \eqref{eqp_011}.
The proof of Lemma~\ref{all_lem4} is now complete, thanks to \eqref{eqp_009}, \eqref{eqp_0010} \eqref{eqp_010},
\eqref{eqp_007}, \eqref{eqp_008} and \eqref{eqp_011}.
\end{proof}
Our next lemma is a simple fact about the `stickout' of sumsets from a set of fixed size:
roughly speaking, it says that if $Y$ is much larger than $X$ then the sum of $X$ with a random few points of
$Y$ is expected to be
much larger than $2|X|$.
\begin{lemma}\label{all_lem7}
Let $X$, $Y$ and $Z$ be subsets of $\mathbb{Z}_p$ and let $1 \leq c_1 \leq |X|$ and $1 \leq c_2 \leq |Y|$ be
integers such that $c_1c_2 \geq 16 |X| $. Suppose that $8|X|, 8|Z| \leq |Y|<p/2$.
Then there exist
$X' \in X^{(c_1)}$ and $Y' \in Y^{(c_2)}$ such that $|(X'+Y')\setminus Z| \geq 2|X|$.
\end{lemma}
\begin{proof}
Suppose the assertion is false, i.e. $\max_{X' \in X^{(c_1)}, Y' \in Y^{(c_2)}} |(X'+Y')\setminus Z| < 2|X|$ which, in particular, implies that $\max_{X' \in X^{(c_1)}, Y' \in Y^{(c_2/2)}} |(X'+Y')\setminus Z| < 2|X|$. Let $X'=\{x_1, \hdots, x_{c_1}\}$ and $Y'=\{y_1, \hdots, y_{c_2/2}\}$ be elements of $X^{(c_1)}$ and $Y^{(c_2/2)}$ chosen uniformly at random. Then ${\mathbb E} |(X'+Y')\setminus Z|$ is bounded from below as follows:
\begin{eqnarray*}
&& \ \ \ \sum_{i,j} {\mathbb E}|\{x_i+y_j\}\, \setminus \, [Z \cup (\{x_1, \hdots, x_{i-1}, x_{i+1}, \hdots, x_{c_1}\} + \{y_1, \hdots, y_{j-1}, y_{j+1}, \hdots, y_{c_2}\} ) ] |\\
&&= \sum_{i,j} \Prob | x_i+y_j \not \in Z \cup (\{x_1, \hdots, x_{i-1}, x_{i+1}, \hdots, x_{c_1}\} + \{y_1, \hdots, y_{j-1}, y_{j+1}, \hdots, y_{c_2}\} ) |\\
&&\geq \sum_{i,j} 1-\max_{ X' \in X^{(c_1)}, Y'\in Y^{(c_2/2)}}\frac{|Z \cup (X'+Y')|}{|Y|-(c_2/2)}\\
&&\geq \sum_{i,j} 1- \frac{|Z|+2|X|}{|Y|/2} \geq \frac{c_1c_2}{2} (1-\frac{3}{4}) \geq 2|X|,
\end{eqnarray*}
so $ {\mathbb E} |(X'+Y')\setminus Z| \geq 2|X|$, completing the proof.
\end{proof}
As an immediate corollary we have the following, obtained by applying the previous lemma inductively on $k$
(increasing $Z$ at each stage).
\begin{corollary}\label{cor_new000}
Let $X_1, \hdots, X_k$, $Y_1, \hdots, Y_k$ and $Z$ be subsets of $\mathbb{Z}_p$ and
let $1 \leq c_1^i \leq |X_i|$ and $1 \leq c_2^i \leq |Y_i|$ be integers such that
for all $i$ we have $c_1^ic_2^i\geq 16|X_i|$. Suppose
that for all $i$ we have $16|X_i|, 16|Z|\leq |Y_i| <p/2$. Then there exist
$X'_i \in (X_i)^{(c_1^i)}$ and $Y'_i \in (Y_i)^{(c_2^i)}$ (for each $i$) such that
\[
\ \ \ \ \ \ |\cup_i(X'_i+Y'_i)\setminus Z|
\geq \min(16^{-1}|Y_1|, \hdots, 16^{-1}|Y_k|, 2\sum_i|X_i|). \ \ \ \ \ \ \square
\]
\end{corollary}
\noindent
The final ingredient we need is a somewhat cumbersome result about partitions into intervals.
\begin{lemma}\label{all_lem6}
For all $1/2^{10}> \alpha, \beta >0$ the following holds. Consider partitions $\mathbb{Z}_p= I_0 \sqcup I_1
\sqcup I_2 \sqcup I_3 =J_0 \sqcup J_1 \sqcup J_2 \sqcup J_3$ into consecutive intervals such that
$|I_2|+|J_2| \leq (1-\beta/2)p$, $(\alpha/2)|J_2| \leq |I_2| \leq (2/\alpha) |J_2|$,
$\min(|I_2|, |J_2|) \ge 24/\beta$ and
\[
\lfloor (\beta/8) p \rfloor \le \min(|I_1|,|I_3|,|J_1|,|J_3|) \le \max(|I_1|, |I_3|,|J_1|,|J_3|) \le (\beta/4) p.
\]
Then there are four families of subsets of $\mathbb{Z}_p$, each of size $k \leq 100/ \alpha \beta$,
\[
\mathcal{I}_0=\{I^1_0, \hdots, I^k_0\}, \ \ \mathcal{I}_2=\{I^1_2, \hdots, I^k_2\}, \ \
\mathcal{J}_0=\{J_0^1, \hdots,J_0^k\}, \ \ \mathcal{J}_2=\{J_2^1, \hdots, J_2^k\}
\]
such that $\cup_i I_0^i = I_0, \cup_i J_0^i = J_0\ \text{ and }\ \cup_i I_2^i \subset I_2, \cup_i J_2^i \subset J_2$.
Furthermore, for every $1 \leq i \leq k$ we have \ $|I_2^i|=|J_2^i|= \lfloor (\beta/24) \min(|I_2|, |J_2|) \rfloor$
and $(I_0^i+J_2^i) \cap (I_2+J_2) = (J_0^i + I_2^i) \cap (I_2+J_2) = \emptyset$.
\end{lemma}
\begin{proof}
We construct the families of intervals $\mathcal{I}_0$ and $\mathcal{J}_2$. The construction of the
$\mathcal{J}_0$ and $\mathcal{I}_2$ is identical.
By the conditions above, for every $x \in I_0$ there is a
subinterval $J_2'$ of $J_2$ of size $\lfloor (\beta/8) \min(|I_2|,|J_2|) \rfloor$ such that
$(x+J_2') \cap (I_2+J_2) =\emptyset.$
Let $\mathcal{J}_2$ be a maximal collection of disjoint intervals of size $\lfloor (\beta/24) \min(|I_2|,|J_2|) \rfloor$
contained inside interval $J_2$.
First note that
\begin{equation*}
|\mathcal{J}_2| \leq |J_2|/\lfloor (\beta/24) \min(|I_2|,|J_2|) \rfloor \leq (48/ \beta)
(2/ \alpha) = 100/ (\alpha \beta).
\end{equation*}
Secondly, note that for any subinterval $J_2'$ of $J_2$ of size $\lfloor (\beta/8) \min(|I_2|,|J_2|) \rfloor$
there exists $J_2'' \in \mathcal{J}_2$
such that $J_2'' \subset J_2'$. Therefore, for any point $x \in I_0$ there exists $J_2^x \in \mathcal{J}_2$ such that
$$(x+J_2^x) \cap (I_2+J_2) =\emptyset.$$
Let $\mathcal{J}_2=\{J_2^1, \hdots, J_2^k\}$. For each $1 \leq i \leq k$
set $I_0^i=\{x \in I_0 \text{ : } J_2^x=J_2^i\}$, and finally put
$\mathcal{I}_0=\{I_0^1, \hdots, I_0^k\}$.
These families $\mathcal{I}_0$ and $\mathcal{J}_2$ have the desired properties.
\end{proof}
We are now ready to prove Theorem~\ref{all_thm1}.
\begin{proof}[Proof of Theorem~\ref{all_thm1}]
Fix $\alpha, \beta>0$ and assume $\alpha, \beta<2^{-30}$. Let $\gamma'$ and $c'$ be the output of
Theorem~\ref{all_thm3} with input $\alpha/2$. Let $\gamma=2^{-30}\min(\gamma',\beta)$.
Let $\epsilon>0$ and $c''>0$ be the
output of Theorem~\ref{all_thm2} with input $\alpha, \beta, \gamma$. Choose
$c=10^{20} \alpha ^{-2} \beta^{-2}\max(c',c'')$.
Let $A$ and $B$ be
subsets of $\mathbb{Z}_p$ and let $1 \leq c_1 \leq |A|$ and $1 \leq c_2 \leq |B|$ such that
$c_1 c_2 \geq c \max(|A|,|B|)$. This forces $\min(c_1,c_2) \geq c$, which, in particular, implies $\min(|A|,|B|) \geq c$. Let $c_1'=c_1 \alpha \beta/ 10^5 $
and $c_2'=c_2 \alpha \beta/ 10^5 $ and note that $c_1'c_2' \geq 10^{10}\max(c',c'') \max(|A|,|B|)$ and
$\min(c_1',c_2') \geq 10^{10}\max(c',c'')$. Observe that we are done unless
\[
\max_{A' \in A^{(c_1')}, B' \in B^{(c_2')}}|A'+B'| < |A|+|B|-1,
\]
so we may assume that this holds. But then we may apply Theorem~\ref{all_thm2} to deduce that
there are arithmetic progressions $I$ and $J$ with the same common difference such that
\begin{equation}\label{main_2}
\max(|A\Delta I|, |B \Delta J|) \leq \gamma \min(|I|,|J|).
\end{equation}
Furthermore, we may and shall assume that $I$ and $J$ are intervals.
But then Lemma~\ref{all_lem4} can be used to deduce that $\mathbb{Z}_p$ has disjoint partitions into
consecutive intervals, $\mathbb{Z}_p=I_0\sqcup I_1\sqcup I_2 \sqcup I_3= J_0\sqcup J_1\sqcup J_2 \sqcup J_3$,
that have the following properties. (Here and elsewhere, the notation $\sqcup$ indicates that we are
taking the union of disjoint sets.) On the one hand we have
\begin{equation}\label{main_3}
\lfloor (\beta/8) p \rfloor \le \min(|I_1|, |I_3|,|J_1|,|J_3|) \le \max(|I_1|, |I_3|,|J_1|,|J_3|) \leq (\beta/4) p
\end{equation}
On the other hand, writing $A_i=A\cap I_i$ and $B_i=B \cap J_i$, we have $|A_1|=|B_1|$, $|A_3|=|B_3|$, and
\begin{equation}\label{main_6}
\max(|A\Delta I_2|, |B\Delta J_2|) \leq 2^{10}\gamma \min(|A|, |B|).
\end{equation}
Moreover, if $i \in \{1,3\}$ and $P$ and $Q$ are arithmetic progressions, then
\begin{equation}\label{main_7}
\max(|A_i \Delta P|, |B_i \Delta Q |) \geq 2^{-10} |A_i|=2^{-10} |B_i|.
\end{equation}
In particular, if $i \in \{1,3\}$ we have $|A_i|=|B_i|=0$ or $|A_i|=|B_i| \geq 2$. Because $\min(c_1',c_2') \geq c''$ and $c_1'c_2' \geq c''|A_i||B_i|$, we either have $\min(c_1',c_2') \geq |A_i|=|B_i|$ or $\min(c_1',|A_i|) \min(c_2', |B_i|) \geq c''|A_i|=c''|B_i|$. By the
contrapositive of Theorem~\ref{all_thm2}, we deduce that
\begin{equation*}
\E_{A' \in A_i^{(c_1')}, \ B' \in B_i^{(c_2')}} |A'+B'| > |A_i|+|B_i|-1.
\end{equation*}
We further deduce that there are sets $A_1' \in A_{1}^{(c_1')},\ B_1' \in B_{1}^{(c_2')}, \
A_3' \in A_{3}^{(c_3')}$\ and \ $B_3' \in B_{3}^{(c_2')}$\ such that
\begin{equation}\label{main_008}
|A_1'+B_1'|\geq |A_1|+|B_1|\ \ \text{and} \ \ |A_3'+B_3'|\geq |A_3|+|B_3|.
\end{equation}
Choose a subset $Z$ of $\cup_{i \in \{1,3\}}(A_i'+B_i') $
with $|A_1|+|B_1|+|A_3|+|B_3|$ elements, which, by \eqref{main_6}, satisfies
\begin{equation}\label{ccc_000}
|Z|=|A_1|+|B_1|+|A_3|+|B_3|\le |A\Delta I_2|+ |B \Delta J_2| \leq 2^{11} \gamma \min(|A|,|B|).
\end{equation}
Now, by \eqref{main_6} we also have
\begin{equation*}
(1- 2^{10} \gamma)|A| \leq |I_2| \leq (1+2^{10}\gamma) |A| \ \ \ \text{and} \ \ \
(1- 2^{10} \gamma)|B| \leq |J_2| \leq (1+2^{10}\gamma) |B|,
\end{equation*}
and so
\begin{equation}\label{main_9}
|I_2|+|J_2| \leq (1-\beta)(1+2^{10}\gamma) p \leq (1-\beta/2)p
\end{equation}
and
\begin{equation}\label{main_10}
(\alpha/2) |I_2| \leq |J_2| \leq (2/ \alpha)|I_2|.
\end{equation}
Furthermore,
we deduce that
\begin{equation}\label{main_11}
\min(|I_2|,|J_2|) \geq 2^{-1}\min(|A|,|B|) \geq 2^{-1}c \ge 24/\beta.
\end{equation}
First, by construction, we have $A_2 \subset I_2 \text{ and } B_2 \subset J_2$.
Hence, recalling the relations \eqref{main_6}, \eqref{main_9} and \eqref{main_10}, and the fact that $\min(c_1', |A_2|) \min(c_2', |B_2|) \geq 4^{-1}c_1'c_2' \geq c'\max(|A_2'|,|B_2'|)$, we may apply Theorem~\ref{all_thm3}
with parameters $\alpha/2$, $\gamma'\geq 2^{30} \gamma$ and $c'$ to obtain the following: there exist $A_2' \in A_2^{(c_1')}$
and $B_2' \in B_2^{(c_2')}$ such that
\begin{equation}\label{ddd_000}
|A_2'+B_2'| \geq |A_2|+|B_2|-1.
\end{equation}
Second, by \eqref{main_3}, \eqref{main_9}, \eqref{main_10} and \eqref{main_11}, we may apply
Lemma~\ref{all_lem6} to find four families of subsets of $\mathbb{Z}_p$, each with
$k\le 100/ \alpha \beta$ sets,
\[
\mathcal{I}_0=\{I^1_0, \hdots, I^k_0\}, \mathcal{I}_2=\{I^1_2, \hdots, I^k_2\}, \mathcal{J}_0=\{J_0^1,
\hdots, J_0^k\}, \mathcal{J}_2=\{J_2^1, \hdots, J_2^k\},
\]
such that
\begin{equation}\label{main_14}
\cup_i I_0^i = I_0,\ \ \cup_i J_0^i = J_0 \ \ \ \text{ and }\ \ \ \cup_i I_2^i \subset I_2, \ \ \ \cup_i J_2^i \subset J_2
\end{equation}
and for every $1 \leq i \leq k$ and we have
\begin{equation}\label{main_15}
|I_2^i|=|J_2^i|= \lfloor (\beta/24) \min(|I_2|, |J_2|) \rfloor
\end{equation}
and
\begin{equation}\label{main_16}
(I_0^i+J_2^i) \cap (I_2+J_2) = (J_0^i + I_2^i) \cap (I_2+J_2) = \emptyset.
\end{equation}
Writing $A_j^i= A \cap I_j^i$ and $B_j^i=B \cap J_j^i$, by \eqref{main_6}, we have
\begin{equation*}\label{ccc_002}
\max \{|A_0^i|, |B_0^i|\} \leq 2^{10} \gamma \min(|A|,|B|) .
\end{equation*}
and by \eqref{main_6}, \eqref{main_11} and \eqref{main_15}, we have
\begin{equation}\label{ccc_001}
\big(\beta/100\big)\ \min(|A|,|B|) \leq \min \{ |A_2^i|, |B_2^i|\} \le \max \{ |A_2^i|, |B_2^i|\} \leq p/2.
\end{equation}
The last two inequalities and inequality \eqref{ccc_000} imply that
\begin{equation}\label{ccc_0025}
10^{10} \max \{|A_0^i|,\ |B_0^i|,\ |Z|\} \leq \min \{ |A_2^i|, |B_2^i|\} \le \max \{ |A_2^i|, |B_2^i|\} \leq p/2.
\end{equation}
As we have $\min(c_1',c_2') \geq 10^{10}$ and $c_1'c_2' \geq 10^{10} \max(|A|,|B|) \geq 10^{10}\max(|A_0^i|,|B_0^i|)$, we further deduce that
\begin{equation}\label{ccc_003}
\min(c_1',|A_0^i|) \min(c_2', |B_2^i|) \geq 16 |A_0^i| \ \ \text{and} \ \ \min(c_1',|B_0^i|) \min(c_2', |A_2^i|) \geq 16 |B_0^i|.
\end{equation}
By Corollary~\ref{cor_new000} together with \eqref{ccc_0025} and \eqref{ccc_003} we deduce that for $i \in [k]$ there exist $A'^i_0 \in (A_0^i)^{(c_1')}, B'^i_2 \in (B_2^i)^{(c_2')}, A'^i_2 \in (A_2^i)^{(c_1')}, B'^i_0 \in (B_0^i)^{(c_2')}$ such that
\begin{equation*}
|\cup_i(A'^i_0+B'^i_2)\cup_i(A'^i_2+B'^i_0)\setminus Z| \geq 16^{-1}\min(|A_2^1|, \hdots, |A_2^k|,|B_2^1|, \hdots, |B_2^k|, 32\sum_i|A^i_0|+|B^i_0|)
\end{equation*}
By \eqref{main_6} and \eqref{ccc_001}, we further deduce
\begin{equation}\label{ccc_005}
|\cup_i(A'^i_0+B'^i_2)\cup_i(A'^i_2+B'^i_0)\setminus Z| \geq |A_0|+|B_0|
\end{equation}
Finally, note that from \eqref{main_3} and \eqref{main_9} it follows that
\[
I_1+J_1, I_2+J_2 \text{ and } I_3+J_3
\]
are disjoint sets, which in particular implies that
$$A_1'+B_1', A_2'+B_2' \text{ and } A_3'+B_3'$$
are disjoint sets. Moreover, by \eqref{main_16}, it follows that
$$\cup_i(A'^i_0+B'^i_2)\cup_i (A'^i_2+B'^i_0) \text{ and } A_2'+B_2' $$
are disjoint sets.
Let $$A'= A_1'\cup A_2'\cup A_3' \cup_i A'^i_0 \cup_i A'^i_2 \text{ and } B'= B_1'\cup B_2'\cup B_3' \cup_i B'^i_0 \cup_i B'^i_2.$$
From \eqref{main_008}, \eqref{ccc_000}, \eqref{ddd_000} and \eqref{ccc_005} we conclude that
$$|A'+B'| \geq (|A_0|+|B_0|)+(|A_1|+|B_1|)+(|A_2|+|B_2|-1)+(|A_3|+|B_3|) = |A|+|B|-1$$
and that
\[
|A'| \leq (3+2\times 100 \alpha^{-1}\beta^{-1})c_1' \leq c_1 \text{ and }|B'| \leq (3+2\times 100 \alpha^{-1}\beta^{-1})c_2' \leq c_2.
\]
This concludes the proof of Theorem~\ref{all_thm1}.
\end{proof}
\section{Open problems}
One very natural question to ask is as follows. Suppose that as usual we are choosing $c_1$ points of $A$ and $c_2$
points of $B$, where $|A|=|B|=n$ and $c_1c_2$ is a fixed multiple of $n$. Are there phenomena that may not occur in
the regime where say we are
choosing $c_1=n$ (in other words, we choose the whole of $A$) and $c_2$ bounded, but might possibly
always hold when both $c_1$ and $c_2$ are of order $\sqrt{n}$?
One example is the following.
{\bf Question 1. }{\em
Is there a constant $c$ such that the following is true? If $A$ and $B$ are non-empty subsets of
${\mathbb Z}_p$ with $|A|=|B|=n \leq (p+1)/2$ then there are subsets $A'\subset A$ and
$B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$.
}
As remarked in the Introduction, this is not true for $c_1=n$ and $c_2$ bounded, in other words for $A'=A$ and
$|B'|$ bounded, as may be seen by taking $A$ and $B$ to be random subsets of ${\mathbb Z}_p$ of size approaching
$p/2$. But it might conceivably be true when we force both $c_1$ and $c_2$ to be large.
In a similar vein, one might ask whether the case of both $c_1$ and $c_2$ being of order $\sqrt{n}$ is
in fact always the `best' case (where $A$ and $B$ are set of size $n$, say). Thus for Theorem~\ref{all_thm1}
we would be asking the following.
{\bf Question 2. }{\em
Let $c>0$ and $c_1=c_1(n)$ be such that whenever $A$ and $B$ are subsets of ${\mathbb Z}_p$ with
$|A|=|B|=n \leq p/3$ there exist subsets $A'\subset A$ and
$B'\subset B$, with $|A'| \leq c_1$ and $|B'| \leq cn/c_1$, such that $|A'+B'|\ge 2n-1$. Does it follow that
whenever $A$ and $B$ are subsets of ${\mathbb Z}_p$ with
$|A|=|B|=n \leq p/3$ there exist subsets $A'\subset A$ and
$B'\subset B$ with $|A'|=|B'|\le \sqrt{cn}$ such that $|A'+B'|\ge 2n-1$?
}
One could also ask what the `worst' case is: is it when $c_1=c$ and $c_2=n$? More generally, is there
`monotonicity' as $c_1$ varies from $c$ to $\sqrt{cn}$?
It would also be very interesting to obtain good bounds on the constants appearing in our various results.
For example, in Theorem~\ref{all_thm1}, what is the form of the dependence of $c$ on $\alpha$ and $\beta$?
Finally, we consider what happens for discrete versions of the Brunn--Minkowski inequality. Green and
Tao~\cite{GreenTao}
showed
that, given a dimension $k$ and a constant $\varepsilon >0$, there exists $t$ such that if $A$ is a subset of
${\mathbb Z}^k$ of size $n$ that is not
contained inside $t$ parallel hyperplanes (intuitively, $A$ `does not look lower-dimensional'),
then $|A+A| \geq (2^k - \varepsilon) n$. We wonder if the following might be true. Although this is a question
about $\mathbb{Z}$ rather than $\mathbb{Z}_p$, we feel that the methods in this paper are likely to be
relevant.
{\bf Question 3. }{\em
For a given dimension $k$, does there exist a constant $c$ such that the
following holds? For any $\varepsilon>0$ there exists $t$ such that if $A$ is a subset of ${\mathbb Z}^k$ of
size $n$ that is not contained in
$t$ parallel hyperplanes, then there exists a subset $A'$ of $A$ of size at most $c \sqrt{n}$ such
that $|A'+A'| \geq (2^k - \varepsilon) n$.
}
\end{document}
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\begin{document}
\newtheorem{definition}{Definition}[section]
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\newtheorem{remark}[definition]{Remark}
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\newtheorem{examples}[definition]{Examples}
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\newtheorem{definitions}[definition]{Definitions}
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{example}[definition]{Example}
\newtheorem{comments}[definition]{Some comments}
\newtheorem{corollary}[definition]{Corollary}
\def\Box{\Box}
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\newtheorem{defsobs}[definition]{Definitions and Observations}
\newenvironment{prf}[1]{ \trivlist
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#1.\hspace*{.3em}}]}{~\hspace{\fill}~$\Box$\endtrivlist}
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\title{Analytic $q$-difference equations}
\author{Marius van der Put \\
\footnotesize Department of Mathematics, University of Groningen,
P.O.Box 800,\\
\footnotesize 9700 AV Groningen,
The Netherlands, [email protected] }
\date{}
\maketitle
\noindent
\section*{Introduction}
A complex number $q$ with $0<|q|<1$ is fixed. By an analytic $q$-difference equation we mean
an equation which can be represented by a matrix equation $Y(z)=A(z)Y(qz)$ where $A(z)$
is an invertible $n\times n$-matrix with coefficients in the field $K=\mathbb{C}(\{z\})$ of the
convergent Laurent series and where $Y(z)$ is a vector of size $n$. The aim of this paper
is to give an overview of our present knowledge of these equations and their solutions.
Definitions and statements are presented in detail. Examples illustrate the main results. For proofs we refer to the cited literature.
For section 1 and part of the following sections, the reference is [P-S]. For the later sections the reference is [P-R]. The last section presents unpublished results. The theory of linear differential equations over
$K$ (see [P-S.2], especially Chapter 10) has many of the features presented in this survey.
The manuscript [R-S]
contains a concise overview of analytic $q$-difference equations and its main purpose is
to develop a theory of $q$-summation leading to, in our terminology, a description of a
universal difference Galois group. As the authors of [R-S] note, this is only partially achieved
and part of their work is still conjectural. However for a certain class of $q$-difference equations,
namely those having at most two slopes and such that the slopes are integral,
they have explicit results. An important part of the extensive literature on $q$-difference equations can be found in the papers cited here.
\section{Difference equations in general}
A difference field $F$ is a field provided with an automorphism $\phi$ of infinite order.
A scalar linear difference equation is an equation of the form
\[\phi ^n(y)+a_{n-1}\phi ^{n-1}(y)+\cdots +a_1\phi (y) +a_0y=0\,\]
with given $a_i\in F$ and, say, $a_0\neq 0$.
As in the case of linear differential equations, one can transform this equation into a matrix difference equation, i.e., an equation of the form $Y=A\phi (Y)$, where $A$ is a given invertible $n\times n$-matrix with coefficients in $F$ and $Y$ denotes a vector of length $n$. On the vector space $M=F^n$ one considers the operator
$\Phi : Y\mapsto A\phi (Y)$. The bijective map $\Phi :M\rightarrow M$ is additive and
$\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$ for $m\in M$ and $f\in F$. This leads to the following definition of a {\it difference module $M=(M,\Phi )$ over $F$}:\\
$M$ is a finite dimensional vector space over $F$ and $\Phi :M\rightarrow M$ is an additive, bijective map
satisfying $\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$ for $m\in M$ and $f\in F$. The equation, corresponding
to a difference module is $Y=\Phi (Y)$. In general this equation has few solutions in $M$ itself. Like ordinary
polynomial equations over fields, one has to construct extensions of $F$ in order to have sufficiently many
solutions.
We will now assume that $F$ has {\it characteristic zero} and that its field of constants
$C: =\{f\in F|\ \phi (f)=f\}$ is {\it algebraically closed}. A {\it Picard-Vessiot ring (or extension) $R$ for a difference module $M$}, say, represented by the matrix equation $Y=A\phi (Y)$ is defined by:\\
(i) $R$ is an $F$-algebra (commutative and with a $1$),\\
(ii) $R$ is provided with an automorphism $\phi$ extending $\phi$ on $F$,\\
(iii) $R$ has only trivial $\phi$-invariant ideals,\\
(iv) there exists an invertible matrix $U$ (called fundamental matrix) with coefficients in $R$ such that
$U=A\phi (U)$,\\
(v) $R$ is generated over $F$ by the coefficients of $U$ and $\frac{1}{\det U}$.\\
Property (iii) translates in terms of $M$ into $V:=\{a\in R\otimes _FM|\ \Phi (a)=a\}$ is a vector space over
$C$ and the natural map $R\otimes _CV\rightarrow R\otimes _FM$ is an isomorphisms.
One constructs $R$ as follows. Let $X$ denote a matrix $(X_{i,j})$ of indeterminates and put $D:=\det X$.
The $F$-algebra $R_0:=F[X,\frac{1}{D}]$ is provided with a $\phi$-action, extending the one on $F$, by
the formula $(\phi X_{i,j} )=A^{-1}(X_{i,j})$. Let $I\subset R_0$ denote an ideal maximal among the ideals
invariant under $\phi$. Then $R=R_0/I$ is a Picard-Vessiot ring for the given equation.
The basic results of difference Galois theory are:\\
(1) A Picard-Vessiot ring $R$ exists and is unique up to a, non unique, isomorphism. \\
(2) $R$ is reduced (i.e., has no nilpotent elements).\\
(3) The set constants of the ring of total fractions of $R$ is $C$.\\
(4) Let $G$ be the group of the $F$-linear automorphism of $R$, commuting with $\phi$.
The natural action of $G$ on $R\otimes _FM$ induces a faithful action of $G$ on $V$, the solution space.
The image of $G$ in ${\rm GL}(V)$ is a linear algebraic subgroup of the latter. This makes $G$ into a linear algebraic group over $C$.\\
(5) The action of $G$ on $Spec(R)$ makes the latter into an $G$-torsor over $F$. In other words, there exists
a finite extension $F^+\supset F$ and a $G$-equivariant isomorphism
$F^+\otimes _CC[G]\rightarrow F^+ \otimes _FR$, where $C[G]$ is the coordinate ring of $G$.\\
Most of the notions and `operations of linear algebra', such as morphisms, kernels, cokernels, direct sums
have an obvious equivalent for difference modules. Let $(M_1,\Phi _1),\ (M_2,\Phi _2)$ denote two difference modules. The tensor product of the two modules is defined as $M_1\otimes _FM_2$ with $\Phi$ given by
$\Phi (m_1\otimes m_2)=(\Phi _1m_1)\otimes (\Phi _2m_2)$. The internal hom of the two modules is
${\rm Hom}_F(M_1,M_2)$ with $\Phi$ defined by $(\Phi (L))(m_1)=\Phi _2^{-1}(L(\Phi _1m_1))$ for
$L\in {\rm Hom}_F(M_1,M_2)$ and $m_1\in M_1$.
This leads to another, more abstract but very useful, formulation of the above Picard-Vessiot theory, namely that of (neutral) Tannakian categories. The category of all difference modules over $F$ is a neutral Tannakian
category. We will return to this in section 6. For a specific difference field $F$ one can
say much more than the above formalism (analogous to the case of ordinary Galois theory for specific fields).
\section{First examples of $q$-difference equations}
This exposition is concerned with the difference field $K=\mathbb{C}(\{z\})$, i.e., the field of convergent Laurent
series over $\mathbb{C}$, provided with the automorphism $\phi$ given by $\phi (z)=qz$, where
$q$ is a fixed complex number satisfying $0<|q|<1$. In order to define $\phi$ on the algebraic closure
$K_\infty =\cup _{n\geq 1} K_n$, with $K_n:=\mathbb{C}(\{z^{1/n}\})$, of
$K$ we choose a $\tau \in \mathbb{C}$ with $\Im (\tau )>0$ such that $q=e^{2\pi i \tau}$. Define $q^\lambda$ for $\lambda \in \mathbb{Q}$ (or any $\lambda \in \mathbb{C}$) as $e^{2\pi i\lambda \tau}$. Then the action of $\phi$ on
$K_\infty$ is given by $\phi (z^\lambda)=q^\lambda z^\lambda $. The action of $\phi$ on
$\widehat{K}=\mathbb{C}((z))$, i.e., the field of the formal Laurent series, and on its algebraic closure $\widehat{K}_\infty =\cup _{n\geq 1}\widehat{K}_n$, with $\widehat{K}_n:=\mathbb{C}((z^{1/n}))$,
is defined in a similar way.
Some $q$-difference rings $F=(F,\phi )$ will be considered, namely $\mathbb{C}[z,z^{-1}]$ and
$O$, the ring of the holomorphic functions on $\mathbb{C}^*$. A difference module $M=(M,\Phi )$ over a $q$-difference ring $F$ will be a {\it free} $F$-module of finite rank provided with a bijective additive map
$\Phi :M\rightarrow M$ such that $\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$ for $f\in F,\ m\in M$.
\begin{examples} {\rm $\ $\\
(a) $Ke, \Phi (e)=ce,\ c\in \mathbb{ C}^*$. There exists a solution ($\neq 0$)
in $K$ $\Leftrightarrow \ c\in q^{\mathbb{Z}}$. Indeed, $\Phi (z^te)=cq^tz^te$. If $c\in q^{\mathbb{Z}}$, then
the Picard-Vessiot ring is $K$ itself and the difference Galois group is $\{1\}$.
There exists a solution ($\neq 0$) in $K_n$ $\Leftrightarrow$ $c\in q^{\frac{1}{n}\mathbb{Z}}$. If
$c\in q^{\frac{1}{n}\mathbb{Z}}$ with $n$ minimal, then $K_n$ is the Picard-Vessiot ring and the difference
Galois group is $\mu _n:=\{a\in \mathbb{C}|\ a^n=1\}$.
In the remaining case, the Picard-Vessiot ring is $K[X,X^{-1}]$ with the action of $\phi$ given by
$\phi (X)=c^{-1}X$. The difference Galois group is $\mathbb{G}_m:=\mathbb{C}^*$. It consists of the automorphisms $\sigma$ given by $\sigma (X)=aX$ (for any $a\in \mathbb{C}^*$). We will use the symbol
$e(c)$ for this $X$. It can be given an interpretation as multivalued function $z^b$ for a $b$ with $q^b=c^{-1}$.
\noindent (b) The difference module $U_n:=Ke_1+\cdots +Ke_n$ with $\Phi$ given by the matrix
\[\left[\begin{array}{cccc}1&1& &\\
&1&1 &\\
&&&1\\
&&& 1 \end{array}\right] \]
is called {\it unipotent of length $n$}. The Picard-Vessiot ring is $K[X]$, with $\phi (X )=X +1$.
The difference Galois group is $\mathbb{G}_a:=\mathbb{C}$. The elements $\sigma$ of this group have the form $\sigma (X )=X +a$ (any $a\in \mathbb{C}$). We will use the symbol $\ell$ for this $X$. It has an interpretation as multivalued function $\frac{\log z}{2\pi i \tau}$. For $n=2$ one easily verifies that the solution space has $\mathbb{C}$-basis $\{e_1-\ell e_2,\ e_2\}$.
$\Box$ }\end{examples}
A difference module $M$ over $K$ is called {\it regular singular} if $M$ has a basis $e_1,\dots ,e_m$ such that
$\mathbb{C}\{z\}e_1+\cdots +\mathbb{C}\{z\}e_m$ is invariant under $\Phi$ and $\Phi ^{-1}$.
\begin{theorem} The following are equivalent\\
{\rm (i)} $M$ regular singular.\\
{\rm (ii)} $M=K\otimes _{\mathbb{C}}W$, $\dim _{\mathbb{C}}W < \infty$ and
$\Phi (f\otimes w)=\phi (f)\otimes A(w)$ for some $A\in {\rm GL}(W)$. Moreover, there is a unique $A$ such that
every eigenvalue $c$ satisfies $|q|<|c|\leq 1$.\\
{\rm (iii)} $M$ is obtained by $\oplus ,\otimes $ from the examples in {\rm 2.1}.
\end{theorem}
The difference module $(Ke,\ \Phi e=(-z)e)$ is the basic example of an {\it irregular singular} difference module. Its Picard-Vessiot
ring is $K[X,X^{-1}]$ with $\phi (X)=(-z)^{-1}X$. The difference Galois group is $\mathbb{G}_m$. We will
use the symbol $e(-z)$ for this $X$. It has the interpretation $ \Theta (z):=\sum _{n\in \bf Z}q^{n(n-1)/2}(-z)^n$
because of the well known formula $(-z)\Theta (qz)=\Theta (z)$.
\section{Towards a classification of modules}
We present here the `classics' by G.D.~Birkhof, P.E.~Guenther, C.R.~Adams et al. and `modern' work
by J.-P.~Ramis, Ch.~Zhang, J.~Sauloy, A.~Duval, M.F.~Singer, M.~van der Put, M.~Reversat et al.,
concerning $q$-difference equations over $K$.
Let $K[\Phi ,\Phi ^{-1}]$ be the skew ring of difference operators. The elements of this ring are the finite
formal sums $\sum _{n\in \mathbb{Z}}a_n\Phi ^n$ and the multiplication is given by the rule $\Phi \cdot f
=\phi (f)\Phi$. This ring is (left and right) Euclidean. Any difference module $(M,\Phi _M)$ can be seen as
a left $K[\Phi ,\Phi ^{-1}]$-module where the action of $\Phi$ on $M$ is just $\Phi _M$. This left module is
{\it cyclic} (i.e., generated by one element) and therefore $M\cong K[\Phi ,\Phi ^{-1}]/K[\Phi ,\Phi ^{-1}]L$ for
some $L=\Phi ^m+a_{m-1}\Phi ^{m-1}+\cdots +a_1\Phi +a_0$ with $a_0\neq 0$.
The usual discrete valuation $v$ on $K$ is given by $v(0)=+\infty$ and for $a\in K^*$, $v(a)$ is the order of
$a$. Using the values $v(a_i)$ one defines the {\it Newton polygon} of $L$, as in the case of an ordinary
polynomial in $K[T]$. This Newton polygon depends on $M$ only. The slopes of the Newton polygon are in
$\mathbb{Q}$. A difference module $M$ is called {\it pure} if there is only one slope.
\begin{examples} {\rm
$M$ is regular singular if and only if $M$ is pure of slope 0.
\noindent
Further $(Ke,\ \Phi e=c(-z)^te)$ with $c\in \mathbb{C}^*,\ t\in \mathbb{Z}$ is pure of slope $t$. }
$\Box$\end{examples}
\begin{theorem}[Adams, Birkhoff, Guenther, Ramis, Sauloy] $\ $\\
$M$ has a unique tower of submodules
$0=M_0\subset M_1\subset \cdots \subset M_r=M$ such that every $M_i/M_{i-1}$
is pure of slope $\lambda _i$ and $\lambda _1<\cdots <\lambda _r$.
\end{theorem}
This filtration is called the {\it slope filtration} of $M$ and one defines the {\it graded module} $gr(M)$ of $M$
by $gr(M)=\oplus _i M_i/M_{i-1}$. In proving Theorem 3.2, one observes that the above operator $L$ has a unique factorization $L_1\cdot L_2\cdots L_r$ with each $L_i\in \widehat{K}[\Phi ,\Phi ^{-1}]$ monic and having only one slope $\lambda _i$. The main step is to show that these $L_i$ are actually convergent, i.e., belong
to $K[\Phi ,\Phi ^{-1}]$. This factorization of $L$ induces the slope filtration.
We note that for obtaining convergence, it is essential that $\lambda _1<\cdots <\lambda _r$. For any order
of the slopes $\lambda _i$ there is a similar factorization of $L$ in the ring $\widehat{K}[\Phi ,\Phi ^{-1}]$.
This has as consequence that $\widehat{K}\otimes _KM$ is in fact equal to the direct sum
$\oplus _i(\widehat{K}\otimes _K M_i/M_{i-1})=\widehat{K}\otimes _Kgr(M)$. In the next section we will study the moduli spaces that describe the modules $M$ with a fixed graded module $gr(M)$. A difference module
$M$ over $K$ is called {\it split} if it is isomorphic to its graded module $gr(M)$.
Here we continue with the classification of pure
modules over $K$. We note that by Theorem 3.2 any irreducible module over $K$ is pure.
\begin{definition} The module $E(cz^{t/n})$.\\{\rm
Given are the data $t/n,\ n\geq 1,\ (t,n)=1, \ c\in \mathbb{C}^*, |q|^{1/n}<|c|\leq 1$. They define
a difference module over $K_n$ of dimension 1, namely $(K_ne, \ \Phi e=cz^{t/n}e)$. This object,
considered as difference module over $K$, has dimension $n$ over $K$ and is called
$E(cz^{t/n})$.}
$\Box$\end{definition}
\begin{theorem} $E(cz^{t/n})$ is pure, irreducible and has slope $t/n$. Further,
$E(c_1z^{t/n})\cong E(c_2z^{t/n})$ if and only if $ c_1^n=c_2^n $. Moreover,
every irreducible $M$ over $K$ is isomorphic to some $E(cz^{t/n})$.
\end{theorem}
A difference module is called {\it indecomposable} if it is not the direct sum of two proper submodules.
We note that an indecomposable module over $K$ need not be pure.
\begin{theorem} The indecomposable pure modules over $K$ are
$E(cz^{t/n})\otimes U_m$ with $|q|^{1/n}<|c|\leq 1$. Further, the triple $(t/n, c^n,m)$ is unique.
\end{theorem}
\begin{definition} Global lattices. \\{\rm
A global lattice $\Lambda$ for a $q$-difference module $M$ over $K$ is a finitely generated
$\mathbb{C}[z,z^{-1}]$-submodule of $M$ (and hence free), invariant under $\Phi$ and $ \Phi ^{-1}$, such that
the natural map $K\otimes _{\mathbb{C}[z,z^{-1}]}\Lambda \rightarrow M$ is an
isomorphism. }
$\Box$\end{definition}
We will see later that {\it any} difference module $M$ has a unique global lattice. This means that the $q$-difference equation, defined locally at $z=0$, is equivalent to an equation on all of $\mathbb{P}^1$ with
at most poles at $z=0$ and $z=\infty$. From the explicit description of the indecomposable pure modules over $K$ it is not hard to deduce the following.
\begin{corollary} Every pure indecomposable difference module $M$ over $K$ has a unique global lattice.
This lattice, {\em denoted by} $M_{global}$, is a difference module over $\mathbb{C}[z,z^{-1}]$. The same holds for {\em split} difference modules over $K$.\\
Moreover, any morphism $f:M\rightarrow N$ between split difference modules over $K$ maps
$M_{global}$ into $N_{global}$. \end{corollary}
Before going on, we remark that the {\it classification of the difference modules over $\widehat{K}$} is remarkably simple. With the same methods used in the proof of Theorem 3.5, one shows that
every pure indecomposable difference module over $\widehat{K}$ has the form
$\widehat{K}\otimes _K(E(cz^{t/n})\otimes U_m)$ (again with unique $(t/n,c^n,m)$). Finally, as remarked before, any difference module over $\widehat{K}$ is a direct sum of pure modules over $\widehat{K}$.\\
It is well known that the elliptic curve $E_q:={\mathbb{C}}^*/q^{\mathbb{Z}}$, which we like to call {\it the Tate curve}, plays an important role for $q$-difference equations. With the help of 3.5, 3.6 and 3.7 one can deduce the following
rather striking result.
\begin{theorem} There is an additive, faithful functor $V$ from the category of the split difference modules over $K$ to the category of the vector bundles on $E_q$. It has the properties:\\
{\rm (i)} $V$ induces a bijection between the (isomorphy classes of) indecomposable modules over
$K$ and the (isomorphy classes of) indecomposable vector bundles on $E_q$.\\
{\rm (ii)} $V$ induces a bijection between (isomorphy classes of) objects. \\
{\rm (iii)} $V$ respects the constructions of linear algebra, i.e., tensor products, exterior powers etc.
\end{theorem}
\begin{proof} We will {\it sketch} a proof. \\
(1). We recall that $O$ denotes the algebra of the holomorphic functions on
${\mathbb C}^*$ and that a difference module $M$ over $O$ is a left module
over the ring $O[\Phi ,\Phi ^{-1}]$, free of some rank $m<\infty $ over
$O$. Further $pr:{\mathbb C}^*\rightarrow E_q:={\mathbb C}^*/q^{\mathbb Z}$
denotes the canonical map. One associates
to $M$ the vector bundle $v(M)$ of rank $m$ on $E_q$ given by
$v(M)(U)=\{f\in O(pr ^{-1}U)\otimes _OM|\ \Phi (f)=f\}$,
where, for any open $W\subset {\mathbb C}^*$, $O(W)$ is the algebra of the
holomorphic functions on $W$.
On the other hand, let a vector bundle $\mathcal M$ of rank
$m$ on $E_q$ be given. Then ${\mathcal N}:=pr^*{\mathcal M}$ is a vector
bundle on ${\mathbb C}^*$ provided with a natural isomorphism
$\sigma _q^*{\mathcal N}\rightarrow {\mathcal N}$, where $\sigma _q$ is the
map $\sigma _q(z)=qz$. One knows that ${\mathcal N}$ is in fact a free
vector bundle of rank $m$ on ${\mathbb C}^*$. Therefore, $M$, the collection
of the global sections of $\mathcal N$, is a free $O$-module
of rank $m$ provided with an invertible action $\Phi$ satisfying
$\Phi (fm)=\phi (f)\Phi (m)$ for $f\in O$ and $m\in M$.
It is easily verified that the above describes an equivalence $v$ of tensor categories.\\
\noindent (2). One associates to any split difference module $M$ over $K$, its global lattice $M_{global}$
and the $q$-difference module $O\otimes _{\mathbb{C}[z,z^{-1}]}M_{global}$ over $O$. The latter
induces by (1) a vector bundle on $E_q$, which we call $V(M)$. For a morphism $f:M\rightarrow N$ between split modules one has $f:M_{global}\rightarrow N_{global}$.Therefore $f$ induces a morphism $V(f):V(M)\rightarrow V(N)$. Thus we found the additive, faithful functor $V$. Clearly $V$ respects the constructions of linear algebra.\\
\noindent (3). As (ii) is an immediate consequence of (i), we are left with proving (i). The indecomposable module $M:=E(cz^{t/n})\otimes U_m$ is producing a vector bundle $V(M)$ of rank $nm$ and degree $tm$.
One can show that $V(M)$ is indecomposable and that $V(M)$ is irreducible if $m=1$. Further one can verify
that non isomorphic indecomposable $M, N$ produce non isomorphism vector bundles
$V(M), V(N)$. It is somewhat more complicated to show that every indecomposable vector bundle is isomorphic to $V(M)$ for a suitable indecomposable $M$. This last step can be avoided by an inspection of Atiyah's paper where the classification of the indecomposable vector bundles on $E_q$ is explicitly
given. \end{proof}
\begin{corollary} Let $B$ be a split module over $K$, then\\
{\rm (i)} ${\rm ker}(\Phi -1,O\otimes B_{global})\cong H^0(E_q,V(B))$.\\
{\rm (ii)} ${\rm coker}(\Phi -1,O\otimes B_{global})\cong H^1(E_q,V(B))$.\\
{\rm (iii)} The two canonical maps ${\rm coker}(\Phi -1,B_{global})
\rightarrow {\rm coker}(\Phi -1,B)$ and ${\rm coker}(\Phi -1,B_{global})
\rightarrow {\rm coker}(\Phi -1,O\otimes B_{global})$ are isomorphisms.
\end{corollary}
\begin{examples}{\rm Consider the difference module
$(B=Ke,\ \Phi e=(-z)^te)$ with $t\in \mathbb{Z}$. The line bundle
$V(B)$ is equal to $O_{E_q}(t\cdot [1])$, where $1$ denotes the neutral element of $E_q$.
For $t\geq 1$, ${\rm coker}(\Phi -1,O\otimes B_{global})=0$ and
${\rm ker}(\Phi -1,O\otimes B_{global})$ is the $t$-dimensional vector space with basis
$\{ \Theta (\zeta z)^te| \zeta ^t=1\}$.
For $t=0$, ${\rm ker}(\Phi -1,O)=\mathbb{C}1$ and
${\rm coker}(\Phi -1,O)$ has dimension 1. Explicitly, $O$ is the vector space of the everywhere
convergent Laurent series and has the formula
$(\Phi -1)(\sum _{n\in \mathbb{Z}}a_nz^n)
=\sum _{n\in \mathbb{Z}}(q^n-1)a_nz^n$. }
$\Box$ \end{examples}
\section{Moduli spaces for difference modules}
Fix a split difference module $S=P_1\oplus \cdots \oplus P_r$ over $K$, where $P_i$ is pure
with slope $\lambda _i$ and $\lambda _1<\cdots <\lambda _r$. The problem is to classify
the difference modules $M$ over $K$ such that $gr(M)$ is isomorphic to $S$. The collection
of the isomorphy classes is a rather ugly object due to the fact that $S$ has many automorphisms.
A better way to formulate the problem is to consider pairs $(M,f)$ consisting
of a difference module $M$ and an isomorphism $f:gr(M)\rightarrow S$. Two pairs
$(M_1,f_1), (M_2,f_2)$ are called {\it equivalent} if there exists an isomorphism
$g:M_1\rightarrow M_2$ such that the induced graded isomorphism $gr(g):gr(M_1)\rightarrow
gr(M_2)$ has the property $f_1=f_2\circ gr(g)$. Let $Equiv(S)$ denote the set of equivalence classes.
This formulation allows us to define a covariant functor $\mathcal F$ from the category of finitely generated $\mathbb{C}$-algebras $R$ (i.e., $R$ is commutative and has a $1$) to the category of sets. For a finitely generated $\mathbb{C}$-algebra $R$ one considers
$K_R:=R\otimes _{\mathbb{C}}K$ and one can define the notion of difference module over
$K_R$. The set ${\mathcal F}(R)$ is the set of equivalence classes of pairs
$(M,f)$ with $M$ a difference module over $K_R$ and $f$ is a $K_R$-isomorphism $f:gr(M)\rightarrow K_R\otimes S$. Equivalence of two pairs is defined as above.
We note that ${\mathcal F}(\mathbb{C})$ is precisely $Equiv(S)$.
One can see the functor $\mathcal F$ as a contravariant functor on the category of affine
$\mathbb{C}$-schemes (of finite type).
\begin{theorem} The contravariant functor $\mathcal F$ on affine $\mathbb{C}$-schemes is representable. In fact, the affine space $\mathbb{A}_{\mathbb{C}}^N$ with
$N=\sum _{i<j}(\lambda _j-\lambda _i)\dim P_i\cdot \dim P_j$, represents $\mathcal F$.
In other words, the covariant
functor $\mathcal F$ is represented by a certain universal $q$-difference module $M$ over
$\mathbb{C}[X_1,\dots ,X_N]$. Further $Equiv(S)$ identifies with $\mathbb{C}^N$.
\end{theorem}
For the case of integer slopes $\lambda _i$, the above result is announced by
J.-P.~ Ramis and J.~Sauloy. The general case is treated in [P-R].
One can normalize the representing space $\mathbb{A}^N_{\mathbb{C}}$ by letting
$0$ correspond to the class of the pair $(S,id_S)$. The vector space structure of
$\mathbb{A}^N_{\mathbb{C}}$ has an interpretation for $s=2$, namely as the vector space
${\rm coker} (\Phi -1,{\rm Hom}(P_2,P_1))$. For $s>2$, the functor $\mathcal F$ and its representing space
still have a weaker structure, namely that of an iterated torsor.
We illustrate Theorem 4.1 by the following {\it basic example}:\\
$S=P_1\oplus P_2$, where $P_1=(Ke_1,\ \Phi e_1=e_1)$ and $P_2=(Ke_2,\ \Phi e_2=(-z)^te_2)$ with $t>0$. The moduli space is $\mathbb{A}^t_{\mathbb{C}}$ and the universal
family above this moduli space is
\[K[x_0,\dots ,x_{t-1}]e_1+K[x_0,\dots ,x_{t-1}]e_2,\ \Phi e_1=e_1,\]
\[ \Phi e_2=(-z)^te_2+(x_0+x_1z+\cdots +x_{t-1}z^{t-1})e_1\ . \]
Surprisingly enough, Theorem 4.1 (for the case of integer slopes) and this example are already present in the work of Birkhoff of Guenther.
\begin{corollary} Every difference module $M$ over $K$ has a unique global lattice.
This lattice will be called, as before, $M_{global}$.
Moreover, every morphism $f:M\rightarrow N$ satisfies $f(M_{global})\subset
N_{global}$. In particular, one can extend the functor $V$ of Theorem {\rm 3.8} to the category
of all $q$-difference modules over $K$.
\end{corollary}
This corollary can be deduced from the Theorem 4.1 and Corollary 3.9.
\section{Difference Galois groups}
In the last section a complete, however complicated, classification of the difference modules over $K$ is given. Using this classification we will be able to give a complete description of the
difference Galois groups. The difference Galois group of a module $M$ will be denoted by
$Gal(M)$.We start with the easiest case and build up to the general case.\\
\noindent (1) {\it Regular singular modules}. \\
We recall that a regular singular module has the form $M=K\otimes _{\mathbb{C}}W$ and
$\Phi (f\otimes w)=\phi (f)\otimes A(w)$ with $A\in {\rm GL}(W)$. We normalize $A$ such that
the eigenvalues of $A$ have absolute values in $(|q|,1]\subset \mathbb{R}$.
Let $L\subset \mathbb{C}^*/q^{\mathbb{Z}}=E_q$ be the group generated by the images of the
eigenvalues of $A$. Then:
\[Gal(M)={\rm Hom}(L,\mathbb {C}^*)\;(\times {\mathbb{C}}). \]
Write $L=L_{free}\oplus L_{torsion}$ with the first summand a free $\mathbb{Z}$-module of
rank $g\geq 0$ and where the second term is a finite commutative group. Then
${\rm Hom}(L,\mathbb{C}^*)$ is a product of $\mathbb{G}_m^g$ with a finite commutative group generated by at most two elements. If $A$ is semi-simple, then this is $Gal(M)$.
If $A$ is not semi-simple, then the term $\mathbb{G}_a=\mathbb{C}$ is also present. \\
\noindent (2) {\it Irreducible modules}.\\
We recall that $M=E(cz^{t/n})$. For $n=1$ and $t\neq 0$, the group $Gal(M)$ is
$\mathbb{G}_m=\mathbb{C}^*$. For $n>1$ one can describe $Gal(M)$ be an exact sequence
$1\rightarrow \mathbb{G}_m\rightarrow Gal(M)\rightarrow (\mathbb{Z}/n\mathbb{Z})^2\rightarrow 0$. The group $Gal(M)$ is not commutative and is not a semi-direct product of
$\mathbb{G}_m$ and $(\mathbb{Z}/n\mathbb{Z})^2$.\\
\noindent (3) {\it Indecomposable modules}.\\
We may suppose $M=E(cz^{t/n})\otimes U_m$ with $t/n\neq 0$ and $m>1$. Then
$Gal(M)=Gal(E(cz^{t/n})\times \mathbb{G}_a$.\\
\noindent (4) {\it Split modules}.\\
A split module $M$ is a direct sum of pure modules $M_i$. An explicit combination of the difference Galois groups of the $M_i$ (described above) yields $Gal(M)$.\\
\noindent (5) {\it The general case}.\\
We recall that $M$ has a slope filtration
$0=M_0\subset M_1\subset \cdots \subset M_r=M$ with $P_i:=M_i/M_{i-1}$ pure
and $gr(M)=S:=P_1\oplus \cdots \oplus P_r$. Let $\xi$ in the moduli space, introduced in section 4, represent $M$. Then there exists an exact sequence
\[1\rightarrow U_\xi \rightarrow Gal(M)\rightarrow Gal(S)\rightarrow 1\ ,\]
with $U_\xi$ a unipotent group, explicitly determined by $\xi$. This sequence is in fact a semi-direct product and the action, by conjugation, of $Gal(S)$ on $U$ is again explicit.\\
\begin{remarks} $\ $\\ {\rm
(1) The above description of $Gal(M)$ implies that $Gal(M)^o$ is a solvable group. This is in contrast with the differential Galois groups that occur for differential equations over $K$.\\
\noindent (2) {\it Difference modules over $\widehat{K}$}.
For a difference module $N$ over $\widehat{K}$, there exists a unique split difference
module $M$ over $K$ such that $N\cong \widehat{K}\otimes _KM$. Further, $N$ and $M$
have the same difference Galois group. \\
\noindent (3)
Let $M$ be a difference module over $K$. The step from $M$ to its global lattice $M_{global}$
is probably not algorithmic since it involves a computation with arbitrary complex numbers.
However, the classification in sections 3 and 4, assuming the knowledge of $M_{global}$,
is algebraic and can be shown to be algorithmic. The computation of $Gal(M)$
(and of the Picard-Vessiot ring for $M$), on the basis of the classification, is algorithmic as well.
We note that for {\it linear differential equations over $K$}, the existence of a theoretical algorithm is proven by E. Hrushovski (2001).
In that case no explicit algorithm is known. }
$\Box$ \end{remarks}
\section{Universal Picard-Vessiot rings and \\
universal difference Galois groups}
We start by explaining some notions and constructions (see also [P-S2]).
An {\it affine group scheme $G$ over $\mathbb{C}$} is given by a
$\mathbb{C}$-algebra $A$ provided with the structure of a Hopf-algebra. The latter is defined by a triple $(m,e,i)$ of
$\mathbb{C}$-algebra morphisms:\\
(a) $m:A\rightarrow A\otimes _\mathbb{C}A$ (the co-multiplication)\\
(b) $e:A\rightarrow \mathbb{C}$ with $e(1)=1$ (the co-unit element),\\
(c) $i:A\rightarrow A$ is an isomorphism (the co-inverse).\\
Put $G:=Spec(A)$. The induced map $m^*:G\times G\rightarrow G$
is the multiplication, further $e^*:Spec(\mathbb{C})\rightarrow G$ is the unit element of $G$ and the induced map $i^*:G\rightarrow G$
is the map $g\mapsto g^{-1}$. The usual rules for a group, expressed
in $m^*,e^*,i^*$, are translated into rules for $m,e,i$. These rules
define a Hopf-algebra.
If $A$ is finitely generated over $\mathbb{C}$, then $G$ is an ordinary linear algebraic group over $\mathbb{C}$. In general, $A$ is the direct limit (in fact
a filtered union) of finitely generated sub-Hopf-algebras. This means that $G$ is the projective limit of linear algebraic groups.
A {\it representation of an affine group scheme $G$ over $\mathbb{C}$} is a morphism of affine group schemes $G\rightarrow {\rm GL}(W)$,
where $W$ is a finite dimensional vector space over $\mathbb{C}$.
Morphisms between representations are defined in the obvious way
and thus we can talk about the category $Repr_G$ of all representations of $G$. In this category one can perform all `constructions of linear algebra', e.g., kernels, co-kernels, direct sums,
tensor products, duals, and the rules that one knows from linear algebra are valid. We note that an equivalence between
$Repr_{G_1}$ and $Repr_{G_2}$, preserving the constructions of
linear algebra, comes from an isomorphism $G_1\rightarrow G_2$
of affine group schemes (this is Tannaka's theorem).
We adopt here the following rather trivial definition of {\it neutral Tannakian category}, namely it is a category $T$ having all
constructions and rules of linear algebra and is, for these structures,
equivalent to $Repr_G$ for a suitable affine group scheme $G$.
Of course there is an {\it intrinsic} definition of neutral Tannakian category. Using that, one can show that $\Delta _K$, {\it the category
of all differential modules over $K$}, is a Tannakian category.
The affine group scheme $G$ such that $\Delta _K$ is isomorphic to
$Repr_G$, is called the {\it universal difference Galois group} for
the category $\Delta_K$. It is this group that we want to describe.
A full subcategory $\Delta$ of $\Delta _K$ (i.e., for any objects $A,B$ of $\Delta$, one has ${\rm Hom}_{\Delta}(A,B)=
{\rm Hom}_{\Delta _K}(A,B)$ ), closed under the operations of linear algebra, is again a neutral Tannakian category.
Consider an object $M$ of $\Delta _K$. Write $\{\{M\}\}$ for the
full subcategory of $\Delta _K$ generated by $M$ and all constructions of linear algebra applied to $M$. Then $\{\{M\}\}$
is isomorphic to some $Repr_H$. Moreover, there is a Picard-Vessiot
ring $PVR(M)$ attached to $M$. It is a {\it general result} that $H$ can be identified with the linear algebraic group consisting of the
$K$-automorphisms of $PVR(M)$ that commute with $\phi$. In other
words, $H$ can be identified with the difference Galois group of
$M$.
The above holds for any full subcategory $\Delta$ of $\Delta _K$,
closed under the operations of linear algebra. There is a Picard-Vessiot ring $PVR(\Delta )$ for $\Delta$, namely the direct limit of the $PVR(M)$ for all objects of $\Delta$. Moreover the affine group scheme $H$, such that $Repr_H$ is equivalent to $\Delta$, identifies with the group of the $K$-linear automorphism of $PVR(\Delta )$,
commuting with $\phi$. An explicit description of the universal
Picard-Vessiot ring for $\Delta _K$ is what we are aiming to produce.
We build up a description of the universal Picard-Vessiot ring and the universal Galois group of $\Delta _K$, by considering suitable
subcategories of $\Delta _K$.\\
\noindent (1) {\it $\Delta _{rs}$, the category of the regular singular
difference modules over $K$}.\\
$PVR(\Delta _{rs})=K[\{e(c)\}_{c\in {\mathbb{C}}^*},\ell ]$ with rules:
\[e(c_1c_2)=e(c_1)\cdot e(c_2),\ e(q)=z^{-1} \mbox{ and }
\phi (e(c))=c^{-1}e(c),\ \phi (\ell )=1+\ell \]
One identifies $z^\lambda =e(q^{-\lambda} )$ for $\lambda \in {\bf Q}$ and thus $PVR(\Delta _{rs})$ contains the {\it algebraic closure} $K_\infty$ of $K$. We can therefore rewrite $PVR(\Delta _{rs})$
as $K_\infty [\{e(c)\},\ell ]$ with the additional relations
$z^\lambda =e(q^{-\lambda })$ for all $\lambda \in \mathbb{Q}$. Further,
the difference Galois group $G_{rs}$ is ${\rm Hom}({\mathbb{C}}^*/q^{\mathbb{Z}},{\mathbb{C}}^*)\times {\mathbb{C}}$.\\
A similar description holds for regular difference modules over
$\widehat{K}$. Let $\widehat{K}_\infty$ denote the {\it algebraic closure} of
$\widehat{K}$. Then the universal Picard-Vessiot ring is
$\widehat{K}[\{e(c)\},\ell ]$ and the universal difference Galois group
coincides with the above group $G_{rs}$.\\
{\it Comments}. This description follows from the observation that
$PVR(\Delta _{rs})$ is generated by the solutions for the modules in Examples 2.1. The given expression for $G_{rs}$ has to be interpreted
as an affine group scheme. The description follows from section 5,
part (1).\\
\noindent (2) {\it $\Delta _{split}$, the category of the split difference
modules over $K$}.
\[PVR(\Delta _{split})=K_\infty [\{e(c)\}_{c\in {\mathbb{C}}^*},\ell ,\{e(z^\lambda )\}_{\lambda \in {\mathbb{Q}}}]\]
with additional rules
$ e(z^{\lambda +\mu})=e(z^\lambda )\cdot e(z^\mu),\
\phi (e(z^\lambda ))=z^{-\lambda} \cdot e(z^\lambda ) $.\\
Let the corresponding universal difference Galois group be denoted by $G_{split}$. From the inclusion $\Delta _{rs}\subset \Delta _{split}$ one obtains an exact sequence of affine group schemes
\[1\rightarrow {\rm Hom}({\mathbb{Q}},{\mathbb{C}}^*)\rightarrow
G_{split}\rightarrow G_{rs}\rightarrow 1\ .\]
The group scheme $G_{split}$ is {\it not} a semi-direct product and
${\rm Hom}({\mathbb{Q}},{\mathbb{C}}^*)$ lies in the center of
$G_{split}$.\\
{\it Comments}. The new universal Picard-Vessiot ring is generated over the one of (1) by solutions for the modules $E(cz^{t/n})$. This explains the terms $e(z^\lambda )$. The exact sequence and its features follow from an explicit calculation of the automorphism
of $PVR(\Delta _{split})$. We note the contrast with the differential case! Finally, the descriptions for the universal Picard-Vessiot ring
and the universal difference Galois group for $\Delta _{\widehat{K}}$,
the category of all difference modules over $\widehat{K}$, is rather
similar.\\
\noindent (3) {\it $\Delta _K$, this is the most interesting and the most complicated case}.\\
What can be proved at present is the following:\\
(a) $PVR(\Delta _K)=\mathcal{D}[\{e(c)\},\ell ,\{e(z^\lambda )\}]$ and the latter is a subalgebra of the explicit universal Picard-Vessiot ring
$PVR(\Delta _{\widehat{K}})=
\widehat{K}_\infty [\{e(c)\},\ell, \{e(z^\lambda )\}]$.
Further $\mathcal{D}$ is the $K_\infty $-subalgebra of
$\widehat{K}_\infty $ consisting of the elements $f\in \widehat{K}_\infty$ satisfying a scalar $q$-differential equation over $K_\infty$.\\
(b) The $K$-algebra $\mathcal D$ is generated over $K_\infty$ by the solutions in $\widehat{K}_\infty $ of all equations of the form
\[ (c_1z^{-\lambda _1}\phi -1)^{m_1}\cdots (c_rz^{-\lambda _r}\phi -1)^{m_r}f=z^\mu \ ,\]
where $ 0<\lambda _1<\cdots <\lambda _r,\ r\geq 1,\ m_1,\dots ,m_r\geq 1,\ \mu \in \mathbb{Q}$.\\
(c) The universal difference group $G$ admits an exact sequence (in
fact is a canonical semi-direct product)
$1\rightarrow N\rightarrow G\rightarrow G_{split}\rightarrow 1$ ,
where $N$ is a (connected) unipotent group scheme. \\
(d) The (pro)-Lie algebra $Lie(N)$ of $N$ consists of the
$K_\infty$-linear
derivations $D$ of $PVR(\Delta _K)$ commuting with $\phi$ and
zero on the elements $e(c), \ell , e(z^\lambda )$. We note that any
such $D$ is determined by its restriction to $\mathcal D$.
\begin{remarks} $\ $\\ {\rm
(1) A standard example for (b) is $f=\sum _{n\geq 1}q^{-n(n+1)/2}z^n$, the only solution of $(z^{-1}\phi -1)f=1$ in $\widehat{K}_\infty $. \\
(2) In [R-S] it is suggested, in analogy with the differential case, that
$Lie(N)$ is a nilpotent completion of a free Lie algebra with a set of free generators derived from the analytic tool of $q$-summation.
}
$\Box$ \end{remarks}
The main obstruction for the determination of $Lie(N)$ is the absence of an explicit description of the algebra $\mathcal D$. Now we present
an {\it intermediate Tannakian category} $\Delta _{2,K}$. It is the Tannakian subcategory of $\Delta _K$ generated by the difference modules having at most two slopes. For this category one has
$PVR(\Delta _{2,K})=\mathcal{D}_2[\{e(c)\},\ell ,\{e(z^\lambda )\}]$
for a certain $K_\infty$-subalgebra
$\mathcal{D}_2 \subset \mathcal{D} \subset \widehat{K}_\infty$ and
a universal difference Galois group $G_2$ which is the semi-direct
product of $G_{split}$ and a (connected) unipotent group scheme
$N_2$. The latter is a quotient of $N$ and the pro-Lie algebra
$Lie(N_2)$ is a quotient of $Lie (N)$.
Any $D\in Lie(N_2)$ is a $K_\infty$-linear derivation
$D:\mathcal{D}_2\rightarrow PVR(\Delta _{2,K})$, commuting
with $\phi$.
\begin{definition} The elements $f_{m,c,\mu}$.
\\ {\rm For $\mu \in \mathbb{Q}$ with $\mu >0$, $c\in \mathbb{C}^*$
with $|q^\mu |<|c|\leq 1$ and $m\geq 1$, the unique solution in
$\widehat{K}_\infty$ of the equation $(c^{-1}z^{-\mu }\phi -1)^my=1$
is called $f_{m,c,\mu}$. }
$\Box$ \end{definition}
\begin{theorem} ${\mathcal D}_2$ is generated over $K_\infty$ by the
elements $f_{m,c,\mu}$. These elements are algebraically independent over $K_\infty$. \end{theorem}
We note that $\phi (f_{m,c,\mu})=cz^\mu (f_{m,c,\mu}+f_{m-1,c,\mu})$,
where we use the notation $f_{0,c,\mu}=1$ for all $c,\mu$. Thus the action of $\phi$ on $\mathcal{D}_2$ is explicit. An element
$D\in Lie(N_2)$ is a $K_\infty$-derivation $\mathcal{D}_2\rightarrow
PVR(\Delta _{2,K})$, commuting with $\phi$. Since the $f_{m,c,\mu}$
are free generators of $\mathcal{D}_2$, the values $D(f_{m,c,\mu})$
have the only restriction that $\phi (D(f_{m,c,\mu}))=
cz^\mu (D(f_{m,c,\mu})+D(f_{m-1,c,\mu}))$.
Choose for every $\mu ,c$ as above, a sequence of complex
numbers\\
$a_0(\mu ,c), a_1(\mu ,c), a_2(\mu ,c),\dots $ . Define $D$ by the formula $D(f_{m,c,\mu}):=$
\[(a_0(\mu ,c){\ell \choose m-1}+a_1(\mu ,c) {\ell \choose m-2}+\cdots + a_{m-1}(\mu ,c){\ell \choose 0})\cdot e(c^{-1})e(z^{-\mu} )\ .\]
One can verify that $D$ commutes with the action of $\phi$ and thus $D$ defines an element of $Lie(N_2)$. Moreover, every element of $Lie(N_2)$ has this form.
One observes that $Lie(N_2)$ is commutative. Now we propose {\it topological generators} for the pro-Lie algebra $Lie(N_2)$ by considering the elements $D_{\mu ,c,n}$ with
$\mu >0,\ |q^\mu |<|c|\leq 1,\ n\geq 0$ defined by the sequences
$\{ a_k(\mu ',c') \}$ with $a_k(\mu ',c')=
\delta _{\mu ,\mu '}\delta _{c,c'}\delta _{k,n}$.
It is not difficult to verify that any element $\xi \in PVR(\Delta _{2,K})$, invariant under $G_{split}$ and satisfying $D _{\mu ,c,n}\xi =0$ for all
$\mu ,c,n$ , lies in $K$. This implies that $N_2$ is connected and that its pro-Lie-algebra is actually topologically generated by
$\{D_{\mu ,c,n}\}$. We {\it conjecture} that $Lie(N_2)$ is actually
$Lie(N)_{ab}:=Lie(N)/[Lie(N),Lie(N)]$.
In [R-S] a set of free topological generators for the pro-Lie-algebra
$Lie(N)$ is proposed. It seems that the restriction of this set
to the quotient $Lie(N_2)$ has a translation into our set $\{D_{\mu ,c, n}\}$. \\
{\bf References}\\
\noindent [P-R] M. van der Put and M. Reversat - {\it Galois theory of $q$-difference equations} -
Ann. Fac. Sci. de Toulouse, vol XVI, no 2, p. 1-54, 2007\\
\noindent [P-S] M. van der Put and M.F. Singer - {\it Galois theory of difference equations} -
Lecture Notes in Mathematics, 1666, Springer Verlag, 1997\\
\noindent [P-S.2] M. van der Put and M.F. Singer - {\it Galois theory of linear differential equations} -
Grundlehren der mathematische Wissenschaften, 328, Springer Verlag, 2003\\
\noindent [R-S] J.-P. Ramis and J. Sauloy -{\it The $q$-analogue of the wild fundamental group} (I) -
arXiv:math.QA/0611521 v1 17 Nov 2006\\
\end{document}
|
\begin{document}
\title{Product-free sets in the free semigroup}
\author{Imre Leader}
\address{Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3\thinspace0WB, UK}
\email{[email protected]}
\author{Shoham Letzter}
\address{ETH Institute for Theoretical Studies, 8092 Zurich, Switzerland}
\email{[email protected]}
\author{Bhargav Narayanan}
\address{Department of Mathematics, Rutgers University, Piscataway NJ 08854, USA}
\email{[email protected]}
\author{Mark Walters}
\address{School of Mathematical Sciences, Queen Mary, University of London, London E1\thinspace4NS, UK}
\email{[email protected]}
\date{6 December 2018}
\subjclass[2010]{Primary 20M05; Secondary 05D05}
\begin{abstract}
In this paper, we study product-free subsets of the free semigroup over a finite alphabet $\mathscr{A}$. We prove that the maximum density of a product-free subset of the free semigroup over $\mathscr{A}$, with respect to the natural measure that assigns a weight of $|\mathscr{A}|^{-n}$ to each word of length $n$, is precisely $1/2$.
\end{abstract}
\maketitle
\section {Introduction}
A subset $S$ of a semigroup is said to be \emph{product-free} if there do not exist $x,y,z \in S$ (not necessarily distinct) such that $x\bigcdot y = z$; it is customary to call $S$ \emph{sum-free} when the underlying semigroup is abelian.
It is a well known fact (and an easy exercise) that any sum-free subset of the integers has upper density at most $1/2$. Sum-free subsets of the integers, and of abelian groups in general, have been studied by very many researchers over the last fifty years. For example, from the work of Green and Ruzsa~\citep{green}, there is now a complete picture of how large a sum-free set we can find in any finite abelian group. We refer the reader to the surveys of Tao and Vu~\citep{tao-s} and Kedlaya~\citep{ked-s} for more information on these questions.
Product-free subsets of finite non-abelian groups were first investigated by Babai and S\'os~\citep{babai}. Following foundational work by Gowers~\citep{gowers} demonstrating so-called `product-mixing' phenomena in groups with no low-dimensional representations, there has been a great deal of recent work in the non-abelian setting; for instance, in a recent breakthrough, Eberhard~\citep{eber} determined how large a product-free subset of the alternating group can be.
In light of these developments, it is natural to ask what one can say about product-free sets in infinite non-abelian structures, a setting in which our knowledge is a bit more limited. Perhaps the first natural place to look among infinite non-abelian structures is among those that are free, so here, we shall investigate how large product-free subsets of the free semigroup can be.
\section{Our results}
Let $\mathscr{A}$ be a finite set. We write $\mathcal{F} = \mathcal{F}_\mathscr{A}$ for the free semigroup over $\mathscr{A}$; in other words, $\mathcal{F}$ is the set of all finite words over the alphabet $\mathscr{A}$ equipped with the associative operation of concatenation. While we state and prove our results for finite alphabets of all possible sizes for the sake of completeness, the reader will lose nothing by supposing that $\mathscr{A}$ is a two-element set in what follows; indeed, this case captures all the difficulties inherent in the questions we study.
Recall that a set $S \subset \mathcal{F}$ is \emph{product-free} if, writing $\bigcdot$ for the operation of concatenation, there do not exist words $x,y,z \in S$ (not necessarily distinct) such that $x\bigcdot y = z$. There is an obvious example of a `large' subset of $\mathcal{F}$ that is product-free: when $\mathscr{A} = \{ a, b\}$ for instance, the set of words which contain an odd number of occurrences of the symbol $a$ (or $b$, for that matter) is easily seen to be a product-free set that contains, roughly, half the words from $\mathcal{F}$.
Our aim in this paper is to prove that these sets are, in a precise sense, the largest product-free subsets of $\mathcal{F}$. We remark in passing that there are several other product-free sets that are `equally large': for any nonempty subset $\Gamma \subset \mathscr{A}$, the \emph{odd-occurrence set} $\mathcal{O}_\Gamma \subset \mathcal{F}$ generated by $\Gamma$, namely the set of words in which the total number of occurrences of symbols from $\Gamma$ is odd, is easily seen to be a product-free set; in the case where $\mathscr{A} = \{ a, b\}$, our earlier example corresponds to taking $\Gamma = \{a\}$, and taking $\Gamma = \{a, b\}$ gives us the set of all words of odd length, for example.
To formally state our results, we need a way to measure the size of a set $S\subset \mathcal{F}$. For an integer $n\in \mathbb{N}$, the \emph{layer} $\mathcal{F}(n) \subset \mathcal{F}$ is the set of words of length $n$, and the \emph{ball} $\mathcal{F}_{\le}(n) \subset \mathcal{F}$ is the set of words of length at most $n$. As a first attempt, one might define the density of a set $S \subset \mathcal{F}$ via its densities in balls, namely as the quantity
\[\limsup_{n \to \infty}\frac{|S \cap \mathcal{F}_{\le}(n)|}{|\mathcal{F}_{\le}(n)|}.\]
However, a little thought should convince the reader that the counting measure is somewhat ill-suited for our purposes. Indeed, when $|\mathscr{A}| > 1$, almost all the words in $\mathcal{F}_{\le}(n)$ are long since $|\mathcal{F}(n)| \ge |\mathcal{F}_{\le}(n)|/2$. Consequently, we may find product-free sets that are intuitively small, and yet have density arbitrarily close to $1$ in the above sense; for example, for any sufficiently large $c \in \mathbb{N}$, the set
\[\bigcup_{n\ge c}(\mathcal{F}_{\le 2^n + c} \setminus \mathcal{F}_{\le 2^n})\]
is product-free and has density at least $1-1/c$ in the above sense, provided $|\mathscr{A}| > 1$.
A more natural approach is to assign a weight of $|\mathscr{A}|^{-n}$ to each word of $\mathcal{F}(n)$, thereby ensuring that the layers $\mathcal{F}(n)$ have the same total weight for all $n\in\mathbb{N}$. To this end, for a subset $S \subset \mathcal{F}$ and an integer $n \in \mathbb{N}$, we define \emph{the density of $S$ in the layer $\mathcal{F}(n)$} by $d_S(n) = |S \cap \mathcal{F}(n)|/|\mathcal{F}(n)|$. With this definition in place, most standard notions of density may now be carried over: we define the \emph{upper asymptotic density} of $S$ by
\[ \bar d(S) = \limsup_{n \to \infty} \frac{ \sum_{i=1}^{n}d_S(i)}{n},\]
and the \emph{upper Banach density} of $S$ by
\[ d^*(S) = \limsup_{n-m \to \infty} \frac{ \sum_{i=m}^{n}d_S(i)}{n-m+1}.\]
Of course, the latter is a weaker notion of density than the former; indeed, it is clear that $\bar d(S) \le d^*(S)$ for any $S \subset \mathcal{F}$.
It is easy to see that any odd-occurrence set has both an upper asymptotic density and an upper Banach density of $1/2$. Our aim in this note is to show that product-free sets cannot be any larger; our main result is as follows.
\begin{theorem}\label{main-res}
Let $\mathscr{A}$ be a finite set. If $S \subset \mathcal{F}_\mathscr{A}$ is product-free, then $d^*(S) \le 1/2$.
\end{theorem}
\begin{comment}
We shall also show the only maximally dense product-free sets are subsets of odd-occurrence sets.
\begin{theorem}\label{unique}
Let $\mathscr{A}$ be a finite set. If $S \subset \mathcal{F}_\mathscr{A}$ is product-free and $d^*(S) = 1/2$, then $S \subset \mathcal{O}_\Gamma$ for some nonempty subset $\Gamma \subset \mathscr{A}$.
\end{theorem}
\end{comment}
Let us mention that product-free sets in cancellative semigroups have been studied by {\L}uczak and Schoen~\citep{semigr}; while their results are sharp for such semigroups in general, these results do not give us any effective bounds on the size of a product-free subset of $\mathcal{F}$.
Before we turn to the proof of Theorem~\ref{main-res}, it is worth pointing out that there is a simple argument that allows us to bound the upper asymptotic density of a product-free subset of $\mathcal{F}$ away from $1$. Indeed, suppose that $S\subset\mathcal{F}$ is product-free. We then have
\[d_S(m)d_S(n) + d_S(m+n) \le 1\]
for any $m,n\in\mathbb{N}$ since the sets $S \cap \mathcal{F}(m+n)$ and $(S \cap \mathcal{F}(m))\bigcdot (S \cap \mathcal{F}(n))$ must be disjoint. Now, consider the set of integers $n \in \mathbb{N}$ for which $d_S(n) > \phi$, where $\phi = (\sqrt 5 - 1)/2 \approx 0.618$ is the unique positive solution to the equation $x^2 + x = 1$. It follows from the inequality above that this set of integers must be sum-free. It is now easy to see that $\bar d(S) \le (1 + \phi)/2 \approx 0.809$.
We shall have to work somewhat harder to prove Theorem~\ref{main-res}, which improves this bound of $(1 + \phi)/2$ for the upper asymptotic density to the optimal bound of $1/2$ for the upper Banach density. The proof of Theorem~\ref{main-res} is given in Section~\ref{sec-proof}. We conclude this note with a discussion of some open problems in Section~\ref{sec-conc}.
\section{Proof of the main result}\label{sec-proof}
We begin by fixing our finite alphabet $\mathscr{A}$. In the sequel, $\mathcal{F}$ will always mean $\mathcal{F}_\mathscr{A}$, the free semigroup over this fixed alphabet $\mathscr{A}$.
It will be helpful to establish some notation. For a pair of words $x, w \in \mathcal{F}$, we say that $x$ is a \emph{prefix} of $w$ if $w = x\bigcdot y$ for some $y \in \mathcal{F}$, and that $x$ is a \emph{suffix} of $w$ if $w = y\bigcdot x$ for some $y \in \mathcal{F}$. For a pair of sets $S_1, S_2 \subset \mathcal{F}$, we write $S_1 \bigcdot S_2$ for their (Minkowski) product; in other words,
\[S_1 \bigcdot S_2 = \{ w_1 \bigcdot w_2 : w_1 \in S_1, w_2 \in S_2\}.\]
For a set $S\subset \mathcal{F}$ and an integer $n \in \mathbb{N}$, we set $S(n) = S \cap \mathcal{F}(n)$. One of the key ideas in the proof of Theorem~\ref{main-res} is the following definition. For any sequence of positive integers $ \ell_1 < \ell_2 <\dots <\ell_k <n$, we define
\[ S(n; \ell_1, \ell_2,\dots,\ell_k) = \mathopen{}\mathclose\bgroup\originalleft\{ w \in S(n): w \text{ has no prefix in } S(\ell_1) \cup S(\ell_2) \cup \dots \cup S(\ell_k) \aftergroup\egroup\originalright \}; \]
in other words,
\[ S(n; \ell_1, \ell_2,\dots,\ell_k) = S(n) \setminus \mathopen{}\mathclose\bgroup\originalleft( \bigcup_{i=1}^{k}S(\ell_i) \bigcdot \mathcal{F}(n-\ell_i) \aftergroup\egroup\originalright).
\]
Let us note, for any $S \subset \mathcal{F}$, that the sets $S(n;m)$ and $S(m) \bigcdot \mathcal{F}(n-m)$ are disjoint for any pair of positive integers $m < n$. Recall that $d_S(n) = |S(n)||\mathcal{F}(n)|^{-1}$; we analogously define
\[d_S(n; \ell_1, \ell_2,\dots,\ell_k) = \frac{|S(n; \ell_1, \ell_2,\dots,\ell_k)|}{|\mathcal{F}(n)|}.\]
When the set $S$ in question is clear, we write $d(n)$ and $d(n; \ell_1, \ell_2,\dots,\ell_k)$ for $d_S(n)$ and $d_S(n; \ell_1, \ell_2,\dots,\ell_k)$, respectively. Recall that for any product-free set $S \subset \mathcal{F}$ and any $m,n \in \mathbb{N}$, we have
\[d(m)d(n) + d(m+n) \le 1.\]
We start by proving a generalisation of this fact.
\begin{proposition}\label{doublecount}
If $S \subset \mathcal{F}$ is product-free, then for any sequence of positive integers $\ell_1 < \ell_2 <\dots <\ell_k <n$, we have
\begin{align*}
& d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n) \\
\le \,\, & d(\ell_1) + d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_k) \le 1.
\end{align*}
\end{proposition}
\begin{proof}
First, consider the products
\[ S(\ell_1) \bigcdot S(n-\ell_1), S(\ell_2;\ell_1)\bigcdot S(n-\ell_2), \dots, S(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) \bigcdot S(n-\ell_k).\]
These subsets of $\mathcal{F}(n)$ are by definition disjoint. Let $L'$ be the union of these $k$ sets. Since $S$ is product-free, $L'$ and $S(n)$ are disjoint as well. Let $L = L' \cup S(n)$; clearly, the density of $L$ in $\mathcal{F}(n)$ is
\[d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n).
\]
Next, consider the Minkowski products
\[
S(\ell_1) \bigcdot \mathcal{F}(n-\ell_1),\, S(\ell_2;\ell_1)\bigcdot \mathcal{F}(n-\ell_2),\, \dots,\, S(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) \bigcdot \mathcal{F}(n-\ell_k).
\]
These subsets of $\mathcal{F}(n)$ are again disjoint by definition; let $R'$ denote their union. Note that $R'$ and $S(n; \ell_1, \ell_2, \dots, \ell_k)$ are disjoint. Let $R = R' \cup S(n; \ell_1, \ell_2, \dots, \ell_k)$; it is easy to see that the density of $R$ in $\mathcal{F}(n)$ is
\[d(\ell_1) + d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_k)\]
and that this quantity is therefore at most $1$.
To finish the proof, it suffices to show that
\[L' \cup S(n) = L \subset R = R' \cup S(n; \ell_1, \ell_2, \dots, \ell_k).\]
It is easy to see that $L' \subset R'$. Therefore, it is sufficient to show that $S(n)$ is a subset of $R' \cup S(n; \ell_1, \ell_2, \dots, \ell_k)$. To see this, note that any word from $S(n)$ which has a prefix in $S(\ell_1) \cup S(\ell_2) \cup \dots \cup S(\ell_k)$ is also contained in $R'$. In other words, $S(n) \setminus S(n; \ell_1, \ell_2, \dots, \ell_k) \subset R'$; the result follows.
\end{proof}
With the above observation in hand, we are now ready to prove Theorem~\ref{main-res}.
\begin{proof}[Proof of Theorem~\ref{main-res}]
We prove by contradiction that the upper Banach density of a product-free set is at most $1/2$.
Suppose that $S \subset \mathcal{F}$ is product-free and that $d^*(S) > 1/2 + \eps$ for some $\eps > 0$. We then claim that we may find an increasing sequence of positive integers $(\ell_k)_{k \in \mathbb{N}}$ such that
\[ d(\ell_1) + d(\ell_2; \ell_1) + \dots + d(\ell_k; \ell_1, \ell_2, \dots, \ell_{k-1}) \ge \frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2^k} = 1 - \frac{1}{2^k}\]
for each $k \in \mathbb{N}$.
We construct this sequence inductively. Since $d^*(S) > 1/2$, it is clear that we may find $\ell_1 \in \mathbb{N}$ such that $d(\ell_1) \ge 1/2$. Having found $\ell_1< \ell_2< \dots< \ell_k$ as required, we choose $\ell_{k+1}$ as follows. Since $d^*(S) > 1/2 + \eps$, there exist arbitrarily long intervals $I \subset \mathbb{N}$ that satisfy
\[
\frac{\sum_{n \in I}d(n)}{|I|} > \frac{1}{2} + \eps.
\]
Choose such an interval $I$ whose length is sufficiently larger than $\ell_k$; we may assume, by passing to a sub-interval if necessary, that $\min I > \ell_k$. We claim that it is possible to choose $\ell_{k+1}$ from $I$; in other words, we claim that there exists an $n \in I$ such that
\begin{multline*}
d(\ell_1) + d(\ell_2; \ell_1) + \dots + d(\ell_k; \ell_1, \ell_2, \dots, \ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_{k}) \ge 1 - \frac{1}{2^{k+1}}.
\end{multline*}
We prove this claim by contradiction. Suppose that there is no such $n \in I$. Then, by Proposition~\ref{doublecount}, we have
\begin{align*}
& d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n) \\
\le \,\, & d(\ell_1) + d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1}) + d(n; \ell_1, \ell_2, \dots, \ell_k) < 1-\frac{1}{2^{k+1}}
\end{align*}
for each $n \in I$. By summing the above inequality over all $n \in I$, we get
\[
\sum_{n \in I'}d(n)\mathopen{}\mathclose\bgroup\originalleft(1+d(\ell_1)+ d(\ell_2;\ell_1) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})\aftergroup\egroup\originalright) < |I|\mathopen{}\mathclose\bgroup\originalleft( 1 -\frac{1}{2^{k+1}}\aftergroup\egroup\originalright),
\]
where $I'\subset I$ is the set of $n \in I$ with $n + \ell_k < \max I$. This implies, by the inductive hypothesis, that
\[\sum_{n \in I'}d(n)\mathopen{}\mathclose\bgroup\originalleft(2 - \frac{1}{2^k}\aftergroup\egroup\originalright) < |I|\mathopen{}\mathclose\bgroup\originalleft(1-\frac{1}{2^{k+1}}\aftergroup\egroup\originalright),\]
or equivalently, $\sum_{n \in I'}d(n) < |I|/2$. Therefore, we have
\[ \sum_{n \in I}d(n) \le \sum_{n \in I'}d(n) + \ell_k + 1 < \frac{|I|}{2} + \ell_k + 1,\]
which contradicts the fact that $\sum_{n \in I}d(n) > |I|/2 + \eps|I|$, provided $|I| > (\ell_k+1)/\eps$.
We now finish the proof of the proposition by showing that the existence of this sequence $(\ell_k)_{k \in \mathbb{N}}$ contradicts our initial assumption that $d^*(S) > 1/2 + \eps$. Fix a $k\in \mathbb{N}$ large enough to ensure that
\[\frac{2^k}{2^{k+1} - 1} < \frac{1 + \eps}{2}\]
and consider any interval $I \subset \mathbb{N}$ with $|I| > 4(\ell_k + 1) / \eps$. We know from Proposition~\ref{doublecount} that
\[ d(\ell_1)d(n-\ell_1) + d(\ell_2;\ell_1)d(n-\ell_2) + \dots + d(\ell_k; \ell_1,\ell_2,\dots,\ell_{k-1})d(n-\ell_k) + d(n) \le 1\]
for each $n \in \mathbb{N}$ with $n > \ell_k$; summing this inequality over such $n \in I$, we get
\[\sum_{n \in I'} d(n)\mathopen{}\mathclose\bgroup\originalleft(2 - \frac{1}{2^k}\aftergroup\egroup\originalright) \le |I|, \]
where $I'$ is the set of $n \in I$ with $n > \ell_k$ and $n + \ell_k < \max I$. Therefore,
\[ \frac{\sum_{n \in I} d(n)}{|I|} \le \frac{2^k}{(2^{k+1} - 1)} + \frac{2(\ell_k + 1)}{|I|} < \frac{1}{2} + \eps, \]
which is a contradiction; this proves the claimed upper bound in Theorem~\ref{main-res}.
\end{proof}
\section{Conclusion}\label{sec-conc}
A common line of enquiry in the study of product-free sets is to ask for `asymmetric' versions of results bounding the upper density of product-free sets. In this spirit, it is natural to ask whether an analogue of Theorem~\ref{main-res} continues to hold when one wishes to solve the equation $x \bigcdot y = z$ with $x$, $y$ and $z$ in specified subsets of $\mathcal{F}$. More precisely, if $X, Y, Z \subset \mathcal{F}$ are such that there are no solutions to $x \bigcdot y = z$ with $x \in X$, $y \in Y$ and $z\in Z$, one might ask if one of $X$, $Y$ or $Z$ has an upper asymptotic density of at most $1/2$. However, it is not hard to construct for any $\eps>0$, three sets $X, Y, Z \subset \mathcal{F}$, each of upper asymptotic density at least $\phi - \eps$, where $\phi = (\sqrt 5 - 1)/2$, such that there are no solutions to $x \bigcdot y = z$ with $x \in X$, $y \in Y$ and $z\in Z$. Indeed, pick a suitably large $n \in\mathbb{N}$ and choose any set $W \subset \mathcal{F}(n)$ such that $||W|/|\mathcal{F}(n)| - \phi| < \eps/3$. Now take $X$ to be the set of all words with a prefix in $W$, $Y$ to be the set of all words with a suffix in $W$, and $Z$ to be the set $\mathcal{F} \setminus (X \bigcdot Y)$. Clearly, there are no solutions to $x \bigcdot y = z$ with $x \in X$, $y \in Y$ and $z\in Z$; it is also not hard to check that each of $X$, $Y$ and $Z$ has an upper asymptotic density at least $\phi - \eps$.
Next, it would be interesting to understand what product-free sets of maximal density look like. As we saw earlier, several non-isomorphic extremal constructions are furnished by the family of odd-occurrence sets. We suspect that these might be the only constructions of maximal density, and make the following conjecture.
\begin{conjecture}\label{unique}
Let $\mathscr{A}$ be a finite set. If $S \subset \mathcal{F}_\mathscr{A}$ is product-free and $d^*(S) = 1/2$, then $S \subset \mathcal{O}_\Gamma$ for some nonempty subset $\Gamma \subset \mathscr{A}$.
\end{conjecture}
Finally, another natural direction is to study product-free subsets of the \emph{free group $\mathbf{F}_\mathscr{A}$} over a finite alphabet $\mathscr{A}$. Similarly to the situation in this paper, the most natural measure to consider in the case of the free group $\mathbf{F}_\mathscr{A}$ would be the one that assigns a weight of $|\mathscr{A}|(|\mathscr{A}|-1)^{-(n-1)}$ to each irreducible word of length $n$. The different notions of density defined here for the free semigroup then have analogous definitions in the free group, and we believe that an analogue of Theorem~\ref{main-res} should hold in the free group as well; concretely, we conjecture the following.
\begin{conjecture}\label{freegrp}
For any finite alphabet $\mathscr{A}$, no product-free subset of the free group $\mathbf{F}_\mathscr{A}$ has upper Banach density exceeding $1/2$.
\end{conjecture}
Note that, in the proof of Theorem~\ref{main-res}, we rely crucially on the fact that there is exactly one way to write a word of length $m+n$ as the concatenation of a word of length $m$ with a word of length $n$; of course, we lose this property when working with free groups, so we believe that some new ideas will be required to understand product-free sets in free groups.
\end{document}
|
\begin{equation}gin{document}
\title{Homogenization of dislocation dynamics}
\alphauthor{Ahmad El Hajj, Hassan Ibrahim and R\'egis Monneau}
\alphaddress{CERMICS, ENPC, 6 \& 8 avenue Blaise
Pascal, Cit\'e Descartes, Champs sur Marne, 77455 Marne-la-Vall\'ee
Cedex 2, France}
\epsilonnd{eqnarray}d{[email protected], [email protected], [email protected]}
\begin{equation}gin{abstract}
In this paper we consider the dynamics of dislocations with the same
Burgers vector, contained in the same glide plane, and moving in a material
with periodic obstacles. We study two cases: i) the
particular case of parallel straight dislocations and ii) the general case of curved
dislocations. In each case, we perform rigorously the homogenization of the
dynamics and predict the corresponding effective macroscopic elasto-visco-plastic flow rule.
\epsilonnd{abstract}
{\bf S}E{Introduction}
In the recent years, an important effort has been done, both to improve the
methods to compute discrete dislocation dynamics (see for instance the book
of Bulatov and Cai
\cite{BC} and the references therein) and also to connect them
to continuum models of plasticity in crystalline solids (see for instance
Fivel et al. \cite{FTRC} and more recently Hoc et
al. \cite{HDK}). Although continuum models of
dislocations are known since the 50's (see Kr\"{o}ner \cite{K,K2}), the
dynamics has been taken into account only recently : see Groma et
al. \cite{GB,GCZ} in 2D (and their mathematical studies in
\cite{FE,IJM}), Hochrainer et al. \cite{HZG}, and Monneau \cite{M} in 3D. The goal
of our work is to present, on a particular
example, a rigorous justification of a continuum model with densities of
dislocations bridging the
gap with dislocation dynamics at the microscale. Indeed for a
very special geometry, we are able to deduce by homogenization, the
macroscopic elasto-visco-plastic flow rule relating the plastic strain
velocity to the shear stress. The full technical details are presented
in \cite{FIM}.
{\bf S}E{Homogenization of straight dislocations}
In this section, we consider the case of parallel straight edge dislocations
with the same Burgers vector $\mbox{\bf b}=be_x$ with $b>0$, where
$(e_x,e_y,e_z)$ is an orthonormal basis with corresponding coordinates
$(x,y,z)$. All these dislocation lines are
assumed to be contained in the same glide plane $(x,y)$ and to move in
this plane.
\sigmaubsection{The microscopic model for straight dislocations}
Because of our assumptions, for every integer $i\inftyn \mathbb{Z}$, we can
simply describe the position of the $i$-th dislocation
by its real abscissa that we call $x_i(t)$ where $t$ is the time. We want
to take into account the interactions of each dislocation with other
defects in the crystal, that constitute obstacles
to their motion. Those obstacles can be for instance other pinned
dislocations or precipitates. In order to simplify the analysis, we will
assume that these obstacles are periodically distributed, of spatial period
$\lambdaambda$. In our model, those obstacles will be simply modeled by a smooth
periodic potential $V^{per}$ satisfying
$V^{per}(x+\lambdaambda)=V^{per}(x)$. Then the energy of the system is the sum
of two contributions: the interactions of each dislocations with the
periodic potential and the sum of the two-body interactions between
dislocations associated to a pair potential $V$.
The energy of a set of
dislocations is then
given by
$$E=\sigmaum_{i} V^{per}(x_i) + \sigmaum_{i<j} V(x_i-x_j)\quad
\mbox{with}\quad V(x)=-\begin{eqnarray}r{\mu} b\lambdan |x| \quad \mbox{and}\quad
\begin{eqnarray}r{\mu}=\frac{\mu }{2\partiali (1-\nu)}$$
where the constants $\mu$ and $\nu$ are respectively the shear modulus
and the Poisson ratio. Remark that
the force $-V'(x)$ is then the usual Peach-Koehler force created at the
point $x$ by an edge dislocation positioned at the origin.
We then consider the fully overdamped dynamics, where the velocity is
proportional to the force, i.e.
\begin{equation}gin{equation}\lambdaabel{eq::1}
B\frac{dx_i}{dt}=-\nabla_{x_i} E + \tau_{ext}
\epsilonnd{equation}
where $B$ is the viscous drag coefficient and the force is
on the right hand side. The first contribution to the force is a term deriving from the
energy and $\tau_{ext}$ is a real exterior applied shear stress, that can be
seen as a driving force of the system.
A natural question is then: what is the macroscopic behavior of this system ?
In order to answer this question (which is done in Theorem \ref{th::1}),
we have to introduce the plastic strain. To each dislocation is associated a three-dimensional
displacement in the crystal, whose plastic strain is localized in the glide
plane $z=0$ and is equal to $\gamma \deltaelta_0(z)$ where $\deltaelta_0$ is the
Dirac mass. For instance, for a dislocation $x_i$, the intensity
$\gamma$ (that we continue to call plastic strain) is equal to $-b
H(x-x_i)$ where the Heaviside
function $H(x)$ is equal to $1$ for positive $x$ and zero otherwise. Here
the sign defining the plastic strain is such that the quantity
$\gamma$ increases when $x_i$ increases. Then the total plastic strain can
be written as
$$\gamma(x,t)=-b \sigmaum_{i} H(x-x_i(t)).$$
\sigmaubsection{The normalization procedure}
We are now interested in the behavior of the system at a macroscopic scale
$\Lambdaambda$ such that $\Lambdaambda >> \lambdaambda =\omegal b$ where $\omegal >1$ is a fixed
ratio. Then we introduce several dimensionless quantities. We call
$\begin{eqnarray}r{x}$ and $\omegat$ the normalized
spatial and time coordinates at the macroscopic level, and introduce a
parameter $\varepsilon$ and the associated
normalized macroscopic plastic strain
${\gamma}^\varepsilon$ such that
\begin{equation}gin{equation}\lambdaabel{eq::0}
\begin{eqnarray}r{x} = \frac{x}{\Lambdaambda}, \quad \omegat =
\frac{\begin{eqnarray}r{\mu}}{B} \frac{t}{\Lambdaambda},\quad \varepsilon = \frac{b}{\Lambdaambda} \quad \mbox{and}\quad
\deltaisplaystyle{{\gamma}^\varepsilon(\omegax,\omegat)=
\frac{1}{\Lambdaambda}\gamma(x,t)} \quad \mbox{with}\quad {\gamma}^\varepsilon(\omegax,0)=\varepsilon\lambdaeft[\frac{1}{\varepsilon}\gamma_0(\omegax)\right]
\epsilonnd{equation}
where $\lambdaeft[\cdot\right]$ is the floor function, $\gamma_0$ is a given function and $B\Lambdaambda/\begin{eqnarray}r{\mu}$ is a
typical macroscopic time deduced from equation (\ref{eq::1}).
Remark that $\varepsilon$ can be very small in our application (for instance $\varepsilon \sigmaimeq 10^{-6}$ if
$b\sigmaimeq 10^{-9}m$ and $\Lambdaambda\sigmaimeq 10^{-3}m$).
We expect that the macroscopic behavior of the model is well described by
{\inftyt the limit macroscopic plastic strain $\gamma^0(\omegax,\omegat)$ of $\gamma^\varepsilon(\omegax,\omegat)$ as
$\varepsilon$ goes to zero}.
\sigmaubsection{Heuristics for the macroscopic stress field}
In this subsection, we want to give heuristic expressions of the normalized
dislocation density and the macroscopic stress field, in terms of the limit
macroscopic plastic strain.
We remark that the gradient of the map $x\mapsto -\gamma^\varepsilon
(x/\Lambdaambda,\omegat) /\varepsilon$
is a sum of Dirac masses, and then the number of dislocations in a large segment
of length ${\cal D}elta x$ is formally given by $-\inftynt_{0}^{{\cal D}elta x}
\frac{1}{\varepsilon \Lambdaambda } \frac{\partialartial \gamma^\varepsilon}{\partialartial \omegax}(x/\Lambdaambda,\omegat)\
dx$. This shows at least formally that the dislocation density can be
estimated as $\rho(x,t)=-\frac{1}{\varepsilon\Lambdaambda} \frac{\partialartial \gamma^0}{\partialartial
\omegax}(\omegax,\omegat)$.
Then the total stress on the right hand side of (\ref{eq::1}) can be formally
described at the macroscopic scale by
\begin{equation}gin{equation}\lambdaabel{eq::2}
\deltaisplaystyle{\tau = \tau_{ext} + \tau_{sc} \quad \mbox{with}\quad
\tau_{sc}(\omegax,\omegat)= -\begin{eqnarray}r{\mu} \inftynt_{-\inftynfty}^{+\inftynfty}
\frac{d\omegax'}{\omegax-\omegax'} \frac{\partialartial \gamma^0}{\partialartial \omegax}(\omegax',\omegat)}
\epsilonnd{equation}
where we take the principal value in the integral defining the
self-consistent field $\tau_{sc}$. This expression can be deduced from the
equation $\tau_{sc}(\omegax,\omegat)= -(V'\sigmatar_x \rho)(x,t)$, where $\sigmatar_x$ denotes the
convolution with respect to the variable $x$.
Remark also that the expression (\ref{eq::2}) of
$\tau_{sc}$
is known to be the resolved shear stress created by the normalized
dislocation density
\begin{equation}gin{equation}\lambdaabel{eq::3}
\rho^0=- \frac{\partialartial \gamma^0}{\partialartial \omegax}
\epsilonnd{equation}
where for instance $\rho^0=1/\begin{eqnarray}r{\lambdaambda}$ when there is one dislocation by spatial period $\lambdaambda$.
In particular, we see that $\tau_{sc}$ keeps the memory of the long range interactions
between dislocations.
\sigmaubsection{The homogenization result}\lambdaabel{s1.3}
We expect that the effective equation satisfied by the limit
$\gamma^0$ can be written
\begin{equation}gin{equation}\lambdaabel{eq::5}
\lambdaeft\{\begin{equation}gin{array}{l}
\deltaisplaystyle{\frac{\partialartial \gamma^0}{\partialartial \omegat} = f(\rho^0, \tau), \quad
\mbox{for all}\quad \omegax\inftyn\mathbb{R},\quad \omegat\inftyn (0,+\inftynfty)},\\
\\
\gamma^0(\omegax,0)=\gamma_0(\omegax) \quad \mbox{for all}\quad \omegax\inftyn \mathbb{R}
\epsilonnd{array}\right.
\epsilonnd{equation}
where $\rho^0$ is given in (\ref{eq::3}) and $\tau$ in (\ref{eq::2}).
Then our main result is:
\begin{equation}gin{theo}\lambdaabel{th::1}{\bf (Homogenization of straight dislocations)}\\
Assume that the initial data $\gamma_0$ is non-decreasing and satisfies
$|\gamma_0|+ |\gamma_0'|+ |\gamma_0''| \lambdae C$ for some constant $C$.
Then for any $C^2$ periodic potential $V^{per}$, there exists
a continuous function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\tau \mapsto
f(\rho^0,\tau)$ is nondecreasing. And there exists a unique viscosity
solution $\gamma^0$ of the equation (\ref{eq::5}).\\
Moreover, under the assumptions and notation of this
section, there exists a unique solution $\gamma^\varepsilon$ associated to
the dynamics (\ref{eq::1}) with initial
data given in (\ref{eq::0}), and $\gamma^\varepsilon$ converges to $\gamma^0$ locally
uniformly on $\mathbb{R} \times [0,+\inftynfty )$.
\epsilonnd{theo}
This result is proven rigorously in \cite{FIM} in the mathematical framework of
viscosity solutions (see for instance Crandall, Ishii, Lions \cite{CIL} for an introduction to
this theory). We explain in the next section how we compute the
function $f$, which keeps the memory of the short range interactions
between the dislocations and the periodic potential $V^{per}$.
\sigmaubsection{Computation of $f$ using Orowan's law}\lambdaabel{ssf}
In this subsection, we briefly explain (without any justifications) how to
compute the function $f$. We refer the reader to \cite{FIM} for the proofs
of those results.\\
\noindent {\inftyt Case A:} $V^{per}\epsilonquiv 0$.\\
In this special case, we can show that
\begin{equation}gin{equation}\lambdaabel{eq::6}
f(\rho^0,\tau)=\rho^0 \begin{eqnarray}r{v} \quad \mbox{with}\quad \begin{eqnarray}r{v}=\frac{\tau}{\begin{eqnarray}r{\mu}}
\epsilonnd{equation}
which is nothing else than the normalized Orowan's law giving, in a
dimensionless form, the
plastic strain velocity as the product of the normalized dislocation
density $\rho^0$ and the normalized mean velocity $\begin{eqnarray}r{v}$ of the
dislocations.\\
\noindent {\inftyt Case B:} General periodic potential $V^{per}$.\\
In that case, the function $f$ can
be computed using the following two steps.\\
\noindent {\underline{Step 1}.}\\
For $i\inftyn\mathbb{Z}$, we look for solutions to (\ref{eq::1}) of the following
special form
$$x_i(t)=b\cdot h\lambdaeft(\frac{vt}{b} + \frac{i}{\rho^0}\right), \quad \mbox{with}\quad
h(a+\begin{eqnarray}r{\lambdaambda})=\begin{eqnarray}r{\lambdaambda}+h(a) \quad \mbox{for all}\quad a\inftyn\mathbb{R}$$
for some constant $v$ and for a function $h$ which is called a {\inftyt hull
function}. Both $v$ and $h$ have to be determined.
Because of the convexity of the two-body potential $V$ outside the origin,
it is possible to show that the constant $v$ exists and is unique. Moreover
this constant $v$ can be interpreted as the mean velocity of each dislocation.\\
\noindent {\underline{Step 2}.}\\
We simply define $f(\rho^0,\tau_{ext})$ using the normalized Orowan's law as in
(\ref{eq::6}), but with the normalized velocity
$\begin{eqnarray}r{v}$ replaced by the constant $\begin{eqnarray}r{v}=\frac{B}{\begin{eqnarray}r{\mu}}v$.
\sigmaubsection{Numerical computation of $f$}
We present numerical simulations for the computation of the function
$f$. We work with dimensionless quantities:
$\lambdaambda=1=\begin{eqnarray}r{\lambdaambda}=b=B=\begin{eqnarray}r{\mu}$.
We put initially $N$ dislocations in
an interval of length $l=10$ which is repeated periodically. Therefore this
interval contains $l$ times the period of
the periodic potential that we choose equal to $V^{per}(x)=\frac{A}{2\partiali}\sigmain(2\partiali x)$
with $A=3$. We discretize the ODE system (\ref{eq::1}), using an
explicit Euler scheme with a time step ${\cal D}elta t =0.01$. We compute
numerically the mean velocity $v$ of the dislocations after a final time
$T=1000$. We then set $f=\rho^0 v$ with $\rho^0=N/l$. We do the computation
with $N=1,...,200$ and $0\lambdae \tau_{ext}\lambdae 9$ with ${\cal D}elta \tau_{ext} =
\frac{9}{200}$. Remark that
we can restrict our computation for positive $\tau_{ext}$, because we have
$f(\rho^0,-\tau_{ext})=-f(\rho^0,\tau_{ext})$, from the symmetry of the
potential $V^{per}$ in our problem. The
level sets of the function $f$ are represented on Figure \ref{F1}. In
order to have a better view of the set where $f=0$, this set is
conventionally represented in Figure \ref{F1} with artificial negative
values of $f$. We remark that this figure shows in particular a
collective behavior of the dislocations: higher is the density of
dislocations, then easier the dislocations move above the obstacles.
Figure \ref{F2} shows the map $\tau_{ext}\mapsto f(\rho^0,\tau_{ext})$ for
$\rho^0=N/l$ with $N=1,10,20$. We see in particular that for $\tau_{ext}$ under a threshold
(that depends on the dislocation density $\rho^0$) the function $f$ vanishes.
\begin{equation}gin{figure}[!h]
\begin{equation}gin{minipage}[b]{.46\lambdainewidth}
\centering\epsilonpsfig{figure=Photos68.eps,width=\lambdainewidth}
\caption{Level sets of the effective $f(N/l,\tau_{ext})$ with $N$ on
abscissas and $\tau_{ext}$ on ordinates\lambdaabel{F1}}
\epsilonnd{minipage}
\begin{equation}gin{minipage}[b]{.46\lambdainewidth}
\centering\epsilonpsfig{figure=Photos32new.eps ,width=\lambdainewidth}
\caption{For $N=1,10,20$, graph of the map $\tau_{ext}\mapsto f(N/l,\tau_{ext})$\\\lambdaabel{F2}}
\epsilonnd{minipage}
\epsilonnd{figure}
{\bf S}E{Homogenization of curved dislocations}
In this section, we very briefly generalize the previous analysis to the
case of curved dislocations all contained in the same plane $(x,y)$ with the
same Burgers vector $\mbox{\bf b}=be_x$ with $b>0$.
\sigmaubsection{The microscopic model for curved dislocations}
For $i\inftyn\mathbb{Z}$, the motion of the $i$-th dislocation curve $\Gammaamma_i(t)$ at the point
$X\inftyn\mathbb{R}^2$ is given by its
normal velocity ${\mathcal V}$ defined by
\begin{equation}gin{equation}\lambdaabel{eq::1bis}
B\cdot{\mathcal V}(X,t)= \tau^{per}(X) + \sigmaum_{j} F_{j}(X,t)
\epsilonnd{equation}
where $F_{j}(X,t)$ is the resolved Peach-Koehler force created by the
dislocation $\Gammaamma_j(t)$ at the point $X$. Here $\tau^{per}$ is a smooth
periodic function satisfying
$\tau^{per}(X +\lambdaambda k)=\tau^{per}(X)$ for all $k\inftyn\mathbb{Z}^2$, which
represents the periodic obstacles to the motion of the dislocations and can also
include the exterior applied stress. To give the expression of this
force, it is convenient
to introduce a continuous function $\tilde{\gamma}(X,t)$
such that each dislocation curve $\Gammaamma_j(t)$ can be seen as the level set
$\tilde{\gamma}(X,t)=jb$ (when this level set is non-degenerated). Then a
good approximation is given by
$$\deltaisplaystyle{F_{j}(X,t)=\frac12 \inftynt_{\mathbb{R}^2} dZ\ J(X-Z)\ \mbox{sign}(\tilde{\gamma}(Z,t)-jb)}$$
where, in the integral, the sign function takes values $-1,0,1$. Here the
kernel $J$ is smooth and satisfies for a cut-off radius $R=\begin{eqnarray}r{R}b$ with $\begin{eqnarray}r{R}>1$ fixed:
$$J(-X)=J(X)\ge 0, \quad \mbox{and}\quad
\deltaisplaystyle{J(X)=J_{\inftynfty}
(X):=\frac{1}{|X|^3}g\lambdaeft(\frac{X}{|X|}\right) \quad \mbox{for}\quad
|X|>R>0}$$
where for isotropic elasticity with $X=(x,y)$, we have
$g\lambdaeft(\frac{X}{|X|}\right)=\frac{\mu b}{4\partiali}\lambdaeft\{\frac{x^2(2\begin{equation}ta -1) +
y^2(2-\begin{equation}ta)}{x^2+y^2}\right\}$ with $\begin{equation}ta=\frac{1}{1-\nu}$. Remark that
this formula allows to describe with the same formalism edge, screw and
mixed dislocations (see for instance \cite{AHLM}).
We also define the plastic strain $\gamma$ as
$$\gamma= b \lambdaeft[
\frac{\tilde{\gamma}}{b}\right]$$
where we recall that $\lambdaeft[\cdot\right]$ is the
floor function. Then we proceed as in the previous section and define
\begin{equation}gin{equation}\lambdaabel{eq::0bis}
\begin{eqnarray}r{X}=\frac{X}{\Lambdaambda},\quad
\begin{eqnarray}r{t}=\frac{\mu}{B}\frac{t}{\Lambdaambda},\quad
\varepsilon=\frac{b}{\Lambdaambda},\quad \mbox{and}\quad
\gamma^{\varepsilon}(\begin{eqnarray}r{X},\omegat)=\frac{1}{\Lambdaambda}\gamma(X,t),\quad \mbox{with}\quad
\gamma^{\varepsilon}(\begin{eqnarray}r{X},0)=\varepsilon
\lambdaeft[\frac{1}{\varepsilon}\gamma_0(\omegaX)\right].
\epsilonnd{equation}
\sigmaubsection{The homogenization result}
We expect that the effective equation satisfied by the limit
$\gamma^0$ of $\gamma^\varepsilon$ can be written
\begin{equation}gin{equation}\lambdaabel{eq::5bis}
\lambdaeft\{\begin{equation}gin{array}{l}
\deltaisplaystyle{\frac{\partialartial \gamma^0}{\partialartial \omegat} = f(-\nabla \gamma^0, \tau_{sc}), \quad
\mbox{for all}\quad \omegaX\inftyn\mathbb{R}^2,\quad \omegat\inftyn (0,+\inftynfty)},\\
\\
\gamma^0(\omegaX,0)=\gamma_0(\omegaX) \quad \mbox{for all}\quad \omegaX\inftyn \mathbb{R}^2
\epsilonnd{array}\right.
\epsilonnd{equation}
with
$$\tau_{sc}(\omegaX,\omegat)=\inftynt_{\mathbb{R}^2} dZ\ J_\inftynfty (\omegaX-Z) \gamma^0(Z,\omegat)$$
where we take the principal value of the integral. Remark that this
expression of $\tau_{sc}$ is consistent with the one given in (\ref{eq::2})
in the special case where $\gamma^{0}(\begin{eqnarray}r{x}, \begin{eqnarray}r{y}, \begin{eqnarray}r{t})$ is
independent of $\begin{eqnarray}r{y}$.
Then we have
\begin{equation}gin{theo}\lambdaabel{th::1bis}{\bf (Homogenization of curved dislocations)}\\
Assume that the initial data satisfies $|\gamma_0|+ |\nabla \gamma_0|+
|D^2 \gamma_0|\lambdae C$ for some constant $C$.
Then for any $C^2$ periodic function $\tau^{per}$, there exists
a continuous function $f: \mathbb{R}^2\times \mathbb{R} \to \mathbb{R}$ such that $\tau \mapsto
f(\cdot,\tau)$ is nondecreasing. And there exists a unique viscosity
solution $\gamma^0$ of the equation (\ref{eq::5bis}).\\
Moreover, under the assumptions and notation of this
section, there exists a unique solution $\gamma^\varepsilon$ associated to
the dynamics (\ref{eq::1bis}) with initial
data given in (\ref{eq::0bis}), and $\gamma^\varepsilon$ converges to $\gamma^0$ locally
uniformly on $\mathbb{R}^2 \times [0,+\inftynfty )$.
\epsilonnd{theo}
{\bf S}E{Conclusion}
The main result of our work is the justification of the
elasto-visco-plastic flow rule by the homogenization of the
dynamics of dislocations with the same
Burgers vector, moving in the same glide plane with periodic obstacles.
Even if this geometry is very particular, this is, up
to our knowledge, the first rigorous result in this direction. We also
explained how to compute the flow rule, and presented numerical results.
The proof of the homogenization for straight dislocations uses strongly the local convexity of the
two-body potential $V$ (which is equivalent to the non-negativity of the
kernel $J$ in the case of curved dislocations).
Remark that for the same dynamics, it is
possible to find non-convex potentials $V$, for which there is no
homogenization. For a general geometry, there is in general no hope to find any
convexity argument to justify homogenization. On the contrary, it seems
reasonable to think that homogenization could arise in general, if we
assume moreover that the dynamics is modified by the addition of a small
random noise. But this is still an open problem to investigate.
\noindent {\bf Acknowledgements}\\
This work was supported by the contract ANR MICA (2006-2009).
\noindent {\bf References}
\begin{equation}gin{thebibliography}{99}
\bibitem{BC}
{Bulatov V V and Cai W},
{\inftyt Oxford University Press}, (2006).
\bibitem{FTRC}
{Fivel M, Tabourot L, Rauch E and Canova G R},
{\inftyt J. Phys.} IV, {\bf 8} (1998), 151-158.
\bibitem{HDK}
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|
\betaegin{document}
\betaegin{abstract} In this paper we construct and study the actions of certain deformations of the Lie
algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the
relative symmetric powers ${\cal C}^{[\betaullet]}$ and the relative Jacobian ${\cal J}$ of a family of curves
${\cal C}/S$. As one of the applications, we show that
in the case of a single curve $C$ this action induces a ${\Bbb Z}$-form of a Lefschetz $\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring ${\Bbb C}H_0(J)$
of $0$-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes
in ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ and in ${\Bbb C}H^*({\cal J})$ and prove for them analogs of the properties established
in the case of the Jacobian of a single curve by Beauville in \cite{Bmain}. We show that the our
algebras of operators preserve the subrings of tautological cycles and act on them via some explicit differential operators.
\operatorname{e}nd{abstract}
\title{Algebraic cycles on the relative symmetric powers and on the
relative Jacobian of a family of curves. I}
\betaigskip
\centerline{\sc Introduction}
\betaigskip
Let ${\cal C}/S$ be a family of smooth projective curves over a smooth quasiprojective base $S$,
and let ${\cal C}^{[N]}$ denote the $N$th relative symmetric power of ${\cal C}$ over $S$.
In this paper we construct and study the natural action of a certain modification of the Lie algebra of differential operators on the line on the direct sum of the Chow groups
$${\Bbb C}H^*({\cal C}^{[\betaullet]}):=\betaigoplus_{N\ge 0}{\Bbb C}H^*({\cal C}^{[N]}),$$
where ${\cal C}^{[0]}=S$ (and on the similar direct sum of cohomology).
We also construct a related action of another algebra on ${\Bbb C}H^*({\cal J})$, where ${\cal J}/S$
is the corresponding relative Jacobian. These constructions are motivated by their potential use in the study of the Chow rings of Jacobians of curves, and in particular, in the study of the tautological subrings (see \cite{Bmain}, \cite{P-univ}, \cite{P-lie}). It was observed that the subalgebra generated by the standard cycles in the Jacobian of a curve depends in an interesting way on the corresponding point in the moduli space. Therefore, it is important to develop the corresponding calculus in the relative situation. On the other hand,
our main construction is reminiscent of the well known construction of the Heisenberg action on
cohomology of Hilbert schemes of surfaces (see \cite{Nak}, \cite{G}), and one can hope that there might
be a direct link between the two actions in the case of a family of curves lying on a surface.
Let us describe our main construction.
Let ${\cal D}={\Bbb Z}[t,\frac{d}{dt}]$ be the algebra of differential operators on the line.
Adjoin an independent variable $h$ and consider the subalgebra
${\cal D}_h\subset {\cal D}\otimesimes{\Bbb Z}[h]$ generated over ${\Bbb Z}[h]$ by $t$ and by $h\frac{d}{dt}$.
We view it as a Lie algebra with the commutator
$$[D_1,D_2]_h=(D_1D_2-D_2D_1)/h.$$
Note that ${\cal D}_h$ is a deformation of the Lie algebra $\widetilde{{\cal HV}}={\Bbb Z}[t,p]$ of polynomial
Hamiltonians on the plane (equipped with the standard Poisson bracket).
Now for any supercommutative ring $A$ and an even element ${\betaf a}_0\in A$ we define the Lie superalgebra
$${\cal D}(A,{\betaf a}_0):={\cal D}_h\otimesimes_{{\Bbb Z}[h]} A,$$
where the homomorphism ${\Bbb Z}[h]\to A$ sends $h$ to ${\betaf a}_0$.
The (super)bracket (resp., ${\Bbb Z}/2{\Bbb Z}$-grading) on ${\cal D}(A,{\betaf a}_0)$ is induced by the bracket on ${\cal D}_h$ and
the product on $A$ (resp., by the ${\Bbb Z}/2{\Bbb Z}$-grading on $A$). More explicitly,
${\cal D}(A,{\betaf a}_0)$ is generated as an abelian group by the elements
$${\betaf P}_{m,k}(a):=t^m(h\frac{d}{dt})^k\otimes a, \ \ a\in A.$$
The supercommutator is given by the formula
\betaegin{equation}\lambdabel{main-com-rel}
[{\betaf P}_{m,k}(a),{\betaf P}_{m',k'}(a')]= \sum_{i\ge 1}(-1)^{i-1}i!\cdotot
\left({k\choose i}{m'\choose i}-{m\choose i}{k'\choose i}\right)
{\betaf P}_{m+m'-i,k+k'-i}(a\cdotot a'\cdotot {\betaf a}_0^{i-1}).
\operatorname{e}nd{equation}
If $R\subset A$ is a subring then we can talk about an $R$-linear action of ${\cal D}(A,{\betaf a}_0)$ on an $R$-module (viewing ${\cal D}(A,{\betaf a}_0)$ as a Lie superalgebra over $R$). Our main construction gives a natural ${\Bbb C}H^*(S)$-linear (resp., $H^*(S)$-linear) action of ${\cal D}({\Bbb C}H^*({\cal C}),K)$ (resp.,
${\cal D}(H^*({\cal C}),cl(K))$) on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ (resp., $\betaigoplus H^*({\cal C}^{[N]})$),
where $K\in{\Bbb C}H^1({\cal C})$ is the relative canonical class,
$cl(K)\in H^2({\cal C})$ is the corresponding cohomology class (see Theorem \ref{action-thm} below).
For every integers $N\ge m\ge 0$ let us consider the morphism
$$s_{m,N}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]},$$
sending a point $(p,D)\in {\cal C}_s\tildemes {\cal C}_s^{[N-m]}$ to $mp+D$, where
${\cal C}_s\subset {\cal C}$ is the fiber of our family over $s\in S$. Here
we identify points of ${\cal C}_s^{[N]}$ with effective divisors of degree $N$ on ${\cal C}_s$.
Note that in the case $m=0$ this map is just the projection to ${\cal C}^{[N]}$.
For a cycle $a\in{\Bbb C}H^*({\cal C})$ and integers $m\ge 0$, $k\ge 0$, we consider the operator
$P_{m,k}(a)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ defined by the formula
\betaegin{equation}\lambdabel{main-oper-eq}
P_{m,k}(a)(x)=s_{m,N-k+m,*}(p_1^*a\cdotot s_{k,N}^*x),
\operatorname{e}nd{equation}
where $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
In the case $a=[{\cal C}]$ we will simply write $P_{m,k}({\cal C})$.
Note that in the case $N<k$ we have $P_{m,k}(a)(x)=0$.
If $a\in{\Bbb C}H^i({\cal C})$ then
$P_{m,k}(a)$ sends ${\Bbb C}H^p({\cal C}^{[N]})$ to ${\Bbb C}H^{p+i+m-1}({\cal C}^{[N-k+m]})$.
\betaegin{thm}\lambdabel{action-thm}
(a) The map ${\betaf P}_{m,k}(a)\mapsto P_{m,k}(a)$ defines a ${\Bbb C}H^*(S)$-linear action of
${\cal D}({\Bbb C}H^*({\cal C}),K)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$,
where $K\in{\Bbb C}H^1({\cal C})$ is the relative canonical class.
More precisely, the commutation relations \operatorname{e}qref{main-com-rel} for the operators $(P_{m,k}(a))$
(with ${\betaf a}_0=K$) hold on the level of relative correspondences over
$S$, i.e., they correspond to certain equalities in ${\Bbb C}H^*({\cal C}^{[\betaullet]}\tildemes_S {\cal C}^{[\betaullet]})$.
\noindent
(b) If we work over ${\Bbb C}$, the same construction defines an action of ${\cal D}(H^*({\cal C},{\Bbb Z}),cl(K))$ on
$H^*({\cal C}^{[\betaullet]})=\betaigoplus_N H^*({\cal C}^{[N]},{\Bbb Z})$.
\operatorname{e}nd{thm}
In the case of a trivial family ${\cal C}=C\tildemes S$ the above relations can be rewritten in a simpler form
(due to the fact that ${\cal D}_h/h^2$ becomes a trivial deformation of $\widetilde{{\cal HV}}$).
Recall that the commutator in the Lie algebra $\widetilde{{\cal HV}}={\Bbb Z}[x,p]$ of polynomial Hamiltonians on the plane is given by
$$\{ x^mp^k, x^{m'}p^{k'} \}=(km'-mk') x^{m+m'-1}p^{k+k'-1}.$$
\betaegin{cor}\lambdabel{triv-base-cor}
Let $C$ be a smooth projective curve over a field $k$.
Choose a theta characteristic $\chi\in{\Bbb C}H^1(C)$ (so that $2\chi=K$) and set
$$L_{m,k}(a)=P_{m,k}(a)-mk P_{m-1,k-1}(p_1^*\chi\cdotot a)$$
for $k\ge 0$, $m\ge 0$, $a\in{\Bbb C}H^*(C\tildemes S)$. Then one has the following relations:
$$[L_{m,k}(a),L_{m',k'}(a')]=(km'-mk')L_{m+m'-1,k+k'-1}(a\cdotot a').$$
In other words, the map $x^mp^k\otimes a\mapsto L_{m,k}(a)$ defines an action of
$\widetilde{{\cal HV}}\otimes{\Bbb C}H^*(C\tildemes S)$ on ${\Bbb C}H^*(C^{[\betaullet]}\tildemes S)$.
Similarly, we can define an action of $\widetilde{{\cal HV}}\otimes H^*(C\tildemes S)$ on $H^*(C^{[\betaullet]}\tildemes S)$.
We have $L_{0,0}([C\tildemes S])=0$, so the operators $(L_{m,k}([C\tildemes S]))$ define the action
of ${\cal HV}=\widetilde{{\cal HV}}/{\Bbb Z}$ on ${\Bbb C}H^*(C^{[\betaullet]}\tildemes S)$.
\operatorname{e}nd{cor}
\betaegin{rem} The operators $P_{n,0}(a)$ (resp., $P_{0,n}(a)$) for $n\ge 0$
are defined by the correspondences that are similar to those defining
Nakajima's operators $q_n(a)$ (resp., $q_{-n}(a)$) for the Hilbert schemes of points on a surface,
where we use the notation of \cite{Lehn}. It is somewhat surprising that in the curve case
the Lie superalgebra generated by these operators is more complicated
(the relations for $q_n(a)$ are simply those of the Heisenberg superalgebra).
\operatorname{e}nd{rem}
Looking at the simplest of the above operators (such as $P_{m,1}(C)$ and $P_{0,1}([p])$, where
$p\in C$ is a point) in the case $S=\operatorname{Spec}(k)$ we will derive the following result.
\betaegin{thm}\lambdabel{curve-thm}
Let $C$ be a smooth projective curve of genus $g\ge 1$ over an algebraically closed field $k$ and let
$J$ be the Jacobian of $C$. Fix a point $p_0\in C(k)$ and consider the embedding $\iota:C\to J$
associated with $p_0$, so that $\iota(p_0)=0\in J(k)$.
Let us denote by $I_C\subset{\Bbb C}H_0(J)$ the subgroup of classes represented by
$0$-cycles of degree $0$ supported on $\iota(C)$.
\noindent
(i) One has a direct sum decomposition
$${\Bbb C}H_0(J)={\Bbb Z}\cdotot [0]\oplus I_C\oplus I_C^{*2}\oplus\ldots\oplus I_C^{*g},$$
where $I_C^{*n}$ denotes the $n$th Pontryagin power of $I_C$.
The associated filtration $(\betaigoplus_{i\ge n}I_C^{*i})$ coincides with the
standard filtration $(I^{*n})$, where $I\subset{\Bbb C}H_0(J)$ is the subgroup of cycles of degree zero.
\noindent
(ii) There exists a family of derivations $(\delta_m)_{m\ge 1}$ of the graded algebra
$${\Bbb C}H_0(J)={\Bbb Z}\oplus \betaigoplus_{n=1}^g I_C^{*n}\sigmameq\betaigoplus_{n=0}^g I^{*n}/I^{*(n+1)},$$
where the multiplication is given by the Pontryagin product, such that for every $x\in I_C$ one has
$$\delta_m(x)=\sum_{i=0}^{m-1}(-1)^i{m\choose i}[m-i]_*x\in I_C^{*m}.$$
Equivalently, $\delta_m|_{I_C}$ can be characterized by the property
$$\delta_m([\iota(p)]-[0])=([\iota(p)]-[0])^{*m} \text{ for all }p\in C(k).$$
These derivations satisfy the commutation relations
$$[\delta_m,\delta_{m'}]=(m'-m)\delta_{m+m'-1},$$
i.e., they define an action of the Lie algebra of polynomial
vector fields on the line vanishing at the origin by $t^m\frac{d}{dt}\mapsto\delta_m$.
\operatorname{e}nd{thm}
We will show (see Remark 2 in the end of section \ref{first-sec})
that for a general curve of genus $\ge 3$ the decomposition in Theorem \ref{curve-thm}(i)
is different from
the decomposition defined by Beauville (see \cite{B1}, p.254; \cite{B2}, Prop.~4).
Theorems \ref{action-thm} and \ref{curve-thm} will be proved in section \ref{first-sec}.
In sections \ref{Jac-sec} and \ref{div-sec}, that are somewhat more technical,
we reprove (and generalize to the relative case) some known results using our algebra of operators.
In section \ref{Jac-sec} we study the relation between our operators $(P_{m,k}(a))$
and the operators $(X_{n,k})_{n+k\ge 2}$ on ${\Bbb C}H(J)_{{\Bbb Q}}$,
where $J$ is the Jacobian of a curve $C$, constructed in \cite{P-lie}.
The latter family of operators satisfies the commutation relations of the
Lie subalgebra ${\cal HV}'\subset{\cal HV}$ spanned over ${\Bbb Z}$ by the elements $x^np^k$ with
$n+k\ge 2$ (it corresponds to Hamiltonian vector fields on the plane vanishing at the origin).
It is natural to ask how this action is related to the one given by Corollary \ref{triv-base-cor} (say,
in terms of the push-forward map ${\Bbb C}H^*(C^{[\betaullet]})_{{\Bbb Q}}\to{\Bbb C}H^*(J)_{{\Bbb Q}}$).
The relation turns out to be not quite straightforward. To work it out we
introduce another family of operators $(T_k(m,a))$,
acting both on ${\Bbb C}H^*(C^{[\betaullet]})$ and on ${\Bbb C}H^*(J)$, and
compatible with the push-forward map. In the case of ${\Bbb C}H^*(C^{[\betaullet]})$ we find an explicit expression of $T_k(m,a)$ in terms of the operators $(P_{k,m}(a))$ (see Proposition \ref{T-P-prop}).
On the other hand, in the
case of ${\Bbb C}H^*(J)_{{\Bbb Q}}$ we find that the operators $T_k(m,a)$ depend polynomially on $m$ and
the corresponding coefficients are closely related to the operators $X_{n,k}$ considered in
\cite{P-lie}. Similar computations work in the case of the relative Jacobian
${\cal J}/S$ of a family of curves ${\cal C}/S$. However, in the relative case the relations
between the operators $X_{n,k}$ get deformed in an interesting way: the corresponding
(quadratic) algebra of operators acting on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$ is not a universal enveloping algebra
of a Lie algebra anymore (see \operatorname{e}qref{X-rel-eq}). We will study this algebra in detail elsewhere.
In section \ref{div-sec} we revisit algebraic Lefschetz $\operatorname{sl}_2$-actions for $C^{[N]}$
using our operators and study related questions of integrality.
The algebraic Lefschetz operators over ${\Bbb Q}$ are easily obtained from the action of
the algebra of (polynomial) differential operators in two variables ${\cal D}_{t,u,{\Bbb Q}}$
on ${\Bbb C}H^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$ generated by operators of the form $P_{10}(a)$ and $P_{01}(a')$
(see Corollary \ref{Heis-K-cor}).
After studying the divided powers of the operators $P_{n,0}({\cal C})$ and $P_{0,n}({\cal C})$ we construct an action of the divided powers subalgebra
${\Bbb Z}[t,u^{[\betaullet]},\partial_t^{[\betaullet]},\partial_u]\subset{\cal D}_{t,u,{\Bbb Q}}$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$. Using this action
we generalize to the relative case
the result of Collino \cite{Col2} on injectivity of the homomorphism
$i_{N*}$ (resp., surjectivity of $i_N^*$), where $i_N:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ is the embedding
associated with $p_0$.
The corresponding $\operatorname{sl}_2$-action given by the operators $e=t\partial_u$, $f=u\partial_t$ and
$h=t\partial_t-u\partial_u$ preserves the Chow groups of the individual symmetric powers.
In the case when $S$ is a point we show that in this way we get a Lefschetz
$\operatorname{sl}_2$-triple for $C^{[N]}$ (see Theorem \ref{Lefschetz-thm}).
Working with divided powers allows us to reprove the fact
(observed by del Ba\~{n}o in \cite{dB2})
that the hard Lefschetz isomorphism for $C^{[N]}$ holds over ${\Bbb Z}$ (see Corollary \ref{Lefschetz-cor}).
Finally, in section \ref{taut-sec} we define the groups of tautological classes in ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
and in ${\Bbb C}H^*({\cal J})$. For tautological classes in ${\Bbb C}H^*({\cal J})$ we establish the properties similar
to those obtained in the case $S=\operatorname{Spec}(k)$ by Beauville in \cite{Bmain}. We also show that tautological
subspaces are preserved by the operators constructed in this paper and by push-forward (resp.,
pull-back) associated with the relative Albanese maps ${\cal C}^{[N]}\to{\cal J}$ (see Theorem \ref{taut-thm}).
As an application of
our techniques we relate the modified diagonal classes in ${\Bbb C}H^{k-1}(C^{[k]})$ introduced by
Gross and Schoen in \cite{GS} to the pull-backs of some tautological classes on $J$ (see
Corollary \ref{pull-back-cor}).
\noindent
{\it Notations and conventions}.
Throughout this paper we work with a family $\pi:{\cal C}\to S$ of smooth projective curves of genus $g$,
where $S$ is smooth quasiprojective over a field $k$ (when we mention cohomology we assume
that $k={\Bbb C}$). We denote by ${\cal J}/S$ the corresponding relative Jacobian.
In the case when $S$ is a point we denote ${\cal C}$ (resp., ${\cal J}$) simply by $C$ (resp., $J$).
We will often use a natural product operation on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ (resp., ${\Bbb C}H^*({\cal J})$)
called the {\it Pontryagin product}. It is defined using the map
$$\alpha_{m,n}:{\cal C}^{[m]}\tildemes_S {\cal C}^{[n]}\to {\cal C}^{[m+n]}:(D_1,D_2)\mapsto D_1+D_2.$$
by the formula
$$x*y=\alpha_{m,n,*}(p_1^*x\cdotot p_2^*y)$$
for $x\in{\Bbb C}H^*({\cal C}^{[m]})$ and $y\in{\Bbb C}H^*({\cal C}^{[n]})$ (where $p_1$ and $p_2$ are the projections).
It is easy to see that this operation makes ${\Bbb C}H^*({\cal C}^{\betaullet})$ into an associative commutative algebra over ${\Bbb C}H^*(S)={\Bbb C}H^*(C^{[0]})$. The Pontryagin product on ${\Bbb C}H^*({\cal J})$ is defined similarly
using the addition map ${\cal J}\tildemes_S{\cal J}\to{\cal J}$.
For every integer $m\in{\Bbb Z}$ we denote by $[m]:{\cal J}\to{\cal J}$ the corresponding map $\xi\mapsto m\xi$.
We usually fix a point $p_0\in{\cal C}(S)$ and consider the corresponding embedding
$\iota:{\cal C}\to{\cal J}$. We denote by ${\cal L}$ the biextension on ${\cal J}\tildemes_S{\cal J}$ corresponding to the autoduality of ${\cal J}$, normalized by the condition that ${\cal L}|_{{\cal C}\tildemes_S{\cal J}}\sigmameq{\cal P}_{{\cal C}}$, where
${\cal P}_{{\cal C}}$ is the Poincar\'e line bundle on ${\cal C}\tildemes_S{\cal J}$ trivialized over $p_0$ and over
the zero section of ${\cal J}$.
We often use results from Fulton's book \cite{Fulton}.
We usually consider Chow groups
only for nonsingular varieties and use the upper grading (by codimension).
For a cartesian diagram
\betaegin{diagram}
X' &\rTo{} &Y'\\
\dTo{} & &\dTo{}\\
X &\rTo{f} & Y
\operatorname{e}nd{diagram}
where $f$ is a locally complete intersection morphism,
we denote by $f^!:{\Bbb C}H^*(Y')\to{\Bbb C}H^*(X')$ the refined Gysin map defined in section 6.6 of
\cite{Fulton}.
When we talk
about $0$-cycles we mean cycles of dimension zero and use the notation ${\Bbb C}H_0$.
Also, on one occasion
in section \ref{Jac-sec} we also use Chow homology groups ${\Bbb C}H_*$ for a possibly singular scheme.
In the relative situation we view Chow groups of an $S$-scheme as a module over ${\Bbb C}H^*(S)$.
The analogs of our results for integral cohomology are based on the formalism developed in
\cite{FM} (that includes in particular Gysin maps $f_*:H^iX\to H^{i-2d}Y$ for proper locally complete
intersection morphisms $f:X\to Y$ such that $\dim X-\dim Y=d$).
The summation variables for which no range is given are supposed to be nonnegative integers.
The symbol $x^i$ for $i<0$ in algebraic formulas should be treated as zero. We denote divided powers
of a variable $x$ by $x^{[d]}=x^d/d!$.
\section{Cycles on symmetric powers}\lambdabel{first-sec}
In this section we will prove Theorems \ref{action-thm} and \ref{curve-thm}.
We start with computing some intersection products.
We will need to work with the closed embedding
$$t_{m,N}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}\tildemes_S {\cal C}^{[N]}:(p,D)\mapsto(p,mp+D).$$
Note that its composition with the projection to the second factor is the
map $s_{m,N}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]}$ considered before.
We will denote by ${\cal D}_N\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the relative universal divisor (the image of
$t_{1,N}$).
\betaegin{lem}\lambdabel{mult-lem2}
Let us consider the cartesian diagram
\betaegin{diagram}
{\Bbb P}i_{m,M,N} &\rTo{} & {\cal C}^{[M]}\tildemes_S {\cal C}^{[N]}\\
\dTo{} & &\dTo{\alpha_{M,N}}\\
{\cal C}\tildemes_S {\cal C}^{[M+N-m]} &\rTo{s_{m,M+N}}& {\cal C}^{[M+N]}
\operatorname{e}nd{diagram}
For every decomposition $m=k+l$ we have a natural closed embedding
$$q_{k,l}:{\cal C}\tildemes_S {\cal C}^{[M-k]}\tildemes_S {\cal C}^{[N-l]}\to {\Bbb P}i_{m,M,N}:
(x,D_1,D_2)\mapsto (x,D_1+D_2,D_1+kx,D_2+lx),$$
where we view ${\Bbb P}i_{m,M,N}$ as a subset of
${\cal C}\tildemes_S {\cal C}^{[M+N-m]}\tildemes_S{\cal C}^{[M]}\tildemes_S {\cal C}^{[N]}$.
Then one has the following identity in ${\Bbb C}H^*({\Bbb P}i_{m,M,N})$:
$$\alpha_{M,N}^![{\cal C}\tildemes_S {\cal C}^{[M+N-m]}]=
\sum_{k+l=m}{m\choose k} q_{k,l,*}[{\cal C}\tildemes_S {\cal C}^{[M-k]}\tildemes_S {\cal C}^{[N-l]}].$$
\operatorname{e}nd{lem}
\noindent {\it Proof} .
We have the following commutative diagram with cartesian squares
\betaegin{diagram}
{\Bbb P}i_{m,M,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[M]}\tildemes_S {\cal C}^{[N]} &\rTo{p_{23}} &
{\cal C}^{[M]}\tildemes_S {\cal C}^{[N]}\\
\dTo{} & & \dTo{\operatorname{id}\tildemes\alpha_{M,N}} & &\dTo{\alpha_{M,N}}\\
{\cal C}\tildemes_S {\cal C}^{[M+N-m]} &\rTo{t_{m,M+N}} & {\cal C}\tildemes_S {\cal C}^{[M+N]} &\rTo{p_2}& {\cal C}^{[M+N]}
\operatorname{e}nd{diagram}
Therefore, we have
$$\alpha_{M,N}^![{\cal C}\tildemes_S {\cal C}^{[M+N-m]}]=(\operatorname{id}\tildemes\alpha_{M,N})^![{\cal C}\tildemes_S {\cal C}^{[M+N-m]}].$$
In the case $m=1$ the image of $t_{1,M+N}$ is exactly the universal divisor ${\cal D}_{M+N}$.
By definition of the map $\alpha_{M,N}$, we have
\betaegin{equation}\lambdabel{div-pullback-eq}
(\operatorname{id}\tildemes\alpha_{M,N})^*[{\cal D}_{M+N}]=p_{12}^*[{\cal D}_M]+p_{13}^*[{\cal D}_N].
\operatorname{e}nd{equation}
This implies the required formula for $m=1$. The general case follows easily by induction in $m$.
\operatorname{e}d
\betaegin{lem}\lambdabel{diag-lem}
Consider the cartesian square
\betaegin{diagram}
\Sigma_{m,N} &\rTo{} & {\cal C}\\
\dTo{} & &\dTo{{\cal D}e_N}\\
{\cal C}\tildemes_S {\cal C}^{[N-m]} &\rTo{s_{m,N}} &{\cal C}^{[N]}
\operatorname{e}nd{diagram}
where ${\cal D}e_N:{\cal C}\to {\cal C}^{[N]}$ is the relative diagonal embedding.
Note that there is a natural isomorphism $\Sigma_{m,N}\sigmameq {\cal C}$ for $m>0$,
while $\Sigma_{0,N}\sigmameq {\cal C}\tildemes_S {\cal C}$.
Then we have
$${\cal D}e_N^!([{\cal C}\tildemes_S {\cal C}^{[N]})=[{\cal C}\tildemes_S {\cal C}]\in {\Bbb C}H^0({\cal C}\tildemes_S {\cal C}), \text{ and}$$
$${\cal D}e_N^!([{\cal C}\tildemes_S {\cal C}^{[N-m]}])=(-1)^{m-1}m!{N\choose m}K^{m-1}\in{\Bbb C}H^{m-1}({\cal C})$$
for $m\ge 1$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . The case $m=0$ is clear since $s_{0,N}$
is simply the projection ${\cal C}\tildemes_S {\cal C}^{[N]}\to {\cal C}^{[N]}$. In the case $m=1$
we should compute the intersection-product for the cartesian diagram
\betaegin{diagram}
{\cal C} &\rTo{{\cal D}e} & {\cal C}\tildemes_S {\cal C}\\
\dTo{} & &\dTo{\operatorname{id}\tildemes{\cal D}e_N}\\
{\cal C}\tildemes_S {\cal C}^{[N-1]} &\rTo{t_{1,N}} &{\cal C}\tildemes_S {\cal C}^{[N]}
\operatorname{e}nd{diagram}
In other words, we have to compute the intersection of the universal divisor
${\cal D}_N\subset {\cal C}\tildemes {\cal C}^{[N]}$ with ${\cal C}\tildemes_S{\cal D}e_N({\cal C})$. We need to check that the corresponding multiplicity with the diagonal
${\cal D}e({\cal C})\subset {\cal C}\tildemes_S {\cal C}$ is equal to $N$.
This is a local problem, so we can pick a local parameter $t$ along the fibers of ${\cal C}\to S$
and think of ${\cal D}_N$ as the set of pairs $(x,f(t))$, where $f$ is a unital polynomial of degree
$N$ in $t$ such that $f(x)=0$.
The diagonal embedding ${\cal D}e_N$ sends a point $y$ to the polynomial $(t-y)^N$. Hence, the restriction of the equation $f(x)=0$ will have form $(x-y)^N$, which gives multiplicity $N$ with the diagonal ${\cal D}e\subset {\cal C}\tildemes_S {\cal C}$.
The case of $m>1$ follows by induction: from the commutative diagram with cartesian squares
\betaegin{diagram}
{\cal C} &\rTo{\operatorname{id}} & {\cal C} &\rTo{\operatorname{id}} & {\cal C}\\
\dTo{} & &\dTo{\operatorname{id}\tildemes{\cal D}e_{N-m+1}} & &\dTo{{\cal D}e_N}\\
{\cal C}\tildemes_S {\cal C}^{[N-m]} &\rTo{t_{1,N-m+1}} &{\cal C}\tildemes_S {\cal C}^{[N-m+1]}&\rTo{s_{m-1,N}}&{\cal C}^{[N]}
\operatorname{e}nd{diagram}
we see that
$${\cal D}e_N^!([{\cal C}\tildemes_S {\cal C}^{[N-m]}])=(\operatorname{id}\tildemes{\cal D}e_{N-m+1})^!([{\cal C}\tildemes_S {\cal C}^{[N-m]}]).$$
Hence, the step of induction follows from the previous computation (for $N-m+1$ instead of $N$)
along with the formula ${\cal D}e^*([{\cal D}e({\cal C})])=K$.
\operatorname{e}d
\betaegin{lem}\lambdabel{inter-mult-lem}
Consider the cartesian square
\betaegin{diagram}
Z_{m,k,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{s_{k,N}}\\
{\cal C}\tildemes_S {\cal C}^{[N-m]} &\rTo{s_{m,N}} & {\cal C}^{[N]}
\operatorname{e}nd{diagram}
We have natural closed embeddings
$$q^0:{\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}\to Z_{m,k,N}: (x,x',D)\mapsto (x,D+kx',x',D+mx),$$
$$q^i:{\cal C}\tildemes_S {\cal C}^{[N-m-k+i]}\to Z_{m,k,N}: (x,D)\mapsto (x,D+(k-i)x,x,D+(m-i)x),$$
where $1\le i\le\min(m,k)$ (we view $Z_{m,k,N}$ as a subset of
${\cal C}\tildemes_S {\cal C}^{[N-m]}\tildemes_S{\cal C}\tildemes_S {\cal C}^{[N-k]}$).
Then we have the following formula for the intersection-product in the above diagram:
$$[{\cal C}\tildemes_S {\cal C}^{[N-m]}]\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
[q^0({\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]})]+
\sum_{i\ge 1}(-1)^{i-1}i!{m\choose i}{k\choose i}q^i_*(K^{i-1}\tildemes [C^{[N-m-k+i]}]).$$
\operatorname{e}nd{lem}
\noindent {\it Proof} .
We can represent $s_{m,N}$ as the composition of ${\cal D}e_m\tildemes\operatorname{id}:{\cal C}\tildemes_S {\cal C}^{[N-m]}\to
{\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}$ followed by $\alpha_{m,N-m}:{\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[N]}$.
Therefore,
$$s_{m,N}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=({\cal D}e_m\tildemes\operatorname{id})^!\alpha_{m,N-m}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}].$$
Using Lemma \ref{mult-lem2} we obtain
$$s_{m,N}^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
\sum_{i+l=k}{k\choose i}z^{i,l}_*({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]}],$$
where for $i+l=k$ we consider a closed subset $Z^{i,l}\stackrel{z^{i,l}}{\hookrightarrow} Z=Z_{m,k,N}$
defined from the cartesian square
\betaegin{diagram}
Z^{i,l} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-m]}\\
\dTo{} & & \dTo{{\cal D}e_m\tildemes\operatorname{id}}\\
{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]} &\rTo{s^{i,l}} & {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}
\operatorname{e}nd{diagram}
with
$$s^{i,l}(x,D_1,D_2)=(D_1+ix,D_2+lx).$$
Note that $s^{i,l}$ factors into the composition of the map
$${\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-l]}\to {\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m]}$$
induced by $t_{l,N-m}$ (identical on the second factor), followed by
$$s_{i,m}\tildemes\operatorname{id}:{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m]}\to {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m]}.$$
Now we can apply Lemma \ref{diag-lem}.
For $i=0$ we immediately get $Z^{0,k}\sigmameq {\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}$ and
$$({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m]}\tildemes_S {\cal C}^{[N-m-k]}]=
[{\cal C}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-m-k]}].$$
Similarly, for $i\ge 1$ we get
$Z^{i,k-i}=q^i({\cal C}\tildemes_S {\cal C}^{[N-m-k+i]})$ and
$$({\cal D}e_m\tildemes\operatorname{id})^![{\cal C}\tildemes_S {\cal C}^{[m-i]}\tildemes_S {\cal C}^{[N-m-k+i]}]=(-1)^{i-1}i!{m\choose i}
p_1^*K^{i-1}\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-m-k+i]}].$$
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{action-thm}.}
The operator $P_{k,m}(a)$ acting on ${\Bbb C}H^*({\cal C}^{[N]})$ is given by the relative correspondence
$f_{k,m*}(p_1^*(a))$, where
$$f_{k,m}:{\cal C}\tildemes_S{\cal C}^{[N-k]}\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N-k+m]}:(p,D)\mapsto(kp+D,mp+D).$$
Therefore, to compute the correspondence inducing
the composition $P_{k,m}(a)\circ P_{k',m'}(a')$ acting on ${\Bbb C}H^*(C^{[N]})$ we have to calculate
the push-forward to ${\cal C}^{[N]}\tildemes_S{\cal C}^{[N'-k+m]}$ of
the intersection product in the following diagram
\betaegin{diagram}
{\cal C}\tildemes_S {\cal C}^{[N-k']}\tildemes_S{\cal C}^{[N'-k+m]} &&&&{\cal C}^{[N]}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N'-k]}\\
&\rdTo{f_{k',m'}\tildemes\operatorname{id}}&&\ldTo{\operatorname{id}\tildemes f_{k,m}}\\
&&{\cal C}^{[N]}\tildemes_S{\cal C}^{[N']}\tildemes_S{\cal C}^{[N'-k+m]}
\operatorname{e}nd{diagram}
multiplied with the pullbacks of $b$ and $a$, where $N'=N-k'+m'$,
It is easy to see that this intersection-product is exactly the one computed in
Lemma \ref{inter-mult-lem} for $Z_{m',k,N'}\subsetset {\cal C}\tildemes_S{\cal C}^{[N-k']}\tildemes_S
{\cal C}\tildemes_S{\cal C}^{[N'-k]}$.
It follows that the above composition is given by the relative correspondence
$$(s_{k',N}\tildemes s_{m,N'-k+m})_*(w\cdotot p_1^*a'\cdotot p_3^*a)\in{\Bbb C}H^*({\cal C}^{[N]}\tildemes_S {\cal C}^{[N'-k+m]}),$$
where $p_1, p_3: Z_{m',k,N'}\to {\cal C}$ are the projections, and
$w\in{\Bbb C}H^*(Z_{m',k,N'})$ is the intersection-product computed in Lemma \ref{inter-mult-lem}.
When we substitute the formula for $w$ in the above equation and
subtract the similar expression for $P_{k',m'}(a')\circ P_{k,m}(a)$ we note that the first
terms (corresponding to the images of $q^0$) will cancel out due to the symmetry exchanging
the two factors ${\cal C}$. The remaining terms will give the required formula for the commutator.
In the case of cohomology we have to work with the supercommutator since switching the order
of $a$ and $a'$ will introduce the standard sign.
\operatorname{e}d
\betaegin{cor}\lambdabel{Heis-K-cor} The operators $(P_{1,0}(a))$ and $(P_{0,1}(a))$ on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ satisfy the following relations:
\betaegin{align*}
&[P_{1,0}(a),P_{1,0}(a')]=[P_{0,1}(a),P_{0,1}(a')]=0,\\
&[P_{0,1}(a),P_{1,0}(a')]=\lambdangle a,a'\rightarrowngle\cdotot\operatorname{id},
\operatorname{e}nd{align*}
where $\lambdangle a,a'\rightarrowngle=\pi_*(a\cdotot a')\in{\Bbb C}H^*(S)$
(recall that we view ${\Bbb C}H^*({\cal C}^{[N]})$ as a ${\Bbb C}H^*(S)$-module
using the product with the pull-back under the projection ${\cal C}^{[N]}\to S$).
In particular, if we are given a pair of divisor classes $\alpha,\beta\in{\Bbb C}H^1({\cal C})$ of nonzero relative degrees
$\deltag(\alpha)$ and $\deltag(\beta)$ then there is an action of the algebra
${\cal D}_{t,u,{\Bbb Q}}={\Bbb Q}[t,u,\partial_t,\partial_u]$ of differential operators in two variables on
${\Bbb C}H^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$ such that
\betaegin{align*}
&t\mapsto P_{1,0}(\alpha)-\frac{\lambdangle\alpha,\beta\rightarrowngle}{\deltag(\beta)}\cdotot P_{1,0}({\cal C}) \ \
(\text{Pontryagin product with }\alpha-\frac{\lambdangle\alpha,\beta\rightarrowngle}{\deltag(\beta)}\cdotot[{\cal C}])\\
&u\mapsto \frac{1}{\deltag(\beta)}P_{1,0}({\cal C}) \ \ (\text{Pontryagin product with } \frac{1}{\deltag(\beta)}[{\cal C}]),\\
&\partial_t\mapsto \frac{1}{\deltag(\alpha)}P_{0,1}({\cal C}),\\
&\partial_u\mapsto P_{0,1}(\beta).
\operatorname{e}nd{align*}
For example, for $g\neq 1$ we can take $\alpha=\beta=K$.
\operatorname{e}nd{cor}
\noindent {\it Proof} . This follows from the relations of Theorem \ref{action-thm} together with the identity
$P_{0,0}(a)=\pi_*(a)\cdotot\operatorname{id}$.
\operatorname{e}d
\betaegin{rem} In the case $S=\operatorname{Spec}({\Bbb C})$
the cohomology $H^*(C^{[\betaullet]},{\Bbb Q})$ can be identified with the
super-symmetric algebra of $H^*(C,{\Bbb Q})$. Then the operators
$P_{1,0}(a)$ and $P_{0,1}(a)$ for $a\in H^*(C,{\Bbb Q})$ are identified with the standard operators
on the super-symmetric algebra (products and contractions).
\operatorname{e}nd{rem}
\noindent
{\it Proof of Theorem \ref{curve-thm}.}
(i) Consider the Abel-Jacobi map $S:{\Bbb C}H_0(J)\to J(k): \sum m_i[a_i]\mapsto \sum m_ia_i$.
It is well known and easy to see that $S$ induces an isomorphism $I/I^{*2}\widetilde{\to} J(k)$
(see sec.0 of \cite{Bl}).
Since the composition ${\Bbb C}H_0(C)\stackrel{\iota_*}{\to}{\Bbb C}H_0(J)\stackrel{S}{\to} J(k)$ is the Abel-Jacobi
map for $C$ that induces an isomorphism of degree zero cycles with $J(k)$, we obtain
a decomposition
\betaegin{equation}\lambdabel{I-eq}
I=I_C\oplus I^{*2}.
\operatorname{e}nd{equation}
By taking the Pontryagin powers we immediately derive that
$$I^{*n}=I_C^{*n}+I^{*(n+1)}.$$
Since $I^{*(g+1)}=0$ by the result of Bloch (see \cite{Bl}), we deduce that
$${\Bbb C}H_0(J)={\Bbb Z}[0]+I_C+I_C^{*2}+\ldots+I_C^{*g}.$$
It remains to prove that this decomposition is direct, i.e., the summands are linearly independent.
The version of Roitman's theorem in
arbitrary characterstic proved by Milne~\cite{Milne} implies that $I^{*2}\subset\operatorname{ker}(S)$ has no torsion.
In view of \operatorname{e}qref{I-eq}, this shows that it is enough to prove our statement after tensoring with
${\Bbb Q}$. A more direct way of getting our decomposition over ${\Bbb Z}$ will be outlined in Remark 3 after
Corollary \ref{module-cor}.
Let us consider the morphisms $C^{[N]}\to J$ induced by $\iota:C\to J$.
It is easy to see that
the induced push-forward map $\sigma_*:{\Bbb C}H_0(C^{[\betaullet]})\to{\Bbb C}H_0(J)$ is compatible with the
Pontryagin products. Also, since $\iota(p_0)=0\in J(k)$, we have $\sigma_*(x*[p_0])=\sigma_*(x)$ for
any $x\in{\Bbb C}H_0(C^{[\betaullet]})$. Let $A_0(C)\subset{\Bbb C}H_0(C)$ denote the subgroup of classes of degree zero. Note that $\iota_*(A_0(C))=I_C$ and the push-forward
map $\sigma_*:{\Bbb C}H_0(C^{[g]})\to{\Bbb C}H_0(J)$ is an isomorphism (since $C^{[g]}\to J$ is birational).
Therefore, it suffices to establish the direct sum decomposition
\betaegin{equation}\lambdabel{AC-dec-eq}
{\Bbb C}H_0(C^{[g]})_{{\Bbb Q}}=
{\Bbb Q}\cdotot [p_0]^{*g}\oplus A_0(C)_{{\Bbb Q}}*[p_0]^{*(g-1)}\oplus \ldots \oplus A_0(C)_{{\Bbb Q}}^{*(g-1)} *[p_0]\oplus
A_0(C)_{{\Bbb Q}}^{*g},
\operatorname{e}nd{equation}
where the Pontryagin products are taken in ${\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}$.
To this end we will use the action of the algebra of differential operators ${\cal D}_{t,{\Bbb Q}}$ in one variable
on ${\Bbb C}H^*(C^{[\betaullet]})_{{\Bbb Q}}$ given by $t\mapsto P_{1,0}([p_0])$,
$\frac{d}{dt}\mapsto P_{0,1}(C)$ (see Corollary \ref{Heis-K-cor}). Note that this
action preserves the subspace of $0$-cycles ${\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}$.
Since $\frac{d}{dt}$ acts locally nilpotently, we have a natural isomorphism of ${\cal D}_{t,{\Bbb Q}}$-modules
\betaegin{equation}\lambdabel{D-mod-eq}
{\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}\sigmameq K_{{\Bbb Q}}[t],
\operatorname{e}nd{equation}
where $K=\operatorname{ker}(P_{0,1}(C))\cap{\Bbb C}H_0(C^{[\betaullet]})$.
Furthermore, this isomorphism is compatible with gradings, where the grading on the right-hand side
is induced by the grading of $K$ and the rule $\deltag(t)=1$.
Next, we observe that for every $a\in{\Bbb C}H_0(C)$ we have the relation
$$[P_{0,1}(C),P_{1,0}(a)]=\deltag(a)\cdotot\operatorname{id}.$$
Hence, $P_{0,1}(C)$ commutes with the Pontryagin product with any $0$-cycle of degree zero on $C$.
Thus, we obtain
$$A_0(C)^{*n}\subset K_n=\operatorname{ker}(P_{0,1}(C))\cap{\Bbb C}H_0(C^{[n]}) \text{ for }n\ge 1.$$
On the other hand, the algebra ${\Bbb C}H_0(C^{\betaullet})_{{\Bbb Q}}$ is generated over ${\Bbb Q}={\Bbb C}H_0(C^{[0]})_{{\Bbb Q}}$
by ${\Bbb C}H_0(C)_{{\Bbb Q}}=A_0(C)_{{\Bbb Q}}\oplus{\Bbb Q}\cdotot [p_0]$.
Therefore, the natural map
$$\betaigoplus_{n\ge 0}A_0(C)_{{\Bbb Q}}^{*n}[t]\to{\Bbb C}H_0(C^{[\betaullet]})_{{\Bbb Q}}$$
is an isomorphism (where we set $A_0(C)_{{\Bbb Q}}^{*0}={\Bbb Q}$).
Looking at the grading component of degree $g$ we get the decomposition \operatorname{e}qref{AC-dec-eq}.
\noindent
(ii) Let us set $A={\Bbb Z}\oplus A_0(C)\oplus A_0(C)^{*2}\oplus\ldots\oplus A_0(C)^{*g}$ (from (i)
we know that this algebra is isomorphic to ${\Bbb C}H_0(J)$) and consider
the natural homomorphism of algebras
\betaegin{equation}\lambdabel{A-hom-eq}
A[t]\to {\Bbb C}H_0(C^{[\betaullet]})
\operatorname{e}nd{equation}
as in part (i). We claim that it is an isomorphism. Indeed, it is surjective, since ${\Bbb C}H_0(C^{[\betaullet]})$
is generated by $[p_0]$ and by $A_0(C)$ as an algebra over ${\Bbb Z}$. Also, from part (i) we know that
\operatorname{e}qref{A-hom-eq} is injective modulo torsion. It remains to show that it is injective
on the torsion subgroup. But the torsion in $A$ is contained in $A_0(C)$ (by Roitman's theorem), so the statement boils down to the fact that the natural map $A_0(C)t^N\to {\Bbb C}H_0(C^{[N]})$ is an embedding.
But this follows from the fact that its composition with the Abel-Jacobi map to $J(k)$ is an isomorphism.
Note that as in part (i) we could have avoided referring to Roitman's theorem and used
the divided powers instead (see Remark 3 after Corollary \ref{module-cor}).
Thus, we can view $(P_{m,1}(C))$ as operators on $A[t]$. For example, $P_{0,1}(C)$ acts by
$\frac{d}{dt}$.
For $m\ge 1$ and $a\in{\Bbb C}H_0(C)$ we have the relation
$$[P_{m,1}(C),P_{1,0}(a)]=P_{m,0}(a).$$
This implies that for every point $p\in C(k)$ one has
$$P_{m,1}(C)([p]*x)=[p]*P_{m,1}(C)(x)+[p]^{*m}*x.$$
Since the classes $[p]$ generate our algebra, this implies that $P_{m,1}(C)$ is a derivation
of $A[t]$ characterized by
$$P_{m,1}(C)([p])=[p]^{*m} \text{ for all }p\in C(k).$$
Setting $x_p=[p]-[p_0]\in A_0(C)$ we derive that $P_{m,1}(C)(t)=t^m$ and
$P_{m,1}(C)(x_p)=(x_p+t)^m-t^m$. Now let us define $\delta_m$ as the following composition
$$A\to A[t]\stackrel{P_{m,1}(C)}{\rightarrow} A[t]\to A,$$
where the first map is the natural embedding and the last map is the evaluation at $t=0$.
Then $\delta_m$ is a derivation of $A$ with the property $\delta_m(x_p)=x_p^m$ for all $p\in C(k)$.
The commutation relations for $\delta_m$ are easily checked on the generators $x_p$.
The formula for $\delta_m|_{I_C}$ follows from the simple identity
$$([a]-[0])^{*m}=\sum_{i=0}^{m-1}(-1)^i{m\choose i}[m-i]_*([a]-[0])$$
in ${\Bbb C}H_0(J)$, where $m\ge 1$, $a\in J(k)$.
\operatorname{e}d
\betaegin{rems} 1. Note that $\delta_1$ is just the grading derivation: it is equal to $n\operatorname{id}$ on the
grading component of degree $n$.
It is easy to see that under the identification \operatorname{e}qref{A-hom-eq} the operators
$P_{m,1}(C)$ are given by
$$P_{m,1}(C)=\delta_m+mt\delta_{m-1}+{m\choose 2}t^2\delta_{m-2}+\ldots+mt^{m-1}\delta_1+t^m\frac{d}{dt},$$
where $\delta_m$ are extended to operators on $A[t]$ commuting with $t$.
\noindent
2. It is natural to compare our decomposition of ${\Bbb C}H_0(J)={\Bbb C}H^g(J)$ (tensored with ${\Bbb Q}$)
with the Beauville's decomposition
$${\Bbb C}H^g(J)_{{\Bbb Q}}=\betaigoplus_{s=0}^g{\Bbb C}H^g_s(J),$$
where ${\Bbb C}H^g_s(J)\subset{\Bbb C}H^g(J)_{{\Bbb Q}}$ is characterized by the condition $x\in{\Bbb C}H^g_s(J)$ if and only
if $[m]_*x=m^s x$ for all $m\in{\Bbb Z}$. The corresponding filtrations
$$\betaigoplus_{s\ge n}{\Bbb C}H^g_s(J)=I^{*n}=\betaigoplus_{s\ge n}I_C^*s$$
are the same (see \cite{B2}). However, the decompositions themselves are different.
Indeed, if they were the same we would have
$(I_C)_{{\Bbb Q}}\subset{\Bbb C}H^g_1(J)$ which would imply that $[2]_*([\iota(p)]-[0])=2[\iota(p)]-2[0]$ in
${\Bbb C}H^g(J)_{{\Bbb Q}}$ for all $p\in C$. But this would mean that $([\iota(p)]-[0])^{*2}=0$ in ${\Bbb C}H^g(J)$, hence
$(p,p)-(p,p_0)-(p_0,p)+(p_0,p_0)$
is a torsion class in ${\Bbb C}H_0(C\tildemes C)$ for all $p$, which is known not to be the case for a general
curve of genus $g\ge 3$ (see \cite{BV}, Prop. 3.2).
\operatorname{e}nd{rems}
\section{Connection with cycles on the relative Jacobian}
\lambdabel{Jac-sec}
Assume that our family $\pi:{\cal C}\to S$ is equipped with a section
$p_0:S\to{\cal C}$, and let $\sigma_N:{\cal C}^{[N]}\to {\cal J}$
denote the corresponding map to the relative Jacobian of ${\cal C}$
sending a divisor $D\in{\cal C}_s^{[N]}$ to the class of the line bundle ${\cal O}_{{\cal C}_s}(D-Np_0)$.
We normalize the relative Poincar\'e line bundle ${\cal P}_{{\cal C}}$ on ${\cal C}\tildemes_S {\cal J}$ so
that its pull-backs under $p_0\tildemes\operatorname{id}_{{\cal J}}$ and under $\operatorname{id}_{{\cal C}}\tildemes e$ are trivial, where
$e:S\to{\cal J}$ is the zero section. Then we have the following equality in ${\Bbb C}H^1({\cal C}\tildemes_S{\cal C}^{[N]})$:
\betaegin{equation}\lambdabel{pull-back-P-b-eq}
(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})=[{\cal D}_N]-Np_1^*[p_0]-p_2^*([{\cal R}_N])-N\psi,
\operatorname{e}nd{equation}
where ${\cal D}_N\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ is
the universal divisor,
$${\cal R}_N=(p_0\tildemes\operatorname{id})^{-1}({\cal D}_N)=s_{1,N-1}(p_0\tildemes \operatorname{id})({\cal C}^{[N-1]})\subset {\cal C}^{[N]}$$
is the divisor in ${\cal C}^{[N]}$ associated with $p_0$, and
$$\psi=p_0^*K\in{\Bbb C}H^1(S)$$
(we view $\psi$ as a divisor class on any $S$-scheme via the pull-back).
We are going to introduce a new family of operators on ${\Bbb C}H^*({\cal J})$ and on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ that are compatible with respect to the push-forward map
$\sigma_*:{\Bbb C}H^*({\cal C}^{[\betaullet]})\to{\Bbb C}H^*({\cal J})$ (that has $\sigma_{N*}$ as components).
It is convenient to consider a more general setup.
Let ${\cal X}_{\betaullet}=(\sigma_N:{\cal X}_N\to{\cal J})_{N\in{\Bbb Z}}$ be a family of proper ${\cal J}$-schemes equipped with
a collection of morphisms
$$s_N:{\cal C}\tildemes_S{\cal X}_{N-1}\to {\cal X}_N,$$
where $N\in{\Bbb Z}$, such that
\noindent
(i) the diagram
\betaegin{equation}
\betaegin{diagram}
{\cal C}\tildemes_S{\cal X}_{N-1} & \rTo{s_N} & {\cal X}_N\\
\dTo{\operatorname{id}\tildemes\sigma_{N-1}} & &\dTo{\sigma_N}\\
{\cal C}\tildemes_S{\cal J} &\rTo{s_N}& {\cal J}
\operatorname{e}nd{diagram}
\operatorname{e}nd{equation}
is commutative, where the lower horizontal arrow is induced
by the map $\iota=\sigma_1:{\cal C}\to{\cal J}$ and by the group law on the Jacobian;
\noindent
(ii) for each $N$ the map ${\cal C}^m\tildemes_S{\cal X}_{N-m}\to{\cal X}_N$
obtained from $(s_N)$ by iteration (where ${\cal C}^m$ is the $m$th cartesian power of ${\cal C}/S$),
factors through a map
${\cal C}^{[m]}\tildemes_S{\cal X}_{N-m}\to{\cal X}_N$.
Two main examples of the above situations are: ${\cal X}_N={\cal C}^{[N]}$ for $N\ge 0$
(where $\sigma_N:{\cal C}^{[N]}\to{\cal J}$ are associated with a point $p_0\in{\cal C}(S)$, and ${\cal X}_N=\operatorname{e}mptyset$ for
$N<0$) and ${\cal X}_N={\cal J}$ for all $N\in{\Bbb Z}$.
Another (singular) example is obtained by taking ${\cal X}_N$
to be the image of the map ${\cal C}^{[N]}\to{\cal J}$.
Restricting the above maps ${\cal C}^m\tildemes_S{\cal X}_{N-n}\to{\cal X}_N$ to the diagonal in ${\cal C}^m$
we get morphisms
$$s_{m,N}:{\cal C}\tildemes_S{\cal X}_{N-m}\to{\cal X}_N.$$
Now, let us define the operator $T_k(m,a)$ on
${\Bbb C}H_*({\cal X}_{\betaullet})=\betaigoplus_{N\in{\Bbb Z}}{\Bbb C}H_*({\cal X}_N)$, where $k\ge 0$, $m\ge 0$,
$a\in{\Bbb C}H^*({\cal C})$, by the formula
$$T_k(m,a)(x)=s_{m,m+N,*}((\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})^k\cdotot p_1^*a\cdotot p_2^*x),$$
where $x\in{\Bbb C}H_*({\cal X}_N)$, $p_1$ and $p_2$ are projections from the product
${\cal C}\tildemes_S {\cal X}_N$ to its factors. In the case $a=[{\cal C}]$ we will simply write $T_k(m,{\cal C})$.
Note that from the projection formula we get
\betaegin{equation}\lambdabel{T-k-0-eq}
T_k(0,a)(x)=\sigma_N^*\tau_k(a)\cdotot x \ \text{ for }x\in{\Bbb C}H_*({\cal X}_N),
\operatorname{e}nd{equation}
where for $a\in{\Bbb C}H^*({\cal C})$ and $k\ge 0$ we set
\betaegin{equation}\lambdabel{tau-eq}
\tau_k(a)=p_{2*}(c_1({\cal P}_{{\cal C}})^k\cdotot p_1^*a)\in{\Bbb C}H^*({\cal J}).
\operatorname{e}nd{equation}
Also, if $(f:{\cal X}_N\to{\cal Y}_N)_{N\in{\Bbb Z}}$ is a morphism of two families as above then
it follows immediately from the definition that the above operators commute with the
push-forward map $f_*:{\Bbb C}H_*({\cal X}_{\betaullet})\to{\Bbb C}H_*({\cal Y}_{\betaullet})$, i.e.,
$$T_k(m,a)\circ f_*=f_*\circ T_k(m,a).$$
\betaegin{thm}\lambdabel{relations-thm}
One has the following relations between operators on ${\Bbb C}H_*({\cal X}_{\betaullet})$:
\betaegin{align*}
&\sum_{i\ge 0}\psi^i\cdotot\left({k\choose i}m^{\operatorname{pr}ime i}
T_{k-i}(m,a)T_{k'}(m',a')-{k'\choose i}m^i T_{k'-i}(m',a')T_k(m,a)\right)=\\
&\sum_{i\ge 1}(-1)^{i-1}
\left({k\choose i}m^{\operatorname{pr}ime i}-{k'\choose i}m^i\right)T_{k+k'-i}(m+m',a\cdotot a'\cdotot
(K+2[p_0(S)])^{i-1})+\\
&\psi^{k'-1}m^{k'}p_0^*(a')T_k(m,a)-\psi^{k-1}m^{\operatorname{pr}ime k}p_0^*(a)T_{k'}(m',a')+\\
&\delta_{k,0}\cdotot\sum_{i\ge 1}{k'\choose i}m^i\psi^{i-1}p_0^*(a)T_{k'-i}(m',a')
-\delta_{k',0}\cdotot\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a') T_{k-i}(m,a)
\operatorname{e}nd{align*}
where $a,a'\in{\Bbb C}H^*({\cal C})$, $k\ge 0$, $k'\ge 0$, $m\ge 0$, $m'\ge 0$.
\operatorname{e}nd{thm}
Let us set $\operatorname{e}ll_N=(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_C)\in{\Bbb C}H^1({\cal C}\tildemes_S {\cal X}_N)$.
Also let us denote $\mu=\operatorname{e}ll_1=(\operatorname{id}\tildemes\sigma_1)^*c_1({\cal P}_C)\in{\Bbb C}H^1({\cal C}\tildemes_S{\cal C})$.
Recall that ${\cal P}_C$ is the pull-back of the biextension ${\cal L}$ of ${\cal J}\tildemes_S{\cal J}$
under the embedding $(\iota\tildemes\operatorname{id}):{\cal C}\tildemes_S {\cal J}\to {\cal J}\tildemes_S {\cal J}$
corresponding to point $p_0$. This implies the following isomorphism in
$CH^1({\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_{N-m})$:
\betaegin{equation}\lambdabel{biext-eq}
(\operatorname{id}_{{\cal C}}\tildemes s_{m,N})^*\operatorname{e}ll_N=m\cdotot p_{12}^*\mu+p_{13}^*\operatorname{e}ll_{N-m}.
\operatorname{e}nd{equation}
\betaegin{lem}\lambdabel{diag-lem2}
One has the following identity in ${\Bbb C}H^*({\cal C}\tildemes_S{\cal C})$ for $n\ge 1$:
$$\mu^n=(-\psi)^n+(-1)^n\psi^{n-1}\cdotot
\left((p_0\tildemes\operatorname{id})_*[{\cal C}]+(\operatorname{id}\tildemes p_0)_*[{\cal C}]\right)+(-1)^{n-1}
\sum_{i\ge 1}{n\choose i}\psi^{n-i}\cdotot{\cal D}e_*(K+2[p_0(S)])^{i-1}.$$
\operatorname{e}nd{lem}
\noindent {\it Proof} . In the case $N=1$ the equality \operatorname{e}qref{pull-back-P-b-eq} gives
$$\mu={\cal D}e_*[{\cal C}]-(p_0\tildemes\operatorname{id})_*[{\cal C}]-(\operatorname{id}\tildemes p_0)_*[{\cal C}]-\psi\cdotot [{\cal C}\tildemes_S{\cal C}],$$
where ${\cal D}e:{\cal C}\to{\cal C}\tildemes_S{\cal C}$ is the diagonal. Now the required identity is easily proved by induction in $n$ using the equalities
$${\cal D}e^*\left({\cal D}e_*[{\cal C}]-(p_0\tildemes\operatorname{id})_*[{\cal C}]-(\operatorname{id}\tildemes p_0)_*[{\cal C}]\right)=-(K+2[p_0(S)]),$$
$$\mu\cdotot (p_0\tildemes\operatorname{id})_*[{\cal C}]=\mu\cdotot (\operatorname{id}\tildemes p_0)_*[{\cal C}]=0.$$
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{relations-thm}.}
The composition $T_k(m,a)\circ T_{k'}(m',a')$ acting on ${\Bbb C}H_*({\cal X}_N)$ is given by
the operator
$$x\mapsto s_{m,m',*}\left((\operatorname{id}_{{\cal C}}\tildemes s_{m',m'+N})^*\operatorname{e}ll_{m'+N}^{k}
p_{23}^*\operatorname{e}ll_N^{k'}\cdotot p_1^*(a)\cdotot p_2^*(a')\cdotot p_3^*(x)\right),$$
where $p_i$ ($i=1,2,3$) are the projections from the product ${\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_N$ to its factors, and $s_{m,m'}$ denotes the following composition
$$
\betaegin{diagram}
{\cal C}\tildemes_S{\cal C}\tildemes_S{\cal X}_N &\rTo{\operatorname{id}_{{\cal C}}\tildemes s_{m',m'+N}}&{\cal C}\tildemes_S{\cal X}_{m'+N}
&\rTo{s_{m,m+m'+N}}&{\cal X}_{m+m'+N}.
\operatorname{e}nd{diagram}
$$
From \operatorname{e}qref{biext-eq} we get
\betaegin{equation}\lambdabel{biext-pow-eq}
(\operatorname{id}_{{\cal C}}\tildemes s_{m',m'+N})^*\operatorname{e}ll_{m'+N}^k=\sum_i {k\choose i}m^{\operatorname{pr}ime i} p_{12}^*\mu^i
\cdotot p_{13}^*\operatorname{e}ll_N^{k-i}.
\operatorname{e}nd{equation}
Let us set
$$S_{k,k';m,m'}(a,a')(x)=s_{m,m',*}\left(p_{12}^*\operatorname{e}ll_N^k\cdotot p_{23}^*\operatorname{e}ll_N^{k'}\cdotot p_1^*(a)\cdotot p_2^*(a')\cdotot p_3^*(x)\right).$$
Then using Lemma \ref{diag-lem2} and \operatorname{e}qref{biext-pow-eq} we derive
\betaegin{align*}
&T_k(m,a)\circ T_{k'}(m',a')=
\sum_{n\ge 0}(-m'\psi)^n{k\choose n}S_{k-n,k';m,m'}(a,a')+\\
&\sum_{n\ge 1,i\ge 1}(-1)^{n-1}m^{\operatorname{pr}ime n}\psi^{n-i}{k\choose n}{n\choose i}
T_{k+k'-n}(m+m',aa'(K+2[p_0(S)])^{i-1})+\\
&(-1)^km^{\operatorname{pr}ime k}\psi^{k-1}p_0^*(a)\cdotot T_{k'}(m',a')+
\delta_{k',0}\sum_{n\ge 1}(-1)^nm^{\operatorname{pr}ime n}\psi^{n-1}{k\choose n}p_0^*(a')\cdotot T_{k-n}(m,a).
\operatorname{e}nd{align*}
From this we deduce that
\betaegin{align*}
&\sum_{p\ge 0}{k\choose p}(m'\psi)^p T_{k-p}(m,a)T_{k'}(m',a')=S_{k,k';m,m'}(a,a')+\\
&\sum_{i\ge 1}(-1)^{i-1}{k\choose i}m^{\operatorname{pr}ime i}T_{k+k'-i}(m+m',aa'(K+2[p_0(S)])^{i-1})
-m^{\operatorname{pr}ime k}\psi^{k-1}p_0^*(a)T_{k'}(m',a')\\
&-\delta_{k',0}\cdotot\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a')T_{k-i}(m,a).
\operatorname{e}nd{align*}
It remains to observe that condition (ii) imposed on $({\cal X}_N)$ implies that
$$S_{k,k';m,m'}(a,a')=S_{k',k;m',m}(a',a).$$
Expressing both sides of this equality in terms of the operators $(T_k(m,a))$ we get the required
relation.
\operatorname{e}d
\betaegin{rems} 1. In the case of ${\cal X}_N={\cal J}$ the definition of the operators $T_k(m,a)$
can be extended to the case of arbitrary $m\in{\Bbb Z}$, so that the relations of Theorem \ref{relations-thm}
still hold. Namely, we define the morphism $s_{m,N}$ for $m\in{\Bbb Z}$, so that it maps $(p,\xi)\in{\cal C}_s\tildemes{\cal J}_s$ to $m\iota(p)+\xi\in{\cal J}_s$.
\noindent
2. Similar relation holds if we replace Chow groups with cohomology
provided we insert the standard sign $(-1)^{\deltag(a)\deltag(a')}$ whenever $a'$ goes before $a$.
\operatorname{e}nd{rems}
We are mostly interested in two cases: $({\cal X}_N={\cal C}^{[N]})$ and $({\cal X}_N={\cal J})$.
First, let us consider the case $({\cal X}_N={\cal C}^{[N]})$. In this case we will deduce the relation
between the operators $(T_k(m,a))$ and $(P_{i,j}(a))$. We need one auxiliary result for this.
Let us denote by $Z_{m,N}\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the image of the closed embedding $t_{m,N}$.
In other words, $Z_{m,N}$
consists of $(p,D)\in{\cal C}_s\tildemes{\cal C}_s^{[N]}$ such that $D-mp\ge 0$. Note that
${\cal D}_N:=Z_{1,N}$ is the universal divisor in ${\cal C}\tildemes_S {\cal C}^{[N]}$.
\betaegin{lem}\lambdabel{Z-lem}
One has the following equality in ${\Bbb C}H^m({\cal C}\tildemes_S {\cal C}^{[N]})$:
$$[Z_{m,N}]=[{\cal D}_N]\cdotot([{\cal D}_N]+p_1^*K)\cdotot\ldots\cdotot([{\cal D}_N]+(m-1)p_1^*K).$$
where $p_1:{\cal C}\tildemes_S {\cal C}^{[N]}\to {\cal C}$ is the projection.
Equivalently,
$$[{\cal D}_N]^m=\sum_{i=0}^m (-1)^{m-i}S(m,i)\cdotot p_1^*K^{m-i}\cdotot [Z_{i,N}],$$
where $S(m,i)=\frac{1}{i!}\sum_{j=0}^i (-1)^j{i\choose j}(i-j)^m$ are the Stirling numbers of
the second kind.
\operatorname{e}nd{lem}
\noindent {\it Proof} . In the case $m=1$ the equality is clear. The general case follows by induction in $m$ using the
identity
\betaegin{equation}\lambdabel{D-res-eq}
[{\cal D}_N]=t_{1,N+1}^*({\cal D}_{N+1}+p_1^*K)
\operatorname{e}nd{equation}
in ${\Bbb C}H^1({\cal C}\tildemes_S {\cal C}^{[N]})$. To obtain this identity one can start with the equality
\operatorname{e}qref{div-pullback-eq} (for $M=1$) in ${\Bbb C}H^1({\cal C}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N]})$ and then
apply the pull-back with respect to the diagonal embedding
${\cal D}e\tildemes\operatorname{id}:{\cal C}\tildemes_S{\cal C}^{[N]}\to{\cal C}\tildemes_S{\cal C}\tildemes_S{\cal C}^{[N]}$
(when the base is a point this was observed in Proposition 19.1 of \cite{P-av}).
\operatorname{e}d
\betaegin{prop}\lambdabel{T-P-prop}
One has the following equality of operators on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$:
\betaegin{align*}
&T_k(m,a)=(-1)^kP_{m,0}(a\cdotot [p_0])P_{1,1}([{\cal C}])^k\psi^{k-1}+\\
&\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)
P_{i+m,i}(a\cdotot K^n)P_{1,1}([p_0]+\psi)^j.
\operatorname{e}nd{align*}
\operatorname{e}nd{prop}
\noindent {\it Proof} . Since $[Z_{k,N}]=t_{k,N,*}([{\cal C}\tildemes_S {\cal C}^{[N-k]}])$,
we can rewrite $P_{k+m,k}(a)$ in the form similar to
that of $T_{k}(m,a)$:
\betaegin{equation}\lambdabel{P-k-m-Z-eq}
P_{k+m,k}(a)(x)=s_{m,N,*}([Z_{k,N}]\cdotot p_1^*a\cdotot p_2^*x),
\operatorname{e}nd{equation}
where $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
Let us use the following shorthand notation for divisors on ${\cal C}\tildemes_S{\cal C}^{[N]}$:
${\cal D}={\cal D}_N$, ${\cal R}=p_2^*{\cal R}_N$, $K=p_1^*K$,
$[p_0]=p_1^*[p_0]$. Then we can write \operatorname{e}qref{pull-back-P-b-eq}
as
$$(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})={\cal D}-{\cal R}-N[p_0]-N\psi.$$
Note that we have the following relations in ${\Bbb C}H^2({\cal C}\tildemes_S{\cal C}^{[N]})$:
$$({\cal D}-{\cal R})\cdotot[p_0]=0, \ \ [p_0]^2=-\psi\cdotot[p_0].$$
It follows that for $j\ge 1$ one has $({\cal D}-{\cal R}-N\psi)^i\cdotot [p_0]^j=N^i(-\psi)^{i+j-1}\cdotot [p_0]$,
so we derive
$$(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})^k=({\cal D}-{\cal R}-N([p_0]+\psi))^k=({\cal D}-{\cal R}-N\psi)^k+(-N)^k[p_0]\psi^{k-1}.
$$
Therefore, using Lemma \ref{Z-lem} we find
$$(\operatorname{id}\tildemes\sigma_N)^*c_1({\cal P}_{{\cal C}})^k=(-N)^k[p_0]\psi^{k-1}+
\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)[Z_{i,N}]\cdotot K^n\cdotot ({\cal R}+N\psi)^j.$$
Taking into the account \operatorname{e}qref{P-k-m-Z-eq}, we get
\betaegin{align*}
&T_k(m,a)(x)=(-N)^kP_{m,0}(a\cdotot[p_0])(x)\cdotot\psi^{k-1}+\\
&\sum_{i+n+j=k}(-1)^{n+j}{k\choose j}S(i+n,i)
P_{i+m,i}(a\cdotot K^n)(({\cal R}+N\psi)^j\cdotot x),
\operatorname{e}nd{align*}
where $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
It remains to use the equalities
$$P_{1,1}([{\cal C}])(x)=Nx,$$
$$P_{1,1}([p_0])(x)={\cal R}\cdotot x$$
for $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
\operatorname{e}d
Now let us specialize to the case ${\cal X}_N={\cal J}$.
In this case we can relate the operators $T_k(m,a)$
to the operators considered in \cite{P-lie} (in the case $S=\operatorname{Spec}(k)$).
Following \cite{P-lie} let us define the operators $A_k(\alpha)$ on ${\Bbb C}H^*({\cal J})$ for $\alpha\in{\Bbb C}H^*({\cal J})$
and $k\ge 0$ by
$$A_k(\alpha)(x)=(p_1+p_2)_*(c_1({\cal L})^k\cdotot p_1^*\alpha\cdotot p_2^*x),$$
where $p_1$ and $p_2$ are projections from the product ${\cal J}\tildemes_S {\cal J}$ to its factors.
\betaegin{lem}\lambdabel{A-T-lem1}
For $a\in{\Bbb C}H^*({\cal C})$ one has
$A_k([m]_*\iota_*a)=m^k T_k(m,a)$ (with the convention that $0^0=1$),
where $[m]:{\cal J}\to {\cal J}:\xi\to m\xi$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . We have
\betaegin{align*}
&A_k([m]_*\iota_*a)=(mp_1+p_2)_*\left(([m]\tildemes\operatorname{id})^*c_1({\cal L})^k\cdotot p_1^*(\iota_*a)\cdotot p_2^*x\right)=\\
&m^k(mp_1+p_2)_*\left(c_1({\cal L})^k\cdotot (\iota\tildemes\operatorname{id}_{{\cal J}})_*(p_1^*a)\cdotot p_2^*x\right).
\operatorname{e}nd{align*}
Now the result follows immediately from the isomorphism ${\cal L}|_{{\cal C}\tildemes_S {\cal J}}\sigmameq{\cal P}_{{\cal C}}$.
\operatorname{e}d
Working with rational coefficients we can
consider the decomposition
$${\Bbb C}H^*({\cal J})_{{\Bbb Q}}=\betaigoplus_{i=0}^{2g}{\Bbb C}H^*({\cal J})_i,$$
where $[m]_*x=m^ix$ for $x\in{\Bbb C}H^*({\cal J})_i$ (see \cite{DM} Thm. 3.1).
It follows that for fixed $k$ and $\alpha\in{\Bbb C}H^*({\cal J})$ the operator valued function
$m\to A_k([m]_*\alpha)$ is a polynomial of degree $\le 2g$. By Lemma \ref{A-T-lem1} the same is true for
$T_k(m,a)$, where $a\in{\Bbb C}H^*({\cal C})$ (more precisely, it is a polynomial in $m$ of degree $\le 2g-k$). Therefore, we can write
$$T_k(m,a)=\sum_{n=0}^{2g-k} \frac{m^n}{n!} \widetilde{X}_{n,k}(a)$$
for some operators $\widetilde{X}_{n,k}(a)$ on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$.
Then the relations of Theorem \ref{relations-thm}
are equivalent to the following relations for $(\widetilde{X}_{n,k}(a))$:
\betaegin{equation}\lambdabel{X-rel-eq}
\betaegin{array}{l}
\sum_{i\ge 0}\psi^i\cdotot i!\left({k\choose i}{n'\choose i} \widetilde{X}_{n,k-i}(a)\widetilde{X}_{n'-i,k'}(a')-
{k'\choose i}{n\choose i} \widetilde{X}_{n',k'-i}(a')\widetilde{X}_{n-i,k}(a)\right)=\\
\sum_{i\ge 1}(-1)^{i-1}i!\left({k\choose i}{n'\choose i}-
{k'\choose i}{n\choose i}\right) \widetilde{X}_{n+n'-i,k+k'-i}(aa'(K+2[p_0(S)])^{i-1})+\\
\delta_{n',0}p_0^*(a')\psi^{k'-1}k'!{n\choose k'}\widetilde{X}_{n-k',k}(a)-
\delta_{n,0}p_0^*(a)\psi^{k-1}k!{n'\choose k}\widetilde{X}_{n'-k,k'}(a')+\\
\delta_{k,0}p_0^*(a)\psi^{n-1}n!{k'\choose n}\widetilde{X}_{n',k'-n}(a')-
\delta_{k',0}p_0^*(a')\psi^{n'-1}n'!{k\choose n'}\widetilde{X}_{n,k-n'}(a).
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
We will denote $\widetilde{X}_{n,k}([{\cal C}])$ simply by $\widetilde{X}_{n,k}({\cal C})$.
In the case $S=\operatorname{Spec}(k)$ the above relations are essentially equivalent to those of
Theorem 2.6 of \cite{P-lie}. Recall that in \cite{P-lie} we showed that
the operators $\widetilde{X}_{n,k}(C)-nk\widetilde{X}_{n-1,k-1}(K/2+[p_0])$
satisfy the commutation relations of the Lie
algebra ${\cal HV}'$ and calculated their Fourier transform. We are going to
present a similar computation in the relative case (i.e., when $S$ is arbitrary).
We refer to \cite{DM} for the basic properties of the Fourier transform on cycles over
abelian schemes (originally introduced in \cite{M} and studied in \cite{B1} and \cite{B2}).
\betaegin{thm}\lambdabel{four-thm}
(i) The operators
\betaegin{equation}
e=\frac{1}{2}\widetilde{X}_{0,2}({\cal C}),\ \
f=-\frac{1}{2}\widetilde{X}_{2,0}({\cal C}),\ \
h=-\widetilde{X}_{1,1}({\cal C})+g\cdotot\operatorname{id}
\operatorname{e}nd{equation}
on ${\Bbb C}H({\cal J})_{{\Bbb Q}}$ define an action of the Lie algebra $\operatorname{sl}_2$.
\noindent (ii) Let us set for $a\in{\Bbb C}H^*({\cal C})$, $n\ge 0$, $k\ge 0$,
$$X_{n,k}(a)=\sum_{i\ge 0}(-1)^i i!{n\choose i}{k\choose i}\widetilde{X}_{n-i,k-i}(a\operatorname{e}ta^i),$$
where $\operatorname{e}ta:=K/2+[p_0(S)]+\psi/2\in{\Bbb C}H^1({\cal C})_{{\Bbb Q}}$.
Consider the Fourier transform defined by
$$F:{\Bbb C}H^*({\cal J})_{{\Bbb Q}}\to{\Bbb C}H^*({\cal J})_{{\Bbb Q}}:x\mapsto p_{2*}(\operatorname{e}xp(c_1({\cal L}))\cdotot p_1^*x).$$
Then one has
$$FX_{n,k}(a)F^{-1}=(-1)^kX_{k,n}(a).$$
$$[e,X_{n,k}(a)]=nX_{n-1,k+1}, \ \ [f,X_{n,k}(a)]=kX_{n+1,k-1}(a), \ \ [h,X_{n,k}(a)]=(k-n)X_{n,k}(a).$$
\operatorname{e}nd{thm}
\betaegin{rem} The $\operatorname{sl}_2$-action of Theorem \ref{four-thm} differs only by a sign from
the relative Lefschetz action associated with the relatively ample class $-\tau_2({\cal C})/2$ on ${\cal J}$
(see \cite{K}).
\operatorname{e}nd{rem}
We start with some preliminary statements.
\betaegin{lem}\lambdabel{four-lem}
One has $F\widetilde{X}_{n,0}(a)F^{-1}=\widetilde{X}_{0,n}(a)$ for every $a\in{\Bbb C}H^*({\cal C})$, $n\ge 0$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . By definition, $\widetilde{X}_{n,0}(a)/n!$ is the Pontryagin product with $a_n\in{\Bbb C}H^*({\cal J})_{{\Bbb Q}}$
where $[m]_*\iota_*a=\sum_{i\ge 0}m^ia_i$ for all $m\in{\Bbb Z}$. On the other hand,
$\widetilde{X}_{0,n}(a)=T_n(0,a)$
is the usual product with $\tau_n(a)$ (see \operatorname{e}qref{T-k-0-eq} and \operatorname{e}qref{tau-eq}).
Since $F(x*y)=F(x)\cdotot F(y)$, it remains to show that
\betaegin{equation}\lambdabel{F-a-eq}
F(a_n)=\frac{1}{n!}\tau_n(a)=p_{2*}(\frac{c_1({\cal L})^n}{n!}\cdotot p_1^*\iota_*a),
\operatorname{e}nd{equation}
where $p_1$ and $p_2$ are the projections of the product ${\cal J}\tildemes_S{\cal J}$ on its factors.
This fact is well known but we will give the proof since it is very short. We have
$$\sum_{i\ge 0}m^i F(a_i)=F([m]_*\iota_*a)=[m]^*F(\iota_*a)=[m]^*p_{2*}(\operatorname{e}xp(c_1({\cal L}))\cdotot
p_1^*\iota_*a)=p_{2*}((\operatorname{id}\tildemes[m])^*\operatorname{e}xp(c_1({\cal L}))\cdotot p_1^*\iota_*a).$$
Using the identity $(\operatorname{id}\tildemes [m])^*c_1({\cal L})=m c_1({\cal L})$, we see that this is equal
to $\sum_{i\ge 0}m^i\tau_i(a)/i!$. Now \operatorname{e}qref{F-a-eq} is obtained by equating the coefficients with $m^n$.
\operatorname{e}d
\betaegin{lem}\lambdabel{sl2-lem}
(i) One has $\widetilde{X}_{0,i}({\cal C})=\widetilde{X}_{i,0}({\cal C})=0$ for $i\le 1$.
\noindent (ii) One has
$$[e,\widetilde{X}_{n,k}(a)]=n\widetilde{X}_{n-1,k+1}(a)-n(n-1)\widetilde{X}_{n-2,k}(a\cdotot\operatorname{e}ta),$$
$$[f,\widetilde{X}_{n,k}(a)]=k\widetilde{X}_{n+1,k-1}(a)-k(k-1)\widetilde{X}_{n,k-2}(a\cdotot\operatorname{e}ta).$$
\operatorname{e}nd{lem}
\noindent {\it Proof} . (i) The operator $\widetilde{X}_{0,0}({\cal C})=T_0(0,{\cal C})$ is the product with the pull-back of
$\pi_*[{\cal C}]=0$, where $\pi:{\cal C}\to S$ is the projection. On the other hand, the operator
$\widetilde{X}_{0,1}({\cal C})=T_1(0,{\cal C})$ is the product with $\tau_1({\cal C})=p_{2*}(c_1({\cal P}_{{\cal C}}))$,
where $p_2:{\cal C}\tildemes_S{\cal J}\to{\cal J}$ is the projection. Since $c_1({\cal P}_{{\cal C}})$ is the divisor
in ${\cal C}\tildemes_S{\cal J}$ that has degree zero on every fiber of $p_2$, we get $\tau_1({\cal C})=0$.
Now the vanishing of $\widetilde{X}_{1,0}({\cal C})$ follows from Lemma \ref{four-lem}.
\noindent
(ii) This is an immediate consequence of relations \operatorname{e}qref{X-rel-eq} and of part (i).
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{four-thm}}.
(i) These relations follow from Lemma \ref{sl2-lem} and from the observation that
$\widetilde{X}_{0,0}(\operatorname{e}ta)=g\cdotot\operatorname{id}$ (since it is the product with the pull-back of $\pi_*(\operatorname{e}ta)=g\cdotot [S]$).
\noindent
(ii) From Lemma \ref{four-lem} we get
\betaegin{equation}\lambdabel{F-f-e-eq}
F f F^{-1}=-e.
\operatorname{e}nd{equation}
Recall that $-f$ is the Pontryagin product with
the class $a_2$ on ${\cal J}$ defined by $[m]_*[{\cal C}]=\sum_i m^i a_i$. Replacing $m$ by $-m$ we see
that $[-1]_*a_2=a_2$. Hence, $f$ commutes with $[-1]^*$. Now the identity
$F^2=(-1)^g[-1]_*$ (see \cite{DM}, Cor.2.22) implies that $F^2$ commutes with $f$. Therefore, from
\operatorname{e}qref{F-f-e-eq} we get
$$FeF^{-1}=-f.$$
On the other hand, from
Lemma \ref{sl2-lem}(ii) we deduce by induction that
\betaegin{equation}\lambdabel{ad-eq}
\betaegin{array}{l}
\operatorname{ad}(f)^k \frac{\widetilde{X}_{0,n+k}(a)}{(n+k)!}=\frac{X_{k,n}(a)}{n!},\\
\operatorname{ad}(e)^k \frac{\widetilde{X}_{n+k,0}(a)}{(n+k)!}=\frac{X_{n,k}(a)}{n!}.
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
Now, combining all of this with Lemma \ref{four-lem} we get
$$F\frac{X_{n,k}(a)}{n!}F^{-1}=\operatorname{ad}(-f)^k(F\frac{\widetilde{X}_{n+k,0}(a)}{(n+k)!}F^{-1})=
(-1)^k\operatorname{ad}(f)^k\frac{\widetilde{X}_{0,n+k}(a)}{(n+k)!}=(-1)^k\frac{X_{k,n}(a)}{n!}$$
as required. The formulas for $[e,X_{n,k}(a)]$ and $[f,X_{n,k}(a)]$ follow immediately
from \operatorname{e}qref{ad-eq}.
\operatorname{e}d
\section{Divided powers}
\lambdabel{div-sec}
In this section we construct and study divided powers of operators
$P_{n,0}({\cal C})$ and $P_{0,n}({\cal C})$. Then we use them to define an action of a certain
${\Bbb Z}$-form of an algebra of differential operators in two variables on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$.
We will also construct a ${\Bbb Z}$-version of the Lefschetz $\operatorname{sl}_2$-action on ${\Bbb C}H^*(C^{[N]})$
(see Theorem \ref{Lefschetz-thm}).
Let us start with divided powers of $P_{n,0}({\cal C})$. By definition, we have
$$P_{n,0}({\cal C})(x)=\delta_n * x,$$
where $\delta_n={\cal D}e_{n*}([{\cal C}])\subset{\Bbb C}H_1({\cal C}^{[n]})$ is the class of the diagonal.
Thus, we can set
\betaegin{equation}\lambdabel{div-eq1}
P_{n,0}({\cal C})^{[d]}(x)=\delta_n^{[d]}*x,
\operatorname{e}nd{equation}
where
$$\delta_n^{[d]}={\cal D}e_{n*}^{[d]}([{\cal C}^{[d]}])\in{\Bbb C}H_d({\cal C}^{[nd]}),$$
$${\cal D}e_n^{[d]}:{\cal C}^{[d]}\to {\cal C}^{[nd]}: D\mapsto nD.$$
Note that $d!\delta_n^{[d]}=\delta_n^{*d}$ --- the $d$th power of $\delta_n$ with the respect to the
Pontryagin product. Hence, $d!P_{n,0}({\cal C})^{[d]}=P_{n,0}({\cal C})^d$.
To describe the divided powers of $P_{0,n}({\cal C})$ let us introduce a new binary operation on
${\Bbb C}H^*({\cal C}^{[\betaullet]})$ as follows. For $a\in{\Bbb C}H^*({\cal C}^{[k]})$ and $x\in{\Bbb C}H^*({\cal C}^{[N]})$ set
$$i_a(x)=p_{2*}(p_1^*a\cdotot\alpha_{k,N-k}^*x)\in{\Bbb C}H^*({\cal C}^{[N-k]}),$$
where $p_1,p_2$ are projections of the product ${\cal C}^{[k]}\tildemes_S {\cal C}^{[N-k]}$ to its factors.
Then it is easy to see that
$$P_{0,n}({\cal C})(x)=i_{\delta_n}(x).$$
Also, it is straightforward to check that
$$i_{a*b}=i_a\circ i_b.$$
Thus, it is natural to set
\betaegin{equation}\lambdabel{div-eq2}
P_{0,n}({\cal C})^{[d]}(x)=i_{\delta_n^{[d]}}(x),
\operatorname{e}nd{equation}
so that we have $d!P_{0,n}({\cal C})^{[d]}=P_{0,n}({\cal C})^d$.
Below we use the notation from the Introduction.
Let $A$ be a supercommutative algebra $A$ with a unit and a distinguished
even element ${\betaf a}_0\in A$. We are going to define two extensions of the universal enveloping algebra
of ${\cal D}(A,{\betaf a}_0)$ by adding two families of divided powers.
\betaegin{lem}\lambdabel{div-sum-lem}
(i) Let ${\frak g}$ be a Lie algebra over ${\Bbb Z}$. Then for $x,y\in{\frak g}$, the following relations hold in
$U({\frak g})$:
$$x^{d}y=\sum_{i=0}^d {d\choose i}(\operatorname{ad} x)^{i}(y)x^{d-i},$$
$$yx^{d}=\sum_{i=0}^d {d\choose i}x^{d-i}(-\operatorname{ad} x)^{i}(y) $$
for all $d\ge 1$.
\noindent (ii) The following relations hold in $U({\cal D}(A,{\betaf a}_0))$:
$${\betaf P}_{m,k}(a){\betaf P}_{n,0}(1)^{d}=\sum_{j\le i}(-1)^{i-j}\frac{i!d!}{j!(d-j)!}{k\choose i}A_j(i,n){\betaf P}_{n,0}(1)^{d-j}
{\betaf P}_{m+nj-i,k-i}(a{\betaf a}_0^{i-j}),$$
$${\betaf P}_{0,n}(1)^{d}{\betaf P}_{m,k}(a)=\sum_{j\le i}(-1)^{i-j}\frac{i!d!}{j!(d-j)!}{m\choose i}A_j(i,n)
{\betaf P}_{m-i,k+nj-i}(a{\betaf a}_0^{i-j}){\betaf P}_{0,n}(1)^{d-j},$$
where
$$A_d(i,n)=\sum_{i_1+\ldots+i_d=i,i_s\ge1}{n\choose i_1}\ldots {n\choose i_d}.$$
We also use the convention $x^i=0$ for $i<0$ (so that $j\le d$ in both sums).
\operatorname{e}nd{lem}
\noindent {\it Proof} . (i) This is easily checked by induction in $d$.
\noindent (ii) Using the commutation relations in ${\cal D}(A,{\betaf a}_0)$ one can check by induction in $d$ that
$$\operatorname{ad}({\betaf P}_{0,n}(1))^d({\betaf P}_{m,k}(a))=\sum_{i\ge d}(-1)^{i-d}i!{m\choose i}A_d(i,n)
{\betaf P}_{m-i,k+nd-i}(a\cdotot {\betaf a}_0^{i-d}),$$
$$\operatorname{ad}(-{\betaf P}_{n,0}(1))^d({\betaf P}_{m,k}(a))=\sum_{i\ge d}(-1)^{i-d}i!{k\choose i}A_d(i,n)
{\betaf P}_{m+nd-i,k-i}(a\cdotot {\betaf a}_0^{i-d}).$$
Now the required relations follow from (i).
\operatorname{e}d
\betaegin{defi}
Let us denote by $\widetilde{U}_1(A,{\betaf a}_0)$ (resp., $\widetilde{U}_2(A,{\betaf a}_0)$)
the superalgebra over ${\Bbb Z}$ with generators
$({\betaf P}_{m,k}(a))$, $m\ge 0, k\ge 0$, depending additively on $a\in A$, and $({\betaf P}_{n,0}(1)^{[d]})$
(resp., $({\betaf P}_{0,n}(1)^{[d]})$), $n\ge 1$,
$d\ge 0$, subject to the following relations:
\noindent
(i) the supercommutator relations of ${\cal D}(A,{\betaf a}_0)$ for $({\betaf P}_{m,k}(a))$;
\noindent
(ii) $d!{\betaf P}_{n,0}(1)^{[d]}={\betaf P}_{n,0}(1)^d$ (resp., $d!{\betaf P}_{0,n}(1)^{[d]}={\betaf P}_{0,n}(1)^d$); \\
${\betaf P}_{n,0}(1)^{[d_1]}{\betaf P}_{n,0}(1)^{[d_2]}={d_1+d_2\choose d_1}{\betaf P}_{n,0}(1)^{[d_1+d_2]}$
(resp.,
${\betaf P}_{0,n}(1)^{[d_1]}{\betaf P}_{0,n}(1)^{[d_2]}={d_1+d_2\choose d_1}{\betaf P}_{0,n}(1)^{[d_1+d_2]}$);
\noindent
(iii)
${\betaf P}_{m,k}(a){\betaf P}_{n,0}(1)^{[d]}=\sum_{j\le i}(-1)^{i-j}\frac{i!}{j!}{k\choose i}A_j(i,n){\betaf P}_{n,0}(1)^{[d-j]}
{\betaf P}_{m+nj-i,k-i}(a{\betaf a}_0^{i-j})$,
$$(\text{resp., }
{\betaf P}_{0,n}(1)^{[d]}{\betaf P}_{m,k}(a)=\sum_{j\le i}(-1)^{i-j}\frac{i!}{j!}{m\choose i}A_j(i,n)
{\betaf P}_{m-i,k+nj-i}(a{\betaf a}_0^{i-j}){\betaf P}_{0,n}(1)^{[d-j]}).$$
\operatorname{e}nd{defi}
\betaegin{thm}\lambdabel{divided-powers-thm} The action of ${\cal D}({\Bbb C}H^*({\cal C}),K)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
extends to the action of $\widetilde{U}_1({\Bbb C}H^*({\cal C}),K)$ (resp., $\widetilde{U}_2({\Bbb C}H^*({\cal C}),K)$), such that
the action of ${\betaf P}_{n,0}({\cal C})^{[d]}$ (resp., ${\betaf P}_{0,n}({\cal C})^{[d]}$)
is given by \operatorname{e}qref{div-eq1} (resp., \operatorname{e}qref{div-eq2}). Furthermore, these relations hold on the level of correspondences. Also, similar statements hold for cohomology.
\operatorname{e}nd{thm}
First, we need to calculate some intersection-products.
\betaegin{lem}\lambdabel{inter-mult-lem2}
Recall that for each $m\ge 0$ we denote by $Z_{m,N}\subset {\cal C}\tildemes_S {\cal C}^{[N]}$ the image of
$t_{m,N}:{\cal C}\tildemes_S{\cal C}^{[N-m]}\to{\cal C}\tildemes_S{\cal C}^{[N]}$.
For $m\le n$ we can consider the fine intersection-product
$[Z_{m,N}]\cdotot [Z_{n,N}]\in{\Bbb C}H^m(Z_{n,N})$. We have the following formula:
$$[Z_{m,N}]\cdotot [Z_{n,N}]=\sum_{i\ge 0}(-1)^i i!{m\choose i}{n\choose i} p_1^*K^i\cdotot [Z_{m+n-i,N}],$$
where $p_1:Z_{n,N}\to {\cal C}$ is the natural projection.
\operatorname{e}nd{lem}
\noindent {\it Proof} . We have an isomorphism $t_{n,N}:{\cal C}\tildemes {\cal C}^{[N-n]}\widetilde{\to} Z_{n,N}$.
Under this isomorphism the intersection-product in question becomes $t_{n,N}^*[Z_{m,N}]$, and the
required formula is equivalent to
$$t_{n,N}^*[Z_{m,N}]=\sum_{i\ge 0}(-1)^i{m\choose i}{n\choose i} p_1^*K^i\cdotot [Z_{m-i,N-n}].$$
For $m=1$ this boils down to the identity
$$t_{n,N}^*[{\cal D}_N]={\cal D}_{N-n}-n p_1^*K$$
that follows easily from \operatorname{e}qref{D-res-eq}. To deduce the case of general $m$ we use Lemma \ref{Z-lem}.
\operatorname{e}d
\betaegin{lem}\lambdabel{Pi-d-lem}
Consider the cartesian diagram
\betaegin{diagram}
{\Bbb P}i_d(i_1,\ldots,i_n) &\rTo{} & {\cal C}^{[d]}\\
\dTo{} & &\dTo{{\cal D}e_n}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]} &\rTo{s_{i_1,\ldots,i_n;d}}& ({\cal C}^{[d]})^n,
\operatorname{e}nd{diagram}
where ${\cal D}e_n$ is the diagonal embedding, while $s_{i_1,\ldots,i_n;d}$
is given by $(p,D_1,\ldots,D_n)\mapsto (i_1p+D_1,\ldots,i_np+D_n)$.
For each $j\ge\max(i_1,\ldots,i_n)$ we have a natural map
$$q^j:{\cal C}\tildemes_S {\cal C}^{[d-j]}\to{\Bbb P}i_d(i_1,\ldots,i_n):(p,D)\mapsto (p,(j-i_1)p+D,\ldots,(j-i_n)p+D,jp+D),$$
where we view ${\Bbb P}i_d(i_1,\ldots,i_n)$ as a subvariety of
${\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}\tildemes_S {\cal C}^{[d]}$.
Set $i=\sum_{s=1}^n i_s$. Then we have
$$s_{i_1,\ldots,i_n;d}^![{\cal C}^{[d]}]=
\sum_{j\ge 0} (-1)^j a(i_1,\ldots,i_n;j)\cdotot p_1^*K^j\cdotot q^{i-j}_*[C\tildemes C^{[d-i+j]}]$$
where the coefficients $a(i_1,\ldots,i_n;j)$ are defined recursively by
\betaegin{equation}\lambdabel{a-rec-eq}
\betaegin{array}{l}
a(i_1,\ldots,i_n;j)=\sum_{k=0}^j k!{i_1\choose k}{i_2+\ldots+i_n-j+k\choose k}a(i_2,\ldots,i_n,j-k),\\
a(i_1;j)=\deltalta_{j,0}
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
(note that $a(i_1,\ldots,i_n;j)=0$ unless $j\le i-\max(i_1,\ldots,i_n)$).
\operatorname{e}nd{lem}
\noindent {\it Proof} . Note that for $n=1$ we have ${\Bbb P}i_d(i_1)={\cal C}\tildemes_S {\cal C}^{[d-i_1]}$, and
the formula holds trivially. For $n>1$ we have the following commutative diagram with
cartesian squares:
\betaegin{diagram}
{\Bbb P}i_d(i_1,\ldots,i_n) &\rTo{} & {\Bbb P}i_d(i_2,\ldots,i_n) &\rTo{} & {\cal C}^{[d]}\\
\dTo{} & & \dTo{} & &\dTo{{\cal D}e_n}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}
&\rTo{t_{i_1,d}\tildemes\operatorname{id}\tildemes\ldots\tildemes\operatorname{id}} &
{\cal C}\tildemes_S {\cal C}^{[d]}\tildemes_S {\cal C}^{[d-i_2]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]} &\rTo{}&
({\cal C}^{[d]})^n\\
\dTo{p_{12}} & &\dTo{p_{12}}\\
{\cal C}\tildemes_S {\cal C}^{[d-i_1]} &\rTo{t_{i_1,d}} & {\cal C}\tildemes_S {\cal C}^{[d]}
\operatorname{e}nd{diagram}
where the second arrow in the second row is induced by $s_{i_2,\ldots,i_n;d}$.
It follows that
$$s_{i_1,\ldots,i_n;d}^![C^{[d]}]=t_{i_1,d}^!s_{i_2,\ldots,i_n;d}^![C^{[d]}].$$
Now the required equality and the recursive formula for the coefficients $a(i_1,\ldots,i_n;j)$ follow easily by induction using Lemma \ref{inter-mult-lem2}.
\operatorname{e}d
\betaegin{lem}\lambdabel{inter-mult-lem3}
Consider the cartesian square
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{s_{k,N}}\\
{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]} &\rTo{s^{[n]}_{d,N}} & {\cal C}^{[N]}
\operatorname{e}nd{diagram}
where $s^{[n]}_{d,N}(D_1,D_2)=nD_1+D_2$.
For every $i,j$ such that $i\le k$ and $i\le nj$ we have a closed embedding
$$q^{i,j}:{\cal C}\tildemes_S {\cal C}^{[d-j]}\tildemes_S {\cal C}^{[N-nd-k+i]}\to Z^{d,[n]}_{k,N}: (p,D_1,D_2)
\mapsto (jp+D_1,(k-i)p+D_2,p, (nj-i)p+nD_1+D_2),$$
where we view $Z^{d,[n]}_{k,N}$ as a subset of
${\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]}\tildemes_S {\cal C}\tildemes_S {\cal C}^{[N-k]}$.
Then we have the following formula for the intersection-product in the above diagram:
$$[{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]}]\cdotot [{\cal C}\tildemes_S {\cal C}^{[N-k]}]=
\sum_{0\le j\le i\le k, i\le nj}(-1)^{i-j}\frac{i!}{j!}{k\choose i} A_j(i,n)
q^{i,j}_*(p_1^*K^{i-j}\cdotot [{\cal C}\tildemes_S {\cal C}^{[d-j]}\tildemes_S {\cal C}^{[N-nd-k+i]}]),$$
where we use the numbers $(A_j(i,n))$ introduced in Lemma \ref{div-sum-lem}(ii).
\operatorname{e}nd{lem}
\noindent {\it Proof} .
Consider the following commutative diagram with
cartesian squares:
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo{} & {\Bbb P}i_{k,(d)^n,N} &\rTo{} & {\cal C}\tildemes_S {\cal C}^{[N-k]}\\
\dTo{} & & \dTo{} & &\dTo{}\\
{\cal C}^{[d]}\tildemes_S {\cal C}^{[N-nd]} &\rTo{{\cal D}e_n\tildemes\operatorname{id}} &
({\cal C}^{[d]})^n\tildemes_S {\cal C}^{[N-nd]} &\rTo{\alpha}& {\cal C}^{[N]}\\
\dTo{} & &\dTo{}\\
{\cal C}^{[d]} &\rTo{{\cal D}e_n}&({\cal C}^{[d]})^n
\operatorname{e}nd{diagram}
where ${\cal D}e_n$ is the diagonal embedding, the map $\alpha$
is given by the addition of divisors. We have to calculate
${\cal D}e_n^!\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]$.
Iterating Lemma \ref{mult-lem2} we obtain
$$\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=\sum_{i_1+\ldots+i_n=i\le k}\frac{k!}{i_1!\ldots i_n!(k-i)!}
[{\cal C}\tildemes_S {\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}\tildemes_S {\cal C}^{[N-nd-k+i]}].$$
Next, by Lemma \ref{Pi-d-lem}
\betaegin{align*}
&{\cal D}e_n^![{\cal C}\tildemes_S{\cal C}^{[d-i_1]}\tildemes_S\ldots\tildemes_S {\cal C}^{[d-i_n]}]=
s_{i_1,\ldots,i_n;d}^![{\cal C}^{[d]}]=\\
&\sum_{l\ge 0}(-1)^l a(i_1,\ldots,i_n;l) p_1^*K^l\cdotot [{\cal C}\tildemes_S{\cal C}^{[d-i+l]}],
\operatorname{e}nd{align*}
where $i=i_1+\ldots+i_n$.
It follows that
$${\cal D}e_n^!\alpha^![{\cal C}\tildemes_S {\cal C}^{[N-k]}]=\sum_{0\le l\le i\le k}(-1)^l{k\choose i}
b(i,l;n) p_1^*K^l\cdotot [{\cal C}\tildemes_S{\cal C}^{[d-i+l]}\tildemes_S{\cal C}^{[N-nd-k+i]}],$$
where
$$b(i,l;n)=\sum_{i_1+\ldots+i_n=i}\frac{i!}{i_1!\ldots i_n!}a(i_1,\ldots,i_n;l).$$
It remains to show that $b(i,l;n)=\frac{i!}{(i-l)!}A_{i-l}(i,n)$. To this end we use the recursive formulas
$$b(i,l;n)=\sum_{0\le i_1\le i,0\le k\le l}\frac{i!(i-i_1-l+k)!}{(i-i_1)!(i_1-k)!(i-i_1-l)!k!}b(i-i_1,l-k;n-1),
\ \ \ b(i,l;1)=\delta_{l,0}$$
that follow immediately from \operatorname{e}qref{a-rec-eq}. Note that from the formula
${n\choose i}={n-1\choose i}+{n-1\choose i-1}$ one can derive a similar recursive formula for
$A_d(i,n)$:
$$A_d(i,n)=\sum_{r+s+t=d}\frac{d!}{r!s!t!}A_{d-r}(i-r-s,n-1).$$
Now the required equality follows easily by induction in $n$.
\operatorname{e}d
\noindent
{\it Proof of Theorem \ref{divided-powers-thm}.}
Relations of type (ii) are easy to check, so we will concentrate on relations of type (iii).
In the notation of Lemma \ref{inter-mult-lem3}
the composition $P_{m,k}(a)\circ P_{n,0}({\cal C})^{[d]}$ acting on ${\cal C}^{[N-nd]}$ is given by
$x\mapsto q_{2*}(w\cdotot p_{{\cal C}}^*a\cdotot q_1^*x)$, where
$w\in{\Bbb C}H^*(Z^{d,[n]}_{k,N})$ is the intersection-product computed in this lemma,
$q_1$ is the composition
$$Z^{d,[n]}_{k,N}\to {\cal C}^{[d]}\tildemes_{{\cal S}} {\cal C}^{[N-nd]}\stackrel{p_2}{\to}{\cal C}^{[N-nd]},$$
$q_2$ is the composition
$$
\betaegin{diagram}
Z^{d,[n]}_{k,N} &\rTo &{\cal C}\tildemes_S{\cal C}^{[N-k]} &\rTo{s_{m,N-k+m}} &{\cal C}^{[N-k+m]},
\operatorname{e}nd{diagram}
$$
and $p_{{\cal C}}:Z^{d,[n]}_{k,N}\to{\cal C}$ is the natural projection.
The formula for $w$ leads to the expression for the $P_{m,k}(a)\circ P_{n,0}({\cal C})^{[d]}$ as the
linear combination of the operators defined by the cycles $p_{{\cal C}}^*(a\cdotot K^{i-j})$ over the correspondences
$$
\betaegin{diagram}
& & {\cal C}\tildemes_S{\cal C}^{[d-j]}\tildemes_S{\cal C}^{[N-nd-k+i]} & \\
&\ldTo{s_{k-i,N-nd}p_{13}} & &\rdTo{q} &\\
{\cal C}^{[N-nd]} & & & &{\cal C}^{[N-k+m]}
\operatorname{e}nd{diagram}
$$
where $q(p,D_1,D_2)=(m+nj-i)p+nD_1+D_2$. It is easy to see that the same correspondence
arises when computing $P_{n,0}({\cal C})^{[d-j]}\circ P_{m+nj-i,k-i}(a\cdotot K^{i-j})$ and that the coefficients
match. The second relation of type (iii) is checked similarly: the operators involved in it are defined
by the transposes of the above correspondences.
\operatorname{e}d
\betaegin{ex} In the case of a trivial family ${\cal C}=C\tildemes S$ the relations established in
Theorem \ref{divided-powers-thm} take form
$$P_{m,k}(a)P_{n,0}(C\tildemes S)^{[d]}=
\sum_{i=0}^d{k\choose i} n^i P_{n,0}(C\tildemes S)^{[d-i]}P_{m+i(n-1),k-i}(a),$$
$$P_{0,n}(C\tildemes S)^{[d]}P_{m,k}(a)=\sum_{i=0}^d{m\choose i}n^i P_{m-i,k+i(n-1)}(a)
P_{0,n}(C\tildemes S)^{[d-i]},$$
where $a\in{\Bbb C}H^*(C\tildemes S)$.
\operatorname{e}nd{ex}
\betaegin{rems} 1. One should be able to establish also some commutation relations between
$P_{n_1,0}({\cal C})^{[d_1]}$ and $P_{0,n_2}({\cal C})^{[d_2]}$. The simplest example of such relations
is given in Proposition \ref{div-com-prop} below.
\noindent 2. For other operators $P_{k,m}({\cal C})$
one should be able construct some modified divided powers.
For example, we have $P_{1,1}({\cal C})(x)=Nx$
for $x\in{\Bbb C}H^*({\cal C}^{[N]})$, so one cannot construct $P_{1,1}({\cal C})^2/2$, however, one can construct
$(P_{1,1}({\cal C})^2-P_{1,1}({\cal C}))/2$.
\operatorname{e}nd{rems}
\betaegin{prop}\lambdabel{div-com-prop}
All the operators in the family $\{P_{1,0}({\cal C})^{[d]}\ |\ d\ge 1\}\cup
\{P_{0,1}({\cal C})^{[d]}\ |\ d\ge 1\}$ commute with each other (on the level of correspondences).
\operatorname{e}nd{prop}
\noindent {\it Proof} . The fact that the operators within each set commute with each other follows from the commutativity
of the Pontryagin product. Now let us check that $P_{1,0}({\cal C})^{[d_1]}$ commutes with
$P_{0,1}({\cal C})^{[d_2]}$.
The composition $P_{0,1}({\cal C})^{[d_2]}\circ P_{1,0}({\cal C})^{[d_1]}$ acting on ${\Bbb C}H^*({\cal C}^{[N]})$
is given by the following correspondence ${\Bbb P}i$ from ${\cal C}^{[N]}$ to ${\cal C}^{[N+d_1-d_2]}$ equipped
with a class of dimension $N+d_1$:
$${\Bbb P}i=\{(D_1,E_1,D_2,E_2)\in{\cal C}^{[d_1]}\tildemes_S{\cal C}^{[N]}\tildemes_S{\cal C}^{[d_2]}\tildemes_S
{\cal C}^{[N+d_1-d_2]}\ |\ D_1+E_1=D_2+E_2\},$$
where the map $q:{\Bbb P}i\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N+d_1-d_2]}$ sends $(D_1,E_1,D_2,E_2)$ to
$(E_1,E_2)$. This correspondence is equipped with the natural intersection-product class of dimension
$N+d_1$
\betaegin{equation}\lambdabel{div-com-inter-class}
[{\cal C}^{[d_1]}\tildemes_S{\cal C}^{[N]}]\cdotot [{\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N+d_1-d_2]}].
\operatorname{e}nd{equation}
On the other hand, the composition $P_{1,0}({\cal C})^{[d_1]}\circ P_{0,1}({\cal C})^{[d_2]}$ is given by the correspondence
$$
\betaegin{diagram}
& &{\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N-d_2]}\tildemes_S{\cal C}^{[d_1]}& &\\
&\ldTo{\alpha_{d_2,N-d_2}p_{12}}& &\rdTo{\alpha_{N-d_2,d_1}p_{23}}&\\
{\cal C}^{[N]}& & & &{\cal C}^{[N+d_1-d_2]}
\operatorname{e}nd{diagram}
$$
The natural map
$${\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N-d_2]}\tildemes_S{\cal C}^{[d_1]}\to{\Bbb P}i:(D_2,E',D_1)\mapsto
(D_1,D_2+E',D_2,D_1+E')$$
is an isomorphism onto an irreducible component ${\Bbb P}i_0\subset{\Bbb P}i$.
Other irreducible components ${\Bbb P}i_i\subset{\Bbb P}i$ are numbered by $i$, such that $1\le i\le\min(d_1,d_2)$.
Namely, ${\Bbb P}i_i$ is the image of the map
$${\cal C}^{[i]}\tildemes_S{\cal C}^{[d_2-i]}\tildemes_S{\cal C}^{[N-d_2+i]}\tildemes_S{\cal C}^{[d_1-i]}\to{\Bbb P}i:
(D_0,D'_2,E',D'_1)\mapsto (D_0+D'_1,D'_2+E', D_0+D'_2,D'_1+E').$$
Since the composition of this map with $q:{\Bbb P}i\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N+d_1-d_2]}$
factors through ${\cal C}^{[d_2-i]}\tildemes_S{\cal C}^{[N-d_2+i]}\tildemes_S{\cal C}^{[d_1-i]}$, we derive
that $q({\Bbb P}i_i)$ has dimension $\le N+d_1-i$. Therefore, the components ${\Bbb P}i_i$ with $i\ge 1$
give zero contribution to the composition $P_{0,1}({\cal C})^{[d_2]}\circ P_{1,0}({\cal C})^{[d_1]}$.
It remains to prove that $[{\Bbb P}i_0]$ appears in the intersection-product \operatorname{e}qref{div-com-inter-class}
with multiplicity $1$. To this end we can replace ${\cal C}^{[d_2]}$ by the cartesian product ${\cal C}^{d_2}$
and then use iteratively Lemma \ref{mult-lem2} (in the case $m=1$).
\operatorname{e}d
Let us denote by ${\cal D}_{t,u,{\Bbb Q}}={\Bbb Q}[t,u,\partial_t,\partial_u]$ the algebra of differential operators in two variables.
Let us also denote by
$${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]\subset{\cal D}_{t,u,{\Bbb Q}}$$
the subalgebra over ${\Bbb Z}$ generated by $t$, $\partial_u$ and by the divided powers of $u$ and $\partial_t$.
We will also consider the ${\Bbb Z}$-subalgebras ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$ and ${\Bbb Z}[u^{[\betaullet]},\partial_u]$ in
this algebra.
\betaegin{cor}\lambdabel{Heis-cor}
Assume that we are given a point $p_0\in{\cal C}(S)$. Then
there is an action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$
on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ such that
\betaegin{align*}
&t\mapsto P_{1,0}([p_0(S)])+\psi\cdotot P_{1,0}({\cal C}) \ \
(\text{Pontryagin product with }[p_0(S)]+\psi\cdotot[{\cal C}])
\\
&u^{[d]}\mapsto P_{1,0}({\cal C})^{[d]} \ \ (\text{Pontryagin product with } [{\cal C}^{[d]}]),\\
&\partial_t^{[d]}\mapsto P_{0,1}({\cal C})^{[d]},\\
&\partial_u\mapsto P_{0,1}([p_0(S)]),
\operatorname{e}nd{align*}
where $\psi=p_0^*K\in{\Bbb C}H^1(S)$.
\operatorname{e}nd{cor}
Here are some simple observations on actions of
${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ (probably well known).
\betaegin{lem}\lambdabel{div-mod-lem}
Let $M$ be a ${\Bbb Z}[u^{[\betaullet]},\partial_u]$-module such that for every $x\in M$ one has
$\partial_u^nx=0$ for all $n\gg 0$ (resp., ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$-module such that for every $x\in M$ one has
$\partial_t^{[n]}x=0$ for all $n\gg 0$).
Set $M_0=\{x\in M\ |\ \partial_ux=0\}$ (resp., $M_0=\{x\in M\ |\ \partial_t^{[n]}x=0 \text{ for all }n>0\}$).
Then the submodule $M_0[u^{[\betaullet]}]\subset M$ (resp., $M_0[t]\subset M$) consisting of elements
of the form $\sum a_i u^{[i]}$ (resp., $\sum a_i t^i$) with $a_i\in M_0$, coincides with the entire $M$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . Assume first that $M$ is a module over ${\Bbb Z}[u^{[\betaullet]},\partial_u]$.
For $x\in M$ let $n$ be the minimal number such that $\partial_u^n(x)\in M_0[u^{[\betaullet]}]$.
Assume that $n>0$. Let
$$\partial_u^n(x)=a_0+a_1u+\ldots+a_ku^{[k]}.$$
where $a_i\in M_0$. Then
$$\partial_u(\partial_u^{n-1}(x)-a_0u-a_1u^{[2]}-\ldots-a_ku^{[k+1]})=0,$$
i.e., $\partial_u^{n-1}(x)-a_0u-a_1u^{[2]}-\ldots-a_ku^{[k+1]}\in M_0$. It follows that
$\partial_u^{n-1}(x)\in M_0[u^{[\betaullet]}]$ contradicting the choice of $n$.
Now, let $M$ be a module over ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$. For $x\in M$ let $n$ be the minimal number
such that $\partial_t^{[k]}(x)\in M_0[t]$ for all $k\ge n$.
Assume that $n>0$ and let us lead this to contradiction.
Replacing $x$ by $\partial_t^{[n-1]}(x)$ we can reduce the proof to the case $n=1$.
In this case set for $k\ge 1$
$$f_k=\partial_t^{[k]}(x)=\sum_{i\ge 0}a_{k,i}t^i\in M_0[t].$$
Note that $f_k=0$ for $k\gg 0$, so we can form the finite sum
$$f=\sum_{k\ge 1}a_{k,0}t^k\in M_0[t].$$
Using the identities $\partial_t^{[m-k]}f_k={m\choose k}f_m$ one can easily check that
$\partial_t^{[k]}f=f_k$ for every $k\ge 1$. Hence, $x-f\in M_0$, i.e., $x\in M_0[t]$.
\operatorname{e}d
For every abelian group ${\Bbb G}a$ there is a natural structure of
a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module on
${\Bbb G}a[t,u^{[\betaullet]}]$ such that $\partial_t^{[n]}(\gamma t^iu^{[j]})=\gamma{i\choose n}t^{i-n}u^{[j]}$.
Note that the operator $\partial_u$ on this module is surjective.
\betaegin{prop}\lambdabel{Heis-mod-prop}
Let $M$ be a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module such that for every $x\in M$ one has
$\partial_t^{[n]}x=\partial_u^nx=0$ for all $n\gg 0$. Then
\noindent (i) $M\sigmameq M_0[t,u^{[\betaullet]}]$ as ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module, where
$$M_0=\{x\in M\ |\ \partial_ux=0, \partial_t^{[n]}x=0 \text{ for all }n>0\};$$
\noindent (ii) the operator $t:M\to M$ is injective, and the operator $\partial_u:M\to M$ is surjective.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) First, it is easy to check that any submodule of an ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module of the form ${\Bbb G}a[t,u^{[\betaullet]}]$ itself
has the form ${\Bbb G}a'[t,u^{[\betaullet]}]$ for a subgroup ${\Bbb G}a'\subset{\Bbb G}a$. Indeed, this follows easily
from the fact that $\partial_t^{[m]}\partial_u^n(\gamma t^m u^{[n]})=\gamma$ for $\gamma\in{\Bbb G}a$.
Therefore, the natural morphism of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-modules
$M_0[t,u^{[\betaullet]}]\to M$ is injective (since $M_0$ embeds into $M$).
It remains to prove that this morphism is also surjective. Viewing $M$ as a
${\Bbb Z}[u^{[\betaullet]},\partial_u]$-module and applying Lemma \ref{div-mod-lem}, we derive that
$M=(\operatorname{ker} \partial_u)[u^{[\betaullet]}]$. Next, applying the same lemma to the ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$-module
$\operatorname{ker}\partial_u$, we obtain $\operatorname{ker}\partial_u=M_0[t]$. Hence, $M=M_0[t,u^{[\betaullet]}]$.
\noindent
(ii) This follows from (i) since for a module of the form ${\Bbb G}a[t,u^{[\betaullet]}]$
injectivity of $t$ (resp., surjectivity of $\partial_u$) is clear.
\operatorname{e}d
In the case when $S$ is a point the following result is due to Collino~\cite{Col2}.
\betaegin{prop}\lambdabel{Col-prop} Let $i_{N}:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ be the closed embedding associated
with a point $p_0\in{\cal C}(S)$. Then the homomorphism $i_{N*}:{\Bbb C}H^*({\cal C}^{[N-1]})\to{\Bbb C}H^*({\cal C}^{[N]})$
(resp., $i_N^*:{\Bbb C}H^*({\cal C}^{[N]})\to{\Bbb C}H^*({\cal C}^{[N-1]})$) is injective (resp., surjective).
\operatorname{e}nd{prop}
\noindent {\it Proof} . Note that $i_{N*}=P_{1,0}([p_0(S)])$ and $i_N^*=P_{0,1}([p_0(S)])$. Now the
surjectivity of $P_{0,1}([p_0(S)])$ follows immediately from Corollary \ref{Heis-cor} and Proposition
\ref{Heis-mod-prop}(ii). To deal with injectivity of $P_{1,0}([p_0(S)])$ we
can modify the action of Corollary \ref{Heis-cor}(i) as follows:
\betaegin{align*}
&t\mapsto P_{1,0}(p_0(S)),\\
&u^{[d]}\mapsto P_{1,0}({\cal C})^{[d]},\\
&\partial_t^{[d]}\mapsto P_{0,1}({\cal C})^{[d]},\\
&\partial_u\mapsto P_{0,1}([p_0(S)])+\psi\cdotot P_{0,1}({\cal C}).
\operatorname{e}nd{align*}
It remains to apply Proposition \ref{Heis-mod-prop}(ii) to this action.
\operatorname{e}d
We also get the following corollary from Proposition \ref{Heis-mod-prop}.
\betaegin{cor}\lambdabel{module-cor}
For a point $p_0\in {\cal C}(S)$ consider the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$
on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ given by Corollary \ref{Heis-cor}. Then
there is an isomorphism of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module
$${\Bbb C}H^*({\cal C}^{[\betaullet]})\sigmameq K_{p_0}[t,u^{[\betaullet]}],$$
where
$$K_{p_0}=\{x\in{\Bbb C}H^*({\cal C}^{[\betaullet]})\ |\ P_{0,1}(p_0(S))x=0,
P_{0,1}({\cal C}))^{[d]}x=0 \text{ for all }d\ge 1\}.$$
\operatorname{e}nd{cor}
\betaegin{rems}
1. The isomorphism of the above corollary is compatible with the bigrading of ${\Bbb C}H^*(C^{[\betaullet]})$:
$t$ (resp., $u$) sends ${\Bbb C}H^i(C^{[N]})$ to ${\Bbb C}H^{i+1}(C^{[N+1]})$ (resp., ${\Bbb C}H^i(C^{[N+1]})$).
In the case $S=\operatorname{Spec}(k)$ the strong stability conjecture (see \cite{KV}, 2.13) is equivalent to the condition that $K_{p_0}\cap{\Bbb C}H^p(C^{[N]})$ is a torsion group for $N>2p$.
\noindent 2. In the case $S=\operatorname{Spec}(k)$ the decomposition of ${\Bbb C}H^*(C^{[\betaullet]})$ into the
direct summands of the form $K_{p_0}t^mu^{[n]}$ is the ${\Bbb Z}$-version of the well known motivic decomposition over ${\Bbb Q}$ obtained by using $\lambdambda$-operations (see \cite{dB1}).
\noindent 3. Using the divided powers of $P_{0,1}(C)$ we can avoid tensoring with ${\Bbb Q}$ in the
proof of Theorem \ref{curve-thm}. Instead one has to use the fact that $P_{0,1}(C)^{[d]}$
commutes with $P_{1,0}(a)$ for $a\in A_0(C)$, and hence the subgroup
$A_0(C)^{*n}\subset{\Bbb C}H_0(C^{[n]})$ is killed by all the operators $P_{0,1}(C)^{[d]}$.
\operatorname{e}nd{rems}
Using Corollary \ref{Heis-cor} we get an interesting $\operatorname{sl}_2$-action on the motive of ${\cal C}^{[N]}$.
In the case when $S$ is a point we obtain in this way a Lefschetz $\operatorname{sl}_2$-action on
${\Bbb C}H^*(C^{[N]})$.
\betaegin{thm}\lambdabel{Lefschetz-thm}
(i) Fix a point $p_0\in {\cal C}(S)$. Then for every $N\ge 0$ the operators
\betaegin{align*}
&e(x)=[{\cal R}]\cdotot x+\psi\cdotot P_{1,0}({\cal C})P_{0,1}([p_0(S)])(x),\\
&f=P_{1,0}({\cal C})P_{0,1}({\cal C}),\\
&h=P_{1,0}([p_0(S)])P_{0,1}({\cal C})-P_{1,0}({\cal C})P_{0,1}([p_0(S)])+\psi\cdotot P_{1,0}({\cal C})P_{0,1}({\cal C})
\operatorname{e}nd{align*}
(given by algebraic correspondences)
define compatible actions of the Lie algebra $\operatorname{sl}_2$ on ${\Bbb C}H^*({\cal C}^{[N]})$ and on
$H^*({\cal C}^{[N]},{\Bbb Q})$,
where ${\cal R}={\cal R}_N\subset{\cal C}^{[N]}$ is the divisor associated with $p_0$ (see section \ref{Jac-sec}).
\noindent
(ii) In the case when $S$ is a point (so we write ${\cal C}=C$)
the operator $h$ acts as $(i-N)\operatorname{id}$ on $H^i(C^{[N]},{\Bbb Q})$. In other words, the
action of $(e,f,h)$ on $H^*(C^{[N]},{\Bbb Q})$ is the Lefschetz action corresponding to the ample divisor
$R\subset C^{[N]}$ (associated with $p_0$).
\operatorname{e}nd{thm}
\noindent {\it Proof} . (i) Consider the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
given by Corollary \ref{Heis-cor}. Then the operators $e=t\partial_u$, $f=u\partial_t$ and $h=t\partial_t-u\partial_u$
satisfy the relations of $\operatorname{sl}_2$.
By definition, we have
$$P_{1,0}([p_0])|_{{\cal C}^{[N-1]}}=i_{N*}, \ \ P_{0,1}([p_0])|_{{\cal C}^{[N]}}=i_{N}^*,$$
where $i_N:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ is
the embedding associated with $p_0$.
Hence, for $x\in{\Bbb C}H^*({\cal C}^{[N]})$ one has
$$P_{1,0}([p_0])P_{0,1}([p_0])(x)=i_{N*}i_N^*x=[{\cal R}]\cdotot x.$$
This implies our formula for $e$.
\noindent
(ii) It is easy to see that two $\operatorname{sl}_2$-triples $(e,f,h)$ and $(e',f',h')$ acting
on the same finite-dimensional space $V$ such that $e=e'$ and $[h,h']=0$ necessarily coincide
(i.e., $h=h'$ and $f=f'$). Indeed, consider the decomposition $V=\betaigoplus_{m,n} V_{m,n}$,
where $h$ (resp., $h'$) acts by $m\cdotot\operatorname{id}$ (resp., $n\cdotot\operatorname{id}$) on $V_{m,n}$.
Consider also the subspace $W=\operatorname{ker}(e)\subset V$ and the induced decomposition
$W=\betaigoplus_{m\ge 0,n\ge 0} W_{m,n}$, where $W_{m,n}=W\cap V_{m,n}$.
Viewing $V$ as a representation of $(e,f,h)$ we deduce that
$$e^iV\cap W=\betaigoplus_{m\ge i,n}W_{m,n}\text{ for all }i\ge 0.$$
On the other hand, using the triple $(e=e',f',h')$ we get
$$e^iV\cap W=\betaigoplus_{m,n\ge i}W_{m,n}\text{ for all }i\ge 0.$$
This immediately implies that $W_{m,n}=0$ for $m\neq n$, i.e., $h=h'$. It is well known that
this implies $f=f'$.
Now comparing the action of our operators $(e,f,h)$ on $H^*(C^{[N]},{\Bbb Q})$ with the Lefschetz action corresponding to $[R]$ and using the above observation we conclude that these two actions coincide.
\operatorname{e}d
\betaegin{rem} It is well known that the standard conjecture B of Lefschetz type for $X$ and
for a projective bundle over $X$ are equivalent (see \cite{Li}).
Also, if it is true for some variety $X$ then
it is true for an ample divisor in $X$. Thus, standard conjecture B for all $C^{[N]}$ and for the Jacobian
$J$ are equivalent.
In particular, from the above theorem we get a new proof of the standard conjecture B for $J$.
\operatorname{e}nd{rem}
Using our methods we easily recover the following result proved by S.~del Ba\~{n}o in
\cite{dB2}.
\betaegin{cor}\lambdabel{Lefschetz-cor}
The hard Lefschetz theorem for the operator $L$ of multiplication by the class
$[R]\in H^2(C^{[N]},{\Bbb Z})$ (associated with a point $p_0\in C$) holds over ${\Bbb Z}$, i.e., the map
$$L^i:H^{N-i}(C^{[N]},{\Bbb Z})\to H^{N+i}(C^{[N]},{\Bbb Z})$$
is an isomorphism for all $i\ge 0$.
\operatorname{e}nd{cor}
\noindent {\it Proof} . Since the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ defined in Corollary \ref{Heis-cor} is given by algebraic correspondences, it can also be defined
for $H^*(C^{[N]},{\Bbb Z})$. The operator $L$ corresponds to the action of $e=t\partial_u$. As we have seen
in Theorem \ref{Lefschetz-thm}, the operator $h=t\partial_t-u\partial_u$ acts as $(i-N)\operatorname{id}$ on $H^i(C^{[N]},{\Bbb Z})$.
On the other hand, by Proposition \ref{Heis-mod-prop}(i) we have an isomorphism of $H^*(C^{[N]},{\Bbb Z})$
with a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module of the form
${\Bbb G}a[t,u^{[\betaullet]}]$. For such a module we have
$h(\gamma t^mu^{[n]})=(m-n)\gamma t^mu^{[n]}$, and our assertion follows from the formula
$$e^{n-m}(\gamma t^mu^{[n]})=\gamma t^nu^{[m]} \text{ for }n\ge m.$$
\operatorname{e}d
\section{Tautological cycles}
\lambdabel{taut-sec}
Similarly to the case of cycles on the Jacobian considered in \cite{Bmain} and \cite{P-lie},
we are going to
define the subalgebra of tautological classes in ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ (resp., ${\Bbb C}H^*({\cal J})$).
We will show that the operators $(P_{m,k}(a))$ (resp., $T_k(m,a)$) act on this subalgebra by some differential operators.
We start with an abstract setup.
Let $R$ be a supercommutative ring,
$A$ a supercommutative $R$-algebra with a fixed even element ${\betaf a}_0\in A$,
${\cal D}(A,{\betaf a}_0)$ the corresponding Lie superalgebra (see the Introduction).
Let us also denote by ${\cal D}_+(A,{\betaf a}_0)\subset{\cal D}(A,{\betaf a}_0)$ the subalgebra generated by the operators
${\betaf P}_{m,k}(a)$ with $m\ge k$.
\betaegin{prop}\lambdabel{diff-op-prop} Let $B$ be a supercommutative $R$-algebra,
and let $x_m:A\to B$, $m\ge 0$, be a family of even $R$-linear maps.
\noindent
(i) Assume that ${\cal D}(A,{\betaf a}_0)$ (resp., ${\cal D}_+(A,{\betaf a}_0)$)
acts $R$-linearly on $B$ in such a way that
$${\betaf P}_{m,0}(a)(b)=x_m(a)\cdotot b$$
for all $m\ge 0$ and $a\in A$.
Assume also that
$${\betaf P}_{m,k}(a)(1)=0 \text{ for }k>0$$
(resp., for $m\ge k>0$).
Let us denote by ${\cal T} B\subset B$
the $R$-subalgebra generated by $(x_m(a))$.
Then ${\cal T} B$ is stable under the action of ${\cal D}(A,{\betaf a}_0)$ (resp., ${\cal D}_+(A,{\betaf a}_0)$).
Furthermore, if we view ${\cal T} B$ as the quotient of the superalgebra of polynomials
$R[x_m(a)\ |\ m\ge 0, a\in G]$,
where $G$ is some set of homogeneous generators of $A$ as an $R$-module,
then the action of this Lie algebra on ${\cal T} B$ is given by the formulas
\betaegin{align*}
&{\betaf P}_{m,k}(a)=\sum_{s\ge 0;k_1+\ldots+k_s=k, k_i\ge 1;
n_1,\ldots,n_s\ge 0; a_1,\ldots,a_s\in G}
(-1)^{k-s}\frac{k!}{s!}{n_1\choose k_1}\cdotot\ldots\cdotot{n_s\choose k_s}\tildemes\\
&x_{m-k+n_1+\ldots+n_s}(a a_s\ldots a_1\cdotot {\betaf a}_0^{k-s})\partial_{x_{n_1}(a_1)}\ldots\partial_{x_{n_s}(a_s)},
\operatorname{e}nd{align*}
where the case $s=0$ occurs only for $k=0$.
\noindent
(ii) Assume in addition that the above action of ${\cal D}(A,{\betaf a}_0)$ on $B$ extends to an action of
$\widetilde{U}_1(A,{\betaf a}_0)$, so that
$${\betaf P}_{m,0}(1)^{[d]}(b)=x_m(1)^{[d]}\cdotot b$$
for some elements $x_m(1)^{[d]}\in B$. Let $\widetilde{{\cal T}} B$ be the subalgebra generated by ${\cal T} B$ and
by all the elements $x_m(1)^{[d]}$. Then $\widetilde{{\cal T}} B$ is stable under the action of $\widetilde{U}_1(A,{\betaf a}_0)$,
and the action of ${\betaf P}_{m,k}(a)$ on it is given by the same formula as above (with $\partial_{x_m(1)}$ extended to the divided powers).
Similarly, if the action of ${\cal D}(A,{\betaf a}_0)$ on $B$ extends to an action of $\widetilde{U}_2(A,{\betaf a}_0)$
such that ${\betaf P}_{0,n}(1)^{[d]}(1)=0$ for $n\ge 1, d\ge 1$, then
${\cal T} B$ is stable under the action of $\widetilde{U}_2(A,{\betaf a}_0)$, and the action of
${\betaf P}_{0,n}(1)^{[d]}$ on ${\cal T} B$ is given by a differential operator of order $nd$.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) For $k=0$ we have ${\betaf P}_{m,0}(a)=x_m(a)$ by assumption.
The general case follows by induction in $k$ using the (super)commutator formula
$$[{\betaf P}_{m,k}(a),x_n(a')]=[{\betaf P}_{m,k}(a),{\betaf P}_{n,0}(a')]=
\sum_{i\ge 1}(-1)^{i-1}i!\cdotot {k\choose i}{n\choose i}{\betaf P}_{m+n-i,k-i}(a\cdotot a'\cdotot {\betaf a}_0^{i-1})$$
together with the assumption that ${\betaf P}_{m,k}(a)(1)=0$ for $k>0$.
Indeed, it is straightforward to check that the similar commutation relation
holds for the differential operators in the right-hand side of the required formula.
\noindent (ii) Same proof as in (i) using the commutation relations in $\widetilde{U}_1(A,{\betaf a}_0)$ (resp.,
$\widetilde{U}_2(A,{\betaf a}_0)$).
\operatorname{e}d
Proposition \ref{diff-op-prop} can be applied to the algebra
$B={\Bbb C}H_*({\cal C}^{[\betaullet]})$ (equipped with Pontryagin product) and the operators
$(P_{m,k}(a))$ acting on it. This leads to a definition of the subalgebra of tautological classes.
\betaegin{defi} The {\it (big) algebra of tautological classes on ${\cal C}^{[\betaullet]}$}
$$\TCH^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$$
is the ${\Bbb C}H^*(S)$-subalgebra with respect to the Pontryagin product
generated by all the classes of the form
${\cal D}e_{n*}(a)\in{\Bbb C}H^*({\cal C}^{[n]})$, where ${\cal D}e_n:{\cal C}\to{\cal C}^{[n]}$ is the diagonal embedding,
$a\in{\Bbb C}H^*({\cal C})$, $n\ge 1$. Let us also denote by
$$\widetilde{\TCH}^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$$
the subalgebra generated by $\TCH^*({\cal C}^{[\betaullet]})$ along with all the classes
$\delta_n^{[d]}\in{\Bbb C}H^{(n-1)d}({\cal C}^{[nd]})$ (see section \ref{div-sec}).
Replacing Chow groups by cohomology we also define the
subalgebra of tautological classes ${\cal T}H^*({\cal C}^{[\betaullet]})\subset H^*({\cal C}^{[\betaullet]})$.
\operatorname{e}nd{defi}
We can also mimic the above definition of tautological classes in the case of the relative Jacobian
${\cal J}$ for a family of curves ${\cal C}/S$ equipped with a point $p_0\in{\cal C}(S)$.
\betaegin{defi} The {\it (big) algebra of tautological classes on ${\cal J}$}
$$\TCH^*({\cal J})\subset{\Bbb C}H^*({\cal J})$$
is the ${\Bbb C}H^*(S)$-subalgebra with respect to the Pontryagin product
generated by all the classes of the form $[n]_*(\iota_*a)$, where
$\iota=\sigma_1:{\cal C}\to{\cal J}$ is the embedding
associated with $p_0$, $a\in{\Bbb C}H^*({\cal C})$, $n\in{\Bbb Z}$.
Here we view ${\Bbb C}H^*({\cal J})$ as a ${\Bbb C}H^*(S)$-algebra via the homomorphism
$e_*:{\Bbb C}H^*(S)\to{\Bbb C}H^*({\cal J})$ associated with the neutral element $e\in{\cal J}(S)$.
Similarly, we define the subalgebra of tautological classes in cohomology.
\operatorname{e}nd{defi}
\betaegin{rem}
One can also consider smaller algebras of tautological classes by choosing a ${\Bbb C}H^*(S)$-subalgebra
${\Bbb A}A\subset{\Bbb C}H^*({\cal C})$ and considering only the classes ${\cal D}e_{n*}(a)$ with $a\in{\Bbb A}A$.
For example, one can take as ${\Bbb A}A$ the subalgebra generated by $[p_0(S)]\in{\Bbb C}H^1({\cal C})$,
or by the relative canonical class $K\in{\Bbb C}H^1({\cal C})$, or by some other divisor class. In \cite{P-lie} we
worked with the subalgebra generated by $\chi+[p_0]$, where $2\chi=K$ (in the case $S=\operatorname{Spec}(k)$).
\operatorname{e}nd{rem}
\betaegin{thm}\lambdabel{taut-thm}
(i) The operators $(P_{m,k}(a))$ from the Introduction preserve the
subalgebra of tautological classes $\TCH^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$
(resp., $\widetilde{\TCH}^*({\cal C}^{[\betaullet]})$)
and act on it by the differential operators given in Proposition \ref{diff-op-prop} with
$x_n(a)={\cal D}e_{n*}(a)$.
\noindent
(ii) The operators $T_k(m,a)$ from section \ref{Jac-sec}
preserve the subalgebra of tautological classes
$\TCH^*({\cal J})\subset{\Bbb C}H^*({\cal J})$ and act on it by differential operators (with respect to the Pontryagin product), so that $T_k(m,a)$ acts by a differential operator of order $k$.
\noindent
(iii) The space of tautological classes with rational coefficients $\TCH^*({\cal J})_{{\Bbb Q}}\subset{\Bbb C}H^*({\cal J})_{{\Bbb Q}}$
is closed under the usual product and under the Fourier transform. It coincides with the
${\Bbb C}H^*(S)_{{\Bbb Q}}$-subalgebra
in ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$ with respect to the usual product, generated by the classes $\tau_k(a)$, $k\ge 0$,
$a\in{\Bbb C}H^*({\cal C})$ (see \operatorname{e}qref{tau-eq}).
\noindent
(iv) For every $N$ the map $\sigma_{N*}:{\Bbb C}H^*({\cal C}^{[N]})\to{\Bbb C}H^*({\cal J})$
(resp., $\sigma_N^*:{\Bbb C}H^*({\cal J})_{{\Bbb Q}}\to{\Bbb C}H^*({\cal C}^{[N]})_{{\Bbb Q}}$) sends
tautological classes to tautological classes (resp., with rational coefficients).
\noindent
(v) Similar statements hold for the cohomology.
\operatorname{e}nd{thm}
\noindent {\it Proof} . (i) This follows immediately from Proposition \ref{diff-op-prop}.
\noindent (ii) The operator $T_0(m,a)$ is simply the Pontryagin product with
$[m]_*\iota_*a\in{\Bbb C}H^*({\cal J})$.
The commutation relation of Theorem \ref{relations-thm} for $k'=0$ and $k\ge 1$ gives
\betaegin{align*}
&[T_k(m,a),T_0(m',a')]+\sum_{i\ge 1}\psi^i{k\choose i}m^{\operatorname{pr}ime i}T_{k-i}(m,a)T_0(m',a')=\\
&\sum_{i\ge 1}(-1)^{i-1}{k\choose i}m^{\operatorname{pr}ime i}T_{k-i}(m+m',aa'(K+2[p_0(S)])^{i-1})-
\psi^{k-1}m^{\operatorname{pr}ime k}p_0^*(a)T_0(m',a')\\
&-\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a')T_{k-i}(m,a).
\operatorname{e}nd{align*}
Since $T_k(m,a)(e_*x)=0$ this implies the assertion by induction in $k$.
\noindent
(iii) First of all, note that $g$-th Pontryagin power of $\sigma_{1*}({\cal C})$ is equal to $g![{\cal J}]$, so
$[{\cal J}]\in\TCH^*({\cal J})_{{\Bbb Q}}$. Next, let us check that $\TCH^*({\cal J})_{{\Bbb Q}}$ is closed under the
Fourier transform $F:{\Bbb C}H^*({\cal J})_{{\Bbb Q}}\to{\Bbb C}H^*({\cal J})_{{\Bbb Q}}$. We have seen in the proof of Lemma
\ref{four-lem} that
$$F([n]_*\iota_*a)=\sum_{k\ge 0}\frac{n^k}{k!}\tau_k(a).$$
Since $F(x*y)=F(x)\cdotot F(y)$, we derive the formula
\betaegin{equation}\lambdabel{F-prod-eq}
F(([n_1]_*\iota_*a_1)*\ldots*([n_s]_*\iota_*a_s))=
\sum_{k_1,\ldots,k_s}\frac{n_1^{k_1}\ldots n_s^{k_s}}{k_1!\ldots k_s!}\tau_{k_1}(a_1)\cdotot\ldots\cdotot
\tau_{k_s}(a_s).
\operatorname{e}nd{equation}
It remains to note that
$$\tau_{k_1}(a_1)\cdotot\ldots\tau_{k_s}(a_s)=T_{k_1}(0,a_1)\ldots T_{k_s}(0,a_s)([{\cal J}]),$$
hence, it is tautological by (ii). Thus, $\TCH^*({\cal J})_{{\Bbb Q}}$ is closed under the Fourier transform.
It follows that it is also closed under the usual product. Note that it contains all the classes
$\tau_k(a)=T_k(a)([{\cal J}])$. Now the fact that it is generated by these classes with respect to the usual product follows from \operatorname{e}qref{F-prod-eq}.
\noindent
(iv) Since the map $\sigma_*:{\Bbb C}H^*({\cal C}^{[\betaullet]})\to{\Bbb C}H^*({\cal J})$ respects the Pontryagin products,
the first assertion follows from the formula
$$\sigma_{n*}{\cal D}e_{n*}(a)=[n]_*\iota_*(a).$$
To check the assertion about the pull-backs we use the fact that the operators
$T_k(0,a)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ are multiplications by the pull-backs of $\tau_k(a)$
(see \operatorname{e}qref{T-k-0-eq}).
Since $[{\cal C}^{[N]}]\in\TCH^0({\cal C}^{[N]})_{{\Bbb Q}}$ and the operators $T_k(0,a)$ preserve
$\TCH^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$, the result now follows from the fact that $\TCH^*({\cal J})_{{\Bbb Q}}$ is generated
by the classes $(\tau_k(a))$ with respect to the usual product (see part (iii)).
\noindent
(v) We can repeat the same proofs changing Chow groups to cohomology (and inserting
appropriate signs where needed).
\operatorname{e}d
\betaegin{rems}
1. Consider the situation of Proposition \ref{diff-op-prop}(i). Assume that the algebra $A$ is equipped with a nonnegative ${\Bbb Z}$-grading (compatible with the ${\Bbb Z}/2{\Bbb Z}$-grading) such that $\deltag({\betaf a}_0)=2$. Let us denote by ${\cal D}'(A,{\betaf a}_0)\subset{\cal D}(A,{\betaf a}_0)$ the subalgebra generated
by ${\betaf P}_{m,k}(a)$ with $m+k+\deltag(a)\ge 2$. Then the analogue of Proposition \ref{diff-op-prop}(i)
holds if we only have an action of the subalgebra ${\cal D}'(A,{\betaf a}_0)$. In particular, this can be applied to
the actions of the Lie algebra ${\cal HV}'({\Bbb Z})\subset{\cal HV}({\Bbb Z})$ (see the Introduction). In the case of the action of
${\cal HV}'({\Bbb Z})$ on ${\Bbb C}H^*(J)_{{\Bbb Q}}$ considered in \cite{P-lie} we recover the differential operators
obtained in {\it loc. cit.}.
\noindent
2. The subalgebra ${\cal D}_+({\Bbb C}H^*({\cal C}),K)\subset{\cal D}({\Bbb C}H^*({\cal C}),K)$
considered in Proposition \ref{diff-op-prop}
is closely related to the algebra generated by the operators $(T_k(m,a))$ (see section \ref{Jac-sec}).
We will study these algebras in a sequel to this paper.
\operatorname{e}nd{rems}
Let us give an example of the calculation involving tautological cycles and using the operators
introduced in this paper.
\betaegin{prop}\lambdabel{pull-back-prop}
(i) Consider the cycles $\tau_k({\cal C})\in{\Bbb C}H^{k-1}({\cal J})$ (see \operatorname{e}qref{tau-eq}).
Their pull-backs under the morphisms $\sigma_N:{\cal C}^{[N]}\to{\cal J}$ are given by
\betaegin{align*}
&\sigma_N^*\tau_k({\cal C})=(-1)^kN^k\psi^{k-1}[{\cal C}^{[N]}]+\\
&\sum_{i+n+m+p+l=k}(-1)^{n+l+m}\frac{k!}{(i+n)!(m+p)!l!}S(i+n,i)S(m+p,m)N^l\psi^{p+l}
{\cal D}e_{i*}(K^n)*[p_0]^{*m}*[{\cal C}^{[N-m-i]}]\\
&-\sum_{i+n+m+p+l+q=k, q\ge 1}(-1)^{n+l+m+q}\frac{k!q!}{(i+n+q)!(m+p)!l!}{i+q\choose i}
{m\choose q} S(i+n+q,i+q)S(m+p,m)N^l\tildemes\\
&\psi^{p+l+n+q-1}[p_0]^{*(m+i)}*[{\cal C}^{[N-m-i]}],
\operatorname{e}nd{align*}
where $[p_0]^{*m}$ denotes the $m$-th power with respect to the Pontryagin product,
$S(\cdotot,\cdotot)$ are the Stirling numbers of the second kind. Note that ${\cal D}e_0:{\cal C}\to S$
is just the projection to the base.
\noindent
(ii) For $k>g+\dim S+1$ one has $\tau_k({\cal C})=0$, and hence, $\sigma_N^*\tau_k({\cal C})=0$.
On the other hand, for $k>2g$ the class $\tau_k({\cal C})$ becomes zero in ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) The idea is to use the formula
$$\sigma_N^*\tau_k({\cal C})=T_k(0,{\cal C})([{\cal C}^{[N]}])$$
(see \operatorname{e}qref{T-k-0-eq}). From Proposition \ref{T-P-prop}
we get the following expression for $T_k(0,{\cal C})$:
$$T_k(0,{\cal C})=(-1)^kP_{1,1}({\cal C})^k\psi^{k-1}+\sum_{i+n+j+l=k}(-1)^{n+j+l}\frac{k!}{(i+n)!j!l!}S(i+n,i)
P_{i,i}(K^n)P_{1,1}([p_0])^jP_{1,1}({\cal C})^l\psi^l.$$
Recall that $P_{1,1}({\cal C})x=Nx$ for $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
Set $t=[p_0]\in{\Bbb C}H^1({\cal C})$, $u=[{\cal C}]\in{\Bbb C}H^0({\cal C})$.
It follows easily from Proposition \ref{diff-op-prop}(ii) that the subalgebra
${\Bbb C}H^*(S)[t,u^{[\betaullet]}]\subset\widetilde{\TCH}^*({\cal C}^{[\betaullet]})$ is preserved by
the operator $P_{1,1}([p_0])$, and its action is given by
$$P_{1,1}([p_0])|_{{\Bbb Z}[t,u^{[\betaullet]}]}=t(\partial_u-\psi \partial_t).$$
From this we deduce by induction in $j$ that
$$P_{1,1}([p_0])^j([{\cal C}^{[N]}])=P_{1,1}([p_0])^j(u^{[N]})=\sum_{m=0}^j(-\psi)^{j-m}S(j,m)t^mu^{[N-m]}$$
for $j\ge 0$.
Next, from Proposition \ref{diff-op-prop}(ii) we see that for an element $f\in{\Bbb C}H^*(S)[t,u^{[\betaullet]}]$
one has
$$P_{i,i}(K^n)f={\cal D}e_{i_*}(K^n)\partial_u^i f-
t^i\cdotot \sum_{q\ge 1}(-1)^q{i\choose q}\psi^{n+q-1}\partial^{i-q}_u\partial^q_t f.$$
From this we get formulas for $P_{i,i}(K^n)P_{1,1}([p_0])^j([{\cal C}^{[N]})$ for all $i,j,n$.
It remains to substitute them
into our expression for $T_k(0,{\cal C})([{\cal C}^{[N]}])$.
\noindent (ii) The first assertion is clear since $\tau_k({\cal C})\in{\Bbb C}H^{k-1}({\cal J})$. The second follows from the equality $\tau_k({\cal C})=T_k(0,{\cal C})[{\cal J}]$ together with the fact that
$T_k(0,{\cal C})=X_{0,k}$ is zero as an operator on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$ for $k>2g$.
\operatorname{e}d
\betaegin{cor}\lambdabel{pull-back-cor} Assume that $S$ is a point.
\noindent
(i) For $k>1$ one has
\betaegin{align*}
&\sigma_N^*\tau_k(C)={\Bbb G}a_{e,k}*[C^{[N-k]}]
-\sum_{i+m=k-1}(-1)^m{k\choose m}{i+1\choose 2}{\cal D}e_{i*}(K)*[p_0]^{*m}*[C^{[N-k+1]}]\\
&-2\delta_{k,2}[p_0]*[C^{[N-1]}],
\operatorname{e}nd{align*}
where
$${\Bbb G}a_{e,k}=\sum_{i+m=k}(-1)^m{k\choose m}[{\cal D}e_i(C)]*[p_0]^{*m}\in{\Bbb C}H^{k-1}(C^{[k]})$$
\noindent
(ii) Assume that $K=(2g-2)[p_0]$. Then for $k>1$ one has
$$\sigma_N^*\tau_k(C)={\Bbb G}a_{e,k}*[C^{[N-k]}]-2g\delta_{k,2}[p_0]*[C^{[N-1]}].$$
Without this assumption one has
$$\sigma_N^*\tau_k(C)\sigmam_{a.e.}{\Bbb G}a_{e,k}*[C^{[N-k]}]-2g\delta_{k,2}[p_0]*[C^{[N-1]}],$$
where $\sigmam_{a.e.}$ denotes algebraic equivalence.
\noindent
(iii) Assume that $K=(2g-2)[p_0]$. Then one has ${\Bbb G}a_{e,k}=0$ for $k>g+1$.
Also, ${\Bbb G}a_{e,k}$ is a torsion class for $k\ge g/2+2$.
\noindent
(iv) One has ${\Bbb G}a_{e,k}\sigmam_{a.e.} 0$ for $k>g+1$.
Furthermore, if $C$ admits a morphism of degree $d\ge 1$ to ${\Bbb P}^1$
then for $k\ge d+1$ some multiple of ${\Bbb G}a_{e,k}$ is algebraically equivalent to zero.
\operatorname{e}nd{cor}
\noindent {\it Proof} . (i) This follows from Proposition \ref{pull-back-prop}(i).
\noindent (ii) This follows from (i) since ${\cal D}e_{i*}[p_0]=[p_0]^{*i}$.
\noindent (iii) Note that under these assumptions we have either $k>2$ or $g=0$. In either case
taking $N=k$ in (ii) we get $\sigma_k^*\tau_k(C)={\Bbb G}a_{e,k}$.
Now the first assertion follows from the trivial vanishing $\tau_k(C)=0$ for $k>g+1$.
The second assertion follows from vanishing of $\tau_k(C)$ in ${\Bbb C}H^*(J)_{{\Bbb Q}}$ for $k\ge g/2+2$ that
is checked as follows. Note that $\tau_k(C)/k!=p_{k-1}$, where $(p_n)$ are the tautological
classes considered in \cite{P-lie}. Now the required vanishing follows from Proposition 4.2 of \cite{P-lie}
(note that the condition $K=(2g-2)[p_0]$ implies the vanishing of all the classes $q_n$).
\noindent (iv) The first statement follows from $K\sigmam_{a.e.}(2g-2)[p_0]$ as in (iii). For the second statement we use the fact that for $k\ge d+1$ some multiple of
$\tau_k(C)$ is algebraically equivalent to zero in this case
by the result of Colombo and Van Geemen in \cite{CG}.
\operatorname{e}d
\betaegin{rem}
The {\it modified diagonal classes} ${\Bbb G}a_{e,k}\in{\Bbb C}H^{k-1}(C^{[k]})$ were introduced
by Gross and Schoen in \cite{GS}. Parts (iii) and (iv) of the above
corollary are related to Propositions 4.5 and 4.8 of {\it loc. cit.}. In the case when $C$ is hyperelliptic
and $p_0$ is stable under the hyperelliptic involution
Gross and Schoen show that ${\Bbb G}a_{e,3}=0$. By
Corollary \ref{pull-back-cor}(ii) this is equivalent to the fact that $\tau_3(C)=0$ in this case
(since $\sigma_N^*$ is injective for sufficiently large $N$).
\operatorname{e}nd{rem}
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\operatorname{e}nd{document}
|
\begin{document}
\date{Version of \today}
\title[Finite dimensional approximation of forward prices]{Approximation of forward curve models in commodity markets with arbitrage-free
finite dimensional models}
\begin{abstract}
In this paper we show how to approximate a Heath-Jarrow-Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have
a finite dimensional state space. Moreover, we recover a closed form representation of the forward price dynamics in the approximation models and derive the rate of convergence uniformly over an interval of time to maturity to the true dynamics under certain additional smoothness conditions. In the Markovian case we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient
Riesz basis on the state space of the term structure dynamics.
\end{abstract}
\author[Benth]{Fred Espen Benth}
\address[Fred Espen Benth]{\\
Department of Mathematics \\
University of Oslo\\
P.O. Box 1053, Blindern\\
N--0316 Oslo, Norway}
\email[]{fredb\@@math.uio.no}
\urladdr{http://folk.uio.no/fredb/}
\author[Kr\"uhner]{Paul Kr\"uhner}
\address[Paul Kr\"uhner]{\\
Financial \& Actuarial Mathematics\\
Vienna University of Technology\\
Wiedner Hauptstr. 8/E105-1\\
AT-1040 Vienna, Austria}
\email[]{[email protected]}
\urladdr{https://fam.tuwien.ac.at/~paulkrue/}
\thanks{F.\ E.\ Benth acknowledges financial support from the project "Managing Weather Risk in Energy Markets (MAWREM)", funded by the ENERGIX program
of the Norwegian Research Council.}
\subjclass[2010]{91B24, 91G20}
\keywords{Energy markets, Heath-Jarrow-Morton, Non harmonic Fourier analysis, arbitrage free approximations}
\maketitle
\section{Introduction}
We develop arbitrage-free approximations to the forward term structure dynamics in commodity
markets. The approximating term structure models have finite dimensional state space, and therefore
tractable for further analysis and numerical simulation. We provide results on the convergence of the
approximating term structures and characterize the speed under reasonable smoothness properties
of the true term structure. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.
In the context of fixed-income markets, Heath, Jarrow and Morton~\cite{HJM} propose to model the entire term structure of interest rates. Filipovi\'c \cite{filipovic.01} reinterprets this approach in the so-called Musiela parametrisation, i.e., studying the so-called forward rates as solutions of first-order stochastic
partial differential equations. This class of stochastic partial differential equations is often referred to as
the Heath-Jarrow-Morton-Musiela (HJMM) dynamics. This highly successful method has been transferred to
other markets, including commodity and energy futures markets (see Clewlow and
Strickland~\cite{CS} and Benth,
Saltyte Benth and Koekebakker~\cite{BSBK-book}), where the term structure of forward and
futures prices are modelled by similar stochastic partial differential equations.
An important stream of research in interest rate modelling has been so-called finite dimensional realizations
of the solutions of the HJMM dynamics (see e.g., Bj\"ork and Svensson~\cite{BjorkSvensson}, Bj\"ork and Landen~\cite{BjorkLanden}, Filipovic and Teichmann~\cite{FT} and Tappe~\cite{Tappe2010}). Starting out with an equation for the forward rates driven by a
$d$-dimensional Wiener process, the question has been under what conditions on the volatility and drift
do we get solutions which belongs to a finite dimensional space, that is, when can the dynamics of the
whole curve be decomposed into a finite number of factors. This property has a close connection with
principal component analysis (see Carmona and Tehranchi~\cite{CT}), but is also convenient when
it comes to further analysis like estimation, simulation, pricing and portfolio management (see
Benth and Lempa~\cite{BL} for the latter).
In energy markets like power and gas, there is empirical and economical evidence for high-dimensional
noise. Moreover, the noise shows clear leptokurtic signs (see Benth, \v{S}altyt\.e Benth and Koekebakker~\cite{BSBK-book} and references therein). These empirical insights motivate the use of
infinite dimensional L\'evy processes driving the noise in the HJMM-dynamics modelling the forward term structure. We refer to Carmona and Tehranchi~\cite{CT} for a thorough analysis of HJMM-models with
infinite dimensional Gaussian noise in interest rate markets. Benth and Kr\"uhner~\cite{BK-stochastics}
introduced a convenient class of infinite dimensional L\'evy processes via subordination of Gaussian
processes in infinite dimensions. These models were used in analysing stochastic partial differential
equations with infinite dimensional L\'evy noise in Benth and Kr\"uhner~\cite{BK-coms}. Further,
pricing and hedging of derivatives in energy markets based on such models were studied in
Benth and Kr\"uhner~\cite{BK-siam}.
The present paper is motivated by the need of an arbitrage-free approximation of Heath, Jarrow, Morton style models -- using the Musiela parametrisation -- in electricity finance. Related research has been carried out by Henseler, Peters and Seydel \cite{Henseler.al.15} who construct a finite-dimensional affine model where a refined principle component analysis (PCA) method does yield an arbitrage free approximation of the term structure model.
Our main result Theorem~\ref{t:main statement} states that the arbitrage-free models for the underlying forward curve
process $f(t,x)$, $x\geq0$ being time to maturity and $t\geq 0$ is current time, can be approximated with processes of the form
$$
f_k(t,x) = S_k(t) + \sum_{n=-k}^k U_n(t)g_n(x) \,,
$$
where $S_k$ denotes the spot prices in the approximating model, $g_{-k},\dots,g_k$ are deterministic functions and $U_{-k},\dots,U_k$ are one-dimensional Ornstein Uhlenbeck type processes. Obviously, models of this type are much easier to handle in applications than general solutions for the HJMM equation. The
approximation $f_k$ is again a solution of an HJMM equation, and as such being an arbitrage-free model
for the forward term structure. We prove a uniform convergence in space of $f_k$ to the "real" forward
price curve $f$, pointwise in time. The convergence rate is of order $k^{-1}$ when the
forward curve $x\mapsto f(t,x)$ is twice continuously differentiable. Our approach
is an alternative to numerical approximations of the HJMM dynamics based on finite difference schemes or finite element
methods, where arbitrage-freeness of the approximating dynamics is not automatically ensured.
We refer to Barth~\cite{Barth} for an analysis of finite element methods
applied to
stochastic partial differential equations of the type we study.
We refine our results to the Markovian case, where the convergence is slightly strengthened to be uniform over time as well. Our approach goes via the explicit construction of
a Riesz basis for a subspace of the so-called Filipovi\'c space (see Filipovi\'c~\cite{filipovic.01}),
a separable Hilbert space of absolutely continuous functions on the positive real line with
(weak) derivative disappearing at a certain speed at infinity. The basis will be the functions $g_n$ in the
approximation $f_k$, and the subspace is defined by concentrating the functions in the Filipovi\'c space
to a finite time horizon $x\leq T$. This space was defined in Benth and Kr\"uhner~\cite{BK-coms},
and we extend the analysis here to accomodate the arbitrage-free finite dimensional approximation of
the HJMM-dynamics. We rest on properties of $C_0$-semigroups and stochastic integration with
respect to infinite dimensional L\'evy processes (see Peszat and Zabczyk~\cite{peszat.zabczyk.07})
in the analysis.
This paper is organised as follows. In Section~\ref{s:the model} we start with the mathematical formulation
of the HJMM dynamics for forward rates set in the Filipovi\'c space. The Riesz basis that will make the foundation for our approximation is defined and analysed in detail in Section~\ref{s:mathematical Preliminaries}. The arbitrage-free finite dimensional approximation to term structure modelling is
constructed in Section~\ref{s:approximation}, where we study convergence properties. The Markovian
case is analysed in the last Section~\ref{s:markovian}.
\section{The model of the forward price dynamics}
\label{s:the model}
Throughout this paper we use the Hilbert space
$$
H_\alpha := \left\{f\in AC(\mathbb R_+,\mathbb C): \int_0^\infty |f'(x)|^2e^{\alpha x} dx <\infty\right\}\,,
$$
where $AC(\mathbb R_+,\mathbb C)$ denotes the space of complex-valued absolutely continuous functions on $\mathbb R_+$. We endow
$H_{\alpha}$ with the scalar product $\langlef,g\rangle_\alpha:=f(0)\overline{g}(0) + \int_0^\infty f'(x)\overline{g}'(x) e^{\alpha x}dx$, and denote
the associated norm by $\|\cdot\|_{\alpha}$. Filipovi\'c~\cite[Section 5]{filipovic.01} shows that $(H_\alpha,\|\cdot\|_\alpha)$ is a separable Hilbert space\footnote{Note that
Filipovi\'c~\cite{filipovic.01} does not consider complex-valued functions. In our context, this minor extension is convenient, as will be clear later.}.
This space has been used in Filipovi\'c~\cite{filipovic.01} for term structure modelling of bonds and many mathematical properties have been derived therein.
We will frequently refer to $H_{\alpha}$ as the {\it Filipovi\'c space}.
We next introduce our dynamics for the term structure of forward prices in a commodity market. Denote by $f(t,x)$ the price at time $t$ of
a forward contract where time to delivery of the underlying commodity is $x\geq 0$. We treat $f$ as a stochastic process in time with values in
the Filipovi\'c space $H_{\alpha}$. More specifically, we assume that the process $\{f(t)\}_{t\geq 0}$ follows the HJM-Musiela model
which we formalize next.
On a complete filtered probability space $(\Omega,\{\mathcal{F}_t\}_{t\geq 0},\mathcal{F},P)$, where the filtration is assumed to be complete and right continuous,
we work with an $H_{\alpha}$-valued L\'evy process $\{L(t)\}_{t\geq 0}$ (cf.\ Peszat and Zabczyk~\cite[Theorem 4.27(i)]{peszat.zabczyk.07} for the construction of $H_\alpha$-valued L\'evy processes). We assume that $L$ has finite variance and mean equal to zero, and denote its covariance operator by $\mathcal Q$.
Let $f_0\in H_\alpha$ and $f$ be the solution of the stochastic partial differential equation (SPDE)
\begin{equation}
\label{e:HJMM-equation}
df(t) = \partial_x f(t) dt + \beta(t) dt + \Psi(t)dL(t),\quad t\geq 0, f(0)=f_0
\end{equation}
where $\beta\in L^1((\Omega\times\mathbb R_+,\mathcal P,P\otimes\lambda),H_\alpha)$, $\mathcal P$ being the predictable $\sigma$-field, and
$\Psi \in \mathcal{L}^2_{L}(H_\alpha):=\bigcup_{T>0}\mathcal{L}^2_{L,T}(H_\alpha) $ where the latter space is defined as in {Peszat and Zabczyk~\cite[page 113]{peszat.zabczyk.07}}. For $t\geq 0$, denote by $\mathcal{U}_t$ the shift semigroup on $H_{\alpha}$ defined by
$\mathcal{U}_t f=f(t+\cdot)$ for $f\in\mathcal H_{\alpha}$. It is shown in Filipovi\'c~\cite{filipovic.01} that $\{\mathcal{U}_t\}_{t\geq 0}$ is a $C_0$-semigroup on
$H_{\alpha}$, with generator $\partial_x$. Recall, that any $C_0$-semigroup admits the bound $\Vert\mathcal U_t\Vert_{\mathrm{op}}\leq Me^{wt}$ for some $w,M>0$ and any $t\geq 0$. Here, $\|\cdot\|_{\mathrm{op}}$
denotes the operator norm. In fact, in Filipovi\'c~\cite[Equation (5.10)]{filipovic.01} and Benth and Kr\"uhner~\cite[Lemma~3.4]{benth.kruehner.15}
it is shown that $\Vert\mathcal U_t\Vert_{\mathrm{op}}\leq C_{\mathcal{U}}$ for any $t\geq 0$ and a constant $C_{\mathcal{U}}:=\sqrt{2(1\wedge\alpha^{-1})}$. Thus $s\mapsto \mathcal U_{t-s}\beta(s)$ is Bochner-integrable and $s\mapsto \mathcal U_{t-s}\Psi(s)$ is integrable with respect to $L$. The unique mild solution of \eqref{e:HJMM-equation} is
\begin{equation}
\label{e:HJMM-equation-mild}
f(t)=\mathcal{U}_tf_0+\int_0^t\mathcal{U}_{t-s}\beta(s)\,ds+\int_0^t\mathcal{U}_{t-s}\Psi(s)\,dL(s)\,.
\end{equation}
If we model the forward price dynamics $f$ in a risk-neutral setting, the drift coefficient $\beta(t)$ will naturally be zero in order to ensure
the (local) martingale property of the process $t\mapsto f(t,\tau-t)$, where $\tau\geq t$ is the time of delivery of the forward. In this
case, the probability $P$ is to be interpreted as the equivalent martingale measure (also called the pricing measure). However, with
a non-zero drift, the forward model is stated under the market probability and $\beta$ can be related to the risk premium in the market.
In energy markets like power and gas, the forward contracts deliver over a period, and forward prices can be expressed
by integral operators on the Filipovi\'c space applied on $f$ (see Benth and Kr\"uhner~\cite{benth.kruehner.14,benth.kruehner.15} for more details).
The dynamics of $f$ can also be considered as a model for the forward rate in fixed-income theory, see Filipovi\'c~\cite{filipovic.01}. This is indeed the
traditional application area and point of analysis of the SPDE in \eqref{e:HJMM-equation}. Note, however, that the original no-arbitrage condition in the HJM approach for interest rate markets is different from the no-arbitrage condition used here. If $f$ is understood as the forward rate modelled
in the risk-neutral setting, there is a no-arbitrage relationship between the drift $\beta$, the volatility $\sigma$ and the covariance of the driving noise $L$. We refer to Carmona and Tehranchi~\cite{CT} for a detailed analysis.
\section{A Riesz basis for the Filipovi\'c space}\label{s:mathematical Preliminaries}
In this section we introduce a Riesz basis for a suitable subspace of $H_\alpha$
defined in Benth and Kr\"uhner~\cite[Appendix A]{benth.kruehner.14} and present various of its
properties. Moreover, we give refined statements for this basis and also identify new properties.
We recall from Young~\cite{Young.80} that any Riesz basis $\{g_n\}_{n\in\mathbb{N}}$ on a separable Hilbert space can
be expressed by $g_n = \mathcal T e_n$ where $\{e_n\}_{n\in\mathbb N}$ is an orthonormal basis and $\mathcal T$ is a bounded invertible linear operator.
For further properties and definitions of Riesz bases, see Young~\cite{Young.80}.
In Section \ref{s:approximation} we want to employ the spectral method to an approximation of the SPDE in \eqref{e:HJMM-equation} involving the differential operator on the Filipovi\'c space $H_\alpha$. Thus, it would be convenient to have available the eigenvector basis for the differential operator. However, its eigenvectors do not seem to have nice basis properties. Instead, we propose to use a system of vectors which forms a Riesz basis which turns out to be almost an eigenvector system for the differential operator. This property will be made precise in Propositions~\ref{p:U and the riesz basis} and \ref{l:commutator of U and projectors}.
Finally, we will identify the convergence speed of the Riesz basis expansion.
Fix $\lambda>0$, $T>0$, and introduce
\begin{equation}
\mathrm{cut}:\mathbb R_+\rightarrow [0,T)\,,\qquad x\mapsto x-\max\{Tz: z\in\mathbb Z:Tz\leq x\}\,,
\end{equation}
and
\begin{equation}
\label{def-A-operator}
\mathcal A:L^2([0,T),\mathbb C)\rightarrow L^2(\mathbb R_+,\mathbb C)\,,\qquad
f\mapsto \left(x\mapsto e^{-\lambda x}f(\mathrm{cut}(x))\right)\,.
\end{equation}
Here, $L^2(A,\mathbb C)$ is the space of complex-valued square integrable functions on the Borel set $A\subset\mathbb R_+$ equipped with the
Lebesgue measure. The inner product of $L^2(A,\mathbb C)$ will be denoted $(\cdot,\cdot)_2$ and the corresponding norm $|\cdot|_2$. We remark that
the set $A$ will be clear from the context and thus not indicated in the notation for
norm and inner product.
We define
\begin{align}
g_*(x) &:= 1, \label{e:g-star-def}\\
g_n(x) &:= \frac{1}{\lambda_n\sqrt{T}}\left(\exp\left(\lambda_nx\right)-1\right)\,, \label{e:g-n-def}
\end{align}
where
\begin{equation}
\label{e:lambda-n-def}
\lambda_n:=\frac{2\pi i}{T}n-\lambda-\frac{\alpha}{2}\,,
\end{equation}
for any $n\in\mathbb Z$, $x\geq0$.
It is simple to verify that $g_n\in H_\alpha$ for any $n\in\mathbb Z$ and $g_*\in H_\alpha$.
As we will see, the system of vectors $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ forms a Riesz basis and we will use this to obtain arbitrage-free finite-dimensional approximations of the forward price dynamics \eqref{e:HJMM-equation}.
We start our analysis with some elementary properties of the operator $\mathcal A$ which have been proven in Benth and Kr\"uhner~\cite{benth.kruehner.14}.
\begin{lem}\label{l:stetige Einbettung}
$\mathcal A$ is a bounded linear operator and its range is closed in $L^2(\mathbb R_+,\mathbb C)$. Moreover,
$$ \frac{e^{-2T\lambda}}{1-e^{-2T\lambda}}| f|_2^2 \leq |\mathcal Af|_2^2 \leq \frac{1}{1-e^{-2T\lambda}}| f|_2^2$$
for any $f\in L^2([0,T),\mathbb C)$.
\end{lem}
\begin{proof}
This proof can be found in Benth and Kr\"uhner~\cite[Lemma A.1]{benth.kruehner.14}.
\end{proof}
In the following Proposition~\ref{p:Riesz basis on L2}, we calculate a Riesz basis of the space $\mathrm{ran}(\mathcal{A})$
and its biorthogonal system. The Riesz basis will be given as the image of an orthonormal basis of $L^2([0,T),\mathbb C)$. Consequently, its biorthogonal
system is given by the image of $(\mathcal A^{-1})^*$, which we calculate in the Lemma below:
\begin{lem}
\label{lem:dual_A}
The dual $(\mathcal A^{-1})^*$ of the inverse of $\mathcal A:L^2([0,T),\mathbb C)\rightarrow \mathrm{ran}(\mathcal{A})$
is given by
\begin{align*}
(\mathcal A^{-1})^*&:L^2([0,T),\mathbb C)\rightarrow \mathrm{ran}(\mathcal{A}) ,\\
(\mathcal A^{-1})^*f(x)&=(1-e^{-2\lambda T})e^{-\lambda x}\left( e^{2\lambda \mathrm{cut}(x)}f(\mathrm{cut}(x)) \right) \\
&= (1-e^{-2\lambda T})e^{2\lambda \mathrm{cut}(x)} \mathcal Af(x),\quad x\geq0\,.
\end{align*}
\end{lem}
\begin{proof}
Let $f,g\in L^2([0,T],\mathbb C)$
and define $h(x):=(1-e^{-2\lambda T})e^{2\lambda \mathrm{cut}(x)} \mathcal Af(x)$ for any $x\geq0$. Then we have
\begin{align*}
(h,\mathcal Ag)_2&= \int_0^\infty h(y) \overline{\mathcal Ag(y)} dy \\
&= (1-e^{-2\lambda T}) \sum_{n=0}^\infty \int_{nT}^{(n+1)T} e^{2\lambda(x-nT)} (e^{-\lambda x}f(x-nT))(e^{-\lambda x}\overline{g(x-nT)}) dx \\
&= (1-e^{-2\lambda T})\sum_{n=0}^\infty e^{-2\lambda nT} \int_{nT}^{(n+1)T} f(x-nT)\overline{g(x-nT)} dx \\
&= \int_0^T f(y)\overline{g(y)} dy\,.
\end{align*}
On the other hand,
\begin{align*}
((\mathcal A^{-1})^*f, \mathcal Ag)_2&=(f,g)_2= \int_0^Tf(y)\overline{g(y)} dy\,.
\end{align*}
Since $g$ is arbitrary, we have $h = (\mathcal A^{-1})^*f$ as claimed.
\end{proof}
Parts of the next proposition can be found in Benth and Kr\"uhner~\cite[Lemma A.3]{benth.kruehner.14}. In that paper there appears to be a gap in the proof which we have filled here.
\begin{prop}\label{p:Riesz basis on L2}
Define
$$ e_n(x) := \frac{1}{\sqrt{T}} \exp\left(\left(\frac{2\pi in}{T}-\lambda\right)x\right),\quad x\geq 0,n\in\mathbb Z.$$
Then $\{e_n\}_{n\in\mathbb Z}$ is a Riesz basis on the closed subspace $\mathrm{ran}(\mathcal{A})$ of $L^2(\mathbb R_+,\mathbb C)$ and
$$ F:=\{ f\in L^2(\mathbb R_+,\mathbb C): f(x)=0,x\in[0,T) \} $$
is a closed vector space compliment of $\mathrm{ran}(\mathcal{A})$. The continuous linear projector $\mathcal P_{\mathcal A}$ with range $\mathrm{ran}(\mathcal{A})$ and kernel $F$ has operator norm $\sqrt{\frac{1}{1-e^{-2\lambda T}}}$ and we have
$$ \mathcal P_{\mathcal A}f(x) = f(x),\quad x\in[0,T], f\in L^2(\mathbb R_+,\mathbb C).$$
The biorthogonal system $\{e_n\}^*_{n\in\mathbb Z}$ for the Riesz basis $\{e_n\}_{n\in\mathbb Z}$ is given by
$$ e_n^*(x) = \left(1-e^{-2\lambda T}\right)e^{2\lambda\mathrm{cut}(x)} e_n(x) $$
\end{prop}
\begin{proof}
Recall that the range of $\mathcal A$ is a closed subspace of $L^2(\mathbb R_+,\mathbb C)$ due to the lower bound given in Lemma \ref{l:stetige Einbettung}. Furthermore, $\{b_n\}_{n\in\mathbb Z}$ with
$$ b_n(x):= \frac{1}{\sqrt{T}}\exp\left(\frac{2\pi i n}{T}x\right),\quad n\in\mathbb Z,x\in[0,T)$$
is an orthonormal basis of $L^2([0,T],\mathbb C)$. Observe, that $e_n = \mathcal Ab_n$ and hence $\{e_n\}_{n\in\mathbb Z}$ becomes a Riesz basis of
$\mathrm{ran}(\mathcal{A})$.
Define the continuous linear operators
\begin{align*}
\mathcal M_{\lambda} &:L^2(\mathbb [0,T),\mathbb C)\rightarrow L^2([0,T),\mathbb C),\mathcal M_{\lambda}f(x):=e^{\lambda x}f(x),\\
\mathcal C &:L^2(\mathbb R_+,\mathbb C)\rightarrow L^2([0,T),\mathbb C),f\mapsto f\vert_{[0,T)}
\end{align*}
and $\mathcal P_{\mathcal A}:=\mathcal A\mathcal M_{\lambda}\mathcal C$. Observe, that $\mathcal M_{\lambda}\mathcal C\mathcal A$ is the identity operator on $L^2([0,T),\mathbb C)$ and hence $\mathcal P_{\mathcal A}^2=\mathcal P_{\mathcal A}$. Therefore, $\mathcal P_{\mathcal A}$ is a continuous linear projection with kernel $F$ and range $\mathrm{ran}(\mathcal{A})$.
Let $f\in L^2(\mathbb R_+,\mathbb C)$ be orthogonal to any element of the kernel of $\mathcal P_{\mathcal A}$. Then $f(x)=0$ Lebesgue-a.e.\ for any $x\geq T$. Hence, we have
\begin{align*}
|\mathcal P_{\mathcal A}f|_2^2 &= \sum_{n\in\mathbb N} \int_{nT}^{nT+T} (e^{-\lambda x} e^{\lambda(x-nT)})^2|f(x-nT)|^2dx \\
&= \sum_{n\in\mathbb N} e^{-2n\lambda T} \vert f\vert_2^2 \\
&= \frac{1}{1-e^{-2\lambda T}} \vert f\vert_2^2
\end{align*}
and it follows that $\Vert\mathcal P_{\mathcal A}\Vert_{\mathrm{op}} = \sqrt{\frac{1}{1-e^{-2\lambda T}}}$.
According to Lemma~\ref{lem:dual_A}, we have
\begin{align*}
e_n^*(x) &= (\mathcal{A}^{-1})^*b_n(x) \\
&= (1-e^{-2\lambda T})e^{-\lambda x}\left( e^{2\lambda \mathrm{cut}(x)}b_n(\mathrm{cut}(x)) \right) \\
&= \left(1-e^{-2\lambda T}\right)e^{2\lambda\mathrm{cut}(x)} e_n(x)\,,
\end{align*}
for any $n\in\mathbb Z$, $x\geq0$, as required.
\end{proof}
The statements collected in this section have been about the space $L^2(\mathbb R_+,\mathbb C)$ so far. However, we are actually interested in the space $H_\alpha$ which has a natural and simple isometry to $\mathbb C\times L^2(\mathbb R_+,\mathbb C)$. The next corollary will translate the $L^2(\mathbb R_+,\mathbb C)$-statements above to $H_\alpha$. Before stating it, we introduce a notation for later use:
Define
\begin{equation}
\label{def:Theta-op}
\mathcal Theta:H_\alpha\rightarrow \mathbb C\times L^2(\mathbb R_+,\mathbb C), f\mapsto (f(0), w_\alpha f')\,,
\end{equation}
where $w_\alpha(x):=e^{x\alpha/2}$ for $x\geq 0$. Then $\mathcal Theta$ is an isometry of Hilbert spaces. Its inverse is given by
\begin{equation}
\label{def:Theta-inv-op}
\mathcal Theta^{-1}:\mathbb C\times L^2(\mathbb R_+,\mathbb C)\rightarrow H_\alpha,(z,f)\mapsto z+\int_0^{(\cdot)} w_{\alpha}^{-1}(y)f(y)dy\,.
\end{equation}
We use these operators to prove:
\begin{cor}\label{k:Riesz basis on H}
The system $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ defined in \eqref{e:g-star-def}-\eqref{e:g-n-def} is a Riesz basis of a closed subspace $H_\alpha^{T}$ of $H_\alpha$.
Indeed, $H_\alpha^{T}$ is the space generated by
$\{g_*,\{g_n\}_{n\in\mathbb Z}\}$.
Moreover, there is a continuous linear projector $\Pi$ with range $H_\alpha^{T}$ and operator norm $\sqrt{\frac{1}{1-e^{-2\lambda T}}}$ such that
$$ \Pi h(x) = h(x), \quad h\in H_\alpha,x\in[0,T]. $$
Consequently, $\Pi\mathcal U_th(x) = \mathcal U_t\Pi h(x) = h(x+t)$ for any $t\in[0,T]$ and any $x\in[0,T-t]$.
The biorthogonal system $\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$ is given by
\begin{align*}
g_*^*(x) &= 1 \\
g_n^*(x) &= \int_0^x e^{-y\frac{\alpha}{2}} e_n^*(y) dy
\end{align*}
where $e_n^*$ is given in Proposition \ref{p:Riesz basis on L2} for any $n\in\mathbb Z$, $x\geq0$.
\end{cor}
\begin{proof}
Let $\{e_n\}_{n\in\mathbb Z}$ be the Riesz basis from Proposition \ref{p:Riesz basis on L2}, $V$ the linear vector space generated by $\{e_n\}_{n\in\mathbb Z}$
(which is in fact $\mathrm{ran}(\mathcal{A})$) and $\mathcal P_{\mathcal{A}}$ the projector from that proposition. Then $\{(1,0),\{(0,e_n)\}_{n\in\mathbb Z}\}$ is a Riesz basis of $\mathbb C\times V$. Furthermore, $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ is a Riesz basis of $\mathcal Theta^{-1}(\mathbb C\times V)$ because $g_*=\mathcal Theta^{-1}(1,0)$ and $g_n=\mathcal Theta^{-1}(0,e_n)$. Define $\Pi:=\mathcal Theta^{-1}(\mathrm{Id},\mathcal P_{\mathcal{A}})\mathcal Theta$. Then $\Pi$ is a linear projector with the same bound as $\mathcal P_{\mathcal A}$ where
$$
(\mathrm{Id},\mathcal P_{\mathcal A})(z,f):=(z,\mathcal P_{\mathcal A}f),\quad z\in\mathbb C,f\in L^2(\mathbb R_+,\mathbb C)\,.
$$
Let $h\in H_\alpha$. Observe that for any $x\in [0,T]$, $\mathrm{cut}(y)=y$ when $0\leq y\leq x$. We have from the definition of the various operators that
\begin{align*}
\Pi h(x)&=\mathcal Theta^{-1}(\mathrm{Id},\mathcal P_{\mathcal{A}})(h(0),\exp(\alpha\cdot/2)h') (x)\\
&=\mathcal Theta^{-1}\left((h(0),(\exp((\lambda+\alpha/2)\cdot)h')\vert_{[0,T)}(\mathrm{cut}(\cdot)\exp(-\lambda\cdot))\right)(x) \\
&=h(0)+\int_0^x e^{-(\lambda+\alpha/2)y} e^{(\lambda+\alpha/2)\mathrm{cut}(y)}
h'(\mathrm{cut}(y))\,dy \\
&=h(0)+\int_0^xh'(y)\,dy=h(x)\,.
\end{align*}
Hence, $\Pi h(x)=h(x)$ for any $x\in[0,T]$.
\end{proof}
We remark in passing that trivially $g_*^*=g_*$.
In the next proposition we compute the action of the shifting semigroup $\{\mathcal U_t\}_{t\geq0}$ on the Riesz basis of Corollary \ref{k:Riesz basis on H} and the dual semigroup on the biorthogonal system.
\begin{prop}\label{p:U and the riesz basis}
For the Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ in \eqref{e:g-star-def}-\eqref{e:g-n-def} and its biorthogonal system $\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$
derived in Corollary~\ref{k:Riesz basis on H}, it holds
\begin{enumerate}
\item $\mathcal U_t g_n = e^{\lambda_n t}g_n + g_n(t)g_*$ and
\item $\mathcal U^*_tg_n^* = e^{\overline{\lambda_n} t}g_n^*$,
\end{enumerate}
for any $n\in\mathbb Z$.
\end{prop}
\begin{proof}
Claim (1) follows from a straightforward computation.
For claim (2), we compute
\begin{align*}
\mathcal U_t^*g_n^* &= g_*\langle\mathcal U_t^*g_n^*,g_*\rangle_{\alpha} + \sum_{k\in\mathbb Z} g^*_k\langle\mathcal U_t^*g_n^*,g_k\rangle_{\alpha} \\
&= g_*\langle g_n^*,\mathcal U_tg_*\rangle_{\alpha} + \sum_{k\in\mathbb Z} g^*_k\langleg_n^*,\mathcal U_tg_k\rangle_{\alpha} \\
&= e^{\overline{\lambda_n}t}g_n^*
\end{align*}
for any $n\in\mathbb Z$, $t\geq 0$. Thus, the Proposition follows.
\end{proof}
A certain Lie commutator plays a crucial role in comparing projected solutions to the SPDE~\eqref{e:HJMM-equation} with solutions to
the approximation. In the next proposition, we derive the essential results for convergence which will be used in the next
Section to analyse approximations of the SPDE~\eqref{e:HJMM-equation}.
\begin{prop}\label{l:commutator of U and projectors}
Let $k\in\mathbb N$, $t\geq0$, $H^{T}_\alpha$ be the closed subspace of $H_\alpha$ generated by the Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$
defined in \eqref{e:g-star-def}-\eqref{e:g-n-def} with biorthogonal system
$\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$ given in Corollary~\ref{k:Riesz basis on H}.
Define the projection
$$ \Pi_k:H^{T}_\alpha\rightarrow \text{span}\{g_*,g_{-k},\dots,g_k\},h\mapsto h(0)g_* + \sum_{n=-k}^k g_n\langleh,g_n^*\rangle_{\alpha}, $$
$c_{k,t}:=\sum_{\vert n\vert >k} g_n(t)g_n^*$, and the operator
$$ \mathcal C_{k,t}:H^{T}_\alpha\rightarrow \text{span}\{g_*\}, h\mapsto \langleh,c_{k,t}\rangle_{\alpha}g_*.$$
Then, $\|\Pi_k\|_{\text{op}}$ is bounded uniformly in $k$, $\Pi_kh\rightarrow h$, $\sup_{s\in[0,t]} \Vert\mathcal C_{k,s}h\Vert_\alpha\rightarrow 0$ for $k\rightarrow\infty$ and any $h\in H_\alpha^{T}$, and $[\Pi_k,\mathcal U_t] =\mathcal C_{k,t}$. Here, $[\Pi_k,\mathcal U_t]$ denotes the Lie commutator of $\Pi_k$ and $\mathcal U_t$, that is $[\Pi_k,\mathcal U_t]=\Pi_k\mathcal U_t-\mathcal U_t\Pi_k$.
Moreover, let $X$ be a stochastic process with values in $H_\alpha^{T}$
such that $X(t)=Y(t)+M(t)$ for some square integrable process $Y$ of finite variation and a square integrable martingale $M$. Then,
$$ \lim_{k\rightarrow\infty}\int_0^t\mathcal C_{k,t-s}dX(s)=0\,,$$
where the convergence is in $L^2(\Omega,H_\alpha)$.\footnote{$L^2(\Omega,H_\alpha)$ denotes the space of $H_{\alpha}$-valued random variables $Z$ with $\mathbb{E}[\Vert Z\Vert_\alpha^2]<\infty$.}
\end{prop}
\begin{proof}
Let $h\in H^T_\alpha$. Since $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ is a Riesz basis of $H^T_\alpha$ we have
$$
h = g_*\langleh,g_*\rangle_{\alpha} + \sum_{n\in\mathbb Z} g_n\langleh,g_n^*\rangle_{\alpha}\,,
$$
and hence we get $\Pi_kh\rightarrow h$ for $k\rightarrow \infty$.
We prove that the operator norm of $\Pi_k$ is uniformly bounded in $k\in\mathbb{N}$.
Recall from Corollary~\ref{k:Riesz basis on H} and \eqref{def:Theta-inv-op}
$g_n=\mathcal Theta^{-1}(0,\mathcal{A}b_n), n\in\mathbb{Z}$ and $g_*=\mathcal Theta^{-1}(1,0)$, where
$\mathcal{A}$ is defined in \eqref{def-A-operator} and $\{b_n\}_{n\in\mathbb{Z}}$ is
an orthonormal basis of $L^2([0,T],\mathbb{C})$. Without loss of generality, we assume
$h(0)=0$ for $h\in H_{\alpha}^T$, and find that
$$
\Pi_kh=\sum_{n=-k}^kg_n\langle h,g_n^*\mathrm{ran}gle_{\alpha}=\sum_{n=-k}^k
\mathcal{T}b_n(\mathcal{T}^{-1}h,b_n)_2=\mathcal {T}\sum_ {n=-k}^kb_n(\mathcal{T}^{-1}h,b_n)_2\,.
$$
Here, $\mathcal{T}f:=\mathcal Theta^{-1}(0,\mathcal{A}f)\in H_{\alpha}$ for
$f\in L^2([0,T],\mathbb{C})$, which is a bounded linear operator. Hence,
since $\sum_{n=-k}^kb_n(\mathcal{T}^{-1}h,b_n)_2$ is the projection of
$\mathcal{T}^{-1}h\in L^2([0,T],\mathbb{C})$ down to its first $2k+1$ coordinates,
$$
\|\Pi_kh\|_{\alpha}\leq|\mathcal{T}\|_{\text{op}}\left|\sum_{n=-k}^kb_n(\mathcal{T}^{-1}h,b_n)_2\right|_2\leq\|\mathcal{T}\|_{\text{op}}|\mathcal{T}^{-1}h|_2
$$
But since $\mathcal{T}^{-1}$ also is a bounded operator, it follows that
$\|\Pi_k\|_{\text{op}}\leq\|\mathcal{T}\|_{\text{op}}\|\mathcal{T}^{-1}\|_{\text{op}}$.
Benth and Kr\"uhner~\cite[Lemma 3.2]{benth.kruehner.14} yields that convergence in $H_\alpha$ implies local uniform convergence. Thus, as we know $h-\Pi_kh \rightarrow 0$,
it holds
$$ \sup_{s\in[0,t]} \vert h(s)-\Pi_{k}h(s)\vert \rightarrow 0\,, $$
for $k\rightarrow \infty$. Hence, we find
$$\sup_{s\in[0,t]}\left\vert\sum_{\vert n\vert>k}g_n(s)\langleh,g^*_n\rangle_{\alpha}\right\vert = \sup_{s\in[0,t]} \vert h(s)-\Pi_{k}h(s)\vert \rightarrow 0\,, $$
for $k\rightarrow\infty$. Therefore, $\sup_{s\in[0,t]}\Vert \mathcal C_{k,s}h\Vert_\alpha\rightarrow 0$ for $k\rightarrow\infty$.
Let $n\in\mathbb Z$. Then, by Proposition~\ref{p:U and the riesz basis}
\begin{align*}
[\Pi_k,\mathcal U_t]g_n &= \Pi_k(e^{\lambda_n t}g_n + g_n(t)g_*) - 1_{\{\vert n\vert\leq k\}}\mathcal U_tg_n \\
&= 1_{\{\vert n\vert\leq k\}}e^{\lambda_n t}g_n + g_n(t)g_* - 1_{\{\vert n\vert\leq k\}}(e^{\lambda_n t}g_n + g_n(t)g_*) \\
&= 1_{\{\vert n\vert>k\}}g_n(t)g_* \\
&= \mathcal C_{k,t}g_n
\end{align*}
for any $t\geq0$. Moreover,
\begin{align*}
[\Pi_k,\mathcal U_t]g_* = \Pi_kg_* - \mathcal U_tg_* = 0 = \mathcal C_{k,t}g_*.
\end{align*}
Let $\langle\langleM,M\rangle\rangle(t) = \int_0^t Q_sd\langleM,M\rangle(s)$ be the quadratic variation processes of the martingale $M$ given in
Peszat and
Zabczyk~\cite[Theorem 8.2]{peszat.zabczyk.07}\footnote{In Peszat and Zabczyk~\cite{peszat.zabczyk.07}, $\langle\langle\cdot,\cdot\rangle\rangle$ is called the operator angle bracket process, while $\langle\cdot,\cdot\rangle$ is the angle bracket process.}. Then, Peszat and Zabczyk~\cite[Theorem 8.7(ii)]{peszat.zabczyk.07} yields
\begin{align*}
\mathbb{E}\left(\Vert \int_0^t \mathcal C_{k,t-s}dM(s) \Vert_\alpha^2\right) &= \mathbb{E} \int_0^t \mathrm{Tr}(\mathcal C_{k,t-s}Q_s\mathcal C^*_{k,t-s})d\langleM,M\rangle(s)\,.
\end{align*}
Recall that for $h\in H_{\alpha}^T$, we find $\mathcal C_{k,t}h=\langleh,c_{k,t}\rangle_{\alpha}g_*$. Thus,
$$
\langleh,\mathcal C_{k,t}^*g_*\rangle_{\alpha}=\langle\mathcal C_{k,t}h,g_*\rangle_{\alpha}=\langleh,c_{k,t}\rangle_{\alpha}\,,
$$
which gives that $\mathcal C_{k,t}^*g_*=c_{k,t}$. For $g\in H_\alpha^T$ orthogonal to $g_*$ we have
$$ \langleh,\mathcal C_{k,t}^*g\rangle_\alpha =\langle\mathcal C_{k,t}h,g\rangle_{\alpha}=\langleh,c_{k,t}\rangle_{\alpha}\langleg_*,g\rangle_{\alpha}= 0 $$
for any $h\in H_\alpha^T$ and hence $\mathcal C_{k,t}^*g=0$. We get
\begin{align*}
\mathrm{Tr}(\mathcal C_{k,t-s}Q_s\mathcal C^*_{k,t-s}) &= \langle\mathcal C_{k,t-s}Q_s\mathcal C_{k,t-s}^*g_*,g_*\rangle_{\alpha} \\
&=\langleQ_sc_{k,t-s},c_{k,t-s}\rangle_{\alpha} \\
&\leq \Vert c_{k,t-s}\Vert_\alpha^2 \mathrm{Tr}(Q_s)\,.
\end{align*}
Hence,
\begin{align*}
\mathbb{E}\left(\left\Vert \int_0^t \mathcal C_{k,t-s}dM(s) \right\Vert_\alpha^2\right) &= \mathbb{E} \int_0^t \mathrm{Tr}(\mathcal C_{k,t-s}Q_s\mathcal C^*_{k,t-s})d\langleM,M\rangle(s) \\
&\leq \sup_{s\in[0,t]}\Vert c_{k,s}\Vert_\alpha^2 \mathbb{E}\left( \int_0^t\mathrm{Tr}(Q_s)d\langleM,M\rangle(s) \right) \\
&= \sup_{s\in[0,t]}\Vert c_{k,s}\Vert_\alpha^2 \mathbb{E}\left(\Vert M(t)-M(0)\Vert_\alpha^2\right) \\
&\rightarrow 0
\end{align*}
for $k\rightarrow\infty$. Similarily, we get
$$ \left\Vert \int_0^t \mathcal C_{k,t-s} dY(s) \right\Vert_\alpha^2 \leq \sup_{s\in[0,t]} \Vert c_{k,s}\Vert_\alpha^2\left(\int_0^t \Vert dY\Vert_\alpha(s)\right)^2 \rightarrow 0$$
as $k\rightarrow 0$, where $\Vert dY\Vert_\alpha$ denotes the total variation measure associated with $dY$ (see Dinculeanu~\cite[Definition \S 2.1]{dinculeanu.00}). The claim follows.
\end{proof}
The projection operator $\Pi_k$ plays an important role in the arbitrage-free approximation of the forward term structure. For notational convenience,
we denote
\begin{equation}
\label{def-H-k-space}
H_{\alpha}^{T,k}:=\text{span}\{g_*,g_{-k},\dots,g_k\}\,,
\end{equation}
for any $k\in\mathbb N$. From the above considerations, we have that $\Pi_k$ projects the space $H_{\alpha}^T$ down to $H_{\alpha}^{T,k}$.
Our next aim is to identify the convergence speed of approximations in $H_{\alpha}^{T,k}$ of certain smooth
elements $f\in H_{\alpha}^T$, that is, how close is $\Pi_kf$ to $f$ in terms
of number of Riesz basis functions. We show a couple of technical results first.
\begin{cor}\label{k:distance to ONB}
Let $f\in H_\alpha^T$. Then, we have
\small{$$ \frac{e^{-2\lambda T}}{1-e^{-2\lambda T}} \left(|f(0)|^2 + \sum_{n\in\mathbb Z} \vert\langlef,g_n^*\rangle_{\alpha}\vert^2\right)\leq \Vert f\Vert_\alpha^2 \leq \frac{1}{1-e^{-2\lambda T}} \left(|f(0)|^2 + \sum_{n\in\mathbb Z} \vert\langlef,g_n^*\rangle_{\alpha}\vert^2\right)\,.$$}
\end{cor}
\begin{proof}
Corollary~\ref{k:Riesz basis on H} states that $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ is a Riesz basis of $H_\alpha^T$. Moreover, it is given by $g_*=\mathcal Theta^{-1}(1,0)$, $g_n=\mathcal Theta^{-1}(0,e_n)$ for any $n\in\mathbb Z$ where $\mathcal Theta$ is the isometry given in \eqref{def:Theta-inv-op} and $\{e_n\}_{n\in\mathbb Z}$ is the Riesz basis given in Proposition \ref{p:Riesz basis on L2}. Moreover, Lemma~\ref{l:stetige Einbettung} yields that $e_n=\mathcal Ab_n$ for any $n\in\mathbb Z$ where $\{b_n\}_{n\in\mathbb Z}$ is an orthonormal basis of
$L^2([0,T],\mathbb C)$ and $\Vert \mathcal A\Vert_{\mathrm{op}}^2\leq \frac{1}{1-e^{-2\lambda T}}$. Thus, we can construct a Hilbert space with orthonormal basis $\{b_*,\{b_n\}_{n\in\mathbb Z}\}$ and a bounded linear operator
$\mathcal B$ with $\Vert \mathcal B\Vert_{\mathrm{op}}^2\leq \frac{1}{1-e^{-2\lambda T}}$, such that $g_*=\mathcal Bb_*$, $g_n=\mathcal Bb_n$.
Thus, we have
\begin{align*}
\Vert f\Vert_\alpha^2 &= \Vert g_*\langlef,g_*\rangle_{\alpha} + \sum_{n\in\mathbb Z}g_n\langlef,g_n^*\rangle_{\alpha} \Vert_\alpha^2 \\
&= \Vert \mathcal Bb_*\langlef,g_*\rangle_{\alpha} + \sum_{n\in\mathbb Z}\mathcal Bb_n\langlef,g_n^*\rangle_{\alpha} \Vert_\alpha^2 \\
&\leq \frac{1}{1-e^{-2\lambda T}} \left(|\langlef,g_*\rangle_{\alpha}|^2 + \sum_{n\in\mathbb Z}|\langlef,g_n^*\rangle_{\alpha}|^2 \right) \\
\end{align*}
where $\{g_*,\{g^*_n\}_{n\in\mathbb Z}\}$ denotes the biorthogonal system to $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ given in Corollary~\ref{k:Riesz basis on H}.
The lower inequality simply uses the lower inequality of Lemma~\ref{l:stetige Einbettung} instead.
\end{proof}
The next technical result connects the inner product of elements in $H_{\alpha}^T$ with the
biorthogonal basis functions to a simple
Fourier-like integral on $[0,T]$:
\begin{cor}
\label{cor:alpha-2-inner-prod}
Assume $f\in H_{\alpha}^T$. Then, for any $n\in\mathbb Z$,
$$
\langlef,g_n^*\rangle_{\alpha}=(1-e^{-2\lambda T})^{-1}T^{-1/2}\int_0^Tf'(x)\exp\left((-\frac{2\pi i}{T}n-\lambda+\frac{\alpha}{2})x\right)\,dx
$$
\end{cor}
\begin{proof}
First, recall that $g_n^*=\mathcal Theta^*(0,e_n)$ for $n\in\mathbb Z$, where $\mathcal Theta$ is defined
in the \eqref{def:Theta-inv-op}. Thus,
\begin{align*}
\langlef,g_n^*\rangle&=\langlef,\mathcal Theta^*(0,e_n)\rangle_{\alpha} \\
&=(\mathcal Theta f,(0,e_n))_{\mathbb C\times L^2(\mathbb R_+)} \\
&=((f(0),e^{\alpha\cdot/2}f'),(0,e_n))_{\mathbb C\times L^2(\mathbb R_+)} \\
&=(e^{\alpha\cdot/2}f',e_n)_2\,.
\end{align*}
Note that $\exp(\alpha\cdot/2)f'$ and $e_n=\mathcal A b_n$ are elements of $\mathrm{ran}(\mathcal A)$.
If $h\in\mathrm{ran}(\mathcal{A})$, then there exists a $\hat{h}\in L^2([0,T],\mathbb C)$ such that
$h=\mathcal A \hat{h}$, or, $h(x)=\exp(-\lambda x)\hat{h}(\mathrm{cut}(x))$. Observe that
for $x\in[0,T]$, $\hat{h}(x)=\exp(\lambda x)h(x)$. Then, if $g\in\mathrm{ran}(\mathcal{A})$, we find
\begin{align*}
(h,g)_2&=\int_0^{\infty}h(x)\overline{g(x)}\,dx \\
&=\int_0^{\infty}e^{-2\lambda x}\hat{h}(\mathrm{cut}(x))\overline{\hat{g}(\mathrm{cut}(x)}\,dx \\
&=\sum_{n=0}^{\infty}e^{-2\lambda n T}\int_{nT}^{(n+1)T}e^{-2\lambda(x-nT)}\hat{h}(\mathrm{cut}(x))
\overline{\hat{g}(\mathrm{cut}(x))}\,dx \\
&=\sum_{n=0}^{\infty}e^{-2\lambda n T}\int_0^Te^{-2\lambda x}\hat{h}(x)\overline{\hat{g}(x)}\,dx \\
&=(1-e^{-2\lambda T})^{-1}\int_0^Th(x)\overline{g(x)}\,dx\,.
\end{align*}
Thus,
\begin{align*}
\langlef,g_n^*\rangle&=(1-e^{-2\lambda T})^{-1}\int_0^Te^{\alpha x/2}f'(x)\overline{e_n(x)}\,dx \\
&=(1-e^{-2\lambda T})^{-1}T^{-1/2}\int_0^Tf'(x)\exp\left((-\frac{2\pi i}{T}n-\lambda+\frac{\alpha}{2})x\right)\,dx
\end{align*}
Hence, the result follows.
\end{proof}
With this results at hand, we can prove a convergence rate of order $1/k$ for sufficiently smooth functions in $H_{\alpha}^T$.
\begin{prop}\label{p:approximation speed}
Assume $f\in H_\alpha^{T}$ is such that $f\vert_{[0,T]}$ is twice continuously differentiable.
Then, we have
$$
\left\Vert f - \Pi_kf\right\Vert^2_\alpha \leq\frac{C_1}{k}\,,
$$
for any $k\in\mathbb N$, where
$$
C_1=\frac{T\left\vert f'(T)e^{T(-\lambda+\alpha/2)}-f'(0)\right\vert^2+(\int_0^T \vert f''(x)\vert e^{x(-\lambda+\alpha/2)}\,dx)^2}{\pi^2(1-e^{-2\lambda T})^3}\,,
$$
and we recall the projection operator $\Pi_k$ from Proposition~\ref{l:commutator of U and projectors}.
\end{prop}
\begin{proof}
Corollary~\ref{k:distance to ONB} yields
\begin{align*}
\Vert f - \Pi_kf\Vert^2_\alpha &= \Vert \sum_{\vert n\vert>k} g_n\langlef,g_n^*\rangle_{\alpha}\Vert_\alpha^2
\leq C \sum_{\vert n\vert>k} \vert \langlef,g_n^*\rangle_{\alpha}\vert^2\,
\end{align*}
where $C:=(1-e^{-2\lambda T})^{-1}$. Define $h_n(x):=\exp(\xi_nx)$, $x\geq 0$, where we
denote $\xi_n=-\frac{2\pi i}{T}n-\lambda+\frac{\alpha}{2}$. Then, by Corollary~\ref{cor:alpha-2-inner-prod}
and integration-by-parts we find
\begin{align*}
\vert \langlef,g_n^*\rangle_{\alpha}\vert^2 &=C^2T^{-1}\left\vert\int_0^Tf'(x)h_n(x)dx\right\vert^2 \\
&= C^2T^{-1}\frac{1}{\vert\xi_n\vert^2}\left\vert f'(T)h_n(T)-f'(0)h_n(0)-\int_0^T f''(x)h_n(x)\,dx \right\vert^2 \\
& \leq \frac{2C^2}{T}\frac{1}{\vert\xi_n\vert^2}A_f\,,
\end{align*}
for any $n\in\mathbb Z\backslash\{0\}$, where the constant $A_f$ is
$$
A_f:=\left\vert f'(T)e^{T(-\lambda+\alpha/2)}-f'(0)\right\vert^2+(\int_0^T \vert f''(x)e^{x(\lambda-\alpha/2)}\, dx)^2\,.
$$
Moreover, we have
\begin{align*}
\sum_{\vert n\vert>k}\frac{1}{\vert\xi_n\vert^2}= 2 \sum_{n>k}\frac{1}{\vert\xi_n\vert^2}\leq \frac{T^2}{2\pi^2 k}.
\end{align*}
Putting the estimates together, we get
$$
\Vert f - \Pi_kf\Vert^2_\alpha \leq A_f\frac{C^3T}{\pi^2k}\,,
$$
as claimed.
\end{proof}
We can find a similar convergence rate for $c_{k,t}$, a result which becomes useful later:
\begin{lem}
\label{lemma:approximation speed_ckt}
Let $c_{k,t}$ be given as in Proposition~\ref{l:commutator of U and projectors}. Then,
$$
\Vert c_{k,t}\Vert_\alpha^2 \leq \frac{C_2}{k}\,,
$$
for any $k\in\mathbb N$, where $C_2=T/\pi^2(1-\exp(-2\lambda T))$.
\end{lem}
\begin{proof}
We appeal to Corollary~\ref{k:distance to ONB}, using $\{g_n^*\}_{n\in\mathbb Z}$ as the Riesz basis
with biorthogonal system $\{g_n\}_{n\in\mathbb Z}$, to find
\begin{align*}
\|c_{k,t}\|_{\alpha}^2&=\|\sum_{|n|>k}g_n(t)g_n^*\|_{\alpha}^2 \\
&\leq C\sum_{|n|>k}|g_n(t)|^2 \\
&=\frac{C}{T}\sum_{|n|>k}\frac{1}{\vert\lambda_n\vert^2}\left\vert e^{\lambda_n t}-1\right\vert^2 \\
&\leq\frac{2C}{T}(1+e^{-(2\lambda+\alpha)t})\sum_{|n|>k}\frac{1}{\vert\lambda_n\vert^2} \\
&\leq \frac{CT}{\pi^2}\frac1k\,,
\end{align*}
for $C=(1-\exp(-2\lambda T))^{-1}$. Hence, the result follows.
\end{proof}
With these results we are now in the position to study arbitrage-free approximations of the forward dynamics in \eqref{e:HJMM-equation}.
\section{Arbitrage free approximation of forward term structure models}\label{s:approximation}
In this section we find an arbitrage-free approximation of a forward term structure model -- stated in the Heath-Jarrow-Morton-type setup -- which lives in a
finite dimensional state space. We furthermore derive the convergence speed of the approximation, and extend the results to account for
forward contracts delivering the underlying commodity over a period which is the case for electricity and gas.
Consider the SPDE \eqref{e:HJMM-equation} with a mild solution $f\in H_{\alpha}$ given by \eqref{e:HJMM-equation-mild}. We recall
from \eqref{e:g-star-def}-\eqref{e:g-n-def} and Corollary~\ref{k:Riesz basis on H} the
Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ on the space $H_{\alpha}^T$ with the biorthogonal system
$\{g_*,\{g_n^*\}_{n\in\mathbb Z}\}$. Furthermore,
$\Pi$ is the projection of $H_{\alpha}$ on $H_{\alpha}^T$, while from \eqref{def-H-k-space} and
Proposition~\ref{p:U and the riesz basis} we have the projection $\Pi_k$ of $H_{\alpha}^T$
on $H_{\alpha}^{T,k}$ and the operator $\mathcal C_{k,t}$ for $k\in\mathbb N$, $t\geq 0$.
Let us define the continuous linear operator $\mathcal Lambda_k:H_{\alpha}\rightarrow H_{\alpha}^{T,k}$ by
\begin{equation}
\mathcal Lambda_k= \Pi_k\Pi
\end{equation}
for any $k\in\mathbb N$. The following theorem is one of the main results of the paper:
\begin{thm}\label{t:main statement}
For $k\in\mathbb N$, let $f_k$ be the mild solution of the SPDE
\begin{equation}
\label{e:approx-f-k}
df_k(t) = \partial_x f_k(t) dt + \mathcal Lambda_k\beta(t) dt + \mathcal Lambda_k\Psi(t) dL(t),\quad t\geq 0, f_k(0) = \mathcal Lambda_kf_0\,.
\end{equation}
Then, we have
\begin{enumerate}
\item $\mathbb{E}\left[\sup_{x\in[0,T-t]}\vert f_k(t,x) - f(t,x)\vert^2\right] \rightarrow 0$ for $k\rightarrow\infty$ and any $t\in [0,T]$,
\item $f_k$ takes values in the finite dimensional space $H_\alpha^{T,k}$, moreover, $f_k$ is a strong solution to the SPDE \eqref{e:approx-f-k}, i.e.\ $f_k\in\mathrm{dom}(\partial_x)$, {$t\mapsto \partial_xf_k(t)$} is $P$-a.s.\ Bochner-integrable and
$$ f_k(t) = f_k(0) + \int_0^t (\partial_xf_k(s)+\mathcal Lambda_k\beta(s))ds + \int_0^t \mathcal Lambda_k\Psi(s) dL(s)\,, $$
\item and,
\begin{align*}
f_k(t) &= S_k(t) + \sum_{n=-k}^k \left(e^{\lambda_n t} \langlef_k(0),g_n^*\rangle_{\alpha} + \int_0^t e^{\lambda_n (t-s)}dX_n(s)\right)g_n \,,
\end{align*}
where $S_k(t)= \delta_0(f_k(t))$ and $X_n(t) := \int_0^t \langle\Pi\beta(s)ds+\Pi\Psi(s)dL(s),g_n^*\rangle_{\alpha}$ for any $n\in\mathbb Z$, $t\geq0$.
\end{enumerate}
\end{thm}
\begin{proof}
(1) Define
$$f_\Pi(t) := \mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s))),\quad t\geq0.$$
Since $f_k$ is a mild solution, we have
\begin{align*}
f_k(t) &= \mathcal U_t\Pi_k\Pi f_0 + \int_0^t \mathcal U_{t-s}\Pi_k(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&= \Pi_k\mathcal U_t\Pi f_0 + \int_0^t \Pi_k\mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&\hspace{0.3cm} - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&= \Pi_k\left(\mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)))\right) \\
&\hspace{0.3cm} - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&= \Pi_k(f_\Pi(t)) - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s))
\end{align*}
for any $t\geq0$. From Benth and Kr\"uhner~\cite[Lemma 3.2]{benth.kruehner.14} the sup-norm is dominated by the $H_{\alpha}$-norm. Thus, there is a constant $c>0$ such that
\begin{align*}
\mathbb{E}\left[\sup_{x\in[0,T-t]}\vert \Pi_k(f_\Pi(t,x)) - f_\Pi(t,x)\vert^2\right] & \leq c\mathbb{E}\left[ \Vert (\Pi_k-\mathcal I)f_\Pi(t) \Vert^2_\alpha \right]
\end{align*}
for any $t\geq 0$ where $\mathcal I$ denotes the identity operator on $H_\alpha$. The dominated convergence theorem yields that the right-hand side converges to $0$ for $k\rightarrow \infty$. Clearly, we have
$$ \sup_{x\in[0,T-t]}\vert \mathcal C_{k,t}f_\Pi(0,x) \vert \leq c\Vert \mathcal C_{k,t}f_\Pi(0) \Vert_\alpha \rightarrow 0\,, $$
for $k\rightarrow \infty$. Proposition~\ref{l:commutator of U and projectors} states that
$$ \mathbb{E}\left\Vert \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \right\Vert_\alpha^2 \rightarrow 0\,,$$
for $k\rightarrow 0$. Hence, we have
$$ \mathbb{E}\left(\sup_{x\in[0,T-t]}\vert f_k(t,x)-f_\Pi(t,x) \vert^2\right) \rightarrow 0\,, $$
for $k\rightarrow \infty$ and any $t\in [0,T]$. Since $f_\Pi(t,x) = f(t,x)$ for any $t\in[0,T]$, $x\in [0,T-t]$ the first part follows.
(2) Note first that $\partial_x g_n(x)=\exp(\lambda_n x)/\sqrt{T}=\lambda_ng_n(x)+g_*(x)/\sqrt{T}$, and hence
$\partial_x g_n\in H_{\alpha}^{T,k}$ whenever $|n|\leq k$. Thus, $H_{\alpha}^{T,k}$ is invariant under the
generator $\partial_x$, and its restriction to $H_{\alpha}^{T,k}$ is continuous and bounded. We find that $f_k$ takes values only in $H_\alpha^{T,k}$ because
\begin{align*}
f_k(t) &= \Pi_k\left(\mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)))\right) \\
&\hspace{0.3cm} - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds+\Pi\Psi(s)dL(s))\,,
\end{align*}
where all summands are clearly in $H_\alpha^{T,k}$.
(3) As $f_k(t)\in H_{\alpha}^{T,k}$, we have the representation
$$
f_k(t)=\langlef_k(t),g_*^*\rangle_{\alpha}g_*+\sum_{n=-k}^k \langlef_k(t),g_n^*\rangle_{\alpha}g_n\,.
$$
Since $g_*^*=1$, we find that $\langlef_k(t),g_*^*\rangle_{\alpha}=f_k(t,0)=\delta_0(f_k(t))$. Thus, from the mild solution of
\eqref{e:approx-f-k} we find, using Proposition \ref{p:U and the riesz basis}
\begin{align*}
f_k(t)&=S_k(t)+\sum_{n=-k}^k \left\langle\mathcal{U}_t f_k(0)+\int_0^t\mathcal U_{t-s}(\mathcal Lambda_k\beta(s)ds + \mathcal Lambda_k\Psi(s)dL(s)),g_n^*\right\rangle_{\alpha}g_n \\
&=S_k(t)+\sum_{n=-k}^{k}\langlef_k(0),\mathcal{U}_t^*g_n^*\rangle_{\alpha}g_n \\
&\qquad\qquad+\sum_{n=-k}^{k}\int_0^t\langle\mathcal Lambda_k\beta(s)ds + \mathcal Lambda_k\Psi(s)dL(s),\mathcal U_{t-s}^*g_n^*\rangle_{\alpha}g_n \\
&=S_k(t)+\sum_{n=-k}^{k}e^{\lambda_n t}\langlef_k(0),g_n^*\rangle_{\alpha}g_n \\
&\qquad\qquad+\sum_{n=-k}^k\int_0^t e^{\lambda_n(t-s)}\langle\mathcal Lambda_k\beta(s)ds + \mathcal Lambda_k\Psi(s)dL(s),g_n^*\rangle_{\alpha}g_n \,.
\end{align*}
Observe that for any $f\in H_{\alpha}$,
$$
\mathcal Lambda_k f=\Pi_k(\Pi f)=(\Pi f)(0)g_*+\sum_{m=-k}^k\langle\Pi f,g_m^*\rangle_{\alpha}g_m\,,
$$
and since $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$, $\{g_*^*,\{g^*_n\}_{n\in\mathbb Z}\}$ are biorthogonal systems
\begin{align*}
\langle\mathcal Lambda_k f,g_n^*\rangle_{\alpha}&=(\Pi f)(0)\langleg_*,g_n^*\rangle_{\alpha}+\sum_{m=-k}^k\langle\Pi f,g_m^*\rangle_{\alpha}\langleg_m,g_n^*\rangle_{\alpha}
=\langle\Pi f,g_n^*\rangle_{\alpha}1_{\{\vert n\vert\leq k\}}\,.
\end{align*}
Hence, the claim follows.
\end{proof}
Another view on Theorem \ref{t:main statement} is that all processes in the $k$-th approximation
of $f$ can be expressed in terms of the factor processes $X_*,X_{-k},\dots,X_k$, as stated below.
\begin{cor}\label{k:state variables}
Under the assumptions and notations of Theorem \ref{t:main statement}, we have for $k\in\mathbb N$,
\begin{align*}
f_k(t,x) &= S_k(t) + \sum_{n=-k}^k U_n(t) g_n(x)\,,
\end{align*}
for any $0\leq t<\infty$ and $x\geq 0$. Here,
\begin{align*}
S_k(t)&= S_k(0) + X_*(t) + \sum_{n=-k}^k \left( g_n(t)U_n(0) +\int_0^tg_n(t-s)dX_n(s)\right) \,,
\end{align*}
with,
\begin{align*}
X_n(t) &:= \left\langle \int_0^t(\Pi\beta(s)ds+\Pi\Psi(s)dL(s)),g_n^*\right\rangle_{\alpha}, \\
X_*(t) &:= \left\langle \int_0^t(\Pi\beta(s)ds+\Pi\Psi(s)dL(s)),g_*\right\rangle_{\alpha}, \\
U_n(t) &:= e^{\lambda_n t} \langlef_k(0),g_n^*\rangle + \int_0^t e^{\lambda_n (t-s)}dX_n(s) \end{align*}
for $n\in\{-k,\dots, k\}$.
\end{cor}
\begin{proof}
The first equation is a restatement of (3) in Theorem \ref{t:main statement}. Proposition \ref{p:U and the riesz basis} yields
$$ \langle\mathcal U_t h,g_*\rangle_{\alpha} = \langleh,g_*\rangle_{\alpha} + \sum_{n=-k}^k g_n(t)\langleh,g_n^*\rangle_{\alpha} $$
for any $h\in H_\alpha^{T,k}$ with $h=\langleh,g_*\rangle_{\alpha}g_*+\sum_{n=-k}^k \langleh,g_n^*\rangle_{\alpha}g_n$. Thus, since $g_*=1$ and $g_n(0)=0$ we have
\begin{align*}
S_k(t) &= f_k(t,0) \\
&=\langle f_k(t),g_*\rangle_{\alpha} \\
&= \langle \mathcal U_{t}f_k(0),g_*\rangle_{\alpha} + \int_0^t \langle\mathcal U_{t-s}(\mathcal Lambda_k\beta(s)\,ds+\mathcal Lambda_k\Psi(s)\,dL(s)),g_*\rangle_{\alpha} \\
&= \langlef_k(0),g_*\rangle_{\alpha}+\sum_{n=-k}^kg_n(t)\langlef_k(0),g_n^*\rangle_{\alpha} \\
&\qquad\qquad+\int_0^t\langle\mathcal Lambda_k\beta(s)\,ds+\mathcal Lambda_k\Psi(s)\,dL(s),g_*\rangle_{\alpha} \\
&\qquad\qquad+\sum_{n=-k}^k\int_0^tg_n(t-s)\langle\mathcal Lambda_k\beta(s)+\mathcal Lambda_k\Psi(s)\,dL(s),g_n^*\rangle_{\alpha}\,.
\end{align*}
As in the proof of Theorem~\ref{t:main statement}, we have $\langle\mathcal Lambda_k f,g_n^*\rangle_{\alpha}=\langle\Pi f,g_n^*\rangle_{\alpha}$ for any $f\in H_{\alpha}$.
Similarly, $\langle\mathcal Lambda_k f,g_*\rangle_{\alpha}=\langle\Pi f,g_*\rangle_{\alpha}$ for $n\in\mathbb Z$ with $\vert n\vert\leq k$. The result follows.
\end{proof}
The processes $S_k, U_{-k},\dots, U_k$ in Corollary~\ref{k:state variables} capture at any time $t$ the whole state of the market in the approximation model.
I.e., the spot price and the forward curve are simple functions of these state variables. As we will see in Corollary \ref{k:Forward prices} below, the forward prices
of contracts with delivery periods can be expressed in these state variables as well. Note that if we assume $\langle\Pi\beta,g_n^*\rangle$, $\langle\Pi\Psi,g_n^*\rangle$ are deterministic and constant, then $(X_{-k},\dots,X_k)$ is a $2k+1$-dimensional L\'evy process and $U_{-k},\dots,U_k$ are Ornstein-Uhlenbeck processes. This corresponds to the spot price model suggested in Benth, Kallsen and Meyer-Brandis~\cite{benth.et.al.05}.
From the proof of Corollary~\ref{k:state variables} we find that $S_k(0)=\langlef_k(0),g_*\rangle_{\alpha}$. But then
$$
S_k(0)=\langle\mathcal Lambda_k f_0,g_*\rangle_{\alpha}=\langle\Pi f_0,g_*\rangle_{\alpha}=(\Pi f_0)(0)=f_0(0)\,.
$$
Obviously, $f_0(0)$ is equal to today's spot price, so we obtain that the starting point of the process $S_k(t)$ in the approximation is today's spot price.
Furthermore, since we have $f_k(t,0)=S_k(t)$ because $g_n(0)=0$ for all $n\in\mathbb Z$, $S_k(t)$ is the approximative spot price dynamics associated
with $f_k(t)$. For $U_n(0)$, $n\in\mathbb Z$ invoking Corollary~\ref{cor:alpha-2-inner-prod} shows that
\begin{align*}
U_n(0)&=\langle\Pi f_0,g_n^*\rangle_{\alpha} \\
&=\frac1{\sqrt{T}(1-e^{-2\lambda T})}\int_0^{T}(\Pi f_0)'(y)
\exp((-\lambda+\alpha/2)x)\exp\left(\frac{2\pi i}{T}nx\right)\,dy\,.
\end{align*}
This is the Fourier transform of the initial forward curve $f_0$ (or, rather its derivative
scaled by an exponential function). In any case, both $S_k(0)$ and $U_n(0)$ are
given by (functionals of) the initial forward curve $f_0$.
Next, we would like to identify the convergence speed of our approximation, that is, the rate for the convergence in part (1) of Theorem \ref{t:main statement}.
\begin{prop}\label{p:convergence speed}
Assume that $x\mapsto f(t,x)$ is twice continuously differentiable and let
$f_k$ be the mild solution of the SPDE
$$ df_k(t) = \partial_x f_k(t) dt + \mathcal Lambda_k\beta(t) dt + \mathcal Lambda_k\Psi(t) dL(t),\quad t\geq 0, f_k(0) = \mathcal Lambda_kf_0\, .$$
Then, we have
$$
\mathbb{E}\left[\sup_{x\in[0,T-t]}\vert f_k(t,x) - f(t,x)\vert^2\right] \leq \frac{A(t)}{k}\,,
$$
for any $k>1$, where
\begin{align*}
A(t)&:=\frac{3T(1+\alpha^{-1})}{(1-e^{-2\lambda T})}\left\{\|\Pi f_0\|_{\alpha}^2+\int_0^T \mathbb{E}[\mathrm{Tr}(\Psi(s)Q\Psi^*(s))]ds+\left(\int_0^T \mathbb{E}\left[\Vert \beta(s)\Vert_\alpha\right]\,ds\right)^2 \right\} \\
&\qquad+\frac{3(1+\alpha^{-1})}{\pi^2(1-e^{-2\lambda T})^3}\left\{T\mathbb{E}\left[|\partial_xf_{\Pi}(t,T)
e^{T(-\lambda+\alpha/2)}-\partial_xf_{\Pi}(t,0)|^2\right] \right. \\
&\qquad\qquad\left.+\left(\int_0^T\mathbb{E}\left[|\partial^2_xf_{\Pi}(t,x)|\right]e^{x(-\lambda+\alpha/2)}\,dx\right)^2\right\}\,.
\end{align*}
\end{prop}
\begin{proof}
In the proof of Theorem~\ref{t:main statement} we have shown that
\begin{align*}
f_k(t) &= \Pi_k(f_\Pi(t)) - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s))\,,
\end{align*}
where $f_\Pi(t) := \mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)))$ for any $t\geq 0$. By Proposition~\ref{p:approximation speed} we have
$$
\left\Vert f_\Pi(t) - \Pi_k(f_\Pi(t))\right\Vert^2_\alpha \leq\frac{C_1(t)}{k}
$$
where $C_1(t)$ is a random variable defined by
$$
C_1(t)=\frac{T|\partial_xf_{\Pi}(t,T)e^{T(-\lambda+\alpha/2)}-\partial_xf_{\Pi}(t,0)|^2+(\int_0^T|\partial^2_xf_{\Pi}(t,x)|e^{x(-\lambda+\alpha/2)}\,dx)^2}{\pi^2(1-e^{-2\lambda T})^3}\,.
$$
Remark that from the proof of Theorem~\ref{t:main statement} we find for any $h\in H_{\alpha}^T$
\begin{align*}
\|\mathcal C_{k,t}h\|_{\alpha}^2&=\|\langleh,c_{k,t}\rangle_{\alpha}g_*\|_{\alpha}^2=|\langleh,c_{k,t}\rangle_{\alpha}|^2\leq \|h\|_{\alpha}^2\|c_{k,t}\|_{\alpha}^2\,,
\end{align*}
and therefore, from Lemma~\ref{lemma:approximation speed_ckt}
$$
\|\mathcal C_{k,t}h\|_{\alpha}^2\leq\|h\|_{\alpha}^2\frac{C_2}{k}\,,
$$
for the constant $C_2=T/\pi^2(1-e^{-2\lambda T})$.
Then, we have
\begin{align*}
\Vert f_k(t) - f_\Pi(t) \Vert_\alpha^2 &\leq 3\Vert \Pi_k(f_\Pi(t))-f_\Pi(t)\Vert_\alpha^2 + 3\Vert \mathcal C_{k,t}\Pi f_0\Vert_\alpha^2 \\
&\qquad\qquad+
3\Vert \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \Vert_\alpha^2 \\
&\leq \frac{3C_1(t)}{k} + \frac{3C_2}{k}\Vert \Pi f_0\Vert_\alpha^2 \\
&\qquad\qquad+ 3\Vert \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \Vert_\alpha^2.
\end{align*}
By Lemma~3.2 in Benth and Kr\"uhner~\cite{benth.kruehner.14}, the supremum norm is bounded by the $H_{\alpha}$-norm with a constant
$c=\sqrt{1+\alpha^{-1}}$. Hence, taking expectations, yield
\begin{align*}
\mathbb{E}&\left[\sup_{x\in[0,T-t]}\vert f_k(t,x) - f(t,x) \vert^2 \right] \\
&\qquad \leq c^2\mathbb{E}\left[ \Vert f_k(t) - f_\Pi(t) \Vert_\alpha^2 \right] \\
&\qquad\leq \frac{3c^2}{k}\left(\mathbb{E}\left[C_1(t)\right]+ C_2\Vert \Pi f_0\Vert_\alpha^2\right) \\
&\qquad\qquad + \frac{3c^2}{k}C_2\left(\int_0^T \mathbb{E}[\mathrm{Tr}(\Psi(s)Q\Psi^*(s))]ds + \left(\int_0^T \mathbb{E}\left[\Vert \beta(s)\Vert_\alpha\right] ds\right)^2 \right) \,.
\end{align*}
The result follows.
\end{proof}
In electricity and gas markets forward contracts deliver over a future period rather than at a fixed time. The holder of the forward contract receives a uniform stream of electricity
or gas over an agreed time period $[T_1,T_2]$. The forward prices of delivery period contracts can be derived from a "fixed-delivery time" forward curve model (see Benth et al.~\cite{BSBK-book})
by
\begin{equation}
\label{e:el-forward-def}
F(t,T_1,T_2) := \frac{1}{T_2-T_1}\int_{T_1}^{T_2} f(t,s-t)\,, ds
\end{equation}
where $f$ is given by the SPDE \eqref{e:HJMM-equation}. The following Corollary adapts Theorem~\ref{t:main statement} to the case of forward
contracts with delivery period.
\begin{cor}\label{k:Forward prices}
Assume the conditions of Theorem~\ref{t:main statement} and define
\begin{align*}
F_k(t,T_1,T_2) &:= \frac{1}{T_2-T_1}\int_{T_1}^{T_2} f_k(t,s-t) ds
\end{align*}
for any $0\leq t \leq T_1\leq T_2\leq T$. Then, we have
$$ F_k(t,T_1,T_2) \rightarrow F(t,T_1,T_2)$$
for $k\rightarrow\infty$ in $L^2(\Omega)$ where $F$ is given in \eqref{e:el-forward-def}. Furthermore,
$$
F_k(t,T_1,T_2) = S_k(t) + \sum_{n=-k}^k G_n(t,T_1,T_2) \left(e^{\lambda_n t} \langleg_n^*,f_k(0)\rangle_{\alpha}
+ \int_0^t e^{\lambda_n (t-s)}dX_n(s)\right)\,,
$$
for any $t\leq T_1\leq T_2\leq T$ where $S_k(t) = \delta_0(f_k(t))$,
$$G_n(t,T_1,T_2) = \frac{\exp(\lambda_n(T_2-t))-\exp(\lambda_n(T_1-t))-\lambda_n(T_2-T_1)}{\lambda_n^2\sqrt{T}(T_2-T_1)}$$
and $X_n(t):=\int_0^t\langle\Pi\beta(s)ds + \Pi\Psi(s)dL(s),g_n^*\rangle_{\alpha}$.
\end{cor}
\begin{proof}
Theorem \ref{t:main statement} yields uniform $L^2$ convergence of the integrands appearing in $F_k$ to the integrand appearing in $F$ and hence the convergence follows. The representation of $F_k$ follows immediately from part (3) of Theorem~\ref{t:main statement}.
\end{proof}
We remark in passing that temperature derivatives market (see e.g. Benth and \v{S}altyt\.{e} Benth~\cite{BSB-weather}) trades in forwards with a "delivery period" as well.
In this market, the forward is cash-settled against an index of the daily average temperature measured in a city over a given period.
\section{Refinement to Markovian forward price models}\label{s:markovian}
In this Section we refine our analysis to Markovian forward price models, making the
additional assumption that the coefficients $\beta$ and $\Psi$ depend on the state of the forward curve.
More specifically, we assume that
\begin{align}
\beta(t) &= b(t,f(t)), \\
\Psi(t) &= \psi(t,f(t)),
\end{align}
where $b:\mathbb R_+\times H_\alpha\rightarrow H_\alpha$, $\psi:\mathbb R_+\times H_\alpha\rightarrow L(H_\alpha)$ are measurable Lipschitz-continuous functions of linear growth
in the sense
\begin{align}
\| b(t,f) - b(t,g)\|_{\alpha} &\leq C_b \| f-g\|_{\alpha}\,, \label{eq:lip-cond-b} \\
\Vert (\psi(t,f) - \psi(t,g))\mathcal Q^{1/2} \Vert_{\mathrm{HS}} &\leq C_\psi \| f-g\|_{\alpha}\,,
\label{eq:lip-cond-psi}
\end{align}
and
\begin{align}
\| b(t,f)\|_{\alpha} &\leq C_b(1+ \|f\|_{\alpha})\,, \label{eq:lingrowth-cond-b} \\
\Vert \psi(t,f)\mathcal Q^{1/2} \Vert_{\mathrm{HS}} &\leq C_\psi (1+ \|f\|_{\alpha})\,,
\label{eq:lingrowth-cond-psi}
\end{align}
for positive constants $C_b$, $C_\psi$. Under these conditions there exists a unique mild solution $f$ of
the semilinear SPDE
\begin{equation}
\label{eq:nonlinear_spde}
df(t) = (\partial_xf(t) + b(t,f(t))) dt + \psi(t,f(t-)) dL(t),\quad f(0) = f_0.
\end{equation}
We would like to note that semilinear SPDEs are treated in the book by Peszat and Zabczyk~\cite{peszat.zabczyk.07} and in Tappe~\cite{tappe.12}. Additionally, we assume that
\begin{align}
b(t,h) &= b(t,g), \label{eq:struct-cond-b}\\
\psi(t,h) &= \psi(t,g) \label{eq:struct-cond-psi}\,,
\end{align}
for any $h,g\in H_\alpha$ such that $h(x)=g(x)$ for any $x\in[0,T-t]$, i.e.\ the structure of the curve beyond our time horizon $T$ does not influence the
dynamics of the curve-valued process $f(t)$.
Before continuing our analysis of the arbitrage-free approximation in the Markovian case, we show a couple of useful lemmas. The first states a
version of Doob's $L^2$ inequality for Volterra-like Hilbert space-valued stochastic integrals with respect to the L\'evy process $L$, and is essentially collected from
Filipovi\'c, Tappe and Teichmann~\cite{FTT}.
\begin{lem}
\label{lem:doob}
Suppose that
$\Phi\in\mathcal L_L^2(H_{\alpha})$. Then,
\begin{align*}
\mathbb{E}\left[\sup_{s\in[0,t]} \Vert
\int_0^s\mathcal{U}_{s-r}\Phi(r)\,dL(r)
\Vert_{\alpha}^2\right] &\leq 4c_t^2 \int_0^t \mathbb{E}\left[\Vert\Phi(r)\mathcal Q^{1/2}\Vert_{\text{HS}}^2\right]\,dr\,,
\end{align*}
for $c_t>0$ being at most exponentially growing in $t$.
\end{lem}
\begin{proof}
Note first that due to Benth and Kr\"uhner~\cite[Lemma 3.5]{benth.kruehner.14} the $C_0$-semigroup $(\mathcal U_t)_{t\geq 0}$ is pseudo-contractive. Filipovi\'c, Tappe and Teichmann~\cite[Prop.~8.7]{FTT} state that there is a Hilbert space extension $H$
of $H_\alpha$ (i.e.\ $H$ is a Hilbert space and $H_\alpha$ is its
subspace and the norm of $H_\alpha$ equals the norm of $H$ restricted to
$H_\alpha$) and a $C_0$-group $(\mathcal V_t)_{t\in\mathbb R}$ on $H$ such that
$\mathcal V_t\vert_{H_\alpha} = \mathcal U_t$ for $t\geq 0$. Then, we have
\begin{align*}
\sup_{s\in[0,t]} \Vert\int_0^s\mathcal{U}_{s-r}\Phi(r)\,dL(r)\Vert_{\alpha}&\leq \sup_{s\in[0,t]}\Vert \mathcal V_{s-t}\Vert_{\mathrm{op}}
\Vert\int_0^s \mathcal U_{t-r}\Phi(r)\,dL(r) \Vert_{\alpha} \\
&\leq \sup_{s\in[0,t]}\Vert \mathcal V_{s}\Vert_{\mathrm{op}}
\sup_{s\in[0,t]}\Vert\int_0^s \mathcal U_{t-r}\Phi(r)\,dL(r) \Vert_{\alpha}\,.
\end{align*}
Thus, by Doob's maximal inequality, Thm.~2.2.7 in
Prevot and R\"ockner~\cite{PR}, we find
\begin{align*}
\mathbb{E}&\left[\sup_{s\in[0,t]} \Vert
\int_0^s\mathcal{U}_{s-r}\Phi(r)\,dL(r)
\Vert_{\alpha}^2\right] \\
&\qquad\qquad\leq \sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\mathbb{E}\left[\sup_{s\in[0,t]} \Vert
\int_0^s\mathcal{U}_{t-r}\Phi(r)\,dL(r)\Vert_{\alpha}^2\right] \\
&\qquad\qquad\leq 4\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\mathbb{E}\left[\Vert\int_0^t\mathcal{U}_{t-r}\Phi(r)\,dL(r)\Vert_{\alpha}^2\right] \\
&\qquad\qquad=4\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\int_0^t\mathbb{E}\left[\|\mathcal U_{t-r}\Phi(r)\mathcal{Q}^{1/2}\Vert^2_{\text{HS}}\right]\,dr \\
&\qquad\qquad\leq 4\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\sup_{s\in[0,t]}\|\mathcal U_{s}\Vert_{\text{op}}^2\int_0^t\mathbb{E}\left[ \Vert\Phi(r)\mathcal{Q}^{1/2}\Vert^2_{\text{HS}}\right]\,dr
\end{align*}
This proves the Lemma by letting $c_t=\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}\sup_{0\leq s\leq t}\Vert\mathcal{U}_{s}\Vert_{\text{op}}$
and recalling that any $C_0$-group is bounded in operator norm by
an exponentially increasing function in $t$. Hence, $c_t\leq c\exp(w t)$ for some
constants $c,w>0$.
\end{proof}
We remark in passing that the above result holds for any pseudo-contractive semigroup
$\mathcal S_t$, $t\geq 0$.
The next lemma is a useful technical result on the distance between processes and the fixed point of an integral operator defined
via the mild solution of \eqref{eq:nonlinear_spde}. The lemma plays a crucial role in showing that certain arbitrage-free approximations of
\eqref{eq:nonlinear_spde} converge to the right limit.
\begin{lem}\label{l:fixpoint estimate}
For an $H_{\alpha}$-valued adapted and c\`adl\`ag stochastic process $h$, define
$$
V(h)(t) := \mathcal U_tf_0 + \int_0^t \mathcal U_{t-s}b(s,h(s))\,ds + \int_0^t \mathcal U_{t-s}\psi(s,h(s-))\,dL(s)\,,
$$
for any $t\geq 0$. Then, $V$ has a fixed point $\widehat{f}$ and it holds
$$
\mathbb{E}\left[\sup_{0\leq s\leq t}\| h(s)-\widehat{f}(s)\|^2_{\alpha} \right] \leq\frac{\pi^2}{6}\exp(4C_t)\mathbb{E}\left[\sup_{0\leq s\leq t}\| V(h)(s)-h(s)\|^2_{\alpha}\right] \,,
$$
for any $t\geq 0$ and any $H_{\alpha}$-valued adapted c\`adl\`ag stochastic processes $h$, with
$C_t$ being a positive constant depending on $t$.
\end{lem}
\begin{proof}
If $h$ is an adapted c\`adl\`ag $H_{\alpha}$-valued
stochastic process such that $\mathbb E[\int_0^t\|h(s)\|_{\alpha}^2\,ds]<\infty$, then from the linear growth assumption~\eqref{eq:lingrowth-cond-b}
on $b$ we find
\begin{align*}
\mathbb E[\int_0^t\|\mathcal U_{t-s}b(s,h(s))\|_{\alpha}\,ds] &\leq C_b e^{wt}(t+\mathbb E[\int_0^t\|h(s)\|_{\alpha}\,ds]) \\
&\leq C_b e^{wt}(t+\sqrt{t}\mathbb E[\int_0^t\|h(s)\|_{\alpha}^2\,ds]^{1/2})\\
&<\infty\,.
\end{align*}
Furthermore, from the linear growth condition~\eqref{eq:lingrowth-cond-psi}
on $\psi$
$$
\mathbb E[\int_0^t\|\mathcal{U}_{t-s}\psi(s,h(s))\|^2_{\alpha}\,ds]\leq 2C^2_{\psi}e^{2wt}\left(t+\mathbb E[\int_0^t\|h(s)\|^2_{\alpha}\,ds]\right)<\infty\,.
$$
Hence, $V(h)$ is well-defined, and it is an adapted c\`adl\`ag process. By a straightforward
estimation using again the linear growth of $b$ and $\psi$, we find similarly that
$$
\mathbb E[\int_0^t\|V(h)(s)\|_{\alpha}^2\,ds]\leq C_t\left(1+\mathbb E[\int_0^t\|h\|_{\alpha}^2\,ds]\right)<\infty\,,
$$
for some constant $C_t>0$
Therefore, $V$ maps into its own domain and, thus, can be iterated.
We note that by general theory, the SPDE
$$
df(t)=\partial_xf(t)\,dt+b(t,f(t))\,dt+\psi(t,f(t-))\,dL(t)
$$
has a unique mild solution $\widehat f$ which has a c\`adl\`ag modification, cf.\ Tappe~\cite[Theorem 4.5, Remark 4.6]{tappe.12}. By definition of mild solutions, we see that $\widehat f$
is a fix point for
$V$, i.e., $V(\widehat f)=\widehat f$.
Let $g,h$ be $H_{\alpha}$-valued adapted c\`adl\`ag stochastic processes and $t\geq 0$. Then, we have
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\|V(h)(s)-V(g)(s)\|^2_{\alpha}\right] \\
&\qquad\qquad\leq 2\mathbb{E}\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(b(r,h(r))-b(r,g(r))\right)\,dr\|_{\alpha}^2\right] \\
&\qquad\qquad\qquad+2\mathbb{E}\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(\psi(r,h(r-))-\psi(r,g(r-))\right)\,dL(r)\|^2_{\alpha}\right] \,.
\end{align*}
Consider the first term on the right hand side of the inequality. By the norm inequality for Bochner integrals and Lipschitz continuity of $b$ in
\eqref{eq:lip-cond-b}, we find
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(b(r,h(r))-b(r,g(r))\right)\,dr\|_{\alpha}^2\right] \\
&\qquad\qquad\leq\mathbb{E}\left[\sup_{0\leq s\leq t}\left(\int_0^s\|\mathcal U_{s-r}\|_{\text{op}}\|b(r,h(r))-b(r,g(r))\|_{\alpha}\,dr\right)^2\right] \\
&\qquad\qquad\leq t\mathbb{E}\left[\sup_{0\leq s\leq t}\int_0^s\|\mathcal U_{s-r}\|^2_{\text{op}}\|b(r,h(r))-b(r,g(r))\|^2_{\alpha}\,dr\right] \\
&\qquad\qquad\leq t^2\sup_{0\leq s\leq t}\|\mathcal{U}_s\|^2_{\text{op}}\mathbb{E}\left[\int_0^t\|b(r,h(r))-b(r,g(r))\|_{\alpha}^2\,dr\right] \\
&\qquad\qquad\leq t^2C_b^2\sup_{0\leq s\leq t}\|\mathcal U_s\|^2_{\text{op}}\int_0^t\mathbb{E}\left[\|h(r)-g(r)\|_{\alpha}^2\right]\,dr\,,
\end{align*}
where we have applied Cauchy-Schwartz' inequality. Recall that since
$\mathcal U_t$ is a pseudo-contractive semigroup, we find for some
$w>0$, it holds that $\sup_{0\leq s\leq t}\|\mathcal U_{s}\|_{\text{op}}^2\leq \exp(2w t)<\infty$.
For the second term, we find by appealing to Lemma~\ref{lem:doob} and the Lipschitz continuity in \eqref{eq:lip-cond-psi} of $\psi$,
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(\psi(r,h(r-))-\psi(r,g(r-))\right)\,dL(r)\|^2_{\alpha}\right] \\
&\qquad\qquad\leq 4c_t^2\int_0^t\mathbb{E}\left[\Vert(\psi(r,h(r))-\psi(r,g(r)))\mathcal Q^{1/2}\Vert^2_{\text{HS}}\right]\,dr \\
&\qquad\qquad\leq 4c_t^2C_{\psi}^2\int_0^t\mathbb{E}\left[\Vert h(r)-g(r)\Vert^2_{\alpha}\right]\,dr
\end{align*}
Here, the constant $c_t$ is from Lemma~\ref{lem:doob}. Denote by $C_t$ the constant
$$
C_t:=2C^2_b t^2 \sup_{s\in[0,t]}\Vert \mathcal U_s\Vert_{\mathrm{op}}+ 8c^2_tC_\psi^2t\,.
$$
Then, we have
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\| V^n(h)(s)-V^n(g)(s)\|^2_{\alpha}\right] \\
&\qquad\qquad\leq C_t\int_0^t \mathbb{E}\left[\|V^{n-1}(h)(s_1)-V^{n-1}(g)(s_1)\|^2_{\alpha}\right] \,ds_1\\
&\qquad\qquad\leq C_t^n \int_0^t\int_0^{s_1}\cdots\int_0^{s_{n-1}}\mathbb{E}\left[\|h(s_n)-g(s_n)\|_{\alpha}^2\right]ds_n\dots ds_1 \\
&\qquad\qquad\leq \frac{C_t^n}{n!}\mathbb{E}\left[\sup_{0\leq s\leq t}\|h(s)-g(s)\|^2_{\alpha} \right]\,,
\end{align*}
for any $n\in\mathbb N$. Denote by $L_{a}^2(\Omega,D([0,t],H_\alpha))$ the space of
$H_{\alpha}$-valued adapted c\`adl\`ag stochastic processes
$\{f(s)\}_{s\in[0,t]}$ for which $\mathbb{E}[\sup_{s\in[0,t]}\|f(s)\|_\alpha^2]<\infty$.
Equip this space with the norm $\|\cdot\|_t$ defined by
$$
\|f\|_t^2 := \mathbb{E}[\sup_{s\in[0,t]}\|f(s)\|_\alpha^2]
$$
for $f\in L_{a}^2(\Omega,D([0,t],H_\alpha))$. From the estimation above, we see
that $V$ operates on the normed space $L_{a}^2(\Omega,D([0,t],H_\alpha))$.
Moreover, $V^n$ is Lipschitz continuous with constant strictly less than $1$ for $n$ sufficiently large. Thus, by Banach's fixed point theorem there is at most one
fixed point for $V$. Hence, $\hat f$ is the unique fix point for $V$. Furthermore, we have
\begin{align*}
\mathbb{E}\left[\sup_{0\leq s\leq t}\|V^n(h)(s)-h(s)\|^2_{\alpha}\right]^{1/2}
& \leq \sum_{k=0}^{n-1} \mathbb{E}\left[\sup_{0\leq s\leq t}\|V^{k+1}(h)(s)-V^k(h)(s)\|^2_{\alpha}\right]^{1/2} \\
& \leq \mathbb{E}\left[\sup_{0\leq s\leq t}\| V(h)(s)-h(s)\|^2_{\alpha} \right]^{1/2}\sum_{k=0}^{n-1} \left(\frac{C_t^k}{k!}\right)^{1/2}\,.
\end{align*}
From Cauchy-Schwartz' inequality and we have that
\begin{align*}
\sum_{k=0}^{n-1} \left(\frac{C_t^k}{k!}\right)^{1/2}&=\sum_{k=0}^{n-1}(k+1)^{-1}\left(\frac{(k+1)^2C_t^k}{k!}\right)^{1/2} \\
&\leq\left(\sum_{k=0}^{n-1}\frac1{(k+1)^2}\right)^{1/2}\left(\sum_{k=0}^{n-1}\frac{(k+1)^2C_t^k}{k!}\right)^{1/2} \\
&\leq \frac{\pi}{\sqrt{6}}\left(\sum_{k=0}^{n-1}\frac{4^kC_t^k}{k!}\right)^{1/2} \\
&\leq \frac{\pi}{\sqrt{6}}\exp(2C_t)\,,
\end{align*}
where we have used the elementary inequality $k+1\leq 2^k$, $k\in\mathbb N$.
\end{proof}
Let us define the Lipschitz continuous functions $b_\Pi:=\Pi\circ b$ and $\psi_\Pi:=\Pi\circ\psi$. Then,
Tappe~\cite[Theorem 4.5]{tappe.12} yields a mild solution $f_\Pi$ for the SPDE
\begin{equation}
df_\Pi(t) = (\partial_xf_\Pi(t) + b_\Pi(t,f_\Pi(t)))\, dt + \psi_\Pi(t,f_\Pi(t-))\,dL(t),\quad f_\Pi(0) = \Pi f_0\,.
\end{equation}
Furthermore, it will be convenient to use the notations
\begin{align}
b_k(t,h) := \mathcal Lambda_k(b(t,h)), \\
\psi_k(t,h) := \mathcal Lambda_k(\psi(t,h))
\end{align}
for any $h\in H_\alpha$, $t\geq0$.
In the proof of Theorem~\ref{t:main statement} we compared the solution $f$ to the projected solution $\Pi f$ which are essentially the same due to properties of $\Pi$. Then we compared $\Pi f$ to $f_\Pi$ which again had been essentially the same. Finally, we compared $\Pi_kf_\Pi$ to solutions of the projected SPDE where the difference was given by a certain Lie-commutator.
However, in the Markovian setting we want to change the dependencies of the coefficients as well, which complicates the proof of the approximation result.
\begin{thm}
\label{thm:main-markovian}
Denote by $\widehat f_k$ be the mild solution of the SPDE
$$
d\widehat f_k(t) = (\partial_x\widehat f_k(t) + b_k(t,\widehat f_k(t)))\,dt + \psi_k(t,\widehat f_k(t-))\,dL(t),\quad \widehat f_k(0) = \mathcal Lambda_k f_0, t\geq0\,.
$$
Then, $\widehat f_k\in H_{\alpha}^{T,k}$ is a strong solution, and we have
$$ \mathbb{E}\left[\sup_{t\in[0,T],x\in[0,T-t]} \vert \hat f_k(t,x)-f(t,x) \vert^2 \right] \rightarrow 0 $$
for $k\rightarrow \infty$.
\end{thm}
\begin{proof}
First we note that a unique mild solution $\widehat{f}_k$ of the SPDE exists due to Tappe~\cite[Theorem 4.5]{tappe.12}.
Define
$$
V_k(h)(t) := \mathcal U_t f_k(0) + \int_0^t \mathcal U_{t-s} (b_k(s,h(s))\,ds + \psi_k(s,h(s-))\,dL(s))\,,
$$
for any $k\in\mathbb N$, $t\geq 0$ and any adapted c\`adl\`ag stochastic process $h$ in $H_{\alpha}$.
Let $f_k$ be
defined as
\begin{align*}
f_k(t): &= \mathcal U_t f_k(0) + \int_0^t \mathcal U_{t-s} (b_k(s,f(s))\,ds + \psi_k(s,f(s))\,dL(s) \\
&= \mathcal U_t f_k(0) + \int_0^t \mathcal U_{t-s} (b_k(s,f_\Pi(s))\,ds + \psi_k(s,f_\Pi(s-))\,dL(s) \\
&= V_k(f_\Pi)(t)\,,
\end{align*}
for $f_k(0)=\mathcal Lambda_kf(0)$. Moreover, $\widehat f_k(t) = V_k(\widehat f_k)(t)$. By Lemma~\ref{l:fixpoint estimate}, it holds
\begin{align*}
\mathbb{E}\left[\sup_{0\leq s\leq t}\| f_\Pi(t) - \hat f_k(t) \|_\alpha^2\right]\leq\frac{\pi^2}{6}\exp(4C_t) \mathbb{E}\left[ \sup_{0\leq s\leq t}\| f_k(s)-f_\Pi(s)\|^ 2_\alpha\right]\,,
\end{align*}
for any $k\in \mathbb N$, $t\geq 0$ and $C_t$ given in the lemma (recall from Section~\ref{s:the model} that the operator norm of
the shift semigroup $\mathcal{U}_t$ is uniformly bounded by the constant $C_{\mathcal{U}}$).
By the definition of $f_k$ and $f_{\Pi}$ we find
\begin{align*}
\|f_k(s)-f_{\Pi}(s)\|_{\alpha}^2&\leq2\|\int_0^s\mathcal U_{s-r}(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\,dr\|_{\alpha}^2 \\
&\qquad+2\|\int_0^s\mathcal U_{s-r}(\psi_k(r,f_{\Pi}(r-))-\psi_{\Pi}(r,f_{\Pi}(r-)))\,dL(r)\|_{\alpha}^2\,.
\end{align*}
Consider the first term on the right-hand side of the inequality. By the norm inequality for Bochner integrals, Cauchy-Schwartz' inequality and
boundedness of the operator norm of
$\mathcal U_t$ we find (for $s\leq t$)
\begin{align*}
\|\int_0^s\mathcal U_{s-r}&(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\,dr\|_{\alpha}^2 \\
&\qquad\qquad\leq\left(\int_0^s\|\mathcal U_{s-r}(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\|_{\alpha}\,dr\right)^2 \\
&\qquad\qquad\leq t\int_0^t\|\mathcal U_{s-r}(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\|^2_{\alpha}\,dr \\
&\qquad\qquad\leq tC^2_{\mathcal U}\int_0^t\|b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\,dr \\
&\qquad\qquad\leq tC_{\mathcal U}^2\int_0^t\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\,dr
\end{align*}
Here, $\mathcal I$ denotes the identity operator on $H_\alpha^T$. Hence, using
Lemma~\ref{lem:doob} and the fact that $\{\mathcal U\}_{t\geq 0}$ is pseudo-contractive,
\begin{align*}
\mathbb{E}&\left[ \sup_{0\leq s\leq t}\Vert f_k(s)-f_\Pi(s)\Vert^ 2_\alpha\right] \\
&\qquad\leq 2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad\qquad + 2\mathbb{E}\left[\sup_{0\leq s\leq t}\Vert\int_0^s \mathcal U_{s-r}(\psi_k(r,f_\Pi(r-))-\psi_\Pi(r,f_\Pi(r-)))\,dL(r)\|_{\alpha}^2\right] \\
&\qquad\leq 2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad\qquad + 8c_t^2\int_0^t\mathbb{E}\left[\|(\psi_k(r,f_\Pi(r))-\psi_\Pi(r,f_\Pi(r)))\mathcal Q^{1/2}\|_{\text{HS}}^2\right]\,dr \\
&\qquad\leq 2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad\qquad + 8c_t^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)\psi_\Pi(r,f_\Pi(r))\mathcal Q^{1/2}\|_{\text{HS}}^2\right]\,dr \,.
\end{align*}
Denote by
\begin{align*}
K_t(k):&=2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad+8c_t^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)\psi_\Pi(r,f_\Pi(r))\mathcal Q^{1/2}\|_{\text{HS}}^2\right]\,dr \,,
\end{align*}
for $k\in\mathbb N$. By standard norm inequalities, we have
\begin{align*}
K_t(k):&=4 tC_{\mathcal U}^2(1+\|\Pi_k\|^2_{\text{op}})\int_0^t\mathbb{E}\left[\|b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad+16c_t^2(1+\|\Pi_k\|^2_{\text{op}})\int_0^t\mathbb{E}\left[\|\psi_\Pi(r,f_\Pi(r))\|_{\text{op}}^2\right]\,dr \,,
\end{align*}
which is seen to be bounded uniformly in $k\in\mathbb N$ from
Proposition~\ref{l:commutator of U and projectors}.
Hence, we have $ K_{t}(k) \rightarrow 0$ for $k\rightarrow \infty$ and any $t\geq 0$ by the dominated convergence theorem because $(\Pi_k-\mathcal I)h\rightarrow 0$ for $k\rightarrow \infty$ and any $h\in H_\alpha^T$.
Thus, we find
$$
\mathbb{E}\left[\sup_{0\leq t\leq T}\Vert f_k(t) - \hat f_k(t) \Vert_\alpha^2\right] \rightarrow 0\,,
$$
for $k\rightarrow \infty$. Finally, $f_\Pi(t,x) = f(t,x)$ for any $t\in[0,T]$, $x\in[0,T-t]$. Moreover,
from Lemma~3.2 in Benth and Kr\"uhner~\cite{benth.kruehner.14} the sup-norm is dominated by the
$H_{\alpha}$-norm, and therefore we have
\begin{align*}
\mathbb{E}\left[\sup_{t\in[0,T],x\in[T-t]}\vert \hat f_k(t,x)- f(t,x)\vert^2\right] &\leq c \mathbb{E}\left[\sup_{0\leq t\leq T}\Vert \hat f_k(t)-f_\Pi(t) \Vert_\alpha^2\right]\rightarrow0\,,
\end{align*}
for $k\rightarrow \infty$.
The Proposition follows.
\end{proof}
The philosophy in Thm.~\ref{thm:main-markovian} is to take $f(t)$ as the actual forward curve dynamics, and study
finite dimensional approximations $\widehat f_k(t)$ of it. By construction, $\widehat f_k$ solves a HJMM dynamics which yields
that the approximating forward curves become arbitrage-free. From the main theorem, the approximations
$\widehat f_k(t)$ converge uniformly to $f(t)$ for $x\in[0,T-t]$. As time $t$ progresses, the times to maturity $x\geq 0$
for which we obtain convergence shrink. The reason is that information of $f$ is transported
to the left in the dynamics of the SPDE. We recall that the approximation of $f$ is constructed by first
localizing $f$ to $x\in[0,T]$ for a fixed time horizon $T$ by the projection operator $\Pi$ down
to $H_{\alpha}^T$, and next creating finite-dimensional approximations of this.
Alternatively, we may use $f_{\Pi}(t)$ as our forward price model. Then, the finite dimensional
approximation $f_k(t)$ will converge uniformly over all $x\in[0,T]$. In practice, there will be a time
horizon for the futures market for which we have no information. For example, in liberalized power markets
like NordPool and EEX, there are no futures contracts traded with settlement beyond 6 years. Hence,
it is a delicate task to model the dynamics of the futures price curve beyond this horizon. The alternative is
then clearly to restrict the modelling perspective to the dynamics with the maturities confined in
$x\in[0,T]$.
Indeed, in such a context the
structural conditions \eqref{eq:struct-cond-b} and \eqref{eq:struct-cond-psi} will be trivially satisfied as
we restrict our model parameters in any case to the behaviour on $x\in[0,T]$.
We end our paper with a short discussion on a possible numerical implementation of $\widehat f_k(t)$, the
finite-dimensional approximation of $f(t)$. Since $\widehat f_k(t)\in H_{\alpha}^{T,k}$, we can express it as
$$
\widehat f_k(t)=\widehat f_{k,*}(t)+\sum_{n=-k}^kg_n\widehat f_{k,n}(t)\,,
$$
where $\widehat f_{k,*}(t)=\widehat f_k(t,0)g_*$ and $\widehat f_{k,n}(t)=\langle\widehat f_k(t),g_n^*\rangle_{\alpha}$ are $\mathbb C$-valued functions. For any $h\in H_{\alpha}^{T,k}$ it follows that $b_k(t,h)\in H_{\alpha}^{T,k}$. Define for $n=-k,\ldots,k$ the functions
\begin{align*}
\overline{b}_{k,n}&:\mathbb R_+\times \mathbb C^{2k+2}\rightarrow \mathbb C\,; \qquad (t,x_*,x_{-k},\ldots,x_k) \mapsto \left\langleb_k(t,x_*g_*+\sum_{j=-k}^kx_jg_j),g_n^*\right\rangle_{\alpha}\,, \\
\overline{b}_{k,*}&:\mathbb R_+\times \mathbb C^{2k+2}\rightarrow \mathbb C\,; \qquad (t,x_*,x_{-k},\ldots,x_k) \mapsto \left\langleb_*(t,x_*g_*+\sum_{j=-k}^kx_jg_j),g_n^*\right\rangle_{\alpha}\,.
\end{align*}
Furthermore, $\psi_k(t,h)\in L_{\text{HS}}(H_{\alpha},H_{\alpha}^{T,k})$. Thus, for any
$g\in H_{\alpha}$ we have that $\psi_k(t,h)(g)\in H_{\alpha}^{T,k}$. We define the mappings
\begin{align*}
\overline{\psi}_{k,n}&:\mathbb R_+\times\mathbb C^{2k+2}\rightarrow H^*_{\alpha};
(t,x_*,x_{-k},\ldots,x_k)\mapsto \left\langle\psi_k(t,x_*g_*+\sum_{j=-k}^kx_jg_j)(\cdot),g_n^*\right\rangle_{\alpha} \\
\overline{\psi}_{k,*}&:\mathbb R_+\times\mathbb C^{2k+2}\rightarrow H^*_{\alpha};
(t,x_*,x_{-k},\ldots,x_k)\mapsto \left\langle\psi_*(t,x_*g_*+\sum_{j=-k}^kx_jg_j)(\cdot),g_n^*\right\rangle_{\alpha}
\end{align*}
for $n=-k,\ldots,k$. Now, since $\partial_x g_*=0$ and $\partial_xg_n=\lambda_ng_n+g_*/\sqrt{T}$, we find
from the SPDE of $\widehat{f}_k$ the following $2k+2$ system of stochastic differential equations (after comparing terms with respect to the Riesz basis functions),
\begin{align*}
d\widehat{f}_{k,*}(t)&=\left(\frac1{\sqrt{T}}\sum_{n=-k}^k\widehat f_{k,n}(t)+
\overline{b}_{k,*}(t,\widehat{f}_{k,*}(t),\widehat{f}_{k,-k}(t),\ldots,\widehat{f}_{k,k}(t))\right)\,dt \\
&\qquad\qquad+
d\overline{\psi}_{k,*}(t,\widehat{f}_{k,*}(t-),\widehat{f}_{k,-k}(t-),\ldots,\widehat{f}_{k,k}(t-))(L(t)) \\
d\widehat{f}_{k,-k}(t)&=\left(\lambda_{-k}\widehat f_{k,-k}(t)+
\overline{b}_{k,-k}(t,\widehat{f}_{k,*}(t),\widehat{f}_{k,-k}(t),\ldots,\widehat{f}_{k,k}(t))\right)\,dt \\
&\qquad\qquad+
d\overline{\psi}_{k,-k}(t,\widehat{f}_{k,*}(t-),\widehat{f}_{k,-k}(t-),\ldots,\widehat{f}_{k,k}(t-))(L(t)) \\
\cdot & \cdots \\
\cdot & \cdots \\
d\widehat{f}_{k,k}(t)&=\left(\lambda_{k}\widehat f_{k,k}(t)+
\overline{b}_{k,k}(t,\widehat{f}_{k,*}(t),\widehat{f}_{k,-k}(t),\ldots,\widehat{f}_{k,k}(t))\right)\,dt \\
&\qquad\qquad+
d\overline{\psi}_{k,k}(t,\widehat{f}_{k,*}(t-),\widehat{f}_{k,-k}(t-),\ldots,\widehat{f}_{k,k}(t-))(L(t))
\end{align*}
In a compact matrix notation, defining $\mathbf{x}(t)=(x_1(t),x_2(t),\ldots,x_{2k+2}(t))'$ and
$$
A=\left[\begin{array}{ccccc} \frac1{\sqrt{T}} & \frac1{\sqrt{T}} & \frac1{\sqrt{T}} &\cdots & \frac1{\sqrt{T}} \\
0 & \lambda_{-k} & 0 & \cdots & 0 \\
0 & 0 & \lambda_{-k+1} & \cdots & 0 \\
\cdot & \cdot & \cdot & \cdots & \cdot \\
\cdot & \cdot & \cdot & \cdots & \cdot \\
0 & 0 & 0 & \cdots & \lambda_k \end{array}
\right]\,,
$$
we have the dynamics
$$
d\mathbf{x}(t)=(A\mathbf{x}(t)+\overline{\mathbf{b}}_k(t,\mathbf{x}(t)))\,dt+d\overline{\mathbf{\psi}}_k(t,\mathbf{x}(t-))(L(t))\,,
$$
with $\widehat{f}_{k,*}=x_1, \widehat{f}_{k,-k}=x_2,\ldots,\widehat{f}_{k,k}=x_k$.
Using for example an Euler approximation, we can derive an iterative numerical scheme for this
stochastic differential
equation in $\mathbb C^{2k+2}$. We refer to Kloeden and Platen~\cite{KP} for a detailed analysis
of numerical solution of stochastic differential equations driven by Wiener noise.
\end{document}
|
\begin{document}
\bstctlcite{IEEEexample:BSTcontrol}
\title{In-memory Realization of In-situ \\ Few-shot Continual Learning with \\ a Dynamically Evolving Explicit Memory
}
\author{\IEEEauthorblockN{G. Karunaratne\IEEEauthorrefmark{1}\IEEEauthorrefmark{4},
M. Hersche\IEEEauthorrefmark{1}\IEEEauthorrefmark{4},
J. Langenegger\IEEEauthorrefmark{1}\IEEEauthorrefmark{4},
G. Cherubini\IEEEauthorrefmark{1},
M. Le Gallo\IEEEauthorrefmark{1},
U. Egger\IEEEauthorrefmark{1},\\
K. Brew\IEEEauthorrefmark{2},
S. Choi\IEEEauthorrefmark{2},
I. Ok\IEEEauthorrefmark{2},
C. Silvestre\IEEEauthorrefmark{2},
N. Li\IEEEauthorrefmark{2},
N. Saulnier\IEEEauthorrefmark{2},
V. Chan\IEEEauthorrefmark{2},
I. Ahsan\IEEEauthorrefmark{2},\\
V. Narayanan\IEEEauthorrefmark{3},
L. Benini\IEEEauthorrefmark{4},
A. Sebastian\IEEEauthorrefmark{1},
A. Rahimi\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}IBM Research,
Z\"{u}rich, Switzerland
\IEEEauthorrefmark{2}IBM Research, Albany, NY, USA\\
\IEEEauthorrefmark{3}IBM T. J. Watson Research Center, NY, USA
\IEEEauthorrefmark{4}ETH Z\"{u}rich, Z\"{u}rich, Switzerland}\\}
\maketitle
\thispagestyle{firststyle}
\begin{abstract}
Continually learning new classes from few training examples without forgetting previous old classes demands a flexible architecture with an inevitably growing portion of storage, in which new examples and classes can be incrementally stored and efficiently retrieved. One viable architectural solution is to tightly couple a stationary deep neural network to a dynamically evolving explicit memory (EM). As the centerpiece of this architecture, we propose an EM unit that leverages energy-efficient in-memory compute (IMC) cores during the course of continual learning operations. We demonstrate for the first time how the EM unit can physically superpose multiple training examples, expand to accommodate unseen classes, and perform similarity search during inference, using operations on an IMC core based on phase-change memory (PCM). Specifically, the physical superposition of few encoded training examples is realized via in-situ progressive crystallization of PCM devices. The classification accuracy achieved on the IMC core remains within a range of 1.28\%--2.5\% compared to that of the state-of-the-art full-precision baseline software model on both the CIFAR-100 and miniImageNet datasets when continually learning 40 novel classes (from only five examples per class) on top of 60 old classes.
\end{abstract}
\begin{IEEEkeywords}
In-memory Computing, Continual Learning, Few-shot Learning, Hyperdimensional Computing, Non-volatile Memory Devices
\end{IEEEkeywords}
\section{Introduction}
Few-shot continual learning (FSCL), aka few-shot class-incremental learning~\cite{FSCIL_CVPR2020,shi_nips2021}, requires a learner to incrementally learn new classes from very few training examples, without forgetting the previously learned classes. The learner is exposed to a series of sessions, whereby each session introduces distinct unseen classes by providing only a few training examples per class. From these few examples, the learner should quickly and incrementally learn novel classes without forgetting the previously learned old classes. After learning novel classes in each session, the learner is evaluated on several query samples from all the classes, to which it was exposed so far (i.e., the union of the classes from the previous and the current sessions). FSCL is a very challenging research problem that could impose significant additional compute and memory costs on the learner.
Very recently, a low-cost solution was proposed that avoids expensive gradient-based computations for learning unseen classes during the course of FSCL~\cite{C-FSCIL_CVPR22}. This solution is inspired by a robust few-shot learner~\cite{kar_ncom_2021} that brings together deep neural networks with hyperdimensional computing~\cite{Kanerva2009,VSA03} to be able to represent raw images with high-dimensional holographic binary or bipolar vectors. Inspired by this powerful combination, the learner in~\cite{C-FSCIL_CVPR22} is composed of a \emph{frozen} controller and a \emph{dynamically evolving} explicit memory (EM) for FSCL. The controller is a deep convolutional network (including a final fully connected layer) that is interfaced with the EM unit to dynamically store or retrieve the acquired knowledge about the classes. The controller interacts with the EM through write and read operations using $d$-dimensional holographic vectors. Although the controller remains stationary, the EM dynamically updates its contents by new examples, and grows its size by storing new classes. Hence, it is desirable to have an EM unit with dynamically evolving contents that retains the acquired knowledge about already-seen classes in both compressed and nonvolatile manner.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:phases}
\end{figure}
\section{Proposed Explicit Memory Unit for FSCL}
Fig.~\ref{fig:phases}(a)(b) illustrates the main stages of FSCL involved in~\cite{C-FSCIL_CVPR22}: a pretraining and metalearning stage, and an inference stage. The first stage is all done in software. The goal of this rather elaborate stage is to train the controller (here, a ResNet-12) to be able to generate $d$-dimensional\footnote{As seen later in Section~\ref{sec:setup}, $d$ is fixed to 256 so that the vectors occupy an entire column of a PCM crossbar array.} quasi-orthogonal real-valued vectors for different classes. By the end of this stage, the controller has learned how to assign 256-dimensional quasi-orthogonal, and thus dissimilar, vectors to novel classes in the EM, which allows the controller to remain stationary afterwards. This pretraining and metalearning stage is based on just the first session's (S1) training samples. Based on the datasets used in our experiments, as explained in Section~\ref{sec:dataset}, the first session contains 60 classes, each including 500 samples, out of the total 100 classes.
Next, the inference stage is composed of two phases, as shown in Fig.~\ref{fig:phases}(c): a continual learning phase of novel classes from very few training/support examples per class (in our results, 5 training examples, hence 5-shot), and a query evaluation phase in which we evaluate the accuracy over a batch of query examples containing a number of samples (100 in the datasets we used) per each class encountered so far. We quantize the controller’s output vector elements to 1-bit, hence the controller generates bipolar support vectors (of training examples) to be written, or accumulated, in the EM unit during the continual learning phase. For the query vectors (of query samples) during the evaluation phase, we quantize the controller’s output vector elements to 8-bit, so an analog input corresponding to the 8-bit query vector element is applied in each row of the crossbar array. This performs a similarity search between query vector and the class vectors. The resulting vector received via the columns of the crossbar array is used to classify the query sample. The class corresponding to the maximum element of the resulting vector is then taken as the predicted class.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:arch}
\end{figure}
For the inference stage, the EM is implemented on an IMC core with a unit-cell array comprising PCM devices (see Fig.~\ref{fig:arch}). In every continual learning phase: (i) when the first example of a new class appears, the EM is expanded by choosing a fully reset column of unit-cells and, based on the sign of the bipolar support vector element, the unit-cell conductance is increased or decreased by the application of single SET pulses; ii) when an example from a previously seen class appears, a similar update is performed on the column of unit cells corresponding to that particular class.
We exploit the in-situ accumulation via progressive crystallization of PCM to realize physical superposition of few related support vectors on a single class vector. This means, as shown in Fig.~\ref{fig:arch}(b), starting from a fully reset pair of PCM devices, fine grained SET pulses are applied on either the positive or the negative device depending on whether the type of accumulation is incremental or decremental based on the input bipolar support vector element. This creates a multibit analog EM, whereby the size of the EM is set just by the number of classes, as opposed to the product of the number of classes and the number of training examples per class~\cite{kar_ncom_2021}. The nonvolatility of the resulting analog states preserves the acquired knowledge about already-seen classes. Finally, during the evaluation phase, the frequent similarity search between the 8-bit query vector (of a query image) and the analog class vectors are computed in-memory by exploiting Kirchhoff’s circuit laws, and the result is directly used for classification.
Moreover, given that the controller requires no parameter updates after the first stage of pretraining and metalearning, it can be treated as a deep neural network with stationary weights that can also benefit from an IMC-based implementation, as demonstrated for example in~\cite{joshi_ncom2020}.
\begin{figure}
\caption{Experimental Setup. On the right: the micrograph of the Hermes chip. The chip is accessed using a sequence of programming and MVM commands prepared and sent by the host computer via a FPGA-based interface.}
\label{fig:exp_setup}
\end{figure}
\begin{figure}
\caption{\textbf{(a)}
\label{fig:programming}
\end{figure}
\section{Experimental Setup}
\label{sec:setup}
The experiments are carried out on the Hermes chip with a 256x256 unit-cell array of PCM devices organized in a differential configuration \cite{Y2022khaddamJSSC} (see Fig.~\ref{fig:exp_setup}), accessed by a host computer via FPGA (Field Programmable Gate Array)-based interface. The bipolar support vectors are sent as SET pulses to the corresponding column of 256 unit cells of the initially fully RESET array. Based on the +1/-1 vector elements, the conductance of the positive/negative polarity devices are increased. The evolution of the conductance distribution of individual PCM devices as a function of the number of applied SET pulses on initially RESET devices is shown in Fig.~\ref{fig:programming}.
For similarity search in the evaluation phase, a 4-quadrant matrix-vector multiply (MVM) is performed between the 8-bit query vector and the set of analog class vectors stored in the unit-cell array.
\begin{figure}
\caption{A 2-D map of the array conductance (of the unit-cells with differential PCM devices) for the FSCL experiment on CIFAR-100 dataset. With each new session, more columns of the array are selected corresponding to the new classes in that session. And with each training example per class, the corresponding class vectors are updated with the application of SET pulses. The unit-cells activated for the update at each panel are highlighted in green. A close up view of the conductance evolution in a 10x10 region of the crossbar is shown in the bottom row.}
\label{fig:condmaps2}
\end{figure}
\begin{figure}
\caption{FSCL classification accuracy for IMC vs. FP32 software implementations for \textbf{(a)}
\label{fig:accuracy}
\end{figure}
\section{Experimental Results}
\subsection{Datasets used for evaluation}
\label{sec:dataset}
For the accuracy evaluation, we use the CIFAR-100 and the miniImageNet dataset, restructured to comply with the FSCL setting~\cite{FSCIL_CVPR2020, shi_nips2021,C-FSCIL_CVPR22}.
Both datasets contain natural images of 100 classes in total, which are divided into a first session (S1) containing 60 classes with 500 training and 100 query examples per class, and eight novel sessions (S2--S9) with 5 novel classes introduced in each session containing 5 support examples and 100 query examples per class.
The updated class vectors occupy 256 rows and 60 columns (classes) on the array in S1, evolving to 100 columns (classes) in the last S9. The conductance evolution of unit cells of the crossbar array corresponding to CIFAR-100 is shown in Fig.~\ref{fig:condmaps2}.
\subsection{Classification accuracy}
The accuracy obtained for each dataset with our EM on the IMC hardware and various full-precision software baselines are illustrated in Fig.~\ref{fig:accuracy}. Compared to the full-precision software baseline in~\cite{C-FSCIL_CVPR22}, the accuracy degradation with our EM realization is at worst 2.5\% and at best 1.28\% across all sessions. This still makes the accuracy of our EM on the IMC hardware from 2.5\% to 11.01\% higher than the other best performing full-precision software methods reported in~\cite{FSCIL_CVPR2020,shi_nips2021}, considering all sessions across CIFAR-100 and miniImageNet datasets.
\subsection{Energy estimation}
The energy consumption during incremental class vector updates is estimated using the programming parameters, which include peak pulse current of 150\,uA, flat pulse duration 5\,ns, trailing edge pulse duration 40\,ns and source voltage 2.34\,V. These parameters yield an energy expense of 8.78\,pJ per PCM device during one programming cycle. Considering the vector dimension of 256, the total programming time and energy spent during an incremental update of one class vector are 11.5\,us and 2.25\,nJ, respectively. Given that 25 (5-shots from 5 classes) class vectors are updated during all subsequent sessions (S2--S9), the time and energy spent on updating the class vectors in these sessions are estimated to be 57.6\,us and 56.2\,nJ, respectively.
In comparison, the time and energy spent on similarity search of a single query (including digital to analog conversion, PCM read and analog to digital conversion) are estimated to be 520\,ns and 7.74\,nJ, respectively, during the last session. This leads to a total similarity search time and energy of 5.2\,ms and 77.3\,uJ respectively, as in this session we evaluate 10,000 queries (100 queries per class) in total, compared to just 25 updates of the class vectors. The limited number of updates and the application of SET pulses with low energy ensure that the durability of PCM is not significantly affected~\cite{Y2016tumaNatureNano,Y2020legalloJPD}.
\renewcommand{1}{1.3}
\begin{table}[!ht]
\centering
\caption{Comparison with the related works}\label{tab:results-perception-new}
\resizebox{\linewidth}{!}{
\begin{NiceTabular}{lcccc}
\toprule
& \begin{tabular}[c]{@{}c@{}}Kazemi\\ \textit{~et~al.}~\cite{kazemi_date2021}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Li \textit{~et~al.}\\ \cite{li_vlsi2021,li_TED2021}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Karunaratne\\ \textit{~et~al.}~\cite{kar_ncom_2021}\end{tabular} & This work \\
\cmidrule(r){1-1} \cmidrule(r){2-5}
Few-shot learning & \checkmark & \checkmark & \checkmark & \checkmark \\
Continual learning & & & & \checkmark \\
In-situ accumulation & & & & \checkmark \\
Holographic rep. in EM & & & \checkmark& \checkmark\\
Analog multibit EM & & & & \checkmark \\
Truly $\mathcal{O}$(1) search\tabularnote{Actual crossbar (no emulation based on individual memory devices) performing all dot product operations in parallel in-memory} & & \checkmark & & \checkmark \\
Query vector dim. ($d$) & 128--192 & 128 & 512 & 256\\
Number of classes & $\leq$20 & 32 & \textbf{$\leq$100} & \textbf{$\leq$100} \\
Similarity search energy\tabularnote{Core energy normalized to one class vector of length 256 ($d=256$)} & - & \textbf{17.5\,pJ} & 25.6\,pJ & 19.1\,pJ\\
Programming energy\tabularnote{per vector element} & - & 99\,pJ & 6240\,pJ & \textbf{8.78\,pJ}\\
Dataset(s) & Omniglot & Omniglot & Omniglot & \begin{tabular}[c]{@{}c@{}}miniImageNet\\ \& CIFAR100\end{tabular} \\
\bottomrule
\end{NiceTabular}
}
\label{tab:comparison}
\end{table}
\renewcommand{1}{1}
\subsection{Comparison}
We compare features of our work against the related works in Table~\ref{tab:comparison}. Our work shares few-shot learning capability with~\cite{kar_ncom_2021,kazemi_date2021,li_vlsi2021,li_TED2021}, holographic representation of vectors with~\cite{kar_ncom_2021}, and truly $\mathcal{O}(1)$ similarity search capability with~\cite{li_vlsi2021,li_TED2021}. However, our work is the first to demonstrate continual learning capability using the in-situ accumulation property. This makes our EM unit the fist multibit analog IMC core, whereas in \cite{li_vlsi2021,li_TED2021} storage and queries use binary vectors. Furthermore, our EM unit can handle up to 100 class vectors for the natural image datasets, making it the largest truly $\mathcal{O}(1)$ similarity search engine with analog states to date.
In Table~\ref{tab:comparison}, we also compare energy saving we gain by programming the support vectors using the in-situ accumulation instead of programming them from scratch in the novel bit lines. Programming with the in-situ accumulation is at least 4.7$\times$ energy efficient than the normal programming, because the in-situ accumulation requires a SET pulse of shorter duration. Our similarity search energy remains on par with other works.
\section{Conclusion}
We present a hardware based on an in-memory compute core consisting of PCM devices to implement the EM operations required in FSCL. We demonstrate for the first time how support vectors are accumulated in-situ using the progressive crystallization property of PCM. The proposed approach leads to physically superposed representations and enhanced energy savings, while retaining the accuracy within 2.5\% from the full-precision software baseline.
\tiny
\end{document}
|
\begin{document}
\begin{center}
{\Large {\bf On the zeros of $\sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s}$}}
\\ Titus HILBERDINK and Eric SAIAS\\
Department of Mathematics, University of
Reading, Whiteknights,\\
PO Box 220, Reading RG6 6AX, UK\\
and\\
Sorbonne Universit\'{e}, LPSM, 4 Place Jussieu\\
F-75005 Paris, FRANCE
\\
\end{center}
\indent
\begin{abstract} Our main result is to answer a question of Michel Balazard by giving a Dirichlet series with only one zero in its half-plane of convergence. At the end of the paper we also give several sufficient conditions for the Generalized Riemann Hypothesis and ask if any of them are true.
\noindent
{\em 2010 AMS Mathematics Subject Classification}: 11M41, 11M06, 11M26.\newline
{\em Keywords and phrases}: zeros of Dirichlet series, completely multiplicative functions
\end{abstract}
In memory of Jean-Pierre Kahane\newline
\begin{center}
TABLE OF CONTENTS
\end{center}
\begin{description}
\item[1.] Introduction
\begin{description}
\item[1(a)] From Euler to Landau.
\item[1(b)] Zeros of Dirichlet series.
\item[1(c)] Generalization of $\lambda(n)$ to $\lambda_\mathcal{P}(n)$ and $\zeta(s)$ to $\zeta_\mathcal{P}(s)$.
\item[1(d)] Easy or known results on the set of zeros $Z_\mathcal{P}$.
\item[1(e)] New results on the multiset of zeros of Dirichlet series.
\item[1(f)] Zeros and abscissae of convergence for Dirichlet series with completely multiplicative coefficients.
\item[1(g)] Zeros of Helson's zeta functions.
\end{description}
\item[2.] Proof of Theorem 3.
\item[3.] Proof of Theorem 0.
\item[4.] Architecture of the proofs of Theorems 1 and 2.
\item[5.] Vocabulary, notations and results for Beurling primes.
\item[6.] Primes in short intervals.
\item[7.] Abscissae of convergence.
\item[8.] Proofs of Theorems 2 and 1.
\item[9.] Open questions related to GRH$\setminus$RH.
\end{description}
\pagebreak
\noindent
{\bf {\large 1. Introduction}\newline
1(a) From Euler to Landau.}\newline
Euler, in his paper \cite{E} of 1737, writes
\[ 1 - \frac{1}{2} - \frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\frac{1}{9}+\frac{1}{10}-\frac{1}{11}-\frac{1}{12}\quad\mbox{etc.} =0.\]
In modern words we define $\lambda(n)$, Liouville's function, to be the completely multiplicative function which is $-1$ at every prime. What Euler writes is then
\[ \sum_{n=1}^\infty \frac{\lambda(n)}{n}=0.\tag{1.1}\]
It is in this paper he uses the ``Euler'' product formula. He applies it first to the completely multiplicative function $\frac{1}{n}$ and obtains
\[ \prod_p \frac{1}{1-\frac{1}{p}} = \sum_{n=1}^\infty \frac{1}{n} = \infty.\tag{1.2}\]
Then he applies it a second time to the completely multiplicative function $\frac{\lambda(n)}{n}$ and obtains
\[ \sum_{n=1}^\infty \frac{\lambda(n)}{n} = \prod_p \frac{1}{1+\frac{1}{p}} = 0\tag{1.3}\]
due to (1.2), and (1.1) follows.
Let us now read formulas (1.2) and (1.3) with our modern eyes, with our definitions of infinite sums. Since the completely multiplicative function $\frac{1}{n}$ is positive, his proof of (1.2) is valid three centuries later. On the other hand the completely multiplicative function $\frac{\lambda(n)}{n}$ is not positive nor summable. Thus the first equality of (1.3) is not proved.
As a matter of fact, it is Riemann \cite{R} in 1859 for the first part, and de la Vall\'{e}e-Poussin \cite{dVP} and Hadamard \cite{H} 37 years later for the second part, by continuing Euler's $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ as a meromorphic function in $\mathbb{C}$ and proving it does not vanish in the closed half-plane Re $s\ge 1$, who brought the tools to prove (1.1). More precisely, von Mangoldt \cite{vM} proved in 1897, just one year after de la Vall\'{e}e-Poussin and Hadamard, that
\[ \sum_{n=1}^\infty \frac{\mu(n)}{n}=0,\tag{1.4}\]
where $\mu(n)$ is the Mobius function. In 1907, Landau \cite{L} deduced (1.1) from (1.4). Thus 180 years separates Euler's claim and its proof! \newline
\noindent
{\bf 1(b) Zeros of Dirichlet series.}\newline
The series
\[ \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}\mbox{ vanishes only at $s=1$}.\tag{1.5}\]
(This result is part of Theorem 0 below, and we recall briefly its proof in section 3.)
In the situation of any Dirichlet series
\[ F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s},\]
we consider the zeros of $F(s)$ from a naive point of view. We denote by $Z(F)$ the set of complex numbers for which the above series converges and its sum is zero. In particular, we have $Z(F)\subset \{ s\in\mathbb{C}: \mathbb{R}e s\ge \sigma_c(F)\}$ where $\sigma_c(F)$ is the abscissa of convergence of $F$. Notice that the series may converge and have zero sum on the line $\sigma=\sigma_c(F)$. In other words, $Z(F)$ may contain points on this line.
\noindent
{\bf 1(c) Generalization of $\lambda(n)$ to $\lambda_\mathcal{P}(n)$ and $\zeta(s)$ to $\zeta_\mathcal{P}(s)$.}\newline
Let $\mathbb{P}$ denote the set of all primes and let $\mathcal{P}\subset\mathbb{P}$ be a subset. We define the generalized Liouville function associated to $\mathcal{P}$ as the completely multiplicative function defined on primes by
\[ \lambda_\mathcal{P}(p) = \left\{ \begin{array}{cl} -1 & \mbox{ if $p\in\mathcal{P}$}\\ 0 & \mbox{ if $p\not\in\mathcal{P}$}\end{array}\right. .\]
In this paper we study the set of zeros
\[ Z_\mathcal{P} = Z\biggl(\sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s}\biggr).\]
Let $\sigma_\mathcal{P}$ denote the abscissa of convergence of the series $\sum_{p\in\mathcal{P}} p^{-s}$. It is easy to see that
\[ \sigma_\mathcal{P}\le 1.\tag{1.6}\]
We generalize the usual $\zeta(s)$ by denoting
\[ \zeta_\mathcal{P}(s):= \prod_{p\in\mathcal{P}}\frac{1}{1-\frac{1}{p^s}}.\tag{1.7}\]
Of course $\lambda_\mathbb{P}=\lambda$ and $\zeta_\mathbb{P}=\zeta$. As in this particular important case $\mathcal{P}=\mathbb{P}$, the general zeta function $\zeta_\mathcal{P}(s)$ is a normally convergent Euler product in every fixed closed half-plane $\sigma\ge \max\{\sigma_\mathcal{P},0\}+\varepsilon$, for any fixed $\varepsilon>0$. If $\mathcal{P}$ is finite, (1.7) defines a non-vanishing meromorphic function in $\mathbb{C}$ whose multiset of poles is the union of $|\mathcal{P}|$ infinite arithmetic progressions of purely imaginary complex numbers. Except perhaps at $s=0$, all those poles are simple. At $s=0$, the multiplicity is $|\mathcal{P}|$.
Let us now study the case where $\mathcal{P}$ is infinite. Then
\[ \zeta_{\mathcal{P}}(s)\ne 0\quad\mbox{ for $\sigma>\sigma_{\mathcal{P}}$}.\tag{1.8}\]
Moreover, as in the basic usual case, if $\zeta_\mathcal{P}(s)$ has a meromorphic continuation in some open set across the line $\sigma=\sigma_\mathcal{P}$, we continue to denote this continuation by $\zeta_\mathcal{P}(s)$.\newline
Let $\mathcal{N}=\{ n\in\mathbb{N}: p|n \implies p\in \mathcal{P}\}$ (i.e. all the positive integers formed from the primes in $\mathcal{P}$) and let $\sigma_\mathcal{N}$ denote the abscissa of convergence of $\sum_{n\in\mathcal{N}}n^{-s}$. It is easy to prove that
\[ \sigma_\mathcal{N} = \left\{ \begin{array}{cl} -\infty & \mbox{ if $\mathcal{P}=\emptyset$}\\ \max\{ \sigma_\mathcal{P}, 0\} & \mbox{ if $\mathcal{P}\ne\emptyset$} \end{array} \right. .
\tag{1.9}\]
By introducing this complex parameter $s$ and generalizing to any set $\mathcal{P}$ of primes, we can interpret the two Euler formulas (1.2) and (1.3) by the absolutely convergent Euler products
\[ \sum_{n\in\mathcal{N}} \frac{1}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1-\frac{1}{p^s}} = \zeta_\mathcal{P}(s), \quad (\sigma>\sigma_\mathcal{N})\]
and
\[ \sum_{n\in\mathbb{N}} \frac{\lambda_\mathcal{P}(n)}{n^s}=\sum_{n\in\mathcal{N}} \frac{\lambda(n)}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1+\frac{1}{p^s}} = \frac{\zeta_\mathcal{P}(2s)}{\zeta_\mathcal{P}(s)}, \quad (\sigma>\sigma_\mathcal{N}).\tag{1.10}\]
\noindent
{\bf 1(d) Easy or known results on the set of zeros $Z_\mathcal{P}$.}\newline
\noindent
{\bf Theorem 0}\newline
{\em Let $\mathcal{P}$ and $\mathcal{P}^\prime$ be sets of primes.}
\begin{enumerate}
\item {\em If $\sigma_c(\sum_{p\in\mathcal{P}\mathbb{D}elta \mathcal{P}^\prime} p^{-s})\le 0$, then $Z_\mathcal{P} = Z_{\mathcal{P}^\prime}$;}
\item {\em $\overline{Z_\mathcal{P}} = Z_\mathcal{P}$;}
\item {\em if $\sigma_\mathcal{P}\le 0$, then $Z_\mathcal{P}=\emptyset$;}
\item {\em if $\sigma_\mathcal{P}>0$, then $\mathbb{R}e Z_\mathcal{P}\subset (0,\sigma_\mathcal{P}]\subset (0,1]$;}
\item $1\in Z_\mathcal{P}\Leftrightarrow \sum_{p\in\mathcal{P}}\frac{1}{p} =\infty$;
\item $Z_\mathbb{P}=\{1\}$;
\end{enumerate}
\noindent
{\bf Remark 1}\, Point (a) shows that the function $\mathcal{P}\to Z_\mathcal{P}$ is, in a way, locally constant.
\noindent
{\bf 1(e) New results on the multiset of zeros of Dirichlet series.}\newline
We know that the maximal open set where a Dirichlet series $F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$ is both convergent and holomorphic is $\mathbb{C}_{\sigma(F)}$ where $\mathbb{C}_\alpha = \{ s\in\mathbb{C}:$ Re $s>\alpha\}$.
Let $\alpha\in [-\infty,\infty)$. Thanks to Weierstrass (\cite{BG}, Theorem 3.3.1), we know that a necessary and sufficient condition for a multiset $Z$ of $\mathbb{C}_\alpha$ to be the multiset $Z(F)$ of some not identically zero holomorphic function $F$ in $\mathbb{C}_\alpha$, is for $Z$ to be locally finite in $\mathbb{C}_\alpha$.
Now for the same question where we ask $Z$ to be equal to the set $Z(F)\cap \mathbb{C}_\alpha$ for some Dirichlet series $F$ with $\sigma(F)=\alpha$, we are far from knowing the necessary and sufficient condition analogue to the Weierstrass theorem. It is even possible that it is impossible to give such a characterization.
In 2000, Balazard (unsolved problem 24 of \cite{MV}) asked the first question for this problem. He asked for an example of a Dirichlet series $F(s)$ for which $Z(F)\cap \mathbb{C}_{\sigma_c(F)}$ has only one element. Notice that under the Riemann Hypothesis, we have an example, namely the series in (1.5). For this series we then have $Z=\{1\}$ and
\[ \sigma_c=\frac{1}{2}. \tag{1.11}\]
This last formula (1.11) is classical under RH. Eighteen years later, we are able to provide an unconditional family of examples. More precisely we get the following result.\newline
\noindent
{\bf Theorem 1}\newline
{\em Let $a$ and $b$ be two real numbers such that
\[ 0<\max\Bigl\{ \frac{a}{2}, a-\frac{19}{40}\Bigr\}<b<a<1.\tag{1.12}\]
Then there exists a set $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that $Z_{\mathcal{P}_{a,b}}=\{a\}$ with $a$ being a simple zero and $\sigma_c=b, \sigma_a=a$, where $\sigma_c$ and $\sigma_a$ are the abscissa of convergence and absolute convergence of $\sum_{n=1}^\infty \lambda_{\mathcal{P}_{a,b}}(n)n^{-s}$ respectively.
Under the Riemann Hypothesis, we can replace $(1.12)$ by $0<\frac{a}{2}<b<a<1$ or $(a,b)=(1,\frac{1}{2})$.}\newline
Let us remark that to provide an example of $\mathcal{P}$ with $|Z_\mathcal{P}|=1$, we were obliged to choose the zero real, because of the symmetry of $Z_\mathcal{P}$ about the real axis ($\overline{Z_\mathcal{P}}=Z_\mathcal{P}$).
Before giving the answer to another question, let us begin with some obvious remarks. It is very easy to construct Dirichlet series with at least two zeros. The simple example is
\[ 1-\frac{1}{2^s}.\]
Now if we want to have at least two zeros with different real part, it is also easy: just choose
\[ \Bigl(1-\frac{1}{2^s}\Bigr)\Bigl(1-\frac{1}{2^{s-1}}\Bigr).\]
But now we ask the following question:
\[ \mbox{\em Find a Dirichlet series with completely multiplicative coefficients and two real zeros.}\tag{1.13}\]
As far as we know, (1.13) is an open question. Once again, some Dirichlet series $\frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}$ with well chosen $\mathcal{P}$ give a family of examples which allow us to answer this question positively. \newline
\noindent
{\bf Theorem 2}\newline
{\em Let $a$ and $b$ be two real numbers such that
\[ 0<\max\Bigl\{ \frac{a}{2}, a-\frac{19}{40}\Bigr\}<b<a<1.\tag{1.14}\]
Then there exists a set $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that $Z_{\mathcal{P}_{a,b}}\supset\{a,b\}$ with $a$ and $b$ being simple zeros and $\sigma_c\le \max\{\frac{3ab}{2(a+b)}, a-\frac{19}{40}\}, \sigma_a=a$, where $\sigma_c$ and $\sigma_a$ are the abscissae of simple and absolute convergence of $\sum_{n=1}^\infty \lambda_{\mathcal{P}_{a,b}}(n)n^{-s}$ respectively.
Under RH, we can replace $\max\{ \frac{a}{2}, a-\frac{19}{40}\}$ by $\frac{a}{2}$ in $(1.14)$, and $ \max\{\frac{3ab}{2(a+b)}, a-\frac{19}{40}\}$ by $\frac{3ab}{2(a+b)}$.}\newline
Notice that the conditions in Theorems 1 and 2 are the same. As a matter of fact, the proofs of the two results have similar structure, as will be seen in sections 4 and 8.
\newline
\noindent
{\bf 1(f) Zeros and abscissae of convergence for Dirichlet series with completely multiplicative coefficients}\newline
We continue with another question on the set of zeros of a Dirichlet series $F(s)=\sum_{n=1}^\infty\frac{f(n)}{n^s}$ with completely multiplicative coefficients $f(n)$. Denote the abscissa of convergence and absolute convergence by $\sigma_c$ and $\sigma_a$ respectively.
Let $V$ denote the set of values of the difference
\[ \mathbb{R}e \rho - \sigma_c\]
when $f$ varies through the completely multiplicative functions and $\rho\in Z(F)$. What can we say about the set $V$?
By Theorem 1, we know that $V\supset (0,\frac{19}{40})$ which is improved under the Riemann Hypothesis to $V\supset (0,\frac{1}{2})$. But, as a matter of fact, a little more is known unconditionally. Before telling our result, let us consider an analogous question for Dirichlet series with completely multiplicative coefficients.
Let $W$ denote the set of values of
\[ \sigma_a - \mathbb{R}e \rho\]
with $\rho$ as before. Now by Theorem 2, we have $W\supset (0,\frac{19}{40})$ which is improved under the Riemann Hypothesis to $W\supset (0,\frac{1}{2})$. But, in contrast to the case for $V$, we are not able to prove this unconditionally. \newline
\noindent
{\bf Theorem 3}\newline
{\em (i)\, $[0,\frac{1}{2}]\subset V\subset [0,1]$.\newline
(ii)\, There exists $v\in [\frac{1}{2},1]$ such that $V=[0,v]$ or $V=[0,v)$.\newline
(iii)\, $[0,\frac{19}{40})\cup \{\frac{1}{2}\}\subset W\subset [0,1]$, and under RH, we can replace (iii) by\newline
(iii$)^\prime$\, $[0,\frac{1}{2}]\subset W\subset [0,1]$.}\newline
Notice that these results cannot be extended to the case where $f$ is only multiplicative. To see this, consider $f$ defined by $f(1)=1$, $f(2) =-1$ and zero otherwise. Then $F(s)=1-2^{-s}$ vanishes at 0 and $\sigma_c(F) = \sigma_a(F)=-\infty$. \newline
\noindent
\noindent
{\bf 1(g) Zeros of Helson zeta functions}\newline
In \cite{S}, Seip studies the multiset of zeros of Helson zeta functions. For each completely multiplicative unimodular function $\chi$, Helson's zeta function $\zeta_\chi(s)$, is the meromorphic continuation (if any) of the Dirichlet series
\[ \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.\]
We shall call a Helson zeta function {\em admissible} if it has a meromorphic continuation to $\mathbb{C}_{\frac{1}{2}}$.
Notice that by the Euler product, $\zeta_\chi(s)$ cannot vanish for $\sigma>1$. Moreover (\cite{S}, Theorem 2.1) an admissible Helson zeta function has at most one zero on the line $\sigma=1$ and, if it has one, it is simple. The function $\zeta(2s)/\zeta(s)$, which we already discussed, is an example where there is one.
Let us look now in the strip $S:=\{ s\in\mathbb{C}:1/2<$ Re$s<1\}$. Let $Z$ be a multiset belonging to $S$. We recall that by Weierstrass' Theorem (\cite{BG}, Theorem 3.3.1), a necessary condition for $Z$ to be the multiset of zeros of an admissible zeta function is:
\[ \mbox{\em $Z$ is locally finite in $\mathbb{C}_{\frac{1}{2}}$.}\tag{1.15}\]
Seip proves that under the Riemann Hypothsesis this necessary condition is also sufficient: if (1.15) is true, then there exists an admissible Helson zeta function whose multiset of zeros in $S$ is equal to $Z$.
The comparison of this result with ours is impressive. Under RH, Seip has found the necessary and sufficient condition. The main result here is only to make unconditional the (conditional on RH) 100 year-old result that there exists an example with a single zero in the half-plane of convergence!
We do not know if it is possible to use some of Seip's results and/or tools in \cite{S} to find new examples of sets of zeros of Dirichlet series with completely multiplicative coefficients themselves, and not of their possible meromorphic continuation.
\noindent
{\bf {\large 2. Proof of Theorem 3}}\newline
{\bf Step 1}\, We prove that $V\subset [0,1]$. Let $f$ be a completely multiplicative function. Let the Dirichlet series
\[ \sum_{n=1}^\infty \frac{f(n)}{n^s}\tag{2.1}\]
have abscissa of convergence and absolute convergence $\sigma_c$ and $\sigma_a$ respectively, and let $\rho$ be a zero of the series.
Then $\sigma_c\le \mathbb{R}e\rho\le \sigma_a$ for the series has to converge at $\rho$ and for $\sigma>\sigma_a$,
\[ \sum_{n=1}^\infty \frac{f(n)}{n^s} = \prod_p\frac{1}{1-\frac{f(p)}{p^s}}\ne 0. \]
Hence
\[ 0\le \mathbb{R}e\rho - \sigma_c\le \sigma_a-\sigma_c \le 1,\tag{2.2}\]
and $V\subset [0,1]$.\newline
\noindent
{\bf Step 2}\, We prove that $0\in V$. Define the completely multiplicative function $f$ on primes $p$ by
\[ f(p) = -\frac{1}{\log\log p}\qquad (p\ge 29)\]
and zero otherwise. As $29>e^e$, we have $-1\le f(p)\le 0$ for all $p$. Moreover, we have
\[ \sum_p \frac{f(p)}{p} = -\sum_{p\ge 29} \frac{1}{p\log\log p} = -\infty.\]
By applying Theorem 9 of \cite{KS1}, it follows that the Dirichlet series in (2.1) vanishes at $s=1$. To finish the proof, it suffices to verify that the abscissa of the series is 1.
Now
\[ \sum_{p\ge 29} \frac{1}{p^\sigma \log\log p} \asymp \log\log \frac{1}{\sigma -1} \quad\mbox{ $(1<\sigma<1+\frac{1}{100})$}.\]
As
\[ F(s) = \prod_{p\ge 29} \frac{1}{1+\frac{1}{p^s \log\log p}},\qquad (\sigma>1)\]
it follows that $\log F(\sigma) \asymp - \log\log\frac{1}{\sigma -1}$ and
\[ F(\sigma)\gg \frac{1}{( \log \frac{1}{\sigma-1})^2}.\]
As $F(1)=0$, $F(s)$ cannot be extended holomorphically to a neighbourhood of 1, and the result follows.\newline
\noindent
{\bf Remark 2}\, Here we have an example where $\sigma_c=\mathbb{R}e\rho = \sigma_a$.\newline
\noindent
{\bf Step 3}\, We prove the following: {\em if $f$ is completely multiplicative and $\beta^*+i\gamma$ is a zero of (2.1), then}
\[ (0,\beta^* - \sigma_c]\subset V.\]
To see this, let $f_1(n) = f(n)n^{-\sigma_c-i\gamma}$. The Dirichlet series for $f_1$ now has abscissa of convergence 0 and a real zero at some $\beta\ge 0$. We have to show that
\[ (0,\beta]\subset V.\tag{2.3}\]
Note that by (2.2), $\beta\le 1$. Let $\mathcal{P} = \{ p_1, p_2, \ldots \}$ where $p_k$ is an increasing sequence of prime numbers such that $p_k\asymp 2^k$ and let $\alpha\in (0,\beta)$.
Define a completely multiplicative function $g_\alpha$ by
\[ g_\alpha(p) = \left\{ \begin{array}{cl} p^\alpha & \mbox{ if $p\in\mathcal{P}$}\\ f_1(p) & \mbox{ if $p\not\in\mathcal{P}$} \end{array} \right. .\]
Let $\sigma >\alpha$. As $f_1(n)\ll n^\alpha$
\[ \sum_{p\in\mathcal{P}} \frac{|f_1(p)+g(p)|}{p^\sigma} \ll \sum_{p\in\mathcal{P}} \frac{1}{p^{\sigma-\alpha}}\asymp \sum_{k=1}^\infty \frac{1}{2^{(\sigma-\alpha)k}}<\infty.\]
It follows that the product
\[ H(s) = \prod_{p\in \mathcal{P}} \frac{1-\frac{f_1(p)}{p^s}}{1-\frac{g_\alpha(p)}{p^s}}\]
is absolutely convergent for $\sigma>\alpha$. Moreover, the Dirichlet series for $f_1$ and $g_\alpha$ are absolutely convergent for $\sigma>1$ and so, using Euler products, we have for $\sigma>1$
\[ \sum_{n=1}^\infty \frac{g_\alpha(n)}{n^s} = H(s) \sum_{n=1}^\infty \frac{f_1(n)}{n^s}.\]
It follows that the series on the left is actually convergent for $\sigma>\alpha$ and hence the above holds for $\sigma>\alpha$. Thus it is zero at $\beta$ and its abscissa of convergence is at most $\alpha$. However, $g_\alpha(p) = p^\alpha$ for $p\in\mathcal{P}$, so this abscissa is at least $\alpha$. Hence it equals $\alpha$. It follows that $\beta-\alpha\in V$. But $\alpha$ has been chosen arbitrarily in $(0,\beta)$, so (2.3) follows.\newline
\noindent
{\bf Step 4}\, $\frac{1}{2}\in V$. This follows immediately from the fact that the $L$-function associated to any non-principal Dirichlet character has zeros on the critical line and abscissa of convergence 0.\newline
These steps prove (i) and (ii) of Theorem 3. Now consider (iii).
By replacing $\mathbb{R}e\rho - \sigma_c$ with $\sigma_a-\mathbb{R}e\rho$ in (2.2) we obtain $W\subset [0,1]$.
Step 2 is also valid in the case of $W$ and shows $0\in W$. Next, the example in Step 4 (with $\sigma_a=1$) shows that $\frac{1}{2}\in W$.
By Theorem 2, we have $(0,\frac{19}{40})\subset W$, which concludes the proof of part (iii).
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 3. Proof of Theorem 0}}\newline
{\em Proof of} (a).\, Let $\rho = a+ib\in Z_{\mathcal{P}}$. Since for all $s$
\[ \sum_{n=1}^\infty \frac{\lambda_\emptyset(n)}{n^s}=1\ne 0,\]
we have $\mathcal{P}\ne\emptyset$.
Since $\lim_{n\to\infty} \frac{\lambda_\mathcal{P}(n)}{n^\rho}=0$, it follows that
\[ a+ib\in Z_{\mathcal{P}}\implies a>0.\tag{3.1}\]
Now we call a completely multiplicative function $h$ CMO if $\sum_{n=1}^\infty h(n)=0$. We suppose again that $\rho = a+ib\in Z_{\mathcal{P}}$. On one hand it means that $f(n):=\frac{\lambda_\mathcal{P}(n)}{n^\rho}$ is CMO. On the other hand, by (3.1), we have $a>0$. Thus for $g(n):=\frac{\lambda_{\mathcal{P}^\prime}(n)}{n^\rho}$, $g$ is completely multiplicative such that for all primes $p$, $|g(p)|<1$. As
$\sigma_c(\sum_{p\in\mathcal{P}\mathbb{D}elta \mathcal{P}^\prime} p^{-s})\le 0$, we have also $\sum_p |g(p)-f(p)|<\infty$. By using th\'{e}or\`{e}me 3 of \cite{KS1}, it follows that $g$ is CMO. In other words, $\rho\in Z_{\mathcal{P}^\prime}$, and (a) is proven.
Part (b) comes from the fact that $\lambda_\mathcal{P}$ is a real function.
Part (c) follows from the combination of $Z_\emptyset=\emptyset$ and part (a).
Part (d) follows from the combination of (3.1), (1.10), (1.9), (1.6) and (1.8).
Part (e) comes from th\'{e}or\`{e}me 9 of \cite{KS1}.
\noindent
{\em Proof of} (f).\, By (e), we know that $1\in Z_\mathbb{P}$.
To show that it contains no other points, let $\rho\in Z_\mathbb{P}$. Since the abscissa of convergence of $\sum_{n=1}^\infty\frac{\lambda(n)}{n^s}$ is at least $\frac{1}{2}$ due to the existence of Riemann zeros, it follows that Re $\rho\ge\frac{1}{2}$. Now, by Abel's Theorem
\[ 0=\sum_{n=1}^\infty\frac{\lambda(n)}{n^\rho} =\lim_{\varepsilon\to 0+}\sum_{n=1}^\infty\frac{\lambda(n)}{n^{\rho+\varepsilon}} = \lim_{\varepsilon\to 0+}\frac{\zeta(2\rho+2\varepsilon)}{\zeta(\rho+\varepsilon)}.\]
But $\zeta(2s)\ne 0$ for $\sigma\ge\frac{1}{2}$, so $\rho$ must be a pole of $\zeta$; i.e. $\rho =1$.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 4. Architecture of the proofs of Theorems 1 and 2}}\newline
Let $a\in (0,1)$. We begin by recalling the proof in \cite{KS2} of the existence of a set of primes $\mathcal{P}_a$ such that $a$ belongs to $Z_{\mathcal{P}_a}$. We know that $1$ is a zero of $F(s):=\frac{\zeta(2s)}{\zeta(s)}$. It follows that $a$ is a zero of $F(s/a)$. But
\[ F\Bigl(\frac{s}{a}\Bigr) = \frac{\zeta_{\mathbb{P}^{1/a}}(2s)}{\zeta_{\mathbb{P}^{1/a}}(s)},\]
where $\zeta_{\mathbb{P}^{1/a}}(s)$ is the zeta function associated to the set of Beurling primes $\mathbb{P}^{1/a} = \{p^{1/a}:p\in\mathbb{P}\}$. This set is sparse. It is the reason why the PNT allows us to approximate $\mathbb{P}^{1/a}$ by a subset $\mathcal{P}_a$ of $\mathbb{P}$ such that
\[ a\mbox{ is a zero of } \frac{\zeta_{\mathcal{P}_a}(2s)}{\zeta_{\mathcal{P}_a}(s)}.\tag{4.1}\]
We have
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_a}(n)}{n^s} = \frac{\zeta_{\mathcal{P}_a}(2s)}{\zeta_{\mathcal{P}_a}(s)}, \qquad (\sigma>a).\tag{4.2}\]
As for the usual case $a=1$ and $\mathcal{P}_1=\mathbb{P}$, we prove that we can deduce the wanted formula
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_a}(n)}{n^a} =0\tag{4.3}\]
from the combination of (4.1) and (4.2).
The architecture of the proofs of Theorems 1 and 2 is similar, but we need to introduce two new tools. Under RH, we recall that we know an example which answers Balazard's question. It is
\[ F(s) = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}\qquad \mbox{ for $\sigma>\frac{1}{2}$}\tag{4.4}\]
for which $s=1$ is the only (simple) zero of $F$ in the open half-plane where $\sigma>\frac{1}{2}$. But we do not know that RH is true. To get an unconditional example, we prove that it is possible to choose a set of $\mathcal{P}$ of primes such that (4.4) is replaced by
\[ F_\mathcal{P}(s) = \sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s} = \frac{\zeta_\mathcal{P}(2s)}{\zeta_\mathcal{P}(s)}\qquad \mbox{ for $\sigma>\sigma_c(F_\mathcal{P})$}\tag{4.5}\]
with $\sigma_c(F_\mathcal{P})<1$, and for which $s=1$ is always a simple zero.
More precisely, instead of working with the usual zeta function, we begin to work with Zhang's zeta function $\zeta_\mathcal{R}(s)$ (see \cite{Z} or \cite{DZ}) associated to an appropriate multiset $\mathcal{R}$ of generalized primes which share some of the properties of $\zeta(s)$ under RH:
\begin{itemize}
\item $\zeta_\mathcal{R}(s) = \prod_{r\in \mathcal{R}} \frac{1}{1-\frac{1}{r^s}}$ is normally convergent in every half-plane $\sigma>1+\varepsilon$ with $\varepsilon>0$;
\item $\zeta_\mathcal{R}(s)$ has a non-vanishing meromorphic continuation of finite order to $\sigma>\frac{1}{2}$ with a unique simple pole at 1 with residue 1.
\end{itemize}
Now, more generally than in \cite{KS2}, we work here with the group of meromorphic functions in some non-empty open vertical half-plane generated by the function $\zeta_\mathcal{R}(\lambda s)$ where $\lambda>0$.
To prove Theorem 1 we use the function
\[ \frac{\zeta_\mathcal{R}(s/b)}{\zeta_\mathcal{R}(s/a)}\]
which has a simple zero at $a$ and a simple pole at $b$. To prove Theorem 2, we use the function
\[ \frac{1}{\zeta_\mathcal{R}(s/a)\zeta_\mathcal{R}(s/b)}\]
which has two simple zeros at $a$ and $b$. To finish off the proofs, we need to approximate these meromorphic functions by functions of the form
\[ \frac{\zeta_{\mathcal{P}_{a,b}}(2s)}{\zeta_{\mathcal{P}_{a,b}}(s)}\]
with $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that this function has the same zeros and poles as described above and such that the formula
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_{a,b}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}_{a,b}}(2s)}{\zeta_{\mathcal{P}_{a,b}}(s)}\]
is valid for $\sigma>b$ in the first case and for $s=a,b$ in the second.
To do this approximation, instead of using the PNT as in \cite{KS2}, we use the Baker, Harman and Pintz result on primes in short intervals
\[ \pi(x+y) - \pi(x) \asymp \frac{y}{\log x}\quad\mbox{ for $x^{\frac{21}{40}}\le y\le x$ and $x$ large enough.}\]
It is this number $\frac{21}{40}$ which explains the number $\frac{19}{40}$ in our results.
\newline
\noindent
{\bf {\large 5. Vocabulary, notations and results for Beurling primes}}\newline
In 1937, Beurling \cite{Be} (see also \cite{DZ}) had the idea to generalize the usual couple $(\mathbb{P}, \mathbb{N})$ formed by the usual sets $\mathbb{P}$ of primes and $\mathbb{N}$ of positive integers in the following way. He considers any multiset $\mathcal{P}$ of $(1,\infty)$ which is locally finite in $[1,\infty)$. The elements of $\mathcal{P}$ are called the {\em generalized} primes. He defines $\mathcal{N}$ to be the multiset of $[1,\infty)$ formed by the finite product of elements of $\mathcal{P}$ (the number 1 occurs as the product indexed by the subset $\emptyset$). We will talk of a discrete generalized prime system or just g-prime system for such a couple $(\mathcal{P}, \mathcal{N})$.
Let us mention that there are two natural generalizations of g-prime systems which we will not use directly here. We refer the interested reader to \cite{Hi}.
Let $(\mathcal{P}, \mathcal{N})$ be a g-prime system. Every element $n$ of the multiset $\mathcal{N}$ has a unique decomposition in generalized primes
\[ n=\prod_{p\in\mathcal{P}} p^{v_p(n)}.\]
We define the generalized M\"{o}bius function $\mu_\mathcal{P}$ on the multiset $\mathcal{N}$ by the formula
\[ \mu_\mathcal{P}(n) = \left\{ \begin{array}{cl} 0 & \mbox{ if $\exists p\in\mathcal{P}$ with $v_\mathcal{P}(n)\ge 2$}\\ (-1)^{\sum_{p\in\mathcal{P}}v_\mathcal{P}(n)} & \mbox{ if not}\end{array} \right. .\]
When $\mathcal{P}=\mathbb{P}$, $\mu_\mathcal{P}$ is the usual M\"{o}bius function.
Notice that for any sequence $(a_n)_{n\in\mathcal{N}}$ of complex numbers defined on $\mathcal{N}$, the function
\[ \sum_{n\in\mathcal{N}} \frac{a_n}{n^s}\]
is a generalized Dirichlet series. We shall use, without referencing anymore, the definitions and properties of generalized Dirichlet series. (See \cite{HR} for the theory of generalized Dirichlet series.)
Notice that in the introduction, we considered the particular example of a g-prime system $(\mathcal{P}, \mathcal{N})$ with $\mathcal{P}\subset\mathbb{P}$. The definition of $\zeta_\mathcal{P}(s)$ generalizes with no difficulty for any g-prime system. If the series $\sum_{n\in\mathcal{N}} n^{-s}$ has a finite abscissa of convergence $\sigma_c$, then
\[ \zeta_\mathcal{P}(s):= \sum_{n\in\mathcal{N}}\frac{1}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1-\frac{1}{p^s}}\]
defines a non-vanishing holomorphic function for $\sigma>\sigma_c$. If $\zeta_\mathcal{P}(s)$ has a meromorphic continuation to some open set across the line $\sigma=\sigma_c$ we continue to write $\zeta_\mathcal{P}(s)$ for this continuation.
If $\mathcal{P},\mathcal{P}_1, \mathcal{P}_2$ are multisets of primes and $a>0$, we have
\[ \zeta_{\mathcal{P}_1\sqcup \mathcal{P}_2}(s) = \zeta_{\mathcal{P}_1} (s)\zeta_{\mathcal{P}_2}(s)\]
and
\[ \zeta_{\mathcal{P}^{1/a}}(s) = \zeta_\mathcal{P}(s/a).\]
Let $Q=\{q_1,q_2,\ldots \}$ be an infinite multiset of generalized primes, $I:Q\to (1,\infty)$ and $\sigma_0\ge 0$. We shall say that
\[ \prod_{q\in Q} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} \tag{5.1}\]
is a {\em simply absolutely convergent quotient of generalized Euler products in} $\sigma>\sigma_0$ if for all $s$ in the half-plane where $\sigma>\sigma_0$ the infinite product
\[ \prod_{k=1}^\infty \frac{1-\frac{1}{I(q_k)^s}}{1-\frac{1}{q_k^s}}\]
is absolutely convergent. Notice that the expression in (5.1) represents a non-vanishing holomorphic function in this half-plane.
We shall denote multisets of generalized integers by calligraphic letters ($\mathcal{A},\mathcal{B},\mathcal{C}, \ldots$) with the corresponding counting function by its capital equivalent.
eg for such a multiset $\mathcal{A}$, let
\[ A(x) = \sum_{\tiny \begin{array}{c} a \le x \\ a\in \mathcal{A}\end{array} } 1.\]
Moreover, if $c>0$, we write
\[ A^{1/c}(x): = | \{ a\in \mathcal{A}:a^{1/c}\le x\}| = A(x^c).\]
For $\mathbb{P}$ however, we shall keep the traditional notation $\pi(x)$ and $[x]$ for the counting funtions of the primes and natural numbers.\newline
\noindent
{\bf Definition.}\, Let $(\mathcal{R}, \mathcal{N})$ be a g-prime system. We say it is {\em good} if it satisfies the following properties:
\begin{align*}
N(x) & = x+O_\varepsilon(x^{1/2+\varepsilon})\tag{5.2}\\
R(x) & = \mbox{\rm{li}}(x)+O_\varepsilon(x^{1/2+\varepsilon})\tag{5.3}
\end{align*}
for all $\varepsilon>0$\footnote{Here li$(x)$ is the usual {\em logarithmic integral}, given by li$(x)=\int_2^x \frac{1}{\log y}\, dy$.}. As such, $\zeta_\mathcal{R}(s)$ has a non-vanishing meromorphic continuation to the half-plane $\sigma>\frac{1}{2}$, with exactly one (simple) pole at $1$ with residue 1.\newline
\noindent
{\bf Remark 3}\, Of course, under the Riemann Hypothesis, the basic g-prime system $(\mathbb{P}, \mathbb{N})$ is good.\newline
\noindent
{\bf Remark 4}\, Comparing to Zhang's work (\cite{Z} or \cite{DZ}) we changed conditions (5.2) and (5.3) a little.\newline
Now Zhang proved that, unconditionally, good systems exist (see \cite{DZ}, Theorem 17.11 and remark 17.12). \newline
\noindent
{\bf Theorem} (Zhang)\newline
{\em A good g-prime system exists.}\newline
\noindent
{\bf Lemma 5.1}\newline
{\em Let $(\cal{R}, \cal{N})$ be a good g-prime system Then for all $\varepsilon>0$, there exists $C>0$ such that }
\begin{align*}
|\zeta_{\cal{R}}(s)| & \le \exp \{ C(\log |t|)^{2(1-\sigma)+\varepsilon}\} \tag{$\frac{1}{2}+\varepsilon\le\sigma\le 1, |t|\ge 2$}\\
|\zeta^{-1}_{\cal{R}}(s)| &\le \exp\{ C(\log (|t|+2))^{2(1-\sigma)+\varepsilon}\} \tag{$\frac{1}{2}+\varepsilon\le\sigma\le 1, t\in\mathbb{R}$}
\end{align*}
This follows from Theorem 2.3 of \cite{HL}.\newline
\noindent
{\bf {\large 6. Primes in short intervals}}\newline
We begin with the usual primes.\newline
\noindent
{\bf Lemma 6.1} (Baker, Harman and Pintz)\newline
{\em We have, for $x^{\frac{21}{40}}\le y\le x$ and $x$ large enough
\[ \pi(x+y)-\pi(x)\ge \frac{y}{12\log x}.\]
Proof.}\, This is a consequence of Theorem 10.8 of \cite{Ha}.\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 6.2} \newline
{\em Let $(\mathcal{R},\mathcal{N})$ be a good g-prime system, $c\in (0,1)$, $h\in (0,1)$ and $\varepsilon>0$. Then
\[ R^{1/c}(x+x^h)-R^{1/c}(x) = \frac{x^{c+h-1}}{\log x} + O_\varepsilon(x^{\max\{\frac{c}{2}+\varepsilon,c+2h-2\}}).\]
Proof.}\, We have
\[ (x+x^h)^c = x^c\Bigl(1+\frac{1}{x^{1-h}}\Bigr)^c = x^c + cx^{c+h-1} + O(x^{c+2h-2}).\]
As $(\mathcal{R},\mathcal{N})$ be a good g-prime system, we have
\begin{align*}
R^{1/c}(x+x^h)-R^{1/c}(x) & = R((x+x^h)^c)-R(x^c) = \int_{x^c}^{(x+x^h)^c}\frac{1}{\log t}\, dt + O_\varepsilon(x^{c/2+\varepsilon})\\
& = \frac{x^{c+h-1}}{\log x} + O_\varepsilon(x^{\max\{\frac{c}{2}+\varepsilon,c+2h-2\}}).
\end{align*}
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 6.3} \newline
{\em Let $\mathcal{Q}$ and $\mathcal{Q}^*$ be two multisets of generalized primes. Let $\delta>0$ and $0<h\le 1$ be two real numbers such that
\[ Q(x)\ll x^\delta\tag{6.1}\]
\[ \lim_{x\to\infty} (Q^*(x)-Q(x))=\infty,\tag{6.2}\]
and for $x$ large enough,
\[ Q^*(x+x^h)-Q^*(x)\ge Q(x+x^h)-Q(x).\tag{6.3}\]
Then there exists an injection $I:\mathcal{Q}\to \mathcal{Q}^*$ such that
\[ \prod_{q\in \mathcal{Q}} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} = \frac{\zeta_{\mathcal{Q}}(s)}{\zeta_{I(\mathcal{Q})}(s)}\]
is a simply absolutely convergent quotient of generalized Euler products in $\sigma>\max\{\delta+h-1,0\}$.}
\newline
\noindent
{\em Proof.}\, Let $T_n$ be an increasing sequence of positive reals defined by $T_1=1$ and
\[ T_{n+1}=T_n+T_n^h \qquad (n\ge 1).\]
By (6.2) and (6.3) there exists a positive integer $n_0$ such that there is an injection $I_0:{\cal{Q}} \cap (1,T_{n_0}] \to {\cal{Q}}^* \cap (1,T_{n_0}]$ and for all $n\ge n_0$, there is also an injection
\[I_n: \mathcal{Q}\cap (T_n,T_{n+1}] \to \mathcal{Q}^*\cap (T_n,T_{n+1}].\]
As the intervals $(T_n,T_{n+1}]$ are disjoint, we get a global injection $I:\cal{Q} \to \cal{Q}^*$ such that $I(q)=q+O(q^h)$.
For fixed $s$ with $\sigma>0$, we have, uniformly in $q$,
\[ I(q)^s = q^s\Big(1+ O\Bigl(\frac{1}{q^{1-h}}\Bigr)\Bigr).\]
Thus
\[ \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} = 1+ O\Bigl(\frac{1}{q^{\sigma+1-h}}\Bigr),\]
and using (6.1)
\[\log\biggl( \prod_{q\in \mathcal{Q}} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}}\biggr) \ll \sum_{q\in\mathcal{Q}}\frac{1}{q^{\sigma+1-h}} = \sum_{n=0}^\infty\sum_{2^n<q\le 2^{n+1}}\frac{1}{q^{\sigma+1-h}} \ll \sum_{n=0}^\infty \frac{2^{\delta n}}{2^{(\sigma+1-h)n}} <\infty\]
if $\sigma>\delta+h-1$, as required.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 7. Abscissae of convergence}}\newline
Let $(\cal{R},\cal{N})$ be a good g-prime system and $c\in (\frac{1}{2},1)$. Let $\mathcal{Q}_c:=\mathcal{R}\cup \mathcal{R}^{1/c}$ and $\mathcal{M}_c$ the multiset of generalized integers associated to $\mathcal{Q}_c$.
\newline
\noindent
{\bf Lemma 7.1}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime system. Then}
\begin{align*}
\sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} &= \frac{x^{1-\alpha}}{1-\alpha} +
\left\{ \begin{array}{cl} O(x^{\frac{1}{2}-\alpha+\varepsilon}) & \mbox{ if $\alpha\in (0,\frac{1}{2}]$}\\ \zeta_{\mathcal{R}}(\alpha) + O(x^{\frac{1}{2}-\alpha+\varepsilon}) & \mbox{ if $\alpha\in (\frac{1}{2},1)$}\end{array} \right. \\
\sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} &= \zeta_{\mathcal{R}}(\alpha) - \frac{1}{(\alpha-1)x^{\alpha-1}} +O(x^{\frac{1}{2}-\alpha+\varepsilon}) \quad\mbox{ if $\alpha>1$.}
\end{align*}
{\em Proof.}\, These follow from writing
\[ \sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} = \int_{1-}^x \frac{1}{t^\alpha}\, dN(t) = \frac{N(x)}{x^\alpha}+\alpha\int_1^x \frac{N(t)}{t^{\alpha+1}}\, dt\]
and using the fact that $N(t)=t+O(t^{\frac{1}{2}+\varepsilon})$. Thus, for $\alpha\ne 1$,
\[ \sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} =x^{1-\alpha} + O(x^{\frac{1}{2}-\alpha+\varepsilon}) +
\frac{\alpha(x^{1-\alpha}-1)}{1-\alpha} +\alpha\int_1^x \frac{N(t)-t}{t^{\alpha+1}}\, dt.\]
If $\alpha\le\frac{1}{2}$, the integral is $O(x^{\frac{1}{2}-\alpha+\varepsilon})$ and the result follows. If $\alpha>\frac{1}{2}$, the integral is
\[ \alpha\int_1^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt -\alpha\int_x^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt = \alpha\int_1^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt+ O(x^{\frac{1}{2}-\alpha+\varepsilon}).\]
Now for $\alpha>1$, the integral on the right is just $\zeta_{\mathcal{R}}(\alpha) - \frac{\alpha}{\alpha-1}$. But by analytic continuation, this still holds for $\alpha\in (\frac{1}{2},1)$.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 7.2}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime system and $c\in (\frac{1}{2},1)$. Then for all fixed $\varepsilon>0$, we have
\[ M_c(x) = \zeta_\mathcal{R}\Bigl(\frac{1}{c}\Bigr)x + \zeta_{\mathcal{R}}(c)x^c
+O_\varepsilon(x^{\frac{3c}{2(1+c)}+\varepsilon}).\]
Proof.}\, Throughout this proof, $k$ and $\ell$ will implicitly be used to denote generalized integers of $\mathcal{N}$. We use Dirichlet's hyperbola method. Thus for every positive reals $x$ and $y$, we have
\begin{align*}
M_c(x) & = \sum_\ell \sum_{k\ell^{1/c}\le x} 1 = \sum_{\ell^{1/c}\le y} \sum_{k\le\frac{x}{\ell^{1/c}}} 1 + \sum_{k\le \frac{x}{y}} \sum_{\ell^{1/c}\le \frac{x}{k}} 1 - \sum_{\ell^{1/c}\le y} \sum_{k\le\frac{x}{y}} 1\\
& = S_1+S_2-S_3,
\end{align*}
where
\[ S_1 = \sum_{\ell^{1/c}\le y} N\Bigl(\frac{x}{\ell^{1/c}}\Bigr), \quad S_2=\sum_{k\le \frac{x}{y}} N\Bigl(\Bigl(\frac{x}{k}\Bigr)^c\Bigr) , \quad S_3 = N(y^c)N\Bigl(\frac{x}{y}\Bigr).\]
Now we use the formula $N(t)=t+O(t^{\frac{1}{2}+\varepsilon})$ to estimate these three sums, and at the end we optimize in $y$.
We have
\[ S_1 = x\sum_{\ell\le y^c} \frac{1}{\ell^{1/c}} +O\biggl(x^{\frac{1}{2}+\varepsilon}\sum_{\ell\le y^c} \frac{1}{\ell^{1/2c}}\biggr).\]
So, by Lemma 7.1, we obtain
\[ S_1 = x\biggl( \zeta_{\mathcal{R}}\Bigl(\frac{1}{c}\Bigr) - \frac{c}{(1-c)y^{1-c}}\biggr) + O\biggl(\frac{x}{y^{1-\frac{c}{2}-\varepsilon}}+x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}\biggr).\]
Next,
\[ S_2 = \sum_{k\le \frac{x}{y}} \Bigl(\frac{x}{k}\Bigr)^c + O\biggl( \sum_{k\le \frac{x}{y}} \Bigl(\frac{x}{k}\Bigr)^{\frac{c}{2}+\varepsilon} \biggr) = x^c\biggl( \frac{(x/y)^{1-c}}{1-c} + \zeta_{\mathcal{R}}(c) + O\Bigl(\Bigl(\frac{x}{y}\Bigr)^{\frac{1}{2}-c+\varepsilon}\Bigr)\biggr) + O\Bigl(x^{\frac{c}{2}+\varepsilon}\Bigl(\frac{x}{y}\Bigr)^{1-\frac{c}{2}}\Bigr).\]
Thus
\[ S_2 = \frac{x}{(1-c)y^{1-c}} + \zeta_{\mathcal{R}}(c)x^c +O\biggl(x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}+\frac{x^{1+\varepsilon}}{y^{1-\frac{c}{2}}}\biggr).\]
Finally,
\[S_3 = (y^c + O(y^{\frac{c}{2}+\varepsilon}))\Bigl(\frac{x}{y} + O\Bigl(\Bigl(\frac{x}{y}\Bigr)^{\frac{1}{2}+\varepsilon}\Bigr)\Bigr)=\frac{x}{y^{1-c}} + O\biggl(\frac{x}{y^{1-\frac{c}{2}-\varepsilon}}+x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}\biggr).\]
Combining these formulas, the different terms in $xy^{c-1}$ disappear and it follows that
\[ M_c(x) = x\zeta_{\mathcal{R}}\Bigl(\frac{1}{c}\Bigr)+ \zeta_{\mathcal{R}}(c)x^c + O\biggl(\frac{x^{1+\varepsilon}}{y^{1-c/2-\varepsilon}}+x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}\biggr).\]
Choosing $y=x^{\frac{1}{1+c}}$ gives the result.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 7.3}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime systems and $c\in (\frac{1}{2},1)$. Then for all fixed $\varepsilon>0$,}
\begin{align*}
(i)\quad \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \mu_{\mathcal{Q}_c}(m) & \ll x^{\frac{3c}{2(1+c)}+\varepsilon}\\
(ii)\quad \sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k) & = \frac{x^c}{\zeta(c)} + O(x^{\frac{3c}{2(1+c)}+\varepsilon}).
\end{align*}
{\em Proof.}\, By Lemma 7.1, the abscissae of convergence of the series
\[ \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^s} \quad \mbox{and}\quad \sum_{k,l\in\mathcal{N}}\frac{\mu_{\mathcal{R}}(k)}{(kl^{1/c})^s}\]
are both at most 1. By the first effective Perron formula, it follows that for $n\ge 3$ and $T\ge 1$,
\begin{align*}
\sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \mu_{\mathcal{Q}_c}(m) &= I_1+O(E) \\
\sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k) &= I_{-1}+O(E)
\end{align*}
where
\begin{align*}
I_1 & = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT}\biggl( \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^s}\biggr) \frac{x^s}{s}\, ds = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT} \frac{x^s}{\zeta(s)\zeta(s/c)}\, \frac{ds}{s}\\
I_{-1} & = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT}\biggl( \sum_{k,l\in\mathcal{N}}\frac{\mu_{\mathcal{R}}(k)}{(kl^{1/c})^s} \biggr) \frac{x^s}{s}\, ds = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT} \frac{\zeta(s/c)x^s}{\zeta(s)}\, \frac{ds}{s},\mbox{ and}\\
E &= x\sum_{m\in\mathcal{M}_c}\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}.
\end{align*}
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Upper bound for $E$}\newline
We divide $E$ into $E=E_1+E_2+E_3$ where $E_1, E_2, E_3$ are as below.
\[ E_1:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ |m-x|\le \frac{x}{T}\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll M_c\Bigl(x+\frac{x}{T}\Bigr) - M_c\Bigl(x-\frac{x}{T}\Bigr)\ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x}{T}\]
by Lemma 7.2. Next $E_2 = E_{2,1}+E_{2,2}$ with
\[ E_{2,1}:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ x(1+\frac{1}{T})\le m\le 2x\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll \frac{x}{T}\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ x(1+\frac{1}{T})\le m\le 2x\end{array} } \frac{1}{m-x}.\]
The sum on the right is, using Lemma 7.2,
\begin{align*}
\int_{x(1+\frac{1}{T})}^{2x} \frac{dM_c(t)}{t-x} & = \frac{M_c(2x)}{x} + \int_{x(1+\frac{1}{T})}^{2x} \frac{M_c(t) - M_c(x(1+1/T))}{(t-x)^2}\, dt\\
& \ll 1+\frac{T}{x}x^{\frac{3c}{2(1+c)}+\varepsilon} + \int_{x(1+\frac{1}{T})}^{2x} \frac{dt}{t-x} =1+\frac{T}{x}x^{\frac{3c}{2(1+c)}+\varepsilon} + \log T.
\end{align*}
Thus
\[ E_{2,1} \ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log T}{T}.\]
We prove in the same way that
\[ E_{2,2}:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ \frac{x}{2}\le m \le x(1-\frac{1}{T})\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log T}{T}.\]
Furthermore,
\[ E_3:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ m<\frac{x}{2}\mbox{ or }m >2x\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll \frac{x\log x}{T}.\]
Putting together these bounds we obtain
\[ E \ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log (T+x)}{T}.\]
\noindent
{\bf Upper bound for $I_1$}\newline
Using the fact that $(\mathcal{R},\mathcal{N})$ is good, the residue theorem and Lemma 5.1, we get
\begin{align*}
I_1 & = \frac{1}{2\pi i}\int_{\frac{1}{2}+\varepsilon -iT}^{\frac{1}{2}+\varepsilon +iT} \frac{x^s}{\zeta(s)\zeta(s/c)}\, \frac{ds}{s} + \frac{1}{2\pi i}\sum_{\gamma\in \{-1,1\} } \gamma \int_{\frac{1}{2}+\varepsilon +i\gamma T}^{1+\frac{1}{\log x} + i\gamma T} \frac{x^s}{\zeta(s)\zeta(s/c)}\, \frac{ds}{s} \\
& \ll x^{\frac{1}{2}+\varepsilon}T^\varepsilon + \frac{x}{T^{1-\varepsilon}}.
\end{align*}
{\bf Approximate formula for $I_{-1}$}\newline
This is almost the same calculation but now we pick up a residue at $c$ because of the pole of $\zeta(s/c)$. We have
\begin{align*}
I_{-1} & = \frac{x^c}{\zeta(c)} + \frac{1}{2\pi i}\int_{\frac{1}{2}+\varepsilon -iT}^{\frac{1}{2}+\varepsilon +iT} \frac{\zeta(s/c)x^s}{\zeta(s)}\, \frac{ds}{s} + \frac{1}{2\pi i}\sum_{\gamma\in \{-1,1\} } \gamma \int_{\frac{1}{2}+\varepsilon +i\gamma T}^{1+\frac{1}{\log x} + i\gamma T} \frac{\zeta(s/c)x^s}{\zeta(s)}\, \frac{ds}{s} \\
& = \frac{x^c}{\zeta(c)} + O(x^{\frac{1}{2}+\varepsilon}T^\varepsilon + \frac{x}{T^{1-\varepsilon}}).
\end{align*}
Finally, we get
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \mu_{\mathcal{Q}_c}(m) \ll x^{\frac{3c}{2(1+c)}+\varepsilon}T^\varepsilon + \frac{x}{T}(T^\varepsilon + \log x)\]
and
\[ \sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k) = \frac{x^c}{\zeta(c)} + O\Bigl(x^{\frac{3c}{2(1+c)}+\varepsilon}T^\varepsilon + \frac{x}{T}(T^\varepsilon + \log x)\Bigr).\]
Choosing $T=x$ concludes the proof.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Remark on the use of Perron's effective formula for Beurling primes}\newline
If we had $M_c(x)=[x]$, the counting function of the usual integers, then we would have had
\[ M_c\Bigl(x+\frac{x}{T}\Bigr) - M_c\Bigl(x-\frac{x}{T}\Bigr)\ll \frac{x}{T} + 1,\]
instead of the upper bound (7.1). But in the case of a general Beurling prime system, we do not always have
\[ N(x+h)-N(x)\ll h+1.\]
It is the reason why, in order to obtain an estimate for for generalized integers in short intervals, we needed first to compute the asymptotic development of the counting function $M_c(x)$ of Lemma 7.2.\newline
\pagebreak
\noindent
{\bf {\large 8. Proofs of Theorems 1 and 2}}\newline
{\bf Comments}\, The two proofs have similar structure. The theorems will follow easily from the two fundamental formulas (see (8.15) and (8.7)),
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_{a,b}}(n)}{n^s} = \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)}H(s),\quad (\sigma>b)\tag{8.1}\]
for Theorem 1, and
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_{a,b}}(n)}{n^s} = \frac{1}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)}H(s),\quad \Bigl(\sigma>\max\Bigl\{\frac{3ab}{2(a+b)},a-\frac{19}{40}\Bigr\}\Bigr)\]
for Theorem 2, where in both cases $(\mathcal{R}, \mathcal{N})$ is a good prime system and $H(s)$ an absolutely convergent product.
The proof of Theorem 1 is longer, mainly because of the presence of the pole at $s=b$ in (8.1) that deserves a special treatment. It is why we begin with the proof of Theorem 2.
\newline
\noindent
{\em Proof of Theorem 2.}\, Let $a$ and $b$ be two real numbers satisfying (1.14) and let $(\mathcal{R},\mathcal{N})$ be a good g-prime system. Thanks to Zhang, we know such a system exists. Let $Q:=Q_{a,b} = \mathcal{R}^{1/a}\cup \mathcal{R}^{1/b}$, choose $\varepsilon$ such that $0<\varepsilon<\frac{a}{4}$, and define
\[ h = \max\Bigl\{ 1-\frac{a}{2}+2\varepsilon, \frac{21}{40}\Bigr\}.\tag{8.2}\]
By using Lemma 6.2, (1.14), (8.2) and Lemma 6.1, we have, for $x$ large enough
\begin{align*}
Q(x+x^h)-Q(x) & = R^{1/a}(x+x^h)-R^{1/a}(x) + R^{1/b}(x+x^h)-R^{1/b}(x)\\
& = \frac{x^{a+h-1}}{\log x} +\frac{x^{b+h-1}}{\log x} + O(x^{\max\{ a/2 + \varepsilon, a+2h-2\} })\\
& \sim \frac{x^{a+h-1}}{\log x} \le \frac{x^h}{12\log x} \le \pi(x+x^h)-\pi (x).
\end{align*}
Moreover, as $\mathcal{R}$ is good, we also have $Q(x)\ll x^a$ and
\[ \lim_{x\to\infty} (\pi(x)-Q(x))=\infty.\]
Put $\sigma(a) = \max\{\frac{a}{2}, a-\frac{19}{40}\}$. By Lemma 6.3, there exists a set $\mathcal{P}:=\mathcal{P}_{a,b}$ of ordinary primes, and a bijection $p:\mathcal{Q}\to\mathcal{P}$ such that
\[ H^*(s):= \frac{\zeta_{\mathcal{Q}}(s)}{\zeta_{\mathcal{P}}(s)} = \frac{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{P}}(s)}\]
is an absolutely convergent product for $\sigma>\sigma(a)+2\varepsilon$.
As $\varepsilon>0$ can be chosen as small as we please, it follows that
\[ \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)} = \frac{H(s)}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)},\quad (\sigma>\sigma(a))\tag{8.3}\]
where $H(s)$ is again an absolutely convergent product for $\sigma>\sigma(a)$.
We denote by $\mathcal{M}_{a,b}$ the multiset of integers associated to the multiset of primes $\mathcal{Q}=\mathcal{Q}_{a,b}$. With the notation of section 7, we have for any $c\in (\frac{1}{2},1)$,
\[ \mathcal{Q}_{1,c}=\mathcal{Q}_c \quad\mbox{ and }\quad \mathcal{M}_{1,c} = \mathcal{M}_c.\]
Let $c=b/a$. For $\sigma>a$, we have the following formula where the generalized Euler products and the generalized Dirichlet series are normally convergent for $\sigma\ge a+\varepsilon$ for any fixed $\varepsilon>0$.
\[ \frac{1}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)} = \frac{1}{\zeta_{\mathcal{Q}_{a,b}}(s)} = \sum_{\tilde{m}\in\mathcal{M}_{a,b}} \frac{\mu_{\mathcal{Q}_{a,b}}(\tilde{m})}{\tilde{m}^s} = \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^{s/a}}.\tag{8.4}\]
Let $A(x) = \sum_{m\le x} \mu_{\mathcal{Q}_c}(m)$ (where $m\in\mathcal{M}_c$). For $\sigma> \frac{3c}{2(1+c)}$, we have by Abel summation
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \frac{\mu_{\mathcal{Q}_c}(m)}{m^\sigma} = \frac{A(x)}{x^\sigma} + \sigma\int_1^x \frac{A(t)}{t^{\sigma+1}}\, dt = \sigma\int_1^\infty \frac{A(t)}{t^{\sigma+1}}\, dt +o(1)\]
by Lemma 7.3(i). It follows that
\[ \sigma_c\Bigl(\sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^{s/a}}\Bigr)\le \frac{3ab}{2(a+b)}.\tag{8.5}\]
We have, normally for $\sigma\ge a+\varepsilon$ for all fixed $\varepsilon>0$
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}.\tag{8.6}\]
Combining (8.6), (8.3), (8.4) and (8.5) gives
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} =\frac{H(s)}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)} , \qquad (\sigma>\max\{ \sigma(a), \frac{3ab}{2(a+b)}\})\tag{8.7}\]
and this proves that the abscissa of absolute convergence of the above series is $a$. By (1.14), we have $\max\{ \sigma(a), \frac{3ab}{2(a+b)}\} = \max\{ \frac{3ab}{2(a+b)}, a-\frac{19}{40}\}<b<a$ and the result follows.
\newline\phantom{a}
$\Box$\newline
\noindent
{\em Proof of Theorem 1.}\, Let $a$ and $b$ be two real numbers satisfying (1.12) and let $(\mathcal{R},\mathcal{N})$ be a good g-prime system. Thanks to Zhang, we know such a system exists. Choose $\varepsilon$ such that $0<\varepsilon<\frac{a}{4}$, and define
\[ h = 1-\frac{a}{2}+2\varepsilon.\tag{8.8}\]
By using Lemma 6.2, (1.12) and (8.8), we have, for $x$ large enough
\begin{align*}
R^{1/b}(x+x^h)-R^{1/b}(x) & = \frac{x^{b+h-1}}{\log x} + O(x^{\max\{ b/2 + \varepsilon, b+2h-2\} })\\
& \le \frac{x^{a+h-1}}{\log x} - O(x^{\max\{ a/2 + \varepsilon, a+2h-2\} }) \le R^{1/a}(x+x^h)-R^{1/a}(x).
\end{align*}
Moreover, as $(\mathcal{R},\mathcal{N})$ is good, we also have $R^{1/b}(x)\ll x^b$ and
\[ \lim_{x\to\infty} (R^{1/a}(x)-R^{1/b}(x))=\infty.\]
By applying Lemma 6.3, there exists an injection $I:\mathcal{R}^{1/b}\to \mathcal{R}^{1/a}$ such that
\[ \frac{\zeta_{\mathcal{R}^{1/b}}(s)}{\zeta_{Im(I)}(s)} = \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{Im(I)}(s)}\tag{8.9}\]
is an absolutely convergent product for $\sigma>\max\{b-\frac{a}{2}+2\varepsilon, 0\}$.
Let $\mathcal{Q} = \mathcal{R}^{1/a} \setminus {\rm Im}(I)$ and define a new $h$ by
\[ h = \max\Bigl\{ 1-\frac{a}{2}+2\varepsilon, \frac{21}{40}\Bigr\}.\tag{8.10}\]
By using Lemma 6.2, (1.12), (8.10) and finally Lemma 6.1, we have, for $x$ large enough
\begin{align*}
Q(x+x^h)-Q(x) & \le R^{1/a}(x+x^h)-R^{1/a}(x) = \frac{x^{a+h-1}}{\log x} + O(x^{\max\{ a/2 + \varepsilon, a+2h-2\} })\\
& \le \frac{x^h}{12\log x} \le \pi(x+x^h)-\pi (x).
\end{align*}
Moreover, as $\mathcal{R}$ is good, we also have $Q(x)\ll x^a$ and
\[ \lim_{x\to\infty} (\pi(x)-Q(x))=\infty.\]
Recall the notation $\sigma(a) = \max\{ \frac{a}{2}, a-\frac{19}{40}\}$. By Lemma 6.3, there exists a set $\mathcal{P}=\mathcal{P}_{a,b}$ of ordinary primes and a bijection
\[ p:\mathcal{Q}\to \mathcal{P}\]
such that the function
\[ \frac{\zeta_{\mathcal{Q}}(s)}{\zeta_{\mathcal{P}}(s)} = \frac{\zeta_{\mathcal{R}}(s/a)}{\zeta_{{\rm Im}(I)}(s)\zeta_{\mathcal{P}}(s)} \]
is an absolutely convergent product for $\sigma>\sigma(a)+2\varepsilon$. We have $\sigma(a)\ge a/2 > b-a/2$ by (1.12). By (8.9), it follows that
\[ \frac{1}{\zeta_{\mathcal{P}}(s)} = \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)}H^*(s), \quad (\sigma>\sigma(a)+2\varepsilon)\]
where $H^*(s)$ is an absolutely convergent product for $\sigma>\sigma(a)+2\varepsilon$. As $\varepsilon>0$ can be chosen as small as we please, it follows that
\[ \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)} = \frac{H(s)\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)},\quad (\sigma>\sigma(a))\tag{8.11}\]
where
\[ H(s) = \frac{H^*(s)\zeta_{\mathcal{R}}(2s/a)}{H^*(2s)\zeta_{\mathcal{R}}(2s/b)} \]
is again an absolutely convergent product for $\sigma>\sigma(a)$.
Let $c = b/a$. We have the following formula where both sides converge normally for $\sigma\ge a+\varepsilon$ for any fixed $\varepsilon>0$.
\[ \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)} = \sum_{k,l\in\mathcal{N}} \frac{\mu_{\mathcal{R}}(k)}{(kl^{1/c})^{s/a}}.\tag{8.12}\]
Let
\[ A(x) = \sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k).\]
By Abel summation, for $\sigma>c$,
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m=kl^{1/c}\in \mathcal{M}_c\end{array} } \frac{\mu_{\mathcal{R}}(k)}{m^\sigma} = \frac{A(x)}{x^\sigma} + \sigma\int_1^x \frac{A(t)}{t^{\sigma+1}}\, dt = \sigma\int_1^\infty \frac{A(t)}{t^{\sigma+1}}\, dt +o(1)\]
by Lemma 7.3(ii). It follows that the abscissa of convergence of the series in (8.12) is at most $b$. As $b$ is a pole, the abscissa is indeed $b$. We have, normally for $\sigma\ge a+\varepsilon$ for all fixed $\varepsilon>0$
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}.\tag{8.14}\]
Combining (8.14), (8.11), (8.12) and (8.13) we get
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} =\frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)}H(s) , \qquad (\sigma>\max\{ \sigma(a), b\})\tag{8.15}\]
and this proves that the abscissa of absolute convergence of the above series is $a$.
But by (1.12), we have $b>\sigma(a)$. Thus (8.15) is actually true for $\sigma>b$. It follows that $s=a$ is the only zero in $\mathbb{C}_b$ and this zero is simple. As $b$ is a pole it follows that the abscissa of convergence is $b$.
Moreover, for $t\in\mathbb{R}$, we have
\[ \lim_{\sigma\to b+} \frac{\zeta_{\mathcal{R}}(\frac{\sigma+it}{b})}{\zeta_{\mathcal{R}}(\frac{\sigma+it}{a})}H(\sigma+it) =
\frac{\zeta_{\mathcal{R}}(1+\frac{it}{b})}{\zeta_{\mathcal{R}}(\frac{b+it}{a})}H(b+it)\ne 0.\]
By Abel's Theorem, it follows that if the series in (8.15) converges at $b+it$, the sum cannot be 0. Thus we have $Z_{\mathcal{P}} = \{a\}$, which concludes the proof of Theorem 1.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 9. Open questions related to GRH-RH}}\newline
Let us recall that one of the classical statements equivalent to the Generalized Riemann Hypothesis (GRH) is the following: {\em for every Dirichlet character $\chi$, the meromorphic function $L_\chi(s)$ does not vanish in $\mathbb{C}_{\frac{1}{2}}$. }
The Dirichlet series defining $\zeta(s)$ and more generally $L_\chi(s)$ with $\chi$ a principal Dirichlet character are not convergent in the critical strip. As only the zeros of Dirichlet series themselves (and not of their meromorphic continuation) are studied here, it leads us to introduce GRH$\setminus$RH: {\em for every non-principal Dirichlet character $\chi$, the Dirichlet series $L_\chi(s)$ does not vanish in $\mathbb{C}_{\frac{1}{2}}$.}\newline
Let us recall the sets $V$ and $W$ of section 1(f). Theorem 3 says $V\supset [0,\frac{1}{2}]$ and, under RH, $W\supset [0,\frac{1}{2}]$. We wonder if these inclusions are equalities. Write
\[ V=\Bigl[0,\frac{1}{2}\Bigr] \tag{$B_c$}\]
and
\[ W=\Bigl[0,\frac{1}{2}\Bigr] \tag{$B_a$}\]
We shall see that either of these implies GRH$\setminus$RH.\newline
Let us also recall the two statements mentioned in \cite{KS1} which imply GRH$\setminus$RH.
\[ \mbox{\em For every completely multiplicative $f$, we have } \sum_{n\le x} f(n) =\Omega\Bigl(\frac{1}{\sqrt{x}}\Bigr).\tag{$A$}\]
Now, let $\sigma_n(f)$ and $\sigma_p(f)$ denote the abscissa of convergence of $\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and
$\sum_p \frac{f(p)}{p^s}$ respectively. Write
\[ \mbox{\em For every completely multiplicative $f$, we have $\sigma_p(f)\le \sigma_n(f)+\frac{1}{2}$}.\tag{$C$}\]
These five statements are related in the following way:
\[ \left. \begin{array}{cl} A \implies & B_c \implies\\ & B_a \implies \\ & C \implies \end{array} \right\} \mbox{ GRH$\setminus$RH}\]
{\em Proof.}\, $A\implies B_c$\newline
Let us suppose $B_c$ is false. Then there exists a completely multiplicative function $f(n)$ and a zero $\beta+i\gamma$ of $\sum_{n=1}^\infty \frac{f(n)}{n^s}$ (with abscissa of convergence $\sigma_c$) such that $\beta-\sigma_c>\frac{1}{2}$.
By writing $f_1(n) = f(n)n^{-\sigma_c-i\gamma}$ we have that
\[ \sum_{n=1}^\infty \frac{f_1(n)}{n^s}\]
has abscissa of convergence zero, and the series vanishes at $s=\beta>\frac{1}{2}$. Defining $A(x) = \sum_{n\le x}f_1(n)$, we have $A(x)\ll x^\varepsilon$ for all $\varepsilon>0$ and by Abel summation that
\[ \sum_{n>x} \frac{f_1(n)}{n^\beta}\ll \frac{1}{x^{\beta-\varepsilon}}.\]
As $\sum_{n=1}^\infty \frac{f_1(n)}{n^\beta}=0$, we also have
\[ \sum_{n\le x} \frac{f_1(n)}{n^\beta}\ll \frac{1}{x^{\beta-\varepsilon}}.\]
Thus $g(n):=f_1(n)n^{-\beta}$ is completely multiplicative but does not satisfy $A$.
\newline
\noindent
$B_c\implies$ GRH$\setminus$RH\newline
Suppose GRH$\setminus$RH is false. Then there exists a non-principal character $\chi$ and a zero $\beta+i\gamma$ of $L_\chi(s)$ with $\beta>\frac{1}{2}$. As $L_\chi(s)$ has $\sigma_c=0$, it follows that $\beta\in V$ and $B_c$ is false. \newline
\noindent
$B_a\implies$ GRH$\setminus$RH\newline
The proof is similar to the above. Suppose GRH$\setminus$RH is false. Then there exists a non-principal character $\chi$ and a zero $\beta+i\gamma$ of $L_\chi(s)$ with $\beta<\frac{1}{2}$. As $L_\chi(s)$ has $\sigma_a=1$, it follows that $1-\beta\in W$ and, since $1-\beta>\frac{1}{2}$, $B_a$ is false. \newline
\noindent
$C\implies$ GRH$\setminus$RH\newline
Suppose $C$ is true. Let $\chi$ be a non-principal character. Note that $\sigma_n(\chi)=0$. By $C$, we have
\[ \sum_{p\le x} \chi(p) \ll x^{\frac{1}{2}+\varepsilon}\quad\mbox{ for all $\varepsilon>0$}.\]
But this is an equivalent form of RH for $L_\chi(s)$; i.e. GRH$\setminus$RH follows.
\newline\phantom{a}
$\Box$\newline
\pagebreak
\noindent
{\bf Question}\, Among the four statements $A,B_c,B_a$ and $C$, which are true and which are false?
\noindent
{\bf Acknowledgements}\newline
We had stimulating discussions on the mathematics in and around this paper with Kristian Seip. We thank him for them.\newline
{\small
}
\end{document}
|
\begin{document}
\title{BDD-Based Algorithm for SCC Decomposition\texorpdfstring{\\}{ }of Edge-Coloured Graphs}
\author[N.~Bene\v{s}]{Nikola Bene\v{s}}
\address{Masaryk University, Brno, Czech Republic}
\email{\{\texttt{xbenes3},\texttt{brim},\texttt{xpastva},\texttt{safranek}\}\texttt{@fi.muni.cz}}
\author[L.~Brim]{Lubo\v{s} Brim}
\author[S.~Pastva]{Samuel Pastva}
\author[D.~Šafránek]{David \v{S}afr{\'a}nek}
\begin{abstract}
Edge-coloured directed graphs provide an essential structure for modelling and analysis of complex systems arising in many scientific disciplines (e.g. feature-oriented systems, gene regulatory networks, etc.). One of the fundamental problems for edge-coloured graphs is the detection of strongly connected components, or SCCs.
The size of edge-coloured graphs appearing in practice can be enormous both in the number of vertices and colours. The large number of vertices prevents us from analysing such graphs using explicit SCC detection algorithms, such as Tarjan's, which motivates the use of a symbolic approach. However, the large number of colours also renders existing symbolic SCC detection algorithms impractical.
This paper proposes a novel algorithm that symbolically computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $O(p\cdot n\cdot\log n)$ symbolic steps, where $p$ is the number of colours and $n$ is the number of vertices.
We evaluate the algorithm using an experimental implementation based on binary decision diagrams (BDDs). Specifically, we use our implementation to explore the SCCs of a large collection of coloured graphs (up to $2^{48}$) obtained from Boolean networks -- a~modelling framework commonly appearing in systems biology.
\end{abstract}
\maketitle
\section*{Introduction}\label{S:one}
In many scientific disciplines, the processing of massive data
sets represents one of the most important computational tasks.
A variety of these data sets can be modelled in terms of very large multi-graphs, augmented by a specific collection of application-dependent edge attributes. These attributes are often abstractly referred to as colours, and the resulting formalism is called an \emph{edge-coloured graph}~\cite{BANGJENSEN1997,bookgraphs79}. Geographic information systems, telecommunications traffic, or internet networks are prime examples of data that are best represented as such edge-coloured graphs.
For instance, in social networks, coloured edges can be used to link together groups of nodes related by some specific criteria (Sports, Health, Technology, Religion, etc.). In software engineering, one often speaks about feature-oriented systems~\cite{classen2010model}. In this case, colours represent possible combinations of features, altering the system's behaviour.
Our interest in processing huge edge-coloured graphs is primarily motivated by applications taken from systems biology~\cite{tcbb,GiacobbeGGHPP17} and genetics~\cite{DORNINGER94} where we have to deal not only with giant graphs as measured by the number of vertices and edges but also with large sets of colours. In this case, the graph colours represent valuations of numerous parameters that influence the dynamics of a~biological system~\cite{tcbb,BattPCGMJ10,Bernot05}.
Fundamental graph algorithms such as breadth-first search, spanning tree construction, shortest paths, decomposition into strongly connected components (SCCs), etc., are building blocks of many practical applications. For the edge-coloured graphs, the primary research focus so far has been on some of the ``classical'' coloured graph problems, like the determination of the chromatic index, finding sub-graphs with a specified colour property (the coloured version of the k-linked problem), alternating edge-coloured cycles and paths, rainbow cliques, monochromatic cliques and cycles, etc.~\cite{Das,Akbari,Alon,BANGJENSEN1997,Thomason,Kano}.
To the best of our knowledge, we are not aware of any work on SCC decomposition specifically for edge-coloured graphs, even though this problem has many important applications. For example, in biological systems, strongly connected components represent the so called attractors of the system. In this case, a specific focus is given to terminal (or bottom) SCCs, but non-terminal (transient) SCCs can also be detrimental to the system's long-term behaviour~\cite{long-lived-transients}. Overall, SCCs play an essential role in determining the system's biological properties, since they may correspond, for example, to the specific phenotypes expressed by a~cell~\cite{choo2016efficient}.
The valuation of parameters (e.g. the presence of certain genes or external stimuli) in such systems is then represented as edge colours in the state-transition graph. The knowledge of SCCs and how their structure depends on parameters is vital for understanding various biological phenomena~\cite{deritei2016principles,Li16}.
Other applications where investigation of attractors is crucial include predictions of the global climate change~\cite{Steffen18} or predictions of spreading of infectious diseases such as COVID-19~\cite{Matouk20}.
There is a serious computational problem related to the processing of massive edge-coloured graphs (or even the non-coloured ones) that significantly affects the tractability of SCC decomposition. The graphs often cannot be handled using standard (explicit) representations, since they are too large to be kept in the main memory. Various approaches have been considered to deal with such giant graphs: distributed-memory computation, symbolic data structures for graph representation, or storing the graphs in external memory. We review these approaches in more detail in the related work section.
In~\cite{cmsb2017,ICFEM19} we have initially attacked the SCC decomposition problem for massive edge-coloured graphs by developing a parallel, semi-symbolic algorithm for detection of bottom SCCs. The algorithm uses symbolic structures to represent sets of parameters, while the graph itself is represented explicitly. However, the results have shown that the parallel semi-symbolic algorithm is often not sufficient to tackle graphs representing real-world problems practically. These findings have motivated us to propose a new, entirely symbolic approach.
In this paper, we consider \emph{edge-coloured multi-digraphs}, i.e., multi-digraphs such that each directed edge has a colour and no two parallel (i.e., joining the same pair of vertices) edges have the same colour. Here, we refer to such graphs simply as \emph{coloured graphs}. For coloured graphs, we can define several notions of strongly connected components involving colours. We consider the simplest case, where the SCCs are \emph{monochromatic}, that is all their edges have the same colour~\cite{Kiraly14}. This choice is motivated by the application in systems biology, as mentioned above.
\paragraph{Contribution}
We propose a novel fully symbolic algorithm for detecting \emph{all}
monochromatic strongly connected components in a coloured graph. This algorithm is in practice significantly faster than what is achievable by na\"ively executing a symbolic SCC decomposition
algorithm for each colour separately. This is because in many applications,
the edges are largely shared among individual
colours~\cite{tcbb} and our algorithm is capable of exploiting this fact.
The algorithm conceptually follows the \emph{lock-step} reachability
approach by Bloem et al.~\cite{bloem2000} for purely monochromatic digraphs. The key
new ingredients behind our algorithm are a~careful orchestration of
the forward and backward reachability for different colours, and
a~colour-aware selection of the pivot set.
\subsection*{Structure of the paper} In Section~1, we recall the notions of strongly connected components and edge-coloured digraphs, and we state the coloured SCC decomposition problem. In Section~2, we first briefly introduce the forward-backward decomposition algorithm and the lock-step algorithm for monochromatic graphs. After that, we present the coloured SCC decomposition algorithm together with the proof of correctness and complexity analysis.
In Section~\ref{section:boolean-networks}, we introduce Boolean networks, discuss their symbolic encoding, and show how they can be translated into coloured graphs suitable for SCC-decomposition. Subsequently, Section~\ref{section:implementation} discusses several practical improvements to the main algorithm (saturation, trimming, and parallelism) which help it scale to larger models, and thus be more practically viable. Finally, Section~4 evaluates the main algorithm (including the improved variants) using a collection of large, real-world Boolean networks. A conclusion is provided in the last section.
This article is an extended version of an article that appeared in the Proceedings of TACAS 2021~\cite{tacas21}. We extend the information provided in the TACAS Proceedings with more in-depth technical details of the algorithm and related proof, including explanation of its key steps. Moreover, we extend the implementation and evaluation sections to give the reader more information on how the performance of the algorithm can be improved, and how the algorithm performs using a variety of real-world case studies.
\subsection*{Related Work}
The detection of SCCs in (monochromatic) digraphs is a well-known problem
computable in linear time.
Best serial (explicit) algorithms are
Kosaraju-Sharir~\cite{SHARIR1981} and Tarjan~\cite{Tarjan}, which are both inherently based on depth-first search. However, these algorithms do not scale for large graphs, e.g., those encountered in model-checking, when using explicit graph representation.
Therefore, alternative approaches to such SCC decomposition have been proposed (e.g. I/O efficient, parallel, or symbolic algorithms).
The algorithm of Jiang~\cite{JIANG1993} gives an I/O-efficient alternative
based on a~combination of depth-first and breadth-first search.
Efficient parallel, distributed-memory algorithms avoid the inherently sequential DFS step~\cite{REIF}
in several different ways. The Forward-Backward algorithm~\cite{FWBW} employs
a~divide-and-conquer approach relying on picking a pivot state and
splitting the graph in three independent (SCC-closed) parts. The
approach of Orzan~\cite{Orzan} uses a different distribution scheme
called a colouring transformation, employing a set of prioritised colours to
split the graph into many parts at once. The OWCTY-Backward-Forward (OBF) approach is
proposed in~\cite{Barnat09}. It recursively splits the graph in a number of independent sub-graphs called OBF slices and applies to each slice the One-Way-Catch-Them-Young (OWCTY) technique. In~\cite{Slota14}, the authors utilise
variants of the Forward-Backward and Orzan's algorithms for optimal
execution on shared-memory multi-core platforms. Finally, Bloemen et
al.~\cite{Bloemen16} present an on-the-fly parallel algorithm utilising a swarm of DFS searches, showing promising speed-up for large graphs containing large SCCs. On another end, GPU-accelerated approaches to computing
SCCs have been addressed for example in~\cite{BarnatBBC11,HRO13,LI2014,Dragan14}.
Computing SCCs of (monochromatic) digraphs symbolically is another way to handle giant graphs and has been thoroughly explored
in literature. As in the case of efficient parallelisation, depth-first
search is not feasible in the symbolic framework~\cite{GentiliniPP08}. In~consequence, many DFS-based algorithms cannot be
easily revised to work with symbolically represented graphs. An algorithm based on forward and backward reachability performing $\mathcal{O}(n^2)$ symbolic steps was presented by Xie and Beerel in~\cite{xie2000}. Bloem et al.~present an improved $\mathcal{O}(n \cdot \log n)$ algorithm in~\cite{bloem2000}. Finally, an $\mathcal{O}(n)$ algorithm was presented by Gentilini et al.~in~\cite{gentilini2003,GentiliniPP08}. This bound has been proven to be tight in~\cite{chatterjee2018}. In~\cite{chatterjee2018}, the authors argue that the algorithm from~\cite{gentilini2003} is optimal even when considering more fine-grained complexity criteria, like the diameter of the graph and the diameters of the individual components. Ciardo et al.~\cite{Ciardo11} use the idea of saturation~\cite{Ciardo06} to speed up state exploration within the Xie-Beerel algorithm, and show a saturation-based technique for computing the transitive closure of the graph's edge relation.
Besides these generic algorithms, there have also been symbolic SCC decomposition methods to deal with large graphs generated specifically by Boolean networks~\cite{MizeraIEEE,YUAN2019}. However, these primarily target detection of bottom SCCs. Methods in this area are also often incomplete, for example focusing on detection of single-state or small bottom SCCs~\cite{zhang-small-attractors}. As such, they generally perform better than an exhaustive symbolic SCC detection in their respective application domains, but are inherently limited in scope.
\section{Problem Definition}\label{S:two}
As we have already stated in the introductory section, the SCC decomposition problem for edge-coloured graphs has remained mostly unexplored until now. We thus start this paper by introducing and formalising the notion of \emph{coloured SCC decomposition} itself and state some of its basic properties.
Before giving exact definitions, it might be instructive
to discuss the substance of the coloured SCC decomposition intuitively. There are
several ways of capturing the notion of a~``coloured connected component''.
One of them is that of a colour-connectivity first introduced by Saad~\cite{Saad92}. It is based on
alternating paths in which successive edges differ in colour. However, there is no
unique, universally acceptable notion of a coloured component.
In the biological applications we have in mind (i.e.~Boolean networks), we want to identify a coloured component as a~coloured collection of SCCs---a~collection where for every colour there is
a set of all relevant monochromatic SCCs. Such a setting leads us to represent SCCs
in the form of a relation. To that end, we first introduce such a relation for
monochromatic graphs (Section~\ref{sec:monographs}) and afterwards extend it
to edge-coloured graphs (Section~\ref{sec:colourgraphs}). The relation-based
approach gives us also the advantage of allowing a feasible symbolic encoding
of the problem.
\subsection{Graphs and Strongly Connected Components}
\label{sec:monographs}
Let us first recall the standard definitions of a~directed graph and its
strongly connected components:
\begin{defi}\label{def:graph}
A \emph{directed graph} is a tuple $G = (V, E)$ where $V$ is a set of graph \emph{vertices} and $E \subseteq V \times V$ is a set of graph \emph{edges}.
\end{defi}
We are going to use the word \emph{graph} to mean \emph{directed graph} in
the following.
We write $u \to v$ when $(u, v) \in E$ and $u \to^* v$ when $(u, v) \in E^*$,
the reflexive and transitive closure of $E$. We say that $v$ is
\emph{reachable} from $u$ if $u \to^* v$.
The reachability relation allows us to decompose a~graph into strongly
connected components, defined as follows:
\begin{defi}
In a~graph $G = (V, E)$, a \emph{strongly connected component} (SCC)
is a~maximal set $W \subseteq V$ such that for all $u, v \in W$, $u
\to^* v$ and $v \to^* u$. For a fixed $v \in V$, we write $SCC(G, v)$
to denote the SCC of $G$ that contains $v$.
\end{defi}
If the graph $G$ is clear from the context, we can simply write $SCC(v)$.
A set of vertices $S \subseteq V$ is said to be \emph{SCC-closed} if every SCC
$W$ is either fully contained inside $S$ ($W \subseteq S$), or in its
complement ($W \subseteq V \setminus S$). Notice that given a vertex $v$, the
set of all vertices reachable from $v$, as well as the set of all vertices
that can reach $v$, are both SCC-closed.
A pivotal problem in computer science is to find the SCC decomposition of~$G$.
As mentioned above, we represent the decomposition in the form of an
\emph{equivalence relation} $\ensuremath{R_\mathit{scc}}$ such that the individual SCCs are exactly
the equivalence classes of $\ensuremath{R_\mathit{scc}}$. The relation-based formulation of the SCC
decomposition problem is the following:
\begin{problem2}[SCC decomposition]
Given a graph $G = (V, E)$, find the~\emph{SCC decomposition relation}
$\ensuremath{R_\mathit{scc}} \subseteq V \times V$ such that $(u,v) \in \ensuremath{R_\mathit{scc}}$ if and only if $SCC(u) = SCC(v)$.
\end{problem2}
Note that $SCC(u)$ can be obtained by fixing the first attribute of $\ensuremath{R_\mathit{scc}}$, i.e.
$SCC(u) = \{ v \mid (u, v) \in \ensuremath{R_\mathit{scc}} \}$.
We refer to such operation as \emph{section} and denote it in the following way: $SCC(u) = \ensuremath{R_\mathit{scc}}(u, \_)$ (the concept is properly formalised later as part of Fig.~\ref{tab:operations}).
Here, $u$ is the specific value of an attribute at which the section is taken,
and $\_$ is used in place of the attributes that remain unchanged. Such
notation naturally extends to arbitrary relations.
\subsection{Coloured SCC Decomposition Problem}\label{sec:colourgraphs}
We now lift the formal framework to the coloured setting. An edge-coloured graph can be
seen as a succinct representation of several different graphs, all sharing the same
set of vertices.
To emphasise the difference from the standard graphs (i.e. Definition~\ref{def:graph}), we sometimes call the standard graphs
\emph{monochromatic}.
\begin{defi}
An \emph{edge-coloured directed multi-graph} (coloured graph for short) is a
tuple $\ensuremath{\mathfrak G} = (V, C, E)$ where $V$ is a set of vertices, $C$ is a set of
colours and $E \subseteq V \times C \times V$ is a~coloured edge relation.
\end{defi}
We also write $u \xrightarrow{c} v$ whenever $(u, c, v) \in E$ and
use $\creach{c}$ to denote the reflexive and transitive closure
of $\xrightarrow{c}$. We say that $v$ is $c$-reachable from $u$ if
$u \creach{c} v$, i.e.~there is a~path from $u$ to $v$ using
only $c$-coloured edges.
By fixing a~colour $c \in C$ and keeping only the $c$-coloured edges (with the
colour attribute removed), we obtain a~monochromatic graph
$\ensuremath{\mathfrak G}(c) = (V, \{(u, v) \mid (u, c, v) \in E\})$. We call this graph the
\emph{monochromatisation of $\ensuremath{\mathfrak G}$ with respect to $c$}.
Intuitively, one can view the elements of $C$ as a type of graph
parametrisation where the edge structure of the graph changes based on the
specific $c \in C$.
The SCC decomposition relation $\ensuremath{R_\mathit{scc}}$ is extended to the coloured SCC
decomposition relation $\ensuremath{\mathfrak R_\mathit{scc}}$ by relating every colour $c\in C$ with all
SCCs of the monochromatisation $\ensuremath{\mathfrak G}(c)$. In consequence, the SCC decomposition
problem is then lifted to the coloured SCC decomposition problem as follows:
\begin{problem2}[Coloured SCC decomposition]
Given a coloured graph $\ensuremath{\mathfrak G} = (V, C, E)$, find the \emph{coloured SCC
decomposition relation} $\ensuremath{\mathfrak R_\mathit{scc}} \subseteq V \times C \times V$ satisfying
$(u,c,v) \in \ensuremath{\mathfrak R_\mathit{scc}}$ if and only if $(u,v) \in \ensuremath{R_\mathit{scc}}$ of $\ensuremath{\mathfrak G}(c)$.
\end{problem2}
From this definition, we can immediately observe the following properties
about the relationship of $\ensuremath{\mathfrak R_\mathit{scc}}$ with the terms which we have defined before:
\begin{itemize}
\item $\ensuremath{R_\mathit{scc}}$ of a monochromatisation $\ensuremath{\mathfrak G}(c)$ is exactly the section
$\ensuremath{\mathfrak R_\mathit{scc}}(\_, c, \_)$;
\item $SCC(\ensuremath{\mathfrak G}(c), v)$ is exactly the section $\ensuremath{\mathfrak R_\mathit{scc}}(v, c, \_)$, or equivalently, $\ensuremath{\mathfrak R_\mathit{scc}}(\_, c, v)$ (since $\ensuremath{\mathfrak R_\mathit{scc}}$ and $\ensuremath{R_\mathit{scc}}$ are symmetric with regards to $V$).
\end{itemize}
From this, it should be immediately apparent that $\ensuremath{\mathfrak R_\mathit{scc}}$ contains all
components of the underlying monochromatisations.
\section{Algorithm}
Conceptually, our algorithm follows the \emph{lock-step} reachability approach by Bloem~\cite{bloem2000} for monochromatic graphs. The lock-step algorithm itself is based on the basic forward-backward decomposition algorithm~\cite{xie2000}.
In this section, we first briefly introduce these two algorithms to explain better the key ideas behind our approach and, in particular, to explain the main difficulties encountered in employing the concepts of these algorithms to edge-coloured graphs.
Although the algorithms were originally presented as producing a~set of SCCs,
we reformulate them slightly using the equivalent relation-based approach as
explained in the previous section.
After that, we present the coloured SCC decomposition algorithm.
However, before we dive into the algorithmics, let us first briefly discuss
the computation model we are using.
\subsection{Symbolic Computation Model}\label{ssec:symb}
As a complexity measure of our algorithm, we consider the number of
symbolic steps, or more specifically, symbolic set and relation operations that the
algorithm performs. As is customary, we assume that sets of vertices ($V$) and colours ($C$) can be represented symbolically (for example, using reduced ordered binary decision diagrams~\cite{bryant86}) as well as any relations over these sets. In particular, we often talk about \emph{coloured vertex sets}, by which we mean the subsets of $V \times C$.
\begin{figure}
\caption{Summary of symbolic operations that appear in the presented algorithms. The derived operations can be implemented using the standard and relational operations. However, typically they also have a slightly more efficient direct implementations.}
\label{tab:operations}
\end{figure}
Aside from normal set operations (union, intersection, difference, product and element selection), we also require some basic relational operations, all of which we outline in Figure~\ref{tab:operations}. These extra operations tend to appear in other applications as well (such as symbolic model checking~\cite{BurchCMDH92}), and are thus typically already available in mature symbolic computation packages.
Finally, there are several derived operators that are partially
specific to our application to coloured graphs. However, these can be
constructed using standard set and relation operations.
The intuitive meaning of the derived operators is as follows:
$\textsc{Colours}$ returns all the colours that appear in the given coloured
vertex set. $\textsc{Pre}$ and $\textsc{Post}$ compute the pre- and post-image
of a (monochromatic or coloured) set of vertices, i.e.~the set of successors
or predecessors of all the vertices in the given set, respectively.
Finally, $\textsc{Join}$ takes a~coloured vertex set $A$ and computes
the set $\{(u, c, v) \mid (u, c) \in A, (v, c) \in A\}$.
\subsection{Forward-Backward Algorithm}
To symbolically compute the SCCs of a~graph $G = (V, E)$, Xie and
Beerel~\cite{xie2000} observed that for any vertex $v \in V$,
the intersection $W = F \cap B$
of the forward reachable vertices $F = \{ v' \in V \mid v \to^* v' \}$ and the
backward reachable vertices $B = \{ v' \in V \mid v' \to^* v \}$ is exactly
the strongly connected component of $G$ which contains $v$.
The algorithm thus picks an arbitrary \emph{pivot} $v \in V$, and divides the
vertices of the graph into four disjoint sets: $W$, $F \setminus W$, $B
\setminus W$ and $V \setminus (F \cup B)$. This is illustrated graphically in Figure~\ref{fig:algorithms}~(left). The set $W$ is then immediately
reported as an SCC of the graph, and added into the component relation: $\ensuremath{R_\mathit{scc}}
\gets \ensuremath{R_\mathit{scc}} \cup (W\times W)$. It is easy to see that every other SCC is
fully contained within one of the three remaining sets (they are SCC-closed), and the algorithm thus
recursively repeats this process independently in each set.
The correctness of the algorithm follows from the initial observation and the
fact that every vertex eventually appears in $W$ (either as a~pivot or as
a~result of $F \cap B$). In the worst case, the algorithm performs $O(|V|^2)$
symbolic steps, since every vertex is picked as a~pivot at most once and the
computation of $F$ and $B$ requires at most $O(|V|)$
\textsc{Pre}/\textsc{Post} operations.
\begin{figure}
\caption{Illustration of the difference between the forward-backward algorithm (left) and the lock-step algorithm (right). On the left, we fully compute both backward ($B$) and forward ($F$) reachable sets from the pivot $v$, identifying $W$ as $F \cap B$. On the right, without loss of generality, assume $F$ is fully computed first. It thus becomes converged ($Con$) and the computation of $B$ ($Non$) is stopped before it is fully explored. }
\label{fig:algorithms}
\end{figure}
\subsection{Lock-step Algorithm}
To improve the efficiency of the forward-backward algorithm, the lock-step
approach~\cite{bloem2000} uses another important observation:
To compute $W$, it is not
necessary to fully compute both $F$ and $B$; only the smaller (in terms of diameter) of the two sets
needs to be entirely known. With this observation, the computation of $F$ and
$B$ can be modified in the following way:
Instead of computing $F$ and $B$ one after the other, the computation is
\emph{interleaved} in a step-by-step manner (dovetailing). When one of the
sets is fully computed, the computation of the second set is stopped.
Let us call the computed set \emph{converged} and denote it by $\ensuremath{\mathit{Con}}$, and the
unfinished set \emph{non-converged} and denote it by $\ensuremath{\mathit{Non}}$. This situation is illustrated in Figure~\ref{fig:algorithms}~(right).
However, even when $\ensuremath{\mathit{Con}}$ is fully known, we still need to finish the
computation of states in $\ensuremath{\mathit{Non}}$ that are inside $\ensuremath{\mathit{Con}}$ to discover the
whole component $W$. This is necessary if there are vertices $w$ in $W$ whose
forward distance from $v$ (i.e.~the length of the path $v \to^* w$) is short
while their backward distance (the length of the path $w \to^* v$) is long,
or vice versa. Such vertices are thus only discovered in one of the two
reachability procedures and still need to be discovered by the other one
to identify the whole component.
However,
an important observation is that only the vertices already inside $\ensuremath{\mathit{Con}}$ need
to be considered in this phase.
After this, the SCC can be identified and reported just as in the
forward-backward algorithm. Finally, the recursion now continues in sets $\ensuremath{\mathit{Con}}
\setminus W$ and $V \setminus \ensuremath{\mathit{Con}}$. This is due to $\ensuremath{\mathit{Non}}$ being not fully
computed; we cannot guarantee that no SCC overlaps outside of $\ensuremath{\mathit{Non}}$ ($\ensuremath{\mathit{Non}}$ is not necessarily SCC-closed).
The algorithm is still correct because every vertex is eventually either
picked as a pivot or discovered in some $W$. Furthermore, due to the way
$\ensuremath{\mathit{Con}}$ and $\ensuremath{\mathit{Non}}$ are computed guarantees that $W$ is still a whole SCC\@.
In terms of complexity, the algorithm performs $O(|V| \cdot \log |V|)$
symbolic steps in the worst case. To see why this is true, we may observe that
every vertex appears in $W$ exactly once, and that the smaller of the two
sets $\ensuremath{\mathit{Con}} \setminus W$ and $V \setminus \ensuremath{\mathit{Con}}$, let us call it $S$,
is always smaller than $\frac{|V|}{2}$. The authors then argue that
the price of every iteration can be attributed (up to a multiplicative
constant) to the vertices in $S \cup W$ and that every vertex appears in $S$
at most $O(\log |V|)$-times.
\subsection{Coloured Lock-step Algorithm}
When developing an algorithm for coloured graphs, one needs to deal with multiple
challenges which do not appear for monochromatic graphs and require
careful consideration. In the following, we refer to the pseudocode in
Algorithm~\ref{algo:symbolic}.
\begin{algorithm}
\SetKwProg{Fn}{Function}{}{}
\Fn{\upshape\textsc{ColouredSCC}$(\ensuremath{\mathfrak G} = (V, C, E))$}{
$\ensuremath{\mathfrak R_\mathit{scc}} \subseteq (V \times C \times V) \gets \emptyset$\;
$\textsc{Decomposition}(\ensuremath{\mathfrak G}, \ensuremath{\mathfrak R_\mathit{scc}}, V \times C)$\;
\Return $\ensuremath{\mathfrak R_\mathit{scc}}$\;
}
\BlankLine
\Fn{\upshape\textsc{Decomposition}$(\ensuremath{\mathfrak G} = (V, C, E), \ensuremath{\mathfrak R_\mathit{scc}} \subseteq (V \times C \times V), \mathcal{V} \subseteq (V \times C))$}{
\lIf{$\mathcal{V} = \emptyset$}{\Return}
$\ensuremath{\mathcal F}, \ensuremath{\mathcal B}, \ensuremath{\mathcal F}f, \ensuremath{\mathcal B}f \subseteq (V \times C) \gets \textsc{Pivots}(\mathcal{V})$\;
$\ensuremath{\mathcal F}u, \ensuremath{\mathcal B}u \subseteq (V \times C) \gets \emptyset$\;
$\ensuremath{F_\mathit{lock}}, \ensuremath{B_\mathit{lock}} \subseteq C \gets \emptyset$\;
\While{$\ensuremath{F_\mathit{lock}} \cup \ensuremath{B_\mathit{lock}} \subset \textsc{Colours}(\mathcal{V})$}{\label{alg:ls-start}
$\ensuremath{\mathcal F}f \gets (\textsc{Post}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal F}f) \cap \mathcal V) \setminus \ensuremath{\mathcal F}$\;\label{alg:lockstep-post}
$\ensuremath{\mathcal B}f \gets (\textsc{Pre}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal B}f) \cap \mathcal V) \setminus \ensuremath{\mathcal B}$\;\label{alg:lockstep-pre}
$\ensuremath{F_\mathit{lock}} \gets \ensuremath{F_\mathit{lock}} \cup (\textsc{Colours}(\mathcal V) \setminus \textsc{Colours}(\ensuremath{\mathcal F}f) \setminus \ensuremath{B_\mathit{lock}})$\;
$\ensuremath{B_\mathit{lock}} \gets \ensuremath{B_\mathit{lock}} \cup (\textsc{Colours}(\mathcal V) \setminus \textsc{Colours}(\ensuremath{\mathcal B}f) \setminus \ensuremath{F_\mathit{lock}})$\;\label{alg:lock}
$\ensuremath{\mathcal F}u \gets \ensuremath{\mathcal F}u \cup (\ensuremath{\mathcal F}f \cap (V \times \ensuremath{B_\mathit{lock}}))$\;
$\ensuremath{\mathcal B}u \gets \ensuremath{\mathcal B}u \cup (\ensuremath{\mathcal B}f \cap (V \times \ensuremath{F_\mathit{lock}}))$\;
$\ensuremath{\mathcal F}f \gets \ensuremath{\mathcal F}f \setminus (V \times \ensuremath{B_\mathit{lock}})$\;
$\ensuremath{\mathcal B}f \gets \ensuremath{\mathcal B}f \setminus (V \times \ensuremath{F_\mathit{lock}})$\;
$\ensuremath{\mathcal F} \gets \ensuremath{\mathcal F} \cup \ensuremath{\mathcal F}f$\;
$\ensuremath{\mathcal B} \gets \ensuremath{\mathcal B} \cup \ensuremath{\mathcal B}f$\;
}\label{alg:ls-end}
$\ensuremath{\mathcal{C}on} \subseteq V \times C \gets (\ensuremath{\mathcal F} \cap (V \times \ensuremath{F_\mathit{lock}})) \cup (\ensuremath{\mathcal B} \cap (V \times \ensuremath{B_\mathit{lock}}))$\;\label{alg:con}
$\ensuremath{\mathcal F}f \gets \ensuremath{\mathcal F}u \cap \ensuremath{\mathcal{C}on}$\;\label{alg:con-apply}
$\ensuremath{\mathcal B}f \gets \ensuremath{\mathcal B}u \cap \ensuremath{\mathcal{C}on}$\;\label{alg:con-apply-2}
\While{$\ensuremath{\mathcal F}f \ne \emptyset \lor \ensuremath{\mathcal B}f \ne \emptyset$}{\label{alg:rest-start}
$\ensuremath{\mathcal F}f \gets (\textsc{Post}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal F}f) \cap \ensuremath{\mathcal{C}on}) \setminus \ensuremath{\mathcal F}$\;
$\ensuremath{\mathcal B}f \gets (\textsc{Pre}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal B}f) \cap \ensuremath{\mathcal{C}on}) \setminus \ensuremath{\mathcal B}$\;
$\ensuremath{\mathcal F} \gets \ensuremath{\mathcal F} \cup \ensuremath{\mathcal F}f$\;
$\ensuremath{\mathcal B} \gets \ensuremath{\mathcal B} \cup \ensuremath{\mathcal B}f$\;
}\label{alg:rest-end}
$\ensuremath{\mathcal W} \subseteq V \times C \gets \ensuremath{\mathcal F} \cap \ensuremath{\mathcal B}$\;\label{alg:scc}
$\ensuremath{\mathfrak R_\mathit{scc}} \gets \ensuremath{\mathfrak R_\mathit{scc}} \cup \textsc{Join}(\ensuremath{\mathcal W})$\;
$\textsc{Decomposition}(\ensuremath{\mathfrak G}, \ensuremath{\mathfrak R_\mathit{scc}}, \mathcal{V} \setminus \ensuremath{\mathcal{C}on})$\;
$\textsc{Decomposition}(\ensuremath{\mathfrak G}, \ensuremath{\mathfrak R_\mathit{scc}}, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$\;
}
\BlankLine
\Fn{\upshape\textsc{Pivots}$(\mathcal{V})$}{
$\mathcal{P} \subseteq (V \times C) \gets \emptyset$; $\mathcal{V}' \subseteq (V \times C) \gets \mathcal{V}$\;
\While{$\mathcal{V}' \ne \emptyset$}{
$(v, c) \gets \textsc{Pick}(\mathcal{V}')$\;
$\mathcal{P} \gets \mathcal{P} \cup ( \{ v \} \times \sigma_1(v, \mathcal{V'})) $\;
$\mathcal{V'} \gets \mathcal{V'} \setminus (V \times \textsc{Colours}(\mathcal{P}))$\;
}
\Return $\mathcal P$\;
}
\caption{Symbolic Coloured SCC Decomposition}\label{algo:symbolic}
\end{algorithm}
An important observation is that the structure of components in the graph can change arbitrarily with
respect to the graph colours. In consequence, our algorithm cannot simply operate with sets of
graph vertices as the normal algorithm would. To that end, we use the
notion of coloured vertex sets as introduced in
Section~\ref{ssec:symb} where the symbolic operations we perform on
these sets have been described.
\paragraph{Pivot selection} Initially, the algorithm starts with all vertices and colours, i.e.~the full
set $V \times C$. However, as the components are discovered, the intermediate results $\mathcal{V}$ may contain different vertices appearing only for certain subsets of $C$. As a result, we often cannot pick a~single pivot vertex that
would be valid for all considered colours. Instead, we aim to pick a~\emph{pivot set} $P \subseteq V \times C$ such that for every colour
that still appears in $\mathcal V$,
the set contains \emph{exactly} one vertex. Alternatively, one can also
view the pivot set as a (partial) function from $C$ to $V$. This is done
in the \textsc{Pivots} function. In the following discussion of the algorithm,
we write $c$-coloured pivot to mean the vertex $u$ such that
$(u, c)$ is found in the coloured set returned by \textsc{Pivots}
in the current iteration (for all colours still present in $\mathcal{V}$).
Please note that the presented \textsc{Pivots} routine is rather naive, as it has to explicitly iterate all the pivot vertices, whose number can be substantial in the worst case. However, as presented, it should be easy to implement for basically any type of coloured graphs, regardless of the underlying representation. In the implementation section, we show \textsc{Pivots} can be re-implemented in the domain of BDDs such that it is guaranteed to always require only $\mathcal{O}(\log|V|)$ symbolic operations.
\paragraph{Coloured lock-step (phase one)} The lock-step reachability procedure also cannot operate as in a standard
graph. First of all, there can be colours where the forward reachability
converges first, as well as colours where this happens for backward
reachability. The algorithm thus has to account for both options
simultaneously. Second, for each colour, the reachability can converge in a
different number of steps. To deal with this problem, we introduce the
$\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ variables. These store the mutually disjoint sets of
colours for which the forward and backward reachability procedures have already
converged. The lock-step procedure then terminates when $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$
contain all the colours that appear in $\mathcal V$.
Throughout the algorithm, we keep track of several coloured-set variables.
The first two, $\ensuremath{\mathcal F}$ and $\ensuremath{\mathcal B}$, represent the forward and backward reachable
sets, respectively.
This means that for every colour $c$ present in $\mathcal V$, if $u$ is the
$c$-coloured pivot, every $(v, c) \in \ensuremath{\mathcal F}$ satisfies $u \creach{c} v$
and every $(v, c) \in \ensuremath{\mathcal B}$ satisfies $v \creach{c} u$.
Furthermore, if $c \in \ensuremath{F_\mathit{lock}}$ then $\ensuremath{\mathcal F}$ contains exactly all such pairs;
similarly for $\ensuremath{B_\mathit{lock}}$ and $\ensuremath{\mathcal B}$.
We say that a coloured vertex pair $(v, c)$ has been \emph{forward expanded}
or \emph{backward expanded} in the current iteration of the algorithm, if
there has been a call to the \textsc{Post} or \textsc{Pre} symbolic operation
with $(v, c)$ being an element of the coloured set argument.
To track which reachable coloured vertices are to be expanded later,
also called the \emph{frontiers} of the reachability sets,
we have the four variables $\ensuremath{\mathcal F}f$, $\ensuremath{\mathcal F}u$, $\ensuremath{\mathcal B}f$, $\ensuremath{\mathcal B}u$.
The frontier of $\ensuremath{\mathcal F}$ is the union $\ensuremath{\mathcal F}f \cup \ensuremath{\mathcal F}u$. The sets $\ensuremath{\mathcal F}f$ and $\ensuremath{\mathcal F}u$
are disjoint: $\ensuremath{\mathcal F}f$ involves those colours for which the
lock-step reachability procedure has not finished yet, i.e.~the colours that
are neither in $\ensuremath{F_\mathit{lock}}$ nor in $\ensuremath{B_\mathit{lock}}$,
while $\ensuremath{\mathcal F}u$ represents
the part of the frontier whose exploration is currently paused due to the fact
that its colours are in $\ensuremath{B_\mathit{lock}}$.
Note that there may be no pair $(v, c)$ of the forward frontier with
$c \in \ensuremath{F_\mathit{lock}}$ as that means that the exploration of the $c$-coloured
forward-reachable set is complete.
A symmetric role is played by the sets $\ensuremath{\mathcal B}f$ and $\ensuremath{\mathcal B}u$.
In the first while loop (lines~\ref{alg:ls-start}--\ref{alg:ls-end}), we
compute the reachability sets in the lock-step manner. Once a~reachability set
is completed for some colours (i.e.,~there are no vertices to expand with those
colours), we add the colours to the corresponding $\ensuremath{F_\mathit{lock}}$ or $\ensuremath{B_\mathit{lock}}$
variable. Note that we ensure that $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ remain disjoint even
if the two reachability procedures converged at the same time for certain
colours---see line~\ref{alg:lock}.
We use $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ to split the newly computed frontier sets into
the parts that are to be expanded in the next iteration ($\ensuremath{\mathcal F}f$, $\ensuremath{\mathcal B}f$)
and the parts currently left unexpanded ($\ensuremath{\mathcal F}u$, $\ensuremath{\mathcal B}u$).
Note that during the computation of $\textsc{Post}$ and $\textsc{Pre}$ on lines~\ref{alg:lockstep-post} and~\ref{alg:lockstep-pre}, we intersect the resulting set with $\mathcal{V}$. This step is not necessary for correctness, but as the algorithm divides $V \times C$ into smaller sets in each recursive call to \textsc{Decomposition}, it can happen that the set of states \emph{reachable} from $\mathcal{V}$ is substantially larger than $\mathcal{V}$ itself. In such cases, this intersection effectively restricts the computation of $\textsc{Post}$ and $\textsc{Pre}$ to the sub-graph of $\ensuremath{\mathfrak G}$ induced by $\mathcal{V}$.
\paragraph{Component identification (phase two)} After the first while loop terminates, we compute the set $\ensuremath{\mathcal{C}on}$ that is an analogue
for the converged set of the original lock-step algorithm
(line~\ref{alg:con}).
As already suggested above and unlike the original
algorithm, this set cannot be just $\ensuremath{\mathcal F}$ or $\ensuremath{\mathcal B}$, but is instead a~mixture of
both, depending on the converged colours. To~compute this set, we use the
$\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ variables.
Once $\ensuremath{\mathcal{C}on}$ is computed, $\ensuremath{\mathcal F}f$ and $\ensuremath{\mathcal B}f$ are restarted using the converged portion of $\ensuremath{\mathcal F}u$ and $\ensuremath{\mathcal B}u$ (lines~\ref{alg:con-apply} and~\ref{alg:con-apply-2}). The second while loop (lines~\ref{alg:rest-start}--\ref{alg:rest-end}) can then
complete the unfinished forward and backward reachability set, now restricted to
the inside of the converged set. The intersection of $\ensuremath{\mathcal F}$ and $\ensuremath{\mathcal B}$ then
forms a~coloured set $\ensuremath{\mathcal W}$ with the property that for all
$c \in \textsc{Colours}(\mathcal V)$, $\ensuremath{\mathcal W}(\_, c)$ is a~strongly
connected component of $\ensuremath{\mathfrak G}(c)$. We create the corresponding relation
using the \textsc{Join} operation, add this relation to the resulting
$\ensuremath{\mathfrak R_\mathit{scc}}$, and recursively call the whole procedure with
$\mathcal V \setminus \ensuremath{\mathcal{C}on}$ and $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$ as the
base sets.
\paragraph{Comments on the coloured approach} Let us note that there is possibly another approach to processing coloured graphs. Instead of trying
to work with all colours still appearing in the coloured vertex set at once,
we could fork a~new recursive procedure whenever the colour set
splits due to the differences in the graph structure. For example, instead
of picking multiple coloured vertices as pivots, one could pick a single
vertex with a valid subset of colours and then address the remaining colours
in a separate recursive call. Similarly, instead of a single recursive $\textsc{Decomposition}$ call with $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$, we could consider two calls, one with the $\mathcal{F}$ portion of $\ensuremath{\mathcal{C}on}$ and the other with the $\mathcal{B}$ portion of $\ensuremath{\mathcal{C}on}$ (note that these are colour-disjoint since each colour can converge only in one of the two sets).
While such an approach could be to some extent
beneficial in a massively parallel environment where each recursive call can
be executed independently on a~new CPU, the amount of forking in large systems
will soon become unreasonable. More importantly, it defeats the purpose of
the symbolic representation, which aims to minimise the number of symbolic
operations.
\begin{figure}
\caption{An illustration of the algorithm execution.
There are two colours: ${\color{red}
\label{fig:example}
\end{figure}
\paragraph{Example} The execution of one iteration of the algorithm is illustrated in
Figure~\ref{fig:example}. Here, we have an edge-coloured graph with six vertices
and two colours (red and blue).
The top-most picture represents the initial situation after we have chosen
the pivots; in this case, $\{(b, \mathit{blue}), (b, \mathit{red})\}$.
The next four rows illustrate the first phase (the first while loop) of the
algorithm. After the second iteration of the loop, the blue colour becomes
locked in $\ensuremath{F_\mathit{lock}}$, and thus $(f, \mathit{blue})$ is not expanded in the
backward reachability procedure. This is illustrated by its dashed outline.
After the third iteration of the loop,
the red colour becomes locked in $\ensuremath{B_\mathit{lock}}$ and thus the first phase ends.
In the second phase, both reachability procedures continue from the paused
coloured vertices (dashed outlines); the result is seen in the fifth row.
The intersection of the two reachable sets (i.e.~the coloured set $\mathcal W$)
is then illustrated in the bottom-most picture.
The algorithm would now continue with the coloured sets
$\mathcal{V} \setminus \ensuremath{\mathcal{C}on} = \{ (a, \mathit{blue}), (d, \mathit{blue}) \}$
and $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W} = \{ (c, \mathit{red}) \}$.
\subsection{Correctness and Complexity of the Coloured Lock-step Algorithm}
\begin{thm}\label{thm:correctness}
Let $\ensuremath{\mathfrak G} = (V, C, E)$ be a~coloured graph. The coloured lock-step
algorithm terminates and computes the coloured SCC decomposition relation~$\ensuremath{\mathfrak R_\mathit{scc}}$.
\end{thm}
\begin{proof}
We first show that the set $\ensuremath{\mathcal W}$ computed in line~\ref{alg:scc}
indeed contains one SCC for every colour $c \in \textsc{Colours}(\mathcal V)$
and that the recursive calls of \textsc{Decomposition} preserve the property
that $\mathcal V$ is SCC-closed with respect to all colours.
Let us assume that $\mathcal V$ is SCC-closed and let us take an arbitrary
$c \in \textsc{Colours}(\mathcal V)$. The function \textsc{Pivots} chooses
a~set that contains exactly one pair whose colour is $c$, let us call this
pair $(v, c)$. Let us further assume that $c$ is assigned into $\ensuremath{F_\mathit{lock}}$
first (the case with $\ensuremath{B_\mathit{lock}}$ is completely symmetric).
Let us now choose an arbitrary vertex $w$ such that $v$ and $w$ are in the
same SCC of $\ensuremath{\mathfrak G}(c)$, i.e.~$v \to^* w$ and $w \to^* v$.
As the first while loop
finishes, $\ensuremath{\mathcal F}$ contains all the pairs of the form $(u, c) \in \mathcal V$
where $u$ is reachable from $v$ in $\ensuremath{\mathfrak G}(c)$. Thus, it also contains $(w, c)$
due to the fact that $\mathcal V$ is SCC-closed.
Now, either $(w, c) \in \ensuremath{\mathcal B}$, or there exists a vertex $x$ such that
$w \to^* x$, $x \to^* v$ in $\ensuremath{\mathfrak G}(c)$ and $x \in \ensuremath{\mathcal B}u$.
This means that $(w, c)$ is added to $\ensuremath{\mathcal B}$ in the second while loop.
In both cases, both $(v, c)$ and $(w, c)$ are then added to
$\ensuremath{\mathcal W}$. As the vertex choices were arbitrary, this proves that
the SCC of $v$ in $\ensuremath{\mathfrak G}(c)$ is contained in $\ensuremath{\mathcal W}$.
Furthermore, if $(y, c) \in \ensuremath{\mathcal W}$ for an arbitrary $y$, then
$v \to^* y$ and $y \to^* v$ in $\ensuremath{\mathfrak G}(c)$, which means that $y$ is in
$SCC(\ensuremath{\mathfrak G}(c), v)$. This proves that $\ensuremath{\mathcal W}$ contains exactly one
SCC for every colour $c \in \textsc{Colours}(\mathcal V)$.
We now argue that $\ensuremath{\mathcal{C}on}$ is SCC-closed with respect to all colours.
This immediately implies that both $\mathcal V \setminus \ensuremath{\mathcal{C}on}$ and
$\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$ are SCC-closed.
Let us assume that there is a~colour $c \in \textsc{Colours}(\mathcal V)$
and two vertices $v$, $w$ in the same SCC of $\ensuremath{\mathfrak G}(c)$ such that
$(v, c) \in \ensuremath{\mathcal{C}on}$, but $(w, c) \not\in \ensuremath{\mathcal{C}on}$.
Let us assume that $c \in \ensuremath{F_\mathit{lock}}$ (as above, the case of $\ensuremath{B_\mathit{lock}}$ is
completely symmetrical).
Then $(v, c) \in \ensuremath{\mathcal F}$ after the first while loop finishes. This also means
that $(w, c) \in \ensuremath{\mathcal F}$ as the forward reachability procedure is completed for
$c$ and thus $(w, c) \in \ensuremath{\mathcal{C}on}$, a~contradiction.
What remains is to show that the algorithm terminates and that every
SCC is eventually found. Termination is trivially proved by the fact
that size of the set $\mathcal V$ always decreases in recursive calls:
both $\ensuremath{\mathcal W}$ and $\ensuremath{\mathcal{C}on}$ are non-empty because they contain the
initial pivot set as a~subset.
Clearly, a representant of every SCC of every monochromatisation $\ensuremath{\mathfrak G}(c)$ is
eventually chosen as a~pivot. Together with the above reasoning, this implies
that the algorithm is correct.
\end{proof}
\begin{thm}\label{thm:complexity}
Let $|V|$ be the number of vertices in the coloured graph and let
$|C|$ be the number of colours. The coloured lock-step algorithm
performs at most $\mathcal O(|C|\cdot |V|\cdot \log |V|)$ symbolic steps.
\end{thm}
\begin{proof}
Let us first note that all the derived operations defined in
Figure~\ref{tab:operations} use only a~constant number of the basic symbolic
operations. As we are considering asymptotic complexity here, we can view all
the operations in Figure~\ref{tab:operations} as elementary symbolic steps.
We first make the observation that each vertex may be chosen as a~part of the
pivot set at most $|C|$ times. Clearly, once a~vertex is included in the pivot
set with a~set of colours $C'$, then, $\{v\} \times C'$ is a subset of first $\ensuremath{\mathcal{C}on}$, and later $\ensuremath{\mathcal W}$
(due to the monotonicity of the construction of $\mathcal F$ and $\mathcal B$). Therefore, the elements of $\{v\} \times C'$ do not appear in subsequent recursive
calls. Since a single vertex-colour pair cannot be returned by \textsc{Pivots} twice, it means that the total cumulative complexity of all the calls to the \textsc{Pivots} routine is bounded by $O(|C|\cdot|V|)$. We can therefore exclude them from the rest of the complexity analysis.
We now consider the complexity of a~single call to \textsc{Decomposition}
without the subsequent recursive calls. Let us now select one of the
colours for which the lock-step reachability procedure
(lines~\ref{alg:ls-start}--\ref{alg:ls-end}) finished last, i.e.,~one of the
colours that have been added to $\ensuremath{F_\mathit{lock}}$ or $\ensuremath{B_\mathit{lock}}$ in the final
iteration of the loop. Let us call this colour $c$.
Recall that $\sigma_2(c, \mathcal X)$ is the set of vertices with colour
$c$ in a~coloured set $\mathcal X$.
Let us denote by $W$ the monochromatic SCC discovered for $c$, i.e. $W := \sigma_2(c, \ensuremath{\mathcal W})$, and let
$S$ be the smaller of $\sigma_2(c, \mathcal V \setminus \ensuremath{\mathcal{C}on})$
and $\sigma_2(c, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$.
Clearly $S$ contains at most $|V|/2$ vertices.
Let $k = |S \cup W|$.
We now argue that the number of symbolic steps in a~given call
(without the recursive calls) is bounded by $\mathcal O(k)$. This is because in a lock-step algorithm, the call to \textsc{Decomposition} must explore the discovered SCC itself (i.e. $W$), and the smaller of the forward or backward reachable sets from this SCC (i.e. $S$) -- intuitively, its complexity should be thus bounded by the size of these two sets.
Assume w.l.o.g.~that $c \in \ensuremath{F_\mathit{lock}}$ (a completely symmetric argument solves
the case $c \in \ensuremath{B_\mathit{lock}}$).
Then after the first while loop finishes, we have $\sigma_2(c, \ensuremath{\mathcal{C}on}) = \sigma_2(c, \mathcal F)$.
If $S$ is $\sigma_2(c, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$
then $k$ is the size of $\sigma_2(c, \mathcal F)$ (and thus also $\sigma_2(c, \ensuremath{\mathcal{C}on})$), since $\sigma_2(c, \mathcal{F})$ consists of $\sigma_2(c, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$ (the set $S$) and $\sigma_2(c, \ensuremath{\mathcal W})$ (the discovered SCC).
Each iteration of the first while loop puts at least one vertex with
colour $c$ into $\mathcal F$; otherwise $c$ would not have finished in the last iteration. This means that the loop runs for at most
$k$ iterations. This also means that the size of $\sigma_2(x, \mathcal X)$ for
all colours $x$ and $\mathcal X \in \{ \mathcal F, \mathcal B \}$ is also
bounded by $k$ after the first while loop finishes, which in turn means that the second while loop cannot make more
than $O(k)$ steps.
If $S$ is $\sigma_2(c, \mathcal V \setminus \ensuremath{\mathcal{C}on})$ instead,
let us define $B := \sigma_2(c, \mathcal B)$ right after the first
while loop has finished. We know that $B \subseteq S \cup W$:
if a vertex $v$ was in $B \setminus S$, then it would have to be in $\ensuremath{\mathcal{C}on}$ (i.e. $(v, c) \in \ensuremath{\mathcal{C}on}$). Due to our initial assumption of $c \in \ensuremath{F_\mathit{lock}}$ (w.l.o.g), we then also have $(v,c) \in \mathcal
F$ which dictates $v \in W$. Consequently, we see that any vertex $v \in B$ must be either in $S$ or in $W$, arriving at $B \subseteq S \cup W$.
Again, each iteration of the first while loop puts at least one vertex
with colour $c$ into $\mathcal B$; otherwise $c$ would have been
in $\ensuremath{B_\mathit{lock}}$ before it appeared in $\ensuremath{F_\mathit{lock}}$. Similarly to the previous
case, this means that both while loops run for at most $O(k)$ steps.
The rest of the argument uses amortised reasoning, in a~way similar to the
proof in~\cite{bloem2000}. Note that each vertex is going to be an element of
the set $W$ as described above at most $|C|$ times (once for each colour).
Furthermore, each vertex is going to be an element of the set $S$ as described
above at most $|C|\cdot\log|V|$ times: for each colour, the vertex can be an
element of the smaller of the two sets at most $\log|V|$ times. As the cost of
each single call can be charged to the vertices in $S \cup W$ as explained
above, it is sufficient to charge each vertex the total cost of
$|C| +|C|\cdot\log|V|$. Together, this means that the total number of symbolic
steps is bounded by $O(|C|\cdot|V|\cdot\log|V|)$.
\end{proof}
Note that the upper bound established by Theorem~\ref{thm:complexity} is
no better than the one we would get if we split the coloured graph into
its monochromatic constituents and processed each
separately using the original lock-step algorithm~\cite{bloem2000}.
We remark, however, that the practical complexity of the coloured approach can be
much smaller. Indeed, the complexity analysis in the previous proof focused
on a~single colour, omitting the fact that SCCs for many other colours are
found at the same time. In cases where the edges are largely shared among
the colours, which is true in many applications, the coloured algorithm has the
potential to significantly outperform the parameter-scan approach. The
situation is similar to that of the coloured model checking; see the
observations made in~\cite{tcbb}.
\section{Symbolic Computation with Boolean Networks}
\label{section:boolean-networks}
The algorithm as presented in the previous section is completely agnostic to the properties of the underlying system, as long one provides an implementation of all the necessary symbolic operations. However, to empirically test its performance, we need to pick such an implementation, which typically entails analysis of a specific class of systems.
In this paper, we consider Boolean networks~\cite{KAUFFMAN,Bernot05,SCHWAB,THOMAS}, specifically asynchronous Boolean networks, which represent a popular discrete modelling framework in systems biology~\cite{Brim2013, Grieb_2015}. Due to incomplete biological knowledge, the dynamics of a Boolean network can by often only partially known. This uncertainty can be then captured using coloured directed graphs. In this section, we introduce Boolean networks and show how they can be translated into coloured graphs suitable for SCC-decomposition.
Asynchronous Boolean networks are especially challenging for symbolic analysis. It is a well-known fact, that using symbolic structures (e.g BDDs) to explore very large state spaces gives good results for synchronous systems, but shows its limits when trying to tackle asynchronicity (see e.g.~\cite{DBLP:conf/forte/CouvreurT05}).
\subsection{Boolean networks with inputs} A Boolean network (BN), as the name suggests, consists of $n$ Boolean \emph{variables} $s_1, \ldots, s_n$ which together describe the state of the network. The dynamics of the network can also depend on additional $m$ Boolean \emph{inputs} $c_1, \ldots, c_m$ (sometimes also called \emph{constants}, or \emph{logical parameters}), whose value is assumed to be fixed, but generally unknown. The valuations of these inputs correspond to the colours of our Kripke structure.
Each network variable $s_i$ is equipped with a Boolean update function $b_i: \{0,1\}^n \times \{0,1\}^m \to \{0,1\}$ that updates the variable based on the state of the network, and the values of its inputs. We assume that the variables are updated \emph{asynchronously}, meaning that during every state transition, exactly one variable is updated.
Such a network with inputs defines a coloured graph where $V = \{0,1\}^n$, $C = \{0,1\}^m$, and for every $c \in C$, we have that $u \xrightarrow{c} v$ if and only if $u \not= v$ and $v = u[u_i \mapsto b_i(u, c)]$ for some $i \in [1,n]$. That is, $v$ is equal to $u$ where the $i$-th variable is updated with the output of function $b_i$. Because all variables and inputs are Boolean, this structure has a~fairly straightforward symbolic representation in terms of binary decision diagrams, as we later demonstrate.
Note that in practice, we often work within a~subset of biologically relevant colours, denoted as~$Valid$ (i.e. not every possible valuation of $c_1, \ldots, c_m$ may be biologically admissible). In the algorithms, this is implicitly reflected such that the set of all possible colours $C$ corresponds to the set $Valid$ instead of the set of \emph{all} possible valuation (i.e. $\{0,1\}^m$) if demanded by the application at hand.
\subsection{Partially specified Boolean networks} A Boolean network with inputs allows us to easily encode a wide range of biochemical systems in a~machine friendly format. However, for systems with a high degree of uncertainty, it often fails to capture this uncertainty in a~way understandable to a human reader.
To mitigate this issue, we consider \emph{partially specified} Boolean networks that allow us to explicitly mark parts of the update functions as unknown. Specifically, let us assume that $f_1^{(a_1)}$, $f_2^{(a_2)}$, $\ldots$ are symbols standing in for some uninterpreted (fixed but arbitrary) Boolean functions (here, $a_i$ denotes their arity). A partially specified Boolean network then consists of $n$ Boolean variables and $p$ uninterpreted Boolean functions. In such a~network, every update function $b'_i$ is specified as a Boolean expression that can use the function symbols $f_1, \ldots, f_p$.
This type of formalism is often easier to comprehend, as the uncertainty in dynamics is tied to the update functions instead of inputs (if desired, input can be still expressed using uninterpreted functions of arity zero). It is not immediately clear how such a~network should be represented symbolically though.
One option is to translate a partially specified network into a~BN with inputs. Any uninterpreted function $f_i^{(a)}$ can be encoded in terms of $2^a$ Boolean inputs $c_1^{i}, \ldots, c_{2^a}^i$ if we consider that input $r_j^i$ denotes the output of $f_i^{(a)}$ in the $j$-th row of its truth table. Formally, this translation can be achieved using a~repeated application of the following expansion rule:
\begin{align*}
f(\alpha_1, \ldots, \alpha_a) \equiv (\alpha_1 \Rightarrow f'_1(\alpha_2, \ldots, \alpha_a)) \land (\neg\alpha_1 \Rightarrow f'_2(\alpha_2, \ldots, \alpha_a))
\end{align*}
Here, $f'_1$ and $f'_2$ are fresh uninterpreted functions of arity $a-1$, and $\alpha_i$ are arbitrary Boolean expressions. Using this rule, we can always convert a partially specified network to a Boolean network with inputs. The number of inputs will be exponential with respect to the arity of the employed uninterpreted functions though (since each application of the rule replaces one uninterpreted function with two, and the depth of the recursive expansion is the arity $a$).
For example, consider the following partially specified Boolean network:
\begin{align*}
b'_1 &:= x_1 \land f_1^{(1)}(x_2)\\
b'_2 &:= \neg x_1 \lor f_2^{(2)}(x_1, x_3)\\
b'_3 &:= (f_3^{(0)} \Leftrightarrow x_3) \land f_2^{(2)}(\neg x_1, x_2)
\end{align*}
It uses three uninterpreted Boolean functions $f_1^{(1)}$, $f_2^{(2)}$, and $f_3^{(0)}$. After performing the aforementioned expansion, and simplifying the resulting expressions slightly for readability, we obtain the following network with logical inputs:
\begin{align*}
b_1(x, c) =&~x_1 \land (x_2 \Rightarrow c^{1}_{[1]}) \land (\neg x_2 \Rightarrow c^{1}_{[0]})\\
b_2(x, c) =&~\neg x_2 \lor (((x_1 \land x_3) \Rightarrow c^{2}_{[1,1]}) \land ((x_1 \land \neg x_3) \Rightarrow c^{2}_{[1,0]})\\&\hspace{15.5pt}\land~((\neg x_1 \land x_3) \Rightarrow c^{2}_{[0,1]}) \land ((\neg x_1 \land \neg x_3) \Rightarrow c^{2}_{[0,0]})) \\
b_3(x, c) =&~(c^{3} \Leftrightarrow x_3) \land ((\neg x_1 \land x_2) \Rightarrow c^{2}_{[1,1]}) \land ((\neg x_1 \land \neg x_2) \Rightarrow c^{2}_{[1,0]})\\&\hspace{57pt}\land((x_1 \land x_2) \Rightarrow c^{2}_{[0,1]}) \land ((x_1 \land \neg x_2) \Rightarrow c^{2}_{[0,0]})
\end{align*}
Here, each $c^i_j$ corresponds to one truth table row of $f_i$, such that $j$ describes the input vector corresponding to said row (i.e. $c^{1}_{[0,1]}$ represents the value of $f_1(0, 1)$).
\subsection{Symbolic Representation of BNs}
As a symbolic representation, a natural choice are Reduced Ordered Binary Decision Diagrams (ROBDD, or simply BDD)~\cite{bryant86}, which can concisely encode Boolean functions or relations of Boolean vectors. Specifically, out implementation leverages the internal tools and libraries provided by the tool AEON~\cite{aeon}.
Since a Boolean network consists of $n$ Boolean variables and $m$ Boolean inputs, any subset of $V$, $C$, or a relation $X \subseteq V \times C$ (a coloured set of vertices) can be seen as a Boolean formula over the network variables and inputs. That is, each network variable and logical input corresponds to one decision variable of the BDD. Here, a pair $(s,c)$ belongs to such a~relation iff it represents a satisfying assignment of this formula $X$. For relations of higher arity, fresh decision variables are created for each component of the relation. Standard set operations as described in Fig.~\ref{tab:operations} then correspond to logical operations on such formulae ($\land \equiv \cap$, $\lor \equiv \cup$, etc.).
Relation operations are similarly implementable using BDD primitives. In particular, existential quantification of a single decision variable (e.g. $\exists s_i . X$ or $\exists c_j . X$) is a native operation on BDDs. Consequently, existential quantification on relations (as well as \textsc{Colours}) is simply a quantification over all decision variables encoding the specific relation component (i.e. all network variables for $V$, or all logical inputs for $C$). Finally, \textsc{Swap} only influences the way in which a BDD is interpreted -- the actual structure of the BDD is unaffected.
To encode the network dynamics, notice that every update function $b_i$ can be directly represented as a separate BDD. From such BDDs, we can build one large BDD describing the whole coloured transition relation, which is traditionally used for the computation of \textsc{Pre} and \textsc{Post}. But the symbolic representation of such relation is often prohibitively complex for asynchronous systems. Instead, we compute \textsc{Pre} and \textsc{Post} using partial results for individual variables, which uses more symbolic operations but is less likely to cause a~blow-up in the size of the BDD:
\begin{align*}
\textsc{VarPost}(\ensuremath{\mathfrak G}, i, \mathcal{X}) & = (\mathcal{X} \land (b_i \centernot\Leftrightarrow s_i))[s_i \mapsto \neg s_i]\\
\textsc{VarPre}(\ensuremath{\mathfrak G}, i, \mathcal{X}) & = \mathcal{X}[s_i \mapsto \neg s_i] \land (b_i \centernot\Leftrightarrow s_i)\\
\textsc{Post}(\ensuremath{\mathfrak G}, \mathcal{X}) & = \bigvee_{i \in [1,n]} \textsc{VarPost}(i, \mathcal{X})\\
\textsc{Pre}(\ensuremath{\mathfrak G}, \mathcal{X}) & = \bigvee_{i \in [1,n]} \textsc{VarPre}(i, \mathcal{X})
\end{align*}
Here, $[s_i \mapsto \neg s_i]$ is the standard substitution operation, which we use to flip the value of variable $s_i$ in the resulting formula if it does not agree with the output of $b_i$. Note that this operation can be also implemented structurally directly on the BDD by exchanging the children of decision nodes conditioning on $s_i$. Also note that sub-formulae that do not depend on $X$ can be pre-computed once for the whole run of the algorithm, and the version of $\textsc{Pre}$ and $\textsc{Post}$ for monochromatic graphs can be implemented in exactly the same way.
\section{Implementation}
\label{section:implementation}
Finally, let us discuss a number of technical improvements which our algorithm employs in practice, and whose impact we consider in the evaluation section.
\subsection{Pivot Selection}
In Algorithm~\ref{algo:symbolic}, we gave a naive implementation of the $\textsc{Pivots}(\mathcal{X})$ function. Here, we show how to implement it for BDDs in a much more concise way. Note that our approach uses the notation we established earlier for Boolean networks, but is generally applicable to any set or relation of bit-vectors represented using BDDs.
First, notice that for a single network variable, we can define a similar operation, which we call $\textsc{Pick}(i, \mathcal{X})$:
\begin{equation*}
\textsc{Pick}(i, \mathcal{X}) = \mathcal{X} \setminus (\mathcal{X} \land \neg s_i)[s_i \gets \neg s_i]
\end{equation*}
Here, we first restrict $\mathcal{X}$ only to the valuations which have $s_i = \mathit{false}$, and then invert the value of $s_i$ (resulting in $s_i$ being always $\mathit{true}$ in the set). Once we subtract these valuations from $\mathcal{X}$, the resulting set then contains a valuation with $s_i = \mathit{true}$ only if it does \emph{not} contain the same valuation with $s_i = \mathit{false}$. Intuitively, for any valuation of the remaining BDD decision variables (i.e. $s_j$ and $c_j$ in our case) that is in $\mathcal{X}$, we just picked a single unique value of $s_i$ (while preferring the value $s_i = \mathit{false}$).
However, observe that we cannot simply apply $\textsc{Pick}$ to every network variable alone to obtain the result of \textsc{Pivots}. Intuitively, the problem lies in the fact that \textsc{Pick} selects a witness for each variable in isolation, while $\textsc{Pivots}$ considers all network variables as interconnected. We resolve this problem using a different equation, one which eliminates the picked variable in the recursive invocation:
\begin{align*}
\textsc{Pivots}(\mathcal{X}) &= \textsc{F}(\mathcal{X}, s_1, \ldots, s_n)\\
\textsc{F}(\mathcal{X}, s_1) &= \textsc{Pick}(1, \mathcal{X})\\
\textsc{F}(\mathcal{X}, s_1, \ldots, s_k) &= \textsc{Pick}(k, \mathcal{X}) \cap \textsc{F}(\exists s_k. \mathcal{X}, s_1, \ldots, s_{k-1})
\end{align*}
In this equation, the final case $\textsc{F}(\mathcal{X}, s_1)$ is clearly correct, since it simply defers to $\textsc{Pick}(i, \mathcal{X})$. However, to understand why the recursive case $\textsc{F}(\mathcal{X}, s_1, \ldots, s_k)$ is correct, observe the following: Assume that the set $\mathcal{Y} = \textsc{F}(\exists s_k. \mathcal{X}, s_1, \ldots, s_{k-1})$ is computed correctly. That is, for any valuation of the remaining variables, $\mathcal{Y}$ contains a single unique \emph{incomplete witness} valuation of variables $s_1, \ldots, s_{k-1}$. Now, since the BDD representing $\exists s_k . \mathcal{X}$ does not depend on $s_k$, each such unique \emph{witness} must be included in $\mathcal{Y}$ twice: once with $s_k = \mathit{true}$ and once with $s_k = \mathit{false}$. In other words, a single \emph{witness} valuation of $s_1, \ldots, s_{k-1}$ must be tied to two different valuations of the remaining variables, and these valuations are differentiated only by the variable $s_k$.
Now, one of these two valuations is necessarily included in the set $\textsc{Pick}(k, \mathcal{X})$. The other is either missing from $\mathcal{X}$ altogether, or is eliminated by $\textsc{Pick}(k, \mathcal{X})$. As such, computing $\textsc{Pick}(k, \mathcal{X}) \cap \mathcal{Y}$ extends the witness from $s_1, \ldots, s_{k-1}$ to $s_1, \ldots, s_{k}$ by eliminating one of the two aforementioned occurrences of the \emph{incomplete witness}.
Observe that, as opposed to the original naive implementation of \textsc{Pivots}, this implementation only requires $\mathcal{O}(n)$ (i.e. $\mathcal{O}(\log |V|)$) symbolic operations in any case.
\subsection{Saturation}
In~\cite{Ciardo06}, and later in greater detail within~\cite{Ciardo11}, Ciardo et al. show that when the system is asynchronous, it may be much easier to compute reachable sets (and consequently SCCs) by applying only one transition (e.g. denoted $t_1$) at a time. Once applying $t_1$ cannot add new states to the reachable set, another transition (e.g denoted $t_2$) can be considered, respecting the order in which the affected variables appear in the symbolic data structure (Ciardo et al. employ multivalued decision diagrams, but the principle also applies to BDDs). If the application of other transitions causes that we can again add new states using $t_1$, the process starts anew and $t_1$ is ``saturated'' again.
In the comparison presented in~\cite{Ciardo11}, only the Xie-Beerel $\mathcal{O}(|V|^2)$ algorithm is used with saturation enabled, while the lock-step algorithm is used as given in~\cite{bloem2000}. However, we argue that saturation can be also beneficial in the lock-step algorithm.
\paragraph{Asymptotic complexity} Unfortunately, combining lock-step with saturation disrupts the $\mathcal{O}(|V| \cdot \log|V|)$ asymptotic complexity of the algorithm. To see why this is the case, observe that classical symbolic reachability (i.e. a fixed-point algorithm iterating the $\textsc{Post}$ procedure) requires $\mathcal{O}(|V|)$ steps to explore a graph. Meanwhile, a reachability procedure employing saturation needs $\mathcal{O}(|V||T|)$ operations, where $|T|$ is the number of distinct transitions.
This is caused by the fact that saturation needs to check up to $|T|$ transitions to discover a vertex. For example, consider an asynchronous graph employing transitions $t_1, \ldots, t_n$ such that $t_1$ and $t_n$ are alternated on a path of length $\mathcal{O}(|V|)$. Between considering $t_1$ and $t_n$, saturation will attempt each of the $|T|$ transitions, which are useless on this path, but still consume a symbolic operation.
Consequently, this $|T|$ factor trickles down into the complexity of both the Xie-Beerel and lock-step algorithms if saturation is used, as both ultimately rely on some form of reachability to discover the graph vertices. The complexity of the coloured algorithms is then similarly affected.
\paragraph{Saturation and lock-step} The main idea of how saturation is applied in a coloured lock-step algorithm (for Boolean networks) is shown in Algorithm~\ref{algo:saturation}. The algorithm presents a helper function which performs \emph{one reachability step}, similar to what is performed by the $\textsc{Post}$ function. However, in this algorithm, only one transition is fired for each colour (we assume the iteration follows the order of variables as they appear in the symbolic representation, which benefits saturation). Additionally, a set $R$ of colours that could not perform a step is computed. A similar procedure can be considered for backwards reachability, simply replacing $\textsc{VarPost}$ with $\textsc{VarPre}$.
Note that there is a slight discrepancy between Algorithm~\ref{algo:saturation} and the intuitive description of saturation that we gave earlier. In particular, we see that during a \textsc{NextStep} operation, a transition for each variable is triggered at most once, as opposed to the original description, where a transitions are fired repeatedly. This is caused by the simple nature of Boolean networks: In a BN, a single transition always modifies a single Boolean variable. Consequently, no new states can be discovered by firing a single transition multiple times in sequence. For other asynchronous systems, $\textsc{VarPost}$ may need to be modify to apply the corresponding transition repeatedly.
Additionally, note that we use the set $R$ to ensure that $\textsc{VarPost}$ (i.e. firing of a single transition) is executed only for colours for which we have not found a successor yet using some of the previously considered transitions. This is necessary to ensure that in each invocation of \textsc{NextStep}, each colour present in $\mathcal{F}$ is either advanced by one step (using exactly one transition), or is reported as converged within the returned set $R$.
Using this process, we can replace the $\textsc{Pre}/\textsc{Post}$ procedures in the main lock-step algorithm (lines 11 and 12 of Algorithm~\ref{algo:symbolic}). The $R$ sets computed here are then used to update $F_{lock}$ and $B_{lock}$ (lines 13 and 14), as they exactly represent the converged colours that do not need further computation. A similar modification is necessary for the second while loop (lines 25-29), but here the sets of remaining colours $R$ are not needed.
\begin{algorithm}
\SetKwProg{Fn}{Function}{}{}
\Fn{\textsc{NextStep}$(\mathfrak{G}, \mathcal{F})$}{
$R \gets \textsc{Colours}(\mathcal{F})$\;
\For{$\var{A} \in \ensuremath{\mathit{Var}}$}{
$\mathcal{S} \gets \textsc{VarPost}(\mathfrak{G}, \var{A}, (\mathcal{F} \cap V) \times R)$\;
$R \gets R \setminus \textsc{Colours}(\mathcal{S})$\;
$\mathcal{F} \gets \mathcal{F} \cup \mathcal{S}$\;
\If{$R = \emptyset$}{\textbf{break}\;}
}
\Return $\mathcal (\mathcal{F}, R)$\;
}
\caption{Main idea of the lock-step-saturation approach. The algorithm extends $\mathcal{F}$ with one additional reachability step, and returns a set of colours locked in this iteration ($R$).}\label{algo:saturation}
\end{algorithm}
\subsection{Trimming and Parallelism}
Most graphs typically contain a large number of trivial SCCs that introduce unnecessary overhead to the main algorithm. To avoid this overhead, we additionally perform a trimming step before each invocation of \textsc{Decomposition}. Trimming consists of repeatedly removing all vertices which have no outgoing or no incoming edges and is employed by most symbolic SCC algorithms on standard directed graphs as well.
The coloured analogue of trimming is straightforward, as it can be achieved using \textsc{Pre} and \textsc{Post} operations just as in the non-coloured case. For a coloured set of vertices $\mathcal{V}$, operation $\textsc{Post}(\ensuremath{\mathfrak G}, \textsc{Pre}(\ensuremath{\mathfrak G}, \mathcal{V}) \cap \mathcal{V}) \cap \mathcal{V}$ returns only the vertices which have at least one predecessor in $\mathcal{V}$. The successor variant simply exchanges the \textsc{Post} and \textsc{Pre} operations.
As such, applying this operation to each $\mathcal{V}$ until a fixed-point is reached before \textsc{Decomposition} is invoked eliminates the undesired trivial SCCs. Since the total number of steps performed collectively by all such fixed-point computations is bounded by $|C||V|$ (the total number of removable vertex-colour pairs), this does not impact the overall asymptotic complexity of the algorithm.
In some cases, we have observed that the symbolic representation is able to handle the SCC computation but explodes during trimming. The algorithm then times-out during trimming, even though useful information about SCCs could be obtained if the trimming was skipped or postponed. To avoid this issue, we enforce an extra condition that a trimming procedure is terminated prematurely if the computed BDDs are more than twice the size (in terms of BDD decision nodes) of the initial set.
Additionally, the lock-step algorithm can be rather trivially parallelised. The recursive \textsc{Decomposition} calls operate on independent coloured vertex sets and can be therefore deferred to separate threads. Since the body of the \textsc{Decomposition} method is rather complex, this can be done easily with a queue guarded by a mutex which is shared between all threads (i.e. the synchronisation overhead is negligible due to the long running time of \textsc{Decomposition}). Finally, a simple termination detection procedure is needed to ensure that idle threads do not terminate prematurely while decomposition is still running.
Note that most BDD packages are not internally thread-safe, as they share decision node memory across different BDD objects. In our experiments, this aspect is handled by cloning the set $\mathcal{V}$ corresponding to each recursive invocation, plus the symbolic representation of the BN necessary to compute $\textsc{Post}$ and $\textsc{Pre}$. As such, the memory used to represent BDDs manipulated by each thread is completely independent from other threads.
\section{Experimental Evaluation}
To test the algorithm, we compiled a benchmark set of Boolean networks from the CellCollective~\cite{helikar2012cell} and GINsim~\cite{chaouiya2012} model databases. Since the models in these databases contain fully specified networks, uninterpreted functions were introduced into existing models by pseudo-randomly erasing parts of the existing update functions.
While this process is to some extent artificial, we believe it to be a good approximation of the model development process, where at some point, the structure of the network is already established, but its dynamics are still not fully determined. Using this process, we obtained a collection of networks ranging between $2^{20}$ and $2^{50}$ in the size of the coloured graph (i.e. $|V \times C|$). Note that for each graph, we consider only a subset of possible input valuations that is biologically relevant with respect to the established network structure. For example, the first model (i.e.~\cite{sanchez2017modeling}) admits $2^{48}$ input valuations, but only $2^{19}$ are biologically relevant due to constraints on function monotonicity.
A complete overview of the employed models is given in Table~\ref{tab:models}. For each model, we give the number of discovered non-trivial components as an interval, because each colour can correspond to a different number of components. We employ a 24h timeout for all experiments.
\begin{table}
\caption{The considered benchmark models. Here, $n$ is the number of BN variables, $m$ is the number of logical inputs (after expansion of uninterpreted functions), $|C|$ is the number of all biologically relevant colours (input valuations), and $|V \times C|$ is the size of the whole biologically relevant coloured state space. Finally, \#SCC gives the number of detected non-trivial SCCs. Note that this number varies depending on input valuation, and is thus given as a range.}
\label{tab:models}
\setlength\tabcolsep{8 pt}
\renewcommand{1.5}{1.5}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
\textbf{Model name} & $n$ & $m$ & $|C|$ & $|V \times C|$ & \#SCC \\\hline
{\small Asymmetric Cell Division~\cite{sanchez2017modeling}} & $5$ & $48$ & $\sim2^{19}$ & $\sim2^{24}$ & 1-13 \\ \hline
{\small Reduced TCR Signalisation~\cite{klamt2006methodology}} & $10$ & $46$ & $\sim2^{14}$ & $\sim2^{24}$ & 36-115 \\ \hline
{\small Budding Yeast (Orlando)~\cite{orlando2008global}} & $9$ & $54$ & $\sim2^{16}$ & $\sim2^{27}$ & 1-16 \\ \hline
{\small Budding Yeast (Irons)~\cite{irons2009logical}} & $18$ & $44$ & $\sim2^{17}$ & $\sim2^{35}$ & 2-5568 \\ \hline
{\small Tumor Cell Migration~\cite{cohen2015mathematical}} & $20$ & $44$ & $\sim2^{15}$ & $\sim2^{35}$ & 436-379308 \\ \hline
{\small T-cell Differentiation~\cite{mendoza2006method}} & $23$ & $40$ & $\sim2^{15}$ & $\sim2^{38}$ & 41728-43264 \\ \hline
{\small WG Signalling Pathway~\cite{mbodj2013logical}} & $26$ & $38$ & $\sim2^{22}$ & $\sim2^{48}$ & 0 \\ \hline
{\small Full TCR Signalisation~\cite{klamt2006methodology}} & $30$ & $48$ & $\sim2^{17}$ & $\sim2^{47}$ & 48-1087 \\ \hline
\end{tabular}
\end{table}
\begin{table}
\caption{Overview of runtime for different version of the SCC detection algorithm. The times (\texttt{hours:minutes:seconds}) refer to the total runtime of the SCC decomposition procedure for the basic lock-step, lock-step with saturation, and lock-step with saturation and parallelism, with \texttt{DNF} representing a time-out after 24-hours. }\label{tab:results}
\centering
\setlength\tabcolsep{8 pt}
\renewcommand{1.5}{1.5}
\begin{tabular} { | c | c | c | c | }
\hline
\textbf{Model Name} & \textbf{Parallel} & \textbf{Satur.} & \textbf{Lock-step} \\ \hline
{\small Asymmetric Cell Division~\cite{sanchez2017modeling}} & \texttt{00:05} & \texttt{00:10} & \texttt{00:15} \\ \hline
{\small Reduced TCR Signalisation~\cite{klamt2006methodology}} & \texttt{00:04} & \texttt{00:45} & \texttt{01:12} \\ \hline
{\small Budding Yeast (Orlando)~\cite{orlando2008global}} & \texttt{06:29} & \texttt{06:50} & \texttt{11:21} \\ \hline
{\small Budding Yeast (Irons)~\cite{irons2009logical}} & \texttt{15:14} & \texttt{2:53:16} & \texttt{3:28:44} \\ \hline
{\small Tumor Cell Migration~\cite{cohen2015mathematical}} & \texttt{40:10} & \texttt{18:34:16} & \texttt{DNF} \\ \hline
{\small T-cell Differentiation~\cite{mendoza2006method}} & \texttt{16:10:41} & \texttt{DNF} & \texttt{DNF} \\ \hline
{\small WG Signalling Pathway~\cite{mbodj2013logical}} & \texttt{1:18:38} & \texttt{1:23:37} & \texttt{1:42:12} \\ \hline
{\small Full TCR Signalisation~\cite{klamt2006methodology}} & \texttt{4:49:04} & \texttt{DNF} & \texttt{DNF} \\ \hline
\end{tabular}
\end{table}
The experiments were performed on a 32-core AMD Threadripper workstation with 64GB of RAM memory. All tested models are available in our source code repository.\footnotemark[3] \footnotetext[3]{\url{https://github.com/sybila/biodivine-lib-param-bn/tree/lmcs}} Note that the smaller models ($<2^{30}$) should be easy to process even on a less powerful machine; however, the larger models can require substantial amount of memory.
For each model, we have tested the lock-step algorithm as presented in the main part of this paper (\emph{Lock-step} in Table~\ref{tab:results}), an enhanced version with saturation enabled (\emph{Satur.} in Table~\ref{tab:results}), and a parallel implementation which also includes saturation (\emph{Parallel} in Table~\ref{tab:results}). In all algorithms, we employ the trimming optimisation.
From the results, we can see that parallelisation improves the performance of the algorithm significantly: in case of models with a large number of SCCs, we see an up-to 30x speed-up, comparing \emph{Parallel} and \emph{Satur.} in Table~\ref{tab:results}. On the other hand, when the number of SCCs is small (such as~\cite{orlando2008global}), the speed-up is understandably minimal, since the number of independent recursive calls is also small.
As expected, the total number of SCCs has a significant impact on the performance of the algorithm (e.g.~\cite{irons2009logical} and~\cite{cohen2015mathematical}) overall, since the number of calls to \textsc{Decomposition} increases. Furthermore, we see that our ``coloured saturation'' indeed provides a performance benefit. However, this improvement is mostly incremental.
After further analysis, we discovered that the whole algorithm is often limited by the performance of the trimming procedure, rather than reachability procedures though. In~particular, the use of saturation has significantly reduced the size of symbolic representation during computation of reachability, however the symbolic representation still performs rather poorly (at least for Boolean networks) during trimming. This limits the performance of the whole method, since all the considered graphs contain a large portion of trivial SCCs. Furthermore, in many cases the number of iterations needed to completely trim a set of states is substantial. This leads us to believe there is still space for improvement in terms of SCC detection in large Boolean networks, even without parameters.
Finally, we examined the benefit of processing all colours simultaneously versus a naive parameter scan approach, where each monochromatic case is handled separately. To do so, we considered various pseudo-random monochromatisations of the studied models and processed these using our algorithm. Here, we observe that for the four models with at least $20$ variables, no computation for any of the monochromatic models finished in under one second (with T-cell differentiation typically requiring more than one minute due to the relatively large number of components).
Consequently, we can extrapolate that computing the full coloured SCC decomposition using such naive parameter scan would require more than 10 ours for each model (and $10+$ days in the case of T-cell differentiation). This approach could be to some extent beneficial in a massively parallel environment (hundreds or thousands of CPUs), but the coloured approach clearly scales better in setups where resources are more limited.
\section{Conclusions}
This paper presents a fully symbolic algorithm for detecting all monochromatic strongly connected components in edge-coloured graphs. The work has been motivated by systems sciences, namely systems biology, where the need for efficient automated analysis of components in large graphs with a large sets of coloured edges is emerging. The algorithm combines several ideas inspired by existing state-of-the-art algorithms for SCC decomposition in a~non-trivial way. We believe this is the first fully symbolic algorithm aiming to solve the problem efficiently.
The experimental evaluation has shown that the algorithm can handle large, real-world systems that would be otherwise too large to fit into the memory of a conventional workstation ($>2^{32}$), and that the performance of the algorithm can be further improved using saturation and parallelisation. Finally, the algorithm has a strong potential to be significantly faster
than iterating a standard algorithm for SCC decomposition executed on all monochromatic sub-graphs one-by-one.
\end{document}
|
\begin{document}
\begin{abstract}
The \emph{Wiener index} of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its \emph{Wiener index} as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets.
For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{Background: the Wiener index of the noncrossing partition lattice}
Let $\mathrm{NC}(n)$ be the lattice of noncrossing partitions
of $n$.
In the paper~\cite{goulden2020asymptotics},
motivated by problems about meanders and meandric systems, Goulden, Nica, and Puder raised the following question: what is the average distance between two (uniform)
random partitions in $\mathrm{NC}(n)$?
The question was answered for large $n$ by Th\'evenin
and the second author in~\cite{feray2022components},
where it was proved that this average distance behaves as $\kappa\, n$ for some constant $\kappa$.
It is natural to ask similar questions for other families of posets, looking either for an exact nice formula or for an asymptotic answer.
When the number of elements is known (which is the case for $\mathrm{NC}(n)$),
one can equivalently ask for the sum of distances between all pairs of elements.
In general, let $G=(V,E)$ be a finite connected graph, and for $p,q\in V$, write $d(p,q)$ for the distance in $G$ from $p$ to $q$. The defn{Wiener index} of $G$ is defined to be
\begin{equation}\label{eq:dist}
d(G):=\sum_{(p,q) \in V \times V} d(p,q).
\end{equation}
This definition has its origin as the \emph{Wiener index} predicting the boiling point of certain organic compounds~\cite{wiener1947structural,wikiwiener,rouvray2002rich}, and it has also been called the \emph{distance} of the graph $G$~\cite{entringer1976distance}. When $P$ is a poset, we define $d(P):=d(G(P))$ for the Wiener index of the Hasse diagram $G(P)$ of $P$ (that is, the vertices of $G(P)$ are the elements of $P$, and there is an edge in $G(P)$ between $p$ and $q$ when there is a cover relation in $P$ between $p$ and $q$).
In the case of the noncrossing partition lattice, the results in~\cite{feray2022components} imply that
\[d(\mathrm{NC}(n)) \sim |\mathrm{NC}(n)|^2 \kappa \, n \sim \frac{\kappa\, 8^n}{\pi n^2},\]
but no exact enumeration appears possible.
(There does not even seem
to be a simple formula for $\kappa$;
see the discussion in~\cite{feray2022components}
and the related open problem in~\cite{ober2022}.)
\subsection{Wiener indices of other lattices}
Computer experiments suggest that there are few nontrivial families of graphs $\{G_n\}_{n\geq 1}$ of combinatorial objects for which it is possible to find \emph{exact} formulas for the Wiener index.
First, there are families with elementary exact solutions, for which $|G_n|$ is a relatively small polynomial in $n$ (for example: path graphs, grid graphs, etc.),
or in which each graph $G_n$ has a transitive underlying symmetry group (for example: the weak order on a finite Coxeter group, a boolean lattice or hypercube, etc.). A short list of examples is given in~\cite{weissteinwiener}.
There is, however, one class of posets in algebraic combinatorics that demonstrates consistently exceptional enumerative behavior: the minuscule lattices~\cite{proctor1984bruhat}. For example, both the number of elements and the number of maximal chains in a minuscule lattice have simple (uniformly stated and proven) product formulas, and the minuscule lattices are well understood from the perspective of dynamical algebraic combinatorics (promotion, rowmotion, etc.)~\cite{striker2012promotion,hopkins2020order}.
It turns out that the Wiener indices of minuscule lattices also admit simple exact formulas,
and one of the goal of the present paper is to establish such formulas.
For completeness, we first recall the definition of minuscule lattices
(note that we will only use here their classification, and not the algebraic definition). Let $\mathfrak{g}$ be a complex simple Lie group with Weyl group $W$. Fix a set $Phi^+$ of positive roots of $\mathfrak g$, and let $\Lambda^+$ be the set of dominant weights. The finite-dimensional irreducible complex representations $V_\lambda$ of $\mathfrak{g}$ are indexed by dominant weights $\lambda \in \Lambda^+$; $\lambda$ is called defn{minuscule} if the $W$-orbit of $\lambda$ is the set of \emph{all} weights in $V_\lambda$. The minuscule weights are exactly those fundamental weights whose corresponding simple roots appear exactly once in the simple root expansion of the highest root. For more information, we refer the reader to~\cite{stanley1980weyl,proctor1984bruhat}.
For $\lambda$ minuscule, define a poset on the weights in $V_\lambda$ by introducing a cover relation $\mu \lessdot \nu$ whenever $\mu + \alpha = \nu$ for some simple root $\alpha \in Phi^+$. This poset on $V_\lambda$ is a distributive lattice, which we call a defn{minuscule lattice}~\cite{proctor1984bruhat}.
There are three infinite families of minuscule lattices---the order ideals in:
\begin{itemize}
\item a rectangle (type $A$; in type $B$, minuscule lattices are chains and thus particular cases of rectangles),
\item a shifted staircase (types $C$ and $D$), and
\item a ``double tailed diamond'' (type $D$)
\end{itemize}
---as well as two exceptional minuscule lattices (of types $E_6$ and $E_7$).
In this paper, we show that the Wiener indices of the infinite families of minuscule lattices admit simple product formulas, although we regrettably have been unable to find a unifying expression for these formulas. We also provide information about the asymptotic distribution
of the distance between random elements in these posets (in the rectangle and shifted staircase cases),
and we give a method for computing the higher moments exactly.
\subsection{Rectangles}
Write $J(P)$ for the lattice of order ideals in a finite poset $P$, ordered by inclusion. By Birhoff's representation theorem, any distributive lattice is of this form. In~\Cref{sec:rect_bijections,sec:proof_rect_thm,sec:proof_rect_cors}, we consider the Wiener index of $P_{m,k}=J([m]\times[k])$, the lattice of order ideals in an $m \times k$ rectangle.
In this case (and, more generally, for
any distributive lattice), it is easy to see that $d(p,q)=|p \vartriangle q|$, where $p \vartriangle q = (p \backslash q) \cup (q \backslash p)$ is the symmetric difference of the order ideals $p$ and $q$.
It will be convenient to draw the elements of $P_{m,k}$ as lattice paths from $(0,0)$ to $(m+k,m-k)$ using steps of the form $U=(1,1)$ and $D=(1,-1)$.
Writing $p_i,q_i$ for the heights of $p$ and $q$ after the ends of their $i$th steps, the number of squares in column $i$ between the lattice paths $p$ and $q$ is given as $\left|\frac{q_i-p_i}{2}\right|$, so that
\begin{equation}\label{eq:distance}
d(p,q) = \left|\frac{q_1-p_1}{2}\right|+\left|\frac{q_2-p_2}{2}\right|+\cdots+\left|\frac{q_{m+k}-p_{m+k}}{2}\right|.
\end{equation}
\begin{example}
The graph $G(P_{2,2})$ is drawn in \Cref{fig:2x2}. Its Wiener index is
\begin{align*}
56 = \frac{4}{18}\binom{10}{5}=&(0{+}1{+}2{+}2{+}3{+}4){+}(1{+}0{+}1{+}1{+}2{+}3){+}(2{+}1{+}0{+}2{+}1{+}2){+}\\&{+}(2{+}1{+}2{+}0{+}1{+}2){+}(3{+}2{+}1{+}1{+}0{+}1){+}(4{+}3{+}2{+}2{+}1{+}0).
\end{align*}
\end{example}
\begin{figure}
\caption{The Hasse diagram $G(P_{2,2}
\label{fig:2x2}
\end{figure}
We have three results that completely describe the Wiener index of the lattice of order ideals of a rectangle.
\begin{theorem}\label{thm:gf}
The generating function for the Wiener index of all posets $P_{m,k}=J([m]\times[k])$ is given by
\begin{equation}\label{eq:total_geo}
\sum_{m=0}^\infty \sum_{k=0}^\infty d(P_{m,k})x^m y^k = \frac{2xy}{(x^2 - 2xy + y^2 - 2x - 2y + 1)^2}.
\end{equation}
\end{theorem}
This theorem is obtained via classical first return decomposition for lattice paths.
The fact that this generating series is rational comes as a surprise,
since several intermediate computation steps involve algebraic non-rational functions.
Extracting the coefficient of $x^m y^k$ from~\Cref{eq:total_geo}, we obtain a formula for $d(P_{m,k})$.
\begin{corollary}\label{cor:coeff}
The Wiener index of $P_{m,k}$ is
\[d(P_{m,k})=\frac{mk}{4m+4k+2}\binom{2m+2k+2}{2k+1}.\]
\end{corollary}
For fixed $\alpha$, we obtain the asymptotic expected value of $d(p,q)$ in a $(\alpha n) \times n$ rectangle as $n \to \infty$.
To keep notation simple, we assume throughout the paper that $\alpha n$ is an integer
(otherwise, it suffices to replace $\alpha n$
by its integer value).
\begin{corollary}\label{cor:asymptotic}
We have \[\frac{d(P_{\alpha n,n})}{|P_{\alpha n,n}|^2} \sim \frac{\sqrt{\pi \alpha (1+\alpha)}}{4} n^{3/2} \text{ as } n \to \infty.\]
\end{corollary}
In~\Cref{sec:asymptotics}, we also describe in this regime the asymptotic distribution of the distance $D_{\alpha,n}$ between
two independent uniform random elements of $P_{\alpha n,n}$.
\begin{proposition}
\label{prop:cv_law_distance}
The random variable $n^{-3/2} D_{\alpha, n}$ converges in distribution
and in moments to $\sqrt{2 \alpha (1+\alpha)} \cdot \int_0^1 |B_0(t)| dt$,
where $B_0(t)$ is a Brownian bridge on $[0,1]$.
\end{proposition}
Informally, a Brownian bridge on $[0,1]$ is a Brownian motion conditioned to have value $0$ at time $1$.
Alternatively,
if $B$ is a Brownian motion, then $B_0(t):=B(t)-tB(1)$ is a Brownian bridge.
Brownian bridges have been extensively studied in the probabilistic litterature.
In particular, much is known on the random variable $\int_0^1 |B_0(t)| dt$;
see \cite[Section 20]{janson2007area} for a survey of results including numerous references.
In particular, a table of the first few moments can be found in \cite[Table 2]{janson2007area}.
We copy here the first three:
\[ \mathbb E\left[\int_0^1 |B_0(t)| dt\right] = \frac14 \sqrt{\frac{\pi}2},
\quad \mathbb E\left[\left(\int_0^1 |B_0(t)| dt\right)^2\right] = \frac7{60},
\quad \mathbb E\left[\left(\int_0^1 |B_0(t)| dt\right)^3\right] = \frac{21}{512} \sqrt{\frac{\pi}2}.\]
Together with \cref{prop:cv_law_distance}, this implies
\begin{align*}
\frac1{|P_{\alpha n,n}|^2} \sum_{p,q \in P_{\alpha n,n}} d(p,q) =\mathbb E[D_{\alpha,n}] &\sim \frac{\sqrt{\pi \alpha (1+\alpha)}}{4} n^{3/2},\\
\frac1{|P_{\alpha n,n}|^2} \sum_{p,q \in P_{\alpha n,n}} d(p,q)^2 =\mathbb E[D_{\alpha,n}^2]&\sim \frac{7}{30} \alpha (1+\alpha) n^{3},\\
\frac1{|P_{\alpha n,n}|^2} \sum_{p,q \in P_{\alpha n,n}} d(p,q)^3 =\mathbb E[D_{\alpha,n}^3]&\sim \frac{21}{256} \sqrt{\pi} \alpha^{3/2} (1+\alpha)^{3/2} n^{9/2}.
\end{align*}
Note that the first estimate is nothing but \cref{cor:asymptotic}. This gives a second derivation
of this asymptotic result, which does not go through the exact expression.
Exact expressions for such higher moments can also be obtained through combinatorial means,
see \cref{sec:higher_moments} for a derivation of the second moment.
\subsection{Shifted staircases}
In~\Cref{sec:bij_stair,sec:proof_stair}, we consider the Wiener index of $Q_n$, the distributive lattice of order ideals in the $n$th defn{shifted staircase} poset. Explicitly, $Q_n$ is the set of order ideals in the poset $\{(i,j):1\leq i\leq j\leq n\}$ under componentwise ordering. The following results determine $d(Q_n)$ exactly.
The elements of $Q_n$ can be represented
as lattice paths starting at $(0,0)$,
ending somewhere on the line $x=n$, and using steps
of the form $U=(1,1)$ and $D=(1,-1)$.
In particular, $|Q_n|=2^n$.
Similarly as for rectangles,
if $p$ and $q$ are elements in $Q_n$,
writing $p_i,q_i$ for the heights of $p$ and $q$ after the ends of their $i$th steps, we have
\begin{equation}
\label{eq:distance_shifted}
d(p,q) = \frac12 \sum_{i=1}^n \left|\frac{q_i-p_i}{2}\right|.
\end{equation}
\begin{example}
The graph $G(Q_{3})$ is plotted in \Cref{fig:ss3}. Its Wiener index is
\begin{align*}
140 = \frac{6\cdot 7}{3}\binom{5}{3}=&24+18+14+14+14+14+18+24.
\end{align*}
\end{example}
\begin{figure}
\caption{The Hasse diagram $G(Q_{3}
\label{fig:ss3}
\end{figure}
\begin{theorem}\label{thm:gf_SS}
The generating function for the Wiener index of all lattices $Q_n$ is given by
\begin{equation}\label{eq:total_geo_SS}
\sum_{n=0}^\infty d(Q_n)x^n = \frac{8x\left(1+\sqrt{1-4x}-x(3+\sqrt{1-4x})\right)}{(1-4x)(1-4x+\sqrt{1-4x})^3}.
\end{equation}
\end{theorem}
\begin{corollary}
\label{cor:Wiener_JSn}
The Wiener index of $Q_n$ is \[d(Q_n)=\frac{2n(2n+1)}{3}\binom{2n-1}{n}.\]
Consequently, as $n$ tends to $+\infty$, we have
$d(Q_n) \sim \frac{2}{3\sqrt \pi} 4^n n^{3/2}$.
\end{corollary}
In~\Cref{sec:asymptotics}, we turn to the asymptotic distribution of the distance $E_n$ between two random order ideals of
$Q_n$.
\begin{proposition}
\label{prop:cv_law_distance_shifted}
The random variable $n^{-3/2} E_n$ converges in distribution
and in moments to $\frac{1}{\sqrt{2}} \cdot \int_0^1 |B(t)| dt$,
where $B(t)$ is a Brownian motion on $[0,1]$.
\end{proposition}
Again, much is known about the random variable
$\int_0^1 |B(t)| dt$, and a comprehensive literature review appears in~\cite[Section 21]{janson2007area}. In particular, the first few moments are given in~\cite[Table 3]{janson2007area}:
\[ \mathbb E\left[\int_0^1 |B(t)| dt\right] = \frac23 \sqrt{\frac2{\pi}},
\quad \mathbb E\left[\left(\int_0^1 |B(t)| dt\right)^2\right] = \frac3{8},
\quad \mathbb E\left[\left(\int_0^1 |B(t)| dt\right)^3\right] = \frac{263}{630} \sqrt{\frac2{\pi}}.\]
Together with \cref{prop:cv_law_distance_shifted}, this implies
\begin{align*}
\frac1{|Q_n|^2} \sum_{p,q \in Q_n} d(p,q) =\mathbb E[E_n]&\sim \frac{2}{3 \sqrt \pi} n^{3/2},\\
\frac1{|Q_n|^2} \sum_{p,q \in Q_n} d(p,q)^2 =\mathbb E[E_n^2] &\sim \frac{3}{16} n^{3},\\
\frac1{|Q_n|^2} \sum_{p,q \in Q_n} d(p,q)^3 =\mathbb E[E_n^3]&\sim \frac{263}{1260 \sqrt{\pi}}
n^{9/2}.
\end{align*}
Again, recalling that $|Q_n|=2^n$,
this allows us
to recover the asymptotic
behaviour of $d(Q_n)$ given in \cref{cor:Wiener_JSn}
without going through its exact expression.
\subsection{The remaining minuscule lattices}
The Wiener indices of the remaining minuscule lattices are simple calculations.
Let $R_n$ be the $n$th ``double tailed diamond''---that is, the distributive lattice of order ideals in the minuscule poset of type $D_n$ corresponding to the first fundamental weight.
\begin{theorem}\label{thm:other_mins}
We have
\[d(R_n)=\frac{2}{3} (n+3) \left(4 n^2+9 n+8\right).\] The minuscule lattices of types $E_6$ and $E_7$ have Wiener indices $3584$ and $24048$, respectively.
\end{theorem}
The proof in the case of $R_n$ is elementary and left to the reader.
The cases of $E_6$ and $E_7$ are treated by computer.
The expressions for Wiener indices given in \Cref{cor:coeff}, \Cref{cor:Wiener_JSn}, and \Cref{thm:other_mins} suggest that there may be a uniform formula for $d(P)$ for $P$ a minuscule lattice---but we regrettably have been unable to find such an expression.
\section{Lattice path bijections}\label{sec:rect_bijections}
We continue to use the steps $U=(1,1)$ and $D=(1,-1)$, but we will also make use of two versions (or colors) of the step $(1,0)$, denoted $O_1$ and $O_2$. For $(p,q) \in P_{k,n-k} \times P_{k,n-k}$, define a lattice path $A(p,q)$ using the following dictionary between the $i$th pair of steps in $(p, q)$ and the $i$th step in $A(p,q)$:
\begin{equation}\label{eq:bija}
\begin{array}{|c|c|c|c|} \hline
(p,q) & A(p,q) & \left|\frac{q_{i+1}-p_{i+1}}{2}\right|-\left|\frac{q_i-p_i}{2}\right| & r_{i+1}-r_i\\ \hline
(D,U) & U & +1 & +1\\
(U,D) & D & -1 & -1\\
(U,U) & O_1 & 0 & 0\\
(D,D) & O_2 & 0 & 0 \\\hline
\end{array}.
\end{equation}
Two examples of this bijection are illustrated in~\Cref{fig:bija}. Given a lattice path $r$ of length $n$ with steps from the set $\{U,D,O_1,O_2\}$, write $r_i$ for the height (i.e.\, the $y$-coordinate) of $r$ at the end of its $i$th step. The unsigned area between $r$ and the $x$-axis is $d(r)=|r_0|+|r_1|+|r_2|+\cdots+|r_{n-1}|+\frac{1}{2}|r_n|$. Let us also write $\overlined(r)=|r_0|+|r_1|+|r_2|+\cdots+|r_{n}|=d(r)+\frac{1}{2}|r_n|$. Then, comparing with~\Cref{eq:distance}, it is clear that $d(p,q) = d(A(p,q))$: certainly $\frac{q_0-p_0}{2}=0=r_0$, so suppose that $\frac{q_i-p_i}{2} = r_i$; then the difference in height at the $(i+1)$st step in $A(p,q)$ matches the difference in height at the $(i+1)$st steps of $p$ and $q$, as shown in the rightmost two columns of~\eqref{eq:bija}.
\begin{figure}
\caption{Illustration of the bijection $A$ from the table in~\eqref{eq:bija}
\label{fig:bija}
\end{figure}
Write the set of all ordered pairs of paths in $P_{k.n-k}$ as \[P^\times_{k,n-k}:=P_{k,n-k} \times P_{k,n-k},\] and denote the restriction of $P^\times_{k,n-k}$ to those pairs $(p,q)$ with $p\leq q$ as \[P^\leq_{k,n-k} := \{(p,q) \in P^\times_{k,n-k} : p \leq q\}.\] We begin by converting the total area between pairs of paths in $P^\times_{k,n-k}$ and $P^\leq_{k,n-k}$ into the (unsigned) area under a single Motzkin path.
\begin{definition}
Write $\mathcal{M}b$ for the set of defn{bilateral Motzkin paths}---that is, lattice paths from $(0,0)$ to $(n,0)$ for some $n \in \mathbb{Z}_{\geq 0}$ that use step set $\{U,D,O_1,O_2\}$. We write $\mathcal{M}b_{n,k}$ for the set of bilateral Motzkin paths that end at $(n,0)$ and use exactly $k$ steps of the form $U$ or $O_1$. A defn{bicolored Motzkin path} is a bilateral Motzkin path that stays weakly above the $x$-axis. Write $\mathcal{M}$ (resp. $\mathcal{M}_{n,k}$) for the set of bicolored Motzkin paths in $\mathcal{M}b$ (resp. $\mathcal{M}b_{n,k}$).
\end{definition}
\begin{proposition}\label{prop:bija}
The map $A\colonP^\times_{n,n-k}\to\mathcal{M}b_{n,k}$ is a bijection satisfying $d(p,q) = d(A(p,q))$, and it restricts to a bijection from $P^\leq_{n,n-k}$ to $\mathcal{M}_{n,k}$.
\end{proposition}
\begin{proof}
The dictionary in~\eqref{eq:bija} shows that $A$ is a bijection. When $p \leq q$, $A(p,q)$ never goes below the $x$-axis, so $A$ maps $P^\leq_{n,n-k}$ onto $\mathcal{M}_{n,k}$. The claim about $d$ was proven immediately after~\eqref{eq:bija}.
\end{proof}
\section{Proof of~\Cref{thm:gf}}\label{sec:proof_rect_thm}
In this section, we use recurrences on Motzkin paths and their associated generating functions to deduce~\Cref{thm:gf}.
\subsection{Bicolored Motzkin paths}
Define the generating function for bicolored Motzkin paths to be
\[\mathcal{M}(x,u) = \sum_{n \geq 0 } \sum_{k \geq 0} |\mathcal{M}_{n,k}| x^{n} u^k,\]
so that the coefficient of $x^n u^k$ counts bicolored Motzkin paths of total length $n$ with exactly $k$ steps of the form $U$ and $O_1$.
\begin{proposition} \label{prop:m_decomp}
The generating function $\mathcal{M}(x,u)$ satisfies the functional equation
\[\mathcal{M}(x,u)=1+x(1+u)\mathcal{M}(x,u)+ux^2 \mathcal{M}(x,u)^2,\]
with the explicit solution
\[\mathcal{M}(x,u) = \frac{1-(u + 1)x - \sqrt{(u^2 - 2u + 1)x^2 - 2(u + 1)x + 1}}{2ux^2}.\]
\end{proposition}
\begin{proof}
The functional equation comes from decomposing a lattice path $r\in \mathcal{M}$ by first return to the $x$-axis: $r$ is empty; or $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by solving this quadratic equation in $\mathcal{M}(x,u)$.
\end{proof}
\begin{figure}
\caption{The decompositions of lattice paths in $\mathcal{M}
\label{fig:m_decomp}
\end{figure}
Define the generating function for the total area of paths in $\mathcal{M}$ by \[\mathcal{M}m(x,u) := \sum_{n \geq 0 } \sum_{k \geq 0} x^n u^k \sum_{p \in \mathcal{M}_{n,k}} d(p).\]
\begin{proposition} \label{prop:mm_decomp}
The generating function $\mathcal{M}m(x,u)$ satisfies the functional equation
\[\mathcal{M}m(x,u)=x(1+u)\mathcal{M}m(x,u)+ux^2(2\mathcal{M}(x,u)\mathcal{M}m(x,u)+\mathcal{M}(x,u) \frac{d}{dx}(x\mathcal{M}(x,u)),\]
with the explicit solution
\[\mathcal{M}m(x,u) = \frac{\left(u^2+1\right) x^2-(u+1) x+((u +1)x-1) \left(\sqrt{(u-1)^2 x^2-2 (u+1) x+1}-1\right)}{2 u x^2 \left((u-1)^2 x^2-2 (u+1) x+1\right)}.\]
\end{proposition}
\begin{proof}
As in~\Cref{prop:m_decomp}, the functional equation comes from decomposing a lattice path $r$ in $\mathcal{M}m$ by first return to the $x$-axis: either $r$ is empty (in which case it contributes zero area); or $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by using the explicit form of $\mathcal{M}(x,u)$ from~\Cref{prop:m_decomp} and solving the linear equation in $\mathcal{M}m(x,u)$.
\end{proof}
\subsection{Bilateral Motzkin paths}
Define the generating function for bilateral Motzkin paths to be
\[\mathcal{M}b(x,u) = \sum_{n \geq 0 } \sum_{k \geq 0} |\mathcal{M}b_{n,k}| x^{n} u^k,\]
so that the coefficient of $x^n u^k$ counts bilateral Motzkin paths of total length $n$ with exactly $k$ steps of the form $U$ or $O_1$.
\begin{proposition}\label{prop:mb_decomp}
The generating function $\mathcal{M}b(x,u)$ satisfies the functional equation
\[\mathcal{M}b(x,u)=1+x(1+u)\mathcal{M}b(x,u)+2u x^2 \mathcal{M}(x,u) \mathcal{M}b(x,u),\]
with the explicit solution
\[\mathcal{M}b(x,u) = (u^2x^2 - 2ux^2 - 2ux + x^2 - 2x + 1)^{-1/2}.\]
\end{proposition}
\begin{proof}
As in~\Cref{prop:m_decomp}, the functional equation comes from decomposing a lattice path $r$ in $\mathcal{M}b$ by first return to the $x$-axis: $r$ is empty; or $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ or $D$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by using the explicit form of $\mathcal{M}(x,u)$ from~\Cref{prop:m_decomp} and solving the linear equation in $\mathcal{M}b(x,u)$.
\end{proof}
\begin{remark}
Because $\mathcal{M}b_{n,k}$ encodes $P^\times_{k,n-k}$ by Proposition~\ref{prop:bija}, we have
\begin{equation}
\label{eq:expansion_bilateral}
\mathcal{M}b(x,u)=\sum_{n,k\geq 0}\binom{n}{k}^2x^nu^k.
\end{equation}
\end{remark}
Define the generating function for the total area of paths in $\mathcal{M}b$ to be
\[\mathcal{M}bm(x,u) := \sum_{n \geq 0 } \sum_{k \geq 0} x^n u^k \sum_{p \in \mathcal{M}b_{n,k}} d(p).\]
\begin{proposition}\label{prop:mbm_decomp}
The generating function $\mathcal{M}bm(x,u)$ satisfies the functional equation
\[\mathcal{M}bm(x,u)=x(1+u)\mathcal{M}bm(x,u)+2ux^2(\mathcal{M}(x,u)\mathcal{M}bm(x,u)+\mathcal{M}m(x,u)\mathcal{M}b(x,u)+\mathcal{M}b(x,u)\frac{d}{dx}(x\mathcal{M}(x,u)),\]
with the explicit solution
\begin{equation}\label{eq:total_geo_sub}
\mathcal{M}bm(x,u) = \frac{2 u x^2}{\left((u-1)^2 x^2-2 (u+1) x+1\right)^2}.
\end{equation}
\end{proposition}
\begin{proof}
As in~\Cref{prop:mm_decomp}, the functional equation comes from decomposing a lattice path $r$ in $\mathcal{M}bm$ by first return to the $x$-axis: if $r$ is empty, then it counts for zero area; otherwise, $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ or $D$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by using the explicit forms of $\mathcal{M}(x,u)$, $\mathcal{M}b(x,u)$, and $\mathcal{M}m(x,u)$ from~\Cref{prop:m_decomp,prop:mb_decomp,prop:mm_decomp} and solving the linear equation in $\mathcal{M}bm(x,u)$.
\end{proof}
Substituting $u=y/x$ into~\Cref{eq:total_geo_sub}, we obtain~\Cref{eq:total_geo} and thus complete the proof of~\Cref{thm:gf} for the generating function for the Wiener indices of the lattices $P_{m,k}=J([m]\times[k])$.
\section{Proofs of~\Cref{cor:coeff,cor:asymptotic}}\label{sec:proof_rect_cors}
We note that
\[x^2 - 2xy + y^2 - 2x - 2y + 1 = (q - t - 1) (q - t + 1) (q + t - 1) (q + t + 1),\]
where $q^2=x$ and $t^2=y$. Then, by performing a partial fraction decomposition with a computer algebra system, we get
\begin{align} \label{eq:qt}
\nonumber \frac{2xy}{(x^2 - 2xy + y^2 - 2x - 2y + 1)^2} &=
\frac{1}{32} \left(
\frac{1}{(-1-t+q)^2}
+\frac{1}{(1-t+q)^2}
+\frac{1}{(-1+t+q)^2}
+\frac{1}{(1+t+q)^2}\right) \\
\nonumber &+\frac{1}{32t(t+1)}\left(
\frac{1+t+t^2}{(1+t-q)}
+\frac{1+t+t^2}{(1+t+q)}
\right)\\
&+\frac{1}{32t(t-1)}\left(
\frac{1-t+t^2}{(1-t+q)}
+\frac{-1+t-t^2}{(-1+t+q)}
\right).
\end{align}
Taking the $n$th coefficient in $q$ from the right-hand side of~\Cref{eq:qt} gives
\begin{align}\label{eq:qt2}
\nonumber \frac{n+1}{32}&\Big((1+t)^{-2-n}+(-1+t)^{-2-n}+(1-t)^{-2-n}+(-1-t)^{-2-n}\Big)\\
\nonumber +
\frac{1}{32t}& \Big(\left(t^2+t+1\right) (t+1)^{-n-2}+\left(t^2+t+1\right) (-t-1)^{-n-2}\Big)\\
+
\frac{1}{32t}& \Big(\left(-t^2+t-1\right) (1-t)^{-n-2}+\left(-t^2+t-1\right) (t-1)^{-n-2}\Big).
\end{align}
Taking the $j$th coefficient in $t$ from~\eqref{eq:qt2} and simplifying gives
\begin{align}\label{eq:qt3}
\frac{\left((-1)^n+1\right) \left((-1)^j+1\right)}{32} \left(
\frac{(n+j+1)!}{n!j!}-
\frac{(n+j)! ((n+1)^2+j^2+j (n+2))}{(n+1)! (j+1)!}\right).
\end{align}
Restricting~\eqref{eq:qt3} to $n=2m$ and $j=2k$ even, we obtain the desired expression in~\Cref{cor:coeff} for the coefficient of $q^{n} t^{m}$, giving the coefficient for $x^{m} y^{k}$:
\begin{align*}
&\frac{1}{8} \left(\frac{(n+j+1)!}{n!j!}-
\frac{ (n+j)! ((n+1)^2+j^2+j(n+2))}{(n+1)! (j+1)!}\right)\\
&=\frac{(n+j)!}{8 n!j!}\left(n+j+1-\frac{j^2+(n+1)^2+j (n+2)}{(n+1)(j+1)}\right)\\
&=\frac{nj}{8(n+j+1)}\binom{n+j+2}{j+1}\\
&=\frac{mk}{4m+4k+2}\binom{2m+2k+2}{2k+1}.
\end{align*}
Given the exact expression for $d(P_{m,k})$,~\Cref{cor:asymptotic} is routine using Stirling's asymptotic equivalent for factorials.
\section{Shifted staircases}\label{sec:bij_stair}
As for rectangles, we can view elements of $Q_n$ as lattice paths of length $n$ that start at $(0,0)$ and use steps of the form $U=(1,1)$ and $D=(1,-1)$. The main difference is that paths representing different order ideals can have different endpoints. Let \[Q_n^\times:=Q_n \times Q_n \quad\text{and}\quad Q_n^{\leq}:=\{(p,q)\in Q_n^\times:p\leq q\}.\]
\begin{definition}
Define a defn{bilateral Motzkin prefix} to be a lattice path that starts at $(0,0)$ and uses the steps of the form $U,D,O_1,O_2$. Let $\mathcal V$ denote the set of bilateral Motzkin prefixes, and let $\mathcal V_n$ be the set of bilateral Motzkin prefixes that use exactly $n$ steps. A defn{bicolored Motzkin prefix} is a bilateral Motzkin prefix that stays weakly above the $x$-axis. Write $\mathcal N$ for the set of bicolored Motzkin prefixes, and let $\mathcal N_n=\mathcal N\cap\mathcal V_n$.
\end{definition}
Throughout this section, we write $d(p,q)$ for the length of a geodesic between bilateral Motzkin prefixes $p$ and $q$ in the Hasse diagram of $Q_n$.
By applying the same rules as in the table in \eqref{eq:bija}, we can transform a pair $(p,q)\in Q_n\times Q_n$ into a path $A(p,q)$ that uses steps $U,D,O_1,O_2$. The path $A(p,q)$ is similar to a bilateral Motzkin path, except it does not necessarily end on the $x$-axis.
\begin{proposition}\label{prop:bija_SS}
The map $A\colonQ_n^\times\to\mathcal V_n$ is a bijection satisfying $d(p,q) = \overlined(A(p,q))$, and it restricts to a bijection from $Q_n^\leq$ to $\mathcal N_n$.
\end{proposition}
\begin{proof}
The proof is essentially the same as that of \Cref{prop:bija}.
\end{proof}
\section{Proof of \Cref{thm:gf_SS} and \Cref{cor:Wiener_JSn}}\label{sec:proof_stair}
\subsection{Bicolored Motzkin prefixes}
Let \[\mathcal N(x):=\sum_{n\geq 0}|\mathcal N_n|x^n\] be the generating function for bicolored Motzkin prefixes.
\begin{proposition}\label{prop:N}
The generating function $\mathcal N(x)$ satisfies the functional equation \[\mathcal N(x)=1+3x\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x).\] Thus, \[\mathcal N(x)=\frac{2}{1-4x+\sqrt{1-4x}}.\]
\end{proposition}
\begin{proof}
The expression $1+x\mathcal N(x)$ counts (possibly empty) bicolored Motzkin prefixes that only touch the $x$-axis at $(0,0)$, while the expression $2x\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x)$ counts bicolored Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$. This is illustrated on the first line of~\Cref{fig:n_decomp}. It is routine to derive the explicit solution from the functional equation and \Cref{prop:m_decomp}.
\end{proof}
\begin{figure}
\caption{The decompositions of lattice paths in $\mathcal N$, $\boldsymbol{\mathcal{N}
\label{fig:n_decomp}
\end{figure}
Define the generating function \[\boldsymbol{\mathcal{N}}(x):=\sum_{n\geq 0}x^n\sum_{p\in\mathcal N_n}\overlined(p).\]
\begin{proposition}\label{prop:Nn}
The generating function $\boldsymbol{\mathcal{N}}(x)$ satisfies the functional equation \begin{align*}
\boldsymbol{\mathcal{N}}(x)&=2x\boldsymbol{\mathcal{N}}(x)+x^2\mathcal{M}m(x,1)\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x)+x^2\mathcal N(x)\frac{\partial}{\partial x}(x\mathcal M(x,1)) \\
&+x\boldsymbol{\mathcal{N}}(x)+x\frac{\partial}{\partial x}(x\mathcal N(x)).
\end{align*}
Thus, \[\boldsymbol{\mathcal{N}}(x)=\frac{4x\left(1+\sqrt{1-4x}-x(1-\sqrt{1-4x})\right)}{\sqrt{1-4x}(1-4x+\sqrt{1-4x})^3}.\]
\end{proposition}
\begin{proof}
Following the same ideas used in the proof of \Cref{prop:mm_decomp}, we find that \[2x\boldsymbol{\mathcal{N}}(x)+x^2\mathcal{M}m(x,1)\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x)+x^2\mathcal N(x)\frac{\partial}{\partial x}(x\mathcal M(x,1))\] counts bicolored Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$, with each path $p$ weighted by $\overline d(p)$. The generating function for bicolored Motzkin prefixes that only touch the $x$-axis at $(0,0)$ (with each path $p$ weighted by $\overline d(p)$) is \[x\boldsymbol{\mathcal{N}}(x)+x\frac{\partial}{\partial x}(x\mathcal N(x)).\] This is illustrated on the second line of~\Cref{fig:n_decomp}.
This yields the functional equation, from which the explicit solution is straightforward to obtain via \Cref{prop:m_decomp,prop:mm_decomp,prop:N}.
\end{proof}
Let \[\mathcal V(x):=\sum_{n\geq 0}|\mathcal V_n|x^n.\]
\begin{proposition}\label{prop:V}
The generating function $\mathcal V(x)$ satisfies the functional equation \[\mathcal V(x)=1+2x\mathcal N(x)+2x\mathcal V(x)+2x^2\mathcal M(x,1)\mathcal V(x).\] Thus, \[\mathcal V(x)=\frac{1+\sqrt{1-4x}}{\sqrt{1-4x}(1-4x+\sqrt{1-4x})}.\]
\end{proposition}
\begin{proof}
The expression $1+2x\mathcal N(x)$ counts (possibly empty) bilateral Motzkin prefixes that only touch the $x$-axis at $(0,0)$, while the expression $2x\mathcal V(x)+2x^2\mathcal M(x,1)\mathcal V(x)$ counts bilateral Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$. This is illustrated on the third line of~\Cref{fig:n_decomp}. The explicit solution can be derived from the functional equation using \Cref{prop:m_decomp,prop:N}.
\end{proof}
Finally, consider the generating function \[\boldsymbol{\mathcal{V}}(x):=\sum_{n\geq 0}x^n\sum_{p\in\mathcal V_n}\overlined(p).\]
\begin{proposition}\label{prop:Vv}
The generating function $\boldsymbol{\mathcal{V}}(x)$ satisfies the functional equation \begin{align*}
\boldsymbol{\mathcal{V}}(x)&=2x\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal{M}m(x,1)\mathcal V(x)+2x^2\mathcal M(x,1)\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal V(x)\frac{\partial}{\partial x}(x\mathcal M(x,1)) \\
&+2x\boldsymbol{\mathcal{N}}(x)+2x\frac{\partial}{\partial x}(x\mathcal N(x)).
\end{align*}
Thus, \begin{equation}\label{eq:Vv}\boldsymbol{\mathcal{V}}(x)=\frac{8x\left(1+\sqrt{1-4x}-x(3+\sqrt{1-4x})\right)}{(1-4x)(1-4x+\sqrt{1-4x})^3}.\end{equation}
\end{proposition}
\begin{proof}
Following the same ideas used in the proof of \Cref{prop:Nn}, we find that \[2x\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal{M}m(x,1)\mathcal V(x)+2x^2\mathcal M(x,1)\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal V(x)\frac{\partial}{\partial x}(x\mathcal M(x,1))\] counts bilateral Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$, with each path $p$ weighted by $\overline d(p)$. Furthermore, the generating function for bilateral Motzkin prefixes that only touch the $x$-axis at $(0,0)$ (with each path $p$ weighted by $\overline d(p)$) is \[2x\boldsymbol{\mathcal{N}}(x)+2x\frac{\partial}{\partial x}(x\mathcal N(x)).\] This is illustrated on the fourth line of~\Cref{fig:n_decomp}.
This yields the functional equation, from which one can derive the explicit solution using \Cref{prop:m_decomp,prop:mm_decomp,,prop:N,,prop:V}.
\end{proof}
We conclude \Cref{cor:Wiener_JSn} by expanding the explicit generating function for $\boldsymbol{\mathcal{V}}(x)$ given in \Cref{eq:Vv} as \[\frac{8x\left(1+\sqrt{1-4x}-x(3+\sqrt{1-4x})\right)}{(1-4x)(1-4x+\sqrt{1-4x})^3} = \sum_{n\geq 0} a_n x^n.\]
Using a computer algebra system, the coefficients $a_n$ satisfy the difference equation
\[(2 n+3)2^{-2n-1} a_n - (4 n +5)2^{-2n-3} a_{n + 1} + (2 + 2 n)2^{-2n-5} a_{n + 2} = 0\]
with initial conditions $a_0=0$ and $a_1=2$. It is easily checked that $\frac{2 n (2 n + 1)}{3}\binom{2 n - 1}{n}$ satisfies this equation and initial conditions.
\section{Asymptotic distributions}\label{sec:asymptotics}
In this section, we prove \cref{prop:cv_law_distance,prop:cv_law_distance_shifted}, which describe the asymptotic distribution
of the distance between 2 random points (also called $2$-point distance) in $P_{\alpha n,n}$ and $Q_n$, respectively.
We start with the case of shifted staircases, which is easier.
\subsection{2-point distance in $Q_n$}
Recall that the elements in $Q_n$
are exactly the lattice paths starting at (0,0),
ending somewhere on the line $x=n$, and using steps
of the form $U=(1,1)$ and $D=(1,-1)$.
Let $p^n$ and $q^n$ be independent uniform random elements in $Q_n$.
Seeing $p^n$ and $q^n$ as lattice paths, we write $p^n_i$
and $q^n_i$
for their heights after $i$ steps.
Clearly, for all $n \ge 1$ and $i \le n$,
one has $p^n_i=X_1+dots+X_i$ and
$q^n_i=Y_1+dots+Y_i$,
where $(X_j)_{j \ge 1}$ and $(Y_j)_{j \ge 1}$ are
independent sequences of i.i.d.~Rademacher random variables of parameter $1/2$.
Using \Cref{eq:distance_shifted}, we write
\[d(p^n,q^n)= \frac12 \sum_{i=1}^n |p^n_i-q^n_i|
= \frac{n}2 \int_0^1 \lvert\, p^n_{\lceil nt \rceil}-q^n_{\lceil nt \rceil}\rvert dt.\]
By Donsker's theorem, the processes
\[\left(\tfrac1{\sqrt n} p^n_{\lceil nt \rceil}\right)_{t \le 1} \text{ and }\left(\tfrac1{\sqrt n} q^n_{\lceil nt \rceil}\right)_{t \le 1} \]
converge in distribution to independent Brownian motions $(B_t)_{t \le 1}$ and $(B'_t)_{t \le 1}$
in Skorokhod space $D[0,1]$
(see \cite[Chapter 3]{billingsley_convergence} for background on Skorokhod space). Since integration is a continuous functional on $D[0,1]$,
we have
\[n^{-3/2} d(p^n,q^n)= \frac12
\int_0^1 \left\lvert n^{-1/2} p^n_{\lceil nt \rceil}-n^{-1/2} q^n_{\lceil nt \rceil}\right\rvert dt \stackrel{d}{\longrightarrow} \frac12
\int_0^1 |B_t-B'_t| dt,\]
where $\stackrel{d}{\longrightarrow}$ means convergence in distribution.
But $B_t-B'_t \stackrel{d}= \sqrt 2 \, B_t$,
proving that $n^{-3/2} d(p^n,q^n)$
converges in distribution to $\frac1{\sqrt 2} \int_0^1 |B_t| dt$, as claimed in \cref{prop:cv_law_distance_shifted}.
It remains to prove moment convergence. By \cite[Corollary of Theorem 25.12]{billingsley_probability}, it suffices to show that for each $s>1$, the sequence of $s$th moments of $n^{-3/2} d(p^n,q^n)$ is bounded as $n$ tends to $+\infty$.
We have
\[n^{-3/2} d(p^n,q^n) \le n^{-1/2} \max_{i \le n} |p^n_i|
+ n^{-1/2} \max_{i \le n} |q^n_i|.\]
Both terms in the upper bound are identically distributed, so we only consider the first one.
By Doob's maximal inequality,
we have
\[ \mathbb{E}\left[\left(\max_{i \le n} |p^n_i|\right)^s\right] \le \left(\frac{s}{s-1}\right)^s
\mathbb{E}\Big[|p^n_n|^s\Big].\]
Since $p^n_n$ is a sum of $n$ i.i.d.~{\em centered}
random variables, we have the following classical bound on its moments
(see, e.g., \cite{petrov1989moments}):
\[ \mathbb{E}\Big[|p^n_n|^s\Big]
\le C(s)\, n^{s/2}\, \mathbb{E}\big[ |X_1|^s \big],\]
where $C(s)$ is a constant depending only on $s$.
In particular the $s$th moment of $n^{-1/2} p^n_n$ is bounded (as $n$ tends to $+\infty$).
Consequently, the $s$th moment of
$n^{-1/2} \max_{i \le n} p^n_i$
is bounded, and that of $n^{-3/2} d(p^n,q^n)$
is as well. This proves that the convergence
of $n^{-3/2} d(p^n,q^n)$ to $\frac1{\sqrt 2} \int_0^1 |B_t| dt$ holds also in moments, concluding the proof of \cref{prop:cv_law_distance_shifted}. \qed
\subsection{2-point distance in $P_{\alpha n,n}$}
We now turn to the case of rectangles.
Let $p^n$ and $q^n$ be independent uniform random elements in $P_{\alpha n,n}$, seen as lattice paths
from $(0,0)$ to ${((\alpha+1)n, (\alpha-1)n)}$.
These paths $p^n$ and $q^n$ can be constructed as partial sums of sequences of i.i.d.~random variables {\em under some conditioning}.
To this end, let $(X_j)_{j \ge 1}$ and $(Y_j)_{j \ge 1}$ be
independent sequences of i.i.d.~Rademacher random variables of parameter $\alpha/(\alpha+1)$.
We also let $(\tilde X^n_j)_{j \ge 1}$ have the distribution
of $(X_j)_{j \ge 1}$ conditioned to the event $\sum_{j \le (\alpha+1)n} X_j=(\alpha-1)n$.
Then one has the equality in distribution
\[\big(p^n_i\big)_{i\le (\alpha+1)n} \stackrel{d}=
\left( \sum_{j \le i} \tilde X^n_j \right)_{i\le (\alpha+1)n}.\]
Recall that we are interested in the quantity
\begin{equation}\label{eq:distance_shifted_integral}
n^{-3/2} D_{\alpha,n} = n^{-3/2} d(p^n,q^n) =
n\frac1{2n^{3/2}} \sum_{i=1}^n |p^n_i-q^n_i|
= \frac{\alpha+1}{2 \sqrt n} \int_0^1 |p^n_{\lceil (\alpha+1)nt \rceil}-q^n_{\lceil (\alpha+1)nt \rceil}| dt.
\end{equation}
A version of Donsker's theorem for conditioned partial sums has been proved by Liggett \cite{liggett1968invariance} (see in particular the corollary of Theorem 4 there).
In our case, the centered process
\[\left(\tfrac1{\sigma \, \sqrt {(1+\alpha)n}} \big(p^n_{\lceil (\alpha+1)nt \rceil} - \lceil nt \rceil (\alpha-1) \big)\right)_{0 \le t \le 1} \]
converges in distribution to $B_0(t)$
in Skorokhod space $D[0,1]$,
where $\sigma^2=\mathrm{Var}(X_1)$ and $B_0(t)$ is a Brownian bridge.
A similar convergence result holds for $q^n$
with an independent Brownian bridge $B'_0(t)$.
Using the continuity of taking integrals on $D[0,1]$, the quantity in \eqref{eq:distance_shifted_integral}
converges in distribution to
\[ \frac12 (\alpha+1) \, \sigma\, \sqrt{\alpha+1} \int_0^1 |B_0(t) -B'_0(t)| dt.\]
An easy computation gives $\sigma=2\sqrt{\alpha}/(\alpha+1)$, while $B_0(t) -B'_0(t)\stackrel{d}=\sqrt{2} B_0(t)$ in distribution.
Consequently, $n^{-3/2} D_{\alpha,n}$ converges in distribution
to $\sqrt{2\alpha(\alpha+1)} \int_0^1 |B_0(t)| dt$, as claimed in \cref{prop:cv_law_distance}.
It remains to prove moment convergence.
As above, we shall prove that for any $s>1$, the random variable $n^{-3/2} D_{\alpha,n}$
has a bounded $s$th moment as $n$ tends to $+\infty$.
Using the convexity of the map $t \mapsto |t|^s$, we obtain
\begin{equation}
\label{eq:bounding_Ens}
n^{-s} D_{\alpha,n}^s = 2^{-s} \left(\frac1{n} \sum_{i \le (\alpha+1)n} |p^n_i-q^n_i|\right)^s
\le \frac{2^{-s}}n \sum_{i \le (\alpha+1)n}
|p^n_i-q^n_i|^s \le \frac{1}{n} \sum_{i \le (\alpha+1)n}
\frac{|\bar p^n_i|^s+|\bar q^n_i|^s}{2},
\end{equation}
where $\bar p^n_i=p^n_i -i \frac{\alpha-1}{\alpha+1}$
is the centered version of $p^n_i$ (and idem for $q$). Writing $\bar X_i=X_i - \frac{\alpha-1}{\alpha+1}$, we have
\[\mathbb{E}\big[ |\bar p^n_i|^s \bar]
= \mathbb{E}\left[ \bigg| \sum_{j \le i} \bar X_j \bigg|^s \Bigg| \sum_{j \le (\alpha+1)n} \bar X_j=0 \right]
= \sum_{k} |k|^s\, \mathbb P\left[ \sum_{j \le i} \bar X_j =k \Bigg| \sum_{j \le (\alpha+1)n} \bar X_j=0 \right],
\]
where the sum runs over possible values $k$ for $\sum_{j \le i} \bar X_j$.
Using the independence of the $\bar X_j$, we have
\begin{align*}
\mathbb P\left[ \sum_{j \le i} \bar X_j =k \Bigg| \sum_{j \le (\alpha+1)n} \bar X_j=0 \right] &=\frac{\mathbb P\left[ \sum_{j \le i} \bar X_j =k \ \wedge \ \sum_{j \le (\alpha+1)n} \bar X_j=0\right]}
{\mathbb P \left[\sum_{j \le (\alpha+1)n} \bar X_j=0 \right]} \\
& = \mathbb P\left[ \textstyle \sum_{j \le i} \bar X_j =k\right] \cdot
\frac{\mathbb P\left[ \sum_{i< j \le (\alpha+1)n} \bar X_j=-k\right]}{\mathbb P \left[\sum_{j \le (\alpha+1)n} \bar X_j=0 \right]}.
\end{align*}
Take $i \le (\alpha+1)n/2$.
The probabilities in the fraction can be evaluated asymptotically---uniformly in $k$---through the local limit theorem (see, e.g., \cite[Theorem 3.5.2]{durrett2019probability}), which yields
\begin{align*}
\mathbb P \left[\sum_{j \le (\alpha+1)n} \bar X_j=0 \right] &\sim \frac{2}{\sigma \sqrt{2 \pi (\alpha+1) n}};\\
\mathbb P\left[ \sum_{i< j \le (\alpha+1)n} \bar X_j=-k\right] &=
\frac{2 e^{-k^2/2((\alpha+1)n-i) \sigma^2}}{\sigma \sqrt{2 \pi ((\alpha+1) n-i)}} + o(n^{-1/2}) \le \frac{2+o(1)}{\sigma \sqrt{\pi (\alpha+1) n}}.
\end{align*}
In particular, the quotient is bounded by $2$ for $n$ large enough, uniformly in $k$.
Bringing everything together, we obtain that for $n$ large enough and $i \le (\alpha+1)n/2$,
\[\mathbb{E}\big[ |\bar p^n_i|^s \bar]
\le \sum_{k} |k|^s \cdot 2 \mathbb P\left[ \textstyle \sum_{j \le i} \bar X_j =k\right]
= 2 \mathbb{E}\left[ \bigg| \sum_{j \le i} \bar X_j \bigg|^s
\right].\]
Since the $\bar X_j$ are i.i.d.~{\em centered} random variables with finite moments, we have (see, e.g., \cite{petrov1989moments})
\[ \mathbb{E}\left[ \bigg| \sum_{j \le i} \bar X_j \bigg|^s
\right] \le C(s) i^{s/2} \mathbb{E}\big[ |X_1|^s \big],\]
where $C(s)$ is a constant depending only on $s$
(and $\alpha$ in the sequel),
which may change from line to line.
Therefore, for $n$ large enough and $i \le (\alpha+1)n/2$, we
have
\[\mathbb{E}\big[ |\bar p^n_i|^s \bar] \le C(s)\, n^{s/2}. \]
By symmetry, this holds also for $i \ge (\alpha+1)n/2$ (we have $p^n_i \stackrel{d}= p^n_{(\alpha+1)n-i}$ for all $i \le (\alpha+1)n$).
Going back to \eqref{eq:bounding_Ens}, we get
\[n^{-s} \mathbb{E} \big[ D_{\alpha,n}^s \big]
\le (\alpha+1)\, C(s)\, n^{s/2}.\]
Thus $n^{-3/2} D_{\alpha,n}$ has bounded moments,
and the convergence to
$\sqrt{2\alpha(\alpha+1)} \int_0^1 |B_0(t)| dt$
holds also in moments.
\cref{prop:cv_law_distance_shifted} is proved. \qed
\section{Higher moments}
\label{sec:higher_moments}
Given a finite graph $G=(V,E)$ and a positive integer $r$, let $d^r(G)$ denote the moment $d^r(G)=\sum_{(p,q)\in V\times V}d(p,q)^r$.
The convergence of the distance between two random elements in distribution and in moments established in the previous section
yield some asymptotic estimates for $d^r(P_{\alpha n,n})$ and $d^r(Q_n)$.
In this section, we give an exact expression of $d^2(P_{k,n-k})$. The same method can, in principle, be used to compute the moments $d^r(P_{k,n-k})$ one by one. Similarly, one could use a similar method, drawing from the ideas in \Cref{sec:bij_stair,sec:proof_stair}, to compute the moments $d^r(Q_n)$. For the sake of brevity, we merely state the explicit formula for $d^2(Q_n)$ and omit the computation.
\begin{proposition} We have
\[d^2(P_{m,k}) = \frac{1}{30} \frac{m+k+1}{m+k} \binom{m+k}{m-1}\binom{m+k}{k-1} \left( 7mk^2 + 7m^2k + 3m^2 + 10mk+ 3k^2 + 3m + 3k + 4 \right)\] and \[d^2(Q_{n}) = 2^{2n-4} n \left(20 + 15 (n - 2) + 3 (n - 2)^2\right).\]
\end{proposition}
\begin{proof}
As mentioned above, we will only prove the first formula. Given a bilateral Motzkin path $p$, let $\mathrm{len}(p)$ denote the length of $p$, and let $\mathcal U(p)$ be the number of steps in $p$ of the form $U$ or $O_1$. Recall that \[\mathcal{W}(x,u)=\sum_{p\in\mathcal{W}}x^{\mathrm{len}(p)}u^{\mathcal U(p)},\quad \mathcal{W}w(x,u)=\sum_{p\in \mathcal{W}}x^{\mathrm{len}(p)}u^{U(p)}d(p),\] \[\mathcal{M}(x,u)=\sum_{p\in\mathcal{M}}x^{\mathrm{len}(p)}u^{\mathcal U(p)}, \quad\mathcal{M}m(x,u)=\sum_{p\in \mathcal{M}}x^{\mathrm{len}(p)}u^{U(p)}d(p).\] Let \[\mathbb{W}(x,u)=\sum_{p\in \mathcal{W}}x^{\mathrm{len}(p)}u^{U(p)}d(p)^2\quad\text{and}\quad\mathcal{M}MM(x,u)=\sum_{p\in \mathcal{M}}x^{\mathrm{len}(p)}u^{U(p)}d(p)^2.\] It follows from \Cref{prop:bija} that $\mathbb{W}(x,u)=\sum_{n\geq 0}\sum_{k\geq 0}d^2(P_{k,n-k})x^nu^k$.
The contribution to $\mathcal{M}MM(x,u)$ coming from paths that start with $O_1$ or $O_2$ is $x(u+1)\mathcal{M}MM(x,u)$. The other paths that contribute to $\mathcal{M}MM(x,u)$ begin with $U$ and have the form $UpDq$ for some $p,q\in\mathcal{M}$. We find that \begin{equation}\label{eq:Sigma}
\mathcal{M}MM(x,u)=x(u+1)\mathcal{M}MM(x,u)+ux^2\Sigma,
\end{equation}
where \[\Sigma=\sum_{p,q\in\mathcal{M}}x^{\mathrm{len}(p)+\mathrm{len}(q)}u^{\mathcal U(p)+\mathcal U(q)}(d(p)+d(q)+\mathrm{len}(p)+1)^2.\] We can write
\begin{align*}
(d(p)+d(q)+\mathrm{len}(p)+1)^2&=(d(p)^2+d(q)^2)+(\mathrm{len}(p)+1)^2+2d(p)(\mathrm{len}(p)+1) \\
&+2d(q)(\mathrm{len}(p)+1)+2d(p)d(q)
\end{align*}
to find that
\begin{align}\label{eq:Sigma2}
\Sigma&=2\mathcal{M}MM(x,u)\mathcal{M}(x,u)+\mathcal{M}(x,u)\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial x}(x\mathcal{M}(x,u))\right)+2\mathcal{M}(x,u)\frac{\partial}{\partial x}(x\mathcal{M}m(x,u)) \nonumber\\
&+2\mathcal{M}m(x,u)\frac{\partial}{\partial x}(x\mathcal{M}(x,u))+2\mathcal{M}m(x,u)^2.
\end{align}
A similar argument yields the functional equation
\begin{equation}\label{eq:Sigma'}
\mathbb{W}(x,u)=x(u+1)\mathbb{W}(x,u)+2ux^2\Sigma',
\end{equation}
where
\begin{align}\label{eq:Sigma'2}
\Sigma'&=\sum_{\substack{p\in\mathcal{M} \\ q\in\mathcal{W}}}x^{\mathrm{len}(p)+\mathrm{len}(q)}u^{\mathcal U(p)+\mathcal U(q)}(d(p)+d(q)+\mathrm{len}(p)+1)^2 \nonumber\\
&=\mathcal{M}MM(x,u)\mathcal{W}(x,u)+\mathbb{W}(x,u)\mathcal{M}(x,u)+\mathcal{W}(x,u)\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial x}(x\mathcal{M}(x,u))\right) \nonumber \\
&+2\mathcal{W}(x,u)\frac{\partial}{\partial x}(x\mathcal{M}m(x,u))+2\mathcal{W}w(x,u)\frac{\partial}{\partial x}(x\mathcal{M}(x,u))+2\mathcal{M}m(x,u)\mathcal{W}w(x,u).
\end{align}
We already computed explicit formulas for $\mathcal{M}(x,u)$, $\mathcal{M}m(x,u)$, $\mathcal{W}(x,u)$, and $\mathcal{W}w(x,u)$ in \Cref{prop:m_decomp,prop:mb_decomp,prop:mm_decomp,prop:mbm_decomp}. Combining those formulas with \Cref{eq:Sigma,eq:Sigma2,eq:Sigma',eq:Sigma'2}, we can derive the explicit formula
\[\mathbb{W}(x,u)=\frac{2 u x^2 \left((u-1)^2 (u+1) x^3-((u-8) u+1) x^2-(u+1) x+1\right)}{\left((u x+x-1)^2-4 u x^2\right)^{7/2}}.\] Setting $u=y/x$ and extracting coefficients yields the desired explicit formula for $d^2(P_{m,k})$.
\end{proof}
\begin{comment}
Generating function for nested lattice paths is
\[\mathcal{M}MM(x,u)=u x^2 \left(2 \mathcal{M}m(x,u) \frac{\partial (x M(x,u))}{\partial x}+2 M(x,u) \frac{\partial (x \mathcal{M}m(x,u))}{\partial x}+2 \mathcal{M}(x,u) \mathcal{M}MM(x,u)+\mathcal{M}(x,u) \frac{\partial }{\partial x}\left(x \frac{\partial (x \mathcal{M}(x,u))}{\partial x}\right)+2 \mathcal{M}m(x,u)^2\right)+(u+1) x \mathcal{M}MM(x,u).\]
Solving gives
\[\mathcal{M}MM(x,u)=\frac{1}{2}\left(\frac{20 u x^2}{\left((u x+x-1)^2-4 u x^2\right)^{5/2}}+\frac{2 u x}{\left((u x+x-1)^2-4 u x^2\right)^{3/2}}+\frac{2 x}{\left((u x+x-1)^2-4 u x^2\right)^{3/2}}+\frac{\sqrt{(u x+x-1)^2-4 u x^2}}{u (x-1) x^2}+\frac{1}{u x^2}+\frac{4}{(u x+x-1)^2-4 u x^2}+\frac{8 (u x+x-1)}{\left((u x+x-1)^2-4 u x^2\right)^2}+\frac{6}{\left((u x+x-1)^2-4 u x^2\right)^{3/2}}-\frac{u}{(x-1) \sqrt{(u x+x-1)^2-4 u x^2}}+\frac{3}{(x-1) \sqrt{(u x+x-1)^2-4 u x^2}}+\frac{1}{(x-1) x \sqrt{(u x+x-1)^2-4 u x^2}}\right.\]
Now the generating function for pairs of lattice paths is
\[\mathbb{W}(x,u)=2 u x^2 \left(2 \left(\mathcal{M}bm(x,u) \frac{\partial (x \mathcal{M}(x,u))}{\partial x}+\mathcal{M}b(x,u) \frac{\partial (x \mathcal{M}m(x,u))}{\partial x}+\mathcal{M}m(x,u) \mathcal{M}bm(x,u)\right)+\mathcal{M}b(x,u) \frac{\partial }{\partial x}\left(x \frac{\partial (x \mathcal{M}(x,u))}{\partial x}\right)+\mathcal{M}(x,u) \mathbb{W}(x,u)+\mathcal{M}MM(x,u) \mathcal{M}b(x,u)\right)+(u+1) x \mathbb{W}(x,u)\]
\[\mathbb{W}(x,u)=\frac{2 u x^2 \left((u-1)^2 (u+1) x^3-((u-8) u+1) x^2-(u+1) x+1\right)}{\left((u x+x-1)^2-4 u x^2\right)^{7/2}}.\]
And then extract coefficients...
\end{comment}
\section{Open problems}
Comparing the results of Proposition~\ref{prop:mb_decomp} and~\ref{prop:mbm_decomp}, we get the intriguing equation
\begin{equation}
\label{eq:interesting_identity}
\mathcal{M}bm=2ux^2\;\mathcal{M}b^4.
\end{equation}
A direct proof of this would be interesting in itself and could lead to a bijective proof of~\Cref{cor:coeff} via the explicit formula \eqref{eq:expansion_bilateral}. Recall that $\mathcal{M}bm$ counts pairs of paths where a cell in the symmetric difference is marked. The connected component where this cell occurs corresponds to a part where the two paths only meet at their beginning and end (this forms a \emph{parallelogram polyomino}), and the generating function $\mathcal{M}b^2$ naturally enumerates the rest of the paths. It follows that a bijective proof of \eqref{eq:interesting_identity} reduces to a bijective proof that $\mathcal{M}b^2$ enumerates the total area of parallelogram polyominoes. The specialization $u=1$ is known \cite{dellungo2004bijection}.
As minuscule lattices arise as the weak order on certain maximal parabolic quotients of finite Coxeter groups, it would be interesting to extend our results to other parabolic quotients. Minuscule lattices also appear as certain crystal graphs; one could also ask about the Wiener indices of more general crystals.
\renewcommand*{\bibliofont}{\normalsize}
\end{document}
|
\begin{document}
\title{Small Entangled Quantum Worlds with a Simple Structure}
\author{E. D. Vol}
\email{[email protected]}
\affiliation{B. Verkin Institute for Low Temperature Physics and Engineering of the
National Academy of Sciences of Ukraine, 47, Nauky Ave., Kharkov 61103,
Ukraine.}
\begin{abstract}
We introduce the notion of a small quantum world (SQW) , which in
our opinion, is very helpful in the situations when an
experimenter's tools for preparing quantum states and (or) for
measuring the observables of a quantum system under study are
restricted due to some kind of reasons. In this case it is advisable
to use as original an appropriate subspace of complete Hilbert space
of states and respectively to utilize observables acting only in
this subspace. If this subspace possesses some additional
symmetries, the structure of a pure states set, as a rule, is
simpler by far than in general case. Moreover in such SQWs some
specific irreducible entangled states may appear. The similar states
could be very helpful in various tasks connected with quantum
informational applications. In the present paper main ideas outlined
above are developed in detail in the simple and instructive case of
a two-qubit system in which the accessible space of states possesses
the additional symmetry structure of permutation group of three
elements.
\end{abstract}
\maketitle
It is well known that there are two fundamental concepts in quantum theory,
namely states and observables, and respectively an experimenter has to deal
with two main procedures that realize these concepts, namely preparation of
the required state and measuring the observable of interest in the end of
the experiment. Note that among all states of quantum system its pure
states, which contain the maximal possible information about the system,
play the most important role. One of the main postulates of quantum theory
claims that there is exact mapping between pure states of a quantum system
under study and vectors of appropriate Hilbert space. For example, in the
case of the most simple quantum system, that is a qubit, the relevant
Hilbert space, representing its states, is two-dimensional and there is a
geometrically descriptive image of the states of such system by means of the
Bloch sphere. As is well known, an arbitrary state of the qubit (mixed or
pure) with a density matrix ${\hat{\rho}}$ can be represented in the form ${
\hat{\rho}}=\frac{1+\vec{P}\vec{\sigma}}{2}$, where $\hat{\sigma}_{i}$ $
(i=1,2,3)$ are the Pauli matrices and $\vec{P}$ is appropriate Bloch vector.
The pure states of a qubit for which the condition ${\hat{\rho}}^{2}={\hat{
\rho}}$ is satisfied are placed on the surface of the Bloch sphere with $
\vec{P}=1$ while all the rest (mixed) states are settled inside this sphere.
Unfortunately, in more complex situations, in particular for composite
quantum systems the problem of description of the set of pure states by
geometrically distinct way is unsolved till now. Furthermore, when one is
operating with the states of composite systems, the important problem of
determination their entanglement ( the quantity that just specifies the
nonlocal informational resource and distinguishes quantum communicational
systems from classical ones), can be solved exactly also only for two-qubit
systems. In spite of enormous number of papers devoted to this problem (see
for example the comprehensive review of the topic \cite{1}), this problem
remains open up to now. In this connection we propose as a first step to
consider more simple problem, namely, to study some special classes of
quantum systems in which both the set of accessible quantum states and the
set of observables that are available for measurement are restricted by some
additional conditions.Evidently we will talking about specific subspaces of
the total Hilbert space and appropriate algebras of observables acting in
these subspaces. From physical point of view it means that, due to various
reasons, capabilities of an experimenter for the quantum states preparation
and for performing arbitrary measurements are restricted. Nevertheless,
since in this approach the space of accessible states of the system remains
linear and closed, all postulates of quantum theory continue to be valid.
Henceforth we will define as a small quantum world (SQW) certain subspace of
states within the complete Hilbert space with the relevant algebra of
observables acting in this subspace. The main goal of the present paper is
to demonstrate on particular examples that the structure of states in such
SQWs could be simpler by far than the structure of states in the enveloping
large quantum world. At first let us briefly remind one of the known
examples of SQW, namely, so called X-states in two-qubit quantum systems
\cite{2}. In this case one assumes that density matrix of any accessible
state of the quantum system under study can be represented as ${\hat{\rho}}=
\begin{pmatrix}
\rho _{11} & 0 & 0 & \rho _{14} \\
0 & \rho _{22} & \rho _{23} & 0 \\
0 & \rho _{32} & \rho _{33} & 0 \\
\rho _{41} & 0 & 0 & \rho _{44}
\end{pmatrix}
$. The relevant basis of the algebra of observables acting on these states
consists of 8 operators (generators of the algebra). Let us write them
explicitly:1) $\hat{1}-$the unit operator ,2) another diagonal operator $
\hat{E}=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
$, and in addition six operators $\hat{\lambda}_{i}$ and ${\hat{\tau}}_{i}$
\ ($i=1,2,3$) that are defined by analogy with the known Pauli
matrices,namely :
\begin{eqnarray*}
&&\hat{\lambda}_{1}=
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0
\end{pmatrix}
,\hat{\lambda}_{2}=
\begin{pmatrix}
0 & 0 & 0 & -i \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
i & 0 & 0 & 0
\end{pmatrix}
, \\
&&\hat{\lambda}_{3}=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}
\end{eqnarray*}
and
\begin{eqnarray*}
&&{\hat{\tau}}_{1}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
,{\hat{\tau}}_{2}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & -i & 0 \\
0 & i & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
, \\
&&{\hat{\tau}}_{3}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
.
\end{eqnarray*}
It is easy to point out the complete system of algebraic relations
connecting the generators of this algebra, namely:
\begin{eqnarray}
&&{\hat{\lambda}_{i}}{\hat{\lambda}_{j}}=\frac{1+\hat{E}}{2}\delta
_{ij}+i\varepsilon _{ijk}{\hat{\lambda}_{k}}, \notag \label{1} \\
&&{\hat{\tau}_{i}}{\hat{\tau}_{j}}=\frac{1-\hat{E}}{2}\delta
_{ij}+i\varepsilon _{ijk}{\hat{\tau}_{k}}, \notag \\
&&{\hat{\lambda}_{i}}{\hat{\tau}_{j}}={\hat{\tau}_{j}}{\hat{\lambda}_{i}}=0,
\notag \\
&&\hat{E}{\hat{\lambda}_{i}}={\hat{\lambda}_{i}}\hat{E}={\hat{\lambda}_{i}}
\text{ and } \notag \\
&&\hat{E}{\hat{\tau}_{i}}={\hat{\tau}_{i}}\hat{E}=-{\hat{\tau}_{i}}
\end{eqnarray}
( where $\varepsilon _{ijk}$ is completely antisymmetric tensor).Any X-state
(that is its density matrix) can be represented as $\hat{\rho}=\frac{1+e\hat{
E}+P_{i}{\hat{\lambda}_{i}}+S_{i}{\hat{\tau}_{i}}}{4}.$(where $e,P_{i},S_{i}$
are appropriate numerical coefficients). It is easy to see that 4
eigenvalues of such matrix can be calculated explicitly and are equal to: $
\rho _{1,2}=\frac{1+e\pm \left\vert P\right\vert }{4}$ and $\rho _{3,4}=
\frac{1-e\pm \left\vert S\right\vert }{4}.$ Evidently, that one can specify
two disjoint classes of pure X-states in such SQW: 1) with $e=1$ , $
\left\vert P\right\vert =2,$ $\left\vert S\right\vert =0$ and 2) with $e=-1$
$\left\vert S\right\vert =2,$ $\left\vert P\right\vert =0$. Thus the
structure of pure states in this SQW turns out to be simpler by far than for
two-qubit states in general case. We will continue to study the properties
of X- states elsewhere, while the present paper will be devoted to another
interesting case, namely the SQWs that can be constructed on the basis of
the permutation group of four elements- $S_{4}$. It is well-known that due
to the principle of indistinguishability the permutation group plays the
fundamental and diverse role in quantum theory, but in this paper it will be
used only for the description of quantum states and their properties. With
reference to group $S_{4}$ it should be noted that among its 30 subgroups
there are 4 subgroups that are isomorphic to group $S_{3}$ (that is
permutation group of 3 elements).Exactly this subgroup can serve as
demonstrative and instructive example of the small (and entangled as we see
further) quantum world. Let us consider now the concrete realization of this
subgroup which leaves as invariant the fourth element (state) of the group
and construct the model of SQW based on this realization. In the standard
basis of two-qubit states the relevant algebra of observables acting in this
SQW consists of six generators, from which three (aside from unit operator)
are Hermitian and the rest two are unitary. Let us write down them in
explicit form. The three Hermitian operators are:
\begin{eqnarray}
{\hat{H}_{1}} &=&
\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
,{\hat{H}_{2}}=
\begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
, \notag \label{2} \\
{\hat{H}_{3}} &=&
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\end{eqnarray}
and two unitary ones are
\begin{equation}
\hat{A}=
\begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\text{ , }{\hat{B}}=
\begin{pmatrix}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
. \label{3}
\end{equation}
Note that unitary operators $\hat{A}$ and ${\hat{B}}$ are conjugate and
reciprocal to each other, that is $\hat{A}={\hat{B}^{+}}$ and $\hat{A}\hat{B}
={\hat{B}}\hat{A}=\hat{1}$. Let us point out the complete system of
relations for hermitian generators of the algebra: ${\hat{H}_{i}}^{2}=\hat{1}
$ ($i=1,2,3$), ${\hat{H}_{1}}{\hat{H}_{2}}={\hat{H}_{2}}{\hat{H}_{3}}={\hat{H
}_{3}}{\hat{H}_{1}}=\hat{A}$ and $\hat{H_{1}}{\hat{H}_{3}}={\hat{H}_{2}}{
\hat{H}_{1}}={\hat{H}_{3}}\hat{H_{2}}={\hat{B}}$. In addition let us give
also the algebraic relations connecting operators ${\hat{H}_{i}}$ with
unitary operators $\hat{A}$ \ which have the next form: ${\hat{H}_{1}}\hat{A}
={\hat{H}_{2}},$ ${\hat{H}_{2}}\hat{A}={\hat{H}_{3}},$ $\hat{H_{3}}\hat{A}={
\hat{H}_{1}}$ and $\hat{A}{\hat{H}_{1}}=\hat{H_{3}},\hat{A}{\hat{H}_{2}}={
\hat{H}_{1}},\hat{A}{\hat{H}_{3}}={\hat{H}_{2}}$ .The similar equations
including the matrix ${\hat{B}}$ can be obtained from these relations by
conjugation. There are also two relations, connecting operators $\widehat{A}$
and $\widehat{B\text{ }}$ : $\hat{A}^{2}={\hat{B}}$,and ${\hat{B}}^{2}=\hat{A
}$. The density matrix of any state belonging to this SQW may be represented
as $\hat{\rho}=\frac{k}{2}\hat{1}+l{\hat{H}_{1}}+m{\hat{H}_{2}}+n\hat{H_{3}}
+p\left( \hat{A}+{\hat{B}}\right) $, where $k,l,m,n,p$ are real numbers
satisfying the normalization condition: $k+l+m+n+p=\frac{1}{2}$. However, it
is easy to see that generators of the algebra ${\hat{H}_{i}}$, $\hat{A}$ and
${\hat{B}}$ are connected by the additional relation : $\hat{A}+{\hat{B}}=
\hat{C}-1$, where $\hat{C}\equiv {\hat{H}_{1}}+{\hat{H}_{2}}+{\hat{H}_{3}}$.
Thus the general representation for the density matrix in given SQW may be
written finally as :
\begin{equation}
\hat{\rho}=\frac{a}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{3}}
\label{4}
\end{equation}
with normalization condition: $a+b+c+d=\frac{1}{2}$.
It is worth noting that operator $\hat{C}$ commutes with all
generators of the algebra and hence is the Casimir operator of the
group $S_{3}$. Let us turn now to the question of clarifying the
structure of pure states in this SQW. Starting from the
representation \eqref{4} and using the defining condition for pure
states: ${\hat{\rho}}^{2}={\hat{\rho}}$, one, after elementary
calculations, can obtain the following restrictions on the
coefficients of the decomposition \eqref{4}, namely$:$
\begin{eqnarray}
&&\frac{a^{2}}{2}=\frac{a^{2}}{4}+b^{2}+c^{2}+c^{2}-(bc+bd+cd); \notag
\label{5} \\
&&b=ab+(bc+bd+cd); \notag \\
&&c=ac+(bc+bd+cd); \notag \\
&&d=ad+(bc+bd+cd).
\end{eqnarray}
It is clear that the only possibility to satisfy all relations \eqref{5} is
to put the coefficient $a$ equal to unit and impose on the coefficients $
b,c,d$ the next restriction: $b^{2}+c^{2}+d^{2}=\frac{1}{4}$ (together with
normalization condition $b+c+d=-\frac{1}{2}$). Thus, in the parameter space
of coefficients $b,c,d$ the set of pure states is the intersection of the
sphere centered in the origin, whose radius is equal to $\frac{1}{2}$, and
the plane satisfying the equation $b+c+d=-\frac{1}{2}$. Evidently it is a
circle. Further, it is well-known that in Hilbert space pure states form the
boundary of a convex set of all quantum states \cite{3}. Hence the result
obtained means that all mixed states of the system in this SQW must be
settled within the circle specified above. Thus the structure of quantum
states in the SQW under study is quite simple and obvious. In addition one
can point out \ the parametrization of pure states in this SQW by writing
the coefficients of \eqref{4} with $a=1$ in the next convenient form:
\begin{eqnarray}
b &=&-\frac{t\left( 1+t\right) }{2\left( 1+t+t^{2}\right) };c=-\frac{\left(
1+t\right) }{2\left( 1+t+t^{2}\right) }; \notag \label{6} \\
d &=&\frac{t}{2\left( 1+t+t^{2}\right) }
\end{eqnarray}
(with the only real parameter $t).$ It is easy to verify directly that two
relations $b+c+d=-\frac{1}{2}$ and $b^{2}+c^{2}+d^{2}=\frac{1}{4}$
characterizing pure states in this SQW are fulfilled automatically. Using
the parametrization \eqref{6} one can write down the normalized vector $
\left\vert \Psi \right\rangle $ corresponding to the density matrix of a
pure state, that is, if ${\hat{\rho}}=\left\vert \Psi \right\rangle
\left\langle \Psi \right\vert $, then the appropriate vector $\left\vert
\Psi \right\rangle $ can be represented as: $\left\vert \Psi \left( t\right)
\right\rangle =\frac{1}{\sqrt{2\left( 1+t+t^{2}\right) }}
\begin{pmatrix}
1+t \\
t \\
1 \\
0
\end{pmatrix}
.$
Since all information contained in quantum state of the system can be
extracted only by making appropriate measurements , it is useful to give a
simple experimental criterion of the state purity. To this end we assume
that the unknown quantum state has the above-mentioned form:
\begin{multline}
{\hat{\rho}}=\frac{1}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d\hat{H_{3}}=
\label{7} \\
=
\begin{pmatrix}
\frac{1}{2}+d & b & c & 0 \\
b & \frac{1}{2}+c & d & 0 \\
c & d & \frac{1}{2}+b & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
.
\end{multline}
First of all let us find the eigenvalues of the density matrix \eqref{7}.
One can verify easily that ${\hat{\rho}}$ has two zero eigenvalues: the
first with eigenvector $\left\vert 0\right\rangle _{1}=
\begin{pmatrix}
0 \\
0 \\
0 \\
1
\end{pmatrix}
$ and the second with eigenvector $\left\vert 0\right\rangle _{2}=\frac{1}{
\sqrt{3}}
\begin{pmatrix}
1 \\
1 \\
1 \\
0
\end{pmatrix}
$. Besides them there are two nonzero eigenvalues $\mu _{1}$and $\mu _{2}$
that satisfy to the quadratic equation: $\mu ^{2}-\mu +3\left(
bc+bd+cd\right) =0$ with the solutions:
\begin{equation}
\mu _{1,2}=\frac{1\pm \sqrt[2]{1-12\left( bc+bd+cd\right) }}{2} \label{8}
\end{equation}
The expression \eqref{8} implies that coefficients $b,c,d$ aside from
normalization condition must satisfy to the inequality $0\leq \left(
bc+bd+cd\right) \leq \frac{1}{12}.$
Let us define now as usual the mean value of an arbitrary observable $\hat{A}
$ as $\left\langle \hat{A}\right\rangle =\mathrm{Tr}\left( \hat{ \rho } \hat{
A}\right)$. Using the representation \eqref{7} one can write down the mean
values of the following three selected observables:
1) $\left\langle H_{1}\right\rangle -1\equiv A_{1}=c+d+4b,$
2) $\left\langle H_{2}\right\rangle -1\equiv A_{2}=b+d+4c,$ and
3) $\left\langle H_{3}\right\rangle -1\equiv A_{3}=b+c+4d.$
Taking into account the condition $b+c+d=-\frac{1}{2}$ one can find that $\
\ \ \left\langle A_{1}+A_{2}+A_{3}\right\rangle =-3$, that is for all states
of SQW (with $a=1$) the mean value of the Casimir operator $\left\langle
C\right\rangle \equiv \left\langle H_{1}+H_{2}+H_{3}\right\rangle =0$. On
the other hand if one considers another useful quantity, namely $R\equiv
A_{1}^{2}+A_{2}^{2}+A_{3}^{2}=\left( c+d+4b\right) ^{2}+\left( b+d+4c\right)
^{2}+\left( b+c+4d\right) ^{2}$, then she(he) obtains that $R=\left( -\frac{1
}{2}+3b\right) ^{2}+\left( -\frac{1}{2}+3c\right) ^{2}+\left( -\frac{1}{2}
+3d\right) ^{2}=9\left( \frac{1}{4}+b^{2}+c^{2}+d^{2}\right) =\frac{9}{2}$.
Thus, for all pure states in SQW the following criterion of purity should be
true:
\begin{equation}
\left( \left\langle H_{1}\right\rangle -1\right) ^{2}+\left( \left\langle
H_{2}\right\rangle -1\right) ^{2}+\left( \left\langle H_{3}\right\rangle
-1\right) ^{2}=\frac{9}{2} \label{9}
\end{equation}
Let us go to the next question which we are interested in: how the
explicit expression for the entanglement of the states (both pure
and mixed) in this SQW looks? For the sake of simplicity as before
we are limited ourselves to studying the states for which the
coefficient $a$ in \eqref{4} is equal to unit and the appropriate
density matrix can be represented as
\begin{equation}
{\hat{\rho}}=\frac{1}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{2}}
\label{10}
\end{equation}
with a normalization condition: $b+c+d=-\frac{1}{2}$. As was explained
above, among these states there are certain pure states (for which $
b^{2}+c^{2}+d^{2}=\frac{1}{4}$) while all the rest are mixed. Let us
determine the entanglement of the state \eqref{10}. To this end in the case
of two-qubit arbitrary mixed state the well-known recipe was proposed by
Wootters in \cite{4}. This recipe reads as follows. First of all one must
construct the auxiliary matrix $\tilde{\rho}=\left( \sigma _{y}\otimes
\sigma _{y}\right) \rho ^{\ast }\left( \sigma _{y}\otimes \sigma _{y}\right)
$, where $\sigma _{y}$ $=
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
$ is the Pauli matrix and ${\hat{\rho}^{\ast }}$ is the matrix
conjugated to given matrix ${\hat{\rho}} $ \eqref{10}. At the next
step one needs
to introduce another auxiliary matrix $\hat{\Omega}$ $=\hat{\rho}{\hat{\rho}}
^{\ast }$ (that is non-Hermitian and positive) and after that to find its
four eigenvalues $\Omega _{i}$ ($i=1,2,3,4$). If one arranges them in
decreasing order $\Omega _{1}\geq \Omega _{2}\geq \Omega _{3}\geq \Omega
_{4}\geq 0$, then according to the paper \cite{4} the entanglement of
formation $E(\widehat{\rho })$ for the state ${\hat{\rho}}$ $\ $ may be
calculated as follows:
\begin{multline}
E\left( {\hat{\rho}}\right) =-\frac{\left( 1+\sqrt{1-C^{2}}\right) }{2}\log
_{2}\frac{1+\sqrt{1-C^{2}}}{2}- \label{11} \\
-\frac{\left( 1-\sqrt{1-C^{2}}\right) }{2}\log _{2}\frac{\left( 1-\sqrt{
1-C^{2}}\right) }{2},
\end{multline}
where the concurrence
\begin{equation*}
C=C\left( {\hat{\rho}}\right) =\max \left\{ 0,\sqrt{\Omega _{1}}-\sqrt{
\Omega _{2}}-\sqrt{\Omega _{3}}-\sqrt{\Omega _{4}}\right\} .
\end{equation*}
Note that since $E\left( {\hat{\rho}}\right) $ is a monotonic and increasing
function of concurrence $C\left( \widehat{\rho }\right) $, we can restrict
ourselves to determination of the value of $C\left( \widehat{\rho }\right) $
that evidently ranges from zero to unit and defines the degree of
entanglement as well as $E\left( {\hat{\rho}}\right) $. Omitting elementary
calculations we give the final expressions for the auxiliary matrix $\hat{
\Omega}$ and its four eigenvalues, namely:
\begin{equation}
\hat{\Omega}=
\begin{pmatrix}
0 & b\left( \frac{1}{2}+b\right) +cd & bd+c\left( \frac{1}{2}+c\right) & -2bc
\\
0 & \left( \frac{1}{2}+b\right) \left( \frac{1}{2}+c\right) +d^{2} & 2\left(
\frac{1}{2}+c\right) d & 0 \\
0 & 2\left( \frac{1}{2}+b\right) d & & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
\label{12}
\end{equation}
The eigenvalues of matrix $\hat{\Omega}$ (in decreasing order) are equal to:
$\Omega _{1}=\left[ d+\sqrt{\left( \frac{1}{2}+b\right) \left( \frac{1}{2}
+c\right) }\right] ^{2},\Omega _{2}=\left[ d-\sqrt{\left( \frac{1}{2}
+b\right) \left( \frac{1}{2}+c\right) }\right] ^{2},\Omega _{3}=\Omega
_{4}=0.$
Using the above-mentioned recipe for determination of entanglement one can
find the required result:
\begin{equation}
C\left( {\hat{\rho}}\right) =2\sqrt{\left( \frac{1}{2}+b\right) \left( \frac{
1}{2}+c\right) }. \label{13}
\end{equation}
It is worth noting that if the state \eqref{6} is pure, that is ${\hat{\rho}}
=\left\vert \Psi \right\rangle \left\langle \Psi \right\vert $, where $
\left\vert \Psi \right\rangle =\frac{1}{2\sqrt{1+t+t^{2}}}
\begin{pmatrix}
1+t \\
t \\
1 \\
0
\end{pmatrix}
$, then the expression \eqref{13} takes the form: $C\left\{ \Psi \right\} =
\frac{\left\vert t\right\vert }{1+t+t^{2}}$ which coincides with the
standard definition of concurrence of a pure state$\left\vert \Psi
\right\rangle $. We see that unlike of general case the entanglement
of states belonging to the SQW considered may be explicitly
expressed in terms of the coefficients of the given density matrix
only.Now we turn to the study another interesting problem, namely,
to clarify how much entanglement may be extracted from the given
state (pure or mixed) belonging to the SQW by means of various
measurements carried out on a system being in the state \eqref{10}.
Clearly, having in hands the simple expression for the amount of
entanglement contained in any quantum state of SQW \eqref{13}, this
problem may be easily solved. We consider here only the cases when
the measured observables are the basic observables that is
generators of the algebra $H_{1},H_{2},H_{3}$
Note that in view of relation ${\hat{H}_{i}}^{2}=1\left( \text{for all }
i=1,2,3\right) $ one can write a simple equation connecting two density
matrices,namely,density matrix ${\hat{\rho}_{0}}$ before the measurement and
the density matrix ${\hat{\rho}_{\infty }}$ after the measurement of the
observable ${\hat{H}_{i}}$. This equation reads as:
\begin{equation}
{\hat{\rho}_{\infty }}=\frac{{\hat{\rho}_{0}}+{\hat{H}_{i}}{\hat{\rho}_{0}}{
\hat{H}_{i}}}{2}. \label{14}
\end{equation}
Using Eq.\eqref{14} and the algebra of operators described above one can
consider separately three cases: the case I when the observable ${\hat{H}_{1}
}$ \ is measured and the initial state is ${\hat{\rho}_{0}}=\frac{1}{2}+b{
\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{3}}$, the case II when the observable
${\hat{H}_{2}}$ is measured and the case III when observable ${\hat{H}_{3}}$
is measured. In the case I after the measurement the state of the system is:
${\hat{\rho}_{\infty }}^{I}=\frac{1}{2}+b{\hat{H}_{1}}+\frac{(c+d)}{2}\left(
{\hat{H}_{2}}+{\hat{H}_{3}}\right) $. Similarly in the case II the final
state is ${\hat{\rho}_{\infty }}^{II}=\frac{1}{2}+\frac{(b+d)}{2}\left( \hat{
H_{1}}+{\hat{H}_{3}}\right) +c{\hat{H}_{2}}$ and in the case III the final
stateof the system after measurement is ${\hat{\rho}_{\infty }}^{III}=\frac{1
}{2}+\frac{\left( b+c\right) }{2}\left( {\hat{H}_{1}}+{\hat{H}_{2}}\right) +d
{\hat{H}_{3}}$. Now we are interested in the maximum amount of entanglement
that can be extracted from initial state by these different measurements.
Let us calculate this maximum. Without loss of generality we may assume that
the initial state of the system is pure and can be represented by
parametrization \eqref{6}. Then the gain of entanglement caused by
measurement of ${\hat{H}_{1}}$ may be written as:
\begin{multline}
\Delta C_{I}=C\left\{ {\hat{\rho}_{\infty }}^{I}\right\} -C\left\{ {\hat{\rho
}_{0}}\right\} = \label{15} \\
=2\sqrt{\left( b+\frac{1}{2}\right) \left( \frac{c+d+1}{2}\right) }-2\sqrt{
\left( b+\frac{1}{2}\right) \left( c+\frac{1}{2}\right) }= \\
=\frac{1}{1+t+t^{2}}\left[ \sqrt{\frac{1+2t+2t^{2}}{2}}-\left\vert
t\right\vert \right] .
\end{multline}
Note that in the derivation of relation \eqref{15} we used the
parametrization \eqref{6} for coefficients $b,c,d$ of the initial pure
state. It is easy to see that maximum \eqref{15} is equal to $\frac{1}{\sqrt{
2}}$ and is reached when $t=0$. The required initial state in this case is $
\left\vert \Psi _{0}\right\rangle _{I}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
0 \\
1 \\
0
\end{pmatrix}
$. In the same way one can find that in the case II the gain of entanglement
can be represented as:
\begin{equation}
\Delta C_{II}=\frac{\left\vert t\right\vert }{1+t+t^{2}}\left[ \sqrt{\frac{
2+2t+t^{2}}{2}}-1\right] . \label{16}
\end{equation}
The maximum of \eqref{16} is reached when $t=\pm \infty $ and is equal to $
\frac{1}{\sqrt{2}}$ as well. The required initial state in this case is $
\left\vert \Psi _{0}\right\rangle _{II}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
\pm 1 \\
0 \\
0
\end{pmatrix}
$. However, in the case III, when the observable ${\hat{H}_{3}}$ is
measured, the result is somewhat distinct, namely, he gain of entanglement
caused by this measurement can be written as:
\begin{equation}
\Delta C_{III}=C\left\{ {\hat{\rho}_{\infty }}^{III}\right\} -C\left\{ {\hat{
\rho}_{0}}\right\} =\frac{1+t^{2}-2\left\vert t\right\vert }{2\left(
1+t+t^{2}\right) }. \label{17}
\end{equation}
It is easy to see that the maximum of \eqref{17} is reached when $t=0$ and
is equal to $\frac{1}{2}$. It is curious that, although optimal initial
states for cases I and III coincide, nevertheless, the extracted amount of
entanglement is larger in the first case. Now let us consider another
important and interesting feature of certain states belonging to the SQWs
that makes them potentially very helpful in various quantum informational
applications. We have in mind the existence of irreducible entangled states
both in SQW under examination and in many others SQWs as well. Really let us
consider the selected mixed state $IE$ (that is irreducible and entangled)
with density matrix ${\hat{\rho}}_{IE}=\frac{1}{2}-\frac{\hat{C}}{6}\equiv
\frac{1}{2}-\frac{\hat{H_{1}}+{\hat{H}_{2}}\text{+}{\hat{H}_{3}}}{6}$. It is
easy to see that this state belongs to the SQW because two necessary
conditions: $b+c+d=-\frac{1}{2}$ and $bc+bd+cd=\frac{1}{12}\leq \frac{1}{12}$
are satisfied. On the other hand, it is clear that this state is irreducible
(that is cannot be changed in time by any dynamical way or by means of
measurements) because the Casimir operator $\hat{C}={\hat{H}_{1}}+{\hat{H}
_{2}}+{\hat{H}_{3}}$ commutes with all generators of the algebra.
In addition note that this selected state is entangled with concurrence $
C_{IE}=\frac{2}{3}$. Thus we come to the conclusion that similar states ( in
the case when realization of the SQW would be possible) may be used as
long-lived keepers of entanglement stored in the system. The mentioned
feature of irreducible entangled states makes them indispensable for various
quantum informational applications. It should also be noted that although
the IE state is dynamically stable it can be achieved easily from other
states by means of appropriate measurements. For example if one takes the
initial state of the system in the form: ${\hat{\rho}_{0}}=\frac{1}{2}-\frac{
{\hat{H}_{1}}}{6}+c{\hat{H}_{2}}+d{\hat{H}_{3}}$ and then performs the
measurement of the observable $\hat{H_{1}}$ in this state then (if the
conditions: 1) $c+d=-\frac{1}{3}$ and 2) $cd\leq \frac{5}{36}$ hold true)
she(he) gets exactly the required IE state after the measurement.
Let us sum up the main results obtained in the present paper. We introduce
the notion of small quantum world (SQW) which allows one to study special
classes of composite quantum systems whose pure states and nonlocal
properties turn out to be simpler by far comparing with large quantum worlds
containing them as a small part. We are demonstrating that some features of
selected states belonging to these SQWs would be very helpful for performing
various quantum information tasks.
\end{document}
|
\begin{document}
\title{Media Theory: Representations and Examples}
\author{Sergei~Ovchinnikov \\
Mathematics Department\\
San Francisco State University\\
San Francisco, CA 94132\\
[email protected]}
\date{\today}
\maketitle
\begin{abstract}
In this paper we develop a representational approach to media theory. We construct representations of media by well graded families of sets and partial cubes and establish the uniqueness of these representations. Two particular examples of media are also described in detail.
\noindent
\emph{Keywords:} Medium; Well graded family of sets; Partial cube
\end{abstract}
\section{Introduction}
A medium is a semigroup of transformations on a possibly infinite set of states, constrained by four axioms which are recalled in Section~2 of this paper. This concept was originally introduced by Jean--Claude Falmagne in his 1997 paper~\cite{jF97} as a model for the evolution of preferences of individuals (in a voting context, for example). As such it was applied to the analysis of opinion polls~\cite{mRjFbG99} (for closely related papers, see~\cite{jF96,jFjD97,jFmRbG97}). As shown by Falmagne and Ovchinnikov~\cite{jFsO02} and Doignon and Falmagne~\cite{jDjF97}, the concept of a medium provides an algebraic formulation for a variety of geometrical and combinatoric objects. The main theoretical developments so far can be found in~\cite{jF97,jFsO02,sOaD00} (see also Eppstein and Falmagne's paper~\cite{dE04} in this volume).
The purpose of this paper is to further develop our understanding of media. We focus in particular on representations of media by means of well graded families of sets and graphs. Our approach utilizes natural distance and betweenness structures of media, graphs, and families of sets. The main results of the paper show that, in some precise sense, any medium can be uniquely represented by a well graded family of sets or a partial cube. Two examples of infinite media are explored in detail in Sections~8 and~9.
\section{Preliminaries}
In this section we recall some definitions and theorems from~\cite{jF97}.
Let ${\cal S}$ be a set of \emph{states}. A \emph{token} (of information) is a function $\tau:{\cal S}\rightarrow{\cal S}$. We shall use the abbreviations $S\tau=\tau(S),$ and
$S\tau_1\cdots\tau_n=\tau_n[\ldots[\tau_1(S)]]$ for the function composition. We denote by $\tau_0$ the identity function on ${\cal S}$ and suppose that $\tau_0$ is not a token. Let ${\cal T}$ be a set of tokens on ${\cal S}$. The pair $({\cal S},{\cal T})$ is called a \emph{token system}. Two distinct states $S,T\in{\cal S}$ are \emph{adjacent} if $S\tau =T$ for some token $\tau\in{\cal T}$. To avoid trivialities, we assume that $|{\cal S}|>1$.
A token $\tau'$ is a \emph{reverse} of a token $\tau$ if for all distinct
$S,V\in{\cal S}$
$$
S\tau=V\quad\Leftrightarrow\quad V\tau'=S.
$$
A finite composition $\boldsymbol m=\tau_1\cdots\tau_n$ of not necessarily distinct tokens $\tau_1,\ldots,\tau_n$ such that $S\boldsymbol m=V$ is called a \emph{message producing} $V$ \emph{from} $S$. We write $\ell(\boldsymbol m)=n$ to denote the \emph{length} of $\boldsymbol m$. The \emph{content} of a message
$\boldsymbol m=\tau_1\ldots\tau_n$ is the set ${\cal C}(\boldsymbol m)$ of its distinct tokens. Thus, $|{\cal C}(\boldsymbol m)|\leq\ell(\boldsymbol m)$. A message $\boldsymbol m$ is \emph{effective} (resp. \emph{ineffective}) for a state $S$ if $S\boldsymbol m\neq S$ (resp. $S\boldsymbol m=S$). A message $\boldsymbol m=\tau_1\ldots\tau_n$ is \emph{stepwise effective} for $S$ if
$$
S\tau_1\ldots\tau_k\neq S\tau_0\ldots\tau_{k-1},\qquad 1\leq k\leq n.
$$
A message is called \emph{consistent} if it does not contain both a token and its reverse, and \emph{inconsistent} otherwise. A message which is both consistent and stepwise effective for some state $S$ is said to be \emph{straight} for $S$. A message $\boldsymbol m=\tau_1\ldots\tau_n$ is \emph{vacuous} if the set of indices $\{1,\ldots,n\}$ can be partitioned into pairs $\{i,j\},$ such that one of $\tau_i,\tau_j$ is a reverse of the other. Two messages $\boldsymbol m$ and $\boldsymbol n$ are \emph{jointly consistent} if $\boldsymbol{mn}$ (or, equivalently, $\boldsymbol{nm}$) is consistent.
The next definition introduces the main concept of media theory.
\begin{definition} \label{D:medium}
A token system is called a \emph{medium} if the following axioms are satisfied.
\begin{enumerate}
\item[]
\begin{enumerate}
\item[{\rm [M1]}] Every token $\tau$ has a unique reverse, which we denote by $\tilde{\tau}$.
\item[{\rm [M2]}] For any two distinct states $S,V,$ there is a consistent message transforming $S$ into $V$.
\item[{\rm [M3]}] A message which is stepwise effective for some state is ineffective for that state if and only if it is vacuous.
\item[{\rm [M4]}] Two straight messages producing the same state are jointly consistent.
\end{enumerate}
\end{enumerate}
\end{definition}
It is easy to verify that \lbrack M2\rbrack\ is equivalent to the following
axiom (cf.~\cite{jF97}, Theorem 1.7).
\begin{enumerate}
\item[]
\begin{enumerate}
\item[{\rm [M2*]}] For any two distinct states $S,V,$ there is a
straight message transforming $S$ into $V$.
\end{enumerate}
\end{enumerate}
We shall use this form of axiom [M2] in the paper.
Various properties of media have been established in~\cite{jF97}. First, we recall the concept of `content'.
\begin{definition}
Let $({\cal S},{\cal T})$ be a medium. For any state $S$, the \emph{content} of $S$ is the set $\widehat{S}$ of all tokens each of which is contained in at least one straight message producing $S$. The family
$\widehat{{\cal S}}=\{\widehat{S}\,|\,S\in{\cal S}\}$ is called the \emph{content family} of ${\cal S}$.
\end{definition}
The following theorems present results of theorems 1.14, 1.16, and 1.17 in~\cite{jF97}. For reader's convinience, we prove these results below.
\begin{theorem} \label{Theorem1.17-1}
For any token $\tau$ and any state $S$, we have either $\tau\in\widehat{S}$ or $\tilde{\tau}\in\widehat{S}$. Consequently, $|\widehat{S}|=|\widehat{V}|$ for any two states $S$ and $V$. {\rm (}$|A|$ stands for the cardinality of the set $A$.{\rm )}
\end{theorem}
\begin{proof}
Since $\tau$ is a token, there are two states $V$ and $W$ such that $W=V\tau$. By Axiom [M2*], there are straight messages $\boldsymbol m$ and $\boldsymbol n$ such that $S=V\boldsymbol m$ and $S=W\boldsymbol n$. By Axiom [M3], the message $\tau\boldsymbol n\widetilde{\boldsymbol m}$ is vacuous. Therefore, $\tilde{\tau}\in{\cal C}(\boldsymbol n)$ or $\tilde{\tau}\in{\cal C}(\widetilde{\boldsymbol m})$. It follows that $\tilde{\tau}\in\widehat{S}$ or $\tau\in\widehat{S}$. By Axiom [M4], we cannot have both $\tilde{\tau}\in\widehat{S}$ and $\tau\in\widehat{S}$.
\end{proof}
\begin{theorem} \label{Theorem1.16}
If $S$ and $V$ are two distinct states, with $S\boldsymbol m=V$ for some straight message $\boldsymbol m$, then \mbox{$\widehat{V}\setminus\widehat{S}={\cal C}(\boldsymbol m)$}.
\end{theorem}
\begin{proof}
If $\tau\in{\cal C}(\boldsymbol m)$, then $\tilde{\tau}\in{\cal C}(\widetilde{\boldsymbol m})$. Thus, $\tau\in\widehat{V}$ and $\tilde{\tau}\in\widehat{S}$. By Theorem~\ref{Theorem1.17-1}, the latter inclusion implies $\tau\notin\widehat{S}$. It follows that ${\cal C}(\boldsymbol m)\subseteq\widehat{V}\setminus\widehat{S}$.
If $\tau\in\widehat{V}\setminus\widehat{S}$, then there is a state $W$ and a straight message $\boldsymbol n$ such that $V=W\boldsymbol n$ and $\tau\in{\cal C}(\boldsymbol n)$. By Axiom [M2*], there is a straight message $\boldsymbol p$ such that $W=S\boldsymbol p$. By Axiom [M3], the message $\boldsymbol p\boldsymbol n\widetilde{\boldsymbol m}$ is vacuous, so $\tilde{\tau}\in{\cal C}(\boldsymbol p\boldsymbol n\widetilde{\boldsymbol m})$, that is, $\tau\in{\cal C}(\boldsymbol m\widetilde{\boldsymbol n}\widetilde{\boldsymbol p})$. But $\tau\notin{\cal C}(\widetilde{\boldsymbol p})$, since $S=W\widetilde{\boldsymbol p}$ and $\tau\notin\widehat{S}$, and $\tau\notin{\cal C}(\widetilde{\boldsymbol n})$, since $\tau\in{\cal C}(\boldsymbol n)$. Hence, $\tau\in{\cal C}(\boldsymbol m)$, that is, $\widehat{V}\setminus\widehat{S}\subseteq{\cal C}(\boldsymbol m)$.
\end{proof}
\begin{theorem} \label{Theorem1.17-2}
For any token $\tau$ and any state $S$, we have either $\tau\in\widehat{S}$ or $\tilde{\tau}\in\widehat{S}$. Moreover,
$$
S=V\quad\Leftrightarrow\quad\widehat{S}=\widehat{V}.
$$
\end{theorem}
\begin{proof}
Let $\widehat{S}=\widehat{V}$ and let $\boldsymbol m$ be a straigt message producing $V$ from $S$. By Theorem~\ref{Theorem1.16},
$$
\varnothing=\widehat{V}\setminus\widehat{S}={\cal C}(\boldsymbol m).
$$
Thus, $S=V$.
\end{proof}
\begin{theorem} \label{Theorem1.14}
Let $\boldsymbol m$ and $\boldsymbol n$ be two distinct straight messages transforming some state $S$. Then $S\boldsymbol m=S\boldsymbol n$ if and only if ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$.
\end{theorem}
\begin{proof}
Suppose that $V=S\boldsymbol m=S\boldsymbol n$. By Theorem~\ref{Theorem1.16},
$$
{\cal C}(\boldsymbol m)=\widehat{V}\setminus\widehat{S}={\cal C}(\boldsymbol n).
$$
Suppose that ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$ and let $V=S\boldsymbol m$ and $W=S\boldsymbol n$. By Theorem~\ref{Theorem1.16},
$$
\widehat{V}\Delta\widehat{S}={\cal C}(\boldsymbol m)\cup{\cal C}(\widetilde{\boldsymbol m})={\cal C}(\boldsymbol n)\cup{\cal C}(\widetilde{\boldsymbol n})=\widehat{W}\Delta\widehat{S},
$$
which implies $\widehat{V}=\widehat{W}$. By Theorem~\ref{Theorem1.17-2}, $V=W$.
\end{proof}
One particular class of media plays an important role in our constructions (cf.~\cite{jFsO02}).
\begin{definition} \label{D:complete-media}
A medium $({\cal S},{\cal T})$ is called \emph{complete} if for any state $S\in{\cal S}$ and token $\tau\in{\cal T}$, either $\tau$ or $\tilde{\tau}$ is effective on $S$.
\end{definition}
An important example of a complete medium is found below in Theorem~\ref{CompleteMedium}.
\section{Isomorphisms, embeddings, and token subsystems}
The purpose of combinatorial media theory is to find and examine those properties of media that do not depend on a particular structure of individual states and tokens. For this purpose we introduce the concepts of embedding and isomorphism for token systems.
\begin{definition} \label{D:embedding}
Let $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ be two token systems. A pair $(\alpha,\beta)$ of one--to--one functions $\alpha:{\cal S}\rightarrow{\cal S}'$ and $\beta:{\cal T}\rightarrow{\cal T}'$ such that
$$
S\tau=T \quad \Leftrightarrow \quad \alpha\left(S\right)\beta\left(\tau\right)=\alpha\left(T\right)
$$
for all $S,T\in{\cal S}$, $\tau\in{\cal T}$ is called an \emph{embedding} of the token system $({\cal S},{\cal T})$ into the token system $({\cal S}',{\cal T}')$.
Token systems $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ are \emph{isomorphic} if there is an embedding $(\alpha,\beta)$ from $({\cal S},{\cal T})$ into $({\cal S}',{\cal T}')$ such that both $\alpha$ and $\beta$ are bijections.
\end{definition}
Clearly, if one of two isomorphic token systems is a medium, then the other one is also a medium.
A general remark is in order. If a token system $({\cal S},{\cal T})$ is a medium and $S\tau_1=S\tau_2\not=S$ for some state $S$, then, by axiom [M3], $\tau_1=\tau_2$. In particular, if $(\alpha,\beta)$ is an embedding of a medium into a medium, then $\beta(\tilde{\tau})=\widetilde{\beta(\tau)}$. We extend $\beta$ to the semigroup of messages by defining $\beta(\tau_1\cdots\tau_k)=\beta(\tau_1)\cdots\beta(\tau_k)$. Clearly, the image $\beta(\boldsymbol m)$ of a straight message $\boldsymbol m$ is a straight message.
Let $({\cal S},{\cal T})$ be a token system and ${\cal Q}$ be a subset of ${\cal S}$ consisting of more than two elements. The restriction of a token $\tau\in{\cal T}$ to ${\cal Q}$ is not necessarily a token on ${\cal Q}$. In order to construct a medium with the set of states ${\cal Q}$, we introduce the following concept.
\begin{definition} \label{D:restriction}
Let $({\cal S},{\cal T})$ be a token system, ${\cal Q}$ be a nonempty subset of ${\cal S}$, and
$\tau\in{\cal T}$. We define a \emph{reduction} of $\tau$ to ${\cal Q}$ by
$$
S\tau_{{\cal Q}} = \begin{cases}
S\tau & \text{if $S\tau\in{\cal Q}$,} \\
S & \text{if $S\tau\notin{\cal Q}$,}
\end{cases}
$$
for $S\in{\cal Q}$. A token system $({\cal Q},{\cal T}_{{\cal Q}})$ where
${\cal T}_{{\cal Q}}=\{\tau_{{\cal Q}}\}_{\tau\in{\cal T}} \setminus\{\tau_0\}$ is the set of all distinct reductions of tokens in ${\cal T}$ to ${\cal Q}$ different from the identity function $\tau_0$ on ${\cal Q}$, is said to be the \emph{reduction} of $({\cal S},{\cal T})$ to ${\cal Q}$.
We call $({\cal Q},{\cal T}_{{\cal Q}})$ a \emph{token subsystem} of $({\cal S},{\cal T})$. If both $({\cal S},{\cal T})$ and $({\cal Q},{\cal T}_{{\cal Q}})$ are media, we call
$({\cal Q},{\cal T}_{{\cal Q}})$ a \emph{submedium} of $({\cal S},{\cal T})$.
\end{definition}
A reduction of a medium is not necessarily a submedium of a given medium. Consider, for instance, the following medium:
{
}
\noindent
The set of tokens of the reduction of this medium to ${\cal Q}=\{P,R\}$ is empty. Thus this reduction is not a medium (Axiom [M2] is not satisfied).
The image $(\alpha({\cal S}),\beta({\cal T}))$ of a token system $({\cal S},{\cal T})$ under embedding $(\alpha,\beta):({\cal S},{\cal T})\rightarrow({\cal S}',{\cal T}')$ is not, generally speaking, the reduction of $({\cal S}',{\cal T}')$ to $\alpha({\cal S})$. Indeed, let ${\cal S}'={\cal S}$, and let ${\cal T}$ be a proper nonempty subset of ${\cal T}'$. Then the image of $({\cal S},{\cal T})$ under the identity embedding is not the reduction of $({\cal S},{\cal T}')$ to ${\cal S}$ (which is $({\cal S},{\cal T}')$ itself).
On the other hand, this is true in the case of media as the following proposition demonstrates.
\begin{proposition}
Let $(\alpha,\beta):({\cal S},{\cal T})\rightarrow({\cal S}',{\cal T}')$ be an embedding of a medium $({\cal S},{\cal T})$ into a medium $({\cal S}',{\cal T}')$. Then the reduction $(\alpha({\cal S}),{\cal T}'_{\alpha({\cal S})})$ is isomorphic to $({\cal S},{\cal T})$.
\end{proposition}
\begin{proof}
For $\tau\in{\cal T}$, we define $\beta'(\tau)=\beta(\tau)_{\alpha({\cal S})}$, the reduction of $\beta(\tau)$ to $\alpha({\cal S})$. Let $S\tau=T$ for $S\not=T$ in ${\cal S}$. Then $\alpha(S)\beta(\tau)=\alpha(T)$ for $\alpha(S)\not=\alpha(T)$ in $\alpha({\cal S})$. Hence, $\beta'$ maps ${\cal T}$ to ${\cal T}'_{\alpha({\cal S})}$.
Let us show that $(\alpha,\beta')$ is an isomorphism from $({\cal S},{\cal T})$ onto $(\alpha({\cal S}),{\cal T}'_{\alpha({\cal S})})$.
\begin{itemize}
\item[(i)] $\beta'$ is onto. Suppose $\tau'_{\alpha({\cal S})}\not=\tau_0$ for some $\tau'\in{\cal T}'$. Then there are $P\not= Q$ in ${\cal S}$ such that $\alpha(P)\tau'_{\alpha({\cal S})}=\alpha(P)\tau'=\alpha(Q)$. Let $Q=P\boldsymbol m$ where $\boldsymbol m$ is a straight message. We have
$$
\alpha(Q)=\alpha(P\boldsymbol m)=\alpha(P)\beta(\boldsymbol m)=\alpha(P)\tau',
$$
implying, by Theorem~\ref{Theorem1.16}, $\beta(\boldsymbol m)=\tau'$, since $\beta(\boldsymbol m)$ is a straight message. Hence, $\boldsymbol m=\tau$ for some $\tau\in{\cal T}$. Thus $\beta(\tau)=\tau'$, which implies
$$
\beta'(\tau)=\beta(\tau)_{\alpha({\cal S})}=\tau'_{\alpha({\cal S})}.
$$
\item[(ii)] $\beta'$ is one--to--one. Suppose $\beta'(\tau_1)=\beta'(\tau_2)$. Since $\beta'(\tau_1)$ and $\beta'(\tau_2)$ are tokens on $\alpha({\cal S})$ and $({\cal S}',{\cal T}')$ is a medium, we have $\beta(\tau_1)=\beta(\tau_2)$. Hence, $\tau_1=\tau_2$.
\item[(iii)]Finally,
$$
S\tau=T \quad \Leftrightarrow \quad \alpha\left(S\right)\beta'\left(\tau\right)=\alpha\left(T\right),
$$
since
$$
S\tau=T \quad \Leftrightarrow \quad \alpha\left(S\right)\beta\left(\tau\right)=\alpha\left(T\right).
$$
\end{itemize}
\end{proof}
We shall see later (Theorem~\ref{FiniteRepresentationTheorem}) that any medium is isomorphic to a submedium of a complete medium.
\section{Families of sets representable as media}
In this section, the objects of our study are token systems which are defined by means of families of subsets of a given set $X$.
Let $\mathfrak P(X)=2^X$ be the family of all subsets of $X$ and let ${\cal G}=\cup\{\gamma_x,\tilde{\gamma}_x\}$ be the family of functions from $\mathfrak P(X)$ into $\mathfrak P(X)$ defined by
\begin{align*}
&\gamma_x:S\mapsto S\gamma_x = S\cup\{x\}, \\
&\tilde{\gamma}_x:S\mapsto S\tilde{\gamma}_x = S\setminus\{x\}
\end{align*}
for all $x\in X$.
It is clear that $(\mathfrak P(X),{\cal G})$ is a token system and that, for a given $x\in X$, the token $\tilde{\gamma}_x$ is the unique reverse of the token $\gamma_x$.
Let ${\cal F}$ be a nonempty family of subsets of $X$. In what follows, $({\cal F},{\cal G}_{\cal F})$ stands for the reduction of the token system $(\mathfrak P(X),{\cal G})$ to ${\cal F}\subseteq\mathfrak P(X)$.
In order to characterize token systems $({\cal F},{\cal G}_{\cal F})$ which are media, we introduce some geometric concepts in $\mathfrak P(X)$ (cf. \cite{kB73,vKsO75,sO83}).
\begin{definition} \label{D:line segment}
Given $P,Q\in \mathfrak P(X)$, the \emph{interval} $[P,Q]$ is defined by
\begin{equation*}
[P,Q]=\{R\in 2^X:P\cap Q\subseteq R\subseteq P\cup Q\}.
\end{equation*}
If $R\in[P,Q]$, we say that $R$ \emph{lies between $P$ and $Q$}.
A sequence $P=P_0,P_1,\ldots,P_n=Q$ of distinct elements of $\mathfrak P(X)$ is a \emph{line segment} between $P$ and $Q$ if
\begin{enumerate}
\item[\emph{L1.}] $P_i\in[P_k,P_m]$ for $k\leq i\leq m,$ and
\item[\emph{L2.}] $R\in[P_i,P_{i+1}]$ implies $R=P_i$ or $R=P_{i+1}$ for all $0\leq i\leq n-1$.
\end{enumerate}
The \emph{distance}
between $P$ and $Q$ is defined by
\begin{equation*}
d(P,Q) = \begin{cases}
|P\Delta Q|, &\text{if $P\Delta Q$ is a finite set,} \\
\infty, &\text{otherwise,}
\end{cases}
\end{equation*}
where $\Delta$ stands for the symmetric difference operation.
A binary relation $\sim$ on $\mathfrak P(X)$ is defined by
\begin{equation*}
P\sim Q\quad\Leftrightarrow\quad d(P,Q)<\infty.
\end{equation*}
The relation $\sim$ is an equivalence relation on $\mathfrak P(X)$. We denote $[S]$ the equivalence class of $\sim$ containing $S\in\mathfrak P(X)$. We also denote $\mathfrak PF(X)=[\varnothing]$, the family of all finite subsets of the set $X$.
\end{definition}
\begin{theorem} \label{DistanceTheorem}
Given $S\in \mathfrak P(X)$,
\begin{itemize}
\item[\emph{(i)}] The distance function $d$ defines a metric on $[S]$.
\item[\emph{(ii)}] $R$ lies between $P$ and $Q$ in $[S]$, that is, $R\in[P,Q],$ if and only if
\begin{equation*}
d(P,R)+d(R,Q)=d(P,Q).
\end{equation*}
\item[\emph{(iii)}] A sequence $P=P_0,P_1,\ldots,P_n=Q$ is a line segment between $P$ and $Q$ in $[S]$ if and only if $d(P,Q)=n$ and $d(P_i,P_{i+1})=1,\;0\leq i\leq n-1$.
\end{itemize}
\end{theorem}
\begin{proof}
(i) It is clear that $d(P,Q)\geq 0$ and $d(P,Q)=0$ if and only if $P=Q$, and that $d(P,Q)=d(Q,P)$ for all $P,Q\in\langle S\rangle$.
It remains to verify the triangle inequality. Let $S_1,S_2,S_3$ be three sets in $[S]$. Since these sets belong to the same equivalence class of the relation $\sim$, the following six sets
$$
V_i = (S_j\cap S_k)\setminus S_i,\;U_i = S_i\setminus(S_j\cup S_k)\quad\text{for $\{i,j,k\}=\{1,2,3\}$,}
$$
are finite. It is not difficult to verify that
$$
S_i\Delta S_j=U_i\cup V_j\cup U_j\cup V_i,
$$
with disjoint sets in the right hand side of the equality. We have
\begin{align} \label{so1 triangle id}
&\phantom{=(}|S_i\Delta S_j|+|S_j\Delta S_k|\notag\\
&=(|U_i|+|V_j|+|U_j|+|V_i|)+(|U_j|+|V_k|+|U_k|+|V_j|)\\
&=|S_i\Delta S_k|+2(|U_j|+|V_j|), \notag
\end{align}
which implies the triangle inequality.
(ii) By~(\ref{so1 triangle id}),
$$
|S_i\Delta S_j|+|S_j\Delta S_k|=|S_i\Delta S_k|
$$
if and only if $U_j=\varnothing$ and $V_j=\varnothing$, or, equivalently, if and only if
$$
S_i\cap S_k\subseteq S_j\subseteq S_i\cup S_j.
$$
(iii) (Necessity.) By Condition L2 of Definition~\ref{D:line segment},
$$
d(P_i,P_{i+1})=|P_i\Delta P_{i+1|}|=1.
$$
Indeed, if there were $x\in P_i\setminus P_{i+1}$ and $y\in P_{i+1}\setminus P_i$, then the set $R=(P_i\setminus\{x\})\cup\{y\}$ would lie strictly between $P_i$ and $P_{i+1}$, a contradiction. By Condition L1 of Definition~\ref{D:line segment} and part (ii) of the theorem, we have
$$
d(P,Q)=1+d(P_1,Q)=\cdots=\underbrace{1+1+\cdots+1}_{n}=n.
$$
(Sufficiency.) Let $P=P_0,P_1,\ldots,P_n=Q$ be a sequence of sets such that $d(P_i,P_{i+1})=1$ and $d(P,Q)=n$. Condition L2 of Definition~\ref{D:line segment} is clearly satisfied.
By the triangle inequality, we have
$$
d(P,P_i)\leq i,\;\;d(P_i,P_j)\leq j-i,\;\;d(P_j,Q)\leq n-j
$$
for $i<j$. Let us add these inequalities and use the triangle inequality again. We obtain
$$
n=d(P,Q)\leq d(P,P_i)+d(P_i,P_j)+d(P_j,Q)\leq n.
$$
It follows that $d(P_i,P_j)=j-i$ for all $i<j$. In particular,
$$
d(P_k,P_i)+d(P_i,P_m)=d(P_k,P_m)\quad\text{for $k\leq i\leq m$.}
$$
By part (ii) of the theorem, $P_i\in[P_k,P_m]$ for $k\leq i\leq m$, which proves Condition L1 of Definition~\ref{D:line segment}.
\end{proof}
The concept of a line segment seems to be similar to the concept of a straight message. The following lemma validates this intuition.
\begin{lemma} \label{StraightMessage}
Let $({\cal F},{\cal G}_{\cal F})$ be a token system and let $P$ and $Q$ be two distinct sets in ${\cal F}$. A message $\boldsymbol m=\tau_1\tau_2\cdots\tau_n$ producing $Q$ from $P$ is
straight if and only if
$$
P_0=P,\;P_1=P_0\tau_1,\;\ldots,\;P_n=P_{n-1}\tau_n=Q
$$
is a line segment between $P$ and $Q$.
\end{lemma}
\begin{proof}
(Necessity.) We use induction on $n=\ell(\boldsymbol m)$. Let $\boldsymbol m=\tau_1$. Since $\tau_1$ is either $\gamma_x$ or $\tilde{\gamma}_x$ for some $x\in X$ and effective, either $Q=P\cup\{x\}$ or $Q=P\setminus\{x\}$ and $Q\not= P$. Therefore, $d(P,Q)=1$, that is $\{P,Q\}$ is a line segment.
Now, let us assume that the statement holds for all straight messages $\boldsymbol n$ with $\ell(\boldsymbol n)=n-1$ and let $\boldsymbol m=\tau_1\tau_2\cdots\tau_n$ be a straight message producing $Q$ from $P$. Clearly, $\boldsymbol m_1=\tau_2\cdots\tau_n$ is a straight message producing $Q$ from $P_1=P\tau_1$ and $\ell(\boldsymbol m_1)=n-1$.
Suppose that $\tau_1=\gamma_x$ for some $x\in X$. Since $\boldsymbol m$ is stepwise effective, $x\notin P$. Since $\boldsymbol m$ is consistent, $x\in Q$. Therefore $P_1=P\tau_1=P\cup\{x\}\in [P,Q]$. Suppose that $\tau_1=\tilde{\gamma}_x$ for some $x\in X$. Since $\boldsymbol m$ is stepwise effective, $x\in P$. Since it is consistent, $x\notin Q$. Again, $P_1=P\setminus\{x\}\in[P,Q]$. In either case, $d(P,P_1)=1$. By Theorem~\ref{DistanceTheorem}(ii) and the induction hypothesis, $d(P,Q)=d(P,P_1)+d(P_1,Q)=n$. By the induction hypothesis and Theorem~\ref{DistanceTheorem}(iii), the sequence
\begin{equation*}
P_0=P,P_1=P_0\tau_1,\ldots,P_n=P_{n-1}\tau_n=Q
\end{equation*}
is a line segment between $P$ and $Q$.
(Sufficiency.) Let $P_0=P,P_1=P_0\tau_1,\ldots,P_n=P_{n-1}\tau_n=Q$ be a line segment between $P$ and $Q$ for some message $\boldsymbol m=\tau_1\cdots\tau_n$. Clearly, $\boldsymbol m$ is stepwise effective. To prove consistency, we use induction on $n$. The
statement is trivial for $n=1$. Suppose it holds for all messages of length less than $n$ and let $\boldsymbol m$ be a message of length $n$. Suppose $\boldsymbol m$ is inconsistent. By the induction hypothesis, this can occur only if either $\tau_1=\gamma_x$, $\tau_n=\tilde{\gamma}_x$ or $\tau_1=\tilde{\gamma}_x$, $\tau_n=\gamma_x$ for some $x\in X$. In the former case, $x\in P_1,\;x\notin P,\;x\notin Q$. In the latter case, $x\notin P_1,\;x\in P,\;x\in Q$. In both cases, $P_1\notin[P,Q]$, a
contradiction.
\end{proof}
The following theorem is an immediate consequence of Lemma~\ref{StraightMessage}.
\begin{theorem} \label{Fin[S]}
If $({\cal F},{\cal G}_{\cal F})$ is a medium, then ${\cal F}\subseteq [S]$ for some $S\in\mathfrak P(X)$.
\end{theorem}
Clearly, the converse of this theorem is not true. To characterize those token systems $({\cal F},{\cal G}_{\cal F})$ which are media, we use the concept of a well graded family of sets \cite{jDjF97}. (See also \cite{vKsO75,sO80} and \cite{sO83} where the same concept was introduced as a ``completeness condition''.)
\begin{definition}
A family ${\cal F}\subseteq\mathfrak P(X)$ is \emph{well graded} if for any two distinct sets $P$ and $Q$ in ${\cal F}$, there is a sequence of sets $P=R_0,R_1,\ldots,R_n=Q$ such that $d(R_{i-1},R_i)=1$ for $i=1,\ldots,n$ and $d(P,Q)=n$.
\end{definition}
In other words, ${\cal F}$ is a well graded family if for any two distinct elements $P,Q\in{\cal F}$ there is a line segment between $P$ and $Q$ in ${\cal F}$.
\begin{theorem} \label{T:wg gamma}
Let ${\cal F}$ be a well graded family of subsets of some set $X$. Then $x\in X$ defines tokens $\gamma_x,\tilde{\gamma}_x\in{\cal G}_{\cal F}$ if and only if $x\in\cup{\cal F}\setminus\cap{\cal F}$.
\end{theorem}
\begin{proof}
It is clear that elements of $X$ that are not in $\cup{\cal F}\setminus\cap{\cal F}$ do not define tokens in ${\cal G}_{\cal F}$.
Suppose that $x\in\cup{\cal F}\setminus\cap{\cal F}$. Then there are sets $P$ and $Q$ in ${\cal F}$ such that $x\in Q\setminus P$. Let $R_0,R_1,\ldots,R_n$ be a line segment in ${\cal F}$ between $P$ and $Q$. Then there is $i$ such that $R_{i+1}=R_i\cup\{x\}$ and $x\notin R_i$. Therefore, $R_{i+1}=R_i\gamma_x$ and $R_i=R_{i+1}\tilde{\gamma}_x$, that is, $\gamma_x,\tilde{\gamma}_x\in{\cal G}_{\cal F}$.
\end{proof}
In the rest of the paper we assume that a well graded family ${\cal F}$ of subsets of $X$ defining a token system $({\cal F},{\cal G}_{\cal F})$ satisfies the following conditions:
\begin{equation} \label{E:wg gamma}
\cap{\cal F}=\varnothing\quad\text{and}\quad\cup{\cal F}=X.
\end{equation}
We have the following theorem (cf.~Theorem~4.2 in~\cite{sOaD00}).
\begin{theorem} \label{WellGradedMedia}
A token system $({\cal F},{\cal G}_{\cal F})$ is a medium if and only if ${\cal F}$ is a well graded family of subsets of $X$.
\end{theorem}
\begin{proof}
(Necessity.) Suppose $({\cal F},{\cal G}_{\cal F})$ is a medium. By axiom [M2*], for given $P,Q\in{\cal F}$, there is a straight message producing $Q$ from $P$. By Lemma~\ref{StraightMessage}, there is a line segment in ${\cal F}$ between $P$ and $Q$. Hence ${\cal F}$ is well graded.
(Sufficiency.) Let ${\cal F}$ be a well graded family of subsets of $X$. We need to show that the four axioms defining a medium are satisfied for $({\cal F},{\cal G}_{\cal F})$.
[M1]. Clearly, $\gamma_x$ and $\tilde{\gamma}_x$ are unique mutual reverses of each other.
[M2*]. Follows immediately from Lemma~\ref{StraightMessage}.
[M3]. (Necessity.) Let $\boldsymbol m$ be a message which is stepwise effective for $P\in{\cal F}$ and ineffective for this state, that is, $P\boldsymbol m=P$.
Let $\tau$ be a token in $\boldsymbol m$ such that $\tilde\tau\notin\boldsymbol m$. If $\tau=\gamma_x$ for some $x\in X$, then $x\notin P$ and $x\in P\boldsymbol m$, since $\boldsymbol m$ is stepwise effective for $P$ and $\tilde\gamma_x=\tilde\tau\notin\boldsymbol m$. We have a contradiction, since $P\boldsymbol m=P$. In a similar way, we obtain a contradiction assuming that $\tau=\tilde\gamma_x$. Thus, for each token $\tau$ in $\boldsymbol m$, there is an appearance of the reverse token $\tilde\tau$ in $\boldsymbol m$.
Because $\boldsymbol m$ is stepwise effective, the appearances of tokens $\tau$ and $\tilde{\tau}$ in $\boldsymbol m$ must alternate. Suppose that the sequence of appearances of $\tau$ and $\tilde{\tau}$ begins and ends with $\tau=\gamma_x$ (the argument is similar if $\tau=\tilde{\gamma}_x$). Since the message $\boldsymbol m$ is stepwise effective for $P$ and ineffective for this state, we must have $x\notin P$ and $x\in P\boldsymbol m=P$, a contradiction. It follows that $\boldsymbol m$ is vacuous.
(Sufficiency.) Let $\boldsymbol m$ be a vacuous message which is stepwise effective for some state $P$. Since $\boldsymbol m$ is vacuous, the number of appearances of $\gamma_x$ in $\boldsymbol m$ is equal to the number of appearances of $\tilde\gamma_x$ for any $x\in X$. Because $\boldsymbol m$ is stepwise effective, the appearances of tokens $\gamma_x$ and $\tilde{\gamma}_x$ in $\boldsymbol m$ must alternate. It follows that $x\in P$ if and only if $x\in P\boldsymbol m$, that is $P\boldsymbol m=P$. Thus the message $\boldsymbol m$ is ineffective for $P$.
[M4]. Suppose two straight messages $\boldsymbol m$ and $\boldsymbol n$ produce $R$ from $P$ and $Q$, respectively, that is, $R=P\boldsymbol m$ and $R=Q\boldsymbol n$. Let us assume that $\boldsymbol m$ and $\boldsymbol n$ are not jointly consistent, that is, that $\boldsymbol{mn}$ is inconsistent. Then there are two mutually reverse tokens $\tau$ and $\tilde{\tau}$ in $\boldsymbol{mn}$. Since $\boldsymbol m$ and $\boldsymbol n$ are straight messages, we may assume, without loss of generality, that $\tau=\gamma_x$ is in $\boldsymbol m$ and $\tilde{\tau}=\tilde{\gamma}_x$ is in $\boldsymbol n$ for some $x\in X$. Since $\boldsymbol m$ is straight, $x\in R$. Since $\boldsymbol n$ is straight, $x\notin R$, a contradiction.
\end{proof}
Clearly, for a given $S\in\mathfrak P(X)$, $[S]$ is a well graded family of subsets of $X$. Hence, $([S],{\cal G}_{[S]})$ is a medium. It is easy to see that any such medium is a complete medium (see Definition~\ref{D:complete-media}). The converse is also true as the following theorem asserts.
\begin{theorem} \label{CompleteMedium}
A medium in the form $({\cal F},{\cal G}_{\cal F})$ is complete if and only if ${\cal F}=[S]$ for some $S\in\mathfrak P(X)$.
\end{theorem}
\begin{proof}
We need to prove necessity only. Suppose that $({\cal F},{\cal G}_{\cal F})$ is a complete medium. By Theorem~\ref{Fin[S]}, ${\cal F}\subseteq[S]$ for some $S\in\mathfrak P(X)$. For a given $P\in[S]$, let $m=d(P,S)$. We prove that $P\in{\cal F}$ by induction on $m$.
Let $m=1$. Then either $P=S\cup\{x\}$ or $P=S\setminus\{x\}$ for some $x\in X$ and $P\not= S$. In the former case, $x\notin S$ implying that $\tilde{\gamma}_x$ is not effective on $S$. By completeness, $S\gamma_x= P$. Thus $P\in{\cal F}$. Similarly, if $P=S\setminus\{x\}$, then $x\in S$ and $S\tilde{\gamma}_x= P$.
Suppose that $Q\in{\cal F}$ for all $Q\in[S]$ such that $d(S,Q)=m$ and let $P$ be an element in $[S]$ such that $d(S,P)=m+1$. Then there exists $R\in{\cal F}$ such that $d(S,R)=m$ and $d(R,P)=1$. Since $[R]=[S]$, it follows from the argument in the previous paragraph that $P\in{\cal F}$.
\end{proof}
The following theorem shows that all media in the form $([S],{\cal G}_{[S]})$ are isomorphic.
\begin{theorem} \label{IsomorphismTheorem}
For any $S',S\in\mathfrak P(X)$, the media $([S'],{\cal G}_{[S']})$ and $([S],{\cal G}_{[S]})$ are isomorphic.
\end{theorem}
\begin{proof}
It suffices to consider the case when $S'=\varnothing$.
We define $\alpha:\mathfrak PF(X)\rightarrow[S]$ and $\beta:{\cal G}_{\mathfrak PF(X)}\rightarrow{\cal G}_{[S]}$ by
\begin{align*}
&\alpha(P)=P\Delta S, \\
&\beta(\tau)= \begin{cases}
\tilde{\tau} &\text{if $\tau=\gamma_x$ or $\tau=\tilde{\gamma}_x$ for $x\in S$,} \\
\tau &\text{if $\tau=\gamma_x$ or $\tau=\tilde{\gamma}_x$ for $x\notin S$.}
\end{cases}
\end{align*}
Clearly, $\alpha$ and $\beta$ are bijections. To prove that $P\tau=Q$ implies $\alpha(P)\beta(\tau)=\alpha(Q)$, let us consider the following cases:
\begin{enumerate}
\item $\tau=\gamma_x,\;x\in S$.
\begin{enumerate}
\item $x\in P$. Then $Q=P$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\setminus\{x\} = P\Delta S = Q\Delta S.$$
\item $x\notin P$. Then $Q=P\cup\{x\}$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\setminus\{x\} = P\Delta(S\setminus\{x\}) = Q\Delta S.$$
\end{enumerate}
\item $\tau=\gamma_x,\;x\notin S$.
\begin{enumerate}
\item $x\in P$. Then $Q=P$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\cup\{x\} = P\Delta S = Q\Delta S.$$
\item $x\notin P$. Then $Q=P\cup\{x\}$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\cup\{x\} = Q\Delta S.$$
\end{enumerate}
\end{enumerate}
A similar argument proves the theorem in the case when $\tau=\tilde{\gamma}_x$. It is also easy to verify the converse implication: $\alpha(P)\beta(\tau)=\alpha(Q)\;\Rightarrow\; P\tau=Q$.
\end{proof}
We summarize the results of this section as follows:
1.~A token system $({\cal F},{\cal G}_{\cal F})$ is a medium if and only if ${\cal F}$ is a well graded family of subsets of $X$.
2.~A complete medium in the form $({\cal F},{\cal G}_{\cal F})$ is $([S],{\cal G}_{[S]})$ for some $S\in\mathfrak P(X)$ and all such media are isomorphic.
3.~Any medium in the form $({\cal F},{\cal G}_{\cal F})$ is a submedium of a complete medium and isomorphic to a submedium of the complete medium $(\mathfrak PF(X),{\cal G}_{\mathfrak PF(X)})$ of all finite subsets of $X$.
\section{The representation theorem}
In this section we show that any medium is isomorphic to a medium in the form $({\cal F},{\cal G}_{\cal F})$ where ${\cal F}$ is a well graded family of finite subsets of some set $X$. In our construction we employ the concept of `orientation'~\cite{jFsO02}.
\begin{definition} \label{D:orientation}
An \emph{orientation} of a medium
$({\cal S},{\cal T})$ is a partition of its set of tokens into two classes ${\cal T}^{+}$ and ${\cal T}^{-}$, respectively called \emph{positive} and \emph{negative}, such that for any $\tau\in{\cal T}$, we have
$$
\tau\in{\cal T}^{+}\quad\Leftrightarrow\quad\tilde{\tau}\in{\cal T}^{-}
$$
A medium $\left({\cal S},{\cal T}\right)$ equipped with an orientation $\{{\cal T}^{+},{\cal T}^{-}\}$ is said to be \emph{oriented} by $\{{\cal T}^{+},{\cal T}^{-}\}$, and tokens from ${\cal T}^{+}$
\emph{(}resp. ${\cal T}^{-}$\emph{)} are called \emph{positive} \emph{(}resp. \emph{negative}\emph{)}. The \emph{positive content} \emph{(}resp.~\emph{negative content}\emph{)} of a state $S$ is the set $\widehat{S}^{+}=\widehat{S}\cap{\cal T}^{+}$ \emph{(}resp. $\widehat{S}^{-}=\widehat{S}\cap{\cal T}^{-}$\emph{)} of its positive \emph{(}resp. negative\emph{)} tokens.
\end{definition}
Let $({\cal S},{\cal T})$ be a medium equipped with an orientation $\{{\cal T}^{+},{\cal T}^{-}\}$. In what follows, we show that $({\cal S},{\cal T})$ is isomorphic to the medium $({\cal F},{\cal G}_{\cal F})$ of the well graded family ${\cal F}=\{\widehat{S}^+\}_{S\in{\cal S}}$ of all positive contents.
\begin{lemma} \label{S+=T+}
\emph{(cf.~\cite{sOaD00})} For any two states $S,T\in{\cal S}$,
\begin{equation*}
\widehat{S}^+=\widehat{T}^+\quad\Leftrightarrow\quad S=T.
\end{equation*}
\end{lemma}
\begin{proof}
By Theorem~\ref{Theorem1.17-2}, it suffices to prove that $\widehat{S}^+=\widehat{T}^+$ implies $\widehat{S}=\widehat{T}$. Let $\tau\in\widehat{S}^-$. Then, by Theorem~\ref{Theorem1.17-1}, $\tilde{\tau}\notin\widehat{S}^+$. Hence, $\tilde{\tau}\notin\widehat{T}^+$ which implies $\tau\in\widehat{T}^-$. Therefore $\widehat{S}^-\subseteq\widehat{T}^-$. By symmetry, $\widehat{S}^-=\widehat{T}^-$. Hence, $\widehat{S}=\widehat{T}$.
\end{proof}
We define
\begin{equation} \label{alpha}
\alpha : S\mapsto \widehat{S}^+,
\end{equation}
for $S\in{\cal S}$. It follows from Lemma~\ref{S+=T+} that $\alpha$ is a bijection.
Suppose that $\tau\in\cap\,{\cal F}=\cap_{S\in{\cal S}}\widehat{S}^+$. There are $S,T\in{\cal S}$ such that $T=S\tau$. Then, by Theorem~\ref{Theorem1.16}, $\widehat{T}\setminus\widehat{S}=\{\tau\}$, that is, $\tau\notin\widehat{S}\supseteq\widehat{S}^+$. Hence, $\cap\,{\cal F}=\varnothing$.
Let $\tau\in{\cal T}^+$. There are $S,T\in{\cal S}$ such that $T=S\tau$. Then $\tau\in\widehat{T}^+$. Hence, $\cup\,{\cal F}={\cal T}^+$.
We define a mapping $\beta:{\cal T}\rightarrow{\cal G}_{\cal F}$ by
\begin{equation} \label{beta}
\beta(\tau)=\begin{cases}
\gamma_\tau &\text{if $\tau\in{\cal T}^+$,} \\
\tilde{\gamma}_{\tilde{\tau}} &\text{if $\tau\in{\cal T}^-$,}
\end{cases}
\end{equation}
and show that the pair $(\alpha,\beta)$, where $\alpha$ and $\beta$ are mappings defined, respectively, by (\ref{alpha}) and (\ref{beta}), is an isomorphism between the token systems $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$.
\begin{theorem} \label{RepresentationTheorem}
Let $({\cal S},{\cal T})$ be an oriented medium. For all $S,T\in{\cal S}$ and $\tau\in{\cal T}$,
\begin{equation*}
T=S\tau\quad\Leftrightarrow\quad\alpha(T)=\alpha(S)\beta(\tau),
\end{equation*}
that is, the token systems $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$ are isomorphic.
\end{theorem}
\begin{proof}
(i) Suppose that $\tau\in{\cal T}^+$. We need to prove that
\begin{equation*}
T=S\tau\quad\Leftrightarrow\quad\widehat{T}^+=\widehat{S}^+\gamma_\tau
\end{equation*}
for all $S,T\in{\cal S}$ and $\tau\in{\cal T}$.
Let us consider the following cases:
\begin{enumerate}
\item $\tau\in\widehat{S}^+$. Suppose $S\tau=T\not=S$. Then, by Theorem~\ref{Theorem1.16}, $\tau\in\widehat{T}\setminus\widehat{S}$, a contradiction. Hence, $T=S$. Clearly,
$\widehat{S}^+=\widehat{S}^+\cup\{\tau\}=\widehat{S}^+\gamma_\tau$.
\item $\tau\notin\widehat{S}^+,\,\widehat{S}^+\cup\{\tau\}\notin{\cal F}$. Suppose that $S\tau=T\not=S$. By Theorem~\ref{Theorem1.16}, $\widehat{T}\setminus\widehat{S}=\{\tau\}$ and $\widehat{S}\setminus\widehat{T}=\{\tilde{\tau}\}$. Since $\tau\in{\cal T}^+$, we have $\widehat{S}^+\cup\{\tau\}=\widehat{T}^+\in{\cal F}$, a contradiction. Hence, in this case, $S=S\tau$ and $\widehat{S}^+=\widehat{S}^+\gamma_\tau$.
\item $\tau\notin\widehat{S}^+,\,\widehat{S}^+\cup\{\tau\}\in{\cal F}$. Then there exists $T\in{\cal S}$ such that $\widehat{T}^+=\widehat{S}^+\cup\{\tau\}$. Thus $\tau\in\widehat{T}\setminus\widehat{S}$. Suppose that there is $\tau'\not=\tau$ which is also in $\widehat{T}\setminus\widehat{S}$. Then $\tau'$ is a negative token. Since $\tau'\notin\widehat{S}$, we have $\tilde{\tau}'\in\widehat{S}^+\subset\widehat{T}^+\subseteq\widehat{T}$. Hence, $\tau'\notin\widehat{T}$, a contradiction. It follows that $\widehat{T}\setminus\widehat{S}=\{\tau\}$. By Theorem~\ref{Theorem1.16}, $T=S\tau$. By the argument in item 2, $\widehat{T}^+=\widehat{S}^+\cup\{\tau\}=\widehat{S}^+\gamma_\tau$.
\end{enumerate}
(ii) Suppose that $\tau\in{\cal T}^-$. Then $\tilde{\tau}\in{\cal T}^+$. We need to prove that
$$
T=S\tau\quad \Leftrightarrow\quad\widehat{T}^+=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}
$$
for all $S,T\in{\cal S}$ and $\tau\in{\cal T}$.
Let us consider the following cases:
\begin{enumerate}
\item $\tilde{\tau}\notin\widehat{S}^+$. Suppose that $S\tau=T\not=S$. Then $S=T\tilde{\tau}$ and, by Theorem~\ref{Theorem1.16}, $\tilde{\tau}\in\widehat{S}\setminus\widehat{T}$, a contradiction since $\tilde{\tau}$ is a positive token. Hence, $S=S\tau$. On the other hand, $\widehat{S}^+=\widehat{S}^+\setminus\{\tilde{\tau}\}=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}$.
\item $\tilde{\tau}\in\widehat{S}^+,\,\widehat{S}^+\setminus\{\tilde{\tau}\}\notin{\cal F}$. Suppose again that $S\tau=T\not=S$. Then $S=T\tilde{\tau}$ and, by Theorem~\ref{Theorem1.16}, $\{\tilde{\tau}\}=\widehat{S}\setminus\widehat{T}$ and $\{\tau\}=\widehat{T}\setminus\widehat{S}$. Since $\tilde{\tau}$ is a positive token, we have $\widehat{S}^+\setminus\{\tilde{\tau}\}=\widehat{T}^+$, a contradiction. Hence, in this case, $S=S\tau$ and $\widehat{S}^+=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}$.
\item $\tilde{\tau}\in\widehat{S}^+,\,\widehat{S}^+\setminus\{\tilde{\tau}\}\in{\cal F}$. There is $T\in{\cal S}$ such that $\widehat{T}^+=\widehat{S}^+\setminus\{\tilde{\tau}\}$. We have $\tau\notin\widehat{S}$, since $\tilde{\tau}\in\widehat{S}^+$, and $\tau\in\widehat{T}$, since $\tilde{\tau}\notin\widehat{T}^+$ and $\tilde{\tau}$ is a positive token. Hence, $\tau\in\widehat{T}\setminus\widehat{S}$. Suppose that there is $\tau'\not=\tau$ which is also in $\widehat{T}\setminus\widehat{S}$. Then $\tau'$ is a negative token. Since $\tau'\notin\widehat{S}$, we have $\tilde{\tau}'\in\widehat{S}^+$. Since $\tau'\not=\tau$, we have $\tilde{\tau}'\in\widehat{T}^+$. Hence, $\tau'\notin\widehat{T}$, a contradiction. It follows that $\widehat{T}\setminus\widehat{S}=\{\tau\}$. By Theorem~\ref{Theorem1.16}, $T=S\tau$. Clearly, $\widehat{T}^+=\widehat{S}^+\setminus\{\tau\}=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}$.
\end{enumerate}
\end{proof}
Since $({\cal S},{\cal T})$ is a medium, the token system $({\cal F},{\cal G}_{\cal F})$ is also a medium. By Theorem~\ref{WellGradedMedia}, we have the following result.
\begin{corollary}
${\cal F}=\{\widehat{S}^+\}_{S\in{\cal S}}$ is a well graded family of subsets of ${\cal T}^+$.
\end{corollary}
Theorem~\ref{RepresentationTheorem} states that an oriented medium is isomorphic to the medium of its family of positive contents. By `forgetting' the orientation, one can say that any medium $({\cal S},{\cal T})$ is isomorphic to some medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family ${\cal F}$ of sets. The following theorem is a stronger version of this result (cf.~Theorem~\ref{IsomorphismTheorem}).
\begin{theorem} \label{FiniteRepresentationTheorem}
Any medium $({\cal S},{\cal T})$ is isomorphic to a medium of a well graded family of finite subsets of some set $X$.
\end{theorem}
\begin{proof}
Let $S_0$ be a fixed state in ${\cal S}$. By Theorem~\ref{Theorem1.17-1}, the state $S_0$ defines an orientation $\{{\cal T}^{+},{\cal T}^{-}\}$ with ${\cal T}^-=\widehat{S}_0$ and ${\cal T}^{+}={\cal T}\setminus{\cal T}^-$. Let $\boldsymbol m$ be a straight message producing a state $S$ from the state $S_0$. By Theorem~\ref{Theorem1.16}, we have ${\cal C}(\boldsymbol m)=\widehat{S}\setminus\widehat{S}_0=\widehat{S}^+$. Thus, $\widehat{S}^+$ is a finite set. The statement of the theorem follows from Theorem~\ref{RepresentationTheorem}.
\end{proof}
\begin{remark}
{\rm An infinite oriented medium may have infinite positive contents of all its states. Consider, for instance, the medium $({\cal F},{\cal G}_{\cal F})$, where
$$
{\cal F}=\{S_n=]-\infty,n]:n\in\mathbb{Z}\}.
$$
Then each $\widehat{S}_n^+=\{\gamma_k\}_{k<n}$ is an infinite set. Nevertheless, by Theorem~\ref{FiniteRepresentationTheorem}, the medium $({\cal F},{\cal G}_{\cal F})$ is isomorphic to a medium of a well graded family of finite sets.
}
\end{remark}
\section{Media and graphs}
In this section we study connections between media and graph theories.
\begin{definition}
Let $({\cal S},{\cal T})$ be a medium. We say that a graph $G=(V,E)$ \emph{represents} $({\cal S},{\cal T})$ if there is a bijection $\alpha:{\cal S}\rightarrow V$ such that an unordered pair of vertices $PQ$ is an edge of the graph if and only if $P\not=Q$ (no loops) and there is $\tau\in{\cal T}$ such that $\alpha^{-1}(P)\tau=\alpha^{-1}(Q)$.
A graph $G$ representing a medium $({\cal S},{\cal T})$ is the \emph{graph of the medium} $({\cal S},{\cal T})$ if the set of states ${\cal S}$ is the set of vertices of $G$, the mapping $\alpha$ is the identity, and the edges of $G$ are defined as above.
\end{definition}
Clearly, any graph which is isomorphic to a graph representing a token system $({\cal S},{\cal T})$, also represents $({\cal S},{\cal T})$, and isomorphic media are represented by isomorphic graphs.
The main goal of this section is to show that two media which are represented by isomorphic graphs are isomorphic (Theorem~\ref{Medium=Graph}).
First, we prove three lemmas about media and their graph representations.
\begin{lemma} \label{Lemma6.1}
Let $(\mathcal{S,T})$ be a medium, and suppose that: $\tau\in{\cal T}$ is a token; $S,T,P$ and $Q$ are states in ${\cal S}$ such that $S\tau=T$ and $P\tau=Q$. Let $\boldsymbol m$ and $\boldsymbol n$ be two straight messages producing $P$ from $S$ and $Q$ from $T$, respectively. Then $\boldsymbol m$ and $\boldsymbol n$ have equal contents and lengths, that is ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$ and $\ell(\boldsymbol m)=\ell(\boldsymbol n)$, and $\boldsymbol m\tau$ and $\tau\boldsymbol n$ are straight messages for $S$.
\end{lemma}
\begin{proof}
The message $\boldsymbol m\tau\tilde{\boldsymbol n}\tilde{\tau}$ is stepwise effective for $S$ and ineffective for that state. By Axiom [M3], this message is vacuous. Hence, ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$ and $\ell(\boldsymbol m)=\ell(\boldsymbol n)$.
Each of two straight messages $\boldsymbol m$ and $\tilde{\tau}$ produces $P$. By Axiom [M4], they are jointly consistent, that is, $\tau\notin{\cal C}(\boldsymbol m)$. Hence, $\boldsymbol m\tau$ is a straight message. Similarly, $\tau\boldsymbol n$ is a straight message.
\end{proof}
\begin{lemma} \label{Lemma6.2}
Let $S,T,P,Q$ be four distinct states such that $S\tau_1=T$, $P\tau_2=Q$ and $\boldsymbol m$ and $\boldsymbol n$ be two straight messages producing $P$ from $S$ and $Q$ from $T$, respectively. If the messages $\tau_1\boldsymbol n,\boldsymbol m\tau_2$, and $\tilde{\tau}_1\boldsymbol m$ are straight, then $\tau_1=\tau_2$.
\end{lemma}
\begin{proof}
Suppose that $\tau_1\not=\tau_2$. By Theorem~\ref{Theorem1.14}, ${\cal C}(\tau_1\boldsymbol n)={\cal C}(\boldsymbol m\tau_2)$. Hence, $\tau_1\in{\cal C}(\boldsymbol m)$, a contradiction, since we assumed that $\tilde{\tau}_1\boldsymbol m$ is a straight message.
\end{proof}
\begin{lemma} \label{Lemma6.3}
Let $({\cal S},{\cal T})$ be a medium, $G=(V,E)$ be a graph representing this medium, and $\alpha$ be the bijection ${\cal S}\rightarrow V$ defining the graph $G$. If $\boldsymbol m=\tau_1\cdots\tau_m$ is a straight message transforming a state $S$ into a state $T$, then the sequence of vertices $(\alpha(S_i))_{0\leq i\leq m}$, where $S_i=S\tau_0\tau_1\cdots\tau_i$, forms a shortest path joining $\alpha(S)$ and $\alpha(T)$ in $G$. Conversely, if a sequence $(\alpha(S_i))_{0\leq i\leq m}$ is a shortest path connecting $\alpha(S_0)=\alpha(S)$ and $\alpha(S_m)=\alpha(T)$, then $S\boldsymbol m= T$ for some straight message $\boldsymbol m$ of length $m$.
\end{lemma}
\begin{proof}
(Necessity.) Let $\alpha(P_0)=\alpha(S),\alpha(P_1),\ldots,\alpha(P_n)=\alpha(T)$ be a path in $G$ joining $\alpha(S)$ and $\alpha(T)$. There is a stepwise effective message $\boldsymbol n=\rho_1\cdots\rho_n$ such that $P_i=T\rho_1\cdots\rho_{n-i}$ for $0\leq i<n$. The message $\boldsymbol{mn}$ is stepwise effective for $S$ and ineffective for this state. By Axiom [M3], this message is vacuous. Since $\boldsymbol m$ is a straight message for $S$, we have $\ell(\boldsymbol m)\leq\ell(\boldsymbol n)$. It follows that $(\alpha(S_i))_{0\leq i\leq m}$ forms a shortest path joining $\alpha(S)$ and $\alpha(T)$ in $G$.
(Sufficiency.) Let $\alpha(S_0)=\alpha(S),\alpha(S_1),\ldots,\alpha(S_m)=\alpha(T)$ be a shortest path connecting vertices $\alpha(S)$ and $\alpha(T)$ in $G$. Then $S_i\tau_{i+1}=S_{i+1}$ for some tokens $\tau_i$, $1\leq i\leq m$. The message $\boldsymbol m=\tau_1\cdots\tau_m$ transforms the state $S$ into the state $T$. By the argument in the necessity part of the proof, $\boldsymbol m$ is a straight message for $S$.
\end{proof}
\begin{definition}\label{so1 cube and partial cube}
{\rm (cf.~\cite{dD73,wI00})} Let $X$ be a set. The graph ${\cal H}(X)$ is defined as follows: the set of vertices is the set $\mathfrak PF(X)$ of all finite subsets of $X$; two vertices $P$ and $Q$ are adjacent if the symmetric difference $P\Delta Q$ is a singleton. We say that ${\cal H}(X)$ is a \emph{cube on} $X$. Isometric subgraphs of the cube ${\cal H}(X)$, as well as graphs that are isometrically embeddable in ${\cal H}(X)$, are called \emph{partial cubes}.
\end{definition}
The proof of the following proposition is straightforward and omitted.
\begin{proposition} \label{cube=wg-family}
An induced subgraph $G=(V,E)$ of the cube ${\cal H}(X)$ is a partial cube if and only if $V$ is a well graded family of finite subsets of $X$. Then a shortest path in $G$ is a line segment in ${\cal H}(X)$ and the graph distance function $d$ on both ${\cal H}(X)$ and $G$ is given by
$$
d(P,Q)=|P\Delta Q|.
$$
\end{proposition}
The following theorem characterizes media in terms of their graphs.
\begin{theorem} \label{MediumGraph}
A graph $G$ represents a medium $({\cal S},{\cal T})$ if and only if $G$ is a partial cube.
\end{theorem}
We give two proofs of this important theorem. The first proof uses the representation theorem.
\begin{proof}
(Necessity.) By Theorem~\ref{FiniteRepresentationTheorem}, we may assume that the given medium is $({\cal F},{\cal G}_{\cal F})$ where ${\cal F}$ is a well graded family of finite subsets of some set $X$. By Proposition~\ref{cube=wg-family}, the graph of this medium is a partial cube.
(Sufficiency.) For a partial cube $G=(V,E)$ there is an isometric embedding $\alpha$ of $G$ into a cube $\mathcal{H}(X)$ for some set $X$. Vertices of $\alpha(G)$ form a well graded family of subsets of $X$. Then the medium $(\alpha(V),{\cal G}_{\alpha(V)})$ has $G$ as its graph.
\end{proof}
The second proof utilizes the Djokovi\'{c}--Winkler relation $\Theta$ on the set of edges of a graph. The definition and properties of this relation are found in the book~\cite{wI00}.
\begin{proof}
(Necessity.) We may assume that $G$ is the graph of $({\cal S},{\cal T})$. Let
$$
S,S_1,\ldots,S_n=S
$$
be a cycle of length $n$ in $G$. There is a stepwise effective message $\boldsymbol m$ such that $S\boldsymbol m=S$. By Axiom [M3], $\boldsymbol m$ is vacuous. Therefore, $\ell(\boldsymbol m)=n$ is an even number. Hence, $G$ is a bipartite graph.
For each edge $ST$ of $G$ there is a unique unordered pair of tokens $\{\tau,\tilde{\tau}\}$ such that $S\tau=T$ and $T\tilde{\tau}=S$. We denote $\sim$ the equivalence relation on the set of edges of $G$ defined by this correspondence. Let $ST\sim PQ$ and the notation is chosen such that $S\tau=T$ and $P\tau=Q$. Then, by lemmas~\ref{Lemma6.1} and~\ref{Lemma6.3},
\begin{equation} \label{Eq6.1}
d=d(S,P)=d(T,Q)=d(S,Q)-1=d(T,P)-1
\end{equation}
By Lemma~2.3 in~\cite{wI00}, $ST\Theta PQ$. On the other hand, if $ST\Theta PQ$ holds, then, by the same lemma, equation~(\ref{Eq6.1}) holds. By lemmas~\ref{Lemma6.2} and~\ref{Lemma6.3}, there is a token $\tau$ such that $S\tau=T$ and $P\tau=Q$, that is, $ST\sim PQ$. Thus $\sim\,=\Theta$, that is, $\Theta$ is an equivalence relation. By Theorem~2.10 in~\cite{wI00}, $G$ is a partial cube.
(Sufficiency.) We have already shown in the first proof that a partial cube is the graph of a medium.
\end{proof}
\begin{theorem} \label{Medium=Graph}
Two media are isomorphic if and only if the graphs representing these media are isomorphic.
\end{theorem}
\begin{proof}
(Necessity.) Clearly, graphs representing isomorphic media are isomorphic.
(Sufficiency.) Let $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ be two media and $G=(V,E)$ and $G'=(V',E')$ be two isomorphic graphs representing these media. Since $G$ and $G'$ are isomorphic, $G$ represents $({\cal S}',{\cal T}')$. Thus we need to show that two media represented by the same graph are isomorphic.
Since $G$ is a partial cube, it also represents a medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family of subsets of some set $X$. We denote $\mu:{\cal S}\rightarrow V$ and $\nu:V\rightarrow{\cal F}$ the two bijections that define the graph representation $G$ of $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$, respectively. Then $\alpha=\nu\circ\mu$ is a bijection ${\cal S}\rightarrow{\cal F}$ such that
$$
S\tau = T\quad\Leftrightarrow\quad |\alpha(S)\Delta\alpha(T)|=1,
$$
for all $S\not= T$ in ${\cal S}$ and $\tau\in{\cal T}$.
Clearly, it suffices to prove that the media $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$ are isomorphic.
Let $\tau$ be a token in ${\cal T}$ and $S$ and $T$ be two distinct states in ${\cal S}$ such that $S\tau= T$. Then either $\alpha(T)=\alpha(S)\cup\{x\}$ for some $x\notin \alpha(S)$ or $\alpha(T)=\alpha(S)\setminus\{x\}$ for some $x\in \alpha(S)$. We define $\beta:{\cal T}\rightarrow{\cal G}_{\cal F}$ by
$$
\beta(\tau) = \begin{cases}
\gamma_x, &\text{if $\alpha(T)=\alpha(S)\cup\{x\}$ for some $x\notin\alpha(S)$,} \\
\tilde{\gamma}_x, &\text{if $\alpha(T)=\alpha(S)\setminus\{x\}$ for some $x\in\alpha(S)$.}
\end{cases}
$$
Let us show that $\beta$ does not depend on the choice of $S$ and $T$. We consider only the case when $\beta(\tau)=\tau_x$. The other case is treated similarly.
Let $P,Q$ be another pair of distinct states in ${\cal S}$ such that $P\tau= Q$, and let $P=S\boldsymbol m$ and $Q=T\boldsymbol n$ for some straight messages $\boldsymbol m$ and $\boldsymbol n$. By Lemma~\ref{Lemma6.1}, $\ell(\boldsymbol m)=\ell(\boldsymbol n)$. Then, by Lemma~\ref{Lemma6.3}, $d(\alpha(S),\alpha(P))=d(\alpha(T),\alpha(Q))$, and, by Lemma~\ref{Lemma6.1},
\begin{align*}
d(\alpha(S),\alpha(Q))&=d(\alpha(S),\alpha(T))+d(\alpha(T),\alpha(Q)),\\
d(\alpha(T),\alpha(P))&=d(\alpha(T),\alpha(S))+d(\alpha(S),\alpha(P)).
\end{align*}
By Theorem~\ref{DistanceTheorem},
\begin{gather*}
\alpha(S)\cap\alpha(Q)\;\subseteq\:\alpha(T)=\alpha(S)\cup\{x\}\;\subseteq\;\alpha(S)\cup\alpha(Q),\\
\alpha(T)\cap\alpha(P)=[\alpha(S)\cup\{x\}]\cap\alpha(P)\;\subseteq\;\alpha(S)\;\subseteq\;\alpha(S)\cup\alpha(P).
\end{gather*}
Since $x\notin\alpha(S)$, it follows that $x\in\alpha(Q)$ and $x\notin\alpha(P)$. Then
\begin{equation*}
\alpha(Q)=\alpha(P)\cup\{x\},
\end{equation*}
since $d(\alpha(P),\alpha(Q))=1$. Hence, the mapping $\beta:{\cal S}\rightarrow{\cal G}_{\cal F}$ is well defined.
Clearly, $\beta$ is a bijection satisfying the condition
$$
S\tau= T\quad\Leftrightarrow\quad\alpha(S)\beta(\tau)=\alpha(T).
$$
Therefore $(\alpha,\beta)$ is an isomorphism from $({\cal S},{\cal T})$ onto $({\cal F},{\cal G}_{\cal F})$.
\end{proof}
We conclude this section with an example illustrating Theorem~\ref{Medium=Graph}.
\begin{example}
{\rm If $(\mathcal{S,T})$ and $(\mathcal{S',T'})$ are two finite isomorphic media, then $|{\cal S}|=|\mathcal{S'}|$ and $|{\cal T}|=|\mathcal{T'}|$. The converse, generally speaking, is not true. Consider, for instance, two media, $({\cal F},{\cal G}_{\cal F})$ and $({\cal F}',{\cal G}_{{\cal F}'})$, of well graded subsets of $X=\{a,b,c\}$ with
\begin{equation*}
{\cal F}=\{a,b,c,ab,ac,bc\}\quad\text{and}\quad{\cal F}'=\{a,c,ab,ac,bc,abc\}.
\end{equation*}
Their graphs, $G$ and $G'$, are shown in Figure~\ref{G and G'}.
{\begin{figure}
\caption{Graphs $G$ and $G'$.}
\label{G and G'}
\end{figure}
}
Clearly, these graphs are not isomorphic. Thus the media $({\cal F},{\cal G}_{\cal F})$ and $({\cal F}',{\cal G}_{{\cal F}'})$ are not isomorphic.
}
\end{example}
\section{Uniqueness of media representations}
Theorem~\ref{FiniteRepresentationTheorem} asserts that any medium $({\cal S},{\cal T})$ is isomorphic to the medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family ${\cal F}$ of finite subsets of some set $X$. In this section we show that this representation is unique in some precise sense.
Let $({\cal F}_1,{\cal G}_{{\cal F}_1})$ and $({\cal F}_2,{\cal G}_{{\cal F}_2})$ be two isomorphic representations of $({\cal S},{\cal T})$ with well graded families ${\cal F}_1$ and ${\cal F}_2$ of subsets of $X_1$ and $X_2$, respectively. By Theorem~\ref{T:wg gamma},
$$
|{\cal T}|=|{\cal G}_{{\cal F}_i}|=2|\cup{\cal F}_i\setminus\cap{\cal F}_i|\quad\text{for $i=1,2$}.
$$
Thus, without loss of generality, we may assume that ${\cal F}_1$ and ${\cal F}_2$ are well graded families of finite subsets of the same set $X$ and that they satisfy conditions~(\ref{E:wg gamma}). The graphs of the media $({\cal F}_1,{\cal G}_{{\cal F}_1})$ and $({\cal F}_2,{\cal G}_{{\cal F}_2})$ are isomorphic partial subcubes of the cube ${\cal H}(X)$. On the other hand, by theorems~\ref{Medium=Graph} and~\ref{MediumGraph}, isometric partial cubes represent isomorphic media.
We formulate the uniqueness problem geometrically as follows:
\begin{quote}
Show that any isometry between two partial subcubes of ${\cal H}(X)$ can be extended to an isometry of the cube ${\cal H}(X)$.
\end{quote}
In other words, we want to show that partial subcubes of ${\cal H}(X)$ are unique up to isometries of ${\cal H}(X)$ onto itself.
\begin{remark}
{\rm Note, that ${\cal H}(X)$ is not a fully homogeneous space~(as defined, for instance, in \cite{dB01}), that is, an isometry between two arbitrary subsets of ${\cal H}(X)$, generally speaking, cannot be extended to an isometry of the cube ${\cal H}(X)$. On the other hand, ${\cal H}(X)$ is a homogeneous metric space.
}
\end{remark}
A general remark is in order. Let $Y$ be a homogeneous metric space, $A$ and $B$ be two metric subspaces of $Y$, and $\alpha$ be an isometry from $A$ onto $B$. Let $c$ be a fixed point in $Y$. For a given $a\in A$, let $b=\alpha(a)\in B$. Since $Y$ is homogeneous, there are isometries $\beta$ and $\gamma$ of $Y$ such that $\beta(a)=c$ and $\gamma(b)=c$. Then $\lambda=\gamma\alpha\beta^{-1}$ is an isometry from $\beta(A)$ onto $\gamma(B)$ such that $\lambda(c)=c$. Clearly, $\alpha$ is extendable to an isometry of $Y$ if and only if $\lambda$ is extendable. Therefore, in the case of the space ${\cal H}(X)$, we may consider only well graded families of subsets containing the empty set $\varnothing$ and isometries between these families fixing this point.
In what follows, we assume that $\varnothing\in{\cal F}$ and $\cup{\cal F}=X$.
\begin{definition}
We define
$$
r_{\cal F}(x)=\min\{|A|: x\in A, A\in{\cal F}\}
$$
and, for $k\geq 1$,
$$
X_k^{\cal F}=\{x\in X: r_{\cal F}(x)=k\}.
$$
\end{definition}
We have $X_i^{\cal F}\cap X_j^{\cal F}=\varnothing$ for $i\not=j$, and $\cup_k X_k^{\cal F}=X$. Note that some of the sets $X_k^{\cal F}$ could be empty for $k>1$, although $X_1^{\cal F}$ is not empty, since, by the wellgradedness property, ${\cal F}$ contains at least one singleton (we assumed that $\varnothing\in{\cal F}$).
\begin{example}
{\rm Let $X=\{a,b,c\}$ and
$$
{\cal F}=\{\varnothing,\{a\},\{b\},\{a,b\},\{a,b,c\}\}.
$$
Clearly, ${\cal F}$ is well graded. We have $r_{\cal F}(a)=r_{\cal F}(b)=1,\;r_{\cal F}(c)=3$, and
$$
X_1^{\cal F}=\{a,b\},\;X_2^{\cal F}=\varnothing,\;X_3^{\cal F}=\{c\}.
$$
}
\end{example}
\begin{lemma}
For $A\in{\cal F}$ and $x\in A$, we have
\begin{equation} \label{OneLess}
r_{\cal F}(x)=|A|\quad\Rightarrow\quad A\setminus\{x\}\in{\cal F}.
\end{equation}
\end{lemma}
\begin{proof}
Let $k=|A|$. Since ${\cal F}$ is well graded, there is a nested sequence $\{A_i\}_{0\leq i\leq k}$ of distinct sets in ${\cal F}$ with $A_0=\varnothing$ and $A_k=A$. Since $r_{\cal F}(x)=k$, we have $x\notin A_i$ for $i<k$. Hence, $A\setminus \{x\}=A_{k-1}\in{\cal F}$.
\end{proof}
Let us recall (Theorem~\ref{DistanceTheorem}(ii)) that
\begin{equation} \label{so1 betweenness}
B\cap C\subseteq A\subseteq B\cup C\;\Leftrightarrow\; d(B,A)+d(A,C)=d(B,C)
\end{equation}
for all $A,B,C\in\mathfrak P(X)$.
It follows that isometries between two well graded families of sets ${\cal F}_1$ and ${\cal F}_2$ preserve the betweenness relation, that is,
\begin{equation} \label{so1 inclusion}
B\cap C\subseteq A\subseteq B\cup C\;\Leftrightarrow\;\alpha(B)\cap \alpha(C)\subseteq \alpha(A)\subseteq \alpha(B)\cup \alpha(C)
\end{equation}
for $A,B,C\in{\cal F}_1$ and an isometry $\alpha:{\cal F}_1\rightarrow{\cal F}_2$.
In the sequel, ${\cal F}_1$ and ${\cal F}_2$ are two well graded families of finite subsets of $X$ and $\alpha:{\cal F}_1\rightarrow{\cal F}_2$ is an isometry such that $\alpha(\varnothing)=\varnothing$.
\begin{definition}
We define a binary relation $\pi$ between on $X$ by means of the following construction. By~(\ref{OneLess}), for a given $x\in X$ there is $A\in{\cal F}_1$ such that $x\in A$, $r_{{\cal F}_1}(x)=|A|$, and $A\setminus\{x\}\in{\cal F}_1$.
Since $\varnothing\subseteq A\setminus\{x\}\subset A$, we have, by~(\ref{so1 inclusion}), $\alpha(A\setminus\{x\})\subset\alpha(A)$. Since $d\,(A\setminus\{x\},A)=1$, there is $y\in X,\;y\notin\alpha(A)$ such that $\alpha(A)=\alpha(A\setminus\{x\})\cup\{y\}$. In this case we say that $xy\in\pi$.
\end{definition}
\begin{lemma}
If $x\in X_k^{{\cal F}_1}$ and $xy\in\pi$, then $y\in X_k^{{\cal F}_2}$.
\end{lemma}
\begin{proof}
Let $A\in{\cal F}_1$ be a set of cardinality $k$ defining $r_{{\cal F}_1}(x)=k$.
Since
$$
|A|=d\,(\varnothing,A)=d\,(\varnothing,\alpha(A))=|\alpha(A)|\quad\text{and}\quad y\in\alpha(A),
$$
we have $r_{{\cal F}_2}(y)\leq k$. Suppose that $m=r_{{\cal F}_2}(y)<k$. Then, by~(\ref{OneLess}), there is $B\in{\cal F}_2$ such that $y\in B$, $|B|=m$, and $B\setminus\{y\}\in{\cal F}_2$. Clearly,
$$
\alpha(A\setminus\{x\})\cap B\subseteq\alpha(A)\subseteq\alpha(A\setminus\{x\})\cup B.
$$
By~(\ref{so1 inclusion}), we have
$$
(A\setminus\{x\})\cap\alpha^{-1}(B)\subseteq A\subseteq (A\setminus\{x\})\cup\alpha^{-1}(B).
$$
Thus, $x\in\alpha^{-1}(B)$, a contradiction, since $r_{{\cal F}_1}(x)=k$ and $|\alpha^{-1}(B)|=m<k$. It follows that $r_{{\cal F}_2}(y)=k$, that is, $y\in X_k^{{\cal F}_2}$.
\end{proof}
We proved that, for every $k\geq 1$, the restriction of $\pi$ to $X_k^{{\cal F}_1}$ is a relation $\pi_k$ between $X_k^{{\cal F}_1}$ and $X_k^{{\cal F}_2}$.
\begin{lemma}
The relation $\pi_k$ is a bijection for every $k\geq 1$.
\end{lemma}
\begin{proof}
Suppose that there are $z\not=y$ such that $xy\in\pi_k$ and $xz\in\pi_k$. Then, by~(\ref{OneLess}), there are two distinct sets $A,B\in{\cal F}_1$ such that
$$
k=r_{{\cal F}_1}(x)=|A|=|B|,\;\;A\setminus\{x\}\in{\cal F}_1,\;B\setminus\{x\}\in{\cal F}_1,
$$
and
$$
\alpha(A)=\alpha(A\setminus\{x\})+\{y\},\;\;\alpha(B)=\alpha(B\setminus\{x\})+\{z\}.
$$
We have
\begin{align*}
d\,(\alpha(A),\alpha(B))&=d\,(A,B)=d\,(A\setminus\{x\},B\setminus\{x\})\\
&=d\,(\alpha(A)\setminus\{y\},\alpha(B)\setminus\{z\}).
\end{align*}
Thus $y,z\in \alpha(A)\cap\alpha(B)$, that is, in particular, that $z\in \alpha(A)\setminus\{y\}$, a contradiction, because $r_{{\cal F}_2}(z)=k$ and $|\alpha(A)\setminus\{y\}|=k-1$.
By applying the above argument to $\alpha^{-1}$, we prove that $\pi_k$ is a bijection.
\end{proof}
It follows from the previous lemma that $\pi$ is a permutation on $X$.
\begin{lemma}
$\alpha(A)=\pi(A)$ for any $A\in{\cal F}_1$.
\end{lemma}
\begin{proof}
We prove this statement by induction on $k=|A|$. The case $k=1$ is trivial, since $\alpha(\{x\})=\{\pi_1(x)\}$ for $\{x\}\in\mathcal{F}_1$.
Suppose that $\alpha(A)=\pi(A)$ for all $A\in{\cal F}_1$ such that $|A|<k$. Let $A$ be a set in ${\cal F}_1$ of cardinality $k$. By the wellgradedness property, there is a nested sequence $\{A_i\}_{0\leq i\leq k}$ of distinct sets in ${\cal F}_1$ with $A_0=\varnothing$ and $A_k=A$. Thus, $A=A_{k-1}\cup\{x\}$ for some $x\notin A_{k-1}$. Clearly, $m=r_{{\cal F}_1}(x)\leq k$.
If $m=k$, then $\alpha(A)=\alpha(A_{k-1})\cup\{\pi(x)\}=\pi(A)$, by the definition of $\pi$ and the induction hypothesis.
Suppose now that $m<k$. There is a set $B\in{\cal F}_1$ containing $x$ such that $|B|=m$. By the wellgradedness property, there is a nested sequence $\{B_i\}_{0\leq i\leq m}$ of distinct sets in ${\cal F}_1$ with $B_0=\varnothing$ and $B_m=B$. We have $x\notin B_i$ for $i<m$, since $m=r_{{\cal F}_1}(x)$. Therefore, $B=B_{m-1}\cup\{x\}$. Clearly,
$$
B_{m-1}\cap A\subseteq B\subseteq B_{m-1}\cup A.
$$
By~(\ref{so1 inclusion}), we have
$$
\alpha(B)\subseteq \alpha(B_{m-1})\cup \alpha(A).
$$
Thus, by the induction hypothesis,
$$
\pi(B_{m-1})\cup\{\pi(x)\}=\pi(B)\subseteq \pi(B_{m-1})\cup \alpha(A).
$$
Hence, $\pi(x)\in\alpha(A)$. Since $\alpha(A)=\pi(A_{k-1})\cup\{y\}$ for $y\notin\pi(A_{k-1})$ and $x\notin A_{k-1}$, we have $y=\pi(x)$, that is, $\alpha(A)=\pi(A)$.
\end{proof}
In summary, we have the following theorem.
\begin{theorem} \label{WG homogeneity}
Any isometry between two partial subcubes of ${\cal H}(X)$ can be extended to an isometry of the cube ${\cal H}(X)$.
\end{theorem}
\begin{remark}
{\rm In the case of a finite set $X$ the previous theorem is a consequence of Theorem~19.1.2 in \cite{mD97}.}
\end{remark}
In the line of our arguments which led to the proof of Theorem~\ref{WG homogeneity} we used two kinds of isometries of ${\cal H}(X)$: isometries that map elements of ${\cal H}(X)$ to the empty set, and isometries defined by permutations on $X$. It is not difficult to show that these isometries generate the isometry group of ${\cal H}(X)$.
\begin{theorem} \label{so1 isometry group of cube}
The isometry group of ${\cal H}(X)$ is generated by permutations on the set $X$ and functions
$$
\alpha_A:S\mapsto S\Delta A,\quad S\in{\cal H}(X).
$$
\end{theorem}
\begin{proof}
Clearly, $\alpha_A$ is an isometry of ${\cal H}(X)$ and $\alpha_A(A)=\varnothing$. A permutation $\pi$ on $X$ defines an isometry $\hat{\pi}:{\cal H}(X)\rightarrow{\cal H}(X)$ by
$$
\hat{\pi}(S)=\{\pi(x): x\in S\}.
$$
Let $\alpha:{\cal H}(X)\rightarrow{\cal H}(X)$ be an isometry of ${\cal H}(X)$ and let $A=\alpha^{-1}(\varnothing)$. Then the isometry $\alpha_A\circ\alpha^{-1}$ fixes $\varnothing\in{\cal H}(X)$ and therefore defines a permutation $\pi:X\rightarrow X$ (singletons are on the distance $1$ from $\varnothing$). Let $\beta=\hat{\pi}^{-1}\circ\alpha_A\circ\alpha^{-1}$. Since $\alpha_A\circ\alpha^{-1}$ fixes $\varnothing$, we have $\alpha_A\circ\alpha^{-1}(\{x\})=\{\pi(x)\}$ for any $x\in X$. Hence, $\beta(\{x\})=\{x\}$ for all $x\in X$. For $S\in{\cal H}(X)$, we have
$$
|\beta(S)|=d(\beta(S),\varnothing)=d(S,\varnothing)=|S|,
$$
since $\beta(\varnothing)=\varnothing$. For any $x\in \beta(S)$, we have
$$
d(\{x\},S)=d(\{x\},\beta(S))=|\beta(S)|-1=|S|-1,
$$
which is possible only if $x\in S$. Thus $\beta(S)\subseteq S$. The same argument shows that $S\subseteq\beta(S)$. Thus $\beta$ is the identity mapping. It follows that $\alpha=\hat{\pi}^{-1}\circ\alpha_A$.
\end{proof}
\section{Linear Media}
The representation theorem (Theorem~\ref{FiniteRepresentationTheorem}) is a powerful tool for constructing media. We illustrate an application of this theorem by constructing a medium of linear orderings on a given finite or infinite countable set $Z$.
Let $Z=\{a_1,a_2,\ldots\}$ be a fixed (finite or infinite) enumeration of elements of $Z$. This enumeration defines a particular irreflexive linear ordering on $Z$ that we will denote by $L_0$.
\begin{definition}
A binary relation $R$ on $Z$ is said to be \emph{locally finite} if there is $n\in\mathbb{N}$ such that the restriction of $R$ to $\{a_{n+1},a_{n+2},\ldots\}$ coincides with the restriction of $L_0$ to the same set.
\end{definition}
Let ${\cal L}{\cal O}$ be the set of all locally finite irreflexive linear orders on the set $Z$. Note that if $Z$ is a finite set, then ${\cal L}{\cal O}$ is the set of all linear orderings on $Z$.
As usual, for a given $L\in{\cal L}{\cal O}$, we say that $x$ \emph{covers} $y$ in $L$ if $yx\in L$ and there is no $z\in Z$ such that $yz\in L,zx\in L$. Here and below $xy$ stands for an ordered pair of elements $x,y\in Z$. In what follows all binary relations are assumed to be locally finite for a given enumeration of $Z$.
\begin{lemma} \label{LO1}
Let $L$ be a linear order on $Z$. Then $L'=(L\setminus yx)\cup xy$ is a linear order if and only if $x$ covers $y$ in $L$.
\end{lemma}
\begin{proof}
Suppose $L'$ is a linear order and there is $z$ such that $yz\in L$ and $zx\in L$. Then $yz\in L'$ and $zx\in L'$ implying $yx\in L'$, a contradiction.
Suppose $x$ covers $y$ in $L$. Let $uv\in L'$ and $vw\in L'$. We need to show that $uw\in L'$. There are three possible cases.
\begin{enumerate}
\item $uv=xy$ and $vw\in L,vw\not=yx$. Then $yw=vw\in L$ which implies $uw=xw\in L$, since $x$ covers $y$ in $L$. We have $uw\in L'$, since $uw=xw\not=yx$.
\item $vw=xy$ and $uv\in L,uv\not=yx$. Then $ux=uv\in L$ which implies $uw=uy\in L$, since $x$ covers $y$ in $L$. We have $uw\in L'$, since $uw=uy\not=yx$.
\item $uv\in L,uv\not=yx$ and $vw\in L,vw\not=yx$. Then $uw\in L$ and $uw\not= yx$, since $x$ covers $y$ in $L$. Therefore, $uw\in L'$.
\end{enumerate}
\end{proof}
We shall also need the following fact.
\begin{lemma} \label{LO2}
Let $P,Q$ and $R$ be complete asymmetric binary relations on $Z$. Then
\begin{equation*}
P\cap R = Q\cap R\quad\Leftrightarrow\quad P=Q.
\end{equation*}
\end{lemma}
\begin{proof}
Suppose that $P\cap R=Q\cap R$ and let $xy\in P$. If $xy\in R$, then $xy\in Q$. Otherwise, $yx\in R$. Since $yx\notin P$, we have $yx\notin Q$ implying $xy\in Q$. Thus $P\subseteq Q$. By symmetry, $P=Q$.
\end{proof}
For $L\in{\cal L}{\cal O}$, we define
\begin{equation*}
\alpha : L \mapsto L\cap L_0
\end{equation*}
By Lemma~\ref{LO2}, $\alpha$ is a one--to--one mapping from ${\cal L}{\cal O}$ onto the set $\alpha({\cal L}{\cal O})$ of partial orders.
Note that for any two locally finite binary relations $R$ and $Q$ on $Z$, the symmetric difference $R\Delta Q$ is a finite set. Thus the distance $d(R,Q)=|R\Delta Q|$ is a finite number. We use this fact in the proof of the following theorem.
\begin{theorem} \label{WGfamilyLO}
The family $\alpha({\cal L}{\cal O})$ is a well graded family of subsets of $L_0$.
\end{theorem}
\begin{proof}
Let $P,P'$ be two distinct partial orders in $\alpha({\cal L}{\cal O})$ and $L,L'$ be corresponding linear orders. It is easy to see that there is a pair $xy\in L$ such that $y$ covers $x$ and $xy\notin L'$. By Lemma~\ref{LO1}, $L''$ defined by
$$
L''=(L\setminus xy)\cup yx
$$
is a linear order. Then
$$
P''=L''\cap L_0 = [(L\cap L_0)\setminus(L_0\cap xy)]\cup(L_0\cap yx)=\begin{cases}
P\setminus xy, & \text{if $xy\in L_0$,} \\
P\cup yx, & \text{if $xy\notin L_0$,}
\end{cases}
$$
where $xy\in P$ if $xy\in L_0$ and $yx\notin P$ if $xy\notin L_0$. Hence, $P''\not= P$ and $d(P,P'')=1$. Clearly,
\begin{equation*}
L\cap L'\subseteq L''\subseteq L\cup L'.
\end{equation*}
Therefore
\begin{equation*}
P\cap P'=L\cap L'\cap L_0\subseteq P''=L''\cap L_0\subseteq (L\cup L')\cap L_0=P\cup P',
\end{equation*}
that is, $P''$ lies between $P$ and $P'$. Thus,
\begin{equation*}
d(P,P')=d(P,P'')+d(P'',P')=1+d(P'',P')
\end{equation*}
and the result follows by induction.
\end{proof}
Since ${\cal F}=\alpha({\cal L}{\cal O})$ is a well graded family of subsets of $X=L_0$, it is the set of states of the medium $({\cal F},{\cal G}_{\cal F})$ with tokens defined by
\begin{equation*}
P\rho_{xy} = \begin{cases}
P\cup xy, & \text{if $P\cup xy\in\alpha({\cal L}{\cal O})$,} \\
P, & \text{otherwise,}
\end{cases}
\end{equation*}
and
\begin{equation*}
P\tilde{\rho}_{xy} = \begin{cases}
P\setminus xy, & \text{if $P\setminus xy\in\alpha({\cal L}{\cal O})$,} \\
P, & \text{otherwise,}
\end{cases}
\end{equation*}
for $xy\in L_0$ and $P=L\cap L_0\in\alpha({\cal L}{\cal O})$.
We have
\begin{align*}
(L\cap L_0)\rho_{xy} & = \begin{cases}
(L\cap L_0)\cup xy, & \text{if $(L\cap L_0)\cup xy\in\alpha({\cal L}{\cal O})$,} \\
L\cap L_0, & \text{otherwise,}
\end{cases} \\
& = \begin{cases}
(L\cup xy)\cap L_0, & \text{if $(L\cup xy)\cap L_0=L'\cap L_0$,} \\
L\cap L_0, & \text{otherwise,}
\end{cases}
\end{align*}
where $L'$ is some linear order. Since $yx\notin L_0$, we have
\begin{equation*}
(L\cup xy)\cap L_0=[(L\setminus yx)\cup xy]\cap L_0 =L'\cap L_0.
\end{equation*}
By Lemma~\ref{LO2}, $(L\setminus yx)\cup xy$ is a linear order. We define
\begin{equation*}
L\tau_{xy}=\begin{cases}
(L\setminus yx)\cup xy & \text{if $x$ covers $y$ in $L$,} \\
L & \text{otherwise.}
\end{cases}
\end{equation*}
Then, for $xy\in L_0$,
\begin{equation*}
(L\cap L_0)\rho_{xy}=L\tau_{xy}\cap L_0.
\end{equation*}
A similar argument shows that, for $xy\in L_0$,
\begin{equation*}
(L\cap L_0)\tilde{\rho}_{xy}=L\tau_{yx}\cap L_0=L\tilde{\tau}_{xy}\cap L_0.
\end{equation*}
We obtained the set of tokens ${\cal T}=\{\tau_{xy}\}_{xy\in L_0}$ by `pulling back' tokens from the set ${\cal G}_{\cal F}$. The medium $({\cal L}{\cal O},{\cal T})$ is isomorphic to the medium $({\cal F},{\cal G}_{\cal F})$. In the case of a finite set $Z$, it is the \emph{linear medium} introduced in \cite{jF97}.
Simple examples show that $\alpha({\cal L}{\cal O})$ is a proper subset of the set of all partial orders contained in $L_0$. In the case of a finite set $Z$ this subset is characterized in the following theorem.
\begin{theorem}
Let $L$ be a linear order on a finite set $Z$ and $P\subseteq L$ be a partial order. Then $P=L\cap L'$, where $L'$ is a linear order, if and only if $P'=L\setminus P$ is a partial order.
\end{theorem}
\begin{proof}
(1) Suppose $P=L\cap L'$. It suffices to prove that $P'=L\setminus P$ is transitive. Let $(x,y),\,(y,z)\in P'$. Then $(x,z)\in L$. Suppose $(x,z)\notin P'$. Then $(x,z)\in P$, implying $(x,z)\in L'$. Since $(x,y),\,(y,z)\in P'$, we have $(x,y)\notin L'$ and $(y,z)\notin L'$, implying $(y,x),\,(z,y)\in L'$, implying $(z,x)\in L'$, a contradiction.
(2) Suppose now that $P$ and $P'=L\setminus P$ are partial orders. We define $L'=P\cup {P'}^{-1}$ and prove that thus defined $L'$ is a linear order.
Clearly, relations $P,P^{-1},P',{P'}^{-1}$ form a partition of $(Z\times Z)\setminus\Delta$. It follows that $L'$ is a complete and antisymmetric binary relation.
To prove transitivity, suppose $(x,y),\,(y,z)\in L'$. It suffices to consider only two cases:
(i) $(x,y)\in P,\;(y,z)\in {P'}^{-1}$. Suppose $(x,z)\notin L'$. Then $(z,x)\in L'$. Suppose $(z,x)\in P$. Since $(x,y)\in P$, we have $(z,y)\in P$, a contradiction, since $(z,y)\in P'$. Suppose $(z,x)\in {P'}^{-1}$. Then $(x,z)\in P'$ and $(z,y)\in P'$ imply $(x,y)\in P'$, a contradiction. Hence, $(x,z)\in L'$.
(ii) $(y,z)\in P,\;(x,y)\in {P'}^{-1}$. Suppose $(x,z)\notin L'$. Then, again, $(z,x)\in L'$. Suppose $(z,x)\in P$. Since $(y,z)\in P$, we have $(y,x)\in P$, a contradiction, since $(y,x)\in P'$. Suppose $(z,x)\in {P'}^{-1}$. Then $(x,z)\in P'$ and $(y,x)\in P'$ imply $(y,z)\in P'$, a contradiction. Hence, $(x,z)\in L'$.
Clearly $P=L\cap L'$.
\end{proof}
We conclude this section with a geometric illustration of Theorem~\ref{WGfamilyLO}.
{\begin{figure}
\caption{The diagram of ${\cal L}
\label{diagram LO}
\end{figure}
}
\begin{example}
{\rm Let $Z=\{1,2,3\}$. We represent linear orders on $Z$ by $3$--tuples. There are $3!=6$ different linear orders on $Z$:
\begin{equation*}
L_0=123,\quad L_1=213,\quad L_2=231,\quad L_3=321,\quad L_4=312,\quad L_5=132.
\end{equation*}
These relations are represented by the vertices of the diagram in Figure~\ref{diagram LO}.
One can compare this diagram with the diagram shown in Figure~5 in~\cite{jF97}.
The elements of ${\cal F}=\alpha({\cal L}{\cal O})$ are subsets of $X=L_0$:
\begin{gather*}
L_0=\{12,13,23\},\quad L_1\cap L_0=\{13,23\},\quad L_2\cap L_0=\{23\}, \\
L_3\cap L_0=\varnothing,\quad L_4\cap L_0=\{12\},\quad L_5\cap L_0=\{12,13\}.
\end{gather*}
These sets are represented as vertices of the cube on the set $L_0=\{12,13,23\}$ as shown in Figure~\ref{(LO,T)}.
}
\end{example}
{\begin{figure}
\caption{Partial cube representing $({\cal L}
\label{(LO,T)}
\end{figure}
}
\section{Hyperplane arrangements}
In this section we consider an example of a medium suggested by Jean--Paul Doignon (see Example~2 in~\cite{jFsO02}).
Let $\mathcal{A}$ be a locally finite arrangements of affine hyperplanes in $\mathbb{R}^r$, that is a family of hyperplanes such that any open ball in $\mathbb{R}^r$ intersects only finite number of hyperplanes in $\mathcal{A}$~\cite[Ch.~V,\;\S 1]{nB02}. Clearly, there are only countably many hyperplanes in $\mathcal{A}$, so we can enumerate them, $\mathcal{A}=\{H_1,H_2,\ldots\}$. Every hyperplane is given by an affine linear function $\ell_i(\boldsymbol{x})=\sum_{j=1}^r a_{ij}x_j+b_i$, that is, $H_i=\{\boldsymbol{x}\in\mathbb{R}^r : \ell_i(\boldsymbol{x})=0\}$.
In what follows, we construct a token system $({\cal S},{\cal T})$ associated with an arrangement $\mathcal{A}$ and show that this system is a medium.
We define the set ${\cal S}$ of states to be the set of connected components of $\mathbb{R}^r\setminus\cup\,\mathcal{A}$. These components are called \emph{regions}~\cite{aB99} or \emph{chambers}~\cite{nB02} of $\mathcal{A}$. Each state $P\in{\cal S}$ is an interior of an $r$--dimensional polyhedron in $\mathbb{R}^r$.
To every hyperplane in $\mathcal{A}$ corresponds an ordered pair $(H,H')$ of open half spaces $H$ and $H'$ separated by this hyperplane. This ordered pair generates a transformation $\tau_{H,H'}$ of the states. Applying $\tau_{H,H'}$ to some state $P$ results in some other state $P'$ if $P\subseteq H,\;P'\subseteq H'$ and regions $P$ and $P'$ share a facet which is included in the hyperplane separating $H$ and $H'$; otherwise, the application of $\tau_{H,H'}$ to $P$ does not change $P$. We define the set ${\cal T}$ of tokens to be the set of all $\tau_{H,H'}$. Clearly, $\tau_{H,H'}$ and $\tau_{H',H}$ are reverses of each other.
\begin{theorem} \label{HyperplaneMedia}
$({\cal S},{\cal T})$ is a medium.
\end{theorem}
\begin{proof}
In order to prove that $({\cal S},{\cal T})$ is a medium, we show that it is isomorphic to a medium of a well graded family of sets. Let $J=\{1,2,\ldots\}$. We denote $H_i^+=\{\boldsymbol{x}\in\mathbb{R}^r : \ell_i(\boldsymbol{x})>0\}$ and $H_i^-=\{\boldsymbol{x}\in\mathbb{R}^r : \ell_i(\boldsymbol{x})<0\}$, open half spaces separated by $H_i$. Each region $P$ is an intersection of open half spaces corresponding to hyperplanes in $\mathcal{A}$. We define $J_P = \{j\in J : P\subseteq H_j^+\}$. Clearly, $P\mapsto J_P$ defines a bijection from ${\cal S}$ to ${\cal S}'=\{J_P:P\in{\cal S}\}$. It is also easy to see that $\cap\,{\cal S}'=\varnothing$ and $\cup\,{\cal S}'=J$.
Given $k\in J$, we define transformations $\tau_k$ and $\tilde{\tau}_k$ of ${\cal S}'$ as follows:
$$
J_P\tau_k = \begin{cases}
J_P\cup\{k\} &\text{if $H_k$ defines a facet of $P$,} \\
J_P &\text{otherwise},
\end{cases}
$$
and
$$
J_P\tilde{\tau}_k = \begin{cases}
J_P\setminus\{k\} &\text{if $H_k$ defines a facet of $P$}, \\
J_P &\text{otherwise}.
\end{cases}
$$
Let $P$ be a region of $\mathcal{A}$ and let $H_k$ be a hyperplane in $\mathcal{A}$ defining a facet of $P$. There is a unique region $P'$ sharing this facet with $P$. Moreover, $H_k$ is the only hyperplane separating $P$ and $P'$. It follows that $J_P\tau_k=J_{P'}$ if $k\notin J_P$ and $J_P\tilde{\tau}_k=J_{P'}$ if $k\in J_P$. Thus transformations $\tau_k$ and $\tilde{\tau}_k$ are well defined.
We denote ${\cal T}'$ the set of all transformations $\tau_i,\;\tilde{\tau}_i,\;i=1,\ldots,n$. Clearly, the correspondences $\tau_{H_i^+,H_i^-}\mapsto\tilde{\tau}_i$ and $\tau_{H_i^-,H_i^+}\mapsto\tau_i$ define a bijection from ${\cal T}$ to ${\cal T}'$. This bijection together with the bijection from ${\cal S}$ to ${\cal S}'$ given by $P\mapsto J_P$ define an isomorphism of two token systems, $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$.
It remains to show that ${\cal S}'$ is a well graded family of subsets of $J$.
Clearly, $k\in J_P\Delta J_Q$ if and only if $H_k$ separates $P$ and $Q$. Since $\mathcal{A}$ is locally finite, there is a finite number of hyperplanes in $\mathcal{A}$ that separate two regions. Thus $J_P\Delta J_Q$ is a finite set for any two regions $P$ and $Q$. Let $d$ be the usual Hamming distance on ${\cal S}'$, i.e, $d(J_P,J_Q)=|J_P\Delta J_Q|$. Thus $d(J_P,J_Q)$ is equal to the number of hyperplanes in $\mathcal{A}$ separating $P$ and $Q$.
Let $\boldsymbol{p}\in P$ and $\boldsymbol{q}\in Q$ be points in two distinct regions $P$ and $Q$. The interval $[\boldsymbol{p},\boldsymbol{q}]$ has a single intersection point with any hyperplane separating $P$ and $Q$. Moreover, a simple topological argument shows that we can always choose $\boldsymbol{p}$ and $\boldsymbol{q}$ in such a way that different hyperplanes separating $P$ and $Q$ intersect $[\boldsymbol{p},\boldsymbol{q}]$ in different points. Let us number these points in the direction from $\boldsymbol{p}$ to $\boldsymbol{q}$ as follows
\begin{equation*}
\boldsymbol{r}_0 = \boldsymbol{p}, \boldsymbol{r}_1,\ldots,\boldsymbol{r}_{k+1}=\boldsymbol{q}.
\end{equation*}
Each open interval $(\boldsymbol{r}_i,\boldsymbol{r}_{i+1})$ is an intersection of $[\boldsymbol{p},\boldsymbol{q}]$ with some region which we denote $R_i$ (in particular, $R_0=P$ and $R_k=Q$). Moreover, by means of this construction, points $\boldsymbol{r}_i$ and $\boldsymbol{r}_{i+1}$ belong to facets of $R_i$. We conclude that regions $R_i$ and $R_{i+1}$ are adjacent, that is, share a facet, for all $i=0,\ldots,k-1$. Clearly, $d(J_{P_i},J_{P_{i+1}})=1$ for all $i=0,\ldots,k-1$ and $d(J_P,J_Q)=k$. Thus, ${\cal S}'$ is a well graded family of subsets of $J$.
\end{proof}
The \emph{region graph} $G$~\cite{aB99} of the arrangement $\mathcal{A}$ has ${\cal S}$ as the set of vertices; edges of $G$ are pairs of adjacent regions in ${\cal S}$. It follows from Theorem~\ref{HyperplaneMedia} that $G$ is a partial cube.
In the case of a finite arrangement $\mathcal{A}$, the graph $G$ is the \emph{tope graph} of the oriented matroid associated with the arrangement $\mathcal{A}$. It follows from Proposition~4.2.3 in~\cite{aB99} that $G$ is an isometric subgraph of the $n$--cube, where $n$ is the number of hyperplanes in $\mathcal{A}$. Thus our Theorem~\ref{HyperplaneMedia} is an infinite dimensional analog of this result.
To give geometric examples of infinite partial cubes, let us consider locally finite line arrangements $\mathcal{A}$ in the plane $\mathbb{R}^2$. The closures of the regions of a given $\mathcal{A}$ form a tiling~\cite{bG87,mS95} of the plane. The region graph of this tiling is the $1$--skeleton of the dual tiling.
\begin{example}
{\rm Let us consider a line arrangement $\mathcal{A}$ shown in Figure~\ref{hex mosaic} by dotted lines.
The regions of this line arrangement are equilateral triangles that form $(3^6)$ mosaic (an edge--to--edge planar tiling by regular polygons; for notations and terminology see, for instance,~\cite{mD02,bG87}). The $1$--skeleton of the orthogonally dual~\cite{mS95} mosaic $(6^3)$ is the region graph of $\mathcal{A}$. This graph is also known as the hexagonal lattice in the plane.
By Theorem~\ref{HyperplaneMedia}, the hexagonal lattice is an infinite partial cube. This lattice is isometrically embeddable into the graph of the cubical lattice $\mathbb{Z}^3$~\cite{mD02}.
}
\end{example}
{\begin{figure}
\caption{Hexagonal lattice ($(6^3)$ mosaic).}
\label{hex mosaic}
\end{figure}
}
\begin{example}
{\rm
Another example of an infinite partial cube is shown in Figure~\ref{truncated mosaic}. There, the region graph is the $1$--skeleton of $(4.8^2)$ mosaic also known~\cite{mD02} as the truncated net $(4^4)$. Like in the previous case, this mosaic is orthogonally dual to the tiling defined by the line arrangement shown in Figure~\ref{truncated mosaic} and the region graph can be isometrically embedded into $\mathbb{Z}^4$~\cite{mD02}.
}
\end{example}
{\begin{figure}
\caption{$(4.8)$ mosaic.}
\label{truncated mosaic}
\end{figure}
}
\begin{example}
{\rm
A more sophisticated example of an infinite partial cube was suggested by a referee. This is one of the Penrose rhombic tilings (see, for instance,~\cite{dB81,mS95}) a fragment of which is shown in Figure~\ref{Penrose}~\cite[Ch.~9]{sW99}. The construction suggested by de~Bruijn~\cite{dB81} demonstrates that the graph of this tiling is the region graph of a particular line arrangement known as a pentagrid. This graph is isometrically embeddable in $\mathbb{Z}^5$~\cite{dB81,mD02}.
}
\end{example}
{\begin{figure}
\caption{A Penrose rhombic tiling.}
\label{Penrose}
\end{figure}
}
\section*{Acknowledgments}
The author is grateful to Jean--Claude Falmagne for his careful reading of the original manuscript and many helpful suggestions, and to Jean--Paul Doignon for his comments on the results presented in Section~7. I also thank the referees for their constructive criticism.
\end{document}
|
\begin{document}
\title[Boundedness of the Gaussian Riesz potentials ] {Boundedness of the Gaussian Riesz potentials on Gaussian variable Lebesgue spaces}
\author{Eduard Navas}
\address{Departamento de Matem\'aticas, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Venezuela.}
\email{[email protected]}
\author{Ebner Pineda}
\address{Departamento de Matem\'{a}tica, Facultad de Ciencias Naturales y Matem\'aticas, ESPOL Guayaquil 09-01-5863, Ecuador.}
\email{[email protected]}
\author{Wilfredo~O.~Urbina}
\address{Department of Mathematics, Actuarial Sciences and Economics, Roosevelt University, Chicago, IL,
60605, USA.}
\email{[email protected]}
\subjclass[2010]{Primary 42B25, 42B35 ; Secondary 46E30, 47G10 }
\keywords{Gaussian harmonic analysis, variable Lebesgue spaces, Ornstein-Uhlenbeck semigroup, Riesz potentials.}
\begin{abstract}
In this paper we prove the boundedness of the Gaussian Riesz potentials $I_{\beta}$, for $\beta\geq 1$ on $L^{p(\cdot)}(\gamma_d)$, the Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$ following \cite{DalSco}. Additionally, this result trivially gives us an alternative proof of the boundedness of Gaussian Riesz potentials $I_\beta$ on Gaussian Lebesgue spaces $L^p(\gamma_d)$.
\end{abstract}
\maketitle
\section{Introduction and Preliminaries}
In the classical case, the Riesz potential of order $\beta>0$ is defined as negative fractional powers of the negative Laplacian $-\Delta = -\displaystyle\sum_{i=1}^d \frac{\partial^2}{\partial x_i^2}$,
\begin{equation}
(-\Delta)^{-\beta/2},
\end{equation}
which means, using Fourier transform, that
\begin{equation}
( (-\Delta)^{-\beta/2} f)\hat{}\;(\xi) = (2\pi |\xi| )^{-\beta} \hat{f}(\xi).
\end{equation}
for more details; see \cite{duo}, \cite{grafak}, \cite{st1}.\\
Analogously, the {\em Gaussian fractional integrals} or {\em Gaussian Riesz potentials} can be also defined as negative fractional powers of the {\em Ornstein-Uhlenbeck operator }
\begin{equation}
(-L)= - \frac12 \Delta + \langle x, \nabla_x \rangle=- \sum_{i=1}^d \Big[\frac{1}{2} \frac{\partial^2}{\partial x_i^2} + x_i \frac{\partial }{\partial x_i}\Big].
\end{equation}
However, since the Ornstein-Uhlenbeck operator has eigenvalue $0,$ the negative powers are not defined on all of $L^2(\gamma_d),$ and therefore we need to be more careful with the definition. Let us consider
$$\Pi_{0}f=f-\displaystyle\int_{\mathbb{R}^{d}}f(y)\gamma_{d}(dy),$$
for $f\in L^{2}(\gamma_{d})$, the orthogonal projection on the orthogonal complement of the eigenspace corresponding to the eigenvalue $0$.
\begin{defi}
The Gaussian Fractional Integral or Gaussian Riesz potential of order $\beta>0$, $I_\beta$, is defined spectrally as,
\begin{equation}\label{i1}
I_\beta=(-L)^{-\beta/2}\Pi_{0},
\end{equation}
which means that for any multi-index $\nu, \; |\nu|>0$ its action on the Hermite polynomial $\vec{H}_\nu$ is given by
\begin{equation}\label{RieszPotAct}
I_\beta \vec{H}_\nu(x)=\frac 1{\left|
\nu \right|^{\beta/2}}\vec{H}_\nu(x),
\end{equation}
and for $\nu=0=(0,...,0), \, I_{\beta}(\vec{H}_{0})=0.$
\end{defi}
By linearity, using the fact that the Hermite polynomials are an algebraic basis of $\mathcal{P}(\mathbb{R}^d),$ $I_\beta$ can be defined for any polynomial function $f(x) = \sum_{\nu} \widehat{f}_{\gamma_{d}}(\nu) \vec{H}_\nu(x),$ where $\widehat{f}_{\gamma_{d}}(\nu) = \frac{1}{\|\vec{H}_\nu\|_2} \int_{\mathbb{R}^d} f(t) \vec{H}_\nu (t) dt,$ as
\begin{equation}\label{RieszPotMult}
I_\beta f(x) = \sum_{\nu} \frac {\widehat{f}_{\gamma_{d}}(\nu)}{\left|\nu \right|^{\beta/2}}\vec{H}_\nu(x) = \sum_{k\geq 1} \frac 1{k^{\beta/2}} {\bf J}_k f(x),
\end{equation}
and similarly for $f \in L^2(\gamma_d),$ as the Hermite polynomials are an orthogonal basis of $L^2(\gamma_d).$\\
It can be proved that the Gaussian Riesz potential $I_\beta,$ $\beta>0,$ has the following integral representations, for $f$ a polynomial function or $f \in C^2_b(\mathbb{R}^d),$
\begin{equation}\label{RieszOUIntRep}
I_\beta f(x) =\frac 1{\Gamma(\beta/2)}\int_0^{\infty}
t^{\beta/2-1} T_t (I-{\bf J}_0)f(x) \,dt,
\end{equation}
with respect to the Ornstein-Uhlenbeck semigroup $\{T_t\}$, and
\begin{equation}\label{RieszPHIntRep}
I_\beta f(x) = \frac 1{\Gamma(\beta)}\int_0^{\infty}
t^{\beta-1}P_t (I-{\bf J}_0)f(x) \,dt,
\end{equation}
with respect to the Poisson-Hermite semigroup, $\{P_t\}$. \\
Therefore, from (\mathbb{R}f{RieszOUIntRep}) we have an explicit integral representation of $I_\beta$ as
\begin{equation}\label{RieszExplntRep}
I_{\beta}f(x)=\int\limits_{\mathbb{R}^{d}}N_{\beta/2}(x,y)f(y)dy\
\end{equation}
where the kernel $N_{\beta/2}$ is defined as
\begin{equation}\label{kernelRiesz}
N_{\beta/2}(x,y)=\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\left(\frac{e^{-\frac{|y-e^{-t}x|^{2}}{1-e^{-2t}}}}{(1-e^{-2t})^{\frac{d}{2}}}-e^{-|y|^{2}}\right)dt
\end{equation}
For more details and background we refer to \cite{urbina2019}.\\
From (\mathbb{R}f{RieszPotMult}) it is clear that the Gaussian Riesz potentials $I_\beta$ are the simplest Meyer's multipliers (see for instance Theorem 6.2 of \cite{urbina2019}), since in this case
\begin{equation}\label{RieszPotMult2}
m(k) = \frac{1}{k^\beta} = h( \frac{1}{k^\beta}),
\end{equation}
with $h(x) =x,$ the identity function and therefore their $L^p(\gamma_d)$-boundedness follows immediately.\\
In this paper we prove that Gaussian Riesz potentials $I_\beta$, for $\beta\geq 1$ are also bounded in $L^{p(\cdot)}(\gamma_d)$, the Gaussian variable Lebesgue spaces for certain exponent functions $p(\cdot)$ that will be determine later. For completeness, we will briefly review the notion of variable Lebesgue spaces.\\
Given $\mu$ a Borel measure, any $\mu_{-}$measurable function $p(\cdot):\mathbb{R}^{d}\rightarrow [1,\infty]$ is an $exponent$ $function$; the set of all the exponent functions will be denoted by $\mathcal{P}(\mathbb{R}^{d},\mu)$. For $E\subset\mathbb{R}^{d}$ we set $$p_{-}(E)=\text{ess}\inf_{x\in E}p(x) \;\text{and}\; p_{+}(E)=\text{ess}\sup_{x\in E}p(x).$$
$\Omega_{\infty}=\lbrace x\in \Omega:p(x)=\infty\rbrace$.\\
We use the abbreviations $p_{+}=p_{+}(\mathbb{R}^{d})$ and $p_{-}=p_{-}(\mathbb{R}^{d})$.
\begin{defi}\label{deflogholder}
Let $E\subset \mathbb{R}^{d}$. We say that $\alpha(\cdot):E\rightarrow\mathbb{R}$ is locally log-H\"{o}lder continuous, and denote this by $\alpha(\cdot)\in LH_{0}(E)$, if there exists a constant $C_{1}>0$ such that
\begin{eqnarray*}
|\alpha(x)-\alpha(y)|&\leq&\frac{C_{1}}{log(e+\frac{1}{|x-y|})}
\end{eqnarray*}
for all $x,y\in E$. We say that $\alpha(\cdot)$ is log-H\"{o}lder continuous at infinity with base point at $x_{0}\in \mathbb{R}^{d}$, and denote this by $\alpha(\cdot)\in LH_{\infty}(E)$, if there exist constants $\alpha_{\infty}\in\mathbb{R}$ and $C_{2}>0$ such that
\begin{eqnarray*}
|\alpha(x)-\alpha_{\infty}|&\leq&\frac{C_{2}}{log(e+|x-x_{0}|)}
\end{eqnarray*}
for all $x\in E$. We say that $\alpha(\cdot)$ is log-H\"{o}lder continuous, and denote this by $\alpha(\cdot)\in LH(E)$ if both conditions are satisfied.
The maximum, $\max\{C_{1},C_{2}\}$ is called the log-H\"{o}lder constant of $\alpha(\cdot)$.
\end{defi}
\begin{defi}\label{defPdlog}
We denote $p(\cdot)\in\mathcal{P}_{d}^{log}(\mathbb{R}^{d})$, if $\frac{1}{p(\cdot)}$ is log-H\"{o}lder continuous and denote by $C_{log}(p)$ or $C_{log}$ the log-H\"{o}lder constant of $\frac{1}{p(\cdot)}$.
\end{defi}
\begin{defi}
For a $\mu_{-}$measurable function $f:\mathbb{R}^{d}\rightarrow \overline{\mathbb{R}}$, we define the modular \begin{equation}
\rho_{p(\cdot),\mu}(f)=\displaystyle\int_{\mathbb{R}^{d}\setminus\Omega_{\infty}}|f(x)|^{p(x)}\mu(dx)+\|f\|_{L^{\infty}(\Omega_{\infty},\mu)},
\end{equation}
\end{defi}
\begin{defi} The variable exponent Lebesgue space on $\mathbb{R}^{d}$, $L^{p(\cdot)}(\mathbb{R}^{d},\mu)$ consists on those $\mu\_$measurable functions $f$ for which there exists $\lambda>0$ such that $\rho_{p(\cdot),\mu}\left(\frac{f}{\lambda}\right)<\infty,$ i.e.
\begin{equation*}
L^{p(\cdot)}(\mathbb{R}^{d},\mu) =\left\{f:\mathbb{R}^{d}\to \overline{\mathbb{R}}: f \; \text{is}\; \mu_{-} \text{measurable and} \; \rho_{p(\cdot),\mu}\left(\frac{f}{\lambda}\right)<\infty, \; \text{for some} \;\lambda>0\right\}
\end{equation*}
and the norm
\begin{equation}
\|f\|_{L^{p(\cdot)}(\mathbb{R}^{d},\mu)}=\inf\left\{\lambda>0:\rho_{p(\cdot),\mu}(f/\lambda)\leq 1\right\}.
\end{equation}
\end{defi}
It is well known that, if $p(\cdot) \in L H\left(\mathbb{R}^{d}\right)$ with $1<p_{-} \leq p^{+}<\infty$ the classical Hardy-Littlewood maximal function $\mathcal{M}$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(\Bbb R^{d}),$ see \cite{dcruz1}. However, it is known that even though these are the sharpest possible point-wise conditions, they are not necessary. In \cite{LibroDenHarjHas} a necessary and sufficient condition is given for the $L^{p(\cdot)}$-boundedness of $\mathcal{M},$ but it is not an easy to work condition. The class $L H(\mathbb{R}^{d})$ is also sufficient for the boundedness on $L^{p(\cdot)}$-spaces of classical singular integrals of Calder\'on-Zygmund type, see \cite[Theorem 5.39]{dcruz}.\\
If $\mathcal{B}$ is a family of balls (or cubes) in $\mathbb{R}^{d}$, we say that $\mathcal {B}$ is $N$-finite if it has bounded overlappings for $N$, i.e., $\displaystyle\sum_{B\in\mathcal{B}}\chi_{B}(x)\leq N$ for all $x\in\mathbb {R}^{d}$; in other words, there is at most $N$ balls (resp. cubes) that intersect at the same time.\\
The following definition was introduced for the first time by Berezhno\v{\i} in \cite{Berez}, defined for a family of disjoint balls or cubes. In the context of variable spaces, it has been considered in \cite{LibroDenHarjHas}, allowing the family to have bounded overlappings.\\
\begin{defi}
Given an exponent $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d})$, we will say that $p(\cdot)\in\mathcal{G}$, if for every family of balls (or cubes) $\mathcal{B}$ which is $N$-finite,
\begin{eqnarray*}
\sum_{B\in\mathcal{B}}||f\chi_{B}||_{p(\cdot)}||g\chi_{B}||_{p'(\cdot)} &\lesssim& ||f||_{p(\cdot)}||g||_{p'(\cdot)}
\end{eqnarray*}
for all functions $f\in L^{p(\cdot)}(\mathbb{R}^{d})$ and $g\in L^{p'(\cdot)}(\mathbb{R}^{d})$. The constant only depends on N.
\end{defi}
\begin{teo}[Teorema 7.3.22 of \cite{LibroDenHarjHas}]\label{implication1}
If $p(\cdot)\in LH(\mathbb{R}^{d})$, then $p(\cdot)\in\mathcal{G}$
\end{teo}
We will consider only variable Lebesgue spaces with respect to the Gaussian measure $\gamma_d,$ $L^{p(\cdot)}(\mathbb{R}^{d},\gamma_d).$ The next condition was introduced by E. Dalmasso and R. Scotto in \cite{DalSco}.
\begin{defi}\label{defipgamma}
Let $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d},\gamma_{d})$, we say that $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$ if there exist constants $C_{\gamma_{d}}>0$ and $p_{\infty}\geq1$ such that
\begin{equation}
|p(x)-p_{\infty}|\leq\frac{C_{\gamma_{d}}}{|x|^{2}},
\end{equation}
for $x\in\mathbb{R}^{d}\setminus\{(0,0,\ldots,0)\}.$
\end{defi}
\begin{obs}\label{obs4.1}
If $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, then $p(\cdot)\in LH_{\infty}(\mathbb{R}^{d})$
\end{obs}
\begin{lemma}[Lemma 2.5 of \cite{DalSco}]\label{lemaequiPgamma}
If $1<p_{-}\leq p_{+}<\infty,$ the following statements are equivalent:
\begin{itemize}
\item [(i)] $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$
\item [(ii)] There exists $p_{\infty}>1$ such that
\begin{eqnarray}
C_{1}^{-1}\leq e^{-|x|^{2}(p(x)/p_{\infty}-1)}\leq C_{1} &\;\;\hbox{and}\;\;& C_{2}^{-1}\leq e^{-|x|^{2}(p'(x)/p'_{\infty}-1)}\leq C_{2},
\end{eqnarray}
for all $x\in\mathbb{R}^{d}$, where $C_{1}=e^{C_{\gamma_{d}}/p_{\infty}}$ and $C_{2}=e^{C_{\gamma_{d}}p'_{-}/p_{\infty}}$.
\end{itemize}
\end{lemma}
Definition \mathbb{R}f{defipgamma} with Observation \mathbb{R}f{obs4.1} and Lemma \mathbb{R}f{lemaequiPgamma} end up strengthening the regularity conditions on the exponent functions $p(\cdot)$ to obtain the boundedness of the Ornstein-Uhlenbeck semigroup $\{T_{t}\}$, see \cite{MorPiUrb}. As a consequence of Theorem \mathbb{R}f{implication1}, we have
\begin{corollary}\label{solapamientoacotadoG}
If $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$, then $p(\cdot)\in\mathcal{G}.$
\end{corollary}
As we have already mentioned, the main result in this paper is the proof that the Gaussian Riesz Potentials $I_\beta$, for $\beta\geq 1$, are bounded on Gaussian variable Lebesgue spaces under the same condition of regularity on \(p(\cdot)\) considered by Dalmasso and Scotto \cite{DalSco}.
\begin{teo}\label{boundLpvarRieszPot}
Let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ with $1<p_{-}\leq p_{+}<\infty$. Then for $\beta\geq 1$ there exists a constant $C>0,$ depending only on $p$, $\beta$ and the dimension $d$ such that
\begin{equation}
\| I_\beta f\|_{p(\cdot),\gamma_{d}} \leq C \| f\|_{p(\cdot),\gamma_{d}},
\end{equation}
for any $f \in L^{p(\cdot)}(\gamma_d).$ \\
\end{teo}
Trivially, Theorem \mathbb{R}f{boundLpvarRieszPot} give us an alternative proof of the boundedness of the Gaussian Riesz Potentials $I_\beta$, for $\beta\geq 1$ on Gaussian Lebesgue spaces $L^p(\gamma_d)$, by simply taking the exponent function constant, but the constant $C$ depends on $\beta$ and the dimension, which is weaker than the estimate obtained using Meyer's multiplier theorem mentioned above.
\section{Proof of the main result.}
In order to prove our main result, Theorem \mathbb{R}f{boundLpvarRieszPot} we need some technical results.
\begin{lemma}\label{lema3.26CU}
Let $\rho(\cdot):\mathbb{R}^{d}\rightarrow[0,\infty)$ be such that $\rho(\cdot)\in LH_{\infty}(\mathbb{R}^{d})$, $0<\rho_{\infty}<\infty$, and let $R(x)=(e+|x|)^{-N}$, $N>d/\rho_{-}$. Then there exists a constant $C$ depending on $d$, $N$ and the $LH_{\infty}$ constant of $\rho(\cdot)$ such that given any set $E$ and
any function $F$ with $0\leq F(y)\leq 1$, for all $y\in E$,
\begin{eqnarray}
\int_{E}F^{\rho(y)}(y)dy &\leq& C\int_{E}F^{\rho_{\infty}}(y)dy + \int_{E}R^{\rho_{-}}(y)dy,\label{3.26.1} \\
\int_{E}F^{\rho_{\infty}}(y)dy &\leq& C\int_{E}F^{\rho(y)}(y)dy + \int_{E}R^{\rho^{-}}(y)dy.\label{3.26.2}
\end{eqnarray}
\end{lemma}
For the proof see Lemma 3.26 of \cite{dcruz}.
\begin{lemma}\label{lemgamma}
If $\alpha>0$, there exists a constant $C>0$ such that
\begin{equation}\label{desigualdadgamma}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\alpha-1}du =
C\Gamma(\alpha)<\infty
\end{equation}
\end{lemma}
\begin{proof}
Taking the change of variables $t=-log(\sqrt{1-u})$ then $u=1-e^{-2t}$\\
and $du=2e^{-2t}dt$. For $\alpha> 0$ we get
\begin{equation*}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\alpha-1}du =
2\int_{0}^{+\infty}t^{\alpha-1}e^{-2t}dt=C\Gamma(\alpha)<\infty
\end{equation*}
\end{proof}
\begin{lemma}\label{leminteg}
For $\beta>0$\\
\begin{enumerate}
\item[ i)]
\begin{equation}
\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{1}{\sqrt{1-u}}du\,<\infty.
\end{equation}
\item[ ii)]
\begin{equation}\label{gammaint}
\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}du\,<\infty.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}\quad\\
\begin{enumerate}
\item[ i)] Using H\"older's inequality with $p=\frac{3}{2}$, $q=3$ and Lemma \mathbb{R}f{lemgamma}, with $\alpha=\frac{3\beta}{2}+1$, we have that
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{du}{\sqrt{1-u}}&\leq
&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{1}{\sqrt{1-u}}du\\
&\leq
&\left(\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{3\beta}{2}}du\right)^{1/3}
\left(\int_{0}^{1}\frac{du}{(1-u)^{3/4}}\right)^{2/3}\\
&=
&\left(\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{(\frac{3\beta}{2}+1)-1}du\right)^{1/3}
\left(\int_{0}^{1}\frac{du}{(1-u)^{3/4}}\right)^{2/3}\,<\infty
\end{eqnarray*}
\item[ii)] Let us rewrite the integral as
\begin{equation*}
\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}du=\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)}{u}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}du,
\end{equation*}
since $\displaystyle\lim_{u\to
0}\frac{\left(-log(\sqrt{1-u})\right)}{u}=\frac{1}{2}$ and
$\displaystyle\frac{\left(-log(\sqrt{1-u})\right)}{u}$ is bounded in $(0,1/2],$ then we have we have by (\mathbb{R}f{desigualdadgamma})
\begin{eqnarray*}
\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}du&=& \int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{\left(-log(\sqrt{1-u})\right)}{u}du\\
&\leq&
C\int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}du\,<
\infty.
\end{eqnarray*}
\end{enumerate}
\end{proof}
We are now ready to prove the main result, Theorem \mathbb{R}f{boundLpvarRieszPot}.
\begin{proof}
As usual, we split the operator $I_\beta$ in its local part and its global part
\begin{equation*}
I_{\beta}f(x)=I_{\beta,L}f(x)+I_{\beta,G}f(x),
\end{equation*}
where
\begin{equation*}
I_{\beta,L}f(x)=I_{\beta}(f\chi_{B_{h}(\cdot)} )(x)
\end{equation*}
is the local part,
\begin{equation*}
I_{\beta,G}f(x)=I_{\beta}(f\chi_{B^{c}_{h}(\cdot)} )(x)
\end{equation*}
is the global part, and for $x\in \Bbb R^{d}$ by taking $m(x)=1\wedge \frac{1}{|x|}$,\\
$B_{h}(x):=\lbrace y\in \Bbb R^{d}:|x-y|<dm(x)\rbrace$ is an $hiperbolic$ $ball$ ($admissible$ $ball$).\\
Let us take
$\omega(s)=\displaystyle\frac{e^{-\frac{|y-e^{-s}x|^{2}}{1-e^{-2s}}}}{(1-e^{-2s})^{\frac{d}{2}}}$,
then, from (\mathbb{R}f{kernelRiesz})
\begin{eqnarray*}
N_{\beta/2}(x,y)&=&\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\left(\omega(t)-\omega(+\infty)\right)dt\\
&=&\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\left(-\int_{t}^{+\infty}\frac{\partial
\omega(s)}{\partial s}ds\right)dt.
\end{eqnarray*}
Thus, using Hardy's inequality, see \cite{st1}
$$
\left|N_{\beta/2}(x,y)\right|\leq \frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\int_{t}^{+\infty}\left|\frac{\partial
\omega(s)}{\partial s}\right|ds\;dt \leq
\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\frac{2}{\beta}\int_{0}^{+\infty}
s^{\frac{\beta}{2}}\left|\frac{\partial
\omega(s)}{\partial s}\right|ds.
$$
Now,
\begin{eqnarray*}
\frac{\partial \omega(s)}{\partial
s}&=&\frac{(1-e^{-2s})^{\frac{d}{2}}e^{-\frac{|y-e^{-s}x|^{2}}{1-e^{-2s}}}}{(1-e^{-2s})^{d}}
\left(\frac{-2(1-e^{-2s})(y-e^{-s}x)\cdot(e^{-s}x)+|y-e^{-s}x|^{2}e^{-2s}}{(1-e^{-2s})^{2}}\right)\\
&&\hspace{4.5cm}-\frac{1}{(1-e^{-2s})^{d}}\left(\frac{d}{2}e^{\frac{|y-e^{-s}x|^{2}}{1-e^{-2s}}}(1-e^{-2s})^{\frac{d}{2}-1}2e^{-2s}\right)\\
&=&\omega(s)\left(\frac{-2(1-e^{-2s})(y-e^{-s}x)\cdot(e^{-s}x)+|y-e^{-s}x|^{2}e^{-2s}}{(1-e^{-2s})^{2}}-\frac{de^{-2s}}{(1-e^{-2s})}\right).
\end{eqnarray*}
Then, taking $u=1-e^{-2s},\; du=2e^{-2s}ds,$ i.e., $e^{-s} = \sqrt{1-u},$ we have
\begin{eqnarray*}
\left|N_{\beta/2}(x,y)\right|&\leq
&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}
\\
&&\hspace{0.2cm}\times\left(\frac{2u|y-\sqrt{1-u}x|\sqrt{1-u}|x|+|y-\sqrt{1-u}x|^{2}(1-u)}{u^{2}}+\frac{d(1-u)}{u}\right)\frac{du}{2(1-u)}\\
&=&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}
\left(\frac{|y-\sqrt{1-u}x||x|}{u\sqrt{1-u}}+\frac{|y-\sqrt{1-u}x|^{2}}{2u^{2}}+\frac{d}{2u}\right)du\\
&=& \int_{0}^{1}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u^{\frac{d}{2}+1}}e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}
\frac{|y-\sqrt{1-u}x||x|}{\sqrt{1-u}}du \\
&& \hspace{1cm}+\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}+1}}\frac{|y-\sqrt{1-u}x|^{2}}{2u}du\\
&& \hspace{1.5cm}+\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}\frac{d}{2u} du\\
&&=I+II+III,
\end{eqnarray*}
where
\begin{equation}
I = \int_{0}^{1}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u^{\frac{d}{2}+1}}e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}
\frac{|y-\sqrt{1-u}x||x|}{\sqrt{1-u}}du,
\end{equation}
\begin{equation}
II= \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}+1}}\frac{|y-\sqrt{1-u}x|^{2}}{2u}du,
\end{equation}
and
\begin{equation}
III = \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}\frac{d}{2u} du.
\end{equation}
\begin{itemize}
\item Let us study the local part first. We need to bound each of the terms $I$, $II$ and $III$ in this part.
For $I$, since we are in the local part
$|x-y| \leq \frac{d}{|x|}$, then we have $|x-y||x| \leq d $ therefore,
\begin{eqnarray}\label{ineqlocpart}
\nonumber |y-\sqrt{1-u}\,x|^{2} & \geq& (|y-x|-|x|(1-\sqrt{1-u}))^{2} \\
& \geq& |y-x|^{2}-2|x||y-x| \frac{u}{1+\sqrt{1-u}} \geq|y-x|^{2}-2 d \; u.
\end{eqnarray}
On the other hand, it is well known that, there exist $C>0$ such that\\
for any $x>0$, $\alpha \geq 0$ and $c>0$,
\begin{equation}\label{expineq}
x^\alpha e^{-cx^2} \leq C \hspace{0.5 cm}\cdot
\end{equation}
Thus, using (\mathbb{R}f{ineqlocpart}) and (\mathbb{R}f{expineq}) twice, with $\alpha =1,\; c= \frac12$ and $\alpha=d-1, c=\frac12$, we get
\begin{eqnarray*}
I&=
&\int_{0}^{1}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u^{\frac{d}{2}}}e^{-\frac{|y-\sqrt{1-u}x|^{2}}{2u}}
\left(\left|\frac{y-\sqrt{1-u}x}{\sqrt{u}}
\right|e^{-\frac{|y-\sqrt{1-u}x|^{2}}{2u}}\right)\frac{|x|}{\sqrt{u}\sqrt{1-u}}du\\
&\leq
&C|x|\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}
\frac{du}{\sqrt{u}\sqrt{1-u}}
\end{eqnarray*}
\begin{eqnarray*}
&=&
C|x|\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d-1}{2}}}
\frac{du}{u\sqrt{1-u}}\\
&\leq
&\frac{C|x|}{|x-y|^{d-1}}\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}.\\
\end{eqnarray*}
By Lemma \mathbb{R}f{leminteg} we get
\begin{eqnarray*}
&&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}\\
&& \hspace{1cm} = \int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}\; +
\;\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}\\
&& \hspace{1cm} \leq C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}
du\; +
\;C\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{\sqrt{1-u}}<\infty,
\end{eqnarray*}
since $\displaystyle\frac{1}{u}$ in bounded on $[1/2,1]$ and
$\displaystyle\frac{1}{\sqrt{1-u}}$ is bounded on $[0,1/2]$. Thus, we have
\begin{equation*}
I\leq \frac{C|x|}{|x-y|^{d-1}}.
\end{equation*}
For $II$, we use again (\mathbb{R}f{ineqlocpart}) and (\mathbb{R}f{expineq}) with $\alpha =2$ and $c=\frac12$ we have
\begin{eqnarray*}
II &=&
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}+1}}\frac{|y-\sqrt{1-u}x|^{2}}{2u}du\\
&\leq &
C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{2u}}}{u^{\frac{d}{2}+1}}du\\
&\leq
& C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}+1}}du\\
&=&C \int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)}{u}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\;+
\;\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}+1}}du.\\
\end{eqnarray*}
Since $\displaystyle\lim_{u\to
0}\frac{\left(-log(\sqrt{1-u})\right)}{u}=\frac{1}{2}$, this function is bounded on $[0,1/2]$ and
$\displaystyle\frac{1}{u}$ is bounded on $[1/2,1]$, thus we get
\begin{eqnarray*}
II&\leq
&C\int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\;+
\;C\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\\
&\leq
&C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\;+
\;C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\\
&=&C\mathcal{K}_{2}(x-y)\;+\;CG_{2}(x-y),
\end{eqnarray*}
where
$$ \mathcal{K}_2(x) := \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|x|^{2}}{2u}}}{u^{\frac{d}{2}}}du,$$
and
$$
G_{2}(x):=\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|x|^{2}}{2u}}}{u^{\frac{d-1}{2}}}\frac{du}{\sqrt{u}}.
$$
Again by (\mathbb{R}f{expineq})
\begin{eqnarray*}
G_{2}(x-y)&=&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d-1}{2}}}\frac{du}{\sqrt{u}}\\
&\leq& \frac{C}{|x-y|^{d-1}}\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{du}{\sqrt{u}}.
\end{eqnarray*}
Now, by H\"older's inequality with $p=\frac{3}{2}$ y $q=3$ we have
\begin{equation*}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{du}{\sqrt{u}}\leq
\left(\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{3\beta}{2}}du\right)^{1/3}
\left(\int_{0}^{1}\frac{du}{u^{3/4}}\right)^{2/3}.
\end{equation*}
Thus, by Lemma (\mathbb{R}f{lemgamma}) we obtain
\begin{equation*}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{3\beta}{2}}du=\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{(\frac{3\beta}{2}+1)-1}du\;<\;\infty
\end{equation*}
and trivially
$\displaystyle{\int_{0}^{1}\frac{du}{u^{3/4}}}< \infty.$\\
Finally for $III$, by analogous arguments as in $II$, we get
$$
III = \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}\frac{d}{2u} du\leq C \mathcal{K}_{2}(x-y)\;+\;CG_{2}(x-y).
$$
Therefore,
\begin{eqnarray*}
\left|N_{\beta/2}(x,y)\right|&\leq & I+II+III\\
&\leq & \frac{C|x|}{|x-y|^{d-1}}\,+\,C\mathcal{K}_{2}(x-y)\,+\,\frac{C}{|x-y|^{d-1}}\\
&\leq & C\frac{|x|+1}{|x-y|^{d-1}}\,+\,C\mathcal{K}_{2}(x-y)= C\mathcal{K}_{3}(x,y)\,+\,C\mathcal{K}_{2}(x-y),
\end{eqnarray*}
where
$$ \mathcal{K}_{3}(x,y) := \frac{|x|+1}{|x-y|^{d-1}}.$$
Thus, using the above estimates, we conclude that the local part
$I_{\beta,L}$ can be bounded as
\begin{eqnarray*}
|I_{\beta,L} f(x)| &=& |I_{\beta} (f\chi_{B_h(\cdot)})(x)| = \Big| \int_{B_h(x)} N_{\beta/2} (x,y) f(y) \;dy \Big|\\
&\lesssim& \int_{B_h(x)} {\mathcal K}_3(x,y) |f(y)| \;dy + \int_{B_h(x)} {\mathcal K}_2(x-y) | f(y) |\;dy \\
&=& IV + V.\\
\end{eqnarray*}
Now, to bound IV and V, we need to take a countable family of admissible balls $ \mathcal{F}$ that satisfies the condition of Lemma 4.3 of \cite{urbina2019}. In particular, \(\mathcal{F}\) verifies
\begin{enumerate}
\item[i)] For each $B \in\mathcal{F}$ let \(\tilde{B}=2 B,\) then, the family of those balls $\tilde{\mathcal{F}}= \{B(0,1),\{\tilde{B}\}_{B \in \mathcal{F}}\}$ is a covering of \(\mathbb{R}^{d}\);
\item[ii)] \(\mathcal{F}\) has a bounded overlaps property;
\item[iii)] Every ball \(B \in \mathcal{F}\) is contained in an admissible ball, and therefore for
any pair \(x, y \in B, e^{-|x|^{2}} \sim e^{-|y|^{2}}\) with constants independent of \(B\)
\item[iv)] There exists a uniform positive constant \(C_{d}\) such that, if \(x \in B \in \mathcal{F}\) then \(B_h(x) \subset C_{d} B:=\hat{B} .\) Moreover, the collection \(\hat{\mathcal{F}}=\{\hat{B}\}_{B \in \mathcal{F}}\) also satisfies the properties ii) and iii).\\
\end{enumerate}
Given $B \in \mathcal{F},$ if $x \in B$ then $B_h(x) \subset
\hat{B}$, we get,
\begin{eqnarray*}
IV &=& (1+|x|) \sum_{k=0}^\infty \int_{2^{-(k+1)}C_d m(x) < |x-y| < 2^{-k}C_d m(x)} \frac{|f(y)| \chi_ {\hat{B}}}{|x-y|^{d-1}} dy\\
&\leq& C_d 2^d \mathcal{M}(f\chi_{\hat{B}})(x) (1+|x|) m(x)
\sum_{k=0}^\infty 2^{-(k+1)} \leq C
\mathcal{M}(f\chi_{\hat{B}})(x)\chi_{B}(x),
\end{eqnarray*}
where $\mathcal{M}(g)$ is the classical Hardy-Littlewood maximal
function of the function $g.$\\
On the other hand, let us consider the function
$\varphi(y) =\frac{1}{\pi^{d/2}} e^{-\frac{1}{2}|y|^2},$
then $\displaystyle\int_{\mathbb{R}^d} \varphi(y) dy =1$. It is well known that
$\varphi$ is a non-increasing radial function, and given $t>0,\,$
we rescale this function as $\varphi_{\sqrt{t}} (y) = t^{-d/2}
\varphi(y/\sqrt{t}),$ and, since $0\leq \varphi \in
L^1(\mathbb{R}^d),$ $\left\{\varphi_{\sqrt{t}}\right \}_{t>0}\;$ is the
classical (Gauss-Weiertrass) approximation of the identity in $\mathbb{R}^d.\,$ Then,
since\\
$$\displaystyle\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt <\infty,$$
we get
\begin{eqnarray*}
V &=& \int_{B_h(x)} \mathcal{K}_2(x-y) | f(y) |\;dy =
\int_{B_h(x)} \Big(\int_0^1 \varphi_{\sqrt{t}}(x-y)
\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt\Big) |f(y)| dy\\
&\leq& \int_{B_h(x)} \Big(\sup_{t>0} \varphi_{\sqrt{t}}(x-y)
\Big)
\Big(\int_0^1\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt\Big) |f(y)| dy\\
&\leq& C \int_{B_h(x)} \Big(\sup_{t>0} \varphi_{\sqrt{t}}(x-y)
\Big) |f(y)| dy.
\end{eqnarray*}
Again, using the family $\mathcal{F}$, if $x \in B$ then $B_h(x)
\subset \hat{B}$. By a similar argument as before and as in as result in Stein's book \cite[Chapter II \S4, Theorem 4]{st1},
\begin{eqnarray*}
V &=& \int_{B_h(x)} \mathcal{K}_2(x-y) | f(y) |\;dy \leq C
\int_{\mathbb{R}^d} \left(\sup_{t>0} \varphi_{\sqrt{t}}(x-y) \right)
|f(y)| \chi_{\hat{B}}(y) dy\\
&\lesssim& \sum_{B \in \mathcal{F}} \left| \left( \sup_{t>0}\varphi_{\sqrt{t}} * |f \chi_{\hat{B}}|\right)(x) \right| \chi_{B}(x)\leq \sum_{B \in \mathcal{F}} \mathcal{M}(f \chi_{\hat{B}})(x)
\chi_{B}(x),
\end{eqnarray*}
which yields,
$|I_{\beta,L}f(x)|\leq\displaystyle\sum_{B \in \mathcal{F}} \mathcal{M}(f \chi_{\hat{B}})(x)
\chi_{B}(x).$\\
Then, for \(f \in L^{p(\cdot)}\left(\mathbb{R}^{d}, \gamma_d\right)\) we will use the characterization of the norm by duality, we get
\begin{equation}\label{ineq2}
\left\|I_{\beta,L}(f)\right\|_{p(\cdot), \gamma_{d}} \leq 2 \sup _{\|g\|_{p^{\prime}(\cdot), \gamma_{d}} \leq 1} \int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_d(dx).
\end{equation}
Using the estimates above, we get
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_d(dx) &\lesssim&
\sum_{B \in \mathcal{F}} \int_{B} \mathcal{M}\left(f \chi_{\hat{B}(\cdot)}\right)(x)|g(x)| e^{-|x|^{2}} d x \\
&& \approx\sum_{B \in \mathcal{F}} e^{-\left|c_{B}\right|^{2}} \int_{B} \mathcal{M}\left(f \chi_{\hat{B}(\cdot)}\right)(x)|g(x)| d x,
\end{eqnarray*}
where \(c_{B}\) is the center of \(B\) and \(\hat{B}\) and we have used property iii) above, i.e. that over each ball of the family \(\mathcal{F},\) the values of \(\gamma_d\) are all equivalent.\\ Applying H\"older's inequality for \(p(\cdot)\) and \(p^{\prime}(\cdot)\) with respect of the Lebesgue measure and the boundedness of \(\mathcal{M}\) on $L^{p(\cdot)}\left(\mathbb{R}^{d}\right),$ we get
\begin{eqnarray}\label{ineq3}
\nonumber \int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) &\lesssim & \sum_{B \in \mathcal{F}} e^{-\left|c_{B}\right|^{2}}\left\|\mathcal{M}\left(f \chi_{\hat{B}(\cdot)}\right) \chi_{B}\right\|_{p(\cdot)}\left\|g \chi_{B}\right\|_{p^{\prime}(\cdot)} \\
\nonumber &\lesssim & \sum_{B \in \mathcal{F}} \mathrm{e}^{-\left|c_{B}\right|^{2}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)}\left\|g \chi_{B}\right\|_{p^{\prime}(\cdot)} \\
&=& \sum_{B \in \mathcal{F}} \mathrm{e}^{-\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)} e^{-\left|c_{B}\right|^{2} / p_{\infty}^{\prime}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot)}.
\end{eqnarray}
Since \(p \in P_{\gamma_{d}}^{\infty}\left(\mathbb{R}^{d}\right)\) and \(p_{-}>1$ then $p'\in P_{\gamma_{d}}^{\infty}\left(\mathbb{R}^{d}\right) .\) Thus, from Lemma \(1.4,\) for
every \(x \in \mathbb{R}^{d}\)
\begin{equation}\label{equiv}
e^{-|x|^{2}\left(p(x) / p_{\infty}-1\right)} \leq C_{1} \text { and } e^{-|x|^{2}\left(p^{\prime}(x) / p_{\infty}^{\prime}-1\right)} \leq C_{2}.
\end{equation}
Moreover, since the values of the Gaussian measure \(\gamma_{d}\) are all equivalent on any ball $\hat{B} \in \tilde{\mathcal{F}}$, we have
\begin{eqnarray*}
\int_{\hat{B}}\left(\frac{|f(y)|}{e^{\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} d y &\lesssim & \int_{\hat{B}}\left(\frac{|f(y)|}{\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} e^{-|y|^{2}\left(p(y) / p_{\infty}-1\right)} \gamma_{d}(dy) \\
&\lesssim &\int_{\hat{B}}\left(\frac{|f(y)|}{\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} \gamma_{d}(dy) \lesssim 1,
\end{eqnarray*}
which yields
$$
e^{-\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)} \lesssim \left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}.
$$
Similarly, by the second inequality of (\mathbb{R}f{equiv}) we also get
$$
e^{-\left|c_{B}\right|^{2} / p_{\infty}^{\prime}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot)} \lesssim \left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot), \gamma_{d}}.
$$
Replacing both estimates in (\mathbb{R}f{ineq3}) we obtain
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) & \lesssim &\sum_{B \in \mathcal{F}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot), \gamma_{d}} \\
&=& \sum_{B \in \mathcal{F}}\left\|f \chi_{\hat{B}} e^{-|\cdot|^{2} / p(\cdot) |}\right\|_{p(\cdot)}\left\|g \chi_{\hat{B}} e^{-|\cdot|^{2} / p^{\prime}(\cdot)}\right\|_{p^{\prime}(\cdot)}.
\end{eqnarray*}
Since the family of balls \(\hat{\mathcal{F}}\) has bounded overlaps, from Corollary 1.1 applied to \(f e^{-|\cdot|^{2} / p(\cdot)} \in L^{p(\cdot)}(\mathbb{R}^{d})\) and \(g e^{-|\cdot|^{2} / p^{\prime}(\cdot)} \in L^{p^{\prime}(\cdot)}(\mathbb{R}^{d}),\) it follows that
$$
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) \leqslant\|f\|_{p(\cdot), \gamma_{d}}\|g\|_{p^{\prime}(\cdot), \gamma_{d}}.
$$
Taking supremum over all functions \(g\) with \(\|g\|_{p^{\prime}(\cdot), \gamma_{d}} \leq 1,\) from (\mathbb{R}f{ineq2}) we get finally
$$
\left\|I_{\beta,L}(f)\right\|_{p(\cdot), \gamma_{d}}\leq C\|f\|_{p(\cdot), \gamma_{d}}.\\
$$
Therefore, the local part $I_{\beta,L}$ is bounded in $L^{p(\cdot)}(\gamma_d)$.\\
\item Now, let us study the global part. Again, since
$$
\left|N_{\beta/2}(x,y)\right|\leq
I+II+III,
$$
we need to estimate each term in this part. As usual for the global part, the arguments are completely different but are based on the following technical result, obtained S. P\'erez \cite[Lemma 3.1]{pe}, see also \cite[\S4.5]{urbina2019}.To simplify the notation, in what follows we denote
$$
a=a(x, y):=|x|^{2}+|y|^{2}, b=b(x, y):=2\langle x, y\rangle \text {, }
$$
$$
u(t)=u(t ; x, y):=\frac{|y-\sqrt{1-t x}|^{2}}{t}=\frac{a}{t}-\frac{\sqrt{1-t}}{t} b-|x|^{2},
$$
$$t_{0}=2 \frac{\sqrt{a^{2}-b^{2}}}{a+\sqrt{a^{2}-b^{2}}},$$
then
$$
u\left(t_{0}\right)=\frac{\sqrt{a^{2}-b^{2}}}{2}+\frac{a}{2}-|x|^{2}=\frac{|y|^{2}-|x|^{2}}{2}+\frac{\sqrt{a^{2}-b^{2}}}{2}
$$
and
$$
t_{0} \sim \frac{\sqrt{a^{2}-b^{2}}}{a} \sim \frac{\sqrt{a-b}}{\sqrt{a+b}}=\frac{|x-y|}{|x+y|}.
$$
It is well known that $t_{0}<1$, the minimum of $u(t)$ is attained at
$t_{0}$ and
$$\frac{1}{t_{0}^{d/2}}\lesssim |x+y|^{d}.$$
For details and other properties of these terms, see \cite{pe}, \cite{TesSon} or \cite{urbina2019}.\\
Let us fix $x\in \Bbb R^{d}$ and consider $E_{x}=\lbrace y\in \Bbb R^{d}:b>0\rbrace$.\\
\begin{itemize}
\item \underline{Case $b\leq 0$}:\\
First, for $0<
{$\Box $}silon <1$, using inequality (\mathbb{R}f{expineq}) we have
\begin{eqnarray*}
I&=&
\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-u(t)}}{t^{\frac{d}{2}}}\frac{\left|y-\sqrt{1-t}x\right||x|}{t\sqrt{1-t}}dt\\
&=&
|x|\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-
{$\Box $}silon
u(t)-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{\left|y-\sqrt{1-t}x\right|}{\sqrt{t}\sqrt{1-t}}dt\\
&\leq &
C_{
{$\Box $}silon}|x|\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}.
\end{eqnarray*}
Since
$\displaystyle\frac{\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}}{\sqrt{1-t}}$
is continuous on $[0,1/2]$ and therefore bounded, we get
\begin{equation*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}
\leq C_{\beta}
\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}dt,
\end{equation*}
and since $0<t<1/2$, $\frac{1}{\sqrt{t}}<\frac{1}{t}$ and then, by (\mathbb{R}f{desint}), we get
\begin{eqnarray*}
\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}dt&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d+1}{2}}}dt\\
&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\\
&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a}= C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{eqnarray*}
Analogously,
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}
&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}\\
&\leq &C_{d}e^{(1-
{$\Box $}silon)|x|^{2}}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{\sqrt{1-t}}dt,
\end{eqnarray*}
since $\displaystyle\frac{1}{t^{\frac{d+1}{2}}}$ is bounded on
$[1/2,1]$, also as $1/2<t<1$ we have $-\frac{a}{t}<-a$ and then by Lemma \mathbb{R}f{leminteg}
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{\sqrt{1-t}}dt&\leq
&e^{-(1-
{$\Box $}silon)a}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{dt}{\sqrt{1-t}}\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)a}.
\end{eqnarray*}
Thus,
\begin{equation*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}
\leq C_{d,\beta}
e^{(1-
{$\Box $}silon)|x|^{2}}e^{-(1-
{$\Box $}silon)a}=Ce^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Therefore
$$I \leq C_
{$\Box $}silon |x| e^{-(1-
{$\Box $}silon)|y|^{2}}.$$
Now, using again inequality (\mathbb{R}f{expineq}), we get
\begin{eqnarray*}
II&=&
\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-u(t)}}{t^{\frac{d}{2}}}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{2t^{2}}dt\\
&=&
\frac{1}{2}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-
{$\Box $}silon
u(t)-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}+1}}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{t}dt\\
&\leq
&C_{
{$\Box $}silon}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}+1}}dt\\
&\leq
&C_{
{$\Box $}silon}e^{(1-
{$\Box $}silon)|x|^{2}}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt.\\
\end{eqnarray*}
Now,
\begin{equation*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt
\leq
C\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt
\leq
C\int_{0}^{1}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt.
\end{equation*}
Therefore, by taking the change of variables $s=a(\frac{1}{t}-1)$, we get
\begin{eqnarray}\label{desint}
\nonumber\int_{0}^{1}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&=
&\int_{0}^{+\infty}e^{-(1-
{$\Box $}silon)(s+a)}\left(\frac{s+a}{a}\right)^{\frac{d}{2}+1}\frac{a}{\left(s+a\right)^{2}}ds\\
\nonumber&=
&\frac{e^{-(1-
{$\Box $}silon)a}}{a^{\frac{d}{2}}}\int_{0}^{+\infty}e^{-(1-
{$\Box $}silon)s}\left(s+a\right)^{\frac{d}{2}-1}ds\\
\nonumber &\leq
&\frac{Ce^{-(1-
{$\Box $}silon)a}}{a^{\frac{d}{2}}}\int_{0}^{+\infty}e^{-(1-
{$\Box $}silon)s}\left(s^{\frac{d}{2}-1}+a^{\frac{d}{2}-1}\right)ds\\
\nonumber &\leq &\frac{C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a}}{a^{\frac{d}{2}}}\left(\Gamma\left(\frac{d}{2}\right)\;+\;a^{\frac{d}{2}-1}\right)\\
\nonumber &=&C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a}\left(\frac{\Gamma\left(\frac{d}{2}\right)}{a^{\frac{d}{2}}}\;+\;\frac{1}{a}\right)\\
&\leq &C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a},
\end{eqnarray}
since $a\geq\frac{d}{2}$. Analogously, by Lemma \mathbb{R}f{lemgamma}
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)}}{t^{\frac{d}{2}+1}}dt
&\leq &C\int_{ 1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-(1-
{$\Box $}silon)\frac{a}{t}}dt\\
&\leq &C\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-(1-
{$\Box $}silon)a}dt\\
&\leq&Ce^{-(1-
{$\Box $}silon)a}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}dt \\
&=& Ce^{-(1-
{$\Box $}silon)a}.
\end{eqnarray*}
Then,
\begin{equation*}
II\leq
C_{
{$\Box $}silon}e^{(1-
{$\Box $}silon)|x|^{2}}e^{-(1-
{$\Box $}silon)a}=C_{
{$\Box $}silon}e^{(1-
{$\Box $}silon)|x|^{2}}e^{-(1-
{$\Box $}silon)\left(|y|^{2}+|x|^{2}\right)}=C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Finally,
\begin{eqnarray*}
III&=&\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-u(t)}}{t^{\frac{d}{2}}}\frac{d}{2}\frac{dt}{t}\leq \frac{d}{2}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-\frac{a}{t}+|x|^{2}}\frac{dt}{t^{\frac{d}{2}+1}}\\
&= &Ce^{|x|^{2}}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt.\\
\end{eqnarray*}
Now,
\begin{equation*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\leq
C\int_{0}^{1/2}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\leq
C\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt
\end{equation*}
since $\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}$ is
bounded in $[0,1/2],$ by taking
$$s=a(\frac{1}{t}-1), \;ds=-\frac{a}{t^{2}}dt,$$
we get
\begin{eqnarray*}
\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&=&
\int_{0}^{+\infty}e^{-(s+a)}\left(\frac{s+a}{a}\right)^{\frac{d}{2}+1}\left(\frac{a}{s+a}\right)^{2}\frac{ds}{a}\\
&=&\frac{e^{-|y|^{2}-|x|^{2}}}{a^{\frac{d}{2}}}\int_{0}^{+\infty}e^{-s}\left(s+a\right)^{\frac{d}{2}-1}ds.\\
\end{eqnarray*}
On the other hand,
\begin{equation*}
\left(s+a\right)^{\frac{d}{2}-1}=\frac{\left(s+a\right)^{\frac{d}{2}}}{s+a}\leq
C\frac{\left(s^{\frac{d}{2}}+a^{\frac{d}{2}}\right)}{s+a}\leq
C\left(\frac{s^{\frac{d}{2}}}{s}+\frac{a^{\frac{d}{2}}}{a}\right)
=C\left(s^{\frac{d}{2}-1}+a^{\frac{d}{2}-1}\right).
\end{equation*}
Therefore,
\begin{eqnarray*}
\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&\leq &
C\frac{e^{-|y|^{2}-|x|^{2}}}{a^{\frac{d}{2}}}\left(\int_{0}^{+\infty}e^{-s}s^{\frac{d}{2}-1}ds\;+
\;\int_{0}^{+\infty}e^{-s}a^{\frac{d}{2}-1}ds\right)\\
&=&C\frac{e^{-|y|^{2}-|x|^{2}}}{a^{\frac{d}{2}}}\left(\Gamma\left(\frac{d}{2}\right)\;+\;a^{\frac{d}{2}-1}\right)=Ce^{-|y|^{2}-|x|^{2}}\left(\frac{\Gamma\left(\frac{d}{2}\right)}{a^{\frac{d}{2}}}\;+\;\frac{1}{a}\right).
\end{eqnarray*}
Thus,
\begin{equation*}
\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\leq
Ce^{-|y|^{2}-|x|^{2}},\hspace{0.7cm}\mbox{as }a\geq\frac{d}{2}.
\end{equation*}
For $\frac{1}{2}<t<1,\hspace{0.3cm}-a>-\frac{a}{t}>-2a$. Hence, by Lemma \mathbb{R}f{lemgamma}
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&\leq
&C\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-a}dt\\
&\leq
&Ce^{-a}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}dt\\
&=&Ce^{-|y|^{2}-|x|^{2}}.
\end{eqnarray*}
Then,
$$
III\leq Ce^{|x|^{2}}e^{-|y|^{2}-|x|^{2}}= Ce^{-|y|^{2}}\leq Ce^{-(1-
{$\Box $}silon)|y|^{2}}.
$$
In other words,
\begin{equation*}
I\leq C_{
{$\Box $}silon}|x|e^{-(1-
{$\Box $}silon)|y|^{2}}
\end{equation*}
and
\begin{equation*}
II, \; III\leq Ce^{-|y|^{2}}\leq Ce^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Thus, for $b\leq 0$
\begin{equation*}
\left|N_{\beta/2}(x,y)\right|\leq
C_{
{$\Box $}silon}(|x|+1)e^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Next, we take $0<
{$\Box $}silon<1/p'_{-}$ and $\tilde{
{$\Box $}silon}=1/p'_{-} -
{$\Box $}silon = 1-
{$\Box $}silon-1/p_{-}$. Then, $\tilde{
{$\Box $}silon}>0$ and $1-
{$\Box $}silon=\tilde{
{$\Box $}silon}+1/p_{-}$\,.Therefore, for $f\in L^{p(\cdot)}(\mathbb{R}^{d},\gamma_{d})$ with $\|f\|_{p,\gamma_{d}}=1$, using H\"older's inequality
\begin{eqnarray*}
\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E^{c}_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx) &\lesssim
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}(|x|+1)e^{-(1-
{$\Box $}silon)|y|^{2}}|f(y)|dy\right)^{p(x)}\gamma_{d}(dx)
\\ &=
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}e^{-(\tilde{
{$\Box $}silon}+1/p_{-})|y|^{2}}|f(y)|dy\right)^{p(x)}(|x|+1)^{p(x)}\gamma_{d}(dx)
\\ &\leq
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}|f(y)|^{p_{-}}e^{-|y|^{2}}dy\right)^{p(x)/p_{-}}\left(\int\limits_{\mathbb{R}^{d}}e^{-\tilde{
{$\Box $}silon}|y|^{2}p'_{-}}dy\right)^{p(x)/p'_{-}}\\
&&\hspace{5.5cm} \times (|x|+1)^{p_{+}}\gamma_{d}(dx)
\\ &\lesssim
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}|f(y)|^{p_{-}}e^{-|y|^{2}}dy\right)^{\frac{p(x)}{p_{-}}}(|x|+1)^{p_{+}}\gamma_{d}(dx),
\end{eqnarray*}
and, since $\rho_{p(\cdot),\gamma_{d}}(f)\leq 1$,
\begin{eqnarray*}
\int\limits_{\mathbb{R}^{d}}|f(y)|^{p_{-}}e^{-|y|^{2}}dy&\lesssim&\int\limits_{|f|\geq 1}|f(y)|^{p_{-}}\gamma_{d}(dy)+\int\limits_{|f|<1}|f(y)|^{p_{-}}\gamma_{d}(dy)\\
&\leq&\int\limits_{|f|\geq 1}|f(y)|^{p(y)}\gamma_{d}(dy)+1\leq\rho_{p(\cdot),\gamma_{d}}(f)+1\leq 2.
\end{eqnarray*}
Thus,
$$\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E^{c}_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx) \lesssim
2^{\frac{p_{+}}{p_{-}}}\int\limits_{\mathbb{R}^{d}}(|x|+1)^{p_{+}}\gamma_{d}(dx)=C_{d,p}.$$
Hence,
$$\|I_{\beta}(f\chi_{B_{h}^{c}(\cdot)\cap E^{c}_{(\cdot)}})\|_{p(\cdot),\gamma_{d}}\leq C_{d,p}.$$
\item \underline{Case $b>0$}:\\
In this case, $I$ is a very problematic term so we will discuss it at the end.\\
Again, by inequality (\mathbb{R}f{expineq})
\begin{eqnarray*}
II&=&\frac{1}{2}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-u(t)}}{t^{d/2+1}}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{t}\,dt\\
&=&\frac{1}{2}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}} e^{-
{$\Box $}silon u(t)}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{t}\,dt\\
&\leq &C_{
{$\Box $}silon}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt.
\end{eqnarray*}
By using inequality (4.44) of \cite{urbina2019}, see also \cite{pe}, we get
\begin{equation}\label{phib0}
\frac{e^{-u(t)}}{t^{d/2}}\leq 2^{d}\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}.
\end{equation}
Thus,
\begin{eqnarray}\label{phib1}
\nonumber \frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2}}&=&\left(\frac{e^{-u(t)}}{t^{d/2}}\right)^{1-
{$\Box $}silon} \frac{1}{t^{
{$\Box $}silon
d/2}}\\
&\lesssim
&\left(\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}\right)^{1-
{$\Box $}silon} \frac{1}{t^{
{$\Box $}silon
d/2}}=\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\frac{1}{t^{
{$\Box $}silon
d/2}}.
\end{eqnarray}
Now, splitting the above integral into two the integrals on $[0,1/2]$ and $[1/2,1]$; we have for the first integral using (\mathbb{R}f{phib1}),
\begin{eqnarray*}
\int^{1/2}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt&\lesssim
&\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\int^{1/2}_{0}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+1}}\,dt.
\end{eqnarray*}
Set $r=\min\{\frac{\beta}{4},\frac{1}{2}\},\,$ then
$0<r<1$ and by taking $
{$\Box $}silon>0$ such that $\frac{
{$\Box $}silon d}{2}<
\frac{\beta}{2}-r\,\,$ we get
\begin{equation*}
\lim_{t\to 0^{+}}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+r}} = \lim_{t\to 0^{+}}\left[\frac{\left(-log(\sqrt{1-t})\right)}{t}\right]^{\beta/2}t^{\beta/2 -(
{$\Box $}silon d/2+r)}= 0,
\end{equation*}
thus, $\displaystyle\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+r}}$ is bounded on $(0,1/2]$,
and hence
\begin{equation*}
\int^{1/2}_{0}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+1}}\,dt=\int^{1/2}_{0}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+r}}\frac{dt}{t^{1-r}}\leq
C_{
{$\Box $}silon,\beta}\int^{1/2}_{0}\frac{dt}{t^{1-r}}=C_{
{$\Box $}silon,\beta}.
\end{equation*}
Then,
\begin{equation}\label{bceromedio}
\int^{1/2}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt\;\leq\;C_{
{$\Box $}silon,\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}.
\end{equation}
For the integral on $[1/2,1]$ we have, using again (\mathbb{R}f{phib1}) and Lemma \mathbb{R}f{lemgamma}
\begin{eqnarray*}
\int^{1}_{1/2}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt&=&
\int^{1}_{1/2}\left(-log(\sqrt{1-t})\right)^{\beta/2}\left(\frac{e^{-u(t)}}{t^{d/2}}\right)^{(1-
{$\Box $}silon)}\,\frac{dt}{t^{
{$\Box $}silon
d/2+1}}\\
&\leq
&C_{
{$\Box $}silon}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\int^{1}_{1/2}\left(-log(\sqrt{1-t})\right)^{\beta/2}\,dt\\
&\leq
&C_{
{$\Box $}silon}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\,dt\\
&=&C_{
{$\Box $}silon,\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}.
\end{eqnarray*}
Therefore, since $t_{0}<1$ and
$\frac{d}{2}(1-
{$\Box $}silon)<\frac{d}{2},$ we get
$$
II\leq C_{
{$\Box $}silon,\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}
\leq C_{
{$\Box $}silon}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d/2}}.
$$
Now, using (\mathbb{R}f{phib0}) and (\mathbb{R}f{gammaint}) we get
\begin{eqnarray*}
III=\frac{1}{4}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-u(t)}}{t^{d/2}}\frac{dt}{t}&\lesssim&
\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{dt}{t}\\
&=&C_{\beta}\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}\leq
C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d/2}}.
\end{eqnarray*}
Now, to estimate $I$, we use again inequality (\mathbb{R}f{expineq})
\begin{eqnarray*}
I&=&\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-u(t)}}{t^{d/2}}\frac{\left|y-\sqrt{1-t}x\right|}{\sqrt{t}}\frac{|x|}{\sqrt{1-t}}\frac{dt}{\sqrt{t}}\\
&\leq
&C_{
{$\Box $}silon}|x|\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2}\sqrt{t}}\frac{dt}{\sqrt{1-t}}.
\end{eqnarray*}
Again, splitting the above integral into two the integrals on $[0,1/2]$ and $[1/2,1]$. For the second integral, using (\mathbb{R}f{phib0}), that $t \geq 1/2$ and Lemma \mathbb{R}f{leminteg}, we get
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}&\lesssim& e^{-(1-
{$\Box $}silon)u(t_{0})}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{dt}{\sqrt{1-t}}\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})},
\end{eqnarray*}
Next, for the integral on $[0,1/2]$ we need to consider two cases:\\
\begin{itemize}
\item Case $\beta>0$ and $d=1$: by Lemma \mathbb{R}f{lemgamma} and the fact that $\frac{-log(\sqrt{1-t})}{t}$ is bounded on $(0, 1/2],$
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{1}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}
&=&\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t}\frac{dt}{\sqrt{1-t}}\\
&=&\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}\frac{\left(-log(\sqrt{1-t})\right)}{t}e^{-(1-
{$\Box $}silon)u(t)}\frac{dt}{\sqrt{1-t}}\\
&\leq&C_{\beta}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}e^{-(1-
{$\Box $}silon)u(t)}\frac{dt}{\sqrt{1-t}}\\
&\leq&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}dt\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\item Case $\beta\geq 1$ and $d\geq 2$: by taking $
{$\Box $}silon <
\frac{2}{d}$
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}
&\leq& C\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}}\frac{dt}{\sqrt{1-t}}\\
&= &C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}}{t^{\frac{(d-2)}{2}}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t\sqrt{1-t}}dt\\
&= &C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}}{t^{\frac{d}{2}\frac{(d-2)}{d}}}e^{-(\frac{d-2}{d})u(t)}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt\\
&\leq &C\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}\frac{(d-2)}{d}}}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt.
\end{eqnarray*}
Since
$\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}$ is continuous
on $[0,1/2]$ for $\beta\geq 1$ and proceeding in analogous way as in Lemma 4.36 of \cite{urbina2019}, we get
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}} &\leq& C_{\beta}\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}\frac{(d-2)}{d}}}\int_{0}^{1}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt\\
&\leq &C_{\beta}\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}-1}}e^{(
{$\Box $}silon-\frac{2}{d})u(t_{0})}\\
&= &C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}-1}}= C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}}}t_{0}.
\end{eqnarray*}
Thus,
$$I\leq C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}.$$
\end{itemize}
Finally, since $\displaystyle\frac{1}{t_{0}^{d/2}}\lesssim |x+y|^{d}$, $t_{0}<1$, and $|x|\leq |x+y|$ as $b>0$, we have
\begin{itemize}
\item For $|x|<1$,
\begin{eqnarray*}
I&\leq& C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}\leq C_{
{$\Box $}silon,\beta}\frac{1}{t_{0}^{\frac{d}{2}}}e^{-(1-
{$\Box $}silon)u(t_{0})}\\
&\leq& C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\item For $|x|\geq 1$, since $b>0$, $|x|\leq |x+y|$
\begin{eqnarray*}
I&\leq & C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}\\
&=& C_{
{$\Box $}silon,\beta}|x|e^{-(1-
{$\Box $}silon)u(t_{0})}\,+\, C_{
{$\Box $}silon,\beta}|x|t_{0}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}}}\\
&\leq &
C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}\,+\, C_{
{$\Box $}silon,\beta}t_{0}|x||x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\end{itemize}
Given that $t_{0}\leq C\frac{|x-y|}{|x+y|}$ and the fact that $|x|\leq |x+y| $ we get, for $|x-y|<1$,
$$
|x|t_{0}\leq C\frac{|x||x-y|}{|x+y|}\leq C.
$$
Thus, for $|x-y|<1$,
$$
I\leq
C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})};
$$
and for $|x-y|\geq 1$,
$$
I\leq
C_{
{$\Box $}silon,\beta}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}.
$$
Hence, we conclude that\\
\begin{itemize}
\item $\left|N_{\beta/2}(x,y)\right|\leq C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})},$ \; for either $|x|\leq 1,$ or for $|x|\geq 1$ with $|x-y|<1$.\\
\item $\left|N_{\beta/2}(x,y)\right|\leq C_{
{$\Box $}silon,\beta}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})},$ \; for \; $|x|\geq 1$ with $|x-y|\geq 1$.\\
\end{itemize}
Now, as $b>0$, for $f\in L^{p(\cdot)}(\mathbb{R}^{d},\gamma_{d})$ with $\|f\|_{p(\cdot),\gamma_{d}}=1$, we have that
\begin{eqnarray*}
&& \int_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)= \int_{|x|<1}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&& \hspace{.5cm}+ \int_{|x|\geq 1}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|<1}\left|N_{\beta/2}(x,y)\right||f(y)|dy+\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)
\end{eqnarray*}
\begin{eqnarray*}
&\leq& \int_{|x|<1}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&+& C\int_{|x|\geq 1}\left(\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|<1}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}+\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\right)\gamma_{d}(dx)\\
&\leq&C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}|f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&& \hspace{0.75cm}+ C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}|f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&=&C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx\\
&& \hspace{0.75cm}+ C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx.
\end{eqnarray*}
Since $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, we obtain that $e^{|y|^{2}/p(y)-|x|^{2}/p(x)}\approx e^{(|y|^{2}-|x|^{2})/p_{\infty}},$ and by the Cauchy-Schwartz inequality we have,
$\left| |y|^{2}-|x|^{2}\right|\leq|x+y| |x-y|,$
for all $x,y\in\mathbb{R}^{d}$. Therefore,
\begin{eqnarray*}
&&\int_{B^{c}_h(x)\cap E_{x}}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy \hspace{3.5cm}\\
&& \hspace{6.5cm} \lesssim \int_{B^{c}_h(x)\cap E_{x}}P(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy,
\end{eqnarray*}
and \begin{eqnarray*}
&&\int_{B^{c}_h(x)\cap E_{x},|x-y|\geq 1}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy \hspace{3.5cm}\\
&& \hspace{5.5cm} \lesssim \int_{B^{c}_h(x)\cap E_{x}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy,
\end{eqnarray*}
where
$$
P(x,y)=|x+y|^{d}e^{-\alpha_{\infty}|x+y||x-y|}, \; Q(x,y)=|x+y|^{d+1}e^{-\alpha_{\infty}|x+y|}$$
and
$$ \alpha_{\infty}=\left(\frac{1-
{$\Box $}silon}{2}-\left|\frac{1}{p_{\infty}}-\frac{1-
{$\Box $}silon}{2}\right|\right).$$
It is easy to see that $\alpha_{\infty}>0$ if $
{$\Box $}silon<1/p'_{\infty}$.
Therefore, in order to make sense of all the estimates above we need to take
$$0<
{$\Box $}silon<\displaystyle\min\left\lbrace \frac{1}{d},\frac{1}{p'_{\infty}},\frac{2}{d}\left(\frac{\beta}{2}-r\right)\right\rbrace.$$
Observe that $P(x,y)$ is the same kernel considered in the proof of Theorem 3.5, page 416 of \cite{DalSco}, so we can conclude that
$$\displaystyle\int_{\Bbb R^{d}}\left(\int_{B^{c}_h(x)\cap E_{x}}P(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx\leq C.$$
On the other hand, it can be proved that $Q(x,y)$ is integrable on each variable and the value of each integral is independent of $x$ and $y$.
Now, we use an analogous argument as in \cite{DalSco} for $Q(x,y)$. Taking,
$$J=\displaystyle\int_{\Bbb R^{d}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy,$$
and using H\"older's inequality, we obtain
$$ J\lesssim\|Q(x,\cdot)\|_{p'(\cdot)}\|fe^{-|\cdot|^{2}/p(\cdot)}\|_{p(\cdot)}\leq \|Q(x,\cdot)\|_{p'(\cdot)}.$$
and,
\begin{eqnarray*}
\int_{\Bbb R^{d}}Q(x,y)^{p'(y)}dy&=&\int_{\Bbb R^{d}}|x+y|^{(d+1)p'(y)}e^{-\alpha_{\infty}|x+y|p'(y)}dy\\
&\leq&\int_{|x+y|<1}|x+y|^{d+1}e^{-\alpha_{\infty}|x+y|}dy +\int_{|x+y|\geq 1}|x+y|^{(d+1)p'_{+}}e^{-\alpha_{\infty}|x+y|}dy\\
&\leq&\int_{\Bbb R^{d}}\left(|z|^{d+1}+|z|^{(d+1)p'_{+}}\right)e^{-\alpha_{\infty}|z|}dz\\
&=&C_{p,d}.
\end{eqnarray*}
Thus, $J\lesssim\|Q(x,\cdot)\|_{p^{'}(\cdot)}\leq C_{p,d}$, and therefore
$$\frac{1}{C_{p,d}}\int_{\Bbb R^{d}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy\leq 1.$$
We set $g(y)=|f(y)|e^{-|y|^{2}/p(y)}=g_{1}(y)+g_{2}(y)$, where $g_{1}=g\chi_{\{g\geq 1\}}$ and $g_{2}=g\chi_{\{g<1\}}$, then
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}J^{p(x)}dx&=& \int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx \\
&\lesssim& \int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g(y)dy\right)^{p(x)}dx\\
&\lesssim& \int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p(x)}dx+\int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p(x)}dx.
\end{eqnarray*}
By H\"{o}lder's inequality and Fubini's theorem
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p(x)}dx&\lesssim&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p_{-}}dx\\
&=&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q^{\frac{1}{p'_{-}}}(x,y)Q^{\frac{1}{p_{-}}}(x,y)g_{1}(y)dy\right)^{p_{-}}dx\\
&\leq&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)dy\right)^{p_{-}/p'_{-}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g^{p_{-}}_{1}(y)dy\right)dx\\
&=&C_{p}\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g^{p_{-}}_{1}(y)dy\right)dx\\
&=&C_{p}\int_{\mathbb{R}^{d}}g^{p_{-}}_{1}(y)\left(\int_{\mathbb{R}^{d}}Q(x,y)dx\right)dy=C_{p}\int_{\mathbb{R}^{d}}g^{p_{-}}_{1}(y)dy.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p(x)}dx&\lesssim&\int_{\mathbb{R}^{d}}g^{p(y)}_{1}(y)dy\leq\int_{\mathbb{R}^{d}}|f(y)|^{p(y)}e^{-|y|^{2}}dy\\
&\lesssim& \rho_{p(\cdot),\gamma_{d}}(f)\leq 1.
\end{eqnarray*}
On the other hand, applying the inequality (\mathbb{R}f{3.26.1}) in Lemma \mathbb{R}f{lema3.26CU}, since\\
$G(x):=\displaystyle\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\leq 1$, we obtain that
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\frac{1}{C_{p,d}}Q(x,y)g_{2}(y)dy\right)^{p(x)}dx&=&\int_{\mathbb{R}^{d}}(G(x))^{p(x)}dx \\
&\leq& \int_{\mathbb{R}^{d}}(G(x))^{p_{\infty}}dx + \int_{\mathbb{R}^{d}}\frac{dx}{(e+|x|)^{dp_{-}}}\\
&\lesssim&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p_{\infty}}+C_{d,p}.
\end{eqnarray*}
Finally, to estimate the integral
$$\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p_{\infty}}dx,$$
we proceed in an analogous way, applying H\"{o}lder's inequality to the exponent $p_{\infty}$, Fubbini's theorem and inequality (\mathbb{R}f{3.26.2}) in Lemma \mathbb{R}f{lema3.26CU}, we get
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p_{\infty}}dx &\lesssim&\int_{\mathbb{R}^{d}}g_{2}^{p_{\infty}}(y)dy\\
&\leq&\int_{\mathbb{R}^{d}}g_{2}^{p(y)}(y)dy+\int_{\mathbb{R}^{d}}\frac{dy}{(e+|y|)^{dp_{-}}}\\
&\leq&\int_{\mathbb{R}^{d}}|f(y)|^{p(y)}e^{-|y|^{2}}dy+C_{d,p}\lesssim 1+C_{d,p}.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)&\leq& C_{d,p}.
\end{eqnarray*}
With this, we obtain that $\|I_{\beta}(f\chi_{B_{h}^{c}(\cdot)\cap E_{(\cdot)}})\|_{p(\cdot),\gamma_{d}}\leq C_{d,p}$.\\
We conclude that
$$
\left\|I_{\beta,G}(f)\right\|_{p(\cdot), \gamma_{d}}=\left\|I_{\beta}\left(f \chi_{B^{c}_h(\cdot)}\right)\right\|_{p(\cdot), \gamma_{d}} \leq C,
$$\\
and by homogeneity of the norm,\\
$
\left\|I_{\beta,G}(f)\right\|_{p(\cdot), \gamma_{d}} \leq C\|f\|_{p(\cdot),\gamma_{d}}$, for all function $f\in L^{p(\cdot)}(\mathbb{R}^{d},\gamma_{d})$.\\
Therefore, the global part $I_{\beta,G}$ is bounded in $L^{p(\cdot)}(\gamma_d)$ and the proof is complete. \\
\end{itemize}
\end{itemize}
\end{proof}
\end{document}
|
\begin{document}
\title{
\leftline{Effective de la Vall\'e Poussin style bounds}
on the first Chebyshev function\\
}
\author{
\Large
Matt Visser\!\orcidMatt\!
}
\affiliation{School of Mathematics and Statistics, Victoria University of Wellington, \\
\null\qquad PO Box 600, Wellington 6140, New Zealand.}
\mathrm{e}mailAdd{[email protected]}
{\mathrm{d}}ef\vartheta{\vartheta}
{\mathrm{d}}ef{\mathcal{O}}{{\mathcal{O}}}
\abstract{
In 1898 Charles Jean de la Vall\'e Poussin, as part of his famed proof of the prime number theorem, developed an ineffective bound on the first Chebyshev function of the form:\\
\[
|\vartheta(x)-x| = {\mathcal{O}}\left(x \mathrm{e}xp(-K \sqrt{\ln x})\right).
\]
This bound holds for $x$ sufficiently large, $x\geq x_0$, and $K$ some unspecified positive constant.
To the best of my knowledge this bound has never been made effective --- I have never yet seen this bound made fully explicit, with precise values being given for $x_0$ and $K$.
Herein, using a number of effective results established over the past 50 years, I shall develop two very simple explicit fully effective bounds of this type:
\[
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 2).
\]
\[
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 3).
\]
Many other fully explicit bounds along these lines can easily be developed. \\
For instance one can trade off stringency against range of validity:
\[
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 29),
\]
\[
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 41).
\]
With hindsight, some of these effective bounds could have been established almost 50 years ago.
\noindent
{\sc Date:} 2 November 2022; \LaTeX-ed \today
\noindent{\sc Keywords}: Chebyshev $\vartheta$ function; effective bounds.
}
\maketitle
{\mathrm{d}}ef{\mathrm{tr}}{{\mathrm{tr}}}
{\mathrm{d}}ef{\mathrm{diag}}{{\mathrm{diag}}}
{\mathrm{d}}ef{\mathrm{cof}}{{\mathrm{cof}}}
{\mathrm{d}}ef{\mathrm{pdet}}{{\mathrm{pdet}}}
{\mathrm{d}}ef{\mathrm{d}}{{\mathrm{d}}}
\parindent0pt
\parskip7pt
{\mathrm{d}}ef{\scriptscriptstyle{\mathrm{Kerr}}}{{\scriptscriptstyle{\mathrm{Kerr}}}}
{\mathrm{d}}ef\mathrm{e}os{{\scriptscriptstyle{\mathrm{eos}}}}
\section{Introduction}
In 1898 Charles Jean de la Vall\'e Poussin developed an ineffective bound on the first Chebyshev function of the form~\cite{Poussin}:
\begin{equation}
\label{E:Poussin}
|\vartheta(x)-x| = {\mathcal{O}}\left(x \mathrm{e}xp(-K \sqrt{\ln x})\right).
\mathrm{e}nd{equation}
This bound holds for $x$ sufficiently large, $x\geq x_0$, and $K$ some unspecified positive constant.
Subsequent work over the last 50 years has developed a large number of related but distinct fully effective bounds of the form~\cite{Schoenfeld,Trudgian,Johnston-Yang,Fiori-et-al}:
\begin{equation}
|\vartheta(x)-x| < a \;x \;(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right);
\qquad (x \geq x_0).
\mathrm{e}nd{equation}
\begin{itemize}
\item
For some widely applicable effective bounds of this type see Table I.\\
(A straightforward elementary numerical computation is required to determine the numerical coefficients in the Schoenfeld~\cite{Schoenfeld} and Trudgian~\cite{Trudgian} bounds.)
\item
For some asymptotically more stringent effective bounds of this type, but valid on significantly more restricted regions, see Table~II (based on reference~\cite{Johnston-Yang}), and the extensive tabulations in reference~\cite{Broadbent-et-al}.
\mathrm{e}nd{itemize}
\mathrm{e}nlargethispage{30pt}
What I have not yet seen is any attempt to take the effective bounds of Tables I and II and use them to
make the original de la Vall\'e Poussin bound fully effective.
Here are two particularly clean fully effective versions of the de la Vall\'e Poussin bound:
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 2).
\mathrm{e}nd{equation}
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 3).
\mathrm{e}nd{equation}
I shall explain how to derive these bounds below.
\begin{table}[!h]
\caption{Some widely applicable effective bounds.}
\begin{center}
\begin{tabular}{||c|c|c|c||c||}
\hline
\hline
$a$ & $b$ & $c$ & $x_0$ & Source \\
\hline
\hline
0.2196138920& 1/4 & 0.3219796502 & 101 & Schoenfeld~\cite{Schoenfeld}\\
\hline
\hline
0.2428127763 & 1/4 &0.3935970880 & 149 & Trudgian~\cite{Trudgian}\\
\hline
\hline
9.220226 & 3/2 & 0.8476836 & 2 & Fiori--Kadiri--Swidinsky~\cite{Fiori-et-al}\\
\hline
\hline
9.40 & 1.515 & 0.8274 & 2 & Johnston--Yang~\cite{Johnston-Yang}\\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
\begin{table}[!htb]
\caption{Asymptotically stringent effective bounds valid on restricted regions~\cite{Johnston-Yang}.}
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
\hline
$a$ & $b$ & $c$ & $x_0$ \\
\hline
\hline
8.87 & 1.514 & 0.8288 & $\mathrm{e}xp(3000)$ \\
8.16 & 1.512 & 0.8309 & $\mathrm{e}xp(4000)$ \\
7.66 & 1.511 & 0.8324 & $\mathrm{e}xp(5000)$ \\
7.23 & 1.510 & 0.8335 & $\mathrm{e}xp(6000)$ \\
7.00 & 1.510 & 0.8345 & $\mathrm{e}xp(7000)$ \\
6.79 & 1.509 & 0.8353 & $\mathrm{e}xp(8000)$ \\
6.59 & 1.509 & 0.8359 & $\mathrm{e}xp(9000)$ \\
6.73 & 1.509 & 0.8359 & $\mathrm{e}xp(10000)$ \\
\hline\hline
23.14 & 1.503 & 0.8659 & $\mathrm{e}xp(10^5)$ \\
38.58 & 1.502 & 1.0318 & $\mathrm{e}xp(10^6)$ \\
42.91 & 1.501 & 1.0706 & $\mathrm{e}xp(10^7)$ \\
44.42 & 1.501 & 1.0839 & $\mathrm{e}xp(10^8)$ \\
44.98 & 1.501 & 1.0886 & $\mathrm{e}xp(10^9)$ \\
45.18 & 1.501 & 1.0903 & $\mathrm{e}xp(10^{10})$ \\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
\section{Strategy}
Note that for any $b>0$, $c>0$, and any $\tilde c\in(0,c)$, elementary calculus implies:
\begin{eqnarray}
(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right)
&=& \left\{ (\ln x)^b \mathrm{e}xp\left(-[c-\tilde c] \; \sqrt{\ln x}\right) \right\} \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right)
\nonumber\\
& \leq & \left\{ \left( 2b\over c-\tilde c \right)^{2b} \mathrm{e}xp(-2 b) \right\} \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right).
\mathrm{e}nd{eqnarray}
The key observation here is that the quantity in braces is explicitly bounded,
and achieves a global maximum at
$x_{peak} = \mathrm{e}xp\left( \left[2b/(c-\tilde c)\right]^2\right)$.
Consequently we have the following lemma.
\paragraph{Lemma:}
\mathrm{e}mph{Any effective bound of the form
\begin{equation}
|\vartheta(x)-x| < a \;x \;(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right);
\qquad (x \geq x_0).
\mathrm{e}nd{equation}
implies the existence of another effective bound of the de la Vall\'e Poussin form
\begin{equation}
|\vartheta(x)-x| < \tilde a \; x\; \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right);
\qquad (x \geq x_*; \;\; x_* \leq x_0).
\mathrm{e}nd{equation}
Here $\tilde c$ is an arbitrary number in the interval $\tilde c \in (0,c)$ and
\begin{equation}
\tilde a = a \left( 2 b\over c-\tilde c \right)^{2b} \mathrm{e}xp(-2 b).
\mathrm{e}nd{equation}
Note that this new bound of the de la Vall\'e Poussin form certainly holds for $x>x_0$, but if $x_0$ is sufficiently small one might be able to widen the range of applicability to some new $x \geq x_*$ with $x_* \leq x_0$ by explicit computation.
}
We now apply this lemma to the various bounds explicated above.
\section{Some effective bounds}
\mathrm{e}nlargethispage{20pt}
First let us consider some widely applicable derived bounds of the de la Vall\'e Poussin form, as presented in Table~III.
Note that the selection of a specific value of $\tilde c$ is a \mathrm{e}mph{choice}, and the computation of $\tilde a$ is then immediate --- there is an infinite number of other
effective
bounds of de la Vall\'e Poussin form that we could develop.
Determining $x_*$ then requires computationally checking low values of $x$.
\begin{table}[!htb]
\caption{Some widely applicable derived bounds of the de la Vall\'e Poussin form.}
\begin{center}
\begin{tabular}{||c|c|c||c||}
\hline
\hline
$\tilde a$ & $\tilde c$ & $x_*$ & Based on \\
\hline
\hline
0.3510691792& 1/4 & 59 & Schoenfeld~\cite{Schoenfeld}\\
\hline
\hline
0.2748124978 & 1/4 & 101 & Trudgian~\cite{Trudgian}\\
\hline
0.4242102935 & 1/3 & 59 & Trudgian~\cite{Trudgian}\\
\hline
\hline
295 & 1/2 & 2 & Fiori--Kadiri--Swidinsky~\cite{Fiori-et-al}\\
\hline
\hline
385 & 1/2 & 2 & Johnston--Yang~\cite{Johnston-Yang}\\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
By now relaxing the prefactor $\tilde a$, one can increase the range of validity of the bound, (ie, decrease $x_*$). In this way, after some computation, one finds
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 2).
\mathrm{e}nd{equation}
Note that this could in principle have been deduced as early as 1976, some 46 years ago,
from the work of Schoenfeld~\cite{Schoenfeld}.
Similarly
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 3).
\mathrm{e}nd{equation}
Note that this particular bound could in principle have been deduced as early as 2016,
some 6 years ago, from the work of Trudgian~\cite{Trudgian}.
For the other two widely applicable bounds, some preliminary experimental investigations \mathrm{e}mph{suggest} that it might be possible to reduce the numerical prefactors (295, 385) significantly --- but doing so would require rather different techniques from the elementary observations made above.
As always one can trade of stringency against range of validity. In this regard let me mention two specific examples
\begin{equation}
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 29),
\mathrm{e}nd{equation}
and
\begin{equation}
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 41).
\mathrm{e}nd{equation}
In contrast, for some asymptotically stringent bounds, based on the Johnston--Yang results presented in reference~\cite{Johnston-Yang}, consider Table~IV. Note that we can again make the exponential factor smaller, (by increasing $\tilde c$), at the cost of making the numerical prefactor $\tilde a$ larger.
For instance one can deduce the ineffective bound
\begin{equation}
|\vartheta(x)-x| = \;
{\mathcal{O}} \left( {x} \;\mathrm{e}xp\left( - \sqrt{\ln x}\right)\right),
\mathrm{e}nd{equation}
which can be made effective as (for instance):
\begin{equation}
|\vartheta(x)-x| = \;
83063 \;\; {x} \;\mathrm{e}xp\left( - \sqrt{\ln x}\right); \qquad \left(x> \mathrm{e}xp(10^{10})\right).
\mathrm{e}nd{equation}
Many variations on this theme can be developed.
\begin{table}[!htb]
\caption{Asymptotically stringent bounds of the de la Vall\'e Poussin form.}
\begin{center}
\begin{tabular}{||c|c|c||}
\hline
\hline
$\tilde a$ & $\tilde c$ & $x_*$ \\
\hline
\hline
357 & 1/2 & $\mathrm{e}xp(3000)$\\
320 & 1/2 & $\mathrm{e}xp(4000)$ \\
295 & 1/2 & $\mathrm{e}xp(5000)$ \\
274 & 1/2 & $\mathrm{e}xp(6000)$ \\
263 & 1/2 & $\mathrm{e}xp(7000)$ \\
252 & 1/2 & $\mathrm{e}xp(8000)$ \\
244 & 1/2 & $\mathrm{e}xp(9000)$ \\
249 & 1/2 & $\mathrm{e}xp(10000)$ \\
\hline
\hline
644 & 1/2 & $\mathrm{e}xp(10^5)$ \\
348 & 1/2 & $\mathrm{e}xp(10^6)$ \\
312 & 1/2 & $\mathrm{e}xp(10^7)$ \\
301 & 1/2 & $\mathrm{e}xp(10^8)$ \\
298 & 1/2 & $\mathrm{e}xp(10^9)$ \\
297 & 1/2 & $\mathrm{e}xp(10^{10})$ \\
\hline
\hline
1642333 & 1 & $\mathrm{e}xp(10^6)$ \\
165152 & 1 & $\mathrm{e}xp(10^7)$ \\
101831 & 1 & $\mathrm{e}xp(10^8)$ \\
87551 & 1 & $\mathrm{e}xp(10^9)$ \\
83063 & 1 & $\mathrm{e}xp(10^{10})$ \\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\label{default}
\mathrm{e}nd{table}
\section{Conclusions}\label{S:discussion}
With some hindsight, deriving effective bounds of the de la Vall\'e Poussin form is, (given various effective results obtained over the last 50 years~\cite{Schoenfeld,Trudgian,Johnston-Yang,Fiori-et-al,Broadbent-et-al}), seen to be almost trivial.
Certainly, (given the effective results reported in~\cite{Schoenfeld,Trudgian,Johnston-Yang,Fiori-et-al,Broadbent-et-al}), nothing deeper than elementary calculus and some slightly tedious numerical checking was required. On the other hand, conceptually it is very pleasant to see simple explicit and effective bounds of the de la Vall\'e Poussin form dropping out so nicely.
\begin{thebibliography}{99}
\newcommand{\arXiv}[1]{arXiv:~{\href{https://arxiv.org/abs/#1}{\color{blue}#1}}}
\bibitem{Poussin}
Charles Jean de la Vall\'e Poussin,\\
``Recherches analytiques sur la th\'eorie des nombres premiers'',\\ Ann. Soc. Scient.
Bruxelles, deuxi\'eme partie {\bf20}, (1896), pp. 183--256
\bibitem{Schoenfeld}
Lowell Schoenfeld,\\
``Sharper bounds
for the Chebyshev functions $\vartheta(x)$ and $\psi(x)$. II'', \\
Mathematics of Computation,
{\bf 30 \#134} (April 1976) 337--360.\\
{\mathrm{d}}oi{10.1090/S0025-5718-1976-0457374-X}
\bibitem{Trudgian}
Tim Trudgian,
``Updating the error term in the prime number theorem'',\\
The Ramanujan Journal {\bf 39} (2016) 225--236,
{\mathrm{d}}oi{10.1007/S11139-014-9656-6}.
\arXiv{1401.2689} [math.NT]
\bibitem{Johnston-Yang}
Daniel R. Johnston, Andrew Yang,\\
``Some explicit estimates for the error term in the prime number theorem'',\\
\arXiv{2204.01980} [math.NT]
\bibitem{Fiori-et-al}
Andrew Fiori, Habiba Kadiri, Joshua Swindisky,\\
``Sharper bounds for the error term in the Prime Number Theorem'',\\
\arXiv{2206.12557} [math.NT]
\bibitem{Broadbent-et-al}
S. Broadbent, H. Kadiri, A. Lumley, N. Ng, and K. Wilk. \\
``Sharper bounds for the Chebyshev function $\vartheta(x)$''. \\
Math. Comp. 90.331 (2021), pp. 2281--2315.
[\arXiv{2002.11068} [math.NT]]
\hrule\hrule\hrule
\mathrm{e}nd{thebibliography}
\mathrm{e}nd{document}
|
\begin{document}
\title{The generalised Oberwolfach problem}
\begin{abstract}
We prove that any quasirandom dense large graph
in which all degrees are equal and even can be
decomposed into any given collection of two-factors
($2$-regular spanning subgraphs).
A special case of this result gives a new solution to
the Oberwolfach problem.
\end{abstract}
\section{Introduction}
At meals in the Oberwolfach Mathematical Institute,
the participants are seated at circular tables.
At an Oberwolfach meeting in 1967, Ringel (see \cite{LR})
asked whether there must exist a sequence of seating plans
so that every pair of participants
sit next to each other exactly once.
We assume, of course, that there are an odd number of participants,
as each participant sits next to two others in each meal.
The tables may have various sizes,
which we assume are the same at each meal.
\nib{Oberwolfach Problem (Ringel).}
Let $F$ be any two-factor (i.e.\ $2$-regular graph)
on $n$ vertices, where $n$ is odd.
Can the complete graph $K_n$ be
decomposed into copies of $F$?
We obtain a new solution of this problem for large $n$,
with a theorem that is more general in three respects:
(a) we can decompose any dense quasirandom graph
that is regular of even degree
(not just $K_n$ for $n$ odd),
(b) we can decompose into any prescribed
collection of two-factors
(not just copies of some fixed two-factor $F$),
(c) our theorem applies to directed graphs (digraphs).
We start by stating our result for undirected graphs.
We require the following quasirandomness definition.
We say that a graph $G$ on $n$ vertices is
$(\varepsilon,t)$-typical if every set $S$ of at most $t$ vertices
has $((1 \pm \varepsilon)d(G))^{|S|} n$ common neighbours, where
$d(G) = e(G) \tbinom{n}{2}^{-1}$ is the density of $G$.
\begin{theo}\leftarrowbel{mainundir}
For all $\alpha>0$ there exist $t,\varepsilon,n_0$ such that
any $(\varepsilon,t)$-typical graph on $n \ge n_0$ vertices
that is $2r$-regular for some integer $r > \alpha n$
can be decomposed into any family of $r$ two-factors.
\end{theo}
Theorem \ref{mainundir} implies some variant forms of the
Oberwolfach problem that have appeared in the literature,
such as the Hamilton--Waterloo Problem (two types of two-factors),
or that if $n$ is even then $K_n$ can be decomposed into a
perfect matching and any specified collection
of $n/2-1$ two-factors. More generally,
with parameters as in Theorem \ref{mainundir},
it is easy to deduce that
any $(\varepsilon,t)$-typical graph on $n \ge n_0$ vertices
that is $(2r+1)$-regular for some integer $r > \alpha n$
can be decomposed into a perfect matching
and any family of $r$ two-factors.
We will deduce Theorem~\ref{mainundir}
from the directed version below.
First we extend our definitions to digraphs.
We say that a digraph $G$ on $n$ vertices is
$(\varepsilon,t)$-typical if for every set $S=S^- \cup S^+$
of at most $t$ vertices
there are $((1 \pm \varepsilon)d(G))^{|S|} n$ vertices which
are both common inneighbours of $S^-$
and outneighbours of $S^+$, where
$d(G) = e(G) \tbinom{n}{2}^{-1}$ is the density of $G$.
We say that $G$ is $r$-regular if
$d^+_G(v)=d^-_G(v)=r$ for all $v \in V(G)$.
A \emph{one-factor} is a $1$-regular digraph;
equivalently, it is a union of
vertex-disjoint oriented cycles.
\begin{theo}\leftarrowbel{main}
For all $\alpha>0$ there exist $t,\varepsilon,n_0$ such that
any $(\varepsilon,t)$-typical digraph on $n \ge n_0$ vertices
that is $r$-regular for some integer $r > \alpha n$
can be decomposed into any family of $r$ one-factors.
\end{theo}
Theorem~\ref{mainundir} follows from Theorem~\ref{main}
and the observation that for any typical graph
that is regular of even degree
there exists an orientation
which is a regular typical digraph.
To see this, one can orient edges independently at random
and make a few modifications to obtain the required orientation.
(See Lemma~\ref{typ:split} below for a similar argument.)
While we were preparing this paper,
the Oberwolfach problem (for large $n$) was solved by
Glock, Joos, Kim, K\"uhn and Osthus~\cite{GJKKO}.
They also obtained a more general result that covers
the other undirected applications just mentioned,
but our result is more general than theirs
in the three respects mentioned above:
(a) we can decompose any dense typical regular graph
(whereas their result only applies to almost complete graphs),
(b) we can decompose into any collection of two-factors
(whereas they can allow for a collection of two-factors
provided that some fixed $F$ occurs $\Omega(n)$ times),
(c) our result also applies to digraphs
(whereas theirs is for undirected graphs).
There is a large literature on the Oberwolfach Problem,
of which we mention just a few highlights
(a more detailed history is given in \cite{GJKKO}).
The problem was solved for infinitely many $n$
by Bryant and Scharaschkin \cite{BS},
in the case when $F$ consists of two cycles by Traetta \cite{T},
and for cycles of equal length by
Alspach, Schellenberg, Stinson and Wagner \cite{ASSW}.
A related conjecture of Alspach that $K_n$ can be
decomposed into any collection of cycles each of length $\le n$
and total size $\tbinom{n}{2}$ was solved
by Bryant, Horsley and Pettersson \cite{BHP}.
There are several recent general results on approximate
decompositions that imply an approximate solution
to the generalised Oberwolfach Problem,
i.e.\ that any given collection of two-factors
can be embedded in a quasirandom graph
provided that a small fraction of the edges
can be left uncovered: we refer to the papers of
Allen, B\"ottcher, Hladk\'y and Piguet \cite{ABHP},
Ferber, Lee and Mousset \cite{FLM}
and Kim, K\"uhn, Osthus and Tyomkyn \cite{KKOT}.
\nib{Notation.}
Given a graph $G = (V,E)$,
when the underlying vertex set $V$ is clear,
we will also write $G$ for the set of edges.
So $|G|$ is the number of edges of $G$. Usually $|V|=n$.
The \emph{edge density} $d(G)$ of $G$ is $|G|/\tbinom{n}{2}$.
We write $N_G(x)$ for the neighbourhood of a vertex $x$ in $G$.
The degree of $x$ in $G$ is $d_G(x)=|N_G(x)|$.
For $A \subseteqseteq V(G)$, we write
$N_G(A) := \bigcap_{x \in A}N_G(x)$;
note that this is the common neighbourhood
of all vertices in $A$, not the neighbourhood of $A$.
In a directed graph $J$ with $x \in V(J)$,
we write $N_J^+(x)$ for the set of out-neighbours of $x$ in $G$
and $N_G^-(x)$ for the set of in-neighbours.
We let $d^\pm_G(A) := |N^\pm_G(A)|$.
We define common out/in-neighbourhoods
$N_J^\pm(A) = \bigcap_{x \in A} N_J^\pm (A).$
We say $G$ is \emph{$(\varepsilon,t)$-typical}
if $d_G(S) = ((1 \pm \varepsilon)d(G))^{|S|} n$
for all $S \subseteqseteq V(G)$ with $|S| \le t$.
We say that an event $E$ holds with high probability (whp)
if $\mb{P}(E) > 1 - \mbox{ex}p(-n^c)$ for some $c>0$ and $n>n_0(c)$.
We note that by a union bound for any fixed collection $\mc{E}$
of such events with $|\mc{E}|$ of polynomial growth
whp all $E \in \mc{E}$ hold simultaneously.
We omit floor and ceiling signs for clarity of exposition.
We write $a \ll b$ to mean $\forall\ b>0 \
\mbox{ex}ists\ a_0>0 \ \forall\ 0<a<a_0$.
We write $a \pm b$ for an unspecified number in $[a-b,a+b]$.
Throughout the vertex set $V$ will come with a cyclic order,
which we usually identify with the natural cyclic order
on $[n]=\{1,\dots,n\}$. For any $x \in V$ we write $x^+$
for the successor of $x$, so if $x \in [n]$ then
$x^+$ is $x+1$ if $x \ne n$ or $1$ if $x=n$.
We define the predecessor $x^-$ similarly. Given $x,y$ in $[n]$
we write $d(x,y)$ for their cyclic distance,
i.e.\ $d(x,y) = \min \{ |x-y|, n-|x-y| \}$.
\section{Overview of the proof} \leftarrowbel{sec:over}
We will illustrate the ideas of our proof by starting
with a special case and becoming gradually more general.
Suppose first that we wish to decompose a typical dense (undirected)
$2r$-regular graph $G$ on $n$ vertices into $r$ triangle-factors
(i.e.\ two-factors in which each cycle is a triangle
-- we require $3 \mid n$ for this question to make sense).
The existence of such a decomposition
(also known as a resolvable triangle-decomposition of $G$)
follows from a recent result of the first author \cite{K2}
generalising the existence of designs (see \cite{Kexist})
to many other `design-like' problems. The proof in \cite{K2} goes
via the following auxiliary decomposition problem,
which also plays an important role in this paper.
Let $J$ be an auxiliary graph with $V(J)$ partitioned
as $V \cup W$, where $V=V(G)$ and $|W|=r$.
Let $J[V]=G$, $J[V,W]=V \times W$ and $J[W]=\emptyset$.
Note that a decomposition of $G$ into triangle-factors
is equivalent to a decomposition of $J$ into copies of $K_4$
each having $3$ vertices in $V$ and $1$ vertex in $W$.
Indeed, given such a decomposition of $J$, for each $w \in W$
we define a triangle-factor of $G$ by removing $w$ from all
copies of $K_4$ containing $w$ in the decomposition;
clearly every edge of $G$ appears in exactly one
of these triangle-factors. Conversely, any decomposition of $G$
into triangle-factors can be converted into a suitable
$K_4$-decomposition of $J$ by adding each $w \in W$ to one of
the triangle-factors (according to an arbitrary matching).
The auxiliary construction described above is quite flexible,
so a similar argument covers many other cases of our problem.
For example, decomposing $G$ into $C_\ell$-factors
(two-factors in which each cycle has length $\ell$)
is equivalent to decomposing $J$ into `wheels' $W_\ell$
with `rim' in $V$ and `hub' in $W$. (We obtain $W_\ell$ from
$C_\ell$, which is called the rim, by adding a new vertex,
called the hub, joined to every other vertex,
by edges that we call spokes.)
Such a decomposition exists by \cite{K2}.
We can encode our generalised Oberwolfach Problem
in full generality by introducing colours on the edges.
For each possible cycle length $\ell$ we introduce a colour,
which we also call $\ell$. For each $w \in W$, we denote its
corresponding factor by $F_w$, and suppose that it has $n^w_\ell$
cycles of length $\ell$ (where $\sum_\ell \ell n^w_\ell = n$).
We colour $J$ so that each $w \in W$ is incident to exactly
$n^w_\ell$ edges of colour $\ell$, and all other edges are uncoloured.
We colour each $W_\ell$ so that exactly one spoke has colour $\ell$
and all other edges are uncoloured. Then a decomposition of $G$ into
$\{F_w: w \in W\}$ is equivalent to a decomposition of $J$
into wheels with this colouring with rim in $V$ and hub in $W$.
Note that this equivalence does not depend on which edges of $J$
we colour, but to apply \cite{K2} we will require the colouring
to be suitably quasirandom. Another important constraint
in applying \cite{K2} is that the number of colours
and the size of the wheels should be bounded by an absolute constant.
Thus our generalised Oberwolfach Problem can only be solved by
direct reduction to \cite{K2} in the case that all factors
have all cycle lengths bounded by some absolute constant.
This now brings us to the crucial issue for this paper:
how can we encode two-factors with cycles of arbitrary length
by an auxiliary construction to which \cite{K2} applies?
Before describing this, we pass to an auxiliary problem
of decomposing a subgraph $G'$ of $G$ into graphs
$(G_w: w \in W)$, where each $G_w$ is a vertex-disjoint
union of paths with prescribed endpoints, lengths and vertex set.
More precisely, for each $w \in W$ we are given specified
lengths $(\ell^w_i: i \in I_w)$, vertex-pairs
$((x^w_i,y^w_i): i \in I_w)$, a forbidden set $Z_w$,
and we want each $G_w$ to be a union of vertex-disjoint
$x^w_i y^w_i$-paths of length $\ell^w_i$ for each $i \in I_w$
with $V(G_w)=V(G) \setminus Z_w$. We will arrive at this problem
having embedded some subgraphs $F'_w \subseteq F_w$ of each $w \in W$,
so the prescribed endpoints will be endpoints of
paths in $F'_w$ that need to be connected up to form cycles,
and $Z_w$ will consist of all vertices of degree $2$ in $F'_w$.
We assume that all lengths $\ell^w_i$ are divisible by $8$
(which is easy to ensure for long cycles).
We will translate the above path factor problem
into an equivalent problem of decomposing a certain
auxiliary two-coloured directed graph $J$,
with $V(J) = V \cup W$ as in the previous construction.
We call the two colours `$0$' (which means `uncoloured')
and `$K$' (which means `special'). Again, $J[W]=\emptyset$.
For now we defer discussion of $J[V,W]$
and describe the arcs of $J[V]$,
which are in bijection with the edges of $G$.
For colour $0$ this bijection simply corresponds
to a choice of orientation for edges, but for
colour $K$ we employ the following `twisting' construction.
We fix throughout a cyclic order of $V$,
and require that each arc $\overlinea{xy}$ of colour $K$ in $J$
comes from an edge $xy^+$ of $G$, where $y^+$ denotes
the successor of $y$ in the cyclic order.
Consider any directed $8$-cycle $C$ in $J$ with vertex sequence
$x_1 \dots x_8$, such that all arcs have colour $0$
except that $\overlinea{x_7 x_8}$ has colour $K$.
The edges in $G$ corresponding to $C$ form a path
with vertex sequence $x_8 x_1 \dots x_7 x_8^+$.
Now suppose we have a family of such cycles
$\mc{C} = (C^i: i \in I)$ where each $C^i$
has vertex sequence $x^i_1 \dots x^i_8$.
Call $\mc{C}$ compatible if
(i) its cycles are mutually vertex-disjoint, and
(ii) if any $(x^i_8)^+$ is used by a cycle in
$\mc{C}$ then it is some $x^j_8$.
Suppose $\mc{C}$ is compatible and let
$([x_j,y_j]: j \in J)$ denote the family of maximal
cyclic intervals contained in $\{x^i_8: i \in I\}$.
Then the edges of $G$ corresponding to the cycles of $\mc{C}$
form a family of vertex-disjoint paths $(P_j: j \in J)$, where
each $P_j$ is an $x_j y_j^+$-path whose vertex sequence is the
concatenation of vertex sequences of the $8$-paths as described
above for each cycle of $\mc{C}$ using a vertex of $[x_j,y_j]$.
The above construction allows us to pass from the path factor problem
to finding certain edge-disjoint compatible cycle families in $J$.
In order for our path factor problem to obey the constraints of
this encoding we require the prescribed vertex-pairs for each $w$
to define disjoint cyclic intervals $([x^w_i,(y^w_i)^-]: i \in I_w)$
of lengths $\ell^w_i/8$ (and also that no successor $y^w_i$
is contained in any of the other intervals for $w$).
We are thus introducing extra constraints into the path factor
problem that may affect up to $n/8$ vertices for each $w$,
but the flexibility on the remaining
vertices will be sufficient.
Now we can complete the description of the auxiliary graph $J$
and the decomposition problem that encodes the compatible cycle
family problem. We define $J[V]$ as above,
and $J[V,W]$ so that all arcs are directed
towards $W$, each in-neighbourhood $N^-_J(w)$ is obtained
from $V(G) \setminus Z_w$ by deleting the interval successors
$\{ y^w_i: i \in I_w \}$, all arcs $\overlinea{xw}$
with $x$ in an interval $[x^w_i,(y^w_i)^-]$ are coloured $K$,
and all other arcs of $J[V,W]$ are coloured $0$.
Finally, the compatible cycle family problem is equivalent
to decomposing $J$ into coloured directed wheels $\ova{W}^K_{\! 8}$,
obtained from $W_8$ by directing the rim cyclically,
directing all spokes towards the hub $w$, giving colour $K$
to one rim edge $\overlinea{xy}$ and one spoke $\overlinea{yw}$,
and colouring the other edges by $0$.
The deduction from \cite{K2} of the existence of wheel
decompositions is given in section \ref{sec:wheel}.
We now describe the strategy for the proof of Theorem \ref{main}.
The goal is to embed some parts of our two-factors
so that the remaining problem is of one of two special types
that has an encoding suitable for applying \cite{K2},
either a path factor problem
encoded as $\ova{W}^K_{\! 8}$-decomposition
or a $C_\ell$-factor problem
encoded as $\overlinea{W}_{\!\ell}$-decomposition
(we take the coloured wheel $W_\ell$ discussed above
for $C_\ell$-factors and introduce directions
as in $\ova{W}^K_{\! 8}$, which are not necessary
but convenient for giving a unified analysis).
We call a factor `long' if it has at least $n/2$ vertices
in cycles of length at least $K$ (as well as denoting the
special colour, $K$ is also used as a large constant length
threshold, above which we treat cycles using the special
twisting encoding as above). We call the other factors `short'.
We start by reducing to the case that all factors are long
or all factors are short. To do so, suppose first that
there are $\Omega(n)$ long factors and $\Omega(n)$ short factors.
Then we can randomly partition $G$ into typical graphs
$G^L$ and $G^S$, each of which is regular of the correct degree
(twice the number of long factors for $G^L$
and twice the number of short factors for $G^S$).
If there are $o(n)$ factors of either type then these can
be embedded one-by-one (by the blow-up lemma \cite{KSS}),
and then the remaining problem still satisfies the conditions
of Theorem \ref{main} (with slightly weaker typicality).
The short factor problem can be further reduced to the case
that there is some length $\ell^*$ such that each factor has
$\Omega(n)$ cycles of length $\ell^*$. Indeed, we can divide the
factors into a constant number of groups according to some
choice of cycle length that appears $\Omega(n)$ times in each factor
of the group. Any group of $o(n)$ factors can be embedded greedily,
so after taking a suitable random partition, it suffices
to show that the remaining groups can each be embedded
in a graph that is typical and regular of the correct degree.
Thus we can assume that we are in one of the following cases.
Case $K$: all factors are long,
our goal is to reduce to $\ova{W}^K_{\! 8}$-decomposition.
Case $\ell^*$: all factors have $\Omega(n)$ cycles of length $\ell^*$,
our goal is to reduce to $\ova{W}_{\! \ell^*}$-decomposition.
In any case, the reduction is achieved by applying an approximate decomposition result in a suitable random subgraph,
in which we embed a subgraph of each of our factors.
At this step, in Case $\ell^*$ we embed
all cycles of length $\ne \ell^*$,
and in Case $K$ we embed all short cycles
and some parts of the long cycles as needed
to reduce to a suitable path factor problem.
This approximate decomposition result is superficially
similar to the maximum degree $2$ case of
the blow-up lemma for approximate decompositions
due to Kim, K\"uhn, Osthus and Tyomkyn \cite{KKOT}.
However, it does not suffice to use their result,
as we require a decomposition that is compatible
with the conditions of our final decomposition problem
(into $\ova{W}^K_{\! 8}$ or $\ova{W}_{\! \ell^*}$),
so the sets of vertices of the partial factors
embedded in this step must be suitably quasirandom
and avoid the intervals needed for Case $K$.
Furthermore, we obtain the required approximate
decomposition by similar arguments to those
for the exact decomposition,
which does not add much extra work.
The technical heart of the paper is a randomised algorithm
(presented in section \ref{sec:alg}), which gives
a unified treatment of the cases described above.
It simultaneously (a) partitions almost all of $G$
into two graphs $G_1$ and $G_2$, and
(b) sets up auxiliary digraphs $J_1$ and $J_2$ such that
(i) an approximate wheel decomposition of $J_2$
gives an approximate decomposition of $G_2$
into the partial factors described above, and
(ii) the graph $G'_1$ of edges that are unused by
the approximate decomposition has an auxiliary digraph
that is a sufficiently small perturbation of $J_1$ that
it can still be used for the exact decomposition step.
The analysis of the algorithm falls naturally into two parts:
the choice of intervals (section \ref{sec:int}),
then regularity properties of an auxiliary
hypergraph defined by wheels (section \ref{sec:reg}).
The results of this analysis are applied
to show the existence of the various partial factor
decompositions discussed above:
the approximate step is in section \ref{sec:approx}
and the exact step in section \ref{sec:exact}.
Section \ref{sec:pf} combines
all the ingredients prepared in the previous sections
to produce the proof of our main theorem.
The final section contains some concluding remarks.
\begin{figure}
\caption{An overview of the proof.}
\end{figure}
\section{Wheel decompositions} \leftarrowbel{sec:wheel}
In this section we describe the results we need on wheel
decompositions and how they follow from \cite{K2}.
We start by recalling the coloured wheels
described in section \ref{sec:over}.
For any $c \ge 3$, the uncoloured $c$-wheel consists
of a directed $c$-cycle (called the rim),
another vertex (called the hub),
and an arc from each rim vertex to the hub.
We obtain the coloured $c$-wheel $\ova{W}_{\! c}$
by giving all arcs colour $0$ except
that one of the spokes has colour $c$.
We obtain the special $c$-wheel $\ova{W}^K_{\! 8}c$
by giving all arcs colour $0$ except
that one rim edge $\overlinea{xy}$
and one spoke $\overlinea{yw}$ have colour $K$.
As discussed in section \ref{sec:over},
we will only use $\ova{W}^K_{\! 8}c$ with $c=8$, but here
we will consider the general configuration
so that the decomposition problems are quite similar.
We start by stating the result for $\ova{W}_{\! c}$.
\begin{center}
\includegraphics{figwheel}
\end{center}
\begin{theo} \leftarrowbel{decompL}
Let $n^{-1} \ll \delta \ll \omega \ll c^{-1}$ and $h=2^{50c^3}$.
Let $J = J^0 \cup J^c$ be a digraph with arcs coloured
$0$ or $c$, with $V(J)$ partitioned as $(V,W)$
where $\omega n \le |V|, |W| \le n$.
Then $J$ has a $\ova{W}_{\! c}$-decomposition such that
every hub lies in $W$ if the following hold:
{\em Divisibility:}
all arcs in $J[V]$ have colour $0$,
all arcs in $J[V,W]$ point towards $W$,
$d_J^-(v,V)=d_J^+(v,V)=d_J^+(v,W)$ for all $v \in V$,
and $d^-_J(w) = cd^-_{J^c}(w)$ for all $w \in W$.
{\em Regularity:}
each copy of $\ova{W}_{\! c}$ in $J$ has a weight
in $[\omega n^{1-c}, \omega^{-1} n^{1-c}]$ such that
for any arc $\overlinea{e}$ there is total weight $1 \pm \delta$
on wheels containing $\overlinea{e}$.
{\em Extendability:}
for all disjoint $A,B \subseteq V$ and $C \subseteq W$
each of size $\le h$ we have
$|N^+_{J^0}(A) \cap N^+_{J^c}(B) \cap W| \ge \omega n$
and $|N^+_{J^0}(A) \cap N^-_{J^0}(B)
\cap N^-_{J^{c'}}(C)| \ge \omega n$
for both $c' \in \{0,c\}$.
\end{theo}
Before stating our result on $\ova{W}^K_{\! 8}$-decompositions,
we recall that $V$ has a cyclic order, which we can
identify with the natural cyclic order on $[n]$,
and define the following separation properties.
\begin{defn} \leftarrowbel{def:sep}
For $1 \le x<y \le n$ the cyclic distance is
$d(x,y) = \min\{y-x,n+x-y\}$.
We say that $S \subseteq [n]$ is $d$-separated
if $d(a,a') \ge d$ for all distinct $a,a'$ in $S$.
For disjoint $S,S' \subseteq [n]$ we say $(S,S')$
is $d$-separated if $d(a,a') \ge d$
for all $a \in S$, $a' \in S'$.
\end{defn}
Now we state our result on $\ova{W}^K_{\! 8}$-decompositions.
We note that it only concerns digraphs $J$ such that
$d(x,y) \ge d$ for all $\overlinea{xy} \in J[V]$,
as this is implied by the regularity assumption.
Our proof of Theorem \ref{main} will require us
to only consider such $J$, so that we
can satisfy the extendability assumption.
\begin{theo} \leftarrowbel{decompK}
Let $n^{-1} \ll \delta \ll \omega \ll c^{-1}$.
Let $h=2^{50c^3}$ and $d \ll n$.
Let $J = J^0 \cup J^K$ be a digraph with arcs coloured
$0$ or $K$, with $V(J)$ partitioned as $(V,W)$
where $\omega n \le |V|, |W| \le n$, such that
all arcs in $J[V,W]$ point towards $W$ and $J[W]=\emptyset$.
Then $J$ has a $\ova{W}^K_{\! 8}c$-decomposition such that
every hub lies in $W$ if the following hold:
{\em Divisibility:}
$d^-_J(w) = cd^-_{J^K}(w)$ for all $w \in W$,
and for all $v \in V$ we have
$d_J^-(v,V)=d_J^+(v,V)=d_J^+(v,W)$
and $d^-_{J^K}(v,V)=d^+_{J^K}(v,W)$.
{\em Regularity:}
each $3d$-separated copy of $\ova{W}^K_{\! 8}c$ in $J$ has a weight
in $[\omega n^{1-c}, \omega^{-1} n^{1-c}]$ such that
for any arc $\overlinea{e}$ there is total weight $1 \pm \delta$
on wheels containing $\overlinea{e}$.
{\em Extendability:}
for all disjoint $A,B \subseteq V$ and $L \subseteq W$
each of size $\le h$,
for any $a, b, \ell \in \{0,K\}$ we have
$|N^+_{J^a}(A) \cap N^-_{J^b}(B)
\cap N^-_{J^\ell}(L)| \ge \omega n$,
and furthermore, if $(A,B)$ is $3d$-separated then
$|N^+_{J^0}(A) \cap N^+_{J^K}(B) \cap W| \ge \omega n$.
\end{theo}
For the remainder of this section we will explain
how Theorem \ref{decompK} follows from \cite{K2}
(we omit the similar and simpler details
for Theorem \ref{decompL}). We follow the exposition
in \cite{K3}, which deduces from \cite{K2} a general
result on coloured directed designs that we will apply here.
\subseteqsection{The functional encoding}
We encode any digraph $J$ by a set of functions $\mf{J}$,
where for each arc $\overlinea{ab} \in J$ we include in $\mf{J}$
the function $(1 \mapsto a, 2 \mapsto b)$, i.e.\ the
function $f:[2] \to V(J)$ with $f(1)=a$ and $f(2)=b$.
We will identify $\mf{J}$ with its characteristic vector,
i.e.\ $\mf{J}_f = 1_{f \in \mf{J}}$; if we want to emphasise
the vector interpretation we write $\underline{\mf{J}}$.
If $J$ has coloured arcs, and $\ell$ is a colour,
we write $J^\ell$ for the digraph in colour $\ell$,
which is encoded by $\mf{J}^\ell$.
We will consider decompositions by a coloured digraph $H$
defined as follows. We start with $\ova{W}^K_{\! 8}c$ on the vertex set
$[c+1]$, where we label the rim cycle by $[c]$ cyclically
(so $c+1$ is the hub) so that, writing $c_-=c-1$ and $c_+=c+1$,
$\overlinea{c_- c}$ and $\overlinea{c c_+}$ have colour $K$
and all other arcs have colour $0$.
We let $\mc{P}$ be the partition $([c],\{c_+\})$ of $[c+1]$.
We introduce new colours $0'$ and $K'$, and change the colours
of $\overlinea{c c_+}$ to $K'$ and of the other spokes to $0'$.
We do this so that $H$ is `$(\mc{P},\text{id})$-canonical'
in the sense of \cite[Definition 7.1]{K3};
specialised to our setting, the relevant properties
are that $H$ is an oriented graph
(with no multiple edges or $2$-cycles)
and that for each colour all of its arcs have one fixed
pattern with respect to $\mc{P}$
(specifically, for colours $0$ and $K$
all arcs are contained in $[c]$,
and for colours $0'$ and $K'$
all arcs are directed from $[c]$ to $\{c_+\}$).
Now we translate the $H$-decomposition problem
for a digraph $J$ into its functional encoding.
We will have a partition $\mc{Q}=(V,W)$ of $V(J)$,
and wish to decompose $J$ by copies $\phi(H)$ of $H$
such that $\phi([c]) \subseteq V$ and $\phi(c_+) \in W$
(i.e.\ wheels with hub in $W$),
and $\phi([c])$ is $3d$-separated
(in which case we will say that the graph $\phi(H)$ is $3d$-separated).
We think of the
functional encoding $\mf{J}$ as living inside a
`labelled complex' $\Phi$ of all possible partial
embeddings of $H$: we define $\Phi = (\Phi_B: B \subseteq [c+1])$,
where each $\Phi_B$ consists of all injections
$\phi:B \to V(J)$ such that $\phi(B \cap [c]) \subseteq V$,
$\phi(B \cap \{c_+\}) \subseteq W$ and $Im(\phi)$ is $3d$-separated.
The set of functional encodings of possible embeddings
of $H$ (if present in $\mf{J}$) is then
\[ H(\Phi) := \{ \phi \mf{H} : \phi \in \Phi_{[c+1]} \},
\quad \text{where }
\phi \mf{H} := \{ \phi \circ \theta: \theta \in \mf{H} \}.\]
The $H$-decomposition problem for $J$ is equivalent
to finding $\mc{H} \subseteq H(\Phi)$ with
$\sum \{ \underline{\mf{H}'}: \mf{H}' \in \mc{H} \} = \underline{\mf{J}}$,
or equivalently $\bigcup \mc{H} = \mf{J}$
(where if $\mf{J}$ has multiple edges
we consider a multiset union).
We call such $\mc{H}$ an $H$-decomposition in $\Phi$.
\subseteqsection{Regularity}
Now we will describe the hypotheses of the theorem
that will give us an $H$-decomposition in $\Phi$.
We start with regularity, which is simply the
functional encoding of the regularity assumption
in Theorem \ref{decompK}. Specifically, we say
$J$ is $(H,\delta,\omega)$-regular in $\Phi$ if there are
weights $y_\phi \in [\omega n^{1-c}, \omega^{-1} n^{1-c}]$
for each $\phi \in \Phi_{[c+1]}$
with $\phi \mf{H} \subseteq \mf{J}$ such that
$\sum_\phi y_\phi \underline{\phi \mf{H}} = (1 \pm \delta)\underline{\mf{J}}$.
\subseteqsection{Extendability}
Next we consider extendability, which we discuss
in a simplified setting that suffices for our purposes,
following \cite[Definition 7.3]{K3}.
The idea is that for any vertex $x$ of $H$
there should be many ways to extend certain sets
of partial embeddings of $H-x$ to embeddings of $H$.
Specifically, we say $(\Phi,J)$ is $(\omega,h,H)$-vertex-extendable
if for any $x \in [c+1]$ and disjoint $A_i \subseteq V \cup W$
for $i \in [c+1] \setminus \{x\}$ each of size $\le h$ such that
$(i \mapsto v_i: i \in [c+1] \setminus \{x\}) \in \Phi$
whenever each $v_i \in A_i$, there are at least $\omega n$
vertices $v$ such that
\begin{enumerate}
\item $(i \mapsto v_i: i \in [c+1]) \in \Phi$
whenever $v_x=v$ and $v_i \in A_i$ for each $i \ne x$, and
\item each $\mf{J}^\ell$ with $\ell \in \{0,K,0',K'\}$
contains all $(1 \mapsto v_1, 2 \mapsto v_2)$
where for some $\theta \in \mf{H}^\ell$ we have
($v_1=v \ \& \ v_2 \in A_{\theta(2)}$) or
($v_2=v \ \& \ v_1 \in A_{\theta(1)}$).
\end{enumerate}
Note that by definition of $\Phi$ this only concerns
maps $\phi$ such that $Im(\phi)$ is $3d$-separated.
To interpret (ii) we consider $4$ cases
according to the position of $x$ in the wheel.
\begin{description}
\item[$x=c+1$.]
For any pairwise $3d$-separated $A_i \subseteq V$, $i \in [c]$ of sizes
$\le h$ there are at least $\omega n$ vertices $v$
such that $\overlinea{v_c v} \in J^{K'}$ for all $v_c \in A_c$
and $\overlinea{v_i v} \in J^{0'}$ for all $v_i \in A_i$, $i \ne c$.
Equivalently, for any disjoint $A,B \subseteq V$
with $|A| \le h$ and $|B| \le (c-1)h$
such that $(A,B)$ is $3d$-separated we have
$|N^+_{J^{K'}}(A) \cap N^+_{J^{0'}}(B)| \ge \omega n$.
\item[$x=c$.]
For any pairwise $3d$-separated $A_i \subseteq V$, $i \in [c-1]$
and $A_{c+1} \subseteq W$ of sizes $\le h$
there are at least $\omega n$ vertices $v$ such that
$\overlinea{v v_{c+1}} \in J^{K'}$ for all $v_{c+1} \in A_{c+1}$,
$\overlinea{v_{c-1} v} \in J^K$ for all $v_{c-1} \in A_{c-1}$,
and $\overlinea{vv_1} \in J^0$ for all $v_1 \in A_1$.
Equivalently, for any disjoint $A,B \subseteq V$ and $C \subseteq W$
of sizes $\le h$ such that $(A,B)$ is $3d$-separated we have
$|N^+_{J^K}(A) \cap N^-_{J^0}(B) \cap N^-_{J^{K'}}(C)| \ge \omega n$.
\item[$x=c-1$.]
For any pairwise $3d$-separated
$A_i \subseteq V$, $i \in [c] \setminus \{c-1\}$
and $A_{c+1} \subseteq W$ of sizes $\le h$
there are at least $\omega n$ vertices $v$ such that
$\overlinea{v v_{c+1}} \in J^{0'}$ for all $v_{c+1} \in A_{c+1}$,
$\overlinea{v v_c} \in J^K$ for all $v_c \in A_c$,
and $\overlinea{v_{c-2} v} \in J^0$ for all $v_{c-2} \in A_{c-2}$.
Equivalently, for any disjoint $A,B \subseteq V$ and $C \subseteq W$
of sizes $\le h$ such that $(A,B)$ is $3d$-separated we have
$|N^-_{J^K}(A) \cap N^+_{J^0}(B) \cap N^-_{J^{0'}}(C)| \ge \omega n$.
\item[$x \in \brak{c-2}$.]
For any pairwise $3d$-separated
$A_i \subseteq V$, $i \in [c] \setminus \{x\}$
and $A_{c+1} \subseteq W$ of sizes $\le h$
there are at least $\omega n$ vertices $v$ such that
$\overlinea{v v_{c+1}} \in J^{0'}$ for all $v_{c+1} \in A_{c+1}$,
$\overlinea{v v_{x+1}} \in J^0$ for all $v_{x+1} \in A_{x+1}$,
and $\overlinea{v_{x-1} v} \in J^0$ for all $v_{x-1} \in A_{x-1}$,
where $A_0 := A_c$.
Equivalently, for any disjoint $A,B \subseteq V$ and $C \subseteq W$
of sizes $\le h$ such that $(A,B)$ is $3d$-separated we have
$|N^-_{J^0}(A) \cap N^+_{J^0}(B) \cap N^-_{J^{0'}}(C)| \ge \omega n$.
\end{description}
All of these conditions follow from the extendability
assumption in Theorem \ref{decompK}
(after renaming colours $0$ and $K$
in $J[V,W]$ as $0'$ and $K'$, and replacing $h$ with $(c-1)h$).
\subseteqsection{Divisibility}
It remains to consider divisibility;
we follow \cite[Definition 7.2]{K3}.
For integers $s \le t$ we write $I^s_t$
for the set of injections from $[s]$ to $[t]$.
We identify $V \cup W$ with $[n']$ for some $n'$.
For $0 \le i \le 2$, $\psi \in I^i_{n'}$, $\theta \in I^i_{c+1}$,
we define index vectors in $\mb{N}^2$ describing types
with respect to the partitions $\mc{P}$ or $\mc{Q}$: we write
$i_{\mc{P}}(\theta) = (|Im(\theta) \cap [c]|,|Im(\theta) \cap \{c_+\}|)$
and $i_{\mc{Q}}(\psi) = (|Im(\psi) \cap V|,|Im(\psi) \cap W|)$.
For example, for $\theta = (1 \mapsto c_-, 2 \mapsto c) \in \mf{H}$
we have $i_{\mc{P}}(\theta) = (2,0)$. We define degree vectors
$\mf{H}(\theta)^*$ and $\mf{J}(\psi)^*$ in $\mb{N}^{C \times I^i_2}$ by
\[ \mf{H}(\theta)^*_{\ell\pi}=|\mf{H}^\ell(\theta\pi^{-1})|
\ \ \text{ and } \ \
\mf{J}(\psi)^*_{\ell\pi}=|\mf{J}^\ell(\psi\pi^{-1})|, \]
where e.g.\ $\mf{H}^\ell(\theta\pi^{-1})$ denotes the set of
$\theta' \in \mf{H}^\ell$ having $\theta\pi^{-1}$ as a restriction.
Letting $\sgen{\cdot}$ denote the integer span of a set of vectors,
we say $J$ is $H$-divisible in $\Phi$ if
\[ \mf{J}(\psi)^* \in \sgen{\mf{H}(\theta)^*:
i_{\mc{P}}(\theta) = i_{\mc{Q}}(\psi) }
\ \ \text{ for all } \psi \in \Phi. \]
We refer to the divisibility conditions for index vectors
$(i_1,i_2)$ with $i_1+i_2=j$ as $j$-divisibility conditions,
where we assume $0 \le j \le 2$, as otherwise they are vacuous.
We describe these conditions concretely as follows.
{\bf $2$-divisibility.}
These conditions simply say that the arcs of $J$
have the same types with respect to $\mc{Q}$
as those of $H$ do with respect to $\mc{P}$,
i.e.\ all arcs of $J[V]$ have colour $0$ or $K$,
all arcs of $J[V,W]$ have colour $0'$ or $K'$, and $J[W]=\emptyset$.
To see this, consider any degree vector $\mf{H}(\theta)^*$
with $\theta \in I^2_{c+1}$. We write
$\text{id} = (1 \mapsto 1, 2 \mapsto 2)$ and
$(12) = (1 \mapsto 2, 2 \mapsto 1)$.
For any $\ell \in C$, $\pi \in \{\text{id},(12)\}$
we have $\mf{H}(\theta)^*_{\ell \pi}$ equal to $1$
if $(\ell,\pi)$ is the pair such that
$\theta \circ \pi^{-1} \in \mf{H}^\ell$
(there is at most one such pair)
or equal to $0$ otherwise.
For example, if $\theta = (1 \mapsto c, 2 \mapsto c_-)$
then $\mf{H}(\theta)^*_{\ell \pi}$ is $1$
if $(\ell,\pi)=(K,(12))$, otherwise $0$.
Thus $\mf{H}\sgen{(i_1,i_2)} :=
\sgen{\mf{H}(\theta)^*: i_{\mc{P}}(\theta) = (i_1,i_2)}$
consists of all integer vectors supported in coordinates
with colours in $\{0,K\}$ if $(i_1,i_2)=(2,0)$
or $\{0',K'\}$ if $(i_1,i_2)=(1,1)$, whereas
$\mf{H}\sgen{(0,2)}$ only contains the all-$0$ vector.
Therefore, the $2$-divisibility conditions say that
$\mf{J}(\psi)^*$ can be non-zero only at coordinates
with colours in $\{0,K\}$ if $i_{\mc{Q}}(\psi)=(2,0)$
or $\{0',K'\}$ if $i_{\mc{Q}}(\psi)=(1,1)$,
and $\mf{J}(\psi)^*=0$ if $i_{\mc{Q}}(\psi)=(0,2)$,
i.e.\ $J$ has the same arc types with respect to $\mc{Q}$
as $H$ with respect to $\mc{P}$.
{\bf $0$-divisibility.}
Writing $\emptyset$ for the function with empty domain,
all $\mf{H}(\emptyset)^*_{\ell \emptyset} = |\mf{H}^\ell|=|H^\ell|$,
and similarly for $J$, so the $0$-divisibility condition
is that for some integer $m$ all $|J^\ell|=m|H^\ell|$.
For our specific $H$, this is equivalent to
$|J[V]|=|J[V,W]|=c|J^c[V]|=c|J^c[V,W]|$.
{\bf $1$-divisibility.}
Given $\theta=(1 \mapsto a) \in I^1_{c+1}$
and $\ell \in C =\{0,K,0',K'\}$,
the two coordinates of $\mf{H}(\theta)^*$
corresponding to colour $\ell$
are the in/outdegrees of $a$ in $H^\ell$:
we have $\mf{H}(\theta)^*_{\ell \text{id}}
= |\mf{H}(1 \mapsto a)| = d^+_{H^\ell}(a)$
and $\mf{H}(\theta)^*_{\ell (12)}
= |\mf{H}(2 \mapsto a)| = d^-_{H^\ell}(a)$.
Similarly, for $\psi=(1 \mapsto v) \in I^1_{n'}$
the coordinates of $\mf{J}(\psi)^*$ corresponding
to colour $\ell$ are $d^\pm_{J^\ell}(v)$. We compute:
\begin{tabular}{LCCCCCCCC}
\mf{H}(1 \mapsto a)^* & d^+_{H^0}(a) & d^-_{H^0}(a)
& d^+_{H^K}(a) & d^-_{H^K}(a) & d^+_{H^{0'}}(a)
& d^-_{H^{0'}}(a) & d^+_{H^{K'}}(a) & d^-_{H^{K'}}(a) \\
a = c_+ & 0 & 0 & 0 & 0 & 0 & c-1 & 0 & 1 \\
a = c & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
a = c_- & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\
a \in [c-2] & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0
\end{tabular}
\begin{align*}
\text{ so } \ \ \sgen{\mf{H}(1 \mapsto c_+)^*}
& = \{ \bm{v} \in \mathbb{Z}^8 : v_1=v_2=v_3=v_4=v_5=v_7=0,
v_6=(c-1)v_8 \}, \text{ and } \\
\sgen{\mf{H}(1 \mapsto a)^*: a \in [c]}
& = \{ \bm{v} \in \mathbb{Z}^8 : v_2=v_5, v_4=v_7,
v_1+v_3=v_2+v_4, v_6=v_8=0 \}.
\end{align*}
For $w \in W$ the $1$-divisibility condition
is $\mf{J}(1 \mapsto w)^* \in \sgen{\mf{H}(1 \mapsto c_+)^*}$,
i.e.\ $d^-_{J^{0'}}(w) = (c-1)d^-_{J^{K'}}(w)$,
or equivalently $d^-_J(w) = cd^-_{J^{K'}}(w)$.
For $v \in V$ the $1$-divisibility condition
is $\mf{J}(1 \mapsto v)^* \in
\sgen{\mf{H}(1 \mapsto a)^*: a \in [c]}$,
which is equivalent to
$d^-_{J^K}(v) = d^+_{J^{K'}}(v)$
and $d^+_J(v,V) = d^-_J(v,V) = d^+_J(v,W)$.
All of these divisibility conditions follow from the
divisibility assumption in Theorem \ref{decompK}
(after renaming colours $0$ and $K$
in $J[V,W]$ as $0'$ and $K'$).
By the above discussion, Theorem \ref{decompK}
follows from the following special case
of \cite[Theorem 7.4]{K3}.
\begin{theo}
Let $n^{-1} \ll \delta \ll \omega \ll c^{-1}$.
Let $h=2^{50c^3}$ and $d \ll n$.
Let $J$ be a digraph with $V(J)$ partitioned as $(V,W)$
where $\omega n \le |V|, |W| \le n$, such that $J[W]=\emptyset$,
all arcs in $J[V,W]$ point towards $W$,
all arcs in $J[V]$ are coloured $0$ or $K$
and all arcs in $J[V,W]$ are coloured $0'$ or $K'$.
Let $\Phi = (\Phi_B: B \subseteq [c+1])$,
where $\Phi_B$ consists of all injections
$\phi:B \to V(J)$ such that $\phi(B \cap [c]) \subseteq V$,
$\phi(B \cap \{c_+\}) \subseteq W$ and $Im(\phi)$ is $3d$-separated.
Suppose $J$ is $H$-divisible in $\Phi$
and $(H,\delta,\omega)$-regular in $\Phi$ and
$(\Phi,J)$ is $(\omega,h,H)$-vertex-extendable.
Then $J$ has an $H$-decomposition in $\Phi$.
\end{theo}
\section{The algorithm} \leftarrowbel{sec:alg}
Suppose we are in the setting of Theorem \ref{main}:
we are given a $(\varepsilon,t)$-typical $\alpha n$-regular digraph $G$
on $n$ vertices, where $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$,
and we need to decompose $G$ into some given
family $\mc{F}$ of $\alpha n$ oriented one-factors on $n$ vertices.
In this section we present an algorithm that partitions
almost all of $G$ into two digraphs $G_1$ and $G_2$,
and each factor $F_w$ into subfactors $F^1_w$ and $F^2_w$,
and also sets up auxiliary digraphs $J_1$ and $J_2$, such that
(i) an approximate wheel decomposition of $J_2$
gives an approximate decomposition of $G_2$
into partial factors that are roughly $\{F^2_w\}$,
(ii) given the approximate decomposition of $G_2$,
we can set up (via a small additional greedy embedding)
the remaining problem to be finding an exact decomposition
of a small perturbation $G'_1$ of $G_1$ into partial factors
that are roughly $\{F^1_w\}$, corresponding to a wheel
decomposition of a small perturbation $J'_1$ of $J_1$.
For most of the section we will describe and motivate
the algorithm; we then conclude with the formal statement.
We fix additional parameters with hierarchy
\begin{equation} \leftarrowbel{hierarchy}
n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll d^{-1}
\ll \eta \ll s^{-1} \ll L^{-1} \ll \alpha.
\end{equation}
For convenient reference later, we also make some comments
here regarding the roles of these additional parameters:
$\eta$ will be used to bound the number of vertices embedded greedily,
we consider a cycle `long' if it has length at least $K$,
and the cyclic intervals used to define the special colour $K$
will have sizes $d_i = d/(2s)^{i-1}$ with $i \in [2s+1]$.
By the reductions in section \ref{sec:red}, we will be able
to assume that we are in one of the following cases:
Case $K$: each $F \in \mc{F}$ has at least $n/2$ vertices
in cycles of length at least $K$,
Case $\ell^*$ with $\ell^* \in [3,L]$: each $F \in \mc{F}$
has $\ge L^{-3} n$ cycles of length $\ell^*$.
We write $\mc{F}=(F_w: w \in W)$, so $|W| = \alpha n$.
We partition each $F_w$ as $F^1_w \cup F^2_w$ as follows.
In Case $\ell^*$ we let $F^1_w$ consist of exactly
$L^{-3} n$ cycles of length $\ell^*$
(and then $F^2_w = F_w \setminus F^1_w$).
In Case $K$ we choose $F^1_w$
with $|F^1_w| - n/2 \in [0,2K]$
to consist of some cycles of length at least $K$
and at most one path of length at least $K$.
To see that this is possible,
consider any induced subgraph $F'_w$ of $F_w$
with $|F'_w|=n/2+K$ obtained by greedily adding cycles
of length at least $K$ until the size is at least $n/2 + K$,
and then deleting a (possibly empty) path from one cycle.
Let $P_1$ and $P_2$ denote the two paths
of the (possibly) split cycle, where $P_1 \in F'_w$.
If $|P_1|, |P_2| \ge K$ we let $F^1_w=F'_w$.
If $|P_1| < K$ we let $F^1_w=F'_w \setminus P_1$.
If $|P_2| < K$ we let $F^1_w=F'_w \cup P_2$.
In all cases, $F^1_w$ is as required.
The algorithm is randomised, so we start by defining
probability parameters. The graphs $G_1$ and $G_2$
are binomial random subdigraphs of $G$ of sizes
that are slightly less than one would expect
(we leave space for a greedy embedding that will occur
between the approximate decomposition step
and the exact decomposition step).
For each $w \in W$ we let
$p^g_w = (1-\eta) n^{-1}|F^g_w| + n^{-.2}$
(so $1-\eta \le p^1_w+p^2_w \le 1-L^{-3}\eta$).
We let $p_g = |W|^{-1}\sum_{w \in W} p^g_w$
(so $1-\eta \le p_1+p_2 \le 1-L^{-3}\eta$).
For each arc $e$ of $G$ independently we will let
$\mb{P}(e \in G_g) = p_g$ for $g \in [2]$.
We introduce further probabilities corresponding
to the cycle distributions of each $F^g_w$.
For $c<K$ we write $q^g_{w,c} n$ for the number
of cycles of length $c$ in $F^g_w$
and let $p^g_{w,c} = (1-\eta) q^g_{w,c}$.
We define $p^g_{w,K}$ so that $F^g_w$
has about $8p^g_{w,K}n$ vertices
not contained in cycles of length $<K$
(for technical reasons, we also ensure that
each $p^g_{w,K} \ge n^{-.2}$, which explains
the term $n^{-.2}$ in the definition of $p^g_w$).
Averaging over $W$ gives the corresponding probabilities
that describe the uses of arcs in each $G_g$:
we let $p^g_c = |W|^{-1} \sum_{w \in W} p^g_{w,c}$
so that for each $c<K$, the number of edges in $G_g$
allocated to cycles of length $c$ will be roughly
$\sum_{w \in W} cp^g_{w,c} n = |W| cp^g_c n
= \alpha cp^g_c n^2 = cp^g_c |G| + O(n)$,
and similarly, roughly $8p^g_K |G| + O(n)$ arcs
in $G_g$ will be allocated to long cycles.
The remainder of the algorithm is concerned
with the auxiliary digraphs $J_g$.
For any colour $c$, we let $J^c_g$
denote the arcs of colour $c$ in $J_g$.
We also write $J^*_g = \cup_{c \ne K} J^c_g$.
First we consider arcs within $J_g[V]$.
Throughout the paper, we fix a cyclic order on $V$,
which we choose uniformly at random.
For $v \in V$, let $v^+$ denote the successor of $v$
and $v^-$ denote the predecessor of $V$.
Arcs of the special colour $K$ should correspond
to $1/8$ of the factor arcs that are not in short cycles,
so should form a graph of density about $p^g_K$.
For each arc $\overlinea{xy} \in G_g$ not of the form
$\overlinea{zz}^+$ (to avoid loops, we don't mind double edges)
independently we assign $\overlinea{xy}$ to
colour $K$ with probability $p^g_K/p_g$
or colour $0$ with probability $p^g_*/p_g$
(where $p^g_K + p^g_*$ is slightly less than $p_g$).
If $\overlinea{xy}$ has colour $K$ we add
$\overlinea{xy}^-$ to $J^K_g$.
Now we consider $J_g[V,W]$.
These arcs are all directed from $V$ to $W$.
For each $w \in W$ and cycle length $c<K$,
there should be about $cp^g_{w,c} n$
vertices available for the $c$-cycles in $F^g_w$.
The colouring of $\ova{W}_{\! c}$ requires $1/c$-fraction
of these to be joined to $w$ in colour $c$, so we
should have $N^-_{J^c_g}(w) \approx p^g_{w,c} n$.
Similarly, there should be about $8p^g_{w,K} n$
vertices available for vertices of $F^g_w$ not in
short cycles, and the colouring of $\ova{W}^K_{\! 8}$ requires
$1/8$ of these to be joined to $w$ in colour $c$,
so we should have $N^-_{J^K_g}(w) \approx p^g_{w,K} n$.
These arcs are chosen randomly, not independently,
but according to a random collection of intervals,
of sizes $d_i = d/(2s)^{i-1}$ with $i \in [2s+1]$,
where $d$ is small enough that the resulting graph
is roughly typical, but large enough to give a good
upper bound on the number of vertices in long cycles
that become unused when they are chopped up into paths,
and so need to be embedded greedily.
These intervals must be chosen quite carefully,
because of the following somewhat subtle constraint.
Recall that in Case $K$ we will reduce to a path factor
problem in some subdigraph $H$ of $G$. This can only have
a solution if each vertex $x$ has degree
$d_H^\pm(x) = d_2(x) - d_{\pm}(x)$, where $d_2(x)$
is the number of path factors that will use $x$
and $d_-(x)$ (respectively $d_+(x)$)
is the number of these
in which $x$ is the start (respectively end).
The path factors will be obtained from
a set of arc-disjoint $\ova{W}^K_{\! 8}$'s, where for each $w \in W$,
its colour $K$ neighbourhood is given by a set of intervals
$([x^w_i,(y^w_i)^-]: i \in I_w)$, so its $\ova{W}^K_{\! 8}$'s will define
paths from $x^w_i$ to $y^w_i$. Thus in the auxiliary
digraph $J$, the degree of $x$ into $W$ must be
$d^+_J(x,W) = d_2(x) - d'_1(x)$, where $d'_1(x)$
is the number of path factors in which
$x$ is some successor $(y^w_i)^+$.
To relate these two formulae,
we note that a wheel decomposition of $J$ requires
$d^+_J(x,W)=d^+_J(x,V)=d^-_J(x,V)$
and $d^+_{J^K}(x,W)=d^-_{J^K}(x,V)$,
and that in the twisting construction,
$d^-_{J^K}(x^-,V)$ arcs of $H$ at $x$
are not counted by $d^-_J(x,V)$,
whereas $d^-_{J^K}(x,V)$ arcs of $H$
not at $x$ are counted by $d^-_J(x,V)$.
Writing $\Delta(x) = d^-_{J^K}(x^-,V) - d^-_{J^K}(x,V)
= d^+_{J^K}(x^-,W) - d^+_{J^K}(x,W)$,
we deduce $d_H^+(x)=d^+_J(x,V)$ and
$d_H^-(x) = d^-_J(x,V) + \Delta(x)$,
so we need $\Delta(x) = d'_1(x) - d_+(x)$
and $d_1'(x)=d_-(x)$.
So $\Delta(x)=d_-(x)-d_+(x)$.
We will ensure that both sides are always $0$
(taking $H$ equal to the digraph $G'_1$ in which we need
to solve the path factor problem), i.e.\
\begin{enumerate}
\item every vertex is used equally often
as a startpoint or as a successor of an interval, and
\item all vertices appear in some interval
for the same number of factors.
\end{enumerate}
To achieve this, we identify $V$ with $[n]$ under the
natural cyclic order, and select our intervals from
canonical sets $\mc{I}^i_j$, $i \in [2s+1]$, $j \in [d_i]$,
where each $\mc{I}^i_j$ is a partition of $[n]$
into $n/d_i \pm 1$ intervals of length at most $d_i$,
we have $\mc{I}^i_j \cap \mc{I}^i_{j'} = \emptyset$ for $j \ne j'$,
and for each $i$, every $v \in [n]$ occurs exactly once as
a startpoint of some interval in $\mc{I}^i = \cup_j \mc{I}^i_j$,
and also exactly once as a successor of some interval in $\mc{I}^i$.
The two conditions discussed in the previous paragraph will then
be satisfied if there are numbers $t_i$, $i \in [2s+1]$ such that
every interval in $\mc{I}^i$ is used by exactly $t_i$ factors.
Each $w$ will select intervals from some $\mc{I}^{i(w)}_{j(w)}$,
and these intervals must be non-consecutive,
so that the paths do not join up into longer paths.
This explains why we use several different interval sizes:
if we only used one size $d$ then a pair of vertices in $V$
at cyclic distance $d$ could never be both used for the same factor,
and so we would be unable to satisfy the conditions of the
wheel decomposition results in section \ref{sec:wheel}.
Now we describe how factors choose intervals.
For each $w \in W$, we start by independently choosing
$i=i(w) \in [2s+1]$ and $j=j(w) \in [d_i]$ uniformly at random.
Given $i$ and $j$, we activate each interval in $\mc{I}^i_j$
independently with probability $1/2$, and select any interval $I$
such that $I$ is activated, and its two neighbouring
intervals $I^\pm$ are not activated.
We thus obtain a random set of non-consecutive intervals where
each interval appears with probability $1/8$ (not independently).
We form random sets of intervals $\mc{X}^g_w$ where each
interval selected for $w$ is included in $\mc{X}^g_w$
independently with probability $8p^g_{w,K}$ (and is included
in at most one of $\mc{X}^1_w$ or $\mc{X}^2_w$).
Thus, given $w \in W_i := \{w': i(w')=i\}$, any interval
$I \in \mc{I}^i$ appears in $\mc{X}^g_w$ with probability
$p^g_{w,K}/d_i$. The events $\{I \in \mc{X}^g_w\}$ for $w \in W_i$
are independent, so whp about $\sum_{w \in W_i} p^g_{w,K}/d_i$
factors use $I$.
Our final sets of intervals $\mc{Y}^g_w$ are obtained from
$\mc{X}^g_w$ by removing a small number of intervals so that
every interval in $\mc{I}^i$ is used exactly $t^g_i$ times,
where $t^g_i$ is about $\sum_{w \in W_i} p^g_{w,K}/d_i$.
(We only need this property when $g=1$, but for uniformity
of the presentation we do the same thing for $g=2$.)
These intervals determine $J^K_g[V,W]$: we let
$N^-_{J^K_g}(w) = Y^g_w := \bigcup \mc{Y}^g_w$,
i.e.~the subset of $V$ which is
the union of the intervals in $\mc{Y}^g_w$.
As each $x$ is the startpoint of exactly one interval
in $\mc{I}^i$ it occurs as the startpoint of an interval
for exactly $t_g := \sum_i t^g_i$ factors; the same
statement holds for successors of intervals.
As each $x \in V$ appears in exactly one interval
in each $\mc{I}^i_j$ we deduce $d^+_{J^K_g}(x,W)
= \sum_{i=1}^{2s+1} \sum_{j=1}^{d_i} t^g_i
\approx \sum_{w \in W} p^g_{w,K} = |W| p^g_K$.
The other arcs of $J$ incident to $w$ will come from
$\overline{Y}_w := V \setminus \big( Y^1_w \cup Y^2_w
\cup (Y^1_w)^+ \cup (Y^2_w)^+ \big)$,
where $(Y^g_w)^+$ is the set of successors of intervals
in $\mc{Y}^g_w$ (these vertices are endpoints of paths so
should be avoided by the short cycles, and also by the $7/8$
of the paths not specified by the intervals).
We define $\overline{J}[V,W]$ by $N^-_{\overline{J}}(w)=\overline{Y}_w$.
For any $x \in V$ we will have
$\mb{P}(x \in Y_w^g) \approx
\mb{P}(x \in X_w^g) = p_{w,K}^g$
and $\mb{P}(x \in Y_w^g \mid w \in W_i) \approx
\mb{P}(x \in X_w^g \mid w \in W_i) = p_{w,K}^g/d_i$,
so $|\overline{Y}_w| \approx \overline{p}_w n$,
where $\overline{p}_w = 1 - \tfrac{d_i+1}{d_i} (p_{w,K}^1+p_{w,K}^2)$.
In $J^*_g = J_g \setminus J^K_g$ we require about $p^g_{w,*} n$
such arcs, where $p^g_{w,*} := p^g_w - p^g_{w,K}$,
and of these, for each cycle length $c<K$
we require about $p^g_{w,c} n$ arcs of colour $c$.
For each $x \in \overline{Y}_w$ independently we include
the arc $xw$ in at most one of the $J^*_g$
with probability $p^g_{w,*}/\overline{p}_w$,
which is a valid probability as
$p^1_{w,*} + p^2_{w,*}
= 1 - L^{-3}\eta - p_{w,K}^1 - p_{w,K}^2 < \overline{p}_w$.
Then we give each $xw \in J^*_g[V,W]$
colour $c$ with probability $p^g_{w,c}/p^g_{w,*}$.
In particular, $xw$ in $J^*_g$ is coloured $0$
with probability $p^g_{w,0}/p^g_{w,*}$, where
$p^g_{w,0} := p^g_{w,*} - \sum_{c=3}^{K-1} p^g_{w,c}$.
\subseteqsection{Formal statement of the algorithm}
The input to the algorithm consists of
an $\alpha n$-regular digraph $G$ on $V$,
a family $(F_w: w \in W)$ of $\alpha n$ oriented one-factors,
each partitioned as $F_w = F^1_w \cup F^2_w$,
and parameters satisfying
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1}
\ll d^{-1} \ll \eta \ll s^{-1} \ll L^{-1} \ll \alpha$.
We identify $V$ with $[n]$ according
to a uniformly random bijection and
adopt the natural cyclic order on $[n]$:
each $x \in [n]$ has successor $x^+=x+1$ (where $n+1$ means
$1$) and predecessor $x^-=x-1$ (where $0$ means $n$).
Let $d_i=d/(2s)^{i-1}$ for $i \in [2s+1]$.
We write $n=r_id_i+s_i$ with $r_i \in \mb{N}$
and $0 \le s_i < d_i$, and let
\[ P^i_j = \left\{ \begin{array}{ll}
\{ kd_i+j: 0 \le k \le r_i \}
& \text{if } j \in [s_i], \\
\{ kd_i+j: 0 \le k \le r_i-1 \}
& \text{if } j \in [d_i] \setminus [s_i].
\end{array} \right. \]
For each $i \in [s+1]$ and $j \in [d_i]$
we define a partition of $[n]$ into a family
of cyclic intervals $\mc{I}^i_j$ defined
as all $[a,b^-]$ where $a \in P^i_j$ and $b$ is
the next element of $P^i_j$ in the cyclic order.
(So $|\mc{I}^i_j|=n/d_i \pm 1$,
each $I \in \mc{I}^i_j$ has $|I| \le d_i$, and
$\mc{I}^i_j \cap \mc{I}^i_{j'} = \emptyset$ for $j \ne j'$.)
We let $\mc{I}^i = \cup_{j \in [d_i]} \mc{I}^i_j$.
(So for every $v \in [n]$,
exactly one $[a,b^-] \in \mc{I}^i$ has $a=v$,
and exactly one $[a,b^-] \in \mc{I}^i$ has $b=v$.)
Each $w \in W$ will be assigned $i(w) \in [2s+1]$.
For $c<K$ write $q^g_{w,c} n$ for the number
of cycles of length $c$ in $F^g_w$. Let
\begin{gather*}
p^g_w = (1-\eta) n^{-1}|F^g_w| + n^{-.2}, \qquad
p^g_{w,c} = (1-\eta) q^g_{w,c}
\ \text{ for } 3 \le c < K, \qquad
p^g_{w,K} = \tfrac{1}{8} \left( p^g_w
- \Sigma_{c=3}^{K-1} cp^g_{w,c} \right), \\
p^g_{w,*} = p^g_w - p^g_{w,K}, \qquad
p^g_{w,0} = p^g_{w,*} - \Sigma_{c=3}^{K-1} p^g_{w,c}, \qquad
p_{w,K} = p^1_{w,K} + p^2_{w,K}, \\
\overline{p}_w = 1 - \tfrac{d_{i(w)}+1}{d_{i(w)}} p_{w,K}, \qquad
p_g = |W|^{-1} \Sigma_{w \in W} p^g_w, \qquad
p^g_c = |W|^{-1} \Sigma_{w \in W} p^g_{w,c}
\ \text{ for } c \in [0,K] \cup \{*\}.
\end{gather*}
We complete the algorithm by applying the
following subroutines INTERVALS and DIGRAPH.
\begin{center}
INTERVALS \end{center}
\begin{enumerate}
\item For each $w \in W$ independently choose
$i(w) \in [2s+1]$ and $j(w) \in [d_{i(w)}]$
uniformly at random. Let $W_i = \{ w: i(w)=i \}$.
\item For each $w \in W$, let $\mc{A}_w$ include each interval
of $\mc{I}^{i(w)}_{j(w)}$ independently with probability $1/2$. \\
Let $\mc{S}_w$ consist of all $I \in \mc{A}_w$ such that
both neighbouring intervals $I^\pm$ of $I$ are not in $\mc{A}_w$.
\item Let $\mc{X}^g_w$, $g \in [2]$ be disjoint
with $\mb{P}(I \in \mc{X}^g_w)=8p^g_{w,K}$
independently for each $I \in \mc{S}_w$.
\item Let $t^g_i = \min \{ |\mc{X}^g(I)|: I \in \mc{I}^i \}$,
where $\mc{X}^g(I) := \{w \in W_i: I \in \mc{X}^g_w\}$,
and obtain $\mc{Y}^g_w \subseteq \mc{X}^g_w$ by deleting
each $I \in \mc{I}^i$, $i \in [2s+1]$ from
$|\mc{X}^g(I)|-t^g_i$ sets $\mc{X}^g_w$ with $w \in \mc{X}^g(I)$,
independently uniformly at random.
Write $\mc{Y}^g(I) := \{w \in W_i: I \in \mc{Y}^g_w\}$
(so $|\mc{Y}^g(I)|=t^g_i$ for $I \in \mc{I}^i$).
\end{enumerate}
\begin{center}
DIGRAPH \end{center}
\begin{enumerate}
\item Let $G_1$ and $G_2$ be arc-disjoint with
$\mb{P}(\overlinea{e} \in G_g) = p_g$ independently for each arc $\overlinea{e}$ of $G$.
\item For each $g \in [2]$ and $\overlinea{xy} \in G_g$ independently,
if $\overlinea{xy}$ is $\overlinea{zz}^-$ or $\overlinea{zz}^+$ for some $z$
add $\overlinea{xy}$ to $J^0_g$, otherwise choose exactly one of
$\mb{P}(\overlinea{xy} \in J^0_g) = p^g_*/p_g$ or
$\mb{P}(\overlinea{xy}^- \in J^K_g) = p^g_K/p_g$.
\item For each $w \in W$,
add $\overlinea{xw}$ to $J^K_g$
for each $x \in Y^g_w := \bigcup \mc{Y}^g_w$,
and add $\overlinea{xw}$ to $\overline{J}$
for each $x \in \overline{Y}_w := V \setminus ( Y^1_w \cup Y^2_w
\cup (Y^1_w)^+ \cup (Y^2_w)^+ )$.
\item For each arc $\overlinea{xw}$ of $\overline{J}[V,W]$ independently,
add $\overlinea{xw}$ to $J^*_g[V,W]$ with probability $p^g_{w,*}/\overline{p}_w$,
and give it exactly one colour $c \ne K$ (including $0$)
with probability $p^g_{w,c}/p^g_{w,*}$.
\end{enumerate}
We conclude this section by recording some estimates
on the algorithm parameters used throughout the paper.
\begin{gather*}
\text{In Case } K, \text{ all }
|F^g_w| = n/2 \pm 2K, \quad p^1_w, p^2_w > .49,
\quad p^1_{w,K} = p^1_w/8 > 1/17, \\
p^1_{w,c}=0 \text{ for } c \in [3,K-1],
\quad p^1_{w,*} = p^1_{w,0} = 7p^1_w/8 > 1/3
\quad \text{ and } \
p^2_{w,*} \ge p^2_{w,0} \ge 2p^2_w/3 > 1/4. \\
\text{In Case } \ell^*, \text{ all }
|F^1_w| = \ell^* L^{-3} n,
\quad |F^2_w| = n - \ell^* L^{-3} n,
\quad p^1_w > (1-\eta)\ell^*L^{-3} > 2L^{-3}, \\
p^2_w > 1 - 2L^{-2} > .9,
\quad p^1_{w,\ell^*} = p^1_w/\ell^* > .9L^{-3},
\quad p^1_{w,K}=n^{-.2}/8,
\quad p^1_{w,c}=0 \text{ for } c \in [3,K-1] \setminus \{\ell^*\}, \\
p^1_{w,*}>2L^{-3},
\quad p^1_{w,0} \ge 2p^1_{w,*}/3 > L^{-3}
\quad \text{ and } \
p^2_{w,*} \ge p^2_{w,0} \ge 2p^2_w/3 > .6.\\
\text{In both cases, } p^2_{w,K} \geq n^{-.2}/8.
\end{gather*}
\section{Analysis I: intervals} \leftarrowbel{sec:int}
In this section we analyse the families of intervals chosen
by the INTERVALS subroutine in section \ref{sec:alg};
our goal is to establish various regularity and extendability
properties of $J^K_g[V,W]$ and $\overline{J}_g[V,W]$
(which are defined in step (iii) of DIGRAPH
but are completely determined by INTERVALS).
We also deduce some corresponding properties that follow
from these under the random choices in DIGRAPH.
Before starting the analysis, we state some standard results
on concentration of probability that will be used throughout
the remainder of the paper. We use the following classical
inequality of Bernstein (see e.g.\ \cite[(2.10)]{BLM})
on sums of bounded independent random variables.
(In the special case of a sum of independent indicator
variables we will simply refer to the `Chernoff bound'.)
\begin{lemma} \leftarrowbel{bernstein}
Let $X = \sum_{i=1}^n X_i$ be a sum of
independent random variables with each $|X_i|<b$.
Let $v = \sum_{i=1}^n \mb{E}(X_i^2)$.
Then $\mb{P}(|X-\mb{E}X|>t)
< 2e^{-t^2/2(v+bt/3)}$.
\end{lemma}
We also use McDiarmid's bounded differences inequality,
which follows from Azuma's martingale inequality
(see \cite[Theorem 6.2]{BLM}).
\begin{defn} \leftarrowbel{def:vary}
Suppose $f:S \to \mb{R}$ where $S = \prod_{i=1}^n S_i$
and $b = (b_1,\dots,b_n) \in \mb{R}^n$.
We say that $f$ is \emph{$b$-Lipschitz} if for any
$s,s' \in S$ that differ only in the $i$th coordinate
we have $|f(s)-f(s')| \le b_i$.
We also say that $f$ is \emph{$v$-varying}
where $v=\sum_{i=1}^n b_i^2/4$.
\end{defn}
\begin{lemma} \leftarrowbel{azuma}
Suppose $Z = (Z_1,\dots,Z_n)$ is a sequence
of independent random variables,
and $X=f(Z)$, where $f$ is $v$-varying.
Then $\mb{P}(|X-\mb{E}X|>t) \le 2e^{-t^2/2v}$.
\end{lemma}
The next lemma records various regularity and extendability
properties of $J^K_g[V,W]$ and $\overline{J}_g[V,W]$.
We recall that each $N^-_{J^K_g}(w) = Y^g_w$
and $N^-_{\overline{J}_g}(w) = \overline{Y}_w$, and also
our notation for common neighbourhoods,
e.g.\ $N^-_{J^K_g}(R) = \bigcap_{w \in R} N^-_{J^K_g}(w)$
in statement (iv). Statements (iv) and (v)
will be applied to $n^{O(1)}$ choices
of set $U$ or function $h$, so their conclusions apply
whp simultaneously to all these choices
(recalling our convention that `whp' refers to events
with exponentially small failure probability).
For $x \in V$ we write
$t^-_g(x)$ or $t^+_g(x)$ for the number of $w$
such that $x$ is the startpoint or successor
of an interval in $\mc{Y}^g_w$.
We also use the separation property
from Definition \ref{def:sep}.
\begin{lemma} \leftarrowbel{lem:int}
Let $g \in [2]$, $U \subseteq V$ and $h:W \to \mb{R}$
with each $|h(w)|<n^{.01}$. Then whp:
\begin{enumerate}
\item
$|\mc{Y}^g(I)| = t^g_i
= \tfrac{|W| p^g_K}{(2s+1)d_i} \pm n^{.51}$
for all $I \in \mc{I}^i$, $i \in [2s+1]$.
\item $d^+_{J^K_g}(x,W) = |W| p^g_K \pm n^{.52}$
and $t^\pm_g(x) = t_g := \sum_i t^g_i$
for each $x \in V$.
\item $d^-_{J^K_g}(w) = |Y^g_w| = p^g_{w,K} n \pm n^{3/4}$ and
$d^-_{\overline{J}}(w) = |\overline{Y}_w| = \overline{p}_w n \pm n^{3/4}$
for all $w \in W$.
\item For any disjoint $R,R' \subseteq W$ of sizes $\le s$ we have
\[ \bsize{U \cap N^-_{J^K_g}(R) \cap N^-_{\overline{J}}(R')}
= |U| \prod_{w \in R} p^g_{w,K}
\prod_{w \in R'} \overline{p}_w \pm 3sn^{3/4}. \]
\item Consider $H :=
\sum \bracc{ h(w): w \in N^+_{J^K_g}(S) \cap N^+_{\overline{J}}(S') }$
for disjoint $S,S' \subseteq V$ of sizes $\le s$.
\begin{align*}
& \text{If } S \cup S' \text{ is } 3d\text{-separated then }
H = \sum_{w \in W} (p^g_{w,K})^{|S|} \overline{p}_w^{|S'|} h(w)
\pm 5sn^{3/4}. \\
& \text{If } (S,S') \text{ is } 3d\text{-separated then }
H \ge 2^{-2s} \sum_{w \in W} (p^g_{w,K})^{|S|} h(w).
\end{align*}
\end{enumerate}
\end{lemma}
Write $X^g_w = \bigcup \mc{X}^g_w$
and $\overline{X}_w = V \setminus (X^1_w \cup X^2_w
\cup (X^1_w)^+ \cup (X^2_w)^+ )$.
In the proof we repeatedly use the observation that
if $S \cup S' \subseteq V$ is $3d$-separated and $w \in W$,
given $i(w)$ and $j(w)$,
the events $\{ \{x \in X^g_w\}: x \in S \}
\cup \{ \{x \in \overline{X}_w\}: x \in S' \}$ are independent,
as they are determined by disjoint sets
of random decisions in INTERVALS.
The weaker assumption that $(S,S')$ is $3d$-separated
only implies independence of $\{S \subseteq X^g_w\}$
and $\{S' \subseteq \overline{X}_w\}$. We also note that for any $S,S'$
the events $\{S \subseteq X^g_w \} \cap \{S' \subseteq \overline{X}_w\}$
are independent over $w \in W$.
\begin{proof}
For (i), consider any $I \in \mc{I}^i_j$
with $i \in [2s+1]$, $j \in [d_i]$.
For each $w \in W_i$ independently
we have $\mb{P}(j(w)=j)=1/d_i$,
$\mb{P}(I \in \mc{S}_w \mid j(w)=j)=1/8$,
$\mb{P}(I \in \mc{X}^g_w \mid I \in \mc{S}_w)=8p^g_{w,K}$,
so $\mb{P}(I \in \mc{X}^g_w) = p^g_{w,K}/d_i$.
As $\mb{P}(w \in W_i) = 1/(2s+1)$ for each $w \in W$
and $\sum_{w \in W} p^g_{w,K} = |W|p^g_K$,
by a Chernoff bound, whp
$|\mc{X}^g(I)| = \tfrac{|W| p^g_K}{(2s+1)d_i} \pm n^{.51}$.
This estimate holds for all such $I$, and so for
$t^g_i = \min \{ |\mc{X}^g(I)|: I \in \mc{I}^i \}$;
thus (i) holds.
For (ii), note that each $x \in V$
appears in exactly one interval in each $\mc{I}^i_j$,
so
$$
d^+_{J^K_g}(x,W) = \sum_{i=1}^{2s+1} \sum_{j=1}^{d_i}
\big( \tfrac{|W| p^g_K}{(2s+1)d_i} \pm n^{.51} \big)
= |W| p^g_K \pm n^{.52}.
$$
Next we recall that
INTERVALS chooses uniformly at random
$\mc{Y}^g(I) \subseteq \mc{X}^g(I)$ of size $t^g_i$.
The statements on $t^\pm_g(x)$ hold as
for each $i$ there is exactly one $[a,b] \in \mc{I}^i$ with $a=x$
and exactly one $[a,b] \in \mc{I}^i$ with $b^+=x$.
For future reference, we note that each
$|\mc{X}^g(I) \setminus \mc{Y}^g(I)| < 2n^{.51}$.
For (iii), consider any $w \in W$.
We start INTERVALS by choosing $i=i(w) \in [2s+1]$
and $j=j(w) \in [d_i]$ uniformly at random.
Given these choices, any $I \in \mc{I}^i_j$ appears in $\mc{S}_w$
if $I \in \mc{A}_w$ and $I^\pm \notin \mc{A}_w$;
this occurs with probability $1/8$, so
$\mb{E}|\mc{S}_w|=|\mc{I}^i_j|/8 = n/8d_i \pm 1$.
As $|\mc{S}_w|$ is a $3$-Lipschitz function of
the events $\{I \in \mc{A}_w\}$, $I \in \mc{I}^i_j$,
by Lemma \ref{azuma} whp $|\mc{S}_w| = n/8d_i \pm n^{.51}$.
Each $I \in \mc{S}_w$ is included in $\mc{X}^g_w$
independently with probability $8p^g_{w,K}$,
so by a Chernoff bound whp
$|\mc{X}^g_w| = p^g_{w,K}n/d_i \pm 2n^{.51}$.
For each $I \in \mc{X}^g_w$ independently we have
$I \in \mc{Y}^g_w$ with probability
$t^g_i / |\mc{X}^g(I)| = 1 \pm n^{-.27}$,
as $p^g_K \ge n^{-.2}$. Thus $d_i \mb{E}|\mc{Y}^g_w|
= p^g_{w,K}n \pm n^{.73}$, so by a Chernoff bound whp
$d^-_{J^K_g}(w) = |Y^g_w| = d_i|\mc{Y}^g_w| \pm d_i
= p^g_{w,K}n \pm 2n^{.73}$. We deduce
$d^-_{\overline{J}}(w) = n - \tfrac{d_i+1}{d_i} (|Y^1_w|+|Y^2_w|)
= \overline{p}_w n \pm n^{3/4}$, so (ii) holds.
We note that each $|Y^g_w| = |X^g_w| \pm n^{3/4}$
and $|\overline{Y}_w|=|\overline{X}_w| \pm n^{3/4}$.
For (iv), we first estimate the number $N$ of $u \in U$
such that $u \in X^g_w$ for all $w \in R$ and
$u \in \overline{X}_w$ for all $w \in R'$.
The actual quantity we need to estimate is obtained
by replacing `X' with `Y', and so differs in size
by at most $2sn^{3/4}$.
For each $u \in U$, we have independently
$\mb{P}(u \in X^g_w) = p^g_{w,K}$ for all $w \in R$
and $\mb{P}(u \in \overline{X}_w) = \overline{p}_w$ for all $w \in R'$,
so $\mb{E}N = |U| \prod_{w \in R} p^g_{w,K}
\prod_{w \in R'} \overline{p}_w$.
Indeed, given choices of $i=i(w)$ and $j=j(w)$,
letting $I$ be the unique interval in $\mc{I}_j^i$
whose successor is $u$, we have
$\mb{P}(u \in \overline{X}_w) = 1-
\sum_{g = 1}^{2}(\mb{P}(u \in X_w^g) + \mb{P}(I \in \mc{X}_w^g))
= \overline{p}_w$.
Now (iv) follows from Lemma \ref{azuma},
as $N$ is a $3d$-Lipschitz function of $\le 2n$
independent random decisions in INTERVALS.
For (v), we will estimate
$H' = \sum \{ h(w) : S \subseteq X^g_w, S' \subseteq \overline{X}_w \}$.
The actual quantity $H$ we need to estimate
is obtained from $H'$ by replacing `X' with `Y'.
We have $|H-H'| < 4sn^{3/4}$, as for each $i,j$ there are
$\le 2s$ intervals $I \in \mc{I}^i_j$
with $I \cap (S \cup S') \ne \emptyset$
each with $<2n^{.51}$ choices of
$w \in \mc{X}^g(I) \setminus \mc{Y}^g(I)$
each with $|h(w)| < n^{.01}$.
If $S \cup S'$ is $3d$-separated then
independently for all $w \in W$ we have
$\mb{P}(x \in X^g_w) = p^g_{w,K}$ for all $x \in S$
and $\mb{P}(x \in \overline{X}_w) = \overline{p}_w$ for all $x \in S'$;
the required estimates on $H'$ and so $H$
follow whp from Lemma \ref{bernstein}.
Finally, we consider (v) when $(S,S')$ is $3d$-separated.
We fix $w \in W$, condition on $i(w)=i$ and $j(w)=j$,
and recall $\mb{P}(S \subseteq X^g_w, S' \subseteq \overline{X}_w)
= \mb{P}(S \subseteq X^g_w) \mb{P}(S' \subseteq \overline{X}_w)$.
We have the bound
$\mb{P}(S' \subseteq \overline{X}_w) \ge 2^{-s}$
from the event $I \notin \mc{A}_w$
for all $I \in \mc{I}^i_j$ with $I \cap S' \ne \emptyset$.
We claim that
$\mb{P}(S \subseteq X^g_w) > (5s)^{-1} (p^g_{w,K})^{|S|}$,
which by Lemma \ref{bernstein} suffices to complete the proof.
To prove the claim, we first note that if for some $\mc{I}^i_j$
no two vertices of $S$ lie in consecutive intervals then
$\mb{P}(S \subseteq X^g_w \mid i(w)=i, j(w)=j)
\ge (p^g_{w,K})^{|S|}$: indeed, the events
$\{I \in \mc{X}^g_w\}$ for $I \in \mc{I}^i_j$
with $I \cap S \ne \emptyset$ are positively correlated.
For $i \in [2s+1]$ let $J^i_s$ be the set of $j \in [d_i]$
for which some pair $x,x'$ of $S$ lie in consecutive intervals
of $\mc{I}^i_j$: we say $j$ is $i$-bad for $x,x'$.
We note that if $j$ is $i$-bad for some pair in $S$
then it is $i$-bad for some consecutive pair $x,x'$ in $S$
(i.e.\ $\{x,x'\} \cap S = \emptyset$).
It suffices to show that some $|J^i_s| < d_i/2$.
For this, we note that as $|S| \le s$ we can fix $i \in [2s+1]$
so that the cyclic distance between any pair of vertices
in $S$ is either $< d_{i+1}$ or $\ge d_{i-1}$.
There are no $i$-bad $j$ for any pair $x,x'$
with $d(x,x') \ge d_{i-1} = 2sd_i$.
Also, if $d(x,x')<d_{i+1}$ then $j$ is $i$-bad for $x,x'$
only if $\mc{I}^i_j$ contains an interval with an endpoint
in the cyclic interval $[x,x']$, so there are at most
$d_{i+1}$ such $j$. We deduce $|J^i_s| < sd_{i+1} = d_i/2$,
which completes the proof of the claim, and so of the lemma.
\end{proof}
The next lemma contains similar statements to those in
the previous one concerning the colours and directions
introduced in DIGRAPH.
In (iii) we define $J^{K'}_g$ by $J^{K'}_g[V,W]=J^K_g[V,W]$ and
$\overlinea{uv} \in J^{K'}_g[V] \Leftrightarrow \overlinea{uv}^- \in J^K_g[V]$,
thus removing the twist: if for some arc $\overlinea{uv}$ of $G_g$
we add $\overlinea{uv}^-$ to $J^K_g$
then we add $\overlinea{uv}$ to $J^{K'}_g$.
\begin{lemma} \leftarrowbel{deg}
Let $g \in [2]$.
Write $q^g_0=p^g_*$, $q^g_{K'}=p^g_K$
and $q^g_c=0$ otherwise. Then whp:
\begin{enumerate}
\item For every $v \in V$
and $c \in [3,K] \cup \{0\}$ we have
$d^\pm_{J_g}(v,V) = p_g (1 \pm \varepsilon)\alpha n \pm n^{.6}$,
$d^\pm_{J^c_g}(v,V) = p^g_c (1 \pm \varepsilon) \alpha n \pm n^{.6}$,
$d^+_{J^c_g}(v,W) = p^g_c \alpha n \pm 2n^{3/4}$.
\item For every $w \in W$
and $c \in [3,K] \cup \{0\}$ we have
$d^-_{J^c_g}(w,V) = p^g_{w,c} n \pm 2n^{3/4}$.
\item For any mutually disjoint sets $R_c \subseteq W$
and $S^+_c, S^-_c \subseteq V$
for $c \in [3,K-1] \cup \{0,K'\}$
with $\sum_c |R_c| \le s$
and $\sum_c |S^\pm_c| \le s$ we have
\begin{align*} &\Big| \bigcap_c
\big( N^-_{J^c_g}(R_c) \cap
N^+_{J^c_g}(S^+_c) \cap
N^-_{J^c_g}(S^-_c) \big) \Big|\\
&= |N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-)|
\prod_c \Big( (q^g_c)^{|S_c^+|+|S_c^-|}
\prod_{w \in R_c} p^g_{w,c} \Big)
\pm 4sn^{3/4}.
\end{align*}
\item Consider $H' :=
\big| W \cap N^+_{J^K_g}(S) \cap
\bigcap_c N^+_{J^c_g}(S_c) \big|$
for disjoint $S,S' \subseteq V$ of sizes $\le s$ with
$S'$ partitioned as $(S_c: c \in [3,K-1] \cup \{0\})$.
\begin{align*}
& \text{If } S \cup S' \text{ is } 3d\text{-separated then }
H' = \sum_{w \in W} (p^g_{w,K})^{|S|} \prod_c (p^g_{w,c})^{|S_c|}
\pm 6sn^{3/4}. \\
& \text{If } (S,S') \text{ is } 3d\text{-separated then }
H' + n^{.6} \ge 2^{-2s} \sum_{w \in W} (p^g_{w,K})^{|S|}
\prod_c (p^g_{w,c})^{|S_c|}.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
All quantities considered are $1$-Lipschitz functions
of the random choices in DIGRAPH, so by Lemma \ref{azuma}
it suffices to estimate the expectations.
For (i), we recall that $G$ has vertex in- and outdegrees
$(1 \pm \varepsilon) \alpha n$, and for each $\overlinea{xy}$ in $G$
we have $\mb{P}(\overlinea{xy} \in J_g) = p_g$,
so $\mb{E}d^+_{J_g}(v,V) = p_g (1 \pm \varepsilon)\alpha n$.
The other expectations are similar, with slightly
modified calculations due to the twisting in colour $K$
and avoiding loops; for example,
$\mb{E}d^-_{J^K_g}(v,V) = p^g_K (d^-_G(v^+) \pm 1)
= p^g_K (1 \pm \varepsilon)\alpha n \pm 1$.
For (ii), we recall $d^-_{\overline{J}}(w) = \overline{p}_w n \pm n^{3/4}$
from Lemma \ref{lem:int}.iii, so for $c \ne K$ we have
$\mb{E}d^-_{J^K_c}(w) = p^g_{w,c}\overline{p}_w^{-1} d^-_{\overline{J}}(w)
= p^g_{w,c} n \pm n^{3/4}$. (The estimate for $c=K$
was already given in Lemma \ref{lem:int}.iii.)
For (iii), we first apply Lemma \ref{lem:int}.iv
with $U = N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-)$,
$R = R_K$
and $R' = \cup_{c \ne K} R_c$ to obtain
\begin{align*} &\phantom{=}\bsize{N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-) \cap N^-_{J^K_g}(R_K)
\cap N^-_{\overline{J}}(\cup_{c \ne K} R_c)}\\
&= |N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-)| \prod_{w \in R_K} p^g_{w,K}
\prod_{w \in \cup_{c \ne K} R_c}
\overline{p}_w \pm 3sn^{3/4}. \end{align*}
For each vertex $v$ counted here independently we have
$\mb{P}(\overlinea{vw} \in J^c_g \mid \overlinea{vw} \in \overline{J})
= p^g_{w,c}/\overline{p}_w$ for all $w \in R_c$,
$\mb{P}(\overlinea{vx} \in J^c_g \mid \overlinea{vx} \in G)
= q^g_{c}$ for all $x \in S_c^-$ and
$\mb{P}(\overlinea{xv} \in J^c_g \mid \overlinea{xv} \in G)
= q^g_{c}$ for all $x \in S_c^+$,
so whp the stated bound for (iii) holds.
For (iv) we first consider
$H := |N^+_{J^K_g}(S) \cap N^+_{\overline{J}}(S')|$.
By Lemma \ref{lem:int}.v with $h(w)=1$,
if $S \cup S'$ is $3d$-separated then
$H = \sum_{w \in W} (p^g_{w,K})^{|S|} \overline{p}_w^{|S'|}
\pm 5sn^{3/4}$, and if $(S,S')$ is $3d$-separated then
$H \ge 2^{-2s} \sum_{w \in W} (p^g_{w,K})^{|S|}$.
For each vertex $w$ counted here independently we have
$\mb{P}(\overlinea{vw} \in J^c_g \mid \overlinea{vw} \in \overline{J})
= p^g_{w,c}/\overline{p}_w$ for all $v \in S_c$,
so whp the stated bound for (iv) holds.
\end{proof}
\section{Analysis II: wheel regularity} \leftarrowbel{sec:reg}
In this section we show how to assign weights to wheels in
each $J_g$ so that for any arc $\overlinea{e}$ there is total weight
about $1$ on wheels containing $\overlinea{e}$, and furthermore
all weights on wheels with $c+1$ vertices are of order $n^{1-c}$.
This regularity property is an assumption in the wheel
decomposition results of section \ref{sec:wheel},
and is also sufficient in its own right for approximate
decompositions by a result of Kahn \cite{KaLP}.
The estimate for the total weight of wheels on an arc will hold
even if we add any new arc to $J_g$, which is useful
as we will need to consider small perturbations of $J_1$
due to arcs of $G$ not allocated to $G_1$ or $G_2$ or not
covered in the approximate decomposition of $G_2$.
We start by considering wheels $\ova{W}_{\! c}$ with $c<K$. Let
\[W^g_{w,c} = n^c p^g_{w,c} (p^g_{w,0})^{c-1} (\alpha p^g_*)^c.\]
The motivation for this formula is that it is about the
expected number of $\ova{W}_{\! c}$'s in $J_g$ using $w$.
For any arc $\overlinea{e}$ let $W^g_c(\overlinea{e})$
be the set of copies of $\ova{W}_{\! c}$ in $J_g$ with hub in $W$ using $\overlinea{e}$.
Let \[ \hat{W}^g_c(\overlinea{e}) = \sum \{
p^g_{w,c} n (W^g_{w,c})^{-1} :
\mc{W} \in W^g_c(\overlinea{e}), w \in V(\mc{W}) \}.\]
(If $p^g_{w,c}=0$ there are no such $\mc{W}$,
so $(W^g_{w,c})^{-1}$ is always defined when used.)
In the following lemma we calculate the total weights on arcs
due to copies of $\ova{W}_{\! c}$, although we note that we do not have
a good estimate for $\overlinea{xy} \in J^0_g[V]$ if $d(x,y)<3d$.
In $J_2$ we can ignore such arcs, as we only need an
approximate decomposition, whereas in $J_1$ we will cover
these by wheels greedily before finding the exact decomposition
-- this forms part of the perturbation referred to above.
\begin{lemma} \leftarrowbel{degWc}
Let $c' \in \{0,c\}$, $N_c=1$ and $N_0 = c-1$.
Then whp:
\begin{enumerate}
\item If $p_{w,c'}^g \neq 0$ and we add $\overlinea{xw}$ to $J^{c'}_g[V,W]$
then $\hat{W}^g_c(\overlinea{xw})
= (1 \pm 4\varepsilon) N_{c'} p^g_{w,c}/p^g_{w,c'} \pm n^{-.2}$.
\item If $d(x,y)\ge 3d$ and we add $\overlinea{xy}$ to $J^0_g[V]$
then $\hat{W}^g_c(\overlinea{xy})
= (1 \pm 4\varepsilon) cp^g_c/p^g_* \pm n^{-.2}$.
\end{enumerate}
\end{lemma}
\begin{proof}
As a preliminary step for counting copies of $\ova{W}_{\! c}$
we count $c$-prewheels, which we define to consist of a wheel
with oriented rim cycle in $G$ and all spokes in $\overline{J}$.
For any arc $\overlinea{e}$ we let $P_c(\overlinea{e})$ be the set of $c$-prewheels
using $\overlinea{e}$; we will estimate $|P_c(\overlinea{e})|$ using the analysis
of INTERVALS in Lemma \ref{lem:int}.
For (i), we estimate $|P_c(\overlinea{xw})|$ as follows.
We let $x=x_c$ and choose the other rim vertices
$x_1,\dots,x_{c-1}$ sequentially in cyclic order.
At $c-2$ steps we choose
$x_{i+1} \in N_G^+(x_i) \cap N^-_{\overline{J}}(w)$:
each has $\alpha n \overline{p}_w \pm 3sn^{3/4}$
options by Lemma \ref{lem:int}.iv
with $U=N_G^+(x_i)$, $R=\emptyset$, $R'=\{w\}$,
using $|N_G^+(x_i)|=\alpha n$ ($G$ is $\alpha n$-regular).
At the last step we choose
$x_{c-1} \in N_G^+(x_{c-2}) \cap N_G^-(x_c) \cap
N^-_{\overline{J}}(w)$, so similarly there are
$|N_G^+(x_{c-2}) \cap N_G^-(x_c)| \overline{p}_w \pm 3sn^{3/4}$
options, where $|N_G^+(x_{c-2}) \cap N_G^-(x_c)|
= ((1 \pm \varepsilon)\alpha)^2 n$ by typicality of $G$.
Thus $|P_c(\overlinea{xw})|
= (1 \pm 3\varepsilon) \alpha^c (\overline{p}_w n)^{c-1}$.
Now consider the case $c'=c$,
i.e.\ $\overlinea{xw}$ is added to $J^c[V,W]$.
For any $c$-prewheel containing $\overlinea{xw}$,
independently we include the cycle arcs in $J^0_g$
with probability $p^g_*$
and give each $\overlinea{x_i w}$ with $i \ne c$ colour $0$
with probability $p^g_{w,0}/\overline{p}_w$, so
$\mb{E}|W^g_c(\overlinea{xw})|
= (1 \pm 3\varepsilon) (\alpha p^g_*)^c (p^g_{w,0} n)^{c-1}
= (1 \pm 3\varepsilon) W^g_{w,c}/p^g_{w,c}n$.
Of these random decisions,
$\le 2n$ concern an arc containing one of $x,w$,
which affect $|W^g_c(\overlinea{xw})|$ by $O(n^{c-2})$,
and the others have effect $O(n^{c-3})$.
Thus $|W^g_c(\overlinea{xw})|$ is $O(n^{2c-3})$-varying,
so by Lemma \ref{azuma} whp $|W^g_c(\overlinea{xw})|
= (1 \pm 4\varepsilon) W^g_{w,c}/p^g_{w,c}n$,
i.e.\ $\hat{W}^g_c(\overlinea{xw}) = 1 \pm 4\varepsilon$.
When $c'=0$ we argue similarly.
Now $x$ can be any $x_i$ with $i \ne c$,
for which we have $c-1$ choices.
The probability factors are the same as in the previous
calculation, except that for $\overlinea{x_c w}$ we replace
$p^g_{w,0}/\overline{p}_w$ by $p^g_{w,c}/\overline{p}_w$.
Again, the stated estimate holds whp
by Lemma \ref{azuma}, so (i) holds.
For (ii), we write $\hat{W}^g_c(\overlinea{xy})
= \sum_{w \in W} \hat{W}^g_c(xyw)$,
where $\hat{W}^g_c(xyw)$ is the sum of $(W^g_{w,c})^{-1}$
over the set $W^g_c(xyw)$ of copies of $\ova{W}_{\! c}$ in $J_g$
using $\overlinea{xy}$, $\overlinea{xw}$ and $\overlinea{yw}$.
Fix $w \in N^+_{\overline{J}}(x) \cap N^+_{\overline{J}}(y)$ and
consider the number $|P_c(xyw)|$ of $c$-prewheels
using $\{\overlinea{xy},\overlinea{xw},\overlinea{yw}\}$. Choosing rim
vertices sequentially as in (i), now there are
$c-3$ steps with $\alpha n \overline{p}_w \pm 3sn^{3/4}$ options
and again $((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$
options at the last step, so $|P_c(xyw)|
= (1 \pm 3\varepsilon) \alpha^{c-1} (\overline{p}_w n)^{c-2}$.
Now we consider which of these $c$-prewheels
extend to wheels in $W^g_c(xyw)$:
there are $c$ choices for the position of $\overlinea{xy}$ on the rim,
then some probabilities determined by independent random decisions:
the $c-1$ rim edges are each correct with probability $p^g_*$,
the spoke of colour $c$ with probability $p^g_{w,c}/\overline{p}_w$,
and the other $c-1$ spokes each with probability $p^g_{w,0}/\overline{p}_w$.
Therefore \[ \mb{E} \hat{W}^g_c(xyw)
= (1 \pm 3\varepsilon) c (\alpha p^g_*)^{c-1} p^g_{w,c} (p^g_{w,0})^{c-1}
\overline{p}_w^{-2} n^{c-2} p^g_{w,c} n (W^g_{w,c})^{-1}
= (1 \pm 3\varepsilon) c (\alpha p^g_*)^{-1} p^g_{w,c} n (\overline{p}_w n)^{-2}. \]
By Lemma \ref{azuma} whp $\hat{W}^g_c(\overlinea{xy})
= (1 \pm 3.1\varepsilon) c (\alpha p^g_* n)^{-1} H$,
with $H = \sum \{ p^g_{w,c} \overline{p}_w^{-2} :
w \in N^+_{\overline{J}}(x) \cap N^+_{\overline{J}}(y) \}$.
We estimate $H$ by Lemma \ref{lem:int}.v with $S=\emptyset$, $S'=\{x,y\}$
and $h(w) = p^g_{w,c}\overline{p}_w^{-2}$ (each $7/8 \le \overline{p}_w \le 1$).
As $S \cup S'$ is $3d$-separated,
whp $H = |W|p^g_c \pm 5sn^{3/4}$, giving
$\hat{W}^g_c(\overlinea{xy}) = (1 \pm 4\varepsilon) cp^g_c/p^g_* \pm n^{-.2}$.
\end{proof}
Now we apply a similar analysis for $\ova{W}^K_{\! 8}$. Let
\[ W^g_{w,K} = n^8 \alpha p^g_K p^g_{w,K}
(\alpha p^g_* p^g_{w,0})^7 .\]
For any arc $\overlinea{e}$ let $W^g_K(\overlinea{e})$
be the set of copies of $\ova{W}^K_{\! 8}$ in $J_g$ using $\overlinea{e}$.
We define $\hat{W}^g_K(\overlinea{e})$ by
setting $c=K$ in $\hat{W}^g_c(\overlinea{e})$.
Now we calculate the total weights on arcs
due to copies of $\ova{W}^K_{\! 8}$. Note that we cannot give
a good estimate for $\overlinea{xy} \in J^K_g[V]$ if $d(x,y)<3d$.
We can ignore such arcs in $J_2$ (as mentioned above),
but in $J_1$ we will replace such arcs
by arcs of colour $0$ (modified by twisting)
-- this also forms part of the perturbation.
\begin{lemma} \leftarrowbel{degWK}
Let $c' \in \{0,K\}$, $N_K=1$,
$N_0 = 7$, $q^g_K = p^g_K$, $q^g_0=p^g_*$. Then whp:
\begin{enumerate}
\item If we add $\overlinea{xw}$ to
$J^{c'}_g[V,W]$ then $\hat{W}^g_K(\overlinea{xw})
= (1 \pm 4\varepsilon) N_{c'} p^g_{w,K}/p^g_{w,c'}$.
\item Suppose we add $\overlinea{xy}$ to $J^{c'}_g[V]$.
If $d(x,y)\ge 3d$ then $\hat{W}^g_c(\overlinea{xy})
= (1 \pm 4\varepsilon) N_{c'} q^g_K/q^g_{c'}$.\\
If $c'=0$ then
$\hat{W}^g_K(\overlinea{xy}) > 2^{-2s-1} p^g_K/p^g_*$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (i), we start by counting $(K,g)$-prewheels, which
we define to consist of a hub $w \in W$ and an oriented $8$-path
in $G$ between $z$ and $z^+$ for some $z$ such that
$\overlinea{zw} \in J^K_g$ and $\overlinea{z'w} \in \overline{J}$
for all internal vertices $z'$ of the path.
For any arc $\overlinea{e}$ we let $P^g_K(\overlinea{e})$ be the set of
$(K,g)$-prewheels using $\overlinea{e}$.
To estimate $|P^g_K(\overlinea{xw})|$, suppose first that $c'=K$.
We require $z=x$. We choose the vertices of the path one by one.
At $6$ steps there are $\alpha n \overline{p}_w \pm 3sn^{3/4}$ options,
and at the last step
$((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$
options of a common outneighbour of some vertex and $z^+$, so
$|P^g_K(\overlinea{xw})| = (1 \pm 3\varepsilon) \alpha^8 (\overline{p}_w n)^7$.
On the other hand, if $c'=0$ then there are $7$ choices
for the position of $x$ as an internal vertex,
dividing the path into two segments.
We construct one segment by choosing its vertices one by one,
and then do the same for the other segment, starting with one of
length $\le 4$ so that $\{z,z^+\}$ is not the last choice.
At the step where we choose $\{z,z^+\}$, there is some vertex $v$
on the path for which we need the arc $\overlinea{vz}$ or $\overlinea{vz}^+$.
We also require $z \in N^-_{J^K_g}(w)$.
The number of options is $\alpha n p^g_{w,K} \pm 3sn^{3/4}$
by Lemma \ref{lem:int}.iv, with
$R=\{w\}$, $R=\emptyset$ and $U=N_G^+(v)$
or $U=N_G^+(v)^- = \{z: \overlinea{vz}^+ \in G\}$.
There are also $5$ steps with
$\alpha n \overline{p}_w \pm 3sn^{3/4}$ options,
and at the last step
$((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$ options,
so $|P^g_K(\overlinea{xw})| = (1 \pm 3\varepsilon)
7 \alpha^8 p^g_{w,K} (\overline{p}_w)^6 n^7$
(as $p^g_{w,K} \ge n^{-.2}/8$).
To estimate $|W^g_K(\overlinea{xw})|$, we first consider $c'=K$.
For any $(K,g)$-prewheel containing $\overlinea{xw}$,
independently we include the last path arc (to $z^+$)
in $J^K_g$ with probability $p^g_K$,
the other $7$ path arcs in $J^0_g$
with probability $p^g_*$,
and give $\overleftarrow{wz}'$ for each internal vertex $z'$ colour $0$
with probability $p^g_{w,0}/\overline{p}_w$, so
$$
\mb{E}|W^g_K(\overlinea{xw})|
= (1 \pm 3\varepsilon) \alpha p^g_K (\alpha p^g_* p^g_{w,0} n)^7
= (1 \pm 3\varepsilon) W^g_{w,K}/p^g_{w,K}n.
$$
As $|W^g_K(\overlinea{xw})|$ is $O(n^{13})$-varying,
by Lemma \ref{azuma} whp $|W^g_K(\overlinea{xw})|
= (1 \pm 3.1\varepsilon) W^g_{w,K}/p^g_{w,K}n \pm n^{6.51}$,
so $\hat{W}^g_K(\overlinea{xw}) = 1 \pm 4\varepsilon$
(using $p^g_K>n^{-.2}$).
For $c'=0$ we have a similar calculation.
Indeed, the path arcs are again correct with
probability $(p^g_*)^7 p^g_K$,
and the arcs $\overleftarrow{wz}'$ (now excluding $z'=x$)
are correct with probability $(p^g_{w,0}/\overline{p}_w)^6$,
so
$$
\mb{E}|W^g_K(\overlinea{xw})|
= (1 \pm 3\varepsilon) 7 \alpha p^g_K p^g_{w,K}
(p^g_{w,0})^6 (\alpha p^g_* n)^7
= (1 \pm 3\varepsilon) 7 W^g_{w,K}/p^g_{w,0}n.
$$
By Lemma \ref{azuma} whp
$|W^g_K(\overlinea{xw})| = (1 \pm 4\varepsilon) 7W^g_{w,K}/p^g_{w,0}n
\pm n^{6.51}$, so $\hat{W}^g_K(\overlinea{xw})
= (1 \pm 4\varepsilon) 7p^g_{w,K}/p^g_{w,0}$.
For (ii), we write $\hat{W}^g_K(\overlinea{xy})
= \sum_{w \in W} |\hat{W}^g_K(xyw)|$,
where $\hat{W}^g_K(xyw)$ is the sum of $(W^g_{w,K})^{-1}$
over the set $W^g_K(xyw)$ of copies of $\ova{W}^K_{\! 8}$ in $J_g$
using $\overlinea{xy}$, $\overlinea{xw}$ and $\overlinea{yw}$. For each $w$
we consider the set $P^g_K(xyw)$ of $(K,g)$-prewheels
using $\{\overlinea{xy},\overlinea{xw},\overlinea{yw}\}$
Suppose first that $\overlinea{xy}$ has colour $c'=K$.
We assume $d(x,y) \ge 3d$ (or there is nothing to prove).
We must have $y=z$ and in our prewheels the oriented $8$-paths
from $z$ to $z^+$ must end with the arc $\overlinea{xz}^+$,
corresponding to $\overlinea{xy} \in J^K$ under twisting.
We need $w \in N^+_{\overline{J}}(x) \cap N^+_{J^K_g}(y)$
so that $\overlinea{yw}$ has colour $K$ and $\overlinea{xw}$ can
receive colour $0$. Choosing rim vertices sequentially,
now $\{z,z'\}$ is already fixed, there are
$5$ steps with $\alpha n \overline{p}_w \pm 3sn^{3/4}$ options,
and at the last step
$((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$ options,
so $|P^g_K(\overlinea{xw})| = (1 \pm 3\varepsilon)
\alpha^7 (\overline{p}_w)^6 n^6$.
Now consider which of these prewheels extend to wheels in $W^g_K(xyw)$,
according to the following independent random decisions:
the other $7$ arcs of the oriented $8$-path excluding $\overlinea{xy}$
are each correct with probability $p^g_*$,
we already have $\overlinea{yw} \in J^K_g$,
and for each of the $7$ internal vertices $z'$ we have
$\overlinea{z'w}$ correct with probability $p^g_{w,0}/\overline{p}_w$.
Therefore \[ \mb{E} \hat{W}^g_K(xyw)
= (1 \pm 3\varepsilon) (\alpha p^g_*)^7 (p^g_{w,0})^7
\overline{p}_w^{-1} n^6 p^g_{w,K} n (W^g_{w,K})^{-1}
= (1 \pm 3\varepsilon) (\alpha p^g_K \overline{p}_w n )^{-1}. \]
By Lemma \ref{azuma} whp $\hat{W}^g_K(\overlinea{xy})
= (1 \pm 3.1\varepsilon) (\alpha p^g_K n)^{-1} H \pm n^{-.2}$,
with $H = \sum \{ \overline{p}_w^{-1} :
w \in N^+_{\overline{J}}(x) \cap N^+_{J^K_g}(y) \}$.
We estimate $H$ by Lemma \ref{lem:int}.v
with $S=\{y\}$ and $S'=\{x\}$. As $d(x,y) \ge 3d$,
whp $H = |W| p^g_K \pm 5sn^{3/4}$, giving
$\hat{W}^g_K(\overlinea{xy}) = 1 \pm 4\varepsilon$.
Now suppose that $\overlinea{xy}$ has colour $c'=0$.
For the hub $w$ we require $\overlinea{yw} \in J^0$
and $\overlinea{xw}$ in $J^K$ or $J^0$.
We first consider the contribution
from $\overlinea{xw} \in J^K$, when the
first vertex of the oriented $8$-path must be $z=x$.
The estimate of $|P^g_K(\overlinea{xw})|$
is the same as when $c'=K$, and the probability factors
are the same except that the factor for the last path
edge (to $z^+$) is now $p^g_K$ instead of $p^g_*$.
If $d(x,y) \ge 3d$ then the same calculation with
Lemma \ref{azuma} and Lemma \ref{lem:int}.v
shows that the contribution to $\hat{W}^g_K(\overlinea{xy})$
from $w \in N^+_{\overline{J}}(x) \cap N^+_{J^K_g}(y) \}$
is $(1 \pm 4\varepsilon) (p^g_* n)^{-1}$.
Now we consider the contribution from $\overlinea{xw} \in J^0$.
There are $6$ positions for $\overlinea{xy}$ on the path
avoiding $\{z,z'\}$. The estimate of $|P^g_K(\overlinea{xw})|$
is the same as before except that one factor of $\overline{p}_w$
is replaced by $p^g_{w,K}$ (at the choice of $\{z,z'\}$).
The probability factors are the same as in the previous
calculation for $\overlinea{xw} \in J^K$, so
$\mb{E} \hat{W}^g_K(xyw) = (1 \pm 3\varepsilon) p^g_{w,K}
(\alpha p^g_* \overline{p}_w^2 n )^{-1}$.
By Lemma \ref{azuma} whp the contribution
to $\hat{W}^g_K(\overlinea{xy})$ from such $w$ is
$(1 \pm 3.1\varepsilon) 6 (\alpha p^g_* n)^{-1} H$,
with $H = \sum \{ h(w) :
w \in N^+_{\overline{J}}(x) \cap N^+_{\overline{J}}(y) \}$,
$h(w) = p^g_{w,K} (\overline{p}_w)^{-2}$.
We estimate $H$ by Lemma \ref{lem:int}.v with $S=\emptyset$, $S'=\{x,y\}$.
As $(S,S')$ is $3d$-separated (vacuously) whp
$H \ge 2^{-2s} \sum_{w \in W} h(w) = 2^{-2s} |W|p^g_K$,
so $\hat{W}^g_K(\overlinea{xy}) > 2^{-2s-1} p^g_K/p^g_*$.
Now suppose $d(x,y) \ge 3d$. Then $S \cup S'$
is $3d$-separated, so whp $H = |W|p^g_K \pm 5sn^{3/4}$.
The contribution here to $\hat{W}^g_K(\overlinea{xy})$ is
$(1 \pm 4\varepsilon) 6 p^g_K/p^g_* $, so altogether
$\hat{W}^g_K(\overlinea{xy}) = (1 \pm 4\varepsilon) 7 p^g_K/p^g_*$.
\end{proof}
We combine the above estimates to deduce the main lemma
of this section, establishing wheel regularity.
Let \[ \hat{W}^g(\overlinea{e}) = \sum \{
\hat{W}^g_c(\overlinea{e}): c \in [3,K] \}.\]
\begin{lemma} \leftarrowbel{reg}
Suppose we add $\overlinea{e}$ to $J$ in any colour,
such that if $\overlinea{e} \in J[V]$
then $\overlinea{e}=\overlinea{xy}$ with $d(x,y) \ge 3d$,
and if $\overlinea{e}$ has a vertex in $W$ then it is an endvertex.
Then $\hat{W}^g(\overlinea{e}) = 1 \pm 5\varepsilon$.
\end{lemma}
\begin{proof}
By Lemmas \ref{degWc} and \ref{degWK} we can
analyse the various cases as follows.
\begin{itemize}
\item
If $\overlinea{e} \in J^c_g[V,W]$ with $c \ne 0$
then $\hat{W}^g(\overlinea{e}) = \hat{W}^g_c(\overlinea{e}) = 1 \pm 5\varepsilon$.
\item
If $\overlinea{xy} \in J^K_g[V]$ with $d(x,y) \ge 3d$ then
$\hat{W}^g(\overlinea{e}) = \hat{W}^g_K(\overlinea{e}) = 1 \pm 5\varepsilon$.
\item
If $\overlinea{e} \in J^0_g[V,W]$ then
$$
\hat{W}^g(\overlinea{e}) =
(1 \pm 4\varepsilon) 7 p^g_{w,K}/p^g_{w,0} + \textstyle\sum_{c=3}^{K-1}
\big( (1 \pm 4\varepsilon) (c-1) p^g_{w,c}/p^g_{w,0} \pm n^{-.2} \big)
= 1 \pm 5\varepsilon,
$$
as
$p^g_{w,0} = 7 p^g_{w,K} + \sum_{c=3}^{K-1} (c-1) p^g_{w,c}$.
\item
If $\overlinea{xy} \in J^0_g[V]$ with $d(x,y) \ge 3d$
then
$$
\hat{W}^g(\overlinea{e}) = (1 \pm 4\varepsilon) 7 p^g_K/p^g_*
+ \textstyle\sum_{c=3}^{K-1} \big(
(1 \pm 4\varepsilon) cp^g_c/p^g_* \pm n^{-.2} \big)
= 1 \pm 5\varepsilon,
$$
as $p^g_* = p_g - p^g_K
= 7p^g_K + \sum_{c=3}^{K-1} cp^g_c$.
\end{itemize}
\end{proof}
\section{Approximate decomposition} \leftarrowbel{sec:approx}
Here we describe the approximate decomposition of $G_2$.
Recall that at the start of section \ref{sec:alg}
we partitioned each factor $F_w$ into subfactors
$F^1_w$ and $F^2_w$, that each $F^g_w$
has $q^g_{w,c} n$ cycles of length $c \in [3,K-1]$,
and $p^g_{w,c} = (1-\eta)q^g_{w,c}$.
We will embed almost all of each $F^2_w$ in $G_2$.
We say $F'_w \subseteq F^2_w$ is valid if it does not have
any independent arcs (i.e.\ arcs $\overlinea{xy}$ such that
both $x$ and $y$ have total degree $1$ in $F'_w$)
and if $F^2_w$ contains a path then $F'_w$
contains the arcs incident to each of its ends.
\begin{lemma} \leftarrowbel{lem:approx}
There are arc-disjoint digraphs $G^2_w \subseteq G_2$ for $w \in W$,
where each $G^2_w$ is a copy of some valid $F'_w \subseteq F^2_w$
with $V(G^2_w) \subseteq N^-_{J_2}(w)$, such that
\begin{enumerate}
\item $G_2^- = G_2 \setminus \bigcup_{w \in W} G^2_w$
has maximum degree at most $5d^{-1/3}n$,
\item the digraph $J_2^-$
obtained from $J_2[V,W]$ by deleting all $\overlinea{xw}$
with $x \in V(G^2_w)$ has maximum degree at most $5d^{-1/3}n$, and
\item any $x \in V$ has degree $1$ in $F'_w$
for at most $n/\sqrt{d}$ choices of $w$.
\end{enumerate}
\end{lemma}
\begin{proof}
Say that an arc $\overlinea{vw}$ with $v \in V$ and $w \in W$ is \emph{bad}
there is some $c \in [3,K-1]$ such that
$\overlinea{vw} \in J^c$ and $p^2_{w,c} < n^{-.1}$,
or $\overlinea{vw} \in J^K$ and $p^2_{w,K} < d^{-1/3}$.
The expected bad degree of $v \in V$
is at most $(Kn^{-.1}+d^{-1/3})n$
so by Chernoff bounds we can assume that
every $v \in V$ has bad degree at most $2d^{-1/3}n$.
Let $J'_2$ be obtained from $J_2$ by deleting
all bad arcs and
all $\overlinea{xy} \in J^K_2[V]$ with $d(x,y)<3d$.
We consider the auxiliary hypergraph $\mc{H}$
whose vertices are all arcs of $J'_2$ and
whose edges correspond to all copies of the
coloured wheels $\ova{W}^K_{\! 8}$ or $\ova{W}_{\! c}$ with $c \in [3,K-1]$.
We recall that
$W^g_{w,c} = n^c p^g_{w,c} (p^g_{w,0})^{c-1} (\alpha p^g_*)^c$
and $W^g_{w,K} = n^8 \alpha p^g_K p^g_{w,K}
(\alpha p^g_* p^g_{w,0})^7 $.
We assign weights
$(1-5\varepsilon)p^g_{w,c} n / (W^g_{w,c})^{-1}$
to each copy of any $\ova{W}_{\! c}$ (and to $\ova{W}^K_{\! 8}$ for $c=K$).
By Lemma \ref{reg}, the total weight
of wheels in $J_2$ on any arc $\overlinea{e}$ satisfies
$1-10\varepsilon < \hat{W}^g(\overlinea{e}) < 1$.
Thus the total weight of wheels in $J_2'$
on any arc $\overlinea{e}$ satisfies
$1-d^{-1/4} < \hat{W}^g(\overlinea{e}) < 1$,
as we deleted at most $2d^{-1/3}n^7$ (say) copies of $\ova{W}^K_{\! 8}$
on $\overlinea{e}$ using a deleted arc.
Note also that for any two arcs the total weight of wheels
containing both is at most $n^{-.7}$ (as $p^g_K \ge n^{-1/4}$).
Thus $\mc{H}$ satisfies the hypotheses of a result of Kahn \cite{KaLP}
on almost perfect matchings in weighted hypergraphs
that are approximately vertex regular and have small codegrees.
A special case of this result (slightly modified) implies that
for any collection $\mc{F}$ of at most $n^{100}$ (say) subsets
of $V(\mc{H})=J$ each of size at least $\sqrt{n}$ (say) we can find
a matching $M$ in $\mc{H}$ such that
$|F \setminus \bigcup M| < d^{-1/5} |F|$ for all $F \in \mc{F}$.
(This is immediate from \cite{KaLP} if $\mc{F}$ has constant size,
and a slight modification using better concentration inequalities
implies the stated version. Alternatively, one can reduce to the
problem to an unweighted version via a suitable random selection
of edges and then apply a result of Alon and Yuster \cite{AY}.)
This is also implied by a recent result of Ehard,
Glock and Joos~\cite{EGJ}.
We choose such a matching $M$ for the family $\mc{F}$
where for each $v \in V \cup W$ we include sets
$F_v = \{ \overlinea{e} \in J_2[V,W]: v \in \overlinea{e} \}$,
$F^K_v = \{ \overlinea{e} \in J^K_2[V,W]: v \in \overlinea{e} \}$,
and $F'_v = \{ \overlinea{e} \in J_2[V]: v \in \overlinea{e} \}$
(the last just for $v \in V$). This $\mc{F}$ is valid as
all $|F|>\sqrt{n}$ by Lemma \ref{deg}.
By construction
for all $c \in [3,K-1]$ every copy of $\ova{W}_{\! c}$ in $M$
with hub $w$ has $p^2_{w,c} \ge n^{-.1}$ and
every copy of $\ova{W}^K_{\! 8}$ in $M$
with hub $w$ has $p^2_{w,K} \ge nd^{-1/3}$.
For each $w$ we define $G^2_w$ to be the subgraph of
$G$ corresponding to the wheels in $M$ containing $w$,
where we take account of the twisting in colour $K$.
Thus $G^2_w$ contains the rim $c$-cycle
of any $c$-wheel in $M$ containing $w$,
and for any copy of $\ova{W}^K_{\! 8}$ in $M$ containing
$\overlinea{xw} \in J^K[V,W]$ we obtain
an oriented path of length $8$ from $x$ to $x^+$.
The maximum degree bounds in (i) and (ii) clearly hold.
Recalling that $N^-_{J_2}(w)$ is disjoint from
the set of interval successors $(Y^2_w)^+$,
we see that these cycles and paths are vertex-disjoint,
except that some paths may connect up to form longer paths,
which can be described as follows. Let $\mc{Y}'_w$ be
the set of maximal cyclic intervals $I$ such that
for every $x \in I$ there is a copy of $\ova{W}^K_{\! 8}$ in $M$
containing $\overlinea{xw} \in J^K[V,W]$. Then for each
$[a,b] \in \mc{Y}'_w$ we have a component of $G^2_w$
that is a path of length $8d(a,b)$ from $a$ to $b^+$.
All these paths have length at most $8d$, as each such $I$
is contained within an interval of $\mc{Y}^2_w$.
Furthermore, if $x \in V$ is an endpoint of some path
in $G^2_w$ then either $x$ is a startpoint or successor of
some interval in $\mc{Y}^2_w$, for which there are
at most $2t_2$ choices of $w$ by Lemma \ref{lem:int},
or $x^+ w \in F^K_{x^+} \setminus \bigcup M$,
or $x^- w \in F^K_{x^-} \setminus \bigcup M$,
giving at most $2n/K$ more choices of $w$,
for a total of at most $n/\sqrt{d}$ (say).
It remains to show that each $G^2_w$ is isomorphic
to some valid $F'_w \subseteq F_w$. First we show
for any $c \in [3,K-1]$ that whp each $G^2_w$
has at most $q^2_{w,c} n$ cycles of length $c$.
The number of $c$-cycles
is in $G^2_w$ is at most $|N^-_{J^c_2}(w)|$, which
by Chernoff bounds is whp $< p^2_{w,c} n + n^{.6}
= (1-\eta) q^2_{w,c} n + n^{.6} < q^2_{w,c} n$,
recalling that $p^2_{w,c} \ge n^{-.1}$.
Next we bound the total length $L_w$ of paths in $G^2_w$.
By Lemma \ref{lem:int} we have
$L_w \le 8|Y^2_w| < 8p^2_{w,K} n + 8n^{3/4}$.
Writing $L'_w$ for the total length of long
(length $\ge K$) cycles and paths in $F^2_w$,
we recall that $8p^2_{w,K} n
= p^2_w n - \sum_{c=3}^{K-1} cp^2_{w,c}n
= (1-\eta)(L'_w+n^{.8})$.
So since $p^2_{w,K} \ge d^{-1/3}n$, we have
$L'_w > 8d^{-1/3}n$ and $L_w < (1-\eta/2)L'_w$.
We embed the paths of $G^2_w$ into the long cycles and
paths in $F^2_w$ according to a greedy algorithm,
where in each step that we embed some path $P$ of $G^2_w$
we delete a path of length $|P|+4$ from $F^2_w$,
which we allocate to a copy of $P$ surrounded
on both sides by paths of length $2$ that we will not
include in $F'_2$ (so that $F'_2$ will be valid).
We choose such a path (if it exists) within a remaining
cycle or path of $G^2_w$, using an endpoint if it is
a path (so that we preserve the number of components).
Recalling that there are at most $n/\sqrt{d}$ endpoints
of paths in $G^2_w$, we thus allocate a total of at most
$2n/\sqrt{d}$ edges to the surrounding paths of length $2$.
Suppose for a contradiction that the algorithm gets stuck,
trying to embed some path $P$ in some remainder $R$.
Then all components of $R$ have size $\le |P|+5 \le 8d+5$.
All components of $G^2_w$ have size $\ge K$,
so $|R| \le (8d+5)|L'_w|/K$. However, we also have
$|R| \ge |L'_w|-|L_w|-2n/\sqrt{d}
\ge \eta |L'_w|/2 - 2n/\sqrt{d}$, which is a contradiction,
as $K^{-1} \ll d^{-1} \ll \eta$ and $L'_w > 8d^{-1/3}n$.
Thus the algorithm succeeds in constructing a valid
copy $F'_w$ of $G^2_w$ in $F^2_w$.
\end{proof}
\section{Exact decomposition} \leftarrowbel{sec:exact}
This section contains the two exact decomposition results
that will conclude the proof in both Case $K$ and Case $\ell^*$.
We start by giving a common setting for both cases.
We say that $G'_1$ is a $\gamma$-perturbation of $G_1$ if
$|N_{G_1}^\pm(x) \bigtriangleup N_{G'_1}^\pm(x)| < \gamma n$ for any $x \in V$.
We say that $J'_1$ is a $\gamma$-perturbation of $J_1$
if $J'_1$ is obtained from $J_1$ by adding, deleting
or recolouring at most $\gamma n$ arcs at each vertex.
We will only consider perturbations which are
compatible in the sense that arcs added between $V$ and $W$
will point towards $W$,
and existing colours will be used.
\begin{set} \leftarrowbel{set}
Let $G'_1$ be an $\eta^{.9}$-perturbation of $G_1$.
Suppose for each $w \in W$ that $Z_w \subseteq V$
with $|Z_w \bigtriangleup (V \setminus N^-_{J^1}(w))| < 5\eta n$.
For $x \in V$ we write $Z(x)=\{w \in W: x \in Z_w\}$.
\end{set}
We start with the exact result for Case $\ell^*$, where we
recall that $F^1_w$ consists of exactly $L^{-3} n$ cycles of length
$\ell^*$, so $p^1_w = (1-\eta)\ell^* L^{-3} + n^{-.2}$,
$p^1_{w,\ell^*} = (1-\eta) L^{-3}$, $p^1_{w,K} = n^{-.2}/8$
and $p^1_{w,c}=0$ for $c \in [3,K-1]$.
\begin{lemma} \leftarrowbel{exactL}
Suppose in Setting \ref{set} and Case $\ell^*$ that
$d_{G'_1}^\pm(x)=|W|-|Z(x)|$ for all $x \in V$
and $\ell^*$ divides $n-|Z_w|$ for all $w \in W$.
Then $G'_1$ can be partitioned into graphs $(G^1_w: w \in W)$,
where each $G^1_w$ is an oriented $C_{\ell^*}$-factor
with $V(G^1_w) = V \setminus Z_w$.
\end{lemma}
\begin{proof}
We will show that there is a perturbation $J'_1$ of $J_1$
such that $J'_1[V] = G'_1$,
each $N^-_{J'_1}(w) = V \setminus Z_w$, and Theorem \ref{decompL}
applies to give a $\ova{W}_{\! \ell^*}$-decomposition of $J'_1$.
This will suffice, by taking each $G^1_w$ to consist of
the rim $\ell^*$-cycles of the copies of $\ova{W}_{\! \ell^*}$ containing $w$.
We construct $J'_1$ by starting with $J'_1=J_1$ and applying
a series of modifications as follows. First we delete all arcs
of $J'_1[V]$ corresponding to arcs of $G_1 \setminus G'_1$ and add
arcs of colour $0$
corresponding to arcs of $G'_1 \setminus G_1$.
Similarly, we delete all arcs $\overlinea{vw} \in J'_1[V,W]$
with $v \in N^-_{J_1}(w) \cap Z_w$ and add arcs $\overlinea{vw}$
of colour $0$ for each $v \in (V \setminus Z_w) \setminus N^-_{J_1}(w)$.
We also recolour any $\overlinea{vw} \in J'_1[V,W]$ of colour $K$
to have colour $0$ and replace any $\overlinea{xy}$ of colour $K$
in $J'_1[V]$ by $\overlinea{xy}^+$ of colour $0$.
As each $p^1_{w,K}=n^{-.2}/8$ in this case,
whp this affects at most $n^{.8}$ arcs at any vertex.
Now $J'_1[V]=G'_1$,
each $N^-_{J'_1}(w) = V \setminus Z_w$ and
$J'_1$ is a $\eta^{.8}$-perturbation of $J_1$.
We note for each $x \in V$ that $d_{J'_1}^\pm(x,V)
= d_{G'_1}^\pm(x) = |W|-|Z(x)| = d^+_{J'_1}(x,W)$,
so the divisibility conditions for $x \in V$ are satisfied.
Finally, to satisfy the divisibility conditions
for all $w \in W$ we recolour so that
$d^-_{(J'_1)^{\ell^*}}(w) = d^-_{J'_1}(w)/\ell^*$,
which is an integer, as $\ell^*$ divides
$d^-_{J'_1}(w) = n-|Z_w|$.
By Lemma \ref{deg} each $d^-_{J_1}(w) = p^1_w n \pm 2n^{3/4}$
and $d^-_{J_1^{\ell^*}}(w) = p^1_{w,\ell^*} n \pm 2n^{3/4}$,
where $p^1_w = \ell^* p^1_{w,\ell^*} + n^{-.2}$ in this case.
As $J'_1$ is an $\eta^{.8}$-perturbation of $J_1$,
we only need to recolour at most $2\eta^{.8} n$ arcs
at any vertex, so our final digraph $J'_1$
is a $3\eta^{.8}$-perturbation of $J_1$.
Next we consider the regularity condition of Theorem \ref{decompK}.
To each copy of $\ova{W}_{\! \ell^*}$ in $J'_1$ with hub $w$
we assign weight $p^1_{w,\ell^*} n/ W^g_{w,\ell^*} =
p^1_{w,0} n (\alpha p^1_{w,0} p^1_* n)^{-\ell^*}$,
which lies in $[n^{1-\ell^*}, L^L n^{1-\ell^*}]$.
We claim that for any arc $\overlinea{e}$ of $P'$ there is total weight
$1 \pm \eta^{.6}$ on wheels containing $\overlinea{e}$.
To see this, we compare the weight to $\hat{W}^1_{\ell^*}(\overlinea{e})$
as defined in section \ref{sec:reg},
which is $1 \pm 4\varepsilon$ by Lemma \ref{degWc}
(as $p^1_{w,0}=(\ell^*-1) p^1_{w,\ell^*}$
and $p^1_*=(\ell^*-1) p^1_{\ell^*}$).
The actual weight on $\overlinea{e}$ differs from this estimate only due
to wheels containing $\overlinea{e}$ that have another arc in $J'_1 \bigtriangleup J_1$.
There are at most $40\eta^{.7} n^{\ell^*-1}$ such wheels,
each affecting the weight by at most $L^L n^{\ell^*-1}$, so the claim holds.
Thus regularity holds with $\delta=\eta^{.6}$ and $\omega=L^{-L}$.
It remains to show that $J'_1$ satisfies the
extendability condition of Theorem \ref{decompL}.
Consider any disjoint $A,B \subseteq V$ and $C \subseteq W$
each of size $\le h$, where $h = 2^{50 (\ell^*)^3}$.
By Lemma \ref{deg}.iii,
for $c \in \{0,\ell^*\}$ we have
$$
|N^+_{J^0_1}(A) \cap N^-_{J^0_1}(B) \cap N^-_{J^c_1}(C)|
= |N_G^+(A) \cap N_G^-(B)| (p_*^1)^{|A|} (p_*^1)^{|B|}
\prod_{w \in C} p^1_{w,c} \pm 4sn^{3/4}
> (L^{-5} \alpha )^{2h} n,
$$
by typicality of $G$.
Also, by Lemma \ref{deg}.iv
(with $S=\emptyset$ and $S'=A \cup B$) we have
$|N^+_{J^0_1}(A) \cap N^+_{J^{\ell^*}_1}(B) \cap W|
\ge 2^{-2s} L^{-7h} |W|$, say.
The perturbation from $J_1$ to $J'_1$ affects these estimates
by at most $6h\eta^{.7}n < \eta^{.6}n$, so $J'_1$ satisfies
extendability with $\omega=L^{-L}$ as above.
Now Theorem \ref{decompL} applies to give
a $\ova{W}_{\! \ell^*}$-decomposition of $J'_1$,
which completes the proof.
\end{proof}
Our second exact decomposition result concerns the path factors
with prescribed ends required for Case $K$.
We recall that each $F^1_w$ consists of cycles of length $\ge K$
and at most one path of of length $\ge K$
with $|F^1_w| - n/2 \in [0,2K]$,
and that $(Y^1_w)^-$ and $(Y^1_w)^+$ are the sets of
startpoints and successors of intervals in $\mc{Y}^1_w$.
We also recall from Lemma \ref{lem:int} that for each $x \in V$,
letting $t^\pm_1(x) = |\{w: x \in (Y^1_w)^\pm\}|$,
we have $t^+_1(x)=t^-_1(x)=t_1$.
After embedding $F^2_w$, and a greedy embedding connecting
the paths to $(Y^1_w)^-$ and $(Y^1_w)^+$, we will need
path factors $G^1_w$ as follows.
\begin{lemma} \leftarrowbel{exactK}
Suppose in Setting \ref{set} and Case $K$ that
$Z_w$ is disjoint from $Y^1_w \cup (Y^1_w)^+$
and $8|Y^1_w| = n-|Z_w|-|(Y^1_w)^+|$ for all $w \in W$,
and $d_{G'_1}^\pm(x)=|W|-t_1-|Z(x)|$ for all $x \in V$.
Then $G'_1$ can be partitioned into graphs $(G^1_w: w \in W)$,
such that each $G^1_w$ is a vertex-disjoint union of oriented paths
with $V(G^1_w) = V\setminus Z_w$,
where for each $[a,b] \in \mc{Y}^1_w$ there
is an $ab^+$-path of length $8d(a,b)$.
\end{lemma}
\begin{proof}
We will show that there is
a perturbation $P$ of $J_1$ such that
each $N^-_P(w) = V \setminus Z_w$ and $P[V]$
corresponds to $G'_1$ under twisting, and
a set $E$ of arc-disjoint copies of $\ova{W}^K_{\! 8}$ in $P$,
such that Theorem \ref{decompK} applies to give
a $\ova{W}^K_{\! 8}$-decomposition of $P' := P \setminus \bigcup E$.
This will suffice, by taking each $G^1_w$ to consist of the
union of the oriented $8$-paths that correspond under twisting to
the rim $8$-cycles of the copies of $\ova{W}^K_{\! 8}$ containing $w$.
We construct $P$ by starting with $P=J_1$ and applying
a series of modifications as follows. First we delete all arcs
of $P[V]$ corresponding to arcs of $G_1 \setminus G'_1$ and add
arcs of colour $0$
corresponding to arcs of $G'_1 \setminus G_1$.
Similarly, we delete all arcs $\overlinea{vw} \in P[V,W]$
with $v \in N^-_{J_1}(w) \cap Z_w$ and add arcs $\overlinea{vw}$
of colour $0$ for each
$v \in V \setminus (Z_w \cup (Y^1_w)^+ \cup N^-_{J_1}(w))$.
We also replace any $\overlinea{xy}$ of colour $K$ with $d(x,y)<3d$
by an arc $\overlinea{xy}^+$ of colour $0$;
this affects at most $6d$ arcs at each vertex.
Now $P[V]$ corresponds to $G'_1$ under twisting,
each $N^-_P(w) = V \setminus (Z_w \cup (Y^1_w)^+)$ and
$P$ is a $2\eta^{.9}$-perturbation of $J_1$.
We note that $P$ now satisfies the divisibility condition
$d^-_P(w) = 8|Y^1_w| = 8d^-_{P^K}(w)$, and for each $v \in V$
that $d^+_P(v,W) = |W|-t_1-|Z(x)| = d_P(v,V)/2$,
so $|P[V,W]| = |P[V]|$. We continue to modify $P$ to obtain
$|P^0[V,W]|=|P^0[V]|$ and $|P^K[V,W]|=|P^K[V]|$.
To do so, we will recolour arcs of $P[V]$ according to a
greedy algorithm, where if $|P^0[V]|>|P^0[V,W]|$ we replace
some $\overlinea{xy} \in P^0[V]$ by $\overlinea{xy}^- \in P^K[V]$,
or if $|P^0[V]|<|P^0[V,W]|$ we replace
some $\overlinea{xy} \in P^K[V]$ by $\overlinea{xy}^+ \in P^0[V]$.
This preserves $P[V]$ corresponding to $G'_1$ under twisting
and $|P[V]| = |P[V,W]|$, so if we ensure $|P^0[V,W]|=|P^0[V]|$,
we will also have $|P^K[V,W]|=|P^K[V]|$.
During the greedy algorithm, we choose the arc to recolour
arbitrarily, subject to avoiding the set $S$ of vertices
at which we have recoloured more than $\eta^{.8} n/2$ arcs.
The total number of recoloured arcs is at most
$||P[V,W]|-|P[V]|| \le ||J_1[V,W]|-|J_1[V]|| + 2\eta^{.9}n^2
< 3\eta^{.9}n^2$ (by Lemma \ref{deg}), so $|S|<12\eta^{.1}n$.
Thus the algorithm can be completed,
giving $P$ that is an $\eta^{.8}$-perturbation of $J_1$
with $|P^0[V,W]|=|P^0[V]|$ and $|P^K[V,W]|=|P^K[V]|$.
We will continue modifying $P[V]$ until it satisfies the
remaining degree divisibility conditions for each $v \in V$,
i.e.\ $d^+_P(v,V) = d^-_P(v,V) = d^+_P(v,W)$
and $d^-_{P^K}(v,V) = d^+_{P^K}(v,W)$.
To do so, we will reduce to $0$
the imbalance $\Delta' = \sum_{v \in V} \Delta'(v)$
with each $\Delta'(v) = |d^+_{P^K}(v,V)-d^+_{P^K}(v,W)|
+ |d^-_{P^K}(v,V)-d^+_{P^K}(v,W)|
$. We do not attempt
to control any $d^\pm_{P^0}(v,V)$, but nevertheless
the divisibility conditions will be satisfied when $\Delta'=0$.
To see this, note that if $\Delta'=0$ then clearly all
$d^+_{P^K}(v,V)=d^-_{P^K}(v,V)=d^+_{P^K}(v,W)$,
so it remains to
show that $d_P^-(v,V)=d_P^+(v,V)=d_P^+(v,W)$.
Here we recall the discussion in section \ref{sec:alg}
relating the choice of intervals to degree divisibility,
where (setting $H=G'_1$ and $J=P$) we noted that
$d_{G'_1}^+(v) = d_P^+(v,V)$ and
$d_{G_1'}^-(v) = d_P^-(v,V) + \Delta(v)$, with
$\Delta(v) = d^-_{P^K}(v^-,V) - d^-_{P^K}(v,V)
= d^+_{P^K}(v^-,W) - d^+_{P^K}(v,W)$.
By our choice of intervals
all $d^+_{P^K}(v,W)$ are equal to $t_1$,
so $\Delta(v)=0$ and $d_P^\pm(v,V) = d_{G'_1}^\pm(v)
= |W|-t_1-|Z(x)| = d^+_P(v,W)$, as required.
We have two types of reduction according to the two types
of term in the definition of $\Delta'(v)$:
\begin{enumerate}
\item If $\sum_v |d^-_{P^K}(v,V)-d^+_{P^K}(v,W)| > 0$
then we can choose $x,y$ in $V$ with
$d^-_{P^K}(x,V) > d^+_{P^K}(x,W)$ and
$d^-_{P^K}(y,V) < d^+_{P^K}(y,W)$.
We will find $z \in V$ such that
$\overlinea{zx} \in P^K$, $\overlinea{zy}^+ \in P^0$
and replace these arcs by
$\overlinea{zx}^+ \in P^0$, $\overlinea{zy} \in P^K$.
\item If $\sum_v |d^+_{P^K}(v,V)-d^+_{P^K}(v,W)| > 0$
then we can choose $x,y$ in $V$ with
$d^+_{P^K}(x,V) > d^+_{P^K}(x,W)$ and
$d^+_{P^K}(y,V) < d^+_{P^K}(y,W)$.
We will find $z \in V$ such that
$\overlinea{xz} \in P^K$, $\overlinea{yz}^+ \in P^0$
and replace these arcs by
$\overlinea{yz} \in P^K$, $\overlinea{xz}^+ \in P^0$.
\end{enumerate}
\begin{center}
\includegraphics{figZ}
\end{center}
Each of these operations preserves $P[V]$
corresponding to $G'_1$ under twisting and
reduces $\Delta'$.
To reduce $\Delta'$ to $0$ we apply a greedy algorithm
where in each step we apply one of the above operations.
We do not allow $z$ with $d(x,z)<3d+2$ or $d(y,z)<3d+2$
(to avoid creating close arcs in colour $K$)
or $z$ in the set $S'$ of vertices
that have played the role of $z$ at $\eta^{.7}n/2$
previous steps. The total number of steps is at most
$2\eta^{.8}n^2$, so $|S'| < 4\eta^{.1} n$.
To estimate the number of choices for $z$ at each step,
we apply Lemma \ref{deg}.iii to
$|N^-_{J^{K'}_1}(x^+) \cap N^-_{J^0_1}(y^+)|$
for operation (i),
$|N^+_{J^{K'}_1}(x) \cap N^+_{J^0_1}(y)|$ to find $z^+$ for (ii).
By typicality of $G$ this gives
at least $\alpha^2 n/9$ choices,
of which at most $5\eta^{.1} n$ are forbidden
by lying in $S$ or too close to $x$ or $y$,
or due to requiring an arc of $J_1 \setminus P$,
so some choice always exists.
Thus the algorithm can be completed, giving $P$
that is an $\eta^{.7}$-perturbation of $J_1$,
satisfies the divisibility conditions, and has
$P[V]$ corresponding to $G'_1$ under twisting.
Next we construct $E$ as a set of arc-disjoint
copies of $\ova{W}^K_{\! 8}$ that cover all $\overlinea{xy} \in P[V]$
with $d(x,y)<3d$. Note that all such $\overlinea{xy}$ have colour $0$.
We apply a greedy algorithm, where in each step that we consider
some $\overlinea{xy}$ we choose a copy of $\ova{W}^K_{\! 8}$ that is arc-disjoint
from all previous choices and does not use any vertex in the
set $S$ of vertices that have been used $.1d^2$ times.
Then $|S|.1d^2 < 27dn$, so this forbids
at most $270n^7/d$ choices of $\ova{W}^K_{\! 8}$.
By Lemma \ref{degWK} we have
$\hat{W}^1_K(\overlinea{xy}) > 2^{-2s-1} p^1_K/p^1_* > 2^{-3s}$,
so the number of choices is at least
$2^{-3s} \min_{w \in W} W^2_{w,K}/p^2_{w,K}n
> 2^{-4s} n^7$, say. Thus there is always some choice
that is not forbidden, so the algorithm can be completed.
We note that $\bigcup E$ has maximum degree
at most $d^2$ by definition of $S$, so $P' := P \setminus \bigcup E$
is a $2\eta^{.7}$-perturbation of $J_1$.
Furthermore, $P'$ satisfies the divisibility conditions,
as $P$ does and so does each $\ova{W}^K_{\! 8}$ in $E$.
Next we consider the regularity condition of Theorem \ref{decompK}.
To each $3d$-separated copy of $\ova{W}^K_{\! 8}$ in $P'$ with hub $w$
we assign weight $p^1_{w,K} n/ W^1_{w,K} =
(\alpha p^1_K (\alpha p^1_* p^1_{w,0} n)^7 )^{-1}$,
which lies in $[n^{-7}, L n^{-7}]$.
We claim that for any arc $\overlinea{e}$ of $P'$ there is total weight
$1 \pm \eta^{.6}$ on wheels containing $\overlinea{e}$.
To see this, we compare the weight to $\hat{W}^1_K(\overlinea{e})$
as defined in section \ref{sec:reg},
which is $1 \pm 4\varepsilon$ by Lemma \ref{degWK}
(as $\overlinea{e}$ is $3d$-separated,
$p^1_{w,0}=7p^1_{w,K}$ and $p^1_*=7p^1_K$).
The actual weight on $\overlinea{e}$ differs from this estimate only due
to wheels containing $\overlinea{e}$ that have another arc in $P' \bigtriangleup J_1$.
There are at most $40\eta^{.7} n^7$ such wheels,
each affecting the weight by at most $Ln^{-7}$, so the claim holds.
Thus regularity holds with $\delta=\eta^{.6}$ and $\omega=L^{-1}$.
It remains to show that $P'$ satisfies the
extendability condition of Theorem \ref{decompK}.
Consider any disjoint $A,B \subseteq V$ and $L \subseteq W$
each of size $\le h$ and $a, b, \ell \in \{0,K\}$.
By Lemma \ref{deg}.iii we have
$|N^+_{J_1^a}(A) \cap N^-_{J_1^b}(B)
\cap N^-_{J_1^\ell}(L)|
= |N_G^+(A) \cap N_G^-(B)| (p_1^a)^{|A|} (p_1^b)^{|B|}
\prod_{w \in L} p^1_{w,\ell} \pm 4sn^{3/4}
> (10^{-3} \alpha )^{2h} n$, say.
Also, if $(A,B)$ is $3d$-separated then
by Lemma \ref{deg}.iv we have
$|N^+_{J_1^0}(A) \cap N^+_{J_1^K}(B) \cap W|
\ge 2^{-2s+10h} |W|$, say.
The perturbation from $J_1$ to $P'$ affects these estimates
by at most $6h\eta^{.7}n < \eta^{.6}n$, so $P'$ satisfies
extendability with $\omega=L^{-1}$ as above.
Now Theorem \ref{decompK} applies to give
a $\ova{W}^K_{\! 8}$-decomposition of $P'$,
which completes the proof.
\end{proof}
\section{The proof} \leftarrowbel{sec:pf}
This section contains the proof of our main theorem.
We give the reduction to cases in the first subsection
and then the proof for both cases in the second subsection.
\subseteqsection{Reduction to cases} \leftarrowbel{sec:red}
In this subsection we formalise the reduction to cases
discussed in section \ref{sec:over}. For Theorem \ref{main},
we are given an $(\varepsilon,t)$-typical $\alpha n$-regular digraph $G$
on $n$ vertices, where $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$,
and we need to decompose $G$ into some given
family $\mc{F}$ of $\alpha n$ oriented one-factors on $n$ vertices.
We prove Theorem \ref{main} assuming that it holds in the
following cases with $t^{-1} \ll K^{-1} \ll \alpha$:
Case $K$: each $F \in \mc{F}$ has at least $n/2$ vertices
in cycles of length at least $K$,
Case $\ell$ for all $\ell \in [3,K-1]$:
each $F \in \mc{F}$ has $\ge K^{-3} n$
cycles of length $\ell$.
We will divide into subproblems via
the following partitioning lemma.
\begin{lemma} \leftarrowbel{typ:split}
Let $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha_0$.
Suppose $G$ is an $(\varepsilon,t)$-typical $\alpha n$-regular digraph
on $n$ vertices and $\alpha = \sum_{i \in I} \alpha_i$
with each $\alpha_i > \alpha_0$. Then $G$ can be decomposed
into digraphs $(G_i: i \in I)$ on $V(G)$ such that
each $G_i$ is $(2\varepsilon,t)$-typical and $\alpha_i n$-regular.
\end{lemma}
\begin{proof}
We start by considering a random partition of $G$
into graphs $(G'_i: i \in I)$ where for each arc $\overlinea{e}$
independently we have $\mb{P}(\overlinea{e} \in G'_i)=\alpha_i/\alpha$.
We claim that whp each $G'_i$ is $(1.1\varepsilon,t)$-typical.
Indeed, this holds by Chernoff bounds,
as $\mb{E}d(G'_i) = \alpha_i d(G)/\alpha$ for each $i$,
so whp $d(G'_i) = \alpha_i \pm n^{-.4}$ (say),
and for any set $S=S_- \cup S_+$
of at most $t$ vertices, by typicality of $G$
we have $\mb{E} |N_{G'_i}^-(S_-) \cap N_{G'_i}^+(S_+)|
= (\alpha_i/\alpha)^{|S|} |N_G^-(S_-) \cap N_G^+(S_+)|
= ((1 \pm \varepsilon)d(G) \alpha_i/\alpha)^{|S|} n$,
so whp $|N_{G'_i}^-(S_-) \cap N_{G'_i}^+(S_+)|
= ((1 \pm 1.1\varepsilon) d(G'_i))^{|S|} n$,
Now we modify the partition to obtain $(G_i: i \in I)$,
by a greedy algorithm starting from all $G_i=G'_i$.
First we ensure that all $|G_i| = \alpha_i n^2$.
At any step, if this does not hold then some
$|G_i| > \alpha_i n^2$ and $|G_j| < \alpha_j n^2$.
We move an arc from $G_i$ to $G_j$, arbitrarily
subject to not moving more than $n^{.7}$ arcs
at any vertex. We move at most $n^{1.6}$ arcs,
so at most $2n^{.9}$ vertices become forbidden
during this algorithm. Hence the algorithm can
be completed to ensure that all $|G_i| = \alpha_i n^2$.
Each $|N_{G'_i}^-(S_-) \cap N_{G'_i}^+(S_+)|$
changes by at most $tn^{.7}$,
so each $G_i$ is now $(1.2\varepsilon,t)$-typical.
Let $\widetilde{G_i}$ be the undirected graph of $G_i$
(which could have parallel edges).
We will continue to modify the partition until each $\widetilde{G_i}$
is $2\alpha_i n$-regular, maintaining all $|G_i|=\alpha_i n^2$.
At each step we reduce the imbalance
$\sum_{i,x} |d_{\widetilde{G_i}}(x)-2\alpha_i n|$.
If some $\widetilde{G_i}$ is not $2\alpha_i n$-regular we have
some $d_{\widetilde{G_i}}(x) > 2\alpha_i n$ and $d_{\widetilde{G_i}}(y) < 2\alpha_i n$.
Considering the total degree of $x$,
there is some $j$ with $d_{\widetilde{G_j}}(x) < 2\alpha_j n$.
We will choose some $z$ with $xz \in \widetilde{G_i}$ and $yz \in \widetilde{G_j}$,
then move $xz$ to $\widetilde{G_j}$ and $yz$ to $\widetilde{G_i}$,
thus reducing the imbalance by at least $2$.
We will not choose $z$ in the set $L$ of vertices that
have played the role of $z$ at $n^{.8}$ previous steps.
We had all $d_{\widetilde{G_i}}(x) = 2(\alpha_i n \pm n^{.7})$
after the first algorithm, so this algorithm will have
at most $2n^{1.7}$ steps, giving $|L| < n^{.9}$.
By typicality, there are at least $3\alpha_i \alpha_j n$
choices of $z$, of which at most $2n^{.9}$ are forbidden
by $L$ or requiring an edge that has been moved,
so the algorithm to make each $\widetilde{G_i}$ be $2\alpha_i n$-regular
can be completed. Each $|N^-_{G_i}(S_-) \cap N^+_{G_i}(S_+)|$
changes by at most
$tn^{.8}$, so each $G_i$ is now $(1.1\varepsilon,t)$-typical.
We will continue to modify the partition until each $G_i$
is $\alpha_i n$-regular, maintaining all $d_{\widetilde{G_i}}(x)=2\alpha_i n$.
At each step we reduce the imbalance
$\sum_{i,x} |d_{G_i}^+(x)-\alpha_i n|$
(if it is $0$ then since total degrees $d_{\widetilde{G_i}}(x)$
are correct, $G_i$ is regular).
If it is not $0$ we have
some $d_{G_i}^+(x) > \alpha_i n$ and $d_{G_i}^+(y) < \alpha_i n$.
Again there is some $j$ with $d_{G_j}^+(x) < \alpha_j n$ and we
choose some $z$ with $\overlinea{xz} \in G_i$ and $\overlinea{yz} \in G_j$,
then move $\overlinea{xz}$ to $G_j$ and $\overlinea{yz}$ to $G_i$,
avoiding vertices $z$ which have played this role
at $n^{.9}$ previous steps.
By typicality we can find such $z$ at every step and
complete the algorithm.
Each $|N_{G_i}^-(S_-) \cap N_{G_i}^+(S_+)|$
changes by at most
$tn^{.9}$, so each $G_i$ is now $(2\varepsilon,t)$-typical.
\end{proof}
Factors of a type that is too rare
will be embedded greedily via the following lemma.
\begin{lemma} \leftarrowbel{typ:greedy}
Let $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$.
Suppose $G$ is an $(\varepsilon,t)$-typical $\alpha n$-regular digraph
on $n$ vertices and $\mc{F}$ is a family of at most $\varepsilon n$ oriented
one-factors. Then we can remove from $G$ a copy of each $F \in \mc{F}$
to leave a $(\sqrt{\varepsilon},t)$-typical $(\alpha n-|\mc{F}|)$-regular graph.
\end{lemma}
\begin{proof}
We embed the one-factors one by one. At each step,
the remaining graph $G'$ is obtained from $G$ by deleting
a graph that is regular of degree at most $2\varepsilon n$,
so is $(\sqrt{\varepsilon},t)$-typical. It is a standard argument
(which we omit) using the blow-up lemma
of Koml\'os, S\'ark\"ozy and Szemer\'edi \cite{KSS}
to show that any one-factor can be embedded in $G'$,
so the process can be completed.
\end{proof}
Now we prove Theorem \ref{main}
assuming that it holds in the above cases.
We introduce new parameters $\alpha_1, \alpha_2, M_1', M_1, M_2, M_3$
with $\varepsilon \ll t^{-1} \ll M_3^{-1} \ll \alpha_2 \ll M_2^{-1}
\ll \alpha_1 \ll (M_1')^{-1} \ll M_1^{-1} \ll \alpha$.
For $\ell \in [3,M_2]$ let $\mc{F}_\ell$ consist of
all factors $F \in \mc{F}$ such that
$F$ has $\ge M_2^{-3} n$ cycles of length $\ell$
but $< M_2^{-3} n$ cycles of each smaller length.
Let $\mc{F}_2$ consist of all remaining factors in $\mc{F}$.
Note that each $F \in \mc{F}_2$ has fewer than $n/M_2$
vertices in cycles of length less than $M_2$,
so at least $(M_2-1)n/M_2$ in cycles of length at least $M_2$.
Let $B$ be the set of $\ell \in [3,M_2]$
such that $|\mc{F}_\ell| < \alpha_2 n$.
Then for $\ell \in I' := [3,M_2] \setminus B$ we have
$\beta_\ell := n^{-1} |\mc{F}_\ell| \ge \alpha_2$. Also,
writing $\mc{F}_B = \bigcup_{\ell \in B} \mc{F}_\ell$,
we have $\beta_B := n^{-1} |\mc{F}_B| < M_2\alpha_2 < \sqrt{\alpha_2}$.
Let $\mc{F}_1$ be the set of $F$ in $\mc{F}$ with
at least $n/2$ vertices in cycles of length $>M_1$.
We first consider the case
$\eta := n^{-1}|\mc{F}_1| \ge \alpha/2$.
Let $B^1 = B \cap [3,M_1]$,
$\mc{F}_{B^1} = \bigcup_{\ell \in B^1} \mc{F}_\ell$,
and $\beta_{B^1} := n^{-1} |\mc{F}_{B^1}| < \beta_B < \sqrt{\alpha_2}$.
We apply Lemma \ref{typ:split} with
$I = (I' \cap [3,M_1]) \cup \{1\}$, letting
$\alpha_\ell = \beta_\ell$ for all $\ell \in I' \cap [3,M_1]$
and $\alpha_1 = \eta + \beta_{B^1}$, thus decomposing $G$ into
$(2\varepsilon,t)$-typical $\alpha_i n$-regular digraphs $G_i$ on $V(G)$.
For each $\ell \in I' \cap [3,M_1]$ we decompose $G_\ell$ into
$\mc{F}_\ell$ by Case $\ell$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll 2\varepsilon \ll t^{-1} \ll M_3^{-1} \ll \alpha_2$.
For $G_1$, we first embed $\mc{F}_{B^1}$ via
Lemma \ref{typ:greedy}, leaving an $\eta n$-regular
digraph $G'_1$ that is $(\varepsilon',t)$-typical
with $\alpha_2 \ll \varepsilon' \ll t{}^{-1} \ll M_2^{-1}$.
We then conclude the proof of this case by decomposing $G'_1$
into $\mc{F}_1$ by Case $K$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll \varepsilon' \ll t{}^{-1} \ll M_1^{-1} \ll \eta$.
It remains to consider the case $\eta < \alpha/2$.
Here there are at least $\alpha n/2$ factors $F \in \mc{F}$
with at least $n/2$ vertices in cycles of length $\le M_1$,
so we can fix $\ell^* \in [M_1] \cap I'$
with $\beta_{\ell^*} > \alpha/2M_1$.
We consider two subcases according to $\beta_2 := n^{-1}|\mc{F}_2|$.
Suppose first that $\beta_2 < \alpha_1 n$.
We apply Lemma \ref{typ:split} with $I = I'$, letting
$\alpha_\ell = \beta_\ell$ for all $\ell \in I \setminus \{\ell^*\}$
and $\alpha_{\ell^*} = \beta_{\ell^*} + \beta_{B^1} + \beta_2$.
For each $\ell \in I \setminus \{\ell^*\}$ we decompose $G_\ell$
into $\mc{F}_\ell$ by Case $\ell$ of Theorem \ref{main},
where (as before) in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll 2\varepsilon \ll t^{-1} \ll M_3^{-1} \ll \alpha_2$.
For $G_{\ell^*}$ we first embed $\mc{F}_B \cup \mc{F}_2$
by Lemma \ref{typ:greedy}, leaving a $\beta_{\ell^*} n$-regular
digraph $G'_{\ell^*}$ that is $(\varepsilon',t)$-typical
with $\alpha_1 \ll \varepsilon' \ll t{}^{-1} \ll M_1^{-1}$.
We then complete the decomposition by decomposing $G'_{\ell^*}$
into $\mc{F}_{\ell^*}$ by Case $\ell^*$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll \varepsilon' \ll t{}^{-1} \ll (M_1')^{-1} \ll \beta_{\ell^*}$.
It remains to consider the subcase $\beta_2 \ge \alpha_1 n$.
We apply Lemma \ref{typ:split} with $I = I' \cup \{2\}$, letting
$\alpha_\ell = \beta_\ell$ for all $\ell \in I \setminus \{\ell^*\}$
and $\alpha_{\ell^*} = \beta_{\ell^*} + \beta_{B^1}$.
The same argument as in the first subcase applies to
decompose $G_\ell$ into $\mc{F}_\ell$
for all $\ell \in I' \setminus \{\ell^*\}$,
and also to embed $\mc{F}_B$ in $G_{\ell^*}$
by Lemma \ref{typ:greedy} and decompose
the leave $G'_{\ell^*}$ into $\mc{F}_{\ell^*}$.
We complete the proof of this case,
and so of the entire reduction, by decomposing
$G_2$ into $\mc{F}_2$ by Case $K$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll 2\varepsilon \ll t^{-1} \ll M_2^{-1} \ll \beta_2$.
\subseteqsection{Proof of Theorem \ref{main}}
We are now ready to prove our main theorem.
We are given an $(\varepsilon,t)$-typical $\alpha n$-regular digraph $G$
on $n$ vertices, where $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$,
and we need to decompose $G$ into some given
family $\mc{F}$ of $\alpha n$ oriented one-factors on $n$ vertices.
By the reductions in section \ref{sec:red},
we can assume that we are in one of the
following cases with $t^{-1} \ll M^{-1} \ll \alpha$:
Case $K$: each $F \in \mc{F}$ has at least $n/2$ vertices
in cycles of length at least $M$,
Case $\ell^*$ with $\ell^* \in [3,M-1]$: each $F \in \mc{F}$
has $\ge M^{-3} n$ cycles of length $\ell^*$.
Here the parameters of section \ref{sec:red} are renamed:
$\ell$ is now $\ell^*$ so that
`$\ell$' is free to denote generic cycle lengths;
$K$ is now $M$, as we want $K$ to take different
values in each case: we introduce $M'$ with
$t^{-1} \ll M'{}^{-1} \ll M^{-1}$ and define
\[ K = \left\{ \begin{array}{ll}
M & \text{ in Case } K, \\
M' & \text{ in Case } \ell^*.
\end{array} \right. \]
We define a parameter $L$ by $L=M$ in Case $\ell^*$
(so $\ell^* \ll L \ll K$), or as a new parameter
with $K^{-1} \ll L^{-1} \ll \alpha$ in Case $K$.
We use these parameters to apply the algorithm
of section \ref{sec:alg} as in~(\ref{hierarchy}), so we can apply the
conclusions of the lemmas in
sections \ref{sec:int} to \ref{sec:exact}.
We recall that each factor $F_w$
is partitioned as $F^1_w \cup F^2_w$,
where $F^1_w$ either consists of exactly
$L^{-3} n$ cycles of length $\ell^*$ in Case $\ell^*$,
or in Case $K$ we have $|F^1_w| - n/2 \in [0,2K]$
and $F^1_w$ consists of cycles of length $\ge K$
and at most one path of length $\ge K$
(and then $F^2_w = F_w \setminus F^1_w$).
By Lemma \ref{lem:approx},
there are arc-disjoint digraphs $G^2_w \subseteq G_2$ for $w \in W$,
where each $G^2_w$ is a copy of some valid $F'_w \subseteq F^2_w$
with $V(G^2_w) \subseteq N^-_{J_2}(w)$, such that
\begin{enumerate}
\item $G_2^- = G_2 \setminus \bigcup_{w \in W} G^2_w$
has maximum degree at most $5d^{-1/3}n$,
\item the digraph $J_2^-$
obtained from $J_2[V,W]$ by deleting all $\overlinea{xw}$
with $x \in V(G^2_w)$ has maximum degree at most $5d^{-1/3}n$,
\item any $x \in V$ has degree $1$ in $F'_w$
for at most $n/\sqrt{d}$ choices of $w$.
\end{enumerate}
(Recall that `valid' means that $F'_w$
does not have any independent arcs,
and if $F^2_w$ contains a path then $F'_w$
contains the arcs incident to each of its ends.)
Note that (ii) implies for each $w \in W$ that
$|F'_w| \ge |N^-_{J_2}(w)| - 5d^{-1/3}n
> p^2_w n - 6d^{-1/3}n$ (by Lemma \ref{deg}),
so as $p^2_w n = (1-\eta)|F^2_w| + n^{.8}$
we have $|F^2_w \setminus F'_w| < \eta n$.
Next we will embed oriented graphs
$R_w = (F^2_w \setminus F'_w) \cup L_w$ for $w \in W$,
where $L_w \subseteq F^1_w$ is defined as follows.
In Case $\ell^*$ we let each $L_w$ consist of
$2\eta L^{-3} n$ cycles of length $\ell^*$.
In Case $K$ we partition each $F^1_w$ as $\mc{P}_w \cup L_w$,
where $\mc{P}_w$ is a valid vertex-disjoint union of paths,
such that for each $[a,b] \in \mc{Y}^1_w$ we have
an oriented path $P^{ab}_w$ in $\mc{P}_w$ of length $8d(a,b)$
(which we will embed as an $ab^+$-path).
To see that such a partition exists, we apply the same
argument as at the end of the proof of Lemma \ref{lem:approx}.
We consider a greedy algorithm, where at each step that we consider
some path $P^{ab}_w$ we delete a path of length $8d(a,b)+4$
from $F^1_w$, which we allocate as $P^{ab}_w$ surrounded on both
sides of paths of length $2$ that we add to $L_w$.
As $|\mc{Y}^1_w| < n/2d_{2s+1} = (2s)^{2s}n/2d$ we thus
allocate $< (2s)^{2s}n/d$ edges to $L_w$.
Suppose for contradiction that the algorithm gets stuck,
trying to embed some path $P$ in some remainder $Q_w$.
Then all components of $Q_w$ have size $\le 8d+5$.
All components of $F^1_w$ have size $\ge K$,
so $|Q_w| \le (8d+5)|F^1_w|/K < 5dn/K$.
However, we also have
$|Q_w| \ge |F^1_w|-|Y^1_w|-|L_w| \ge \eta n/3$,
as $|F^1_w| \ge n/2$ and
$|Y^1_w| = (1-\eta)n/2 \pm 2n^{3/4}$ by Lemma \ref{lem:int}.
This is a contradiction, so the algorithm finds a partition
$F^1_w = \mc{P}_w \cup L_w$ with $\mc{P}_w$ valid.
We note that each $|R_w| < 2\eta n$.
Now we apply a greedy algorithm to construct arc-disjoint
embeddings $(\phi_w(R_w): w \in W)$ in $G_1$.
At each step we choose some $\phi_w(x) \in N^-_{J^1}(w)$
(which is disjoint from $G^2_w \subseteq N^-_{J_2}(w)$).
We require $\phi_w(x)$ to be an outneighbour of some
previously embedded $\phi_w(x_1)$ or both an outneighbour of
$\phi_w(x_1)$ and an inneighbour of $\phi_w(x_2)$
for some previously embedded images;
the latter occurs when we finish a cycle or a path
(the image under $\phi_w$ of the ends of the paths in $R_w$
have already been prescribed:
they are either images of endpoints of paths in $F'_w$
or startpoints / successors of intervals in $\mc{Y}^1_w$).
We also require $\phi_w(x)$ to be distinct from all
previously embedded $\phi_w(x_1)$ and not to lie
in the set $S$ of vertices that are already in the image
of $\phi_{w'}$ for at least $\eta^{.9} n/2$ choices of $w'$.
As $\eta^{.9} n |S|/2 \le \sum_{w \in W} |R_w| < 2\eta n^2$
we have $|S| < 4\eta^{.1} n$.
To see that it is possible to choose $\phi_w(x)$,
first note for any $v,v'$ in $V$ and $w \in W$ that
$|N_{G_1}^+(v) \cap N_{G_1}^-(v') \cap N^-_{J^1}(w)| > \alpha^2 n/3$,
by Lemma \ref{deg}.iii and typicality of $G$.
At most $|R_w|+|S| < 5\eta^{.1} n$ choices of $\phi_w(x)$
are forbidden due to using $S$ or some previously embedded
$\phi_w(x_1)$. Also, by definition of $S$, we have used
at most $\eta^{.9} n$ arcs at each of $v$ and $v'$ for
other embeddings $\phi_{w'}$, so this forbids
at most $2\eta^{.9} n$ choices of $\phi_w(x)$.
Thus the algorithm never gets stuck, so we can construct
$(\phi_w(R_w): w \in W)$ as required.
Let $G'_1 = G \setminus \bigcup_{w \in W} (G^2_w \cup R_w)$.
For each $w \in W$ let $Z_w$ be the set
of vertices of in- and outdegree $1$ in $G^2_w \cup R_w$.
We claim that $G'_1$ and $Z_w$ satisfy Setting \ref{set}.
To see this, first note that by definition of $S$ above
each $|N_{G_1}^\pm(x) \setminus N_{G'_1}^\pm(x)| < \eta^{.9} n/2$.
As $d_{G_2^-}^\pm(x) < 5d^{-1/3}n$ by (i) above and
(by Lemma~\ref{deg}) $d_G^\pm(x)-d_{G_1}^\pm(x)-d_{G_2}^\pm(x)
< (1-p_1-p_2)d_G^\pm(x) + n^{.6} < 2\eta n$
we have $|N_{G_1}^\pm(x) \bigtriangleup N_{G'_1}^\pm(x)| < \eta^{.9} n$,
so $G'_1$ is an $\eta^{.9}$-perturbation of $G_1$.
Also, as $|N^-_{J_2}(w) \setminus F'_w| \le 5d^{-1/3}n$,
$|R_w| < 2\eta n$ and $|V \setminus N^-_J(w)| < 2\eta n$
(the last by Lemma \ref{deg}) we have
$|Z_w \bigtriangleup (V \setminus N^-_{J^1}(w))| < 5\eta n$, as claimed.
In Case $\ell^*$, every vertex
has equal in- and outdegrees $0$ or $1$ in $G^2_w \cup R_w$
(it is a vertex-disjoint union of cycles) so
$d_{G'_1}^\pm(x)=|W|-|Z(x)|$ for all $x \in V$
and $\ell^*$ divides $n-|Z_w|$ for all $w \in W$.
Thus Lemma \ref{exactL} applies to partition $G'_1$
into graphs $(G^1_w: w \in W)$, where each $G^1_w$
is a $C_{\ell^*}$-factor with $V(G^1_w) = V \setminus Z_w$,
thus completing the proof of this case.
In Case $K$, a vertex $x$ has indegree (respectively outdegree)
$1$ in $G^2_w \cup R_w$
exactly when $x \in (Y^1_w)^-$
(respectively $(Y^1_w)^+$),
for which there are each $t_1$ choices of $w$,
so $d_{G'_1}^\pm(x)=|W|-t_1-|Z(x)|$ for all $x \in V$.
By construction, $Z_w$ is disjoint from $(Y^1_w)^- \cup (Y^1_w)^+$,
and the total length of paths required in the remaining path
factor problem satisfies
$8|Y^1_w| = n-|Z_w|-|(Y^1_w)^+|$ for all $w \in W$.
Thus Lemma \ref{exactK} applies to partition $G'_1$
into graphs $(G^1_w: w \in W)$,
such that each $G^1_w$ is a vertex-disjoint union of oriented paths
with $V(G^1_w) = V\setminus Z_w$,
where for each $[a,b] \in \mc{Y}^1_w$ there
is an $ab^+$-path of length $8d(a,b)$.
This completes the proof of this case,
and so of Theorem \ref{main}.
\section{Concluding remarks} \leftarrowbel{sec:con}
As mentioned in the introduction,
our solution to the generalised Oberwolfach Problem
is more general than the result of \cite{GJKKO}
in three respects: it applies to any typical graph
(theirs is for almost complete graphs)
and to any collection of two-factors
(they need some fixed $F$ to occur $\Omega(n)$ times),
and it applies also to directed graphs.
Although there are some common elements
in both of our approaches (using \cite{K2}
for the exact step and some form of twisting),
the more general nature of our result
reflects a greater flexibility in our approach
that has further applications.
One such application is our recent proof \cite{KSringel}
that every quasirandom graph with
$n$ vertices and $rn$ edges can be
decomposed into $n$ copies of any fixed
tree with $r$ edges.
The case of the complete graph solves
Ringel's tree-packing conjecture~\cite{ringel}
(solved independently via different
methods by Montgomery, Pokrovskiy
and Sudakov~\cite{MPS3}).
A natural open problem raised in \cite{GJKKO}
is whether the generalised Oberwolfach problem
can be further generalised to decompositions of $K_n$
into any family of regular graphs of bounded degree
(where the total of the degrees is $n-1$).
\end{document}
|
\begin{document}
\begin{frontmatter}
\title{A supplement on feathered gyrogroups \tnoteref{t1}}
\tnotetext[t1]{This research was supported by the National Natural Science Foundation of China (Nos. 12071199, 11661057), the Natural Science Foundation of Jiangxi Province, China (No. 20192ACBL20045).}
\author[M. Bao]{Meng Bao}
\ead{[email protected]}
\address[M. Bao]{College of Mathematics, Sichuan University, Chengdu 610064, China}
\author[X. Ling]{Xuewei Ling}
\ead{[email protected]}
\address[X. Ling]{Institute of Mathematics, Nanjing Normal University, Nanjing 210046, China}
\author[X. Xu]{Xiaoquan Xu\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author.}
\ead{[email protected]}
\address[X. Xu]{Fujian Key Laboratory of Granular Computing and Applications, Minnan Normal University, Zhangzhou 363000, China}
\begin{abstract}
A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup $G$, based on the characterization of feathered strongly topological gyrogroups, we show that if $G$ has countable $cs^{*}$-character, then it is metrizable; and it is also shown that $G$ has a compact resolution swallowing the compact sets if and only if $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space.
\end{abstract}
\begin{keyword}
Topological gyrogroups, metrizable, $\omega^{\omega}$-base, feathered
\MSC 22A22; 54A20; 20N05; 18A32.
\end{keyword}
\end{frontmatter}
\section{Introduction}
In the field of Topological Algebra, topological groups are standard researching objects and have been studied for many years, see \cite{AA}. Moreover, the combination of topology and non-associative algebra has attracted the attention of many scholars. For example, in \cite{CZ}, Cai, Lin and He introduced and investigated the concept of paratopological left Bol loops and proved some results of paratopological groups can be extended to paratopological left Bol loops. Banakh and Repov\v s \cite{Banakh} studied many generalized metric properties in rectifiable spaces and topological lops and showed that a rectifiable space $X$ is metrizable if and only if it is sequential, has
countable $cs^{*}$-character, and contains no closed copy of the Fr\'echet-Urysohn fan $S_{\omega}$. In \cite{Shen2020}, Shen introduced paratopological left-loops and showed that every weakly first-countable paratopological left-loop is first-countable. As we all known, a gyrogroup as an important type of non-associative algebra has many applications in Geometry and Physics, in particular, in the study of the $c$-ball of relativistically admissible velocities with the Einstein velocity addition, see \cite{UA1988,UA2002,UA2005,UA}. Therefore, topological gyrogroups are very important topological spaces which were posed by Atiponrat \cite{AW}. Clearly, every topological group is a topological gyrogroup and each topological gyrogroup is a rectifiable space. The readers may consult \cite{AW1,AW2020,BL,BL2,BL3,BX2022,BZX,BZX2,WAS2020,ZBX} for more details about topological gyrogroups.
In this paper, we show that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, we mainly research some weakly first-countable properties in feathered strongly topological gyrogroups. Indeed, for further study on M\"{o}bius gyrogroups, Bao and Lin posed the concept of strongly topological gyrogroups and showed that a strongly topological gyrogroup $G$ is feathered if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable. Based on the characterization of feathered strongly topological gyrogroups, they proved that each feathered strongly topological gyrogroup is paracompact. Also based on the characterization, we give some equivalent relationships of metrizability for strongly topological gyrogroups, such as every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable, and so on. It is also shown that for a feathered strongly topological gyrogroup $G$, $G$ has a compact resolution swallowing the compact sets if and only if $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space. Some problems about topological gyrogroups with countable $cs^{*}$-character are posed.
\section{Preliminary}
Throughout this paper, all topological spaces are assumed to be Hausdorff, unless otherwise is explicitly stated. Let $\mathbb{N}$ be the set of all positive integers and $\omega$ the first infinite ordinal. The readers may consult \cite{AA, E, linbook1, UA} for notation and terminology not explicitly given here. Next we recall some definitions and facts.
\begin{definition}\cite{AW}
Let $G$ be a nonempty set, and let $\oplus: G\times G\rightarrow G$ be a binary operation on $G$. Then the pair $(G, \oplus)$ is called a {\it groupoid}. A function $f$ from a groupoid $(G_{1}, \oplus_{1})$ to a groupoid $(G_{2}, \oplus_{2})$ is called a {\it groupoid homomorphism} if $f(x\oplus_{1}y)=f(x)\oplus_{2} f(y)$ for any elements $x, y\in G_{1}$. Furthermore, a bijective groupoid homomorphism from a groupoid $(G, \oplus)$ to itself will be called a {\it groupoid automorphism}. We write $\mbox{Aut}(G, \oplus)$ for the set of all automorphisms of a groupoid $(G, \oplus)$.
\end{definition}
\begin{definition}\cite{UA}
Let $(G, \oplus)$ be a groupoid. The system $(G,\oplus)$ is called a {\it gyrogroup}, if its binary operation satisfies the following conditions:
$(G1)$ There exists a unique identity element $0\in G$ such that $0\oplus a=a=a\oplus0$ for all $a\in G$.
$(G2)$ For each $x\in G$, there exists a unique inverse element $\ominus x\in G$ such that $\ominus x \oplus x=0=x\oplus (\ominus x)$.
$(G3)$ For all $x, y\in G$, there exists $\mbox{gyr}[x, y]\in \mbox{Aut}(G, \oplus)$ with the property that $x\oplus (y\oplus z)=(x\oplus y)\oplus \mbox{gyr}[x, y](z)$ for all $z\in G$.
$(G4)$ For any $x, y\in G$, $\mbox{gyr}[x\oplus y, y]=\mbox{gyr}[x, y]$.
\end{definition}
Notice that a group is a gyrogroup $(G,\oplus)$ such that $\mbox{gyr}[x,y]$ is the identity function for all $x, y\in G$. The definition of a subgyrogroup is given as follows.
\begin{definition}\cite{ST}
Let $(G,\oplus)$ be a gyrogroup. A nonempty subset $H$ of $G$ is called a {\it subgyrogroup}, denoted
by $H\leq G$, if $H$ forms a gyrogroup under the operation inherited from $G$ and the restriction of $\mbox{gyr}[a,b]$ to $H$ is an automorphism of $H$ for all $a,b\in H$.
Furthermore, a subgyrogroup $H$ of $G$ is said to be an {\it $L$-subgyrogroup}, denoted
by $H\leq_{L} G$, if $\mbox{gyr}[a, h](H)=H$ for all $a\in G$ and $h\in H$.
\end{definition}
\begin{definition}\cite{AW}
A triple $(G, \tau, \oplus)$ is called a {\it topological gyrogroup} if the following statements hold:
(1) $(G, \tau)$ is a topological space.
(2) $(G, \oplus)$ is a gyrogroup.
(3) The binary operation $\oplus: G\times G\rightarrow G$ is jointly continuous while $G\times G$ is endowed with the product topology, and the operation of taking the inverse $\ominus (\cdot): G\rightarrow G$, i.e. $x\rightarrow \ominus x$, is also continuous.
\end{definition}
Obviously, every topological group is a topological gyrogroup. However, every topological gyrogroup whose gyrations are not identically equal to the identity is not a topological group.
\begin{example}\cite[Example 3]{AW}
The Einstein gyrogroup with the standard topology is a topological gyrogroup but not a topological group.
\end{example}
Then we recall some weakly first-countable concepts which are important in the following researches.
\begin{definition}\cite{BT,GK,LPT}
A point $x$ of a topological space $X$ is said to have a {\it neighborhood $\omega^{\omega}$-base} or a {\it local $\mathfrak{G}$-base} if there exists a base of neighborhoods at $x$ of the form $\{U_{\alpha}(x):\alpha \in \mathbb{N}^{\mathbb{N}}\}$ such that $U_{\beta}(x)\subset U_{\alpha}(x)$ for all elements $\alpha \leq \beta$ in $\mathbb{N}^{\mathbb{N}}$, where $\mathbb{N}^{\mathbb{N}}$ consisting of all functions from $\mathbb{N}$ to $\mathbb{N}$ is endowed with the natural partial order, ie., $f\leq g$ if and only if $f(n)\leq g(n)$ for all $n\in \mathbb{N}$. The space $X$ is said to have an {\it $\omega^{\omega}$-base} or a {\it $\mathfrak{G}$-base} if it has a neighborhood $\omega^{\omega}$-base or a local $\mathfrak{G}$-base at every point $x\in X$.
\end{definition}
\begin{definition}
Let $X$ be a topological space.
$(1)$\, $X$ is called a {\it sequential space} \cite{FS} if for each non-closed subset $A\subseteq X$, there are a point $x\in X\setminus A$ and a sequence in $A$ converging to $x$ in $X$.
$(2)$\, $X$ is called a {\it Fr\'{e}chet-Urysohn space} \cite{FS} if for any subset $A\subseteq X$ and $x\in \overline{A}$, there is a sequence in $A$ converging to $x$ in $X$.
$(3)$\, $X$ is called an {\it $\alpha_{7}$-space} \cite{BZ}, if for every point $x\in X$ and each sheaf $\{S_{n}:n\in\omega\}$ with the vertex $x$, there exists a sequence converging to some point $y\in X$ which meets infinitely many sequences $S_{n}$.
\end{definition}
A family $\mathcal{N}$ of subsets of a topological space $X$ is called a {\it $cs^{*}$-network at a point $x\in X$} \cite{GMZ} if for each sequence $(x_{n})_{n\in \mathbb{N}}$ in $X$ convergent to $x$ and for each neighborhood $O_{x}$ of $x$ there is a set $N\in \mathcal{N}$ such that $x\in N\subset O_{x}$ and the set $\{n\in \mathbb{N}:x_{n}\in N\}$ is infinite.
Then we give the concept of $cs^{*}$-character of a topological gyrogroup.
\begin{definition}\cite[Theorem 3.7]{BZX2}
Let $G$ be a topological gyrogroup, the $cs^{*}$-character of $G$ is the least cardinality of $cs^{*}$-network at the identity element $0$ of $G$.
\end{definition}
\section{Weakly first-countable properties of topological gyrogroups}
In this section, it is shown that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup $G$, based on the characterization of feathered strongly topological gyrogroups, that is, a strongly topological gyrogroup is feathered if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable, we give some equivalent relationships of metrizability for strongly topological gyrogroups, such as every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\begin{definition}\cite[Definition 3.1]{GKL}
A family $\{K_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ of compact sets of a topological space $X$ is called a {\it compact resolution} if $X=\bigcup \{K_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ and $K_{\alpha}\subseteq K_{\beta}$ for all $\alpha \leq \beta$. In additionally, every compact set in $X$ is contained in some $K_{\alpha}$, we say that $\{K_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ {\it swallows the compact sets} of $X$.
\end{definition}
It is well-known that each Polish space $X$ has a compact resolution swallowing the compact sets of $X$. Moreover, it was proved in \cite[Theorem 3.3]{CRJP} that if $X$ is a metrizable topological space, then $X$ is a Polish space if and only if $X$ has a compact resolution swallowing the compact sets of $X$. Then, it follows from \cite[Proposition 3.3]{TVV} that each hemicompact topological space has a compact resolution swallowing the compact sets and the property of having a compact resolution swallowing the compact sets is closed-hereditary and is closed under countable products.
\begin{theorem}\label{3compact}
Let $G$ be a topological gyrogroup which has an $\omega^{\omega}$-base, $K$ an arbitrary compact subset of $G$. Then $K$ is metrizable.
\end{theorem}
\begin{proof}
By the hypothesis, let $\{U_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ be an open $\omega^{\omega}$-base in $G$. Without loss of generality, we assume that all sets $U_{\alpha}$ are symmetric. By \cite[Theorem 1]{CBOJ}, a compact space $K$ is metrizable if and only if $(K\times K)\setminus \Delta$ has a compact resolution swallowing its compact sets, where $\Delta =\{(x,x):x\in K\}$. Therefore, it suffices to show that the set $W=(K\times K)\setminus \Delta$ has a compact resolution which swallows its compact sets.
For each $\alpha\in \mathbb{N}^{\mathbb{N}}$, set $W_{\alpha}=\{(x,y)\in W,x\oplus (\ominus y)\not\in U_{\alpha}\}$. Then $W_{\alpha}$ is closed in $K\times K$, and hence it is compact for each $\alpha\in \mathbb{N}^{\mathbb{N}}$.
For each compact subset $C$ of $W$, $q(C)=\{x\oplus (\ominus y):(x,y)\in C\}$ is compact and does not contain the identity element $0$ of $G$. Since $\{U_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a local base at $0$, for some $\alpha\in \mathbb{N}^{\mathbb{N}}$, we obtain $U_{\alpha}\cap q(C)=\emptyset$. Then $C\subseteq W_{\alpha}$. Thus, $\{W_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution swallowing the compact sets in $W$. We conclude that $K$ is metrizable.
\end{proof}
In \cite[Theorem 1.1]{Banakh}, Banakh showed that each non-metrizable sequential rectifiable space $X$ of countable $cs^{*}$-character contains a clopen rectifiable submetrizable $k_{\omega}$-subspace. Indeed, during the process of proof, it is not difficult to see that if $X$ is a non-metrizable sequential topological gyrogroup which has countable $cs^{*}$-character, then it contains an open and closed subgyrogroup which is a submetrizable $k_{\omega}$-space. For a topological space $X$, Chasco, Mart\'{i}n and Tarieladze in \cite[Lemma 1.5]{CMT} showed that if $X$ is sequential, then it is a $k$-space and if $X$ is a Hausdorff $k$-space and its compact subsets are sequential (in particular first countable or metrizable), then $X$ is sequential. Furthermore, it was proved in \cite[Theorem 3.8]{BZX2} that if a topological gyrogroup $G$ has an $\omega^{\omega}$-base, then it has countable $cs^{*}$-character. Therefore, by Theorem \ref{3compact} and these results, we obtain:
\begin{corollary}\label{k-sequential}
If a topological gyrogroup $G$ has an $\omega^{\omega}$-base, then the following conditions are equivalent.
\begin{enumerate}
\item $G$ is a $k$-space;
\item $G$ is sequential;
\item $G$ is metrizable or contains an open submetrizable $k_{\omega}$-subgyrogroup.
\end{enumerate}
\end{corollary}
In \cite[Theorem 3.5]{ZBX}, the authors showed that if $G$ is a sequential topological gyrogroup with an $\omega^{\omega}$-base, then $G$ has the strong Pytkeev property. Therefore, Corollary \ref{k-sequential} poses the following result directly.
\begin{theorem}\label{k-Pytkeev}
Let $G$ be a topological gyrogroup with an $\omega^{\omega}$-base. If $G$ is a $k$-space, then $G$ has the strong Pytkeev property.
\end{theorem}
Since for a Baire topological gyrogroup $G$, $G$ is metrizable if and only if it has the strong Pytkeev property, see \cite[Theorem 3.10]{ZBX}, Theorem \ref{k-Pytkeev} provides the following corollary.
\begin{corollary}
Let $G$ be a Baire topological gyrogroup. Then $G$ is metrizable if and only if $G$ is a $k$-space and has an $\omega^{\omega}$-base.
\end{corollary}
Theorem \ref{k-Pytkeev} also provides the other type of proof about the following result.
\begin{corollary}\cite[Corollary 3.6]{BZX2}
A topological gyrogroup $G$ is metrizable if and only if $G$ has an $\omega^{\omega}$-base and $G$ is also Fr\'echet-Urysohn.
\end{corollary}
\begin{proof}
The necessity is trivial, it suffices to prove the sufficiency.
Suppose that $G$ is a Fr\'echet-Urysohn topological gyrogroup and has an $\omega^{\omega}$-base. By Theorem \ref{k-Pytkeev}, $G$ has the strong Pytkeev property, hence has countable $cs^{*}$-character. It follows from \cite[Corollary 3.6]{BX2022} that every Fr\'echet-Urysohn topological gyrogroups with countable $cs^{*}$-character is metrizable.
\end{proof}
\begin{remark} A topological gyrogroup $G$ with an $\omega^{\omega}$-base has countable $cs^{*}$-character, see \cite[Theorem 3.8]{BZX2}, hence it is natural to consider the following question. If a topological gyrogroup $G$ is of countable $cs^{*}$-character and it is a $k$-space, then is $G$ sequential? Indeed, Shen in \cite[Example 4.5]{Shen2014} showed that there is a non-metrizable $snf$-countable topological group $X$ which is a $k$-space. Clearly, $X$ is not sequential. Furthermore, this example gives a negative answer to the question whether the $k$-property and sequentiality are equivalent for topological groups with countable $cs^{*}$-character posed in \cite{GKL} under Corollary 3.13.
\end{remark}
Then, we introduce the concept of strongly topological gyrogroups, which was first posed by Bao and Lin in \cite{BL}, and we investigate some weakly first-countable properties in feathered strongly topological gyrogroups.
\begin{definition}{\rm (\cite{BL})}\label{d11}
Let $G$ be a topological gyrogroup. We say that $G$ is a {\it strongly topological gyrogroup} if there exists a neighborhood base $\mathscr U$ of $0$ such that, for every $U\in \mathscr U$, $\mbox{gyr}[x, y](U)=U$ for any $x, y\in G$. For convenience, we say that $G$ is a strongly topological gyrogroup with neighborhood base $\mathscr U$ of $0$.
\end{definition}
For each $U\in \mathscr U$, we can set $V=U\cup (\ominus U)$. Then, $$\mbox{gyr}[x,y](V)=\mbox{gyr}[x, y](U\cup (\ominus U))=\mbox{gyr}[x, y](U)\cup (\ominus \mbox{gyr}[x, y](U))=U\cup (\ominus U)=V,$$ for all $x, y\in G$. Obviously, the family $\{U\cup(\ominus U): U\in \mathscr U\}$ is also a neighborhood base of $0$. Therefore, we may assume that $U$ is symmetric for each $U\in\mathscr U$ in Definition~\ref{d11}. Moreover, in the classical M\"{o}bius, Einstein, or Proper Velocity gyrogroups, we know that gyrations are indeed special rotations, however for an arbitrary gyrogroup, gyrations belong to the automorphism group of $G$ and need not be necessarily rotations.
In \cite{BL}, the authors proved that there is a strongly topological gyrogroup which is not a topological group.
\begin{example}\cite[Example 3.1]{BL}
The M\"{o}bius gyrogroup with the standard topology is a strongly topological gyrogroup but not a topological group.
\end{example}
A topological gyrogroup $G$ is {\it feathered} if it contains a non-empty compact set $K$ of countable character in $G$. It was proved in \cite[3.1 E(b) and 3.3 H(a)]{E} that every locally compact topological gyrogroup is feathered. Moreover, by \cite[Theorem 3.14]{BL}, we know that a strongly topological gyrogroup $G$ is feathered if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable.
\begin{theorem}\label{feathered}
Let $G$ be a feathered strongly topological gyrogroup. Then $G$ has an $\omega^{\omega}$-base if and only if $G$ is metrizable.
\end{theorem}
\begin{proof}
Suppose that $G$ is a strongly topological gyrogroup and has an $\omega^{\omega}$-base. Then $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable. It follows from Theorem \ref{3compact} that the subgyrogroup $H$ is metrizable. Since each compact subset of a Hausdorff space is closed, it is clear that $H$ is a closed $L$-subgyrogroup of $G$. Then, by \cite[Corollary 4.3]{BZX}, if $G$ is a topological gyrogroup and $H$ is a closed $L$-subgyrogroup of $G$ and if the spaces $H$ and $G/H$ are metrizable, then the space $G$ is also metrizable. Therefore, we obtain that $G$ is a metrizable space.
\end{proof}
\begin{corollary}
Let $G$ be a locally compact strongly topological gyrogroup. Then $G$ has an $\omega^{\omega}$-base if and only if $G$ is metrizable.
\end{corollary}
Since every topological gyrogroup with an $\omega^{\omega}$-base has countable $cs^{*}$-character, it is natural to pose the following question.
\begin{question}\label{question-cs}
Let $G$ be a feathered strongly topological gyrogroup with countable $cs^{*}$-character. Is $G$ metrizable?
\end{question}
Then, we give a affirmative answer to Question \ref{question-cs}. We note that Uspenski\v\i \cite{UVV, UVV89} proved that compact rectifiable spaces are dyadic. Since every topological gyrogroup is a rectifiable space, it is trivial that each compact topological gyrogroup is dyadic. Moreover, Banakh and Zdomskyy in \cite[Proposition 7]{BZ} claimed that a dyadic compactum is metrizable if and only if it has countable $cs^{*}$-character.
\begin{proposition}
Every compact topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{proposition}
\begin{theorem}\label{csf-feath}
Every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{theorem}
\begin{proof}
Since $G$ is a feathered strongly topological gyrogroup, there exists a compact $L$-subgyrogroup $H$ of $G$ such that the quotient space $G/H$ is metrizable. By the hypothesis, $G$ has countable $cs^{*}$-character, then $H$ also has countable $cs^{*}$-character, which deduces that the compact subgyrogroup $H$ with countable $cs^{*}$-character is metrizable, and it follows from \cite[Corollary 4.3]{BZX} that $G$ is metrizable.
\end{proof}
\begin{corollary}
Every locally compact strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{corollary}
\begin{theorem}
Let $G$ be a strongly topological gyrogroup. Then the following conditions are equivalent:
\begin{enumerate}
\item $G$ is metrizable;
\item $G$ is Fr\'echet-Urysohn and has countable $cs^{*}$-character;
\item $G$ is Fr\'echet-Urysohn and has an $\omega^{\omega}$-base;
\item $G$ is feathered and has countable $cs^{*}$-character;
\item $G$ is feathered and has an $\omega^{\omega}$-base.
\end{enumerate}
\end{theorem}
In the research of rectifiable spaces, Banakh and Repov\v s \cite[Lemma 5.1]{Banakh} showed that suppose that $G$ is a topological lop and $F\subseteq G$ is a subset containing the unit $e$ of $G$, then put $F_{1}=F$ and $F_{n+1}=F_{n}^{-1}F_{n}$ for $n\in \mathbb{N}$. If $F$ is a sequential space containing no closed topological copy of the Fr\'echet-Urysohn fan $S_{\omega}$ and each space $F_{n}$, $n\in \mathbb{N}$, has a countable $cs^{*}$-network at $e$, then $F$ has a countable $sb$-network at $e$; if $F$ is sequential and each space $F_{n},n\in \mathbb{N}$, has countable $sb$-network at $e$, then $F$ is first-countable at $e$. Then, Shen \cite[Proposition 2.6 and Theorem 2.7]{Shen2020} showed that every paratopological left-loop with $sb$-network is $sof$-countable and a sequential, regular paratopological left-loop $G$ with countable $cs^{*}$-network is first-countable if and only if $G$ contains no closed copy of $S_{\omega}$. These results in both of two articles can obtain the following results immediately.
\begin{proposition}\label{csf-snf}
If $G$ is a sequential topological gyrogroup with countable $cs^{*}$-network containing no closed copy of $S_{\omega}$, then $G$ has countable $sb$-network.
\end{proposition}
\begin{proposition}\label{snf-sof}
Every topological gyrogroup with countable $sb$-network is $sof$-countable.
\end{proposition}
\begin{theorem}
A strongly topological gyrogroup $G$ is metrizable if and only if $G$ is a $k$-space of countable pseudocharacter with countable $sb$-network.
\end{theorem}
\begin{proof}
The necessity is trivial, it suffices to claim the sufficiency.
Let a strongly topological gyrogroup $G$ be a $k$-space of countable pseudocharacter with countable $sb$-network. It follows from \cite[Theorem 4.3]{BL1} that every strongly topological gyrogroup with countable pseudocharacter is submetrizable. We obtain that every compact subset of $G$ is metrizable. Since $G$ is a $k$-space, it is easy to see that $G$ is sequential. Indeed, a space $X$ is first-countable if and only if $X$ is sequential and $sof$-countable, which deduces that $G$ is first-countable by Proposition \ref{snf-sof}, hence $G$ is metrizable.
\end{proof}
Recall that a continuous mapping $q:G\rightarrow H$ is called {\it compact-covering} if for every compact subset $K$ of $H$ there exists a compact subset $C$ of $G$ such that $q(C)=K$. Indeed, it was claimed in \cite[Theorem 3.8]{BL} that if $G$ is a topological gyrogroup and $H$ is a compact L-subgyrogroup of $G$, then the quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is perfect. Furthermore, if $f:X\rightarrow Y$ is a perfect mapping, then for every compact subspace $Z\subseteq Y$, the inverse image $f^{-1}(Z)$ is compact by \cite[Theorem 3.7.2]{E}. Therefore, if $H$ is a compact $L$-subgyrogroup of a topological gyrogroup $G$, then the quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is a compact covering mapping.
\begin{theorem}\label{Polish}
If $G$ is a feathered strongly topological gyrogroup, the followings are equivalent.
\begin{enumerate}
\item $G$ has a compact resolution swallowing the compact sets of $G$;
\item $G$ has a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space.
\end{enumerate}
If (1) holds, then $G$ is $\check{C}$ech-complete.
\end{theorem}
\begin{proof}
(1)$\Rightarrow$(2). Let $\pi$ be the natural homomorphism from $G$ to its quotient topology on $G/H$. Since $G$ is a feathered strongly topological gyrogroup, it follows from \cite[Lemma 3.14]{BL} that $G$ contains a compact $L$-subgyrogroup $H$ such that $G/H$ is metrizable. Let $G$ have a compact resolution swallowing the compact sets of $G$, say $\mathcal{K}=\{K_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$. Then put $\mathcal{K}'=\{\pi (K_{\alpha}):\alpha\in \mathbb{N}^{\mathbb{N}}\}$. Then $\mathcal{K}'$ swallows the compact subsets of $G/H$. Indeed, if $K'$ is compact in $G/H$, then $\pi^{-1}(K')$ is compact in $G$. So there exists $\alpha\in \mathbb{N}^{\mathbb{N}}$ such that $\pi^{-1}(K')\subseteq K_{\alpha}$ and hence $K'\subseteq \pi (K_{\alpha})$. We know that $G/H$ is a Polish space by \cite[Theorem 3.3]{CRJP}.
(2)$\Rightarrow$(1). Since the space $G/H$ is a Polish space, it follows from \cite[Theorem 3.3]{CRJP} that $G/H$ has a compact resolution swallowing the compact sets of $G/H$, say $\mathcal{K}'=\{K'_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$. For every $\alpha\in \mathbb{N}^{\mathbb{N}}$, put $K_{\alpha}=\pi^{-1}(K'_{\alpha})$. Then $K_{\alpha}$ is a compact subset of $G$. Hence, $\mathcal{K}=\{K_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution. Let $C$ be a compact subset of $G$. Then there exists $\alpha\in \mathbb{N}^{\mathbb{N}}$ such that $\pi (C)\subseteq K'_{\alpha}$. Therefore, $C\subseteq K_{\alpha}$, and hence $\mathcal{K}$ swallows the compact sets of $G$. We conclude that $G$ has a compact resolution swallowing the compact sets of $G$.
By \cite[Theorem 3.17]{BL}, a strongly topological gyrogroup $G$ is $\check{C}$ech-complete if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable by a complete metric. Since each Polish space is a complete metric space, we know that $G$ is $\check{C}$ech-complete.
\end{proof}
\begin{proposition}
Let $G$ be a topological gyrogroup and have a compact resolution swallowing the compact sets of $G$. If $q:G\rightarrow H$ is a quotient compact-covering map, then $H$ also has a compact resolution swallowing the compact sets of $H$.
\end{proposition}
\begin{proof}
Indeed, the result is trivial. If $\{K_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution swallowing the compact sets of $G$, it is clear that $\{q(K_{\alpha}):\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution swallowing the compact sets of $H$.
\end{proof}
\begin{proposition}
Let $G$ be a topological gyrogroup and a $k$-space. If $G$ has an $\omega^{\omega}$-base and also has a compact resolution swallowing the compact sets of $G$, then $G$ is either a Polish space or contains a submetrizable $k_{\omega}$-subgyrogroup.
\end{proposition}
\begin{proof}
Since topological gyrogroup $G$ is a $k$-space and has an $\omega^{\omega}$-base, we have that $G$ is metrizable or contains an open submetrizable $k_{\omega}$-subgyrogroup. Moreover, by \cite[Theorem 3.3]{CRJP}, in a metrizable space $X$, $X$ is a Polish space if and only if $X$ has a compact resolution swallowing the compact sets of $X$. Therefore, if $G$ is metrizable, we know that $G$ is a Polish space.
\end{proof}
In Theorems \ref{feathered}, \ref{csf-feath} and \ref{Polish}, it is clear that the characterization of feathered strongly topological gyrogroup playes an important role in the proof. However, we do not know whether the characterization of feathered holds in topological gyrogroups. If it holds in topological gyrogroups, all of Theorems \ref{feathered}, \ref{csf-feath} and \ref{Polish} can be extended to topological gyrogroups immediately.
\begin{question}
If $G$ is a feathered topological gyrogroup, is there a compact $L$-subgyrogroup of $G$ such that the quotient space $G/H$ metrizable?
\end{question}
A space $X$ is called {\it hemicompact} if $X=\bigcup_{n\in \mathbb{N}}X_{n}$, where $X_{n}$ is compact for any $n\in \mathbb{N}$ and for any compact $K\subseteq X$, there is $n\in \mathbb{N}$ such that $K\subseteq X_{n}$.
\begin{corollary}
Let $G$ be a locally compact strongly topological gyrogroup. Then $G$ has a compact resolution swallowing the compact sets of $G$ if and only if $G$ is hemicompact space.
\end{corollary}
\begin{proof}
Since each hemicompact topological space has a compact resolution swallowing the compact sets, it suffices to claim the necessity.
Suppose that $G$ has a compact resolution swallowing the compact sets of $G$. Since each locally compact topological gyrogroup is feathered, by Theorem \ref{Polish}, $G$ contains a compact $L$-subgyrogroup $H$ such that the locally compact space $G/H$ is second countable. Therefore, $G/H$ is hemicompact, and $G$ is also hemicompact.
\end{proof}
\begin{proposition}
Every Fr\'echet-Urysohn hemicompact topological gyrogroup is locally compact.
\end{proposition}
\begin{proof}
Let $G=\bigcup_{n\in \mathbb{N}}K_{n}$, where $\{K_{n}\}_{n}$ is an increasing sequence of compact subsets of $K$ containing the identity element $0$ such that every compact set in $G$ is contained in some $K_{n}$. Then we can find $n\in \mathbb{N}$ such that $K_{n}$ is a neighborhood of $0$. Suppose on the contrary that there is no $n$ such that $K_{n}$ is a neighborhood of $0$ for each $n\in \mathbb{N}$. Then for each $n\in \mathbb{N}$ and each neighborhood $U$ of $0$, there exists $x_{U,n}\in U\setminus K_{n}$. For each $n\in \mathbb{N}$, set $B_{n}=\{x_{U,n}:U\mbox{ is an open neighborhood of 0}\}$. Then $0\in \overline{B_{n}}$. Since $G$ is Fr\'echet-Urysohn, for each $n\in \mathbb{N}$, we can find an open neighborhood sequence $\{U_{n}(k)\}_{k}$ of $0$ such that $x_{U_{n}(k),n}\rightarrow 0$ at $k\rightarrow \infty$. Since every Fr\'echet-Urysohn topological gyrogroup is a strong $\alpha_{4}$-space by \cite[Lemma 3.3]{BZX2}, there exists strictly increasing sequences $(k_{p})_{p}$ and $(n_{p})_{p}$ such that $x_{U_{n_{p}}(k_{p}),n_{p}}\rightarrow 0$ at $p\rightarrow \infty$. Since the set $B=\{x_{U_{n_{p}}(k_{p}),n_{p}}:p\in \mathbb{N}\}\cup \{0\}$ is compact in $G$, we can find $m\in \mathbb{N}$ such that $B\subseteq K_{m}$, which is a contradiction . Therefore, $G$ is locally compact.
\end{proof}
\section{metrizability of strongly topological gyrogroups}
In \cite{AW}, Atiponrat posed a question that is it true that the first-countability axiom implies that $G$ is metrizable for a topological gyrogroup $G$? Then Cai, Lin and He showed that every topological gyrogroup is a rectifiable space, which deduces that every first-countable (strongly) topological gyrogroup is metrizable. Indeed, Alexandra S. Gul'ko \cite[Theorem 3.2]{Alexan} proved that every first-countable $T_{0}$ rectifiable space is metrizable by the tool of strong development. Here, we give a direct construction to show that every first-countable strongly topological gyrogroup is metrizable.
\begin{lemma}\label{3.3.10}\cite[Lemma 3.12]{BL}
Let $G$ be a strongly topological gyrogroup with the symmetric neighborhood base $\mathscr{U}$ at $0$, and let $\{U_{n}: n\in\mathbb{N}\}$ and $\{V(m/2^{n}): n, m\in\mathbb{N}\}$ be two sequences of open neighborhoods satisfying the following conditions (1)-(5):
(1) $U_{n}\in\mathscr{U}$ for each $n\in \mathbb{N}$.
(2) $U_{n+1}\oplus U_{n+1}\subseteq U_{n}$, for each $n\in \mathbb{N}$.
(3) $V(1)=U_{0}$;
(4) For any $n\geq 1$, put $$V(1/2^{n})=U_{n}, V(2m/2^{n})=V(m/2^{n-1})$$ for $m=1,...,2^{n-1}$, and $$V((2m+1)/2^{n})=U_{n}\oplus V(m/2^{n-1})=V(1/2^{n})\oplus V(m/2^{n-1})$$ for each $m=1,...,2^{n-1}-1$;
(5) $V(m/2^{n})=G$ when $m>2^{n}$;
Then there exists a prenorm $N$ on $G$ that satisfies the following conditions:
(a) for any fixed $x, y\in G$, we have $N(\mbox{gyr}[x,y](z))=N(z)$ for any $z\in G$;
(b) for any $n\in \mathbb{N}$, $$\{x\in G: N(x)<1/2^{n}\}\subseteq U_{n}\subseteq\{x\in G: N(x)\leq 2/2^{n}\}.$$
\end{lemma}
\begin{theorem}\label{new-metric}
Every first-countable strongly topological gyrogroup is metrizable.
\end{theorem}
\begin{proof}
Let $G$ be a strongly topological gyrogroup with a symmetric neighborhood base $\mathscr U$. Since $G$ is first-countable, put $\{W_{n}:n\in \mathbb{N}\}$ a countable base at the identity element $0$. By induction, we obtain a sequence $\{U_{n}:n\in \mathbb{N}\}\subseteq \mathscr{U}$ such that $U_{n}\subseteq W_{n}$ and $U_{n+1}\oplus U_{n+1}\subseteq U_{n}$, for each $n\in \mathbb{N}$. It is easy to see that $\{U_{n}:n\in \mathbb{N}\}$ is also a base of $G$ at $0$. By Lemma \ref{3.3.10}, there exists a continuous prenorm $N$ on $G$ which satisfies $$N(\mbox{gyr}[x, y](z))=N(z)$$ for any $x, y, z\in G$ and $$\{x\in G: N(x)<1/2^{n}\}\subseteq U_{n}\subseteq \{x\in G: N(x)\leq 2/2^{n}\},$$ for each integer $n\geq 0$. Put $B_{N}(\varepsilon)=\{x\in G:N(x)<\varepsilon\}$, where $\varepsilon$ is a positive number. It is easy to see that $B_{N}(1/2^{n})$ also forms a base of $G$ at $0$.
Now, for arbitrary $x$ and $y$ in $G$, put $\varrho _{N}(x, y)=N(\ominus x\oplus y)$. Let us show that $\varrho _{N}$ is a metric on $G$.
(1) Clearly, $\varrho _{N}(x, y)=N(\ominus x\oplus y)\geq 0$, for every $x, y\in G$. At the same time, $\varrho _{N}(x, x)=N(0)=0$, for each $x\in G$. Assume that $$\varrho _{N}(x, y)=N(\ominus x\oplus y)=0.$$ Then, for each $n\in\mathbb{N}$, $$\ominus x\oplus y\in \{x\in G: N(x)<1/2^{n}\}\subseteq U_{n}.$$ Since $\{0\}=\bigcap _{n\in \mathbb{N}}U_{n}$, it follows that $\ominus x\oplus y=0$, that is, $x=y$.
(2) For every $x, y\in G$, $\varrho _{N}(y, x)=N(\ominus y\oplus x)=N(gyr[\ominus y,x](\ominus x\oplus y))=N(\ominus x\oplus y)=\varrho _{N}(x, y)$.
(3) For every $x, y, z\in G$, it follows from \cite[Theorem 2.11]{UA2005} that
\begin{eqnarray}
\varrho _{N}(x, y)&=&N(\ominus x\oplus y)\nonumber\\
&=&N((\ominus x\oplus z)\oplus \mbox{gyr}[\ominus x, z](\ominus z\oplus y))\nonumber\\
&\leq&N(\ominus x\oplus z)+N(\mbox{gyr}[\ominus x, z](\ominus z\oplus y))\nonumber\\
&=&N(\ominus x\oplus z)+N(\ominus z\oplus y)\nonumber\\
&=&\varrho _{N}(x, z)+\varrho _{N}(z, y)\nonumber
\end{eqnarray}
Thus, $\varrho _{N}$ is a metric on $G$.
Since $B_{N}(1/2^{n})$ forms a base of $G$ at $0$ and $G$ is homogeneous, for each $x\in G$, $B_{N}(1/2^{n})\oplus x$ constitutes a base of $G$ at $x$. Therefore, it is easy to see that the topology generated by metric $\varrho _{N}$ is coincide with the original topology of $G$. Hence, $G$ is metrizable.
\end{proof}
However, we do not know whether each topological gyrogroup has the similar result like Lemma \ref{3.3.10}, therefore, we can not give the direct construction that every first-countable topological gyrogroup is metrizable. Moreover, we pose the following questions.
\begin{question}
Let $G$ be a metrizable (strongly) topological gyrogroup and $H$ a closed $L$-subgyrogroup of $G$. Is the quotient space $G/H$ metrizable?
\end{question}
\begin{remark}
It was posed a question in \cite{BL} that if $G$ is a (strongly) topological gyrogroup with a countable pseudocharacter, is $G$ submetrizable? then Bao and Lin gave an affirmative answer to this question in \cite[Theorem 4.3]{BL1} by constructing a metric $\varrho _{N}(x, y)=N(\ominus x\oplus y)+N(\ominus y\oplus x)$ when $G$ is a strongly topological gyrogroup. Notice that the proof of Theorem \ref{new-metric} can be applied to show that every strongly topological gyrogroup with a countable pseudocharacter is submetrizable, that is, the metric in \cite[Theorem 4.3]{BL1} can be replaced by $\varrho _{N}(x, y)=N(\ominus x\oplus y)$.
\end{remark}
A subgyrogroup $H$ of a topological gyrogroup $G$ is called {\it inner (outer) neutral} if for every open neighborhood $U$ of $0$ in $G$, there exists an open neighborhood $V$ of $0$ such that $H\oplus V\subseteq U\oplus H$ ($V\oplus H\subseteq H\oplus U$).
\begin{question}
Let $G$ be a (strongly) topological gyrogroup and $H$ a closed inner neutral $L$-subgyrogroup of $G$. If the quotient space $G/H$ is first-countable, is it metrizable?
\end{question}
\begin{question}
Let $G$ be a feathered (strongly) topological gyrogroup and $H$ a closed $L$-subgyrogroup of $G$. If the quotient space $G/H$ is first-countable, is it metrizable?
\end{question}
\end{document}
|
\begin{equation}gin{document}
\title{The power of symmetric extensions for
entanglement detection}
\author{Miguel Navascu\'es, $^{1}$
Masaki Owari,$^{1,2}$
Martin B. Plenio,$^{1,2}$}
\address{$^1${\it Institute for Mathematical Sciences, 53 Prince's Gate,
Imperial College London, London SW7 2PG, UK}\\
$^2${\it QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK }}
\begin{equation}gin{abstract}
In this paper, we present new progress on the study of the symmetric extension criterion for separability.
First, we show that a perturbation of order $O(1/N)$ is sufficient and, in general, necessary to destroy the entanglement of
any state admitting an $N$ Bose symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from $N$ Bose symmetric extensions with Positive Partial Transpose (PPT) decreases at least as fast as $O(1/N^2)$. From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT symmetric extensions.
Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) $N$-extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.
\end{abstract}
\maketitle
\section{Introduction}
\label{intro}
The separability problem, that is, the problem to determine whether a given quantum state is separable or entangled,
is one of the most fundamental problems in Entanglement Theory \cite{intro_measures}. Starting from the famous PPT (Positive Partial Transpose) criterion \cite{P96}, nowadays we have an enormous number of
different separability criteria to choose from (see the citation lists
of review papers in this topic \cite{intro_measures,B02,T02,SSLS05,I07,HHHH07,GT08}).
Among all known separability criteria, those based on ``symmetric extensions'' and ``PPT symmetric extensions'' (i.e., symmetric extensions with an additional PPT constraint), as conceived by Doherty et al. \cite{doherty,doherty2},
are considered to be among the most powerful \cite{I07}.
These criteria rely on the fact that any
set of $N$-symmetrically extendable states (PPT or not) converges
to a set of separable states in the limit of $N {\mbox r}ightarrow \infty$, as first noticed by Raggio and Werner \cite{raggio,werner}, although it also follows from the Quantum de Finetti theorem \cite{CFS02}.
Since both the set of $N$-symmetrically extendable states
and the set of $N$-PPT symmetrically extendable states can be characterized by Semidefinite Programming \cite{sdp},
a well-known optimization problem for which many free solvers are available (like the MATLAB toolbox SeDuMi\cite{sedumi}),
these tests are not only powerful, but also easy to implement.
This explains why, over all known numerical methods, the algorithms created by Doherty, Parrilo and Spedalieri (DPS) are the most popular in the Quantum Information community (notice, however, that there exist other methods for entanglement detection based on Semidefinite Programming besides the DPS criterion \cite{relaxations,brandao}).
This family of schemes has, though, an important drawback:
in this approach, in order to conclude that a given state ${\mbox r}ho$ is entangled,
it is enough to find an $N$ such that ${\mbox r}ho$ does not belong to the set of $N$-(PPT) symmetric extendable states.
On the other hand, in order to show that a given state is separable,
we would have to prove that it admits an $N$-(PPT) symmetric extension {\it for all natural numbers $N$}.
The DPS method then becomes useless: since we always operate under finite time and memory constraints, all we can do in practice is to check for the existence of $N$-(PPT) symmetric extensions for $N$ less or equal than some finite number $N_0$. If the state ${\mbox r}ho$ under analysis happened to admit an $N_0$ (PPT) symmetric extension, we could thus not conclude anything about its separability.
Hulke and Bruss \cite{florian} tried to solve the issue by providing a complementary criterion designed to detect separability instead of entanglement, to be implemented at the same time as the DPS criterion. Unfortunately, the time complexity of that other method scales superexponentially with the dimension of the subsystems involved \cite{I07}. The reduced speed of convergence of the resulting two-way algorithm (much smaller than that of the DPS criterion) thus makes it unsuitable to study quantum correlations in high dimensional systems.
Besides, there is a more elegant way to approach the problem.
In a recent work, Ioannou observed that, even if a state happens to have an $N_0$-(PPT) symmetric extension,
we can at least bound the distance between such state and the set of separable states in terms of $N_0$ \cite{I07}.
In the language of Computer Science, this means that the ``truncated'' DPS criterion allows to solve an instance of an approximate separability problem, the weak membership problem of separability ({\it WMEM}($\bar{S}$)). Ioannou therefore
provided an upper bound on the full time-complexity of the algorithm for WMEM based on
symmetric extension criteria.
But even after Ioannou's work, an open question remains to be solved.
The PPT symmetric extension criterion is considered to be stronger
than the symmetric extension criterion \cite{doherty, doherty2}.
By definition, it is actually at least as strong as the symmetric extension criterion
in the sense that a $N$-PPT symmetrically extendable state is $N$-symmetrically extendable.
However, so far, there are no results that quantify {\it how strong} the additional PPT constraint makes the DPS criterion.
In particular, since the additional PPT constraint increases quadratically the size of the matrices that define the Semidefinite Programming problem,
there still remains the possibility that the PPT criterion just makes the DPS algorithm slower for {\it WMEM}($\bar{S}$).
In order to make this point clear, a similar analysis as Ioannou's should be done for the PPT-symmetric extension criteria.
Since Ioannou's analysis is based on the finite quantum de Finetti theorem \cite{KR05,CKMR07} and
there exists no similar theorem for states satisfying the PPT constraint, there is no straightforward extension of Ioannou's work to the PPT symmetric extension criterion.
In this paper, by analyzing these criteria in more detail,
we extend Ioannou's result to account for the PPT condition.
The structure of this article is as follows: in Section {\mbox r}ef{DPS} we will give the reader a detailed explanation of the DPS criterion and introduce the basic notation that will be used in the paper. Then we will move on to present the main result of this article, namely, an upper bound on the amount of noise needed to make the DPS states separable. This will allow us to compute upper bounds on the entanglement robustness of these states, and on their distance to the set of separable states. We will also briefly discuss how close our bounds are to being optimal. In Section {\mbox r}ef{sec: complexity}, we will use the previous results to analyze the computational complexity of solving the weak membership problem of separability through the DPS criterion. In particular, we will show that the PPT constraint in the DPS criterion reduces the dominant factor of the upper bound on the time complexity from $\left ( k_1 /\delta {\mbox r}ight )^{6d_B}$ to $\left ( k_2/\delta {\mbox r}ight )^{4d_B}$, where $\delta$ is the accuracy parameter of {\it WMEM}($\bar{S}$). In Section {\mbox r}ef{sec: estimation} we will bound the speed of convergence of the DPS criterion when applied to compute the optimal fidelity in state estimation problems, the output purity of quantum channels and the geometric entanglement of arbitrary states. There we will perform some numerical tests to have a grasp at the actual speed of convergence of the DPS criterion, as opposed to our analytical upper bounds on it. In Sections {\mbox r}ef{prueba}, {\mbox r}ef{sec: multi} we will give the proof of the main theorem and explain how it can be extended to deal with the multipartite case. Afterwards, we will also show a very simple method to bound the entanglement of general PPT states. Finally, Section {\mbox r}ef{conclusion} will present our conclusions.
\section{The DPS criterion}
\label{DPS}
The Doherty-Parrilo-Spedalieri (DPS) criterion for entanglement detection \cite{doherty2}
is a numerical algorithm that, combining the aforementioned results \cite{raggio,werner,CFS02} on $N$-extendibility with convex optimization methods, allows to characterize the set $S$ of separable operators up to arbitrary precision.
The criterion arises from the following observation: if ${\cal L}ambda_{AB}\in S$, then, by definition, it belongs to the cone of bipartite product states, i.e.,
\begin{equation}
{\cal L}ambda_{AB}=\sum_i p_i\overline{\pi}roj{u_i}\otimes \overline{\pi}roj{v_i},
\end{equation}
\noindent with $p_i\geq 0$ for all $i$.
Once this decomposition is known, we can define a uniparametric family of operators ${\cal L}ambda_{AB^N}\in B({\cal H}_A\otimes {\cal H}_B^{\otimes N})$ by tensoring $N$ times the last part:
\begin{equation}
{\cal L}ambda_{AB^N}\equiv\sum_i p_i\overline{\pi}roj{u_i}\otimes \overline{\pi}roj{v_i}^{\otimes N}.
\label{exten}
\end{equation}
Let us study the properties of the newly defined operators: first of all, from the above definition it is clear that they are all positive semidefinite.
Also, from ({\mbox r}ef{exten}) it can be seen that tracing out the last $N-1$ systems we recover the initial operator, i.e., $\mbox{tr}_{B^{N-1}}({\cal L}ambda_{AB^N})={\cal L}ambda_{AB}$, and that the last $N$ systems are invariant under the action of the permutation group.
Finally, when viewed as an $N+1$-partite system, ${\cal L}ambda_{AB^N}$ is multiseparable, and therefore must remain positive semidefinite under the partial transposition of any bipartition of these systems.
For simplicity, we will incorporate all these properties in a single definition:
\begin{equation}gin{defin}{Bose symmetric extensions (BSE)} \\
Let ${\cal L}ambda_{AB}\in {\cal B}({\cal H}_A\otimes{\cal H}_B)$ be a non-negative operator.
We will say that ${\cal L}ambda_{AB^N}\in {\cal B}({\cal H}_A\otimes{\cal H}_B^{\otimes N})$ is an $N$ Bose symmetric extension (BSE) of ${\cal L}ambda_{AB}$ iff:
\begin{equation}gin{enumerate}
\item ${\cal L}ambda_{AB^N}\geq 0$.
\item $\mbox{tr}_{B^{N-1}}({\cal L}ambda_{AB^N})={\cal L}ambda_{AB}$.
\item ${\cal L}ambda_{AB^N}$ is Bose symmetric, i.e., ${\cal L}ambda_{AB^N}({\mathbb I}_A\otimes P_{\mbox{sym}}^N)={\cal L}ambda_{AB^N}$, where $P_{\mbox{sym}}^N$ denotes the symmetric projector of $N$ particles.
\end{enumerate}
\end{defin}
\noindent In case ${\cal L}ambda_{AB^N}$ is PPT with respect to all or some of its bipartitions $AB^K|B^{N-K}$, we will call it a \emph{PPT Bose symmetric extension} (PPT BSE) of ${\cal L}ambda_{AB}$.
From what we have seen, it is clear that, if ${\cal L}ambda_{AB}$ is a separable operator, then there exists an $N$ (PPT) BSE of ${\cal L}ambda_{AB}$ for any $N$.
Since (PPT) Bose symmetric extensions are defined through linear matrix inequalities, the problem of determining whether a given state ${\cal L}ambda_{AB}$ admits one or not can be cast as a semidefinite program (SDP) \cite{sdp}, and therefore can be solved efficiently for fixed $N$ and varying dimensions.
The DPS criterion consists precisely in, given an operator ${\cal L}ambda_{AB}$ whose separability is at stake, check for the existence of $N$ (PPT) Bose symmetric extensions for different values of $N$.
A hierarchy of separability tests arises then naturally: if some operator ${\cal L}ambda_{AB}$ does not admit a (PPT) Bose symmetric extension for some $N$ (i.e., it does not pass the $N^{th}$ test), then it has to be entangled.
If, on the contrary, such extension exists, then we would go for the $(N+1)^{th}$ test, that is, we would search for $N+1$ (PPT) Bose symmetric extensions of ${\cal L}ambda_{AB}$.
This last test would be in general more restrictive than the previous one, since for any $N+1$ (PPT) Bose symmetric extension ${\cal L}ambda_{AB^{N+1}}$ of ${\cal L}ambda_{AB}$ we can obtain an $N$ (PPT) Bose symmetric extension by tracing out the last system.
Doherty et al. \cite{doherty} showed that the previous hierarchy completely characterizes the set of separable operators, in the sense that for any entangled positive operator ${\cal L}ambda_{AB}$ there exists an $N$ such that ${\cal L}ambda_{AB}$ does not admit an $N$ Bose symmetric extension.
We will now introduce a notation that will be used for the rest of the article:
$S^N$ will denote the \emph{cone} of all bipartite operators that have an $N$ BSE, and $S_p^N$ will be understood as the set of all unnormalized quantum states that admit an $N$ BSE that is \emph{PPT with respect to the bipartition $AB^{\lceil N/2{\mbox r}ceil}|B^{\lfloor N/2{\mbox r}floor}$}. In case we also demand normalization, we will be dealing with the sets of states $\bar{S}^N, \bar{S}^N_p$. The elements of the previous four sets will be called $N$-(PPT) symmetrically extendable operators, or states, if normalized, or just DPS operators or states. Our previous discussion can then be summarized as
\begin{equation}gin{eqnarray}
& S^1\supset S^2\supset S^3\supset... \supset S,
& \nonumber\\ &S_p^1\supset S_p^2\supset S_p^3\supset... \supset S,& \nonumber\\
&\lim_{N\to\infty}S^N,S^N_p=S.&
\label{secuencia}
\end{eqnarray}
\noindent Note that $S^1=S^1_p$ $(\bar{S}^1=\bar{S}^1_p)$ is the set of all positive semidefinite operators (states).
Before ending this section, we would like to point out one additional fact.
As we already explained in the introduction, when we use the DPS criterion in practice,
it is not possible to conclude with certainty that a given state is separable.
However, in the PPT case,
by checking some rank constraints on the density matrices output by the computer,
we can \emph{sometimes} conclude separability in a finite number of steps. In that case, we will say that the PPT BSE presents a \emph{rank loop}.
We will make use of rank loops in Section {\mbox r}ef{sec: estimation} in order to estimate the accuracy of our upper bounds on the error we introduce when we perform linear optimizations over the sets $S^N$ or $S^N_p$ instead of $S$ in state estimation problems. A detailed explanation of this criterion for optimality can be found in Appendix {\mbox r}ef{optim}.
\section{Characterization of $S^N$ and $S_p^N$}
\label{charac}
We have seen that the sequences of sets $(S^N)$, $(S_p^N)$ tend to the set $S$ in the limit $N\to\infty$. Intuitively, this means that, for $N>>1$, any state ${\mbox r}ho_{AB}$ belonging to one of these sets must be either separable, or, at least, very close to a separable state. It seems thus plausible that the little entanglement such states may possess could be destroyed by some very attenuated local noise. One of the most simple noise models one can think of is depolarization, where a quantum state is turned into white noise with probability $p$. The action of the depolarizing channel $\Omega^{(p)}$ over some state ${\mbox r}ho\in B({\cal H})$ is given by
\begin{equation}
\Omega^{(p)}({\mbox r}ho)=(1-p){\mbox r}ho+p\frac{{\mathbb I}}{d},
\label{depolarizing}
\end{equation}
\noindent where $d$ is the dimension of the Hilbert space ${\cal H}$. Given any bipartite quantum state ${\mbox r}ho_{AB}$, shared by Alice and Bob, we could thus define its \emph{critical disentangling probability} $p_c({\mbox r}ho_{AB})$ as the minimum probability with which one of the parties, say Bob, would have to prepare the maximally mixed state in his subsystem in order to disentangle it from Alice's. That is,
\begin{equation}
p_c({\mbox r}ho_{AB})=\min\{p:{\mathbb I}_A\otimes\Omega^{(p)}_B({\mbox r}ho_{AB})\in \bar{S}\}.
\end{equation}
\noindent Similarly, we can define the critical disentangling probability of a set of states $W$ as the maximum of all $p_c({\mbox r}ho)$ for all ${\mbox r}ho\in W$. Clearly, $p_c\leq 1$ for all states, although this bound can be greatly improved if the dimensionality of Bob's system is small, as we shall see.
In this section, we will give upper bounds on this critical probability valid for
any state in $\bar{S}^N$ (or $\bar{S}_p^N$).
Then, by means of these results,
we will provide several upper bounds on the speed of convergence of $\bar{S}^N$ and $\bar{S}_p^N$ to $\bar{S}$.
Before proceeding, though, a remark on notation: in this article, we will be mainly concerned with linear operators or quantum states acting over a bipartite Hilbert space ${\cal H}_A\otimes {\cal H}_B$, and all the formulas and bounds that we will derive in this section and the following three will involve the dimension of the Hilbert space ${\cal H} _B$ where the symmetric extensions are to be made. For the sake of clarity, we will therefore introduce the notation $d \stackrel{{\mbox r}m def}{=} \dim {\cal H}_B$.
The following theorems will play a key role in deriving most of the results of this paper.
\begin{equation}gin{theo}
\label{bosesym}
\begin{equation}
p_c(\bar{S}^N)\leq \frac{d}{N+d}.
\end{equation}
\noindent In other words: for any operator ${\cal L}ambda_{AB}\in S^N$, the positive semidefinite operator
\begin{equation}gin{equation}
\tilde{{\cal L}ambda}_{AB}\equiv\frac{N}{N+d}{\cal L}ambda_{AB}+\frac{1}{N+d}{\cal L}ambda_A\otimes{\mathbb I}_B
\label{canonic1}
\end{equation}
\noindent is separable.
\end{theo}
\begin{equation}gin{theo}
\label{egregium}
Define $g_N$ (or $g_N^{(d)}$ in case $d$ is ambiguous) as
\begin{equation}gin{eqnarray}
g_N=& &\min\{1-x:P_{N/2+1}^{(d-2,0)}(x)=0\}\mbox{ for } N \mbox{ even},\nonumber\\
& &\min\{1-x:P_{(N+1)/2}^{(d-2,1)}(x)=0\}\mbox{ for } N \mbox{ odd},
\end{eqnarray}
\noindent with $P_n^{(\alpha,\begin{equation}ta)}(x)$ being the Jacobi Polynomials \cite{abramo}.
\noindent Then,
\begin{equation}
p_c(\bar{S}^N_p)\leq \frac{d}{2(d-1)}g_N.
\end{equation}
\noindent That is, for any ${\cal L}ambda_{AB}\in S^N_p$, the positive semidefinite operator
\begin{equation}gin{equation}
\tilde{{\cal L}ambda}_{AB}\equiv(1-\frac{d}{2(d-1)}g_N){\cal L}ambda_{AB}+\frac{1}{2(d-1)}g_N{\cal L}ambda_A\otimes{\mathbb I}_B
\label{canonic2}
\end{equation}
\noindent is separable.
\end{theo}
\noindent The proof of these two theorems is given in Section {\mbox r}ef{prueba},
where a separable decomposition for the states ({\mbox r}ef{canonic1}), ({\mbox r}ef{canonic2}) is also provided. Also, it is worth mentioning that, in both cases, $\tilde{{\cal L}ambda}_A={\cal L}ambda_A$.
Notice that, in Theorem {\mbox r}ef{egregium}, $g_N$ is defined in terms of the greatest root of Jacobi polynomials.
The properties of the roots of Jacobi polynomials have been studied for quite time \cite{abramo}.
This allows us to derive an expression for the asymptotic behavior of $g_N$:
\begin{equation}gin{eqnarray}
g_N &\approx & 2\left(\frac{j_{d-2,1}}{N}{\mbox r}ight)^2, \mbox{ for } N>>1,\\
& \approx & 2\left(\frac{d +1.856d^{1/3}+O(d^{-1/3}) }{N}{\mbox r}ight)^2, \nonumber \\
&\quad & \qquad \mbox{ for } N\gg d\gg 1,
\label{bessel}
\end{eqnarray}
where $j_{n,1}$ is the first positive zero of the Bessel function $J_n(y)$.
How far can then the states in $\bar{S}^N,\bar{S}^N_p$ be from the set $\bar{S}$ of separable states? A way to answer this question could be to bound the maximum possible entanglement of such states.
The \emph{robustness of entanglement} of a state ${\mbox r}ho$ is defined as the minimum amount of separable noise needed to destroy the entanglement of such a state \cite{vidal}:
\begin{equation}gin{equation}
R({\mbox r}ho) \stackrel{{\mbox r}m def}{=}
\min_{\lambda}\{\lambda :\exists \sigma \in \bar{S}, \ \mbox{s.t. } \frac{{\mbox r}ho+\lambda \sigma}{1+\lambda }\in S\}.
\end{equation}
\noindent The robustness of entanglement is also an upper bound on the \emph{global robustness of entanglement} $R_{G}({\mbox r}ho)$ \cite{vidal}, defined by allowing $\sigma$ to be an arbitrary normalized quantum state in the above expression. And the global robustness of entanglement is, in turn, lower bounded by several other entanglement measures, like the negativity, the geometric measure of entanglement and the relative entropy of entanglement \cite{vidal, computable,hayashishash,daniel,nature}. Any non trivial upper bound on the entanglement robustness of the states in $\bar{S}^N$ and $\bar{S}^N_p$ could thus retrieve a lot of information.
The following corollary follows straightforwardly from theorems {\mbox r}ef{bosesym} and {\mbox r}ef{egregium}.
\begin{equation}gin{cor}
Any ${\mbox r}ho \in \bar{S}^N$ satisfies
\begin{equation}gin{equation}
R({\mbox r}ho ) \le \frac{d-1}{N} \label{hulk}.
\end{equation}
Similarly, any ${\mbox r}ho \in \bar{S}_p^N$ satisfies
\begin{equation}gin{equation}\label{hulkppt}
R({\mbox r}ho) \le \frac{g_N}{2-\frac{d}{d-1}g_N} \approx \left(\frac{d}{N}{\mbox r}ight)^2.
\end{equation}
\end{cor}
To see why, suppose that ${\mbox r}ho$ is normalized and use formulas ({\mbox r}ef{canonic1}), ({\mbox r}ef{canonic2}) to express $\tilde{{\mbox r}ho}$ (i.e., $\tilde{{\cal L}ambda}_{AB}$) in each case as a convex sum of the non negative operators ${\mbox r}ho$ and $\sigma\equiv\frac{1}{d-1}({\mbox r}ho_{A}\otimes {\mathbb I}_B-\tilde{{\mbox r}ho})$. Then, notice that, since $\tilde{{\mbox r}ho}_A={\mbox r}ho_A$ and $\tilde{{\mbox r}ho}$ is separable, then $\sigma$ must also be a separable operator \footnote{To understand why, write $\tilde{{\mbox r}ho}$ as a convex combination of product states, i.e., $\tilde{{\mbox r}ho}=\sum p_i\overline{\pi}roj{u_i}\otimes\overline{\pi}roj{v_i}$. Then, $\tilde{{\mbox r}ho}_A\otimes{\mathbb I}-\tilde{{\mbox r}ho}=\sum p_i\overline{\pi}roj{u_i}\otimes({\mathbb I}-\overline{\pi}roj{v_i})$. That is, $\tilde{{\mbox r}ho}_A\otimes{\mathbb I}-\tilde{{\mbox r}ho}={\mbox r}ho_A\otimes{\mathbb I}-\tilde{{\mbox r}ho}$ is a separable operator.}
Theorems {\mbox r}ef{bosesym} and {\mbox r}ef{egregium} also allow to obtain bounds on the distance between the states in ${\mbox r}ho_{AB} \in S^N,S^N_p$ and the set of separable states $\bar{S}$.
\begin{equation}gin{cor}\label{precisebounds}
For any ${\mbox r}ho \in \bar{S}^N$, there exist $\tilde{{\mbox r}ho} \in \bar{S}$ such that
\begin{equation}gin{eqnarray}
&& \| {\mbox r}ho - \tilde{{\mbox r}ho} \|_{1} \le \frac{2(d-1)}{N+d-1}, \label{trace1}\\
&& \| {\mbox r}ho - \tilde{{\mbox r}ho} \|_{\infty} \le \frac{d-1}{N+d-1},\\
&& \| {\mbox r}ho - \tilde{{\mbox r}ho} \|_F = \frac{d}{N+d}\sqrt{\mbox{tr}({\mbox r}ho^2)-\frac{\mbox{tr}({\mbox r}ho_A^2)}{d}},
\end{eqnarray}
where $\| \cdot \|_{1}$, $\| \cdot \|_{\infty}$ and $\| \cdot \|_F$ are the trace, the operator and the Frobenius norm, respectively.
Similarly, for any ${\mbox r}ho \in \bar{S}_p^N$ (and $N\geq 2$), there exists a state $\tilde{{\mbox r}ho} \in \bar{S}$ such that
\begin{equation}gin{eqnarray}
&& \| {\mbox r}ho - \tilde{{\mbox r}ho} \|_{1} \le g_N, \\
&& \| {\mbox r}ho - \tilde{{\mbox r}ho} \|_{\infty} \le g_N/2,\\
&& \| {\mbox r}ho - \tilde{{\mbox r}ho} \|_F = \frac{dg_N}{2d-2}\sqrt{\mbox{tr}({\mbox r}ho^2)-\frac{\mbox{tr}({\mbox r}ho_A^2)}{d}}.
\end{eqnarray}
\end{cor}
\begin{equation}gin{proof}
Here we give the proof for the bounds on the trace and operator norm.
The proof for the Frobenius norm is omitted, since it is similar and simpler.
Let ${\mbox r}ho\in \bar{S}^N$. Them Theorem {\mbox r}ef{bosesym} implies that there exists $\tilde{{\mbox r}ho}\in \bar{S}$, with $\tilde{{\mbox r}ho}_A={\mbox r}ho_A$, such that:
\begin{equation}
{\mbox r}ho-\tilde{{\mbox r}ho}=\frac{d-1}{N+d-1}{\mbox r}ho-\frac{1}{N+d-1}({\mbox r}ho_A\otimes{\mathbb I}_B-\tilde{{\mbox r}ho}).
\end{equation}
\noindent Using the triangle inequality, we have that
\begin{equation}gin{eqnarray}
& &\|{\mbox r}ho-\tilde{{\mbox r}ho}\|_1\leq\frac{d-1}{N+d-1}\|{\mbox r}ho\|_1+\nonumber\\
& &+\frac{1}{N+d-1}\|({\mbox r}ho_A\otimes{\mathbb I}_B-\tilde{{\mbox r}ho})\|_1=\frac{2(d-1)}{N+d-1},
\end{eqnarray}
\noindent where in the last step we used once more the fact that ${\mbox r}ho_A\otimes{\mathbb I}_B-\tilde{{\mbox r}ho}$ is separable (and, therefore, positive). Relation ({\mbox r}ef{trace1}) is thus proven.
For the operator norm, let $u_+(u_-)$ be the eigenvector corresponding to the maximum (minimum)
eigenvalue of ${\mbox r}ho-\tilde{{\mbox r}ho}$. It follows that
\begin{equation}
\|{\mbox r}ho-\tilde{{\mbox r}ho}\|_\infty=\mbox{max}(\mbox{tr}\{({\mbox r}ho-\tilde{{\mbox r}ho})\overline{\pi}roj{u_+}\},\mbox{tr}\{(\tilde{{\mbox r}ho}-{\mbox r}ho)\overline{\pi}roj{u_-}\}).
\end{equation}
\noindent On the other hand,
\begin{equation}gin{eqnarray}
&\mbox{tr}\{({\mbox r}ho-\tilde{{\mbox r}ho})\overline{\pi}roj{u_+}\}=\frac{d-1}{N+d-1}\mbox{tr}\{{\mbox r}ho \overline{\pi}roj{u_+}\}-\nonumber\\
&-\frac{1}{N+d-1}\mbox{tr}\{({\mbox r}ho_A\otimes{\mathbb I}_B-\tilde{{\mbox r}ho})\overline{\pi}roj{u_+}\}\leq \frac{d-1}{N+d-1},
\end{eqnarray}
\noindent and
\begin{equation}gin{eqnarray}
&\mbox{tr}\{(\tilde{{\mbox r}ho}-{\mbox r}ho)\overline{\pi}roj{u_-}\}=-\frac{d-1}{N+d-1}\mbox{tr}\{{\mbox r}ho \overline{\pi}roj{u_-}\}+\nonumber\\
&+\frac{1}{N+d-1}\mbox{tr}\{({\mbox r}ho_A\otimes{\mathbb I}_B-\tilde{{\mbox r}ho})\overline{\pi}roj{u_-}\}\leq \frac{d-1}{N+d-1}.
\end{eqnarray}
\noindent The first part of the corollary has been proven.
If ${\mbox r}ho\in \bar{S}^N_p$ and $N\geq 2$, then ${\mbox r}ho$ can be seen to be PPT. Since the PPT criterion implies
the reduction criterion \cite{reduction, reduction2}, we have that ${\mbox r}ho_A\otimes{\mathbb I}_B-{\mbox r}ho\geq 0$.
This observation, combined with the techniques used to derive the first set of relations, allows to prove the second one.
\end{proof}
The above corollaries can be reformulated as:
\begin{equation}gin{cor}\label{trnorm}
Suppose $\bar{S}(\delta)$ is a $\delta$-neighbor of the set of all separable states $\bar{S}$ in terms of the trace distance:
\begin{equation}gin{equation}
\bar{S}(\delta) \stackrel{{\mbox r}m def}{=} \bigcup _{{\mbox r}ho \in \bar{S}} \left \{ \sigma \in \bar{S}^1 \ | \ \| {\mbox r}ho - \sigma\| \le \delta {\mbox r}ight \}
\end{equation}
\noindent(remember that $\bar{S}^1$ is the set of all quantum states in ${\cal H}_A\otimes {\cal H}_B$).
\noindent Then, the following relations hold:
\begin{equation}gin{eqnarray}
\bar{S}^N & \subset & \bar{S} \left( \frac{2(d-1)}{N+d-1} {\mbox r}ight) \approx \bar{S}\left( 2\frac{d}{N}{\mbox r}ight ), \label{eq:convergence sym} \\
\bar{S}_p^N & \subset & \bar{S}(g_N) \approx \bar{S}\left( 2\left( \frac{d}{N} {\mbox r}ight)^2 {\mbox r}ight),
\end{eqnarray}
\noindent where the approximations are granted to hold in the limit $N\gg d \gg 1$.
\end{cor}
\noindent This corollary suggests that the upper bounds for $\bar{S}_p^N$ converge quadratically faster than those for $\bar{S}^N$.
In other words, if these bounds were optimum, then we would have proven that the additional PPT constrain gives the DPS criterion a quadratic speed-up.
It is then natural to wonder if such bounds are indeed optimal. We will argue that at least the scaling of the upper bounds for $\bar{S}^N$ is correct, i.e., fixing $d_A$ and $d$, the maximum possible entanglement robustness of any bipartite state ${\mbox r}ho_{AB}$ arising
from an $N$ Bose symmetric extension scales with $N$ as $O(1/N)$.
To see this, let $N=2K-1$, and consider the $N+1$ bipartite state given by
\begin{equation}
\ket{{\cal P}si_{AB^N}}\equiv\frac{1}{C_K}\sum _{{\mbox r}m perm} \ket{\overbrace{0 \cdot 0}^{K} \overbrace{1 \cdot 1}^{K}},
\end{equation}
\noindent where $C_K$ is a normalization factor. Define now ${\mbox r}ho_{AB}\equiv\mbox{tr}_{B^{N-1}}(\overline{\pi}roj{{\cal P}si_{AB^N}})$. Clearly, ${\mbox r}ho_{AB}\in S^N$. Now, it can be shown that
\begin{equation}gin{eqnarray}
& &{\mbox r}ho_{AB}=\frac{K-1}{2(2K-1)}(\overline{\pi}roj{00}+\overline{\pi}roj{11})+\nonumber\\
& &+\frac{K}{2(2K-1)}(\ket{01}+\ket{10})(\bra{01}+\bra{10}).
\label{ejemplo}
\end{eqnarray}
\noindent The partially transposed operator ${\mbox r}ho_{AB}^{T_B}$ has a negative eigenvalue $-1/2(2K-1)$
corresponding to the eigenvector $(\ket{00}-\ket{11})/\sqrt{2}$, whose maximum Schmidt coefficient is $1/\sqrt{2}$.
According to \cite{vidal}, this implies that $R({\mbox r}ho_{AB})=1/(2K-1)=1/N$.
The bound ({\mbox r}ef{hulk}) is, therefore, tight for $d_A=d=2$.
Since for any pair for Hilbert spaces ${\cal H}_A,{\cal H}_B$ of dimensions greater than 1 we can embed the previous family of states in $B({\cal H}_A\otimes {\cal H}_B)$,
it follows that the optimal upper bound on the entanglement robustness of partial traces of Bose symmetric extensions must scale as $O(1/N)$.
On the other hand, the bound ({\mbox r}ef{hulkppt}) guarantees that the corresponding value for $\bar{S}_p^N$
at least scales as $O(1/N^2)$, i.e., Theorem {\mbox r}ef{egregium} allows to derive an upper bound for the entanglement robustness that decreases asymptotically faster than the optimal upper bound in the general Bose symmetric case.
Note that the above considerations also allow us to obtain a dimension-dependent lower bound on the maximum possible entanglement robustness $R^N_{\sup}$ of a state in $\bar{S}^N$. Following the lines of \cite{shor}, consider the state $\sigma\equiv{\mbox r}ho_{AB}^{\otimes M}$, with ${\mbox r}ho_{AB}$ given by equation ({\mbox r}ef{ejemplo}). Clearly, $\sigma\in \bar{S}^N$, with $d_A=d_B=d=2^M$. As $-1/(2N)$ is the only negative eigenvalue of ${\mbox r}ho_{AB}^{T_B}$ and, therefore, the sum of its positive eigenvalues adds up to $1+1/(2N)$, the negativity of $\sigma$ \cite{neg1} (i.e., minus the sum of the negative eigenvalues of $\sigma^{T_B}$) can be seen equal to
\begin{equation}gin{eqnarray}
& &{\cal N}(\sigma)=\sum_{j=0}^{\lfloor (M-1)/2{\mbox r}floor}\left(\begin{equation}gin{array}{c}M\\2j+1\end{array}{\mbox r}ight)\frac{\left(1+\frac{1}{2N}{\mbox r}ight)^{M-2j-1}}{(2N)^{2j+1}}=\nonumber\\
& &=\frac{[(1+\frac{1}{2N})+\frac{1}{2N}]^M-[(1+\frac{1}{2N})-\frac{1}{2N}]^M}{2}=\nonumber\\
& &=\frac{(1+\frac{1}{N})^M-1}{2}\approx\frac{M}{2N},
\end{eqnarray}
\noindent where the last approximation is valid in the limit of large $N$. Since $R(\sigma)\geq {\cal N}(\sigma)$ \cite{computable}, it follows that $R(\sigma)\gtrapprox O(\log(d)/N)$. That is, for fixed dimension $d$, $R^N_{\sup}$ satisfies $O(\log(d)/N)\leq R^N_{\sup}\leq O(d/N)$.
\section{Computational complexity of WMEM($\bar{S}$)}\label{sec: complexity}
In this section, we will analyze the consequences of the previous results on separability from the point of view of Computer Science.
Actually, there are several different ways to describe the separability problem as a computational problem \cite{I07}.
We chose to focus our attention in
an approximated separability problem called the \emph{weak membership problem of separability}.
This {\it ``promise''} problem (as opposed to a {\it ``decision'' } problem) roughly consists on
deciding the separability of a given state, but allowing an uncertainty parameterized by $\delta$.
In this Section we will derive upper bounds on the time and space complexity when we attack this problem via the DPS criterion.
The ``{\it In-biased}'' weak membership problem is defined as follows \cite{I07}: \\
\begin{equation}gin{defin}{Weak membership problem of separability (WMEM($\bar{S}$))} \\
Given a bipartite quantum state ${\mbox r}ho \in \bar{S}^1$ and rational $\delta >0$, assert either that
\begin{equation}gin{eqnarray}
{\mbox r}ho &\in& \bar{S}(\delta ) \ or \label{eq: wmem1}\\
{\mbox r}ho &\not\in& \bar{S},\label{eq: wmem2}
\end{eqnarray}
\noindent where $\bar{S}(\delta )$ is a $\delta$ neighbor of $\bar{S}$, i.e., $\bar{S}(\delta )=
\{ \sigma \in \bar{S}^1 :\tilde{\sigma} \in \bar{S}\subset \bar{S}^1, \| \tilde{\sigma}-\sigma\|_1\leq \delta \}$.
\end{defin}
In the above definition, $\| \omega \|_1=\mbox{tr}(\sqrt{\omega\omega^\dagger})$, the trace norm of the operator $\omega$,
although, in principle, we could have chosen other norms or distance measures as an accuracy parameter.
WMEM($S$) is, thus, an approximation of the conventional separability problem in the sense
that an algorithm solving WMEM($\bar{S}$) may assert equation ({\mbox r}ef{eq: wmem1}) for a state ${\mbox r}ho_{AB}$ having just a small amount of entanglement.
This approximated formalism is more practical than a non-approximated or exact formalism like EXACT-QSEP \cite{I07}, because of
the inevitable errors we incur in both numerical and experimental studies, that should somehow be accounted for in our analysis of separability.
A fair amount of effort has been devoted to the study of the time complexity of WMEM($\bar{S}$), the most remarkable result being that,
if $d_A \geq d_B$,
then WMEM($S$) is NP-hard whenever $1/\delta$ increases exponentially \cite{G03} or polynomially \cite{sevag} with respect to $d_B$.
We will now proceed to evaluate the time complexity of WMEM($\bar{S}$) when solved through the DPS criterion.
First, following the discussion of Doherty et al. \cite{doherty},
$S^N$ can be characterized by a semidefinite program with $\left ( ( \dim {\cal H}_{\mbox{sym}}^N )^2 - d_B^2 {\mbox r}ight )d_A^2$ free variables
and a matrix of size $ ( \dim {\cal H}_{\mbox{sym}}^N ) d_A$ on which we will impose the positivity constraint.
On the other hand, for $\bar{S}_p^N$, the PPT constraint implies demanding positivity from an additional matrix of size
$ ( \dim {\cal H}_{\mbox{sym}}^{N/2} )^2 d_A$.
Since the time-complexity of an SDP with $m$ variables and of matrix size $n$
is $O(m^2n^2)$ (with a small extra cost coming from an iteration of algorithms),
the dominant factors for the asymptotic time-complexity of these tests can be written as
\begin{equation}gin{eqnarray}\label{eq: complexity}
&{{\mbox r}m Symmetric}:d_A^6(\dim {\cal H}_{\mbox{sym}}^{\overline{N_{{\mbox r}m sym}}} )^6 \label{eq: complexity} \\
&{{\mbox r}m PPT\ symmetric}:d_A^6( \dim {\cal H}_{\mbox{sym}}^{\overline{N_{{\mbox r}m ppt}}} )^4( \dim {\cal H}_{\mbox{sym}}^{\overline{N_{{\mbox r}m ppt}}/2} )^4,
\label{eq: complexity PPT}
\end{eqnarray}
where $\overline{N_{{\mbox r}m sym}}$ and $\overline{N_{{\mbox r}m ppt}}$ are the sizes of the extensions
needed to achieve a given accuracy parameter $\delta$.
Thus, at this stage, even though $\bar{S}^N_p$ converges to $\bar{S}$ faster than $\bar{S}^N$,
there still remains the possibility that the algorithm based on
the sets $\{\bar{S}^N_p\}$ is slower than the one based on the sets $\{\bar{S}^N\}$, because of the increase in time complexity that arises from imposing positivity on the partially transposed operator.
The following calculation will rule out this possibility.
From Eq. ({\mbox r}ef{eq:convergence sym}) of Corollary {\mbox r}ef{trnorm}, we have that
\begin{equation}gin{eqnarray}
&\overline{N_{{\mbox r}m sym}} \leqq \frac{(2-\delta)(d_B-1)}{\delta},\nonumber\\
&\overline{N_{{\mbox r}m ppt}}\lessapprox \frac{\sqrt{2}j_{d_B-2,1}}{\sqrt{\delta}}.
\end{eqnarray}
\noindent Taking into account that $j_{d,1}\approx d+O(d^{1/3})$ \cite{abramo},
the final expressions for upper bounds of the time complexity with respect to one method and the other are
\begin{equation}gin{eqnarray}
O\left(d_A^6\left[\frac{2e}{\delta}{\mbox r}ight]^{6d_B}{\mbox r}ight), & &\mbox{ for } \bar{S}^N \label{eq: complexity sym}\nonumber\\
O\left(d_A^6\left[\frac{e^2}{\delta}{\mbox r}ight]^{4d_B}{\mbox r}ight), & &\mbox{ for } \bar{S}_p^N, \label{eq: complexity ppt}
\end{eqnarray}
where we just wrote the dominant (exponential) terms and omitted all polynomially growing terms. Note that the scaling law derived for the non PPT DPS criterion is valid as long as the optimal bounds on the trace distance to the set of separable states scale as $d_B/N$. We conjecture that such is the case, although all our attempts to derive an analytical proof have failed so far. Under this assumption, the above formula thus shows that the criterion based on PPT BSEs indeed requires less steps than the one based on plain BSEs in order to solve WMEM($\bar{S}$) for a given accuracy $\delta$.
The space complexity of both the plain DPS criterion and the PPT DPS criterion, though, is of the same type. This is because, although the PPT condition imposes (at least) a quadratic speedup in the speed of convergence, it also increases quadratically the size of the matrices involved in the SDP. Thus one effect cancels the other, and the size of the matrices needed in both cases to solve WMEM($\bar{S}$) up to a given precision $\delta$ is comparable for any value of $d_B$. It follows that, according to our bounds, in some situations it may be more convenient not to use the PPT condition in order to save memory space.
Our experience with the DPS method suggests, however, that this expectation is not realistic,
but rather a consequence of the non optimality of the bounds implicit in Theorem {\mbox r}ef{egregium}.
Actually, in practice, the algorithm based on PPT BSEs seems to have smaller space complexity than the one based on general BSEs.
A big underestimation of the role of the PPT condition in the DPS criterion could also explain why the bound ({\mbox r}ef{eq: complexity ppt}) behaves much worse than the asymptotic expressions $(k/\delta)^{2d_B}$ derived in \cite{I07} for the performance of the algorithm conceived by Ioannou et al. for entanglement detection \cite{witness,witness2}. Indeed, as we will see, our bounds on the distance between the sets $\bar{S}^N_p$ and the set of separable states are far from optimal, at least for small values of $d_A$. Therefore, a more refined analysis could in principle end up with a different scaling law for this distance, that would eventually lead to a much better estimate of the time complexity of methods based in PPT BSEs.
\section{Approximate algorithms for state estimation, maximum output purity, and geometric measure of entanglement} \label{sec: estimation}
There are many relevant quantities in quantum information whose definition involves a linear optimization over a set of separable operators. The maximum average fidelity in state estimation problems \cite{H76,H82},
the output purity of a quantum channel \cite{output} or the geometric measure of entanglement \cite{intro_measures} are examples of such quantities.
In order to compute these functions, we could think of an approximate algorithm that optimized over the sets $S^N$ or $S_p^N$
instead of $S$, and it is easy to see that such an algorithm would give the correct answer in the limit of large $N$.
So far, we have seen how Theorems {\mbox r}ef{bosesym} and {\mbox r}ef{egregium} can be used to derive bounds related to the separability problem. In this Section we will show how to use these same theorems to bound the precision of the approximate linear optimizations over the cone of separable operators mentioned above.
\subsection{State Estimation Problems}
In a \emph{general} state estimation scenario, a source chooses
with probability $p_i$ a virtual quantum state ${\cal P}si_i$ that is encoded
afterwards into another quantum state ${\cal P}si'_i$, to which we are given
full access. The goal of the game is to measure our given state by means of a Positive Operator Valued Measure (POVM) $\{M_x\}_x$ and thus
obtain a classical value $x$ that we will use to make a guess $\overline{\pi}hi_x$ on
the original state ${\cal P}si_i$, which from now on we will assume to be pure.
In conventional estimation theory, we usually restrict the guess $\overline{\pi}hi_x$ to be one of the original states $\{{\cal P}si_i\}_i$ \cite{H76,H82}.
In this section, however, we will consider the more general setting in which we are allowed to choose arbitrary states
as a guess.
Being ${\cal P}si_i$ a pure state, the efficiency of the protocol as a whole can be parametrized in terms of the \emph{average fidelity} $f$:
\begin{equation}gin{equation}
0\leq f\equiv\sum_{i,x}p_i\mbox{tr}({\cal P}si_i'M_x)\mbox{tr}(\overline{\pi}hi_x{\cal P}si_i)\leq 1.
\end{equation}
\noindent And the state estimation problem consists on determining $F$, the maximum fidelity among all possible measure-and-prepare schemes $(M_x,\overline{\pi}hi_x)$.
Since $F$ can be used as well to determine whether a given quantum channel can be simulated or not
by an entanglement breaking channel, this problem is also referred to as the \emph{Quantum benchmark problem} \cite{HWPC2005,SDP07,AC08,CAMB08,OPPSW08}.
In \cite{pasado}, it is explained how to map the SE problem into a linear optimization over the set $S$ of separable states, via the relation
\begin{equation}gin{equation}
F=\max\{\mbox{tr}({\mbox r}ho_{AB}{\cal L}ambda_{AB}):{\cal L}ambda_{AB}\in S,{\cal L}ambda_A={\mathbb I}\},
\label{funda}
\end{equation}
\noindent where ${\mbox r}ho_{AB}=\sum_i p_i {\cal P}si'_i\otimes{\cal P}si_i$ is given by the particular SE problem.
There it is also shown that any separable decomposition of the optimal operator ${\cal L}ambda_{AB}=\sum_x M_x\otimes \overline{\pi}hi_x$ corresponds to the optimal strategy $(M_x, \overline{\pi}hi_x)$.
Now, consider the sequence of optimization problems:
\begin{equation}gin{eqnarray}
\hspace*{-1cm}& &F^N\equiv\max\{\mbox{tr}({\mbox r}ho_{AB}{\cal L}ambda_{AB}):{\cal L}ambda_{AB}\in
S^N ,{\cal L}ambda_A={\mathbb I}\},\nonumber\\
\hspace*{-1cm}& &F_p^N\equiv\max\{\mbox{tr}({\mbox r}ho_{AB}{\cal L}ambda_{AB}):{\cal L}ambda_{AB}\in
S^N_p ,{\cal L}ambda_A={\mathbb I}\},
\label{ideacentral}
\end{eqnarray}
\noindent From ({\mbox r}ef{secuencia}), it is immediate that $F^1\geq F^2\geq F^3\geq ... \geq F$, with $\lim_{N\to\infty}F^N=F$. An analogous property holds for the bounds $F^N_p$. Note that these maximizations are SDPs and therefore can be easily computed.
Unfortunately, given limited computational (and specially memory) resources, it is only possible to compute these bounds up to some index $N$.
In spite of the asymptotic convergence of the sequence, $F^N$ or $F^N_p$ could very well be far away from the actual solution of the problem.
Is there any way to estimate the error of the truncation?
Take ${\cal L}ambda_{AB}\in S^N (S_p^N)$ to be the operator that maximizes equation ({\mbox r}ef{ideacentral}).
Theorem {\mbox r}ef{bosesym} ({\mbox r}ef{egregium}) then implies that $\tilde{{\cal L}ambda}_{AB}$, as defined by equation ({\mbox r}ef{canonic1}) (({\mbox r}ef{canonic2})), corresponds to a feasible state estimation strategy, since it is separable and $\tilde{{\cal L}ambda}_A={\cal L}ambda_A={\mathbb I}$.
Moreover, we can use the separable decomposition of $\tilde{{\cal L}ambda}_{AB}$ that appears in Section {\mbox r}ef{prueba} to express it as a measure-and-prepare protocol $(M_x,\overline{\pi}hi_x)$.
The fidelities $\tilde{F}^N$ or $\tilde{F}^N_p$ associated to these strategies, although non trivial, will not be optimal in general, but they should provide a lower bound for $F$.
From ({\mbox r}ef{funda}), it is easy to see that
\begin{equation}gin{eqnarray}
& &\tilde{F}^N=\frac{N}{N+d}F^N+\frac{1}{N+d},\nonumber\\
& &\tilde{F}_p^N=\left(1-\frac{dg_N}{2(d-1)}{\mbox r}ight)F_p^N+\frac{g_N}{2(d-1)}.
\end{eqnarray}
\noindent Notice that both lower bounds asymptotically converge to $F$.
That is, from the solutions of the semidefinite programs ({\mbox r}ef{ideacentral}) it is possible to obtain a sequence of state estimation strategies that converges to the optimal measure-and-prepare scheme.
To have a grasp on the efficiency of the method, consider the following state estimation problem: suppose we have a device that outputs two copies of one of the 4 qubit states $\{\ket{{\cal P}si_k}\}_{k=1}^4\equiv\{\ket{0},\ket{1},\ket{+},\ket{-}\}$ with equal probabilities. Our task is to estimate the state produced by the device. However, due to the environmental noise, once we are ready to measure the copies, those have degraded into ${\mbox r}ho_k\equiv \Omega^{(\epsilon)}(\overline{\pi}roj{{\cal P}si_k})=(1-\epsilon)\overline{\pi}roj{{\cal P}si_k}+\epsilon{\mathbb I}/2$. The results for $\epsilon=0.3$ are shown in Figure {\mbox r}ef{fidelitas}, for both the PPT and non PPT case and different values of $N$.
\begin{equation}gin{figure}
\centering
\includegraphics[width=8 cm]{Fidelidad2_paper.eps}
\caption{Upper (squares) and lower (circles) bounds for the maximum fidelity $F$ as a function of $N$. The dashed line indicates the value of the exact solution, attained exactly by the PPT upper bounds on $F$ from $N=2$ and onwards. The minimum difference between the upper and lower bounds is of the order of $10^{-2}$ in both plots.}
\label{fidelitas}
\end{figure}
We used the MATLAB package \emph{YALMIP} \cite{yalmip} in combination with \emph{SeDuMi} \cite{sedumi} to perform the numerical calculations. Note that the curve corresponding to the upper bounds is constant, i.e., $F^N=F^M=F^*$, for all $M,N$. This suggested that $F^*$ could be equal to $F$, the solution of the problem, although we did not observe any rank loop in the matrices output by the computer. We thus had to \emph{force} the rank loop to occur. Using rank minimization heuristics \cite{lmirank} we checked for the existence of low rank PPT BSEs of ${\cal L}ambda_{AB}$ such that $\mbox{tr}({\cal L}ambda_{AB}{\mbox r}ho_{AB})\geq F^*-\delta$. Taking $\delta=10^{-4}$, the computer returned a matrix with a rank loop, therefore proving the optimality of $F^*$ up to this precision.
We performed a similar analysis for $d=3$, this time considering the problem where a degraded copy of one of the states
\begin{equation}gin{eqnarray}
\ket{\overline{\pi}si_{ij}}=& &\cos\left(\frac{j\overline{\pi}i}{6}{\mbox r}ight)\ket{0}+\sin\left(\frac{j\overline{\pi}i}{6}{\mbox r}ight)\cos\left(\frac{i\overline{\pi}i}{6}{\mbox r}ight)\ket{1}+\nonumber\\
& &+\sin\left(\frac{j\overline{\pi}i}{6}{\mbox r}ight)\sin\left(\frac{i\overline{\pi}i}{6}{\mbox r}ight)\ket{2},
\end{eqnarray}
\noindent (where $i$ and $j$ run from 0 to 5) is sent to us with probability 1/36 through a depolarizing channel ${\mbox r}ho\to \Omega^{(0.2)}({\mbox r}ho)$. In this case we were also able to force a rank loop in the PPT BSEs, so we again knew the optimal solution. Figure {\mbox r}ef{fidelitas2} illustrates our numerical results.
\begin{equation}gin{figure}
\centering
\includegraphics[width=8 cm]{Fidelidad3_paper.eps}
\caption{Upper (squares) and lower (circles) bounds for the maximum fidelity $F$ as a function of $N$ in dimension 3. This time, the minimum difference between our lower bounds and the exact solution is around 0.03 (and it is attained in the non PPT case).}
\label{fidelitas2}
\end{figure}
Note that, in both cases, the lower bounds on the solution behave very similarly as the upper bounds given by the DPS criterion, as long as we are considering the non PPT case. In the PPT case, however, our bounds prove to be terrible, since the second available upper bound obtained through the DPS criterion already seems to attain the optimal solution. We will discuss briefly this topic in Section {\mbox r}ef{conclusion}.
The main features of the practical performance of the DPS criterion have already been illustrated above. Therefore, in the following two problems we will just stick to analytical results.
\subsection{Maximal output purity of quantum channels}
Let $\omega$ be a quantum channel. The \emph{maximal output purity} \cite{output} $\nu$ of $\omega$ is defined as
\begin{equation}
\nu=\max_{{\mbox r}ho}\|\omega({\mbox r}ho)\|_{\infty},
\label{output}
\end{equation}
\noindent where the maximization is to be performed over all normalized quantum states ${\mbox r}ho$.
At first sight this quantity may seem extremely non linear. We will show that, actually, ({\mbox r}ef{output}) can be reformulated as a linear optimization over the set of separable states.
Denote by $\Omega_{AB}$ the Choi operator corresponding to $\omega$, i.e., $\omega({\mbox r}ho)=\mbox{tr}_A(\Omega_{AB}\cdot{\mathbb I}_A\otimes{\mbox r}ho)$. It follows that
\begin{equation}
\nu=\max_{{\mbox r}ho}\|\omega({\mbox r}ho)\|_{\infty}=\max_{{\mbox r}ho,\sigma}\mbox{tr}(\Omega_{AB}\cdot\sigma\otimes{\mbox r}ho),
\end{equation}
\noindent with $\sigma, {\mbox r}ho\geq 0, \mbox{tr}({\mbox r}ho)=\mbox{tr}(\sigma)=1$.
Or, equivalently,
\begin{equation}
\nu=\max\{\mbox{tr}(\Omega_{AB}{\cal L}ambda_{AB}):{\cal L}ambda_{AB}\in \bar{S}\}.
\end{equation}
As in the state estimation case, it is possible to define decreasing sequences $(\nu^N)_N, (\nu_p^N)_N$ of upper bounds on $\nu$ that converge asymptotically to the optimal output purity of the channel.
Using Theorems {\mbox r}ef{bosesym} and {\mbox r}ef{egregium}, together with the fact that $\mbox{tr}_B(\Omega_{AB})={\mathbb I}_A$, we have that there exist sequences $(\tilde{\nu}^N)_N, (\tilde{\nu}_p^N)_N$ of lower bounds on $\nu$ given by
\begin{equation}gin{eqnarray}
\tilde{\nu}^N=& &\frac{N}{N+d}\nu^N+\frac{1}{N+d},\nonumber\\
\tilde{\nu}^N_p=& &\left(1-\frac{dg_N}{2(d-1)}{\mbox r}ight)\nu_p^N+\frac{g_N}{2(d-1)}.
\end{eqnarray}
\subsection{Geometric entanglement of tripartite pure states}
Let $\ket{{\cal P}si}_{ABC}$ be a pure tripartite state.
A popular entanglement measure for this kind of systems is the so called \emph{geometric entanglement} \cite{S95,geometric} (that some mathematicians may recognize as the square of the \emph{$\epsilon$-norm} \cite{tensor}), defined as
\begin{equation}
E=\max_{\overline{\pi}hi_A,\overline{\pi}hi_B,\overline{\pi}hi_C}|\bra{\overline{\pi}hi_A}\bra{\overline{\pi}hi_B}\bra{\overline{\pi}hi_C}\ket{{\cal P}si_{ABC}}|^2.
\end{equation}
\noindent Notice, though, that, if we fix $\overline{\pi}hi_A$ and $\overline{\pi}hi_B$, the state $\overline{\pi}hi_C$ maximizing the overlap will have to be proportional to $\bra{\overline{\pi}hi_A}\bra{\overline{\pi}hi_B}\ket{{\cal P}si_{ABC}}$.
This overlap will be therefore equal to
\begin{equation}gin{eqnarray}
& &\mbox{tr}_C(\bra{\overline{\pi}hi_A}\bra{\overline{\pi}hi_B}\ket{{\cal P}si_{ABC}}\bra{{\cal P}si_{ABC}}\ket{\overline{\pi}hi_A}\ket{\overline{\pi}hi_B}=\nonumber\\
& &=\mbox{tr}({\mbox r}ho_{AB}\overline{\pi}roj{\overline{\pi}hi_A}\otimes\overline{\pi}roj{\overline{\pi}hi_B}),
\end{eqnarray}
\noindent where ${\mbox r}ho_{AB}=\mbox{tr}_C(\overline{\pi}roj{{\cal P}si_{ABC}})$. It follows that $E$ can also be reformulated as a linear optimization over $S$, i.e.,
\begin{equation}
E=\max\{\mbox{tr}({\cal L}ambda_{AB}{\mbox r}ho_{AB}):{\cal L}ambda_{AB}\in \bar{S}\}.
\end{equation}
As before, converging and decreasing sequences $(E^N)_N,(E^N_p)_N$ of upper bounds on $E$ can be derived via the DPS criterion, and Theorems {\mbox r}ef{bosesym}, {\mbox r}ef{egregium} allow us to obtain complementary increasing sequences of lower bounds $(\tilde{E}^N)_N,(\tilde{E}^N_p)_N$, given by
\begin{equation}gin{eqnarray}
\tilde{E}^N=& \frac{N}{N+d}E^N+\frac{1}{N+d}\lambda_{A},\nonumber\\
\tilde{E}^N_p=& \left(1-\frac{dg_N}{2(d-1)}{\mbox r}ight)E_p^N+\frac{g_N}{2(d-1)}\lambda_{A}.
\end{eqnarray}
\noindent Here $\lambda_{A}$ denotes the smallest eigenvalue of ${\mbox r}ho_A$.
\section{Proof of Theorems {\mbox r}ef{bosesym}, {\mbox r}ef{egregium}}
\label{prueba}
The purpose of this section is to derive Theorems {\mbox r}ef{bosesym}, {\mbox r}ef{egregium}. But first, a few words on notation.
Given a unitary operator $U$, by $\ket{U}$ we will denote the state $U\ket{0}$. Also, for any permutation $\overline{\pi}i\in P_N$, $V_\overline{\pi}i\in B({\cal H}^{\otimes N})$ will represent the corresponding permutation operator. $V$ alone must be understood as the SWAP operator acting over a bipartite system ${\cal H}^{\otimes 2}$, i.e.,
\begin{equation}
V=\sum_{i,j=0}^d \ket{i}\ket{j}\bra{j}\bra{i}.
\end{equation}
\noindent To finish, ${\cal H}^N_{\mbox{sym}}$ will denote the symmetric subspace of ${\cal H}^{\otimes N}$ (the dimension of ${\cal H}$ will be clear from the context).
We will now proceed to proof Theorems {\mbox r}ef{bosesym}, {\mbox r}ef{egregium}.
\noindent The basic idea for both proofs is to notice that the original problem of finding a separable state \footnote{Along this proof, we will consider the operator ${\cal L}ambda_{AB}$ to be a quantum state rather than a quantum operator, i.e., to be normalized. It is easy to see that, if ({\mbox r}ef{canonic1}) and ({\mbox r}ef{canonic2}) are separable states when ${\cal L}ambda_{AB}$ is normalized, the very same expressions have to lead to separable operators with $\mbox{tr}(\tilde{{\cal L}ambda}_{AB})=\mbox{tr}(\tilde{{\cal L}ambda}_{AB})$ if ${\cal L}ambda_{AB}$ is not normalized.} $\tilde{{\cal L}ambda}_{AB}$ very close to ${\cal L}ambda_{AB}$ from its BSE ${\cal L}ambda_{AB^N}$ can be viewed
as a \emph{probabilistic state estimation problem} \cite{fiurasek}.
Consider the following protocol, in which Alice plays a passive part:
\begin{equation}gin{enumerate}
\item A copy of ${\cal L}ambda_{AB^N}$ is distributed to two parties, Alice and Bob.
\item Bob performs performs an incomplete measurement over ${\cal H}_B^{\otimes N}$, described by the POVM $\{M_x\geq 0\}_x$, with $\sum_x M_x\leq {\mathbb I}$.
As a result, he obtains
either an outcome $x$ or a \emph{FAIL message}, indicating that his measurement
has failed to produce an outcome.
\item If Bob receives a FAIL message, then he makes it public. Otherwise, he prepares a state $\sigma_x\in B({\cal H}_B)$, and both Alice and Bob would output the state $\frac{\mbox{tr}_{B^N}(M_x{\cal L}ambda_{AB^N})\otimes\sigma_x}{p_x}$
with probability $p_x=\mbox{tr}_{B^N}(M_x{\cal L}ambda_{B^N})$.
\end{enumerate}
The state Alice and Bob will produce conditioned on a non FAIL message will be then given by
\begin{equation}
\tilde{{\cal L}ambda}_{AB}=\sum_x \frac{\mbox{tr}_{B^N}(M_x{\cal L}ambda_{AB^N})\otimes\sigma_x}{\sum_y
p_y},
\end{equation}
\noindent and is, therefore, a separable state. Moreover, since any entanglement breaking map can be decomposed as a measurement followed by the preparation of a state, this is the most general linear map we can apply over ${\cal H}_B^{\otimes N}$ in order to return a separable state $\tilde{{\cal L}ambda}_{AB}$.
But how to find a measure-and-prepare strategy for Bob such that $\tilde{{\cal L}ambda}_{AB}$ is close to ${\cal L}ambda_{AB}$?
A possible scheme could be that Bob \emph{pretended} that his subsystems are $N$ identical copies of an unknown pure state, performed tomography over each of these subsystems independently and then prepared a state consistent with the average values he would measure.
This strategy should give good results in the particular case where ${\cal L}ambda_{AB^N}$ can be approximated by a state of the form
\begin{equation}
\int p(U)dU {\mbox r}ho_U\otimes \overline{\pi}roj{U}^{\otimes N}.
\label{convexco}
\end{equation}
However, supposing that the state had the form above, an even better strategy would be to allow Bob to perform \emph{collective} measurements over his subsystems and then prepare the most convenient state.
In conclusion, Bob should apply a POVM that allows him to efficiently identify
the state $U\overline{\pi}roj{0}U^\dagger$ out of $N$ copies of it. Because in
principle Bob has no a priori knowledge of $p(U)dU$, it is reasonable
that he assumes that $p(U)dU=dU$, the Haar measure.
In this particular case,
the best state estimation strategy and the best probabilistic state estimation
strategy coincide \cite{fiurasek}.
This implies that Bob should apply the POVM $\{\overline{\pi}roj{U}^{\otimes N}dU\}_U$ and prepare the state $\overline{\pi}roj{U}$ whenever he gets the result $U$. Therefore,
\begin{equation}
\tilde{{\cal L}ambda}_{AB}=\frac{\int dU \mbox{tr}_{B^N}\left({\mathbb I}_A\otimes \overline{\pi}roj{U}^{\otimes N+1}{\cal L}ambda_{AB^N}\otimes
{\mathbb I}_B{\mbox r}ight)}{\int dU \mbox{tr}(\overline{\pi}roj{U}^{\otimes N}{\mbox r}ho_{B^N})}.
\label{canonic}
\end{equation}
\noindent To evaluate these integrals it is enough to notice that
\begin{equation}gin{enumerate}
\item For any operator $C$,
\begin{equation}
\int dU U^{\otimes N}C(U^\dagger)^{\otimes N}=\sum_{\overline{\pi}i\in P_N} c_\overline{\pi}i
V_\overline{\pi}i,
\end{equation}
\noindent for some coefficients $c_\overline{\pi}i$. In particular,
\begin{equation}gin{eqnarray}
& &\int dU \overline{\pi}roj{U}^{\otimes N}=\frac{(d-1)!N!P_{\mbox{sym}}^N}{(N+d-1)!}=\nonumber\\
& &=\frac{(d-1)!\sum_{\overline{\pi}i\in
P_N}V_\overline{\pi}i}{(N+d-1)!}.
\end{eqnarray}
\item Due to the fact that ${\cal L}ambda_{AB^N}$ acts over ${\cal H}_A\otimes{\cal H}_{\mbox{sym}}^N$, for any $\overline{\pi}i\in P_{N+1}$,
\begin{equation}gin{eqnarray}
& &\mbox{tr}_{B^N}\{({\cal L}ambda_{AB^N}\otimes {\mathbb I}_B) {\mathbb I}_A\otimes V_\overline{\pi}i\}=\nonumber\\
& &=\Big\{\begin{equation}gin{array}{l}{\cal L}ambda_A\otimes {\mathbb I}_B,\mbox{ if } \overline{\pi}i(N+1)=N+1;\\
{\cal L}ambda_{AB}, \mbox{ otherwise}.\end{array}
\end{eqnarray}
\end{enumerate}
\noindent Finally, we arrive at the expression
\begin{equation}
\tilde{{\cal L}ambda}_{AB}=\frac{N}{N+d}{\cal L}ambda_{AB}+\frac{1}{N+d}{\cal L}ambda_A\otimes{\mathbb I}_B.
\end{equation}
\noindent We have just proven Theorem {\mbox r}ef{bosesym}.
The next step is to extend the previous ideas to account for the PPT condition, and a possible way is to modify the previous bipartite protocol to give Bob the ability
to transpose part of his state before proceeding with any measure-and-prepare scheme. Suppose then that Bob partially transposes a partition $B'$,
corresponding to half of Bob's systems
in ${\cal L}ambda_{AB^N}$ (we will take $N$ even for simplicity). Following the previous arguments, Bob could pretend that he and Alice are sharing a state ${\cal L}ambda_{AB}^{T_{B'}}$
very similar to
\begin{equation}gin{equation}
\int p(U)dU {\mbox r}ho_U\otimes (\overline{\pi}roj{U}\otimes \overline{\pi}roj{U^*})^{\otimes N/2}.
\end{equation}
\noindent The benefits of this apparently useless step become evident when
we take into account the well established fact that it is easier to estimate a state from a copy and its complex conjugate than from two identical copies \cite{spinflip,fiurasek}. In the case of $N=2$, the optimal POVM has the form $\{U\otimes
U^*\overline{\pi}roj{\varphi}(U\otimes U^*)^\dagger dU\}$, where $\ket{\varphi}$ is a linear combination
of $\ket{00}$ and $\ket{{\cal P}si^+}=\sum_i \ket{ii}$, the (non normalized) maximally entangled state. The optimal
strategy for general $N$ is not known, but we suggest the measurement
\begin{equation}
\overline{\pi}hi_UdU\equiv(U\otimes U^*)^{\otimes N/2}\overline{\pi}roj{\overline{\pi}hi}(U^\dagger\otimes (U^*)^\dagger)^{\otimes N/2}
dU,
\end{equation}
\noindent followed by the preparation of $\overline{\pi}roj{U}$.
Here $\ket{\overline{\pi}hi}$ is an arbitrary linear combination of the states \footnote{Note that a further symmetrization of these states over the particles in $B'$ and $B\\B'$ would be more intuitive, but irrelevant, since the support of the state ${\cal L}ambda_{AB^N}^{T_{B'}}$ is in ${\cal H}_A\otimes{\cal H}_{\mbox{sym}}^{N/2}\otimes{\cal H}_{\mbox{sym}}^{N/2}$.
That is, such symmetrization is automatically performed when we apply this POVM over ${\cal L}ambda_{AB^N}^{T_{B'}}$.} $\ket{\overline{\pi}hi_n}\equiv\ket{00}^{\otimes n}\ket{{\cal P}si^+}^{N/2-n}$, i.e.,
\begin{equation}
\ket{\overline{\pi}hi}=\sum_{n=0}^{N/2}c_n \ket{\overline{\pi}hi_n}.
\end{equation}
\noindent Of course, applying the POVM $\overline{\pi}hi_U$ over ${\cal L}ambda_{AB}^{T_{B'}}$ is equivalent to apply
the (non positive!) map associated to $U^{\otimes N}\overline{\pi}roj{\overline{\pi}hi}^{T_{B'}}(U^\dagger)^{\otimes N/2}$ over our state ${\cal L}ambda_{AB^N}$. That way, we can use the same tricks
employed in the computation of ({\mbox r}ef{canonic}).
A fast way to perform these calculations is to notice that, for $m>n$,
\small
\begin{equation}
\ket{\overline{\pi}hi_n}\bra{\overline{\pi}hi_m}^{T_{B'}}=\overline{\pi}roj{00}^{\otimes n}\otimes ({\mathbb I}\otimes\overline{\pi}roj{0})^{\otimes m-n}\otimes V^{\otimes N/2-m}.
\end{equation}
\normalsize
\noindent Therefore, there exists a pair of permutations $\overline{\pi}i,\overline{\pi}i'\in P_N$ such that
\begin{equation}
V_\overline{\pi}i\ket{\overline{\pi}hi_n}\bra{\overline{\pi}hi_m}^{T_{B'}}V_{\overline{\pi}i'}^\dagger=\overline{\pi}roj{0}^{\otimes m+n}\otimes {\mathbb I}^{\otimes N-m-n}.
\end{equation}
But ${\mathbb I}_A\otimes V_\overline{\pi}i^\dagger {\cal L}ambda_{AB^N}= {\cal L}ambda_{AB^N}{\mathbb I}_A\otimes V_\overline{\pi}i={\cal L}ambda_{AB^N}$, so
\begin{equation}gin{eqnarray}
&\mbox{tr}_{B^N}({\cal L}ambda_{AB^N}{\mathbb I}_A\otimes U^{\otimes N}\ket{\overline{\pi}hi_n}\bra{\overline{\pi}hi_m}^{T_{B'}}(U^\dagger)^{\otimes N})=\nonumber\\
&=\mbox{tr}_{B^N}({\cal L}ambda_{AB^N}{\mathbb I}_A\otimes\overline{\pi}roj{U}^{\otimes m+n}\otimes {\mathbb I}^{\otimes N-m-n}).
\end{eqnarray}
\noindent In the end, we have that
\begin{equation}
\tilde{{\cal L}ambda}_{AB}=\left(1-d\frac{\vec{c}^\dagger\tilde{A}\vec{c}}{\vec{c}^\dagger\tilde{B}\vec{c}}{\mbox r}ight){\cal L}ambda_{AB}+\frac{\vec{c}^\dagger\tilde{A}\vec{c}}{\vec{c}^\dagger\tilde{B}\vec{c}}{\cal L}ambda_A\otimes{\mathbb I}_B,
\label{PPT}
\end{equation}
\noindent where $\tilde{A}$ and $\tilde{B}$ are square matrices given by
\begin{equation}gin{eqnarray}
&\tilde{B}_{nm}=\frac{(n+m)!}{(n+m+d-1)!},\tilde{A}_{nm}=\frac{(n+m)!}{(n+m+d)!},\nonumber\\
&n,m=0,1,...,N/2.
\end{eqnarray}
In case of odd $N$, we would make Bob partially transpose $(N-1)/2$ parts
of his state and then use the following (incomplete) POVM:
\begin{equation}
U^{\otimes N}\overline{\pi}roj{\overline{\pi}hi}^{T_{B'}}\otimes\overline{\pi}roj{0}(U^\dagger)^{\otimes N}dU.
\end{equation}
\noindent After the appropriate computations, we again arrive at expression
({\mbox r}ef{PPT}), but the form of $\tilde{A}$ and $\tilde{B}$ changes to:
\begin{equation}gin{eqnarray}
&\tilde{B}_{nm}=\frac{(n+m+1)!}{(n+m+d)!},\tilde{A}_{nm}=\frac{(n+m+1)!}{(n+m+d+1)!},\nonumber\\
&n,m=0,1,...,(N-1)/2.
\end{eqnarray}
Obviously, in order to guarantee that ${\cal L}ambda_{AB}$ is close to $\tilde{{\cal L}ambda}_{AB}$,
it is in our interest to minimize the quantity
\begin{equation}
f_N(\vec{c})\equiv\frac{\vec{c}^\dagger\tilde{A}\vec{c}}{\vec{c}^\dagger\tilde{B}\vec{c}}
\label{complicaciones}
\end{equation}
\noindent over all possible vectors $\vec{c}$. Details on how to calculate the minimum of ({\mbox r}ef{complicaciones}), together with the expression of the
optimal $\vec{c}$ can be found in Appendix {\mbox r}ef{minimum}. The result is:
\begin{equation}
\min_{\vec{c}}f_N(\vec{c})=\frac{1}{2(d-1)}g_N.
\end{equation}
\noindent This concludes the proof of Theorem {\mbox r}ef{egregium}.
Notice that in both cases the given separable decomposition of the states $\tilde{{\cal L}ambda}_{AB}$ is continuous. Because of the presence of the Haar measure, however, via Design Theory it is possible to arrive at an approximate \cite{emerson} or exact \cite{hayashi} finite separable decomposition for these operators.
\section{Extensions to multiseparability} \label{sec: multi}
So far, we have only been considering separability in \emph{bipartite} systems.
In this section, we show that almost all the results we have derived
can be easily extended to deal with separability in $m$-partite scenarios.
More concretely, we will show how to generalize Theorems {\mbox r}ef{bosesym} and {\mbox r}ef{egregium} to the multipartite case, since, as we have already seen, most of the other results are just corollaries of these two theorems.
In this case, we will be interested in sets $S^N$ of states that derive from an $N$ \emph{locally} (PPT) Bose-symmetric extension\cite{trisep}.
\begin{equation}gin{defin}{$N$ locally Bose-symmetric extension}\\
Let ${\cal L}ambda_{123...}\in {\cal B}({\cal H}_1\otimes{\cal H}_2\otimes{\cal H}_3\otimes...)$ be a non negative operator. We will say that ${\cal L}ambda_{12^{N}3^{N}...}\in {\cal B}({\cal H}_1\otimes{\cal H}_2^{\otimes N}\otimes{\cal H}_3^{\otimes N}\otimes...)$ is an $N$ locally Bose symmetric extension of ${\cal L}ambda_{123...}$ iff:
\begin{equation}gin{enumerate}
\item ${\cal L}ambda_{12^{N}3^{N}...}\geq 0$.
\item $\mbox{tr}_{2^{N-1}3^{N-1}...}({\cal L}ambda_{12^{N}3^{N}...})={\cal L}ambda_{123...}$.
\item ${\cal L}ambda_{12^N3^N...}$ is independently Bose symmetric in systems $2,3,4...$.
\end{enumerate}
\end{defin}
As before, in case such extension is PPT with respect to some partition, we will denote it as an \emph{$N$ PPT locally Bose-symmetric extension}.
How close is ${\cal L}ambda_{123...}$ to the set of separable states? Consider a triseparable system, for instance, and suppose that we have an $N$ locally Bose-symmetric extension ${\cal L}ambda_{AB^{N}C^{N}}$ for ${\cal L}ambda_{ABC}$. In order to estimate the distance of ${\cal L}ambda_{ABC}$ to the set of triseparable states we could conceive a protocol where the state ${\cal L}ambda_{AB^{N}C^{N}}$ is distributed between Alice, Bob and Charlie. As before, Bob and Charlie could then independently apply probabilistic state estimation over their subsystems and prepare both a quantum state depending on their measurement outcomes.
From what we already have, the derivation of the final expression of the triseparable state $\tilde{{\cal L}ambda}_{ABC}$ is straightforward. Equation ({\mbox r}ef{canonic1}) describes the action of Bob's strategy over \emph{any} bipartite state. Considering the partition $AC^N|B^N$, it follows that the resulting tripartite state after Bob performs state estimation will be:
\begin{equation}
\frac{N}{N+d_B}{\cal L}ambda_{ABC^N}+\frac{1}{N+d_B}{\cal L}ambda_{AC^N}\otimes{\mathbb I}_B.
\end{equation}
Now it is Charlie's turn. This time we will take the partition $AB|C^N$. The final result is that
\small
\begin{equation}gin{eqnarray}
&\tilde{{\cal L}ambda}_{ABC}=\frac{N^2}{(N+d_B)(N+d_C)}{\cal L}ambda_{ABC}+\frac{N}{(N+d_B)(N+d_C)}{\cal L}ambda_{AB}\otimes{\mathbb I}_C+\nonumber\\
&+\frac{N}{(N+d_B)(N+d_C)}{\cal L}ambda_{AC}\otimes{\mathbb I}_B+\frac{1}{(N+d_B)(N+d_C)}{\cal L}ambda_{A}\otimes{\mathbb I}_{BC}
\end{eqnarray}
\normalsize
\noindent is a triseparable state.
The generalization to more parties is immediate. Invoking again the definition of depolarizing channels ({\mbox r}ef{depolarizing}), in $m$-partite separability the expression for $\tilde{{\cal L}ambda}_{1234...}$ would be
\begin{equation}
\tilde{{\cal L}ambda}_{1234...}=({\mathbb I}_1\bigotimes_{i=2}^m\Omega_{(p_i)})({\cal L}ambda_{1234...}),
\label{multisep}
\end{equation}
\noindent where
\begin{equation}
p_i=\frac{d_i}{N+d_i}.
\end{equation}
The corresponding expression for $\tilde{{\cal L}ambda}_{123...}$ when it arises from an $N$ locally Bose-symmetric extension, PPT with respect to the partition $12^{\lceil N/2{\mbox r}ceil} 3^{\lceil N/2{\mbox r}ceil}...|2^{\lfloor N/2{\mbox r}floor} 3^{\lfloor N/2{\mbox r}floor}...$, is still
({\mbox r}ef{multisep}), but this time
\begin{equation}
p_i=\frac{d_i}{2(d_i-1)}g_N^{(d_i)}.
\end{equation}
\section{The power of PPT alone}
\label{sec: PPT}
The Peres-Horodecki criterion, aka the PPT (Positive Partial Transpose) criterion \cite{P96}, is one of the most popular existent criteria for entanglement detection. It is simple, it provides a very good approximation to the set of separable states in small dimensional cases and it usually leads to analytical results when applied over families of quantum states. Actually, some entanglement measures, like the negativity \cite{neg1} or the PPT entanglement robustness \cite{intro_measures} are based on the PPT condition.
It is interesting, thus, to try to determine how good the PPT criterion is for entanglement detection \emph{alone}, i.e., not in combination with Doherty et al.'s method. Here, through a very simple argument, we show what we believe is the first result in this direction after the seminal paper of the Horodeckis \cite{PPThoro}.
The main idea of our derivation stems from the fact that positivity under partial transposition is equivalent to separability in ${\mathbb C}^3\otimes{\mathbb C}^2$ systems \cite{PPThoro}. Suppose, then, that we have a PPT state ${\mbox r}ho_{AB}\in B({\cal H}_A\otimes {\cal H}_B)$, with $d_A\geq 3$, and $d_B\geq 2$, and consider the (non normalized) state $\tilde{{\mbox r}ho}_{AB}$ given by
\begin{equation}
\tilde{{\mbox r}ho}_{AB}\overline{\pi}ropto\int dUdW P_{U}^3\otimes P_{W}^2{\mbox r}ho_{AB}P_{U}^3\otimes P_{W}^2,
\label{decomp}
\end{equation}
\noindent where $dU$ and $dW$ denote the Haar measures corresponding to $S(d_A)$ and $SU(d_B)$, respectively, and
\begin{equation}
P_{U}^3\equiv U\sum_{k=0}^2\overline{\pi}roj{k} U^\dagger,P_{W}^2\equiv W\sum_{k=0}^1\overline{\pi}roj{k} W^\dagger.
\end{equation}
It follows that $\tilde{{\mbox r}ho}_{AB}$ is a convex combination of unnormalized states ${\mbox r}ho_{U,W}\equiv P_{U}^3\otimes P_{W}^2{\mbox r}ho_{AB}P_{U}^3\otimes P_{W}^2$, with ${\mbox r}ho_{U,W}\in B({\mathbb C}^3\otimes{\mathbb C}^2)$. Notice, also, that each ${\mbox r}ho_{U,W}$ is PPT, since
\begin{equation}gin{eqnarray}
{\mbox r}ho_{U,W}^{T_B}=& &(P_{U}^3\otimes P_{W}^2{\mbox r}ho_{AB}P_{U}^3\otimes P_{W}^2)^{T_B}=\nonumber\\
=& &P_{U}^3\otimes P_{W^*}^2{\mbox r}ho^{T_B}_{AB}P_{U}^3\otimes P_{W^*}^2\geq 0.
\end{eqnarray}
Since PPT equals separability in ${\mathbb C}^3\otimes {\mathbb C}^2$ systems, it follows that each ${\mbox r}ho_{U,W}$ is separable, and so is $\tilde{{\mbox r}ho}_{AB}$, since by construction it is a convex combination of these states.
It only rests to find an analytical expression for $\tilde{{\mbox r}ho}_{AB}$. Using the previous techniques it is straightforward to arrive at
\begin{equation}gin{theo}
Let ${\mbox r}ho_{AB}\in B({\cal H}_A\otimes{\cal H}_B)$ be a PPT normalized quantum state, with $d_A\geq 3, d_B\geq 2$. Then, for
\begin{equation}
p_A=\frac{d_A(d_A-3)}{d_A^2-1},p_B=\frac{d_B(d_B-2)}{d_B^2-1},
\end{equation}
\noindent the state $\Omega^{(p_A)}\otimes\Omega^{(p_B)}({\mbox r}ho_{AB})$ is separable.
\end{theo}
\noindent Note that, in the particular case $d_A=3,d_B=2$, $\tilde{{\mbox r}ho}_{AB}={\mbox r}ho_{AB}$.
By simple application of the tools already developed, we end up with the following Corollary.
\begin{equation}gin{cor}
For any PPT state ${\mbox r}ho_{AB}$, with $d_A\geq 3,d_B\geq2$,
\begin{equation}
R_G({\mbox r}ho_{AB})\leq \frac{1}{12}(d_A+1)(d_B+1)-1,
\end{equation}
\noindent and there exists a separable state $\sigma$ such that
\begin{equation}
\|{\mbox r}ho_{AB}-\sigma\|_1\leq 2-\frac{24}{(d_A+1)(d_B+1)}.
\end{equation}
\end{cor}
\begin{equation}gin{figure}
\centering
\includegraphics[width=8 cm]{robust_PPT.eps}
\caption{Optimum bound on the global robustness of entanglement $R$ for generic states (dashed line), as opposed to the upper bound for PPT states (solid line). In this plot, we assume that $d_A=d_B=d$. Note that the new bound becomes trivial as soon as $d>9$.}
\label{PPT}
\end{figure}
To get an idea on how good these bounds are, have a look at Figure {\mbox r}ef{PPT}. There the maximum possible \emph{global} robustness of entanglement of a ${\mathbb C}^d\times {\mathbb C}^d$ state is compared with our upper bound for PPT states. We see that, although our upper bound becomes useless for $d>9$, it is very powerful in the small dimensional case. For instance, for ${\mathbb C}^3\times {\mathbb C}^3$ systems, the bound is equal to $1/3$ as opposed to $2$. This means that we would have to apply the non PPT version of the DPS method up to $N=6$ in order to characterize likewise the set of separable states.
\section{Conclusion}
\label{conclusion}
In this paper, we have studied the efficiency of the DPS criterion for entanglement detection. First, we showed that it is enough to subject the DPS states to some local noise in order to deprive them from their entanglement properties. It turned out that, while the minimal amount of noise necessary to turn an arbitrary state in $\bar{S}^N$ into a separable state decreases as $O(1/N)$, the corresponding amount of noise needed to disentangle states in $\bar{S}^N_p$ decreases at least as $O(1/N^2)$. We used these expressions to estimate the time complexity of both methods when applied to solve the Weak Membership Problem of Separability, and concluded that the PPT condition is worth imposing provided that the \emph{optimal} bounds on the speed of convergence of the method based on plain BSEs scale as $O(d/N)$, as our own bounds suggest. We therefore hope to have shed some light on the question of how much the DPS criterion owes its strength to the PPT condition.
We also derived bounds on the error we incur when we substitute the set of separable operators by $S^N$ or $S^N_p$ in linear optimization problems, like the state estimation problem, the problem of determining the maximal output purity of an arbitrary quantum channel and the computation of the geometric entanglement. We performed numerical calculations of the first of these problems to test the accuracy of our analytical bounds. In order to compare our uncertainty with the actual solution of the problem, we developed a new technique that allows to prove in some cases the optimality of the DPS relaxations. We observed that, although the bounds for the non PPT case seem to be very accurate, the bounds for the PPT case are too big when compared with reality.
This disagreement between theory and practice may be explained in part by the fact that our bounds do not take into account the dimensionality of Alice's system, a crucial fact when dealing with the PPT constraint \cite{PPThoro}. For all we know, our PPT bounds could be exact in the limit $d_A\to \infty$. Our intuition, nevertheless, is that better bounds could be found by applying linear maps over the initial state ${\mbox r}ho_{AB}$ in order to obtain a separable state $\tilde{{\mbox r}ho}_{AB}$, as we did, but whose separable decomposition would be given by a \emph{non linear map}, unlike in our examples. Actually, we already used that approach in Section {\mbox r}ef{sec: PPT} to bound the entanglement of PPT states. That kind of schemes, together with state estimation considerations, may allow in the future to obtain such better bounds.
\begin{equation}gin{appendix}
\section{Minimization of ({\mbox r}ef{complicaciones})}
\label{minimum}
Take $N$ even. Then it can be checked that
\begin{equation}gin{eqnarray}
& &\tilde{A}_{mn}=\int_0^1x^{m+n}\cdot\frac{(1-x)^{d-1}}{(d-1)!}dx,\nonumber\\
& &\tilde{B}_{mn}=\int_0^1x^{m+n}\cdot\frac{(1-x)^{d-2}}{(d-2)!}dx.
\label{idea1}
\end{eqnarray}
\noindent Combining this relation with ({\mbox r}ef{complicaciones}), it follows that
\begin{equation}
f(\vec{c})=\frac{1}{d-1}\frac{\int_0^1|\sum_{n=0}^{N/2}c_nx^n|^2(1-x)(1-x)^{d-2}dx}{\int_0^1|\sum_{n=0}^{N/2}c_nx^n|^2(1-x)^{d-2}dx}.
\end{equation}
That way, we can see the minimization of $f(\vec{c})$ as a minimization over
the set of all polynomials $Q_{N/2}(x)=\sum c_nx^n$ of degree $N/2$. Making the
change of coordinates $y=2x-1$ we find that the above minimization is equivalent
to
\begin{equation}
\min_{Q_{N/2}}\frac{1}{2(d-1)}\frac{\int_{-1}^1|Q_{N/2}(y)|^2(1-y)^{d-1}dy}{\int_{-1}^1|Q_{N/2}(y)|^2(1-y)^{d-2}dy},
\end{equation}
\noindent where $Q_{N/2}(y)$ is an arbitrary polynomial of order $N/2$. This
problem can be solved by means of the \emph{Jacobi polynomials}.
The Jacobi polynomials $P_n^{(\alpha,\begin{equation}ta)}(y)$ are a complete set of functions orthogonal upon integration in the interval $[-1,1]$ under the weight $(1+y)^\begin{equation}ta(1-y)^\alpha$
\cite{abramo}.
Now, define the \emph{normalized Jacobi polynomials} $p_n(y)$ as
\begin{equation}
p_n(y)\equiv \frac{P^{(d-2,0)}_n(y)}{\|P^{(d-2,0)}_n\|},
\end{equation}
\noindent with
\begin{equation}
\|P^{(d-2,0)}_n\|= \sqrt{\int_{-1}^1|P^{(d-2,0)}_n(y)|^2(1-y)^{d-2}dy}.
\end{equation}
It is
clear that we can express any $Q_{N/2}(y)$ as a linear combination of normalized
Jacobi polynomials
of order less or equal than $N/2$. That is,
\begin{equation}
Q_{N/2}(y)=\sum_{n=0}^{N/2} e_np_n(y),
\end{equation}
\noindent for some coefficients $e_n$. Because of the orthogonality of the
$p_n$'s, when we input this expression in the
integral of the denominator, we end up with
\begin{equation}
\int_{-1}^1|Q_{N/2}(y)|^2(1-y)^{d-2}dy=\sum_n |e_n|^2.
\end{equation}
To calculate the integral on the numerator, we can make use of the recurrence
relation
\begin{equation}
(1-y)p_n(y)=\alpha_np_n(y)+\begin{equation}ta_np_{n+1}(y)+\gamma_np_{n-1}(y),
\label{recursion}
\end{equation}
\noindent that holds for some coefficients $\alpha_n,\begin{equation}ta_n,\gamma_n$, with $\gamma_0=0$ and
$\gamma_{n+1}=\begin{equation}ta_n$ \cite{abramo}. Invoking again the orthogonality of the Jacobi
polynomials, we have that
\begin{equation}
\min_{\vec{c}}f(\vec{c})=\min_{|\vec{e}|^2=1}\frac{1}{2(d-1)}\vec{e}^\dagger
\tilde{C}\vec{e},
\end{equation}
\noindent where $\tilde{C}$ is an $(N/2+1)\times (N/2+1)$ tridiagonal hermitian matrix given by
\begin{equation}gin{eqnarray}
\tilde{C}_{m,n}=& &\alpha_n, \mbox{ if } m=n,\nonumber\\
& &\begin{equation}ta_n, \mbox{ if } m=n+1,\nonumber\\
& &\gamma_n, \mbox{ if } m=n-1,\nonumber\\
& &0 \mbox{ elsewhere}.
\end{eqnarray}
\noindent Now we will proceed to diagonalize $\tilde{C}$.
Let $\lambda$ be an eigenvalue
of $\tilde{C}$. This means that there exists a vector $\{v_i\}_{i=0}^{N/2+1}$ such that
\begin{equation}gin{equation}
(\alpha_n-\lambda)v_n+\begin{equation}ta_nv_{n+1}+\gamma_nv_{n-1}=0,
\label{eigenvector}
\end{equation}
\noindent with $v_{N/2+1}=0$.
Choose a real number $y_0$ and try the ansatz $v_n=p_n(y_0)$. From ({\mbox r}ef{recursion}), it is clear that $v_n$ will satisfy ({\mbox r}ef{eigenvector}),
provided that
\begin{equation}gin{eqnarray}
&\lambda=1-y_0,\nonumber\\
&p_{N/2+1}(y_0)=0.
\end{eqnarray}
\noindent That is, any root of
the polynomial $p_{N/2+1}(y)$ corresponds to an eigenvalue of $\tilde{C}$.
But $p_{N/2+1}(y)$ has $N/2+1$ \emph{simple} roots \cite{abramo}, so all the eigenvalues of $\tilde{C}$ are obtained using this strategy. It follows that
\begin{equation}
\min_{\vec{c}}f_N(\vec{c})=\frac{1}{2(d-1)}\min\{1-x:P_{N/2+1}^{(d-2,0)}(x)=0\}.
\end{equation}
\noindent Let us remark that this is not the first time the zeros of the Jacobi polynomials naturally appear in state estimation problems \cite{jacobino}.
The expression for the case of odd $N$ can be derived in an analogous way taking into account that, this time,
\begin{equation}gin{eqnarray}
& &\tilde{A}_{mn}=\int_0^1x^{m+n}\cdot\frac{x(1-x)^{d-1}}{(d-1)!}dx,\nonumber\\
& &\tilde{B}_{mn}=\int_0^1x^{m+n}\cdot\frac{x(1-x)^{d-2}}{(d-2)!}dx.
\end{eqnarray}
\section{Optimality criterion (rank loops)}
\label{optim}
For some problems involving linear optimizations over the set $S$, it may happen (see \cite{pasado})
that a particular relaxation of the problem $F^N$ turns out to coincide with $F$. In this appendix we will show how this optimality can sometimes be detected.
We will take inspiration from optimality detection in other hierarchies of semidefinite programs that appear in scientific literature.
Consider the hierarchy of semidefinite programs used in \cite{qcorr} for the calculation of the maximal violation of linear Bell inequalities.
There the optimality of a relaxation is detected when the rank of the matrix generated by the computer is equal to that of some of its submatrices.
Remarkably, we can find similar results in the hierarchies of semidefinite programs defined by Henrion and Lasserre
to minimize real polynomials in a bounded region of $\mathbb{R}^n$ \cite{lasserre}.
The corresponding result in this scenario is the following:
\begin{equation}gin{lemma}
\label{rankloop}
Let ${\cal L}ambda_{AB^N}$ be a BSE of ${\cal L}ambda_{AB}$, PPT with respect to the partition $AB^K|B^{N-K}$. If
\begin{equation}
\mbox{rank}({\cal L}ambda_{AB^N})\leq \max\{\mbox{rank}({\cal L}ambda_{AB^K}),\mbox{rank}({\cal L}ambda_{B^{N-K}})\}
\label{optimality}
\end{equation}
\noindent then ${\cal L}ambda_{AB}$ is a separable operator.
\end{lemma}
Following \cite{qcorr}, we will say that ${\cal L}ambda_{AB^N}$ presents a \emph{rank loop} when it fulfills condition ({\mbox r}ef{optimality}).
The proof of Lemma {\mbox r}ef{rankloop} follows trivially from an old result by Horodecky et al. \cite{lowrank}:
\begin{equation}gin{theo}
Let ${\mbox r}ho_{AB}$ be a PPT bipartite quantum state. If
\begin{equation}
\mbox{rank}({\mbox r}ho_{AB})\leq \mbox{rank}({\mbox r}ho_{A}),
\end{equation}
\noindent then ${\mbox r}ho_{AB}$ is a separable state.
\end{theo}
\noindent See \cite{lowrank} for a proof.
The possibility of finding a rank loop in practice in cases where the optimization over the set $S_p^N$ coincides with the optimization over $S$ should not be surprising. Note that any (finite dimensional) separable state ${\cal L}ambda_{AB}$ can be expressed as a finite convex combination of product states, i.e.,
\begin{equation}
{\cal L}ambda_{AB}=\sum_{i=1}^K p_i {\mbox r}ho_i\otimes \overline{\pi}roj{\overline{\pi}si_i}, \mbox{ with } p_i>0,\forall i,
\end{equation}
\noindent with $\overline{\pi}roj{\overline{\pi}si_i}\not=\overline{\pi}roj{\overline{\pi}si_j}$, for $i\not=j$. Now, consider the PPT Bose symmetric extension of ${\cal L}ambda_{AB}$ given by
\begin{equation}
{\cal L}ambda_{AB^N}=\sum_{i=1}^K p_i {\mbox r}ho_i\otimes \overline{\pi}roj{\overline{\pi}si_i}^{\otimes N},
\end{equation}
Clearly, as $N$ tends to infinity, the vectors $\{\ket{\overline{\pi}si_i}^{\otimes N}\}_i$ become orthogonal. It follows that $K^*\equiv\lim_{N\to\infty}\mbox{rank}({\cal L}ambda_{AB^N})$ exists and is equal to $\sum_i\mbox{rank}({\mbox r}ho_i)$. Being the rank a natural number, this implies that there is an $M$ such that, for any $N> M$, $\mbox{rank}({\cal L}ambda_{AB^M})=\mbox{rank}({\cal L}ambda_{AB^N})=K^*$. That is, for any finite dimensional separable state there exists a PPT Bose symmetric extension with a rank loop.
Of course, the fact that for any separable state ${\mbox r}ho_{AB}$ there exists a PPT BSE with a rank loop does not mean that our computer is going to return such an extension. Note, though, that, if at the same time we set our computer to the task of finding PPT BSEs of ${\mbox r}ho_{AB}$ we also demand a rank minimization of these matrices (i.e., we look for PPT BSEs with minimal rank), at some point we will find a rank loop.
Unfortunately, rank minimization of positive semidefinite matrices with linear constraints is in general an NP-hard problem \cite{PhD,sdp}. There are, however, heuristics \cite{maryam} that have proven to be very efficient for solving small-scale problems (that is, for small $d$).
\end{appendix}
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\end{document}
\end{document}
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\begin{document}
\thispagestyle{plain}
\begin{center}
\Large
\textsc{Resolutions of surfaces with big cotangent bundle and $A_2$ singularities}
\end{center}
\begin{center}
\textit{Bruno De Oliveira}, \textit{Michael Weiss}
\\
\begin{tabular}{l}
\small University of Miami \\
\small 1365 Memorial Dr \\
\small Miami, FL 33134 \\
\small e-mail: \texttt{[email protected]} \\
\small \phantom{e-mail: }\texttt{[email protected]}
\end{tabular}
\end{center}
\noindent
\textbf{Abstract} We give a new criterion for when a resolution of a surface of general type with canonical singularities has big cotangent bundle and a new lower bound for the values of $d$ for which there is a surface with big cotangent bundle that is deformation equivalent to a smooth hypersurface in $\Bbb P^3$ of degree $d$. This preprint is the base of the article to appear in the Boletim da SPM volume 77, December 2019 (special collection of the work of Portuguese mathematicians working abroad).
\noindent\textbf{keywords:}
big cotangent bundle; surfaces of general type; canonical singularities.
\section{Introduction and general theory}
\label{sec:a}
Symmetric differentials, i.e. sections of the symmetric powers of the cotangent bundle $S^m\Omega^1_X$, of a projective manifold $X$ play a role in obtaining hyperbolicity properties of $X$. Symmetric differentials give constraints on the existence of rational, elliptic and even entire curves in $X$ (nonconstant holomorphic maps from $\Bbb C$ to $X$), see for example \cite{demailly2015recent} and \cite{debarre2004hyperbolicity}.
\
The cotangent bundle of a projective manifold is said to be big if the order of growth of $h^0(X, S^m \Omega_X^1)$ with $m$ is maximal (i.e., $=2\dim X-1$). The work of Bogomolov \cite{bogomolov_finiteness} and McQuillan \cite{mcquillan} gives that if a surface of general type has big $\Omega^1_X$, then $X$ satisfies the Green-Griffiths-Lang conjecture, i.e., there exists a proper subvariety $Z$ of $X$ such that any entire curve is contained in $Z$.
\
Smooth hypersurfaces $X \subset \mathbb{P}^3$ with degree $d \geq 5$ have $\Omega_X^1$ with strong positivity properties, such as $K_X$ being ample, but they have trivial cotangent algebra \cite{Brjukman_1971},
$$S(X) : = \bigoplus_{m = 0}^\infty H^0(X, S^m\Omega_X^1) = H^0(X, S^0 \Omega_X^1) = \mathbb{C}$$
\noindent see also \cite{bogomolov2008symmetric}. The absence of symmetric differentials on smooth hypersurfaces of $\mathbb{P}^3$ a priori prevents them from playing a role in obtaining hyperbolicity properties on smooth hypersurfaces of $\mathbb{P}^3$.
\
Previous work of the 1st author and Bogomolov \cite{bogomolov_nodes} showed that there are smooth surfaces $X$ with big $\Omega^1_X$ that are deformation equivalent to smooth hypersurfaces in $\mathbb{P}^3$. Hence symmetric differentials can still play a role in obtaining hyperbolicity properties for hypersurfaces of $\mathbb{P}^3$. In \cite{bogomolov_nodes} it was shown that there are nodal hypersurfaces $X\subset \mathbb{P}^3$ whose resolutions $\tilde X$ have big cotangent bundle. The simultaneous resolution result of Brieskorn \cite{brieskorn1970singular} implies that minimal resolutions $\tilde X$ of hypersurfaces $X \subset \mathbb{P}^3$ with only rational double points, i.e. canonical singularities, are deformation equivalent to smooth hypersurfaces of the same degree.
\
The results in this presentation are:
\begin{theorem}
Let $X$ be a surface of general type with canonical singularities. Then the minimal resolution $\tilde{X}$ of $X$ has big cotangent bundle if
$$\sum_{x \in \text{Sing}X} h^1(x) \geq - \frac{s_2(\tilde{X})}{3!}$$
\end{theorem}
See (\ref{sec:2.1}) for the definition of $h^1(x)$, it is an invariant of the singularity. Note that the left side encodes only information about the germs of the singularities of $X$, so it is local in nature. This result is stronger than the result in \cite{roulleau2014} stating that $\Omega^1_{\tilde{X}}$ is big if $s_2(\tilde{X}) + s_2(\mathcal{X}) > 0$, $s_2(\tilde{X})$ and $s_2(\mathcal{X})$ respectively the 2nd Segre number of $\tilde{X}$ and of the orbifold $\mathcal{X}$ associated to $X$, see section \ref{sec:2} for more details.
\
In section \ref{sec: 2.2} we give a method to find $h^1(x)$ where $(X, x)$ is the germ of an $A_2$-singularity. In a later work \cite{future-paper} we show how to extend this method to calculate $h^1(x)$ for other $A_n$ singularities. Then using theorem 1 and information on the possible number of canonical singularities of prescribed types allowed in a hypersurface $X \subset \mathbb{P}^3$ of degree $d$, we obtain
\begin{theorem}
For $d=9$ and $d\ge 11$, there are minimal resolutions of hypersurfaces $X\subset \mathbb{P}^3$ with canonical singularities and degree $d$ which have big cotangent bundle.
\end{theorem}
The condition $s_2(\tilde{X}) + s_2(\mathcal{X}) > 0$ of \cite{roulleau2014} gives only $d\ge 13$ and there nodes are the best singularities. The above theorem uses $A_2$ singularities which due to theorem 1 are unexpectedly better than nodes, see \ref{eq:2.2} for more details.
\subsection{Big Cotangent Bundle}
\label{sec:1.1}
The cotangent bundle $\Omega_X^1$ on a complex manifold of dimension $n$ is said to be big if $$\lim_{m \rightarrow \infty} \dfrac{h^0(X, S^m \Omega_X^1)}{m^{2n - 1}} \neq 0$$ (i.e., $h^0(X, S^m \Omega_X^1)$ has the maximal growth order possible with respect to $m$ for $\dim X= n$). The property of $\Omega_X^1$ being big is birational.
\
In the case of surfaces of general type there is a topologically sufficient condition for bigness of $\Omega^1_X$, $s_2(X)>0$, where $s_2(X) = c_1^2(X) - c_2(X)$ is the 2nd Segre number of $X$. This follows from the asymptotic Riemann-Roch theorem for symmetric powers of $\Omega^1_X$:
\begin{equation}
h^0(X, S^m\Omega_X^1) - h^1(X, S^m\Omega_X^1) + h^2(X, S^m\Omega_X^1) = \frac{s_2(X)}{3!} m^3 + O(m^2)\tag{1.1}\label{eq:1.1}
\end{equation}
\noindent and Bogomolov's vanishing for surfaces of general type, $h^2(X, S^m\Omega_X^1) = 0$ for $m > 2$ \cite{bogomolov_1979}.
\
Very few examples of minimal surfaces with $s_2(X) \leq 0$ are known to have $\Omega_X^1$ big, they appear in \cite{bogomolov_nodes} and \cite{roulleau2014}. In these examples, bigness of $\Omega_X^1$ follows from complex analytic and not topological properties of $X$. The complex analytic conditions are the presence of enough configurations of $(-2)$-curves associated with canonical singularities. In fact, these surfaces with big $\Omega^1_X$ are diffeomorphic to surfaces with trivial cotangent algebra, $S(X)\simeq\Bbb C$.
\
If $X$ is a smooth surface of general type, it follows from \ref{eq:1.1} and $h^2(X, S^m \Omega_X^1) = 0$ that $\Omega_X^1$ is big if and only if:
\begin{align}
\lim_{m \rightarrow \infty} \frac{h^1(X, S^m\Omega_X^1)}{m^3} >- \frac{s_2(X)}{3!} \tag{1.2}\label{eq:1.2}
\end{align}
\subsection{Quotient singularities and local asymptotic Riemann-Roch for orbifold $\hat{S}^m\Omega_X^1$}
\label{sec:1.2}
In this section we present the local asymptotic Riemann-Roch for the orbifold symmetric powers of the cotangent bundle of a normal surface with only quotient singularities. For references on this topic, see \cite{wahl_chernclasses}, \cite{blache}, \cite{kawamata}, \cite{miyaoka_orbibundle}.
\
The germ of a normal surface singularity $(X,x)$ is a quotient singularity germ if it is biholomorphic to $(\mathbb{C}^2, 0)/G_x$, with $G_x \subset GL_2(\mathbb{C})$ finite and small, where $G_x$ is the local fundamental group. Canonical surface singularities are quotient singularities with $G_x \subset SL_2(\mathbb{C})$. Consider
\begin{equation*}
\begin{tikzcd}
& (\mathbb{C}^2, 0) \arrow[dl, "\varphi"', dashed] \arrow[d, "\pi"] \\
(\tilde{X}, E) \arrow[r, "\sigma"] & (X, x) \\
\end{tikzcd}
\end{equation*}
\noindent with $\pi: (\mathbb{C}^2, 0) \rightarrow (X, x)$, the quotient map by the local fundamental group, called the local smoothing of $(X, x)$ and $\sigma:(\tilde{X}, E) \rightarrow (X,x)$ a good resolution of $(X,x)$ where $(\tilde{X}, E)$ is the germ of a neighborhood of the exceptional locus $E$ with $E$ consisting of smooth curves intersecting transversally.
\
A reflexive coherent sheaf $\mathcal{F}$, i.e. $\mathcal{F}^{\vee \vee} = \mathcal{F}$, on $(X, x)$ is a locally free sheaf away from the singularity and satisfies $\mathcal{F} = i_*(\mathcal{F}|_{X\setminus \{x\}})$, $i: X\setminus \{x\} \lhook\joinrel\longrightarrow X$. Associated to a reflexive sheaf $\mathcal{F}$ on the quotient surface germ $(X,x)$ there are locally free sheaves $\tilde{\mathcal{F}}$ on $(\tilde{X}, E)$ (not uniquely determined) and $\hat{\mathcal{F}}$ on $(\mathbb{C}^2, 0)$ (uniquely determined) satisfying $\mathcal{F}\cong(\sigma_*\tilde{\mathcal{F}})^{\vee \vee}\cong (\pi_*^{G_x}) \hat{\mathcal{F}}$, where $(\pi_*^{G_x})\hat{\mathcal{F}}$ is a maximal subsheaf of $\pi_* \hat{\mathcal{F}}$ on which $G_x$ acts trivially, (\cite{blache} section 2).
\
The previous paragraph implies that reflexive coherent sheaves on normal surfaces with only quotient singularities $X$ are orbifold vector bundles on $X$ (also called $\mathbb{Q}$-vector bundles or locally $V$-free bundles over $X$). The orbifold $m$-symmetric power of the cotangent bundle on a normal surface $X$ with only quotient singularities is $\hat{S}^m \Omega_X^1 := (S^m\Omega_X^1)^{\vee \vee}$ with $\Omega_X^1 = i_*(\Omega_{X_{reg}}^1)$. If $\tilde{X} \xrightarrow{\sigma} X$ is a good resolution $\hat{S}^m\Omega_X^1 = (\sigma_* S^m \Omega_{\tilde{X}}^1)^{\vee \vee}$.
\
In the proof of theorem 1 a lower bound for $h^1(\tilde{X}, S^m\Omega_{\tilde{X}}^1)$ is given using only information on the singularities of $X$. Each $x_i$ contributes with $h^1(\tilde{U}_{x_i}, S^m \Omega_{\tilde{X}}^1)$ where $\tilde{U}_{x_i}$ is the minimal resolution of an affine neighborhood $U_{x_i}$ of $x_i$ with $U_{x_i}\cap \text{Sing}(X)=\{x_i\}$. The bigness of $\Omega_{\tilde{X}}^1$ depends on the asymptotics of $h^1(\tilde{X}, S^m\Omega_{\tilde{X}}^1)$, see section (\ref{sec:1.2}), and hence on the combined asymptotics of the $h^1(\tilde{U}_{x_i}, S^m\Omega_{\tilde{X}}^1)$.
\
Let $(\tilde{X}, E) \xrightarrow{\sigma} (X, x)$ be a good resolution of the germ of a quotient surface singularity and $\tilde{\mathcal{F}}$, $\mathcal{F}$ be sheaves such that $\tilde{\mathcal{F}}$ is locally free of rank $r$ on $\tilde{X}$ and $\mathcal{F} = (\sigma_* \tilde{\mathcal{F}})^{\vee \vee}$ a reflexive sheaf on $X$. In comparing the Euler characteristics $\chi(X, \mathcal{F})$ and $\chi(\tilde{X}, \tilde{\mathcal{F}})$ one has $\chi(X, \mathcal{F}) = \chi(\tilde{X}, \tilde{\mathcal{F}}) + \chi (x, \tilde{\mathcal{F}})$ with
\begin{equation}
\chi (x, \tilde{\mathcal{F}}) = \dim (H^0(\tilde{X}\setminus E, \tilde{\mathcal{F}})/H^0(\tilde{X}, \tilde{\mathcal{F}})) + h^1(\tilde{X}, \tilde{\mathcal{F}}) \tag{1.3}\label{eq:1.3}
\end{equation}
called the modified Euler characteristic of $\tilde{\mathcal{F}}$ (\cite{wahl_chernclasses}, \cite{blache} 3.9). The asymptotics of \ref{eq:1.3} are described via a local asymptotic Riemann-Roch theorem (\cite{blache} 4.1)
\begin{equation}
\lim_{m\rightarrow \infty} \frac{\chi(x, S^k\tilde{\mathcal{F}})}{m^{2 + r - 1}} = - \frac{1}{(2 + r -1)!}s_2(x, \tilde{\mathcal{F}})\tag{1.4}\label{eq:1.4}
\end{equation}
with $s_2(x, \tilde{\mathcal{F}}):=c_1^2(x, \tilde{\mathcal{F}}) - c_2(x, \tilde{\mathcal{F}})$, the local 2nd Segre number of $\tilde{\mathcal{F}}$ and $c_i(x, \tilde{\mathcal{F}}) \in H_{dRc}^{2i}(\tilde{X}, \mathbb{C})$ the $i$-th local Chern class of $\tilde{\mathcal{F}}$. The local Chern classes appear when comparing the pullback of orbifold Chern classes of an orbifold vector bundle $\mathcal{F}$ on an orbifold $X$ and the Chern classes of the vector bundle $\tilde{\mathcal{F}}$ on $\tilde{X}$, a good resolution $\sigma: \tilde{X} \rightarrow X$ of $X$, satisfying $\mathcal{F} = (\sigma_* \tilde{\mathcal{F}})^{\vee \vee}$.
\
We are only concerned with good resolutions $\sigma:(\tilde{X}, E) \rightarrow (X, x)$ of canonical surface singularities and $\tilde{\mathcal{F}} = \Omega_{\tilde{X}}^1$, one has $c_1^2(x, \Omega_{\tilde{X}}^1) = 0$ and:
\begin{equation}
s_2(x, {\Omega}_{\tilde{X}}^1) = -c_2(x, \Omega_{\tilde{X}}^1) = -(e(E) - \frac{1}{|G_x|}) \tag{1.5} \label{eq:1.5}
\end{equation}
with $e(E)$ the topological Euler characteristic of the exceptional locus and $|G_x|$ the order of the local fundamental group (\cite{blache} 3.18). We will use the invariant of the singularity:
\begin{equation}
s_2(x,X):=s_2(x, {\Omega}_{\tilde{X}_{min}}^1)\tag{1.6} \label{eq:1.6}
\end{equation}
\noindent where $\sigma:(\tilde{X}_{min}, E) \rightarrow (X, x)$ is the minimal good resolution.
\section{Theorems}
\label{sec:2}
\subsection{Resolutions with big cotangent bundle}
\label{sec:2.1}
We consider minimal resolutions $\sigma:\tilde{X} \rightarrow X$ of normal surfaces $X$ with only canonical singularities. The minimality condition has several advantages: i) the local 2nd Segre numbers $s_2(x, \tilde{\Omega}_{\tilde{X}}^1)$ being considered are $s_2(x,X)$ which depend only on the singularity (since the resolution is fixed); ii) in section \ref{sec: 2.2} the simultaneous resolution results used involve minimal resolutions of canonical singularities. Also, blowing up $b: \hat{X} \rightarrow X$ a smooth surface $X$ at a point does not affect inequality \eqref{eq:1.2} determining bigness of the cotangent bundle, since
$$\lim_{m \rightarrow \infty} \frac{h^1(\hat{X}, S^m\Omega_{\hat{X}}^1)}{m^3} + \frac{s_2(\hat{X})}{3!}=\lim_{m \rightarrow \infty} \frac{h^1(X, S^m\Omega_X^1)}{m^3} + \frac{s_2(X)}{3!} $$
Let $\sigma:\tilde{U}_x \rightarrow U_x$ be the minimal resolution of an affine normal surface $U_x$ with a single canonical singularity at the point $x \in U_x$. Set:
\begin{align*}
h^1(x) &:= \lim_{m\rightarrow \infty}
\frac{h^1\left(\tilde{U}_x, S^m \Omega_{\tilde{X}}^1\right)}{m^3} \tag{2.1}\label{eq:2.1}\\
&\\
h^0(x) &:= \lim_{m \rightarrow \infty}
\frac{\left[H^0\left(\tilde{U}_x\setminus E, S^m\Omega_{\tilde{U}_x}^1\right)/ H^0\left(\tilde{U}_x, S^m \Omega_{\tilde{U}_x}^1\right)\right]}{m^3} \tag{2.2}\label{eq:2.2}
\end{align*}
The local asymptotic Riemann-Roch equation (\ref{eq:1.4}) for the local modified Euler characteristic (\ref{eq:1.3}) for $\tilde{U}_x$ and $S^m\Omega_{\tilde{U}_x}^1$ gives:
\begin{equation}
h^1(x) = -\frac{1}{3!}s_2(x, X) - h^0(x). \tag{2.3}\label{eq:2.3}
\end{equation}
with $s_2(x, \Omega_{\tilde{U}_x}^1)$ an invariant of the canonical singularity $(U_x, x)$, since $\tilde{U}_x$ is its minimal resolution (and hence unique). In \cite{future-paper} using local duality and local cohomology for the pair $(\tilde{X}, E)$, it is shown that $h^0(x)\le h^1(X)$ holds, hence:
\begin{equation}
h^1(x) \geq -\frac{s_2(x, X)}{2\cdot3!} \tag{2.4}\label{eq:2.4}
\end{equation}
\
\
\setcounter{theorem}{0}
\begin{theorem}
Let $X$ be a normal projective surface of general type with only canonical singularities and $\sigma: \tilde{X} \rightarrow X$ a minimal resolution. Then $\Omega_{\tilde{X}}^1$ is big if and only if:
\begin{equation}
\sum_{x \in \text{Sing}X} h^1(x) \geq -\frac{s_2(\tilde{X})}{3!} \tag{2.5}\label{eq:2.5}
\end{equation}
\end{theorem}
\begin{proof}
We saw in section \ref{sec:1.1} that $\Omega_{\tilde{X}}^1$ is big if and only if $\lim_{m\rightarrow \infty} \frac{h^1(\tilde{X}, S^m\Omx)}{m^3} > -\frac{s_2(\tilde{X})}{3!}$.
From the Leray spectral sequence for $\sigma_*$ and Bogomolov's vanishing $H^2(\tilde{X}, S^m \Omega_{\tilde{X}}^1) = 0$ for $m>2$, we obtain for $m>2$:
\begin{equation}
\begin{tikzcd}[cramped,sep=small]
0 \rar &H^1(X, \sigma_*S^m\Omega_{\tilde{X}}^1) \rar & H^1(\tilde{X}, S^m \Omega_{\tilde{X}}^1) \rar & H^0(X, R^1 \sigma_* S^m \Omega_{\tilde{X}}^1) \\
\rar& H^2(X, \sigma_* S^m \Omega_{\tilde{X}}^1) \rar & 0 \tag{2.6}\label{eq:2.6}
\end{tikzcd}
\end{equation}
\
The 1st direct image sheaf $R^1\sigma_* S^m \Omega_{\tilde{X}}^1$ has support on the zero-dimensional singularity locus $\text{Sing}(X) = \{x_1, \dots, x_k\}$ of $X$. Each $x_i$ has an affine neighborhood $U_{x_i}$ such that $U_{x_i} \cap \text{Sing}(X) = \{x_i\}$. Using the Leray spectral sequence again for each $\tilde{U}_x = \sigma^{-1}(U_x)$, $\sigma: \tilde{U}_x \rightarrow U_{x_i}$ we obtain:
$$H^0\left(X, R^1 \sigma_* S^m \Omega_{\tilde{X}}^1\right) = \bigoplus_{i=1}^k H^1\left(\tilde{U}_x, S^m\Omega_{\tilde{U}_x}^1\right)$$
Hence using the notation of section \ref{sec:2.1}:
\begin{equation}
\sum_{x \in \text{Sing}(X)} h^1(x)= \lim_{m\rightarrow \infty}\frac{h^0\left(X, R^1 \sigma_* S^m \Omega_{\tilde{X}}^1\right)}{m^3} \tag{2.7}\label{eq:2.7}
\end{equation}
\noindent $\bf {Claim}$: $H^2(X, \sigma_*S^m \Omega_{\tilde{X}}^1) = 0$
\begin{proof}
Recalling that $\hat{S}^m\Omega_{\tilde{X}}^1 := (\sigma_* S^m \Omega_{\tilde{X}}^1)^{\vee \vee}$, consider the short exact sequence:
$$0 \rightarrow \sigma_* S^m \Omega_{\tilde{X}}^1 \rightarrow \hat{S}^m \Omega_{\tilde{X}}^1 \rightarrow Q_m \rightarrow 0.$$
Left injectivity holds since $\sigma_* S^m \Omega_{\tilde{X}}^1$ is torsion free. The support of $Q_m = \frac{(\sigma_* S^m \Omega_{\tilde{X}}^1)^{\vee \vee}}{\sigma_* S^m \Omega_{\tilde{X}}^1}$
is again $\text{Sing}(X)$, hence $H^2(X, \sigma_*S^m \Omega_{\tilde{X}}^1) \cong H^2(X, \hat{S}^m \Omega_{\tilde{X}}^1)$.
\
The surface $X$ is an orbifold surface of general type with canonical singularities and $\hat{S}^m\Omega_{\tilde{X}}^1$ is the orbifold $m$-th symmetric power of the cotangent bundle of $X$. Bogomolov's vanishing $H^2(X, \hat{S}^m \Omega_{\tilde{X}}^1) = 0$ for $m>2$ also holds in this setting, due to the existence of orbifold K\"ahler-Einstein metrics \cite{kobayashi1985}, \cite{Tian1986ie}, see also \cite{roulleau2014}.
\end{proof}
The vanishing of $H^2\left(X, \sigma_* S^m \Omega_{\tilde{X}}^1\right) = 0$ for $m > 0$, \eqref{eq:2.6} and \eqref{eq:2.7} give:
\begin{equation}
\lim_{m\rightarrow \infty} \frac{h^1(\tilde{X}, S^m \Omega_{\tilde{X}}^1)}{m^3} \geq \sum_{x\in \text{Sing}(X)} h^1(x) \tag{2.8}\label{eq:2.8}
\end{equation}
and the result follows from \eqref{eq:1.2}.
\end{proof}
\begin{remark} theorem 1 is stronger than the main theorem in \cite{roulleau2014} which states that $\Omega_{\tilde{X}}^1$ is big if $s_2(\tilde{X}) + s_2(X) > 0$. We have that $s_2(\tilde{X}) = s_2(X) + \sum_{x \in \text{Sing}X} s_2(x,X)$, (\cite{blache} 3.14), hence the condition $s_2(\tilde{X}) + s_2(X) > 0$ can be reexpressed as:
\end{remark}
\begin{equation}
-\sum_{x \in \text{Sing}X} \frac{s_2(x, X)}{2} > - s_2(\tilde{X}) \tag{2.9}\label{eq:2.9}
\end{equation}
It follows from \eqref{eq:2.4} that the condition \eqref{eq:2.5} in theorem 1 implies \eqref{eq:2.9}.
In fact it gives much stronger results. In the next section we will show that if $(X, x)$ is the germ of an $A_2$ singularity, then $h^1(x) = \frac{67}{216}$ while $-\frac{s_2(x, X)}{2\cdot 3!} = \frac{48}{216}$. This implies that our inequality \eqref{eq:2.5} guarantees $\Omega_{\tilde{X}}^1$ is big for surfaces of general type $X$ with only $\frac{48}{67} \cdot \ell$ $A_2$-singularities, where $\ell$ is the number needed to satisfy inequality \eqref{eq:2.9}.
\subsection{Deformations of smooth hypersurfaces with big $\Omega_X^1$}
\label{sec: 2.2}
In this section we study for which $d$ there are (smooth) surfaces with big cotangent bundle that are deformation equivalent to smooth hypersurfaces in $\mathbb{P}^3$ of degree $d$. We do this by considering minimal resolutions $\tilde{X}$ of hypersurfaces $X\subset \mathbb{P}^3$ of degree $d$ with only $A_2$ singularities. A simultaneous resolution result of Brieskorn \cite{brieskorn1970singular} gives that $\tilde{X}$ is deformation equivalent to a smooth hypersurface of $\mathbb{P}^3$ of degree $d$. In \cite{future-paper} other canonical singularities are also considered.
\begin{prop}
Let $\sigma: (\tilde{X}, E) \rightarrow (X,x)$ be the minimal resolution of the germ of an $A_2$ surface singularity. Then:
\begin{equation}
h^0(x):=\lim_{m\rightarrow \infty}\frac{\dim[H^0(\tilde{X}\setminus E_i, S^m\Omega_{\tilde{X}}^1)/H^0(X, S^m \Omega_{\tilde{X}}^1))]}{m^3} = \frac{29}{216} \tag{2.10}\label{eq:2.10}
\end{equation}
\label{prop:1}
\end{prop}
\begin{proof}
For the full proof see \cite{future-paper}.
\
We give here an extended description of what is involved in the proof. We use the affine model of an $A_2$-singularity $X= \{xz-y^3=0\}\subset \mathbb{C}^3$ with the minimal resolution $\tilde{X}$ obtained as the strict preimage of $X$ under $\sigma: \hat{\mathbb{C}}^3 \rightarrow \mathbb{C}^3$, the blow up of $\mathbb{C}^3$ at $(0,0,0)$.
\begin{center}
\begin{tikzpicture}
\begin{scope}
\node (tC) at (-6.7, 1.5) {$\mathbb{C}^2$};
\node (C) at (-12.2,0) {$\mathbb{C}^2 $};
\node (U) at (-11.2,0) {$\cong U^1\subset$};
\node (tX) at (-10.3,0) {$\tilde{X}$};
\node (X) at (-6.7, 0) {$X = \{xz - y^3 = 0\} \subset \mathbb{C}^3$};
\draw[->, to path={-| (\tikztotarget)}] (tX) -- node[above] {$\sigma$} (X);
\draw[->, to path={-| (\tikztotarget)}] (tC) -- node[right] {$\pi$,$(z_1^3,z_1z_2,z_2^3)$} (X);
\draw[dashed, ->, to path={-| (\tikztotarget)}] (tC) to [out=160, in=35] node[above left] {$\phi_1$,$(\frac{z_1^2}{z_2}, \frac{z_2^2}{z_1})$} (C);
\draw[dashed, ->, to path={-| (\tikztotarget)}] (tC) to [out=200, in=30] node[above left] {$\phi$} (tX);
\end{scope}
\end{tikzpicture}
\end{center}
\noindent where $\pi: \mathbb{C}^2 \rightarrow X$ gives the smoothing as in section \ref{sec:1.2}. Let $U^1= \tilde{X} \cap p^{-1}(U_1)$ with $p: \hat{\mathbb{C}^3} \rightarrow \mathbb{P}^2$ the canonical projection and $U_1 = \{ y \neq 0\} \subset \mathbb{P}^2$, $[x:y:z]$ as homogeneous coordinates of $\mathbb{P}^2$. The exceptional locus of $\sigma$ is $E = E_1 + E_2$, $E_i$ $(-2)$-curves intersecting transversally. On $U^1$ put coordinates $(u_1,u_2)$ with $\phi_1^* u_1 = \frac{z_1^2}{z_2}$ and $\phi_1^*u_2 = \frac{z_2^2}{z_1}$ and $E\cap U^1 = \{u_1 u_2 = 0\}$.
\
The isomorphism $\phi^*:H^0(\tilde{X}\setminus E, S^m\Omega_{\tilde{X}}^1)\to H^0(\mathbb{C}^2, S^m\Omega_{\mathbb{C}^2}^1)^{\mathbb{Z}_3}$ will be used to move the setting for finding $h^0(x)$ from $\tilde{X}\setminus E$ to $\mathbb{C}^2$. We need a good description of $G(m) := \phi^*(H^0(\tilde{X}, S^m \Omega_{\tilde{X}}^1))$. We use:
$$G(m) = \phi_1^*(H^0(\mathbb{C}^2, S^m \Omega_{\mathbb{C}^2}^1))\cap H^0(\mathbb{C}^2, S^m \Omega_{\mathbb{C}^2}^1)$$
\
We call $z_1^{i_1}z_2^{i_2}dz_1^{m_1}dz_2^{m_2}$ a z-monomial of full type (f-type) $(i_1,i_2,m_1,m_2)_z$ and type $(i,m)_z$ with $i=i_1+i_2$ the order and $m=m_1+m_2$ the degree of the monomial. A monomial is holomorphic if $i_1,i_2\ge 0$ and $\mathbb{Z}_3$-invariant if $i_1+2i_2+m_1+2m_2\equiv 0$ mod 3.
\vspace {.1in}
For each triple $(k,i,m)$ with $k\equiv -(m+i)$ mod 3 there is a collection of z-monomials:
\vspace {-.01in}
$$B(k,i,m)_z=\{(k-m+l,i+m-k-l,m-l,l)_z\}_{l=0,...,m} \hspace {.3in}(2.11)$$
These collections give
a partition of the set of all $\mathbb{Z}_3$-invariant z-monomials of type $(i,m)$. Set $V(k,i,m)_z=$Span$(B(k,i,m)_z)$.
Let $B_h(k,i,m)_z$ be the subcollection of holomorphic z-monomials of $B(k,i,m)_z$. Set $V_h(k,i,m)_z:=$ $\text{Span}(B_h(k,i,m)_z)$= $H^0(\mathbb{C}^2, S^m \Omega_{\mathbb{C}^2}^1)\cap V(k,i,m)$. Set $h_z(k,i,m):=\dim V_h(k,i,m)_z$=$\# B_h(k,i,m)_z$, from (2.11) it follows that $h_z(k,i,m)=\min(m+1,k+1, i + 1, m-k+i+1)$. Note that $h_z(k,i,m)=0$ unless $0\le k\le m+i$.
Set $G(k,i,m)$:= $G(m)\cap V(k,i,m)=G(m)\cap V_h(k,i,m)$. All the above gives (we will see below that $I(m)=2m$):
\vspace {-.08in}
$$
\dim[H^0(\tilde{X}\setminus E, S^m\Omega_{\tilde{X}}^1)/H^0(X, S^m \Omega_{\tilde{X}}^1))]=\dim [H^0(\mathbb{C}^2, S^m\Omega_{\mathbb{C}^2}^1)^{\mathbb{Z}_3}/G(m)]$$ $$\hspace {.9in}=\sum_{i=0}^{I(m)}\sum_{\substack{0\le k \le m+i \\ k \equiv -(m+i) \, \text{mod} 3}} h_z(k,i,m) -\dim G(k,i,m)\hspace {.4in}(2.12)$$
\
The reason to consider the collections $B(k,i,m)$ will now be examined. The rational map $\phi_1:(\mathbb{C}^2,z_1,z_2) \dashrightarrow (\mathbb{C}^2,u_1,u_2)$ pulls back holomorphic u-monomials of type $(i,m)$ to rational $\mathbb{Z}_3$-invariant z-monomials of type $(i,m)$:
\begin{equation}
\phi_1^*(p,i-p,q,m-q)_u=\sum_{l=0}^m c_{ql}(3(p+q)-(i+2m)+l,-3(p+q)+2(i+m)-l,m-l,l)_z \tag{2.13}\label{eq:2.13}
\end{equation}
\noindent with the $c_{ql}$ given by $(2x-y)^q(-x+2y)^{m-q}=\sum_l c_{ql}x^{m-l}y^l$.
\vspace {.08in}
From \eqref{eq:2.13} and (2.11) it follows that the pullback of a u-monomial of type $(i,m)$ lies in a single $V(k,i,m)$ and that the u-monomials whose pullback lie in $V(k,i,m)$ themselves form the collection $B(k,i,m)_u:=\{(k'-m+l,i+m-k'-l,m-l,l)_u\}_{l=0,...,m}$ with $k'=\frac{i+m+k}{3}$. Let $B_h(k,i,m)_u$ be the subcollection of holomorphic u-monomials of $B(k,i,m)_u$ and set $V_h(k,i,m)_u=$Span$(B_h(k,i,m)_u)$. Set $h_u(k,i,m) :=\dim V_h(k,i,m)_u$, we have $h_u(k,i,m)= \min(m+1, \frac{k + (i+m)}{3}+1, i + 1, \frac{2(i+m)-k}{3}+1 )$.
\
We proceed to find $I(m)$ and $\dim G(k,i,m)$ and calculate (2.12). We have that $G(k,i,m)=\phi_1^*(V_h(k,i,m)_u)\cap V_h(k,i,m)_z$. By using information on the rank of relevant subblocks of matrix $[c_{ql}]$, with $c_{ql}$ as in (2.12) (see \cite{future-paper} for details), we obtain that:
\vspace {-.05in}
$$\dim G(k,i,m) = \max{(h_z(k,i,m) + h_u(k,i,m) - (m+1), 0})$$
From the formula for $h_u(k,i,m)$ above, it follows that $h_u(k,i,m)=m+1$ and hence $G(k,i,m)=h_z(k,i,m)$ for all $0\le k\le m+1$ if $i\ge 2m$. This implies that all the terms in (2.12) for $i\ge 2m$ vanish, hence by setting $I(m)=2m$ we can write the full sum and obtain:
\vspace {-.2in}
$$h^0(x)=\lim_{m\to \infty}\frac{1}{m^3}\sum_{i=0}^{2m}\sum_{\substack{0\le k \le m+i \\ k \equiv -(m+i) \, \text{mod} 3}} \min(m+1 - h_u(k,i,m), h_z(k,i,m)) = \frac{29}{216}$$
\end{proof}
\begin{remark} For $A_1$ singularities using the set up described in \cite{bogomolov2008symmetric} by the 1st author the method to find $h^0(x)$ is substantially simpler and $h^0(x)=\frac{11}{108}$, see Jordan Thomas' thesis \cite{thomas}. For an approach along the lines of proposition 2.1 and valid for all $A_n$ singularities see \cite{future-paper}.
\end{remark}
\begin{theorem}
For $d = 9$ and $d \geq 11$ there are minimal resolutions of hypersurfaces in $\mathbb{P}^3$ with canonical singularities and degree $d$ which have big cotangent bundle.
\end{theorem}
\begin{proof}
Let $X_{d, \ell}\subset \mathbb{P}^3$ denote a hypersurface of degree $d$ with $\ell$ $A_2$-singularities as its only singularities and $\tilde{X}_{d, \ell}$ its minimal resolution. The Brieskorn simultaneous resolution theorem, \cite{brieskorn1970singular} and Ehresmann's fibration theorem give that $\tilde{X}_{d, \ell}$ is diffeomorphic to a smooth hypersurface of degree $d$ in $\mathbb{P}^3$, hence $s_2(\tilde{X}_{d, \ell})=-4d^2 + 10d$.
From sections \ref{sec:1.2} and \ref{sec:2.1} we have that $h^1(x) = -\frac{1}{3!}s_2(x, X)- h^0(x)=\frac {1}{3!}(e(E) - \frac{1}{|\mathbb{Z}_3|}) - h^0(x)$, where $(\tilde{X}, E)$ is a minimal resolution of the germ of the $A_2$-singularity $(X,x)$ ($e(E)=3$). Using proposition \ref{prop:1}, it follows that:
\begin{equation}
h^1(x) = \frac{67}{216} \tag{2.14}\label{eq:2.14}
\end{equation}
In Labs \cite{labs} it is shown how to construct hypersurfaces in $\mathbb{P}^3$ with only $A_n$ singularities with $n$ fixed using Dessins d'Enfants. For $A_2$ singularities one has that there are hypersurfaces $X_{d,\ell}$ if:
\begin{equation}
\ell = \begin{cases}
\frac{1}{2}d(d-1)\cdot\floor{\frac{d}{3}} + \frac{1}{3} d(d-3)(\floor{\frac{d-1}{2}}) - \floor{\frac{d}{3}}) & d \equiv 0 \mod 3 \\
\frac{1}{2} d(d-1)\cdot\floor{\frac{d}{3}} + \frac{1}{3}(d(d-3) + 2)(\floor{\frac{d-1}{2}}) - \floor{\frac{d}{3}}) & \text{otherwise}\\
\end{cases}
\tag{2.15}\label{eq:2.15}
\end{equation}
Theorem 1 and \ref{eq:2.14} give that $\Omega_{\tilde{X}_{d, \ell}}^1$ is big if $\frac{67}{216}\ell > s_2(\tilde{X}_{d, \ell})$ or equivalently if:
\begin{equation}
\ell>\frac {72}{67}(2d^2-5d) \tag{2.16}\label{eq:2.16}
\end{equation}
By \ref{eq:2.15} there are hypersurfaces $X_{d, \ell}\subset \mathbb{P}^3$ with $d$ and $\ell$ satisfying (2.16) if $d=9$ or $d \geq 11$.
\end{proof}
\
\begin{remark} 1) In Theorem 2 we can see the strength of theorem 1 when compared to the criterion for the cotangent bundle $\Omega_{\tilde{X}_{d, \ell}}^1$ to be big of \cite{roulleau2014}, $s_2(\tilde{X}_{d, \ell})+s_2({X}_{d, \ell})>0$. The criterion of \cite{roulleau2014} needs $\ell>\frac {3}{2}(2d^2-5d)$ instead of (2.16). The known upper bounds by Miyaoka or Varchenko, (see \cite{varchenko1983semicontinuity}, \cite{miyaoka1984maximal}, and also \cite{labs}), for the number of $A_2$ singularities possible on a hypersurface in $\mathbb{P}^3$ of degree $d$ prevent $\ell>\frac {3}{2}(2d^2-5d)$ for $d\le 11$. Moreover, one has to go to degree $d=14$ for the known constructions to give enough $A_2$ singularities for the criterion of \cite{roulleau2014}.
\vspace {.1in}
2) Following the method of theorem 2, if instead of using hypersurfaces in $\mathbb{P}^3$ with only $A_2$ singularities, one used hypersurfaces with only $A_1$ singularities (nodes), then one would need $\ell>\frac {9}{4}(2d^2-5d)$ nodes for the minimal resolution of an hypersurface with $\ell$ nodes to have big cotangent bundle. This would give surfaces with big cotangent bundle deformation equivalent to smooth hypersurfaces in $\mathbb{P}^3$ of degree $d\ge 10$. The known upper bounds for the number of nodes possible in hypersurfaces of a given degree, see \cite{labs}, give that for degree 9 you can not have more than 246 nodes, our criterion needs 264. So $A_2$ singularities give a better result.
\end{remark}
\vspace {-.2in}
\end{document}
|
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{document}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{abstract}
\end{abstract}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{abstract}
In this paper, we construct a countable partition ${\mathscr A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of ${\mathscr A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time.
This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction and statement of results}
\subsection{Singular flows}
{\em Singular flows} are flows that exhibit {\em equilibria}, or {\em singularities}. Such flows have been proven to be very resistant to rigorous mathematical analysis,
from both conceptual (existence of the equilibrium accumulated by regular orbits
prevents the flow to be hyperbolic) as well numerical (solutions slow down as
they pass near the equilibrium, which means unbounded return times and, thus,
unbounded integration errors and derivative) point of view.
For non-singular flows, the construction of {\em cross sections}, or {\em Poincar\'e sections}, is an important tool to study the dynamics of such flows, as it allows one to reduce the system to a discrete-time map (the {\em Poincar\'e map}) on the cross sections. See, for example, the celebrated work of Ratner~\cite{Ra} on Anosov flows, and the recent work by Lima and Sarig~\cite{LS} on three dimensional non-singular flows. However, the construction becomes far more difficult when the flow has a singularity. Often-times one has to construct several sections in order to capture flow orbits that approach, and leave the singularity.
See for example~\cite{APPV} and~\cite{GP}, where the authors constructed a family of cross sections for three-dimensional singular hyperbolic attractors, and~\cite{PT} for contracting Lorenz flows. Their construction requires a priori knowledge on how regular points approach singularities. Furthermore, they need linearization in a neighborhood of the singularity (thus putting assumptions on the eigenvalues of the tangent flow), require the stable foliation to have sufficient regular, and $\delta} \def\De{\Deltaim E^{cu}=2$ in order to reduce the dynamics on the cross sections to a one-dimensional system. Those assumptions significantly limit the situations where such strategy can be applied. As a result, as far as the authors are aware, there is no general construction of cross sections for singular flows on higher-dimensional manifolds.
In fact, the difficulty caused by the presence of equilibria shows up not only in the construction of cross sections, but also in the ergodic theory for flows.
It is a well accepted fact that for flows with singularities, the topological entropy, as well as metric entropies, can behave in a rather bizarre way. For example, in~\cite{SYZ} the authors constructed $C^\infty$ equivalent flows, such that one has zero entropy while the other has positive entropy. Even with those cross sections in~\cite{GP} and~\cite{PT}, the unbounded return time, which results in the unbounded derivative for the return map, has been proven to be the main obstruction for the ergodic theory of singular flows.
\subsection{Entropy theory for flows}
Entropy theory for flows not only is interesting by itself, but also has been proven to be a useful tool to classify the topological structure for flows.
In~\cite{PYY}, the authors use the entropy expansiveness to obtain a dichotomy on the chain recurrent classes of generic star flows, showing that every chain recurrent class with positive topological entropy must be isolated. More recently in~\cite{GYZ}, SRB-like measures (measures that are defined by Pesin's entropy formula) are used to classify the periodic orbit in the chain recurrent class for flows away from homoclinic tangencies.
However,
the entropy (both topological and measure-theoretical) for a flow is defined through its time-one map, whose dynamics is quite different from that of the Poincar\'e map. As a result, the cross sections constructed in the classical way (like those in~\cite{APPV}) does not work well for the entropy theory. Also due to the difficult caused by singularities, there has been little development in the entropy theory of singular flows for many years. One of the recent breakthrough
is the aforementioned work~\cite{PYY}, where it is proven that Lorenz-like flows are entropy expansive in any dimension. This, in particular, shows that the metric entropy is upper semi-continuous. However, the proof there strongly relies on the singularities being Lorenz-like and the entire flow being sectional hyperbolicity, therefore cannot be applied to singular flows in general.
\subsection{Statement of results: local dynamics near a hyperbolic singularity}
The goal of this paper is to give a complete description for the dynamics near a hyperbolic singularity $\sigma$, without making any extra assumption on the global structure of the flow itself. We will introduce a cross section $D_\sigma$ that contains the singularity,\footnote{Recall that in~\cite{APPV}, the cross sections are chosen away from the singularities.} and construct two countable measurable partitions, ${\mathscr C}_\sigma$ and ${\mathscr A}_\sigma$, using this cross section.
Below we let $X$ be a $C^1$ vector field and $\phi_t$ the associated flow on a Riemannian manifold $M$ without boundary. $\sigma\in{\rm Sing}(X)$ will be a hyperbolic singularity of $X$. When we take a neighborhood of $\sigma$, we will always assume that $\sigma$ is the only singularity in this neighborhood.
Given a neighborhood $B_r(\sigma)$ for a singularity $\sigma$, we will take the cross section to be:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}
D_\sigma = \exp_\sigma \left(\{v\in T_\sigma M: |v|\le\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta, |v^s| = |v^u|\}\right);
\end{equation}
One can think of it as the place where the flow speed is the ``slowest'', and orbit segments near $\sigma$ is ``making the turn''.
For each point $x\in D_\sigma$, we will write
$$
t^+_x = \inf\{\tau>0:\phi_\tau(x)\in\partial B_{r}(\sigma)\}
$$
and
$$
t^-_x = \inf\{\tau>0:\phi_{-\tau}(x)\in\partial B_{r}(\sigma)\}.
$$
for the first time that the orbit of $x$ exits the ball $B_r(\sigma)$ under the flow $\phi_t$ and $\phi_{-t}$.
Our first theorem is on the cross section $D_\sigma$ and the coarse partition ${\mathscr C}_\sigma$, which gives an accurate estimate on how long each orbit spend in the neighborhood of $\sigma$. More importantly, despite ${\mathscr C}$ being a countable partition, its metric entropy w.r.t. any invariant probability measure is uniformly bounded.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{figure}
\centering
\delta} \def\De{\Deltaef\columnwidth{\columnwidth}
\includegraphics[scale = 1]{Pic1.pdf}
\caption{The partition ${\mathscr C}_\sigma$.}
\label{f.partitionC}
\end{figure}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{main}\label{m.C}[The coarse partition ${\mathscr C}_\sigma$]
For every hyperbolic singularity $\sigma$ of a $C^1$ vector field $X\in\mathscr X^1(M)$ and every $r>0$ small enough, there is a cross section $D_\sigma\subset B_r(\sigma)$ containing $\sigma$ and a countable measurable partition ${\mathscr D}_\sigma = \{D_n\}_{n>n_0(r)}$ on $D_\sigma$, with the following properties:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}[label={({\cal R}oman*).}]
\item every orbit segment in $B_r(\sigma)$ intersects $D_\sigma$ only once;
\item there is $0<L_0<L_1$ such that for every $n>n_0$ and $x\in D_n$, we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.flowspeed}
|X(x)|\in [L_0 e^{-n-1}, L_1 e^{-n}];
\end{equation}
\item there is $0<K_0<K_1$ such that for every $n>n_0$ and $x\in D_n$,
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.timeinB}
\frac{t^\pm_x}{n} \in [K_0, K_1];
\end{equation}
\item the closure of the set
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.O(s)}
O(\sigma) = \bigcup_{n>n_0}\bigcup_{x\in D_n} \phi_{[-t^-_x,t^+_x]}(x)\subset B_{r}(\sigma)
\end{equation}
contains an open ball of $\sigma$ with diameter $\exp(-n_0)$;
\item the countable measurable partition ${\mathscr C}_\sigma$ defined by:
$$
{\mathscr C}_\sigma = \{C_n = \phi_{[0,1)}(D_n): n>n_0\}\cup\{(\cup_{n>n_0} C_n)^c
$$
forms a cone near $\sigma$, with $\sigma$ being the end point (see Figure~\ref{f.partitionC} and Figure~\ref{f.lorenz});
\item there exists $H_1>0$ such that for any probability measure $\mu$, we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.entropyC}
H_\mu({\mathscr C}_\sigma)<H_1<\infty.
\end{equation}
\end{enumerate}
Furthermore, the above properties hold robustly in a $C^1$ neighborhood of $X$ and for the continuation of $\sigma$, with the same constants $L_0, L_1, K_0, K_1, H_1$.
\end{main}
Recall that for a diffeomorphism $f$ on a Riemannian manifold, the hyperbolicity (or the dominated splitting) of the tangent map $Df|_x$ determines the dynamics in a neighborhood of $x$ with uniform size. The same holds for non-singular flows. However, for flows with singularity, the situation is quite different: as discovered by Liao~\cite{Liao96}, the tangent flow governs the dynamics only in a {\em tubular neighborhood} along the orbit of $x$, and the size of this neighborhood is proportional to the flow speed. This means that near singularities, the size of such neighborhoods become much smaller (usually exponentially small if the singularity is hyperbolic), since the flow speed slows down exponentially. However, both the topological theory and the entropy theory for flows require estimates on a uniform size under the time-one map. This turns out to be the main obstruction for the study of singular flows.
To solve this issue, we will construct a countable partition ${\mathscr A}_\sigma$, by taking any $L>0$ large enough and refining each element $C_n$ of ${\mathscr C}_\sigma$ into ${\mathcal O}(L^n)$ many elements, such that the diameter of each $A\in C_n$ is at most ${\mathcal O}(L^{-n})$. Combine this with the estimation on the flow speed~\eqref{e.flowspeed}, we will show that each partition element of ${\mathscr A}_\sigma$ controls the dynamics in a long tubular neighborhood. Moreover, the metric entropy of ${\mathscr A}_\sigma$ is still uniformly bounded.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{main}\label{m.A}[The refined partition ${\mathscr A}_\sigma$]
For every hyperbolic singularity $\sigma$ of a $C^1$ vector field $X\in\mathscr X^1(M)$ and $r>0$ small enough, there exists $N_0>0$ such that for every $L\ge N_0$, $0<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$, there is a measurable partition ${\mathscr A}_\sigma$ refining ${\mathscr C}_\sigma$, with the following properties:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}[label={({\cal R}oman*).}]
\item for every $n>n_0 = n_0(r)$ and $C_n\in{\mathscr C}_\sigma$, the collection ${\mathscr B}_n : =\{A\in{\mathscr A}_\sigma : A\subset C_n\}$ is a finite partition of $C_n$
\item there exists $c_0>0$ independent of $L$, such that for $L'=L^{K_1}e$ and every $n>n_0$, if $A\in{\mathscr A}_\sigma$ satisfies $A\subset C_n$, then
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.Adiameter}
\operatorname{diam}(A)\le c_0\cdot \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta (L')^{-n};
\end{equation}
\item there exists $L''>0$ depending explicitly on $L$ and $c_1>0$, such that for every $n>n_0$, we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.Bnumber}
\#{\mathscr B}_n\le c_1 (L'')^n;
\end{equation}
\item for two points $x,y\in A\in{\mathscr A}_\sigma$, $y$ is in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of $x$ (for the precise definition, see the next section) until $x$ leaves $B_r(\sigma)$;
\item there exists $H_2>0$ depending on $L$, such that for any invariant probability measure $\mu$, we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.entropyA}
H_\mu({\mathscr A}_\sigma)<H_2<\infty.
\end{equation}
\end{enumerate}
Furthermore, the above properties hold robustly in a $C^1$ neighborhood of $X$ and for the continuation of $\sigma$, with the same constants $N_0, L', L'', c_0, c_1$ and $H_2$.
\end{main}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}\label{r.tower}
For readers who are familiar with the language of {\em Rokhlin-Kakutani towers}, the set:
$$
\left\{\phi_j(A): A\in{\mathscr A}, A\subset C_n \mbox{ for some }n>n_0, j \in [-K_1n, K_1n]\right\}
$$
forms a tower which contains the neighborhood $B_r(\sigma)$. The {\em base of the tower} is the cone
$\Omega_0 = \phi_{[0,1)}(D_\sigma)$, which consists of elements of ${\mathscr A}_\sigma$. Also note that our partition ${\mathscr A}_\sigma$ treats the complement of the base $\Omega_0$ as a single element.
Recall that in the classical definition of Rokhlin towers, the top floor is mapped back to the base of the tower. However, in our setting, the top floor of the tower is mapped to $(B_r(\sigma))^c$.
\end{remark}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{figure}
\centering
\delta} \def\De{\Deltaef\columnwidth{\columnwidth}
\includegraphics[scale=0.9]{Pic2.pdf}
\caption{The partitions ${\mathscr B}_n$ and ${\mathscr A}_\sigma$.}
\label{f.partitionB}
\end{figure}
\subsection{Statement of the result: when all the singularities are hyperbolic}
Observe that in the previous two theorems, we do not impose any hypothesis on other singularities of $X$; in fact, both theorems only deal with the local dynamics near $\sigma$. However, if one makes the assumption that all the singularities of $X$ are hyperbolic, then the construction of ${\mathscr A}_\sigma$ can be carried out near each singularity. This leads to the next theorem:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{main}\label{m.3}
Let $X$ be a $C^1$ vector field, such that every singularity of $X$ is hyperbolic. Then for every $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta>0$ small enough and $L\ge N_0$, there exists a countable, measurable partition ${\mathscr A}$ with the following property:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}[label={({\cal R}oman*).}]
\item for two points $x,y$ in the same element of the partition ${\mathscr A}^\infty = \vee_{j\in{\mathbb Z}} \phi_j({\mathscr A})$, the orbit of $y$ stays in the infinite $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of $x$, under both $X$ and $-X$.
\item there exists $H>0$, such that for any invariant probability measure $\mu$, we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.entropy}
H_\mu({\mathscr A})<H<\infty.
\end{equation}
\end{enumerate}
Furthermore, the partition ${\mathscr A}$ can be made continuous for nearby $C^1$ vector fields, in the sense that if $X_n\xrightarrow{C^1} X$, then there is a sequence of partitions $\{{\mathscr A}_n\}$ satisfying the above properties, such that for each element $A\in {\mathscr A}$, there is $\{A_n: A_n\in{\mathscr A}_n\}$ such that $\operatorname{C}l (A_n)\to \operatorname{C}l (A)$ in Hausdorff topology.
\end{main}
Regarding the notation: in this paper, a partition with an index, such as ${\mathscr A}_\sigma$, ${\mathscr B}_n$ and and ${\mathscr C}_\sigma$, are constructed locally near the singularity $\sigma$; the partition ${\mathscr A}$ without any index is defined for the entire flow. The only exception to this rule is in Section~\ref{s.8}, where we need to take a sequence of partitions ${\mathscr A}_n$ and define a family of finite partitions ${\mathscr A}_{n,N}$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}\label{r.musupp}
In all the theorems throughout this paper, unless otherwise specified, the measure $\mu$ may assign positive weight to some singularity $\sigma$. It is easy to check that this does not affect the estimation on the entropy $H_\mu$. See the proof of Proposition~\ref{p.Centropy} and~\ref{p.Aentropy} below.
\end{remark}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}
Item (I) in Theorem~\ref{m.3} says that the set ${\mathscr A}^\infty(x)$ is contained in the infinite scaled Bowen-ball of $x$ with size $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$. For the precise definition, see~\cite{WW}.
\end{remark}
\subsection{Applications: star flows, and flows away from homoclinic tangencies}
Next, we will state several applications for star flows, and for flows that are away from homoclinic tangencies. Recall that a vector field $X$ is said to be {\em star}, if there exists a $C^1$ neighborhood ${\mathcal U}$ of $X$, such that for every $Y\in {\mathcal U}$, all the critical elements (singularities, periodic orbits) of $Y$ are hyperbolic.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{maincor}\label{mc.star1}
Let $X$ be a star vector field. Then for $L$ large enough, the partition ${\mathscr A}$ given by Theorem~\ref{m.3} is ``almost'' generating, in the sense that for every ergodic, invariant probability measure $\mu$ and $\mu$-almost every $x\in M$, there exists $s(x)>0$ such that ${\mathscr A}^\infty(x)$ is contained in the finite orbit segment $\phi_{(-s(x), s(x))}(x)$. In particular, for any ergodic, invariant probability measure $\mu$, we have
$$
h_\mu(X) = h_\mu(\phi_1, {\mathscr A}).
$$
\end{maincor}
Next, we turn our attention to flows away from tangencies, where the situation is more subtle. $X$ is said to exhibit {\em homoclinic tangency}, if $X$ has a hyperbolic periodic orbit with non-transverse homoclinic intersection. We denote by ${\mathcal T}$ the collection of $C^1$ vector fields with homoclinic tangency.
Unfortunately, the partition ${\mathscr A}$ may not be (almost) generating when the flow is away from tangencies but not star. However, we will prove that ${\mathscr A}$ can be used to compute the metric entropy for any invariant measure $\mu$. For this purpose, we need the following theorem which generalizes the entropy expansiveness by Bowen~\cite{B72} and the almost entropy expansiveness in~\cite{LVY}.
Let $f: M\to M$ be a homeomorphism, $\mu$ an invariant probability such that $h_\mu(f)<\infty$.
Let ${\mathscr A}$ be a measurable partition such that $H_\mu({\mathscr A})<\infty$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{definition}\label{d.1}
For any $x\in M$, we define its \emph{$\infty$ ${\mathscr A}$-ball} by
$${\mathscr A}^\infty(x)=\bigcap_{n\in \mathbb{Z}} f^{-n} {\mathscr A}(f^n(x)).$$
We denote the \emph{${\mathscr A}$-tail entropy} of $x\in M$ by
$$h_{tail}(f,x,{\mathscr A})=h_{top}({\mathscr A}^\infty(x),f).$$
An invariant probability measure $\mu$ is called {\em ${\mathscr A}$-expansive}, if for $\mu$ almost every $x$,
$h_{tail}(f,x,{\mathscr A})=0$.
\end{definition}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{main}\label{m.tailestimate}
Let $f$ be a homeomorphism over a compact manifold $M$ with finite dimension. Suppose $\mu$ is an invariant probability of $f$ that is ${\mathscr A}$-expansive. Then we have
$h_\mu(f)=h_\mu(f,{\mathscr A})$.
\end{main}
A similar result holds for finite partitions where the tail entropy is defined by infinite Bowen-balls, see~\cite[THeorem 1.2]{CGY}. Note that if ${\mathscr A}$ is a generating partition, then every measure is automatically ${\mathscr A}$-expansive. On the other hand, if $f$ is $\varepsilon$-entropy expansive, then for every partition ${\mathscr A}$ with $\operatorname{diam}{\mathscr A}<\varepsilon$, every measure is ${\mathscr A}$-expansive. We do not know if the converse is true.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{question}
If ${\mathscr A}$ is a finite or countable partition such that every invariant measure $\mu$ is ${\mathscr A}$-expansive and $H_\mu({\mathscr A})<\infty$. Does this imply that
$$
h_{top}({\mathscr A}^\infty(x), f) = 0
$$
for {\em every} point $x\in M$? Does it imply $\varepsilon$-entropy expansiveness for some $\varepsilon>0$?
\end{question}
The following theorem generalizes Corollary~\ref{mc.star1} to flows away from tangencies:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{main}\label{m.tangency}
Let $X\in {\mathscr X}^1(M)\setminus \operatorname{C}l({\mathcal T})$ be a $C^1$ vector fields with all the singularities hyperbolic. For $L\ge N_0$ and $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta>0$ small enough, let ${\mathscr A}$ be the partition given by Theorem~\ref{m.3}. Then every invariant probability measure $\mu$ is ${\mathscr A}$-expansive. In particular, we have
$$
h_\mu(X) = h_\mu(\phi_1, {\mathscr A}).
$$
\end{main}
Furthermore, we obtain the upper semi-continuity for the metric entropy with respect to the flows in $C^1$ topology, and with respect to the measure in weak*-topology.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{main}\label{m.continuous}
For $X \in {\mathscr X}^1(M)\setminus \operatorname{C}l({\mathcal T})$ with all singularities hyperbolic, there exists $L_2>0$ with the following property: if $X_n \in {\mathscr X}^1(M)\setminus \operatorname{C}l({\mathcal T})$ is a sequence of $C^1$ vector fields such that $X_n\xrightarrow{C^1}X$. Let $\mu_n,\mu$ be invariant measures of $X_n$ and $X$, respectively, with $\mu_n\to\mu$ in weak*-topology. Then we have
$$
\lim_{n\to\infty}h_{\mu_n}(X_n)\le h_\mu(X) + L_2\mu({\rm Sing}(X)).
$$
In particular, if $\mu({\rm Sing}(X))=0$ then
$$
\lim_{n\to\infty}h_{\mu_n}(X_n)\le h_\mu(X).
$$
\end{main}
This theorem shows that, if $\mu({\rm Sing}(X))=0$ then $\mu$ is a point of upper-semi continuity for the metric entropy in the space $\{\mu: \mu \mbox{ is invariant for some } Y\in{\mathcal U}\subset{\mathscr X}^1(M)\}$.
Let us make some remarks on the condition $\mu({\rm Sing}(X))=0$. It is proven in~\cite{LGW} that for diffeomorphisms away from tangencies, the metric entropy is upper semi-continuous. The proof requires one to consider a {\em finite} partition; for this purpose, it is natural to glue the tail of ${\mathscr A}$ into a large element, and expect the entropy to remain approximately the same. However this is not the case for singular flows: we in fact prove that the loss of the metric entropy is (at most) proportional to the measure of a small region $O^N(\sigma)$ near the singularity; this region is, in fact, the image of $\cup_{n>N} C_n$ in $B_r(\sigma)$ under the flow. See Figure~\ref{f.finitepartition} and the precise statement in Theorem~\ref{t.finitepartitionentropy}.
This is possibly a new mechanism on how metric entropy is lost when a sequence of measures $\mu$ converges to $\mu$: if $\mu({\rm Sing}(X))>0$, then we may lose metric entropy by (at most) a constant multiple of $\mu({\rm Sing}(X))$. This phenomenon does not exist for non-singular flows or diffeomorphisms.
Also note that in~\cite{PYY} it is proven that the metric entropy is upper semi-continuous for Lorenz-like flows. However, the proof there comes from the entropy expansiveness, which relies on the sectional hyperbolic structure on the entire class. However, there are examples where this structure does not exist, even for star flows. See~\cite{BD}, an example of a chain recurrent class without dominated splitting.
On the other hand, the metric entropy for star flows may still be upper-semi continuous after all. This is due to the strong hyperbolicity of star systems. It is proven in~\cite{GSW} that every {\em Lyapunov stable} chain recurrent class of generic star vector field must be Lorenz-like, therefore entropy expansive (by~\cite{PYY}). It is also known from~\cite{PYY} that the example of Bonatti and da Luz~\cite{BD} is an isolated homoclinic class. This invites us to make the following conjectures:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{conjecture}
For (generic) star flows, the metric entropy varies upper-semi continuously.
\end{conjecture}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{conjecture}
There exists $X\in {\mathscr X}^1(M)\setminus \operatorname{C}l({\mathscr X}^*(M))$ and a sequence of invariant measures $\mu_n\to\mu$, along which the metric entropy is not upper semi-continuous. Here ${\mathscr X}^*(M)$ is the space of $C^1$ star vector fields.
\end{conjecture}
\subsection{Structure of the paper}
The study of the local dynamics in $B_r(\sigma)$ is carried out in Section~\ref{s.3} and~\ref{s.4}.
Section~\ref{s.3} contains the construction of the coarse partition ${\mathscr C}_\sigma$, and the proof of Theorem~\ref{m.C}, while Section~\ref{s.4} contains the construction of the refined partition ${\mathscr A}_\sigma$, and the proof of Theorem~\ref{m.A}.
Then in Section~\ref{s.5} we combine the construction of ${\mathscr A}_\sigma$ at each $\sigma\in{\rm Sing}(X)$ together and prove Theorem~\ref{m.3}.
The proof of Theorem~\ref{m.tailestimate} can be found in Section~\ref{s.6}. Then in Section~\ref{s.7} we show that the partition ${\mathscr A}$ can be used to compute the metric entropy for star flows and flows away from tangencies, proving Corollary~\ref{mc.star1}, Theorem~\ref{m.tangency}. Finally we show the upper semi-continuity of the metric entropy in Section~\ref{s.8}.
We also include a comparison between our cross section $D_\sigma$ and those in~\cite{APPV} at the end of Section~\ref{s.3}.
\section{Preliminaries}\label{s.2}
Throughout this article, all the singularities of $X$ are assumed to be hyperbolic. Whenever we take a neighborhood ${\mathcal U}$ of $X$ in ${\mathscr X}^1(M)$, we will always assume that all the singularities of every $Y\in{\mathcal U}$ are hyperbolic, and are exactly the continuation of those in ${\rm Sing}(X)$.
\subsection{The scaled linear Poincar\'e flow}
For a regular point $x$ and $v\in T_xM$, the {\em linear Poincar\'e flow} $\psi_t: {\mathcal N}_x\to {\mathcal N}_{\phi_t(x)}$ is the projection of $\Phi_t(v)$ to ${\mathcal N}_{\phi_t(x)}$, where ${\mathcal N}_x$ is the orthogonal complement of $X(x)$. To be more precise, we denote the normal bundle
of $\phi_t$ over $\Lambda} \def\e{\varepsilonmbda$ by
$$
{\mathcal N}_\Lambda} \def\e{\varepsilonmbda = \bigcup_{x\in\Lambda} \def\e{\varepsilonmbda\setminus{\rm Sing}(X)}{\mathcal N}_x,
$$
where ${\mathcal N}_x$ is the orthogonal complement of the flow direction $X(x)$, i.e.,
$$
{\mathcal N}_x = \{v \in T_xM: v \operatorname{per}p X(x)\}.
$$
Denote the orthogonal projection of $T_xM$ to ${\mathcal N}_x$ by $\pi_x$. Given $v \in {\mathcal N}_x$ for a regular point $x \in
M \setminus {\rm Sing}(X)$ and recall that $\Phi_t$ is the tangent flow, we can define $\psi_t(v)$ as the orthogonal projection of $\Phi_t(v)$ onto ${\mathcal N}_{\phi_t(x)}$, i.e.,
$$
\psi_t(v) = \pi_{\phi_t(x)}(\Phi_t(v)) = \Phi_t(v) -\frac{< \Phi_t(v), X(\phi_t(x)) >}{\|X(\phi_t(x))\|^2}X(\phi_t(x)),
$$
where $<\cdot,\cdot >$ is the inner product on $T_xM$ given by the Riemannian metric.
The {\em scaled linear Poincar\'e flow}, which we denote by $\psi^*_t$, is defined as
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.scaledpoincare}
\psi^*_t(v) = \frac{\|X(x)\|}{\|X(\phi_t(x))\|}\psi_t(v) = \frac{\psi_t(v)}{\|\Phi_t|_{<X(x)>}\|}.
\end{equation}
It is introduced by Liao~\cite{Liao} to study flows with singularities.
Whenever necessary, we will write $\psi_{X,t}$ and $\psi^*_{X,t}$ to emphasis the dependence of $\psi$ and $\psi^*$ on the initial vector field $X$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.C1norm}
For every $\tau>0$ and a $C^1$ neighborhood ${\mathcal U}$ of $X$, there exists $L_{\tau,{\mathcal U}}>0$ such that for every $Y\in{\mathcal U}$ and $t\in[-\tau,\tau]$, we have
$$
\|\psi^*_{Y,t}\|\le L_{\tau,{\mathcal U}}
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
For fixed $\tau$ and ${\mathcal U}$, $\Phi_{X,t}$ is uniformly bounded above and away from zero in both $X\in{\mathcal U}$ and $t\in[-\tau,\tau]$. As a result, $\psi_{X,t}$ has uniformly bounded norm, so is $\psi^*$.
\end{proof}
\subsection{A scaled tubular neighborhood theorem}
For each regular point $x$ and $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta>0$, we denote by $N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$ to be the submanifold given by
$$
N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta) = \exp_x({\mathcal N}_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)),
$$
where ${\mathcal N}_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta) = \{v\in {\mathcal N}_x: |v|< \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta\}$, and $\exp_x$ is the exponential map from $T_xM$ to $M$. We may take $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0>0$ small enough (but uniformly for $Y$ in a small $C^1$ neighborhood of $X$), such that $\exp_x$ is a diffeomorphism from ${\mathcal N}_{Y,x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta|X(x)|)$ to $N_{Y,x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta|X(x)|)$ for every $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$. For such $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$, we define ${\mathcal P}_{x,t}(y)$ to be the Poincar\'e map from $N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$ to $N_{\phi_t(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$. In other words, ${\mathcal P}_{x,t}(y)$ is the first point of intersection between the orbit of $y$, and the submanifold $N_{\phi_t(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$. As before, ${\mathcal P}_{X,x,t}(y)$ highlights the dependence of ${\mathcal P}$ on the vector field.
For a regular point $x\in M\setminus {\rm Sing}(X)$ and $T>0$, $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta>0$, we denote by
$$
B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta} (x,T) = \bigcup_{t\in[0,T]} N_{X,\phi_t(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta |X(\phi_t(x))|)
$$
to be the {\em $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood} of the orbit segment $\phi_{[0,T]}(x)$. We will refer to $T$ as the {\em length} of this tubular neighborhood. When $T=+\infty$, we call it {\em the infinite $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood}.
By continuity, $B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta} (x,T)$ contains an open neighborhood of the orbit segment $\phi_{[\varepsilon,T-\varepsilon]}(x)$, for every $\varepsilon>0$. One should note that the size of the neighborhood at $y\in \phi_{[0,T]}(x)$ depends on the flow speed at $y$. Therefore, the neighborhood becomes smaller as the orbit gets closer to some singularity.
The next proposition provides a scaled tubular neighborhood theorem for flows with singularities. Most importantly, it gives a uniform size for the tubular neighborhood when normalizing with the flow speed. Such estimates played an important role in the work of Liao~\cite{Liao96} and~\cite{GY} on singular flows.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.tubular}\cite[Lemma 2.2]{GY}
There exists $L=L(X)>1$ and a small $C^1$ neighborhood ${\mathcal U}$ of $X$, such that for every $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$, $Y\in{\mathcal U}$ and every regular point $x$ of $Y$, ${\mathcal P}_{Y,x,1}$ is well-defined and injective from $N_{Y,x}((\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta/L) |Y(x)|)$ to $N_{Y,\phi_{Y,1}(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta|Y(\phi_{Y,1}(x))|)$. Moreover, for $y\in N_{Y,x}((\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta/L) |Y(x)|)$, the orbit segment from $y$ to ${\mathcal P}_{Y,x,1}(y)$ stays in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of the orbit segment $\phi_{[0,1]}(x)$.
\end{proposition}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
This is essentially Lemma 2.2 in~\cite{GY} with $T=1$. One only need to check that the constants there can be made uniform for $Y\in{\mathcal U}$.
\end{proof}
For simplicity, below we will assume that $T>0$ is an integer. Apply the previous proposition recursively, we obtain the following proposition:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.tubular1}
There exists $L=L(X)>1$ and a small $C^1$ neighborhood ${\mathcal U}$ of $X$, such that for every $T\in{\mathbb N}$, $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$, $Y\in{\mathcal U}$ and every regular point $x$ of $Y$, ${\mathcal P}_{Y,x,T}$ is well defined and injective from $N_{Y,x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-T} |Y(x)|)$ to $N_{Y,\phi^Y_T(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta|Y(\phi_{Y,T}(x))|)$. Moreover, the orbit segment from $y\in N_{Y,x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-T} |Y(x)|)$ to ${\mathcal P}_{Y,x,T}(y)$ stays in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of the orbit segment $\phi_{[0,T]}(x)$.
\end{proposition}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}\label{r.flytime}
From the construction, we see that for $y\in N_{x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-T} |X(x)|) $, if we denote by $\tau_{x,T}(y)>0$ to be the first time that the orbit of $y$ hits the normal manifold $N_{\phi_T(x)}$, then Proposition~\ref{p.tubular} gives:
$$
\phi_{\tau_{x,1}(y)}(y) = {\mathcal P}_{x,1}(y),
$$
In fact, more can be said regarding the hitting time $\tau_{x,T}(y)$. Note that in Proposition~\ref{p.tubular}, for any given $\varepsilon>0$, we can increase $L$ to obtain
$$\tau_{x,1}(y)\le 1+\varepsilon\mbox{ for all }y\in N_{x}((\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta/L) |X(x)|).$$
Then the recursive argument gives
$$\tau_{x,T}(y)\le (1+\varepsilon)^T\mbox{ for all }y\in N_{x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-T} |X(x)|).$$
\end{remark}
The map ${\mathcal P}$ can be lifted to a map on the normal bundle using the exponential map in a natural way:
$$
P_{X,x,T} = \exp_{\phi_T(x)}^{-1}\circ{\mathcal P}_{X,x,T}\circ\exp_x.
$$
The fact that the orbit segment in $N_{Y,x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-T} |Y(x)|)$ remains in the scaled tubular neighborhood $B_\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta(x,T)$ guarantees that $P_{X,x,T}$ and ${\mathcal P}_{X,x,T}$ are semi-conjugate by $\exp_{(\cdot)}$, and the previous proposition remains true for $P_{X,x,T}$ in ${\mathcal N}_{X,x}( \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-T} |X(x)|)$ (in fact, this is how the scaled tubular neighborhood theorem is stated in~\cite{GY}). Furthermore, we have:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\cite[Lemma 2.3]{GY}\label{p.tubular2}
$DP_{X,x,1}$ is uniformly continuous in the following sense: for every $\varepsilon>0$ and $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta>0$, there exists $0<\delta} \def\De{\Deltaelta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-1}$ such that for every $Y\in {\mathcal U}$ and a regular point $x$ of $Y$, $y,y'\in N_{Y,x}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-1}|Y(x)|)$, if $d(y,y')<\delta} \def\De{\Deltaelta L^{-1} |Y(x)|$, then we have
$$
|DP_{Y,x,1}(y) - DP_{Y,x,1}(y')|<\varepsilon.
$$
As a result, there exists $K>0$, independent of $Y\in{\mathcal U}$ and $x\in M\setminus {\rm Sing}(Y)$, such that
$$
|DP_{Y,x,T}|\le K.
$$
\end{proposition}
Note that the previous propositions remain valid if one replaces $L$ by a larger constant (we already used this fact in Remark~\ref{r.flytime}). This observation will play an important role in the construction of the sections near singularities.
\subsection{The entropy theory for countable partitions}
In his famous paper~\cite{Ma81}, Ma\~n\'e gave a very useful criterion for a countable, measurable partition to have finite entropy:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\cite[Lemma 1]{Ma81}\label{l.finiteentropy}
For every $N>0$, there exists $H>0$ such that if $\sum_{n=1}^\infty x_n$ is a series with $x_n\in (0,1)$, such that $\sum_{n=1}^{\infty} n x_n<N$, then
$$
-\sum_{n=1}^\infty x_n\log x_n<H.
$$
\end{lemma}
Note that the version that we stated here is slightly stronger than Ma\~n\'e's original statement; however, one can easily prove it using the same argument in~\cite{Ma81},
To put it in a more modern context, the previous lemma says that: if $\mu$ is a probability measure (not necessarily invariant) over a set $\Omega_0$, and $\Omega$ is a discrete time suspension over $\Omega_0$ with roof function $R:\Omega_0 \to {\mathbb N}$ satisfying $\mu(R)<\infty$, then the partition of $\Omega_0$ into level sets of $R$:
$$
{\mathscr A} = \{\Omega_k=R^{-1}(k):k\in{\mathbb N}\}
$$
has finite entropy w.r.t. $\mu\mid_{\Omega_0}$. In this case, the suspension $\Omega$ can be seen as a Rokhlin-Kakutani tower over $\Omega_0$, and the lift of $\mu$ to $\Omega$ via
$$
\tilde\mu(A) := \sum_{k=1}^\infty \sum_{j=0}^{k-1} \mu(\Omega_k\cap T^{-j}(A))
$$
is a finite measure.
\section{Construction of the coarse partition ${\mathscr C}_\sigma$}\label{s.3}
In this section, we will define the coarse partition ${\mathscr C}_\sigma$. The construction consists of three steps:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}
\item first, we will identify a cross section $D_\sigma$ in the neighborhood $B_r(\sigma)$. Unlike the previous construction in~\cite{APPV}, this new cross section, in fact, contains the singularity $\sigma$. One can think of it as the place where the flows speed is the slowest;
\item we will cut $D_\sigma$ into countably many layers $\{D_n\}_{n>n_0}$, each of which is roughly $e^{-n}$ close to $\sigma$; we will show that the estimation on the flow speed~\eqref{e.flowspeed} holds on each $D_n$;
\item finally, we define the partition ${\mathscr C}_n$ in the cone $\phi_{[0,1)}(D_\sigma)$ by pushing each $D_n$ along the flow by time one; we will show that this partition has finite entropy for any probability measure.
\end{enumerate}
\subsection{The cross section $D_\sigma$}
To start with, we fix $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1>0$ small enough, such that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{itemize}
\item for every $\sigma'\in{\rm Sing}(X), \sigma'\ne\sigma$, we have $B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma)\cap B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma')=\emptyset$;
\item the exponential map $\exp_\sigma$ is well defined on $\{v\in T_\sigma{M}: |v|\le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1\}$;
\item the flow speed $|X(x)|$ is a Lipschitz function of $d(\sigma,x)$ on $\operatorname{C}l(B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma))$: there exists $0<L_0<L_1$, such that for every $\sigma\in{\rm Sing}(X)$ and every $x\in \operatorname{C}l(B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma))$, we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.flowlip}
\frac{|X(x)|}{d(x,\sigma)}\in[L_0,L_1].
\end{equation}
In particular, we have
$|X(y)|\in [L_0 \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1,L_1\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1]$ for all $y\in\partial B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma)$.
\item the flow in $B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma)$ is a $C^1$ small perturbation of the linear flow
$$
\tilde\phi_t(x) = e^{At}x,
$$
where $A$ is a matrix with non-zero eigenvalues;
\item for $x\in B_{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1}(\sigma)$, the tangent maps $D\phi_1(x)$ are small perturbations of the hyperbolic matrix $e^A$, with eigenvalues bounded away from $1$.
\end{itemize}
The second requirement is possible since the vector field $X$ is $C^1$, and the singularities are non-degenerate. Moreover, $L_0$ and $L_1$ can be made uniform in a $C^1$ neighborhood of $X$.
We treat the hyperbolic splitting $E^s\oplus E^u$ as orthogonal in $T_\sigma M$, in which we use the box norm. For $r\le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1$, we will also think of $B_r(\sigma)$ as a box, whose sides are ``parallel'' to the stable and unstable manifolds of $\sigma$.\footnote{In fact, whether $B_r(\sigma)$ is a box or not does not affect our construction, as long as the flow speed on the boundary is bounded above and below.} For each $v\in T_\sigma M$, we write $v=(v^s, v^u)$ for the components of $v$ along $E^s$ and $E^u$, respectively.
We define $${\mathcal D}_\sigma(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1) = \{v\in T_\sigma M: |v|\le\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta, |v^s| = |v^u|\},$$
and
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.D}
D_\sigma(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1) = \exp_\sigma \left({\mathcal D}_\sigma(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1)\right)
\end{equation}
for its projection to the manifold.
\subsection{The partition ${\mathscr C}_\sigma$}
Below we will fix $r\le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1$. We take $n_1$ large enough, such that $e^{-n_1}<r$ (below we will enlarge $n_1$ once to obtain $n_0$, see Lemma~\ref{l.tx}). For $n>n_1$, define
$$
D_n = D_\sigma(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1) \cap \left(B_{e^{-n}}(\sigma)\setminus B_{e^{-(n+1)}}(\sigma)\right),
$$
and note that $D_n$ and $D_m$ are disjoint if $n\ne m$. Furthermore, \eqref{e.flowlip} immediate gives
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.speedDn}
|X(x)|\in[L_0e^{-(n+1)},L_1e^{-n}],
\end{equation}
as required by (II) of Theorem~\ref{m.C}.
Following~\cite[Section 5.3.1]{PYY}, for each $x\in B_{r}(\sigma)$, we write $x^s = d(x,W^u(x))$ and $x^u=d(x,W^s(\sigma))$, where $W^s(\sigma)$ and $W^u(\sigma)$ are the stable and unstable manifolds of $\sigma$, respectively. Then we define the {\em $\alpha$-cone on the manifold}, denote by $D^{i}_\alpha(\sigma)$, $i=s,u$, as:
$$
D^s_\alpha(\sigma) = \{x\in B_{r}(\sigma): x^u<\alpha x^s\},\pitchforkspace{1cm}D^u_\alpha(\sigma) = \{x\in B_{r}(\sigma): x^s<\alpha x^u\}.
$$
Clearly, the stable and unstable manifold of $\sigma$ are contained in the $\alpha$-cones, for all $\alpha>0$.
We also extend the hyperbolic splitting $E^s\oplus E^u$ on $T_\sigma M$ to $ B_{r}(\sigma)$ and define the $\alpha$-cone ${\mathcal C}_\alpha(E^s)$ and ${\mathcal C}_\alpha(E^u)$ on the tangent bundle. The next lemma shows that the cones on the manifold and the cones on the tangent bundle are naturally related.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\cite[Lemma 5.1]{PYY}\label{l.cones}
There exists $K\ge1$, such that for all $\alpha>0$ small enough,
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}
\item for every $x\in D^s_\alpha(\sigma)$, we have $X(x)\in {\mathcal C}_{K\alpha}(E^s)$;
\item for every $x\in B_{r}(\sigma)$, if $X(x)\in {\mathcal C}_\alpha(E^s)$, we have $x\in D^s_{K\alpha}(\sigma)$.
\end{enumerate}
Moreover, the same holds for $D^u_\alpha(\sigma)$ and ${\mathcal C}_\alpha(E^u)$.
\end{lemma}
The proof of this lemma easily follows from the fact that $\phi_t$ in $B_r(\sigma)$ is a small perturbation of the linear flow $e^{At}x$.
Note that for $x\in B_{r}(\sigma)\setminus (D^s_\alpha(\sigma)\cup D^u_\alpha(\sigma))$, we lose control on the direction of $X(x)$. One can think of the region $ B_{r}(\sigma)\setminus (D^s_\alpha(\sigma)\cup D^u_\alpha(\sigma))$ to be the place where the flow is `making the turn' from the $E^s$ cone to the $E^u$ cone. The key observation is that, once $\alpha$ is fixed, the time it takes from $D^s_\alpha(\sigma)$ to $D^u_\alpha(\sigma)$ is uniformly bounded. See~\cite[Lemma 5.2]{PYY}.
For each $n$ and $x\in D_n$, we write
$$
t^+_x = \inf\{\tau>0:\phi_\tau(x)\in\partial B_{r}(\sigma)\},
$$
and
$$
t^-_x = \inf\{\tau>0:\phi_{-\tau}(x)\in\partial B_{r}(\sigma)\}.
$$
The next lemma provide the estimate on $t^\pm_x$ for $x\in D_n$. which is the key for our construction.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.tx}
There exists constants $K_1>K_0>0$ independent of $r$, and $n_0\gg n_1$ depending on $r$, such that for each $n>n_0$ and $x\in D_n$, we have
$$
\frac{t^\pm_x}{n}\in[K_0,K_1].
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
We only need to estimate $t^+_x$, then the same argument applied on the vector field $-X$ will give the desired result for $t^-_x$.
Since the flow in $B_{r}(\sigma)$ is a small perturbation of the flow $e^{At}$, we can take $\alpha_0>0$ small enough, such that the flow speed grows exponentially fast in ${\mathcal C}_{K\alpha_0}(E^u)$, where $K>0$ is the constant given by Lemma~\ref{l.cones}. To be more precise, there exists $C,C'>0,1<\lambda<\lambda'$, such that for all $x\in D^u_{\alpha_0}(\sigma)$, we must have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.exponential}
C\lambda^t\le\frac{|X(\phi_t(x))|}{|X(x)|}\le C'(\lambda')^t,
\end{equation}
provided that $\phi_{[0,t]}(x)\subset \operatorname{C}l(B_r(\sigma))$.
We write
$$
t^u_x = \inf\{t>0:\phi_t(x)\in D^u_{\alpha_0}(\sigma)\}.
$$
By Lemma 5.2 of~\cite{PYY}, there exists $T^{\alpha_0}>0$, such that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation*}\label{e.turn}
t^u_x<T^{\alpha_0} ,\mbox{ for all } x\in B_{r}(\sigma).
\end{equation*}
In particular, the above estimate holds uniformly on every $D_n$. As an immediate corollary, we get
$$
\frac{|X(\phi_{t^u_x}(x))|}{|X(x)|}\in[d_0,d_1]\mbox{ for some } d_1>d_0>0 \mbox{ independent of } n.
$$
Combine this with~\eqref{e.flowlip} and~\eqref{e.speedDn}, we see that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.du}
|X(\phi_{t^u_x}(x))|\in [d_0L_0e^{-n-1},d_1L_1e^{-n}]\mbox{ for all } x\in D_n.
\end{equation}
We are left to control $t^+_x - t^u_x$ for $x\in D_n$. Note that for all $x\in D_\sigma$, we have
$$
d(\phi_{t^+_x}(x),\sigma)=r;
$$
consequently,
$$
|X(\phi_{t^+_x}(x))|\in[L_0r, L_1r].
$$
This combined with~\eqref{e.speedDn},~\eqref{e.exponential} and~\eqref{e.du} gives
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation*}
\frac{n+\log\frac{L_0r}{C'd_1L_1}}{\log\lambda'}\le t^+_x\le \frac{n+\log\frac{L_1r}{Cd_0L_0}}{\log\lambda}+T^{\alpha_0}.
\end{equation*}
In particular, there exists $n_0=n_0(r)$, such that if $n>n_0$ then we must have
$$
K_0 = \frac{1}{2\log\lambda'}\le \frac{t^+_x}{n} \le \frac{2}{\log\lambda}= K_1.
$$
We conclude the proof of this lemma.
\end{proof}
We make the following observation on the choice of the constants $K_0$ and $K_1$, which will be used in the next section.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}\label{r.uniformconstant}
Note that the constants $L_0,L_1, C,C',d_0,d_1$ depends continuously on the vector field, thus can be made uniform in a $C^1$ neighborhood of $X$. On the other hand, the constants $\lambda$, $\lambda'$ and $T^\alpha$ depends on the hyperbolicity of $\sigma$, therefore can be made uniform for nearby $C^1$ vector fields.
\end{remark}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.intersect}
Each orbit segment $\phi_{[-t^-_x,t^+_x]}(x)$ intersect with $ D_\sigma$ at exactly one point, which is $x$.
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
For each $x\in B_{r}(\sigma)$, we write $\exp_\sigma^{-1}(x) = (v^s(x), v^u(x))$. Then this lemma easily follows from the fact that for $x\in D_\sigma$, $|v^s(\phi_t(x))|$ is strictly decreasing along the forward orbit of $x$, and $|v^u(\phi_t(x))|$ is strictly increasing (thanks to $\phi_t$ being a small perturbation of the linear flow $e^{At}$). Since points on $ D_\sigma$ satisfies $|v^s(x)|=|v^u(x)|$, $\phi_{[-t^-_x,t^+_x]}(x)$ and $ D_\sigma$ can only intersect at $x$.
\end{proof}
The next lemma deals with the measure of the flow box $\phi_{[-t^-_x,t^+_x]}(D_n)$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.measure.dn}
For every probability measure $\mu$ that is invariant under $\phi_t$, we have
$$\sum_{n>n_0}\mu\left(\bigcup_{x\in D_n}\phi_{[-t^-_x,t^+_x]}(x)\right)\le 1.$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
Recall that $\{D_n\}$ are pairwise disjoint. Set
$$
\tilde D_n = \bigcup_{x\in D_n}\phi_{[-t^-_x,t^+_x]}(x),
$$
we claim that $\{\tilde D_n\}$ are also pairwise disjoint.
We prove by contradiction. Assume there exists $m\ne n$, $x\in D_m$, $y\in D_n$ such that
$$
\phi_t(x) = \phi_s(y),
$$
for $0<t<t_x^+$ and $0<s<t_y^+$. Then $x$ and $y$ are on the same orbit, which intersects with $ D_\sigma$ at two different points, a contradiction with the previous lemma.
As a result, we have
$$\sum_{n>n_0}\mu\left(\bigcup_{x\in D_n}\phi_{[t^-_x,t^+_x]}(x)\right)= \mu\left(\bigsqcup_{n>n_0}\tilde{ D}_n\right)\le\mu(B_{r}(\sigma))\le 1.$$
\end{proof}
From now on, we write, for each $n>n_0,$
$$
C_n = \phi_{[0,1)}D_n.
$$
Then $\cup_{n>n_0}C_n \subset \phi_{[0,1)} D_\sigma$ is contained in a fundamental domain of the time-one map $\phi_1$.
The next lemma shows that for $N>n_0$, the set $\cup_{n>N}C_n$ have uniformly small measure:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.measure.triangle}
Let $K_0$ be the constant given by Lemma~\ref{l.tx}. For every $N>n_0$ and every invariant probability $\mu$, we have
$$
\mu\left(\bigcup_{n>N}C_n\right)\le \frac{1}{K_0N}.
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
For every $x\in D_n$, $n\ge N$, Lemma~\ref{l.tx} gives
$$
t^+_x\ge K_0 n \ge K_0 N.
$$
Also note that for each $j,k\in{\mathbb N}\cup\{0\}$ with $j\ne k\le \min_{x\in D_n}\{t^+_x\}$, the sets $\phi_{j}(C_n)$ and $\phi_{k}(C_n)$ are disjoint and have the same measure.
Therefore, we get
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
N\cdot \mu\left(\bigcup_{n>N}C_n\right)=&\frac{1}{K_0}\mu\left(\bigcup_{n>N}\bigcup_{k=0}^{NK_0-1}\phi_k(C_n)\right)\\
=&\frac{1}{K_0}\sum_{n>N}\mu\left(\phi_{[0,NK_0)}(D_n)\right)\\
\le&\frac{1}{K_0}\sum_{n>N}\mu\left(\bigcup_{x\in D_n} \phi_{[0,t^+_x))} (x)\right) \le \frac{1}{K_0}.
\end{align*}
\end{proof}
Now we are ready to construct the coarse partition ${\mathscr C}_\sigma$. One can think of $\{C_n\}_{n>n_0}$ as a (one-sided) infinite cylinder, with the singularity $\sigma$ sitting at the end. See Figure~\ref{f.partitionB}.
We define:
$$
{\mathscr C}_\sigma = \{C_n: n>n_0\}\cup\{M\setminus (\cup_{n>n_0}C_n)\}
$$
as a measurable, countable partition of $M$. See Figure~\ref{f.partitionC}.
\subsection{Finite entropy}
The next proposition states that the metric ${\mathscr C}_\sigma$ w.r.t. any invariant measure is uniformly bounded.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.Centropy}
There exists $H_1>0$, such that for every invariant probability measure $\mu$,
we have $$H_\mu({\mathscr C}_\sigma)<H_1<\infty.$$
\end{proposition}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
By Lemma~\ref{l.finiteentropy}, we only need to verify that $\sum_n n\mu(C_n)<N$ for some constant $N>0$.
By Lemma~\ref{l.tx}, we have $t^+_x \ge K_0 n$ on $D_n$. Then it follows that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
n\mu(C_n) = &\frac{n}{K_0 n} \sum_{j=0}^{K_0n-1}\mu(C_n) = \frac{1}{K_0 } \sum_{j=0}^{K_0n-1}\mu(\phi_j(C_n))\\
=&\frac{1}{K_0 }\mu\left(\bigsqcup_{j=0}^{K_0n-1}\phi_j (C_n)\right)
\le \frac{1}{K_0}\mu\left(\bigcup_{x\in D_n}\phi_{[0,t^+_x]}(x)\right).
\end{align*}
Now we can sum over $n$ and obtain:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align}\label{e.sumC}
\sum_n n\mu(C_n)\le \sum_n \frac{1}{K_0}\mu\left(\bigcup_{x\in D_n}\phi_{[0,t^+_x]}(x)\right)\le \frac{1}{K_0},
\end{align}
where we apply Lemma~\ref{l.measure.dn} to obtain the last inequality.
\end{proof}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}[Proof of Theorem~\ref{m.C}]
(I) is proved as Lemma~\ref{l.intersect}. (II) and (III) are obtained in Lemma~\ref{l.tx}. Item (IV) easily follows from the continuity of the flow and the choice of $n_0$. (V) is the definition of ${\mathscr C}_\sigma$, and (VI) is precisely Proposition~\ref{p.Centropy}. The proof of Theorem~\ref{m.C} is finished.
\end{proof}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}
It is important to observe that the proof of Proposition~\ref{p.Centropy} does not depend on how small $\mu(B_r(\sigma))$ is, or whether $\mu(\sigma)=0$ or not. In fact, when $\mu(\sigma)>0$, one can obtain a better bound in both Lemma~\ref{l.measure.dn} and Proposition~\ref{p.Centropy}.
\end{remark}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}
Note that the construction of ${\mathscr C}_\sigma$ depends continuously on the flow $X$ in $C^1$ topology. Furthermore, the constants in Lemma~\ref{l.measure.dn},~\ref{l.measure.triangle} and Proposition~\ref{p.Centropy} can be made uniform in a $C^1$ neighborhood.
\end{remark}
As can be seen from the proof of Lemma~\ref{l.tx}, $H_1$ can be chosen arbitrarily close to the largest Lyapunov exponent at $\sigma$ (by taking $n_0$ large enough). At first glance, this may seem to contradict with Ruelle's inequality; however, it is due to the fact that the partition ${\mathscr C}_\sigma$ is not expansive in the sense of Definition~\ref{d.1}. This problem will be partially solved by the refined partition ${\mathscr A}_\sigma$, which will be constructed in the next section.
\subsection{Comparison with the conventional sections}
Here we will relate our new section $D_\sigma$ to the cross sections $\Sigma^{i/o,\pm}$ constructed in~\cite{APPV}.
In~\cite{APPV} the authors considered {singularly hyperbolic flows}, that is, flows on a three-dimensional manifold with an attractor $\Lambda} \def\e{\varepsilonmbda$, on which there is a dominated splitting $E^s\oplus E^{cu}$, such that the tangent flow on $E^s$ is uniformly contracting, and $E^{cu}$ is volume expanding. If $\Lambda} \def\e{\varepsilonmbda$ is singular hyperbolic without any singularity, then it must be Anosov. On the other hand, if $\Lambda} \def\e{\varepsilonmbda$ contains a singularity $\sigma$ that is accumulated by regular orbits, then $\sigma$ is {\em Lorenz-like}. Here being Lorenz-like means that the eigenvalues of $DX|_\sigma$ must satisfy
$$
\lambda_1>0>\lambda_2>\lambda_3, \mbox{ and } \lambda_1+\lambda_2>0.
$$
See~\cite{ArPa10} for more detail.
More importantly, it is proven that the strong stable manifold $W^{ss}(\sigma)$ (which is given by the dominated splitting $E^s\oplus E^{cu}$) is tangent to the eigenspace $E^3$ of $\lambda_3$, and intersects with $\Lambda} \def\e{\varepsilonmbda$ only at the singularity $\sigma$. Combine this with~\cite{LGW} and~\cite{GSW}, we see that regular orbit in $\Lambda} \def\e{\varepsilonmbda$ can only approach $\sigma$ in a very small cone around $W^{cu}(\sigma)$.\footnote{In fact, in a small cone around $E^2$. See the discussion in~\cite[Section 5.2]{PYY}.} Assuming linearization in a neighborhood of $\sigma$,\footnote{Note that this imposes certain conditions on the eigenvalue $\lambda_i$, $i=1,2,3$, especially if one requires the linearization to be sufficiently smooth.} the authors constructed four cross sections, $\Sigma^{i/o,\pm}$, for each singularity. Here $\Sigma^{i,\pm}$ are used to capture orbits that approaches $\sigma$, and $\Sigma^{o,\pm}$ tracks those whose are leaving $\sigma$. See Figure~\ref{f.lorenz}. Using the smoothness of the linearization, they show that the fly time $\tau$ from $\Sigma^i$ to $\Sigma^o$ satisfies
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.16}
\tau(x) = -\frac{\log x_1}{\lambda_1},
\end{equation}
where $x_1$ is the distance $x_1 = d(x,W^{cs}(\sigma)).$ In particular, $\tau$ is integrable w.r.t.\, the Lebesgue measure on $\Sigma^{i,+}$.
Now let us describe the relation between $D_\sigma$ and $\Sigma^{i/o}$. Assuming that $\sigma$ is a Lorenz-like singularity for some vector field $X$ on a three-dimensional manifold $M$ (and note that we do not need the singular-hyperbolicity outside a neighborhood of $\sigma$), we may further assume that $\Sigma^{i/o,\pm}$ are taken on the set $\partial B_r(\sigma)$. Here we can take $B_r(\sigma)$ to be a cube around $\sigma$ which does not affect our construction, as we only need $B_r(\sigma)$ to be small and the flow speed on $\partial B_r(\sigma)$ to be bounded above and below.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{figure}
\centering
\delta} \def\De{\Deltaef\columnwidth{\columnwidth}
\includegraphics[scale=0.9]{Pic4.pdf}
\caption{The image and pre-image of $\{D_n\}$ on the sections $\Sigma^{i/o,+}\subset \partial B_r(\sigma)$.}
\label{f.lorenz}
\end{figure}
Then we can construct $D_\sigma$ and $\{D_n\}_{n>n_0}$ as before. For simplicity, we will only focus on the right half-space which contains $\Sigma^{o,+}$. If we define
$$
D_n^o = \bigcup_{x\in D_n} \phi_{t^+_x}(x),\mbox{ and }D_n^i = \bigcup_{x\in D_n} \phi_{-t^-_x}(x)
$$
for the image and the pre-image of $D_n$ on $\partial B_r(\sigma)$ under the flow $\phi_t$, then $\{D_n^i\cap \Sigma^{i,+}\}$ is a family of countably many strips (the yellow strips in Figure~\ref{f.lorenz}) on $\Sigma^{i,+}$, which becomes closer to the curve $W^{cs}(\sigma)\cap\Sigma^{i,0}$ as $n$ gets larger. Their forward image under the flow:
$$
\phi_{\tau}(D_n^i\cap \Sigma^{i,+})
$$
are contained in a triangular region inside $D_n^o\cap\Sigma^{o,+}$.
Recall that Lemma~\ref{l.tx} shows $t_x^- = {\mathcal O}(n)$ on $D_n$, which means that for $x\in D_n^i\cap \Sigma^{i,+}$, the distance $d(x,W^{cs}(\sigma)) $ is of order $e^{-(\lambda_1+1)n}$. Combined with~\eqref{e.16}, this shows that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation*}
\tau(x) \approx \frac{\lambda_1+1}{\lambda_1}n\,\,\mbox{ for } x\in D_n^i\cap\Sigma^{i,+}.
\end{equation*}
In other words, the partition $\{D_n\}$ on $D_\sigma$ induces a countable partition $\{D_n^i\cap\Sigma^{i,+}\}$ on $\Sigma^{i,+}$, which can be seen as the level sets of $\tau$.
In a later work~\cite{GP}, the authors considered the return map $T$ on the cross sections $\Sigma^{i,+}$. They showed that the return map $T$ can be reduced to a one-dimensional, uniformly expanding map $T_{L}$ on the interval $[-1/2,1/2]$ with unbounded derivative at zero, known as the Lorenz-map. Then our partition $\{D_n\}$ naturally induces a countable partition on $[-1/2,1/2]$, which is of the form
$$
\left\{(-e^{-{(\lambda_1+1)n}},0), (0, e^{-(\lambda_1+1)n}): n>n_0\right\}.
$$
Note that partitions of this form has been widely used to study unimodal maps, namely interval maps with zero derivative at the point $0$.
Similarly, the same treatment can be applied to the contracting Lorenz-attractors in \cite{Rov}, resulting in the same partition $\left\{(-e^{-{(\lambda_1+1)n}},0), (0, e^{-(\lambda_1+1)n}): n>n_0\right\}$ for the one-dimensional Rovella maps. Such maps can be seen as unimodal maps with discontinuity at zero, and our partition coincide with the the classical partitions for the Rovella maps. See~\cite{PT} for more detail.
Finally, we would like to emphasis that, unlike in~\cite{APPV} and~\cite{GP}, our construction for the cross section $D_\sigma$ and the countable partition $\{D_n\}$:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}
\item does not require knowledge on know regular orbits approaches $\sigma$;
\item does not need information on the hyperbolicity of $X$ at regular points;
\item avoids linearization altogether, thus does not require any condition on the eigenvalues at $\sigma$;
\item can be applied in any dimension.
\end{enumerate}
In fact, our estimation in Lemma~\ref{l.tx} is enough to show that $\tau$ is integrable w.r.t.\,the Lebesgue measure on $\Sigma^{i,+}$, which is a crucial step in~\cite{APPV} ,~\cite{GP} and~\cite{PT}.
\section{Construction of the refined partition ${\mathscr A}_\sigma$}\label{s.4}
As we have discussed before, both the topological theory (size of the invariant manifold, transverse homoclinic intersections, etc.) and the entropy theory (entropy expansiveness, upper semi-continuity of the metric entropy, etc.) requires estimation on a uniform size. However, for singular flows, the tangent map determines the underlying dynamics only in the scaled tubular neighborhood along the orbit. From Proposition~\ref{p.tubular1} which goes back to the classical work of Liao~\cite{Liao96}, the size of such neighborhoods depend on the length and the flow speed at each point. Combine this with Lemma~\ref{l.tx}, we see that for points in $C_n$, the size of such neighborhoods must be exponentially small.
This observation forces us to construct a new partition ${\mathscr A}_{ \sigma}$ by refining each element of ${\mathscr C}_\sigma$ with a finite partition ${\mathscr B}_n$ (due to the observation above, the cardinality of ${\mathscr B}_n$ must be exponential in $n$), such that on each element of ${\mathscr B}_n$, the scaled tubular neighborhood is sufficiently long. For this purpose, we need a sharp control over the flow speed $|X(x)|$, and the time it takes for the point $x$ to leave $B_r(\sigma)$, which
is already given by Theorem~\ref{m.C}. The main difficulty here is to show that ${\mathscr A}_\sigma$ still has finite entropy.
\subsection{The partition ${\mathscr B}_n$ and ${\mathscr A}_\sigma$}
Recall that $L_{1,{\mathcal U}}$ is an upper bound of the scaled linear Poincar\'e flow $\psi_t^*$ given by Lemma~\ref{l.C1norm}. Let $L(X)$ and $L(-X)$ be the constants given by Proposition~\ref{p.tubular1} for the vector field $X$ and $-X$, respectively. We define
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.L}
N_0 =\max\{L(X), L(-X),L_{1,{\mathcal U}}\},
\end{equation}
where $K_1$ and $K_0$ are the constants given by Lemma~\ref{l.tx}. For any given $L\ge N_0$ and $0<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$ with $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$ given by Proposition~\ref{p.tubular1}, we consider balls with center in $D_n$ and radius:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.rn}
r_n := \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta L^{-K_1 n}L_0e^{-(n+1)},
\end{equation}
where $L_0$ is given by~\eqref{e.flowlip}.
Fix $r<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1$, then Theorem~\ref{m.C} gives a countable partition ${\mathscr C}_n = \{C_n\}_{n>n_0}$ in the neighborhood $B_r(\sigma)$. Also recall that each $C_n$ is the image of some $D_n\subset D_\sigma$ under the flow by time one. For each $n>n_0$, we take a $r_n$-separated set in $\overline{D_n}$ with maximal cardinality, denote by $E_n$. Here $E_n$ being $r_n$-separated means that for every $x,y\in E_n$, we have $d(x,y)> r_n$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.diam}
There exists a finite partition $\tilde {\mathscr B}_n$ of $D_n$, such that for every $B\in\tilde {\mathscr B}_n$, there exists $x\in E_n$ with $B_{r_n/2}(x)\cap D_n\subset B \subset B_{r_n}(x)\cap D_n$. In particular, we have
$$
\operatorname{diam} B \le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta \cdot \frac{L_0 }{e}\left(L^{K_1}e\right)^{-n} ,\,\forall B\in\tilde{\mathscr B}_n
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
Since $E_n$ is $r_n$-separated with maximal cardinality, we have
$$
B_{r_n/2}(x)\cap B_{r_n/2}(y)=\emptyset \mbox{ for } x,y\in E_n, x\ne y,
$$
and
$$
D_n\subset \bigcup_{x\in E_n}B_{r_n}(x).
$$
Furthermore, the same hold when restricted to $D_n$. Then the existence of such partition immediately follows from the above properties.
\end{proof}
The choice of $L$ in~\eqref{e.L} together with Proposition~\ref{p.tubular1} lead to the following proposition:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.tubular.sing}
For the partition $\{ \tilde{\mathscr B}_n\}$ given by the previous lemma and for every $x,y$ that are contained in the same element of $\tilde {\mathscr B}_n$, the orbit of $y$ from $-t^-_y$ to $t^+_y$ is contained in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of $x$, until the orbit of $x$ leaves $B_{r}(\sigma)$.
\end{proposition}
The next lemma shows that the cardinality of ${\mathscr B}_n$ grows exponentially in $n$:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.Bcard}
There exists constants $c_1>0$ such that for $L''=(L^{K_1}e)^{\delta} \def\De{\Deltaim M}$ and for every $n>n_0$, we have
$$
\# \tilde{\mathscr B}_n\le c_1(L'')^{n}.
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
We write $c_0 = L_0/e$ and $L'=L^{K_1}e$. Then for each $n>n_0$, \eqref{e.rn} becomes
$$
r_{n}= c_0 \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta \left(L'\right)^{-n},
$$
which means
$$
\operatorname{vol}(B_{r_{n}/2}(x))\ge c L'^{ - n\cdot \delta} \def\De{\Deltaim M}
$$
for some constant $c>0$ depending on $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$ and the Riemannian metric. Since for $x\ne y\in E_n$ we have $B_{r_n/2}(x)\cap B_{r_n/2}(y)=\emptyset$, it follows that
$$
\#\tilde {\mathscr B}_n\le \# E_n \le \frac{\operatorname{vol} M}{c L'^{ - n\cdot \delta} \def\De{\Deltaim M}}.
$$
Then the lemma follows with $c_1=\operatorname{vol} (M)/c$, and $L'' = L'^{\delta} \def\De{\Deltaim M} = (L^{K_1}e)^{\delta} \def\De{\Deltaim M}$.
\end{proof}
Now we write
$$
{\mathscr B}_n = \{\phi_{[0,1)}(\tilde{B}): \tilde{B}\in\tilde {\mathscr B}_n\}.
$$
Then $ {\mathscr B}_n$ is a partition of $C_n$ for each $n>n_0$.
Recall that the closure of the set
$$
O(\sigma) = \bigcup_{n>n_0}\bigcup_{x\in D_n} \phi_{[-t^-_x,t^+_x]}(x)\subset B_{r}(\sigma)\\
$$
contains a neighborhood of $\sigma$. We also define:
$$
B^-(\sigma) =O(\sigma)\cap \exp_\sigma\left(\{v\in T_\sigma M: |v|\le\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta, |v^s|<|v^u|\}\right),\\
$$
$$
B^+(\sigma) = O(\sigma) \setminus\left(B_\sigma^-\cup \bigcup_{n>n_0}C_n\right).
$$
$B^\pm(\sigma)$ can be seen as the regions in $O(\sigma)$ that sit ``above'' and ``below'' the set $\bigcup_{n>n_0}C_n$, respectively. One should note that $\sigma\in\operatorname{C}l( B^-(\sigma))\cap \operatorname{C}l(B^+(\sigma))$.
We then define the partition ${\mathscr A}_\sigma$ as:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.partition}
{\mathscr A}_\sigma = \{B : B\in {\mathscr B}_n \mbox{ for some }n>n_0\}\cup\{B^-(\sigma),B^+(\sigma),O(\sigma)^c\}.
\end{equation}
Then ${\mathscr A}_\sigma$ is a countable partition of $M$ which refines ${\mathscr C}_\sigma$. Note that the partition is constructed locally inside the neighborhood $O(\sigma)$ of $\sigma$, and treat the complement of this neighborhood as a single partition element.
\subsection{Finite entropy}
Next, we show that the metric entropy of ${\mathscr A}_\sigma$ is uniformly bounded from above:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.Aentropy}
There exists $H_2>0$ depending on $L$, such that for every invariant probability measure $\mu$,
we have $$H_\mu({\mathscr A}_\sigma)<H_2<\infty.$$
\end{proposition}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
We use the following inequality for the conditional entropy:
$$
H_\mu({\mathscr A}_\sigma)\le H_\mu({\mathscr A}_\sigma|{\mathscr C}_\sigma) + H_\mu({\mathscr C}_\sigma).
$$
Note that the second term is bounded by $H_1$ due to Proposition~\ref{p.Centropy}. For the first term, recall that ${\mathscr A}_\sigma$ is obtained by refining each element of ${\mathscr C}_\sigma$ with the partition $ {\mathscr B}_n$. In the mean time, Lemma~\ref{l.Bcard} gives
$$
\# {\mathscr B}_n = \# \tilde{\mathscr B}_n\le c_1(L'')^{n}.
$$
We have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
H_\mu({\mathscr A}_\sigma|{\mathscr C}_\sigma) &= -\sum_n \mu(C_n)\sum_{B\in {\mathscr B}_n} \mu_{C_n}(B)\log \mu_{C_n}(B)\\
&\le \sum_n\mu(C_n)\log \left(\# {\mathscr B}_n\right)\\
&\le \sum_n(n\log L''+\log c_1)\mu(C_n)\\
&\le \log c_1 + \log L''\sum_n n\mu(C_n)\\
&\le \log c_1 + \frac{1}{K_0}\log L''<\infty,
\end{align*}
where the last line follows from~\eqref{e.sumC}. Now the proposition holds with $H_2=\log c_1+\frac{1}{K_0}\log L'' + H_1$.
\end{proof}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}\label{r.uniform}
Following Remark~\ref{r.uniformconstant}, we see that the constants $c_0, c_1, L', L''$ and $H_2$ can be made uniform for nearby $C^1$ vector fields.
\end{remark}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}[Proof of Theorem~\ref{m.A}]
Item (I) and (III) follows from Lemma~\ref{l.Bcard}, while (II) is given by Lemma~\ref{l.diam}. (IV) is precisely Proposition~\ref{p.tubular.sing}, and (V) is proven as Proposition~\ref{p.Aentropy}.
For the continuity of the partition ${\mathscr A}_n$, note that (all the continuity is in Hausdorff topology and $C^1$ topology): (1) the cross sections $D_\sigma$ varies continuously for nearby $C^1$ vector fields; the same holds for each $D_n$; (2) for each $n$, the partition $\tilde{\mathscr B}_n$ can be made continuous; in particular, this means that ${\mathscr B}_n$ is continuous; (3) the (finitely many) neighborhoods $O(\sigma)$ varies continuously, therefore $C_{reg}$ varies continuously; (4) the finite partition ${\mathscr A}_{reg}$ can be made continuous w.r.t. nearby flows.
We conclude the proof of Theorem~\ref{m.A}.
\end{proof}
\section{The partition ${\mathscr A}$}\label{s.5}
In this section we will prove Theorem~\ref{m.3}. We assume that $X$ is a $C^1$ vector field such that all the singularities of $X$ are hyperbolic; in particular, $X$ has only finitely many singularities.
\subsection{Near each singularity}
First, note that $N_0$ in Theorem~\ref{m.A} is given by~\eqref{e.L}, which is defined for the entire flow. The same can be said about $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$ in Proposition~\ref{p.tubular1}. On the other hand, the constants $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1,L_0, L_1, K_, K_1, L', L'', c_0, c_1$ in both Theorem~\ref{m.C} and \ref{m.A} depends on the hyperbolicity of each singularity. Since there are only finitely many singularities, such constants can be made uniform for the vector field $X$ (also robust in a $C^1$ neighborhood).
Now, we can fix some $L\ge N_0$, $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$, $r<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_1$ and apply Theorem~\ref{m.A} to obtain a countable partition ${\mathscr A}_\sigma$ for each $\sigma\in{\rm Sing}(X)$. Each partition ${\mathscr A}_\sigma$ also comes with a set $O(\sigma)\subset B_r(\sigma)$, and $n_0^\sigma\in{\mathbb N}$. We will write
$$
n_0 = \max_{\sigma\in {\rm Sing}(X)} n_0^\sigma.
$$
\subsection{Away from singularities}
The set
$$
C_{reg} =\operatorname{C}l\left( M\setminus (\cup_{\sigma\in{\rm Sing} X} \operatorname{C}l(O(\sigma))) \right)
$$
consists only of regular points of $X$. Furthermore, by (IV) of Theorem~\ref{m.C}, points in $C_{reg}$ satisfies
$$
d(x,{\rm Sing}(X)) \ge e^{-n_0}.
$$
For $L\ge N_0$ and $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0$ as above, the set
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.reg}
B(x): = \phi_{(-\frac14,\frac14)}\left(N_x\left(\frac{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta}{2L} \cdot |X(x)|\right)\right)
\end{equation}
is the $\frac{\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta}{2}$-scaled tubular neighborhood starting at $\phi_{-\frac14}(x)$, with length $\frac12$. Moreover, if $B(x)\cap B(y)\ne\emptyset$, then both $B(x)$ and $B(y)$ are contained in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood at $\phi_{-1/2}(z)$ with length one, for every $z\in B(x)\cap B(y)$ (the choice of $\phi_{-1/2}(z)$ makes $z$ the ``center'' of this tubular neighborhood).
The collection $\{B(x): x\in C_{reg}\}$ forms an open covering of $C_{reg}$. Since $C_{reg}$ is compact, we can take a finite sub-covering $\{B(x_i): i=1,\ldots,k\}$, and obtain a finite partition of $C_{reg}$, whose elements are given by the intersection of elements in the sub-covering. We denote this partition by ${\mathscr A}_{reg}$.
Then for each $A\in{\mathscr A}_{reg}$, $\partial A$ consists of flow lines with bounded length, and the normal manifold $N_x$ at some regular point.\footnote{To obtain ${\mathscr A}_{reg}$, one can follow the language of {\em Bowen-Sinai refinement} for Markov partition, See~\cite{B08, Si}. This set-theoretical procedure refines a finite open covering into a finite partition, without destroying the local product structure. In our case, the local product structure is given by the normal manifolds and the flow lines (and recall that $C_{reg}$ is uniformly away from singularities).}
\subsection{The partition ${\mathscr A}$}
Finally, we define the partition ${\mathscr A}$ as
$$
{\mathscr A} = {\mathscr A}_{reg}\vee\bigvee_{\sigma\in{\rm Sing}(X)}{\mathscr A}_\sigma.
$$
Then we have:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.entropy}
For any invariant probability measure $\mu$, $h_\mu(\phi_1, {\mathscr A})$ is finite.
\end{proposition}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
We have
$$
H_\mu({\mathscr A})\le H_\mu({\mathscr A}_{reg})+\sum_{\sigma\in{\rm Sing}(X)}H_\mu({\mathscr A}_\sigma).
$$
The first term is finite since ${\mathscr A}_{reg}$ is a finite partition. Each term in the second summation is finite, thanks to Proposition~\ref{p.Aentropy}; also note that the summation itself has only finitely many terms since $X$ has only finitely many singularities.
\end{proof}
Note that for given $x\in M$ and $y\in {\mathscr A}^\infty(x)$, by Proposition~\ref{p.tubular.sing} and the construction at regular points by~\eqref{e.reg}, we see that the orbit of $y$ must stay in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of the orbit of $x$ forever. This finishes the proof of Theorem~\ref{m.3}.
In fact, more can be said: generally, given a regular point $x\in C_{reg}$,
the map:
$$
P_x(y):{\mathscr A}(x)\to N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)
$$
which maps the point $y\in{\mathscr A}(x)$ to the unique point of intersection $\{P_x(y)\} = \phi_{[-1,1]}(y)\cap N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$ is well-defined, since the partition ${\mathscr A}(x)$ is contained in a scaled tubular neighborhood of the orbit of $x$.
Note that if $y\in{\mathscr A}^\infty(x)$, then we have $\phi_j(y)\in A(\phi_j(x))$ for every $j\in{\mathbb Z}$. In particular, if $j\in{\mathbb Z}$ satisfies $\phi_j(x)\in C_{reg}$, then the construction of ${\mathscr A}$ at regular points means that $\phi_j(x),\phi_j(y)$ are in the same tubular neighborhood with length $1$, and
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.reg.partition}
P_{\phi_j(x)}(\phi_j(y))\in N_{\phi_j(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta).
\end{equation}
On the other hand, if $j\in{\mathbb Z}$ is such that $\phi_j(x)\in O(\sigma)$ for some singularity $\sigma$, then Proposition~\ref{p.tubular.sing} states that the orbit of $y$ and the orbit of $x$ are in the same scaled tubular neighborhood, as long as they are both in $O(\sigma)$; moreover, $y\in{\mathscr A}^\infty(x)$ guarantees that ${\rm Orb}(y)$ and ${\rm Orb}(x)$ must hit the same element of $ {\mathscr B}_n$ at the same iterate. Furthermore, ${\rm Orb}(y)$ and ${\rm Orb}(x)$ must enter and leave $O(\sigma)$ at the same iterates under the time-one map $\phi_1$. This is because, once $x$ leaves $B^+(\sigma)$, it enters the partition at a regular point, which is contained in a scaled tubular neighborhood with length $1$. Since $y\in{\mathscr A}^\infty(x)$, $y$ must enter the same element at the same iterate. In particular, this means that ${\rm Orb}(y)$ and ${\rm Orb}(x)$ spend the same amount of time in $B^\pm(\sigma)$; however, we lose control (in the sense that~\eqref{e.reg.partition} may not hold) for the orbit segment in $B^\pm(\sigma)$, since we treat each of them as a single partition element.
\section{A general result on the expansiveness w.r.t. a partition}\label{s.6}
In this section we will prove Theorem~\ref{m.tailestimate}, which gives a criterion for partitions whose entropy is equal to the metric entropy. To put our result in a more general context, let $f: M\to M$ be a homeomorphism, $\mu$ an invariant probability such that $h_\mu(f)<\infty$.
Let ${\mathscr A}$ be a measurable partition such that $H_\mu({\mathscr A})<\infty$.
Before we dive into the proof, let us make some remark regarding our notion of expansiveness w.r.t. a partition. Following Bowen~\cite{B72}, the infinite Bowen ball is defined by
$$B_{\infty,\varepsilon}(x)=\bigcap_{n\in \mathbb{Z}} f^{-n} B_\varepsilon(f^n(x)),$$
and the $\varepsilon$-tail entropy at $x$ is defined as
$$h_{tail}(f,x,\varepsilon)=h_{top}(B_{\infty,\varepsilon}(x),f).$$
The system $f$ is $\varepsilon$-entropy expansive if $h_{tail}(f,x,\varepsilon) = 0 $ for all $x$. A measure $\mu$ is called $\varepsilon$-almost entropy expansive, if $h_{tail}(f,x,\varepsilon) = 0$ for $\mu$ a.e. $x$. Bowen proved that if $f$ is $\varepsilon$-entropy expansive, then every finite partition ${\mathscr A}$ with $\operatorname{diam}{\mathscr A}<\varepsilon$ satisfies
$$
h_\mu(f) = h_\mu(f,{\mathscr A})
$$
for every invariant measure $\mu$. On the other hand, it is proven in~\cite{LVY} that $f$ is $\varepsilon$-entropy expansive if and only if every $f$ invariant measure $\mu$ is $\varepsilon$-almost entropy expansive.
For any $x\in M$, recall that the $\infty$ ${\mathscr A}$-ball is defined
$${\mathscr A}^\infty(x)=\bigcap_{n\in \mathbb{Z}} f^{-n} {\mathscr A}(f^n(x)),$$
and the ${\mathscr A}$-tail entropy of $x\in M$ is given by
$$h_{tail}(f,x,{\mathscr A})=h_{top}({\mathscr A}^\infty(x),f).$$
In other words, we are replacing the geometric balls $B_\varepsilon$ in Bowen's definition of tail entropy by ``partition balls'' ${\mathscr A}(x)$. Similarly, $\mu$ being ${\mathscr A}$-expansive can be seen as the equivalence of $\varepsilon$-almost expansiveness defined using the partition ${\mathscr A}$.
Also note that if $\operatorname{diam} {\mathscr A}<\varepsilon$ and if $\mu$ is $\varepsilon$-almost entropy expansive, then ${\mathscr A}(x)\subset B_{\infty,\varepsilon}(x)$ must have zero topological entropy. In other words, $\varepsilon$-almost entropy expansive implies ${\mathscr A}$-expansive when $\operatorname{diam}{\mathscr A}<\varepsilon$.
The key advantage of ${\mathscr A}$-expansiveness is that, it allows us to obtain
$$
h_\mu(f) = h_\mu(f,{\mathscr A})
$$
for a particular measure $\mu$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}[Proof of Theorem~\ref{m.tailestimate}]
Since $M$ is finite dimensional, there is $m$ determined by the dimension of $M$, such that we can take finite partition
${\mathscr B}=\{B_1,\cdots, B_m\}$ of the ambient manifold with arbitrarily small diameter, such that each point $x\in M$
lies in at most $m$ elements of $\overline{{\mathscr B}}=\{\operatorname{C}l(B_i)\}_{i=1,\cdots,m}$.
For any $E\subset M$, let
$$F(E,{\mathscr B})=\{B\in {\mathscr B}: B\cap E \neq \emptyset\}.$$
Denote by $r_n(\delta} \def\De{\Deltaelta,E)$ the minimal cardinality of $(n,\delta} \def\De{\Deltaelta)$-spanning sets on $E$. The next lemma is due to Bowen:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\cite{B72}
Let $\overline{{\mathscr B}}=\{\operatorname{C}l(B_i)\}_{i=1,\cdots,m}$ be a compact cover of $M$. There is a $\delta} \def\De{\Deltaelta>0$ such that
$$\#(F(E,\overline{{\mathscr B}}^n))\leq r_n(\delta} \def\De{\Deltaelta,E) m^n$$
for all $E\subset M$ and $n\geq 0$.
\end{lemma}
Let us continue the proof. We have
$$h_\mu({\mathscr B})\leq h_\mu({\mathscr B}\bigvee {\mathscr A})=\lim\frac{1}{n}H_\mu(\vee_{i=0}^{n-1} f^{-i}({\mathscr A})\bigvee \vee_{i=0}^{n-1} f^{-i}({\mathscr B}))$$
$$\leq \limsup \frac{1}{n}[H_\mu(\vee_{i=0}^{n-1}f^{-i}({\mathscr A}))+H_\mu(\vee_{i=0}^{n-1}f^{-i}({\mathscr B})\mid \vee_{i=0}^{n-1}f^{-i}({\mathscr A}))$$
$$=h_\mu({\mathscr A})+ \limsup \frac{1}{n}H_\mu(\vee_{i=0}^{n-1}f^{-i}({\mathscr B})\mid \vee_{i=0}^{n-1}f^{-i}({\mathscr A}))$$
Observe that
$$H_\mu(\vee_{i=0}^{n-1}f^{-i}({\mathscr B})\mid \vee_{i=0}^{n-1}f^{-i}({\mathscr A}))\leq \sum_{i=0}^{n-1} H_\mu(f^{-i}({\mathscr B})\mid \vee_{j=0}^{n-1}f^{-j}({\mathscr A}))$$
Fix $n_0>0$, for $i\geq n_0$.
Because
$$H_\mu(f^{-i}({\mathscr B})\mid \vee_{j=0}^{n-1}f^{-i}({\mathscr A}))=H_\mu({\mathscr B}\mid \vee_{j=-i}^{n-i-1}f^{-i}({\mathscr A}))\leq H_\mu({\mathscr B}\mid \vee_{j=-n_0}^{n-i-1}f^{-i}({\mathscr A}))$$
is decreasing to $H_\mu({\mathscr B}\mid \vee_{j=-n_0}^{\infty}f^{-j}({\mathscr A}))$, and because for $0\leq i <n_0$,
$$H_\mu(f^{-i}({\mathscr B})\mid \vee_{j=0}^{n-1}f^{-i}({\mathscr A}))\leq H_\mu(f^{-i}({\mathscr B}))=H_\mu({\mathscr B}),$$
we have
$$h_\mu({\mathscr B})\leq h_\mu({\mathscr A})+H_\mu({\mathscr B}\mid\vee_{j=-n_0}^{\infty}f^{-j}({\mathscr A})).$$
Since the above inequality holds for any $n_0$, and because $H_\mu({\mathscr B}\vee_{j=-n_0}^{\infty}f^{-j}({\mathscr A}))\searrow H_\mu({\mathscr B}\mid \vee_{-\infty}^\infty f^{-j}{\mathscr A})$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{eq.onesteptail}
h_\mu({\mathscr B})\leq h_\mu({\mathscr A})+H_\mu({\mathscr B}\mid \vee_{j=-\infty}^{\infty}f^{-j}({\mathscr A})).
\end{equation}
Now for any $n>0$ we consider $F=f^n$, ${\mathscr B}^n=\vee_{i=0}^{n-1} f^{-i}({\mathscr B})$ instead of ${\mathscr B}$ and ${\mathscr A}^n=\vee_{i=0}^{n-1} f^{-i}({\mathscr A})$,
we have
$$h_\mu(f^n,{\mathscr B}^n)\leq h_\mu(f^n,{\mathscr A}^n)+H_\mu({\mathscr B}^n\mid \vee_{j=-\infty}^{\infty}f^{-j}({\mathscr A})).$$
Because
$h_\mu(f^n,{\mathscr B}^n)=nh_\mu(f,{\mathscr B})$ and $h_\mu(f^n,{\mathscr A}^n)=nh_\mu(f,{\mathscr A})$, we have
$$h_\mu(f,{\mathscr B})\leq h_\mu(f,{\mathscr A})+\lim \frac{1}{n}H_\mu({\mathscr B}^n\mid {\mathscr A}^\infty)$$
$$\leq h_\mu(f,{\mathscr A})+\lim \frac{1}{n}\int \log \#(F({\mathscr A}^\infty(x),\overline{{\mathscr B}}^n))d\mu(x)$$
$$\leq h_\mu(f,{\mathscr A})+\lim \int \frac{1}{n} (\log r_n(\delta} \def\De{\Deltaelta,{\mathscr A}^\infty(x))+ nm) d\mu(x).$$
Since $\frac{1}{n} \log r_n(\delta} \def\De{\Deltaelta,{\mathscr A}^\infty)\leq r_1(\delta} \def\De{\Deltaelta, M)$, by dominate convergence,
$$h_\mu(f,{\mathscr B})\le h_\mu(f,{\mathscr A})+\int h_{top}({\mathscr A}^\infty(x),f)d\mu(x)+m=h_\mu(f,{\mathscr A})+m.$$
To get rid of $m$, for each $n>0$ we take $f^n$, $\bigvee_{i=0}^{n-1}f^{-i}({\mathscr A})$ and ${\mathscr B}$. Then we have
$h_\mu(f^n,{\mathscr B})\leq h_\mu(f^n,\bigvee_{i=0}^{n-1}f^{-i}({\mathscr A}))+m$.
Taking the diameter of ${\mathscr B}$ converging to $0$, we have
$h_\mu(f^n)\leq h_\mu(f^n,\bigvee_{i=0}^{n-1}f^{-i}({\mathscr A}))+m$,
thus $$h_\mu(f)\leq h_\mu(f,{\mathscr A})+m/n,$$
let $n\to \infty$, we finish the proof.
\end{proof}
\section{Application 1: entropy theory for flows away from tangencies}\label{s.7}
The rest of this paper is devoted to the entropy theory for star flows and flows away from homoclinic tangencies, using the partition ${\mathscr A}$. In this section, we will show that the partition ${\mathscr A}$ can be used to compute the metric entropy for any invariant measure.
The key idea of this proof is to relate the images of ${\mathscr A}^\infty$ with a family of one-dimensional curves, whose length are well-controlled. For this purpose, we use an argument similar to~\cite{LVY}. However, we will see that the argument here is much more involved. This is because, in~\cite{LVY} when one considers a diffeomorphism away from tangencies, there exists a dominated splitting on the tangent bundle given by \cite{W04}. As we have discussed, such splitting controls a neighborhood of the invariant set with uniform size. However this is not the case for singular flows. As we will see below, the fake foliations are only defined for the scaled linear Poincar\'e flow; as a result, the size of such foliation is exponentially small when the orbit approaches a singularity.
\subsection{Fake foliations}
The following lemma is borrowed from \cite[Lemma 3.3]{LVY} (see also \cite[Proposition 3.1]{BW}), which shows that given a dominated splitting, one can always construct local fake foliations. Moreover, these fake foliations have local product structure, and this structure is preserved as long as they stay in a neighborhood.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.fakefoliation}
Let $K$ be a compact invariant set of a diffeomorphism $f$. Suppose $K$ admits a dominated splitting $T_K M = E^1 \oplus E^2 \oplus E^3$. Then there are $\rho > r_0 > 0$, such that the neighborhood $B_\rho(x)$ of every $x \in K $ admits foliations ${\mathcal F}^1_x, {\mathcal F}^2_x, {\mathcal F}^3_x, {\mathcal F}^{12}_x$ and ${\mathcal F}^{23}_x$, such that for every $y \in B_{r_0}(x)$ and $* \in \{1, 2, 3, 12, 23\}:$
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{enumerate}[label=(\roman*)]
\item ${\mathcal F}^*_x(y)$ is $C^1$ and tangent to the respective cone.
\item Forward and backward invariance: $f({\mathcal F}^*_x(y, r_0)) \subset {\mathcal F}^*_{f(x)}(f(y))$, and\\ $f^{-1}({\mathcal F}^*_x(y, r_0)) \subset {\mathcal F}^*_{f^{-1}(x)}(f^{-1}(y))$.
\item ${\mathcal F}^1_x$ and ${\mathcal F}^2_x$ sub-foliate ${\mathcal F}^{12}_x$; ${\mathcal F}^2_x$ and ${\mathcal F}^3_x$ sub-foliate ${\mathcal F}^{23}_x$.
\end{enumerate}
\end{lemma}
Note that such foliations are constructed locally. In particular, following their construction, one can show that if there is a dominated splitting $E_1\oplus E_2$ on the normal bundle ${\mathcal N}_\Lambda} \def\e{\varepsilonmbda$ for the scaled linear Poincar\'e flow $\psi^*_t$, which can be extended to a neighborhood by Proposition~\ref{p.tubular2}, then near every $x\in\Lambda} \def\e{\varepsilonmbda$ one has fake foliations ${\mathcal F}^i$ on $N_x$, and tangent to $E_i$, $i=1,2$. Furthermore, the size of such foliations are at least $r_0$ after scaling with the flow speed at $x$. In other words, for every regular point $x\in\Lambda} \def\e{\varepsilonmbda$, there is fake foliation with size $r_0|X(x)|$.
From now on, we will assume that $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\frac12\min\{r_0,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0\}$. This makes the size of the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood less than the size of the fake foliation at every point.
\subsection{Control the tail entropy of ${\mathscr A}^\infty$}
Recall that the constant $N_0$ is defined by~\eqref{e.L}, and $K_0, K_1$ are the constants in Theorem~\ref{m.C}.
Below, we will prove that if $L\ge N_0$, then the partition ${\mathscr A}$ given by Theorem~\ref{m.3} for the constants $L$ and $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\frac12\min\{r_0,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0\}$ satisfies Theorem~\ref{m.tangency}.
The main result of this section is the following:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.Atailentropy}
Let $X\in {\mathscr X}^1(M)\setminus \operatorname{C}l({\mathcal T})$ be a $C^1$ vector field such that all the singularities of $X$ are hyperbolic, and ${\mathscr A}$ be the partition given by Theorem~\ref{m.3} for $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\frac12\min\{r_0,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0\}$ and $L\ge N_0$. Then $$h_{tail}(\phi_1,x,{\mathscr A})=0$$ for every invariant probability measure $\mu$ and $\mu$-a.e.\,$x$.
In particular, we have
$$
h_\mu(\phi_1) = h_\mu(\phi_1,{\mathscr A}).
$$
\end{proposition}
It is proven in~\cite[Corollary 2.11]{GY} that for vector fields away from homoclinic tangencies, there is a dominated splitting for both $\psi_t$ and $\psi^*_t$ on the normal bundle ${\mathcal N}_\Lambda} \def\e{\varepsilonmbda = \cup_{x\in\Lambda} \def\e{\varepsilonmbda\setminus {\rm Sing}(X)} {\mathcal N}_x$ over any invariant set $\Lambda} \def\e{\varepsilonmbda$. Furthermore, following the proof of~\cite[Proposition 3.4]{LVY}, which uses the result of~\cite{W04} for diffeomorphisms away from homoclinic tangencies, we have the following lemma.
Here the notation $\phi_{Y,t}$ and $\psi^*_{Y,t}$ represents the flow and the scaled linear Poincar\'e flow defined using the vector field $Y$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.7.3}
Let $X$ be a $C^1$ vector field away from tangencies. Then there exist $\lambda_0 > 0$, $J_0 \ge 1$, and a $C^1$ neighborhood ${\mathcal U}_0$ of $X$ , such that, given any vector field $Y \in {\mathcal U}_0$, the support of any ergodic $Y$-invariant measure $\mu$ admits an $L_0$-dominated splitting for both $\psi_t$ and $\psi_t^*$ over the normal bundle:
$${\mathcal N}_{\operatorname{supp} \mu} = E^1 \oplus E^2 \oplus E^3\mbox{, with }\delta} \def\De{\Deltaim(E^2)\le 1,$$
and, for $\mu$-almost every point $x$, we have
$$
\lim_{n\to\infty}\frac1n\sum^n_{i=1}\log \|\psi^*_{Y,J_0} | E^1_{\phi_{Y,?iL_0}(x)}\| \le ?\lambda_0,\mbox{ and }
$$
$$
\lim_{n\to\infty}\frac1n\sum^n_{i=1}\log \|\psi^*_{Y,-J_0} | E^3_{\phi_{Y,iL_0}(x)}\| \le ?\lambda_0.
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{remark}\label{r.7.3}
The proof of the previous lemma exploits the fact that if $f$ is away from tangencies, then every periodic point of nearby diffeomorphism $g$ can have at most one eigenvalue with modulus one, which has to be real and has multiplicity one (if such eigenvalue exists). As a result, the constant $\lambda_0>0$, which is given by~\cite[Lemma 3.6]{W02}, can be made arbitrarily close to $0$.
\end{remark}
From now on, to simplify notation, we will fix any $Y\in{\mathcal U}_0$ where ${\mathcal U}_0$ is given by Lemma~\ref{l.7.3}, and write $g=\phi_{Y,1}$ for the time-one map of $Y$.
Following the proof of Theorem~3.1 in~\cite{LVY}, we see that for $\mu$ almost every $x$, the projection of ${\mathscr A}^\infty(x)$ and its image along the flow to the normal manifold $N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)= \exp_x({\mathcal N}_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta))$ must be contained in the fake foliation tangent to $E^2$. To be more precise, for $\mu$ almost every point $x$, the map:
$$
P_x(y):{\mathscr A}(x)\to N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)
$$
which projects ${\mathscr A}(x)$ to the normal manifold at $x$ along the flow must satisfy
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.22}
P_{g^j(x)}\circ g^j({\mathscr A}^\infty(x))\subset {\mathcal F}^2_{g^j(x)}(g^j(x), r_1), \forall j\in{\mathbb Z},
\end{equation}
where ${\mathcal F}^2_\cdot(\cdot,r_1)$ is the fake foliation in $N_x(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$ associated to the dominated splitting $E^1\oplus E^2\oplus E^3$ on ${\mathcal N}_{\operatorname{supp}\mu}$, given by Lemma~\ref{l.fakefoliation}.
Observe that in the case $\delta} \def\De{\Deltaim E^2 = 0$, there is nothing to prove since ${\mathcal F}^2$ reduces to a point. In the case $\delta} \def\De{\Deltaim E^2 = 1$, the relation in~\eqref{e.22} significantly improves~\eqref{e.reg.partition}: the projection of ${\mathscr A}^\infty$ to the normal manifolds of $g^j(x)$ is, in fact, contained in a family of one-dimensional curves with bounded length.
This in particular shows that ${\mathscr A}^\infty(x)$ is contained in a two-dimensional strip, which is the image of the one-dimensional curves on $N_{g^j(x)}$ under the flow. This invites us to give the following general mechanism for a set to have zero topological entropy:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{definition}
We say that a set $A$ is {\em $(Y,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$-shadowed} by a family of one-dimensional compact curves $\{I_j\}_{j\in{\mathbb Z}}$, if for every $j\in {\mathbb Z}$, $g^j(A)\subset \phi_{[-1,1]}(I_j)$.
\end{definition}
Note that we do not require $I_j$ to be contained in a tubular neighborhood of length $1$ at some point. Such requirement is only possible near regular points in $C_{reg}$, as we lose control in the region $B^\pm(\sigma)$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.shadow.entropy}
Let $A$ be a set that is $(Y,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$-shadowed by $\{I_j\}_{j\in{\mathbb Z}}$, for $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\frac12\min\{r_0,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0\}$. If
$$\lim_{n\to\infty}\frac1n\log \left(\sum_{j=-n}^n \operatorname{length}(I_j)\right) = 0,$$ then we have
$
h_{top}(A,\phi_1) = 0.
$
\end{proposition}
The proof of this proposition is left to the appendix. We continue the proof of Theorem~\ref{m.tangency}.
Below we will construct the family of one-dimensional curves $\{I_j\}$ that $(Y,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$-shadows ${\mathscr A}^\infty(x)$, and control the length of $I_j$.
For each singularity $\sigma$, $n>n_0$ and $x\in B\in{\mathscr B}_n$, we write $x^D$ for the unique point on $D_n$ such that $x = \phi_{a}(x^D)$ for some $a\in[0,1)$. We also define $\pitchforkat I(x)$ for the connected component of ${\mathcal F}^2_x(x)\cap B$ that contains $x$, and $I(x)$ for the image of $\pitchforkat I(x)$ under the flow to $D_n$ (in fact, pre-image since $\pitchforkat I(x) \in \phi_{[0,1)}(D_n)$). It then follows that
$$
x^D\in I(x)\subset {\mathscr B}(x^D).
$$
The following lemma gives a natural selection of $I_j$ near each singularity
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{lemma}\label{l.epsilon}
There exists $C>0$, $\tilde\lambda>1$, such that For every $\sigma\in{\rm Sing}(Y)$, every $n>n_0$ and $x\in{\mathscr B}_n$, we have
$$
\operatorname{length}(g^j(I(x)))\le C\tilde\lambda^{-(t^+_x-j)}, \mbox{ for every } j\in [0, t^+_x],
$$
and
$$
\operatorname{length}(g^j(I(x)))\le C\tilde\lambda^{-(t^-_x+j)}, \mbox{ for every } j\in [-t^-_x,0].
$$
\end{lemma}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
We only need to consider the case $j\ge0$. The case $j\le0$ follows by considering the vector field $-Y$.
First, recall that Lemma~\ref{l.diam} gives the estimate on the length of $I(x)$ as:
$$
\operatorname{length}(I(x)) \le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta \frac{L_0}{e} (L^{K_1}e)^{-n}.
$$
Also recall that $\lambda'>1$ in Lemma~\ref{l.tx} is such that $\|Dg\mid_{B_r(x)}\|\le \lambda'$, and $K_0$ is chosen to be $\frac{1}{2\log \lambda'}$. Then we see that:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
\operatorname{length}(g^j(I(x))) \le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta \frac{L_0}{e} (L^{K_1}e)^{-n} \cdot \lambda'^j.
\end{align*}
We set
$$
\tilde\lambda : = (L^{K_1}e)^\frac{1}{K_0} >1.
$$
Our choice of $L\ge N_0$ guarantees that $\lambda'\le \tilde\lambda$.
Also note that $t^+_x \in [K_0n, K_1n]$ by Lemma~\ref{l.tx}.
As a result, we obtain
$$
\operatorname{length}(g^j(I(x))) \le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta \frac{L_0}{e} \tilde\lambda^{-K_0n}\tilde\lambda^{j}\le \bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta \frac{L_0}{e} \tilde\lambda^{-t^+_x}\tilde\lambda^{j},
$$
as required.
\end{proof}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}[Proof of Proposition~\ref{p.Atailentropy}]
For each ${\mathscr A}^\infty(x)$, we will only construct the family of one-dimensional curves $\{I_j\}$ for $j\ge 0$. The case $j\le 0$ can be done using the same argument on the flow $-Y$.
For each $j\ge 0$, we consider two cases:
\noindent Case 1. $g^j(x)\in C_{reg}$. In this case, we take $I_j$ to be the connected component of $${\mathcal F}^2_{g^j(x)}(g^j(x), r_1)\cap N_{g^j(x)}(\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta|Y(g^j(x))|)$$ that contains $g^j(x)$. Then $I_j$ is in the $\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta$-scaled tubular neighborhood of $g^j(x)$, and $g^j({\mathscr A}^\infty(x))\subset \phi_{[-1,1]}(I_j)$ by~\eqref{e.22}. Note that in this case, we have $\operatorname{length}(I_j)\le r_1$.
\noindent Case 2. $g^j(x)\in O(\sigma)$ for some $\sigma\in {\rm Sing}(Y)$. Due to the construction inside $O(\sigma)$, there is $n>n_0$ and $j'\in{\mathbb N}$ such that
$$
g^{j'}(x)\in C_n, \,\, |j-j'|\le K_1n.
$$
Then we take $I_{j'} = I(g^{j'}(x))$, and $I_j = g^{j-j'}(I_{j'})$. In other words, we mark the nearest $j'$ such that the point $g^{j'}(x)$ is in the base $\cup C_n$, and define $I_{j'}$ to be the projection of ${\mathcal F}^2(g^{j'}(x))$ to $D_n$, and iteration $I_{j'}$ to obtain $I_j$. This construction is consistent as long as the orbit of $x$ remains in $O(\sigma)$.
Then it is straight forward to verify that ${\mathscr A}^\infty(x)$ is $(Y,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta)$-shadowed by $\{I_j\}$. To control the total length, for each $n>0$ we parse the orbit segment from $0$ to $n$ into:
$$
0 \le n_1< n_1' < n_2 < n_2' <\ldots < n_m\le n,
$$
where $n_i$ is the $i$th times where the orbit of $x$ enters $O(\sigma)$ for some $\sigma\in{\rm Sing}(Y)$, and $n_i'$ is the $i$th time that the orbit leaves $O(\sigma)$. For convenience we set $n_0' = 0$ and $n_m' = n$.\footnote{That is, if $g^n(x)\in O(\sigma)$. If instead we have $g^n(x)\in C_{reg}$, then we have $n_m' < n$ and let $n_{m+1} = n$.}
Now we write
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
\sum_{j=0}^n \operatorname{length}(I_j)\le& \sum_{j= 1}^m\sum_{k=n_{j-1}'}^{n_j-1}\operatorname{length}(I_k) + \sum_{j= 1}^m\sum_{k=n_{j}}^{n_j'}\operatorname{length}(I_k).
\end{align*}
Observe that first summation is taken along the orbit segment that is in $C_{reg}$; as a result, each term is bounded by $r_1$. As for the second sum, by Lemma~\ref{l.epsilon} there exists $\tilde{C}>0$, such that for each $j$, we have
$$
\sum_{k=n_{j}}^{n_j'}\operatorname{length}(I_k)\le \tilde{C}.
$$
Therefore, we obtain
$$
\sum_{j=0}^n \operatorname{length}(I_j)\le\sum_{j= 1}^m\sum_{k=n_{j-1}'}^{n_j-1}r_1 + \sum_{j= 1}^m \tilde{C}\le n(r_1+\tilde{C}).
$$
In particular, we have
$$
\frac1n\log \left(\sum_{j=0}^n \operatorname{length}(I_j)\right) \xrightarrow{n\to\infty} 0.
$$
By Proposition~\ref{p.shadow.entropy}, this shows that $h_{top}({\mathscr A}^\infty(x), g) = 0.$
\end{proof}
Now Theorem~\ref{m.tangency} follows from Proposition~\ref{p.Atailentropy} and Theorem~\ref{m.tailestimate}.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}[Proof of Corollary~\ref{mc.star1}]
Let $X$ be a star vector field. We want to show that for every measure $\mu$ and $\mu$ almost every $x$, the set ${\mathscr A}^\infty(x)$ reduces to a flow segment.
Since $X$ is star, every critical element of $X$ is hyperbolic. Therefore it suffices to consider those $\mu$ that are non-trivial, that is, $\mu$ is not supported on a singularity or a periodic orbit.
By~\cite[Theorem 5.6]{GSW}, every ergodic invariant measure $\mu$ is hyperbolic. In fact, following the proof of~\cite[Theorem 5.6]{GSW}, we see that there exists $\eta>0$, such that every non-trivial measure $\mu$ does not have any Lyapunov exponent in $(-\eta,\eta)$. By Remark~\ref{r.7.3}, we may take $\lambda_0<\eta$ in Lemma~\ref{l.7.3}, making the bundle $E^2$ trivial.
This in particular means that ${\mathscr A}^\infty(x)$ reduces to a flow segment containing $x$.
\end{proof}
\section{Application 2: upper semi-continuity}\label{s.8}
In this section we will prove Theorem~\ref{m.continuous}. The main result here is Theorem~\ref{t.finitepartitionentropy}, which estimates the drop in the metric entropy when approximating ${\mathscr A}$ with a finite partition.
Let $X_n$ be a sequence of $C^1$ vector fields, approaching $X$ in $C^1$ topology. Let $\mu_n$ be a sequence of probability measures, invariant under $T_n$. We assume that $\mu_n\xrightarrow{weak^*}\mu$ where $\mu$ is an invariant probability measure of $X$
To simplify notation, we will write $X=X_0$ and $\mu = \mu_0$. We will make the standard assumption that the sequence $\{X_n\}$ is contained in the $C^1$ neighborhood of $X$ described in Theorem~\ref{m.3}.
Let ${\mathscr A}_n$ be the partition defined in Section~\ref{s.5} for $L=N_0,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta<\frac12\min\{r_0,\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}ta_0\}$ using the flow $X_n$, for $n=0,1,\ldots$. Then Theorem~\ref{m.3} shows that ${\mathscr A}_n\to{\mathscr A}$. We denote by ${\mathscr A}_{n,\sigma}, {\mathscr A}_{n,reg}, {\mathscr C}_{n,\sigma}, {\mathscr B}_{n,m}$ for the partitions defined in Theorem~\ref{m.C} and~\ref{m.A} for the flow $X_n$.
Note that the index $\sigma$ refers to the continuation of $\sigma$ for the flow $X_n$, and in general is different from $\sigma\in {\rm Sing}(X)$ itself.
By Proposition~\ref{p.Atailentropy}, we have
$$
h_{\mu_n}(X_n) = h_{\mu_n}(\phi_{X_n,1},{\mathscr A}_n) , n=0,1,\ldots,
$$
where $\phi_{X,1}$ is the time-one map of the flow $X$.
The key idea in the proof of Theorem~\ref{m.continuous} is that, we need to obtain a {\em finite} partition by glueing certain elements of ${\mathscr A}_n$ together. To this end, we fix some $N>n_0$ and define:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
{\mathscr A}_{n, \sigma,N} = &\{B: B\in {\mathscr B}_{n,m} \mbox{ for some }n_0<m\le N\}\\&\cup\{B^-(\sigma),B^+(\sigma),O(\sigma)^c\}\cup\left\{\cup_{m>N} C_{n,m}\right\}.
\end{align*}
In other words, ${\mathscr A}_{n, \sigma,N}$ is a finite partition obtained by taking the partition ${\mathscr A}_{n, \sigma}$ defined by~\eqref{e.partition} for the flow $X_n$, and glueing all the partition elements of $ {\mathscr B}_{n,k}$, $k>N$, into one set (the last term). See Figure~\ref{f.finitepartition}.
For each $\sigma$, we have thus obtained a sequence of finite partitions $\{{\mathscr A}_{n,\sigma,N}\}_{n\ge 0}$. Next, we write, for $n=0,1,\ldots,$
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}
{\mathscr A}_{n,N} = {\mathscr A}_{n,reg}\vee\bigvee_{\sigma\in{\rm Sing}(X_n)}{\mathscr A}_{n,\sigma,N}.
\end{equation}
Then for each $n$, ${\mathscr A}_{n,N}$ is a {\em finite} partition obtained by glueing all the partition elements of ${\mathscr A}_n$ near each singularity into one element. It is clear that ${\mathscr A}_n$ refines ${\mathscr A}_{n,N}$ for every $N>n_0, n=0,1,\ldots$.
Next, we define, for each $\sigma$ (and its continuation):
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}
O^N(\sigma) = \bigcup_{n>N}\bigcup_{x\in D_n} \phi_{[-t^-_x,t^+_x]}(x).
\end{equation}
Clearly we have $O^N(\sigma)\subset O(\sigma)$ for each $N>n_0$, and $\cap_{k>N} O^k(\sigma) = \emptyset$. Also note that
$$
\bigcap_{k>N} \operatorname{C}l(O^k(\sigma)) = \sigma\cup W_{\operatorname{loc}}^s(\sigma )\cup W_{\operatorname{loc}}^u(\sigma),
$$
where $W^{s/u}_{\operatorname{loc}}(\sigma)$ is the stable and the unstable manifold of $\sigma$ contained in $B_r(\sigma)$.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{figure}
\centering
\delta} \def\De{\Deltaef\columnwidth{\columnwidth}
\includegraphics[scale=1.2]{Pic3.pdf}
\caption{The finite partition ${\mathscr A}_{\sigma,N}$ for the original flow $X$. The yellow region is $O^N(\sigma)$.}
\label{f.finitepartition}
\end{figure}
The next theorem controls the loss of the metric entropy during this glueing process. In particular, it shows that the loss of the metric entropy is proportional to the measure of $O^N(\sigma)$:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{theorem}\label{t.finitepartitionentropy}
Let $X$ be a $C^1$ vector fields, with all the singularities hyperbolic. Let ${\mathscr A}$ be the partition given by Theorem~\ref{m.3} for the constants $L=N_0$. Then there exists a constant $L_2>0$, such that for any invariant probability measure $\mu$ of $X$ and every $N>n_0$, we have
$$
h_{\mu}(\phi_{1},{\mathscr A})-L_2\sum_{\sigma\in{\rm Sing}(X)}\mu(O^N(\sigma)) - u_{X,\mu}(N)\le h_{\mu}(\phi_{1},{\mathscr A}_{0,N})\le h_{\mu}(\phi_{1},{\mathscr A}),
$$
for every $n$. Here $u_{X,\mu}(N)$ is a function of $N$ that converges to zero as $N\to\infty$, uniformly in $\mu$ and in a neighborhood of $X$.
Furthermore, $L_2$ can be made uniform for nearby $C^1$ vector fields.
\end{theorem}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
The second inequality follows from the fact that ${\mathscr A}_{0,N}$ is coarser than ${\mathscr A}$. To obtain the first inequality, we write:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
&h_{\mu}(\phi_{1},{\mathscr A})-h_{\mu}(\phi_{1},{\mathscr A}_{0,N})\\
=&\lim_{k\to\infty} H_{\mu}({\mathscr A}\,\big|\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A})-H_{\mu}({\mathscr A}_{0,N}\big|\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A}_{0,N})\\
\le&\lim_k\Bigg(H_{\mu}({\mathscr A}\,\big|{\mathscr A}_{0,N}) +H_\mu({\mathscr A}_{0,N}\big|\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A}_{0,N}) + H(\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A}_{0,N}\Big|\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A})\\& -H_{\mu}({\mathscr A}_{0,N}\big|\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A}_{0,N})\Bigg).
\end{align*}
Note that the second term is cancelled with the forth, and the third term is zero since $\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A}$ is a refinement of $\bigvee_{j=1}^{k}\phi_{1}^{-j}{\mathscr A}_{0,N}$. The only remaining term,which is the first term, does not depend on $k$. We thus conclude that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.26}
h_{\mu}(\phi_{1},{\mathscr A})-h_{\mu}(\phi_{1},{\mathscr A}_{0,N})\le H_{\mu}({\mathscr A}\,\big|{\mathscr A}_{0,N}).
\end{equation}
It remains to show that
$$
H_{\mu}({\mathscr A}\,\big|{\mathscr A}_{0,N})\le L_2\sum_{\sigma\in{\rm Sing}(X)}\mu(O^N(\sigma)) + u_{X,\mu}(N)
$$
for some $L_2>0$ and some function $u_{X,\mu}(N)$, which holds if we can prove that
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{equation}\label{e.27}
H_{\mu}({\mathscr A}_\sigma\,\big|{\mathscr A}_{0,\sigma,N})\le L_2\mu(O^N(\sigma))+u_{X,\mu,\sigma}(N),
\end{equation}
for some function $u_{X,\mu,\sigma}(N)$ that converges to zero as $N\to \infty$, uniformly in $\mu, X$ and $\sigma$. Since ${\rm Sing}(X)$ only contains finitely many singularities, we then set
$$
u_{X,\mu}(N) =\sum_{\sigma\in{\rm Sing}(X)}u_{X,\mu,\sigma}(N)
$$
which also goes to zero uniformly.
It remains to prove~\eqref{e.27}. The proof is, in fact, hidden in the proof of Proposition~\ref{p.Aentropy}. We define
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
\tilde {\mathscr A}_{0,\sigma,N} = &\{B: B\in {\mathscr B}_{m} \mbox{ for some }n_0<m\le N\}\\&\cup\{B^-(\sigma),B^+(\sigma),O(\sigma)^c\}\cup\left\{C_{m}: m>N\right\}.
\end{align*}
Then $\tilde {\mathscr A}_{0,\sigma,N}$ is a countable partition, obtained by glueing {\em each} ${\mathscr B}_m$
with $m>N$ into $C_m$. Then ${\mathscr A}_{0,\sigma,N}$ can be seen as glueing {\em all the} $C_m, m>N$ into one element $C^{N} = \cup_{k>N}C_{k}$. We immediately see that ${\mathscr A}_\sigma$ refines $\tilde {\mathscr A}_{0,\sigma,N}$, while the latter refines ${\mathscr A}_{0,\sigma,N}$.
Now we write
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
H_{\mu}({\mathscr A}_\sigma\,\big|{\mathscr A}_{0,\sigma,N})\le& H_{\mu}({\mathscr A}_\sigma\,\big|\tilde{\mathscr A}_{0,\sigma,N}) + H_{\mu}(\tilde{\mathscr A}_{0,\sigma,N} | {\mathscr A}_{0,\sigma,N})\\
=& I+II.
\end{align*}
First, note that all three partitions coincide outside $C^N$. Therefore, we can estimate $I$ as:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
I&= H_{\mu}({\mathscr A}_\sigma\,\big|\tilde{\mathscr A}_{0,\sigma,N}) \\
&\le -\sum_{m>N}\mu(C_m) \sum_{B\in{\mathscr B}_m} \mu_{C_m} (B)\log \mu_{C_m}(B)\\
&\le \sum_{m>N}\mu(C_m)\log \left(\# {\mathscr B}_m\right)\\
&\le \sum_{m>N}(m\log L''+\log c_1)\mu(m)\\
&\le \mu(C^N)\log c_1 + \log L''\sum_{m>N} m\mu(C_m)\\
&\le \frac{\log c_1}{K_0N} +\log L''\sum_{m>N} \frac{1}{K_0}\mu\left(\bigcup_{x\in D_n}\phi_{[0,t^+_x]}(x)\right)\\
&\le \frac{\log c_1}{K_0N} + \frac{\log L''}{K_0}\mu(O^N(\sigma)).
\end{align*}
Here we used Lemma~\ref{l.Bcard} for $\# {\mathscr B}_m$, Lemma~\ref{l.measure.triangle} for the measure of $C^N$, and~\eqref{e.sumC} to control $\sum_{m>N} m\mu(C_m)$.
On the other hand, $II$ can be controlled as:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
II=& H_{\mu}(\tilde{\mathscr A}_{0,\sigma,N} | {\mathscr A}_{0,\sigma,N}) \\
\le& - \sum_{m>N} \mu (C_m)\left(\log \mu(C_m) - \log\mu(C^N)\right)\\
\le& \, \mu(C^N)\log \mu(C^N) + \sum_{m>N} \mu (C_m)|\log \mu(C_m)|.
\end{align*}
Thanks to the uniform estimate on the measure of $\mu(C^N)$ by Lemma~\ref{l.measure.triangle}, we see that the first term goes to zero uniformly in $\mu$ and $X$.
For the second term, we use Ma\~n\'e's proof of~\ref{l.finiteentropy} in~\cite{Ma81}. We write $a_n = \mu(C_n)$, and define the set
$$
{\mathcal G} = \{n: a_n>e^{-n}\} = \{n: |\log a_n|<n\}.
$$
Then we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
\sum_{m>N} \mu (C_m)|\log \mu(C_m)| =&\sum_{m>N, m\in{\mathcal G}} a_m|\log a_m| + \sum_{m>N, m\in{\mathcal G}^c} a_m|\log a_m|\\
\le& \sum_{m>N}m a_m + \sum_{m>N, m\in{\mathcal G}^c}\sqrt{e^{-m}} \cdot \sqrt{a_m}|\log a_m|.
\end{align*}
It is easy to see that the second term in the last line is of order ${\mathcal O}(e^{-N/2})$ with the hidden constant uniformly bounded in $\mu$ and $X$. Therefore we have
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
\sum_{m>N} \mu (C_m)\log \mu(C_m)
\le& \sum_{m>N}m \mu(C_m)+ {\mathcal O}(e^{-N/2})\\
\le &\sum_{m>N} \frac{1}{K_0}\mu\left(\bigcup_{x\in D_n}\phi_{[0,t^+_x]}(x)\right) + {\mathcal O}(e^{-N/2})\\
\le& \frac{1}{K_0}\mu(O^N(\sigma))+{\mathcal O}(e^{-N/2}),
\end{align*}
where we used~\eqref{e.sumC} again to control the sum over $m\mu(C_m)$.
Now we collect $I$, $II$ and obtain
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{align*}
H_{\mu}({\mathscr A}_\sigma\,\big|{\mathscr A}_{0,\sigma,N})\le& \frac{\log c_1}{K_0N} + \frac{\log L''}{K_0}\mu(O^N(\sigma))+ \mu(C^N)\log \mu(C^N)\\&+\frac{1}{K_0}\mu(O^N(\sigma))+{\mathcal O}(e^{-N/2}).
\end{align*}
In particular, \eqref{e.27} follows with $L_2=\frac{\log L''+1}{K_0}$, which is uniform in a $C^1$ neighborhood of $X$.
With that we conclude the proof of Theorem~\ref{t.finitepartitionentropy}.
\end{proof}
As an immediate corollary, we have:
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{corollary}\label{c.1}
Under the assumptions of Theorem~\ref{t.finitepartitionentropy}, if $\mu({\rm Sing}(X)) = 0$, then for every $\varepsilon>0$ we can take $N>n_0$ such that
$$
h_{\mu}(\phi_{1},{\mathscr A})-\varepsilon\le h_{\mu}(\phi_{1},{\mathscr A}_{0,N})\le h_{\mu}(\phi_{1},{\mathscr A}),
$$
\end{corollary}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
Observe that $\sum_{\sigma\in{\rm Sing}(X)}\mu(O^N(\sigma)) \le \sum_{\sigma\in{\rm Sing}(X)}\mu(\operatorname{C}l(O^N(\sigma)))$, and
$$\bigcap_{k>N} \operatorname{C}l(O^k(\sigma))=\sigma\cup W_{\operatorname{loc}}^s(\sigma )\cup W_{\operatorname{loc}}^u(\sigma),$$ which has measure zero. So we can take $N>n_0$ large enough, such that $u_{X,\mu}(N)<\varepsilon/2$ and $\sum_{\sigma\in{\rm Sing}(X)}\mu(O^N(\sigma)) <\varepsilon/2$.
\end{proof}
For each given $N$, we have obtained a sequence of finite partitions $\{{\mathscr A}_{n,N}\}_{n=0}^\infty$. The next proposition is well known in the classical entropy theory. See for example~\cite{B08}.
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proposition}\label{p.finitecontinuous}
Under the assumptions of Theorem~\ref{m.continuous}, for each $N\in{\mathbb N}$ large enough, we have
$$
\lim_{n\to\infty}h_{\mu_n}(\phi_{X_n,1},{\mathscr A}_{n,N})\le h_{\mu_0}(\phi_{X_0,1},{\mathscr A}_{0,N}).
$$
\end{proposition}
Now Theorem~\ref{m.continuous} is a direct consequence of Theorem~\ref{t.finitepartitionentropy} and Proposition~\ref{p.finitecontinuous}, and the observation that for every $\varepsilon>0$, one can take $N$ large enough such that $$\sum_{\sigma\in{\rm Sing}(X)}\mu(O^N(\sigma)) < \sum_{\sigma\in{\rm Sing}(X)}\mu({\rm Sing}(X))+\varepsilon.$$ The case $\mu({\rm Sing}(X))=0$ follows from Corollary~\ref{c.1}. \quad} \def\qq{\qquaded
\appendix
\section{ Proof of Proposition~\ref{p.shadow.entropy}}
\bb{equation}} \delta} \def\De{\Deltaef\ee{\end{equation}gin{proof}
First, note that if we had $g^j(A)\subset I$ instead of $g^j(A)\subset \phi_{[-1,1]}(I_j)$, then this proposition is immediate (in fact, this argument is already used in~\cite{LVY}). This is because for each $\varepsilon>0$, the number of $\varepsilon$-balls needed to cover $I_j$ is of the order ${\mathcal O}(\frac{1}{\varepsilon}\operatorname{length}(I_j))$. Also note that the set $A$ induces a natural order on each $I_j$. As a result, the sub-exponential growth of $\sum_{j=-n}^n \operatorname{length}(I_j)$ implies the sub-exponential growth of the cardinality of a $(\varepsilon,n)$-spanning set.
In the case $g^j(A)\subset \phi_{[-1,1]}(I_j)$, we define
$$
\tilde I_j = P_{x_{j}}(I_j).
$$
Then we have $\operatorname{length}(\tilde I_j) \le I_j$. The set $\phi_{[-2,2]}(\tilde I_j)$ contains $h^j(A)$. Furthermore, there exists a constant $C$ determined by the vector field $X$, such that $\phi_{[-2,2]}(\tilde I_j)$ can be covered by no more than $\frac{C}{\varepsilon^2}\operatorname{length}(\tilde I_j)$ many $\varepsilon$-balls. In the meantime, $A$ induces a natural order on each $\tilde{I_j}$. As a result, the minimal cardinality of a $(n,\varepsilon)$-spanning set is bounded from above by
$$
\frac{C}{\varepsilon^2}\sum_{j=-n}^n\operatorname{length}(\tilde I_j)\le \frac{C}{\varepsilon^2}\sum_{j=-n}^n\operatorname{length}( I_j).
$$
This shows that
$$
h_{top}(A,g) \le \lim_{\varepsilon\to 0}\lim_n\frac{1}{n}\log \left(\frac{C}{\varepsilon^2}\sum_{j=-n}^n\operatorname{length}( I_j)\right)=0.
$$
\end{proof}
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|
\begin{document}
\begin{abstract}
In this paper, we consider stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient.
We establish a necessary and sufficient stochastic maximum principle. To achieve this, we first derive an explicit representation of the first variation process (in Sobolev sense ) of the controlled diffusion.
Since the drift coefficient is not smooth, the representation is given in terms of the local time of the state process. Then we construct a sequence of optimal control problems with smooth coefficients by an approximation argument. Finally, we use Ekeland's variational principle to obtain an approximating adjoint process from which we derive the maximum principle by passing to the limit.
\varepsilonnd{abstract}
\maketitle
\section{Introduction}
Let $T \in (0,\infty)$ be a given deterministic time horizon and $d \in \mathbb{N}$, let $\Omega := C([0,T],\mathbb{R}^d)$ be the canonical space of continuous paths.
We denote by $B$ the canonical process and by $\mathbb{P}$ the Wiener measure.
Equip $\Omega$ with $(\mathcal{F}_t)_{t\in [0,T]}$, the $\mathbb{P}$-completion of the canonical filtration of $B$.
Given a $d$-dimensional vector $\sigma$ and a function $b: [0,T]\times \mathbb{R}\times \mathbb{R}^m\to \mathbb{R}$, we consider a controlled diffusion of the form
\begin{align}\label{eqSpro1}
\,\mathrm{d} X^\alpha (t) = b(t,X^\alpha(t),\alpha(t))\,\mathrm{d} t +\sigma\,\mathrm{d} B(t) ,\quad t \in [ 0,T] ,\quad X^\alpha(0) = x_0
\varepsilonnd{align}
and the control problem
\begin{align} \label{eqconpb1}
V(x_0) := \sup_{\alpha \in \mathcal{A}}J(\alpha).
\varepsilonnd{align}
Hereby, the performance functional $J$ is given by
\begin{align*}
J(\alpha):= \mathbb{E}\Big[ \int_0^T f(s,X^\alpha(s),\alpha(s))\,\mathrm{d} s + g(X^\alpha(T))\Big],\label{perfunct1}
\varepsilonnd{align*}
where, $f$ and $g$ may be seen as profit and bequest functions, respectively.
The set $\mathcal{A}$ is the set of admissible controls and is defined as the set of progressively measurable processes $\alpha$ valued in a closed convex set $\mathbb{A}\subseteq \mathbb{R}^m$ such that \varepsilonqref{eqSpro1} admits a unique strong solution.
The goal of the present article is to derive the maximum principle for the above control problem when the drift $b$ is merely measurable is the state variable $x$.
The stochastic maximum principle is arguably one of the most prominent ways to tackle stochastic control problems as \varepsilonqref{eqconpb1} by fully probabilistic methods.
It is the direct generalization to the stochastic framework of the maximum principle of Pontryagin \cite{PontryaginBook} in deterministic control.
It gives a necessary condition of optimality in the form of a two-point boundary value problem and a maximum condition on the Hamiltonian.
More precisely let the Hamiltonian $H$ be defined as
$$H(t, x, y, a): = f(t,x,a) + b(t,x,a)y$$
and assume just for a moment the functions $b,f$ and $g$ to be continuously differentiable.
Then, if $\hat\alpha \in \mathcal{A}$ is an optimal control, then according to the stochastic maximum principle, it holds $H(t, X^{\hat\alpha}(t), Y(t), \hat\alpha(t)) \ge H(t, X^{\hat\alpha}(t), Y(t), a) $ $P\otimes dt$-a.s. for every $a \in \mathbb{A}$ where $(Y,Z)$ are adapted processes solving the so-called adjoint equation
\begin{equation*}
dY(t) = - \partial_xf(t, X^{\hat\alpha}(t),\hat\alpha(t)) - \partial_xb(t, X^{\hat\alpha}(t),\hat\alpha(t))Y(t)\,\mathrm{d} t + Z(t)\,\mathrm{d} B(t),\quad Y(T) = \partial_xg(X^{\hat\alpha}(T)).
\varepsilonnd{equation*}
Under additional convexity conditions, this necessary condition is sufficient.
The interest of the maximum principle is that it reduces the solvability of the control problem \varepsilonqref{eqconpb1} to that of a (scalar) variational problem, and therefore allows to derive (sometimes explicit) characterizations of optimal controls.
We refer for instance to \cite{MR3629171,YZ99} for proofs and historical remarks.
The maximum principle has far-reaching consequences and is widely used in the stochastic control and stochastic differential game literature \cite{Car-Del15,MR3325083,Pen90,Pontryagin,optimierung}.
Its use also fueled by recent progress on the theory of forward backward SDEs.
We refer the reader for instance to, \cite{Delarue,FbsdeRough,Ma-Zhang11,Zhang_Book17,dqFBSDE} and the references therein.
The maximum principle roughly presented above naturally requires differentiability of the coefficients of the control problem, which precludes the applicability of this method to control problems with non-smooth coefficients.
The effort to extend the stochastic maximum principle to problems with non-smooth coefficients started with the work of Merzedi \cite{Mer88} who derived a necessary condition of optimality for a problem with a Lipschitz continuous drift, but not necessarily differentiable everywhere in the state and the control variable.
His result was further extended, notably to degenerate diffusion cases and singular control problems in \cite{Bah-Dje-Mer07,Bah-Dje-Mer-AMO07,Bah-Chi-Dje-Mer}. See also \cite{KoMe15} for the infinity horizon case.
The present work considers the case where $b$ is Borel measurable in $x$ and bounded, and we will derive both necessary and the sufficient conditions of optimality.
At this point, an immediate natural question is: What form should the adjoint equation take in this case?
The starting point of our argument is the following simple observation:
When $b$ is differentiable, the adjoint equation is explicitly solvable, with the solution given by
\begin{equation*}
Y(t) = \mathbb{E}\Big[\Phi^{\hat\alpha}(t,T)\partial_xg(X^{\hat\alpha}(T)) + \int_t^T\Phi^{\hat\alpha}(t,s)\partial_xf(s, X^{\hat\alpha}(s),\hat\alpha(s))\,\mathrm{d} s\mid \mathcal{F}_t \Big],
\varepsilonnd{equation*}
where the process
\begin{equation}
\label{eq:flow.smooth}
\Phi^{\hat\alpha}(t,s) = e^{\int_t^s\partial_xb(u, X^{\hat\alpha}(u),\hat\alpha(u))\,\mathrm{d} u}\quad 0\le t\le s\le T
\varepsilonnd{equation}
is the first variation process (in the Sobolev sense) of the dynamical system $X^{\hat\alpha,x}$ solving \varepsilonqref{eqSpro1} with initial condition $X^{\hat\alpha,x}_0 = x$.
This suggests the form of the adjoint process when $b$ is not differentiable, since it is well-known that despite the roughness of the drift $b$, the dynamical system $X^{\hat\alpha,x}$ is still differentiable (at least in the Sobolev sense), due to Brownian regularization \cite{MNP2015} and therefore admits a flow.
The crux of our argument will be to make use of this Sobolev differential stochastic flow to define the \varepsilonmph{adjoint process} (rather than the adjoint equation) in the non-smooth case to prove necessary and sufficient conditions of optimality.
Throughout this work the functions $f$ and $g$ are assumed to be continuously differentiable with bounded first derivatives.
In particular, we will assume
\begin{equation*}
\sigma \in \mathbb{R}^d \text{ satisfies } |\sigma|^2>0 \quad \text{and}\quad |f(t, x, a)| + |g(x)| \le C(1 + |x|)\quad \text{for all $(t,x,a)$ and some $C>0$.}
\varepsilonnd{equation*}
The main results of this work are the following necessary and sufficient conditions in the Pontryagin stochastic maximum principle.
\begin{thm}
\label{thm:necc}
Assume that $b$ satisfies $b(t,x,a):= b_1(t,x) + b_2(t,x,a)$ where $b_1$ is a bounded, Borel measurable function and $b_2$ is bounded measurable, and continuously differentiable in its second and third variables with bounded derivatives.
Let $\hat\alpha \in \mathcal{A}$ be an optimal control and let $X^{\hat\alpha}$ be the associated optimal trajectory.
Then the flow $\Phi^{\hat\alpha}$ of $X^{\hat\alpha}$ is well-defined and it holds
\begin{equation}
\label{eq:nec.cond}
\partial_{\alpha}H(t, X^{\hat\alpha}(t),Y^{\hat\alpha}(t),\hat\alpha(t) )\cdot(\beta - \hat\alpha(t)) \ge 0 \quad \mathbb{P}\otimes \,\mathrm{d} t\text{-a.s. for all } \beta \in \mathcal{A},
\varepsilonnd{equation}
where $Y^{\hat\alpha}$ is the adjoint process given by
\begin{equation}
\label{eq:adj.proc}
Y^{\hat\alpha}(t) := \mathbb{E}\Big[\Phi^{\hat\alpha}(t, T) g_x( X^{\hat\alpha}(T)) + \int_t^T\Phi^{\hat\alpha}(t,s) f_x(s, X^{\hat\alpha}(s), \hat\alpha(s))\mathrm{d}s\mid \mathcal{F}_t \Big].
\varepsilonnd{equation}
\varepsilonnd{thm}
\begin{thm}
\label{thm:suff}
Let the conditions of Theorem \ref{thm:necc} be satisfied, further assume that $g$ and $(x,a)\mapsto H(t,x,y,a)$ are concave.
Let $\hat\alpha\in \mathbb{A}$ satisfy
\begin{equation}
\label{eq:suff.con}
\partial_\alpha H(t, X^{\hat\alpha}(t), Y^{\hat\alpha}(t), \hat\alpha_t)=0 \quad \mathbb{P}\otimes \,\mathrm{d} t\text{-a.s.}
\varepsilonnd{equation}
with $Y$ given by \varepsilonqref{eq:adj.proc}.
Then, $\hat\alpha$ is an optimal control.
\varepsilonnd{thm}
We will elaborate on the conditions imposed in the above theorems in section \ref{subsec.conclusion}.
Let us at this point remark that these results correspond exactly to the classical version of the stochastic maximum principle when $b$ is smooth.
The only difference here being the fact that the process $\Phi^{\hat\alpha}$ seems abstract, as it is obtained from an existence result (of the flow).
As noted by \cite{BMBPD17}, it turns out that when the drift is not smooth, the flow $\Phi^{\hat\alpha}$ still admits an explicit representation much similar to \varepsilonqref{eq:flow.smooth}.
This representation will be extended to the present controlled case (see Theorem \ref{Thmexpliflowder}) and will be used in the proof of the maximum principle.
The remainder of the article is dedicated to the proofs of Theorem \ref{thm:necc} and \ref{thm:suff}.
The necessary condition is proved in the next section and the sufficient condition is proved in section \ref{sec:sufficient}.
The paper ends with an appendix on explicit representations of the flow of SDEs with measurable and random drifts.
\section{The necessary condition for optimality}
\label{sec:neccessary}
The goal of this section is to prove Theorem \ref{thm:necc}.
Let us first precise the definition of the set of admissible controls.
Let $\mathbb{A}\subseteq \mathbb{R}^m$ be a closed convex subset of $\mathbb{R}^m$.
The set of admissible controls is defined as:
\begin{multline*}
\mathcal{A} := \Big\{\alpha:[0,T]\times \Omega\to \mathbb{A}, \text{ progressive, \varepsilonqref{eqSpro1} has a unique strong solution and }\\ \mathbb{E}\big[\sup_{t\in[0,T]}|\alpha(t)|^2\big]< M \Big\}
\varepsilonnd{multline*}
for some $M>0$.
The difficulty in the existence and uniqueness of \varepsilonqref{eqSpro1} is the fact that the drift $b$ is both non-smooth and depends on the random term $\alpha$.
Such equations were treated in \cite{MenTan19}.
In fact, consider the set $\mathcal{A}'$ defined as:
The set of progressively measurable processes $\alpha:[0,T]\times \Omega\to \mathbb{A}$ which are Malliavin differentiable (with Malliavin derivative $D_s\alpha(t)$), with
\begin{equation}\label{eqcondal1}
\mathbb{E}\Big[\int_0^T|\alpha(t)|^2\,\mathrm{d} t \Big] + \sup_{s\in [0,T]}\mathbb{E}\Big[\Big(\int_0^T|D_s\alpha(t)|^2\,\mathrm{d} t\Big)^4 \Big] <\infty
\varepsilonnd{equation}
and such that there are constants $C,\varepsilonta>0$ (possibly depending on $\alpha$) such that
\begin{equation}\label{eqcondal2}
\mathbb{E}[|D_s \alpha(t) - D_{s'}\alpha(t)|^4] \le C|s-s'|^\varepsilonta.
\varepsilonnd{equation}
It follows from \cite[Theorem 1.2]{MenTan19} that if the drift satisfies the conditions of Theorem \ref{thm:necc}, then the SDE \varepsilonqref{eqSpro1} is uniquely solvable for every $\alpha \in \mathcal{A}'$.
Since we do not make use of Malliavin differentiability in the present article we restrict ourselves to the set of admissible controls $\mathcal{A}$.
For later reference, note that for every $\alpha\in \mathcal{A}$ it holds $E[\sup_{t\in [0,T]}|X^\alpha(t)|^p]<\infty$ for every $p\ge1$.
In the rest of the article, we let $b_n$ be a sequence of functions defined by $b_n:= b_{1,n} + b_{2}$ such that $b_{1,n}: [0, T ] \times \mathbb{R} \rightarrow \mathbb{R}, n \geq 1$ are smooth functions with compact support and converging a.e. to $b_1$.
Since $b_1$ is bounded, the sequence $b_{1,n}$ can also be taken bounded.
We denote by $X^{\alpha}_n$ the solution of the SDE \varepsilonqref{eqSpro1} with drift $b$ replaced by $b_n$.
This process is clearly well-defined since $b_n$ is a Lipschitz continuous function.
Similarly, we denote respectively by $J_n$ and $V_n$ the performance and the value function of the problem when the drift $b$ is replaced by $b_n$.
That is, we put
\begin{equation*}
J_n(\alpha) := \mathbb{E}\Big[\int_0^Tf(s, X^\alpha_n(s), \alpha(s))\,\mathrm{d} s + g(X_n^\alpha(T)) \Big],\quad V_n(x_0) := \sup_{\alpha \in \mathcal{A}}J_n(\alpha)
\varepsilonnd{equation*}
and
\begin{equation*}
\,\mathrm{d} X_n^\alpha (t) = b_n(t,X_n^\alpha(t),\alpha(t))\,\mathrm{d} t +\sigma\,\mathrm{d} B(t) ,\quad t \in [ 0,T] ,\quad X^\alpha(0) = x_0.
\varepsilonnd{equation*}
Furthermore, we denote by $\delta$ the distance
\begin{equation*}
\delta(\alpha_1, \alpha_2) : = \mathbb{E}\big[\sup_{t \in[0,T]}|\alpha_1(t) - \alpha_2(t)|^{2} \big]^{1/2} .
\varepsilonnd{equation*}
The general idea of the proof will be to start by showing that an optimal control for the problem \varepsilonqref{eqconpb1} is also optimal for an appropriate perturbation of the approximating problem with value $V_n(x_0)$.
This is due to the celebrated variational principle of Ekeland.
This maximum principle for control problems with smooth drifts will involve the state process $X_n^{\hat\alpha_n}$ and its flow $\Phi^{\hat\alpha_n}_n$.
The last and most demanding step is to pass to the limit and show some form of "stability" of the maximum principle.
We first address this limit step by a few intermediary technical lemmas that will be brought together to prove Theorem \ref{thm:suff} at the end of this section.
\begin{lemm}
\label{lem:conv.Xnn}
We have the following bounds:
\begin{itemize}
\item[(i)] For every $\alpha_1,\alpha_2 \in \mathcal{A}$ it holds that
$$
\mathbb{E}\big[| X^{\alpha_1}_n(t) - X^{\alpha_2}(t)| \big] \le C\Big( \delta(\alpha_1,\alpha_2) + \Big(\int_0^T\varphirac{1}{\sqrt{2\pi s}}e^{\varphirac{|x_0|^2}{2s}}\int_{\mathbb{R}^d}\big|b_{1,n} (s,\sigma y)-b_{1} (s,\sigma y)\big|^4e^{-\varphirac{|y|^2}{4s}}\,\mathrm{d}ns y\,\mathrm{d} s\Big)^{1/2}\Big).
$$
\item[(ii)]Given $k \in \mathbb{N}$, for every sequence $(\alpha_n)$ in $\mathcal{A}$ converging to some $\alpha\in \mathcal{A}$ it holds that
$$ \mathbb{E}\big[| X^{\alpha_n}_k(t) - X^{\alpha}_k(t)|^2 \big] \to 0.$$
\varepsilonnd{itemize}
\varepsilonnd{lemm}
\begin{proof}
Adding and subtracting the same term and then using the fundamental theorem of calculus, we arrive at
\begin{align*}
&X_n^{\alpha_1}(t) - X^{\alpha_2}(t)
= \int_0^t\int_0^1\partial_xb_{1,n}(s, \Lambda_n(\lambda,s)) + \partial_xb_2\big(s,\Lambda_n(\lambda,s),\alpha_1(s)\big)\mathrm{d}\lambda(X^{\alpha_1}_n(s) - X^{\alpha_2}(s))\mathrm{d}s\\
& + \int_0^t b_{1,n}(s, X^{\alpha_2}(s)) - b_1(s, X^{\alpha_2}(s))\mathrm{d}s + \int_0^tb_2(s, X^{\alpha_2}(s),\alpha_1(s)) - b_2(s, X^{\alpha_2}(s), \alpha_2(s))\,\mathrm{d} s,
\varepsilonnd{align*}
where $\Lambda_n(\lambda,t)$ is the process given by $\Lambda_n(\lambda,t):= \lambda X^{\alpha_1}_n(t) + (1 - \lambda)X^{\alpha_2}(t)$.
Therefore, we obtain that $X^{\alpha_1}_n - X^{\alpha_2}$ admits the representation
\begin{align*}
&X^{\alpha_2}(t) - X_n^{\alpha_2}(t) = \int_0^t\varepsilonxp\Big(\int_{s}^t\int_0^1\partial_xb_{1,n}(r, \Lambda_n(\lambda,r)) + \partial_xb_2(r,\Lambda_n(\lambda,r), \alpha_1(r))\mathrm{d}\lambda\mathrm{d}r \Big)\\
&\times \Big(b_{1,n}(s, X^{\alpha_2}(s)) - b_1(s, X^{\alpha_2}(s)) +b_2(s, X^{\alpha_2}(s),\alpha_1(s)) - b_2(s, X^{\alpha_2}(s),{\alpha_2}(s))\Big)\mathrm{d}s.
\varepsilonnd{align*}
Hence, taking expectation on both sides above and then using twice Cauchy-Schwarz inequality, we have that
\begin{align}
\notag
&\mathbb{E}\big[|X^{\alpha_1}_n(t) - X^{\alpha_1}(t)|\big] \le \mathbb{E}\Big[\int_0^t \varepsilonxp\Big(2\int_{s}^t\int_0^1\partial_xb_{1,n}(r, \Lambda_n(\lambda,r)) + \partial_xb_2(r,\Lambda_n(\lambda,r),\alpha_1(r))\mathrm{d}\lambda\mathrm{d}r \Big)\mathrm{d} s\Big]^{1/2}\\
\label{eq:estim.diff.x}
&\times \mathbb{E}\Big[\int_0^{t}|b_1(s, X^{\alpha_2}(s)) - b_{1,n}(s, X^{\alpha_2}(s))|^2 + |b_2(s, X^{\alpha_2}(s),\alpha_1(s)) - b_2(s, X^{\alpha_2}(s),{\alpha_2}(s))|^2\,\mathrm{d} s\Big]^{1/2}.
\varepsilonnd{align}
By Lipschitz continuity of $b_2$, the last term on the right hand side is estimated as
\begin{equation}
\label{eq:estim.alpha12}
\mathbb{E}\Big[\int_0^T|b_2(s, X^{\alpha_2}(s),\alpha_1(s)) - b_2(s, X^{\alpha_2}(s),{\alpha_2}(s))|^2\,\mathrm{d} s \Big] \le C\mathbb{E}\Big[\int_0^T|\alpha_1(s) - \alpha_2(s)|^2\,\mathrm{d} s \Big]\le C(\delta(\alpha_1,\alpha_2))^{2}.
\varepsilonnd{equation}
Moreover, denoting $$\mathcal{E}\Big(\int_0^Tq(s)\,\mathrm{d} B(s) \Big) = \varepsilonxp\Big(\int_0^Tq(s)\,\mathrm{d} B(s) - \varphirac12\int_0^T|q(s)|^2\,\mathrm{d} s \Big),$$ the second integral on the right hand side of \varepsilonqref{eq:estim.diff.x} can be further estimated as follows:
\begin{align*}
&\mathbb{E}\Big[\int_0^{T}|b_1(s, X^{\alpha_2}(s)) - b_{1,n}(s, X^{\alpha_2}(s))|^2\mathrm{d}s\Big]\\
& = \mathbb{E}\Big[\mathcal{E}\Big(\varphirac{\sigma^\top}{|\sigma|^2}\int_0^Tb(s, X^{\alpha_2}(s),{\alpha_2}(s))\mathrm{d}B(s) \Big)^{1/2}\mathcal{E}\Big(\int_0^T\varphirac{\sigma^\top}{|\sigma|^2}b(s, X^{\alpha_2}(s),{\alpha_2}(s))\mathrm{d}B(s) \Big)^{-1/2}\\
&\quad \times\int_0^{T}|b_1(s, X^{\alpha_2}(s)) - b_{1,n}(s, X^{\alpha_2}(s))|^2\mathrm{d}s \Big]\\
& \le C\mathbb{E}_{\mathbb{Q}}\Big[ \int_0^{T}|b_1(s, X^{\alpha_2}(s)) - b_{1,n}(s, X^{\alpha_2}(s))|^4\mathrm{d}t \Big]^{1/2}
\varepsilonnd{align*}
for some constant $C>0$ and the probability measure $\mathbb{Q}$ is the measure with density
\begin{equation}
\label{eq:def.probab.Q}
\varphirac{\,\mathrm{d} \mathbb{Q}}{\,\mathrm{d} \mathbb{P}} := \mathcal{E}\Big(\int_0^T\varphirac{\sigma^\top}{|\sigma|^2}b(s, X^{\alpha_2}(s),{\alpha_2}(s))\mathrm{d}B(s) \Big).
\varepsilonnd{equation}
Note that we used Cauchy-Schwarz inequality and then the fact that $b$ is bounded to get $\mathbb{E}[(\varphirac{\,\mathrm{d} \mathbb{Q}}{\,\mathrm{d}\mathbb{P}})^{-1}]\le C$.
By Girsanov's theorem, under the measure $\mathbb{Q}$, the process $(X^{\alpha_2}(t) - x_0)\sigma^\top/|\sigma|^2 $ is a Brownian motion.
Thus, it follows that
\begin{align}
\notag
\mathbb{E}_{\mathcal{Q}}\Big[\int_0^{T}|b_1(s, X^{\alpha_2}(s))& - b_{1,n}(s, X^{\alpha_2}(s))|^4 \mathrm{d}s\Big]^{1/2} \le C\mathbb{E}\Big[ \int_0^{T}|b_1(s, x_0+ \sigma B(s)) - b_{1,n}(s,x_0+ \sigma B(s))|^4\mathrm{d}s \Big]^{1/2}
\varepsilonnd{align}
and using the density of Brownian motion, we have for every $p\ge 1$
\begin{align*}
\mathbb{E}\Big[\Big| b_{1}(s,x_0+ \sigma B(s))- b_{1,n}(s,x_0+\sigma B(s))&\Big|^p\Big]= \varphirac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}^d}\Big|b_{1,n} (s,x_0+\sigma y)-b_{1} (s,x_0+\sigma y)\Big|^pe^{-\varphirac{|y|^2}{2s}}\,\mathrm{d}ns y\\
=&\varphirac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}^d}\Big|b_{1,n} (s,\sigma y)-b_{1} (s,\sigma y)\Big|^pe^{-\varphirac{|y-x_0|^2}{2s}}\,\mathrm{d}ns y\\
=&\varphirac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}^d}\Big|b_{1,n} (s,\sigma y)-b_{1} (s,\sigma y)\Big|^pe^{-\varphirac{|y-2x_0|^2}{4s}}e^{-\varphirac{|y|^2}{4s}}e^{\varphirac{|x_0|^2}{2s}}\,\mathrm{d}ns y\\
\leq &\varphirac{1}{\sqrt{2\pi s}}e^{\varphirac{|x_0|^2}{2s}}\int_{\mathbb{R}^d}\big|b_{1,n} (s,\sigma y)-b_{1} (s,\sigma y)\big|^pe^{-\varphirac{|y|^2}{4s}}\,\mathrm{d}ns y.
\varepsilonnd{align*}
By Fubini's theorem, this shows that
\begin{multline}
\label{eq:estim.bnb}
\mathbb{E}\Big[\int_0^{T}|b_1(s, X^{\alpha_2}(s)) - b_{1,n}(s, X^{\alpha_2}(s))|^2\mathrm{d}s\Big]\\ \le C\Big(\int_0^T\varphirac{1}{\sqrt{2\pi s}}e^{\varphirac{|x_0|^2}{2s}}\int_{\mathbb{R}^d}\big|b_{1,n} (s,\sigma y)-b_{1} (s,\sigma y)\big|^4e^{-\varphirac{|y|^2}{4s}}\,\mathrm{d}ns y\,\mathrm{d} s\Big)^{1/2}.
\varepsilonnd{multline}
Let us now turn our attention to the first term in \varepsilonqref{eq:estim.diff.x}.
Since $\Lambda(\lambda,t)$ takes the form
\begin{align*}
\Lambda(\lambda, t) &= x+ \int_0^t\Big\{\lambda b_{n}(s, X^{\alpha_1}_n(s),\alpha_1(s)) + (1-\lambda)b(s, X^{\alpha_2}(s),{\alpha_2}(s))\Big\}\,\mathrm{d} s + \sigma B(t)\\
& = x +\int_0^tb^{\lambda,\alpha_2}(s)\mathrm{d}s + \sigma B(t).
\varepsilonnd{align*}
we use Jensen inequality, Girsanov's theorem as above and Lipschitz continuity of $b_2$ to get
\begin{align}
\notag
\mathbb{E}\Big[&\varepsilonxp\Big(2\int_{s}^t\int_0^1\partial_{x}b_{1,n}(r, \Lambda_n(\lambda,r))+\partial_xb_2(r,\Lambda_n(\lambda,r),\alpha_1(r))\,\mathrm{d}\lambda\mathrm{d}r\Big) \Big]\\\notag
&\le C\int_0^1 \mathbb{E}_{\mathbb{Q}^\lambda}\Big[\varepsilonxp\Big(4\int_{s}^t\partial_xb_{1,n}(r, \Lambda_n(\lambda,r))\mathrm{d}r \Big) \Big]^{1/2}\,\mathrm{d}\lambda\\
\label{eq:estime.bprime}
&\le C\int_0^1\mathbb{E}\Big[\varepsilonxp\Big(4\int_{s}^t\partial_xb_{1,n}(r, x_0+ \sigma B(r))\mathrm{d}r \Big) \Big]^{1/2}\,\mathrm{d}\lambda,
\varepsilonnd{align}
with $\,\mathrm{d} \mathbb{Q}^\lambda = \mathcal{E}\big(\varphirac{\sigma^\top}{|\sigma|^2}\int_0^Tb^{\lambda,\alpha_2}(s)\mathrm{d}B(s) \big)\,\mathrm{d} \mathbb{P}$, and where we used the fact that $b^{\lambda,\alpha_2}$ is bounded.
Since the sequence $(b_{1,n})_n$ is uniformly bounded, it follows from Lemma \ref{lemmaexpoloc} that
\begin{align}
\label{eq:bound.bprime}
\sup_nE\Big[\varepsilonxp\Big(4\int_{s}^t\partial_xb_{1,n}(r, x_0+ \sigma \cdot B(r))\mathrm{d}r \Big) \Big]
\leq C.
\varepsilonnd{align}
Therefore, putting together \varepsilonqref{eq:estim.diff.x}, \varepsilonqref{eq:estim.alpha12}, \varepsilonqref{eq:estim.bnb} and \varepsilonqref{eq:bound.bprime} concludes the proof.
Since $b_k$ is Lipschitz continuous the convergence (ii) follows by classical arguments, the proof is omitted.
\varepsilonnd{proof}
\begin{lemm}
\label{lem:J.continuous}
Let $\alpha\in \mathcal{A}$ and let $\alpha_n$ be a sequence of admissible controls such that $\delta(\alpha_n,\alpha)\to 0$.
Then, it holds
\begin{itemize}
\item[(i)] $| J_k(\alpha_n) - J_k(\alpha) | \to 0$ as $n\to \infty$ for every $k \in \mathbb{N}$ fixed. In particular, the function $J_k:(\mathcal{A},\delta) \to \mathbb{R}$ is continuous.
\item[(ii)] $|J_n(\alpha) - J(\alpha)| \le \varepsilon_n$ for some $C>0$ with $\varepsilon_n\downarrow 0$.
\varepsilonnd{itemize}
\varepsilonnd{lemm}
\begin{proof}
(i) The continuity of $J_k$ easily follows by Lipschitz continuity of $f$ and $g$.
In fact, we have
\begin{align*}
|J_k(\alpha_n) - J_k(\alpha)|& \le \mathbb{E}\Big[|g(X^{\alpha_n}_k(T)) - g(X^{\alpha}_k(T))| + \int_0^T|f(t, X^{\alpha_n}_k(t), \alpha_n(t)) - f(t, X^{\alpha}_k(t), \alpha(t)) |\,\mathrm{d} t \Big]\\
&\le C\mathbb{E}\Big[|X^{\alpha_n}_k(T) - X^{\alpha}_k(T)| + \int_0^T|X^{\alpha_n}_k(t) - X_k^{\alpha}(t)| + |\alpha_n(t) - \alpha(t)| \,\mathrm{d} t\Big] \to 0,
\varepsilonnd{align*}
where the convergence follows by dominated convergence and Lemma \ref{lem:conv.Xnn}.
(ii) is also a direct consequence of Lemma \ref{lem:conv.Xnn} since Fubini's theorem and Lipschitz continuity of $f$ and $g$ used as in part (i) above imply
\begin{align*}
| J_n(\alpha) - J(\alpha) | &\le C\sup_{t\in [0,T]}\mathbb{E}[|X^{\alpha}_n(t) - X^{\alpha}(t)|]
\le \varepsilon_n,
\varepsilonnd{align*}
where the second inequality follows from Lemma \ref{lem:conv.Xnn}.
\varepsilonnd{proof}
The next lemma pertains to the stability of the adjoint process with respect to the drift and the control process.
This result is based on similar stability properties for stochastic flows.
Given $x \in \mathbb{R}$ and the solution $X^{\alpha,x}$ of the SDE \varepsilonqref{eqSpro1} with initial condition $X^{\alpha,x}_t = x$, the first variation process of $X^{\alpha,x}$ is the derivative $\Phi^\alpha(t,s)$ of the function $x\mapsto X^{\alpha,x}(s)$.
Existence and properties of this Sobolev differentiable flow have been extensively studied by Kunita \cite{Kun90} for equations with sufficiently smooth coefficients.
In particular, when the drift $b$ is Lipschitz and continuously differentiable, the function $\Phi^\alpha(t,s)$ exists and, for almost every $\omega$, is the (classical) derivative of $x\mapsto X^{\alpha,x}(s)$.
The case of measurable (deterministic) drifts is studied by Mohammed et. al. \cite{MNP2015} and extended to measurable and random drifts in \cite{MenTan19}.
These works show that, when $b$ is measurable, then $X^{\alpha,\cdot}(s)\in L^2(\Omega, W^{1,p}(U))$ for every $s \in [t,T]$ and $p>1$, where $W^{1,p}(U)$ is the usual Sobolev space and $U$ an open and bounded subset of $\mathbb{R}$.
That is, $\Phi^\alpha(t,s)$ exists and is the weak derivative of $X^{\alpha,\cdot}$.
The proof of the stability result will make use of an explicit representation of the process $\Phi^\alpha$ with respect to the time-space local time.
Recall that for $a\in \mathbb{R}$ and $X=\{X(t),t\geq 0\}$ a continuous semimartingale, the local time $L^{X}(t,a)$ of $X$ at $a$ is defined by the Tanaka-Meyer formula as
$$
|X(t)-a|=|X(0)-a|+\int_0^t\sgn(X(s)-a)\,\mathrm{d}ns X(s) +L^{X}(t,a) ,
$$
where $\sgn(x)=-1_{(-\infty,0]}(x)+1_{(0,+\infty)}(x)$.
The local time-space integral plays a crucial role in the representations of the Sobolev derivative of the flows of the solution to the SDE \varepsilonqref{eqSpro1}.
It is defined for functions in the space $(\mathcal{H}_x, \|\cdot\|^x)$ defined (see e.g. \cite{Ein2000}) as the space of Borel measurable functions $f:[0,T]\times \mathbb{R}\rightarrow \mathbb{R} $ with the norm
\begin{align*}
\left\|f\right\|_x&:=2\Big(\int_0^1\int_{\mathbb{R}}f^2(s,z)\varepsilonxp(-\varphirac{|z-x|^2}{2s})\varphirac{\,\mathrm{d}ns s \,\mathrm{d} z}{\sqrt{2\pi s}}\Big)^{\varphirac{1}{2}}\notag
+\int_0^1\int_{\mathbb{R}}|z-x| |f(s,x)|\varepsilonxp(-\varphirac{|z-x|^2}{2s})\varphirac{\,\mathrm{d}ns s \,\mathrm{d} z}{s\sqrt{2\pi s}}.
\varepsilonnd{align*}
Since $b_1$ is bounded, we obviously have $b_1 \in \mathcal{H}^x$ for every $x$.
Moreover, it follows from \cite{Eisen07} (see also \cite{BMBPD17}) that for every continuous semimartingale $X$ the local time-space integral of $f\in \mathcal{H}^x$ with respect to $L^{X}(t,z)$ is well defined and satisfies
\begin{align}\label{eqtransLT1}
\int_0^t\int_{\mathbb{R}}f(s,z) L^{X}(\,\mathrm{d}ns s,\,\mathrm{d}ns z) = - \int_0^t\partial_xf(s,X(s))\,\mathrm{d}ns \langle X\rangle_ s,
\varepsilonnd{align}
for every continuous function (in space) $f \in \mathcal{H}^x$ admitting a continuous derivative $\partial_xf(s,\cdot)$, see \cite[Lemma 2.3]{Eisen07}.
This representation allows to derive the following:
\begin{lemm}
\label{lem:bound.int.local.time}
For every $\alpha \in \mathcal{A}$ and $c\ge0$, it holds
\begin{equation*}
\mathbb{E}\Big[e^{c\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big] <\infty.
\varepsilonnd{equation*}
\varepsilonnd{lemm}
\begin{proof}
First observe that for every $n \in \mathbb{N}$, it follows by Cauchy-Schwarz inequality that
\begin{align*}
&\mathbb{E}\Big[e^{c\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]\\
& = \mathbb{E}\Big[\mathcal{E}\Big(\varphirac{\sigma^\top}{|\sigma|^2}\int_0^Tb(s, X^{\alpha}(s),{\alpha}(s))\mathrm{d}B(s) \Big)^{1/2}\mathcal{E}\Big(\int_0^T\varphirac{\sigma^\top}{|\sigma|^2}b(s, X^{\alpha}(s),{\alpha}(s))\mathrm{d}B(s) \Big)^{-1/2}\\
&\quad \times e^{6\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)} \Big]\\
& \le C\mathbb{E}_{\mathbb{Q}}\Big[ e^{2c\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)} \Big]^{1/2}
\varepsilonnd{align*}
where $\mathbb{Q}$ is the probability measure given as in \varepsilonqref{eq:def.probab.Q} with $\alpha_2$ therein replaced by $\alpha$.
Hence, since $(X^{\alpha,x}-x_0)\sigma^\top/|\sigma|^2$ is a Brownian motion under $\mathbb{Q}$, it follows by \varepsilonqref{eqtransLT1} that
\begin{align*}
E\Big[e^{c\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]&\le C\mathbb{E}_{\mathbb{Q}}\Big[ e^{-2c\|\sigma\|^2\int_t^{s}\partial_xb_{1,n}\left(u,X^{\alpha,x}(u)\right)\mathrm{d}u} \Big]^{1/2}\\
&= C\mathbb{E}\Big[ e^{-2c\|\sigma\|^2\int_t^{s}\partial_xb_{1,n}\left(u,x_0 + \sigma B(u)\right)\mathrm{d}u} \Big]^{1/2}\le \overline C
\varepsilonnd{align*}
for some constant $\overline C>0$ which does not depend on $n$, where this latter inequality follows by Lemma \ref{lemmaexpoloc}.
Since $b_1$ is bounded and $b_{1,n}$ converges to $b_1$ pointwise, it follows by \cite[Theorem 2.2]{Eisen07} that $\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z) \to \int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z) $ as $n$ goes to infinity.
Thus, it follows by dominated convergence that
\begin{equation*}
E\Big[e^{c\int_t^{s}\int_{\mathbb{R}}b_{1}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big] = \lim_{\to \infty}E\Big[e^{c\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]<\overline C.
\varepsilonnd{equation*}
\varepsilonnd{proof}
We are now ready to prove stability of the follow and of the adjoint processes.
\begin{lemm}
\label{lem:conv.y.phi}
Let $\alpha\in \mathcal{A}$ and $\alpha_n$ be a sequence of admissible controls such that $\delta(\alpha_n,\alpha)\to 0$.
Then, the processes $X^{\alpha_n}_n$ and $X^{\alpha}$ admit Sobolev differentiable flows denoted $\Phi^{\alpha_n}_n$ and $\Phi^{\alpha}$, respectively and for every $0\le t\le s\le T$ it holds
\begin{itemize}
\item[(i)] $\mathbb{E}\big[|\Phi^{\alpha_n}_n(t,s) - \Phi^\alpha(t,s) |^2 \big] \to 0$ as $n\to \infty$,
\item[(ii)] $\mathbb{E}\big[| Y^{\alpha_n}_n(t) - Y^\alpha(t)| \big] \to 0$ as $n\to \infty$,
\varepsilonnd{itemize}
where $Y^\alpha$ is the adjoint process defined as
\begin{equation*}
Y^{\alpha}(t) := \mathbb{E}\Big[\Phi^{\alpha}(t,T) \partial_xg( X^{\alpha}(T)) + \int_t^T\Phi^{\alpha}(t,s) \partial_xf(s, X^{\alpha}(s), \alpha(s))\mathrm{d}s\mid \mathcal{F}_t \Big],
\varepsilonnd{equation*}
and $Y^{\alpha_n}_n$ is defined similarly, with $(X^{\alpha},\alpha, \Phi^\alpha)$ replaced by $(X^{\alpha_n}_n,\alpha_n, \Phi^{\alpha_n}_n)$.
\varepsilonnd{lemm}
\begin{proof}
The existence of the process $\Phi^{\alpha_n}_n$ is standard, it follows for instance by \cite{kunita01}.
The existence of the flow $\Phi^{\alpha}$ follows by \cite[Theorem 1.3]{MenTan19}.
We start by proving the first convergence claim.
As explained above, these processes admit explicit representations in terms of the space-time local time process.
It fact, it follows from Theorem \ref{Thmexpliflowder} that $\Phi^\alpha$ admits the representation
\begin{equation*}
\Phi^\alpha(t,s) = e^{\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}e^{\int_t^{s}\partial_xb_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u}
\varepsilonnd{equation*}
and $\Phi_n^{\alpha_n}$ admits the same representation with $(b_1, X^{\alpha,x},\alpha)$ replaced by $(b_{1,n}, X^{\alpha_n,x}, \alpha_n)$.
Using these explicit representations and H\"older inequality we have
\begin{align*}
&\mathbb{E}\Big[\Big|\Phi^{\alpha}(t,s)- \Phi_n^{\alpha_n}(t,s)\Big|^2\Big]\notag\\
\le& 2\mathbb{E}\Big[\Big|e^{\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big\{e^{\int_t^{s}\partial_xb_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u}
-e^{\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big\}\Big|^2\Big]\notag\\
& +2\mathbb{E}\Big[\Big|e^{\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big\{e^{\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}
-e^{\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X_n^{\alpha_n,x}}(\mathrm{d}u,\mathrm{d}z)}\Big\}\Big|^2\Big]\notag\\
\leq &2 \mathbb{E}\Big[e^{4\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[\Big\{e^{\int_t^{s}\partial_xb_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u}
-e^{\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big\}^4\Big]^{\varphirac{1}{2}}\notag\\
& +2\mathbb{E}\Big[e^{4\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[\Big\{e^{\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}
-e^{\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X_n^{\alpha_n,x}}(\mathrm{d}u,\mathrm{d}z)}\Big\}^4\Big]^{\varphirac{1}{2}}.
\varepsilonnd{align*}
Splitting up the terms in power 4, then applying H\"older and Young's inequality we continue the estimations as
\begin{align}
&\mathbb{E}\Big[\Big|\Phi^{\alpha}(t,s)- \Phi_n^{\alpha_n}(t,s)\Big|^2\Big]\notag\\
\leq & 2^7\mathbb{E}\Big[e^{4\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{2}} \mathbb{E}\Big[\Big\{e^{6\int_t^{s}\partial_xb_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u}
+e^{6\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big\}\Big]^{\varphirac{1}{4}}
\notag \\
&\times \mathbb{E}\Big[\Big\{e^{\int_t^{s}\partial_xb_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u}
-e^{\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big\}^2\Big]^{\varphirac{1}{4}}\notag\\
& +2^7 \mathbb{E}\Big[\Big|e^{4\int_t^{s}\partial_xb_{2}\left(u,X_n^{\alpha_n,x}(u),\alpha_n(u)\right) \mathrm{d}u}\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[\Big\{e^{6\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}
+e^{6\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X_n^{\alpha_n,x}}(\mathrm{d}u,\mathrm{d}z)}\Big\}\Big]^{\varphirac{1}{4}}
\notag\\
&\times \mathbb{E}\Big[\Big\{e^{\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}
-e^{\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X_n^{\alpha_n,x}}(\mathrm{d}u,\mathrm{d}z)}\Big\}^2\Big]^{\varphirac{1}{4}}\notag\\
=&CI_1^{\varphirac{1}{2}}\times I^{\varphirac{1}{2}}_{2,n}\times I^{\varphirac{1}{4}}_{3,n} +CI^{\varphirac{1}{2}}_{4,n}\times I^{\varphirac{1}{4}}_{5,n}\times I^{\varphirac{1}{4}}_{6,n}.
\varepsilonnd{align}
It follows from Lemma \ref{lem:bound.int.local.time} that $I_1$ and $I_{5,n}$ are bounded.
Since $\partial_xb_2$ is bounded, it follows that $I_{2,n}$ and $I_{4,n}$ are also bounded with bounds independent on $n$. Let us now show that $I_{3,n}$ and $I_{6,n} $ converge to zero.
We show only the convergence of $I_{6,n}$ since that of $I_{3,n}$ will follow (at least for a subsequence) from Lemma \ref{lem:conv.Xnn} and dominated convergence since $\partial_xb_{2}$ is continuous and bounded.
To that end, further define the processes $A_n^{\alpha_n}$ and $A^{\alpha}$ by
$$
A_n^{\alpha_n}(t,s):=e^{\int_t^{s}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X_n^{\alpha_n,x}}(\mathrm{d}u,\mathrm{d}z)}\quad \text{and} \quad A^{\alpha}(t,s):=e^{\int_t^{s}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}.
$$
In order to show that $A_n^{\alpha_n}$ converges to $A^{\alpha}$ in $L^2$, we will show that $A_n^{\alpha_n}$ converges weakly to $A^{\alpha}$ in $L^2$ and that $E[|A_n^{\alpha_n}|^2]$ converges to $E[|A^{\alpha}|^2]$ in $\mathbb{R}$.
We first prove the weak convergence.
Since the set
$$
\Big\{\mathcal{E}\Big(\int_0^1\dot{\varphi}(s)\mathrm{d}B(s)\Big):\varphi\in C^{1}_b([0,T],\mathbb{R}^d)\Big\}
$$
spans a dense subspace in $L^2(\Omega)$, in order to show weak convergence, it is enough to show that
$$
E\Big[A_n^{\alpha_n}(t,s) \mathcal{E}\Big(\int_0^1\dot{\varphi}(s)\mathrm{d}B(s)\Big)\Big]\rightarrow E\Big[A^{\alpha}(t,s) \mathcal{E}\Big(\int_0^1\dot{\varphi}(s)\mathrm{d}B(s)\Big)\Big]\quad \text{for every}\quad \varphi\in C^{1}_b([0,T],\mathbb{R}^d).
$$
Denote by $\tilde X_n^{\alpha_n, x}$ and $\tilde X^{\alpha,x}$ the processes given by
\begin{align}\label{eqxntilde10}
\,\mathrm{d}ns \tilde X^{\tilde \alpha_n,x}_n(t)= \Big(b_{1,n}(t,\tilde X^{\tilde \alpha_n,x}_n(t))+ b_{2}(t,\tilde X^{\tilde \alpha_n,x}_n(t),\tilde \alpha_n)+\sigma \dot\varphi(t) \Big)\,\mathrm{d}ns t
+\sigma \,\mathrm{d}ns B(t),
\varepsilonnd{align}
and
\begin{align}\label{eqxntilde11}
\,\mathrm{d}ns \tilde X^{\tilde \alpha,x}(t) = \Big(b_{1}(t,\tilde X^{\tilde \alpha,x}(t))+ b_{2}(t,\tilde X^{\tilde \alpha,x}(t),\tilde \alpha_n)+\sigma \dot\varphi(t)\Big)\,\mathrm{d}ns t
+\sigma \,\mathrm{d}ns B(t).
\varepsilonnd{align}
Observe that these processes are well-defined, since we have $\tilde X^{\tilde \alpha,x}(t,\omega) = X^{\alpha,x}(t,\omega + \varphi)$ and $\tilde X_n^{\tilde \alpha_n,x}(t,\omega) = X_n^{\alpha_n,x}(t,\omega + \varphi)$.
Using the Cameron-Martin-Girsanov theorem as in the proof of Lemma \ref{lem:bound.int.local.time}, we have
\begin{align*}
& \Big|E\Big[\mathcal{E}\Big(\int_0^T\dot{\varphi}(s)\mathrm{d}B(s)\Big)\Big\{A_n^{\alpha_n}(t,s)-A^{\alpha}(t,s)\Big\}\Big]\Big|\notag\\
=&\Big|E\Big[e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde\alpha_n,x}}(\mathrm{d}u,\mathrm{d}z)}-e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\tilde X^{\tilde\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]\Big|\notag\\
=&\Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\Big\{\tilde u_n(s,x+\sigma \cdot B(s),\alpha_n (s))+\sigma \cdot \dot{\varphi}(s)\Big\}\mathrm{d}B(s)\Big)e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\notag\\
&\quad -\mathcal{E}\Big(\int_0^T\Big\{\tilde u(s,x+\sigma \cdot B(s),\alpha (s))+\sigma \cdot \dot{\varphi}(s)\Big\}\mathrm{d}B(s)\Big)e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]\Big| ,
\varepsilonnd{align*}
where $\tilde u(s, x,\alpha(\omega)): = u(s, x,\alpha(\omega+\varphi))$.
Next, add and subtract the same term and then use the inequality $|e^x-e^y|\leq |x-y||e^x+e^y|$ and then H\"older inequality and putting
\begin{equation*}\label{eqnewbm1}
u(s, x,\alpha(\omega)): = (\varphirac{\sigma^1b}{|\sigma|^2}, \dots, \varphirac{\sigma^db}{|\sigma|^2})(t,x,\alpha(\omega))\quad \text{and}\quad B^x_\sigma:=x+\sum_{i=1}^d\varphirac{\sigma_i}{\|\sigma\|}B^i,
\varepsilonnd{equation*}
we obtain
\begin{align*}
&\Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\dot{\varphi}(s)\mathrm{d}B(s)\Big)\Big\{A_n^{\alpha_n}(t,s)-A^{\alpha}(t,s)\Big\}\Big]\Big|\notag\\
\leq & \Big|E\Big[\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma \cdot B(s),\alpha (s,\omega+\varphi))+\sigma \cdot \dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\notag\\
&\Big|\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)-\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)\Big|\\
&\times\Big(e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}+e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big)\Big]\Big|\notag\\
&+\Big|E\Big[e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big\{\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma \cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma \cdot \dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\notag\\
&-\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma \cdot B(s),\alpha (s,\omega+\varphi))+\sigma \cdot \dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\Big\}\Big]\Big|.
\varepsilonnd{align*}
Therefore, another application of H\"older's inequality yields the estimate
\begin{align}\label{eq:estimates.J}
\notag
& \Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\dot{\varphi}(s)\mathrm{d}B(s)\Big)\Big\{A_n^{\alpha_n}(t,s)-A^{\alpha}(t,s)\Big\}\Big]\Big|\\\notag
\leq & 4\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma \cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma \cdot \dot{\varphi}(s)\}\mathrm{d}B(s)\Big)^4\Big]^{\varphirac{1}{4}}\notag\\
&\mathbb{E}\Big[\Big|\int_s^{t}\int_{\mathbb{R}}\Big(b_{1,n}\left(u,z\right)-b_1\left(u,z\right)\Big)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)\Big|^2\Big]^{\varphirac{1}{2}}\notag\\
&\times \mathbb{E}\Big[e^{4\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}+e^{4\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{4}}\notag\\
&+\mathbb{E}\Big[e^{2\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[\Big\{\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma \cdot B(s),\alpha_n (s,\omega+\varphi))+\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\notag\\
&-\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma \cdot B(s),\alpha (s,\omega+\varphi))+\sigma \cdot \dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\Big\}^2\Big]^{\varphirac{1}{2}}\notag\\
=&J_{1,n}^{\varphirac{1}{4}}\times J_{2,n}^{\varphirac{1}{2}}\times J_{3,n}^{\varphirac{1}{4}}+J_{4,n}^{\varphirac{1}{2}}\times J_{5,n}^{\varphirac{1}{2}}.
\varepsilonnd{align}
Using Lemma \ref{Lemmbound1}, it follows that $J_{2,n}$ converge to zero, and by dominated convergence $J_{5,n}$ also convergences to zero.
Thanks to Lemma \ref{lemmaexpoloc} and boundedness of $b_{1,n}$ (respectively $b_1$), the term $J_{3,n}$ (respectively $J_{4,n}$) is bounded.
The bound of $J_{1,n}$ follows by the uniform boundedness of $u_n$.
It remains to show that $\mathbb{E}[|A_n^{\alpha_n}(t)|^2]$ converges to $\mathbb{E}[|A^{\alpha}(t)|^2]$ in $\mathbb{R}$. Using Girsanov transform as in the proof of Lemma \ref{lem:bound.int.local.time}, we have
\begin{align}
\mathbb{E}[|A_n^{\alpha_n}(t)|^2]=&\mathbb{E}\Big[e^{2\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X_n^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]\notag\\
=& \mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma\cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)e^{2\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]
\varepsilonnd{align}
and
\begin{align}
\mathbb{E}[|A^{\alpha}(t)|^2]=&\mathbb{E}\Big[e^{2\int_s^{t}\int_{\mathbb{R}}b_{1}\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]\notag\\
=& \mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma\cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)e^{2\int_s^{t}\int_{\mathbb{R}}b_{1}\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big].
\varepsilonnd{align}
Therefore using once more $|e^x-e^y|\leq |x-y||e^x+e^y|$ and Cauchy-Schwarz inequality
\begin{align}
&|\mathbb{E}[|A_n^{\alpha_n}(t)|^2]-\mathbb{E}[|A^{\alpha}(t)|^2]|\notag\\
=&\Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma\cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)e^{2\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^{x}_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]\notag\\
&-\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma\cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)e^{2\int_s^{t}\int_{\mathbb{R}}b_{1}\left(u,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]\Big|\notag\\
\leq& \Big|\mathbb{E}\Big[e^{4\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^{x}_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[
\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma\cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\notag\\
&-\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma\cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)^2\Big]^{\varphirac{1}{2}}\Big|\notag\\
&+C\Big|\mathbb{E}\Big[\Big(\int_s^{t}\int_{\mathbb{R}}\{b_{1,n}\left(u,z\right)-b_{1}\left(u,z\right)\}L^{\|\sigma\|B^{x}}(\mathrm{d}u,\mathrm{d}z)\Big)^2\Big]^{\varphirac{1}{2}}\notag\\
&\times \Big(\mathbb{E}\Big[e^{8\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^{x}_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{4}}+\mathbb{E}\Big[e^{8\int_s^{t}\int_{\mathbb{R}}b_{1}\left(u,z\right)L^{\|\sigma\|B^{x}_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{4}}\Big)\notag\\
&\times \mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma\cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)^4\Big]^{\varphirac{1}{4}}\Big|\notag.
\varepsilonnd{align}
Now, introducing the random variables
\begin{align*}
V_n := &\int_0^T\Big(u_n(s,x+\sigma \cdot B(s),\alpha_n (s,\omega+\varphi))-u(s,x+\sigma \cdot B(s),\alpha (s,\omega+\varphi))\Big)\mathrm{d}B(s)\\
&-\varphirac{1}{2}\int_0^T\Big(|u_n(s,x+\sigma \cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)|^2\notag\\
&-|u(s,x+\sigma \cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)|^2\Big)\mathrm{d} s
\varepsilonnd{align*}
and
\begin{align*}
F_{1,n} := \int_s^{t}\int_{\mathbb{R}}\{b_{1,n}\left(u,z\right)-b_{1}\left(u,z\right)\}L^{\|\sigma\|B^{x}}(\mathrm{d}u,\mathrm{d}z)
\varepsilonnd{align*}
we continue the above estimations as
\begin{align}
|\mathbb{E}[|A_n^{\alpha_n}(t)|^2] &- \mathbb{E}[|A^{\alpha}(t)|^2]|\notag \\
&\leq CE\Big[V_n^2\Big\{\mathcal{E}\Big(\int_0^T\{u_n(s,x+\sigma \cdot B(s),\alpha_n (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\notag\\
&\quad + \mathcal{E}\Big(\int_0^T\{u(s,x+\sigma \cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)\Big\}^2\Big]\notag\\
&\quad+C\Big|\mathbb{E}\Big[|F_{1,n}|^2\Big]^{\varphirac{1}{2}} \Big(E\Big[e^{8\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\|\sigma\|B^{x}_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{4}}+ \mathbb{E}\Big[e^{8\int_s^{t}\int_{\mathbb{R}}b_{1}\left(u,z\right)L^{\|\sigma\|B^{x}_\sigma}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{4}}\Big)\notag\\
&\quad \times \mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u(s,x+\sigma\cdot B(s),\alpha (s,\omega+\varphi))+\sigma\cdot\dot{\varphi}(s)\}\mathrm{d}B(s)\Big)^4\Big]^{\varphirac{1}{4}}\Big|.
\varepsilonnd{align}
By Lemma \ref{Lemmbound1}, $F_{1,n}$ converges to zero in $L^2(\Omega)$.
Using similar arguments as in \cite[Lemma A.3]{BMBPD17}, one can show that $V_n$ converges to zero in $L^2(\Omega)$ by the boundedness of $u_n$ and the definition of the distance $\delta$. Observe however that in this case, $u_n$ depends on $\alpha_n$ and not on $\alpha$ as in \cite[Lemma A.3]{BMBPD17}. Nevertheless using the fact that $b_{1,n}$, $b_1$ and $b_2$ are bounded and Lipschitz in the second variable, one can show by dominated convergence theorem and similar reasoning as in \varepsilonqref{eq:estim.alpha12} that the overall term converges to zero. It is also worth mentioning that the other terms are uniformly bounded by application of either Girsanov theorem and/or Lemma \ref{lemmaexpoloc} to the uniformly bounded senquences $(u_n)_{n\geq 1},(b_{1,n})_{n\geq 1}$ and the bounded functions $u, b_{1}$.
Let us now turn our attention to the proof of (ii).
Compute the difference $Y_{n}^{\alpha_n}(t)-Y^\alpha(t)$, add and subtract the terms $\Phi^{\alpha}(t,T) \partial_xg(X_n^{\alpha_n}(T))$ and $\int_t^T\Phi^{\alpha}(t,u) \partial_xf(u,X_n^{\alpha_n}(u), \alpha_n(u))\,\mathrm{d} u$ and then apply H\"older's inequality to obtain
\begin{align}\label{eq:conv.Y}
\notag
&\mathbb{E}[|Y_{n}^{\alpha_n}(t)-Y^\alpha(t)|]\\\notag
\leq & C_T\Big\{\mathbb{E}\Big[\Big|\Phi^{\alpha}(t,T)\Big|^2\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[|\partial_xg( X_n^{\alpha_n}(T)) - \partial_xg( X^{\alpha}(T))|^2\Big]^{\varphirac{1}{2}}\\
&+\mathbb{E}\Big[|\partial_xg( X_n^{\alpha_n}(T))|^2\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[\Big|\Phi_n^{\alpha_n}(t,T) - \Phi^{\alpha}(t,T)\Big|^2\Big]^{\varphirac{1}{2}}\notag\\\notag
&+\mathbb{E}\Big[\int_t^T|\Phi^{\alpha}(t,u)|^2\,\mathrm{d} u\Big]^{\varphirac{1}{2}} \mathbb{E}\Big[\int_0^T|\partial_xf(u, X^{\alpha}(u), \alpha(u))-\partial_xf(u, X_n^{\alpha_n}(u), \alpha_n(u))|^2\,\mathrm{d} u\Big]^{\varphirac{1}{2}}\notag\\
&+\mathbb{E}\Big[\int_0^T|\partial_xf(u, X_n^{\alpha_n}(u), \alpha_n(u))|^2\,\mathrm{d} u\Big]^{\varphirac{1}{2}}\mathbb{E}\Big[\int_0^T|\Phi_n^{\alpha_n}(u)-\Phi^{\alpha}(u)|^2\,\mathrm{d} u\Big]^{\varphirac{1}{2}}\Big\}
\varepsilonnd{align}
for some constant $C_T$ depending only on $T$.
Since the process $\Phi^{\alpha}$ is square integrable, (see \cite[Theorem 1.3]{MenTan19}) it follows by boundedness and continuity of $\partial_xg,\partial_xf$ as well as Lemma \ref{lem:conv.Xnn} that the first and third terms converge to zero as $n$ goes to infinity.
Moreover, by boundedness of $\partial_xf$ and $\partial_xg$ and the $L^2$ convergence of $\Phi_n^{\alpha_n}(t,u)$ to $\Phi^{\alpha}(t,u)$ given in part (i), we conclude that the second and last terms in \varepsilonqref{eq:conv.Y} converge to zero, which shows (ii).
\varepsilonnd{proof}
\begin{proof}(of Theorem \ref{thm:necc})
Let $\hat\alpha$ be an optimal control and $n\ge 1$ fixed.
Observe that by the linear growth assumption on $f,g$ the function $J_n$ is bounded from above.
By Lemma \ref{lem:J.continuous} the function $J_n$ is also continuous on $(\mathcal{A},\delta)$ and there exists $\varepsilon_n$ such that
\begin{equation*}
J(\hat\alpha) - J_n(\hat\alpha)\le \varepsilon_n \text{ and } J_n(\alpha) - J(\alpha) \le \varepsilon_n\quad \text{for all } \alpha \in \mathcal{A}.
\varepsilonnd{equation*}
That is, $J_n(\hat\alpha) \le \inf_{\alpha \in \mathcal{A}}J_n(\alpha) + 2\varepsilon_n$.
Thus, by Ekeland's variational principle, see e.g. \cite{Ekeland79}, there is a control $\hat\alpha_n \in \mathcal{A}$ such that $\delta(\hat\alpha, \hat\alpha_n)\le (2\varepsilon_n)^{1/2}$ and
\begin{equation*}
J_n(\hat\alpha_n) \le J_n(\alpha) + (2\varepsilon_n)^{1/2}\delta(\hat\alpha_n,\alpha)\quad \text{for all}\quad \alpha \in \mathcal{A}.
\varepsilonnd{equation*}
In other words, putting $J^\varepsilon_n(\alpha):= J_n(\alpha) + (2\varepsilon_n)^{1/2}\delta(\hat\alpha_n,\alpha)$, the control process $\hat\alpha_n$ is optimal for the problem with cost function $J^\varepsilon_n$.
Now, let $\beta \in \mathcal{A}$ be an arbitrary control and $\varepsilon>0$ a fixed constant.
By convexity of $\mathbb{A}$, it follows that $\hat\alpha_n + \varepsilon\varepsilonta \in \mathcal{A}$, with $\varepsilonta := \beta - \hat\alpha_n$.
Thus, since $b_n$ is sufficiently smooth, it is standard that the functional $J_n$ is G\^ateau differentiable (see
\cite[Lemma 4.8]{MR3629171}) and its G\^ateau derivative in the direction $\varepsilonta$ is given by
\begin{align*}
\varphirac{d}{d\varepsilon}J_n(\alpha + \varepsilon \varepsilonta)_{|_{\varepsilon = 0}}& = \mathbb{E}\Big[\int_0^T\partial_xf(t, X_n^{\hat\alpha_n}(t), \hat\alpha_n(t))V_n(t) + \partial_{\alpha}f(t, X_n^{\hat\alpha_n}(t), \hat\alpha_n(t))\varepsilonta(t)\mathrm{d}t\\
&\qquad + \partial_xg(X_n^{\hat\alpha_n}(T))V_n(T) \Big] ,
\varepsilonnd{align*}
where $V_n$ is the stochastic process solving the linear equation
\begin{equation*}
dV_n(t) = \partial_xb_n(t, X_n^\alpha(t),\alpha(t))V_n(t)\mathrm{d}t + \partial_\alpha b_n(t, X_n^\alpha(t),\alpha(t))\varepsilonta(t)\mathrm{d}t,\quad V_n(0) = 0.
\varepsilonnd{equation*}
On the other hand, we have
\begin{equation*}
\lim_{\varepsilon\downarrow 0}\varphirac{1}{\varepsilon}\big(\delta(\hat\alpha_n, \alpha + \varepsilon\varepsilonta) - \delta(\hat\alpha_n, \alpha) \big) \le C_M\mathbb{E}\big[ \sup_{t \in [0,T]}|\varepsilonta(t)|^2 \big]^{1/2}.
\varepsilonnd{equation*}
for a constant $C_M>0$ depending on the constant $M$ (introduced in the definition of $\mathcal{A}$).
Therefore, $J^\varepsilon_n$ is also G\^ateau differentiable and since $\hat\alpha_n$ is optimal for $J^\varepsilon_n$, we have
\begin{align*}
0\le \varphirac{\mathrm{d}}{\mathrm{d}\varepsilon}J^\varepsilon_n(\hat\alpha_n + \varepsilon \varepsilonta)_{|_{\varepsilon = 0}} &= \varphirac{\mathrm{d}}{\mathrm{d}\varepsilon}J_n(\hat\alpha_n + \varepsilon \varepsilonta)_{|_{\varepsilon = 0}} + \lim_{\varepsilon\downarrow 0} (2\varepsilon_n)^{1/2}\varphirac{1}{\varepsilon}\delta(\hat\alpha_n,\hat\alpha_n + \varepsilon\varepsilonta) \\
& = \mathbb{E}\Big[\int_0^T\partial_xf\big(t, X_n^{\hat\alpha_n}(t), \hat\alpha_n(t) \big)V_n(t) + \partial_{\alpha}f\big( t, X_n^{\hat\alpha_n}(t), \hat\alpha_n(t) \big)\varepsilonta(t)\mathrm{d}t\\
&\qquad + \partial_xg(X_n^{\hat\alpha_n}(T))V_n(T) \Big] + C_M\big(2\varepsilon_nE[\sup_t|\varepsilonta(t)|^2] \big)^{1/2}\\
&\le \mathbb{E}\Big[\int_0^T\partial_\alpha H_n\big(t, X_n^{\hat\alpha}, Y_n^{\hat\alpha_n}(t), \hat\alpha_n(t) \big)\varepsilonta(t)\mathrm{d}t \Big] + C_M\varepsilon_n^{1/2},
\varepsilonnd{align*}
for some constant $M>0$.
The inequality following since $\hat\alpha_n\in \mathcal{A}$, and where $H_n$ is the Hamiltonian of the problem with drift $b_n$ given by
\begin{equation*}
H_n(t,x,y,a) := f(t, x,a) + b_n(t,x,a)y
\varepsilonnd{equation*}
and $(Y^{\hat\alpha_n}_n, Z^{\hat\alpha_n}_n)$ the adjoint processes given by
\begin{equation*}
\mathrm{d}Y^{\hat\alpha_n}_n(t) = -\partial_xH_n(t, X_n^{\hat\alpha}, Y^{\hat\alpha_n}_n(t), \hat\alpha_n(t))\mathrm{d}t + Z^{\hat\alpha_n}_n(t)\mathrm{d}B(t).
\varepsilonnd{equation*}
By standard arguments, we can thus conclude that
\begin{equation*}
C_M\varepsilon_n^{1/2} +\partial_\alpha H_n(t, X_n^{\hat\alpha_n}(t), Y^{\hat\alpha_n}_n(t), \hat\alpha_n(t))\cdot (\beta - \hat\alpha_n(t)) \ge 0 \quad \mathbb{P}\otimes \mathrm{d}t \mathrm{-a.s}.
\varepsilonnd{equation*}
Recalling that $b_{1,n}$ does not depend on $\alpha$, this amounts to
\begin{equation*}
C_M\varepsilon_n^{1/2} + \Big\{ \partial_{\alpha}f(t, X_n^{\hat\alpha_n}(t), \hat\alpha_n(t)) + \partial_{\alpha}b_2\big(t, X_n^{\hat\alpha_n}(t), \hat\alpha_n(t) \big)Y^{\hat\alpha_n}_n(t) \Big\}\cdot(\beta - \hat\alpha_n(t)) \ge 0 \quad \mathbb{P} \otimes dt\text{-a.s.}
\varepsilonnd{equation*}
We will now take the limit on both sides above as $n$ goes to infinity.
It follows by Lemma \ref{lem:conv.Xnn} and Lemma \ref{lem:conv.y.phi} respectively that $X_n^{\hat\alpha_n}(t) \to X^{\hat\alpha}(t)$ and $Y^{\hat\alpha_n}_n(t) \to Y^{\hat\alpha}(t)$ $\mathbb{P}$-a.s. for every $t\in [0,T]$.
Since $\hat\alpha_n\to \alpha$, we therefore conclude that
\begin{equation*}
\Big\{ \partial_{\alpha}f(t, X^{\hat\alpha}(t), \hat\alpha(t)) + \partial_{\alpha}b_2\big(t, X^{\hat\alpha}(t), \hat\alpha(t) \big)Y^{\hat\alpha}(t) \Big\}\cdot(\beta - \hat\alpha(t)) \ge 0 \quad \mathbb{P}\otimes \mathrm{d}t\text{-a.s.}
\varepsilonnd{equation*}
This shows \varepsilonqref{eq:nec.cond}, which concludes the proof.
\varepsilonnd{proof}
\section{The sufficient condition for optimality}
\label{sec:sufficient}
Let us now turn to the proof of the sufficient condition of optimality.
Since we will need to preserve the concavity of $H$ assumed in Theorem \ref{thm:suff} after approximation, we specifically assume that the function $b_n$ is defined by standard mollification.
Therefore, $H_n(t,x,y,a):= f(t,x,a)+ b_n(t,x,a)y$ is a mollification of $H$ and thus remains concave.
\begin{proof}(of Theorem \ref{thm:suff})
Let $\hat\alpha \in \mathcal{A}$ satisfy \varepsilonqref{eq:suff.con} and $\alpha'$ an arbitrary element of $\mathcal{A}$.
We would like to show that $J(\hat\alpha) \ge J(\alpha')$.
Let $n \in \mathbb{N}$ be arbitrarily chosen.
By definition, we have
\begin{align*}
&J_n(\hat\alpha) - J_n(\alpha')\\
& = \mathbb{E}\Big[g(X^{\hat\alpha}_n(T)) - g(X^{\alpha'}_n(T)) + \int_0^Tf(u, X_n^{\hat\alpha}(u), \hat\alpha(u)) - f(u, X_n^{\alpha'}(u), \alpha'(u))\,\mathrm{d} u \Big] \\
&\ge \mathbb{E}\Big[\partial_xg(X^{\hat\alpha}_n(T))\big\{X^{\hat\alpha}(T) -X^{\alpha'}_n(T)\big\} + \int_0^T\big\{ b_n(u, X_n^{\alpha'}(u), \alpha'(u)) - b_n(u, X_n^{\hat\alpha}(u),\hat\alpha(u))\big\} Y_n^{\hat\alpha}(u)\,\mathrm{d} u\\
&\quad + \int_0^T H_n(u, X_n^{\hat\alpha}(u), Y_n^{\hat\alpha}(u), \hat\alpha(u)) - H_n(u, X_n^{\alpha'}(u),Y_n^{\hat\alpha}(u), \alpha'(u))\,\mathrm{d} u \Big],
\varepsilonnd{align*}
where we used the definition of $H_n$ and the fact that $g$ is concave.
Since $Y_n^{\hat\alpha}$ satisfies
\begin{equation*}
Y^{\hat\alpha}_n(t) = \mathbb{E}\Big[\Phi_n^{\hat\alpha}(t,T) \partial_xg( X^{\hat\alpha}_n(T)) + \int_t^T\Phi_n^{\hat\alpha}(t,u) \partial_xf(u, X_n^{\hat\alpha}(u), \hat\alpha(u))\mathrm{d}u\mid \mathcal{F}_t \Big],
\varepsilonnd{equation*}
it follows by martingale representation and It\^o's formula that there is a square integrable progressive process $(Y^{\hat\alpha}_n,Z^{\hat\alpha}_n)$ such that $Y_n^{\hat\alpha}$ satisfies the (linear) equation
\begin{equation*}
Y^{\hat\alpha}_n(t) = \partial_xg(X^{\hat\alpha}_n) + \int_t^T\partial_xH_n(u, X^{\hat\alpha}_n(u), Y_n^{\hat\alpha}(u),\hat\alpha(u))\,\mathrm{d} u - \int_t^TZ_n^{\hat\alpha}(u)\,\mathrm{d} W(u).
\varepsilonnd{equation*}
Recall that since $b_n$ is smooth, so is $H_n$.
Therefore, by It\^o's formula once again we have
\begin{align*}
&Y^{\hat\alpha}_n(T)\big\{X_n^{\hat\alpha}(T) - X_n^{\alpha'}(T)\big\} = \int_0^TY^{\hat\alpha}_n(u)\big\{b_n(u, X^{\hat\alpha}_n(u),\hat\alpha(u)) - b_n(u, X^{\alpha'}_n(u),\alpha'(u)) \big\}\,\mathrm{d} u\\
&\quad - \int_0^T\big\{X^{\hat\alpha}_n(u) - X^{\alpha'}_n(u) \big\}\partial_xH_n(u, X^{\hat\alpha}_n(u), Y_n^{\hat\alpha}(u),\hat\alpha(u))\,\mathrm{d} u + \int_0^T\big\{X^{\hat\alpha}_n(u) - X^{\alpha'}_n(u) \big\} Z^{\hat\alpha}_n(u)\,\mathrm{d} W(u).
\varepsilonnd{align*}
Since the stochastic integral above is a local martingale, a standard localization argument allows to take expectation on both sides to get that
\begin{align*}
J_n(\hat\alpha) - J_n(\alpha') &\ge \mathbb{E}\Big[- \int_0^T\big\{X^{\hat\alpha}_n(u) - X^{\alpha'}_n(u) \big\}\partial_xH_n(u, X^{\hat\alpha}_n(u), Y_n^{\hat\alpha}(u),\hat\alpha(u))\,\mathrm{d} u \\
&\quad + \int_0^T H_n(u, X_n^{\hat\alpha}(u), Y_n^{\hat\alpha}(u), \hat\alpha(u)) - H_n(u, X_n^{\alpha'}(u),Y_n^{\hat\alpha}(u), \alpha'(u))\,\mathrm{d} u \Big]\\
&\ge \mathbb{E}\Big[\int_0^T \partial_\alpha H_n(u, X_n^{\hat\alpha}(u), Y_n^{\hat\alpha}(u), \hat\alpha(u))\cdot(\hat\alpha(u) - \alpha'(u))\,\mathrm{d} u \Big],
\varepsilonnd{align*}
where the latter inequality follows by concavity of $H_n$.
Coming back to the expression of interest $J(\hat\alpha) - J(\alpha')$, we have
\begin{align*}
J(\hat\alpha) - J(\alpha') & = J(\hat\alpha) - J_n(\hat\alpha) + J_n(\hat\alpha) - J_n(\alpha') + J_n(\alpha') - J(\alpha')\\
&\ge J(\hat\alpha) - J_n(\hat\alpha) + \mathbb{E}\Big[\int_0^T \partial_\alpha H_n(u, X_n^{\hat\alpha}(u), Y_n^{\hat\alpha}(u), \hat\alpha(u))\cdot(\hat\alpha(u) - \alpha'(u))\,\mathrm{d} u \Big]\\
&\quad + J_n(\alpha') - J(\alpha').
\varepsilonnd{align*}
Since $b_{1,n}$ does not depend on $\alpha$, we have
$\partial_\alpha H_n(u, X_n^{\hat\alpha}(u), Y_n^{\hat\alpha}(u), \hat\alpha(u)) = \partial_\alpha b_2(u, X^{\hat\alpha}_n(u),\hat\alpha(u))Y^{\hat\alpha}_n(u) + \partial_\alpha f(u, X^{\hat\alpha}_n(u),\hat\alpha(u))$.
Therefore, taking the limit as $n$ goes to infinity, it follows by Lemmas \ref{lem:conv.Xnn}, \ref{lem:J.continuous} and \ref{lem:conv.y.phi} that it holds
\begin{align*}
J(\hat\alpha) - J(\alpha') \ge E\Big[\int_0^T \partial_\alpha H(u, X^{\hat\alpha}(u), Y^{\hat\alpha}(u), \hat\alpha(u))\cdot(\hat\alpha(u) - \alpha'(u))\,\mathrm{d} u \Big].
\varepsilonnd{align*}
Since $\hat\alpha$ satisfies \varepsilonqref{eq:suff.con}, we therefore conclude that $J(\hat\alpha) \ge J(\alpha')$.
\varepsilonnd{proof}
\subsection{Concluding remarks}
\label{subsec.conclusion}
Let us conclude the paper by briefly discussing our assumptions.
The condition $b=b_1+b_2$ seems essential to derive existence and uniqueness results of the controlled system.
For instance, the crucial bound \varepsilonqref{eq:bound.bprime} derived in \cite{BMBPD17,MMNPZ13} is unknown when $b_1$ depends on $\alpha$.
This condition is also vital in obtaining the explicit representation of the Sobolev derivative of the flows of the solution to the SDE in terms of its local time.
This representation cannot be expected in multidimensions due to the non commutativity of matrices and the local time.
Therefore, much stronger (regularity) conditions are needed to derive the maximum principle in this case (see for example \cite{Bah-Chi-Dje-Mer, Bah-Dje-Mer-AMO07, Bah-Dje-Mer07}).
Note in addition that the boundedness assumption on $b$ is made mostly to simplify the presentation.
The results should also hold with $b$ of linear growth in the spacial variable, albeit with more involved computations and with $T$ small enough, since the flow in this case is expected to exist in small time.
Given the drift $b$, some known conditions on the control $\alpha$ that guaranty existence and uniqueness of the strong solution to the SDE \varepsilonqref{eqSpro1} satisfied by the controlled process are given by \varepsilonqref{eqcondal1} and \varepsilonqref{eqcondal2}.
These conditions involve the Malliavin derivative of $\alpha$.
Let us remark that the Malliavin differentiability of the control is not an uncommon assumption. This condition appears implicitly in the works \cite{Menou20142, MOZ12, OS09} on the stochastic maximum principle where the coefficients are required to be at least two times differentiable with bounded derivatives.
\begin{appendix}
\section{Representation of the differential flow by time-space local time}
It is well-known that solutions of stochastic differential equations admit a stochastic differential flow.
Such flows have been extensively investigated in the work of Kunita \cite{Kun90} for equations with sufficiently smooth coefficients.
When the drift merely measurable, it turns out (see e.g. \cite{MMNPZ13,MNP2015,XichZhang16}) that flows still exists, at least in the Sobolev sense.
The study of existence of such flows is extended to the case of random coefficients in \cite{MenTan19}.
In this appendix, we show that the stochastic differential flow admits an explicit representation.
The difficulty here is the lack of regularity of the drift, around which we get using local time integration.
This representation has been obtained in \cite{BMBPD17} assuming that the drift $b=b_1+b_2$ is deterministic with $b_1$ bounded and measurable and $b_2$ Lipschitz--continuous.
\begin{thm}\label{Thmexpliflowder}
Suppose that $b$ is as in Theorem \ref{thm:necc} and $\alpha \in \mathcal{A}$.
For every $0\leq s\leq t\leq T$, the stochastic flows $\Phi^{\alpha, x}(t,s)$ of the unique strong solution to the SDE \varepsilonqref{eqSpro1} admits the representation
\begin{align}\label{eqflow11}
\Phi^{\alpha,x}(t,s)=&\varepsilonxp\Big(-\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)+\int_s^{t}b'_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u\Big).
\varepsilonnd{align}
Here $\int_s^t\int_{\mathbb{R}}b_1(u,z)L^{X^x}(\,\mathrm{d}ns u,\,\mathrm{d}ns z)$ is the integration with respect to the time-space local time of $X^x$ and $b'_2$ is the derivative with respect to the second parameter.
\varepsilonnd{thm}
\begin{proof}
We know from \cite{MenTan19}, \cite{BMBPD17} that under the condition of the Theorem, the SDE \varepsilonqref{eqSpro1} has a Sobolev differentiable flow denoted $\Phi^{\alpha,x}$.
In particular, it is shown in these references that $\Phi^{\alpha,x}_n(t,s)$ converges to $\Phi^{\alpha,x}(t,s)$ weakly in $L^2(U\times\Omega)$.
Thus, in order to show the representation \varepsilonqref{eqflow11}, it suffices to show that $\Phi^{\alpha,x}_n(t,s)$ converges to
$$
\Gamma^{\alpha,x}(t,s):=e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}e^{\int_s^{t}b'_2\left(u,X^{\alpha,x}(u),\alpha(u)\right) \mathrm{d}u}
$$
weakly in $L^2(U\times\Omega)$.
Since the set
$$
\Big\{h\otimes \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big):\varphi\in C^{1}_b(\mathbb{R}),h\in C^\infty_0(U)\Big\}
$$
spans a dense subspace in $L^2(U\times\Omega)$, it is therefore enough to show that
$$
\int_{\mathbb{R}}h(x)E\Big[\Phi^{\alpha,x}_n(t,s) \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big)\Big]\mathrm{d}x\rightarrow \int_{\mathbb{R}}h(x)E\Big[\Gamma^{\alpha,x}(t,s) \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big)\Big]\mathrm{d}x.
$$
Recall that for $\varphi\in C^1_b([0,T],\mathbb{R}^d)$, for every $n$, the process $\tilde X^{\tilde \alpha,x}_n:=X^{\tilde \alpha,x}_n(\omega+\varphi)$, with $\tilde\alpha(\omega)=\alpha(\omega+\varphi)$ satisfies the SDE
\begin{align}\label{eqxntilde1}
\,\mathrm{d}ns \tilde X^{\tilde \alpha,x}_n(t)=(b_{1,n}(t,\tilde X^{\tilde \alpha,x}_n(t))+ b_{2}(t,\tilde X^{\tilde \alpha,x}_n(t),\tilde \alpha)+\sigma \dot\varphi)\,\mathrm{d}ns t+\sigma \,\mathrm{d}ns B(t).
\varepsilonnd{align}
We have by using Cameron-Martin theorem, the fact that $|e^x-e^y|\leq |x-y||e^x+e^y|$, H\"older inequality and boundedness of $b_2^\prime$ that
\begin{align*}
&\Big|\int_{\mathbb{R}}h(x)\mathbb{E}\Big[\Phi^{\alpha,x}_n(t,s) \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big)\Big]\mathrm{d}x - \int_{\mathbb{R}}h(x)\mathbb{E}\Big[\Gamma^{\alpha,x}(t,s) \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big)\Big]\mathrm{d}x\Big|\notag\\
=&\Big|\int_{\mathbb{R}}h(x)\mathbb{E}\Big[
e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{X^{\alpha,x}_n}(\mathrm{d}u,\mathrm{d}z)}e^{\int_s^{t}b'_2\left(u,X^{\alpha,x}_n(u),\alpha(u)\right)\,\mathrm{d}ns u} \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big)\Big]\mathrm{d}x\\
&-\int_{\mathbb{R}}h(x)\mathbb{E}\Big[e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{X^{\alpha,x}}(\mathrm{d}u,\mathrm{d}z)}e^{\int_s^{t}b'_2\left(u,X^{\alpha,x}(u),\alpha(u)\right)\,\mathrm{d}ns u} \mathcal{E}\Big(\int_0^1\dot{\varphi}(u)\mathrm{d}B(u)\Big)\Big]\mathrm{d}x\Big|\notag\\
=&\Big|\int_{\mathbb{R}}h(x)\mathbb{E}\Big[
e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}e^{\int_s^{t}b'_2\left(u,\hat X_n^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u} \Big]\mathrm{d}x\\
&-\int_{\mathbb{R}}h(x)\mathbb{E}\Big[e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\tilde X^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}e^{\int_s^{t}b'_2\left(u,\tilde X^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u} \Big]\mathrm{d}x\Big|\\
=&\Big|\int_{\mathbb{R}}h(x)\mathbb{E}\Big[
e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big(e^{\int_s^{t}b'_2\left(u,\tilde X_n^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u}-e^{\int_s^{t}b'_2\left(u,\tilde X^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u}\Big) \Big]\mathrm{d}x\\
&+\int_{\mathbb{R}}h(x)\mathbb{E}\Big[\Big(e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}-e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\tilde X^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big)e^{\int_s^{t}b'_2\left(u,\tilde X^{\tilde\alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u} \Big]\mathrm{d}x\Big|\\
\leq &\int_{\mathbb{R}}|h(x)|\mathbb{E}\Big[
e^{2\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{2}}\mathbb{E}\Big|e^{\int_s^{t}b'_2\left(u,\tilde X_n^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u}-e^{\int_s^{t}b'_2\left(u,\tilde X^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u}\Big|^2 \Big]^{\varphirac{1}{2}}\mathrm{d}x\\
&+C\int_{\mathbb{R}}|h(x)|\mathbb{E}\Big[\Big|e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}-e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\tilde X^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big|^2 \Big]^{\varphirac{1}{2}}\mathbb{E}\Big[e^{2\int_s^{t}b'_2\left(u,\tilde X^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)\,\mathrm{d}ns u} \Big]^{\varphirac{1}{2}}\mathrm{d}x\\
\leq &C\int_{\mathbb{R}}|h(x)|\Big\{\mathbb{E}\Big[
e^{2\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big]^{\varphirac{1}{2}}\int_s^{t}\mathbb{E}\Big[\Big|b'_2\left(u,\tilde X_n^{\tilde \alpha,x}(u),\tilde \alpha(u)\right)-b'_2\left(u,\tilde X^{\tilde \alpha,x}(u),\tilde \alpha(s)\right)\Big|^2 \Big]^{\varphirac{1}{4}}\,\mathrm{d}ns s\Big\}\,\mathrm{d}ns x\\
&+C\int_{\mathbb{R}}|h(x)|\mathbb{E}\Big[\Big|e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}-e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\tilde X^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}\Big|^2 \Big]^{\varphirac{1}{2}}
\mathrm{d}x,
\varepsilonnd{align*}
where the last inequality follows from the boundedness of $b_2$ and $b'_2$.
By Lemma \ref{lem:bound.int.local.time}, we have that $\mathbb{E}[ e^{2\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}]$ is bounded.
The second term on the right side of the above converges to zero since one can show as in Lemma \ref{lem:conv.Xnn} that $\tilde X^{n,\tilde \alpha,x}(s)$ converges strongly to $\tilde X^{\tilde \alpha,x}(s)$ in $L^2$ and $b_2^\prime$ is bounded and continuous.
We now show that the second term converges to zero.
We will show weak convergence and convergence in mean square.
Using the Cameron-Martin-Girsanov theorem as above, for every $\varphi_1 \in C^1_b([0,T],\mathbb{R}^d)$ we have
\begin{align}
&\Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\dot{\varphi_1}(v)\mathrm{d}B(v)\Big)\Big\{e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(v,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}v,\mathrm{d}z)}-e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\tilde X^{\tilde \alpha,x}}(\mathrm{d}v,\mathrm{d}z)}\Big\}\Big]\Big|\notag\\
=&\Big|\mathbb{E}\Big[e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(v,z\right)L^{\tilde {\tilde X}_n^{\tilde {\tilde \alpha},x}}(\mathrm{d}v,\mathrm{d}z)}-e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\tilde{\tilde X}^{\tilde {\tilde \alpha},x}}(\mathrm{d}v,\mathrm{d}z)}\Big]\Big|\notag\\
=&\Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u_n(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}\notag\\
&-\mathcal{E}\Big(\int_0^T\{u(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}\Big]\Big|.
\varepsilonnd{align}
Therefore, using the inequality $|e^x-e^y|\leq |x-y||e^x+e^y|$ and H\"older's inequality we have
\begin{align}
\leq & \Big|\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u_n(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)\notag\\
&\times\Big|\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)-\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)\Big|\notag\\
&\times \Big(e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}+e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}\Big)\Big]\Big|\notag\\
&+\Big|E\Big[e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}\notag\\
&\times \Big\{\mathcal{E}\Big(\int_0^T\{u_n(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)\notag\\
&-\mathcal{E}\Big(\int_0^T\{u(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)\Big\}\Big]\Big|\notag\\
\leq & 4\mathbb{E}\Big[\mathcal{E}\Big(\int_0^T\{u_n(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)^4\Big]^{\varphirac{1}{4}}\notag\\
&\times \mathbb{E}\Big[\Big|\int_s^{t}\int_{\mathbb{R}}\Big(b_{1,n}\left(v,z\right)-b_1\left(v,z\right)\Big)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)\Big|^2\Big]^{\varphirac{1}{2}} \notag\\
&\times \mathbb{E}\Big[e^{4\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}+e^{4\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}\Big]^{\varphirac{1}{4}}\notag\\
&+\mathbb{E}\Big[e^{2\int_s^{t}\int_{\mathbb{R}}b_1\left(v,z\right)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}v,\mathrm{d}z)}\Big]^{\varphirac{1}{2}}\notag\\
&\times\mathbb{E}\Big[\Big\{\mathcal{E}\Big(\int_0^T\{u_n(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)\notag\\
&-\mathcal{E}\Big(\int_0^T\{u
(v,x+\sigma\cdot B(v),\alpha (v,\omega+\varphi+\varphi_1))+\sigma\cdot(\dot{\varphi}(v)+\dot{\varphi_1}(v))\}\mathrm{d}B(v)\Big)\Big\}^2\Big]^{\varphirac{1}{2}}\notag\\
=&J_{1,n}^{\varphirac{1}{4}}\times J_{2,n}^{\varphirac{1}{2}}\times J_{3,n}^{\varphirac{1}{4}}+J_{4,n}^{\varphirac{1}{2}}\times J_{5,n}^{\varphirac{1}{2}}.
\varepsilonnd{align}
Lemma \ref{Lemmbound1}, shows that $J_{2,n}$ converges to zero, and convergence to zero of $J_{5,n}$ follows by dominated convergence.
Thanks to Lemma \ref{lemmaexpoloc} and boundedness of $b_{1,n}$ and $b_1$, respectively, the term $J_{3,n}$ (respectively $J_{4,n}$) is bounded. The bound of $J_{1,n}$ follows by the uniform boundedness of $u_n$.
Set $A_n^{\alpha}(t)=e^{\int_s^{t}\int_{\mathbb{R}}b_{1,n}\left(u,z\right)L^{\tilde X_n^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}$ and $A^{\alpha}(t)=e^{\int_s^{t}\int_{\mathbb{R}}b_1\left(u,z\right)L^{\tilde X^{\tilde \alpha,x}}(\mathrm{d}u,\mathrm{d}z)}$.
It remains to show convergence of the second moment, i.e. that $\mathbb{E}[|A_n^{\alpha}(t)|^2]$ converges to $\mathbb{E}[|A^{\alpha}(t)|^2]$ in $\mathbb{R}$. This follows as in the proof of Lemma \ref{lem:conv.y.phi}.
The desired result follows.
\varepsilonnd{proof}
We know from \cite[Theorem 2.1]{Ein2006} that the local time-space integral of $f \in {\mathcal H}^0$ admits the decomposition
\begin{align}\label{eqslocalt1}
&\int_0^t\int_{\mathbb{R}}f(s,z) L^{B_a^x}(\,\mathrm{d}ns s,\,\mathrm{d}ns z)\notag\\
=&a\int_0^t f (s,B_a^{x}(s))\,\mathrm{d}ns B(s)+a\int_{T-t}^T f (T-s,\widehat{B}_a^{x}(s))\,\mathrm{d}ns W(s)-a\int_{T-t}^T f (T-s,\widehat{B}_a^x(s))\varphirac{\widehat{B}(s)}{T-s}\,\mathrm{d}ns s,
\varepsilonnd{align}
$0\leq t\leq T$, a.s., where $\widehat{B}$ is the time-reversed Brownian motion, that is
\begin{align}\label{eqstimrevbm1}
\widehat{B}(t):=B(T-t),\,\,0\leq t\leq T.
\varepsilonnd{align}
In addition, the process $W=\{W(t),\,\,\,0\leq t\leq T\}$ is an independent Brownian motion with respect to the filtration $\mathcal{F}_t^{\widehat{B}}$ generated by $\widehat{B}_t$, and satisfies:
\begin{align}\label{eqstimrevbm2}
W(t)= \widehat{B} (t)-B(T)+\int_t^T\varphirac{\widehat{B}(s)}{T-s}\,\mathrm{d}ns s.
\varepsilonnd{align}
\begin{lemm}\label{Lemmbound1}
Let $\varphi\in C^1_b([0,T],\mathbb{R}^d)$ and define $F_{1,n}$ and $F_{2,n}$ by
\begin{align}
F_{1,n}:=&\int_s^{t}\int_{\mathbb{R}}\Big(b_{1,n}(u,z)-b_1(u,z)\Big)L^{\|\sigma\|B^x_\sigma}(\mathrm{d}u,\mathrm{d}z),\label{eqF1n}
\varepsilonnd{align}
Then $\mathbb{E}[|F_{1,n}|^2]$ converges to zero as $n$ goes to $\infty$.
\varepsilonnd{lemm}
\begin{proof}
Using the local time-space decomposition \varepsilonqref{eqslocalt1}, the Minkowski integral inequality with the measure $\nu(\sigma)=\int_{\sigma}\varphirac{\,\mathrm{d}ns s}{2\sqrt{T-s}}$, the H\"older and the Burkholder-Davis-Gundy inequalities, we get
\begin{align*}
\mathbb{E}[|F_{1,n}|^2]
\leq &4\|\sigma\|^2\mathbb{E}\Big[\Big\{\int_t^s \Big(b_{1,n} (u,B^{x}_\sigma(u))-b_{1} (u,B_\sigma^{x}(u))\Big)\,\mathrm{d}ns B(s)\Big\}^2\Big]\\
& +4\mathbb{E}\Big[\Big\{\int_{T-t}^{T-s} \Big(b_{1,n}(T-u,\widehat{B}_\sigma^{x}(u))-b_{1}(T-u,\widehat{B}_\sigma^{x}(u))\Big)\,\mathrm{d}ns W(u)\Big\}^2\Big]\\
&+4\mathbb{E}\Big[\Big\{\int_{T-t}^{T-s} \Big(b_{1,n}(T-u,\widehat{B}_\sigma^x(u))-b_{1}(T-u,\widehat{B}_\sigma^x(u))\Big)\varphirac{\widehat{B}(u)}{\sqrt{T-u}}\varphirac{\,\mathrm{d}ns u}{\sqrt{T-u}}\Big\}^2\Big]\\
\leq &C_\sigma\Big\{ \int_t^s \mathbb{E}\Big[\big| b_{1,n} (u,B_\sigma^{x}(u))-b_{1} (u,B_\sigma^{x}(u))\big|^2\Big]\,\mathrm{d}ns u\\
& +\int_{T-t}^{T-s} \mathbb{E}\Big[\big|b_{1,n}(T-u,\widehat{B}^{x}_\sigma(u))-b_{1}(T-u,\widehat{B}_\sigma^{x}(u))\big|^2\Big]\,\mathrm{d}ns u\\
&+\Big(\int_{T-t}^{T-s} \mathbb{E}\Big[\Big(b_{1,n}(T-u,\widehat{B}_\sigma^x(u))-b_{1}(T-u,\widehat{B}_\sigma^x(u))\Big)^2\Big(\varphirac{\widehat{B}(u)}{\sqrt{T-u}}\Big)^2\Big]^{\varphirac{1}{2}}\varphirac{\,\mathrm{d}ns s}{\sqrt{T-u}}\Big)^2\Big\}.
\varepsilonnd{align*}
Now using the Cauchy-Schwartz inequality and the fact that $E[B^4(t)]=3t^2$, we can continue the estimation as
\begin{align*}
\mathbb{E}[|F_{1,n}|^2] \leq &C_\sigma\Big\{ \int_t^s \mathbb{E}\Big[\big|b_{1,n} (u,B_\sigma^{x}(u))-b_{1} (u,B_\sigma^{x}(u))\big|^2\Big]\,\mathrm{d}ns u\\
& +\int_{T-t}^{T-s} \mathbb{E}\Big[\big|b_{1,n}(T-u,\widehat{B}^{x}_\sigma(u))-b_{1}(T-u,\widehat{B}_\sigma^{x}(u))\big|^2\Big]\,\mathrm{d}ns u\\
&+\Big(\int_{T-t}^{T-s} \mathbb{E}\Big[\big|b_{1,n}(T-u,\widehat{B}_\sigma^x(u))-b_{1}(T-u,\widehat{B}_\sigma^x(u))\big|^4\Big]^{\varphirac{1}{4}}\varphirac{\,\mathrm{d}ns s}{\sqrt{T-u}}\Big)^2\Big\}.
\varepsilonnd{align*}
Each term above converges to zero.
We give the detail only for the first term.
The treatment of the two oder terms is analogous.
Given $p>1$, using the density of the Brownian motion, we have as in the proof of Lemma \ref{lem:conv.Xnn} (see \varepsilonqref{eq:estim.bnb})
\begin{align*}
\mathbb{E}\Big[\big|b_{1,n} (s,B^{x}(s))-b_{1} (s,B^{x}(s))\big|^p\Big]
\leq &\varphirac{1}{\sqrt{2\pi s}}e^{\varphirac{x^2}{2s}}\int_{\mathbb{R}}\big|b_{1,n} (s,y)-b_{1} (s,y)\big|^pe^{-\varphirac{y^2}{4s}}\,\mathrm{d}ns y.
\varepsilonnd{align*}
Since $b_{1,n}$ converges to $b_1$, it follows from the dominated convergence theorem that each term in the above inequality converge to zero.
\varepsilonnd{proof}
The following Lemma corresponds to \cite[Lemma A.2]{BMBPD17} and it gives the exponential bound of the local time-space integral of a bounded function
\begin{lemm}\label{lemmaexpoloc}
Let $b:[0,T]\times \mathbb{R} \rightarrow \mathbb{R}$ be a bounded and measurable function. Then for $t\in [0,T],\, \lambda \in \mathbb{R}$ and compact subset $K\subset \mathbb{R}$, we have
$$
\underset{x\in K}{\sup} \mathbb{E}\Big[\varepsilonxp\Big(\lambda \int_0^t\partial_xb(s,B^x)\,\mathrm{d}ns s\Big) \Big]=\underset{x\in K}{\sup} \mathbb{E}\Big[\varepsilonxp\Big(\lambda \int_0^t\int_{\mathbb{R}}b(s,y)L^{B^x}(\,\mathrm{d}ns s,\,\mathrm{d}ns y)\Big) \Big]<C(\|b\|_{\infty}),
$$
where $C$ is an increasing function and $L^{B^x}(\,\mathrm{d}ns s,\,\mathrm{d}ns y)$ denotes integration with respect to the local time of the Brownian motion $B^x$ in both time and space. In addition, if $b_n$ is an approximating sequence of $b$ such that the $b_n$ are uniformy bounded by $\|b\|_{\infty}$ then the above bound still hold true with the bound independent of $n$.
\varepsilonnd{lemm}
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\noindent Olivier Menoukeu-Pamen: University of Liverpool Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences,
L69 7ZL, United Kingdom and African Institute for Mathematical Sciences, Ghana. [email protected]\\
Financial support from the Alexander von
Humboldt Foundation, under the program financed by the German Federal Ministry of Education and Research
entitled German Research Chair No 01DG15010 is gratefully acknowledged.
\noindent Ludovic Tangpi: Department of Operations Research and Financial Engineering, Princeton University, Princeton, 08540,
NJ; USA. [email protected]\\
Financial suupport by NSF grant DMS-2005832 is gratefully acknowledged.
\varepsilonnd{document}
|
\begin{document}
\title{Adiabatic Theorem in the Case of Continuous Spectra}
\author{M. Maamache and Y. Saadi \\
\\
\textit{Laboratoire de Physique Quantique et Syst\`{e}mes Dynamiques,}\\
\textit{{Facult\'{e} des Sciences,Universit\'{e} Ferhat Abbas de S\'{e}tif},
S\'{e}tif 19000, Algeria}}
\date{}
\maketitle
\begin{abstract}
In this paper,we present a rigorous demonstration and discussion of the
quantum adiabatic theorem for systems having a non degenerate continuous
spectrum. A new strategy is initiated by defining a kind of gap, "\textit{a
virtual gap}", for the continuous spectrum through the notion of
eigendifferential (Weyl's packet) and using the differential projector
operator. Finally we obtain the validity condition of the adiabatic
approximation. \newline
PACS: 03.65.Ca, 03.65.Ta
\end{abstract}
The adiabatic theorem is one of the basic results in quantum theory \cite{1,
2}. It is concerned with quantum systems described by an explicitly, but
slowly, time-dependent Hamiltonian. There has been a sudden regain of
interest in the adiabatic theorem for itself among physicists when in 1984
M.\ V. Berry \cite{3}\ pointed out that if it was applied to Hamiltonians
satisfying $H\left( t_{1}\right) =H\left( t_{2}\right) $, it could generate
a phase factor having non trivial geometrical meaning. And more recently,
the adiabatic theorem has renewed its importance in the context of quantum
control \cite{4}, for example, concerning adiabatic passage between atomic
energy levels, as well as for adiabatic quantum computation \cite{5}.
There are several points of view for a discussion of the quantum adiabatic
theorem; each one offers interesting insight. As T. Kato \cite{2}\ has
pointed out, the contents of the adiabatic theorem embody two parts: first,
the existence of a virtual change of the system which may be called an
adiabatic transformation, and, second, the dynamical transformation of the
system goes over to an adiabatic transformation in the limit when the change
of the Hamiltonian is infinitely slow. The adiabatic theorem proof given by
M. Born and V. Fock \cite{1}, although very general, is still restricted by
the assumption of considering the purely discrete and non-degenerate
Hamiltonian's spectrum, except for accidental degeneracy caused by crossing.
These limitations are rather artificial from the physical point of view and
should be removed from Kato's derivation of the adiabatic theorem. Several
authors \cite{6}\ had formally extended the Kato's results on the
approximate validity of the adiabatic theorem when the time $T$, during
which the approximation takes place, is large but finite. G. Nenciu \cite{9}
demonstrated the adiabatic theorem for bounded Hamiltonians. Later, J. E.
Avron and A. Elgart \cite{10} showed that the adiabatic theorem holds for
unbounded Hamiltonians as well and applied it to deal with the quantum Hall
effect.
Let us simply recall here that the works following that of Born and Fock
\cite{1} by Kato \cite{2}, Garrido \cite{6}, Nenciu \cite{9} and J. E. Avron
et al. \cite{9} have led to a formulation of the adiabatic theorem under the
usual gap assumption $g_{nm}\left( t\right) =E_{n}\left( t\right)
-E_{m}\left( t\right) $, between level $n$ and $m$. One may then state that
a general validity condition for adiabatic behavior is well controlled as
follows: the larger is the quantity $\underset{0\leq t\leq T,\ m}{\min }
\left[ g_{nm}\left( t\right) \right] $ the smaller will be the transition
probability.
Despite the existence of extensive literature on rigorous proofs of
estimates needed to justify the adiabatic approximation \cite{2, 9, 10, 12},
doubts have been raised about its validity \cite{14} leading to confusion
about the precise condition needed to use it \cite{15}. In part, this is
because some papers emphasize different aspects, such as the asymptotic
expansion, the replacement of the requirement of non-degenerate ground state
by a spectral projection separated from the rest of the spectrum, dependence
of first order estimates on the spectral gap, and even extensions to systems
without a gap. Adiabatic theorem without gap conditions is know to be true
\cite{10}, however, in general, no estimates on the error terms are
available. J. E. Avron and A. Elgart have shown in ref.\cite{10}\ that the
adiabatic theorem holds provided the spectral projection is of finite rank
independently of any spectral considerations. A similar result was proven by
F. Bornemann \cite{18} for discrete Hamiltonian when the set of eigenvalues
crossings is of measure zero\ in time. The limitation of these approaches is
that, in general, no estimate can be made on the rate at which the adiabatic
regime is attained \cite{10}. The gap condition is generally associated to
spectral stabilities\textbf{.} Consequently, the situation where the gap
does not exist will led to spectral instabilities. Thus it is difficult to
establish smooth spectral projections which is a necessary condition for the
validity of the adiabatic theorem in the practical applications. In fact,
the generalized adiabatic theorem, according to J. E. Avron and A. Elgart's
approach \cite{10}, is much more appropriate for the systems without a gap
condition and which have a discrete origin.
In this letter, we present a straightforward, yet rigorous, proof of the
adiabatic theorem and adiabatic approximation for systems whose Hamiltonian
has a completely continuous spectrum supposed non-degenerated for reasons of
simplicity and which checks a certain number of conditions which will be
given later on.
In the case of continuous spectrum we cannot numerate eigenvalues and
eigenfunctions, they are characterised by the value of the physical quantity
in the corresponding state. Althoug the eigenfunctions $\varphi \left(
k;t\right) $ of the operators with continuous spectra cannot be normalised
in the usual manner as is done for the functions of discrete spectra, one
can construct with the $\varphi \left( k;t\right) $ new quantities -
theWeyl's\ \textit{eigendifferentials (wave packets)- }\cite{19} which
possess the properties of the eigenfunction of discrete spectrum. The
eigendifferentials are defined by the equation
\begin{equation}
\left\vert \delta \varphi \left( k;t\right) \right\rangle =\overset{k+\delta
k}{\underset{k}{\int }}\left\vert \varphi \left( k^{\prime };t\right)
\right\rangle dk^{\prime }. \label{eigendiff}
\end{equation}
They divide up the continuous spectrum of the eigenvalues into finite but
sufficiently small discrete regions of size $\delta k$ (see Fig.\ref
{Decomposition}).\FRAME{ftbphFUX}{0.8769in}{1.785in}{0pt}{\Qcb{\textbf{
Decomposition of the continuous spectrum. }}}{\Qlb{Decomposition}}{Figure}{
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "T";width 0.8769in;height 1.785in;depth
0pt;original-width 0.8441in;original-height 1.7478in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";tempfilename
'JZ5FR500.wmf';tempfile-properties "XPR";}}
The eigendifferential (\ref{eigendiff}) is a special wave packet which has
only a finite extension in space; hence, it vanishes at infinity and
therefore can be seen in analogy to bound states. Furthermore, because the $
\delta \varphi $ have finite spatial extension, they can be normalized. Then
in the limit $\delta k\rightarrow 0$,\ a meaningful normalization of the
function $\varphi $ themselves follows: the normalization on $\delta $
functions.
For $\delta k$, a small connected range of value of the parameter $k$ (this
corresponds to a group of "neighboring" states, see Fig.\ref{Decomposition}
), the operator
\begin{equation}
\delta P\left( k;t\right) =\underset{k}{\overset{k+\delta k}{\dint }}
\left\vert \varphi \left( k^{\prime };t\right) \right\rangle \left\langle
\varphi \left( k^{\prime };t\right) \right\vert dk^{\prime }
\label{projector}
\end{equation}
represents the projector (the differential projection operator \cite{20})
onto those states contained in the interval and characterized by the values
of the parameter $k$\ within the range of values $\delta k$. The action of $
\delta P\left( k;t\right) $ on a wavefunction $\left\vert \psi \left(
t\right) \right\rangle $ is defined by
\begin{equation}
\delta P\left( k;t\right) \left\vert \psi \left( t\right) \right\rangle =
\underset{k}{\overset{k+\delta k}{\dint }}C\left( k^{\prime };t\right)
\left\vert \varphi \left( k^{\prime };t\right) \right\rangle dk^{\prime }.
\end{equation}
The application of the differential projection operator $\delta P\left(
k;t\right) $ causes thus the projection of the wavefunction onto the domain
of states $\varphi \left( k;t\right) $ which is characterized by $k$\ values
within the $\delta k$ interval .Before proceeding further, we give the
statement of the adiabatic theorem.
Let us call $U_{T}\left( s\right) $ the evolution operator where $s$ is the
fictitious time and $T$ is the time interval during which the evolution of
the system takes place
\begin{equation}
i\hbar \frac{\partial }{\partial s}U_{T}\left( s\right) =TH\left( s\right)
U_{T}\left( s\right) , \label{OperEvolution}
\end{equation}
and the slowly time-dependent Hamiltonian $H\left( s\right) =\int E\left(
k,s\right) \left\vert \varphi \left( k,s\right) \right\rangle \left\langle
\varphi \left( k,s\right) \right\vert dk\ $, $0\leq s\leq 1$, has a purely
continuous spectrum $E\left( k,s\right) $.
If the following conditions are fulfilled
\begin{description}
\item[(i)] As it is mentioned earlier (see Fig.\ref{Decomposition}) the
continuous spectrum is divided into discrete regions of size $\delta k$, we
must define or create a gap of energy for the continuous spectrum, in other
words, the size $\delta k$ is chosen so that
\begin{equation}
E\left( k;s\right) -E\left( k^{\prime };s\right) >>\frac{1}{T}\ ,\ \forall
k^{\prime }\notin \left[ k,k+\delta k\right] . \label{GapWidth}
\end{equation}
\item[(ii)] We assume that the eigenvalues are piecewise differentiable in
the parameter $s$, and there is no level crossing throughout the transition
(see Fig.\ref{evolution}), in other words:
\begin{equation}
E\left( k^{\prime };s\right) \neq E\left( k^{\prime \prime };s\right) \ \
/s\in \left[ 0,1\right] ,k^{\prime }\in \left[ k,k+\delta k\right] ,\
k"\notin \left[ k,k+\delta k\right] . \label{Noncrossing}
\end{equation}
\end{description}
\FRAME{ftbpFU}{3.0779in}{2.0358in}{0pt}{\Qcb{Evolution of a range of energy
of width $\protect\delta k$ as a function of time. }}{\Qlb{evolution}}{Figure
}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "T";width 3.0779in;height 2.0358in;depth
0pt;original-width 3.0338in;original-height 1.9969in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";tempfilename
'JZ5FR501.wmf';tempfile-properties "XPR";}}
\begin{description}
\item[(iii)] The derivatives $\frac{\partial }{\partial s}\delta P\left(
k;s\right) $ and $\frac{\partial ^{2}}{\partial s^{2}}\delta P\left(
k;s\right) $ are well defined and continuous in the interval $0\leq s\leq 1$.
\end{description}
Under these conditions it is possible to prove the adiabatic theorem:
\begin{theorem}
If the quantum system with time-dependent Hamiltonian having a non
degenerate continuous spectrum is initially in an eigenstate $\left\vert
\varphi \left( k,0\right) \right\rangle $ of $H\left( 0\right) $ and if $
H\left( s\right) $ evolves slowly enough then the state of the system at any
time $s$ will remain in the interval $\left[ k,k+\delta k\right] $.
\end{theorem}
The adiabatic theorem can be formally written, at the first order, in terms
of the evolution operator as
\begin{equation}
\forall k:\underset{T\rightarrow \infty }{\ \lim }U\left( s\right) \delta
P\left( k;0\right) =\delta P\left( k;s\right) \underset{T\rightarrow \infty }
{\lim }U\left( s\right) +O\left( \frac{1}{T}\right) . \label{Theorem}
\end{equation}
Notice that if, initially, the system is in the state $\left\vert \varphi
\left( k,0\right) \right\rangle $ so that $H\left( 0\right) \left\vert
\varphi \left( k,0\right) \right\rangle =E\left( k,0\right) \left\vert
\varphi \left( k,0\right) \right\rangle $ and expanding an arbitrary state
vector on the basis of the instantaneous quasi-eigenfunction, then (\ref
{Theorem}) implies
\begin{equation}
\underset{T\rightarrow \infty }{\ \lim }U\left( s\right) \left\vert \varphi
\left( k,0\right) \right\rangle =\delta P\left( k;s\right) \underset{
T\rightarrow \infty }{\lim }U\left( s\right) \left\vert \varphi \left(
k;0\right) \right\rangle
\end{equation}
and in the limit $T\rightarrow \infty $ the state $U\left( s\right)
\left\vert \varphi \left( k;0\right) \right\rangle =\underset{k}{\overset{
k+\delta k}{\dint }}C\left( k^{\prime };s\right) \left\vert \varphi \left(
k^{\prime };s\right) \right\rangle dk^{\prime },$ belongs to the subspace
generated by the states $\left\vert \varphi \left( k;s\right) \right\rangle $
pertaining to the interval $\left[ k,k+\delta k\right] .$
\begin{proof}
The demonstration that we present follows the same approach developed in ref.
\cite{20} for the discrete case. To this effect, we introduce a unitary
operator $A\left( s\right) $ having the property
\begin{equation}
\delta P\left( k,s\right) =A\left( s\right) \delta P\left( k,0\right)
A^{+}\left( s\right) \qquad \forall k\in \Re . \label{Rotating}
\end{equation}
It is completely defined by the initial condition $A\left( 0\right) =I$ and
the differential equation
\begin{equation}
i\hbar \frac{\partial }{\partial s}A\left( s\right) =K\left( s\right)
A\left( s\right) . \label{EqGene}
\end{equation}
The operator $K\left( s\right) $ obeys the following commutation relation:
\begin{equation}
i\hbar \frac{\partial }{\partial s}\delta P\left( k,s\right) =\left[ K\left(
s\right) ,\delta P\left( k,s\right) \right] , \label{Commut}
\end{equation}
and is determined \ without ambiguity if we add the following supplementary
condition:
\begin{equation}
\left\langle \varphi \left( k;t\right) \left\vert K\left( t\right)
\right\vert \varphi \left( k^{\prime };t\right) \right\rangle =0,\ \forall
k^{\prime }\in \left[ k,k+\delta k\right] , \label{SuppCond}
\end{equation}
equation that yields the following expression
\begin{equation}
K\left( t\right) =i\hbar \int \left[ 1-\delta P\left( k;t\right) \right]
\left\vert \dot{\varphi}\left( k;t\right) \right\rangle \left\langle \varphi
\left( k;t\right) \right\vert dk. \label{Generator}
\end{equation}
The unitary transformation $A^{+}\left( s\right) $, applied to the operators
and the vectors of the Schr\"{o}dinger's picture, produces a new picture:
the rotating axis picture:
\begin{equation}
H^{\left( A\right) }\left( s\right) =A^{+}\left( s\right) H\left( s\right)
A\left( s\right) =\int E\left( k,s\right) \left\vert \varphi \left(
k,0\right) \right\rangle \left\langle \varphi \left( k,0\right) \right\vert
dk, \label{HamiltA}
\end{equation}
similarly $K^{\left( A\right) }\left( s\right) $ becomes
\begin{equation}
K^{\left( A\right) }\left( s\right) =A^{+}\left( s\right) K\left( s\right)
A\left( s\right) . \label{GeneratA}
\end{equation}
The evolution operator in this new "representation" is $U^{\left( A\right)
}\left( s\right) =A\left( s\right) U_{T}\left( s\right) .$It is defined by
\begin{equation}
i\hbar \frac{\partial }{\partial s}U^{\left( A\right) }\left( s\right) =
\left[ TH^{\left( A\right) }\left( s\right) -K^{\left( A\right) }\left(
s\right) \right] U^{\left( A\right) }\left( s\right) ,\text{ \ \ \ \ \ \ \ }
U^{\left( A\right) }\left( 0\right) =I. \label{OperEvolutionA}
\end{equation}
Since $H^{\left( A\right) }\left( s\right) $ and $K^{\left( A\right) }\left(
s\right) $ are $T$-independent, it is to be expected that in the $
T\rightarrow \infty $ limit the first term of the right hand side in (\ref
{OperEvolutionA}) dominates. We can approximately solve such an equation in
which we go over to a time-dependent reference frame following the axis
which diagonalize $H^{\left( A\right) }\left( s\right) $. We define $\Phi
_{T}\left( s\right) $ via
\begin{equation}
i\hbar \frac{\partial }{\partial s}\Phi _{T}\left( s\right) =TH^{\left(
A\right) }\left( s\right) \Phi _{T}\left( s\right) , \label{EquPhi}
\end{equation}
in which the solution may be written, with the initial condition $\Phi
_{T}\left( 0\right) =I$, as
\begin{equation}
\Phi _{T}\left( s\right) =\int \exp \left[ -\frac{iT\alpha \left( k,s\right)
}{\hbar }\right] \left\vert \varphi \left( k,0\right) \right\rangle
\left\langle \varphi \left( k,0\right) \right\vert dk, \label{Phi}
\end{equation}
where $\alpha \left( k,s\right) =\int_{0}^{s}E\left( k,s^{\prime }\right)
ds^{\prime }.$ If, as we will see immediately, $U^{\left( A\right) }\left(
s\right) $ tends toward $\Phi _{T}\left( s\right) $ for large $T$, we will
have approximately
\begin{equation}
U_{T}\left( s\right) \underset{T\rightarrow \infty }{\simeq }A\left(
s\right) \Phi _{T}\left( s\right) . \label{AdiaTheo}
\end{equation}
We go over to a second picture and we show that the remaining evolution
operator differs from the identity by terms $O\left( \frac{1}{T}\right) $.
Thus, we change to a last picture with operator$W\left( s\right) \equiv \Phi
_{T}^{+}\left( s\right) A^{+}\left( s\right) U_{T}\left( s\right) $, and $
\left( -\right) $ generator $\bar{K}$ $\left( s\right) =\Phi _{T}^{+}\left(
s\right) A^{+}\left( s\right) K\left( s\right) A\left( s\right) \Phi
_{T}\left( s\right) $:
\begin{equation}
i\hbar \frac{\partial }{\partial s}W\left( s\right) =\bar{K}\left( s\right)
W\left( s\right) ,\qquad W\left( 0\right) =I, \label{Lastpicture}
\end{equation}
equivalent to the integral equation
\begin{equation}
W\left( s\right) =I+\frac{i}{\hbar }\int_{0}^{s}\bar{K}\left( s^{\prime
}\right) W\left( s^{\prime }\right) ds^{\prime }. \label{IntegrEqu}
\end{equation}
Now, we prove that in the limit $T\rightarrow \infty $ , $W\left( s\right)
=I+O\left( \frac{1}{T}\right) .$We begin by considering the operator $
F\left( s\right) =\int_{0}^{s}\bar{K}\left( s^{\prime }\right) ds^{\prime }.$
Any operator (and in particular $F\left( s\right) $) admits the following
decomposition
\begin{eqnarray}
F\left( s\right) &=&\int \int F\left( k,k^{\prime },s\right) dkdk^{\prime }
\notag \\
&=&\int_{0}^{s}\int \int \left\langle \varphi \left( k,0\right) \left\vert
\bar{K}\left( s^{\prime }\right) \right\vert \varphi \left( k^{\prime
},0\right) \right\rangle \left\vert \varphi \left( k,0\right) \right\rangle
\left\langle \varphi \left( k^{\prime },0\right) \right\vert dkdk^{\prime
}ds^{\prime }. \label{Decompos}
\end{eqnarray}
Using (\ref{Phi}) we obtain:
\begin{equation}
F\left( k,k^{\prime },s\right) =\int_{0}^{s}\exp \left[ \frac{iT\left(
\alpha \left( k,s^{\prime }\right) -\alpha \left( k^{\prime },s^{\prime
}\right) \right) }{\hbar }\right] K^{\left( A\right) }\left( k,k^{\prime
}s^{\prime }\right) ds^{\prime }\qquad k^{\prime }\notin \left[ k,k+\delta k
\right] , \label{Fdecompos}
\end{equation}
an expression in which we have introduced the condition $k^{\prime }\notin
\left[ k,k+\delta k\right] $ because, from (\ref{SuppCond}), we deduce $
F\left( k,k^{\prime },s\right) =0$ for $k^{\prime }\in \left[ k,k+\delta k
\right] .$
Let $k^{\prime }\notin \left[ k,k+\delta k\right] $, since $K^{\left(
A\right) }\left( k,k^{\prime }s^{\prime }\right) $ is a continuous function
of $s$, our assumption implies that $\alpha \left( k,s^{\prime }\right)
-\alpha \left( k^{\prime },s^{\prime }\right) $ is a continuous nonvanishing
monotonic function of $s$; after integrating (\ref{Fdecompos}) by parts we
obtain
\begin{eqnarray}
F\left( k,k^{\prime },s\right) &=&\frac{\hbar }{iT}\left[ \left. \exp \left[
\frac{iT\left( \alpha \left( k,s^{\prime }\right) -\alpha \left( k^{\prime
},s^{\prime }\right) \right) }{\hbar }\right] \frac{K^{\left( A\right)
}\left( k,k^{\prime }s^{\prime }\right) }{E\left( k,s^{\prime }\right)
-E\left( k^{\prime },s^{\prime }\right) }\right\vert _{0}^{s}\right. -
\notag \\
&&-\left. \int_{0}^{s}\exp \left[ \frac{iT\left( \alpha \left( k,s^{\prime
}\right) -\alpha \left( k^{\prime },s^{\prime }\right) \right) }{\hbar }
\right] \frac{\frac{\partial }{\partial s^{\prime }}K^{\left( A\right)
}\left( k,k^{\prime }s^{\prime }\right) }{E\left( k,s^{\prime }\right)
-E\left( k^{\prime },s^{\prime }\right) }ds^{\prime }\right] ,
\label{PartIntegr}
\end{eqnarray}
hence, according to the condition (\ref{GapWidth}),$\ F\left( k,k^{\prime
},s\right) $, $k^{\prime }\notin \left[ k,k+\delta k\right] $,
asymptotically converges toward $0$ as $\frac{1}{T}$. Summarizing, as $
T\rightarrow \infty $ we have:
\begin{equation}
F\left( s\right) =O\left( \frac{1}{T}\right) . \label{Order}
\end{equation}
Using (\ref{Lastpicture}), integration by parts turns (\ref{IntegrEqu}) into:
\begin{equation}
W\left( s\right) =I+\frac{i}{\hbar }F\left( s\right) W\left( s\right) +\frac{
1}{\hbar ^{2}}\int_{0}^{s}F\left( s^{\prime }\right) \bar{K}\left( s\right)
W\left( s^{\prime }\right) ds^{\prime },
\end{equation}
since the last two terms in this equation contain the factor $F\left(
s\right) $, then for $T\rightarrow \infty $ and from $U_{T}\left( s\right)
=A\left( s\right) \Phi _{T}\left( s\right) W\left( s\right) $ we obtain
\begin{equation}
U_{T}\left( s\right) \simeq A\left( s\right) \Phi _{T}\left( s\right) \left[
I+O\left( \frac{1}{T}\right) \right] \label{AdiabaticTheo}
\end{equation}
Finally (\ref{Phi}) implies $\Phi _{T}\left( s\right) \delta P\left(
k,0\right) =\delta P\left( k,0\right) \Phi _{T}\left( s\right) $ and hence $
A\left( s\right) \Phi _{T}\left( s\right) \delta P\left( k,0\right) =A\left(
s\right) \delta P\left( k,0\right) \Phi _{T}\left( s\right) =\delta P\left(
k,s\right) A\left( s\right) \Phi _{T}\left( s\right) $. This concludes the
proof of the adiabatic theorem (\ref{Theorem}).
\end{proof}
If $T$ is sufficiently large, we can, \ in first approximation, replace $
U\left( t_{1},t_{0}\right) $ by its asymptotic form:
\begin{equation}
U\left( t_{1},t_{0}\right) =U_{T}\left( 1\right) \simeq A\left( 1\right)
\Phi _{T}\left( 1\right) .
\end{equation}
This is called the adiabatic approximation. If the initial normalized state
is $\left\vert \varphi \left( k_{0},0\right) \right\rangle $, under this
approximation $U\left( t_{1},t_{0}\right) $ $\left\vert \varphi \left(
k_{0},0\right) \right\rangle \approx A\left( 1\right) \Phi _{T}\left(
1\right) \left\vert \varphi \left( k_{0},0\right) \right\rangle $. To
determine the validity of the adiabatic approximation for a given process,
we can estimate the error by computing the probability $\eta $ of finding
the system at time $t_{1}$ in a state different from $A\left( 1\right) \Phi
_{T}\left( 1\right) \left\vert \varphi \left( k_{0},0\right) \right\rangle $:
\begin{equation}
\eta =\left\langle \varphi \left( k_{0},0\right) \left\vert U^{+}\left(
t_{1},t_{0}\right) A\left( 1\right) \Phi _{T}\left( 1\right)
Q_{0}A^{+}\left( 1\right) \Phi _{T}^{+}\left( 1\right) U\left(
t_{1},t_{0}\right) \right\vert \varphi \left( k_{0},0\right) \right\rangle ,
\label{Probabilty}
\end{equation}
where $Q_{0}=I-\delta P\left( k_{0},0\right) $. This quantity may be
rewritten as
\begin{equation}
\eta =\left\langle \varphi \left( k_{0},0\right) \left\vert W^{+}\left(
1\right) Q_{0}W\left( 1\right) \right\vert \varphi \left( k_{0},0\right)
\right\rangle . \label{Probabilty1}
\end{equation}
Solving (\ref{IntegrEqu}) iteratively and keeping only the first order term,
we find
\begin{eqnarray}
\eta &\approx &\frac{1}{\hbar ^{2}}\left\langle \varphi \left(
k_{0},0\right) \left\vert F^{+}\left( 1\right) Q_{0}F\left( 1\right)
\right\vert \varphi \left( k_{0},0\right) \right\rangle \notag \\
&=&\frac{1}{\hbar ^{2}}\int_{k\notin \left[ k_{0},k_{0}+\delta k_{0}\right]
}\left\vert \left\langle \varphi \left( k_{0},0\right) \left\vert F\left(
1\right) \right\vert \varphi \left( k,0\right) \right\rangle \right\vert
^{2}dk. \label{Probabilty2}
\end{eqnarray}
Now, let us define a normalized time through the variable transformation $
t=t_{0}+sT$ $\left( 0\leq s\leq 1\right) $, and the initial normalized state
$\left\vert \varphi \left( k_{0},t_{0}\right) \right\rangle $ of $H\left(
t_{0}\right) $ with the eigenvalue $E\left( k_{0},t_{0}\right) $. Then,
using (\ref{Generator}), (\ref{Fdecompos}) and performing the change $
s\rightarrow t$ in eq. (\ref{Probabilty2}) yields
\begin{eqnarray}
\left\langle \varphi \left( k_{0},t_{0}\right) \left\vert F\left(
t_{1}\right) \right\vert \varphi \left( k,t_{0}\right) \right\rangle
&=&i\hbar \int_{t_{0}}^{t_{1}}\exp \left\{ \frac{i}{\hbar }\int_{t_{0}}^{t}
\left[ E\left( k_{0},t^{\prime }\right) -E\left( k,t^{\prime }\right) \right]
dt^{\prime }\right\} \notag \\
&&\left[ \left\langle \varphi \left( k_{0},t\right) |\dot{\varphi}\left(
k,t\right) \right\rangle -\left\langle \varphi \left( k_{0},t\right)
\left\vert \delta P\left( k_{0},t\right) \right\vert \dot{\varphi}\left(
k,t\right) \right\rangle \right] dt.
\end{eqnarray}
Since $\left\langle \varphi \left( k_{0},t\right) \left\vert \delta P\left(
k_{0},t\right) \right\vert \dot{\varphi}\left( k,t\right) \right\rangle =0$
for $k\notin \left[ k_{0},k_{0}+\delta k_{0}\right] $ equation (\ref
{Probabilty2}) may be recast as
\begin{equation}
\eta \approx \frac{1}{\hbar ^{2}}\int_{k\notin \left[ k_{0},k_{0}+\delta
k_{0}\right] }\left\vert i\hbar \int_{t_{0}}^{t_{1}}\exp \left\{ \frac{i}{
\hbar }\int_{t_{0}}^{t}\left[ E\left( k_{0},t^{\prime }\right) -E\left(
k,t^{\prime }\right) \right] dt^{\prime }\right\} \left\langle \varphi
\left( k_{0},t\right) |\dot{\varphi}\left( k,t\right) \right\rangle
dt\right\vert ^{2}dk,
\end{equation}
and the adiabatic approximation for $\left\vert \varphi \left(
k_{0},t_{0}\right) \right\rangle $ holds only if $\eta \ll 1$ which requires
\begin{equation}
\delta \wp _{\left( k_{0}\rightarrow k,t\right) }=\left\vert i\hbar
\int_{t_{0}}^{t_{1}}\exp \left\{ \frac{i}{\hbar }\int_{t_{0}}^{t}\left[
E\left( k_{0},t^{\prime }\right) -E\left( k,t^{\prime }\right) \right]
dt^{\prime }\right\} \left\langle \varphi \left( k_{0},t\right) |\dot{\varphi
}\left( k,t\right) \right\rangle dt\right\vert ^{2}\ll 1,\quad \forall
k\notin \left[ k_{0},k_{0}+\delta k_{0}\right] , \label{Transition}
\end{equation}
the integral of eq. (\ref{Transition}) will be sufficiently small, if the
phase of the integrated function vibrates fast enough and the amplitude of
the integrated function is small enough, thus $\delta \wp _{\left(
k_{0}\rightarrow k,t\right) }$ is at the maximum
\begin{equation}
\delta \wp _{\left( k_{0}\rightarrow k,t\right) }\approx \underset{t\in
\left[ t_{0},t_{1}\right] }{\max }\left\vert \frac{\hbar \left\langle
\varphi \left( k_{0},t\right) |\dot{\varphi}\left( k,t\right) \right\rangle
}{E\left( k_{0},t^{\prime }\right) -E\left( k,t^{\prime }\right) }
\right\vert ^{2},\quad \forall k\notin \left[ k_{0},k_{0}+\delta k_{0}\right]
. \label{max1}
\end{equation}
The condition $\eta \ll 1$ is therefore, in most cases, certainly satisfied
if $\underset{k\notin \left[ k_{0},k_{0}+\delta k_{0}\right] }{\max }\delta
\wp _{\left( k_{0}\rightarrow k,t\right) }\ll 1$ or equivalently,
\begin{equation}
\underset{k\notin \left[ k_{0},k_{0}+\delta k_{0}\right] }{\max }\left\vert
\left\langle \varphi \left( k_{0},t\right) |\dot{\varphi}\left( k,t\right)
\right\rangle \right\vert \ll \underset{k\notin \left[ k_{0},k_{0}+\delta
k_{0}\right] }{\min }\left\vert E\left( k_{0},t^{\prime }\right) -E\left(
k,t^{\prime }\right) \right\vert ,\quad \forall t\in \left[ t_{0},t_{1}
\right] , \label{AdiaApproximation}
\end{equation}
with $\max $ and $\min $ taken over all $k\notin \left[ k_{0},k_{0}+\delta
k_{0}\right] $. Condition (\ref{AdiaApproximation}) may be taken as a
criterion for the validity of the adiabatic approximation in the case of a
continuous spectrum. This estimate of the adiabatic approximation could not
be made in the Avron-Elgart's approach \cite{10} as mentioned earlier.
\end{document}
|
\begin{document}
\date{\today}
\author{Christian~Kraglund~Andersen}
\email{[email protected]}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Ants~Remm}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Stefania~Lazar}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Sebastian~Krinner}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Nathan~Lacroix}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Graham~J.~Norris}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Mihai~Gabureac}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Christopher~Eichler}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\author{Andreas~Wallraff}
\affiliation{Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland}
\title{Repeated Quantum Error Detection in a Surface Code}
\date{\today}
\begin{abstract}
The realization of quantum error correction is an essential ingredient for reaching the full potential of fault-tolerant universal quantum computation. Using a range of different schemes, logical qubits can be redundantly encoded in a set of physical qubits. One such scalable approach is based on the surface code. Here we experimentally implement its smallest viable instance, capable of repeatedly detecting any single error using seven superconducting qubits, four data qubits and three ancilla qubits. Using high-fidelity ancilla-based stabilizer measurements we initialize the cardinal states of the encoded logical qubit with an average logical fidelity of 96.1\%. We then repeatedly check for errors using the stabilizer readout and observe that the logical quantum state is preserved with a lifetime and coherence time longer than those of any of the constituent qubits when no errors are detected. Our demonstration of error detection with its resulting enhancement of the conditioned logical qubit coherence times in a 7-qubit surface code is an important step indicating a promising route towards the realization of quantum error correction in the surface code.
\end{abstract}
\maketitle
\section*{Introduction}
The feasibility of quantum simulations and computations with more than 50 qubits has been demonstrated in recent experiments~\cite{Zhang2017k, Bernien2017, Arute2019}. Many near-term efforts in quantum computing are currently focused on the implementation of applications for noisy intermediate-scale quantum devices~\cite{Preskill2018}. However, to harness the full potential of quantum computers, fault tolerant quantum computing must be implemented. Quantum error correction and fault-tolerance have been explored experimentally in a variety of physical platforms such as nuclear magnetic resonance~\cite{Cory1998}, trapped ions~\cite{Chiaverini2004, Schindler2011, Lanyon2013, Linke2017a}, photonics~\cite{Yao2012a, Bell2014}, NV-centers~\cite{Cramer2016}, and superconducting circuits~\cite{Reed2012, Shankar2013a, Riste2015, Kelly2015, Corcoles2015, Ofek2016}. In particular, recent experiments have demonstrated quantum state stabilization~\cite{Riste2013, Negnevitsky2018, Andersen2019, Bultink2019}, simple error correction codes~\cite{Schindler2011, Riste2015, Kelly2015, Nigg2014b, Gong2019a} and the fault-tolerant encoding of logical quantum states~\cite{Linke2017a, Takita2017}. Quantum error correction with logical error rates comparable or below that of the physical constituents has also been achieved encoding quantum information in continuous variables using superconducting circuits~\cite{Ofek2016, Hu2019a, Campagne-Ibarcq2019}. These bosonic encoding schemes take advantage of high quality factor microwave cavities which are predominantly limited by photon loss. However, so far no repeated detection of both amplitude and phase errors on an encoded logical qubit has been realized in any qubit architecture. In this work, we present such a demonstration using the surface code~\cite{Fowler2012}, which, due to its high error-threshold, is one of the most promising architectures for large-scale fault-tolerant quantum computing.
\begin{figure*}
\caption{Seven qubit surface code. (a) The surface code consists of a two-dimensional array of qubits. Here the data qubits are shown in red an the ancilla qubits for measuring $X$-type ($Z$-type) stabilizers in blue (green). The smallest surface code consists of seven qubits indicated by the data qubits D1-D4 and the ancilla qubits A1-A3. (b) Gate sequence for quantum error detection using the seven qubit surface code. Details of the gate sequence are discussed in the main text.}
\label{fig:fig1}
\end{figure*}
In stabilizer codes for quantum error correction~\cite{Lidar2013, Terhal2015n}, a set of commuting multi-qubit operators is repeatedly measured, which projects the qubits onto a degenerate eigenspace of the stabilizers referred to as the code space. Thus, the experimental realization of quantum error detection crucially relies on high-fidelity entangling gates between the data qubits and the ancilla qubits and on the simultaneous high-fidelity single-shot readout of all ancilla qubits. For superconducting circuits, multiplexed readout has recently been implemented for high-fidelity simultaneous readout in multi-qubit architectures~\cite{Groen2013, Schmitt2014a, Jeffrey2014} with small crosstalk~\cite{Heinsoo2018}. Small readout crosstalk leading to minimal unwanted dephasing of data qubits when performing ancilla readout has been key enabler of recent experiments in superconducting circuits realizing repeated acilla-based parity detection~\cite{Andersen2019, Bultink2019}. Moreover, repeatable high-fidelity single- and two-qubit gates~\cite{Barends2014, Rol2019}, required for quantum error correction, have also been demonstrated for superconducting qubits.
Here, we utilize low-crosstalk multiplexed readout and a sequential stabilizer-measurement scheme~\cite{Versluis2017} for implementing a seven qubit surface code with superconducting circuits.
In the surface code, as in any stabilizer code, errors are detected by observing changes in the stabilizer measurement outcomes. Such syndromes are typically measured by entangling the stabilizer operators with the state of ancilla qubits, which are then projectively measured to yield the stabilizer outcomes.
The surface code consists of a $d\times d$ grid of data qubits with $d^2{-}1$ ancilla qubits, each connected to up to four data qubits~\cite{Fowler2012}. The code can detect $d-1$ errors and correct up to $\lfloor(d-1)/2\rfloor$ errors per cycle of stabilizer measurements. In particular, the stabilizers of the $d=2$ surface code, see Fig.~\ref{fig:fig1}, are given by
\begin{align}
X_{D1} X_{D2} X_{D3} X_{D4},\qquad Z_{D1} Z_{D3}, \qquad Z_{D2} Z_{D4}.
\end{align}
For the code-distance $d=2$, it is only possible to detect a single error per round of stabilizer measurements and once an error is detected, the error can not be unambiguously identified, e.g. one would obtain the same syndrome outcome for an $X$-error on D1 and on D3.
Here, we use the following logical qubit operators
\begin{align}
Z_L = Z_{D1} Z_{D2}, \quad\text{or}\quad Z_L = Z_{D3} Z_{D4}, \label{eq:zl}\\
X_L = X_{D1} X_{D3}, \quad\text{or}\quad X_L = X_{D2} X_{D4}, \label{eq:xl}
\end{align}
such that the code space in terms of the physical qubit states is spanned by the logical qubit states
\begin{align}
\ket{0}_L = \frac{1}{\sqrt{2}}(\ket{0000} + \ket{1111}), \\
\ket{1}_L = \frac{1}{\sqrt{2}}(\ket{0101} + \ket{1010}).
\end{align}
To encode quantum information in the logical subspace, we initialize the data qubits in a separable state, chosen such that after a single cycle of stabilizer measurements and conditioned on ancilla measurement outcomes being $\ket{0}$, the data qubits are encoded into the target logical qubit state.
In this work, we demonstrate this probabilistic preparation scheme for the logical states $\ket{0}_L$, $\ket{1}_L$, $\ket{+}_L = (\ket{0}_L + \ket{1}_L)/\sqrt{2}$ and $\ket{-}_L = (\ket{0}_L - \ket{1}_L)/\sqrt{2}$ and we perform repeated error detection on these states.
\begin{figure*}
\caption{Seven-qubit device. (a) False colored micrograph of the seven-qubit device used in this work. Transmon qubits are shown in yellow, coupling resonators in cyan, flux lines for single-qubit tuning and two-qubit gates in green, charge lines for single-qubit drive in pink, the two feedlines for readout in purple, transmission line resonators for readout in red and Purcell filters for each qubit in blue. (b) Enlarged view of the center qubit (A2) which connects to four neighboring qubits.}
\label{fig:chip}
\end{figure*}
\section*{Implementation}
The seven qubit surface code, as discussed above, can be realized with a set of qubits laid out as depicted in Fig.~\ref{fig:fig1}(a). The logical qubit is encoded into four data qubits, D1-D4, and three ancilla qubits, A1, A2 and A3 are used to measure the three stabilizers $Z_{D1} Z_{D3}$, $X_{D1} X_{D2} X_{D3} X_{D4}$ and $Z_{D2} Z_{D4}$, respectively. We initially herald all qubits in the $\ket{0}$-state~\cite{Johnson2012, Riste2012} and subsequently prepare the data qubits in a product state using single qubit rotations around the $y$-axis. These initial states are then projected onto the code space after the initial stabilizer measurement cycles.
We perform the $X_{D1} X_{D2} X_{D3} X_{D4}$ stabilizer measurement by first applying basis change pulses ($R_Y^{\pi/2}$) on the data qubits to map the $X$ basis to the $Z$ basis. Then we perform the entangling gates as in Fig.~\ref{fig:fig1}(b) and finally we revert the basis change. The measurement of A2 will therefore yield the $\ket{0}$-state ($\ket{1}$-state) corresponding to the eigenvalue $+1$ ($-1$) of the stabilizer $X_{D1} X_{D2} X_{D3} X_{D4}$. While the measurement pulse for A2 is still being applied, we perform the $Z_{D1} Z_{D3}$ and $Z_{D2} Z_{D4}$ stabilizer measurements simultaneously using the ancilla qubits A1 and A3, respectively. To avoid unwanted interactions during entangling gate operations, we operate the surface code using a pipelined approach similar to the scheme introduced by Versluis et.al.~\cite{Versluis2017}, for which we perform $X$-type and $Z$-type stabilizer measurements sequentially, see Fig.~\ref{fig:fig1}(b) and Appendix~\ref{app:pulse_seq}. The cycle is repeated after this step, and, after $N$ stabilizer measurement cycles, we perform state tomography of the data qubits.
The gate sequence described above is implemented on the seven qubit superconducting quantum device shown in Fig.~\ref{fig:chip}(a), see Appendix~\ref{app:device} for device parameters. Each qubit (yellow) is a single-island transmon qubit~\cite{Koch2009} and features an individual flux line (green) for frequency tuning and an individual charge line (pink) for single qubit gates. Additionally, each qubit is coupled to a readout resonator (red) combined with an individual Purcell filter (blue). The Purcell filters protect against qubit decay into the readout circuit~\cite{Reed2010} and suppress readout crosstalk such that multiplexed ancilla measurements can be performed without detrimental effects on the data qubits~\cite{Heinsoo2018}. Each Purcell filter is coupled to a feedline and we perform all measurements by probing each feedline with a frequency-multiplexed readout pulse~\cite{Heinsoo2018}, see Appendix~\ref{app:readout} for a complete characterization of the readout. The qubits are coupled to each other via 1.5~mm long coplanar waveguide segments (cyan) as displayed in Fig.~\ref{fig:fig1}(a). The seven qubit surface code requires the central ancilla qubit to connect to four neighbors. The qubit island shape, shown Fig.~\ref{fig:chip}(b), is designed to facilitate coupling to a readout resonator and four two-qubit couplers. To ensure a closed ground plane around the qubit island, each coupler element crosses the ground plane with an airbridge (white). We install the device in a cryogenic measurement setup~\cite{Krinner2019}, see Appendix~\ref{app:setup}, and we characterize and benchmark the device using time-domain and randomized benchmarking methods as detailed in Appendix~\ref{app:device}.
\section*{Results}
Changes in the outcome of repeated stabilizer measurements, also referred to as syndromes, signal the occurrence of an error. It is, thus, critical to directly verify the ability to measure the multi-qubit stabilizers using the ancilla readout~\cite{Takita2016}. We characterize the performance of the stabilizer measurements by preparing the data qubits in each of the computational basis states and measure the $Z$-stabilizers, see Fig.~\ref{fig:parity}. For each stabilizer, the other ancilla qubits and unused data qubits are left in the ground state.
We correctly assign the ancilla measurement outcome corresponding to the prepared basis state with success probabilities of $95.0\%$, $83.5\%$ and $91.8\%$ for the stabilizers $Z_{D1} Z_{D3} $, $Z_{D1} Z_{D2} Z_{D3} Z_{D4}$ and $Z_{D2} Z_{D4}$ calculated as the overlap between the measured probabilities and the ideal case (gray wireframe in Fig.~\ref{fig:parity}). Master equation simulations, which include decoherence and readout errors, are shown by the red wireframes in Fig.~\ref{fig:parity}. The parity measurements are mainly limited by the relaxation of the data qubits, which directly leads to worse results for states with multiple excitations such as the $\ket{1111}$-state when measuring $Z_{D1} Z_{D2} Z_{D3} Z_{D4}$. Further variations in the correct parity assignment probability arise due to the differences in qubit lifetimes and two-qubit gate durations (see Appendix~\ref{app:device}).
\begin{figure}
\caption{Stabilizer measurements of the data qubits. In (a) we show the outcomes of the measurement of $Z_{D1}
\label{fig:parity}
\end{figure}
In a next step, we prepare logical states by projecting the data qubits onto the desired code space. We use a probabilistic encoding scheme, where we initialize the data qubits in a given product state and perform one cycle of stabilizer measurements. Then, in the events where all syndrome results are $\ket{0}$, the data qubits are projected onto the desired logical state. We can use this probabilistic scheme to prepare any logical state by initializing the state $\ket{0}(a\ket{0}+b\ket{1})\ket{0}(a\ket{0}+be^{i\phi}\ket{1})$, which will be projected onto the (unnormalized) logical state $\ket{\psi}_L = a^2 \ket{0}_L + b^2 e^{i\phi} \ket{1}_L$. Here, we specifically initialize the logical states $\ket{0}_L$, $\ket{1}_L$, $\ket{+}_L$ and $\ket{-}_L$ by performing one cycle of stabilizer measurements on the states $\ket{0000}$, $\ket{0101}$, $\ket{0{+}0{+}}$ and $\ket{0{+}0{-}}$, respectively, with $\ket{\pm} = (\ket{0} \pm \ket{1})/\sqrt{2}$.
\begin{figure}
\caption{Preparation of logical states. (a) Real and (b) imaginary part of the density matrix of the four physical data qubits prepared in the $\ket{0}
\label{fig:density}
\end{figure}
First, we consider the preparation of $\ket{0}_L$ for which the data qubit state $\ket{0000}$ after one cycle of stabilizer measurements is projected onto the state $\ket{\psi_{0}} = (\ket{0000} + \ket{1111})/\sqrt{2}$ when all ancilla qubits are measured in $\ket{0}$. We measure all ancilla qubits to be in the $\ket{0}$ state with a success-probability of 25.1\%, compared to an expected probability of 50\% in the ideal case. To verify the state preparation, we perform full state tomography of the four data qubits after the completion of one cycle of stabilizer measurements and construct the density matrix based on a maximum likelihood estimation taking readout errors into account. The measured density matrix of the physical data qubits has a fidelity of $F_{\mathrm{phys}}=\bra{\psi_{0}} \rho \ket{\psi_{0}} = 70.3\%$ to the target state, see Fig.~\ref{fig:density}(a-b). While the infidelity is dominated by qubit decoherence, we also observe small residual coherent phase errors as seen by the finite imaginary matrix elements in Fig.~\ref{fig:density}(b) corresponding to a phase error of 5 degrees accumulated over the cycle time of 1.92~$\mu$s or, equivalently, a frequency drift of 7~kHz for any qubit.
Given access to the full density matrix, we can project it onto the logical qubit subspace $\rho_{L,ji} = \bra{j}\rho\ket{i} / P_L$ for $\ket{i },\ket{j} \in \lbrace \ket{0}_L, \ket{1}_L \rbrace$. Here. $P_L = \sum_{i} \bra{i}\rho\ket{i}$ is the probability of the prepared state to be within the logical subspace, which is also referred to as~the acceptance probability~\cite{Takita2017} or yield~\cite{Linke2017a}. The state $\rho_L$ is the logical qubit state, conditioned on the prepared state residing in the code space at the end of the cycle. In general, the physical fidelity of the data qubits can be expressed in the form $F_{\mathrm{phys}} = F_L P_L$, where $F_L$ is the fidelity of $\rho_L$ compared to the ideal logical state. We experimentally find the probability $P_L = 0.717$ of the prepared state to be within the logical subspace. From simulations, we understand that the reduced $P_L$ mainly arises from decoherence during the stabilizer measurement cycle. After the projection onto the code space, the logical qubit state $\ket{0}_L$ is described by a single qubit density matrix, see Fig.~\ref{fig:density}(c), which has a fidelity of $F_L = 98.2\%$ to the ideal logical state. Similarly, we prepare the logical states $\ket{+}_L$, $\ket{-}_L$ and $\ket{1}_L$, shown in Fig.~\ref{fig:density}(d-f), with logical state fidelities of 94.2\%, 94.8\% and 97.3\%, respectively.
The corresponding logical fidelities of the four logical states from master equation simulations are 98.5\%, 96.6\%, 96.4\% and 98.1\%, see Appendix~\ref{app:sim}. The slightly lower fidelities for the $\ket{+}_L$ and $\ket{-}_L$ states arise from the pure dephasing of the data qubits making $Z$-errors during the encoding more likely than $X$-errors.
\begin{figure}
\caption{Repeated quantum error detection. The expectation values of (a) the logical $Z_L$ operator and (b) the logical $X_L$ operator as a function of $N$, the number of stabilizer measurement cycles. The expectations values are shown for the prepared $\ket{0}
\label{fig:logical}
\end{figure}
Next, we demonstrate repeated quantum error detection of any single error, which is a key ingredient of quantum error correction schemes such as the surface code. We do so by repeatedly measuring the expectation value of the encoded qubit's logical $Z_L$ ($X_L$) operator conditioned on having detected no error in any repetition of the stabilizer measurement and on having the final measurement of the data qubits satisfy $Z_{D1}Z_{D3}=Z_{D2}Z_{D4}=1$ ($X_{D1} X_{D2} X_{D3} X_{D4} = 1$). This latter condition ensures that the qubits have remained in the logical subspace during the last detection cycle. We find that the expectation value $\langle Z_L \rangle$ (green and blue data points) decays in good approximation exponentially from unity with a logical life time of $62.7\pm9.4$~$\mu$s from this exponential fit, which exceeds the life time, 16.8~$\mu$s, of the best physical qubit (dashed lines) of the device, see Fig.~\ref{fig:logical}(a). The logical expectation values are evaluated after the $N$th cycle at time $T=(1.92N+0.3)\,\mu$s shown at the top axis of Fig.~\ref{fig:logical}(a,b). The approximately exponential decay of the logical qubit expectation value $\langle X_L \rangle$ (brown and purple points) indicates a logical coherence time $72.5\pm32.9$~$\mu$s, also exceeding that of the best physical qubit, 21.5~$\mu$s, on the device (dashed lines), Fig.~\ref{fig:logical}(b). However, the fits to $\langle X_L \rangle$ show larger error bars due to the finite fidelity of preparing the logical $\ket{+}_L$ and $\ket{-}_L$ states, limited by the pure dephasing of the qubits as also seen in Fig.~\ref{fig:density}(d-e). Converting the measured decay times into an error per stabilizer measurement cycle, we find a logical $X_L$ error probability of $3.1\%\pm0.45\%$ and a logical $Z_L$ error probability of $2.6\pm 1.3\%$.
Generally, we find good agreement between the measured expectation values of the logical qubit operators and the ones calculated using numerical simulations, solid lines in Fig.~\ref{fig:logical}(a,b), accounting for finite physical qubit life- ($T_1$) and coherence times ($T_2$), residual-$ZZ$ coupling and readout errors, see Appendix~\ref{app:sim} for details. From the numerical simulations, we extract logical decay times of $44.2$~$\mu$s and $59.6$~$\mu$s for $Z_L$ and $X_L$ operators when no errors are detected, which are smaller than the experimentally obtained times, but within the experimental error bars. The simulated decay times correspond to a logical $X_L$ error probability of 4.2\% and a logical $Z_L$ error probability of 3.1\% per error detection cycle. We suspect that for the $\ket{+}_L$-state coherent errors from qubit frequency drifts during the data collection cause the deviations between data and simulations.
Finally, we discuss the probability to observe $k$ ancilla qubits simultaneously in the $\ket{1}$ state per error detection cycle when no errors were detected in previous cycles.
We find that the probability to observe no errors slowly increases with $N$ from about 40\% to 50\%, see Fig.~\ref{fig:logical}(d). From numerical simulations, we find that the probability to observe no additional errors after one cycle is between 49.9\% and 50.3\% per cycle, slightly larger than the experimentally observed values. We also observe experimentally that the probability of detecting more than a single ancilla qubit in the $\ket{1}$ state per cycle is approximately suppressed exponentially. Consistent with this analysis, we find that the measured probability of not detecting an error (blue data points) decreases exponentially with $N$, Fig.~\ref{fig:logical}(c). After $N=10$ cycles, the success probability, i.e.~the total probability that the state remained in the code space, approaches~$10^{-4}$, around a factor of 6 smaller than the simulated value. The difference between the simulated (dashed line) and experimentally determined success probabilities stems from the smaller simulated error probability per cycle discussed above.
\section*{Discussion}
In conclusion, we have implemented a seven qubit surface code for repeated quantum error detection. In particular, our experiment was enabled by fast and low-crosstalk readout for ancilla measurements. Using the seven qubit surface code, we demonstrated preparation of the logical states $\ket{0}_L$, $\ket{1}_L$, $\ket{+}_L$ and $\ket{-}_L$ with an average fidelity in the logical subspace of 96.1\%. The probability to be within the logical subspace was found to be around 70\% due to the accumulated errors during the stabilizer measurement cycle in good agreement with the corresponding numerical simulations. When executing the quantum error detection sequence for multiple cycles, we find an extended lifetime and coherence time of the logical qubit conditioned on detecting no errors. The data presented here is postselected on the ancilla measurement outcomes and on the condition that the final measurement of the data qubits satisfies the stabilizer conditions of the code. Crucially, since we found both extended logical life- and coherence time, we verified that neither the syndrome measurements nor the postselection extract information about the logical quantum state. The techniques used in this work for high-fidelity gates~\cite{Rol2019} and low-crosstalk qubit readout~\cite{Heinsoo2018} are directly applicable to a range of error correction codes~\cite{Bacon2006, Bombin2006, Chamberland2019, Li2019s} which also critically require repeated measurements of ancilla qubits with minimal detrimental effects on the data qubits.
Our implementation uses a gate sequence that is extensible to large surface codes~\cite{Versluis2017} and, thus, our work represents a key demonstration towards using superconducting quantum devices for fault-tolerant quantum computing.
\section*{Data availability statement}
The data produced in this work is available from the corresponding author upon reasonable request.
\section*{Acknowledgments}
The authors are grateful for valuable feedback from K.~Brown and A.~Darmawan. The authors acknowledge contributions to the measurement setup from S.~Storz, F.~Swiadek and T.~Zellweger.
The authors acknowledge financial support by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office grant W911NF-16-1-0071, by the National Centre of Competence in Research Quantum Science and Technology (NCCR QSIT), a research instrument of the Swiss National Science Foundation (SNSF), by the EU Flagship on Quantum Technology H2020-FETFLAG-2018-03 project 820363 OpenSuperQ, by the SNFS R'equip grant 206021-170731 and by ETH Zurich. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government.
\section*{Author Contributions}
C.K.A. designed the device and A.R., S.K., G.N. and M.G fabricated the device. C.K.A., A.R., S.L. and N.L. developed the experimental control software. C.K.A., A.R., S.K. and N.L. installed the experimental setup. C.K.A., A.R. and S.L. characterized and calibrated the device and the experimental setup. C.K.A. carried out the main experiment and analyzed the data. C.K.A. performed the numerical simulations. C.E. and A.W. supervised the work. C.K.A., A.R. and S.L. prepared the figures for the manuscript. C.K.A. wrote the manuscript with input from all co-authors.
\section*{Competing interests}
The authors declare no competing interests.
\begin{appendix}
\section*{Supplementary Information}
\section{Pulse sequence}
\label{app:pulse_seq}
\begin{figure*}
\caption{AWG waveforms for two cycles of the stabilizer measurement. Solid lines represent the microwave pulses for single qubit gates, dark solid pulses the readout pulses, and dashed lines zero-area flux pulses. The shaded areas indicate which two qubits interact during each flux pulse. }
\label{fig:ps}
\end{figure*}
We physically implement the gate sequence shown in Fig.~\ref{fig:fig1} by waveforms on the arbitrary waveform generators (AWGs) of the experimental setup, see Fig.~\ref{fig:ps} for an example with two cycles of stabilizer measurements. The pulse sequence includes dynamical decoupling pulses on the ancilla qubits during the stabilizer measurements and dynamical decoupling pulses on the data qubits in between each stabilizer cycle. All qubits are parked at their upper sweetspot which enable us to use the net-zero flux pulse shape as introduced by Rol et.al.~\cite{Rol2019}. The net-zero pulse is shaped such that the integral of the pulse is zero which serves to limit memory effects on the two-qubit gates e.g. due to charge accumulation in the flux lines. Beyond the flux pulses that enable the two-qubit gates (indicated with the shaded background), we apply additional flux pulses to non-interacting qubits. These additional flux pulses serves the purpose of pushing the frequency of the qubit down in frequency such that we avoid frequency collisions during the gate.
\begin{table*}[t]
\centering
\begin{tabular}{lccccccc}
\hline
\noalign{\vskip 1mm}
& D1 & D2 & D3 & D4 & A1 & A2 & A3\\
\hline
\hline
Qubit frequency, $\omega_q/2\pi$ (GHz) & 5.494 & 5.712 & 4.108 & 4.222 & 4.852 & 4.963 & 5.190 \\
Lifetime, $T_1$ ($\mu$s) & 11.2 & 8.7 & 8.7 & 16.3 & 5.7 & 16.8 & 11.8 \\
Ramsey decay time, $T_2^*$ ($\mu$s) & 18.2 & 14.4 & 4.3 & 21.5 & 8.5 & 16.7 & 9.9 \\
Readout frequency, $\omega_r/2\pi$ (GHz) & 6.611 & 6.838 & 5.832 & 6.063 & 6.255 & 6.042 & 6.299\\
Readout linewidth, $\kappa_{\mathrm{eff}}/2\pi$ (MHz) & 7.5 & 10.6 & 6.0 & 7.2 & 17.3 & 10.9 & 11.0 \\
Purcell filter linewidth, $\kappa_{P}/2\pi$ (MHz) & 47.6 & 46.4 & 13.6 & 49.2 & 56.3 & 68.1 & 46.4 \\
Purcell-readout coupling, $J_{PR}/2\pi$ (MHz) & 20.0 & 22.2 & 17.5 & 18.4 & 18.8 & 18.7 & 19.0 \\
Purcell-readout detuning, $\Delta_{PR}/2\pi$ (MHz) & 33.8 & 25.7 & 19.4 & 32.3 & 11.3 & 25.6 & 20.6 \\
Dispersive shift, $\chi/2\pi$ (MHz) & -2.5 & -2.5 & -0.75 & -1.0 & -1.25 & -2.4 & -2.0 \\
Thermal population, $P_{\mathrm{th}}$ ($\%$) & 0.06 & 0.04 & 0.8 & 0.8 & 0.08 & 0.4 & 0.6 \\
Individual readout assignment prob. (\%) & 99.4 & 99.2 & 97.8 & 98.2 & 98.7 & 98.8 & 98.8 \\
Multiplexed readout assignment prob. (\%) & 98.9 & 99.1 & 98.2 & 97.4 & 97.7 & 98.4 & 98.6 \\
Measurement efficiency, $\eta$ & 0.30 & 0.24 & 0.15 & 0.15 & 0.20 & 0.27 & 0.22 \\
\hline
\end{tabular}
\caption{Measured parameters of the seven qubits.}
\label{tab:qb_params}
\end{table*}
\section{Device Fabrication and Characterization}
\label{app:device}
The device in Fig.~\ref{fig:chip} consists of seven qubits coupled to each other in the geometry showed in Fig.~\ref{fig:fig1}(a).
The resonator, coupling and qubit structures are defined using photolithography and reactive ion etching from a 150$\,$nm thin niobium film sputtered onto a high-resistivity intrinsic silicon substrate. To establish a well-connected ground plane, we add airbridges to the device. Airbridges are also used to cross signal lines, i.e., for the flux and charge lines to cross the feedlines. The aluminum-based Josephson junctions of the qubits are fabricated using electron beam lithography.
We extract the qubit parameters, see Table~\ref{tab:qb_params}, using standard spectroscopy and time domain methods. In addition to the parameters characterizing individual qubits, we measure the residual $ZZ$-coupling between all qubit pairs, see results in Fig.~\ref{fig:resid_zz}, by performing a Ramsey experiment on the measured qubit with the pulsed qubit in either the $\ket{0}$ or $\ket{1}$ state. To characterize the gate performance, we implement randomized benchmarking on all qubits to find the error per single qubit Clifford and we perform interleaved randomized benchmarking for the characterization of errors per conditional-phase gate. The resulting gate errors are shown in Fig.~\ref{fig:rb}. By directly measuring the $\ket{2}$-state population after the randomized benchmarking sequences, we further extract leakage per gate~\cite{Wood2017, Chen2016}. For single qubit gates, we find a leakage per Clifford operation to be $0.025\%$ on average while the leakage per conditional-phase gate is between $0.1\%$ and $0.7\%$.
\begin{figure}
\caption{Residual $ZZ$-coupling measured between all pairs of qubits on our device. The pulsed qubit is prepared in either ground or excited state and a Ramsey experiment is performed on the measured qubit to extract its frequency. The qubit pairs with gray label indicate pairs with no direct coupling.}
\label{fig:resid_zz}
\end{figure}
\begin{figure}
\caption{Single qubit errors per Clifford for each qubit (circles), and average CZ gate errors from interleaved randomized benchamrking (lines).}
\label{fig:rb}
\end{figure}
\section{Readout Characterization}
\label{app:readout}
\begin{figure*}
\caption{Single-shot readout assignment error, $E$, for the simultaneous readout of all 7 qubit.}
\label{fig:readout}
\end{figure*}
\begin{figure}
\caption{Average dephasing rate (left axis) of qubit Q$i$ during a 200~ns long readout pulse on qubit Q$j$ and the corresponding probability of a phase error (right axis) on qubit Q$i$. }
\label{fig:dephase}
\end{figure}
We perform multiplexed readout as detailed in Ref~\cite{Heinsoo2018}. Our readout scheme allows us to selectively address any subset of qubits. The readout is performed with a 200~ns readout pulse and a 300~ns integration window for qubits D1, D2, A1, A2 and A3 and a 300~ns readout pulse with a 400~ns integration window for qubits D3 and D4 due to the smaller dispersive shifts for these qubits. In Fig.~\ref{fig:readout}, we show single-shot readout errors for all computational basis states of the seven qubits with an average assignment error of 11\%.
To characterize measurement induced dephasing on the data qubits when reading out the ancilla qubits, we perform a Ramsey experiment on each of the data qubits. We interleave the Ramsey pulses with a readout on qubit Q$j$~\cite{Heinsoo2018}, and in Fig.~\ref{fig:dephase} we show the resulting additional dephasing rates, $\Gamma_{ij}$, on the data qubits introduced by readout pulses. We can convert the dephasing rates to a probability for introducing a phase error by $P_\phi = [1-\text{exp}(-\Gamma_{ij}\tau_r)]/2$, where $\tau_r$ is the readout time. We find that measurements of the ancilla qubits induce less than 0.3\% phase error on any data qubits.
\begin{figure*}
\caption{Experimental setup described in the text.}
\label{fig:setup}
\end{figure*}
\section{Experimental setup}
\label{app:setup}
The seven qubit device is installed at the base plate of a cryogenics setup~\cite{Krinner2019}, see Fig.~\ref{app:setup}. Here, the qubits (indicated by their labels) are controlled by flux and control AWGs through a series of microwave cables each with attenuators and filters, such as bandpass filters (BP), lowpass filters (LP), high pass fiters (HP) and eccosorb filters, installed as indicated. The flux pulses and microwave drive pulses are generated using arbitrary waveform generators (AWG) with 8 channels and a sampling rate of 2.4~GSa/s. The flux pulses are combined with a DC current using a bias-tee. The baseband microwave control pulses are generated at an intermediate frequency (IF) of $100$~MHz and then upconverted to microwave frequencies using IQ mixers installed on upconversion boards (UC). The multiplexed readout pulses, see also Appendix~\ref{app:readout}, are generated and detected using an FPGA based control system (\emph{Zurich Instruments UHFQA}) with a sampling rate of 1.8~GSa/s. The measurement signals at the outputs of the sample are amplified using a wide bandwidth near-quantum-limited traveling wave parametric amplifier (TWPA)~\cite{Macklin2015} connected to isolators at its input and output. Moreover, we installed bandpass filters in the output lines to suppress amplifier noise outside the bandwidth of interest. The output signals are further amplified by high-electron-mobility transistor (HEMT) amplifiers and additional amplifiers at room temperature (WAMP). After amplification, the signals are downconverted (DC) and processed using the weighted integration units of the UHFQAs.
\section{Numerical Simulations}
\label{app:sim}
We model the dynamics of our seven qubit quantum system by a master equation given by
\begin{align}
\dot{\rho} = -\frac{i}{\hbar} [ H(t), \rho ] + \sum_i \Big[ \hat{c}_i \rho \hat{c}_i^{\dagger} - \frac{1}{2} \Big( \hat{c}_i^{\dagger}\hat{c}_i \rho + \rho \hat{c}_i^{\dagger} \hat{c}_i \Big) \Big], \label{eq:mastereq}
\end{align}
where $\rho$ is the density matrix describing the system at time $t$ and $H(t)$ is the Hamiltonian, the time-dependence of which models the applied gate sequence. The collapse operators $\hat{c}_i$ model incoherent processes. We solve the master equation numerically~\cite{Johansson2013a}. To simplify the description of the system's time evolution, we consider the Hamiltonian to be piece-wise constant, see details in Ref.~\cite{Andersen2019}. In addition, we include the Hamiltonian
\begin{align*}
H_{ZZ}/\hbar &= \sum_{i,j} \alpha_{i,j} \ket{11}_{i,j}\bra{11}
\end{align*}
modeling the residual $ZZ$ coupling $\alpha_{ZZ}$ shown in Fig.~\ref{fig:resid_zz}.
The incoherent errors are described by the Lindblad terms in Eq.~\eqref{eq:mastereq} with
\begin{align*}
\hat{c}_{T_1,i} &= \sqrt{\frac{1}{T_{1,i}}} \sigma_{-,i}, \\
\hat{c}_{T_{\phi,i}} &= \sqrt{ \frac{1}{2} \Big( \frac{1}{T_{2,i}} - \frac{1}{2T_{1,i}} \Big) } \sigma_{z,i},
\end{align*}
where $T_{1,i}$ and $T_{2,i}$ are the lifetime and decoherence time (Ramsey decay time) of qubit $i$.
To simulate the ancilla measurement, we consider the POVM operators:
\begin{align}
M_0 &= \sqrt{P(0|0)} \ket{0}\bra{0}_A + \sqrt{P(0|1)} \ket{1}\bra{1}_A, \\
M_{1} &= \sqrt{P(1|0)} \ket{0}\bra{0}_A + \sqrt{P(1|1)} \ket{1}\bra{1}_A,
\end{align}
for the outcomes 0 and 1 respectively, where $P(i|j)$ are the experimentally determined probabilities for measuring the state $i$ when preparing the state $j$. We choose for simplicity the POVM operators corresponding to minimal disturbance measurements~\cite{Wiseman2010} as these POVM operators will mostly remove coherences similar to the real physical measurements. We evaluate the probability for each ancilla measurement outcome by $p_i = \text{Tr}(M_i \rho M_i^{\dagger})$ for $i\in\lbrace0,1\rbrace$. The resulting density matrix given a certain measurement outcome $i$ is calculated as $\rho \rightarrow M_i \rho M_i^{\dagger} / p_i$.
\end{appendix}
\end{document}
|
\begin{document}
\title{{f The Distribution of Argmaximum or \a Winner Problem}
\begin{quote}
{\bf Abstract.}
We consider a limit theorem for the distribution of a r.v. $Y_n=\arg\max_{i: 1\dots n}\{X_i\}$, where $X_i$'s are independent continuous non-negative random variables. The r.v.'s $X_i, i=1.\dots, n$, may be interpreted as the gains of $n$ players in a game, and the r.v. $Y_n$ itself as the number of a ``winner". In the case of i.i.d.r.v.'s, the distribution of $Y_n$ is, clearly, uniform on $\{1,\dots, n\}$, while when the $X$'s are non-identically distributed, the problem requires some calculations.
\noindent AMS 1991 Subject Classification:
Primary 60F17, Secondary 60G15.
\noindent Keywords: limit theorem, maximum of random variables, argmaximum.
\end{quote}
\section{Introduction and a Basic Formula}
\renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
Let $X_{1},X_{2},...$ be positive and independent r.v.'s. We will deal with $\max\{X_1,...,X_n\}$. For the case of identically distributed r.v.'s, the theory of limiting distribution for the maximum was developed in papers by Fisher\&Tippett \cite{fisher}, von Mises \cite{mises} and Gnedenko \cite{gnedenko}; see also systematic presentations in \cite{feller}, \cite{haan}, \cite{rotar1}, \cite{rotar2}. The case of non-identically distributed r.v.'s was considered in \cite{rotar1}.
This paper concerns the probability
\[
p_{in}=P(X_{i}= \max\{X_1,...,X_n\}), \,\,\,\, i=1,\dots ,n.
\]
If the r.v.'s $X_i, i=1.\dots, n$, are interpreted as the gains of $n$ players in a game, then $p_{in}$ is the probability that the $i$-th player is a ``winner".
Below, we assume the $X$'s to be continuous, and in this case, the winner is unique with probability one.
In the case of i.i.d. r.v.'s, the probability $p_{in}$ is, clearly, equal to $1/n$; if the $X$'s are non-identically distributed, the problem requires some calculations.
Let $F_{i}(x)=P(X_{i}\leq x), \,\,\, F(0)=0, F(x)>0$ for $x>0$.
For $x>0$, set
\[
\nu_i(x)=-\ln F_i(x). \,\,\,\,
\]
and $\nu_i(0)=\infty$.
So, for all $i$,
\begin{eqnarray}
&& F_i(x)=\exp\{-\nu_i(x)\},\label{nu0} \\
&& \nu_i(x) \,\,\,\text{ is non-increasing}, \,\,\, \nu_i(0)=\infty, \,\,\, \nu_i(\infty)=0. \label{nu1}
\end{eqnarray}
The asymptotic behavior of $\nu_i(x) $ as $x\to\infty$ is equivalent to that of $1-F_i(x)$.
Below, we assume all $\nu_i(x)$'s to be continuous for $x>0$.
\textit{\textbf{Example} (\textit{Weibull's distrinution}) }
\[
F_i(x)=\exp\left\{-\frac{c_i}{x^\alpha}\right\}, \text{and} \,\,\, c_i, \,\,\alpha>0,
\]
a well known distribution stable with respect to maximization. $\square$ \footnote{the symbol $\square$ means the end of an example; the symbol $\blacksquare$ below will mean the end of a proof. }
We have
\begin{eqnarray*}
p_{in}&=& \int_{0}^{\infty }\prod_{j=1,\, j\neq i}^{n}\,F_{j}(x)dF_{i}(x)\\ \nonumber
&=& - \int_0^\infty \exp\left\{-\sum_{j=1, j\neq i}^n \nu_j(x) \right\}\exp\{-\nu_i(x)\}d\nu_i(x) \nonumber \\
&=& - \int_0^\infty \exp\left\{-\sum_{j=1}^n \nu_j(x) \right\}d \nu_i(x).
\end{eqnarray*}
Integrating by parts and taking into account (\ref{nu0})--(\ref{nu1}), we have
\begin{equation}
p_{in}= - \int_0^\infty \nu_i(x) \exp\left\{-\sum_{j=1}^n \nu_j(x) \right\}d\left(\sum_{j=1}^n \nu_j(x)\right). \label{nu2}
\end{equation}
Consider substitution
\begin{equation}\label{sub}
\sum_{i=1}^n \nu_i(x)=y.
\end{equation}
For any non-increasing function $r(x)$, we define its inverse as
\[
r^{-1}(y)=\sup\{x: r(x)\geq y\}.
\]
Let $x_n(y)$ be the inverse of the function $\sum_{i=1}^n \nu_i(x)$; in other words, a solution (in the above sense) to equation (\ref{sub}).
Then from (\ref{nu2})--(\ref{sub}), it follows that
\begin{equation}\label{p-in}
p_{in}= \int_0^\infty \nu_i(x_n(y)) e^{-y}dy.
\end{equation}
This may serve as a basic formula.
\textbf{Remark. } Condition $F_i(x)>0$ for all $x>0$ is not necessary; we imposed it just to make the proof of (\ref{p-in}) more explicit. As a matter of fact, it is easy (though a bit long) to prove that the same is true, for example, if for all $n$ and a finite $a\geq0$
\[
a_n=: \max_{i=1,\dots n}\sup\{x: F_i(x)=0\}\leq a.
\]
\section{A Basic Example}
\renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
Suppose
\begin{equation}\label{r1}
\nu_i(x)=c_ir(x),
\end{equation}
where $r(x)$ is a non-negative, continuous, and non-increasing function; $r(0)=\infty$, \, $r(\infty)=0$, and $c_i$'s are non-negative.
Then
\[
\sum_{i=1}^n \nu_i(x)=r(x)\sum_{i=1}^n c_i,
\]
and a solution to equation (\ref{sub}) is
\begin{equation}\label{x_n-example}
x_n(y)=r^{-1}\left(\frac{y}{\sum_{i=1}^nc_i}\right).
\end{equation}
So,
\begin{equation}\label{r2}
\nu_i(x_n(y))=c_ir\left(r^{-1}\left(\frac{y}{\sum_{i=1}^nc_i}\right)\right) =\frac{c_i}{\sum_{i=1}^nc_i}\,y.
\end{equation}
Thus, in this case,
\[
p_{in}=\frac{c_i}{\sum_{i=1}^nc_i}\int_0^\infty ye^{-y}dy=\frac{c_i}{\sum_{i=1}^nc_i}.
\]
\section{A General Scheme}
\renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
When looking at (\ref{x_n-example}), one may suppose that for large $n$, the asymptotic behavior of $p_{in}$ is based just on the asymptotics of $r^{-1}(x)$ at zero, which is connected with that of $r(x)$ at infinity (or tails $1-F_i(x))$.
Assume the following.
\begin{enumerate}
\item
\begin{equation}\label{cond1}
\nu_i(x)=c_i r(x) (1+\delta_i(x)),
\end{equation}
where $r(x)$ is defined as above, $\delta_i(x)$ are continuous, uniformly in $i$
\begin{equation}\label{cond1+}
\delta_i(x)\to 0 \,\,\, \text{as}\,\,\, x\to\infty,
\end{equation}
and for positive constants $M<\infty$ and $m<1$, and for all $i$ and $x$,
\begin{equation}\label{cond1++}
-m\leq \delta_i(x)\leq M.
\end{equation}
\item
\begin{equation}\label{cond2}
b_n=:\sum_{i=1}^nc_i \to\infty \,\,\, \text{ as }\,\,\, n\to \infty.
\end{equation}
\end{enumerate}
\begin{Proposition}
\label{pr1}
Set
\[
\alpha_{in}=\frac{c_{i}}{b_n}.
\]
Then
\begin{equation} \label{p1}
p_{in}\sim \alpha_{in} \,\,\,\text{as}\,\,\,n\to\infty, \,\,\,\text{uniformly in}\,\,\,i. \footnote{The symbol $\sim$ means that the ratio of the left- and right-hand sides converges to one.}
\end{equation}
\end{Proposition}
\textbf{Proof}
Let $x_n(y)$ be a solution to equation
\begin{equation}\label{eq1}
\sum_{i=1}^n \nu_i(x)=y,
\end{equation}
that is,
\begin{equation}\label{eq1+}
r(x)\sum_{i=1}^n c_i(1+\delta_i(x))=y.
\end{equation}
So,
\begin{equation}\label{x_n}
x_n(y) =r^{-1}\left(\frac{y}{\sum_{i=1}^n c_i(1+\delta_i(x_n(y)))}\right) .
\end{equation}
From (\ref{cond1++}), it follows that
\begin{equation}\label{x_n1}
x_n(y) \geq r^{-1}\left(\frac{y}{(1-m)\sum_{i=1}^n c_i}\right).
\end{equation}
Hence, since $r^{-1}(0)=\infty$, and in view of (\ref{cond2}),
\begin{equation}\label{xtoinfty}
x_n(y)\to \infty
\end{equation}
as $n\to\infty$.
Furthermore, in view of (\ref{x_n}),
\[
\nu_i(x_n(y))=c_i\cdot\frac{y(1+\delta_i(x_n(y)))}{\sum_{j=1}^nc_j(1+\delta_j(x_n(y)))}.
\]
Thus,
\begin{eqnarray*}
p_{in}&=&\int_0^\infty \nu_i(x_n(y))e^{-y}dy \\
&=& \frac{c_i}{\sum_{j=1}^n c_j} \int_0^\infty y\cdot \frac{(1+\delta_i(x_n(y)))\sum_{j=1}^n c_j}{\sum_{j=1}^n c_j(1+\delta_j(x_n(y)))}\cdot e^{-y}dy.
\end{eqnarray*}
For each $y>0$, in view of (\ref{xtoinfty}) and (\ref{cond1+}),
\[
\frac{(1+\delta_i(x_n(y)))\sum_{j=1}^n c_j}{\sum_{j=1}^nc_j(1+\delta_j(x_n(y))}\to 1 \,\,\,\text{as}\,\, n\to\infty,
\]
uniformly in $i$. On the other hand,
\[
\frac{(1+\delta_i(x_n(y)))\sum_{j=1}^n c_j}{\sum_{j=1}^nc_j(1+\delta_j(x_n(y)))}\leq \frac{1+M}{1-m}.
\]
Hence,
\[
\frac{p_{in}}{\alpha_{in}}\to \int_0^\infty ye^{-y}dy.\,\,\,\, \,\,\,\,\blacksquare
\]
\section{The case of ``regularly" varying $c_i$'s}
\renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
In the case where the coefficients $c_i$'s are varying -- in a sense -- regularly, we can present the result in a nicer form.
Consider the segment $[0,1]$ and identify a point $i/n, \,\,\,i=1.\dots,n$, with a r.v. $X_i$; so to speak, with the $i$-th ``player". Let us assign to this point probability $\alpha_{in}$, and suppose that the measure so defined weakly converges to a probability measure $\alpha$ on $[0,1]$. In other terms,
\begin{equation}\label{alpha}
\alpha_n=: \sum_{i=1}^n\delta_{\{i/n\}}\alpha_{in}\Rightarrow\alpha,
\end{equation}
where $\delta_{\{x\}}$ is a measure concentrated at point $x$.
\begin{Proposition}
\label{pr2}
Suppose that together with conditions of Proposition \ref{pr1}, (\ref{alpha}) holds. Then discrete measure
\begin{equation}\label{alpha1}
\mu_n=: \sum_{i=1}^n\delta_{\{i/n\}}p_{in}\Rightarrow\alpha,
\end{equation}
\end{Proposition}
\textbf{Proof} is straightforward. Since the convergence in (\ref{p1}) is uniform in $i$, for any continuous bounded function $h$,
\begin{eqnarray*}
\int_0^1hd\mu_n &=&\sum_1^nh\left(\frac{i}{n}\right)p_{in}=\sum_1^nh\left(\frac{i}{n}\right)\alpha_{in}(1+o(1)) \\
&=& (1+o(1))\int_0^1hd\alpha_n \to\int_0^1hd\alpha. \,\,\,\, \,\,\,\,\blacksquare
\end{eqnarray*}
\pagebreak
\textbf{ Examples }
\begin{enumerate}
\item Set $c_i=i^s$, for $s\geq0$, and let $x\in(0,1]$. Let $k=k_n$ be such that $\frac{k}{n}\leq x<\frac{k+1}{n}$. Then, as is easy to verify,
\begin{equation}\label{cond3}
\frac{\sum_{i=1}^{k_n}c_i}{\sum_{i=1}^nc_i}\to x^{s+1}.
\end{equation}
(For $k_n=0$, we set $\sum_{i=1}^{k_n}=0$.) In other words, if $F(x)$ is the distribution function (d.f.) of $\alpha$, then $F(x)=x^{s+1}$.
Say, if $c_i=i$, then for large $n$, the distribution of the winner numbers may be well presented by a distribution on $[0,1]$ with d.f. $F(x)=x^2$.
\item Let $c_i=2^i$, Then in the same notations, for any $x<1$,
\begin{equation}\label{cond3}
\frac{\sum_{i=1}^{k_n}c_i}{\sum_{i=1}^nc_i}\to 0,
\end{equation}
and measure $\alpha$ is concentrated at point $1$.
\item Let $c_i=1/i$, Then, as is easy to verify, for any $x\in (0,1]$,
\begin{equation}\label{cond3}
\frac{\sum_{i=1}^{k_n}c_i}{\sum_{i=1}^nc_i}\to 1,
\end{equation}
and measure $\alpha$ is concentrated at point $0$. \,\,\, $\square$
\end{enumerate}
As a matter of fact, the class of possible limiting distributions $\alpha$ is narrow because, as we will see, in (\ref{alpha}) we deal with regularly varying functions (reg.v.f.'s).\footnote{A positive function $H(x)$ on $[0,\infty)$ is regular varying in the sense of Karamata with an order of $\rho, \,\,\,-\infty<\rho<\infty $, iff for any $x>0$
\[
\frac{H(tx)}{H(x)}\to x^\rho \,\,\, \text{as}\,\,\, t\to\infty.
\]
A function $L(x)$ is called slowly varying if it is regularly varying with $\rho=0$. Any reg.v.f.\ $H(x)=x^\rho L(x)$, where $L(\cdot)$ is slowly varying. A detailed presentation of reg.v.f.'s is given, for example, in Feller, \cite{feller}, Chapter VIII, Section 8. Some definitions and examples may be also found in \cite{rotar1}, Ch,15. }
\begin{Proposition}
\label{pr3}
\begin{description}
\item[(A)] Suppose (\ref{alpha}) holds, and
\begin{equation}\label{regular}
\frac{b_{n+1}}{b_n}\to 1\,\,\,\text{as}\,\,\,n\to\infty.
\end{equation}
Then the d.f. of $\alpha$ is
\begin{equation}\label{rho}
F(x)=x^\rho, \,\,\,\,\, x\in[0,1],
\end{equation}
where $0\leq\rho\leq\infty$, and $b_n=b(n)$, where $b(t)$ is a non-decreasing reg.v.f.
(In (\ref{rho}), if $\rho=0$, then $F(x)=1$ for all $x\in[0,1]$; if $\rho=\infty$, then $F(x)=0$ for all $x<1$.)
\item[(B)] Vice versa, let $b_n=b(n)$, where $b(t)$ is a non-decreasing positive reg.v.f. Then (\ref{regular}) holds automatically, and (\ref{alpha}) is true with the d.f.\ $F(x)$ of $\alpha$ defined in (\ref{rho}),
\end{description}
\end{Proposition}
\textbf{Proof}
\textbf{(A)} Let $F_n(x)$ and $F(x)$ be the d.f.'s of measures $\alpha_n$ and $\alpha$, respectively. Then
\begin{equation}\label{f1}
F_n(x)\to F(x)
\end{equation}
as $n\to \infty$ for all $x$'s that are continuity points of $F(x)$.
Let $b_0=0$, and for all $t\geq0$ function $b(t)=b_n$ if $t\in[n,n+1)$. We will prove that $ b(t)$ is a reg.v.f.
Let us fix a continuity point $x$, and let an integer $k=k_n$ be such that $\frac{k}{n}\leq x <\frac{k+1}{n}$. Then from (\ref{f1}) it follows that
\[
\frac{b_{k_n}}{b_n}\to F(x)\,\,\,\text{as} \,\,\, n\to\infty.
\]
On the other hand, by definition, $b_{k_n}=b(k_n)=b(nx)$, and hence
\begin{equation}\label{f2}
\frac{b(nx)}{b(n)}\to F(x)\,\,\,\text{as} \,\,\, n\to\infty.
\end{equation}
Together with (\ref{regular}), this implies that
\begin{equation}\label{f3}
\frac{b(tx)}{b(t)}\to F(x)\,\,\,\text{as}\,\,\, t\to\infty,
\end{equation}
where $t$'s are arbitrary positive numbers.
Indeed, let $n=n_t$ be such that\\ $t\in[n, n+1)$. Then
\begin{equation}\label{f4}
\frac{b(nx)}{b(n+1)}\leq \frac{b(tx)}{b(t)}\leq \frac{b((n+1)x)}{b(n)}.
\end{equation}
\
Furthermore, if $t\to\infty$, then $n=n_t\to\infty$, and
\[
\frac{b(nx)}{b(n+1)}=\frac{b(n)}{b(n+1)}\cdot\frac{b(nx)}{b(n)}\to F(x)
\]
in view of (\ref{regular}) and (\ref{f2}). Similarly, the same is true for the very right fraction in (\ref{f4}).
So, function $b(t)$ is a regularly varying function, and the limit in (\ref{f3}) must be equal to a power function $x^\rho$; see, for instance, Lemma 1 from Feller \cite{feller}, VIII, 8.
\textbf{(B)} Let $b_n=b(n)$ where $b(t)$ is a reg.v.f.\ (that may be different from the piecewise constant function $b(x)$ defined in part (A) of the proof). Let us fix an $x\in(0,1]$, and let again an integer $k=k_n$ be such that $\frac{k}{n}\leq x <\frac{k+1}{n}$.
First, since $b(x)$ is non-decreasing,
\begin{equation}\label{f5}
F_n(x)=\frac{b_{k_n}}{b_n}=\frac{b(k_n)}{b(n)}\leq \frac{b(nx)}{b(n)}\to x^\rho,
\end{equation}
where $0\leq\rho<\infty$.
On the other hand,
\begin{equation}
F_n(x)=\frac{b(k_n)}{b(n)}\geq \frac{b(xn-1)}{b(n)} = \frac{b(xn-1)}{b(xn)}\frac{b(xn)}{b(n)}. \label{f6}
\end{equation}
Let us note that for any non-decreasing reg.v.f.\ $b(x)$
\begin{equation}\label{f8}
\frac{b(x-1)}{b(x)}\to 1 \,\,\, \text{as} \,\,\, x\to\infty.
\end{equation}
Indeed, for $s<1$ and sufficiently large $x$'s
\begin{equation*}
1 \geq \frac{b(x-1)}{b(x)}\geq \frac{b(sx)}{b(x)}\to s^\rho,
\end{equation*}
and the right-hand side can be made arbitrary close to $1$.
By virtue of (\ref{f8}), the first factor in (\ref{f6}), converges to $1$, and the whole product converges to $x^\rho$.
Relation (\ref{f8}) also implies (\ref{regular}). \,\,\,
$\blacksquare$.
\textbf{Remarks and Examples }
\begin{enumerate}
\item When considering examples, it is more convenient to deal directly with sequences $b_n$ rather than coefficients $c_i$'s. In particular, if $b_n$ are asymptotically exponential, (\ref{regular}) is not true but it is easy to show that the limiting measure $\alpha $ exists and concentrated at point $1$ (see also Example 2 above). On the other hand, if for instance, $b_n\sim e^{c\sqrt{n}}$ for a positive $c$, (\ref{regular}) is true though the limiting measure is again concentrated a $1$.
\item To specify a particular $\rho$, we may, for example, use the fact that, under conditions of Proposition 3,
\[
\frac{b_n}{b_{2n}}\to \left(\frac{1}{2}\right)^\rho.
\]
So, if we know $\lim \frac{b_n}{b_{2n}}$, then we may find $\rho$. In particular, if $ \frac{b_n}{b_{2n}}\to 0$, then $\rho=\infty$, and the distribution $\alpha$ is concentrated at $1$, while if $ \frac{b_n}{b_{2n}}\to 1$, then $\rho=0$, and the distribution $\alpha$ is concentrated at $0$.
\item We may deal with a triangular array, that is, set $c_i=c_{in}$. Then a limiting distribution, if any, may be practically arbitrary. As an example, consider an integrable, non-negative function $g(x)$ on $[0,1]$ and set the coefficient $c_{in}=g(i/n)$. Then, the limiting distribution will be that with the density
\[
f(x)=\frac{g(x)}{\int_0^1g(x)dx}.
\]
A proof of (\ref{alpha}) in this case may run similarly to what we did above. Note that when considering a counterpart of (\ref{x_n-example}), we may take into account that in this case
\[
b_n=:\sum_{i=1}^n c_{in}\sim n\cdot\int_0^1 g(x)dx .
\]
\item Clearly, $\arg \max\{X_1,\dots.X_n\}\stackrel{d}{=}\arg \max\{\tilde{X}_1,\dots.\tilde{X}_n\}$, where $\tilde{X}_i=f(X_i)$, and $f(x)$ is a continuous strictly increasing function. It is easy to verify that the corresponding function $\tilde{r}(x)=r(f^{-1}(x))$. This is a way to ``improve'' $r(x)$.
\item In the case where the distributions of the $X$'s are not continuous, the above technique needs to be improved. Regarding the fact that in this case there may be several ``winners", one can conjecture that the situation may be fixed if we select from winners one at random (throw lots). On the other hand, in this case probability $p_{in}\neq 1/n$ even if the $X_i$'s are identically distributed.
Consider the simplest\\
\textbf{ Example}. Let all $X_i=1$ or $0$ with probabilities $p$ and $q$, respectively.
Then
\[
p_{in}=p\cdot1+q\cdot q^{n-1}=p+q^n,
\]
However, in the case of selecting a winner at random, $ p_{in}=1/n$ just by symmetry, though the same may be also proved directly.
\end{enumerate}
We thank professor Vadim Ponomarenko (SDSU) a bygone conversation with whom
helped us to come to the statement of this paper problem. The problem we discussed with professor Ponomarenko, above else, may serve as a good application example. It concerns a complex machine consisting of a large number of parts with random and non-identically distributed lifetimes. The question is which part will break down first. Certainly, we deal here with $\arg\min $ but it can be easily reduced to $\arg\max$.
\end{document}
|
\begin{document}
\title[Torsion and ground state maxima]{Torsion and ground state maxima: \\ close but not the same}
\author[]{B. A. Benson, R. S. Laugesen, M. Minion, B. A. Siudeja}
\address{Department of Mathematics, Kansas State Univ., Manhattan,
KS 66506, U.S.A.}
\varepsilonmail{babenson\@@math.ksu.edu}
\address{Department of Mathematics, Univ.\ of Illinois, Urbana,
IL 61801, U.S.A.}
\varepsilonmail{Laugesen\@@illinois.edu}
\address{Lawrence Berkeley National Lab, Berkeley, CA 94720, U.S.A.}
\varepsilonmail{mlminion\@@lbl.gov}
\address{Department of Mathematics, Univ.\ of Oregon, Eugene,
OR 97403, U.S.A.}
\varepsilonmail{Siudeja\@@uoregon.edu}
\date{\today}
\keywords{Semilinear, Poisson, maximum point, torsion, landscape function, Dirichlet eigenfunction.}
\subjclass[2010]{\text{Primary 35B09. Secondary 35B38,35P99}}
\begin{abstract}
Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture.
We construct counterexamples on the half-disk, by working with the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has merit, since the maxima of the torsion function and eigenfunction are unexpectedly close together. It is an open problem to quantify this closeness in terms of the domain and the nonlinearity.
\varepsilonnd{abstract}
\maketitle
\section{\bf Introduction}
\label{sec:intro}
Suppose the Poisson equation
\[
\begin{cases}
-{\mathbb D}elta u = f(u) & \text{in $\Omega$,} \\
\quad \ \ u = 0 & \text{on $\partial \Omega$,}
\varepsilonnd{cases}
\]
has a positive solution on the bounded convex plane domain $\Omega$. Here the nonlinearity $f$ is assumed to be Lipschitz and \varepsilonmph{restoring}, which means $f(z)>0$ when $z>0$. Cima and Derrick \cite{CD11,CDK14} have conjectured that the location of the maximum point of $u$ is independent of the form of the nonlinearlity $f$.
This conjecture sounds impossible, since the graph of the solution must vary with the nonlinearity. Numerical computations by Cima and co-authors give surprising support for the conjecture, though, and \autoref{fig:levelcurves} provides further food for thought by considering a triangular domain and plotting the level curves and maximum point for the choices $f(z)=1$ and $f(z)=\lambda z$. The corresponding linear Poisson equations describe the \varepsilonmph{torsion function} and the \varepsilonmph{ground state of the Laplacian} (see below). Our solutions were computed numerically by the finite element method on a mesh with approximately $10^6$ triangles. The maximum points for the two solutions in \autoref{fig:levelcurves} appear to coincide, even though the level curves differ markedly near the boundary.
\begin{figure}[t]
\hspace{\fill}
\includegraphics[width=3cm]{exit.png}
\hspace{\fill}
\includegraphics[width=3cm]{eig.png}
\hspace{\fill}
\caption{Level curves and the maximum point on a triangular domain, for solutions of two different Poisson type equations: the torsion function (left) and the first eigenfunction (right).}
\label{fig:levelcurves}
\varepsilonnd{figure}
We disprove the conjecture on a half-disk in \autoref{sec:half-disk}, and again on the right isosceles triangle in \autoref{sec:rightisos}. Interestingly, the conjecture is remarkably close to being true in these counterexamples, with the maximum points occurring in almost but not quite the same location. We cannot explain this unexpected closeness.
A fascinating open problem is to bound the difference in location of the maximum points of two semilinear Poisson equations in terms of the difference between their nonlinearity functions and geometric information on the shape of the domain. Also, note that for both the half-disk and right isosceles triangle, our results show that the maximum point of the torsion function lies to the left of the maximum for the ground state (when oriented as in \autoref{fig:levelcurves}), which perhaps hints at a general principle for a class of convex domains.
\subsection*{Notation}
The \varepsilonmph{torsion} or \varepsilonmph{landscape} function is the unique solution of the Poisson equation
\[
\begin{cases}
-{\mathbb D}elta u = 1 & \text{in $\Omega$,} \\
\quad \ \ u = 0 & \text{on $\partial \Omega$.}
\varepsilonnd{cases}
\]
Here we have chosen $f(z)=1$. Clearly $u$ is positive inside the domain, by the maximum principle.
The \varepsilonmph{Dirichlet ground state} or \varepsilonmph{first Dirichlet eigenfunction of the Laplacian} is the unique positive solution of
\begin{equation*}
\begin{cases}
-{\mathbb D}elta v = \lambda v& \text{in $\Omega$,} \\
\quad \ \ v = 0 & \text{on $\partial \Omega$,}
\varepsilonnd{cases}
\varepsilonnd{equation*}
where $\lambda>0$ is the first eigenvalue of the Laplacian on the domain under Dirichlet boundary conditions. Here we have chosen $f(z)=\lambda z$.
\section{\bf The half-disk}
\label{sec:half-disk}
The maximum points for the torsion function and ground state can lie so close together that one cannot distinguish them by the naked eye, as the following Proposition reveals. Yet the two points are not the same.
\begin{proposition} \label{pr:half-disk}
Take $\Omega = \{ (x,y) : x> 0, x^2 + y^2 < 1 \}$ to be the right half-disk. On this domain the torsion function $u$ attains its maximum at $(0.48022,0)$ while the ground state $v$ attains its maximum at $(0.48051,0)$. Here the $x$-coordinates have been rounded to $5$ decimal places.
\varepsilonnd{proposition}
\begin{proof}
(i) The ground state is given in polar coordinates by
\[
v(r,\theta) = J_1(j_{1,1}r) \cos \theta
\]
where $J_1$ is the first Bessel function and $j_{1,1} \simeq 3.831706$ is its first positive zero. Clearly the maximum is attained on the $x$-axis, where $\theta=0$, and the function is plotted along this line in \autoref{fig:Bessel}. By setting $J_1^\prime(j_{1,1}r)=0$ and solving, we find $r = j^\prime_{1,1}/j_{1,1} \simeq 0.48051$, rounded to five decimal places, where $j^\prime_{1,1} \simeq 1.841184$ is the first zero of $J_1^\prime$.
\begin{figure}
\includegraphics[width=4cm]{Bessel-halfdisk.pdf}
\caption{The radial part of the ground state on the right half-disk: $v(r,0) = J_1(j_{1,1}r)$.}
\label{fig:Bessel}
\varepsilonnd{figure}
(ii) The torsion function is more complicated \cite[Section 4.6.2]{W}, and is given by
\begin{align*}
u(x,y) & = \frac{1}{4\pi} \Big[ -2 \pi x^2 - 2 x \Big( (x^2 + y^2)^{-1} - 1\Big) \\
& \qquad \quad + \Big( 2 + (x^2 - y^2) \big( (x^2 + y^2)^{-2} + 1 \big) \Big) \arctan \Big( \frac{2x}{1 - (x^2 + y^2)} \Big) \\
& \qquad \quad + xy \Big( (x^2 + y^2)^{-2} - 1 \Big) \log \frac{x^2 + (1 + y)^2}{x^2 + (1 - y)^2} \, \Big] .
\varepsilonnd{align*}
One verifies the Dirichlet boundary condition on the right half-disk by examining four cases: (i) $u=0$ if $x=0$ and $0<|y|<1$, (ii) $u \to 0$ as $(x,y) \to (0,0)$, (iii) $u \to 0$ as $(x,y) \to (0,\pm 1)$, and (iv) $u \to 0$ as $(x,y) \to (x_1,y_1)$ with $x_1>0$ and $x_1^2+y_1^2=1$.
To check $u$ satisfies the Poisson equation $-{\mathbb D}elta u=1$, a lengthy direct calculation suffices.
We claim $u$ attains its maximum at a point on the horizontal axis. For this, first notice $u$ is even about the $x$-axis by definition, meaning $u(x,y)=u(x,-y)$. Hence the harmonic function $u_y$ equals zero on the $x$-axis for $0<x<1$. Further, $u_y \leq 0$ at points on the unit circle lying in the open first quadrant, since $u>0$ in the right half-disk and $u=0$ on the boundary. Also, one can compute that $u_y(x,y)$ approaches $0$ as $(x,y) \to (0,0)$ or $(x,y) \to (1,0)$ or $(x,y) \to (0,1)$ from within the first quadrant of the unit disk. Lastly $u_y$ vanishes on the $y$-axis for $0 < y <1$ (since $u=0$ there). Hence we conclude from the maximum principle that $u_y \leq 0$ in the first quadrant of the unit disk, and so $u$ attains its maximum somewhere on the $x$-axis.
On the $x$-axis we have
\[
u(x,0) = \frac{1}{4\pi} \Big[ -2 \pi x^2 - 2 x^{-1} + 2x + (2 + x^{-2} +x^2) \arctan \Big( \frac{2x}{1 - x^2} \Big) \, \Big]
\]
for $0<x<1$. Clearly $u(0,0)=u(1,0)=0$, and
\[
u_x(x,0) = \frac{1}{\pi x^3} \big[ x + x^3 - \pi x^4 + \frac{1}{2}(x^4 - 1) \arctan \Big( \frac{2x}{1 - x^2} \Big) \big] .
\]
One can show by taking another derivative and applying elementary estimates that $u(x,0)$ is concave. Calculations show $u_x(x,0)$ is positive at $x=0.480219$ and negative at $x=0.480220$, and so the maximum of $u$ lies between these two points, that is, at $x=0.48022$ to $5$ decimal places.
\varepsilonnd{proof}
\section{\bf The right isosceles triangle}
\label{sec:rightisos}
\begin{proposition} \label{pr:rightisos}
Take $\Omega = \{ (x,y) : 0 < x < 1, |y|< 1-x \}$, which is an isosceles right triangle. On this domain the torsion function $u$ attains its maximum at $(0.39168,0)$ while the ground state $v$ attains its maximum at $(0.39183,0)$. Here the $x$-coordinates have been rounded to $5$ decimal places.
\varepsilonnd{proposition}
\begin{proof}
(i) Rotate the triangle by 45 degrees clockwise about the origin and scale up by a factor of $\pi/\sqrt{2}$, then translate by $\pi/2$ to the right and upwards, so that the triangle becomes
\[
T = \{ (x,y) : 0<y<x<\pi \}.
\]
This new triangle has ground state
\[
v(x,y) = \sin x \sin 2y - \sin 2x \sin y = 2 \sin x \sin y (\cos y - \cos x) > 0
\]
with eigenvalue $1^2 + 2^2 =5$. One checks easily that $v=0$ on the boundary of $T$, where $y=0$ or $x=\pi$ or $y=x$. To find the maximum point, set $v_x=0$ and $v_y=0$ and deduce $\cos 2x = \cos x \cos y = \cos 2y$. Therefore the maximum lies on the line of symmetry $y=\pi-x$ of the triangle $T$. A little calculus shows that $v(x,\pi-x)$ attains its maximum when $x=\arcsin(1/\sqrt{3})+\pi/2$. Hence the ground state of the original triangle attains its maximum at $\big( (2/\pi) \arcsin(1/\sqrt{3}),0 \big) = (0.39183,0)$ to $5$ decimal places.
(ii) The torsion function on the triangle $T$ is
\begin{align*}
& \! \! \! u(x,y) \\
& = - \frac{1}{4}(x-y)^2 + \sum_{n=1}^\infty \frac{n^2 \pi^2 -2\big( 1 - (-1)^n \big)}{2\pi n^3 \sinh n\pi} \Big[ \sinh nx \sin ny - \sin nx \sinh ny \\
& \qquad \qquad \qquad + \sin n(\pi-x) \sinh n(\pi-y) - \sinh n(\pi-x) \sin n(\pi-y) \Big] ,
\varepsilonnd{align*}
as we now explain. Observe that $-{\mathbb D}elta u = 1$ because the infinite series is a harmonic function, and $u=0$ on the boundary of $T$ by simple calculations with Fourier series when $0<x<\pi,y=0$, and when $x=\pi,0<y<\pi$; also $u=0$ on the hypotenuse where $y=x$.
The torsion function is known to attain its maximum somewhere on the line of symmetry $y=\pi-x$, either by general symmetry results \cite{CD11,CDK14} or else by arguing as in the proof of \autoref{pr:half-disk} part (ii). On that line of symmetry we evaluate
\begin{align*}
& u(x,\pi-x) \\
& = - (x-\pi/2)^2 + \sum_{n=1}^\infty \frac{n^2 \pi^2 -2\big( 1 - (-1)^n \big)}{\pi n^3 \sinh n\pi} \big[ (-1)^{n+1} \sinh nx - \sinh n(\pi-x) \big] \sin nx .
\varepsilonnd{align*}
The series converges exponentially on each closed subinterval of $(0,\pi)$, and so we may differentiate term-by-term to find
\begin{align}
& \frac{d\ }{dx} u(x,\pi-x) \label{eq:derivseries} \\
& = - 2(x-\pi/2) + \sum_{n=1}^\infty \frac{n^2 \pi^2 -2\big( 1 - (-1)^n \big)}{\pi n^2 \sinh n\pi} \Big\{ \big[ (-1)^{n+1} \cosh nx + \cosh n(\pi-x) \big] \sin nx \notag \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \big[ (-1)^{n+1} \sinh nx - \sinh n(\pi-x) \big] \cos nx \Big\} , \notag
\varepsilonnd{align}
where once again the series converges exponentially on closed subintervals of $(0,\pi)$.
The absolute value of the $n$-th term in series \varepsilonqref{eq:derivseries} is bounded by
\[
\frac{\pi(e^{nx}+e^{n(\pi-x)})}{\sinh(n\pi)} < 3\pi (e^{-n(\pi-x)}+e^{-nx}) ,
\]
as we see by bounding the $\sin$ and $\cos$ terms with $1$, adding the $\sinh$ and $\cosh$ terms having the same arguments, and using that $\sinh(n \pi) > e^{n\pi}/3$ for $n \geq 1$. Hence the infinite series \varepsilonqref{eq:derivseries} is bounded term-by-term by $3\pi$ times the sum of two geometric series having ratios $e^{-(\pi-x)}$ and $e^{-x}$.
The derivative of $u$ along the line of symmetry is positive at $x=2.1860525$ and negative at $x=2.1860530$, as one finds by evaluating the first 20 terms of the series in \varepsilonqref{eq:derivseries} and then estimating the remainder with the geometric series as above. Hence $u$ has a local maximum at $x=2.186053$ to 6 decimal places. This local maximum is a global maximum because $\sqrt{u}$ is concave (see \cite[Example 1.1]{B85} or \cite{K84}). Translating to the left and downwards by $\pi/2$ and then scaling down by a factor of $\sqrt{2}/\pi$ and rotating counterclockwise by $45$ degrees, we find the torsion function on the original triangle has a maximum at
\[
x= \frac{2}{\pi} (2.186053-\pi/2) = 0.39168
\]
to 5 decimal places.
\varepsilonnd{proof}
\section{\bf Concluding remarks}
\label{sec:remarks}
The counterexamples in this paper concern Poisson's equation for $f(z)=1$ and $f(z)=\lambda z$. One can find a whole family of counterexamples using $f(z)=a+bz$, where $a>0$ and $0<b \leq \lambda$. Note the maximum point depends on $b$ but not $a$, as one checks by rescaling the solution $u$ to $u/a$. To study this maximum point as $b$ varies, one starts with the eigenfunctions of $-{\mathbb D}elta-b$ on the half-disk or right isosceles triangle and notes that the eigenfunctions are the same as for $-{\mathbb D}elta$, just with eigenvalues shifted by $b$. The corresponding torsion function can be computed in terms of an eigenfunction expansion, and then the position of the maximum point can be carefully numerically located. We leave such investigations to the interested reader.
Finally, while our counterexamples involve linear Poisson equations, our choices of $f$ could presumably be perturbed to obtain genuinely nonlinear counterexamples.
\section*{Acknowledgments}
This work was partially supported by grants from the Simons Foundation (\#204296 to Richard Laugesen) and Polish National Science Centre (2012/07/B/ST1/03356 to Bart\-{\l}omiej Siudeja). We are grateful to the Institute for Computational and Experimental Research in Mathematics (ICERM) for supporting participation by Benson and Minion in IdeaLab 2014, where the project began. Thanks go also to the Banff International Research Station for supporting participation by Laugesen and Siudeja in the workshop ``Laplacians and Heat Kernels: Theory and Applications (March 2015), during which some of the research was conducted.
\newcommand{\doi}[1]{
\href{http://dx.doi.org/#1}{doi:#1}}
\newcommand{\arxiv}[1]{
\href{http://front.math.ucdavis.edu/#1}{ArXiv:#1}}
\newcommand{\mref}[1]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#1}}
\begin{thebibliography}{99}
\bibitem{B85}
C. Borell. \textit{Greenian potentials and concavity.}
Math. Ann. 272 (1985), no. 1, 155--160. \mref{MR0794098}
\bibitem{CD11}
J. A. Cima and W. Derrick. \textit{A solution of ${\mathbb D}elta u+f(u)=0$ on a triangle.} Irish Math. Soc. Bull. No.\ 68 (2011), 55--63 (2012). \mref{MR2964026}
\bibitem{CDK14}
J. A. Cima, W. R. Derrick and L. V. Kalachev. \textit{The maximum of ${\mathbb D}elta u+f(u)=0$ on an isosceles triangle.} Preprint, 2014.
\bibitem{K84}
B. Kawohl. \textit{When are superharmonic functions concave? Applications to the St. Venant torsion problem and to the fundamental mode of the clamped membrane.}
Z. Angew. Math. Mech. 64 (1984), no. 5, 364--366. \mref{MR0754534}
\bibitem{W}
Wolfram Mathematica Structural Mechanics Documentation, Chapter 4 Torsional Analysis \url{http://reference.wolfram.com/applications/structural/TorsionalAnalysis.html}
\varepsilonnd{thebibliography}
\varepsilonnd{document}
|
\begin{document}
\title{One Useful Logic\ That Defines Its Own Truth}
\begin{abstract}
Existential fixed point logic (EFPL) is a natural fit for some
applications, and the purpose of this talk is to attract attention to
EFPL. The logic is also interesting in its own right as it has
attractive properties. One of those properties is rather unusual:
truth of formulas can be defined (given appropriate syntactic
apparatus) in the logic. We mentioned that property elsewhere, and
we use this opportunity to provide the proof.
\end{abstract}
\begin{quote}\raggedleft\small\it
Believe those who are seeking the truth. Doubt those who find it.\\[1ex]
---Andr\'{e} Gide
\end{quote}
\section{Introduction}
First-order logic lacks induction but first-order formulas can be used to define the steps of an induction. Consider a first-order (also called elementary) formula $\varphi(P,x_1,\ldots,x_j)$ where a $j$-ary relation $P$ has only positive occurrences. The formula may contain additional individual variables, relation symbols, and function symbols. In every structure whose vocabulary is that of $\varphi$ minus the symbol $P$ and where the additional individual variables are assigned particular values, we have an operator
\[
\Gamma(P) = \{ \bar{x}:\ \varphi(P,\bar{x}) \}.
\]
A relation $P$ is a \varnothingh{closed point} of $\Gamma$ if $\Gamma(P) \subseteq P$, and $P$ is a \varnothingh{fixed point} of $\Gamma$ if $\Gamma(P) = P$. Since $P$ has only positive occurrences in $\varphi(P,\bar{x})$, the operator is monotone: if $P\subseteq Q$ then $\Gamma(P)\subseteq\Gamma(Q)$. By the Knaster-Tarski Theorem, $\Gamma$ has a least fixed point $P^*$ which is also the least closed point of $\Gamma$ \cite{tarski}.
There is a standard way to construct $P^*$ from the empty set by iterating the operator $\Gamma$. Let $P^0 = \varnothingtyset$, $P^{
\noindent\textbf{A:\ }lpha+1} = \Gamma(P^
\noindent\textbf{A:\ }lpha)$ and $P^\lambda = \bigcup_{
\noindent\textbf{A:\ }lpha<\lambda} P^
\noindent\textbf{A:\ }lpha$ if $\lambda$ is a limit ordinal. There is an ordinal $
\noindent\textbf{A:\ }lpha$ such that $P^
\noindent\textbf{A:\ }lpha = P^{
\noindent\textbf{A:\ }lpha+1} = P^*$. The least such ordinal $
\noindent\textbf{A:\ }lpha$ is the \varnothingh{closure ordinal} of the iteration. Such elementary inductions have been extensively studied in logic \cite{ynm,aczel}.
Notice that we have not really used the fact that $\varphi(P,\bar{x})$ is first-order. One property of $\varphi(P,\bar{x})$ that we used was that $\varphi(P,\bar{x})$ is monotone in $P$, that is that, in every structure of the appropriate vocabulary with fixed values for the additional individual variables, $\Gamma$ is a monotone operator. $\varphi(P,\bar{x})$ could be e.g.\ a second-order formula monotone in $P$.
The least fixed point $P^*$ can be denoted $\mbox{LFP}_{P,\bar{x}}\varphi(P,\bar{x})$ and viewed as a $j$-ary relation, so that $[\mbox{LFP}_{P,\bar{x}}\varphi(P,\bar{x})](y_1,\ldots,y_j)$ functions semantically as a formula. This observation gives rise to an idea to use LFP as a new formula constructor, in addition to propositional connectives and quantifiers. Aho and Ullman \cite{au} indeed suggested to enrich first-order logic with the LFP constructor. The new logic became known as FOL+LFP.
Model checking is polynomial time for any FOL+LFP formula $\psi$. In other words, it can be checked in time polynomial in the size of a finite structure $X$ of the vocabulary of $\psi$ whether $X$ with some values for the free individual variables of $\psi$ is a model of $\psi$. Immerman \cite{immerman} and Vardi \cite{vardi} proved that, over ordered finite structures, the converse is true: every property that model checks in polynomial time is expressible in FOL+LFP. In that sense, FOL+LFP captures polynomial time.
Existential fixed point logic (EFPL) is essentially an extension of the existential fragment of first-order logic with the LFP construct. It does not have the universal quantifier and lacks means to simulate universal quantification; see the definition of EFPL in the next section. As far as we know, it was first introduced --- in a different guise --- by Chandra and Harel \cite{ch} in the context of database theory where vocabularies are relational, that is, consist of relation symbols and constants and do not have function symbols of positive arity. Chandra and Harel observed that relational EFPL is equi-expressive with Datalog, a popular database query language.
Existential fixed point logic (EFPL) was further developed by the present authors in \cite{efp}; see Section~\ref{sec:properties}. The motivation came from program verification. We noticed that EFPL was appropriate for formulating pre- and post-conditions in Hoare's logic of asserted programs \cite{hoare}. In particular, the heavy expressivity hypothesis needed for Cook's completeness theorem \cite{cook} in the context of first-order logic is automatically satisfied in the context of EFPL.
More recent developments include a deductive system for EFPL introduced by Compton \cite{compton} and a normal form for EFPL formulas discovered by Grohe \cite{grohe}, who also studied connections between EFPL and other logics. One of the present authors found connections with topos theory and showed that these connections imply some of the other, previously known, nice properties of EFPL \cite{ab-topoi1,ab-topoi2}. The other of the present authors, together with Neeman, applied a logic equivalent to EFPL, called liberal Datalog, to develop a powerful authorization language \cite{dkal}; the equivalence between liberal Datalog and EFPL is shown in detail in \cite{g193}.
In this note, we recall the definition and known properties of EFPL, and then we prove that the truth definition of EFPL formulas can be given in EFPL.
\begin{rmk}
Nikolaj Bj{\o}rner \cite{nikolaj} observed that writing a truth definition for EFPL in EFPL is related to writing an interpreter for EFPL in EFPL. Indeed. But the interesting issue is out of scope here, in this paper, and will have to be addressed elsewhere.
\end{rmk}
\section{Existential fixed-point logic: Definition}
\label{sec:definitions}
As indicated in the introduction, existential fixed-point logic differs from first-order logic in two respects, the absence of the universal quantifier and the presence of the least-fixed-point operator. Both of these deserve some clarification.
First we define existential logic EL. Notice that mere removal of the
universal quantifier $\forall$ has no real effect on first-order
logic, since $\forall x\,\varphi$ can be expressed as $\neg\exists
x\,\neg\varphi$. To correctly define the existential fragment of
first-order logic, one must prevent such surreptitious reintroduction
of the universal quantifier. A traditional way to do that is to
insist that all formulas have the prenex existential form $\exists x_1
\ldots \exists x_n \varphi(x_1,\ldots,x_n)$ where $\varphi$ is
quantifier-free.
But there is an alternative and more convenient form of the existential fragment proposed in \cite{efp}: Allow as propositional connectives only conjunction, disjunction, and negation; use only the existential quantifier; and apply negation only to atomic formulas. It is easy to see that every formula in this alternative fragment is equivalent to one in prenex existential form, and the other way round.
With an eye on the forthcoming introduction of recursion, we stipulate that all relation symbols are divided into two categories: \varnothingh{negatable} and \varnothingh{positive}. And we restrict further the use of negation in the alternative existential fragment of first-order logic: negation can be applied only to atomic formulas with negatable relation symbols. The resulting fragment of first-order logic will be called \varnothingh{existential logic} and denoted EL.
Now we extend existential logic by adding a new formula constructor. As usual, formulas are built by induction from atomic formulas by means of formula constructors. In the case of EFPL, the formula constructors are those of existential logic --- the three propositional connectives and the existential quantifier --- and one additional LET-THEN constructor that is used to construct induction assertions. We explain how the new constructor works.
Let ${\cal F}$ be the collection of formulas constructed so far. A \varnothingh{logic rule} has the form $P(x_1,\ldots,x_j)\leftarrow \delta(P,x_1,\ldots,x_j)$ where $P$ is a positive relation symbol of arity $j$, the $x_i$'s are distinct variables and $\delta$ is any formula in ${\cal F}$. We wrote $\delta$ as $\delta(P,x_1,\ldots,x_j)$ to emphasize that it is allowed to contain the relation symbol $P$ and the individual variables $x_1,\ldots,x_j$, but it may also contain additional individual variables, relation symbols, and function symbols. $P$ is the \varnothingh{head symbol} of the rule and $\delta$ is its \varnothingh{body}. Note that the arrow $\leftarrow$ in a logic rule is not the (reverse) implication connective but a special symbol whose only use, in our syntax, is in forming logic rules. A \varnothingh{logic program} is a finite collection of logic rules. (To write a program as text, one needs to order its rules, but the choice of ordering will never matter.) To be compatible with \cite{efp}, we require that different rules have different head symbols; we could remove this restriction. If $\Pi$ is a program and $\varphi$ is a formula in ${\cal F}$ then
\[
\mbox{LET}\ \Pi\ \mbox{THEN}\ \varphi
\]
is an EFPL formula, an \varnothingh{induction assertion}.
If $P(x_1,\ldots,x_j) \leftarrow \delta$ is a rule in $\Pi$ then all occurrences of the variables $x_1,\ldots,x_j$ in the rule are bound occurrences in the induction assertion. And $P$ is a bound relation variable in the induction assertion.
In general, an occurrence of an individual variable $v$ in a formula $\psi$ is bound if it belongs to a subformula of the form $\exists v\,
\noindent\textbf{A:\ }lpha$ or to a rule of the form $P(\ldots,v,\ldots) \leftarrow \delta$; otherwise the occurrence is free. The free individual variables of $\psi$ are those with free occurrences in $\psi$. An occurrence of relation symbol $P$ in $\psi$ is bound if it belongs to subformula LET $\Pi$ THEN $\varphi$ of $\psi$ and $P$ is a head symbol of $\Pi$; otherwise the occurrence is free. The vocabulary of $\psi$ consists of all the function symbols in $\psi$ and all relation symbols with free occurrences in $\psi$.
It remains to define the semantics of the induction assertion $\psi$ = LET $\Pi$ THEN $\varphi$. To simplify the exposition, we presume that the program $\Pi$ consists of two rules, $P(x_1,\ldots,x_j)\leftarrow
\noindent\textbf{A:\ }lpha$ and $Q(y_1,\ldots,y_k)\leftarrow \beta$. In every structure of the vocabulary of $\psi$ with fixed values for the free individual variables of $\psi$, the program gives rise to an operator
\[
\Gamma(P,Q) \leftarrow (\{\bar{x}:\
\noindent\textbf{A:\ }lpha\},\{\bar{y}:\ \beta\}).
\]
Since $P$ and $Q$ are positive relation symbols, $\Gamma$ is monotone and thus has a least fixed point $(P^*,Q^*)$. To evaluate $\psi$, evaluate $\varphi$ using $P^*$ and $Q^*$ as the values of relations $P$ and $Q$.
\section{EFPL: Some properties}
\label{sec:properties}
We describe some properties of EFPL. The default reference is \cite{efp}.
\subsection*{Capturing polynomial time}
EFPL captures polynomial time computability over structures of the form $\{0,1,\dots,n\}$ with (at least) the successor relation and names for the endpoints. In contrast to the corresponding result for FOL+LFP mentioned above, we use the successor relation here rather than the ordering relation $<$. In fact, both proofs depend on the successor relation rather than the order, but in FOL one can define successor in terms of order (but not vice versa), whereas in EFPL one can define order in terms of successor (but not vice versa).
\subsection*{Validity is r.e.\ complete}
The set of logically valid EFPL formulas is recursively enumerable (in short r.e.). Furthermore, every r.e. set reduces, by means of a recursive function, to the set of valid EFPL formulas. Thus the set of valid EFPL formulas is a complete r.e.\ set.
\subsection*{Satisfiability is r.e.\ complete}
The set of satisfiable EFPL formulas is a complete r.e. set.
\subsection*{Finite validity is co-r.e.\ complete}
The set of EFPL formulas that hold in all finite structures is a complete co-r.e.\ set. In other words, the set of EFPL formulas $\psi$ such that $\psi$ fails in some finite structure is a complete r.e.\ set.
\subsection*{Finite model property}
When an EFPL formula $\psi$ is satisfied in a structure $X$, this fact depends on only a finite part of the structure $X$. More precisely, there is a finite subset $D$ of the elements of $X$ such that $\psi$ is satisfied in every structure $X'$ of the vocabulary of $X$ that coincides with $X$ on $D$. Note that $X'$ can be always chosen to be finite. If we allow basic functions of a structure to be partial, then the property in question can be formulated in a particularly simple way: If an EFPL formula is satisfied in a structure then it is satisfied in a finite substructure.
\subsection*{No transfinite induction is needed}
The closure ordinal of any monotone induction
\[
P \mapsto \{\bar{x}: \varphi(P,\bar{x}) \},
\]
where $\varphi$ is EFPL is at most $\omega$, the first infinite ordinal. The definition of the closure ordinal generalizes in a straightforward way to simultaneous monotone induction. The closure ordinal of the induction given by any logic program is at most $\omega$.
\subsection*{Truth is preserved by homomorphisms}
Truth of EFPL formulas is preserved by homomorphisms. Here a homomorphism is a function $h$ from one structure to another such that
\begin{itemize}
\item $h$ commutes with (the interpretations of) function symbols,
\item $P(a_1,\ldots,a_j)$ implies $P(ha_1,\ldots,ha_j)$\\ for every positive relation symbol $P$ of any arity $j$, and
\item $P(a_1,\ldots,a_j)$ if and only if $P(ha_1,\ldots,ha_j)$\\ for every negatable relation symbol $P$ of any arity $j$.
\end{itemize}
\subsection*{EFPL $\cap$ FOL $\subseteq$ EL
\noindent\textbf{Q:\ }uad}
If an EFPL formula $\varphi$ is expressible in first-order logic then $\varphi$ is equivalent to an existential formula. Only a limited form of this result survives in finite model theory. If an EFPL formula $\varphi$ without function symbols and without negations is equivalent, on finite structures, to a first-order formula, then $\varphi$ is equivalent, on finite structures, to an existential formula without negations \cite{ajtai,rossman}. This fails even if $\varphi$ has no function symbols and only the equality relation is negatable \cite[Section~10]{ajtai}.
\section{Prerequisites for truth}
Our objective in the rest of the article is to show that EFPL can formalize its own truth definition. That is, we shall define, in EFPL with suitable vocabulary, truth of EFPL sentences (that is formulas with no free variables) of the same vocabulary.
We use the term predicate to mean a relation symbol or a relation depending on the context.
Since sentences are built from subformulas that may have free variables, we shall actually define the slightly more general concept of satisfaction of formulas by assignments of values to the free variables. The need to define the more general notion of satisfaction of formulas in order to obtain truth for sentences is familiar from first-order logic.
\footnote{A few authors, notably Shoenfield \cite{shoenfield}, define truth directly. To do so, they expand the vocabulary by adding constants for all elements of the structure under consideration, and instead of assigning values to variables they substitute constants for variables. We could have used this approach for EFPL, but we chose to parallel the more widely used approach in FOL, via satisfaction.}
A new complication, of the same general nature, arises in EFPL. The bound predicates of a sentence $\varphi$ are free in some subformulas of $\varphi$. We should define satisfaction of $\varphi$ in a structure whose vocabulary does not include those predicates. But the definition will pass through subformulas of $\varphi$ whose satisfaction will depend on the interpretations of those predicates. As a result, we need to define satisfaction of $\varphi$ in a context that includes not only a structure (for the vocabulary of $\varphi$) and an assignment of values to the free variables of $\varphi$ (as in FOL) but also the logic rules that provide the meaning of all other predicates that occur in $\varphi$ --- or that occur in the bodies of those rules.
Let $\Upsilon$ be a vocabulary and $X$ a structure of vocabulary $\Upsilon$. Any predicate that does not occur in $\Upsilon$ will be called an \varnothingh{extra predicate}. We shall define satisfaction in $X$ for $\Upsilon$-formulas. Requirements will be imposed shortly on $\Upsilon$ and $X$, but for now $\Upsilon$ is just some vocabulary and $X$ some $\Upsilon$-structure. We intend to define, in EFPL, a ternary predicate Sat such that, when
\begin{ls}
\item the value of its first argument is a formula $\varphi$, of vocabulary $\Upsilon$ plus (possibly) some extra predicates,
\item the value of its second argument is a logic program $\Pi$ whose head predicates include all extra predicates that occur in $\varphi$ or $\Pi$, and
\item the value of its third argument is an assignment $s$ of elements of $X$ to (at least) all individual variables that are free in $\varphi$ or in $\Pi$,
\end{ls}
then the truth value of $\text{Sat}(\varphi,\Pi,s)$ in $X$ is the same as the truth value, in $X$, of $\varphi$ with values for its variables given by $s$ and with the extra predicates interpreted by the least fixed point of (the monotone operator defined by) $\Pi$.
Furthermore, we do not intend to use any clever tricks in our definition of Sat. It will be a formalization of the explanation given above (and in \cite{efp}) of the meaning of EFPL formulas. The point of this work is to show that this formalization can be carried out in EFPL itself.
For all this to make sense, the structure $X$ must contain the formulas $\varphi$ of EFPL, the logic programs $\Pi$, and the assignments $s$. Furthermore, the vocabulary must be adequate to express the basic syntactic properties of formulas and to allow basic constructions of assignments, rules, and programs. We do not, however, wish to specify the exact syntactic nature of formulas --- for example, are they sequences of symbols, or are they parse trees, or are they G\"odel numbers? Our work is independent of such details. So we shall merely assume that certain notions (e.g., the operation of forming the conjunction of two formulas) are expressible; the details of how they are expressed (and which notions are primitive and which are derived) are irrelevant.
\footnote{We shall occasionally indicate how certain notions can be defined from others in EFPL. Those indications can help to reduce the assumptions needed about $\Upsilon$.}
In the rest of this section, we list what we require of our vocabulary $\Upsilon$ and structure $X$, occasionally adding some comments about the reasons for particular requirements.
$\Upsilon$ should be finite. The reason is that the definition of satisfaction must, in the clauses for atomic formulas, use all the relation and function symbols of $\Upsilon$.
The equality predicate should be negatable. The reason is that the notion of EFPL formula requires some things to be distinct, for example the variables in the head of a rule and the head symbols of different rules in a program.
$X$ should contain a copy \bbb N of the natural numbers, and $\Upsilon$ should have a constant symbol for 0 and a unary function symbol $S$ for successor. \bbb N itself, as a unary relation, is definable:
\[
\bbb N(x) :\equiv \text{LET }N(z)\leftarrow z=0\ \lor\ \exists y\,(N(y)\land z=S(y))\text{ THEN }N(x).
\]
We could also define addition and multiplication as ternary relations, and the ordering, and similarly for other primitive recursive functions and relations.
We need \bbb N primarily to index elements of lists, for example the list of terms that serves as the arguments of a relation or function symbol. Since $\Upsilon$ is finite, we could handle the argument lists of its own relation and function symbols in an ad hoc manner, without a general notion of natural number or of list. But EFPL imposes no bound on the arities of the head symbols of logic rules, so atomic formulas can involve arbitrarily long argument lists, and natural numbers are needed for treating these.
Although EFPL does not allow universal quantification in general, it
can simulate universal quantification over finite initial segments of
\bbb N, as shown by the following lemma from \cite{efp}.
\begin{la} \label{bdd-all-N}
For any EFPL formula $\varphi(x)$, there is an EFPL formula $\psi(y)$ equivalent, for all $y\in\bbb N$, to $(\forall x<y)\,\varphi(x)$.
\end{la}
\begin{pf}
The most natural choice of $\psi(y)$ describes a search from 0 up to $y$:\\[1ex]
\begin{minipage}{\textwidth}
\mbox{}
\noindent\textbf{Q:\ }uad
LET $K(x)\leftarrow x=0\ \lor\ \exists w\, \big( x=S(w)\ \land\ K(w)\ \land\ \varphi(w) \big)$ THEN $K(y)$.
\noindent\textbf{Q:\ }ed
\end{minipage}
\end{pf}
\begin{conv}
Consider the definition of \bbb N exhibited above, and notice that its
essential content is contained in the rule
\[
N(z)\leftarrow z=0\, \lor\, \exists y\,(N(y)\land z=S(y)),
\]
which makes the bound predicate symbol $N$ denote the set of natural
numbers. The rest of the definition,
\[
\bbb N(x) :\equiv \text{LET }\dots\text{ THEN }N(x),
\]
merely transfers this denotation to the defined notation \bbb N.
Instead of introducing a bound predicate variable $N$ to, in effect,
duplicate the desired predicate \bbb N, we could convey the same
information by writing
\[
\bbb N(z):\leftarrow z=0\, \lor\, \exists y\,(\bbb N(y)\land z=S(y)).
\]
Although this is not an EFPL formula, we adopt the convention that it
is to serve as an abbreviation of the definition of \bbb N displayed
earlier. In general, when we write a rule with a colon before the
$\leftarrow$, it is to be interpreted as defining a formula. Thus,
\[
\bbb P(\bar x):\leftarrow\delta(\bbb P,\bar x)
\]
means that $\bbb P(\bar x)$ is defined as the formula
\[
\text{LET }Q(\bar z)\leftarrow\delta(Q,\bar z)\text{ THEN }Q(\bar x).
\]
\end{conv}
\begin{conv}
Later, we shall also need to deal with definitions of this sort in which the body $\delta$ is a disjunction of many subformulas. For example, our ultimate goal, the definition of Sat, will have several disjuncts, covering the different syntactic constructs of EFPL. In such cases, it is convenient to present one disjunct (or a small number of them) at a time. Thus, for a small example, the definition of \bbb N above could be broken into two parts:
\begin{align*}
\bbb N(z)&;\leftarrow z=0\\
\bbb N(z)&;\leftarrow \exists y\,(\bbb N(y)\land z=S(y)).
\end{align*}
We use a semicolon before $\leftarrow$ (instead of a colon) to indicate that the full definition involves more disjuncts. (This use of a semicolon as a partial colon is suggested by the word ``semicolon.") In general, if we write several semicolon definitions $\bbb P(\bar x);\leftarrow\delta_i$ for the same $P(\bar x)$, then they are to be understood as meaning $\bbb P(\bar x): \leftarrow \bigvee_i \delta_i$.
\end{conv}
Returning to the requirements on $X$ and $\Upsilon$, we require $X$ to contain the variables and the assignments. The latter are finite partial functions from the variables into (the universe of) $X$. $\Upsilon$ should define a predicate Vbl for the set of variables, a constant symbol $\varnothing$ for the empty assignment, and a ternary function symbol Modify for the function defined as follows: Given an assignment $s$, a variable $v$, and an element $a$ of $X$, $\text{Modify}(s,v,a)$ is the assignment $t$ that sends $v$ to $a$ and otherwise agrees with $s$ (whether or not $a$ is in the domain of $s$).
\begin{conv}
Here and in what follows, we use the terminology ``$\Upsilon$ should
define a predicate for'' some relation on $X$ to mean that there
should be an EFPL formula in vocabulary $\Upsilon$ whose truth set
in $X$ is the desired relation. Of course, the easiest way to
arrange this would be for the given relation to be one of the basic
relations of $X$, so that the required EFPL formula would be
atomic. But it will never matter whether the formula is atomic or
not.
Similarly, when we ask that $\Upsilon$ should have certain function
symbols, we could weaken that to require only some terms, possibly
involving nesting of function symbols, and our proofs would be
unchanged.
\end{conv}
We also need to express ``$s$ is an assignment,'' ``$v$ is in the
domain of $s$,'' and ``$s(v)=a$,'' but we need not assume these
separately, as they are definable from $\varnothing$ and Modify. They are
given, using our conventions above and the familiar convention of
(existentially) quantifying several variables at once, by
\begin{align*}
\text{Assgt}(s)&;\leftarrow s=\varnothing\\
\text{Assgt}(s)&;\leftarrow\exists t,v,a\,(\text{Assgt}(t)\land
\text{Vbl}(v)\land s=\text{Modify}(t,v,a))\\
v\text{ inDom }s&:\leftarrow\exists t,a\,(s=\text{Modify}(t,v,a)).\\
s(v)=a&:\leftarrow\exists t\,(s=\text{Modify}(t,v,a))
\end{align*}
Note that here $s(v)=a$ is defined as a ternary relation, not as an
instance of equality.
We shall also need to have, among the elements of $X$, the relation and function symbols of $\Upsilon$ as well as the extra predicates available as head symbols of rules. Each relation symbol $P$ or function symbol $f$ of $\Upsilon$, should be denoted by a closed term $\dot P$ or $\dot f$ of $\Upsilon$. (We remain flexible as to what the symbols of $\Upsilon$ should be. For example, they could be G\"odel numbers, and then their names $\dot P$ and $\dot f$ could be terms of the form $SS\dots S(0)$. But there are many other options, and all will work. Note, however, that we cannot take all the $\dot f$'s to be simple constant symbols, as they would then be among the $f$'s, and there would not be enough room in a finite $\Upsilon$ for all of these names to have names.)
The extra predicates available as head symbols of rules should have specified numbers of arguments. That is, there should be an $\Upsilon$-definable predicate Arity such that $\text{Arity}(a,n)$ holds in $X$ (for elements $a,n\in X$) if and only if $a$ is one of these head predicate symbols and $n\in\bbb N$ is the number of its argument places.
As mentioned earlier, we shall need lists, so we require that $X$
contain all lists (i.e., finite sequences) of elements of $X$. The
vocabulary $\Upsilon$ should contain at least the constant Nil,
denoting the empty list, and the binary function symbol Append, for
the function that lengthens a list by adding one element at the end.
Thus, for example,
\[
\sq{a,b,c} =\text{Append}(\text{Append}(\text{Append}(\text{Nil},a)
,b),c).
\]
Other predicates and functions that we shall need for dealing with
lists can be defined in terms of Nil and Append.
\begin{align*}
\text{List}(l);\leftarrow&\ l=\text{Nil}\\
\text{List}(l);\leftarrow&\ \exists x,a\,(\text{List}(x)\land l=\text{Append}(x,a))\\
l \text{ hasLength } n;\leftarrow&\ l=\text{Nil}\land n=0\\
l \text{ hasLength } n;\leftarrow&\ \exists x,a,m\,
\big(l=\text{Append}(x,a)\land x\text{ hasLength }m\land n=S(m)\big)\\
(l)_i=a;\leftarrow&\ \exists x\,
\big( x\text{ hasLength }i\land l=\text{Append}(x,a) \big)\\
(l)_i=a;\leftarrow&\ \exists x,b\,
\big( (x)_i=a\land l=\text{Append}(x,b) \big)\\
\text{Cat}(a,b,l);\leftarrow&\ b=\text{Nil} \land l=a\\
\text{Cat}(a,b,l);\leftarrow&\ \exists c,x,m\, \big(\text{Cat}(a,c,m)\ \land\\
&\ b=\text{Append}(c,x) \land (l=\text{Append}(m,x) \big).
\end{align*}
Here $(l)_i=a$, though it looks like an equation, is really a defined ternary relation, whose meaning is that $a$ is the $i\ensuremath{{}^{\text{th}}}$ component of the list $l$, where we start counting with 0, and where the length of $l$ must be at least $i+1$ so that there is an $i\ensuremath{{}^{\text{th}}}$ term. And ``Cat" alludes to ``concatenation". If $a,b,l$ are lists and $\text{Cat}(a,b,l)$ holds, then $l$ is the concatenation $a*b$ of $a$ and $b$.
We note the following consequence of Lemma~\ref{bdd-all-N}, allowing
universal quantification over the elements of a list.
\begin{cor} \label{bdd-all-list}
For any EFPL formula $\varphi(x)$, there is an EFPL formula $\psi(y)$
that holds, when the value of $y$ is a list, if and only if $\varphi$
holds of all elements of that list. That is, $\psi(y)$ is the
result of universally quantifying $\varphi(x)$ over all elements $x$ of
the list $y$.
\end{cor}
\begin{pf}
Use Lemma~\ref{bdd-all-N} to express\\[1ex]
\begin{minipage}{\textwidth}
\mbox{}
\noindent\textbf{Q:\ }quad
$\exists n\, \big( y\text{ hasLength }n\, \land\, (\forall i<n)\,\exists z\, ((y)_i=z\, \land\, \varphi(z)) \big)$.
\noindent\textbf{Q:\ }ed
\end{minipage}
\end{pf}
It will be convenient to write $(\forall x\in y)\,\varphi(x)$ for the
formula $\psi$ given by this corollary.
Finally, $X$ must contain the syntactic entities relevant to EFPL,
such as terms, logic rules, logic programs, and formulas. The precise
nature of these entities depends on arbitrary choices of how to
represent syntax. We require merely that some representation be
present and that $\Upsilon$ be able to describe fundamental syntactic
relationships.
First, $\Upsilon$ should have a binary function symbol Apply, used to
form a compound term $f(t_1,\dots,t_n)$ from an $n$-ary function
symbol $f$ and a list \sq{t_1,\dots,t_n} of $n$ terms, and also used
similarly to form atomic formulas $P(t_1,\dots,t_n)$. Depending on
how syntax is represented, Apply could, for example, be simply a
pairing function, or it could be the operation of prepending an
element to a list, or it could produce a tree from a root and its
immediate subtrees, or it could be an arithmetical operation on
G\"odel numbers.
There should also be a unary function symbol Neg and binary function symbols Conj, Disj, Quant, and IndAsrt for the operations of negating a formula, forming conjunctions, forming disjunctions, forming existential quantifications, and forming induction assertions LET $\Pi$ THEN $\varphi$. The arguments of these operations are intended to be formulas, except that the first argument of Quant is the variable being quantified and the first argument of IndAsrt is the program that goes between LET and THEN.
There should also be a binary function symbol Rule for the operation building a logic rule from its head and its body. We shall take logic programs to be (certain) lists of rules, so we do not need additional capabilities in $\Upsilon$ to handle these. (We could have used sets of rules instead, but then $\Upsilon$ would need additional capabilities.) Finally, there is a ternary relation RenameAway such that, if $\Pi$ is a program and $\varphi$ is a formula and $\text{RenameAway}(\varphi,\Pi,\varphi')$ holds, then $\varphi'$ is a formula obtained from $\varphi$ by renaming the bound predicates of $\varphi$ away from the head predicates of $\Pi$, so that the formula $\varphi'$ is equivalent to $\varphi$, and no head predicate of $\Pi$ is bound in $\varphi'$.
This completes our requirements on $\Upsilon$ and $X$. They can be summarized thus: EFPL syntax and basic combinatorial ingredients for EFPL semantics (like assignments) are available in $X$ and expressible in EFPL in vocabulary $\Upsilon$.
\section{Semantics of terms}
Terms are built, as in FOL, by starting with variables and iteratively
applying function symbols. The definition is formalized as follows.
\begin{align*}
\text{Term}(t)&;\leftarrow \text{Vbl}(t)\\
\text{Term}(t)&;\leftarrow \exists l \big( t=\text{Apply}(\dot f,l)\land
\text{List}(l)\land l\text{ hasLength }\hat n \land(\forall x\in
l)\text{Term}(x) \big).
\end{align*}
Here the second line is to be repeated for each function symbol $f$ of
$\Upsilon$, $n$ is the arity of $f$, and $\hat n$ is the numeral for
$n$, namely $SS\dots S(0)$ with $n$ occurrences of $S$. Recall that
the universal quantification $\forall x \in l$ was introduced after
Corollary~\ref{bdd-all-list} as an abbreviation of an EFPL formula.
Recall also that $\Upsilon$ is finite, so there is no difficulty
writing the appropriate line for each $f$.
Semantically, a term gets a value (in the given structure $X$) once an assignment provides values for all the variables in $t$. So the values of terms are given by a binary function, whose arguments are a term and an assignment. To define it recursively, we regard this binary function as a ternary relation, and we define it as follows.
\begin{align*}
\text{Val}(t,s,a)&;\leftarrow \text{Vbl}(t)\land\text{Assgt}(s)\land s(t)=a\\
\text{Val}(t,s,a)&;\leftarrow \exists l,u_0,\dots,u_{n-1},b_0,\dots, b_{n-1}\\
& \big(t=\text{Apply}(\dot f,l)\, \land\, \text{List}(l)\, \land\,
l\text{ hasLength }\hat n\, \land\, \text{Assgt}(s)\\
& \land \bigwedge_{i<n}((l)_i=u_i\, \land\, \text{Val}(u_i,s,b_i))\,
\land\, a=f(b_1,\dots,b_n) \big).
\end{align*}
The explanatory comments after the definition of Term apply here as
well.
\begin{rmk}
In principle, we could do without the definition of Term. The
definition of Val assigns values only to terms in any case. But it
would do no harm if Val were defined in some extraneous cases, as
long as it worked correctly for terms.
\end{rmk}
\section{Semantics of formulas}
As indicated earlier, the semantics of a formula involves not only the structure $X$ and an assignment $s$ but also a collection $\Pi$ of logic rules to determine the meaning of any extra predicates used in the formula but not bound by LET-THEN constructions in the formula. Ultimately, when we deal with $\Upsilon$-formulas, there will be no such extra predicates, so $\Pi$ will be irrelevant, but in the recursive construction of an $\Upsilon$-formula (and in the recursive definition of its satisfaction), subformulas can occur that do use extra predicates. So we shall define Sat as a ternary predicate, where the intended meaning of $\text{Sat}(\varphi,\Pi,s)$ is that the formula $\varphi$ is true, in our given structure $X$, when the extra predicates are interpreted by the least fixed point of $\Pi$ and the free variables are assigned values by $s$.
The definition of Sat will have numerous clauses, according to the
last constructor used in building $\varphi$, so we shall make much use of
the ``;$\leftarrow$'' convention. This way, we can present the
clauses one (or a few) at a time and insert comments and even other
definitions between them.
We begin with the case of atomic formulas whose predicates are from $\Upsilon$. The definition is quite analogous to the earlier definition of the values of terms.
\begin{equation}
\begin{split}
\text{Sat}(\varphi,\Pi,s)&;\leftarrow \exists l,u_0,\dots,u_{n-1},b_0,\dots,b_{n-1}\\
& \big( \varphi=\text{Apply}(\dot P,l)\, \land\, \text{List}(l)\, \land\,
l\text{ hasLength }\hat n\, \land\, \text{Assgt}(s)\\
& \land
\bigwedge_{i<n}((l)_i=u_i\, \land\, \text{Val}(u_i,s,b_i))\,
\land\, P(b_1,\dots,b_n) \big).
\end{split}
\end{equation}
This is to be repeated for all of the (finitely many) predicates $P$ of $\Upsilon$ with $n$ being the arity of $P$. As before, $\hat n$ is the numeral for $n$.
The case of negated atomic formulas is almost the same; of course it
is to be repeated only for negatable $P$.
\begin{equation}
\begin{split}
\text{Sat}(\varphi,\Pi,s)&;\leftarrow \exists l,u_0,\dots,u_{n-1},b_0,\dots,b_{n-1}\\
& \big(\varphi=\text{Neg}(\text{Apply}(\dot P,l))\, \land\, \text{List}(l)\, \land\,
l\text{ hasLength }\hat n\, \land\, \text{Assgt}(s)\\
&\land
\bigwedge_{i<n}((l)_i=u_i\, \land\, \text{Val}(u_i,s,b_i))\, \land\,
\neg P(b_1,\dots,b_n) \big).
\end{split}
\end{equation}
Rather than continuing with the remaining atomic formulas, those that
use extra predicates, let us first dispose of the remaining ``easy''
clauses, those not involving $\Pi$.
\begin{equation}
\begin{split}
\text{Sat}(\varphi,\Pi,s)&;\leftarrow
\exists
\noindent\textbf{A:\ }lpha,\beta\, \big(\varphi=\text{Conj}(
\noindent\textbf{A:\ }lpha,\beta)\, \land\,
\text{Sat}(
\noindent\textbf{A:\ }lpha,\Pi,s)\, \land\, \text{Sat}(\beta,\Pi,s) \big)\\
\text{Sat}(\varphi,\Pi,s)&;\leftarrow
\exists
\noindent\textbf{A:\ }lpha,\beta\, \big(\varphi=\text{Disj}(
\noindent\textbf{A:\ }lpha,\beta)\, \land\,
(\text{Sat}(
\noindent\textbf{A:\ }lpha,\Pi,s)\, \lor\, \text{Sat}(\beta,\Pi,s) \big)\\
\text{Sat}(\varphi,\Pi,s)&;\leftarrow
\exists
\noindent\textbf{A:\ }lpha,v,a\, \big(\varphi=\text{Quant}(v,
\noindent\textbf{A:\ }lpha)\, \land\,
\text{Sat}(
\noindent\textbf{A:\ }lpha,\Pi,\text{Modify}(s,v,a)) \big)
\end{split}
\end{equation}
This completes the easier part of the definition of Sat, the part
concerning just EL. To complete the definition for EFPL, we must deal
carefully with programs in both of their roles --- as the second
argument of Sat and as a constituent of induction assertions.
This will require some preliminaries. First, we need the notion of a
list with no repetitions.
\begin{align*}
\text{1-1-List}(l)&:\equiv \exists n\, \big(l\text{ hasLength }n\, \land \\
& (\forall i,j<n)\,\exists x,y\,
((l)_i=x\, \land\, (l)_j=y\land(i=j\, \lor\, \neg(x=y))) \big).
\end{align*}
We also need a construction that amounts to applying a unary function to each element of a list, producing a new list. The situation is complicated by the fact that our unary functions are often given as binary relations. We therefore adopt the following notation. If we have defined a binary relation $R$, then we write $R^+$ for the relation defined as follows.
\begin{align*}
R^+(l,m)&:\equiv\exists n\, \big(l\text{ hasLength }n\,
\land\, m\text{ hasLength }n\, \land \\
& (\forall i<n)\, \exists u,v\,
((l)_i=u\, \land\, (m)_i=v\, \land\, R(u,v)) \big).
\end{align*}
For example, let us define HS (abbreviating ``head symbol'') by
\[
HS(r,p):\equiv\exists y,z\,(r = Rule(Apply(p,y),z)).
\]
Then when $\Pi$ is a list of rules, $HS^+(\Pi,m)$ means that $m$ is the
list of their head symbols. One of the requirements for a program is
that this list $m$ be one-to-one, so there will be a clause $\exists
m\, (HS^+(\Pi,m)\land\text{1-1-List}(m))$ in the definition of
program.
We shall also use the plus-notation with a parameter. Specifically,
we think of $\text{Val}(u,s,b)$ as the graph of a function $u\mapsto
b$ with $s$ fixed, so the plus-notation makes $\text{Val}^+(\bar
u,s,\bar b)$ the relation between a list of terms and their values, all
for the same assignment $s$. We refrain from writing out the
definition, since it's just like the definition of $R^+$ above, with
the extra argument $s$ inserted into both $R$ and $R^+$.
We need an improved version of the function Modify, to modify an
assignment by mapping all the variables in a list $l$ to the
corresponding values in another list $q$ (of the same length).
\begin{align*}
\text{Change}(s,l,q,r)&;\leftarrow
l=\text{Nil}\, \land\, q=\text{Nil}\, \land\, s=r\\
\text{Change}(s,l,q,r)&;\leftarrow\exists l',q',r',v,a\,
\big (l=\text{Append}(l',v)\, \land\, q=\text{Append}(q',a)\\
& \land\, \text{Change}(s,l',q',r')\, \land\, r=\text{Modify}(r',v,a) \big).
\end{align*}
With these preliminaries, we can write down the definition of
satisfaction for atomic formulas that begin with one of the extra
predicates. The idea is to find, in $\Pi$, the rule having
that symbol as its head symbol, and to use the body of that rule as
the criterion of truth for our atomic formula. It will be useful
later to make sure that the $\Pi$ in the second argument place of Sat
has no repeated head symbols, so we include that in the definition.
\begin{equation}
\begin{split}
\text{Sat}(\varphi,\Pi,s)&;\leftarrow\exists p,t,k,i,m,l,r,q,\delta\\
& \big(\varphi=\text{Apply}(p,t)\, \land\, t\text{ hasLength }k\, \land\,
\text{Arity}(p,k)\ \land\\
& (\forall x\in t)\ \text{Term}(x)\, \land\, \text{HS}^+(\Pi,m)\, \land\,
\text{1-1-List}(m)\ \land\\
& (\Pi)_i=\text{Rule}(\text{Apply}(p,l),\delta)\, \land\, \text{1-1-List}(l)\ \land\\
& l\text{ hasLength }k\, \land\, (\forall x\in l)\,\text{Vbl}(x)\, \land\,
\text{Val}^+(t,s,q)\ \land\\
& \text{Change}(s,l,q,r)\, \land\, \text{Sat}(\delta,\Pi,r) \big).
\end{split}
\end{equation}
In prose, the essential part of this says that $\varphi$ has the form $p(\bar t)$ for an extra predicate of arity $k$, with $\bar t$ being a $k$-tuple of terms; that $\Pi$ contains a rule $p(\bar l)\leftarrow\delta$ with head $p$, $\bar l$ being a $k$-tuple of distinct variables; and that $\delta$ is satisfied by the assignment $r$ obtained from $s$ by replacing each of the variables in the list $\bar l$ by the value of the corresponding element of $\bar t$. This replacement amounts, intuitively, to taking the definition of $p(\bar l)$ as $\delta(\bar l)$ and applying it to $p(\bar t)$, the terms $\bar t$ replacing the variables $\bar l$. Instead of doing a syntactic substitution of $\bar t$ for $\bar l$ in $\delta$, we have made the corresponding semantic change, assigning to the variables in $\bar l$ the values of the terms in $\bar t$.
It may seem strange that this clause in the definition of Sat says
nothing about iterating the operator defined by $\delta$. After all,
$p$ should be interpreted as the least fixed point of that operator.
But the desired iteration is automatically accomplished by the
iteration involved in the definition of Sat. That is, if $p$ occurs
in $\delta$, then the true instances of $p$ can contribute to the
true instances of $\delta$ and can thereby contribute to additional
true instances of $p$.
We must still provide the clause for induction assertions in our definition of Sat. Fortunately, this is relatively easy, since iteration is already implicitly done in the preceding clause.
\begin{equation}
\begin{split}
\text{Sat}(\varphi,\Pi,s)\ &;\leftarrow\ \exists \varphi',\Sigma,
\noindent\textbf{A:\ }lpha,\Theta\\
& \big( \text{RenameAway}(\varphi,\Pi,\varphi')\, \land\, \varphi'=\text{IndAsrt}(\Sigma,
\noindent\textbf{A:\ }lpha)\\
& \land\, \text{Cat}(\Pi,\Sigma,\Theta)\, \land\, \text{Sat}(
\noindent\textbf{A:\ }lpha,\Theta,s) \big).
\end{split}
\end{equation}
Here $\varphi'$ is equivalent to $\varphi$ and so $\text{Sat}(\varphi,\Pi,s)$ should be equivalent to $\text{Sat}(\varphi',\Pi,s)$. Further, $\varphi'$ = LET $\Sigma$ THEN $
\noindent\textbf{A:\ }lpha$, and no head predicate of $\Pi$ is bound in $\varphi'$. It follows that the head predicates of $\Pi$ are disjoint from the head predicates of $\Sigma$, so that the concatenation $\Theta$ of $\Pi$ and $\Sigma$ is a legitimate program. Accordingly $\text{Sat}(\varphi',\Pi,s)$ should be equivalent to $\text{Sat}(
\noindent\textbf{A:\ }lpha,\Theta,s)$.
That concludes the definition of $\text{Sat}(\varphi,\Pi,s)$. It is easy to see that it works as intended. In the case when $\varphi$ is a sentence and when both $\Pi$ and $s$ are empty, $\text{Sat}(\varphi,\Pi,s)$ holds in the structure $X$ if and only $\varphi$ does.
\end{document}
|
\begin{document}
\begin{abstract}
Let $X$ be a projective, equidimensional, singular scheme over an algebraically closed field. Then the existence of a geometric smoothing (i.e. a family of deformations of $X$ over a smooth base curve whose generic fibre is smooth) implies the existence of a formal smoothing as defined by Tziolas.
In this paper we address the reverse question giving sufficient conditions on $X$ that guarantee the converse, i.e. formal smoothability implies geometric smoothability.
This is useful in light of Tziolas' results giving sufficient criteria for the existence of formal smoothings.
\end{abstract}
\title{From formal smoothings to geometric smoothings}
{\hypersetup{linkcolor=black}
\tableofcontents}
\section{Introduction}
Let $X$ be a proper $k$-scheme of finite type over an algebraically closed field $k$ of characteristic $0$. A \emph{geometric smoothing} of $X$ is a Cartesian diagram
\begin{equation}
\begin{tikzcd}
X \arrow[d]\arrow[r, hook] & \mathcal{X} \arrow[d, "p"]\\
\Spec k \arrow[r, hook, "c"] & C
\end{tikzcd}
\end{equation}
where $C$ is a smooth curve, $c\in C$ is a closed point and $p$ is a flat, proper morphism such that $p^{-1}(\eta_C)=:\mathcal{X}_{\text{gen}}$ is smooth, where $\eta_C$ is the generic point of $C$. We say that $X$ is \emph{geometrically smoothable} if it has a geometric smoothing. Following \cite[Definition~11.6]{tziolas2010smoothings}, we define a \emph{formal smoothing} of $X$ to be a formal deformation
\[
\begin{tikzcd}
X \arrow[d] \arrow[r, hook] & \mathfrak{X} \arrow[d, "\mathfrak{p}"]\\
\Spf k \arrow[r, hook] & \Spf k\llbracket t\rrbracket
\end{tikzcd}
\]
such that there exists a $b\in\mathbb{N}$ with $\mathfrak{I}^b\subset\Fitt_{\dim X}(\Omega^1_{\mathfrak{X}/\Spf k\llbracket t\rrbracket})$, where $\mathfrak{I}$ is an ideal of definition of $\mathfrak{X}$ and $\Fitt_{a}(\Omega^1_{\mathfrak{X}/\Spf k\llbracket t\rrbracket})$ is the $a^{\text{th}}$ Fitting sheaf of ideals (see \cite[\href{https://stacks.math.columbia.edu/tag/0CZ3}{Tag 0CZ3}]{stacks-project}). We say that $X$ is \emph{formally smoothable} if it admits a formal smoothing.
Note that if $X$ is smooth then it is geometrically (hence formally) smoothable. Furthermore, Tziolas proved that geometrical smoothability implies formal smoothability. The main result of this paper is the following:
\begin{thm}\label{theorem: A}
If $X$ is a projective, equidimensional and singular scheme over $k$ such that one of the following assumptions hold:
\begin{enumerate}
\item $\coho^2(X,\mathcal{O}_X)=0$,
\item if $X$ Gorenstein, then either the dualising sheaf $\omega_X$ or its dual $\omega_X^{\vee}$ is ample,
\end{enumerate}
then the formal smoothability of $X$ is equivalent to its geometrical smoothability.
\end{thm}
The above theorem also extends Grothendieck's algebraisation theorem, see \cite[\href{https://stacks.math.columbia.edu/tag/089A}{Tag 089A}]{stacks-project}, since we have found a way to enlarge the parameter space from the spectrum of a local complete k-algebra to an affine curve.
\subsection{Motivation}
This result is motivated by the study of moduli spaces of surfaces of general type and their higher-dimensional analogues.
Moduli spaces of surfaces of general type are well studied and it is known that stable surfaces lie within the compactification of these moduli spaces.
A \emph{stable surface}, see \cite{Kollarbook}, is a proper two-dimensional reduced connected scheme satisfying one local and one global condition. The local condition bounds the badness of singularities that such surfaces can have, requiring them to be semi-log-canonical (see \cite[Definition~1.40]{Kollarbook}).
The global condition requires the dualising sheaf to be ample.
Since stable surfaces appear as points on the boundary of the moduli space of surfaces of general type, it is of great interest to understand which stable surfaces are geometrically smoothable.
In order to understand which surfaces can be smoothed, it is important to know which singularities among the semi-log-canonical ones can be smoothed. The class of such singularities is very broad since it admits both isolated and non-isolated singularities. If $X$ has isolated singularities, it is known \cite[Proposition~2.4.6]{sernesi2007deformations} that $\coho^2(X,\mathcal{T}_X)$ is an obstruction space to the extension of local smoothings to global ones.
The study of non-isolated singularities is not so easy. In \cite{Persson1983SomeEO}, they gave examples of non-smoothable singularities with normal crossing divisors, showing that not all non-isolated singularities are smoothable. Another difficulty that one has to face studying non-isolated singularities is that the Schlessinger's cotangent sheaf $\mathcal{T}^1$ (and its higher analogues), which is a sheaf supported on the singular locus, is difficult to describe and sometimes not finite dimensional, as shown in \cite{Fantechi2017OnTR}. An application to Godeaux surfaces of \Cref{theorem: A} is given in \cite{fantechi2021smoothing}.
\subsection{Structure of the paper}
This paper is an expository article on formal schemes, formal deformation and smoothing. It organized in four sections: in the first one it is collected an introduction to formal schemes, following the treatment of Illusie in \cite{FGAIll} and of Alonso, Jerem\'ias and P\'erez in \cite{tlr1} and \cite{tlr2}. The second section contains a discussion on formal deformation theory with, what we hope, a clear treatment on the differences between the various type of definition of deformations. We decide to add this information in order fix the terminology and better clarify what is the main point of this article. This section ends with a discussion of two different notions of smoothing of a scheme; in particular, in there we motivate the definition of formal smoothing as given by Tziolas in \cite{tziolas2010smoothings}. The third section is an overview of the Gorenstein condition and its behaviour under deformation, mostly following \cite{stacks-project}. Since we were not able to find a reference in the literature, in this section we include a proof of a classical result on good behaviour of the Gorenstein property under infinitesimal deformations. The fourth and last part contains the main result, its proof and an example of application to a real moduli problem.
\subsection{Conventions}\label{conventions}
All schemes are defined over an algebraically closed field $k$ of characteristic $0$. We will assume that all schemes will be of finite type and separated and we will denote by $\text{FTS}_k$ (or simply by FTS) the category whose objects are separated, finite type $k$-schemes and whose morphisms are morphisms of $k$-schemes.
\subsection{Acknowledgment}
I would like to thank the algebraic geometry group at SISSA for useful mathematical discussions and precious suggestions. A very special thanks is due to my PhD advisor, prof. Barbara Fantechi, for her constant patience, support and precious advices.\\
I would also like to thank the algebraic geometry group at Universt\'e du Luxembourg.
\section{Locally Noetherian formal schemes}
We recall for the reader's convenience some basic results on formal schemes. We follow Illusie's and Grothendieck's language and presentation in \cite{FGAIll} and \cite{MR0217083} respectively. At some points we will also refer to articles \cite{tlr1} and \cite{tlr2} by Alonso, Jerem\'ias and P\'erez.
\subsection{The category of locally Noetherian formal schemes}
\begin{definition}\label{def: adic ring}
An \emph{adic} (or \emph{$I$-adic}) \emph{Noetherian ring} is a topological Noetherian ring $A$ that admits an ideal $I$, called an \emph{ideal of definition}, such that
\begin{itemize}
\item $\{I^n\}_{n\in\mathbb{N}}$ is a fundamental system of neighbourhoods of $0$ in $A$;
\item the topology induced on $A$ turns $A$ into a separated and complete topological space.
\end{itemize}
\end{definition}
In general an ideal of definition is not unique. Indeed for another ideal $J$ to be an ideal of definition it is necessary and sufficient that there are two non-negative integers $n,m$ such that $J\supset I^m\supset J^n$.
We remark that $A$ is an $I$-adic Noetherian ring if and only if $A=\varprojlim_n A/I^{n}=:\hat{A}$, where $\hat{A}$ denotes the formal completion of $A$ along the ideal $I$.
Examples of adic Noetherian rings are the \emph{ring of formal power series} and the \emph{ring of restricted power series}, see \cite[Example 1.6]{tlr1}, in the following denoted respectively by $k\llbracket t\rrbracket$ and $A\{T_1,\dots,T_n\}$, with $A$ an $I$-adic Noetherian ring and $t$, $T_1,\dots,T_n$ indeterminates.
We wish to introduce the notion of an affine formal scheme, needed for the definition of a formal scheme. If $A$ is an $I$-adic Noetherian ring $A$ and $n$ a non-negative integer, we denote by $A_n$ the quotient $A/I^{n+1}$ and by $X_n$ the affine scheme $\Spec A_n$. We then have a chain of closed subschemes
\[
X_0\subset X_1\subset\cdots\subset X_n\subset\cdots
\]
and all these subschemes have the same underlying topological space $|\Spec A/I|$.
\begin{definition}\label{def: formal affine spectrum}
Let $A$ be an adic Noetherian ring with $I$ an ideal of definition. The \emph{affine formal spectrum of $A$} is the topologically ringed space $(\Spf A, \mathcal{O}_{\Spf A})$ where
\begin{itemize}
\item the topological space is
\[
\Spf A:=\{\mathfrak{p}\in\Spec A\colon I\subset \mathfrak{p}\}
\]
which is naturally homeomorphic to $|\Spec A/I |$. Equivalently, we could have defined $\Spf A$ to be the topological space made by open primes ideals of $A$;
\item the structure sheaf is
\[
\begin{aligned}
\mathcal{O}_{\Spf A}:=\varprojlim_n \mathcal{O}_{X_n}
\end{aligned}
\]
\end{itemize}
and is a sheaf of topological rings. Its topology is given by
\[
\Gamma(U,\mathcal{O}_{\Spf A})=\varprojlim_n\Gamma(U,\mathcal{O}_{X_n})
\]
for every open subset $U$ of $\Spf A$, where $\Gamma(U,\mathcal{O}_{X_n})$ has the discrete topology.
\end{definition}
The definition above does not depend on the ideal of definition. Indeed if a prime ideal $\mathfrak{p}$ of $A$ contains $I$ it also contains all of its powers, in particular it contains $I^m$ and hence $J^n$. Since $\mathfrak{p}$ is prime, it follows that it contains also $J$.
Since the topology of $\Spf A$ admits a base of neighbourhoods made by quasi-compact open subsets, it is enough to require that for every quasi-compact open subset $U$ of $\Spf A$,
\[
\Gamma(U,\mathcal{O}_{\Spf A})=\varprojlim_n\Gamma(U,\mathcal{O}_{X_n}),
\]
where $\Gamma(U,\mathcal{O}_{X_n})$ has the discrete topology (see \cite[(\textbf{1}.10.1.1)]{MR0217083}).
\begin{remark}\label{remark: canonical inclusion}
For an $I$-adic Noetherian ring $A$, the canonical morphism $A\to\hat{A}$ is an isomorphism and it induces a morphism of ringed spaces from $\Spf\hat{A}=\Spf A$ to $\Spec A$.
\end{remark}
We can now define what are affine Noetherian formal schemes and locally Noetherian formal schemes.
\begin{definition}\label{def: affine Noetherian formal scheme}
An \emph{affine Noetherian formal scheme} is a topologically ringed space isomorphic to an affine formal spectrum as in \Cref{def: formal affine spectrum}.
\end{definition}
\begin{definition}\label{def: formal scheme}
A \emph{locally Noetherian formal scheme} is a topologically ringed space $(\mathfrak{X}, \mathcal{O}_{\mathfrak{X}})$ such that every point has an open neighbourhood which is isomorphic to an affine Noetherian formal scheme.
A \emph{Noetherian formal scheme} is a quasi-compact locally Noetherian formal schemes.
\end{definition}
Since affine formal schemes are locally topologically ringed spaces, locally Noetherian formal schemes are locally topologically ringed spaces.
As in the classical case, we denote the locally Noetherian formal scheme $(\mathfrak{X}, \mathcal{O}_{\mathfrak{X}})$ by $\mathfrak{X}$.
\begin{notation}
For the rest of the article we will abbreviate ``locally Noetherian formal scheme'' by LNFS.
\end{notation}
Example of locally Noetherian formal schemes, which are in particular affine Noetherian formal schemes, are $\Spf k\llbracket t\rrbracket$ and $\Spf A\{T_1,\dots,T_n\}$. In what follows, we will denote the formal scheme $\Spf A\{T_1,\dots,T_n\}$ by $\mathbb{A}^{n}_{\Spf A}$ and we will call it the formal affine $n$-space. Observe that the underlying topological space of $\mathbb{A}^{n}_{\Spf A}$ is $\Spec\left( (A/I)[T_1,\dots, T_n]\right)$.
\begin{notation}
In what follows we will denote $\Spf k\llbracket t\rrbracket$ by $\mathfrak{S}$ and, for every non-negative integer $n$,
$S_n$ will denote $\Spec\frac{k\llbracket t\rrbracket}{(t^{n+1})}=\Spec\frac{k[ t]}{(t^{n+1})}$.
\end{notation}
Now we define morphisms between LNFSs.
\begin{definition}\label{def: morphism of formal schemes}
Let $\mathfrak{X}$ and $\mathfrak{Y}$ be two LNFSs. A \emph{morphism of LNFSs} is a morphism $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ of locally ringed spaces such that for every open subset $\mathfrak{V}$ of $\mathfrak{Y}$ the induced map
\[
\Gamma(\mathfrak{V}, \mathcal{O}_{\mathfrak{Y}})\to\Gamma(\mathfrak{f}^{-1}(\mathfrak{V}), \mathcal{O}_{\mathfrak{X}})
\]
is continuous.
\end{definition}
As in the classical case of schemes, there is an equivalence of categories between adic Noetherian rings and affine Noetherian formal schemes, for more see \cite[(\textbf{1}.10.2.2)]{MR0217083}. Furthermore, the classical adjunction holds also in the case of LNFSs.
\begin{proposition}[{\cite[(\textbf{1}.10.4.6)]{MR0217083}}]
Let $\mathfrak{X}$ be a LNFS and let $A$ be a Noetherian adic ring. Then there is a natural bijection between morphisms of locally Noetherian formal schemes from $\mathfrak{X}$ to $\Spf A$ and continuous ring homomorphisms from $A$ to $\Gamma(\mathfrak{X}, \mathcal{O}_{\mathfrak{X}})$.
\end{proposition}
As a further example of formal schemes, we can consider the \emph{completion of a scheme along a closed subscheme}.
\begin{example}\label{example: formal completion along a closed subscheme}
Suppose that $X$ is a locally Noetherian scheme and consider a closed subscheme $Y$ of $X$ with sheaf of ideals given by $\mathcal{I}$. Then we can consider the schemes $X_n:=(Y, \mathcal{O}_{X}/\mathcal{I}^{n+1})$, for every $n\in\mathbb{N}$, which gives rise to the sequence of thickenings
\[
X_0\hookrightarrow X_1\hookrightarrow\cdots X_n\hookrightarrow\cdots
\]
Taking now the colimit we get a LNFS, denoted by $\hat{X}_{/Y}$ and called \emph{the formal completion of $X$ along $Y$}.
\end{example}
We point out that, if $Y=X$, then $\hat{X}_{/X}=X$. Therefore the category of LNFSs contains the category of Noetherian schemes.
However, Hironaka and Matsumura in \cite[Theorem~(5.3.3) page 81]{hironaka1968formal} and independently Hartshorne in \cite[Example 3.3 page 205]{hartshorne2006ample} constructed two examples showing that not all formal schemes appear as the completion of a single scheme along a closed subscheme. This consideration motivates the following definition.
\begin{definition}\label{def: algebraisable formal scheme}
A LNFS $\mathfrak{X}$ is called \emph{algebraisable} if there are a scheme $X$ and a closed subscheme $Y$ of $X$ such that $\mathfrak{X}=\hat{X}_{/Y}$.
\end{definition}
\subsection{Sheaves on LNFSs}
We now define the notion of a coherent formal sheaf on a LNFS. In the classical case of affine Noetherian schemes there is the functor $\widetilde{(-)}$ that associates to any finitely generated module its coherent sheaf. Similarly, in the formal case there is the functor $(-)^{\Delta}$ which associates to any finitely generated module its formal coherent sheaf.
Note that if $A$ is an adic Noetherian ring, then every $A$-module $M$ has an induced $I$-adic topology where a system of fundamental neighbourhoods of $0$ is given by $\{I^n\cdot M\}_{n\in\mathbb{N}}$.
\begin{notation}\label{notation: homomorphisms instead of continuous homomorphisms}
If $A$ is a Noetherian $I$-adic ring and $M$ and $N$ are finitely generated $A$-modules that are separated and complete in the induced $I$-adic topology, then, by \cite[(\textbf{0}.7.8.1)]{MR0217083} it follows that every $A$-module homomorphism is automatically continuous. Therefore, in what follows, we will write $\Hom_{A}(M,N)$ in place of $\Hom_{A-\text{cont}}(M,N)$.
Furthermore, by \cite[(\textbf{0}.7.8.2)]{MR0217083} we have a canonical isomorphism
\[
\Hom_{A}(M,N)\stackrel{\cong}{\rightarrow}\varprojlim_{n}{}{}\Hom_{\frac{A}{I^{n+1}}}\left(\frac{M}{I^{n+1}M},\frac{N}{I^{n+1}N}\right).
\]
From this we conclude that, if $A$ is a Noetherian $I$-adic ring and $M$ is a finitely generated $A$-module, then
\[
M^{\vee}:=\Hom_{A}(M, A)=\varprojlim_{n}{}{}\Hom_{\frac{A}{I^{n+1}}}\left(\frac{M}{I^{n+1}M},\frac{A}{I^{n+1}}\right)=\varprojlim_{n}\left(\frac{M}{I^{n+1}M}\right)^{\vee}.
\]
\end{notation}
\begin{definition}\label{def: Delta construction for modules}
Let $A$ be an $I$-adic Noetherian ring and let $M$ be a finitely generated $A$-module. Then we define the \emph{coherent formal sheaf} $M^{\Delta}$ on $\Spf A$ to be the completion of $\widetilde{M}$ along the ideal sheaf $\widetilde{I}$ of the closed embedding $\Spec A/I\hookrightarrow\Spec A$:
\[
M^{\Delta}:=\varprojlim_n \frac{\widetilde{M}}{\widetilde{I^n}\cdot\widetilde{M}}.
\]
\end{definition}
The functor $(-)^{\Delta}$ satisfies similar properties of the functor $\widetilde{(-)}$, for more see \cite[(\textbf{1}.10.10.2)]{MR0217083}.
\begin{definition}\label{def: ideal sheaf of definition}
An \emph{ideal of definition} of a LNFS $\mathfrak{X}$ is a formal coherent sheaf of ideals $\mathfrak{I}$ of $\mathcal{O}_{\mathfrak{X}}$ such that for any point $x\in\mathfrak{X}$ there exists a formal affine neighbourhood $\Spf A$ of $x$ in $\mathfrak{X}$ and there exists an ideal of definition $I$ of $A$ such that $\mathfrak{I}|_{\Spf A}=I^{\Delta}$.
\end{definition}
A formal coherent sheaf $\mathfrak{I}$ on a LNFS $\mathfrak{X}$ is an ideal of definition if and only if $(\mathfrak{X},\frac{\mathcal{O}_{\mathfrak{X}}}{\mathfrak{I}})$ is a scheme.
Actually, for any LNFS $\mathfrak{X}$ there exists a maximal ideal of definition $\mathfrak{I}$ which is the unique ideal of definition such that $(\mathfrak{X},\frac{\mathcal{O}_{\mathfrak{X}}}{\mathfrak{I}})$ is a reduced scheme.
In a Noetherian formal scheme the ideal of definition is not unique; indeed, any other formal coherent sheaf of ideals $\mathfrak{J}$ on the LNFS $\mathfrak{X}$ is an ideal of definition if and only if there are positive integers $m,n$ such that the chain of inclusions
$\mathfrak{J}\supset\mathfrak{I}^{m}\supset\mathfrak{J}^n$ holds.
\begin{remark}\label{rem: definition of formal scheme as collection of infinitesimal neighbourhoods}
As in the affine formal case, it is also possible to define LNFSs as a collection of all of their infinitesimal neighbourhoods (or thickenings).
More precisely, let $\mathfrak{X}$ be a LNFS and let $\mathfrak{I}$ be an ideal of definition. For every $n\in\mathbb{N}$, define $(X_n,\mathcal{O}_{X_n})$ to be the ringed space $(|\mathfrak{X}|, \mathcal{O}_{\mathfrak{X}}/\mathfrak{I}^{n+1})$ which is a locally Noetherian scheme. This induces a sequence of closed embeddings
\[
X_0\hookrightarrow X_1\hookrightarrow X_2\hookrightarrow \cdots\hookrightarrow X_n\hookrightarrow \cdots
\]
whose ideals of definition are nilpotent and all the maps on the underlying topological spaces are the identity. Then $\mathfrak{X}$ can be recovered from the above sequence of thickenings by passing through the direct limit in the category of locally Noetherian topologically ringed spaces, i.e.
\[
\mathfrak{X}=\varinjlim_n X_n.
\]
In particular there are natural morphisms of ringed spaces
\[
\alpha_n\colon X_n\to\mathfrak{X},
\]
where $\alpha_n$ is the identity on the underlying topological space and the map of sheaves of topological rings is just the quotient map
\[
\alpha_n^{\natural}\colon\mathcal{O}_{\mathfrak{X}}\to\mathcal{O}_{X_n}=\frac{\mathcal{O}_{\mathfrak{X}}}{\mathfrak{I}^{n+1}}.
\]
Conversely, see \cite[(\textbf{1}.10.6.3)]{MR0217083}, given a collection $\{X_n\}_{n\in\mathbb{N}}$ of locally Noetherian schemes satisfying:
\begin{itemize}
\item[(i)] for every $n$, there are morphisms of schemes $\psi_{n+1,n}\colon X_n\to X_{n+1}$ such that they are homeomorphisms on the underlying topological spaces and induce surjective morphisms of sheaves $\mathcal{O}_{X_{n+1}}\to\mathcal{O}_{X_{n}}$;
\item[(ii)] if $\mathcal{J}_n:=\ker(\mathcal{O}_{X_n}\to\mathcal{O}_{X_0})$, then $\ker(\mathcal{O}_{X_n}\to\mathcal{O}_{X_m})=\mathcal{J}_n^{m+1}$, for $m\leq n$;
\item[(iii)] $\mathcal{J}_1\in\coh(X_0)$;
\end{itemize}
then the topologically ringed space $\mathfrak{X}:=\varinjlim_n X_n$ obtained by taking the direct limit is a LNFS. Moreover, denoting by $\mathfrak{I}:=\ker(\mathcal{O}_{\mathfrak{X}}\to\mathcal{O}_{X_0})$, then $\mathfrak{I}$ is an ideal of definition of $\mathfrak{X}$ and satisfies the following properties
\[\mathfrak{I}=\varprojlim_n\mathcal{J}_n\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ and }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathfrak{I}^{n+1}=\ker(\mathcal{O}_{\mathfrak{X}}\to\mathcal{O}_{X_n}).
\]
\end{remark}
\begin{definition}\label{def: coherent formal sheaf}
A \emph{coherent formal sheaf} on a LNFS $\mathfrak{X}$ is a sheaf $\mathfrak{F}$ such that, for every open Noetherian formal affine subset $\mathfrak{U}=\Spf A$ of $\mathfrak{X}$, there exists a finitely generated $A$-module $M$ with $\mathfrak{F}|_{\mathfrak{U}}=M^{\Delta}$.
\end{definition}
Next we give an interpretation of coherent formal sheaves on a LNFS as the limit of coherent sheaves on all thickenings.
\begin{remark}\label{remark: formal coherent sheaf as collection of coherent sheaves on the thickenings}
Let $\mathfrak{X}$ be a LNFS, let $\mathfrak{I}$ be an ideal of definition and let $\mathfrak{F}$ be a coherent formal sheaf of $\mathcal{O}_{\mathfrak{X}}$-modules. For every $n$, let us denote by $X_n$ the locally Noetherian scheme as defined in \Cref{rem: definition of formal scheme as collection of infinitesimal neighbourhoods}. If, for every $n$, we define
\[
\mathscr{F}_n:=\frac{\mathfrak{F}}{\mathfrak{I}^{n+1}\mathfrak{F}},
\]
then we have that $\mathscr{F}_n\in\coh(X_n)$ and we recover $\mathfrak{F}$ by considering $\varprojlim_n\mathscr{F}_n$.
Conversely, see \cite[(\textbf{1}.10.11.3)]{MR0217083}), let $\mathfrak{X}$ be a locally Noetherian scheme and $\mathfrak{I}$ and ideal of definition of $\mathfrak{X}$. Let $\{X_n\}_{n\in\mathbb{N}}$ be a collection of locally Noetherian schemes defining $\mathfrak{X}$ as in \Cref{rem: definition of formal scheme as collection of infinitesimal neighbourhoods} and, for $m\leq n$, let $\psi_{n,m}\colon X_m\to X_n$ denote the canonical maps. Suppose that for every $n\in\mathbb{N}$, $\mathscr{F}_n$ is a coherent sheaf on $X_n$ together with morphisms, for $m\leq n$
\[
\phi_{n,m}\colon\mathscr{F}_{m}\to(\psi_{n,m})_*\mathscr{F}_{n},
\]
such that for every $l\geq m\geq n$ we have $\phi_{n,m}\circ\phi_{m,l}=\phi_{n,l}$\footnote{The conditions listed here are equivalent to requiring that the system $\{\mathscr{F}_n, \phi_{m,n}\}_{n,m\in\mathbb{N}}$ be a projective system.}. Then the limit $\mathfrak{F}:=\varprojlim_n\mathscr{F}_n$ is a coherent formal sheaf on $\mathfrak{X}$.
\end{remark}
\begin{definition}\label{def: locally free formal sheaf} Let $\mathfrak{X}$ be a LNFS and $r\in\mathbb{N}$. We say that a formal coherent sheaf $\mathfrak{F}$ on $\mathfrak{X}$ is locally free of rank $r$ if for every open Noetherian affine subset $\mathfrak{U}=\Spf A$ of $\mathfrak{X}$, the finitely generated $A$-module $M$ (which exists since $\mathfrak{F}$ is coherent) is free of rank $r$.
\end{definition}
We can give an equivalent definition of formal coherent sheaf on a LNFS based on the infinitesimal thickening description of LNFSs. It is done as follows: a formal coherent sheaf $\mathfrak{F}$ on a LNFS $\mathfrak{X}$ is locally free of finite rank $r$ if each sheaf $\mathscr{F}_n:=\frac{\mathfrak{F}}{\mathfrak{I}^{n+1}\mathfrak{F}}$ is locally free of the same rank $r$, for all natural numbers $n$, where $\mathfrak{I}$ is an ideal of definition of the formal scheme $\mathfrak{X}$.
\subsection{Adic morphisms between LNFSs}
In order to give a description in terms of thickenings for morphisms of formal schemes, we need to restrict our interest to a particular kind of morphisms: the adic morphisms.
\begin{definition}\label{def: adic morphism}
A morphism $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ of LNFSs is called an \emph{adic morphism} if there exists an ideal of definition $\mathfrak{J}$ of $\mathfrak{Y}$ such that $\mathfrak{f}^*\mathfrak{J}\cdot\mathcal{O}_{\mathfrak{X}}$ is an ideal of definition of $\mathfrak{X}$.
\end{definition}
The definition of an adic morphism does not depend on the choice of the ideal of definition; indeed one could equivalently ask that the condition $\mathfrak{f}^*\mathfrak{J}\cdot\mathcal{O}_{\mathfrak{X}}$ holds \textit{for all} ideals of definition of $\mathfrak{Y}$ (see \cite[(\textbf{1}.10.12.1)]{MR0217083}). Observe that if $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ is an adic morphism bethween LNFSs, then the topology on $\mathcal{O}_{\mathfrak{Y}}$ determines the topology on $\mathcal{O}_{\mathfrak{X}}$.
\begin{remark}\label{rem: morphism of formal schemes as a collection of compatible morphisms of schemes}
Suppose that $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ is an adic morphism of LNFSs and let $\mathfrak{J}$ and $\mathfrak{I}:=\mathfrak{f}^*\mathfrak{J}\cdot\mathcal{O}_{\mathfrak{X}}$ be ideals of definition of $\mathfrak{Y}$ and $\mathfrak{X}$ respectively.
Then we can consider the sequences of thickenings
\[
X_0\hookrightarrow X_1\hookrightarrow\cdots X_n\hookrightarrow\cdots
\,\,\,\,\text{ and }\,\,\,\,
Y_0\hookrightarrow Y_1\hookrightarrow\cdots Y_n\hookrightarrow\cdots
\]
as in \Cref{rem: definition of formal scheme as collection of infinitesimal neighbourhoods}.
Since the morphism was supposed to be adic, we get that for every $n\in\mathbb{N}$, $\mathfrak{f}^*(\mathfrak{J}^{n+1})\cdot\mathcal{O}_{\mathfrak{X}}=\mathfrak{I}^{n+1}$. Therefore we have induced morphisms
\[
f_n\colon X_n\to Y_n
\]
such that all the squares
\begin{equation}\label{eq: adic morphism have Cartesian squares}
\begin{tikzcd}
X_n \arrow[r, "f_n"] \arrow[d, hook] & Y_n \arrow[d, hook]\\
X_{n+1} \arrow[r, "f_{n+1}"] & Y_{n+1}
\end{tikzcd}
\end{equation}
are Cartesian. Then $\mathfrak{f}$ can be recovered by the collection of morphisms $\{f_n\}_{n\in\mathbb{N}}$ by considering the colimit, i.e. $\mathfrak{f}=\varinjlim_n f_n$.
Conversely (see \cite[(8.1.5)]{FGAIll}), any system of morphisms of locally Noetherian schemes $\{f_n\colon X_n\to Y_n\}_{n\in\mathbb{N}}$ such that all squares \cref{eq: adic morphism have Cartesian squares} are Cartesian induces an adic morphism of LNFSs by considering the colimit.
\end{remark}
\subsection{Properties of adic morphisms}
Now we introduce the notions of finite type, properness and flatness for a morphism of formal schemes.
\begin{definition}\label{def: morphism of finite type of formal scheme}
Let $\mathfrak{X}$ and $\mathfrak{Y}$ be LNFSs. A morphism $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ is said to be \emph{of finite type} if $\mathfrak{f}$ is an adic morphism and the induced morphism $f_0\colon X_0\to Y_0$ is of finite type.
\end{definition}
\begin{definition}\label{def: proper morphism of formal schemes}
A morphism $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ of LNFSs is \emph{proper} if it is of finite type and $f_0\colon X_0\to Y_0$ is proper.
\end{definition}
\begin{definition}\label{def: flat morphism of formal schemes}
Let $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ be a morphism of LNFSs. We say that $\mathfrak{f}$ is \emph{flat} if it is adic and for every $x\in\mathfrak{X}$, $\mathcal{O}_{\mathfrak{X},x}$ is a flat $\mathcal{O}_{\mathfrak{Y}, \mathfrak{f}(x)}$-module.
\end{definition}
\begin{proposition}
Let $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ be an adic morphism of LNFSs, let $\{f_n\colon X_n\to Y_n\}_{n\in\mathbb{N}}$ be a compatible collection associated to $\mathfrak{f}$ and $\mathcal{P}$ be one of the following properties of morphisms: of finite type, proper, flat. Then the following conditions are equivalent:
\begin{enumerate}
\item $\mathfrak{f}$ has $\mathcal{P}$;
\item $f_n$ has $\mathcal{P}$, for every $n\in\mathbb{N}$.
\end{enumerate}
\end{proposition}
We point out that we could have defined a flat morphism of LNFSs $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ without assuming it to be adic. However, with that choice, we would not be able to deduce the flatness of $\mathfrak{f}$ from the flatness of all $\{f_n\}_{n\geq0}$ and vice versa. See \cite[Proposition~3.3]{tlr2} for the local criterion of flatness for formal schemes, and \cite[Example~3.2]{tlr2} gives a counter example.
We conclude the section by presenting one result needed in the proof of the main result.
\begin{theorem}[{\cite[II - Ex. 9.6(c)]{hartshorne1977algebraic}}]\label{teo: locally free sheaves on each nilpotent subscheme induce a locally free sheaf on the completion}
Let $\mathfrak{X}$ be a LNFS, let $\mathfrak{I}$ be an ideal of definition of $\mathfrak{X}$ and, for each $n\in\mathbb{N}$, let us denote by $X_n$ the scheme $(\mathfrak{X}, \mathcal{O}_{\mathfrak{X}}/\mathfrak{I}^n)$. Suppose that, for every $n\in\mathbb{N}$, we are given invertible sheaves $\mathscr{L}_n$ on $X_n$ together with isomorphisms $\mathscr{L}_{n+1}\otimes_{\mathscr{O}_{X_{n+1}}}\mathscr{O}_{X_n}\cong\mathscr{L}_n$. Then the sheaf
\[
\mathfrak{L}:=\varprojlim_n\mathscr{L}_n
\]
is an invertible sheaf on $\mathfrak{X}$.
\end{theorem}
\section{On deformations and smoothings}
In this section we introduce various definitions of deformations of a scheme and we discuss their relationship. Then we present and explain the two different definitions of smoothing of a scheme used in this paper.
\subsection{Introducing formal deformations}
\begin{definition}\label{def: formal deformation}
Let $X$ be a scheme and let $(R, \mathfrak{m})$ be a complete local ring. A \emph{formal deformation} of $X$ over $R$ is a Cartesian diagram
\begin{equation}\label{eq: formal deformation}
\begin{tikzcd}
X \arrow[d] \arrow[r, hook] & \mathfrak{X} \arrow[d, "\mathfrak{f}"]\\
\Spf(\frac{R}{\mathfrak{m}}) \arrow[r, hook] & \Spf R
\end{tikzcd}
\end{equation}
with $\mathfrak{f}$ a flat morphism.
\end{definition}
\begin{notation}
In the future, in order to ease the notation, we will denote any deformation (either classical or formal) by its flat morphism. For example, we will refer to the formal deformation of \cref{eq: formal deformation} only by $\mathfrak{f}\colon\mathfrak{X}\to\Spf R$.
\end{notation}
As we have seen before in \Cref{rem: definition of formal scheme as collection of infinitesimal neighbourhoods}, formal schemes can be equivalently described as compatible collection of infinitesimal thickenings. A similar description can be given for formal deformations.
\begin{remark}\label{rem: formal deformations and thickenings}
Fix a formal deformation of a scheme $X$ as in \cref{eq: formal deformation} and, for any non-negative integer $n$, let us denote by $R_n$ the quotient ring $R/\mathfrak{m}^{n+1}$. Then, for any $n\geq0$, we have diagrams
\[
\begin{tikzcd}
& \mathfrak{X} \arrow[d, "\mathfrak{f}"]\\
\Spec R_n \arrow[r, hook] & \Spf R.
\end{tikzcd}
\]
Pulling back $\mathfrak{f}$ along the closed immersion $\Spec R_n\hookrightarrow\Spf R$, we obtain a collection of deformations $\{f_n\colon\mathcal{X}_n\to\Spec R_n\}_{n\geq0}$ of $X$ over $\Spec R_n$. Moreover, by construction, all these deformations of $X$ are compatible, i.e. for every non negative integer $n$, we have Cartesian diagrams
\[
\begin{tikzcd}
\mathcal{X}_n \arrow[r, hook] \arrow[d, "f_n"] & \mathcal{X}_{n+1} \arrow[d, "f_{n+1}"]\\
\Spec R_n \arrow[r, hook] & \Spec R_{n+1}.
\end{tikzcd}
\]
\end{remark}
The converse also holds true, as stated in the following proposition.
\begin{proposition}[{\cite[Proposition~21.1]{hartshorne2009deformation}}]\label{prop: formal deformation as compatible collection of deformations}
Let $(R,\mathfrak{m})$ be an adic local Noetherian ring with residue field $k$, let $X$ be a scheme and define and, for every non-negative integer $n$, let $R_n:=R/\mathfrak{m}^{n+1}$. Suppose that for every $n\in\mathbb{N}$ we are given a family $\{f_n\colon\mathcal{X}_{n}\to\Spec R_{n}\}_{n\geq0}$ of infinitesiaml deformations such that $\mathcal{X}_0 = X$, the morphisms $f_n$ are flat, of finite type and the following compatibility condition holds: for all $n\geq0$, the diagrams
\begin{equation}\label{eq: compatibility of deformation to induced a formal deformation}
\begin{tikzcd}
\mathcal{X}_{n} \arrow[r, hook] \arrow[d, "f_{n}"] & \mathcal{X}_{n+1} \arrow[d, "f_{n+1}"]\\
\Spec R_{n} \arrow[r, hook] & \Spec R_{n+1}
\end{tikzcd}
\end{equation}
are all Cartesian.
Then there exists a (Noetherian) formal scheme $\mathfrak{X}$, flat over $\Spf R$, such that $\mathcal{X}_n\cong\mathfrak{X}\times_{\Spf R}\Spec R_n$, for every natural number $n$.
\end{proposition}
Concluding, \Cref{rem: formal deformations and thickenings} together with \Cref{prop: formal deformation as compatible collection of deformations} imply that a formal deformation $\mathfrak{f}\colon\mathfrak{X}\to\Spf R$ is uniquely determined by a family of infinitesimal deformations $\{f_n\colon\mathcal{X}_n\to\Spec R_n\}_{n\geq0}$ satisfying the compatibility condition expressed by asking that all diagrams of \cref{eq: compatibility of deformation to induced a formal deformation} must be Cartesian.
Next we explain how to construct a formal deformation starting from a deformation over the spectrum of an algebra essentially of finite type.
\begin{remark}\label{remark: constructing formal deformation from classic deformation}
Let $X$ be a scheme, let $(A,\mathfrak{m})$ be a $k$-algebra essentially of finite type, i.e. a localisation of a $k$-algebra of finite type. Consider a deformation of $X$ over $A$
\[
\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & \mathcal{X} \arrow[d, "f"]\\
\Spec k \arrow[r, hook] & \Spec A.
\end{tikzcd}
\]
Let $\hat{A}$ be the formal completion of $A$ at $\mathfrak{m}$; for every $n\geq0$, define $A_n$ to be the quotient ring $A/\mathfrak{m}^{n+1}$ and note that we have canonical isomorphisms $\hat{A}/\mathfrak{m}^{n+1}\hat{A}\cong A_n$ (see \cite[Theorem~7.1~b)]{eisenbud1995commutative}). Now, for every natural number $n$, consider the following diagram of solid arrows
\[
\begin{tikzcd}
\mathcal{X}_n \arrow[d, dashed, "f_n"] \arrow[r, hook, dashed] & \mathcal{X} \arrow[d, "f"]\\
\Spec A_n \arrow[r, hook] & \Spec A
\end{tikzcd}
\]
and complete it to a Cartesian one. For every non-negative integer $n$, we have that $f_n\colon\mathcal{X}_n\to\Spec A_n$ is a deformation of $X$ and all these deformations satisfy the compatibility condition of \cref{eq: compatibility of deformation to induced a formal deformation}. By applying \Cref{prop: formal deformation as compatible collection of deformations} we have constructed a formal deformation $\mathfrak{f}\colon\mathfrak{X}\to\Spf\hat{A}$.
\end{remark}
We call the formal deformation $\mathfrak{f}$ constructed in \cref{remark: constructing formal deformation from classic deformation} the \emph{formal deformation associated} to $f$.
\subsection{Relations among different types of deformations}
It is now a good time to exploit the relationships among the deformations we will find in this article. We start by recalling a few definitions taken from \cite{sernesi2007deformations}.
\begin{definition}
Let $X$ be a proper scheme over an algebraically closed field $k$ and consider the following Cartesian diagram of schemes
\begin{equation}\label{eq: deformation}
\begin{tikzcd}
X \arrow[d] \arrow[r, hook] & \mathcal{X} \arrow[d, "f"]\\
\Spec k \arrow[r, hook, "b"'] & B
\end{tikzcd}
\end{equation}
with $f$ flat, proper and surjective morphism, $b\in B$ a closed point inducing the closed embedding $b\colon\Spec k\hookrightarrow B$. We say \cref{eq: deformation} is
\begin{itemize}
\item[(a)] a \textit{family of deformations} of $X$ iff $B$ is a connected $k$-scheme;
\item[(b)] an \textit{algebraic deformation} of $X$ iff $B$ is a $k$-scheme (essentially) of finite type;
\item[(c)] a \textit{local deformation} of $X$ iff $B$ is the affine spectrum of a local Noetherian $k$-algebra with residue field $k$;
\item[(d)] an \textit{infinitesimal deformation} of $X$ iff $B=\Spec A$ with $A$ a local Artinian $k$-algebra with residue field $k$;
\item[(e)] a \textit{first-order deformation} of $X$ iff $B=\Spec k[\varepsilon]/(\varepsilon^2)$.
\item[(f)] We say that a Cartesian diagram
\begin{equation*}
\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & \mathfrak{X} \arrow[d, "\mathfrak{f}"]\\
\Spec k \arrow[r, hook] & \Spf A
\end{tikzcd}
\end{equation*}
of formal schemes is a \textit{formal deformation} iff $A$ is a local complete Noetherian $k$-algebra with residue field $k$ and $\mathfrak{f}$ is a flat proper morphism of finite type of formal schemes. As we have shown, this is equivalent to give a collection of infinitesimal deformations $\{f_n\colon X_n\to B_n\}_{n\in\mathbb{N}}$, where $B_n:=\Spec A/\mathfrak{m}_{A}^{n+1}$, such that the following diagram is Cartesian
\[
\begin{tikzcd}
X_n\arrow[d, "f_n"] \arrow[r, hook] & X_{n+1}\arrow[d, "f_{n+1}"]\\
B_n\arrow[r, hook] & B_{n+1}.
\end{tikzcd}
\]
\end{itemize}
\end{definition}
We remark that in cases (c), (d), (e) and (f) the underlying topological spaces of $X$ and $\mathcal{X}$ (respectively $\mathfrak{X}$) are the same and what is changing is the scheme (respectively formal scheme) structure. In particular it follows that the properness condition of $X$ is equivalent to $f$ (respectively $\mathfrak{f}$) being proper.
In the same hypotheses and notations used in the previous definition, we have the following properties:
\begin{enumerate}
\item any algebraic deformation induces a local one by taking the closed point $b\in B$ and considering the pull-back of $f\colon\mathcal{X}\to B$ along the closed embedding $\Spec\mathcal{O}_{B,b}\hookrightarrow B$; \item any infinitesimal deformation is in particular a local deformation since every Artinian ring is Noetherian too;
\item any first order deformation is an infinitesimal one because the ring of dual numbers $k[\varepsilon]/(\varepsilon^2)$ is an example of Artinian ring;
\item since, by definition, a formal deformation is a (numerable) collection of infinitesimal deformations, we get that any formal deformation induces countably many infinitesimal deformation;
\item on the other hand, any local deformation induces a formal one. To see this, let $\mathfrak{m}_{b}$ denotes the maximal ideal of the local ring $\mathcal{O}_{B,b}$ and, for any $n\in\mathbb{N}$, consider the following diagram made by Cartesian faces
\[
\begin{tikzcd}
& X \arrow[dd] \arrow[dl, hook] \arrow[dr, hook] &\\
\mathcal{X}_n \arrow[dd, "\pi_n"'] \arrow[rr, hook] & & \mathcal{X}\arrow[dd, "\pi"]\\
& \Spec k \arrow[dl, hook] \arrow[dr, hook] &\\
\Spec\frac{\mathcal{O}_{B,b}}{\mathfrak{m}_{b}^{n+1}}\arrow[rr, hook] & & \Spec\mathcal{O}_{B,b}.
\end{tikzcd}
\]
Doing this for every $n\in\mathbb{N}$ we get a collection of compatible deformations of $X$, which defines a formal deformation.
\end{enumerate}
In this work there are more steps to be aware of. To explain them, let us consider the following diagram of deformations of a $k$-scheme $X$ (this simply means that each vertical arrow is a deformation of $X$):
\[
\begin{tikzcd}
Y\arrow[d, "l"']\arrow[r, hook] & \mathfrak{X} \arrow[d, "\mathfrak{f}"']\arrow[r] &\mathcal{T} \arrow[d, "f"'] \arrow[rrdd, bend left = 10] &&&\\
\Spec C \arrow[r, hook] & \Spf A \arrow[r] & \Spec A \arrow[rdd, dashed] \arrow[rrdd, bend left = 10] &&&\\
&&& \arrow[from=luu, crossing over, dashed] \mathcal{Z} \arrow[d, "g"] \arrow[r, crossing over] &\mathcal{Y} \arrow[d, "h"] \arrow[r, hook] & \mathcal{X} \arrow[d, "w"]\\
&&& \Spec D \arrow[r] &\Spec E \arrow[r, hook] & B\\
\end{tikzcd}
\]
where $B$ is a $k$-scheme of finite type, $E$ is a $k$-algebra (essentially) of finite type (essentially of finite type means that it is the localization of a $k$-algebra of finite type), $D$ is a $k$-algebra which is also a DVR, $A$ is a local complete Noetherian $k$-algebra and $C$ is a local Artinian $k$-algebra.
We will say that a morphism defining a deformation is induced by another if the second deformation is isomorphic (as deformations, see \cite[page $21$]{sernesi2007deformations}) to the pull-back of the first along the closed embedding on the base. We point out that, in general, there is not a natural arrow from $f$ to $g$, hence the dashed arrow, unless $A$ is taken to be the completion of the DVR $D$ along its maximal ideal. Now, $h$ is induced by $w$ since the closed embedding $b\colon\Spec k\to B$ factors through the spectrum of a $k$-algebra (essentially) of finite type. Passing from $h$ to $f$ can be done as follows: since $E$ is the localization of a $k$-algebra of finite type, it has a maximal ideal $\mathfrak{m}_{E}$ and we can complete $E$ along such maximal ideal. Similarly, from $g$ we can deduce $f$ by considering the completion of the DVR along the powers of its maximal ideal. $f$ induces $\mathfrak{f}$ since the formal spectrum has a natural map to the affine spectrum. Since the quotient of a DVR by powers of its maximal ideal is an Artinian ring, it follows that $g$ induces $l$. Similarly, $f$ induces $l$. Lastly, the formal deformation $\mathfrak{f}$ induces a infinitesimal deformation $l$ since the quotient of a local complete Noetherian ring by a power of the maximal ideal is an Artinian ring.
\subsection{Reversing some constructions on deformation}
Reversing some constructions above is usually a hard problem and without further hypotheses on the scheme $X$ is a very hard one. For example, passing from a formal deformation of a $k$-scheme $X$ to a deformation of the same scheme over an affine spectrum of a $k$-algebra (essentially) of finite type means to find ``an algebraisation of the formal deformation''. By an algebraisable formal deformation we mean the following:
\begin{definition}\label{def: algebraic deformation}
Let $X$ be a scheme and let $(A,\mathfrak{m})$ be a complete local Noetherian ring. A formal deformation $\mathfrak{f}\colon\mathfrak{X}\to\Spf A$ is called \emph{algebraisable} if there exist
\begin{itemize}
\item a $k$-algebra essentially of finite type $(R, \mathfrak{n})$,
\item a deformation $g\colon\mathcal{Y}\to\Spec R$ of $X$,
\item an isomorphism $A\cong\hat{R}_{\mathfrak{n}}$,
\item an isomorphism between $\mathfrak{f}$ and the formal deformation $\mathfrak{g}\colon\mathfrak{Y}\to\Spf\hat{R}$ associated to $g$.
\end{itemize}
The deformation $g\colon\mathcal{Y}\to\Spec A$ is called an \emph{algebraisation} of $\mathfrak{f}$.
\end{definition}
The existence of an algebraisation is a very difficult problem. To solve it, Artin introduced in \cite{artin1969algebraization} a weaker condition than algebraisation, ``effectivity of a formal deformation'', which we introduce next.
\begin{definition}\label{def: effective deformation}
Let $X$ be a scheme and let $(A,\mathfrak{m})$ be a complete local Noetherian ring. A formal deformation $\mathfrak{f}\colon\mathfrak{X}\to\Spf A$ is called \emph{effective} if there exists a deformation
\[
\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & \mathcal{X} \arrow[d, "f"]\\
\Spec k \arrow[r, hook] & \Spec A
\end{tikzcd}
\]
with $f$ a flat morphism of finite type such that $\mathfrak{X}=\hat{\mathcal{X}}_{/X}$.
\end{definition}
The idea of Artin was to split the problem of algebraisation in two subproblems:
\begin{itemize}
\item[(i)] to prove the effectivity of the formal deformation: in other words, using notations above, find conditions on $X$ to extend the formal deformation $\mathfrak{f}$ to the deformation $f$;
\item[(ii)] to find hypotheses on the formal deformation to extend it to the spectrum of an (essentially) of finite type $k$-algebra.
\end{itemize}
For step (ii), a sufficient criterion was given by Artin in \cite{artin1969algebraization} and goes by the name of Artin algebraisation theorem, see \cite[Theorem 2.5.14]{sernesi2007deformations}. In there, under the hypotheses that the central fibre $X$ is a projective scheme, Artin showed that if the formal deformation is versal, see \cite[Definition~2.2.6]{sernesi2007deformations}, and effective then it is algebraisable.
However, step (i) above can not be always achieved: for instance, the universal formal deformation of a $K3$ surface is not effective, see \cite[Example 2.5.12]{sernesi2007deformations}.
Recall from \Cref{def: algebraisable formal scheme} that a LNFS $\mathfrak{X}$ is called algebraizable if there are a scheme $Y$ and a closed subscheme $X$ of $Y$ such that $\mathfrak{X}=\hat{Y}_{/X}$. It also makes sense to define algebraisable schemes in the relative setting. For this, suppose we have a formal scheme $\mathfrak{X}$ over the affine formal scheme $\Spf A$, with $A$ a local adic Noetherian $k$-algebra $A$ with residue field $k$.
\begin{definition}
We say that $\mathfrak{X}$ is \emph{algebraisable} over $\Spf A$ if there exists a scheme $\mathcal{X}$ over $\Spec A$ such that $\mathfrak{X}$ is isomorphic to the formal completion $\widehat{\mathcal{X}}_{/X_{0}}$, where $X_{0}:=\mathfrak{X}\times_{\Spf A}\Spec\frac{A}{I}$.
When the formal scheme $\mathfrak{X}$ is proper over $\Spf A$, we say that it is algebraisable if there exists a proper scheme $\mathcal{X}$ over $\Spec A$ such that $\mathfrak{X}\cong\widehat{\mathcal{X}}_{/X_0}$, where again $X_{0}:=\mathfrak{X}\times_{\Spf A}\Spec\frac{A}{I}$.
\end{definition}
Note that the ring $A$ is left fixed but we are changing the locally ringed space structure induced by it.
At this point one wonders if there are conditions to ensure algebraisability of a formal scheme and, in the case an algebraisation exists, how unique it is. We first address the latter problem. Assume we have found an algebraisation of a locally Noetherian formal scheme; then in general it is not unique and a counterexample is given in \cite[Remark~8.4.8.]{FGAIll}. However, in \cite[Corollary~8.4.7.]{FGAIll}, Illusie proved that if we restrict to proper formal schemes then an algebraisation is unique up to a unique isomorphism inducing the identity on $\mathfrak{X}$.
\begin{theorem}[{\cite[(\textbf{3}.5.4.5)]{MR0163910} or \cite[Theorem~8.4.10]{FGAIll}}]\label{Gro: algebrization theorem}
Let $A$ be a Noetherian $I$-adic ring, let $T=\Spec A$, $\mathfrak{T}:=\Spf A$, let $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{T}$ be a proper morphism of formal schemes. For any $l\in\mathbb{N}$, let $T_l:=\Spec (A/I^{l+1})$, $X_l:=\mathfrak{X}\times_{\mathfrak{T}}T_l$. Suppose that there is an invertible formal sheaf $\mathfrak{L}$ such that $\mathfrak{L}_0:=\mathfrak{L}/I\mathfrak{L}$ is an ample invertible sheaf on $X_0$. Then $\mathfrak{X}$ is algebraisable.
Furthermore, if $\mathcal{X}$ is its algebraisation, which is proper over $T$ since $\mathfrak{f}$ was supposed proper, then there exists a unique ample invertible sheaf $\mathcal{M}$ on $\mathcal{X}$ such that $\mathfrak{L}\cong\widehat{\mathcal{M}}_{/X_0}$.
\end{theorem}
We present now a few remarks on the above theorem. The first one is that the hypotheses $\mathfrak{f}$ proper, which is equivalent to $X_0$ proper over $k$, and $\mathfrak{L}_{0}$ ample on $X_{0}$ together imply that $X_0$ is projective over $T_0$. Furthermore, since $\mathfrak{f}$ was proper, the algebraisation of $\mathfrak{X}$ is proper over $T$ by the above definition; in particular it is unique up to a unique isomorphism inducing the identinty on the formal scheme. We also remark that the existence of an ample invertible sheaf $\mathcal{M}$ on $\mathcal{X}$ together with the fact that $\mathcal{X}$ is proper, imply that $\mathcal{X}$ is projective over $T$.
A corollary of the above theorem, is the classical result by Grothendieck:
\begin{theorem}[{\cite[Th{\'e}or{\`e}me~4]{GAGF1960}}]\label{theorem: effectivisation of deformation II}
Let $A$ be a local adic Noetherian ring with residue field $k$, let $\mathfrak{X}$ be a proper formal scheme over $\Spf A$ and suppose that
\begin{enumerate}
\item the local rings of $\mathcal{O}_{\mathfrak{X}}$ are flat $A$-modules (in other words $\mathfrak{f}$ is flat);
\item $X_0:=\mathfrak{X}\otimes_{A}k$ satisfies $\coho^2(X_0,\mathcal{O}_{X_0})=0$;
\item $X_0$ is projective.
\end{enumerate}
Then $\mathfrak{X}$ is algebraisable and its algebraisation is projective over $\Spec A$.
\end{theorem}
We can interpret \Cref{theorem: effectivisation of deformation II} as a theorem on deformations; it says that, in the same notations as above, if the structure morphism $\mathfrak{f}\colon\mathfrak{X}\to\Spf A$ is proper and a formal deformation of a projective scheme $X_0$ with $\coho^2(X_0, \mathcal{O}_{X_0})=0$, then the formal deformation $\mathfrak{f}$ is effective. Therefore \Cref{theorem: effectivisation of deformation II} gives sufficient conditions to achieve step (i) above.
The difference between the algebraisation of a formal scheme over a formal affine scheme, say $\Spf A$ with $A$ as above, and the algebraisation of a formal deformation over $\Spf A$ with proper central fibre lies in the base affine scheme: in the algebraisation of the formal scheme, the $k$-algebra is required to be complete, while in the algebraisation of the formal deformation the $k$-algebra is required to be (essentially) of finite type.
Examples of formal schemes that are not algebraizable (resp. formal deformations that are not effective) are $K3$ surfaces and Abelian varieties, see \cite[Example~2.5.12 ]{sernesi2007deformations} or \cite[Remark~8.5.24(b) and remark~8.5.28(a)]{FGAIll}. Even though in both cases we are able to extend (resp. deform) the scheme at all infinitesimal neighbourhoods, there are ample line bundles that do not lift to the whole formal scheme. This is the consequence of the fact that the space of all deformation of pairs Abelian variety together with an ample line bundle on it (or $K3$ surface together with an ample line bundle) is a proper subspace of the space of all deformations of Abelian varieties (or of $K3$ surfaces).
\subsection{Motivating formal smoothness}
In the next part we introduce two definitions of smoothing that will be relevant in the following. In particular we motivate why the definition of formal smoothing given by Tziolas in \cite{tziolas2010smoothings} is the most natural and, in some sense, the only one possible in our framework.
This section was motivated by the following result of Tziolas, which is key to our argument.
\begin{proposition}[{\cite[Proposition~11.8]{tziolas2010smoothings}}]\label{Tziolas: smoothing over dvr is the same as formal smoothing}
Let $Y$ be a proper, equidimensional scheme and let $A$ be a $k$-algebra which is a DVR. Let $g\colon\mathcal{Y}\to\Spec A$ also be a deformation of $Y$ over $A$ and let $\mathfrak{g}\colon\mathfrak{Y}\to\Spf\hat{A}$ be the associated formal deformation. Then $g$ is a smoothing if and only if $\mathfrak{g}$ is a formal smoothing.
\end{proposition}
The importance of the above result is that it gives a criterion to recognise if a one-parameter deformation is a smoothing by checking if the associated formal deformation is a formal smoothing. Let us first introduce the two definitions of smoothings.
\begin{definition}\label{def: smoothing}
Let $Y$ be a proper, equidimensional scheme and let $A$ be a $k$-algebra which is a DVR. We say that a deformation $g\colon\mathcal{Y}\to\Spec A$ of $Y$ over $A$ is a \emph{smoothing} if the generic fibre $\mathcal{Y}_{\text{gen}}:=\mathcal{Y}\times_{\Spec A}\Spec\kappa(A)$ is smooth.
\end{definition}
Following \cite{tziolas2010smoothings}, we now recall the notion of formal smoothing. Such definition requires the knowledge of the sheaf of Fitting ideals, which can be found either in \cite[Chapter~20.2]{eisenbud1995commutative} or in \cite[\href{https://stacks.math.columbia.edu/tag/0C3C}{TAG 0C3C}]{stacks-project}. We will not introduce it but we will just give an interpretation of what the Fitting ideal is. Let $\mathfrak{X}$ be a formal scheme, let $\mathfrak{F}$ be a formal coherent sheaf and let $a\in\mathbb{N}$; we denote by $\Fitt_a(\mathfrak{F})$ the $a^{\text{th}}$ Fitting ideal sheaf of $\mathfrak{F}$. This ideal measures the obstructions for the sheaf $\mathfrak{F}$ to be locally generated by $a$ elements. For example, $\mathfrak{F}$ is locally generated by $a$ elements if and only if $\Fitt_a(\mathfrak{F})=\mathcal{O}_{\mathfrak{X}}$.
\begin{definition}\label{def: formal smoothing}
Let $X$ be a proper, equidimensional scheme. A formal deformation of $X$ over $\mathfrak{S}$
\[
\begin{tikzcd}
X \arrow[d] \arrow[r, hook] & \mathfrak{X} \arrow[d, "\pi"]\\
\Spf k \arrow[r, hook] & \mathfrak{S}
\end{tikzcd}
\]
is called a \emph{formal smoothing} of $X$ if and only if there exists a natural number $a$ such that $\mathfrak{I}^a\subset\Fitt_{\dim X}(\Omega^1_{\mathfrak{X}/\mathfrak{S}})$, where $\mathfrak{I}$ is an ideal of definition of $\mathfrak{X}$ and $\Fitt_{\dim X}(\Omega^1_{\mathfrak{X}/\mathfrak{S}})$ is the Fitting sheaf of ideals.
We say that $X$ is \emph{formally smoothable} if it admits a formal smoothing.
\end{definition}
We point out that \Cref{Tziolas: smoothing over dvr is the same as formal smoothing} establishes an equivalence among two different notions of smoothing that, apparently, are very different. Indeed, in \Cref{def: smoothing}, the condition uses strongly the existence of a generic point, while in \Cref{def: formal smoothing} the same condition is ``forced'' to be algebraic since there is not a generic point in $\Spf k\llbracket t\rrbracket$.
As mentioned above the two notions differ only apparently as we are going to explain next.
First observe that any DVR which is a $k$-algebra is a local Noetherian ring; in particular we have that its completion with respect to the adic topology induced by its maximal ideal is isomorphic to the formal power series in one variable, $k\llbracket t\rrbracket$. We also remark that the (classical) spectrum of a DVR contains two points: the closed and the generic one. On the other hand, the formal spectrum of the formal power series ring is made of one point only. Therefore it is natural to define the notion of smoothing of a scheme over a DVR as a deformation of $X$ whose general fibre, i.e. the fibre over the open generic point, is smooth. On the other hand, in the case of formal deformation over $\Spf k\llbracket t \rrbracket$ such idea is not possible. However, Tziolas come up with a definition of formal smoothing that does not need the generic point, as we are going to explain now. Let us suppose that $\pi\colon\mathcal{X}\to B$ is a locally of finite type, flat of relative dimension $r$ morphism of schemes and define
\[
U_{r}=\left\{x\in\mathcal{X}\colon \pi\text{ is smooth at $x$ of relative dimension }r\right\}.
\]
By \cite[\href{https://stacks.math.columbia.edu/tag/02G2}{TAG 02G2}]{stacks-project}, it is an open subset of $\mathcal{X}$ and by \cite[407]{eisenbud1995commutative} or \cite[\href{https://stacks.math.columbia.edu/tag/0C3K}{TAG 0C3K}]{stacks-project} we have that
\[
U_{r}=\mathcal{X}\setminus\text{V}(\Fitt_{r}(\Omega^1_{\pi}))\,\,\,\,\,\,\,\,\text{ and }\,\,\,\,\,\,\,\,\text{Sing}_{r}(\pi)=\text{V}(\Fitt_r(\Omega^1_{\pi})).
\]
If we assume that $\pi$ is proper, then $\pi(U_r)\subset B$ is open too and $\pi|_{U_{r}}\colon U_r\to A_{r}$ is smooth of relative dimension $r$, where $A_r:=B\setminus\pi(\text{V}(\Fitt_r(\Omega^1_{\pi})))$. Doing a base change, we can always find a smoothing from the family over $B$ if and only if $A_r$ is not empty.
Suppose that $B$ is affine smooth curve over $k$, let $p\in B$ be a closed point and let $R:=\mathcal{O}_{B,p}$; it is known that $R$ is a DVR with residue field $k$. $\pi\colon\mathcal{X}\to B$ is a smoothing (according to \Cref{def: smoothing}) if and only if the pullback deformation $\mathcal{X}_{R}\to\Spec R$ along the localization morphism $\Spec R\to B$ is a smoothing (again in the sense of \Cref{def: smoothing}). We then have following diagram:
\[
\begin{tikzcd}
\mathcal{X}_{n} \arrow[d, "\pi|_{\mathcal{X}_n}"'] \arrow[r, hook] & \mathcal{X}_{\widehat{R}} \arrow[d, "\pi|_{\mathcal{X}_{\widehat{R}}}"'] \arrow[r, hook, "\beta"] & \mathcal{X}_R \arrow[d, "\pi|_{\mathcal{X}_R}"] \arrow[r, hook] & \mathcal{X} \arrow[d, "\pi"]\\
S_n \arrow[r, hook] & \Spec \widehat{R} \arrow[r, hook, "\alpha"'] & \Spec R \arrow[r, hook] & B
\end{tikzcd}
\]
where all squares are Cartesian, $\widehat{R}$ denotes the completion of $\mathcal{O}_{B,p}$ along its maximal ideal $\mathfrak{m}_{p}$ and, for every $n\in\mathbb{N}$, $R_n:=\frac{k\llbracket t\rrbracket}{(t^{n+1})}=\frac{k[t]}{(t^{n+1})}$ and $S_n:=\Spec R_n$. As previously mentioned, the completion of $\mathcal{O}_{B,p}$ along the maximal ideal is isomorphic to $k\llbracket t \rrbracket$.
In order to lighten the notation, let us denote $\pi_n:=\pi|_{\mathcal{X}_n}$, $\widehat{\pi}:=\pi|_{\mathcal{X}_{\widehat{R}}}$, $\widetilde{\pi}:=\pi|_{\mathcal{X}_R}$.
Observe now that $\alpha$ is a homeomorphism, hence $\beta$ is at least a bijective function on the sets; by \cite[Corollary~20.5]{eisenbud1995commutative} we have that
\[
\text{Sing}_r(\widehat{\pi})=\beta^{-1}(\text{Sing}_r(\widetilde{\pi})).
\]
Therefore, $\widetilde{\pi}$ is smooth of relative dimension $r$ along $\widetilde{\pi}^{-1}(\eta)$ if and only if $\widehat{\pi}$ is smooth of relative dimension $r$ along $\widehat{\pi}^{-1}(\widehat{\eta})$, where $\eta$ and $\widehat{\eta}$ are the generic points of $\Spec R$ and $\Spec\widehat{R}$ respectively. Now $\Spec R$ has only two points: the closed one, $Y$ with ideal sheaf $\mathcal{I}_{Y/\Spec R}=(t)$, and the open one, $\eta$. Let $C_r:=\text{Sing}_{r}(\widetilde{\pi})$. Now $\widetilde{\pi}(C_{r})\subset Y$ as schemes if and only if there exists a structure of closed $Spec R$-subscheme $\widetilde{Y}$ on $Y$ with $\widetilde{Y}_{\text{red}}=Y$ and such that $\widetilde{\pi}(C_{r})\subset\widetilde{Y}$ as sets. We are then reduced to classify all closed subscheme structures on $\Spec k\llbracket t\rrbracket$. These are given by $Y_{k}:=\text{V}((t^{k+1}))$, for every $k\in\mathbb{N}$. In particular we have a chain of closed subschemes
\[
Y=\text{V}((t))=Y_{0}\subset Y_1=\text{V}((t^2))\subset Y_3\subset\cdots.
\]
Hence, $\widetilde{Y}$ is a closed $\Spec R$-subscheme structure on $Y$ satisfying $\widetilde{Y}_{\text{red}}=Y$ and $\widetilde{\pi}(C_{r})\subset\widetilde{Y}$ if and only if there exists a non-negative integer $k$ such that $\widetilde{Y}=Y_{k}$.
Concluding, we have proven that the following statements are equivalent:
\begin{itemize}
\item[(a)] $\pi\colon\mathcal{X}\to B$ is smooth of relative dimension $r$;
\item[(b)] $\widetilde{\pi}\colon\mathcal{X}_{R}\to\Spec R$ is smooth of relative dimension $r$;
\item[(c)] $\widehat{\pi}\colon\mathcal{X}_{\widehat{R}}\to\Spec\widehat{R}$ is smooth of relative dimension $r$;
\item[(d)] there is a closed subscheme $\widetilde{Y}$ of $\Spec R$ such that $\widetilde{Y}_{\text{red}}=Y$ and $\widetilde{\pi}(C_{r})\subset\widetilde{Y}$;
\item[(e)] there exist a $k\in\mathbb{N}$ such that $\widetilde{\pi}(C_{r})\subset Y_{k}$;
\item[(f)] there exists a $k\in\mathbb{N}$ such that $C_{r}\subset\widetilde{\pi}^{-1}(Y_{k})$;
\item[(g)] there exists a $k\in\mathbb{N}$ such that
\[
\Fitt_r(\Omega^1_{\widetilde{\pi}})=\mathcal{I}_{C_{r}/\mathcal{X}}\supseteq\widetilde{\pi}^{-1}((t^{k+1}))=\widetilde{\pi}^{-1}(\mathcal{I}_{Y_{k}/\Spec R}).
\]
\end{itemize}
Observing that the condition we have found is independent of the ideal of definition, we have reached the definition of formal smoothing as given in \cite[Definition~11.6]{tziolas2010smoothings}.
\section{Gorenstein schemes, morphisms and their deformations}
In this part we will review, following \cite[\href{https://stacks.math.columbia.edu/tag/08XG}{Tag 08XG}]{stacks-project} and \cite[\href{https://stacks.math.columbia.edu/tag/0DWE}{Tag 0WDE}]{stacks-project}, the notions of dualising complexe and of Gorenstein morphisms. We then discuss how the Gorenstein property behaves under infinitesimal deformations. The main result of this section is that the relative dualising sheaf extends to every infinitesimal deformation. In the way to prove this result, we also present a proof of the classical result that deformation of a Gorenstein morphism is still Gorenstein, for which we were not able to find a proof in the literature.
\subsection{Gorenstein schemes and morphisms}
We start the section introducing the notions of dualising sheaf, Gorenstein scheme and Gorenstein morphism.
\begin{definition}\label{def: dualising complex}
Let $A$ be a Noetherian ring. A \emph{dualising complex} is a complex of $A$ modules $\omega_A^{\bullet}$ such that
\begin{enumerate}
\item $\omega_A^{\bullet}$ has finite injective dimension;
\item $\coho^i(\omega_A^{\bullet})$ is a finite $A$-module, for every $i$;
\item $A\to\mathbf{R}\Hom_A(\omega_A^{\bullet},\omega_A^{\bullet})$ is a quasi-isomorphism in the derived category of $A$-modules.
\end{enumerate}
\end{definition}
We remark that the dualising complex thus defined is not unique. Indeed, according to \cite[\href{https://stacks.math.columbia.edu/tag/0A7F}{TAG 0A7F}]{stacks-project}, if $\omega_{A}^{\bullet}$ and $\nu_{A}^{\bullet}$ are two dualising complexes for $A$, then there exists an invertible object $L^{\bullet}\in\text{D}(A)$ such that $\nu_{A}^{\bullet}$ is quasi-isomorphic to $\omega_{A}^{\bullet}\otimes_{A}^{\textbf{L}}L^{\bullet}$.
\begin{definition}\label{def: Gorenstein local ring}
Let $A$ be a local Noetherian ring. We say that $A$ is a \emph{Gorenstein local ring} if $A[0]$ is a dualising complex.
\end{definition}
\begin{definition}\label{def: Gorenstein scheme}
A scheme $X$ is called \emph{Gorenstein} if it is locally Noetherian and for every $x\in X$, $\mathcal{O}_{x, X}$ is a Gorenstein local ring according to \Cref{def: Gorenstein local ring}.
\end{definition}
\begin{definition}\label{def: Gorenstein morphism}
Let $f\colon X\to Y$ be a morphism of schemes such that for every $y\in Y$, the fibre $X_y$ is a locally Noetherian scheme.
\begin{enumerate}
\item Let $x\in X$ and $y:=f(x)$. We say that $f$ is \emph{Gorenstein at $x$} if $f$ is flat at $x$ and $\mathcal{O}_{X_y,x}$ is a Gorenstein local ring.
\item We say that $f$ is \emph{Gorenstein} if it is Gorenstein at $x$, for all $x\in X$.
\end{enumerate}
\end{definition}
\begin{lemma}[{\cite[\href{https://stacks.math.columbia.edu/tag/0C12}{Tag 0C12}]{stacks-project}}]\label{lemma: Gorenstein scheme and flat morphism imply Gorenstein morphism}
Let $f\colon X\to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Gorenstein, then $f$ is Gorenstein.
\end{lemma}
\begin{proposition}[{\cite[\href{https://stacks.math.columbia.edu/tag/0C07}{Tag 0C07}]{stacks-project}}]\label{prop: base change of Gorenstein morphism is Gorenstein}
Let $f\colon X\to Y$ be a morphism of schemes such that for every $y\in Y$ the fiber $X_y$ is locally Noetherian and let $g\colon Y'\to Y$ be a locally of finite type morphism of schemes. Consider the following Cartesian diagram
\begin{equation}\label{eq: fibre product diagram}
\begin{tikzcd}
X' \arrow[d, "f'"] \arrow[r, "g'"] & X \arrow[d, "f"]\\
Y' \arrow[r, "g"] & Y.
\end{tikzcd}
\end{equation}
If $f'$ is Gorenstein at $x'\in X'$ and $f$ is flat at $g'(x')$, then $f$ is Gorenstein at $g'(x')$.
\end{proposition}
From this it follows that being a Gorenstein is local in the flat topology on the category of schemes.
\subsection{Right adjoint to the pushforward and relative dualising complex}
Now we introduce the derived pushforward functor and its right adjoint. This machinery will be used to define a relative dualising complex and to show that it behaves well under pullbacks.
\begin{definition}
Let $f\colon X\to Y$ be a morphism of scheme with $Y$ quasi-compact. By \cite[\href{https://stacks.math.columbia.edu/tag/0A9E}{Tag 0A9E}]{stacks-project}, $\textbf{R}f_*\colon\text{D}_{\text{QCoh}}(X)\to\text{D}_{\text{QCoh}}(Y)$ admits a right adjoint and we denote it by $\Psi\colon\text{D}_{\text{QCoh}}(Y)\to\text{D}_{\text{QCoh}}(X)$.
\end{definition}
\begin{definition}\label{def: relative dualising complex}
Let $Y$ be a quasi-compact scheme, let $f\colon X\to Y$ be a proper, flat morphism of finite presentation and let $\Psi$ be the right adjoint for $\mathbf{R}f_*$. We define the \emph{relative dualising complex} $\omega_f^{\bullet}$ of $f$ (or of $X$ over $Y$) as follows
\[
\omega_f^{\bullet}:=\Psi(\mathcal{O}_Y)\in\text{D}_{\text{QCoh}}(X).
\]
\end{definition}
The following proposition explains the behaviour of the relative dualising complex under base change.
\begin{proposition}[{\cite[\href{https://stacks.math.columbia.edu/tag/0AAB}{Tag 0AAB}]{stacks-project}}]\label{coro: base change for dualising complex}
Let $X$ be a scheme, let $Y$ and $Y'$ be quasi-compact schemes, let $g\colon Y'\to Y$ also be any morphism and let $f\colon X\to Y$ be a proper, flat morphism of finite presentation. Consider the fibre diagram as in \cref{eq: fibre product diagram}. Then we have a canonical isomorphism
\[
\omega_{f'}^{\bullet}\cong\mathbf{L}(g')^*\omega_f^{\bullet}\in\text{D}_{\text{QCoh}}(X'),
\]
where $X':=X\times_{Y}Y'$.
\end{proposition}
\subsection{Upper shriek functor and Gorenstein morphisms}
We now introduce the upper shriek functor and explain its relationships with the right adjoint functor for the derived pushforward functor and with Gorenstein morphisms.
Remember from \Cref{conventions} that FTS is the category whose objects are separated, algebraic schemes over the field $k$ and whose morphisms are morphisms of $k$-schemes.
\begin{definition}\label{def: upper shriek functor}
Let $f\colon X\to Y$ be a morphism in the category of FTS schemes. We define the upper shriek functor
\[
f^!\colon\text{D}^+_{\text{QCoh}}(\mathcal{O}_Y)\to\text{D}^+_{\text{QCoh}}(\mathcal{O}_X)
\]
as follows. We choose a compactification $X\to\bar{X}$ of $X$ over $Y$. Such a compactification always exists by \cite[\href{https://stacks.math.columbia.edu/tag/0F41}{Tag 0F41}]{stacks-project} and \cite[\href{https://stacks.math.columbia.edu/tag/0A9Z}{Tag 0A9Z}]{stacks-project}. Let denote by $\bar{f}\colon\bar{X}\to Y$ the structure morphism and consider its right adjoint functor $\bar{\Psi}$; we then let $f^! K:=\bar{\Psi}(K)|_{X}$ for $K\in\text{D}^+_{\text{QCoh}}(\mathcal{O}_Y)$.
\end{definition}
According to \cite[\href{https://stacks.math.columbia.edu/tag/0AA0}{Tag 0AA0}]{stacks-project}, the definition of the upper shriek functor is, up to canonical isomorphism, independent of the choice of the compactification of $X$.
\begin{remark}\label{rem: relationship between upper shriek and right derived functor in case of proper morphism}
We point out that if $f\colon X\to Y$ is a proper morphism in the category FTS, then $\bar{\Psi}=\Psi$, implying that the upper shriek functor is the restriction to $\text{D}_{\text{QCoh}}(\mathcal{O}_Y)$ of $\Psi$, the right adjoint functor of $\mathbf{R}f_*$ (see \cite[\href{https://stacks.math.columbia.edu/tag/0AU3}{Tag 0AU3}]{stacks-project}).
\end{remark}
We are now ready to present the link between the Gorenstein condition and the upper shriek functor.
\begin{proposition}[{\cite[\href{https://stacks.math.columbia.edu/tag/0C08}{Tag 0C08}]{stacks-project}}]\label{prop: Gorenstein iff dualising complex is invertible in an open subset}
Consider $f\colon X\to Y$ a flat morphism of schemes in FTS and let $x\in X$. Then the following conditions are equivalent:
\begin{enumerate}
\item $f$ is Gorenstein at $x$;
\item $f^!\mathcal{O}_Y$ is isomorphic to an invertible object (of the derived category) in a neighbourhood of $x$.
\end{enumerate}
In particular the set $\{x\in X\colon f\text{ is Gorenstein at }x\}$ is open in $X$.
\end{proposition}
If we assumed that $f$ were proper, then $\{y\in Y\colon f\text{ is Gorenstein at }x\in f^{-1}(y)\}$ is open in the target.
\subsection{Relative dualising sheaf and dualising complex}
The aim of this subsection is to show that all the definitions given until now, under mild hypotheses, converge. In particular, the next proposition introduces the notion of relative dualising sheaf for a morphism in the category FTS and describes its relationships with the relative dualising complex and with the Gorenstein morphisms.
\begin{proposition}[{\cite[\href{https://stacks.math.columbia.edu/tag/0BV8}{Tag 0BV8}]{stacks-project}}]\label{prop: existence of the relative dualising sheaf}
Let $X$ and $Y$ be separated schemes and let $f\colon X\to Y$ be a Gorenstein morphism of schemes. Then there exists a coherent, invertible sheaf, called the relative dualising sheaf of $f$ and denoted by $\omega_f$, which is flat over $Y$ and satisfies
\[
f^{!}\mathcal{O}_{Y}\cong\omega_f[-d],
\]
where $d$ is the locally constant function on $X$ which gives the relative dimension of $X$ over $Y$.
If $f$ is also proper, flat and of finite presentation, then $\omega_f^{\bullet}=\omega_f[-d]$.
\end{proposition}
If $Y=\Spec k$, then we denote the relative dualising sheaf of $X$ over $k$ by $\omega_{X}$.
\begin{proposition}\label{prop: infinitesimal deformation of Gorenstein scheme is Gorenstein}
Let $X$ be a Gorenstein scheme and let $A$ be an Artinian local $k$-algebra with residue field $k$. Consider now a deformation of $X$ over $A$; that is a Cartesian diagram
\[
\begin{tikzcd}
X \arrow[d] \arrow[r, hook] & \mathcal{X} \arrow[d, "f"]\\
\Spec k \arrow[r, hook] & \Spec A
\end{tikzcd}
\]
with $f$ flat (see \cite{sernesi2007deformations}).
Then $f$ is a Gorenstein morphism.
\end{proposition}
\begin{proof}
Since $X$ is Gorenstein and $X\to\Spec k$ is flat, by \Cref{lemma: Gorenstein scheme and flat morphism imply Gorenstein morphism} it follows that $X\to\Spec k$ is Gorenstein. Applying now \Cref{prop: base change of Gorenstein morphism is Gorenstein}, we deduce that $f\colon\mathcal{X}\to\Spec A$ is Gorenstein.
\end{proof}
\begin{remark}
The result can be improved to obtain that the scheme $\mathcal{X}$ is Gorenstein. This is true as soon as we require that the affine base scheme $\Spec A$ is the spectrum of a local, Artinian, Gorenstein $k$-algebra $A$. This result and its proof can be found in \cite{nobile2021}.
\end{remark}
Now we present the first result that will help us to deduce the existence of a geometric smoothing.
\begin{proposition}\label{dualising sheaf of a Gorenstein proper scheme always extends}
Let $X$ be a proper, Gorenstein scheme. If $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{S}$ is a formal deformation of $X$, then there exists a unique invertible formal sheaf $\mathfrak{L}$ on $\mathfrak{X}$ such that $\mathfrak{L}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{O}_{X}\cong\omega_X$ and $\mathfrak{L}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{O}_{\mathcal{X}_{n}}\cong\omega_{f_n}$, for every $n\in\mathbb{N}$, where $\omega_{f_n}$ is the relative dualising sheaf. In particular, every morphism $f_{n}$ is Gorenstein.
\end{proposition}
\begin{proof}
By \Cref{prop: formal deformation as compatible collection of deformations}, the formal deformation $\mathfrak{f}$ is equivalent to a collection of deformations $\left\{f_n\colon\mathcal{X}_n\to S_n\right\}_{n\in\mathbb{N}}$ satisfying the compatibility condition of \Cref{eq: compatibility of deformation to induced a formal deformation}, with $f_n$ flat, proper morphisms. Since $X$ is Gorenstein, applying \Cref{prop: infinitesimal deformation of Gorenstein scheme is Gorenstein} we deduce that for every natural number $n$, the morphism $f_n$ is Gorenstein. Now consider the following Cartesian diagram
\[
\begin{tikzcd}
\mathcal{X}_n \arrow[d, "f_n"] \arrow[r, hook, "j_{n}"] & \mathcal{X}_{n+1} \arrow[d, "f_{n+1}"]\\
S_n \arrow[r, hook] & S_{n+1};
\end{tikzcd}
\]
we have, for every natural number $n$, the following chain of equalities and natural isomorphisms
\[
\begin{aligned}
j_{n}^*\omega_{f_{n+1}}&=\coho^{-\dim X}(j_n^*\omega_{f_{n+1}}[-\dim X])\,\,\,\,\,(\text{\Cref{prop: existence of the relative dualising sheaf}})\\
&=\coho^{-\dim X}(\mathbf{L}j_n^*\omega_{f_{n+1}}^{\bullet})\\
&\cong\coho^{-\dim X}(\omega_{f_{n}}^{\bullet})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{\Cref{coro: base change for dualising complex}})\\
&=\coho^{-\dim X}(\omega_{f_n}[-\dim X])\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{\Cref{prop: existence of the relative dualising sheaf}})\\
&=\omega_{f_n}.
\end{aligned}
\]
\Cref{teo: locally free sheaves on each nilpotent subscheme induce a locally free sheaf on the completion} then implies that there exists an invertible formal sheaf $\mathfrak{L}$ on $\mathfrak{X}$ such that $\mathfrak{L}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{O}_{X}\cong\omega_X$.
\end{proof}
As a consequence of this last proposition, we get that if $X$ is a proper, local complete intersection scheme over a field $k$ and we have a formal deformation $\mathfrak{f}\colon\mathfrak{X}\to\Spf k\llbracket t\rrbracket$, then the relative dualising sheaf $\omega_X$ always extends to the formal deformation $\mathfrak{f}$. To see this, it is enough to observe that l.c.i. schemes/morphisms are in particular Gorenstein schemes/morphisms and then apply the previous proposition.
This result for l.c.i. schemes can be achieved only by using properties of the naive cotangent complex; this second way is described in length in \cite{nobile2021}.
\section{From formal smoothing to geometric smoothing}
In this last section we use all the previous results to show how pass from a formal smoothing to a geometric one. We start by recalling the definition of geometric smoothing.
\begin{definition}\label{def: geometric smoothing}
Let $X$ be a proper scheme. A \emph{geometric smoothing} is a Cartesian diagram
\begin{equation}\label{eq: geometric smoothing diagram}
\begin{tikzcd}
X \arrow[d]\arrow[r, hook] & \mathcal{X} \arrow[d, "\pi"]\\
\Spec k = \Spec \frac{\mathcal{O}_{c, C}}{\mathfrak{m}_{c}} \arrow[r, hook] & C
\end{tikzcd}
\end{equation}
where $C$ is a smooth curve, $c\in C$ is a closed point and $\pi$ is a flat and proper morphism, such that $\pi^{-1}(\eta_C)=:\mathcal{X}_{\text{gen}}$ is smooth, where $\eta_C$ is the generic point of $C$. We say that $X$ is \emph{geometrically smoothable} if it has a geometric smoothing.
\end{definition}
We remark that, if $X$ is smooth over $\Spec k$, then $X$ is geometrically smoothable in a trivial way by considering the trivial family $\text{pr}_2\colon X\times_k C\to C$ of deformations.
We now present some results that will be needed in the proof of the main theorem.
\begin{lemma}[{\cite[Lemma 7.2.1 page 87]{kempf_1993}}]\label{lemma: existence of a curve}
Let $X$ be a scheme, let $U$ be an open, dense subset of $X$ and let $p\in X$ be a closed point. Then there exists an affine curve $C$ in $X$ such that $C$ intersects $U$ and passes through $p$.
\end{lemma}
\begin{remark}\label{remark: completion of local ring at a curve}
Let $C$ be a smooth curve over $k$ and let $c\in C(k)$ be a closed point. Denote by $l$ a local parameter of the maximal ideal $\mathfrak{m}_{c}$ in $\mathcal{O}_{C,c}$. Then there is a isomorphism of topological rings
\[
\widehat{\mathcal{O}_{C,c}}\cong k\llbracket t\rrbracket
\]
such that $l$ is sent to $t$.
\end{remark}
\begin{proposition}\label{prop: morphism from an irreducible and reduced scheme factors trough an irreducible and reduced component of the target}
Let $f\colon X\to Y$ be a morphism of schemes such that $X$ is reduced and irreducible. Then there exists an irreducible and reduced component $Y'$ of $Y$ such that $f$ factors trough $Y'$, i.e. the following diagram commutes
\[
\begin{tikzcd}
X\arrow[rr, "f"] \arrow[rd] & & Y\\
& Y' \arrow[ru, hook]
\end{tikzcd}.
\]
\end{proposition}
\begin{proof}
Since $X$ is irreducible, by \cite[\href{https://stacks.math.columbia.edu/tag/0379}{Tag 0379}]{stacks-project}, $f(X)$ is an irreducible subset of $Y$. Then $Y':=\overline{f(X)}$ is an irreducible component of $Y$ and $f$ factors through $Y'$ by construction. By \cite[II-Ex. 2.3(c)]{hartshorne1977algebraic}, we can always assume $Y'$ to be a reduced scheme.
\end{proof}
\begin{notation}
From now on, we will denote by $\mathfrak{S}$ the formal scheme $\Spf k\llbracket t\rrbracket$ and by $S$ the scheme $\Spec k\llbracket t\rrbracket$. Moreover, for any non-negative integer $n$, we denote by $S_n$ the scheme $\Spec \frac{k\llbracket t \rrbracket}{(t^{n+1})}$.
\end{notation}
The next lemma shows that geometrical smoothability implies formal smoothability.
\begin{lemma}\label{lem: geometric smoothing implies formal smoothing}
Let $X$ be a projective, equidimensional scheme. If $X$ is geometrically smoothable, then it is also formally smoothable.
\end{lemma}
\begin{proof}
Suppose $X$ has a geometric smoothing like \cref{eq: geometric smoothing diagram}, where $c$ is the closed point of $C$ such that the fibre of $\pi$ over $c$ is $X$. Consider the pullback $\tilde{\pi}$ of $\pi$ along the composite morphism $\Spec\widehat{\mathcal{O}_{C,c}}\to\Spec\mathcal{O}_{C,c}\to C$; since $\pi$ is a smoothing of $X$, so is $\tilde{\pi}$. By \Cref{remark: completion of local ring at a curve} we have that the completion of the regular local ring $\mathcal{O}_{C,c}$ is continuously isomorphic to $\mathfrak{S}$. Now using \Cref{remark: constructing formal deformation from classic deformation}, we can construct the associated formal deformation $\mathfrak{p}\colon\mathfrak{X}\to\mathfrak{S}$. We end the argument by invoking \Cref{Tziolas: smoothing over dvr is the same as formal smoothing}.
\end{proof}
At this point we are ready to restate and prove our main result.
\begin{theorem}\label{main result}
Let $X$ be a projective, equidimensional scheme such that one of the following hypotheses hold:
\begin{enumerate}
\item $\coho^2(X,\mathcal{O}_X)=0$,
\item if $X$ Gorenstein, then either the dualising sheaf $\omega_X$ or its dual $\omega_X^{\vee}$ is ample.
\end{enumerate}
Then $X$ is formally smoothable if and only if $X$ is geometrically smoothable.
\end{theorem}
\begin{proof}
One implication is proved in \Cref{lem: geometric smoothing implies formal smoothing}
Suppose we are given a formal smoothing $\mathfrak{p}\colon\mathfrak{X}\to\mathfrak{S}$. Now,
\begin{enumerate}
\item if $\coho^2(X, \mathcal{O}_X)=0$, then by \cite[Theorem~2.5.13]{sernesi2007deformations}, we get that every formal deformation of $X$ is effective; that is to say that there exists a deformation of schemes $p\colon\mathcal{X}\to S$ such that $\mathfrak{X}\cong\hat{\mathcal{X}}_{/X}$. In particular, from the proof, we also deduce that the morphism $p$ is projective.
\item By \Cref{dualising sheaf of a Gorenstein proper scheme always extends} the dualising sheaf $\omega_X$ (or $\omega_X^{\vee}$) extends to an invertible formal sheaf $\mathfrak{L}$ on the formal scheme $\mathfrak{X}$. \Cref{Gro: algebrization theorem} then gives us a deformation $p\colon\mathcal{X}\to S$ of $X$ such that the completion of $\mathcal{X}$ along the central fibre is $\mathfrak{X}$. Moreover, as bonus point of the aforementioned theorem, we deduce that $\mathcal{X}$ is projective over $S$.
\end{enumerate}
Concluding, from either hypothesis, if we start with a formal deformation
\[
\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & \mathfrak{X} \arrow[d, "\mathfrak{p}"]\\
\Spf k \arrow[r, hook] & \mathfrak{S}
\end{tikzcd}
\]
then we can construct a deformation of schemes
\begin{equation}\label{eq: deformazione di schemi}
\begin{tikzcd}
X \arrow[d]\arrow[r, hook] & \mathcal{X} \arrow[d, "p"]\\
\Spec k \arrow[r, hook] & S
\end{tikzcd}
\end{equation}
such that $\mathfrak{X}\cong\hat{\mathcal{X}}_{/X}$. Since $\mathfrak{p}$ is assumed to be a formal smoothing and since $k\llbracket t\rrbracket$ is a DVR, we use \Cref{Tziolas: smoothing over dvr is the same as formal smoothing} to conclude that \cref{eq: deformazione di schemi} is a smoothing of $X$. Moreover, in \cref{eq: deformazione di schemi}, the scheme $\mathcal{X}$ is projective over $S$; i.e. there is a non-negative integer $d$ such that $p$ factors as a closed embedding $\iota\colon\mathcal{X}\hookrightarrow\mathbb{P}^d_S=S\times_k\mathbb{P}^d_k$ followed by the first projection $\text{pr}_1\colon\mathbb{P}^d_S\to S$.
Now we use the fact that the Hilbert functor $\mathfrak{Hilb}_{\mathbb{P}^d}$ is representable to deduce the existence of an isomorphism
\[
\alpha_{S}\colon\mathfrak{Hilb}_{\mathbb{P}^d}(S)\to\Hom_{(\text{Sch})}(S,\text{Hilb}_{\mathbb{P}^d}):=\text{h}_{\text{Hilb}_{\mathbb{P}^d}}(S).
\]
Therefore there exists a unique morphism $\psi\colon S\to\text{Hilb}_{\mathbb{P}^d}$ such that both the following diagrams are Cartesian
\[
\begin{tikzcd}
\mathcal{X} \arrow[rr, bend left, "p"] \arrow[r, hook, "\iota"] \arrow[d, "(\id\times\psi)|_{\mathcal{X}}"] & S\times_k\mathbb{P}^d_k \arrow[r, "\text{pr}_1"] \arrow[d, "\psi\times\id"] & S \arrow[d, "\psi"]\\
\text{Univ}_{\mathbb{P}^d} \arrow[rr, bend right, "\text{pr}_1"] \arrow[r, hook] & \text{Hilb}_{\mathbb{P}^d}\times_k\mathbb{P}^d_k \arrow[r, "\text{pr}_1"]& \text{Hilb}_{\mathbb{P}^d}
\end{tikzcd}.
\]
Recall that $\text{Univ}_{\mathbb{P}^d}$ is by definition a closed subscheme of $\mathbb{P}^d_k\times_k\text{Hilb}_{\mathbb{P}^d}$. Inside the Hilbert scheme we consider the smooth locus, defined as follows
\[
\text{H}_{\text{smooth}}:=\left\{[Z]\in\text{Hilb}_{\mathbb{P}^d}(\Spec k)\colon Z\text{ is smooth }\right\}
\]
By \cite[\href{https://stacks.math.columbia.edu/tag/01V5}{Tag 01V5}]{stacks-project}, $\text{H}_{\text{smooth}}$ is an open subset of the Hilbert scheme $\text{Hilb}_{\mathbb{P}^d}$.
Now we study the map $\psi\colon S\to\text{Hilb}_{\mathbb{P}^d}$.
To do so, we first observe that, since $k\llbracket t\rrbracket$ is a DVR, its spectrum $S$ is made of two points: the closed point, $q$, and the generic point, $\eta$. According to our results so far we have that
\begin{itemize}
\item $\psi(\eta)=[\mathcal{X}_{\text{gen}}]\in\text{H}_{\text{smooth}}$, since (\cref{eq: deformazione di schemi}) is a smoothing;
\item $\psi(q)=[X]\in\text{Hilb}_{\mathbb{P}^d}\setminus\text{H}_{\text{smooth}}$, since $X$ was singular.
\end{itemize}
Since $S$ is connected, there exists a polynomial $\Phi\in\mathbb{Q}[m]$ such that the image of $\psi$ is contained in the connected component $\text{Hilb}_{\mathbb{P}^d}^{\Phi}$ of the Hilbert scheme. By \Cref{prop: morphism from an irreducible and reduced scheme factors trough an irreducible and reduced component of the target} there exists a reduced, irreducible component $Y$ of $\text{Hilb}^{\Phi}_{\mathbb{P}^d}$ such that $\psi$ factors through it:
\[
\begin{tikzcd}
S \arrow[dr, "\widetilde{\psi}"] \arrow[rr, "\psi"] && \text{Hilb}_{\mathbb{P}^d}^{\Phi}\\
& Y \arrow[ur, hook, "i"] &
\end{tikzcd}.
\]
Observe now that if we define $Y_{\text{smooth}}:=Y\cap\text{H}_{\text{smooth}}$ and denote $\overline{Y_{\text{smooth}}}$ the schematic closure of $Y_{\text{smooth}}$, then $\widetilde{\psi}(\eta)\in Y_{\text{smooth}}$ and $\widetilde{\psi}(q)\in\overline{Y_{\text{smooth}}}$. Since $Y_{\text{smooth}}$ is a non-empty open, and therefore dense, subset of $\overline{Y_{\text{smooth}}}$ and $\widetilde{\psi}(q)\in\overline{Y_{\text{smooth}}}$, then we can apply \Cref{lemma: existence of a curve} concluding that there exists a curve $C$ inside $\overline{Y_{\text{smooth}}}$ such that $\widetilde{\psi}(q)\in C$ and $C\cap Y_{\text{smooth}}\neq\emptyset$.
Now let $\nu\colon\tilde{C}\to C$ be the normalisation morphism, and $\tilde{p}\colon\widetilde{\mathcal{X}}\to\widetilde{C}$ be the pullback under the normalisation morphism $\nu$ of the universal family over $Y$. Since $\nu$ is surjective, let $\tilde{c}\in\tilde{C}$ be such that $\nu(\tilde{c})=\widetilde{\psi}(q)$. This completes the proof since we have that the fibre $\tilde{p}^{-1}(\tilde{c})$ is isomorphic to $X$ and $\widetilde{\mathcal{X}}$ is smooth.
\end{proof}
\subsection{Applications of the theorem}
In this section we present an application of our result: smoothability of local complete intersection schemes. We start by recalling the definitions of local complete intersection (l.c.i.) schemes and of complete intersection morphisms.
\begin{definition}\label{def: lci morphism}
Let $f\colon X\to Y$ be a morphism of schemes. We say that $f$ is a local complete intersection morphism, or l.c.i. morphism for short, if it is of finite type and for every point $x\in X$ there are an open neighbourhood $x\in U\subset X$, a scheme $P$ together with a regular immersion $i\colon U\to P$, a smooth morphism of finite type $s\colon P\to Y$ such that $f|_{U}=s\circ i$. We say that a $k$-scheme $X$ is a l.c.i. scheme if the structure morphism $X\to\Spec k$ is a l.c.i. morphism.
\end{definition}
The first remark is that the definition of l.c.i. morphisms does not depend on the factorisation chosen, see \cite[\href{https://stacks.math.columbia.edu/tag/069E}{Tag 069E}]{stacks-project}.
Moreover, if $f\colon X\to Y$ is any morphism of schemes, then the locus $X_{\text{l.c.i.}}$ of points of $X$ such that $f$ is a l.c.i. morphism at $x$, is open in $X$. If we further assume that $f$ is proper, then the locus of points
\[
Y_{\text{l.c.i.}}:=\{y\in Y\colon f\text{ is l.c.i. at }x, \forall x\in f^{-1}(y)\}
\]
is open in $Y$.
\begin{definition}\label{def: complete intersection morphism}
We say that the morphism $f\colon X\to Y$ is a complete intersection morphism if there exists a scheme $P$ together with a global factorisation $s\circ i$ of $f$, with $i\colon X\to P$ a regular immersion and $s\colon P\to Y$ a smooth morphism. We also say that a scheme $X$ is a complete intersection scheme if the structure morphism $X\to\Spec k$ is a complete intersection.
\end{definition}
We now present a theorem of Tziolas \cite[Theorem~12.5]{tziolas2010smoothings} which gives a sufficient condition for the existence of a formal smoothing. We start by introducing the following notation.
\begin{notation}\label{notation: formal neighbourhood Schlessinger sheaf and tangent sheaf}
Let $f\colon X\to Y$ be a morphism of schemes. We denote the relative tangent sheaf by
$\mathcal{T}_{X/Y}:=\shom_{\mathcal{O}_X}(\Omega^1_{X/Y},\mathcal{O}_X)$ and for $i\in\mathbb{N}$, the \emph{$i^{\text{th}}$ relative cotangent sheaf}\index{$i^{\text{th}}$ Schlessinger's relative cotangent sheaf} in the sense of Schelessinger, see \cite{LiS1967}, by $\mathcal{T}^i_{X/Y}:=\sext_{\mathcal{O}_X}^{i}(\Omega^1_{X/Y},\mathcal{O}_X)$. In case $Y$ is the spectrum of the ground field $k$, we let $\mathcal{T}_X:=\mathcal{T}_{X/k}$ and $\mathcal{T}^{i}_X:=\mathcal{T}^{i}_{X/k}$ be the tangent sheaf and the $i^{\text{th}}$ cotangent sheaf respectively.
\end{notation}
\begin{theorem}[{\cite[Theorem~12.5]{tziolas2010smoothings}}]\label{Tziolas: existence of formal smoothing}
Let $X$ be a proper, reduced, pure dimensional scheme. If the following conditions hold
\begin{itemize}
\item[(\text{a})] $X$ has complete intersection singularities;
\item[(\text{b})] $\coho^2(X,\mathcal{T}_X)=0$;
\item[(\text{c})] $\coho^1(X,\mathcal{T}^1_X)=0$;
\item[(\text{d})] $\mathcal{T}^1_X$ is finitely generated by its global sections;
\end{itemize}
then $X$ is formally smoothable, i.e. it admits a formal smoothing.
\end{theorem}
As a corollary we would like to mention the following result that can be found in \cite[Corollary~12.9]{tziolas2010smoothings}.
\begin{corollary}\label{coro: lci scheme with normal sheaf satisfy Tziolas conditions}
Let $X$ be a projective, lci scheme such that there exists a regular embedding in a smooth scheme $Y$. If the normal sheaf $\mathcal{N}_{X/Y}$ is finitely generated by its global sections, $\coho^1(X,\mathcal{T}^1_X)=\coho^2(X,\mathcal{T}_X)=0$, then $X$ admits a formal smoothing.
\end{corollary}
Putting together \Cref{coro: lci scheme with normal sheaf satisfy Tziolas conditions} and \Cref{main result} we get the following.
\begin{proposition}
Let $X$ be a singular, projective, l.c.i. variety (i.e. an integral Noetherian scheme of finite type over $k$) over $k$ satisfying conditions $(\text{a})$, $(\text{b})$ and $(\text{c})$ of \Cref{Tziolas: existence of formal smoothing} and such that either its dualising sheaf or its dual is ample. Then $X$ is geometrically smoothable.
\end{proposition}
The above result can be used to get information about points on the moduli space in the following sense.
\begin{theorem}
Let $X$ be a projective, l.c.i. variety with $\omega_{X}$ (respectively $\omega_{X}^{\vee}$) ample. Assume that $X$ satisfies also hypotheses (b), (c) and (d) of \cref{Tziolas: existence of formal smoothing}. Then we have that
\begin{enumerate}
\item $X$ represents a point in closure of the open subset of the (algebraic) moduli stack $\mathcal{M}$ of all projective smooth Gorenstein varieties with ample canonical (respectively anti-canonical) sheaf;
\item the general point of the unique irreducible component of $\overline{\mathcal{M}}$ containing $X$ is smooth.
\end{enumerate}
\end{theorem}
\begin{proof}
The hypothesis of \cref{Tziolas: existence of formal smoothing} and of \cref{main result} are satisfied; hence $X$ is geometrically smoothable. In other words, $X$ represents a point that lies in the closure of the open subset of the moduli stack of projective l.c.i. varieties with ample canonical (respectively anticanonical) sheaf. This proves (1) and (2) above.
\end{proof}
The above theorem has been proved in \cite{fantechi2021smoothing} for the specific case of Godeaux stable surfaces. More precisely, in there the authors verified the hypotheses of Tziolas' \Cref{Tziolas: existence of formal smoothing} and then apply \cref{main result} to show that stable semi-smooth complex Godeaux surfaces appear in the closure of the smooth locus of the moduli stack of stable surfaces of general type and such moduli stack at the point representing the surface has dimension equal to the expected dimension.
\printbibliography
\noindent
\textit{Alessandro Nobile}, \texttt{[email protected]}\\
\textsc{Department of Mathematics, Université du Luxembourg}, 6, av. de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
\end{document}
|
\begin{equation}} \def\ee{\end{equation}gin{document}
\title{Differentiable Quantum Architecture Search}
\author{Shi-Xin Zhang}
\affiliation{Institute for Advanced Study, Tsinghua University, Beijing 100084, China}
\affiliation{Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China, 518057}
\author{Chang-Yu Hsieh}
\epsilonmail{[email protected]}
\affiliation{Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China, 518057}
\author{Shengyu Zhang}
\affiliation{Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China, 518057}
\author{Hong Yao}
\epsilonmail{[email protected]}
\affiliation{Institute for Advanced Study, Tsinghua University, Beijing 100084, China}
\affiliation{State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China}
\begin{equation}} \def\ee{\end{equation}gin{abstract}
Quantum architecture search (QAS) is the process of automating architecture engineering of quantum circuits.
It has been desired to construct a powerful and general QAS platform
which can significantly accelerate current efforts to identify quantum advantages of error-prone and depth-limited quantum circuits in the NISQ era.
Hereby, we propose a general framework of differentiable quantum architecture search (DQAS), which enables automated designs of quantum circuits in an end-to-end differentiable fashion.
We present several examples of circuit design problems to demonstrate the power of DQAS.
For instance, unitary operations are decomposed into quantum gates, noisy circuits are re-designed to improve accuracy, and circuit layouts for quantum approximation optimization algorithm are automatically discovered and upgraded for combinatorial optimization problems.
These results not only manifest the vast potential of DQAS being an essential tool for the NISQ application developments, but also present an interesting research topic from the theoretical perspective as it draws inspirations from the newly emerging interdisciplinary paradigms of differentiable programming, probabilistic programming, and quantum programming.
\epsilonnd{abstract}
\downarrowte{\today}
\mathrm{e}xtrm{matter}aketitle
\renewcommand\thetaesection{\arabic{section}}
{\sigma}ection{Introduction}
In the noisy intermediate-scale quantum technology (NISQ) era \cite{Preskill2018}, the hybrid quantum-classical (HQC) computational scheme, combining quantum hardware evaluations with classical optimization outer loops, is widely expected to deliver the first instance of quantum advantages (for certain non-trivial applications) in the absence of fault-tolerant quantum error corrections.
Several prototypical examples in this category include finding the ground state of complex quantum systems by variational quantum eigensolver (VQE) \cite{Peruzzo2014, McClean2016,McArdle2020}, exploring better approximation for NP hard combinatorial optimization problems by quantum approximation optimization algorithms (QAOA) \cite{Farhi2014,Hadfield2017,Zhou2020}, and solving some learning tasks in either the classical or quantum context by the quantum machine learning (QML) setup \cite{Farhi2018,Verdon2019, Benedetti2019a,Benedetti2019b,Cong2019,Verdon2019}.
Under the typical setting in the HQC computational paradigm, the structure of variational ansatz is held fixed and only trainable parameters are optimized to satisfy an objective function.
This lack of flexibility is rather undesirable as different families of parametrized circuits may differ substantially in their expressive power and entangling capability \cite{Akshay2020,Farhi2020}.
Moreover, in the NISQ era, a thoughtful circuit design should minimize the consumption of quantum resources due to decoherence and limited connectivity among qubits in current quantum hardwares.
For instance, the number of two-qubits gates (or the circuit depth) should be minimized to reduce noise-induced errors.
Additional error mitigation strategy should be conducted without using extra qubits if possible.
With these requirements in mind, the design of an effective circuit ansatz should take into account of the nature of the computational problems and the specifications of a quantum hardware as well.
We term the automated design of parameterized circuits, in the aforementioned setting, as quantum {\it ansatz} search (QAS).
In a broader context, we denote QAS as quantum {{\it architecture} search, which covers all scenarios of quantum circuit design and goes beyond the design of a variational ansatz for HQC algorithms.
QAS can facilitate a broad range of tasks in quantum computations.
Its applications include but not limited to decomposing arbitrary unitary \cite{Kiani2020} into given quantum gates, finding possible shortcuts for well-established quantum algorithms \cite{Li2017c, Cincio2018a}, exploring optimal quantum control protocols \cite{Yang2017, Fosel2018, Lin2019a}, searching for powerful and resource-efficient variational ansatz \cite{Rattew2019}, and designing end-to-end and task-specific circuits which also incorporate considerations on quantum error mitigation (QEM), native gate set, and topological connectivity of a specific quantum hardware \cite{Chivilikhin2020, Cincio2020}.
Neural architecture search (NAS) \cite{Yao2018}, devoted to the study and design of neural networks shares many similarities with designing parameterized quantum circuits.
The common approaches for NAS include greedy algorithms \cite{Huang2018},
evolutionary or genetic algorithms \cite{Stanley2019, Real2017a, Xie2017, Liu2018a, Real2019a}, and reinforcement learning (RL) based methods \cite{Zoph2017, Baker2017, Cai2018, Zoph2018}. It is interesting to witness that the progress in QAS follows closely the ideas presented in NAS.
Recent works on quantum circuit structure or ansatz design also exploited greedy methods \cite{Ostaszewski2019, Grimsley2019,Li2020}, evolutional or genetic methodologies \cite{LasHeras2016, Li2017c,Cincio2018a, Rattew2019,Cincio2020,Chivilikhin2020} and RL engine based approaches \cite{Fosel2018, Niu2019} for tasks such as quantum control, QEM or circuit ansatz searching.
Recently, differentiable neural architecture search (DARTS) has been proposed \cite{Liu2019a} and further refined with many critical improvements and generalizations \cite{Xie2019,Liang2019, Zela2019,Hundt2019,Chen2019,Casale2019}.
The key idea of a differentiable architecture search is to relax the discrete search space of neural architectures onto a continuous and differentiable domain, rendering much faster end-to-end NAS workflow than previous methods.
Due to the close relation between NAS and QAS, it is natural to ask
whether it is possible to devise a differentiable quantum architecture search (DQAS) incorporating DARTS-like ideas.
Our answer is affirmative;
as presented in this work, we constructed a general framework of DQAS that works very well as a universal and fully automated design tool for quantum circuits.
As a general framework sitting at the intersection of newly emerging interdisciplinary paradigms of differentiable programming, probabilistic programming and quantum programming, DQAS is of both high theoretical and practical values across various fields in quantum computing and quantum information processing.
The organization of this work goes as follows. In Background and Related Work section, we review backgrounds and relevant works on fields including NAS, QAS and QAOA. In Methods section, we introduce the setup of the DQAS algorithm, where the overall workflow and the main components are both discussed. In Applications section, we demonstrate various applications in quantum computing domain enabled by DQAS, including QEM and variational quantum algorithm design examples. We conclude with a brief Discussion section. The Appendix contains more details and further applications of DQAS \footnote{See Appendix for more information on: A. Glossary for ingredients of DQAS. B. The connection betweeen DARTS and DQAS. C. Gradients derivation from Monte Carlo expectation in DQAS. D. General hyperparameters for DQAS training E. Training techniques that stabilize and improve DQAS. F. DQAS applications on state preparation and unitary learning. G. Relevant hyperparameter and ingredient settings on experiments in this work. H. More results and comparisons on QEM for QFT circuit. I. More DQAS results for QAOA ansatz search including instance learning, block encoding, etc.}.
{\sigma}ection{Background and Related Work} \lambdabel{sec:background}
\nonumberindent{\bf Differentiable Neural Architecture Search.}
NAS \cite{Ren2020} is a burgeoning and active field in AutoML, and the ultimate goal of NAS is to automate the search for a top-performing neural network architectures for any given task.
Popular approaches to implement NAS include reinforcement learning \cite{Zoph2017}, in which an RNN controller chooses an action on building the network structure layerwise from a discrete set of options;
and evolutionary or genetic algorithms \cite{Real2017a, Xie2017,Real2019a}, in which a population of network architectures is kept, evaluated, mutated for the fittest candidates.
Such RL or evolutionary algorithms are rather resource intensive and time consuming,
since the core task involves searching through an exponentially large space of discrete choices for different elementary network components.
Recently, differentiable architecture search \cite{Liu2019a} and its variants have been proposed and witnessed a surge in the number of related NAS studies \cite{Xie2019,Liang2019, Zela2019,Hundt2019,Chen2019,Casale2019, Dong2019,Yao2019,Noy2019,Cai2019, Li2019,Xu2019, Chang2019,Chen2020, Hu2020}.
Under the DARTS framework, the network architecture space of discrete components is relaxed into a continuous domain that facilitates search by differentiation and gradient descent.
The relaxed searching problem can be efficiently solved with noticeably reduced training time and hardware requirements.
In the original DARTS, the search space concerns with choices of distinct microstructures within one cell. Two types of cell are assumed for the networks: normal cell and reduction cell.
The NAS proceeds by first determining the microstructures within these two types of cell, then a large network is built by stacking these two cell types up to a variable depth with arbitrary input and output size.
Within each cell, two inputs, four intermediate nodes and one output (concatenation of four intermediate nodes) are presented as nodes in a directional acylic graph.
For each edge between nodes, one needs to determine optimal connection layers, eg. conv with certain kernel size, or max/average pooling with given window size, zero/identity connections and so on.
To make such search process differentiable, we assume each edge is actually the weighted sum of all these primitive operations from the pool, i.e. $o(x) = {\sigma}um_i \mathrm{e}xt{softmax}(\alpha_i) o_i(x)$ where $o_i$ stands for i-th type of layers primitives and $\alpha_i$ is the continuous weights which determines the structure of neural network as structural parameters.
Therefore, we have two sets of continuous parameters: structure weights $\alpha$ which determines the optimal network architecture by pruning in evaluation stage, and conventional parameters in neural network $ \omega$.
Via DARTS setup, neural architecture search turns into a bi-optimization problem where differentiation is carried out end-to-end.
DARTS requires the evaluation of the whole super network where each edge is composed of all types layers.
This is memory intensive and limits its usage on large dataset or enriched cell structures.
Therefore, there are works extending DARTS idea while enabling forward evaluation on sub network, usually using only one path \cite{Casale2019,Hu2020} or two \cite{Cai2019}.
Specifically, in \cite{Casale2019}, the authors viewed the super network as a probabilistic ensemble of subnetworks and thus variational structural parameters enter into NAS as probabilistic model parameters instead.
So we can sample subnetworks from such probabilistic distribution and evaluate one subnetwork each time.
This is feasible as probabilistic model parameters can also be updated from general theory for Monte Carlo expectations' gradient in a differentiable approach \cite{Mohamed2019a}.
There are additional follow-up works that focus on improving drawbacks of DARTS with various training techniques.
In general, these DARTS-related techniques are also illuminating and inspirational for further DQAS developments in our work.
~\newline
\nonumberindent{\bf Related works on QAS.} Quantum architecture search, though no one brand it as this name before, is scattered in the literature with different contexts.
These works are often specific to problem setup and denoted as quantum circuit structure learning \cite{Ostaszewski2019}, adaptive variational algorithm \cite{Grimsley2019}, ansatz architecture search \cite{Li2020}, evolutional VQE \cite{Rattew2019}, multipleobjective genetic VQE \cite{Chivilikhin2020} or noise-aware circuit learning \cite{Cincio2020}.
The tasks they focused are mainly in QAOA \cite{Li2020} or VQE \cite{Ostaszewski2019, Grimsley2019, Rattew2019, Chivilikhin2020} settings.
From higher theoretical perspective, some quantum control works can also be classified as QAS tasks, where optimal quantum control protocol is explored using possible machine learning tools \cite{Fosel2018, Niu2019} .
These QAS relevant works are closed related to NAS methodologies.
And this relevance is as expected, since quantum circuit and neural network structure share a great proportion of similarities.
The mainstream approach of QAS is evolution/genetic algorithms with different variants on mutation, crossover or tournament details \cite{LasHeras2016, Li2017c,Cincio2018a, Rattew2019,Cincio2020,Chivilikhin2020}.
There are also works exploiting simple greedy/locality ideas \cite{Ostaszewski2019, Grimsley2019,Li2020} and reinforce learning ideas \cite{Fosel2018, Niu2019}.
All of the QAS works mentioned above are still searching quantum ansatz/architecture in discrete domain, which increases the difficulty on search and is in general time consuming.
Due to the close relation between QAS and NAS together with the great success of differentiable NAS ideas in machine learning, we here introduce a framework of differentiable QAS that enable end-to-end automatic differentiable QAS (DQAS).
This new approach unlocks more possibilities than previous works with less searching time and more versatile capabilities.
It is designed with general QAS philosophy in mind, and DQAS framework is hence universal for all types of circuit searching tasks, instead of focusing only on one type of quantum computing tasks as previous work.
~\newline
\nonumberindent{\bf Brief review on QAOA.}
As introduced in \cite{Farhi2014}, QAOA is designed to solve classical combinatorial optimization (CO) problems.
These problems are often NP complete, such as MAX CUT or MIS in the graph theory.
The basic idea is that we prepare a variational quantum circuit by alternately applying two distinct Hamiltonian evolution blocks.
Namely, a standard QAOA anstaz reads
\epsilonq{
\vert \partialrtialsi\rightarrowngle =\partialrtialrod_{j=0}^{P} (e^{iH_c \mathrm{e}xtrm{gauge}ammamma_j}e^{iH_b\begin{equation}} \def\ee{\end{equation}ta_j})\vert \partialrtialsi_0\rightarrowngle,
}{vanilla-qaoa}
where $\vert\partialrtialsi_0\rightarrowngle$ should be prepared in the space of feasible solutions (better as even superposition of all possible states, and in MAX CUT case $\vert\partialrtialsi_0\rightarrowngle = \otimes H^n\vert 0^n\rightarrowngle$, where n is the number of qubits and H is transversal Hadamard gates.)
In general, $H_c$ is the objective Hamiltonian as $H_c\vert\partialrtialsi\rightarrowngle = f(\partialrtialsi)\vert\partialrtialsi\rightarrowngle$, where $f(\partialrtialsi)$ is the CO objective. For MAX CUT problem on weighted graph with weight $\omega_{ij}$ on edge $ij$, $H_c=-{\sigma}um \omega_{ij}Z_iZ_j$ up to some unimportant phase.
(We use the notation $X/Y/Z_i$ for Pauli operators on i-th qubit throughout this work)
$H_b$ is the mixer Hamiltonian to tunnel different feasible solutions, where $H_b={\sigma}um_{i=0}^n X_i$ is the most common one when there is no limitation on feasible Hilbert space.
The correctness of such ansatz is guaranteed when p approach infinity as it can be viewed as quantum annealing (QA), where we start from the ground state of Hamiltonian $H_b$ as $\vert +^n\rightarrowngle$ and go through adiabatically to another Hamiltonian $H_c$, then it is expected that the final output state is the ground state of $H_c$ which of course has the minimum energy/objective and thus solve the corresponding CO problems.
If we relax the strong restrictions from the QA limit and just take QAOA as some form of variational ansatz, then there are four Hamiltonians instead of two defining the ansatz.
\begin{equation}} \def\ee{\end{equation}gin{itemize}
\item $H_p$ the preparation Hamiltonian: we should prepare the initial states from zero product to the ground state of $H_p$. In original case, $H_p$ is the same as $H_b$.
\item $H_b$ the mixer Hamiltonian: responsible to make feasible states transitions happen.
\item $H_p$ the phase/problem Hamiltonian: time evolution under the phase Hamiltonian and the mixer Hamiltonian alternately makes the bulk of the circuit, in original QAOA, $H_p$ is the same as $H_c$.
\item $H_c$ the cost Hamiltonian: the Hamiltonian used in objectives and measurements where $\lambdangle \partialrtialsi \vert H_c\vert \partialrtialsi\rightarrowngle$ is optimized.
\epsilonnd{itemize}
Moreover, such four Hamiltonian generalization of original QAOA can be further extended.
For example, $H_b, H_p$ are not necessarily the same Hamiltonian for each layer of the circuit.
Nonetheless, the essence of such ansatz is that: the number of variational parameters is of order the same as layer number $P$ which is much less than other variational ansatz of the same depth such as typical hardware efficient VQE or quantum neural network design.
This fact renders QAOA easier to train than VQE of the same depth and suffers less from barren plateus \cite{McClean2018}.
And as QAOA ansatz has some reminiscent from QA, the final ansatz has better interpretation ability than typical random circuit ansatz.
It is an interesting direction to automatically search for the four definition Hamiltonians or even more general layouts beyond vanilla QAOA, to see whether there are similar quantum architectures that can outperform vanilla QAOA in CO problems, this is where DQAS plays a role.
The physical intuition behind such QAOA type ansatz relaxation and searching originates from the close relation between QAOA and quantum adiabatic annealing.
In particular, we draw inspirations from efforts to optimize annealing paths and boost performance for quantum annealers.
We reckon at least two fronts to search for better ansatz for the hybrid quantum-classical algorithm. The first case is to actually inspect the standard QAOA (which typically uses two alternating Hamiltonians to build the ansatz) and inquire if any ingredient may be improved. For instance, given the four Hamiltonians for the quantum-adiabatic inspired ansatz, one may search for a better initial-state-preparation Hamiltonian, or find better mixer Hamiltonians than the plain ${\sigma}um_i X_i$ for specific problems.
Another inspiration derives from attempts to speed up quantum adiabatic annealing via ideas like catalyst Hamiltonians \cite{Crosson2014, Albash2019}, counter diabatic Hamiltonians \cite{Sels2017, Hartmann2019}, and other ideas in shortcut to adiabacity.
These ultrafst annealing methods would entail design of complex annealing schedules that deviate from the simple linear schedule interpolating between an initial Hamiltonian and the target Hamiltonian.
When these complex annealing paths are digitalized and projected onto the quantum gate model with variational approximations, they may just live in the form of XX Hamiltonians or local Y Hamiltonians. With these extra Hamiltonians, catalyst or counter diabatic, we anticipate better performances with shallower QAOA-like circuits layout may be achieved.
{\sigma}ection{Methods}
\nonumberindent{\bf Overview.} The task of DQAS is to select several unitaries to compose the circuit that minimize the corresponding objective for a given task. The aim of DQAS is two-fold: one the one hand, DQAS determines a potentially optimal circuit layout, on the other hand, it also identifies suitable variational parameters for the circuit. To achieve the two goals at the same time, we regard DQAS as a bi-optimization problem, where both the parameters determining the quantum structure and trainable weights on the parameterized circuit are optimized via some gradient-based optimizers.
To enable gradient descent search on the quantum structure, we relax the discrete structure parameters into continous domain, where quantum architecture are viewed as the sample from some parameterized probabilistic model.
DQAS is presented as Algorithm \ref{alg:dqas} with a visualized workflow in Fig. \ref{fig:workflow}. We introduce the ingredients for DQAS and the general workflow below. (See Appendix A for more details and the glossary of DQAS algorithm \cite{Note1}) .
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.47\mathrm{e}xtwidth]{workflow.pdf}
\caption{Schematic illustration of DQAS.
We sample a batch of circuit configurations for each epoch from probabilistic model $P(\vect{k}, \vect{\alpha})$.
We then compose corresponding quantum circuits by filling in operations and parameters from two pools.
We can evaluate quantum circuits and compute final objective $\mathrm{e}xtrm{matter}athcal{L}$ where $\vect{\alpha}, \vect{\thetaeta}$ can be adjusted accordingly with gradient based optimization method.
}\lambdabel{fig:workflow}
\epsilonnd{figure}
~\newline
\nonumberindent{\bf Circuit encoding and operation pool.} Any quantum circuit is composed of a sequence of unitaries with and without trainable parameters, i.e.
\epsilonq{U=\partialrtialrod_{i=0}^p U_i(\vect{\thetaeta}_i), }{circuit}
where $\vect{\thetaeta}_i$ can be of zero length corresponding to the case that $U_i$ is a fixed unitary gate. Hence, we formulate the framework to cover circuit-design tasks beyond searching of variational ansatz.
In the most general term, these $U_i$ can represent an one-qubit gate, a two-qubit gate or a higher-level block encoding, such as $e^{iH\thetaeta}$ with a pre-defined Hermitian Hamiltonian $H$.
This set of possible unitaries $U_i$ constitutes the operation pool for DQAS, and the algorithm attempts
to assemble a quantum circuit by stacking $U_i$ together in order to optimize a task-dependent objective function.
We call the choice of primitive unitary gates in the operation pool along with the circuit layout of these gates a circuit encoding.
In the operation pool, there are $c$ different unitaries $V_j$, and each placeholder $U_i$ should be assigned one of these $V_j$ by DQAS.
In this work, we refer to the placeholder $U_i$ as the $i$-th layer of the circuit $U$, no matter such placeholder actually stands for layers or other positional labels.
The circuit design comes with replacement: one $V_j$ from the operation pool can be used multiple times in building a single circuit $U$.
~\newline
\nonumberindent{\bf Objectives.} To enable an end-to-end circuit design, a suitable objective should be specified.
Such objectives are typically just sum of expectation values of some observables for hybrid quantum-classical scenarios, such as combinatorial optimization problems or quantum simulations.
Namely, the objective in these cases reads
\epsilonq{L=\lambdangle 0\vert U^\downarrowgger H U\ket{0},}{objective}
where $H$ is some Pauli strings such as $H=-{\sigma}um_{\lambdangle ij\rightarrowngle} Z_iZ_j$ for MAX CUT problems and $\ket{0}$ represents the direct product state . This loss function $L$ can be easily estimated by performing multiple shots of sample measurements in quantum hardwares.
However, the objectives can assume more general forms for a HQC algorithm. For instance, one may define more sophisticated objectives that not only depend on the mean value of measurements but also depend on distributions of different measurements.
Examples include CVaR \cite{Barkoutsos2020} and Gibbs objective \cite{Li2020}, proposed to improve quality of solutions in QAOA. In general, DQAS-compatible objectives for HQC algorithms assume the following form,
\epsilonq{L={\sigma}um_i g_i(\bra{0} U^\downarrowgger f_i(H_i) U\ket{0}),}{gen-fg}
where $f_i$ and $g_i$ are differentiable functions and $H_i$ are Hermitian observables.
Extending the HQC algorithm to supervised machine learning setup that is commonly used in classification tasks, the objective function has to be further generalized to incorporate quantum-encoded data $\ket{\partialrtialsi_j}$ with corresponding label $y_j$,
\epsilonq{L={\sigma}um_j\left ({\sigma}um_i g_i(\bra{\partialrtialsi_j} U^\downarrowgger f_i(H_i) U\ket{\partialrtialsi_j})- y_j\right )^2.}{obj-gen}
Beyond ansatz searching for HQC algorithms, DQAS can be used to design circuits for state preparation or circuit compilations.
In these scenarios, the objective is often taken as the fidelity between the proposed circuit design and a reference circuit, and the objective for pure states now reads
\epsilonq{L={\sigma}um_j\bra{\partialrtialhi_{j} }U\ket{\partialrtialsi_j},}{obj-st}
where $\ket{\partialrtialhi_j} = U_{\mathrm{e}xt{ref}}\ket{\partialrtialsi_j}$ is the expected output of a reference circuit when $\ket{\partialrtialsi_j}$ is the input state.
For a state-preparation setup, the objective above is reduced to $L=\bra{\partialrtialhi} U\ket{0}$, where $\ket{\partialrtialhi}$ is the target state. It is worth noting that the overlap objective can induce barren plateau issues and the local version of Hilbert-Schmidt Test can be used as objectives to avoid barren plateaus \cite{Khatri2019, Sharma2020}.
For a general task of unitary learning or compilation, the dimension of $\ket{\partialrtialsi_j}$ can be as large as $2^n$ where $n$ is the qubits number, such condition may be relaxed by sampling random inputs from Haar measure \cite{Nakata2017}, which follows the philosophy of machine learning, especially stochastic batched gradient descent.
~\newline
\nonumberindent{\bf Sampling the structures.} With circuit encoding and operation pool, the task of DQAS is reduced to assign $p$ unitaries (selected from the operation pool) to the placeholder $U_i$ in order to construct a circuit $U$ that minimizes an objective $L(U)$.
To facilitate the architecture search, it is tempting to relax the combinatorial problem into a continuous domain, amenable to optimization via gradient descent.
We thus propose to embed the discrete structural choices into a continuously-parameterized probabilistic model.
For instance, we consider a probabilistic model $P(\vect{k}, \vect{\alpha})$ where $\vect{k}$ is the discrete structural parameter determining the quantum circuits structure and hence $\vect{k}$ is often denoted as an intermediate representation (IR) for quantum circuit structure.
For example, if IR $\vect{k}=[1,3,1]$ then it implies that the circuit structure $U(\vect{k}) = V_1 V_3 V_1$ where $V_1$ and $V_3$ refer to elements in the predefined operation pool introduced earlier. In the context of Eq. \epsilonqref{circuit}, $U_1=V_1$, $U_2=V_3$ and $U_3=V_1$.
$\vect{\alpha}$ is the continuous variable characterizing the distribution of the probabilistic model $P$.
For na\"ive mean field probabilistic model, $\alpha_{ij}$ stands for the logarithmic probability to place $V_j$ operator on the position of $U_i$ placeholder.
By such a design, we replace the intimidating task of searching for optimal structure in discrete IR space $\vect{k}$ with the easier task of optimizing continuous model parameters $\vect{\alpha}$.
In short, discrete random variables $\vect{k}$ are sampled from a probabilistic model characterized by parameters $\vect{\alpha}$.
A particular $\vect{k}$ determines the structure of the circuit $U(\vect{k})$, and this circuit is used to evaluate the objectives $L(U)$. The final end-to-end objectives for DQAS reads
\epsilonq{\mathrm{e}xtrm{matter}athcal{L}={\sigma}um_{\vect{k}{\sigma}im P(\vect{k}, \vect{\alpha})} L(U(\vect{k}, \vect{\thetaeta})).}{arg2}
And $\mathrm{e}xtrm{matter}athcal{L}$ depends indirectly on both variational circuit parameters $\vect{\thetaeta}$ and probabilistic model parameters $\vect{\alpha}$, which can be both trained via gradient descent using automatic differentiation.
\begin{equation}} \def\ee{\end{equation}gin{algorithm}[H]
\caption{{\sigma}mall Differentiable Quantum Architecture Search.}
\begin{equation}} \def\ee{\end{equation}gin{algorithmic}[1]
\REQUIRE $p$ as the number of components to build the circuit; operation pool with $c$ distinct unitaries; probabilistic model and its parameters $\vect{\alpha}$ with shape $p\times c$ initialized all to zero; resuing parameter pool $\vect{\thetaeta}$ initialized with given initializer with shape $p\times c\times l$, where $l$ is the max number of parameters of each op in operation pool.
\WHILE{not converged}
\STATE ~~~~Sample a batch of size K circuits from model $P(k, \vect{\alpha})$.
\STATE ~~~~Compute the objective function for each circuit in the batch in the form of Eq. \epsilonqref{objective}, Eq. \epsilonqref{obj-gen}, Eq. \epsilonqref{obj-st} depending on different problem settings.
\STATE ~~~~Compute the gradient with respect to $\vect{\thetaeta}$ and $\vect{\alpha}$ according to Eq. \epsilonqref{nablatheta} and Eq. \epsilonqref{nablaalpha}, respectively.
\STATE ~~~~ Update $\vect{\thetaeta}$ and $\vect{\alpha}$ using given gradient based optimizers and learning schedules.
\ENDWHILE
\STATE Get the circuit architecture $\vect{k}^*$ with the highest probability in $P(\vect{k}, \vect{\alpha})$; and fine tuning the circuit parameters as $\vect{\thetaeta}^*$ associated with this circuit if necessary.
\RETURN final optimal circuit structure labeled by $\vect{k}^*$ and the associating weights $\vect{\thetaeta}^*$.
\epsilonnd{algorithmic}
\lambdabel{alg:dqas}
\epsilonnd{algorithm}
~\newline
\nonumberindent{\bf Filling the circuit parameters.} Since DQAS needs to sample multiple circuits $U$ before deciding whether the current probabilistic model is ideal, we adopt the circuit parameter sharing mechanism for parametrized operators in the operation pool.
We store a tensor of parameters $\vect{\thetaeta}$ with size $p\times c\times l$, where $p$ is the total number/layer of unitary placeholders to build the circuit, $c$ is the size of the operation pool and $l$ is the largest number of parameters for each unitaries in the operation pool, we denoted this tensor as a circuit parameter pool.
For example, if we place the $j$-th operator $V_j$ on the position of placeholder $U_i$ as defined in Eq. \epsilonqref{circuit}, then we should fill such parameterized operator of $l$ parameters with $l$ values from parameter pool: $\thetaeta[i, j, :]$.
Therefore, every sampled parametrized $V_j$ should be initialized with $l$ parameters taken from the circuit parameter pool depending on the placeholder index $i$ and its operation-pool index $j$. With this circuit parameter sharing mechanism, the variational parameters we need to maintain in architecture search is reduced from $lc^p$ to $lcp$, i.e. an exponential reduction of trainable weights in total. This is the key to enabling a large scale quantum architecture search in terms of the operation pool size and the depth of the circuit. The number of possible quantum architectures is still exponential as $c^p$. However, this exponential scaling in terms of operation pool size is not a severe issue as: (1) the operation pool can be highly customizable and small enough by considering high-level encodings and (2) the exponential space can still be efficiently reached via Monte Carlo sampling from a informed probabilistic model. Therefore, the introduction of parameter sharing and architecture sampling render DQAS as a highly scalable approach for architecture search with moderate resources.
~\newline
\nonumberindent{\bf Quantum and Monte Carlo gradients.} DQAS needs to optimize two sets of parameters, $\vect{\alpha}$ and $\vect{\thetaeta}$, in order to identify a potentially ideal circuit for the task at hand.
The gradients with respect to trainable circuit parameters $\vect{\thetaeta}$ are easy to determine
\epsilonq{\nabla_{\vect{\thetaeta}} \mathrm{e}xtrm{matter}athcal{L}= {\sigma}um_{\vect{k}{\sigma}im P(\vect{k}, \vect{\alpha})}\nabla_{\vect{\thetaeta}} L(U(\vect{k}, \vect{\thetaeta})).}{nablatheta}
$\nabla_{\vect{\thetaeta}} L(U)$ can be obtained with automatic differentiation in a classical simulation and from parameter shift \cite{Crooks2019} or other analytical gradient measurements \cite{Harrow2019} in quantum experiments.
As explained in Algorithm \ref{alg:dqas}, not all $\thetaeta$ parameters would be present in a circuit
which are sampled according to the probability $P(\vect{k},\vect{\alpha})$ at every iteration. For missing parameters in a particular circuit, the gradients are simply set to $0$ as anticipated.
Calculations of gradients for $\vect{\alpha}$ should be treated more carefully, since these parameters are directly related to the outcomes of the Monte Carlo sampling from $P(\vect{k},\vect{\alpha})$. The calculation of gradient for the Monte Carlo expectations is an extensively studied problem \cite{Mohamed2019a} with two possible mainstream solutions: score function estimator \cite{Kleijnen1996} (also denoted as REINFORCE \cite{Williams1992}) and pathwise estimator (also denoted as reparametrization trick \cite{Kingma2013}).
In this work, we utilize the score function approach as it is more general and bears the potential to support calculations of higher order derivatives if desired \cite{Foerster2018b, Zhang2019b}.
For unnormalized probabilistic model, the gradient with respect to $\vect{\alpha}$ is given by \cite{Note1}
\al{\nabla_{\vect{\alpha}} \mathrm{e}xtrm{matter}athcal{L}={\sigma}um_{\vect{k}{\sigma}im P}\nabla_{\vect{\alpha}} \ln P(\vect{k}, \vect{\alpha})\, L(U(\vect{k}, \vect{\thetaeta}))- \nonumbernumber \\
{\sigma}um_{\vect{k}{\sigma}im P}L(U(\vect{k}, \vect{\thetaeta})) {\sigma}um_{\vect{k}{\sigma}im P}\nabla_{\vect{\alpha}} \ln P(\vect{k}, \vect{\alpha}).\lambdabel{nablaalpha}}
For normalized probability distributions, $\lambdangle \nabla_{\vect{\alpha}} \ln P\rightarrowngle_P = 0$ and we may simply focus on the first term.
Gradient of $\ln P$ can be easily evaluated via backward propagations on the given well-defined probabilistic model. By considering baseline trick to reduce the estimation variance, a batch size in the order of $10$ is enough for a success DQAS training.
~\newline
\nonumberindent{\bf Probabilistic models.} Throughout this work, we utilize the simplest probabilistic models: independent category probabilistic model, also known as na\"ive mean field model in energy model context.
We stress that more complicated models such as the energy based models \cite{Hinton2012, Carleo2017, Verdon2019} and autoregressive models \cite{Germain2015, Wu2019,Sharir2019, Liu2019} may yield better performances under certain settings where explicit correlation between circuit layers is important.
Such sophisticated probabilistic models can be easily incorporated into DQAS, and we leave this investigation as a future work.
The independent categorical probabilistic model we utilized is described as:
\epsilonq{P(\vect{k}, \vect{\alpha}) = \partialrtialrod_{i=1}^{p} p(k_i, \vect{\alpha_i}),}{arg2}
where the probability $p$ in each layer is given by a softmax
\epsilonq{p(k_i=j, \vect{\alpha_i)}=\frac{e^{\alpha_{ij}}}{{\sigma}um_{k} e^{\alpha_{ik}}},}{arg2}
where $k_i=j$ means that we pick $U_i=V_j$ from the operation pool, and the parameters $\vect{\alpha}$ are of the dimension $p\times c$.
The gradient of such a probabilistic model can be determined analytically,
\epsilonq{\nabla_{\alpha_{ij}} \ln P(k_i=m) = -P(k_i=m) + \deltata_{jm}.}{arg2}
{\sigma}ection{Applications}
\nonumberindent DQAS is a versatile tool for near-term quantum computations. In the following, we present several concrete examples to illustrate DQAS's potential to accelerate research and development of quantum algorithms and circuit compilations in the NISQ era \footnote{Code implementation of DQAS and its applications can be found at \url{https://github.com/refraction-ray/tensorcircuit/tree/master/tensorcircuit/applications}}. Our implementation are based on quantum simulation backend of either Cirq \footnote{See \url{https://github.com/quantumlib/Cirq}}/TensorFlow Quantum \footnote{See \url{https://github.com/tensorflow/quantum}} stack or TensorNetwork \footnote{See \url{https://github.com/google/tensornetwork}}/TensorCircuit \footnote{See \url{https://github.com/refraction-ray/tensorcircuit}} stack.
Firstly, it is natural to apply DQAS to quantum circuits design for state preparation as well as unitary decomposition. For example we can use DQAS to construct exact quantum circuit for GHZ state preparation or Bell circuit unitary decomposition \cite{Note1}. We focus on QEM and HQC context in details below to demonstrate the power of DQAS for NISQ-relevant tasks.
~\newline
\nonumberindent{\bf Quantum error mitigation on QFT circuit.}
Next, we demonstrate that DQAS can also be applied to design noise resilient circuits that mitigate quantum errors during a computation.
The strategy we adopt in this work is to insert single qubit gates (usually Pauli gates) into the empty slots in a quantum circuit, where the given qubit are supposed to be found in idle/waiting status. Such gate-inserting technique can mitigate quantum errors since these extra unitaries (collectively act as an identity operation) can turn coherent errors into stochastic Pauli errors, which are easier to handle and effectively reduce the final infidelity. Similar QEM tricks are reported in related studies \cite{Wallman2016, Zlokapa2020}.
The testbed is the standard circuit for quantum Fourier transformation (QFT), as shown in Fig.~\ref{fig:qem}{\sigma}ubref{fig:qft3}.
We assume the following error model for an underlying quantum hardware. In between two quantum gates, there is a $2\%$ chance of bit flip error incurred on a qubit. When a qubit is in an idle state (with much longer waiting time), there is a much higher chance of about $20\%$ for bit flip errors.
Although the error model is ad-hoc, it does not prevent us from demonstrating how DQAS can automatically design noise-resilient circuits.
Looking at Fig.~\ref{fig:qem}{\sigma}ubref{fig:qft3}, there are six empty slots in the standard QFT-3 circuit. Hence, we specify these slots as $p=6$ placeholders for a search of noise-resilient circuits with DQAS. The search ends when DQAS fills each placeholder with a discrete single-qubit gate such that the fidelity of the circuit's output (with respect to the expected outcome) is maximized in the presence of noises.
If the operation pool is limited to Pauli gates and identity, $\{I, X, Y, Z\}$, then DQAS recommends a rather trivial circuit for error mitigation.
In short, DQAS fills the pair gaps (of qubit 0 and qubit 2) with the same Pauli gate twice, which together yields an identity, in order to reduce the error in the gap. As for qubit 1, where a single gap occurs at the beginning and the end of the circuit as shown in Fig.~\ref{fig:qem}{\sigma}ubref{fig:qft3}, DQAS simply fills these gaps with nothing (identity placeholder). However, if we allow more variety of gates in the operation pool, such as S gate and T gate, then more interesting circuits can be found by DQAS. For instance, Fig.~\ref{fig:qem}{\sigma}ubref{fig:qft3qem} is one such example. In this case, DQAS fills the two gaps of qubit 1 with a $T$ gate each. This circuit cannot be found by the simpler strategy of inserting unitaries into consecutive gaps. Thus, DQAS provides a systematic and straightforward approach to identify this kind of long-range correlated gate assignments that should effectively reduce detrimental effects of noise.
We also carried out DQAS on QFT circuit for 4 qubits with $p=12$ circuit gaps as shown in Fig.~\ref{fig:qft4}{\sigma}ubref{fig:textbookqft4}.
DQAS automatically finds better QEM architecture which outperforms na\"ive gate inserting policies again. Fig.~\ref{fig:qft4}{\sigma}ubref{fig:qemqft4} displays one such example.
The interesting patterns of long-range correlated gate insertions are obvious for quibt 2. It is also clear that DQAS learns that more than two consecutive gates can combined collectively to render identity such as the three inserted gates for qubit 0.
Further details on the search for optimal QEM architectures and comprehensive comparison on experiment values of final fidelities can be found in the Appendix H \cite{Note1}.
In summary, DQAS not only learns about inserting pairs of gates as identity into the circuit to mitigate quantum error, but also picks up the technique of the long-range correlated gate assignment to further reduce the noise effects.
This result is encouraging and shows how instrumental DQAS as a tool may be used for designing noise-resilient circuits with moderate consumption of computational resource.
In this study, we only adapt the simple gate-insertion policy to design QEM within DQAS framework. We expect more sophisticated QEM methods may also be adapted to work along with DQAS to identify novel types of noise-resilient quantum circuits. This is a direction that we will actively explore in follow-up studies.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
{\sigma}ubfloat[]{\lambdabel{fig:qft3}
\includegraphics[width=0.39\mathrm{e}xtwidth]{qft3.pdf}}
{\sigma}ubfloat[]{\lambdabel{fig:qft3qem}
\includegraphics[width=0.39\mathrm{e}xtwidth]{qft3qem.pdf}}
\caption{(a) The basic circuit for QFT on 3 qubits, T gate and H gate in the middle of the circuit can be easily arranged in the same vertical moment with no gap. And there are six gaps left (two on each qubit) in this setup. (b) The QEM circuit for QFT automatically found by DQAS. All slots are filled, DQAS is powerful enough to learn long range correlations so that it can fill the gaps on qubit 1 which are seperately located. The fidelity between the two circuits on noisy hardwares and the ideal circuit are $0.33$ and $0.6$, respectively.
}\lambdabel{fig:qem}
\epsilonnd{figure}
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
{\sigma}ubfloat[]{\lambdabel{fig:textbookqft4}
\includegraphics[width=0.43\mathrm{e}xtwidth]{qft4.pdf}}
{\sigma}ubfloat[]{\lambdabel{fig:qemqft4}
\includegraphics[width=0.46\mathrm{e}xtwidth]{qft4qem.pdf}}
\caption{(a) The basic circuit for QFT on 4 qubits, some of the gates can be easily arranged in the same vertical moment with no gap. And there are $12$ gaps left in this arrangement. (b) The QEM circuit for QFT automatically found by DQAS. The fidelity between the two circuits on noisy hardwares and the ideal circuit are $0.13$ and $0.46$, respectively.
}\lambdabel{fig:qft4}
\epsilonnd{figure}
~\newline
\nonumberindent{\bf QAOA ansatz searching.} QAOA introduces the adiabatic-process inspired ansatz that stacks alternating Hamiltonian evolution blocks as $e^{-i\thetaeta H}$, where $H$ could be different Hermitian Hamiltonians. QAOA can obtain better approximation ratio with increasing number of repetitive circuit blocks $P$ as its infinite $P$ limit is equivalent to quantum adiabatic evolution. To the end of employing DQAS to design parametrized quantum circuits within the hybrid quantum-classical paradigm for algorithmic developments, we adopt a higher-level circuit encoding scheme as inspired by QAOA. More specifically, the operation pool consists of $e^{-i\thetaeta H}$ blocks with different Hermitian Hamiltonians and also parameter free layers of traversal Hadamard gates $\otimes^n H$. In comparison to assembling a circuit by specifying individual quantum gates, this circuit encoding scheme allows a compact and efficient description of large-scale and deep circuits. For simplicity, we dub the circuit-encoding scheme above as the layer encoding.
For illustrations, we apply DQAS to design parametrized circuit for the MAXCUT problem in this subsection in QAOA-like fashion.
Aiming to let DQAS find ansatz without imposing strong QAOA-type assumptions on the circuit architecture, we expand the operation pool with additional Hamiltonians of the form $\hat{H} = -{\sigma}um_{\lambdangle ij \rightarrowngle} O_iO_j$ and $\hat{H} = {\sigma}um_i O_i$, where $O \in \{X,Y,Z\}$; and we refer to these operations as the xx-layer, rx-layer, rz-layer and so on.
In addition, we also add the transversal Hadamard gates and denote it as the H-layer. All these primitive operations can be compiled into digital quantum gates exactly.
Next, let us elaborate on an interesting account that DQAS automatically re-discovers the standard QAOA circuit for the MAXCUT problem. To begin, we distinguish two settings: instance learning (for a single MAXCUT problem) and ensemble learning (for MAXCUT problems on ensemble of graphs). As noted in \cite{Brandao2018}, the expected outputs by an ensemble of QAOA circuits (defined by graph instances from, say, Erd\"os-R\'enyi distributions or regular graph distributions) with fixed variational parameters $\thetaeta$ are highly concentrated. The implication of such concentration is that the optimal parameters (for an arbitrary instance in the ensemble) can be quite close to being optimal for the entire ensemble of graph instances.
This fact not only increases the stability of the learning process with an ensemble of data inputs, but also makes QAOA more practical when the outer optimization loop can be done in this once-for-all fashion.
In this work, we apply DQAS to both instance learning task and regular graph ensemble learning task \cite{Note1}.
For an ensemble learning on regular graph ensemble (node 8, degree 3), we let DQAS search for an optimal circuit design with $p=5$. By using the aforementioned operation pool comprising the H-layer, rx/y/z-layer and zz-layer with the expected energy for the MAXCUT Hamiltonian as objective function, DQAS recommends the optimal circuit with the following layout: H, zz, rx, zz, rx layers, which coincides exactly with the original QAOA circuit. For metrics in the searching stage, see Fig.~\ref{fig:loss}.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.35\mathrm{e}xtwidth]{loss.pdf}
\caption{Metrics on DQAS training (depth $p=5$) for MAX CUT problem of degree-3 regular graph ensemble with 8 nodes. The upper plot shows the expected energy/averaged cut value in the training process, the loss is approaching $-8.8$ which reflects the result from $P=2$ QAOA with H, zz, rx, zz, rx layer arrangement. The lower plot indicates how the probability of such optimal layout is increased when the probabilistic model underlying is updated.
}\lambdabel{fig:loss}
\epsilonnd{figure}
We also carried out DQAS on QAOA ansatz searching with multiple objective consideration on hardware details as well as double-layer block encoding for operations. For details, see the Appendix I \cite{Note1}.
~\newline
\nonumberindent{\bf Reduced graph ansatz searching.} To the end of designing circuits shallower than QAOA, another approach worth attempt is to re-define the primitive circuit layers in the operation pool.
For instance, the zz-layer block is usually generated by the Ising Hamiltonian with the full connectivity of the MAXCUT problem.
However, if the underlying graph of a zz-layer is only a subgraph then the number of gates would be reduced. Suppose we now replace the standard zz-layer (with full connectivity of the original problem) with a set of reduced zz-layers (each generated by a subgraph containing at most half of all edges in the original graph), then a circuit comprising 2 such reduced zz-layers is shallower than the standard $P=1$ QAOA circuit.
As summarized below, ansatz built from such reduced zz-layers is more resource efficient and outperforms the vanilla QAOA layout using the same number of quantum gates. Fig.~\ref{fig:reducedansatz} summarizes the DQAS workflow in searching ansatz with reduced zz-layers.
To demonstrate the effectiveness of this strategy, we consider the circuit design under instance learning setup in which reduced zz-layers in the operation pool are induced by the graph connectivity of a particular instance. In this numerical experiment, we again set out to design a $p=5$ circuit for $n=8$ qubits. More specifically, we generate 10 subgraph with edge density lower than half of the base graph and substitute the base zz-layer with these 10 newly introduced reduced zz-layers in the operation pool.
In such a setup, DQAS is responsible for finding (1) an optimal circuit layout of different types of layers, (2) best reduced graphs that give rise to the zz-layer in circuit, and (3) optimal parameters $\thetaeta$ for rx/y/z-layer and zz-layer.
Here we give a concrete example. For an arbitrary graph instance drawn from the Erd\"os-R\'enyi distribution with a MAX CUT of 12, DQAS automatically design a circuit that exactly predicts the MAX CUT of 12. This $p=5$ circuit is composed of following layers: rx-layer, zz-layer, zz-layer, ry-layer and rx-layer. Note the two zz-layers are induced by distinct sets of underlying subgraphs with only four edges each. As a comparison, the $P=1$ vanilla QAOA gives expected MAX CUT of $10.39$, while $P=2$ vanilla QAOA predicts $11.18$.
In terms of overlap with exact MAX CUT configuration state, the reduced ansatz found by DQAS has nearly $100\%$ success probability for one-shot measurement to get the MAX CUT value while $P=2$ vanilla QAOA has $47\%$ success probability to get the correct MAX CUT value.
The reduced ansatz designed by DQAS consumes about the same amount of quantum resources as the $P=1$ QAOA circuit yet even outperforms the vanilla $P=2$ QAOA circuit. We stress that such an encouraging result is not a special case. By using the reduced ansatz layers, we can consistently find reduced ansatz that outperforms vanilla QAOA of the same depth for MAX CUT problems on a variety of unweighted and weighted graphs \cite{Note1}.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.47\mathrm{e}xtwidth]{reducedansatz.pdf}
\caption{Schematic workflow for reduced graph ansatz search on MAX CUT setup.
In reduced ansatz searching, there are various reduced graph backend zz-layer in the operation pool.
These reduced graph are sub graph instances from the problem graph instance. DQAS can not only find the optimal layout and optimal parameters $\vect{\thetaeta}$, but also find the best reduced graph for these zz-layers.
}\lambdabel{fig:reducedansatz}
\epsilonnd{figure}
DQAS not only can learn QAOA from scratch, but also can easily find better alternatives with shorter circuit depth with an operation pool using slightly tweaked Hamiltonian evolution blocks as primitive circuit layers. This last achievement is of paramount importance in the NISQ era where circuit depth is a key limitation.
{\sigma}ection{Discussions}
DQAS is a versatile and useful tool in the NISQ era. Not only can DQAS handle the design of a quantum circuit, but it can also be seamlessly tailored for a specific quantum hardware with customized noise model and native gate set in order to get best results for error mitigation.
We have demonstrated the potential of DQAS with the following examples: circuit design for state preparing and unitary decomposition (compilation), and noiseless and noisy circuit design for the hybrid quantum classical computations.
In particular, we also introduce the reduced ansatz design that proposes shallower circuits that outperforms the conventional QAOA circuits that are inherently more resource intensive ansatz.
In conclusion, we re-formulate the design of quantum circuits and hybrid quantum-classical algorithm as an automated differentiable quantum architecture search. Inspired by DARTS-like setup in NAS, DQAS works in a differentiable search space for quantum circuits.
By tweaking multiple ingredients in DQAS, the framework is highly flexible and versatile. Not only can it be used to design optimal quantum circuits under different scenarios but it also does the job in a highly customized fashion that takes into account of native gate sets, hardware connectivity, and error models for specific quantum hardwares. The theoretical framework itself offers a fertile ground for further study as it draws advanced concepts and techniques from the newly emerged interface of differential, probabilistic, and quantum programming paradigms.
~\newline
~\newline
\nonumberindent{\bf Note added}\\
\nonumberindent After this work was posted on arXiv, a relevant paper \cite{Du2020a} was also posted. This paper also proposed the idea of using quantum architecture search as a promising strategy for designing hardware-specific and noise-resilient quantum circuits. Conceptually, this work shares some similarities with our work. The approach utilized in their work is of random search and evolutionary nature, where the circuit sampling process stays evenly distributed (i.e. a fixed probabilistic model in our context) while our DARTS-inspired workflow iteratively updates both circuit parameters and the circuit-structure probabilistic model. Together, these two works validate the benefits of using QAS framework to optimizie quantum circuits and should help substantially in establishing quantum advantage in the NISQ era.
~\newline
\nonumberindent{\bf Acknowledgments}\\
\nonumberindent SXZ and HY are supported in part by the NSFC under Grant No. 11825404. HY is also supported in part by the MOSTC under Grant Nos. 2016YFA0301001 and 2018YFA0305604, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDB28000000, Beijing Municipal Science and Technology Commission under Grant No. Z181100004218001, and Beijing
Natural Science Foundation under Grant No. Z180010.
\input{DQAS_2007.bbl}
\begin{equation}} \def\ee{\end{equation}gin{widetext}
\renewcommand{\thetaeequation}{S\arabic{equation}}
{\sigma}etcounter{equation}{0}
\renewcommand{\thetaefigure}{S\arabic{figure}}
{\sigma}etcounter{figure}{0}
{\sigma}ubsection{Glossary for ingredients of DQAS}
The main components for DQAS are summarized below.
\begin{equation}} \def\ee{\end{equation}gin{table}[htbp]
\begin{equation}} \def\ee{\end{equation}gin{tabular}{cc} \toprule Glossary & Explanation \\
\mathrm{e}xtrm{matter}athrm{i}drule
circuit encoding & The specific arrangement of circuit ``blanks" to be filled by DQAS. \\
operation pool & The set contains all possible parameterized unitaries to construct the circuit. \\
probabilistic model & Circuit candidates are sampled from this model. \\
parameter pool & Circuit parameters organized in the form of (position, operator type) tuple. \\
intermediate representation & The discrete valued vector determining the quantum circuit structure.\\
\bottomrule
\epsilonnd{tabular}
\caption{\lambdabel{tab:glossary} The glossary table summarizing DQAS components.}
\epsilonnd{table}
{\sigma}ubsection{Connection to DARTS}
We illustrate how DQAS is related to DARTS in the neural architecture search.
In particular, we draw attention to a specific variant of DARTS, the probabilistic neural architecture search \cite{Casale2019} that employs a probabilistic model as the backend NAS. Both frameworks represent the super network (during an architecture search) in terms of a probabilistic model, and relies on Monte Carlo sampling along with the score-function method to evaluate gradients for structural variables etc.
Different from DARTS, the probabilistic description of the super network is not just an optional approach for avoiding memory-intensive operations to deterministically evaluate the super network but rather an indispensable ingredient of DQAS for circuit design in the NISQ era.
More precisely, the super network analogy of quantum circuit dictates an implementation of a complex (and potentially non-unitary) operation comprising elementary unitraies $U_j$ present in the operation pool,
\epsilonq{U=\partialrtialrod_{i=0}^p {\sigma}um_{j}\alpha_{ij}U_j.}{arg2}
This complex operation may be implemented in a quantum circuit via methods like linear combination of unitaries \cite{Childs2012}, which is expensive in the NISQ era.
The alternative based on the probabilistic model tremendously reduces the near-term implementation challenges by sampling a batch of simpler quantum circuits, each only carrying out an easily implementable unitary transformation. Other than this implementation issue in quantum circuits to motivate a probabilistic solution, DQAS and probabilistic DARTS are highly similar. The comparison is more succinctly summarized in Fig.~\ref{fig:analogy}.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.26\mathrm{e}xtwidth]{analogy.pdf}
\caption{Comparison between setup of (a) DARTS and (b) DQAS in this work. In DARTS, the super network with all paths are evaluated at the same time, while in DQAS, to be in accordance with quantum circuit, only one path is evaluated once as a quantum circuit simulation (indicated by the solid line as shown in (b)), and different path choices are determined by the underlying probabilistic model.
}\lambdabel{fig:analogy}
\epsilonnd{figure}
{\sigma}ubsection{Derivation on gradients of probabilistic model parameters}
The forward pass evaluation on the objective reads as:
\epsilonq{\mathrm{e}xtrm{matter}athcal{L} = {\sigma}um_{\vect{k}\in P} L(\vect{k}) = {\sigma}um_{\vect{k}} \frac{P(\vect{k}, \vect{\alpha})}{Z(\vect{\alpha})} L(\vect{k}), }{eq:mcf}
where we have use $L(\vect{k})$ for the shortcut of $L(U(\vect{k}, \vect{\thetaeta}))$ and we consider unnormalized probability distribution for the general case, where the implicit normalization factor $Z(\vect{\alpha})={\sigma}um_{\vect{k}} P(\vect{k}, \vect{\alpha})$
Directly apply gradient to \Eq{eq:mcf} with score function approach in mind, we have:
\newcommand{{\sigma}xp}{P(\vect{k}, \vect{\alpha})}
\newcommand{{\sigma}xz}{Z(\vect{\alpha})}
\newcommand{{\sigma}xl}{L(\vect{k})}
\newcommand{{\sigma}xn}{\nabla_{\alpha}}
\newcommand{{\sigma}xs}{{\sigma}um_{\vect{k}\in P}}
\al{\nabla_{\alpha}\mathrm{e}xtrm{matter}athcal{L} &= {\sigma}um_{\vect{k}} \nabla_{\alpha} \frac{P(\vect{k}, \vect{\alpha})}{Z(\vect{\alpha})} L(\vect{k})
\nonumbernumber\\ &=
{\sigma}um_{\vect{k}} \frac{P(\vect{k}, \vect{\alpha})}{Z(\vect{\alpha})} (\frac{\nabla_\alpha{{\sigma}xp}}{{\sigma}xp}{\sigma}xl)-
\frac{{\sigma}xp}{{\sigma}xz}(\frac{{\sigma}xn {\sigma}xz}{{\sigma}xz}{\sigma}xl)\nonumbernumber\\
&= {\sigma}um_{\vect{k}} \frac{P(\vect{k}, \vect{\alpha})}{Z(\vect{\alpha})} (\frac{\nabla_\alpha{{\sigma}xp}}{{\sigma}xp}{\sigma}xl)-({\sigma}um_{\vect{k}} \frac{{\sigma}xp}{{\sigma}xz}{\sigma}xl)({\sigma}um_{\vect{k}}\frac{{\sigma}xp}{{\sigma}xz}\frac{{\sigma}xn {\sigma}xp}{{\sigma}xp}) \nonumbernumber\\
&={\sigma}xs {\sigma}xn \ln {\sigma}xp {\sigma}xl -{\sigma}xs{\sigma}xn\ln{\sigma}xp {\sigma}xs {\sigma}xl .
}
This concludes the derivation of gradient formula for model parameters utilized in the DQAS.
{\sigma}ubsection{General hyperparameters for DQAS training.}
We summarize some of the most important ingredients for DQAS below, and leave the extensive investigation on the effects of these adjustable ingredients as well as other ones to future works.
\begin{equation}} \def\ee{\end{equation}gin{enumerate}
\item Ingredients in common machine learning setup: optimizers, learning rate and schedule for both trainable parameters $\thetaeta$ and structural parameters $\alpha$ of probabilistic models. Since one epoch of evaluation for DQAS is more expensive than conventional neural network evaluations, we may need to find better learning schedules to boost training efficiency for DQAS.
\item Batch sizes: This factor plays a very important role in DQAS since score function estimators is in general of high variance. In practice, batch size of $O(100)$ shows good performance in circuit structure searching.
\item Baselines: there is no theoretical guarantee that running average of objective is the best baseline to lower the variance in Monte Carlo estimations. Therefore, new baselines and even new methods to control variance are worth exploring.
\item Encoding scheme: as we have seen in examples, different encoding schemes of basic unitary blocks matters in QAS. Therefore, domain specific and expressive encoding scheme beyond simple gate sets, such as the layer and block encoding discussed in the main text, are highly desired for a broader set of applications.
\item Probabilistic model: The probabilistic model for DQAS can be more sophisticated to better characterize the correlation between layers of circuits. Exploring energy-based models or autoregressive models is a promising future direction for DQAS.
\item Regularization terms: It is interesting to try and add other regularizations and reward terms into the objectives in order to address multiple objectives such as hardware restriction and quantum noise reduction in the circuit design.
\item Circuit parameter reusing mechanism: Since the theoretical framework for DQAS is general and can easily go beyond DARTS, we may also explore novel parameter reusing mechanisms beyond the na\"ive ones based on the vanilla super network viewpoint.
\epsilonnd{enumerate}
The following hyperparameter settings are assumed unless explicitly stated otherwise,
\begin{equation}} \def\ee{\end{equation}gin{itemize}
\item No prethermalization for circuit parameters $\thetaeta$.
\item Optimizer for probabilistic model parameters $\alpha$ and circuit parameter $\thetaeta$: Adam optimizer with learning step $0.1$.
\item Initializations: Standard normal distribution for circuit parameters and all zero for probabilistic parameters.
\item Other techniques: No regularization terms or noise for circuit parameters by default.
\epsilonnd{itemize}
Note that we do not carry out any extensive search for optimal hyperparameter settings. Hence, there is no guarantee that hyperparameters listed below are optimal for corresponding tasks.
For individual tasks and applications in this work, please see the Appendix G for their specific setup and hyperparameter choice.
{\sigma}ubsection{Training techniques implemented in DQAS}
There are various training ingredients that can be incorporated into the DQAS framework and many of these tricks and/or improvements are inspired by the works devoted to developing DARTS. In this subsection, we elaborate on some of these advanced techniques that we have tested.
~\newline
\nonumberindent{\bf Multiple starts.} Since the energy landscape for objective functions may be very rugged, parallel training on multiple instances with different fractions of dataset, initialization or randomization scheme may be necessary, where the best candidate circuit with optimal objective value amongst all training instances is returned as the final result.
~\newline
\nonumberindent{\bf Parameters prethermalization.} Pretraining and updating circuit trainable parameters $\thetaeta$ from the parameter pool for several epochs as the prethermalization process. A related topic is parameters initializations.
Since we would like to search quantum structure without bias, we use all zero initializer for structural probabilistic model parameters $\alpha$.
For trainable parameters $\thetaeta$, sometimes we can apply domain specific knowledge on the initialization. See QAOA applications in the main text for an example.
~\newline
\nonumberindent{\bf Early stopping.} To avoid overfitting and reduce runtime, some forms of early stopping may be adopted during the training of DQAS.
Following the common practices in training DARTS, we may consider typical criteria, such as the standard deviation in a batch of objective evaluations $L$ or the standard deviation in probability of each layer $P$, to decide when to invoke the early stopping.
The performance gains of DARTS, due to early stopping, were documented in \cite{Liang2019,Zela2019}.
~\newline
\nonumberindent{\bf Top-k grid search or beam search.} If the energy landscape is sufficiently complex, DQAS may settle for a local optimal solution instead of a global one.
In such cases, early stopping can be combined with the so called top-k grid search to avoid trapping into a local minimum.
Namely, for each layer of the ansatz, we keep the top k (usually $k=2$) most probable operations instead of top-1.
Therefore, we have $p^k$ candidates for the optimal circuit ansatz. One can easily train these candidate circuits, benchmark their performance , and pick the top performing one as the optimal circuit architecture for a given problem.
Similarly, we can utilize beam search for optimal structure search, which is the common strategy from text decoder in NLP community. Beam search always maintains $k$ most probable circuit structures and thus avoids the exponential scaling as in grid search.
~\newline
\nonumberindent{\bf Baseline for score function estimators.} It is well known that the score function for the Monte-Carlo estimated gradients suffers high variance; although, the situation can somehow be alleviated by the baseline or control variate approach.
Namely, for the normalized probability distribution $P$, ${\sigma}um_{k{\sigma}im P} \nabla \ln P(k) = 0$, one can add any constant in the objectives' gradient as a baseline in order to reduce the variance.
For instance, during the training of DQAS, we use the running average of $L$ as the baseline. Namely, the loss in Eq. \epsilonqref{nablaalpha} is actually $L = L-\begin{array}r{L}$, where $\begin{array}r{L}$ is the average of objective from the last evaluated batch.
~\newline
\nonumberindent{\bf Layer-by-layer learning.} For deep quantum circuits with large $p$, DQAS may be hard to train from bootstrap.
Inspired by the progressive training \cite{Chen2019, Skolik2020} for DARTS and quantum neural network training, we apply similar ideas to DQAS.
Namely, one first find optimal quantum structure with small $p$ and then adaptively increase $p$ by adding more layers to be trained. In this process, one can also reduce the number of candidate operations in the pool based on the knowledge gained from training instances with smaller $p$.
~\newline
\nonumberindent{\bf Random noise on parameters $\thetaeta$.} The high-dimensional energy landscape can be very rugged in theory.
However, based on some numerical evidences, the optimal quantum circuit tends to consistently output similar objective values even when trainable parameters $\thetaeta$ deviate slightly from the optimal values.
This observation suggests the landscape for optimal quantum structure in terms of trainable parameters is expected to be more flatten than expected. Therefore, to facilitate the search for an optimal circuit architecture, one may add random noise to the trainable parameters $\thetaeta$ to escape the local trapping.
It is worth noting that the random noises are only added onto trainable/network parameters in our setup instead of structural parameters as in \cite{Chen2020} which tried to bridge the gap of performance between two stages in DARTS.
~\newline
\nonumberindent{\bf Regularization and penalty terms.}
Similar to conventional practices in training neural networks, regularization and penalty terms may be introduced in DQAS to avoid overfitting or induce sparsity etc. The applicability of regularization on tunable parameter $\thetaeta$ (with respect to a fixed circuit design) may be easily understood given the high similarity between neural networks and parameterized quantum circuits.
In this section, instead, we focus on the aspect of imposing regularizations on the structural parameters $\alpha$ and manifest the benefits of regularization on searching for a resource-efficient architecture.
We provide two concrete examples for illustrations.
The first example deals with the issue of block merging in DQAS.
For the simplest probabilistic model for $P(\vec{k},\vec{\alpha})$, each circuit layer is independently sampled. There is a high probability that the same parametrized gates are picked in a consecutive order.
Namely, the final architecture may contain snippets like $rx(\thetaeta_1)rx(\thetaeta_2)$, which can be easily merged into one layer.
To address this issue, we propose to add the following terms in the final objective $\mathrm{e}xtrm{matter}athcal{L}$ to punish such trivial arrangement of circuit layers,
\epsilonq{\Deltata \mathrm{e}xtrm{matter}athcal{L}_1= \lambdambda_1{\sigma}um_{i=1}^p {\sigma}um_{k\in c} p(k_i=k, \alpha) p(k_{i-1}=k, \alpha).}{penalty-merge}
Secondly, since two-qubit gates are primarily responsible for infidelity and errors of quantum computations, it is desirable to select a circuit architecture comprising fewer number of two-qubit gates. This kind of resource considerations are encouraged during the architecture search if penalty terms of the following form are explicitly added,
\epsilonq{\Deltata \mathrm{e}xtrm{matter}athcal{L}_2=\lambdambda_2{\sigma}um_{i=1}^p {\sigma}um_{k\in c} p(k_i=k)\times\mathrm{e}xt{\# of two-qubits gates in k}.}{}
We note similar regularization for achieving better training performance \cite{Zela2019} and the multiple objective considerations specifically on computation complexity \cite{Lu2018} have also been reported in the NAS literature.
~\newline
\nonumberindent{\bf Proxy tasks and transfer learning.}
DARTS heavily relies on the idea of proxy tasks to boost performance.
In DARTS training, one first trains and identifies a suitable network architecutre on the simpler CIFAR-10 (image) dataset; subsequently, one uses the same block topology to build neural networks classifiers for the large-scale ImageNet dataset.
Same technique may be adapted to DQAS: finding some structure or patterns in quantum circuits for small size problems with small number of qubits or layers and try to apply similar pattern on larger problems.
For the quantum circuit design, we can even classically simulate the training for small proxy tasks and transfer optimal structures to large problem beyond classical computation power.
We recommend various training techniques, inspired by DARTS-related studies, to obtain more robust and versatile DQAS.
Due to the close relation between architecture search in the context of quantum circuit and neural network, more interesting ideas may be borrowed from NAS to further improve QAS and innovations in QAS may also inspire developments in NAS.
{\sigma}ubsection{DQAS application in state preparation and circuit compiling}
\nonumberindent{\bf State preparation circuit.}
We set out to design a quantum circuit for generating GHZ states $\vert \mathrm{e}xt{GHZ}_n\rightarrowngle = \frac{1}{{\sigma}qrt{2}}(\vert 0^n\rightarrowngle+\vert 1^n\rightarrowngle)$ from $\vert 0^n\rightarrowngle$.
To find an optimal structure with less redundancy, we may progressively reduce the layer number $p$ in DQAS until the objective can no longer be accomplished.
In this case, the operation pool is composed of primitive gates such as single qubit gates and CNOT gates on any pair of qubit.
In principle, the availability of CNOT gates in the operation pool may be further subjected to the connectivity map of an actual hardware.
The objective we choose for this problem is the final states distance given by ${\sigma}um_i\vert\partialrtialsi_i-\partialrtialhi_i\vert$ where $\ket{\partialrtialhi}$ is the target GHZ state. Such a metric is better to optimize than the typical fidelity or state-overlap objective $\lambdangle \partialrtialsi\vert \partialrtialhi\rightarrowngle$.
The optimal circuit found by DQAS is shown in Fig.~\ref{fig:circuit}{\sigma}ubref{fig:ghz3}.
It is interesting to observe that DQAS tries to optimize $R_y(\thetaeta)$ by tuning $\thetaeta$ to approximate the behavior of Hadamard gate when Hadamard gate is not given in the operation pool.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
{\sigma}ubfloat[]{\lambdabel{fig:ghz3}
\includegraphics[width=0.3\mathrm{e}xtwidth]{ghz3.pdf}}
{\sigma}ubfloat[]{\lambdabel{fig:bell}
\includegraphics[width=0.4\mathrm{e}xtwidth]{bell.pdf}}
\caption{(a) The minimal circuit for preparation of GHZ$_3$ states automatically found by DQAS, where $R_y(\thetaeta) = e^{-i\frac{\thetaeta}{2}}$. ~(b) The circuit with $p=5$ found by DQAS for Bell states transformation.
}\lambdabel{fig:circuit}
\epsilonnd{figure}
~\newline
\nonumberindent{\bf Unitary decomposition.}
For state-preparation example, we only care about how the circuit acts on the input state $\vert 0^n\rightarrowngle$ and ignore how other input states could be transformed. This lack of consideration is obvious in the chosen objective above. For the current example, we aim to decompose arbitrary unitray operation into a set of primitive quantum gates and this implies all inputs transformation are considered.
For a concrete illustration, we use DQAS to design a quantum circuit for 2-qubits Bell state generation, which is useful for superdense coding.
We need $2^2=4$ independent input-output pairs to fully characterize the two-qubit unitary under investigation. For instance, the Bell state preparation circuit needs to conform to the following input/output relations, convert the input $\ket{00}$ and $ \ket{11}$ to the Bell states $\re{{\sigma}qrt{2}}(\ket{00}\partialrtialm\ket{11})$, and convert the input $\ket{01}$ and $\ket{10}$ to $\re{{\sigma}qrt{2}}(\ket{01}\partialrtialm\ket{10})$, respectively.
In the next example, we illustrate how to assemble this Bell-state preparation circuit with a finite set of quantum gates. In other words, we purposely restrict the operation pool to a finite number of discrete gates without any trainable parameters such as rotation angles. To apply DQAS to search for a Bell-state preparation circuit, we use the input/output relations in Table.~\ref{table:bell} to build the objective function,
\al{\mathrm{e}xtrm{matter}athcal{L} = -\lambdangle Z_0Z_1\rightarrowngle_{U\ket{00}} -\lambdangle X_0X_1\rightarrowngle_{U\ket{00}}\nonumbernumber\\
+\lambdangle Z_0Z_1\rightarrowngle_{U\ket{01}} +\lambdangle X_0X_1\rightarrowngle_{U\ket{01}}\nonumbernumber\\
+\lambdangle Z_0Z_1\rightarrowngle_{U\ket{10}} -\lambdangle X_0X_1\rightarrowngle_{U\ket{10}}\nonumbernumber\\
-\lambdangle Z_0Z_1\rightarrowngle_{U\ket{11}} +\lambdangle X_0X_1\rightarrowngle_{U\ket{11}}
,}
where $U$ is the tentative circuit proposed by DQAS.
The final ($p=5$-depth) circuit obtained via DQAS is presented in Fig.~\ref{fig:circuit}{\sigma}ubref{fig:bell}.
\begin{equation}} \def\ee{\end{equation}gin{table}[]
\begin{equation}} \def\ee{\end{equation}gin{tabular}{|c|c|c|c|}
\hline
input & output & $Z_0Z_1$ & $X_0X_1$ \\ \hline
$\vert 00\rightarrowngle$ & $\frac{1}{{\sigma}qrt{2}}(\vert 00\rightarrowngle+\vert 11\rightarrowngle)$ & +1 & +1 \\ \hline
$\vert 01\rightarrowngle$ & $\frac{1}{{\sigma}qrt{2}}(\vert 10\rightarrowngle-\vert 01\rightarrowngle)$ & -1 & -1 \\ \hline
$\vert 10\rightarrowngle$ & $\frac{1}{{\sigma}qrt{2}}(\vert 10\rightarrowngle+\vert 01\rightarrowngle)$ & -1 & +1 \\ \hline
$\vert 11\rightarrowngle$ & $\frac{1}{{\sigma}qrt{2}}(\vert 00\rightarrowngle-\vert 11\rightarrowngle)$ & +1 & -1 \\ \hline
\epsilonnd{tabular}
\caption{Specification of a Bell circuit.}
\lambdabel{table:bell}
\epsilonnd{table}
~\newline
\nonumberindent{\sigma}ubsection{Hyperparameter settings and training ingredients in experiments}
\nonumberindent{\bf State preparation circuit for GHZ$_3$.} Primitive operation pools: parameterized $R_y$ gate on qubits 0,1,2, CNOT on (0,1); (1,0); (1,2); (2;1) since we consider circuit topology with the nearest neighbor connections. Batch size is 128. Initializer for circuit parameters are zero initializer. Optimizer for $\alpha$: Adam with $0.15$ learning rate.
~\newline
\nonumberindent{\bf Bell state circuit.} Primitive operator pool: $X, Y, H$ and CNOT gates on each qubit. Note the whole set of operators in the pool is trainable parameter free. Batch size: 128. Optimizer for $\alpha$: Adam with 0.15 learning rate.
~\newline
\nonumberindent{\bf QEM on QFT-3 circuit.} Primitive operation pool includes discrete gates: X, Y, Z, S, T, and I (the identity gate). $S = diag(1,i)$ and $T=diag(1, e^{\frac{\partialrtiali}{4}i})$. The I gate must always be in the pool as it stands for leaving the qubits in an idle state. Batch size is 256.
The objective for this QEM task is to maximize the fidelity between noisy output of DQAS-designed quantum circuit and the ideal output.
In principle , it should be evaluated from a batch of different input states for each circuit. However, as observed in numerical tests, the standard deviation of the fidelity between noisy and ideal circuit for different input states is small. Therefore, following the spirit of the stochastic gradient descent, the fidelity of such circuit is only evaluated for {\bf one} random input state of each circuit in one epoch. Such a random input state is drawn from the random Haar measure, and this can be partially achieved by a short-depth circuit denoted as the unitary 2-design \cite{Nakata2017}. We utilize 4 blocks repetition of unitary 2-design as input states preparation circuit by default. Since we only want to evaluate the noise in the QFT circuit, we assume the preparation circuit for the random input states noiseless.
~\newline
\nonumberindent{\bf QEM on QFT-4 circuit.} For the QFT-4 circuit in the main text, there are 12 slots in total. Therefore, we set $p=12$ for this DQAS design. Since the search space is very large, the search tends to be trapped in a local minimum. Nevertheless, the designed (and potentially sub-optimal) circuits usually outperforms the bare QFT-4 circuit in terms of the fidelity. To reduce the multi-start number, we can restrict the search space by limiting the number of single-qubit gates in the pool. Knowledges on the relevant set of single-qubit gates for such a task can be learned from similar examples such as a QFT-3 case. This procedure utilizing prior knowledge observed in smaller systems and pruning of possible operators follows the philosophy of the progressive training for DARTS \cite{Chen2019} as well as the idea of transfer learning.
In particular, in this study, the operation pool contains the I placeholder, Z gate, $Z^{\frac{2}{3}}$, $Z^{\frac{4}{3}}$, S gate and T gate.
Note that there is another trick that may further increase the probability of finding highly nontrivial QEM circuits.
The idea is to exclude the I placeholder from the operation pool. Without the I placeholder, we force the DQAS engine to fill every gap in the circuit while attempting to maintain a high fidelity. In this way, we encourage DQAS to find nontrivial filling pattern containing long-range correlations of added gates in a QEM circuit and we can recover the nontrivial QEM circuits more easily as shown in the main text. The batch size is $256$. Optimizer for $\alpha$ is Adam with learning rate $0.03$ to $0.06$.
~\newline
\nonumberindent{\bf Typical setups for QAOA ansatz searching.}
We consider both small and large operation pools for the layer encoding. The small one includes rO-layer with $O=x,y,z$, zz-layer and H-layer. The large one also includes the xx-layer and yy-layer. We have also tested an extra large operation pool with NNN-layers. Namely, we also consider Hamiltonians in the form of ZZ, XX, and YY that couples pairs of next nearest neighbors on the underlying graph. We have reproduce QAOA type layout successfully in such extra large pool. For block encoding scheme, the operator pool includes H-layer, rx-zz-block, zz-ry-block, zz-rx-block, zz-rz-block, xx-rz-block, yy-rx-block, rx-rz-block. For example, zz-ry-block represents for operation on the circuits as $e^{i\thetaeta_m {\sigma}um_{\lambdangle ij\rightarrowngle}Z_iZ_j}e^{-i\thetaeta_n{\sigma}um_i X_i}$.
For the setup of reduced ansatz searching, the operation pool includes H-layer and r-O layers as well as zz-layers but now with different sets of edges.
We often include 8 to 12 different subgraphs from the problem graph instance.
And each of them only have a small part of edges as the original graph.
These subgraphs can be chose randomly and in general $O(10)$ of them is well enough to search for some better ansatz than plain QAOA.
However, the caveat is that: the number of such reduced graph based zz-layer may have to be larger with graph of more nodes/problems involving more qubits.
It is more interesting that the reduced graph for these new zz-layers are not necessarily exact subgraph of the problem under investigation.
These reduced graph can also have random edges and random weights on them, and the result found by DQAS can be as good as subgraph ansatz sometimes.
The ablation study on reduced graph instances design is an interesting future direction.
We have tried various combinations of ingredient in QAOA ansatz searching. Some are of particular value including: large batch size, typically $64$ to $512$; noise on circuit parameters in simulation, typically independent noise on each parameters in the pool as zero centered Gaussian distribution with standard deviation $0.2$.
Different objectives, of which CVaR gives promising result apart from conventional energy expectation objectives.
And when CVaR is concerned, these is no need to add trainable parameter noise in general.
Initializers for circuit parameters, Gaussian initializer with narrow width and the mean value around $0.2$ to $0.3$ which is near the QAOA optimal parameters region.
For reduced ansatz search which goes far beyond vanilla QAOA layout, some initializer with larger standard deviation around $1.0$ is utilized.
SGD optimizer may be better in some cases for updating $\alpha$ with learning rate $0.15$ to $0.3$ and even larger when the gradient is small.
The learning rate for circuit parameters should be small as $0.005$ to $0.05$ that depends on batch size.
We also tried L-BGFS and Nelder-Mead optimizers for circuit parameters updates and see no obvious improvements in terms of final objective values.
Fixing header operator as traversal Hadamard gates boost the training but it is not necessary: the initialization with $\otimes^n H$ can also be auto found via DQAS itself.
Penalty terms as in the main text with $\lambdambda_1$ around $0.1$ to $0.2$ may help to alleviate block merging in the training.
The resource efficiency regularization terms as mentioned in the main text has a smaller $\lambdambda_2=0.01$. Such term is particularly useful to avoid early attraction by xx-layer and yy-layer (or one can simply drop xx-layer and yy-layer from the beginning as they are shown to be redundancy in the main text).
Other techniques include those discussed in the main text: top-k grid search postprocessing and multi start may be necessary to find optimal structure as the energy landscape of such search is rather complicated and lots of lower p QAOA layouts serve as local minimum traps.
For example, if we carry out DQAS for 5 layers, we will often end in an architecture equivalent to $P=1$ QAOA instead of $P=2$.
The reason behind that is the performance improvement with deeper QAOA layout is slight and the global minima is vagued by lots of local minimum with similar objective values.
Block encoding with two layers combination as primitive operators in the pool is much better to train than layer encoding scheme and mitigate the above problem. On the contrary, novel objectives such as Gibbs objective that shows sharper energy landscape in circuit parameter space are not suitable in DQAS when bi-optimization dominates and flatten energy landscape helps.
It is also worth mentioning that, in ensemble learning setup, the dataset/graph instances set is not pre-determined. Instead the graph instance is generated on the fly in DQAS training and in princinple one can experience all graph instances of one ensemble as long as the searching epochs is large enough. This design is better than training on fixed dataset of ensemble of graph instances, which tends to over fitting.
Construction of QAOA primitive layers with native quantum gates set as follows:
The single-qubit layer specified in the form of $e^{-i \thetaeta{\sigma}um_i O_i }$ is just a single-qubit rotation gate $r_O$.
The layer of two-qubit gates is of the form $e^{i \thetaeta{\sigma}um_{\lambdangle ij\rightarrowngle} O_i O_j}=\partialrtialrod_{\lambdangle ij\rightarrowngle}e^{i\thetaeta O_iO_j}$, and each term $e^{i \thetaeta O_iO_j }$ can be implemented as in Fig.~\ref{fig:zz}.
If $O$ is not Z, then basis-rotation gates (Hardmard $H$ for $O=x$, $R_x(\partialrtialm \frac{\partialrtiali}{4})$ for $O=Y$) are attached on both sides of the circuit.
From this perspective, xx-layer and yy-layer are actually redundant since they can be exactly implemented by using H-layer+zz-layer+H-layer or rx-layer+zz-layer+rx-layer, respectively. Therefore we can safely drop the xx-layer and yy-layer from the operation pool.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.3\mathrm{e}xtwidth]{zzgate.pdf}
\caption{The circuit construction for $e^{-\frac{i\thetaeta}{2}Z_0Z_1}$ by CNOT and $R_z$ gate. For implementation of $e^{-\frac{i\thetaeta}{2}X_0X_1}$ and $e^{-\frac{i\thetaeta}{2}Y_0Y_1}$, the only change is to pretend and append Hadamard gate or $R_x(\partialrtialm \frac{\partialrtiali}{4})$ gate on both sides and each qubit lines. This is the key building block for QAOA layers.
}\lambdabel{fig:zz}
\epsilonnd{figure}
{\sigma}ubsection{Further results on QEM of QFT circuit}
\nonumberindent{\bf Fidelity results for bare circuit, na\"ive QEM circuit and nontrivial QEM circuit by DQAS.}
The noise model for the backend quantum simulation is described in the main text. The bit-flip errors are randomly inserted between adjacent circuit layers. Whenever there is an empty slot (idel state) in the circuit, the error rate is larger. Next, we not only present a comparative study between QEM circuits found by DQAS and bare circuit, but also present a comparative study of QEM circuit designed by DQAS and QEM circuit inspired by theoretical methods.
For QFT-3 circuit, the typical textbook circuit only gives a fidelity of $0.33$ in the presence of the bit-flip errors;
naive QEM circuit with additions of pairs of Pauli gates on qubit 0 and 2 gives an ameliorated fidelity of $0.55$ (different choices of Pauli gates only yield a small differences in the fidelity). QEM circuit found by DQAS gives a further improved fidelity of $0.6$.
For QFT-4 circuit, the typical textbook circuit gives a fidelity of $0.13$. Again, the na\"ive QEM circuit, where as many pairs of Pauli gates as possible are added to fill empty slots (see Fig.~\ref{fig:policy}{\sigma}ubref{fig:naive}), gives a fidelity of $0.3$. There is another type of circuit-filling policy (see Fig.~\ref{fig:policy}{\sigma}ubref{fig:advance}), where we do nothing to single empty slots in a circuit, and insert a series of $Z^{\alpha_i}$ that collectively give an identity into contiguous empty slots (spanning across more than 2 layers). This strategy allows one to fill empty slots in the circuit except those isolated ones restricted to a single circuit layer. QEM circuit designed under this policy gives a fidelity around $0.41$. On the other hand, DQAS discovers many distinct configurations of QEM circuit for the QFT-4 case, and the associated fidelities are usually found to be in the range of $0.45{\sigma}im 0.46$. Clearly, automated circuit designs outperform the ones recommended by human-designed and sophisticated empty-slot-filling policy .
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
{\sigma}ubfloat[]{\lambdabel{fig:naive}
\includegraphics[width=0.38\mathrm{e}xtwidth]{naiveqemqft4.pdf}}
{\sigma}ubfloat[]{\lambdabel{fig:advance}
\includegraphics[width=0.45\mathrm{e}xtwidth]{advanceqemqft4.pdf}}
\caption{Some human design gate inserting policies for QEM on QFT-4 circuit. (a) The na\"ive Pauli pair inserting whenever possble. (b) The advance inserting which tries to collectively return identity in each gap except from single holes.
DQAS found QEM circuit can outperform these circuits in terms of fidelity.
}\lambdabel{fig:policy}
\epsilonnd{figure}
~\newline
\nonumberindent{\bf More QEM circuits with similar fidelity for QFT-4.}
In QFT-4 case, DQAS also finds various circuits of similar fidelity as the optimal one in the main text, we present some of them in Fig.~\ref{fig:moreqem}.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
{\sigma}ubfloat[]{
\includegraphics[width=0.43\mathrm{e}xtwidth]{qemvariant1.pdf}}
{\sigma}ubfloat[]{
\includegraphics[width=0.4\mathrm{e}xtwidth]{qemvariant2.pdf}}
\caption{Some optimal QEM circuit found by DQAS. They shows similar performance in fidelity as the one given in the main text.
}\lambdabel{fig:moreqem}
\epsilonnd{figure}
{\sigma}ubsection{Further results on QAOA ansatz searching}
\nonumberindent{\bf Multiple objectives with hardware consideration.}
Additional considerations may be taken into account during the search for an ideal circuit design for the MAXCUT problem. Suppose that we still work with the layer-encoding operation pool given above but with xx-layer, yy-layer explicitly considered. Furthermore, we suppose the backend quantum hardware is equipped with the primitive gates of rO, H, and CNOT. Therefore, every circuit layer has to be translated into this native gate set. To design resource-efficient quantum circuit by DQAS, we may add the following penalty term to incorporate consideration of resource limitation and quantum error mitigation into the mix,
\epsilonq{\nabla \mathrm{e}xtrm{matter}athcal{L} =\lambdambda_2 {\sigma}um_{i=1}^p {\sigma}um_{k\in c} p(k_i=k) \omega(k),}{qaoa-penalty}
where $\omega(\mathrm{e}xt{CNOT}) = 2$, $\omega(\mathrm{e}xt{rO}) = 1$ and $\omega(\mathrm{e}xt{H}) = 1$. Given the above costs for each gate, one can easily show that $\omega(\mathrm{e}xt{xx-layer}) = 27/2$ for regular graph of degree 3.
These weights can be further adjusted as hyperparameters to reach better multiple objective frontier on both performance and resource efficiency. Within such a large operation pool and regularization terms as Eq. \epsilonqref{qaoa-penalty}, we can again resume typical QAOA layout as H, zz, rx, zz, rx.
Furthermore, we can find other competitive structures beyond the vanilla QAOA layout as shown in the following information.
~\newline
\nonumberindent{\bf Block encoding for QAOA search.} When dealing with large-scale circuits, it may become progressively more challenging to discover an optimal structure for MAX CUT problems with DQAS.
This is partially related to our decision to utilize the simplest probabilistic model, where no explicit correlations about choices of consecutive layers are taken into account of. It might be useful to add these correlations into the design. For instance, in the standard QAOA circuit, an rx-layer is always followed by a zz-layer.
Acknowledging the usefulness of such a complex block that contains at least two primitive layers, we introduce the block encoding. Namely, the primitives in the operation pool are of the form $e^{i\thetaeta_1 ZZ}e^{-i\thetaeta_2 X}$. In other words, the block encoding deals with various combinations of basic operations like zz-rx-block, yy-rz-block and so on. Same as before, it is useful to keep the Hadamard H-layer in the pool.
Via this block encoding, we easily discover the standard QAOA layout for $P=3$ as H-layer, zz-rx-block$* 3$ in 8 qubit system. The emergence of a clear pattern of circuit layout indicates that zz-x-block holds the key for making highly accurate predictions on the MAXCUT problems. Thus, DQAS learns the essence of QAOA circuit purely by exploring and assessing different possibilities without imposing strict assumptions.
~\newline
\nonumberindent{\bf Proxy tasks and transfer learning.} We mainly apply DQAS to design QAOA-like ansatz for systems consisting of $8$ or $10$ qubits.
We can think of these automated designs, conducted for small systems, as proxy tasks since large systems (containming more qubits) often share the same optimal circuit patterns with small systems of the same family of problem like MAXCUT. Then we can fix the circuit architecture and only optimize circuit parameters for larger systems as in the standard QAOA algorithm.
~\newline
\nonumberindent{\bf Other competitive and near optimal layouts found by DQAS.}
In the study of a bunch of random regular graphs, DQAS not only finds QAOA circuits but also discovers other circuit designs with comparable or slightly inferior performance. More specifically, via an ensemble made of $n=8$ regular graphs with degree $3$, we found an alternative architecture comprising of H-layer, yy-layer, rx-layer, zz-layer, rx-layer, which gives an expectation value of MAX CUT around $8.8$ which is similar to the performance of QAOA layout of the same depth.
~\newline
\nonumberindent{\bf Layerwise training.} It becomes increasingly challenging to find an optimal ansatz for deeper circuit, since there are multiple local minima on this rugged energy landscape for MAXCUT problems. One way to understand the challenge is to note that all QAOA circuits that effectively are constituted of $P-1$ blocks are local minima for $P$ blocks QAOA.
When the $P$-block optimal circuit delivers a marginal performance gain with respect to the $(P-1)$-block optimal circuits, the difficulty of doing a brute force search for an optimal circuit layout increases with DQAS depth $p$. To resolve this challenge, the idea of layer-wise progressive training should be adopted \cite{Skolik2020}.
Note that multiple-start technique is necessary due to the hardness of finding optimal circuits when the circuit depth is large. Early stopping as well as top-2 grid search are also found to be useful.
~\newline
\nonumberindent{\bf Erd\"os-R\'enyi graph results.} In this study, we also try out DQAS design for MAXCUT problems with the graphs drawn from the Erd\"os - R\'enyi ensemble. Moe specifically, we consider two ensembles characterized, respectively, with $n=10, p=0.3$ and $n=8, p=0.4$ . Interestingly, DQAS rediscover QAOA layout as an optimal architecture under the setting of ensemble learning in this case too. Note that an ensemble of Erd\"os - R\'enyi graph usually suffers greater variance, in terms of the exact MAX CUT value and the optimal circuit parameters across each graph instance, than that of a regular-graph ensemble.
~\newline
\nonumberindent{\bf Different objectives beyond an average value.} Since the aim of QAOA is to estimate the lowest energy instead of an average one, it seems to be natural to use objectives that may more accurately reflect the overlap between the prepared quantum state and the ground state instead of the average energy. To this end, there are some newly proposed objectives such as the CVaR \cite{Barkoutsos2020} and Gibbs objective \cite{Li2020}. These objectives can be easily handled within the framework of DQAS with customized objectives. For example, for Gibbs objective, we simply set $f(x)=e^{-\lambdambda x}$ and $g(x) = \ln x$. Gibbs objective is supposed to reward the prepared quantum state to have a higher overlap with lower energy states. However, Gibbs objective is found to introduce a very steep landscape with respect to circuit parameters. Gibbs objective may be useful when the task only requires to search for optimal circuit parameters for a given circuit architecture. However, in a typical DQAS-task, we need to identify an optimal circuit design (i.e. simultaneously finding an architecture and related circuit parameters), the sharp landscape of Gibbs objective presents a non-trivial challenge.
In particular, the circuit parameters in DQAS-designed circuits frequently deviate from the optimal circuit parameters in the searching process due to the complications of the existence of a super network structure, the minimum of Gibbs objective is hence vague.
On the contrary, CVaR seems to be a good objective to try with DQAS. CVaR actually measures the mean energy of only a proportion of samples (say 20\%) having the lowest energies. This objective gives a much smoother energy landscape than that of Gibbs objective by nature.
In short, Gibbs objective is not compatible with the DQAS framework, but CVaR objective shows promising potential in our study.
The reduced ansatz search on weighted graphs, mentioned in the main text and further described in the next section of this supporting information, is actually based on CVaR objective. We leave detailed comparison and ablation study on different choices of objectives as a future work.
~\newline
\nonumberindent{\bf Instance learnings.} As we have discussed in the main text, we may design an optimal circuit for individual problems (dubbed as the instance learning) or an ensemble of problems. It is quite obvious that highly customized circuit architecture, adapting specifics of a particular graph instance, may outperform a generic QAOA layout when the circuit depth is restricted. In this section, we report our study on using DQAS to design ideal circuits for individual instances of the MAXCUT problem.
For example, DQAS recommends an optimal circuit composed of yy-layer, zz-layer and yy-layer that gives an expected energy of $-8.0$, whereas the vanilla QAOA circuit of the same depth only gives an average energy of $-7.75$. Therefore, such an example demonstrates the effectiveness of using DQAS to design customized circuits in an end-to-end fashion. By applying DQAS, we can both discover a universally optimal architecture for problems under the setting of the ensemble learning, and recommends customized optimal designs for specific problems under the setting of the instance learning.
Different graph instances tend to present distinct searching difficulty for DQAS. For some graph instance, the performance of an optimal quantum circuit is way better than that of other candidates. There is a significant ``performance gap" among circuit designs; therefore, DQAS can easily identify the optimal circuit in these cases. However, there exists other graph instances in which the performances are rather similar. These scenarios are usually accompanied with a training landscape having many local minima. DQAS may easily get trapped into one local minimum and recommends a sub-optimal circuit in these cases.
Nonetheless, this is totally acceptable for problems like MAXCUT. For more challenging problem, such as the quantum simulations where the exact ground state is strictly desired, we anticipate additional tweaks to be implemented to enhance DQAS. For instance, one may adopt the idea of an imaginary-time evolution (or natural gradient descent) for the training procedure.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.28\mathrm{e}xtwidth]{graph8b.pdf}
\caption{One of the instance learning unweighted graph for MAX CUT problem. The MAX CUT value for this graph is $10$. We can found better quantum architecture than QAOA of the same depth to approximate MAX CUT value in such instances.
}\lambdabel{fig:graph8b}
\epsilonnd{figure}
~\newline
\nonumberindent{\bf Reduced graph ansatz searching on instance learning.}
Finally, we elucidate the study of ansatz searching with reduced graphs introduced in the main text. We provide two examples.
The first task is to design optimal ansatz circuit for an unweighted graph (shown in Fig.~\ref{fig:unweightedreduce}), and adopt the mean energy of the proposed circuits as the training objective. An optimal circuit, found by DQAS, for this specific graph is rx-layer, zz-layer, zz-layer, ry-layer and rx-layer. Recall the zz-layers are generated with Hamiltonian with restricted connectivity sampled from the original graph as explained in the main text. Figure \ref{fig:unweightedreduce} also gives the two subgraphs used to generate the 2nd and 3rd layer of ZZ Hamiltonians in this optimal design. Finally, we quote the MAX CUT estimated by the optimal circuit and the vanilla QAOA for comparison. For this specific graph, the exact MAXCUT is 12. The optimal reduced ansatz by DQAS and the $P=1,2$ vanilla QAOA estimate the MAXCUT to be 12, 10.39, 11.18, respectively. In terms of overlap with correct MAX CUT ground state, optimal reduced ansatz has nearly identity overlap while $P=1,2$ QAOA ansatz has overlap $0.250, 0.471$ in value. Note the edge number in these reduced graph is way less than one vanilla zz-layer, the obtained structure is actually much shallower than $P=1$ vanilla QAOA while the performance is better than deeper QAOA layouts.
The second example is to design circuits for a weighted graph with weights distributed as Gaussian distribution $\mathrm{e}xtrm{matter}athcal{N}(1, 0.2)$. For this case, we adopt the CVaR objective. This specific graph instance, shown in Fig.~\ref{fig:weightedreduce}, has 8 nodes. The optimal reduced ansatz recommended by DQAS is ry-layer, rx-layer, zz-layer, zz-layer, rx-layer. Again, the two zz-layers are induced by two subgraphs shown in Fig.~\ref{fig:weightedreduce}. Note the subgraphs inherit the exact weight value of corresponding edges. For this graph, the CVaR results for the exact evaluation, optimal reduced ansatz and $P=1,2$ vanilla QAOA circuits are $10.20587$, $10.18$, $9.63$, and $10.20587$, respectively.
For this instance, $P=2$ QAOA circuit essentially gives the exact result while the ``optimal" DQAS circuit does not. However, we remind readers that this ``optimal" DQAS circuit is only about the same circuit depth as the $P=1$ QAOA circuit.
We remark that the reduced subgraphs, in both examples above, have edge density far below half of the base graphs. Therefore, DQAS can identify suitable circuit to give highly accurate estimations of MAX CUT based on very spare graph connectivity for zz-layers in these illustrations.
There are also some arguments or physical intuitions behind such reduced graph ansatz. It is reminiscent to the random attention or the global attention mechanism \cite{Beltagy2020, Zaheer2020} recently developed for transformer models for the natural language processing. Even the number of edge is way less than base graph, each node can still be visited via random walk as long as there are enough layers.
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.6\mathrm{e}xtwidth]{unweightedreduce.pdf}
\caption{Reduced graph ansatz for unweighted graph case (all weights are unity). (a) Base graph for MAX CUT and (b)(c) reduced graph found by DQAS in reduced ansatz layout which outperforms $P=2$ plain QAOA.
}\lambdabel{fig:unweightedreduce}
\epsilonnd{figure}
\begin{equation}} \def\ee{\end{equation}gin{figure}[t]
\includegraphics[width=0.54\mathrm{e}xtwidth]{weightedreduce.pdf}
\caption{Reduced graph ansatz for weighted graph case. (a) Base graph for MAX CUT and (b)(c) reduced graph found by DQAS in reduced ansatz layout which outperforms $P=1$ plain QAOA.
}\lambdabel{fig:weightedreduce}
\epsilonnd{figure}
\epsilonnd{widetext}
\epsilonnd{document}
|
\begin{equation}gin{document}
\font\titlefont=cmbx14 scaled\magstep1
\title{\titlefont Geometry of the uniform spanning forest components in high dimensions}
\author{
M. T. Barlow\footnote{Research partially supported by NSERC (Canada)},
A. A. J\'arai
}
\maketitle
\begin{equation}gin{abstract}
In this note we study the geometry of the component of the origin
in the Uniform Spanning Forest of $\mathbb{Z}^d$, as well as
in the Uniform Spanning Tree of wired subgraphs of $\bZ^d$,
when $d \ge 5$. In particular, we study connectivity properties with
respect to the Euclidean and the intrinsic distance.
We intend to supplement these with further estimates
in the future. We are making this preliminary note
available, as one of our estimates is used in work of
Bhupatiraju, Hanson and J\'arai \cite{BHJ15} on sandpiles.
\end{abstract}
\section{ Introduction } \label{sec:intro}
The Uniform Spanning Tree (UST) on a finite graph $G$ is a random spanning
tree of $G$, chosen uniformly among all spanning trees of $G$.
Motivated by questions of Lyons, Pemantle \cite{Pem91} considered
the weak limit of the USTs on a growing sequence of subgraphs of $\bZ^d$,
induced by sets $V_n \uparrow \bZ^d$, and showed that the limit
exists. The limiting random object, that is a random spanning forest
of $\bZ^d$, is called the Uniform Spanning Forest (USF).
Implicit in Pemantle's work is the result that an alternative
choice of boundary condition yields the same limit. Namely, form the
``wired'' graph $G_n^W = (V_n \cup \{ r_n \}, E_n)$, by collapsing all
vertices in $\bZ^d \setminus V_n$ into $r_n$, and removing self-loops created
at $r_n$. Then the weak limit of the USTs on $G_n^W$ coincides with the USF.
One of Pemantle's results was that the USF is connected a.s.~in dimensions $1 \le d \le 4$,
but it consists of infinitely many (infinite) trees a.s.~in dimensions $d \ge 5$.
Fundamental to the study of the UST/USF is Wilson's algorithm \cite{W}, \cite{LP:book}
that allows one to build the UST/USF from Loop-Erased Random Walks (LERWs),
and thereby analyze it in terms of random walk. All the necessary background
about the UST/USF, that we do not detail in this note, can be found in
the book \cite{LP:book}.
Masson \cite{Mas} and Barlow and Masson \cite{BM1,BM2} studied
the geometry of the LERW and the UST in two dimensions. This led
to a detailed understanding of random walk on the UST.
The purpose of this note is to prove estimates on the geometry
of the LERW and the USF in dimensions $d \ge 5$. We are interested
in properties such as the length of paths and volumes of balls,
both with respect to Euclidean distance and the intrinsic
metric of the tree components. On the one hand we are interested
in extending results from 2D to high dimensions, where the geometry
is very different. On the other hand, our Theorem \ref{T:cyclepop}
is used in work of Bhupatiraju, Hanson and J\'arai \cite{BHJ15}
on sandpiles.
\section{Notation}
\label{sec:notation}
Let $\sU=\sU_{\bZ^d}$ be the USF in $\bZ^d$, viewed as a random
subgraph of the nearest neighbour integer lattice.
Write $\sU(x)$ for the connected component of $\sU$ containing $x$.
We extend this notation to $D \subset \bZ^d$ as follows.
When $D$ is finite, $\sU = \sU_D$ denotes the UST on the wired graph
$G_D^W = (D \cup \{ r_D \}, E_D)$, where $E_D$ can be
identified with those edges of $\bZ^d$ that have at least one
endpoint in $D$. For $x \in D$, we denote by $\sU(x)$ the
connected component of $x$ in the graph obtained from $\sU$
by splitting all edges at $r_D$. In other words, $\sU(x)$ is the
union of those paths in $\sU$ that do not contain $r_D$ as an
interior vertex. We write $\sU_0$ for $\sU(0)$, when $0 \in D$.
When $D \subset \bZ^d$ is infinite, we let $\sU$ denote the weak
limit, as $n \to \infty$, of the USTs on the wired graphs
$G_{D_n}^W$, where $D_n = \{ x \in D : |x| \le n \}$.
The limit exists due to monotonicity; see \cite{LP:book}.
Wilson's algorithm rooted at infinity \cite{BLPS}, \cite{LP:book} can be
easily adapted to sample $\sU$. We let $\sU(x)$ denote the union of
those paths in $\sU$ that do not contain $r_D$ as an interior vertex,
and $\sU_0 = \sU(0)$.
For any of the cases of $\bZ^d$, or $D \subset \bZ^d$ finite or infinite,
we let
\eqnst
{ d_\sU(x,y)
:= \text{graph distance between $x$ and $y$ in $\sU$}, }
where, if $y \not\in \sU(x)$, we set $d_\sU(x,y) = \infty$.
The meaning of $\sU$ will always be clear from context.
\med
{\bf Notation for sets:}
We denote balls in different metrics as follows:
\begin{equation}gin{align*}
B_E(x,r) &=\{ y \in \bZ^d : |x-y| \le r \}, \\
B_n &= B_E(0,n)\\
Q(x,n) &=\{ y\in \bZ^d: ||x-y||_\infty \le n \}, \\
Q_n &= Q(0,n), \\
B_\sU(x,r) &=\{ y \in \bZ^d : d_\sU(x,y) \le r \},
\end{align*}
For $A \subset \bZ^d$ we denote:
\begin{equation}gin{align*}
\pd A &= \{ x \in \bZ^d - A: x \sim y \hbox{ for some } y \in A \}, \\
\pd_i A &= \{ x \in A: x \sim y \hbox{ for some } y \in A^c \}, \\
A^o &= A - \pd_i A.
\end{align*}
Let $\pi_i$ be projection onto the $i$th coordinate axis, and
$\bH_n$ be the hyperplane
$$ \bH_n=\{ x : \pi_1(x) = n \}. $$
Let $\sR_n = \{ n \} \times [-n,n]^{d-1}$ denote the
``right-hand face'' of $[-n,n]^d$, in the
first coordinate direction.
\med
{\bf Notation for processes.} $S^x=(S^x_k, k \ge 0)$ is simple random walk with $S^x_0 =x$, and
$\bP^x$ is its law. We let $S=S^0$, and $\bP = \bP^0$.
If we discuss random walks $S^x$ and $S^y$ with $x \not= y$,
then they will always be independent.
A path $\gam$ is a (non-necessarily self avoiding) sequence of adjacent
vertices in $\bZ^d$ -- ie $\gam=(\gam_0, \gam_1, \dots )$ with $\gam_{i-1}\sim \gam_i$.
(Sometimes we will write $\gam(i)$ for $\gam_i$.)
Paths can be either finite or infinite.
We will often need to consider the beginning or final portions
of paths with respect to the first or last hit on a set.
To this end, we define a number of operations on paths.
Let $\gam= (\gam_0, \gam_1, \dots )$ be a path.
Given a set $A \subset \bZ^d$ define
$k_1 = \min\{ k \ge 0: \gam_k \in A \}$,
$k_2 = \max \{ k \ge 0: \gam_k \in A \}$,
and set
\begin{equation}gin{align*}
\sB^F_A \gam &= ( \gam_{k_1}, \gam_{k_1 +1}, \dots, ), \\
\sB^L_A \gam &= ( \gam_{k_2}, \gam_{k_1 +1}, \dots, ), \\
\sE^F_A \gam &= (\gam_0, \dots, \gam_{k_1}),\\
\sE^L_A \gam &= (\gam_0, \dots, \gam_{k_2}), \\
\Th_k \gam &= (\gam_k, \dots ), \\
\mathbf{P}hi_k \gam &= (\gam_0, \dots, \gam_k), \\
H_A(\gam) &= \sum_i 1_{( \gam_i \in A )}.
\end{align*}
Thus $\sB^F_A \gam$ is the path $\gam$ `\underline{B}eginning' at the `\underline{F}irst' hit
on $A$, and $\sE^L_A \gam$ is the path $\gam$ `\underline{E}nded' at the `\underline{L}ast' hit
on $A$, etc. If $\gam$ is a finite path we write $|\gam|$ for the
length of $\gam$. $H_A(\gam)$ is the number of hits by $\gam$ on the set $A$.
Let $\sL \gam$ be chronological loop erasure of $\gam$, and
if $\gam=(\gam_0, \dots, \gam_n)$ is a finite path let
$\sR \gam = (\gam_n , \gam_{n-1}, \dots , \gam_0)$ be the time reversal of
$\gam$.
We define hitting times
\eqnsplst
{ \tau_A &= \inf \{ j \ge 0 : S_j \not\in A \}, \\
T_A &= \inf \{ j \ge 0 : S_j \in A \}, \\
T_A^+ &= \inf \{ j \ge 1 : S_j \in A \}. }
When we need to specify the process we write $T_A[S]$ etc.
Given a domain $D \subset \bZ^d$, we denote the Green functions
\eqnsplst
{ G_D(x,y)
&= \bE^x \Big( \sum_{0 \le k < \tau_D} I[ S^x = y ] \Big) \\
G(x,y)
&= G_{\bZ^d}(x,y). }
\med
{\bf A note on constants.} Throughout, $c$ and $C$ will denote positive
finite constants that only depend on the dimension $d$, and whose value may change
from line to line, and even within a single string of inequalities.
\section{Properties of the LERW}
\label{sec:lew}
In this section we derive a number of auxiliary estimates on LERW
in dimensions $d \ge 5$. Some of these will be used in
Sections \ref{sec:ub} and \ref{sec:vol-lb}, where we give upper and
lower bounds on the volume of balls in the intrinsic metric.
Two results of this section that are of interest
in themselves are: (i) Proposition \ref{P:len-lb}, that gives a
large deviation upper bound on the lower tail of the number of
steps in a LERW up to its exit from a large box; and (ii)
Theorem \ref{T:Ass2}, that gives an upper bound on the probability that
$x, y \in \bZ^d$ are in the same component of $\sU$ and the
path between them has length at most $n$.
The papers \cite{Mas, BM1} give a number of properties of LERW
in $\bZ^2$, some of which hold for more general graphs.
A fundamental fact about LERWs is the following ``Domain Markov property''
--- see \cite{La2}.
\begin{equation}gin{lemma} \label{L:dmp}
Let $D \subset \bZ^d$,
let $\gam=(\gam_0, \dots , \gam_n)$ be a path from $x=\gam_0$ to $D^c$.
Set $\al = \mathbf{P}hi_k \gam$, $\begin{equation}ta = \Th_k \gam$.
Let $Y$ be a random walk started at $\gam_k$ conditioned
on the event $\{ \tau_D(Y) < T^+_\al(Y) \}$. Then
\begin{equation}
\bP\big( \sL ( \sE^F_{D^c} S ) = \gam | \mathbf{P}si_k ( \sL ( \sE^F_{D^c} S ))=\al \big)
= \bP( \sL( \sE^F_{D^c} Y ) = \begin{equation}ta ).
\end{equation}
\end{lemma}
A key result in \cite{Mas} is a `separation lemma' when $d=2$ -- see
\cite[Theorem 4.7]{Mas}.
Let $S, S'$ be independent SRW in $\bZ^d$ with $S_0=S'_0=0$,
and $T_n, T'_n$ be the hitting times of $\pd Q_n$. Set
\begin{equation}gin{align*}
F_n &= \{ S[1, T_n] \cap S'[1, T'_n] = \emptyset \}, \\
Z_n &= d( S(T_n), S'[1, T'_n]) \vee d( S'(T'_n), S[0, T_n]).
\end{align*}
\begin{equation}gin{lem}\label{L:sep} (`Separation lemma').
Let $d\ge 5$. There exists $c_1>0$ such that
$$ \bP( Z_n \ge \half n | F_n) \ge c_1. $$
\end{lem}
\proof
Let $e_1=(1,0, \dots,0)$. Let $X$ be a SRW started at $2k e_1$, and $A_k=\{ je_1, k\le j \le 2k\}$.
Since $d \ge 5$ two independent SRWs intersect with probability less than 1, and thus
there exists $k$ (depending on $d$) such that
$$ \bP^0( \hbox{$S$ hits } X \cup A_k ) \le \fract{1}{16} d^{-2}. $$
Now fix this $k$, and let
$$ G_1 =\{ S_i = -i e_1, S'_i = ie_1, 0\le i \le k\}. $$
So $\bP(G_1) = (2d)^{-2k}$.
Then writing $G_2= \{ S[1, T_{n/2}] \cap S'[1, T'_{n/2}] \neq \emptyset \}$,
\bas
\bP(G_2|G_1) &\le
\bP( S[k+1, T_{n/2}] \cap S'[1, T'_{n/2}] \neq \emptyset | G_1)
+ \bP( S[1, T_{n/2}] \cap S'[k, T'_{n/2}] \neq \emptyset | G_1) \\
&\le \fract18 d^{-2}.
\end{align*}
Let $H_\pm$ be the left and right faces (in the $e_1$ direction) of the cube
$Q_{n/2}$. We have
$$ \bP( S_{T_{n/2}} \in H_- |G_1 ) \ge (2d)^{-1}. $$
So if $G_3= G_2^c \cap \{ S_{T_{n/2}} \in H_-, S'_{T'_{n/2}} \in H_+\}$,
\begin{equation}gin{align*}
\bP( G_3 | G_1) &\ge
\bP( S_{T_{n/2}} \in H_-, S'_{T'_{n/2}} \in H_+ |G_1 ) -
\bP( G_2 | G_1) \\
&\ge (2d)^{-2} - (8d^2)^{-1} = (8d^2)^{-1}.
\end{align*}
If $G_3$ occurs then let $G_4$ be the event that $S'$ then (i.e. after
time $T'_{n/2}$ leaves $Q_n$ before it hits hits $\bH_0$,
and $S$ leaves $Q_n$ before it hits
$\bH_0$. By comparison with a one-dimensional SRW each of these events
has probability at least $1/3$, so $\bP(G_4|G_3) \ge 1/9$.
On the event $G_1 \cap G_3 \cap G_4$ the path $S[0, T_n]$ is contained in
$[-n,0]\times [-n,n]^{d-1} \cup Q_{n/2}$, and $\pi_1(S'_{T'_n}=n)$, so that
$d(S'_{T'_n} ,S[0, T_n]) \ge n/2$. The same bound holds if we interchange
$S'$ and $S$, and so we deduce that
$$ \bP(Z_n \ge \half n|F_n ) \ge \bP(\{Z_n \ge \half n\} \cap F_n)
\ge \bP( G_1 \cap G_3 \cap G_4) \ge (2d)^{-2k} (8d^2)^{-1} 9^{-1}. $$
\qed
\sm {\bf Remark.}
The result in $d \ge 5$ is much easier than $d =2$, since with high probability
$S$ and $S'$ do not interesect. The
proof for $d=2$ uses the fact that if the two processes get too close,
then by the Beurling estimate they hit with high probability.
\ms
In the remainder of this section we give some estimates on the length of
LERW paths in $\bZ^d$ with $d \ge 5$. We fix $D \subset \bZ^d$ and
$N \ge 1$ such that $Q_N = Q(0,N) \subset D$. We will be interested in
the number of steps the LERW from $0$ to $\pd D$ takes up to its
first exit from $Q_N$. Let $S$ be SRW on $\bZ^d$ with $S_0=0$.
Let $$ L= \sL ( \sE^F_{D^c} (S)). $$
In words, $L$ is the loop erasure of $S$ up to its first hit
on the boundary of $D$.
Our estimate will be broken down into studying $L$ in `shells'
$Q_{n+m} \setminus Q_n$. For this purpose, let us fix $n,m$ such that
$16 \le n < n+m \le N$, with $m \le n/8$.
Let $$ \al = \sE^F_{ \pd_i Q_n } L , \q L' = \sB^F_{\pd_i Q_n}L. $$
So $\al$ is the path $L$ up to its first hit on $\pd_i Q(0,n)$, and
$L'$ is the path of $L$ from this time on.
See Figure \ref{fig:box-setup}.
Let us condition on $\al$. Let $x_0 \in \pd_i Q_n$ be the endpoint
of $\al$. When $x_0 \in \bH_n$, we let $ x_1 = x_0 + (m/2)e_1 $
and set
$$ A=A(x_0) = Q(x_1, m/4 ),
\quad, A^* = Q(x_1, 3m/8 ). $$
When $x_0$ lies on one of the other
faces of $Q_n$, we replace $e_1$ by the unit vector pointing
towards that faces to define $x_1$ and $A(x_0)$.
See Figure \ref{fig:box-setup}.
\begin{equation}gin{figure}
\centerline{\includegraphics{box-setup.pdf}}
\caption{\label{fig:box-setup} Setup and notation for the piece
of the LERW in the shell $Q_{n+m} \setminus Q_n$.}
\end{figure}
Set $$ \begin{equation}ta = \sE^F_{\pd_i Q(x_0,m)} L' . $$
Let $\wt X^z$ be $S^z$ conditioned on $\{ \tau_D < T^+_\al\}$.
While the process $\wt X^z$ depends on $\al$, our notation will not
emphasize this point.
Write $\wt X$ for $\wt X^{x_0}$, and $\wt G_D(x,y)$ for the Green
function for $\wt X^x$. By the domain Markov property, Lemma \ref{L:dmp},
we have (conditional on $\al$) that
\begin{equation}
L' \, \, {\buildrel (d) \over =} \, \,
\sL( \sE^F_{\pd D} \wt X ).
\end{equation}
We write $\wt T$, $\wt \tau$, etc.~for hitting and exit times by $\wt X$.
Set
$$ h(x) = \bP^x( \tau_D < T_\al). $$
Then
\begin{equation} \label{e:wtG}
\wt G_D(x,y) = \frac{ h(y)}{h(x)} G_D(x,y), \, x,y \in D- \alpha.
\end{equation}
The standard Harnack inequality
(see \cite{La2}) gives
\begin{equation} \label{e:hiA}
h(y) \asymp h(x_1), \q y \in A^*,
\end{equation}
and thus
\begin{equation} \label{e:hGn}
\wt G_D(x,y) \asymp G_D(x,y), \, x,y \in A^*.
\end{equation}
\begin{equation}gin{lemma} \label{L:Mub}
Let $d \ge 3$. For any $\al$ we have
\begin{equation}gin{align}
\bE( H_A(\begin{equation}ta)| \al) &\le c_1 m^2, \\
\bE( H_A(\begin{equation}ta)^2|\al) &\le c_1 m^4.
\end{align}
\end{lemma}
\proof
This is a standard computation with Green functions. Let
$B = Q(x_0, m)$. Then, since $\begin{equation}ta$ is a subset of the
path of $\wt X$, we have
$$ H_A(\begin{equation}ta) \le \sum_{k=0}^{\wt \tau_{B}} 1_{ (\wt X \in A) }
= H_A( \sE^F_{\pd_i B} \wt X) =: \wt H. $$
Then for $p=1,2$,
$$ \bE^{x_0} (\wt H^p| \al)
= \bE^{x_0}\Big( 1_{( \wt T_{A^*} <\wt \tau_B)} \bE^{\wt X_{\wt T_{A^*}}}( {\wt H}^p) \Big)
\le \max_{ z\in \pd_i A^*} \bE^z \wt H^p. $$
Let $z \in \pd_i A^*$. Then using \eqref{e:hGn}
\begin{equation}gin{align*}
\bE^z (\wt H|\al) &= \sum_{y \in A} \wt G_B(z, y)
\le c |A| \max_{y \in A} G_B(z,y) \le c' m^2 m^{2-d} = c' m^2.
\end{align*}
Also since on $A^*$ we have
$\wt G_B \asymp G_B \le G$,
\begin{equation}gin{align*}
\bE^z (\wt H^2|\al) &\le 2 \sum_{k=0}^\infty \sum_{k=j}^\infty
1_{( k\le \wt \tau_B)} 1_{( j \le \wt \tau_B)} 1_{(\wt X_k \in A)}
1_{(\wt X_j \in A)} \\
&\le 2 \sum_{x \in A} \sum_{ y \in A} \wt G_B(z,x)\wt G_B(x,y) \\
&\le c |A| m^{2-d} \max_{x \in A} \sum_{ y \in A} \wt G(x,y)
\le c' m^4.
\end{align*}
\qed
\begin{equation}gin{remark}
The same argument works if we consider
$\bE( H_{Q(x_1, \lam m)}(\begin{equation}ta)^p |\al)$, $p=1,2$, for any $\lam \in (0, \frac12)$.
\end{remark}
We now turn to the harder problem of obtaining a lower bound on $\bE H_A(\begin{equation}ta)$,
and begin with a boundary Harnack inequality which extends
\cite[Proposition 3.5]{Mas} to higher dimensions.
See \cite{BK} for further extensions.
In what follows $\sR_m = \bH_m \cap Q_m$ is the `right hand face' of $Q_m$.
\begin{equation}gin{lemma} \label{L:ubh}
Assume $d \ge 1$. Let $\sK$ be an arbitrary nonempty subset of
$[-m+1,0] \times [-m+1,m-1]^{d-1}$. For all $m \ge 1$
and all $\sK$ we have
\begin{equation} \bP^0 \big( S(\tau_{Q(0,m-1)}) \in \sR_m \,\big|\, \tau_{Q(0,m-1)} < T^+_\sK \big) \ge (2d)^{-1}.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Let $h(z) = \bP^z \big( S_{\tau_{Q(0,m-1)}} \in \sR_m \big)$, $z \in Q(0,m-1)$.
By symmetry we have $h(0)= 1/2d$. We first show that
\eqn{e:max-property}
{ h(z)
\le h(0) \text{ for all
$z \in ([-m+1,0] \times [-m+1,m-1]^{d-1}) \cap \bZ^d$}. }
Let $z' = (0,z_2,\dots,z_d)$.
Let $X^z$ and $X^{z'}$ be simple random walks with
starting points $z$ and $z'$ respectively; we have
$ h(z) = \bP( X^z_{\tau_A} \in \sR_n)$, with a similar expression
for $h(z')$.
We couple these random walks
by taking $X^z = z+ S$, $X^{z'}=z'+S$, where $S$ is a
SRW with $S_0=0$. Then
$\{ X^{z}_{\tau_A} \in \sR_n\} \subset \{ X^{z'}_{\tau_A} \in \sR_n\}$,
and so $h(z) \le h(z')$.
To prove that $h(z') \le h(0)$ we use a coupling of continuous
time random walks $Y$, $Y'$ with $Y_0=0$, $Y'_0=z'$; these have the
same exit distribution as the discrete time walk $S$.
Recall that $\pi_j$ is the projection onto the $j$th coordinate
axis, so that $\pi_j(Y_t)$ gives the $j$th coordinate of $Y_t$;
each coordinate is a continuous time simple random walk (run at rate $1/d$) on
$\bZ$.
The coupling is as follows. If at time $t$ we have
$\pi_j (Y_t)=\pi_j(Y'_t)$ then we run the two $j$th coordinate
processes together, so $\pi_j (Y_{t+s})=\pi_j(Y'_{t+s})$ for
all $s \ge 0$
Note that we have $|\pi_j (Y_t)| \le |\pi_j(Y'_t)|$ when $t=0$;
the coupling will preserve this inequality for all $t \ge 0$.
If $|\pi_j (Y_t) - \pi_j(Y'_t)|\ge 2$ then we use reflection coupling,
so that $\pi_j (Y_t)$ and $\pi_j(Y'_t)$ jump at the same time,
and in opposite directions.
Finally, suppose that $|\pi_j (Y_t) - \pi_j(Y'_t)| =1$,
and let $a=\pi_j (Y_t)$, $a+1= \pi_j (Y'_t)$.
We take three independent Poisson processes on $\bR_+$,
$\sP_1, \sP_2, \sP_3$; each with rate $1/2d$, and make the first
jump of either $\pi_j (Y)$ or $\pi_j(Y')$ after time $t$ to be
at time $t+T$, where
$T$ is the first point in $\sP_1 \cup \sP_2 \cup \sP_3$.
If $T \in \sP_1$ we set
$\pi_j (Y_{t+T}) = a-1$, $\pi_j (Y'_{t+T}) = a+2$.
If $T \in \sP_2$ then we set
$\pi_j (Y_{t+T}) = a+1$, $\pi_j (Y'_{t+T}) = a+1$, and if $T \in \sP_3$ then
$\pi_j (Y_{t+T}) = a$, $\pi_j (Y'_{t+T}) = a$.
With this coupling we have
$\{ Y'_{\tau_A(Y') } \in \sR_n\} \subset \{ Y_{\tau_A(Y)} \in \sR_n\}$,
and so $h(z') \le h(0)$.
Stopping the bounded martingale $h(S(k))$ at $\tau_{Q(0,m-1)} \wedge T_\sK$, and
using \eqref{e:max-property} we get
\eqnsplst
{ h(0)
&= \sum_{y \in \sK} h(y) \bP^0 \big( S(\tau_{Q(0,m-1)} \wedge T^+_\sK) = y \big)
+ \bP^0 \big( \tau_{Q(0,m-1)} < T^+_\sK,\, S(\tau_{Q(0,m-1)}) \in \sR_m \big) \\
&\le h(0) \bP \big( \tau_{Q(0,m-1)} > T^+_\sK )
+ \bP( \tau_{Q(0,m-1)} < T^+_\sK, \, S(\tau_{Q(0,m-1)}) \in \sR_m \big). }
Rearranging gives the statement of the lemma. \qed
\end{proof}
\sms
We will also need two extensions of Lemma \ref{L:ubh} that we
prove next.
\begin{equation}gin{lemma} \label{L:ubh-ext}
Assume $d \ge 3$. Let $N \ge 1$ and $Q_{4N} \subset D \subset \bZ^d$.
Let $8 \le m \le N/2$ and $n \le N$.
Suppose that $\sK$ is an arbitrary nonempty subset of $Q_n$, and
$x_0 \in \sK \cap \bH_n$. Let $z_0 = x_0 + m e_1$.
There exists a constant $c = c(d) > 0$ such that
\begin{equation} \bP^{z_0}( T_{Q(x_0,m/2)} > \tau_D \,|\, T_{\sK} > \tau_D )
\ge c.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
It is easy to see that the statement holds when $m \ge n/8$, since then
$\bP^{z_0} ( T_{Q_{n+m/2}} > \tau_D ) \ge \bP^{z_0} ( T_{Q_{n+m/2}} = \infty ) \ge c$.
Henceforth we assume that $m < n/8$.
Let $f(z) = \bP^z ( T_{\sK} > \tau_D )$ and
$g(z) = \bP^z ( T_{\sK} \wedge T_{Q(x_0,m/2)} > \tau_D)$, so that we have to
prove $f(z_0) \le C g(z_0)$. Let $z_1 = x_0 + 8 m e_1$. Due to the
Harnack principle, it is sufficient to show that $f(z_1) \le C g(z_1)$.
We first show that for all $y \in \pd Q(x_0, 8 m)$ we have
$g(y) \le C g(z_1)$. Let us write $\bH$ for
the hyperplane $\bH_{n + 4 m}$, and $\bH'$ for the hyperplane
$\bH_{n + 2 m}$. Observe that $\bH$ and $\bH'$ are both disjoint
from $\sK \cup Q(x_0,m/2)$, and they both separate
$\sK \cup Q(x_0,m/2)$ from $z_1$.
If $y \in \pd Q(x_0, 8 m)$ lies on the same side of $\bH'$
as $z_1$, then $y$ is at least distance $m$ from $\sK \cup Q(x_0,m/2)$,
and this is comparable to the distance between $y$ and $z_1$.
Hence for such $y$, the Harnack principle
implies $g(y) \le C g(z_1)$.
Suppose now that $\bH'$ separates $y$ from $z_1$. Let $Q^{(1)}$ and $Q^{(2)}$
be cubes that are both translates of $Q_{2N}$, such that:\\
(i) the right hand face of $Q^{(1)}$ and the left hand face of $Q^{(2)}$ coincide;\\
(ii) the common set $\mathcal{R} = Q^{(1)} \cap Q^{(2)}$, is contained in $\bH$; \\
(iii) the center of $\mathcal{R}$ (viewed as a $(d-1)$-dimensional cube),
is the point $x_0 + 4 m e_1$. \\
Since $g(S(n \wedge \tau_{Q^{(1)}}))$ is a submartingale under $\bP^y$, we have
\eqn{e:submart}
{ g(y)
\le \bE^y ( g(\tau_{Q^{(1)}}) )
= \sum_{w \in \pd Q^{(1)} \setminus \mathcal{R}} g(w) \, \bP^y ( S(\tau_{Q^{(1)}}) = w )
+ \sum_{u \in \mathcal{R}} g(u) \, \bP^y ( S(\tau_{Q^{(1)}}) = u ). }
Since $g(S(n \wedge \tau_{Q^{(2)}}))$ is a martingale under $\bP^{z_1}$,
we also have
\eqn{e:mart}
{ g(z_1)
= \bE^{z_1} ( g(\tau_{Q^{(2)}}) )
= \sum_{w' \in \pd Q^{(2)} \setminus \mathcal{R}} g(w') \, \bP^{z_1} ( S(\tau_{Q^{(2)}}) = w' )
+ \sum_{u \in \mathcal{R}} g(u) \, \bP^{z_1} ( S(\tau_{Q^{(2)}}) = u ). }
The mirror symmetry between $Q^{(1)}$ and $Q^{(2)}$, as well as the Harnack
principle implies that
\eqnsplst
{ \bP^y ( S(\tau_{Q^{(1)}}) = u )
&\le C \bP^{z_1} ( S(\tau_{Q^{(2)}}) = u ) \\
\bP^y ( S(\tau_{Q^{(1)}}) = w )
&\le C \bP^{z_1} ( S(\tau_{Q^{(2)}}) = w' ), }
where $w'$ is the mirror image of $w \in \pd Q^{(1)} \setminus \mathcal{R}$
in the hyperplane $\bH$. We also have $g(w) \le 1$, $w \in \pd Q^{(1)} \setminus \mathcal{R}$,
and $g(w') \ge c$, $w' \in \pd Q^{(2)}$. These observations and \eqref{e:submart}
and \eqref{e:mart} together imply $g(y) \le C g(z_1)$.
We now show the desired inequality $f(z_1) \le C g(z_1)$.
Let $1 \le R < \infty$ denote the random variable that counts the
number of times $S^{z_1}$ makes a crossing from $\pd Q(x_0, 8 m)$
to $Q(x_0,m/2)$ before $T_\sK \wedge \tau_D$. We have
\eqnst
{ \bP^{z_1} ( R \ge \ell )
\le \left( \max_{y \in \pd Q(x_0, 8 m)} \bP^{y} ( T_{Q(x_0,m/2)} < \infty ) \right)^\ell
\le \gamma^\ell }
with some $0 < \gamma = \gamma(d) < 1$.
Using the strong Markov property at the time when the $\ell$-th crossing
has occurred, we can write
\eqnsplst
{ f(z_1)
&= \sum_{\ell = 0}^\infty \bP^{z_1} ( R = \ell,\, T_\sK > \tau_D )
= g(z_1) + \sum_{\ell = 1}^\infty \bP^{z_1} ( R = \ell,\, T_\sK > \tau_D ) \\
&\le g(z_1) + \sum_{\ell = 1}^\infty \bP^{z_1} ( R \ge \ell ) \,
\max_{z \in Q(x_0,m/2)} \bP^z ( (T_{Q(x_0,m/2)} \wedge T_\sK) \circ \Theta_{\tau_{Q(x_0, 8 m)}} > \tau_D ) \\
&\le g(z_1) + \sum_{\ell = 1}^\infty \gamma^\ell \max_{y \in \pd Q(x_0, 8 m)} g(y) \\
&\le g(z_1) + C g(z_1). }
This completes the proof of the Lemma.
\end{proof}
\begin{equation}gin{lemma} \label{L:ubh-tauD}
Assume $d \ge 3$. Let $N \ge 1$ and $Q_{4N} \subset D \subset \bZ^d$.
Let $8 \le m \le N/2$ and $n \le N$.
Suppose that $\sK$ is an arbitrary nonempty subset of $Q_n$, and
$x_0 \in \sK \cap \bH_n$. Let $\mathcal{R}_{n,m}$ denote the
right hand face of $Q(x_0,m)$. There exists a constant $c = c(d) > 0$
such that
\begin{equation} \bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m} \,\big|\, T^+_\sK > \tau_D \big)
\ge c.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Let $\sK_0 = \sK \cap Q(x_0, 2 m)$ and
$\sK_1 = \sK \setminus \sK_0 = \sK \setminus Q(x_0, 2 m)$.
Due to the boundary Harnack inequality, Lemma \ref{L:ubh}, we have
\eqn{e:simple-ubh}
{ \bP^{x_0} \big( S(\tau_{Q(x_0,m)} \in \mathcal{R}_{n,m} \,\big|\, T^+_\sK > \tau_{Q(x_0,m)} \big)
\ge (2d)^{-1}. }
Let $Z$ denote the process that is $S$ conditioned on
$T_{\sK_1} > \tau_D$. Then \eqref{e:simple-ubh} and an application of the
Harnack principle implies that
\eqn{e:cond-simple-ubh}
{ \bP^{x_0} \big( Z(\tau_{Q(x_0,m)} \in \mathcal{R}_{n,m} \,\big|\, T^+_\sK[Z] > \tau_{Q(x_0,m)}[Z] \big)
\ge c. }
This in turn implies that
\eqnspl{e:cond-ubh}
{ &\bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m},\, T^+_\sK > \tau_{Q(x_0,m)},\, T_{\sK_1} > \tau_D \big) \\
&\qquad \ge c \bP^{x_0} \big( T^+_\sK > \tau_{Q(x_0,m)},\, T_{\sK_1} > \tau_D \big) \\
&\qquad \ge c \bP^{x_0} \big( T^+_\sK > \tau_D \big). }
Let $z_0 = x_0 + 4 m e_1$. Using the Harnack principle, the left hand side
of \eqref{e:cond-ubh} can be bounded from above by
\eqnspl{e:away}
{ &\bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m},\, T^+_{\sK} > \tau_{Q(x_0,m)} \big) \,
\max_{z \in \mathcal{R}_{n,m}} \bP^z \big( T_{\sK_1} > \tau_D \big) \\
&\qquad \le C\, \bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m},\, T^+_{\sK} > \tau_{Q(x_0,m)} \big) \,
\bP^{z_0} \big( T_{\sK_1} > \tau_D \big). }
An application of Lemma \ref{L:ubh-ext} (with $2 m$ playing the role of $m/2$)
shows that
\eqnst
{ \bP^{z_0} \big( T_{\sK_1} > \tau_D \big)
\le C \, \bP^{z_0} \big( T_{\sK_1 \cup Q(x_0, 2 m)} > \tau_D \big)
\le C \, \bP^{z_0} \big( T_{\sK} > \tau_D \big). }
Substituting this into \eqref{e:away}, and using the Harnack principle again,
we get that the right hand side of \eqref{e:away} is bounded above by
\eqnspl{e:bound-away}
{ &C\, \bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m},\, T^+_{\sK} > \tau_{Q(x_0,m)} \big) \,
\bP^{z_0} \big( T_{\sK} > \tau_D \big) \\
&\qquad \le C \, \bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m},\, T^+_{\sK} > \tau_{Q(x_0,m)} \big) \,
\min_{z \in \mathcal{R}_{n,m}} \bP^z \big( T_{\sK} > \tau_D \big) \\
&\qquad \le C \, \bP^{x_0} \big( S(\tau_{Q(x_0,m)}) \in \mathcal{R}_{n,m},\, T^+_{\sK} > \tau_D \big). }
The inequalities \eqref{e:cond-ubh}, \eqref{e:away} and \eqref{e:bound-away}
together imply the claim of the Lemma.
\end{proof}
\sms
We now return to the task of giving a lower bound for $\bE ( H_A(\begin{equation}ta) )$.
We will need the following lower bound on $\wt G$.
\begin{equation}gin{lemma} \label{L:wtG}
Assume $d \ge 3$. Let $z \in A$. Then
$$ \wt G_D(x_0, z) \ge c m^{2-d}. $$
\end{lemma}
\proof
This uses the extension of the boundary Harnack inequality, Lemma \ref{L:ubh-tauD}.
Let $V_z$ be the number of hits on $z$ by $\wt X$ before $\wt \tau_D$.
Let $\wt T = \wt T_{\pd_i Q(x_0, m/8)}$. Note that $Q(x_0, m/8)$ and
$A^*$ intersect on one of the faces of $Q(x_0, m/8)$. Then
since $\wt T < \wt \tau_D$,
\begin{equation}gin{align*}
\wt G_D(x_0, z) &= \bE^{x_0} V_z
= \bE^{x_0}\Big( \bE^{\wt X_{\wt T}} V_z \Big)
\ge \bE^{x_0}\Big( 1_{ (\wt X_{\wt T} \in A^*)}
\min_{y \in \pd_i A^*} \bE^y V_z\Big)\\
&= \bP^{x_0}( \wt X_{\wt T} \in A^*)
\min_{y \in \pd_i A^*} \wt G_D(y,z).
\end{align*}
Using \eqref{e:wtG} and \eqref{e:hiA} we have
$\wt G_D(y,z) \asymp G_D(y,z) \asymp m^{2-d}$ if $y \in \pd_i A^*$.
Let $T= T_{\pd_i Q(x_o, m/8)}$ (for $S$). Lemma \ref{L:ubh-tauD}
implies
\begin{equation}gin{align*}
\bP^{x_0}(\wt X_{\wt T} \in A^*)
= \bP^{x_0}(S_T \in A^*| T^+_\al > \tau_D ) \ge c,
\end{align*}
and the Lemma follows.
\qed
\ms
The key estimate is the following.
\begin{equation}gin{lemma} \label{L:ptlb}
Assume $d \ge 5$. Then
\begin{equation}
\bE( H_A(\begin{equation}ta) | \al) \ge c m^2.
\end{equation}
\end{lemma}
\proof
It is enough to prove that if $z \in A$ then
\begin{equation} \label{e:zhit}
\bP( z \in \begin{equation}ta | \al ) \ge c m^{2-d}.
\end{equation}
Let $Y$ be $\wt X$ conditioned to hit $z$ before
$\wt T^+_\al \wedge \wt \tau_D$, and let $\wt X^{z}$ be independent
of $Y$. Let
$$ Y'= \sE^L_{z}( \sE^F_{\pd D}( Y)), $$
so $Y'$ is the path of $Y$ up to its last hit on $z$ before its
first exit from $D$. Let also $X'= \Th_1 \sE^F_{\pd D} \wt X^z$.
(We need to apply $\Th_1$ since the last point of $Y'$ and
the first point of $X'$ are both $z$.)
Then as in Lemma 6.1 of \cite{BM1} we have
\begin{equation} \label{e:3pg}
\bP( z \in \begin{equation}ta | \al ) = \wt G_D(x_0,z)
\bP\big( \sL Y' \cap X' =\emptyset, \sL Y' \subset Q(x_0,m) \big).
\end{equation}
Due to Lemma \ref{L:wtG}, it remains to show that the probability
on the right hand side is bounded away from $0$.
We will in fact prove the stronger statement:
\eqn{e:avoid}
{ \bP\big( Y' \cap X' =\emptyset, Y' \subset Q(x_0,m) \big)
\ge c > 0. }
This result is not surprising, since two independent SRW in $\bZ^d$
(with $d\ge 5$) intersect with probability strictly less than 1.
Let us denote $A_z = Q(z,m/16)$, $B = Q(x_0,m)$ and $B' = Q(x_0,m/16)$.
Note that $Y'$ starts at $x_0$ and ends at $z$.
We decompose $Y'$ into four subpaths, defined below, and give separate
estimates for these subpaths that together will imply the lower bound
on the probability in \eqref{e:avoid}. We define:
\eqnsplst
{ Y'_1
= \sE^F_{\pd B'} (Y') \qquad
Y'_2
= \sE^L_{\pd A_z} ( \sB^F_{\pd B'} (Y') ) \qquad
Y'_3
= \sB^L_{\pd A_z} (Y'). }
That is, $Y'_1$ ends at the first exit from $B'$,
$Y'_3$ begins at the last entrance to $A_z$ and $Y'_2$ is
the portion in between. We let $y_1 = Y'_1(|Y'_1|) = Y'_2(0)$ and
$y_2 = Y'_2(|Y'_2|) = Y'_3(0)$.
We further decompose $Y'_2$ into the pieces:
\eqnsplst
{ Y'_{2,1}
= \sE^F_{y_2} (Y'_2) \qquad
Y'_{2,2}
= \sB^F_{y_2} (Y'_2). }
That is, $Y'_{2,1}$ is the piece from $y_1$ to the first hit on $y_2$,
and $Y'_{2,2}$ is the remaining loop at $y_2$.
Observe that conditional on $y_1$ and $y_2$, the paths
$Y'_1, Y'_{2,1}, Y'_{2,2}, Y'_3$ are independent.
We now state our estimates for each piece.
Our notation will assume that $x_0 \in \bH_n$; trivial modification
can be made when this is not the case.
\ms
\emph{Claim 1.}
There is constant probability that $Y'_1$ exits
$B'$ on the right hand face. That is,
we have $\bP ( y_1 \in \mathcal{R}_{n,m/16} ) \ge c > 0$,
where $\mathcal{R}_{n,m/16} = \bH_{n+m/16} \cap Q(x_0,m/16)$.
\emph{Proof of Claim 1.}
Using Lemma \ref{L:ubh-tauD} we have
\eqnsplst
{ \bP ( y_1 \in \mathcal{R}_{n,m/16} )
&= \frac{\bP^{x_0} ( \wt X(\wt \tau_{B'}) \in \mathcal{R}_{n,m/16},\, \wt T_{z} < \wt \tau_D )}{\bP^{x_0} ( \wt T_{z} < \wt \tau_D )} \\
&\ge \frac{\wt G_D(z,z)}{\wt G_D(x_0,z)} \, \bP^{x_0} ( \wt X(\wt \tau_{B'}) \in \mathcal{R}_{n,m/16} ) \,
\min_{w \in \mathcal{R}_{n,m/16}} \bP^{w} ( \wt T_{z} < \tau_D ) \\
&\ge c \min_{w \in \mathcal{R}_{n,m/16}} \frac{\wt G_D(w,z)}{\wt G_D(x_0,z)}
\ge c. }
\ms
In the next three claims we will use the notation
$B'' = x_0 + ([0, z_1 + m/32] \times [-m,m]^{d-1}) \cap \bZ^d$.
\ms
\emph{Claim 2.}
There is constant probability that the following six events occur:\\
(i) $Y'_3$ starts on the left hand face of $A_z$;\\
(ii) $Y'_3 \subset z + ([-m/16,m/32] \times [-m/16,m/16]^{d-1}) \cap \bZ^d$;\\
(iii) $X'$ exits $A_z$ on the right hand face;\\
(iv) $X' \cap A_z \subset z + ([-m/32,m/16] \times [-m/16,m/16]^{d-1}) \cap \bZ^d$;\\
(v) $Y'_3 \cap (X' \cap A_z) = \emptyset$;\\
(vi) $\sB^F_{\pd A_z} (X')$ is disjoint from $B''$.
\emph{Proof of Claim 2.}
Let $\wt S^{z}$ be the process defined as $S^{z}$ conditioned to
hit on $x_0$ before $T_{\al \setminus \{x_0\}} \wedge \tau_D$.
The time-reversal of $Y'$ has the law of $\wt S^{z}$.
Therefore, the time-reversal of $Y'_3$ has the law of $\sE^F_{\pd A_z} (\wt S^{z})$.
The proof of Lemma \ref{L:sep} (Separation Lemma), shows that
for independent simple random walks $S^{z}$ and $S'^{z}$ there is
probability $\ge c > 0$ that the analogues of the events
(i)--(v) all hold. An application of the
Harnack principle then shows that in fact (i)--(v) hold
with constant probability.
It is left to show that conditionally on (i)--(v), we also
have (vi) with constant probability. Since $X'$ is
$S$ conditioned on $T_\al > \tau_D$, this can be proved
in the same way as Lemma \ref{L:ubh-ext}. For this
we merely have to replace $Q(x_0,m/2)$ in that lemma
by $B''$, and make straightforward adjustments. Hence Claim 2 follows.
\ms
\emph{Claim 3.}
Conditional on $y_1$ being in the right hand face of $B'$ and
$y_2$ being in the left hand face of $A_z$, there is constant probability that
$Y'_{2,1} \subset B''$.
\emph{Proof of Claim 3.}
Condition on $y_1$ and $y_2$. Then $Y'_{2,1}$ has the
law of $S^{y_1}$ conditioned to hit on $y_2$ before
$T_\al \wedge \tau_D$ (stopped at the first hit on $y_2$).
Since $y_1$ and $y_2$ are at least distance $c m$ from
the boundary of $B''$, such a path has constant
probability to stay inside $B''$. (One way to see this
is to use an argument similar to that of Lemma \ref{L:ubh-ext},
where we let $R$ count the number of crossings by the walk
from $Q(z,m/64)$ to $\pd B''$ before time
$T_z \wedge T_\al \wedge \tau_D$.) Hence the claim
follows.
\ms
\emph{Claim 4.}
Conditional on $y_2$ being in the left hand face of $A_z$,
there is constant probability that $Y'_{2,2} \subset B''$.
\emph{Proof of Claim 4.}
Condition on $y_2$. The probability that $Y'_{2,2}$ consists
of a single point is
$G_{D \setminus \al} (y_2,y_2)^{-1} \ge G(y_2,y_2)^{-1} \ge c > 0$.
\ms
When all the events in Claims 1--4 occur, the event in \eqref{e:avoid}
occurs. Hence the Lemma follows.
\ms
An application of Lemmas \ref{L:Mub} and \ref{L:ptlb} and the
one-sided Chebyshev inequality give the following corollary.
\begin{equation}gin{cor} \label{C:HAlb}
When $d \ge 5$, there exists a constant $c_0>0$ such that
$$ \bP( H_A(\begin{equation}ta) \ge c_0 m^2| \al ) \ge c_0. $$
\end{cor}
\begin{equation}gin{prop}
\label{P:len-lb}
Assume $d \ge 5$. Let $N \ge 1$ and $Q_{4N} \subset D \subset \bZ^d$.
Let $L = \sL \sE^F_{\pd D} S$ be a loop erased walk
from $0$ to $\pd D$, and $M_N = |\sE^F_{\pd_i Q_N} L|$
be the number of steps in $L$ until its first hit on $\pd_i Q_N$.
Then for all $\lambda > 0$ we have
\begin{equation} \label{e:Llb}
\bP( M_N < \lam N^2) \le C \exp( -c \lam^{-1} ).
\end{equation}
\end{prop}
\proof
Suppose $k \ge 1$ and $m \ge 4$ such that $N/2 \le k m < N-m$.
For $j=1, \dots, k$ let
$$ \al_j = \sE^F_{\pd_i Q(0, jm) } L, \q \sF_j = \sigma( \al_j). $$
Let $Y_j = \al_j ( |\al_j|)$ be the last point in $\al_j$, and
$$ \begin{equation}ta_j = \sE^F_{\pd_i Q(Y_j,m)}( \sB^F_{\pd_i Q(0, jm) }L) $$
be the path $L$ between $Y_j$ and its first hit after $Y_j$ on
$\pd_i( Y_j, m)$. We have
$$ M_N \ge \sum_{i=1}^k |\begin{equation}ta_j|, $$
Let $G_j = \{ |\begin{equation}ta_j| < c_0 m^2 \}$; then by Corollary \ref{C:HAlb}
$$ \bP( G_j | \sF_j) \le 1 - c_0. $$
Therefore, $M_N$ stochastically dominates a sum of $k$
independent random variables that take the values $c_0 m^2$ and $0$
with probabilities $c_0$ and $1 - c_0$, respectively. Hence
\begin{equation}gin{align*}
\bP( M_N \le (1/2) k c_0^2 m^2 ) &\le C \exp ( - c k ).
\end{align*}
We now take $k \asymp \lambda^{-1}$ and $m \asymp \lambda N$
and we obtain \eqref{e:Llb}.
\qed
In the following theorem, we obtain a lower bound on the length
of paths in the USF. We define the event:
\begin{equation} \label{e:Fndef}
F(y,x,n) = \left\{ \text{$T_x[S^y] < \infty$ and $|\sL \sE^F_x (S^y)| \le n$} \right\}.
\end{equation}
\begin{equation}gin{thm} \label{T:Ass2}
For every $x, y \in \bZ^d$ we have
\begin{equation} \label{e:clb}
\bP(F(y,x,n))
\le C (1 + |x-y|)^{2-d} \exp \left[ - c \frac{|x-y|^2} {n} \right].
\end{equation}
\end{thm}
\proof
For notational convenience, we assume $y = 0$ (otherwise translate $x, y$ by $-y$).
If $|x|^2/n \le 1$ then the term in the exponential in \eqref{e:clb}
is of order 1, so
$$ \bP( F(0,x,n)) \le \bP( T_x < \infty) \le (1+|x|)^{2-d}
\le e^c (1+|x|)^{2-d} e^{- c |x|^2 / n}. $$
Now assume $|x|^2 > n$, and let $N= ||x||_\infty/4$, and $Q=Q(0,N)$.
Let $X'$ be $S$ conditioned on $\{ T_x < \infty\}$. Then if
$h(z) = \bP^z( T_x[S] < \infty)$, we have $h(z) \asymp N^{2-d}$ on $Q(0,N)$, and
thus the processes $S$ and $X'$ have comparable laws inside $Q(0,N)$.
The explicit law of a section of the loop erased random path
given in \cite{Law99} (see also (5) in \cite{Mas})
then implies that the loop erasures of $S$ and $X'$
also have comparable laws inside $Q$.
Let
\begin{equation}
F_1(x,n) = \left \{ |\sE^F_{\pd_i Q} ( \sL \sE^F_x S) | \le n,
T_x< \infty \right \}.
\end{equation}
Thus $F(0,x,n) \subset F_1(x,n)$. Then
\begin{equation}gin{align}
\nn
\bP( F(0,x,n)) &\le \bP( F_1(x,n)) \\
\nn
&= \bP( | \sE^F_{\pd_i Q} \sL(\sE^F_x S) | \le n | T_x<\infty) \, \bP(T_x < \infty)\\
\nn
&\le C |x|^{2-d} \bP( |\sE^F_{\pd_i Q} \sL( \sE^F_x X' )| \le n ) \\
\nn
&\le C |x|^{2-d} \bP( |\sE^F_{\pd_i Q} \sL( \sE^F_x S)| \le n ).
\end{align}
Taking $n=\lam N^2$, so that $\lam^{-1} \ge c |x|^2 n^{-1}$,
and using Proposition \ref{P:len-lb} completes the proof. \qed
\section{Upper bound on $| B_\sU(0,n) |$} \label{sec:ub}
Recall that $\sU(x)$ is the component of the USF containing $x \in \mathbb{Z}^d$.
It is well-known \cite[Theorem 4.2]{Pem91} that
for $d \ge 5$ and $x \not= y \in \mathbb{Z}^d$ we have
\eqn{e:cnctd-bnd}
{ c |x - y|^{4-d}
\le \bP ( y \in \sU(x) )
\le C |x - y|^{4-d}. }
A corollary of this bound is that the volume of $\sU_0 \cap B(r)$
grows as $r^4$ in expectation. Our main result in the previous section,
Theorem \ref{T:Ass2}, is a variant of the upper bound in \eqref{e:cnctd-bnd}
that gives control over the length of the path connecting $x$ and $y$.
Since that bound was formulated in terms of a single LERW, the exponent $4-d$
changes to $2-d$. In this section we extend Theorem \ref{T:Ass2} to control
the volume of balls in the intrinsic metric.
\begin{equation}gin{theorem}
\label{thm:B_U-moments}
Assume $d \ge 5$, and let $\sU = \sU_{\bZ^d}$. There exists a constant $C_1$ such that
for all $k \ge 0$ we have
\begin{equation} \label{e:ubk}
\bE \big( |B_\sU(0,n)|^k \big) \le C_1^k k! n^{2k}.
\end{equation}
Hence there are constants $c_1 > 0$ and $C_2$ such that
\begin{equation} \label{e:ub-exp}
\bP ( |B_\sU(0,n)| \ge \lam n^2 ) \le C_2 e^{-c_1 \lam},
\quad \lambda > 0,\, n \ge 1.
\end{equation}
\end{theorem}
\begin{equation}gin{proof}
The bound \eqref{e:ub-exp} follows easily from \eqref{e:ubk} using Markov's
inequality and the power series for $e^x$.
We prove \eqref{e:ubk} by induction on $k$. The case $k = 0$ holds trivially.
We fix $k \ge 1$ and $y_1, \dots, y_k \in \mathbb{Z}^d$, and
estimate the probability
\eqnst
{ \bP \big( y_1, \dots, y_k \in B_\sU(0,n) \big). }
This can be done similarly to the ``tree-graph inequalities'' known
in percolation \cite{AN84}. To facilitate notation, we write $y_0 = 0$. On the event
$y_1, \dots, y_k \in \sU_0$ consider the minimal subtree
$T(y_0, \dots, y_k) \subset \sU_0$ that contains
the vertices $y_0, \dots, y_k$. This tree is finite.
Since $\sU_0$ has one end \cite{BLPS}, \cite{LP:book},
there is a unique infinite path in $\sU_0$, whose only vertex in $T(y_0, \dots, y_k)$
is its starting vertex.
Let us write $T(y_0, \dots, y_k, \infty)$ for the infinite
subtree of $\sU_0$ obtained by adding this infinite path to
$T(y_0, \dots, y_k)$.
Now let us consider the ``topology'' of
$T(y_0, \dots, y_k, \infty)$. In the case $k = 1$, it is
easy to see that there exists a vertex $z_1 \in T(y_0, y_1, \infty)$
such that the paths $T(y_0,z_1)$, $T(y_1,z_1)$ and $T(z_1,\infty)$
(some of which may degenerate to a single vertex) are edge-disjoint.
In the general case $k \ge 1$, we have $k$ ``branch points''
$z_1, \dots, z_k$. We use a fixed rule for indexing
the $z_i$'s, in requiring that for every $i \ge 1$
the path $T(y_i,z_i)$ is edge-disjoint from
$T(y_0, \dots, y_{i-1}, \infty)$. See Figure \ref{fig:taus}.
\begin{equation}gin{figure}
\includegraphics{k23.pdf}
\caption{\label{fig:taus}
All three labelled tree graphs with $k = 2$,
and two of the five possible labelled tree graphs with $k = 3$.}
\end{figure}
We can formalize the construction via the following
recursive procedure. Let $\mathcal{T}(0)$ denote the set containing
the unique tree with vertex set $\{ 0, \infty \}$. Assume
that the collection $\mathcal{T}(k-1)$ of trees with vertex set
$\{ 0, \dots, k-1 \} \cup \{ \infty \} \cup \{ \bar{1}, \dots, \overline{k-1} \}$
has been defined for some $k \ge 1$. Let $\mathcal{T}(k)$ denote
the collection of trees with vertex set
$\{ 0, \dots, k \} \cup \{ \infty \} \cup \{ \bar{1}, \dots, \bar{k} \}$
that can be obtained in the following way. Pick some
$\tau' \in \mathcal{T}(k-1)$, and pick one of the edges of $\tau'$.
Split this edge into two by introducing a new vertex
$\bar{k}$ on the edge, and add the new edge $\{ k, \bar{k} \}$
to $\tau'$.
It is easy to see that any $\tau \in \mathcal{T}(k)$ has the following
properties (see Figure \ref{fig:taus}):
\begin{equation}gin{itemize}
\item[(i)] $\deg_\tau(\infty) = 1 = \deg_\tau(y_i)$,
$i = 0, \dots, k$.
\item[(ii)] $\deg_\tau(\bar{i}) = 3$, $i = 1, \dots, k$.
\end{itemize}
With the above definitions, the event
$\{ y_1, \dots, y_k \in \sU_0 \}$ implies that there exist
$z_1, \dots, z_k \in T(y_0, \dots, y_k, \infty)$ and
$\tau \in \mathcal{T}(k)$ such that $T(y_0, \dots, y_k, \infty)$ is the
edge-disjoint union of paths $T(\varphi(r),\varphi(s))$, where
$\{ r, s \} \in E(\tau)$, and $\varphi : V(\tau) \to \mathbb{Z}^d \cup \{ \infty \}$
is defined by
\eqn{e:def-varphi}
{ \begin{equation}gin{cases}
\varphi(i) = y_i & i = 0, \dots, k; \\
\varphi(\infty) = \infty; \\
\varphi(\bar{i}) = z_i & i = 1, \dots, k.
\end{cases} }
Note that the choice of $\tau$ is not unique, due to possible
coincidences between the vertices $y_0, \dots, y_k, z_1, \dots, z_k$.
We neglect the overcounting resulting from this, for an upper bound.
If the additional restriction $d_\sU(0,y_i) \le n$, $i = 1, \dots, k$
is in place, we must also have $d_\sU(\varphi(r), \varphi(s)) \le n$
for all $\{ r, s \} \in E(\tau)$ such that
$r, s \not= \infty$. We define the event
\eqnsplst
{ &E(y_1, \dots, y_k, z_1, \dots, z_k, \tau, n) \\
&\qquad= \left\{ \parbox{8.5cm}{$T(y_0, \dots, y_k, \infty)
= \cup_{\{ r, s \} \in E(\tau)}
T(\varphi(r),\varphi(s))$ as an edge-disjoint union and
$d_\sU(\varphi(r),\varphi(s)) \le n$ for all
$\{ r, s \} \in E(\tau)$ such that
$r, s \not= \infty$} \right\}. }
Considering all possible choices of $\tau$ and $z_1, \dots, z_k$,
we get
\eqnsplst
{ \bE \big( | B_\sU(0,n) |^k \big)
&= \sum_{y_1, \dots, y_k \in \mathbb{Z}^d}
\bP \big( y_1, \dots, y_k \in B_\sU(0,n) \big) \\
&\le \sum_{\tau \in \mathcal{T}(k)} \sum_{y_1, \dots, y_k \in \mathbb{Z}^d}
\sum_{z_1, \dots, z_k \in \mathbb{Z}^d}
\bP \big( E(y_1, \dots, y_k, z_1, \dots, z_k, \tau, n) \big). }
We use Wilson's algorithm \cite{W,LP:book} to replace the complicated
event $E(y_1, \dots )$ by a slightly larger event that is easier to handle.
For this, enumerate the edges of $\tau$ as
\eqnst
{ \{ r_0, s_0 \}, \{ r_1, s_1 \}, \dots, \{ r_{2k}, s_{2k} \}, }
where the labelling is chosen in such a way that the following two
properties are satisfied (see Figure \ref{fig:enum}(a)):
\begin{equation}gin{figure}
\includegraphics{k3enum.pdf}
\caption{\label{fig:enum}(a) A possible enumeration of edges
for the application of Wilson's method.
(b) A possible enumeration of edges for performing the
summations using \eqref{e:conv-bnd} in the order
$j = 1, 2, \dots, 2k$. Summing over the spatial location
$\varphi(s'_1)$ eliminates the factor involving the edge
$\{ s'_1, r'_1 \}$. Following this, it is possible to sum
over $\varphi(s'_2)$, etc.}
\end{figure}
\begin{equation}gin{itemize}
\item[(a)] $s_0 = \infty$.
\item[(b)] For every $j = 1, \dots, 2k$, the set of edges
$\{ \{ r_\ell, s_\ell \} : \ell = 0, \dots, j-1 \}$ spans
a subtree of $\tau$, and $s_j$ is a vertex of this subtree.
\end{itemize}
Using Wilson's method with random walks started at
$\varphi(r_0), \dots, \varphi(r_{2k})$, we see that
\eqn{e:E-incl}
{ E(y_1, \dots, y_k, z_1, \dots, z_k, \tau, n)
\subset \bigcap_{j=1}^{2k} F(\varphi(s_j),\varphi(r_j),n). }
Here $F(\cdot, \cdot, n)$ are the events defined in \eqref{e:Fndef}.
Importantly, the events on the right hand side are independent.
Theorem \ref{T:Ass2} and the inclusion \eqref{e:E-incl} imply that
\eqnspl{e:E-bnd}
{ &\bP \big( E(y_1, \dots, y_k, z_1, \dots, z_k, \tau, n) \big) \\
&\qquad\le \prod_{j=1}^{2k}
C (1 + |\varphi(s_j) - \varphi(r_j)|)^{2-d}
\exp \left[ - c \frac{|\varphi(s_j) - \varphi(r_j)|^2}{n} \right]. }
It remains to estimate the sum of the right hand side
of \eqref{e:E-bnd} over all choices of the $y_i$'s and
$z_i$'s. For this it will be convenient to use a different
enumeration of $E(\tau)$. Suppose that
\eqnst
{ \{ r'_0, s'_0 \}, \{ r'_1, s'_1 \}, \dots,
\{ r'_{2k}, s'_{2k} \} }
satisfies the following properties (see Figure \ref{fig:enum}(b)).
\begin{equation}gin{itemize}
\item[(a')] $s'_0 = \infty$ and $r'_{2k} = 0$.
\item[(b')] For every $j = 1, \dots, 2k$ the set
$\{ \{ r'_\ell, s'_\ell \} : \ell = j, \dots, 2k \}$
induces a connected subtree of $\tau$, and $s'_j$ is a
leaf of this subtree.
\end{itemize}
For ease of notation, let us write
$u_j = \varphi(r'_j)$ and $w_j = \varphi(s'_j)$.
With the new enumeration the right hand side of
\eqref{e:E-bnd} takes the following form:
\eqnspl{e:E-bnd2}
{ &\bP \big( E(y_1, \dots, y_k, z_1, \dots, z_k, \tau, n) \big) \\
&\qquad\le \prod_{j=1}^{2k}
C (1 + |w_j - u_j|)^{2-d}
\exp \left[ - c \frac{|w_j - u_j|^2}{n} \right]. }
Note again that the $w_j$'s and $u_j$'s are
$z_i$'s and $y_i$'s, determined implicitly
by $\tau$. Importantly, property (b') of the enumeration
implies that if $w_j = \varphi(s'_j) = z_i$ for some $i, j$,
then the variable $z_i$ does not occur in the product
\eqnst
{ \prod_{\ell=j+1}^{2k}
C (1 + |w_j - u_j|)^{2-d}
\exp \left[ - c \frac{|w_j - u_j|^2}{n} \right]. }
Similar considerations apply if $w_j = \varphi(s'_j) = y_i$
for some $i, j$. The summation over
$y_1, \dots, y_k$ and $z_1, \dots, z_k$
can be accomplished by the following lemma.
\begin{equation}gin{lemma}
\label{lem:conv-bnd}
For any $u \in \mathbb{Z}^d$, we have
\eqn{e:conv-bnd}
{ \sum_{w \in \mathbb{Z}^d} (1 + |w - u|)^{2-d}
\exp \left[ - c \frac{|w - u|^2}{n} \right]
\le C n. }
\end{lemma}
We apply Lemma \ref{lem:conv-bnd} successively
to the factors with $j = 1, \dots, 2k$ on the
right hand side of \eqref{e:E-bnd2}. See Figure \ref{fig:enum}(b)
for an example of how the edges of $\tau$ are
successively removed by the summations.
We obtain
\eqn{e:B_U-moments}
{ \bE \big( | B_\sU(0,n) |^k \big)
\le \sum_{\tau \in \mathcal{T}(k)} (C n)^{2k}. }
Since the number of trees in $\mathcal{T}(k)$ is
$1 \cdot 3 \cdot \cdots (2k-1) \le 2^k k!$, this proves \eqref{e:ubk}.
\qed
\end{proof}
\begin{equation}gin{remark}
The statements of Theorem \ref{thm:B_U-moments} still hold, with
essentially the same proof, when $\sU = \sU_D$, with any $D \subset \bZ^d$.
Note that $\sU_0$ still has one end. This follows from \cite[Proposition 3.1]{LMS},
and the fact that the component of $0$ under the measure
$\mathsf{WSF}_o$ in the domain $D$ is stochastically smaller
then it is in $\bZ^d$. Therefore, a decomposition into events
$E(y_1,\dots,y_k,z_1,\dots,z_k,n)$ still holds (with $\sU = \sU_D$),
where now all vertices are in $D$. The inclusion \eqref{e:E-incl} still holds,
with the events $F$ having the same meaning as before. This allows
to bound the summations in exactly the same way as in $\bZ^d$.
\end{remark}
\section{Lower bounds on volumes} \label{sec:vol-lb}
In this section we return to the setup of Section \ref{sec:lew},
in order to give a lower bound on the volume of $\sU_0$.
We first estimate the number of vertices of $\sU_0$ in shells
$Q_{n+m} \setminus Q_n$. Recall that $Q_N \subset D \subset \bZ^d$,
and $n, m$ satisfy $16 \le n < n+m \le N$, with $m \le n/8$.
We have $L = \sL ( \sE^F_{D^c} (S))$, $\al = \sE^F_{ \pd_i Q_n } L$,
and $x_0 \in \pd_i Q_n$ is the endpoint of $\al$.
The remaining piece of $L$ is $L' = \sB^F_{\pd_i Q_n}L$, and
$\begin{equation}ta = \sE^F_{\pd_i Q(x_0,m)} L'$.
See Figure \ref{fig:boxes-cycle-pop}.
\begin{equation}gin{figure}
\centerline{\includegraphics{boxes-cycle-pop.pdf}}
\caption{\label{fig:boxes-cycle-pop} Boxes for the cycle popping argument.}
\end{figure}
Recall that when $x_0 \in \bH_n$, we defined
$A = A(x_0) = Q(x_0 + (m/2) e_1, m/4)$ and $x_1 = x_0 + (m/2) e_1$,
with appropriate rotations applied when $x_0$ was on a different
face of $Q_n$. We will now also need a point $x_2 \in Q_{n+m} \setminus Q_n$
of order $m$ away from $A$, and further boxes contained in
$Q_{n+m} \setminus Q_n$ that we define as follows.
If $x_0 \in \bH_n$ and the second coordinate of $x_0$ is negative, let
\eqnspl{e:A'-A''-def}
{ x_2
&= x_1 + m e_2 \\
A'
&= A'(x_0)
= Q( x_1 + 2 m e_2, m/4 ) \\
A''
&= A''(x_0)
= x_1 + [-3m/8,3m/8] \times [-m,3m] \times [-m,m]^{d-2} \cap \bZ^d. }
If $x_0 \in \bH_n$ and the second coordinate of $x_0$ is positive,
we replace $e_2$ by $-e_2$ and $[-m,3m]$ by $[-3m,m]$. If
$x_0$ is on a different face of $Q_n$, we replace $e_1$ and $e_2$
by two other suitable unitvectors.
The key technical estimate is to show that $\begin{equation}ta \cap A$ has capacity
of order $m^2$ with probability bounded away from $0$, which we
do in the next section.
\subsection{A capacity estimate}
Let $S^{x_2}$ be a random walk with $S^{x_2}(0) = x_2$, independent of $S$,
$\tilde{X}$, etc.
\begin{equation}gin{proposition} \label{P:capacity}
Assume $N \ge 1$, $Q_{4N} \subset D \subset \bZ^d$, and the setup of Section \ref{sec:lew}.\\
(a) There exists $c_1 = c_1(d) > 0$ such that
\eqnst
{ \bP \big( \text{$S^{x_2}$ hits $(A \cap \begin{equation}ta)$} \,\big|\, \alpha \big)
\ge c_1 m^{4-d}. }
(b) We have
\begin{equation} \label{e:capest1}
\bP \big( c m^2 \le \mathrm{Cap} (A \cap \begin{equation}ta) \le C_1 m^2 \,\big|\, \alpha \big)
\ge c > 0.
\end{equation}
\end{proposition}
\begin{equation}gin{proof}
(a) For ease of notation, we omit the conditioning on $\alpha$.
Let
\eqnst
{ U := \sum_{ z \in A} I [z \in \begin{equation}ta]
I [\text{$S^{x_2}$ hits $z$}], }
so that
$$ \bP \big( \text{$S^{x_2}$ hits $(A \cap \begin{equation}ta)$} \big)
= \bP ( U > 0 ). $$
Using Lemma \ref{L:ptlb}, we have
\eqnsplst
{ \bE ( U )
= \sum_{z \in A} \bP ( z \in \begin{equation}ta )
\bP \big( T_z[S^{x_2}] < \infty \big)
\ge c m^d m^{2-d} m^{2-d}
= c m^{4-d}. }
On the other hand,
\eqn{e:U^2}
{ \bE \big( U^2 \big)
= \sum_{x, y \in A} \bP ( x, y \in \begin{equation}ta ) \,
\bP \big( T_x[S^{x_2}] < \infty,\, T_y[S^{x_2}] < \infty \big). }
Since the process $\wt X$ generating $L'$ must pass through
$\pd A^*$ in order for the event $x, y \in \begin{equation}ta$ to occur,
we have
\eqnsplst
{ \bP ( x, y \in \begin{equation}ta )
&\le \max_{z \in \pd A^*} [ \wt G_D(z,x) \wt G_D(x,y) + \wt G_D(z,y) \wt G_D(y,x) ] \\
&\le C m^{2-d} G(x,y). }
For the other term in the right hand side of \eqref{e:U^2} we have
\eqnsplst
{ \bP \big( T_x[S^{x_2}] < \infty,\, T_y[S^{x_2}] < \infty \big)
&\le [ G(x_2,x) G(x,y) + G(x_2,y) G(y,x) ] \\
&\le C m^{2-d} G(x,y). }
Since $d \ge 5$, we have $\sum_{x, y \in A} G(x,y)^2 \le C m^d$, which gives
$\bE \big( U^2 \big) \le C m^{4-d}$.
The Paley-Zygmund inequality then gives
\eqnst
{ \bP \big( \text{$S^{x_2}$ hits $(A \cap \begin{equation}ta)$} \big)
= \bP ( U > 0 )
\ge \frac{\bE ( U )^2}{\bE \big( U^2 \big)}
\ge c m^{4-d}. }
(b) Since $\mathrm{Cap}(A \cap \begin{equation}ta) \le C |A \cap \begin{equation}ta|$,
and $m^{2-d} \mathrm{Cap}(A \cap \begin{equation}ta) \asymp
\bP \big( T_{A \cap \begin{equation}ta}[S^{x_2}] < \infty \,\big|\, \begin{equation}ta \big)$,
combining (a) with Lemma \ref{L:Mub} gives (b).
\qed
\end{proof}
Assume now, similarly to Proposition \ref{P:len-lb}, that
$k \ge 1$ and $m \ge 4$ such that $N/2 \le k m < N-m$.
Recall that for $j=1, \dots, k$ we denote
$\al_j = \sE^F_{\pd_i Q(0, jm) } L$. Let $Y_j = \al_j ( |\al_j|)$
be the last point in $\al_j$, and
$\begin{equation}ta_j = \sE^F_{\pd_i Q(Y_j,m)}( \sB^F_{\pd_i Q(0, jm) }L)$
be the path $L$ between $Y_j$ and its first hit after $Y_j$ on
$\pd_i( Y_j, m)$. Let $Y_{j,1}$ and $Y_{j,2}$ be the points $x_1$ and
$x_2$ defined with respect to $x_0 = Y_j$, respectively.
Define the following event, measurable with respect to $L$:
\eqn{e:G-event}
{ G(c_1, c_2, C_1)
= \left\{ \parbox{9cm}{there are at least $c_2 k$ indices
$j$ with $1 \le j \le k$ such that
$\bP \big( T_{A(Y_j) \cap \begin{equation}ta_j}[S^{Y_{j,2}}] < \infty \,\big|\, L \big)
\ge c_1 m^{4-d}$ and $|Q(Y_j,m) \cap \begin{equation}ta_j| \le C_2 m^2$} \right\}. }
Proposition \ref{P:capacity} and an argument similar to that of
Proposition \ref{P:len-lb} gives the following corollary.
\begin{equation}gin{corollary}
\label{C:enough boxes}
Under the assumptions of Proposition \ref{P:capacity},
there exist $c_1, c_2 > 0$ and $C_2$ such that we have
\eqnspl{e:good boxes}
{ \bP \left[ G(c_1, c_2, C_2) \right]
\ge 1 - \exp( -c k ). }
\end{corollary}
\begin{equation}gin{remark}
\label{R:cond-exit}
We note the following minor extension of Corollary \ref{C:enough boxes}.
Assuming still that $Q_{4N} \subset D$, let $w \in \partial D$
be fixed, condition $S$ to exit $D$ at $w$, and let $L' = \sL (\sE^F_{D^c} S)$
be the loop-erasure. Masson \cite{Mas} proves that the law of
$\sE^F_{Q_N^c} L'$ is comparable, up to constants factors, to the
law of $\sE^F_{Q_N^c} L$. Since the event $G(c_1,c_2,C_2)$
is measurable with respect to $\sE^F_{Q_N^c} L$, the statement
of the corollary follows also for $L'$.
\end{remark}
\subsection{Lower bound on $| Q_N \cap \sU_0 |$}
\label{ssec:cycle-pop}
We continue with the setup of the previous section.
Our argument will use the cycle popping idea of Wilson \cite{W};
see also \cite{LP:book}.
\begin{equation}gin{theorem}
\label{T:cyclepop}
Assume $N \ge 1$, $Q_{4N} \subset D \subset \bZ^d$, and
let $\sU = \sU_D$. There exist constants $C, c$, such that
\eqnst
{ \bP \big( |Q_N \cap \sU_0| \le \lambda N^4 \big)
\le C \exp ( - c \lambda^{-1/3} ). }
\end{theorem}
\begin{equation}gin{proof}
Condition on $L$, and assume that the event \eqref{e:G-event} occurs.
Let $J$ be the set of indices $1 \le j \le k$ (a $\sigma(L)$-measurable
random set) satisfying the requirements in this event.
For each $j \in J$, let
\eqnst
{ A'(j)
= A'(Y_j) \qquad
A''(j)
= A''(Y_j). }
The definitions of $A'$ and $A''$ made in \eqref{e:A'-A''-def}
ensure that $A''(j)$, $j \in J$ are disjoint.
We will need two coupled collections of stacks.
Associate to each $z \in (\cup_{j \in J} A''(j)) \setminus L$
a stack of arrows, and let us call these $\mathsf{Stacks\ I}$.
For each $j \in J$ and each $z \in A''(j) \cap L \setminus \begin{equation}ta_j$,
pick a new independent stack leaving the rest of the stacks
unchanged. Call this second collection of stacks
$\mathsf{Stacks\ II}$. In both $\mathsf{Stacks\ I}$ and
$\mathsf{Stacks\ II}$, and for every $j \in J$,
pop all cycles that are entirely contained in $A''(j)$.
That is, if a cycle starts in $A''(j)$, but part of it
lies outside $A''(j)$, we do not pop it.
It is important to note that the order of popping cycles
is irrelevant for determining the final configuration
on the top of the stacks.
For each $j \in J$, let
\eqnsplst
{ V^I_j
&= \left\{ y \in A'(j) :
\parbox{5.5cm}{cycle popping using $\mathsf{Stacks\ I}$
reveals a path from $y$ to $L$} \right\} \\
V^{II}_j
&= \left\{ y \in A'(j) :
\parbox{6.3cm}{cycle popping using $\mathsf{Stacks\ II}$ \\
reveals a path from $y$ to $A''(j) \cap \begin{equation}ta_j$} \right\} }
Note that $(V^I_j,V^{II}_j)_{j \in J}$ are conditionally
independent, given $L$, $J$.
\begin{equation}gin{lemma}
\label{L:coupling}
We have $V^I_j \supset V^{II}_j$ for all $j \in J$.
\end{lemma}
\begin{equation}gin{proof}
Let $y \in V^{II}_j$, and consider $\mathsf{Stacks\ II}$. Starting from $y$,
follow the arrows in $\mathsf{Stacks\ II}$, until $A''(j) \cap \begin{equation}ta_j$
is hit. Removing cycles chronologically from this path
pops some cycles entirely contained in $A''(j)$, and reveals a
path from $y$ to $A''(j) \cap \begin{equation}ta_j$. Now if we follow the
arrows in $\mathsf{Stacks\ I}$ instead, then the same arrows
are used until the first time $L$ is hit. This guarantees
that a path from $y$ to $L$ is revealed, that does not
leave $A''(j)$, and hence $y \in V^I_j$.
\end{proof}
\begin{equation}gin{lemma}
\label{L:VII-lb}
Assume $d \ge 5$. For some $c_3 > 0$ we have
\eqnst
{ \bP \big( |V^{II}_j| \ge c_3 m^4 \,\big|\, L,\, j \in J \big)
\ge c
> 0. }
\end{lemma}
\begin{equation}gin{proof}
We estimate the first and second moments of $|V^{II}_j|$.
Fix $y \in A'(j)$. Following the arrows from $y$ in
$\mathsf{Stacks\ II}$ we perform a random walk until
either we exit $A''(j)$, or we hit $A''(j) \cap \begin{equation}ta_j$.
Therefore,
\eqnspl{e:hit-from-y}
{ \bP \big( y \in V^{II}_j \,\big|\, L,\, j \in J \big)
&= \bP \big( T_{A''(j) \cap \begin{equation}ta_j}[S^y] < \tau_{A''(j)}[S^y]
\,\big|\, L,\, j \in J \big) \\
&\ge \bP \big( T_{A(j) \cap \begin{equation}ta_j}[S^y] < \tau_{A''(j)}[S^y]
\,\big|\, L,\, j \in J \big). }
The last expression is
\eqn{e:hit-first}
{ \ge c \bP \big( T_{A(j) \cap \begin{equation}ta_j}[S^y] < \infty
\,\big|\, L,\, j \in J \big). }
(One way to see this is by an argument similar to
that of Lemma \ref{L:ubh-ext}, where we let $R$ count
the number of crossings by the walk from a box
$A^{**} \subset A''(j)$ to $\pd A''(j)$ before hitting $\begin{equation}ta_j \cap A(j)$,
where each face of $\pd A^{**}$ is at distance $m/16$ away
from the corresponding face of $\pd A''(j)$.)
The Harnack inequality and Proposition \ref{P:capacity} now implies,
after summing over $y$ in \eqref{e:hit-from-y}--\eqref{e:hit-first}, that
\eqnst
{ \bE \big( |V^{II}_j| \,\big|\, L,\, j \in J \big)
\ge c c_1 m^d m^{4-d}
= c m^4. }
We now bound the second moment of $|V^{II}_j|$.
If $x, y \in V^{II}_j$ occurs, then there exists
a unique $w \in A''(j)$ with the property that cycle popping
reveals three edge-disjoint paths:
one from $w$ to $A''(j) \cap \begin{equation}ta_j$, a second from
$x$ to $w$ and a third from $y$ to $w$. (We allow to have
$x = w$ or $y = w$ or both.) When this event
happens with a fixed $w$, we can reveal the paths
by first following the arrows starting from $w$
until $A''(j) \cap \begin{equation}ta_j$ is hit, then following
the arrows starting from $x$ until $w$ is hit, then
following the arrows starting from $y$ until $w$
is hit. This shows that
\eqnspl{e:VII2ndm-ub}
{ &\bP \big( x, y \in V^{II}_j \,\big|\, L,\, j \in J \big) \\
&\qquad \le \sum_{w \in A''(j)}
\bP \big( T_{A''(j) \cap \begin{equation}ta_j}[S^w] < \infty
\,\big|\, L,\, j \in J \big) \,
\bP \big( T_w[S^x] < \infty \big) \, \bP \big( T_w[S^y] < \infty \big). }
Let
$\tilde{A}(j) = Q(Y_{j,1}, (3m/2))$, and note that
$\partial \tilde{A}(j)$ has distance at least $c m$ from
$A''(j) \cap \begin{equation}ta_j$, and also distance at least
$c m$ from $A'(j)$.
We estimate separately the cases:\\
(a) $w \in A''(j) \setminus \tilde{A}(j)$; and \\
(b) $w \in A''(j) \cap \tilde{A}(j)$.\\
The sum of the terms in the right hand side of \eqref{e:VII2ndm-ub}
corresponding to case (a) is at most:
\eqnsplst
{ &C m^{2-d} \mathrm{Cap}(A''(j) \cap \begin{equation}ta_j)
\sum_{w \in A''(j) \setminus \tilde{A}(j)}
\sum_{x, y \in A'(j)} G(x,w) G(y,w) \\
&\qquad \le C m^{2-d} m^2 m^2 m^2 m^d
= C m^8. }
The sum for case (b) is at most:
\eqnsplst
{ &C m^{2-d} m^{2-d} m^d m^d
\sum_{w \in A''(j) \cap \tilde{A}(j)}
\bP \big( T_{A''(j) \cap \begin{equation}ta_j}[S^w] < \infty \big) \\
&\qquad \le C m^4 \sum_{w \in \tilde{A}(j)}
\bP \big( T_{A''(j) \cap \begin{equation}ta_j}[S^w] < \tau_{\tilde{A}(j)} \big) \\
&\qquad \le C m^4 m^2 \mathrm{Cap}(A''(j) \cap \begin{equation}ta_j) \\
&\qquad \le C m^8. }
Here the last line follows from $j \in J$ and Proposition \ref{P:capacity}.
The moment estimates for $|V^{II}_j|$ and the one-sided Chebyshev
inequality yield:
\eqnst
{ \bP \big( V^{II}_j \ge c m^4 \,\big|\, L,\, j \in J \big)
\ge c
> 0. }
This completes the proof of the Lemma.
\end{proof}
We can now complete the proof of Theorem \ref{T:cyclepop}.
Choose $k \asymp \lambda^{-1/3}$ so that $\lambda N^4 \asymp k m^4$.
Then using Corollary \ref{C:enough boxes}, the conditional
independence of $(V^{II}_j)_{j \in J}$, and Lemma \ref{L:coupling},
for a suitably small $c_4 > 0$ we have
\eqnsplst
{ \bP \big( |Q_N \cap \sU_0| \le \lambda N^4 \big)
&\le C \exp( - c k )
+ \bE \bigg( \bP \bigg( \parbox{4cm}{$V^{II}_j \ge c_3 m^4$ for less than $c_4 k$ indices $j \in J$}
\,\bigg|\, L \bigg) I[G(c_1,c_2,C_2)] \bigg) \\
&\le C \exp ( -c \lambda^{-1/3} ). }
This completes the proof of the Theorem.
\end{proof}
\begin{equation}gin{theorem}
\label{T:UST-lb}
Assume $d \ge 5$ and let $\sU = \sU_{\bZ^d}$. There exist
$c > 0$ and $C$ such that for all $\lambda > 0$ we have
\eqnst
{ \bP \big( |B_\sU(0,n)| \le \lambda n^2 \big)
\le C \exp ( - c \lambda^{-1/5} ). }
\end{theorem}
For the proof of this theorem, we assume the setting of
Proposition \ref{P:len-lb}, with $D = \mathbb{Z}^d$.
Recall that $M_N = |\sE^F_{\partial_i Q_N} L|$.
\begin{equation}gin{lemma}
\label{L:len-box-ub}
We have
\eqnst
{ \bE \big( M_N^k \big)
\le C_2^k k! N^{2k}. }
Consequently, there exist $c > 0$ and $C$ such that
for all $\lambda > 0$ we have
\begin{equation} \label{e:MNexp}
{ \bP \big( M_N \ge \lambda N^2 \big)
\le C \exp ( - c \lambda ). }
\end{equation}
\end{lemma}
\begin{equation}gin{remark}
If $M^S_N$ is the length of a simple random walk path
run until its first exit from $Q_N$ then it is well known that
$M^S_N/N^2$ has an exponential tail. However
we do not have $M_N \le M^S_N$, so need an alternative argument
to obtain the bound \eqref{e:MNexp}.
\end{remark}
\begin{equation}gin{proof}[Proof of Lemma \ref{L:len-box-ub}.]
We have
\eqnsplst
{ \bE \big( M_N^k \big)
&\le \bE \big( |S[0,\infty) \cap Q_N|^k \big) \\
&= k! \sum_{x_1, \dots, x_k \in Q_N}
G(0,x_1) G(x_1,x_2) \dots G(x_{k-1},x_k) \\
&\le k! \Big( \sum_{z \in Q_{2N}} G(0,z) \Big)^k \\
&= C_2^k k! N^{2k}. }
To see the second statement:
\eqnst
{ \bP \big( M_N \ge \lambda N^2 \big)
\le \exp (- \lambda t N^2) \bE \big( e^{t M_N} \big)
\le \exp ( - \lambda t N^2 ) \frac{1}{1 - C_2 t N^2}. }
Choosing $t = 1/(2 C_2 N^2)$ completes the proof of the Lemma.
\end{proof}
\begin{equation}gin{proof}[Proof of Theorem \ref{T:UST-lb}]
It is sufficient to prove the statement for
$0 < \lambda < \lambda_0$ for some fixed $\lambda_0$.
Let us choose $N = \lambda^{\alpha} \sqrt{n}$ with some
exponent $\alpha > 0$, that we will optimize over at the
end of the proof. We have
\eqnst
{ \bP ( M_N \ge n/2 )
\le C \exp \Big( - c \frac{n}{2 N^2} \Big)
= C \exp ( - c \lambda^{-2 \alpha} ). }
Condition on $L$, as in the proof of Theorem \ref{T:cyclepop},
and assume the event
\eqnst
{ \widetilde{G}
= G(c_1,c_2,C_1) \cap \{ M_N < n/2 \}. }
We set
\eqnst
{ \lambda n^2
= c_3 k m^4
\asymp N m^3, }
which means we pick $m$ to be
\eqnst
{ m
\asymp \sqrt{n} \lambda^{(1-\alpha)/3}. }
Hence $N / m \asymp k \asymp \lambda^{(4 \alpha - 1)/3}$.
Note that this implies that
\eqnst
{ \bP \big( G(c_1,c_2,C_1)^c \big)
\le C \exp ( -c (N/m) )
= C \exp ( -c \lambda^{(4 \alpha - 1)/3} ). }
Since we want $N / m \gg 1$, we impose the
condition $0 < \alpha < 1/4$ on $\alpha$.
For each $j \in J$, let
\eqnsplst
{ \widetilde{V}^I_j
&= \left\{ y \in A'(j) :
\parbox{8cm}{cycle popping using $\mathsf{Stacks\ I}$
reveals a path from $y$ to $L$ of lenght $\le n/2$} \right\} \\
\widetilde{V}^{II}_j
&= \left\{ y \in A'(j) :
\parbox{8cm}{cycle popping using $\mathsf{Stacks\ II}$
reveals a path from $y$ to $A''(j) \cap \begin{equation}ta_j$
of length $\le n/2$} \right\} }
Notice that $(\widetilde{V}^I_j,\widetilde{V}^{II}_j)_{j \in J}$
are again conditionally independent, given $L$.
The same proof as in Lemma \ref{L:coupling} shows that
we have $\widetilde{V}^I_j \supset \widetilde{V}^{II}_j$
for all $j \in J$.
In estimating $\bE \big( \widetilde{V}^{II} \big)$ from below,
we write
\eqnspl{e:first-moment-lb}
{ \bP \big( y \in \widetilde{V}^{II}_j \,\big|\, L,\, j \in J \big)
&\ge \bP \big( T_{A''(j) \cap \begin{equation}ta_j}[S^y] < \tau_{A''(j)}[S^y]
\,\big|\, L,\, j \in J \big) \\
&\qquad - \bP \big( | \sE^F_{\pd A''(j)}(S^y) | > n/2,\, T_{A''(j) \cap \begin{equation}ta_j}[S^y] \circ \Theta_{n/2} < \infty \big). }
The first term on the right hand side is $\ge c m^{4-d}$ due to \eqref{e:hit-first} and $j \in J$.
We now show that the subtracted term is $\le C \exp ( - c n/m^2 ) m^{4-d}$.
Note that we may restrict to $n/2 > 2 m^2$ for convenience (although not needed for the claim),
since our choice of $m$ implies that $n \asymp m^2 \lambda^{-2(1 - \alpha)/3}$, and we
are considering small $\lambda$.
Using the Markov property at time $n/2 - m^2$, the second term
in the right hand side of \eqref{e:first-moment-lb} is at most
\eqnst
{ \bP^y \big( \tau_{A''(j)} > n/2 - m^2 \big) \,
\sum_{z \in A''(j)} \bP^z \big( T_{A''(j) \cap \begin{equation}ta_j} < \infty \big) \,
\bP^y \big( S(n/2) = z \,\big|\, \tau_{A''(j)} > n/2 - m^2 ). }
The first probability can be bounded by $C \exp ( - c n/m^2 )$, by considering
stretches of the walk of length $m^2$, in each of which there is probability
$\ge c > 0$ of exit from $A''(j)$. The conditional distribution of $z$ is
bounded above by $c m^{-d}$, due to the local CLT applied to
$S(n/2-m^2), \dots, S(n/2)$. Hence we are left to show that
\eqnst
{ \sum_{z \in A''(j)} \bP^z \big( T_{A''(j) \cap \begin{equation}ta_j} < \infty \big)
\le m^4. }
Let us write $\wt \begin{equation}ta_j = A''(j) \cap \begin{equation}ta_j$, and
$h(z) = \bP^z ( T_{\wt \begin{equation}ta_j} < \infty )$. By a last exit decomposition
$h(z) = \sum_{u \in \wt \begin{equation}ta_j} G(z,u) e_{\wt \begin{equation}ta_j}(u)$, where
$e_{\wt \begin{equation}ta_j}(u) = \bP^u ( T^+_{\wt \begin{equation}ta_j} = \infty )$. Therefore,
we have
\eqnsplst
{ \sum_{z \in A''(j)} h(z)
&= |\wt \begin{equation}ta_j| + \sum_{z \in A''(j) \setminus \wt \begin{equation}ta_j} h(z)
\le C m^2 + \sum_{u \in \wt \begin{equation}ta_j} \sum_{z \in A''(j)} G(z,u) e_{\wt \begin{equation}ta_j}(u) \\
&\le C m^2 + C m^2 \sum_{u \in \wt \begin{equation}ta_j} e_{\wt \begin{equation}ta_j}(u)
= C m^2 + C m^2 \Cap(\wt \begin{equation}ta_j)
\le C m^4, }
using that $|\wt \begin{equation}ta_j|, \Cap(\wt \begin{equation}ta_j) \le C m^2$ when $j \in J$.
Hence we obtain that there exists $\lambda_0 = \lambda_0(d) > 0$, such that
when $0 < \lambda \le \lambda_0$, the right hand side of \eqref{e:first-moment-lb}
is at least
\eqnsplst
{ c m^{4-d} - C \, \exp ( -c n/m^2 ) \, m^{4-d}
\ge c m^{4-d} - C \, \exp ( -c \lambda^{- 2 (1 - \alpha)/3} ) \, m^{4-d}
\ge c m^{4-d}. }
It follows that $\bE \big( |\wt V^{II}_j| \,\big|\, L,\, j \in J \big) \ge c m^4$.
For the second moment, we simply estimate
\eqnst
{ \bE \big( (\widetilde{V}^{II}_j)^2 \,\big|\, L,\, j \in J \big)
\le \bE \big( (V^{II}_j)^2 \,\big|\, L,\, j \in J \big)
\le C m^8. }
The one-sided Chebyshev inequality yields that for
some $c_4 = c_4 (d) > 0$ we have
\eqnst
{ \bP \big( \widetilde{V}^{II}_j \ge c_4 m^4 \,\big|\, L,\, j \in J \big)
\ge c
> 0. }
This allows us to complete the proof as follows.
\eqnsplst
{ &\bP \big( |B_\sU(0,n)| \le \lambda n^2 \big) \\
&\qquad \le \bP \big( \widetilde{G}^c \big)
+ \bP \Big( \widetilde{G},\,
\sum_{j \in J} \widetilde{V}^{I}_j \le \lambda n^2 \Big) \\
&\qquad \le \bP ( M_N > n/2 ) + \bP \big( G(c_1,c_2,C_1)^c \big)
+ \bE \Big( \bP \Big( \sum_{j \in J} \widetilde{V}^{II}_j < c_3 k m^4
\,\Big|\, L \Big) ; \widetilde{G} \Big) \\
&\qquad \le C \exp ( -c \lambda^{-2 \alpha} )
+ C \exp ( -c \lambda^{(4 \alpha - 1)/3} )
+ \exp ( -c \lambda^{(4 \alpha - 1)/3} ). }
We choose $\alpha$, so that $- 2 \alpha = (4 \alpha - 1)/3$,
so $\alpha = 1/10$. This completes the proof of the Theorem.
\end{proof}
\begin{equation}gin{remark}
We note the following minor extension of Theorem \ref{T:cyclepop}, that is
needed in \cite{BHJ15}. Similarly to Remark \ref{R:cond-exit}, since the
arguments of Theorem \ref{T:cyclepop} only rely on properties of
$\sE^F_{Q_N^c} L$, the result extends to the case when the
component of the origin is connected to a fixed vertex $w \in \partial D$.
\end{remark}
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\end{document}
|
\begin{document}
\title{The learnability of Pauli noise}
\author{Senrui Chen}
\thanks{S.C. and Y.L. contributed equally to this work (alphabetical order). Correspondence and requests for materials should be addressed to S.C. (\href{mailto:[email protected]}{[email protected]}), Y.L. (\href{mailto:[email protected]}{[email protected]}) or L.J. (\href{mailto:[email protected]}{[email protected]}).}
\affiliation{Pritzker School of Molecular Engineering, University of Chicago, IL 60637, USA}
\author{Yunchao Liu}
\thanks{S.C. and Y.L. contributed equally to this work (alphabetical order). Correspondence and requests for materials should be addressed to S.C. (\href{mailto:[email protected]}{[email protected]}), Y.L. (\href{mailto:[email protected]}{[email protected]}) or L.J. (\href{mailto:[email protected]}{[email protected]}).}
\affiliation{Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA}
\author{Matthew Otten}
\affiliation{HRL Laboratories, LLC, 3011 Malibu Canyon Rd., Malibu, CA 90265, USA}
\author{Alireza Seif}
\affiliation{Pritzker School of Molecular Engineering, University of Chicago, IL 60637, USA}
\author{Bill Fefferman}
\affiliation{Department of Computer Science, University of Chicago, IL 60637, USA}
\author{Liang Jiang}
\thanks{S.C. and Y.L. contributed equally to this work (alphabetical order). Correspondence and requests for materials should be addressed to S.C. (\href{mailto:[email protected]}{[email protected]}), Y.L. (\href{mailto:[email protected]}{[email protected]}) or L.J. (\href{mailto:[email protected]}{[email protected]}).}
\affiliation{Pritzker School of Molecular Engineering, University of Chicago, IL 60637, USA}
\date{\today}
\begin{abstract}
Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today’s quantum devices. A fundamental issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question what information is learnable and what is not, which is unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates using graph theoretical tools. Our results reveal the optimality of cycle benchmarking in the sense that it can extract all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM’s CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we show that an attempt to extract unlearnable information by ignoring state preparation noise yields unphysical estimates, which is used to lower bound the state preparation noise.
\end{abstract}
\maketitle
\section{Introduction}
Characterizing quantum noise is an essential step in the development of quantum hardware~\cite{eisert2020quantum,preskill2018quantum}. Remarkably, despite recent progress in both gate-level and scalable noise characterization methods~\cite{Emerson2005scalable,Knill2008randomized,Dankert2009exact,Magesan2011scalable,Magesan2012characterizing,helsen2020general,erhard2019characterizing,flammia2020efficient,harper2020efficient,harper2020fast,flammia2021pauli,liu2021benchmarking,flammia2021averaged,chen2022quantum}, the full characterization of the noise channel of a single CNOT/CZ gate remains infeasible. This is unlikely to be caused by limitations of existing benchmarking algorithms. Instead, it is believed to be related to the fundamental question of what information about a quantum system can be learned, in a setting where initial states, gates, and measurements are all subject to unknown quantum noise. It is well-known that \emph{some} information about quantum noise can be learned (such as the gate fidelity learned by randomized benchmarking~\cite{Emerson2005scalable,Knill2008randomized,Dankert2009exact,Magesan2011scalable,Magesan2012characterizing} or cycle benchmarking~\cite{erhard2019characterizing}), but \emph{not everything} can be learned (due to the gauge freedom in gate set tomography~\cite{Merkel2013Self-consistent,Blume-Kohout2013Robust,nielsen2021gate}). The boundary of learnability of quantum noise -- a precise understanding of what information is learnable and what is not, still remains an open question.
Recently, there has been an interest in formulating noise characterization as learning unknown gate-dependent Pauli noise channels~\cite{erhard2019characterizing,harper2020efficient}. This is motivated by randomized compiling, a technique that has been proposed to suppress coherent errors via inserting random Pauli gates~\cite{wallman2016noise,hashim2020randomized}. As an added benefit, randomized compiling twirls the gate-dependent CPTP noise channel into Pauli noise, thus reducing the number of parameters to be learned. Note that the twirled Pauli noise channel corresponds to the diagonal of the process matrix of the CPTP map, so Pauli noise learning is a necessary step for characterizing the CPTP map, regardless of whether randomized compiling is performed.
However, even under this simplified setting of Pauli noise learning, all prior experimental attempts can only partially characterize the noise channel of a single CNOT/CZ gate~\cite{hashim2020randomized,berg2022probabilistic,Ferracin2022Efficiently}, which only has 15 degrees of freedom. A natural question is whether this limitation is caused by the fundamental unlearnability of the noise channel, and if so, which part of the noise channel and how many degrees of freedom among the 15 are unlearnable?
In this paper, we give a precise characterization of what information in the Pauli noise channel attached to Clifford gates is learnable, in a way that is robust against state preparation and measurement (SPAM) noise.
We develop a systematic method for characterizing learnable degrees of freedom of a Clifford gate set using notions from algebraic graph theory
and show that learnable information exactly corresponds to the cycle space of the Pauli pattern transfer graph, while unlearnable information exactly
corresponds to the cut space. This characterization can be used to write down a list of linear functions of the noise model that corresponds to all independent learnable degrees of freedom. As an example, we show that the Pauli noise channel of an arbitrary 2-qubit Clifford gate has at most 2 unlearnable degrees of freedom. We perform an experimental characterization of a CNOT gate on IBM Quantum hardware~\cite{ibmquantum} up to 2 unlearnable degrees of freedom. Although the unlearnable information cannot be estimated with high precision, we can determine a feasible region of those freedoms using the constraint that the noise model must be physical (\textit{i.e.}, all Pauli error rates are nonnegative).
A corollary of our result is that cycle benchmarking is optimal in the setting we consider, in the sense that it can learn all the information that is learnable. This reveals a fundamental fact about noise benchmarking, namely that cycle benchmarking -- the idea of repeatedly applying the same gate sequence interleaved by single qubit gates, is the ``right'' algorithm for benchmarking Clifford gates, because of the fact that learnable information forms a cycle space.
As an interesting side remark, the term ``cycle'' in cycle benchmarking originally refers to parallel gates applied in a clock cycle. Here we show that the term can also be understood in a graph-theoretical context.
In addition, we also explore ways to overcome the unlearnability barrier. It has been recognized that the unlearnability does not apply if the initial state $\ket{0}^{\otimes n}$ can be prepared perfectly~\cite{flammia2021averaged,Ferracin2022Efficiently}, and it has been suggested that state preparation noise could be much smaller than gate and/or measurement noise in practice~\cite{Maciejewski2020mitigationofreadout,Bravyi2021Mitigating,Ferracin2021Experimental}, which would make gate noise fully learnable up to small error. We develop an algorithm based on cycle benchmarking that fully learns gate-dependent Pauli noise channel assuming perfect initial state preparation, and experimentally demonstrate the method on IBM's CNOT gate. Based on the experiment data, we conclude that this assumption is unlikely to be correct in our experiment as it gives unphysical estimates that are outside the feasible region we determined. Furthermore, we use the data to obtain a lower bound on the state preparation noise and conclude that it has the same order of magnitude as gate noise on the device we used. Therefore, the issue of unlearnability is a practically relevant concern, for which the noise on initial states is an important factor that cannot be neglected on current quantum hardware.
\section{Results}
\subsection{Theory of learnability}
We start by considering the learnability of the Pauli noise channel of a single $n$-qubit Clifford gate. A Pauli channel can be written as
\begin{equation}
\Lambda(\cdot) = \sum_{a\in{{\sf P}^n}}p_a P_a(\cdot)P_a,
\end{equation}
where $\{p_a\}$ is a probability distribution on ${\sf{ P}}^n=\{I,X,Y,Z\}^n$. The goal is to learn this distribution, which has $4^n-1$ degrees of freedom. Considering $\Lambda$ as a linear map, its eigenvectors exactly correspond to all $n$-qubit Pauli operators, as
\begin{equation}
\Lambda(P_a) = \lambda_a P_a,\quad\forall a\in{{\sf P}^n}
\end{equation}
where $\lambda_a = \sum_{b\in{{\sf P}^n}}p_b(-1)^\expval{a,b}$ is the Pauli fidelity associated with the Pauli operator $P_a$. Therefore $\Lambda$ is a linear map with known eigenvectors and unknown eigenvalues, so a natural way to learn $\Lambda$ is to first learn all the Pauli fidelities $\lambda_a$, and then reconstruct the Pauli errors via $p_a = \frac{1}{4^n}\sum_{b\in{{\sf P}^n}}\lambda_b(-1)^\expval{a,b}$.
\begin{figure}
\caption{Cycle benchmarking for learning the Pauli noise channel of a CNOT gate. (a) Standard CB circuits, where CNOT gates are interleaved by random Pauli gates (green boxes), with initial stabilizer states and Pauli basis measurements (red boxes). (b) CB circuits with additional interleaved single qubit Clifford gates (blue boxes).}
\label{fig:main_cb}
\end{figure}
The convenience of working with Pauli fidelities is further demonstrated by the fact that some Pauli fidelities can be directly learned by cycle benchmarking, even with noisy state preparation and measurement. For example, consider the CNOT gate which maps the Pauli operator $IX$ to itself. Fig.~\ref{fig:main_cb} (a) shows the cycle benchmarking circuit. Imagine that we put the Pauli operator $IX$ after the left red box and evolve it with the circuit, then the evolved operator (before the right red box) equals $\lambda_{IX}^3\cdot IX$, up to a $\pm$ sign (which comes from the random Pauli gates and can always be accounted for during post-processing).
Here we use the convention that the noise channel happens before each CNOT gate. In experiments, we prepare a $+1$ eigenstate of $IX$ (such as $\ket{+}\ket{+}$), measure the expectation value of $IX$ at the end, and average over random Pauli twirling sequences. These SPAM operations are noisy and are represented as the red boxes. It is shown~\cite[Theorem~1 in Supplementary Information]{erhard2019characterizing} that the measured expectation value equals
\begin{equation}
{\mathop{\mbb{E}}} \expval{IX}=A_{IX}\cdot \lambda_{IX}^d
\end{equation}
where the expectation is over random Pauli twirling gates and randomness of quantum measurement, and $A_{IX}$ depends on SPAM noise but is independent of circuit depth $d$. From this $\lambda_{IX}$ can be learned by estimating the observable $IX$ at several different depths and perform a curve fitting.
The Pauli operator $IX$ is special as it is invariant under CNOT. Consider another example: CNOT maps $XZ$ to $YY$ and vice versa. Consider Fig.~\ref{fig:main_cb} (b) where we insert additional layers of single-qubit Clifford gates $\sqrt{Z}\otimes\sqrt{X}$ that also maps $XZ$ to $YY$ and vice versa (up to a minus sign that can always be accounted for during post-processing). After $XZ$ picks up a coefficient $\lambda_{XZ}$ in front of the CNOT gate, it gets mapped to $\lambda_{XZ}\cdot YY$ by CNOT but then rotated back to $\lambda_{XZ}\cdot XZ$ by $\sqrt{Z}\otimes\sqrt{X}$. Following the same argument we conclude that both $\lambda_{XZ}$ and $\lambda_{YY}$ are learnable. For simplicity here we make an assumption that single qubit gates are noiseless, motivated by the fact that single qubit gates are 1-2 magnitudes less noisy than 2-qubit gates on today's quantum hardware~\cite{ibmquantum}. In practice, it is a standard assumption to model noise on single-qubit gates as gate-independent (\textit{e.g.}~\cite[Sec. II A]{Ferracin2022Efficiently}), and our noise characterization result can be interpreted as the noise channel induced by a dressed cycle which consists of a CNOT gate and two single-qubit gates~\cite{wallman2016noise}.
The main challenge comes with the next example: CNOT maps $IZ$ to $ZZ$ and vice versa. By directly applying cycle benchmarking as in Fig.~\ref{fig:main_cb} (a) (with even depth $d$) we obtain
\begin{equation}
{\mathop{\mbb{E}}}\expval{IZ}=A_{IZ}\cdot \lambda_{IZ}\lambda_{ZZ}\lambda_{IZ}\lambda_{ZZ}\cdots = A_{IZ}\left(\lambda_{IZ}\lambda_{ZZ}\right)^{d/2},
\end{equation}
and curve fitting gives $\sqrt{\lambda_{IZ}\lambda_{ZZ}}$ (similar results have been obtained in~\cite{erhard2019characterizing,hashim2020randomized,berg2022probabilistic,Ferracin2022Efficiently}). To learn $\lambda_{IZ}$, we may consider applying the same technique in Fig.~\ref{fig:main_cb} (b). However, the problem is that once $IZ$ gets mapped to $ZZ$, it cannot be rotated back to $IZ$ because $I$ is invariant under single qubit unitary gates. The main difference between this example and previous examples is that here the \emph{Pauli weight pattern} (an $n$-bit binary string with 0 indicating identity and 1 indicating non-identity) changes from 01 to 11, thus making the single qubit rotation tool inapplicable.
In fact we can go on to prove that $\lambda_{IZ}$ (as well as $\lambda_{ZZ}$) is unlearnable. Here unlearnable means that there exists two noise models such that the parameter $\lambda_{IZ}$ is different, but the two noise models are indistinguishable by any quantum experiment, meaning that any quantum experiment generates exactly the same output statistics with the two noise models. The result also generalizes to arbitrary $n$-qubit Clifford gates.
\begin{theorem}\label{thm:mainpaulifidelity}
Given an $n$-qubit Clifford gate $\mathcal G$ and an $n$-qubit Pauli operator $P_a$,
the Pauli fidelity $\lambda_a$ of the noise channel attached to $\mathcal G$ is learnable if and only if $\mathrm{pt}(\mathcal G(P_a))= \mathrm{pt}(P_a)$. Here $\mathrm{pt}$ denotes the Pauli weight pattern.
\end{theorem}
The ``if'' part follows directly from cycle benchmarking as discussed above. For the ``only if'' part, when $\mathrm{pt}(\mathcal G(P_a))\neq \mathrm{pt}(P_a)$, we construct a gauge transformation to prove the unlearnability of $\lambda_a$, following ideas from gate set tomography~\cite{Merkel2013Self-consistent,Blume-Kohout2013Robust,nielsen2021gate}. A gauge transformation is an invertible linear map $\mathcal M$ that converts a noise model (initial states $\rho_i$, POVM operators $E_j$, noisy gates $G_k$) to a new noise model as
\begin{equation}
\rho_i\mapsto \mathcal M(\rho_i),\quad E_j\mapsto (\mathcal M^{-1})^\dagger (E_j),\quad G_k\mapsto \mathcal M\circ G_k \circ\mathcal M^{-1},
\end{equation}
with the constraint that the new noise model is physical. Note that the old and new noise models are indistinguishable by definition. To construct such a gauge transformation, as $\mathrm{pt}(\mathcal G(P_a))\neq \mathrm{pt}(P_a)$, there exists a bit on which the two Pauli weight patterns differ. We then define $\mathcal M$ as a single-qubit depolarizing noise channel on the corresponding qubit. In this way we can show that the old and new noise models assign different values to $\lambda_a$, which means $\lambda_a$ is unlearnable. This proof naturally implies that using other noisy gates from the gate set (that are subject to different unknown noise channels) does not change the learnability of Pauli fidelities.
More details of the proof are given in Supplementary Section II B.
As a side remark, it is known that under the stronger assumption of gate-independent noise (where different multi-qubit gates are assumed to have the same noise channel), the noise channel is fully learnable~\cite{kimmel2014robust,helsen2021estimating,huang2022foundations}.
Theorem~\ref{thm:mainpaulifidelity} provides a simple condition for determining the learnability of individual Pauli fidelities, but it is not sufficient for characterizing the learnability of joint functions of different Pauli fidelities. In the CNOT example, we know that both $\lambda_{IZ}$ and $\lambda_{ZZ}$ are unlearnable, but we also know that their product $\lambda_{IZ}\lambda_{ZZ}$ is learnable. This means that there is only one unlearnable degree of freedom in the two parameters $\{\lambda_{IZ},\lambda_{ZZ}\}$. In the following we show how to determine learnable and unlearnable degrees of freedom of Pauli noise, and also generalize the discussion from a single gate to a gate set.
We start by defining learnable information. Consider a Clifford gate set with $m$ gates, where we model each gate as an $n$-qubit gate associated with an $n$-qubit Pauli noise channel. This model is applicable to both individual gates (\textit{e.g.} a 2-qubit system where each 2-qubit gate is implemented by a different physical process and subject to a different noise channel) as well as parallel applications of gates (\textit{e.g.} an $n$-qubit system where each ``gate'' in the gate set is implemented by a layer of 2-qubit gates; the $n$-qubit noise channel models the crosstalk among the 2-qubit gates). The goal is to characterize the learnable degrees of freedom among the $m\cdot 4^n$ parameters.
Recall that the output of cycle benchmarking is a product of Pauli fidelities (including SPAM noise). We further show that without loss of generality this is the only type of information that we need to obtain from quantum experiments for the purpose of noise learning. This is because in general the output probability of any quantum experiment can be expressed as a sum of products of Pauli fidelities, and each individual product can be learned by cycle benchmarking (Supplementary Section IV). We therefore consider learning functions of the noise model that can be expressed as a product of Pauli fidelities (also see below Eq.~\eqref{eq:mainapproximation} for a related discussion). This can be reduced to considering functions of the form $f=\sum_{a,\mathcal G}v_{a}^{\mathcal G}\cdot l_a^{\mathcal G}$, where $l_a^{\mathcal G}:=\log \lambda_a^{\mathcal G}$ is the log Pauli fidelity, $v_{a}^{\mathcal G}\in\mathbb{R}$, and the superscript $\mathcal G$ denotes the corresponding Clifford gate. In the CNOT example $l_{IZ}+l_{ZZ}$ is a learnable function. The idea of learning log Pauli fidelities in benchmarking has also been considered in~\cite{flammia2021averaged,nielsen2022first}. The advantage of considering log Pauli fidelities here is that the set of all learnable functions $f$ forms a vector space. Therefore to characterize all independent learnable degrees of freedom, we only need to determine a basis of the vector space.
\begin{figure}
\caption{Pattern transfer graph of CNOT, SWAP, and a gate set consisting of CNOT and SWAP. Here, multiple edges are represented by a single edge with multiple labels.
The labels on the first two graphs are gate dependent, though we omit the superscripts of CNOT or SWAP.
The labels on the last graph are a combination of the first two graphs and are omitted for clarity.}
\label{fig:main_patterntransfer}
\end{figure}
Recall that the reason that $l_{IZ}+l_{ZZ}$ is learnable in the CNOT example is because the path of Pauli operator in the cycle benchmarking circuit forms a cycle $IZ\to ZZ\to IZ\to\cdots$, and the product of Pauli fidelities along the cycle ($\lambda_{IZ}\lambda_{ZZ}$) can be learned via curve fitting. In general, as we can also insert single qubit Clifford gates in between, we do not need to differentiate between $X,Y,Z$. We therefore consider the \emph{pattern transfer graph} associated with a Clifford gate set where vertices corresponds to binary Pauli weight patterns and each edge is labeled by the Pauli fidelity of the incoming Pauli operator. The graph has $2^n$ vertices and $m\cdot 4^n$ directed edges. They can also be merged to form the pattern transfer graph of the gate set $\{\mathrm{CNOT},\mathrm{SWAP}\}$. Fig.~\ref{fig:main_patterntransfer} shows the pattern transfer graph of CNOT, SWAP, and the gate set of $\{\text{CNOT}, \text{SWAP}\}$. Consider an arbitrary cycle in the pattern transfer graph $C=(e_1,\dots,e_k)$ where each edge $e_i$ is associated with some Pauli fidelity $\lambda_i$. Following Fig.~\ref{fig:main_cb} (b), a cycle benchmarking circuit can be constructed which learns the product of the Pauli fidelites along the cycle, or equivalently the function $f_C:=\sum_{e_i\in C}\log \lambda_i$ can be learned. This implies that the set of functions defined by linear combination of cycles $\{\sum_{C\in\text{cycles}}\alpha_C f_C:\alpha_C\in\mathbb{R}\}$ are learnable. In the following we show that this in fact corresponds to all learnable information about Pauli noise.
We label the edges of the pattern transfer graph as $e_1,\dots,e_M$ where $M=m\cdot 4^n$ and each edge $e_i$ is a variable that represents some log Pauli fidelity. The goal is to characterize the learnability of linear functions of the edge variables $f=\sum_{i=1}^M v_i e_i$, $v_i\in\mathbb{R}$. The set of linear functions can be equivalently understood as a vector space of dimension $M$, called the \emph{edge space} of the graph, where $f$ corresponds to a vector $(v_1,\dots,v_M)$ and we think of $e_1,\dots,e_M$ as the standard basis. Following the above discussion, the \emph{cycle space} of the graph is defined as $\mathrm{span}\{\sum_{e\in C}e:C\text{ is a cycle}\}$, which is a subspace of edge space. We also define another subspace, the \emph{cut space}, as $\mathrm{span}\{\sum_{e\in C}(-1)^{e\text{ from }V_1\text{ to }V_2}e:C\text{ is a cut between a partition of vertices }V_1,V_2\}$. It is known that the edge space is the orthogonal direct sum of cycle space and cut space for any graph~\cite{bollobas1998modern}. Interestingly, we show that the complementarity between cycle and cut space happens to be the dividing line that determines the learnability of Pauli noise.
\begin{theorem}\label{thm:mainpaulilearnability}
The vector space of learnable functions of the Pauli noise channels associated with an $n$-qubit Clifford gate set is equivalent to the cycle space of the pattern transfer graph. In other words,
\begin{equation}
\begin{aligned}
\text{All information}\quad &\equiv \quad \text{Edge space},\\
\text{Learnable information}\quad &\equiv \quad \text{Cycle space},\\
\text{Unlearnable information}\quad &\equiv \quad \text{Cut space}.\\
\end{aligned}
\end{equation}
This implies that the number of unlearnable degrees of freedom equals $2^n - c$, where $c$ is the number of connected components of the pattern transfer graph.
\end{theorem}
The learnability of cycle space follows from cycle benchmarking as discussed above. To prove the unlearnability of cut space, we use a similar argument as in Theorem~\ref{thm:mainpaulifidelity} and show that a gauge transformation can be constructed for each cut in the pattern transfer graph. By linearity, this implies that any vector in the cut space corresponds to a gauge transformation. By definition, a learnable function must be orthogonal to all such vectors and thus orthogonal to the entire cut space. More details of the proof are given in Supplementary Section II C.
It is a well-known fact in graph theory that the cycle space of a directed graph $G=(V,E)$ has dimension $|E|-|V|+c$ while the cut space has dimension $|V|-c$, where $c\geq 1$ is the number of connected components in $G$~\cite{bollobas1998modern}
(a (weakly) connected component is a maximal subgraph in which every vertex is reachable from every other vertex via an undirected path).
Theorem~\ref{thm:mainpaulilearnability} implies that among the $m\cdot 4^n$ degrees of freedom of the Pauli noise associated with a Clifford gate set, there are $2^n -c$ unlearnable degrees of freedom. This shows that while the number of unlearnable degrees of freedom can be exponentially large, they only occupy an exponentially small fraction of the entire space. In addition, a cycle and cut basis can be efficiently determined for a given graph, though in our case this takes exponential time because the pattern transfer graph itself is exponentially large. However, computing the cycle/cut basis is not the bottleneck as the information to be learned also grows exponentially with the number of qubits.
For small system sizes such as 2-qubit Clifford gates, we can write down a cycle basis as shown in Table~\ref{tab:main:CNOT_full} (a) for the CNOT and SWAP gates, which represents all learnable information about these gates. The CNOT gate has 2 unlearnable degrees of freedom while the SWAP gate has 1 unlearnable degree of freedom. As the pattern transfer graph has at least 2 connected components, we conclude that the Pauli noise channel of a 2-qubit Clifford gate has at most 2 unlearnable degrees of freedom. Note that when treating $\{\mathrm{CNOT},\mathrm{SWAP}\}$ together as a gate set, there are only 2 unlearnable degrees of freedom according to Theorem~\ref{thm:mainpaulilearnability} instead of $2+1=3$, because there is one additional learnable degree of freedom (such as $l_{IZ}^{\mathrm{CNOT}}+l_{XX}^{\mathrm{CNOT}}+l_{XI}^{\mathrm{SWAP}}$) that is a joint function of the two gates.
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|}
\hline
Gate & CNOT & SWAP \\
\hline
\makecell{(a) Cycle basis}
&\makecell{ $l_{II},l_{ZI},l_{IX},l_{ZX},
l_{XZ},l_{YY},l_{XY},l_{YZ},$ \\ $l_{IZ}+l_{ZZ},l_{IY}+l_{ZY},l_{IZ}+l_{ZY},$\\$l_{XI}+l_{XX},l_{YI}+l_{YX},l_{XI}+l_{YX}$ } & \makecell{$l_{II},l_{XX},l_{XY},l_{XZ},l_{YX},l_{YY},l_{YZ},l_{ZX},l_{ZY},$\\$l_{ZZ},l_{IX}+l_{XI},l_{IY}+l_{YI},l_{IZ}+l_{ZI},$\\$l_{XI}+l_{IY},l_{XI}+l_{IZ}$}
\\ \hline
\makecell{(b) Learnable\\ Pauli fidelities}
&\makecell{ $\lambda_{II},\lambda_{ZI},\lambda_{IX},\lambda_{ZX},
\lambda_{XZ},\lambda_{YY},\lambda_{XY},\lambda_{YZ},$
\\$\lambda_{IZ}\cdot\lambda_{ZZ},\lambda_{IY}\cdot\lambda_{ZY},\lambda_{IZ}\cdot\lambda_{ZY},$\\$\lambda_{XI}\cdot\lambda_{XX},\lambda_{YI}\cdot\lambda_{YX},\lambda_{XI}\cdot\lambda_{YX}$} &\makecell{$\lambda_{II},\lambda_{XX},\lambda_{XY},\lambda_{XZ},\lambda_{YX},\lambda_{YY},\lambda_{YZ},\lambda_{ZX},\lambda_{ZY},$\\$\lambda_{ZZ},\lambda_{IX}\cdot\lambda_{XI},\lambda_{IY}\cdot\lambda_{YI},\lambda_{IZ}\cdot\lambda_{ZI},$\\$\lambda_{XI}\cdot\lambda_{IY},\lambda_{XI}\cdot\lambda_{IZ}$}
\\ \hline
\makecell{(c) Learnable\\ Pauli errors}
&\makecell{ $p_{II},p_{ZI},p_{IX},p_{ZX},
p_{XZ},p_{YY},p_{XY},p_{YZ},$
\\$p_{IZ}+p_{ZZ},p_{IY}+p_{ZY},p_{IZ}+p_{ZY},$\\$p_{XI}+p_{XX},p_{YI}+p_{YX},p_{XI}+p_{YX}$} &\makecell{$p_{II},p_{XX},p_{XY},p_{XZ},p_{YX},p_{YY},p_{YZ},p_{ZX},p_{ZY},$\\$p_{ZZ},p_{IX}+p_{XI},p_{IY}+p_{YI},p_{IZ}+p_{ZI},$\\$p_{XI}+p_{IY},p_{XI}+p_{IZ}$} \\
\hline
\makecell{(d) Unlearnable\\ degrees of freedom}
& $\lambda_{XI},\lambda_{IZ}$ & $\lambda_{XI}$ \\
\hline
\end{tabular}
\caption{A complete basis for the learnable linear functions of log Pauli fidelities and Pauli error rates for a single CNOT/SWAP gate.
}
\label{tab:main:CNOT_full}
\end{table}
Finally, the learnability of Pauli errors can be determined by the learnability of Pauli fidelities according to the Walsh-Hadamard transform $p_a = \frac{1}{4^n}\sum_{b\in{{\sf P}^n}}\lambda_b(-1)^\expval{a,b}$. An issue here is that Pauli errors are linear functions of $\{\lambda_b\}$ instead of $\{\log \lambda_b\}$. Here we make a standard assumption in the literature~\cite{erhard2019characterizing,flammia2020efficient} that the total Pauli error is sufficiently small. In this case all individual Pauli errors are close to 0 while all individual Pauli fidelities are close to 1. Therefore the Pauli errors can be estimated via
\begin{equation}\label{eq:mainapproximation}
p_a = \frac{1}{4^n}\sum_{b\in{{\sf P}^n}}\lambda_b(-1)^\expval{a,b}\approx\frac{1}{4^n}\sum_{b\in{{\sf P}^n}}(-1)^\expval{a,b}\left(1+\log\lambda_b\right),
\end{equation}
which means that their learnability can be determined by Theorem~\ref{thm:mainpaulilearnability}. In fact it has been suggested~\cite{nielsen2022first} that any function of Pauli fidelities can be estimated in this way (as a linear function of log Pauli fidelities) up to a first-order approximation, which means that the learnability of any function of Pauli fidelities can be determined by Theorem~\ref{thm:mainpaulilearnability}. In Table~\ref{tab:main:CNOT_full} (c) we show the learnable Pauli errors for CNOT and SWAP, where ``learnable'' is in an approximate sense up to Eq.~\eqref{eq:mainapproximation}.
Interestingly, for these two gates, the learnable functions of Pauli errors have the same form as the cycle basis, \textit{i.e.} the cycle space is invariant under Walsh-Hadamard transform.
We calculate the learnable Pauli errors for up to 4-qubit random Clifford gates and this seems to be true in general.
We leave a rigorous investigation into this phenomenon for future work.
\subsection{Experiments on IBM Quantum hardware}
We demonstrate our theory on IBM quantum hardware~\cite{ibmquantum} using a minimal example -- characterizing the noise channel of a CNOT gate. In our experiments both the gate noise and SPAM noise are twirled into Pauli noise using randomized compiling. In the following we show how to extract all learnable information of Pauli noise SPAM-robustly, and also attempt to estimate the unlearnable degrees of freedom by making additional assumptions.
First, we conduct two types of cycle benchmarking (CB) experiments, the standard CB and CB with interleaving single-qubit gates (called \emph{interleaved CB}), as shown in Fig.~\ref{fig:main_cb}.
The results are shown in Fig.~\ref{fig:main_exp_cbraw}. Here a set of two Pauli labels in the $x$-axis (\textit{e.g.}, $\{IZ,ZZ\}$) corresponds to the geometric mean of the Pauli fidelity (\textit{e.g.}, $\sqrt{\lambda_{IZ}\lambda_{ZZ}}$).
Comparing to Table~\ref{tab:main:CNOT_full}, we see that all learnable information of Pauli fidelities (including learnable individual and 2-product) are successfully extracted.
Also note from Fig.~\ref{fig:main_exp_cbraw} that the two types of CB experiments give consistent estimates, in terms of both the process fidelity and individual Pauli fidelities (\textit{e.g.}, $\sqrt{\lambda_{XZ}\lambda_{YY}}$ estimated from standard CB is consistent with $\lambda_{XZ}$ and $\lambda_{YY}$ from interleaved CB).
\begin{figure}
\caption{Estimates of Pauli fidelities of IBM's CNOT gate via standard CB (left) and CB with interleaved gates (right), using circuits shown in Fig.~\ref{fig:main_cb}
\label{fig:main_exp_cbraw}
\end{figure}
We have shown that all 13 learnable degrees of freedom (excluding the trivial $\lambda_{II}=1$) are extracted in Fig.~\ref{fig:main_exp_cbraw} by comparing with Table~\ref{tab:main:CNOT_full}, and there remain 2 unlearnable degrees of freedom. We can bound the feasible region of the 2 unlearnable degrees of freedom using physical constraints, \textit{i.e.}, the reconstructed Pauli noise channel must be completely positive.
This is equivalent to requiring $p_a\ge 0$ for all Pauli error rates $p_a$.
We choose $\lambda_{XX}$ and $\lambda_{ZZ}$ as a representation of the unlearnable degrees of freedom, and plot the calculated feasible region in Fig.~\ref{fig:main_exp_cbfeasible} (a), which happens to be a rectangular area. We also calculate the feasible region for each unlearnable Pauli fidelity and Pauli error rate, which are presented in Fig.~\ref{fig:main_exp_cbfeasible} (b), (c). In particular, we choose two extreme points (blue and green dots in Fig.~\ref{fig:main_exp_cbfeasible} (a)) in the feasible region and plot the corresponding noise model in Fig.~\ref{fig:main_exp_cbfeasible} (b), (c). Note that the (approximately) learnable Pauli error rates (on the left of the red vertical dashed line) are nearly invariant under change of gauge degrees of freedom, but they can be estimated to be negative due to statistical fluctuation. Thus, when we calculate the physical constraints, we only require those unlearnable Pauli error rates (on the right of the red vertical dashed line) to be non-negative.
\begin{figure}
\caption{Feasible region of the learned Pauli noise model, using data from Fig.~\ref{fig:main_exp_cbraw}
\label{fig:main_exp_cbfeasible}
\end{figure}
Next, we explore an approach to estimate the unlearnable information with additional assumptions.
Suppose that one can prepare $\ket{0}^{\otimes n}$ perfectly.
Since we assume noiseless single-qubit gates, this means we can prepare a set of perfect tomographically complete states $\{\ket{0/1},\ket{\pm},\ket{\pm i}\}$.
In this case, all the unlearnable degrees of freedom become learnable, as one can first perform a measurement device tomography, and then directly estimate the process matrix of a noisy gate with measurement error mitigated~\cite{Maciejewski2020mitigationofreadout}.
Following this general idea, we propose a variant of cycle benchmarking for Pauli noise characterization, which we call \emph{intercept CB} as it uses the information of intercept in a standard cycle benchmarking protocol. Given an $n$-qubit Clifford gate $\mathcal G$, let $m_0$ be the smallest positive integer such that $\mathcal G^{m_0} = \mathcal I$.
For any Pauli fidelity $\lambda_a$ (regardless of whether learnable or not according to Theorem~\ref{thm:mainpaulifidelity}), consider the following two CB experiments using the standard circuit as in Fig.~\ref{fig:main_cb} (a). First, prepare an eigenstate of $P_a$, run CB with depth $l m_0+1$ for some non-negative integer $l$, and estimate the expectation value of $P_b\mathrel{\mathop:}\nobreak\mkern-1.2mu=\mathcal G( P_a )$. The result equals
\begin{equation}
{\mathop{\mbb{E}}}\expval{P_b}_{l m_0+1}=\lambda^S_{P_a}\lambda^M_{P_b}\lambda_{a}\left(\prod_{k=1}^{m_0}\lambda_{\mathcal G^k(P_a)}\right)^l,
\end{equation}
where $\lambda_{P_{a/b}}^{S/M}$ is the Pauli fidelity of the state preparation and measurement noise channel, respectively (earlier we have absorbed these two coefficients into a single coefficient $A$ for simplicity). Second, prepare an eigenstate of $P_b$, run CB with depth $l m_0$, and estimate the expectation value of $P_b$. The result equals
\begin{equation}
{\mathop{\mbb{E}}}\expval{P_b}_{l m_0}=\lambda^S_{P_b}\lambda^M_{P_b}\left(\prod_{k=1}^{m_0}\lambda_{\mathcal G^k(P_a)}\right)^l.
\end{equation}
By fitting both ${\mathop{\mbb{E}}}\expval{P_b}_{l m_0+1}$ and ${\mathop{\mbb{E}}}\expval{P_b}_{l m_0}$ as exponential decays in $l$, extracting the intercepts (function values at $l=0$), and taking the ratio,
we obtain an estimator $\widehat{\lambda}^{\text{ICB}}_a$
that is asymptotically unbiased to $\lambda_{a}\cdot{\lambda^{S}_{P_a}}/{\lambda^{S}_{P_b}}$.
This estimator is robust against measurement noise. Note that $\lambda^S_{P_a}=\lambda^S_{P_b}=1$ if we assume perfect initial state preparation, and in this case the above shows that $\lambda_a$ is learnable, and thus the entire Pauli noise channel is learnable.
We note that, instead of fitting an exponential decay in $l$, one could in principle just take $l=0$ and estimate the ratio of ${\mathop{\mbb{E}}}\expval{P_b}_{0}$ and ${\mathop{\mbb{E}}}\expval{P_b}_{1}$, which also yields a consistent estimate for $\lambda_a\cdot\lambda^S_{P_a}/\lambda^S_{P_b}$.
If one has already obtained all the learnable information from previous experiments, this could be a more efficient approach.
However, if one has not done those experiments, the intercept CB with multiple depths can estimate the intercept (unlearnable information) and slope (learnable information) simultaneously, which is more sample efficient.
We numerically simulate intercept CB for characterizing the CNOT gate under different state preparation (SP) and measurement (M) noise. As shown in Fig.~\ref{fig:main_sim_intercept}, this method yields relatively precise estimate when there is only measurement noise even if the noise is orders of magnitude stronger than the gate noise, but will have large deviation from the true noise model even under small state preparation noise. We refer the reader to Supplementary Section III for more details about the numerical simulation.
Finally, we experimentally implement intercept CB to estimate $\lambda_{XX}$ and $\lambda_{ZZ}$, which are the two unlearnable degrees of freedom of CNOT, allowing us to determine all the Pauli fidelities and Pauli error rates.
One challenge in interpreting the results is that we do not know in general whether the low SP noise assumption holds, therefore it is unclear if the learned results should be trusted.
However, for the estimate to be correct, it should at least lie in the physically feasible region we obtained earlier in Fig.~\ref{fig:main_exp_cbfeasible}.
In Fig.~\ref{fig:main_exp_intercept}, we present our experimental results of intercept CB. It turns out that certain Pauli fidelities are far away from the physical region by several standard deviations.
This gives strong evidence that the low SP noise assumption was \emph{not} true on the platform we used.
The data collected here can further be used to give a lower bound for the SP noise. Suppose we obtain the physical region of $\lambda_a$ to be $[\widehat{\lambda}_{a,\mathrm{min}},\widehat{\lambda}_{a,\mathrm{max}}]$. Combining with the expression of intercept CB, we have
\begin{equation}
{\widehat{\lambda}^\mathrm{ICB}_a}/{\widehat{\lambda}_{a,\mathrm{max}}}\le{\lambda^S_{P_a}}/{\lambda^S_{P_b}}\le{\widehat{\lambda}^\mathrm{ICB}_a}/{\widehat{\lambda}_{a,\mathrm{min}}}.
\end{equation}
Applying this to the data of $IZ$ and $ZZ$ in Fig.~\ref{fig:main_exp_intercept} (a), we have $\lambda^S_{IZ}/\lambda^S_{ZZ} \le 0.9879(23)$.
If we make a physical assumption that the state preparation noise is a random bit-flip during the qubit initialization, one can conclude the bit-flip rate on the first qubit is lower bounded by $0.61(12)\%$.
One can in principle bound the bit-flip rate on the second qubit by looking at $\lambda^S_{XX}/\lambda^S_{XI}$. Unfortunately, our estimate of $\lambda^S_{XX}$ from intercept CB falls in the physical region within one standard deviation, so there is no nontrivial lower bound. One could expect obtaining a useful lower bound by looking at a CNOT gate with reversed control and target.
The lower bound of SP noise obtained here is completely independent of the measurement noise and does not suffer from the issue of gauge freedom~\cite{nielsen2021gate}, as long as all of our noise assumptions are valid, \textit{i.e.}, there is no significant contribution from time non-stationary, non-Markovian, or single-qubit gate-dependent noise.
\begin{figure}
\caption{Simulation of intercept CB on CNOT under different SPAM noise rate. The simulated noise channel is a $2$-qubit amplitude damping channel with effective noise rate $5\%$, and SPAM noise are modeled as bit-flip errors. For the blue (green) lines, we introduce random bit-flip errors to the measurement (state preparation). The solid lines show the $l_1$-distance of the estimated Pauli fidelities from the true Pauli fidelities. The solid lines show the $l_1$-distance of the (individually) learnable Pauli fidelities from the ground truth.
}
\label{fig:main_sim_intercept}
\end{figure}
\begin{figure}
\caption{The learned Pauli noise model using intercept CB. The feasible region (blue bars) are taken from Fig.~\ref{fig:main_exp_cbfeasible}
\label{fig:main_exp_intercept}
\end{figure}
\section{Discussion}
We have shown how to characterize the learnability of Pauli noise of Clifford gates and discussed a method to extract unlearnable information by assuming perfect initial state preparation. It is also interesting to consider other physically motivated assumptions on the noise model to avoid unlearnability. For example, we can write down a parameterization of the noise model based on the underlying physical mechanism which may have fewer than $4^n$ parameters. The main issue here is that these assumptions are highly platform-dependent and should be decided case-by-case. Moreover, it is unclear to what extent should the learned results be trusted when additional assumptions are made, since in general we cannot test whether the assumptions hold due to unlearnability.
Another direction to overcome the unlearnability is to change the model of quantum experiments. Here we have been working with the standard model as in gate set tomography, where a quantum measurement decoheres the system and only outputs classical information. However, some platforms might support quantum non-demolition (QND) measurements, and in this case measurements can be applied repeatedly, which could potentially allow more information to be learned~\cite{laflamme2022algorithmic}.
Recently, Ref.~\cite{huang2022foundations} considered similar issues of noise learnability. They studied a different Pauli noise model with perfect initial state $\ket{0}$, perfect computational basis measurement, and noisy single qubit gates, and showed the existence of unlearnable information. In contrast, here we focus on the learnability of Pauli noise of multi-qubit Clifford gates assuming perfect single-qubit gates (with noisy SPAM), and in practice we make the standard assumption that noise on single-qubit gates is gate-independent (\textit{e.g.}~\cite[Sec. II A]{Ferracin2022Efficiently}), in which case our noise learning results are interpreted as characterizing a dressed cycle.
This work leaves open the question of noise learnability for non-Clifford gates. An issue here is that randomized compiling is not known to work with non-Clifford gates in general, so it is unclear if the general CPTP noise learnability problem can be reduced to Pauli noise. Recent work~\cite{liu2021benchmarking} shows that random quantum circuits can effectively twirl the CPTP noise channel into Pauli noise and can be used to learn the total Pauli error. The question of whether more information can be learned still remains open.
Another issue to address is the scalability in noise learning. It is impossible to estimate all learnable degrees of freedom efficiently as there are exponentially many of them (an exponential lower bound on the sample complexity is shown in~\cite{chen2022quantum}). One way to avoid the exponential scaling issue is to assume the noise model has certain special structure (such as sparsity or low-weight) such that the noise model only has polynomially many parameters~\cite{harper2020efficient,harper2021fast,flammia2020efficient,berg2022probabilistic}. It is an interesting open direction to study the characterization of learnability under these assumptions, and we give some related discussions in Supplementary Section II D.
\section*{Data availability}
The data generated in this study is available at \url{https://github.com/csenrui/Pauli_Learnability}
\section*{Code availability}
The code that supports the findings of this study is available at \url{https://github.com/csenrui/Pauli_Learnability}
\begin{acknowledgments}
We thank Ewout van den Berg, Arnaud Carignan-Dugas, Robert Huang, Kristan Temme and Pei Zeng for helpful discussions.
We thank the anonymous reviewer \#2 for suggesting an alternative approach to intercept cycle benchmarking.
S.C. and L.J. acknowledge support from the ARO (W911NF-18-1-0020, W911NF-18-1-0212), ARO MURI (W911NF-16-1-0349, W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209), AFRL (FA8649-21-P-0781), DoE Q-NEXT, NSF (OMA-1936118, EEC-1941583, OMA-2137642), NTT Research, and the Packard Foundation (2020-71479).
Y.L. was supported by DOE NQISRC QSA grant \#FP00010905, Vannevar Bush faculty fellowship N00014-17-1-3025, MURI Grant FA9550-18-1-0161 and NSF award DMR-1747426. A.S. is supported by a Chicago Prize Postdoctoral Fellowship in Theoretical Quantum Science. B.F. acknowledges support from AFOSR (YIP number FA9550-18-1-0148 and FA9550-21-1-0008). This material is based upon work partially supported by the National Science Foundation under Grant CCF-2044923 (CAREER) and by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.
\end{acknowledgments}
\section*{Author contributions}
S.C. and Y.L. developed the theory and performed the experiments. B.F. and L.J. supervised the project. All authors contributed important ideas during initial discussions and contributed to writing the manuscript.
\section*{Competing interests}
The authors declare no competing interests.
\begin{appendix}
\date{\today}
\maketitle
\tableofcontents
\section{Preliminaries}\label{sec:pre}
Define ${\sf P}^n$ to be the $n$-qubit Pauli group modulo its center.
We can label any Pauli operator $P_a\in{\sf P}^n$ with a $2n$-bit string $a$.
Specifically, we define $P_{\bm 0}$ to be the identity operator $I$.
We will use the notations $P_a$ and $a$ interchangeably when there is no confusion.
The \emph{pattern} of an $n$-qubit Pauli operator $P_a$, denoted as $\mathrm{pt}(P_a)$, is an $n$-bit string that takes $0$ at the $j$th bit if $P_a$ equals to $I$ at the $j$th qubit and takes $1$ otherwise.
For example, $\mathrm{pt}(XYIZI)=\mathrm{pt}(XXIXI)=11010$.
An $n$-qubit \emph{Pauli diagonal map} $\Lambda$ is a linear map of the following form
\begin{equation}
\Lambda(\cdot) = \sum_{a\in{{\sf P}^n}}p_a P_a(\cdot)P_a,
\end{equation}
where $\bm{p}\mathrel{\mathop:}\nobreak\mkern-1.2mu= \{p_a\}_a$ are called the \emph{Pauli error rates}.
If $\Lambda$ is further a CPTP map, which corresponds to the condition $p_a\ge 0$ and $\sum_a p_a = 1$, then it is called a \emph{Pauli channel}.
An important property of Pauli diagonal maps is that their eigen-operators are exactly the $4^n$ Pauli operators. Thus, an alternative expression for $\Lambda$ is
\begin{equation}
\Lambda(\cdot) = \frac{1}{2^n}\sum_{b\in{{\sf P}^n}}\lambda_b\Tr(P_b(\cdot))P_b,
\end{equation}
where $\bm\lambda \mathrel{\mathop:}\nobreak\mkern-1.2mu= \{\lambda_b\}_b$ are called the \emph{Pauli fidelities} or \emph{Pauli eigenvalues} ~\cite{flammia2020efficient,flammia2021pauli,chen2020robust}.
These two sets of parameters, $\bm p$ and $\bm \lambda$, are related by the Walsh-Hadamard transform
\begin{equation}
\begin{aligned}
\lambda_b = \sum_{a\in{{\sf P}^n}}p_a(-1)^\expval{a,b},\quad
p_a = \frac{1}{4^n}\sum_{b\in{{\sf P}^n}}\lambda_b(-1)^\expval{a,b},
\end{aligned}
\end{equation}
where $\expval{a,b}$ equals to $0$ if $P_a,P_b$ commute and equals to $1$ otherwise.
For a general linear map $\mathcal E$, define its \emph{Pauli twirl} as
\begin{equation}
\mathcal E^{P}\mathrel{\mathop:}\nobreak\mkern-1.2mu= \sum_{a\in{{\sf P}^n}} \mathcal P_a\mathcal E \mathcal P_a.
\end{equation}
Here we use the calligraphic $\mathcal P_a$ to represent the unitary channel of Pauli gate $P_a$, $\mathcal P_a(\cdot) := P_a(\cdot)P_a$.
The Pauli twirl of any linear map (quantum channel) is a Pauli diagonal map (Pauli channel). When we talk about the Pauli fidelities of a non-Pauli channel, we are effectively referring to the Pauli fidelities of its Pauli twirl.
\section{Theory on the learnability of Pauli noise}
In this section, we give a precise characterization of what information in the Pauli noise channel associated with Clifford gates can be learned in the presence of state-preparation-and-measurement (SPAM) noise.
Our results show that certain Pauli fidelities of a noisy multi-qubit Clifford gate cannot be learned in a SPAM-robust manner, even with the assumption that single-qubit gates can be perfectly implemented.
The proof is related to the notion of \emph{gauge freedom} in the literature of gate set tomography~\cite{nielsen2021gate}.
We note that the results presented in this section emphasizes on the no-go part, \textit{i.e.}, some information about the Pauli noise is (SPAM-robustly) unlearnable even with many favorable assumptions on the experimental conditions.
As shown in the main text, the learnable information about Pauli noise can be extracted in a much more practical setting using cycle benchmarking~\cite{erhard2019characterizing} and its variant.
\subsection{Assumptions and definitions}\label{sec:noisemodelAssumptions}
We focus on an $n$-qubit quantum system. Below are our assumptions on the noise model.
\begin{itemize}
\item \textbf{Assumption 1.} All single qubit unitary operation can be perfectly implemented.
\item \textbf{Assumption 2.} A set of multi-qubit Clifford gates $\mathfrak G \mathrel{\mathop:}\nobreak\mkern-1.2mu= \{\mathcal G\}$ can be implemented and are subject to gate-dependent Pauli noise, \textit{i.e.}, $\widetilde{\mathcal G} = \mathcal G\circ \Lambda_{\mathcal G}$ where $\Lambda_\mathcal G$ is some $n$-qubit Pauli channel.
\item \textbf{Assumption 3.} Any state preparation and measurement can be implemented, up to some fixed Pauli noise channel $\mathcal E^S$ and $\mathcal E^M$, respectively.
\item\textbf{Assumption 4.} The Pauli noise channels appearing in the above assumptions satisfy that all Pauli fidelities and Pauli error rates are strictly positive.
\end{itemize}
Assumption 1 is motivated by the fact that the noise of single-qubit gates are usually much smaller than that of multi-qubit gates on today's hardware. Such approximation is widely adopted in the literature~\cite{erhard2019characterizing,wallman2016noise} with slight modifications.
In Assumption 2, we view every Clifford gate as an $n$-qubit gate, and allow the noise to be $n$-qubit. This means we are taking all crosstalk into account.
A Clifford gate acting on a different (ordered) subset of qubits is viewed as a different gate and can thus have a different noise channel (\textit{e.g.}, CNOT$_{12}$, CNOT$_{21}$, CNOT$_{23}$ have different noise channels.)
We will discuss the no-crosstalk situation in Sec.~\ref{sec:no_crosstalk}.
The rationale for assuming Pauli noise in Assumption 2 and 3 is that we can always use randomized compiling~\cite{wallman2016noise,hashim2020randomized} to tailor general noise into Pauli channels.
Finally, Assumption 4 is mostly for technical convenience. The requirement of positive Pauli error rates roughly implies the Pauli channels are at the interior of the CPTP polytope, and will be useful later in constructing valid gauge transformations. The requirement of positive Pauli fidelities is also reasonable for any physically interesting noise model.
Specifying a Clifford gate set $\mathfrak G$, a \emph{noise model} satisfying our assumptions is determined by the Pauli channels describing gate noise and SPAM noise.
We can thus view a noise model as a collection of Pauli fidelities, denoted as $\mathcal N = \{\mathcal E^S,\mathcal E^M,\Lambda\}$, where $\mathcal E^{S/M} = \{\lambda_a^{S/M}\}_a$ describes the SPAM noise and $\Lambda = \{\lambda_a^{\mathcal {G}}\}_{a,\mathcal G}$ describes the gate noise.
We note that this is an example of \emph{parametrized gate set} in the language of gate set tomography~\cite{nielsen2021gate}.
In order to gain information about an unknown noise model, one needs to conduct \emph{experiments}. In the circuit model, any experiment can be described by some state preparation, a sequence of quantum gates, and some POVM measurements.
An experiment conducted with different underlying noise model would yield different measurement outcome distributions.
Explicitly, consider an (ideal) experiment with initial state $\rho_0$, gate sequence $\mathcal C$, POVM measurements $\{E_o\}_o$. Denote the noisy implementation of these objects within a certain noise model $\mathcal N$ with a tilde. Then the experiment effectively maps $\mathcal N$ to a probability distribution $p_{\mathcal N}(o) = \Tr(\widetilde E_o(\widetilde{\mathcal C}(\widetilde \rho_0)))$.
We call two noise models $\mathcal N_1$, $\mathcal N_2$ \emph{indistinguishable} if for all possible experiments we have $p_{\mathcal N_1}=p_{\mathcal N_2}$, and distinguishable otherwise.
\begin{definition}[Learnable and unlearnable function]\label{de:learnability}
A function $f$ of noise models is learnable if
\begin{equation}
f(\mathcal N_1)\ne f(\mathcal N_2) \implies \mathcal N_1, \mathcal N_2~\text{are distinguishable},
\end{equation}
for any noise models $\mathcal N_1$, $\mathcal N_2$.
In contrast, $f$ is unlearnable if there exist indistinguishable noise models $\mathcal N_1$, $\mathcal N_2$ such that $f(\mathcal N_1)\ne f(\mathcal N_2)$.
\end{definition}
Note that the above definition of ``learnable'' does not necessarily mean that the value of the function can be learned. However, throughout this paper whenever some function is ``learnable'' according to Definition~\ref{de:learnability}, it is also learnable in the stronger sense that we can design an experiment to estimate it up to arbitrarily small error with high success probability.
In the language of gate set tomography, an unlearnable function is a \emph{gauge-dependent} quantity of the gate set~\cite{nielsen2021gate}.
On the other hand, any learnable function can in principle be learned to arbitrary precision.
In the following, we will focus the learnability of the functions of the gate noise, including individual and multiplicative combinations of Pauli fidelities.
\subsection{Learnability of individual Pauli fidelity}
We first study the learnability of individual Pauli fidelities associated with a Clifford gate. This has been an open problem in recent study of quantum benchmarking.
Perhaps surprisingly, we obtain the following simple criteria on the learnability of Pauli fidelities with any Clifford gate.
\begin{theorem}\label{th:nogo}
With Assumptions 1-4, for any $n$-qubit Clifford gate $\mathcal G$ and Pauli operator $P_a$, the Pauli fidelity $\lambda_a^{\mathcal G}$ is unlearnable if and only if $\mathcal G$ changes the pattern of $P_a$, \textit{i.e.,} $\mathrm{pt}(\mathcal G(P_a))\ne \mathrm{pt}(P_a)$.
\end{theorem}
The fact that certain Pauli fidelities are SPAM-robustly unlearnable is observed in some recent works~\cite{erhard2019characterizing,hashim2020randomized,berg2022probabilistic,Ferracin2022Efficiently}, described as ``degeneracy'' of the noise model. Our work is the first to give a rigorous argument for this by establishing connections to gate set tomography.
As an example, for the CNOT and SWAP gates, we can immediately list its learnable and unlearnable Pauli fidelities in Table~\ref{tab:cnot_swap_individual}.
We note that, the no-go theorem holds even under the no-crosstalk assumption as will be discussed in Sec.~\ref{sec:no_crosstalk},
so introducing ancillary qubits or other multi-qubit Clifford gates cannot help resolve the unlearnability.
\begin{table}[!htp]
\centering
\begin{tabular}{|c|c|c|}
\hline
Gate & Learnable & Unlearnable \\
\hline
CNOT&$\lambda_{II},\lambda_{ZI},\lambda_{IX},\lambda_{ZX},
\lambda_{XZ},\lambda_{YY},\lambda_{XY},\lambda_{YZ}$ &
$\lambda_{IZ},\lambda_{XI},\lambda_{ZZ},\lambda_{XX},
\lambda_{IY},\lambda_{YI},\lambda_{ZY},\lambda_{YX}$\\
\hline
SWAP&$\lambda_{II},\lambda_{XX},\lambda_{XY},\lambda_{XZ},
\lambda_{YX},\lambda_{YY},\lambda_{YZ},\lambda_{ZX},\lambda_{ZY},\lambda_{ZZ}$ &
$\lambda_{IX},\lambda_{IY},\lambda_{IZ},
\lambda_{XI},\lambda_{YI},\lambda_{ZI}$\\
\hline
\end{tabular}
\caption{Learnability of individual Pauli fidelity of CNOT and SWAP.}
\label{tab:cnot_swap_individual}
\end{table}
Before going into the proof, we make several remarks about Theorem~\ref{th:nogo}.
The correct interpretation of the no-go result in Theorem~\ref{th:nogo} is that certain Pauli fidelities cannot be learned in a fully SPAM-robust manner.
If one has some pre-knowledge that the SPAM noise is much weaker than the gate noise, there exist methods to give a pretty good estimate of those unlearnable Pauli fidelities, according to physical constraints.
See the discussions in the main text.
On the other hand, it is observed that the product of certain unlearnable Pauli fidelities can be learned in a SPAM-robust manner, such as $\lambda_{XI}\cdot\lambda_{XX}$ for the CNOT gate~\cite{erhard2019characterizing}. We will characterize the learnability of this kind of products of Pauli fidelities in the next subsection.
\begin{proof}[Proof of Theorem~\ref{th:nogo}]
We start with the ``only if'' part, which is equivalent to saying that $\mathrm{pt}(P_a)=\mathrm{pt}(\mathcal G(P_a))$ implies $\lambda_a^{\mathcal G}$ being learnable.
The condition $\mathrm{pt}(\mathcal G(P_a))=\mathrm{pt}(P_a)$ implies $\mathcal G(P_a)$ is equivalent to $P_a$ up to some local unitary transformation, \textit{i.e.}, there exists a product of single-qubit unitary gates $\mathcal U \mathrel{\mathop:}\nobreak\mkern-1.2mu= \bigotimes_{j=1}^n\mathcal U_j$ such that
\begin{equation}
\mathcal U\circ\mathcal G(P_a) = P_a.
\end{equation}
Now we design the following experiments parameterized by a positive integer $m$,
\begin{itemize}
\item Initial state: $\rho_0 = (I+P_a)/2^n$,
\item POVM measurement: $E_{\pm 1} = (I\pm P_a)/{2}$,
\item Circuit: $\mathcal C^m = \left(\mathcal U\circ\mathcal G\right)^m$.
\end{itemize}
Consider the measurement probability by running these experiments within a noise model $\mathcal N$.
\begin{equation}
\begin{aligned}
p^{(m)}_{\pm 1}(\mathcal N) &= \Tr\left( \widetilde{E}_{\pm 1} \widetilde{\mathcal C}^m (\widetilde{\rho}_0)\right) \\
&= \Tr \left( \frac{I\pm P_a}{2} \cdot\left( \mathcal E^M \circ \left(
\mathcal U\circ\mathcal G
\right)^m\circ\mathcal E^S \right)
\left(\frac{I+P_a}{2^n}\right) \right)\\
&= \Tr\left(\frac{I\pm P_a}{2} \cdot\frac{I+\lambda^M_{a}
\left(\lambda_a^{\mathcal G}\right)^m
\lambda^S_{a}P_a}{2^n} \right)\\
&= \frac{1\pm\lambda^M_{a}
\left(\lambda_a^{\mathcal G}\right)^m
\lambda^S_{a}}{2}.
\end{aligned}
\end{equation}
Recall that $\lambda_a^{S/M}$ is the Pauli fidelity of the SPAM noise channel for $P_a$.
The expectation value is
\begin{equation}
\mathbfb E^{(m)}(\mathcal N) = \lambda^M_{a}
\left(\lambda_a^{\mathcal G}\right)^m
\lambda^S_{a}.
\end{equation}
If we take the ratio of expectation values of two experiments with consecutive $m$, we obtain (recall that all these Pauli fidelities are strictly positive by Assumption 4)
\begin{equation}
\mathbfb E^{m+1}(\mathcal N)/\mathbfb E^{m}(\mathcal N) = \lambda_a^{\mathcal G}.
\end{equation}
This implies that if two noise model assign different values for $\lambda_a^{\mathcal G}$, the above experiments would be able to distinguish between them. By definition~\ref{de:learnability}, we conclude $\lambda_a^{\mathcal G}$ is learnable.
Next we prove the ``if'' part. Fix an $n$-qubit Clifford gate $\mathcal G$.
Let $P_a$ be any Pauli operator such that $\mathrm{pt}(\mathcal G(P_a))\neq \mathrm{pt}(P_a)$. We will show that $\lambda_a^{\mathcal G}$ is unlearnable by explicitly constructing indistinguishable noise models that assign different values to $\lambda_a^{\mathcal G}$.
Recall that any experiment involves a noisy initial state $\tilde{\rho}_0$, a noisy measurement $\{\widetilde{E}_l\}_l$, and a quantum circuit consisting of noiseless single-qubit gates $\mathcal U \mathrel{\mathop:}\nobreak\mkern-1.2mu= \bigotimes_{j=1}^n\mathcal U_j$ and noisy multi-qubit Clifford gates $\widetilde{\mathcal T}$.
Now, introduce an invertible linear map $\mathcal M:\mathcal L(\mathcal H_{2^n})\to \mathcal L(\mathcal H_{2^n})$, and consider the following transformation
\begin{equation}\label{eq:gauge_trans}
\begin{aligned} &\widetilde{\rho}_0\mapsto \mathcal M(\widetilde{\rho}_0),\quad
\widetilde{E}_l\mapsto (\mathcal M^{-1})^\dagger (\widetilde{E}_l),\\
&\bigotimes_{j=1}^n\mathcal U_j \mapsto \mathcal M\circ\bigotimes_{j=1}^n\mathcal U_j\circ\mathcal M^{-1},\\
&\widetilde{\mathcal T} \mapsto \mathcal M\circ \widetilde{\mathcal T}\circ\mathcal M^{-1}.
\end{aligned}
\end{equation}
One can immediately see that any measurement outcome distribution $p_l\mathrel{\mathop:}\nobreak\mkern-1.2mu=\Tr(\widetilde{E}_l\widetilde{\mathcal C}(\widetilde{\rho}_0))$ remains unchanged via such transformation. Therefore the noise models related by this transformation are indistinguishable. This is called a \emph{gauge transformation} in the literature of gate set tomography~\cite{nielsen2021gate}.
To use this idea for the proof, we start with a noise model $\mathcal N$ and construct a map $\mathcal M$ such that
\begin{enumerate}
\item The transformation yields a physical noise model $\mathcal N'$ satisfying Assumptions 1-5 in Sec.~\ref{sec:noisemodelAssumptions}.
\item The two noise models $\mathcal N$, $\mathcal N'$ assign different values to $\lambda_a^{\mathcal G}$.
\end{enumerate}
Starting with a generic noise model $\mathcal N = \{\mathcal E^S,\mathcal E^M,\Lambda\}$ satisfying the assumptions,
we construct the gauge transform map $\mathcal M$ as follows.
Since $\mathrm{pt}(\mathcal G(P_a))\ne \mathrm{pt}(P_a)$, there exists an index $i\in[k]$ such that one and only one of $(P_a)_i$ and $\mathcal G(P_a)_i$ equals to $I$. Let $\mathcal M$ be the single-qubit depolarizing channel on the $i$-th qubit,
\begin{equation}\label{eq:dep_trans}
\mathcal M\mathrel{\mathop:}\nobreak\mkern-1.2mu=\mathcal D_{i}\otimes\mathcal I_{[n]\backslash i},
\end{equation}
where the single-qubit depolarizing channel is defined as
\begin{equation}
\forall P\in\{I,X,Y,Z\},\quad \mathcal D(P) = \left\{
\begin{aligned}
P,\quad&~\text{if}~ P=I,\\
\eta P,\quad&~\text{otherwise},
\end{aligned}
\right.
\end{equation}
for some parameter $0<\eta<1$. We will specify the value of $\eta$ later.
Now we calculate the transformed noise model $\mathcal N'=\{\mathcal E^{S'},\mathcal E^{M'},\Lambda'\}$. The SPAM noise channels are transformed as
\begin{equation}\label{eq:SPAMupdate}
\mathcal E^{S'} = \mathcal M\mathcal E^S,\quad \mathcal E^{M'} = \mathcal E^M\mathcal M^{-1},
\end{equation}
both of which are still Pauli diagnoal maps. Thanks to our Assumption 4, as long as $\eta$ is sufficiently close to $1$, they can be shown to be Pauli channels.
Next, the single-qubit unitary gates are transformed as
\begin{equation}
\mathcal M \left(\bigotimes_{j=1}^n \mathcal U_j \right)\mathcal M^{-1}
=\mathcal D_i\mathcal U_i\mathcal D_i^\dagger \otimes \bigotimes_{j\ne i}\mathcal U_j
= \bigotimes_j \mathcal U_j,
\end{equation}
since the single-qubit deplorizing channel commutes with any single-qubit unitary. This implies the single-qubit unitary gates are still noiseless.
Finally, consider an arbitrary $n$-qubit Clifford gate ${\mathcal T}$. We show that the transformed noisy gate takes the form $\widetilde{\mathcal T}'=\widetilde{\mathcal T}\circ \Lambda_{\mathcal T}'$ where $\Lambda_{\mathcal T}'$ is still a Pauli channel, with the Pauli fidelities updated as follows.
\begin{equation}\label{eq:paulifidelityupdate}
{\lambda_b^{\mathcal T}}'=\begin{cases}
\eta \lambda_b^{\mathcal T}, & \text{if }\mathrm{pt}(P_b)_i=0\text{ and }\mathrm{pt}(\mathcal T(P_b))_i=1,\\
\eta^{-1}\lambda_b^{\mathcal T}, & \text{if }\mathrm{pt}(P_b)_i=1\text{ and }\mathrm{pt}(\mathcal T(P_b))_i=0,\\
\lambda_b^{\mathcal T}, &\text{if }\mathrm{pt}(P_b)_i=\mathrm{pt}(\mathcal T(P_b))_i.
\end{cases}
\end{equation}
We give a proof for the first case. Note that
\begin{equation}\label{eq:gate_noise_trans}
\begin{aligned}
\mathcal M\circ \widetilde{\mathcal T}\circ\mathcal M^{-1}
&= \mathcal D_{i}\circ \widetilde{\mathcal T}\circ \mathcal D_{i}^{-1}\\
&= \mathcal D_{i}\circ {\mathcal T}\circ\Lambda_{\mathcal T}\circ \mathcal D_{i}^{-1}\\
&= {\mathcal T}\circ({\mathcal T}^{-1}\circ\mathcal D_{i}\circ {\mathcal T}\circ\Lambda_{\mathcal T}\circ \mathcal D_{i}^{-1}) \\
&\mkern-1.2mu=\nobreak\mathrel{\mathop:}\mathcal T\circ \Lambda'_{\mathcal T},
\end{aligned}
\end{equation}
where we use $\mathcal D_i$ as a shorthand for $\mathcal D_i\otimes\mathcal I_{[n]\backslash i}$.
The transformed noise channel can be written as
\begin{equation}\label{eq:noisechannelupdate}
\Lambda'_{\mathcal T}={\mathcal T}^{-1}\circ\mathcal D_{i}\circ {\mathcal T}\circ\Lambda_{\mathcal T}\circ \mathcal D_{i}^{-1}.
\end{equation}
Let us calculate its action on arbitrary $P_b$.
\begin{equation}
\begin{aligned}
\Lambda_{\mathcal T}'
({P_{b}}) &= (\mathcal T^{-1}\circ \mathcal D_{i}\circ \mathcal T\circ\Lambda_{\mathcal T}\circ \mathcal D_{i}^{-1})(P_b)\\
&= (\eta^{-1})^{\mathrm{pt}(P_b)_i} (\mathcal T^{-1}\circ \mathcal D_{i}\circ \mathcal T\circ\Lambda_{\mathcal T})(P_b)\\
&= \lambda_b^{\mathcal T}(\eta^{-1})^{\mathrm{pt}(P_b)_i} (\mathcal T^{-1}\circ \mathcal D_{i}\circ \mathcal T)(P_b)\\
&=\eta^{\mathrm{pt}(\mathcal T(P_b))_i}\lambda_b^{\mathcal T}(\eta^{-1})^{\mathrm{pt}(P_b)_i} ~P_b.
\end{aligned}
\end{equation}
Thus, $\Lambda_{\mathcal T}'$ is indeed a Pauli diagonal map with Pauli fidelities given by Eq.~\eqref{eq:paulifidelityupdate}. The fact that $\Lambda_{\mathcal T}'$ is guaranteed to be a CPTP map by choosing appropriate $\eta$ will be verified later.
Specifically, if we take $\mathcal T$ to be the Clifford gate $\mathcal G$ that we are interested in, we have $\lambda_a^{\mathcal G'} = \eta\lambda_a^{\mathcal G}$ or $\lambda_a^{\mathcal G'} = \eta^{-1}\lambda_a^{\mathcal G}$. In either case, $\lambda_a^{\mathcal G'} \ne \lambda_a^{\mathcal G}$. This means the two indistinguishable noise model $\mathcal N$, $\mathcal N'$ indeed assign different values to $\lambda_a^{\mathcal G}$.
We now verify that $\mathcal N'$ is indeed a physical noise model and satisfies Assumptions 1-4. We have already shown that single-qubit unitary gates remain noiseless and that all gate noise and SPAM noise are described by Pauli diagonal maps. The only thing left is to make sure all these Pauli diagonal maps are CPTP and satisfy the positivity constraints in Assumption 4.
According to Eq.~\eqref{eq:SPAMupdate} and \eqref{eq:paulifidelityupdate}, any Pauli fidelity $\lambda_b$ of either SPAM noise or gate noise is transformed to one of the following $\lambda_b'\in\{\lambda_b,\eta\lambda_b,\eta^{-1}\lambda_b\}$, so $\lambda_b>0$ implies $\lambda_b'>0$. On the other hand, any transformed Pauli error rate can be bounded by
\begin{equation}\label{eq:positivity}
\begin{aligned}
p_c' &= \frac{1}{4^n}\sum_{b\in{\sf P}^n}(-1)^\expval{b,c}\lambda_b'\\
&\ge \frac{1}{4^n}\sum_{b\in{\sf P}^n}\left((-1)^\expval{b,c}\lambda_b - (\eta^{-1}-1)\lambda_b\right)\\
&\ge p_c - (\eta^{-1}-1).
\end{aligned}
\end{equation}
To ensure every $p'_c>0$, we can choose $1>\eta>(p_{\min}+1)^{-1}$ with $p_{\min}$ being the minimum Pauli error rate among all Pauli channels of both SPAM and gate noise, which is possible since $p_{\min}>0$ by Assumption 4. This means each transformed Pauli diagonal maps are completely positive (CP). To see they are also trace-preserving (TP), just notice from Eq.~\eqref{eq:SPAMupdate},~\eqref{eq:paulifidelityupdate} that $\lambda_{\bm 0}'=\lambda_{\bm 0} = 1$ always holds.
Now we conclude that $\mathcal N'$ is indeed a physical noise model satisfying all the assumptions.
Combining with the reasoning in the last paragraph, we see $\lambda_a^{\mathcal G}$ is unlearnable.
This completes our proof.
\end{proof}
\subsection{Characterization of learnable space via algebraic graph theory}\label{sec:space}
We have characterized the learnability of individual Pauli fidelities associated with any Clifford gates in Theorem~\ref{th:nogo}.
Here, we want to understand the learnablity for a general function of the gate noise.
We first show that, in our setting, any measurement outcome probability in experiment can be expressed as a polynomial of Pauli fidelities of gate and SPAM noise, and each term in the polynomial can be learned via a CB experiment (see Sec.~\ref{sec:justification} for details).
Therefore, it suffices to study the monomials, \textit{i.e.}, products of Pauli fidelities.
For each Pauli fidelity $\lambda_a^{\mathcal G}$, we define the \emph{logarithmic Pauli fidelity} as $l_a^{\mathcal G}\mathrel{\mathop:}\nobreak\mkern-1.2mu= \log \lambda_a^{\mathcal G}$ ($\lambda_a^{\mathcal G}>0$ by Assumption 4).
It then suffices to study the learnability of linear functions of the logarithmic Pauli fidelities.
An alternative reason to only study this class of function is that, under a weak noise assumption, we have $l_a\to 0$, so we can express any function of the noise model as a linear function of $l_a$ under a first order approximation. Note that similar approaches have been explored in the literature~\cite{nielsen2022first,flammia2021averaged}.
Since we are working with Assumption 1-4 which takes all crosstalk into account, we treat the noise channel for each gate in $\mathfrak G$ as $n$-qubit.
The number of independent Pauli fidelities we are interested in is thus
\begin{equation}
|\Lambda| = |\mathfrak G|\cdot 4^n.
\end{equation}
Denote the space of all (real-valued) linear function of logarithmic Pauli fidelities as $F$, then we have $F\cong \mathbfb R^{|\Lambda|}$. A function $f\in F$ uniquely corresponds to a vector $\bm v\in \mathbfb R^{|\Lambda|}$ by $f(\bm l) = \bm v\cdot \bm l = \sum_{a,\mathcal G}v_{a,\mathcal G}l_a^{\mathcal G}$. We will use the vector to refer to the linear function when there is no ambiguity.
Denote the set of all learnable function in $F$ as $F_L$ (in the sense of Def.~\ref{de:learnability}). As shown in the following lemma, $F_L$ forms a linear subspace in $F$, so we call $F_L$ the \emph{learnable space}.
\begin{lemma}\label{le:learnable_is_space}
$F_L$ is a linear subspace of $F$.
\end{lemma}
\begin{proof}
Given $\bm v_1,\bm v_2 \in F_L$, consider the learnability of $\bm v_1+\bm v_2$. For any noise models $\mathcal N_1,~\mathcal N_2$,
\begin{equation}
\begin{aligned}
(\bm v_1+\bm v_2)\cdot\bm l_{\mathcal N_1} \ne (\bm v_1+\bm v_2)\cdot\bm l_{\mathcal N_2} &\implies \bm v_1\cdot\bm l_{\mathcal N_1} \ne \bm v_1\cdot\bm l_{\mathcal N_2} ~\text{or}~\bm v_2\cdot\bm l_{\mathcal N_1} \ne \bm v_2\cdot\bm l_{\mathcal N_2}\\
&\implies \mathcal N_1, \mathcal N_2~\text{are distinguishable}.
\end{aligned}
\end{equation}
Thus $\bm v_1+\bm v_2\in F_L$.
We also have $\bm v\in L \implies k\bm v\in F_L$ for all $k\in\mathbfb R$. Therefore, $F_L$ forms a vector space in $\mathbfb R^{|\Lambda|}$.
\end{proof}
Our goal is to give a precise characterization of the learnable space $F_L$.
For example, we may want to know the dimension of $F_L$, which represents the learnable degrees of freedom for the noise.
This is also the maximum number of linearly-independent equations about the logarithmic Pauli fidelities we can expect to extract from experiments.
Conversely, the unlearnable degrees of freedom roughly correspond to the number of independent gauge transformations.
We summarize these definitions as follows.
\begin{definition}
Given a Clifford gate set $\mathfrak G$, the learnable degrees of freedom $\mathrm{LDF}(\mathfrak G)$ and unlearnable degrees of freedom $\mathrm{UDF}(\mathfrak G)$ are defined as, respectively,
\begin{equation}
\mathrm{LDF}(\mathfrak G) \mathrel{\mathop:}\nobreak\mkern-1.2mu= \mathrm{dim}(F_L),\quad
\mathrm{UDF}(\mathfrak G) \mathrel{\mathop:}\nobreak\mkern-1.2mu= |\Lambda| - \mathrm{dim}(F_L).
\end{equation}
\end{definition}
\noindent Our approach is to relate $F_L$ to certain properties of a graph defined as follows.
\begin{definition}[Pattern transfer graph]
The pattern transfer graph associated with a Clifford gate set $\mathfrak G$ is a directed graph $G=(V,E)$ constructed as follows:
\begin{itemize}
\item $V(G) = \{0,1\}^n$.
\item $E(G) = \{e_{a,\mathcal G} \mathrel{\mathop:}\nobreak\mkern-1.2mu= (\mathrm{pt}(P_a),~\mathrm{pt}(\mathcal G(P_a)) ~|~\forall~ P_a\in{\sf P}^n,~\mathcal G\in\mathfrak{G} \}$.
\end{itemize}
\end{definition}
The $2^n$ vertices each corresponds to a possible Pauli pattern.
The $|E| = |\Lambda| = |\mathfrak G|\cdot 4^n$ edges each corresponds to a Pauli operator and a Clifford gate, describing how the Clifford gate evolves the pattern of the Pauli operator. One can also think each edge corresponds to a unique Pauli fidelity ($e_{a,\mathcal G}\leftrightarrow \lambda_{a}^{\mathcal G}$).
The rationale for only tracking the Pauli pattern is that we assume the ability to implement noiseless single-qubit unitaries, which makes the actual single-qubit Pauli operators unimportant. Fig.~2 of main text shows the pattern transfer graphs for a CNOT gate, a SWAP gate, and a gate set of CNOT and SWAP, respectively.
Next, we give some definitions from graph theory (see~\cite{gleiss2003circuit,bollobas1998modern}). A \emph{chain} is an alternating sequences of vertices and edges $z=(v_0,e_1,v_1,e_2,v_2,...,v_{q-1},e_q,v_q)$ such that each edge satisfies $e_k=(v_{k-1},v_k)$ or $e_k=(v_{k},v_{k-1})$.
A chain is \emph{simple} if it does not contain the same edge twice.
A closed chain (\textit{i.e.}, $v_0=v_q$) is called a \emph{cycle}. If an edge $e_k$ in a chain satisfies $e_k = (v_{k-1},v_k)$, it is called an \emph{oriented edge}. A chain consists solely of oriented edges is called a \emph{path}. A closed path is called a \emph{oriented cycle} or a \emph{circuit}.
A graph is called \emph{strongly connected} if there is a path from every vertex to every other vertex. A graph is called \emph{weakly connected} if there is a chain from every vertex to every other vertex. The number of (strongly or weakly) \emph{connected components} is the minimum number of partitions of the vertex set $V=V_1\cup\cdots\cup V_c$ such that each subgraph generated by a vertex partition is (strongly or weakly) connected.
We can equip a graph with vector spaces.
Following the notations of~\cite[Sec. II.3]{bollobas1998modern},
the \emph{edge space} $C_1(G)$ of a directed graph $G$ is the vector space of all linear functions from the edges $E(G)$ to $\mathbfb R$.
By construction, $C_1(G)\cong \mathbfb R^{|\Lambda|} \cong F$.
Every linear function of the logarithmic Pauli fidelities naturally corresponds to a linear function of the edges according to the label of the edges ($l_{a}^{\mathcal G} \leftrightarrow e_{a,\mathcal G}$).
Again, we use vectors in $\mathbfb R^{|\Lambda|}$ to refer to elements of $C_1(G)$. The inner product on $C_1(G)$ is defined as the standard inner product on $\mathbfb R^{|\Lambda|}$.
There are two subspaces of $C_1(G)$ that is of special interest.
For a simple cycle
$z$ in $G$, we assign a vector $\bm v_z\in C_1(G)$ as follows
\begin{equation}
\bm v_z(e) = \left\{
\begin{aligned}
+1,\quad& e\in z,~\text{$e$ is oriented.}\\
-1,\quad& e\in z,~\text{$e$ is not oriented.}\\
0,\quad& e\notin z.
\end{aligned}
\right.
\end{equation}
The \emph{cycle space} $Z(G)$ is the linear subspace of $C_1(G)$ spanned by all cycles $\bm v_z$ in $G$.
Given a partition of vertices $V=V_1\cup V_2$ such that there is at least one edge between $V_1$ and $V_2$, a \emph{cut} is the set of all edges $e = (u,v)$ such that one of $u,v$ belongs to $V_1$ and the other belongs to $V_2$. For each cut $p$ we assign an vector $\bm v_p\in C_1(G)$ as follows
\begin{equation}\label{eq:cut}
\bm v_p(e) = \left\{
\begin{aligned}
+1,\quad& e\in p,~\text{$e$ goes from $V_1$ to $V_2$.}\\
-1,\quad& e\in p,~\text{$e$ goes from $V_2$ to $V_1$.}\\
0,\quad& e\notin p.
\end{aligned}
\right.
\end{equation}
The \emph{cut space} $U(G)$ is the linear subspace of $C_1(G)$ spanned by all cuts $\bm v_p$ in $G$.
Note that different partition of vertices may result in the same cut vector if $G$ is unconnected.
\begin{lemma}\cite[Sec. II.3, Theorem~1]{bollobas1998modern}\label{le:complement}
The edge space $C_1(G)$ is the orthogonal direct sum of the cycle space $Z(G)$ and the cut space $U(G)$, whose dimensions are given by
\begin{equation}
\mathrm{dim}(Z(G)) = |E|-|V|+c(G),\quad
\mathrm{dim}(U(G)) = |V|-c(G),
\end{equation}
where $c(G)$ is the number of weakly connected components of $G$.
\end{lemma}
In some cases, we are more interested in circuits (oriented cycles) instead of general cycles. The following lemma gives a sufficient condition when the cycle spaces have a circuit basis, \textit{i.e.} the cycle space is spanned by oriented cycles.
\begin{lemma}\cite[Theorem~7]{gleiss2003circuit}\label{le:circuit}
A directed graph has a circuit basis if it is strongly connected, or it is a union of strongly connected subgraphs.
\end{lemma}
With all the graph theoretical tools introduced above, we are ready to present the main result of this section.
\begin{theorem}\label{th:space}
Under the Assumptions 1-4.
For any $\mathfrak G$,
$F_L \cong Z(G)$.
Explicitly, a linear function $f_{\bm v}(\bm l) = \bm v\cdot\bm l$ is learnable if and only if $\bm v$ belongs to the cycle space $Z(G)$.
\end{theorem}
We give the proof at the end of this section.
The proof involves two parts.
The first is to show that every cycle is learnable using a variant of cycle benchmarking~\cite{erhard2019characterizing}, thus the cycle space belongs to the learnable space. The second part is to show that every cut induces a gauge transformation~\cite{nielsen2021gate}, and thus the learnable space must be orthogonal to the cut space, which implies it lies in the cycle space.
We remark that Theorem~\ref{th:nogo} can be viewed as a corollary of Theorem~\ref{th:space}.
This is because an individual Pauli fidelity $\lambda_a^{\mathcal G}$ whose Pauli pattern changes (\textit{i.e.}, $\mathrm{pt}(P_a)\ne\mathrm{pt}(\mathcal G(P_a))$) corresponds to an simple edge in the pattern transfer graph, which does not belong to the cycle space and is thus unlearnable. On the other hand, a Pauli fidelity without Pauli pattern change corresponds to a self-loop in the pattern transfer graph, which belongs to the cycle space by definition, and is thus learnable.
Combing Theorem~\ref{th:space} with Lemma~\ref{le:complement} leads to the following.
\begin{corollary}\label{co:udf}
The learnable and unlearnable degrees of freedom associated with $\mathfrak G$ are given by
\begin{equation}
\mathrm{LDF}(\mathfrak G) = |\mathfrak G|\cdot 4^n-2^n+c(\mathfrak G),\quad
\mathrm{UDF}(\mathfrak G) = 2^n-c(\mathfrak G),
\end{equation}
where $c(\mathfrak G)$ is the number of connected components of the pattern transfer graph associated with $\mathfrak G$.
\end{corollary}
Note that the unlearnable degrees of freedom always constitute an exponentially small portion, though they can grow exponentially.
Examples of some gate sets are given in Table~\ref{tab:UDF} and Figure~\ref{fig:more_gate_sets}. One can notice some interesting properties. The UDF of CNOT and SWAP equals to $2$ and $1$, respectively, but a gate set containing both has $\mathrm{UDF}=2$. This means UDF is not ``additive''. The interdependence between different gates can give us more learnable degrees of freedom.
However, Corollary~\ref{co:udf} implies that the UDF of a gate set cannot be smaller than the UDF of any of its subset.
This is because adding new gates can only decrease the number of connected components $c(\mathfrak G)$ of the pattern transfer graph.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Number of qubits $n$ & Gate set $\mathfrak G$ & Number of parameters $|\Lambda|=4^n|\mathfrak G|$ & $\mathrm{UDF}(\mathfrak G)$\\
\hline
2 & CNOT & 16 & 2\\
2 & SWAP & 16 & 1\\
2 & \{CNOT, SWAP\} & 32 & 2\\
3 & $\mathrm{\{CNOT_{12},CNOT_{23},CNOT_{31}\}}$ & 192 & 6\\
3 & $\mathrm{CIRC_3}$ & 64 & 4\\
\hline
\end{tabular}
\caption{The unlearnable degrees of freedom of some gate sets. Here $\mathrm{CIRC_3}$ is the circular permutation on $3$ qubits. UDF is calculated by applying Corollary~\ref{co:udf} to the corresponding pattern transfer graph in Fig.~2 of main text and Fig.~\ref{fig:more_gate_sets}.}
\label{tab:UDF}
\end{table}
\begin{figure}
\caption{Pattern transfer graphs for $\mathrm{\{CNOT,~SWAP\}
\label{fig:more_gate_sets}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{th:space}]
The proof is divided into showing $Z(G)\subseteq F_L$ and $F_L\subseteq Z(G)$ (up to the natural isometry between $F$ and $C_1(G)$).
$Z(G)\subseteq F_L$: Roughly, this is equivalent to saying that all cycles are learnable.
We will first show that the pattern transfer graph always has a circuit basis, and then show that the linear function associated with each circuit can be learned using a variant of cycle benchmarking protocol~\cite{erhard2019characterizing}.
We begin by showing that the pattern transfer graph $G$ associated with a gate set $\mathfrak G$ is a union of strongly connected subgraphs. This is equivalent to saying that for any vertices $u,v\in V(G)$, if there is a path from $u$ to $v$, there must be a path from $v$ to $u$.
It suffices to show that for each edge $e=(u,v)$ there is a path from $v$ to $u$, since any path is just concatenation of edges.
By definition, the existence of $e=(u,v)$ implies there exists $P\in {\sf P}^n$ and $\mathcal G\in\mathfrak G$ such that $\mathrm{pt}(P) = u$ and $\mathrm{pt}(Q) = v$ where $Q\mathrel{\mathop:}\nobreak\mkern-1.2mu= \mathcal G(P)$. Since a Clifford gate is a permutation on the Pauli group, there must exist some integer $d>0$ such that $\mathcal G^d = \mathcal I$, thus $P = \mathcal G^{d-1}(Q)$, which induces the following path from $v$ to $u$:
\begin{equation*}
(
\mathrm{pt}(Q),~e_{Q,\mathcal G},~\mathrm{pt}(\mathcal G(Q)),~e_{\mathcal G(Q),\mathcal G},~ \mathrm{pt}(\mathcal G^2(Q)),~\cdots,~ \mathrm{pt}(\mathcal G^{d-2}(Q)),~ e_{\mathcal G^{d-1}(Q),\mathcal G},~ \mathrm{pt}(\mathcal G^{d-1}(Q))
).
\end{equation*}
One can verify this is a path according to the definition of $G$. This shows that $G$ is indeed a union of strongly connected subgraphs. According to Lemma~\ref{le:circuit}, $G$ has a circuit basis that spans the cycle space $Z(G)$.
Now we show that every circuit in $G$ represents a learnable function.
Consider an arbitrary circuit $z = (v_0,e_1,v_1,e_2,v_2,...,v_{q-1},e_q,v_q\equiv v_0)$. For each $k=1...q$, the edge $e_k$ corresponds to a Pauli operator $P_k\in{\sf P}^n$ and a Clifford gate $\mathcal G_k \in \mathfrak G$ such that $\mathrm{pt}(P_k) = v_{k-1}$ and $\mathrm{pt}(Q_k) = v_{k}$ where $Q_k\mathrel{\mathop:}\nobreak\mkern-1.2mu= \mathcal G_k(P_k)$.
On the other hand, since $\mathrm{pt}(Q_k)=\mathrm{pt}(P_{k+1})$, there exists a product of single qubit unitaries $\mathcal U_k$ such that $P_{k+1} = \mathcal U_k(Q_k)$ for $k=1...q$ (where we define $P_{q+1}\mathrel{\mathop:}\nobreak\mkern-1.2mu= P_1$, as $\mathrm{pt}(Q_q)=\mathrm{pt}(P_1)$ by assumptions).
Consider the following gate sequence,
\begin{equation}
\mathcal C \mathrel{\mathop:}\nobreak\mkern-1.2mu= \mathcal U_q\mathcal G_q\mathcal U_{q-1}\mathcal G_{q-1}\cdots\mathcal U_1\mathcal G_1
\end{equation}
One can see that $\mathcal C(P_1) = P_1$. Now we design the following experiments parameterized by a positive integer $m$,
\begin{itemize}
\item Initial state: $\rho_0 = (I+P_1)/2^n$,
\item POVM measurement: $E_{\pm 1} = (I\pm P_1)/{2}$,
\item Circuit: $\mathcal C^m = \left(\mathcal U_q\mathcal G_q\mathcal U_{q-1}\mathcal G_{q-1}\cdots\mathcal U_1\mathcal G_1\right)^m$.
\end{itemize}
Consider the outcome distribution generated by running these experiments within a noise model $\mathcal N$.
\begin{equation}
\begin{aligned}
p^{(m)}_{\pm 1}(\mathcal N) &= \Tr\left( \widetilde{E}_{\pm 1} \widetilde{\mathcal C}^m (\widetilde{\rho}_0)\right) \\
&= \Tr \left( \frac{I\pm P_1}{2} \cdot\left( \mathcal E^M \circ \left(\mathcal U_q\widetilde{\mathcal G}_q
\cdots\mathcal U_1\widetilde{\mathcal G}_1\right)^m\circ\mathcal E^S \right)
\left(\frac{I+P_1}{2^n}\right) \right)\\
&= \Tr\left(\frac{I\pm P_1}{2} \cdot\frac{I+\lambda^M_{P_1}
\left(\lambda^{\mathcal G_q}_{P_q}\cdots\lambda^{\mathcal G_2}_{P_2}\lambda^{\mathcal G_1}_{P_1}\right)^m
\lambda^S_{P_1}P_1}{2^n} \right)\\
&= \frac{1\pm\lambda^M_{P_1}
\left(\lambda^{\mathcal G_q}_{P_q}\cdots\lambda^{\mathcal G_2}_{P_2}\lambda^{\mathcal G_1}_{P_1}\right)^m
\lambda^S_{P_1}}{2}.
\end{aligned}
\end{equation}
The expectation value is
\begin{equation}
\mathbfb E^{(m)}(\mathcal N) = \lambda^M_{P_1}
\left(\lambda^{\mathcal G_q}_{P_q}\cdots\lambda^{\mathcal G_2}_{P_2}\lambda^{\mathcal G_1}_{P_1}\right)^m
\lambda^S_{P_1}.
\end{equation}
If we take the ratio of expectation values of two experiments with consecutive $m$, we obtain (recall that all these Pauli fidelities are strictly positive by Assumption 4)
\begin{equation}
\mathbfb E^{m+1}(\mathcal N)/\mathbfb E^{m}(\mathcal N) = \lambda^{\mathcal G_q}_{P_q}\cdots\lambda^{\mathcal G_2}_{P_2}\lambda^{\mathcal G_1}_{P_1}.
\end{equation}
This implies that if two noise models have different values for the product of Pauli fidelities $\lambda^{\mathcal G_q}_{P_q}\cdots\lambda^{\mathcal G_2}_{P_2}\lambda^{\mathcal G_1}_{P_1}$, the above experiments would be able to distinguish between them. Therefore, $\lambda^{\mathcal G_q}_{P_q}\cdots\lambda^{\mathcal G_2}_{P_2}\lambda^{\mathcal G_1}_{P_1}$ is a learnable function.
By taking the logarithm of this expression, we see that $f(\bm l)\mathrel{\mathop:}\nobreak\mkern-1.2mu=\sum_{k=1}^q l_{P_q}^{\mathcal G_q}$ is a learnable linear function of the logarithmic Pauli fidelities.
Notice that $f(\bm l)$ exactly corresponds to the circuit of $z$ according to the natural isometry between $F$ and $C_1(G)$. This tells us that every circuit in $G$ indeed corresponds to a learnable linear function.
Combining with the fact that the circuits in $G$ span the cycle space $Z(G)$, and the fact that learnable functions are closed under linear combination (Lemma~\ref{le:learnable_is_space}), we conclude that $Z(G)\subseteq F_L$.
$F_L\subseteq Z(G)$:
For this part, we just need to show that $F_L$ is orthogonal to the cut space $U(G)$, which is the orthogonal complement of the cycle space $Z(G)$. To show this, we will construct a gauge transformation for each element of $U(G)$. The definition of learnability then requires a learnable linear function to be orthogonal to all gauge transformations, thus orthogonal to the entire cut space.
Consider a cut $V = V_1 \cup V_2$ (such that there is at least one edge between $V_1$ and $V_2$). We define the \emph{gauge transform map} $\mathcal M$ as the following Pauli diagonal map,
\begin{equation}
\mathcal M(P) \mathrel{\mathop:}\nobreak\mkern-1.2mu= \left\{
\begin{aligned}
\eta P, \quad& \text{if}~\mathrm{pt}(P)\in V_1,\\
P, \quad& \text{if}~\mathrm{pt}(P)\in V_2,
\end{aligned}
\right.\quad\forall P\in{\sf P}^n,
\end{equation}
for a positive parameter $\eta\ne 1$. The gauge transformation induced by $\mathcal M$ is defined in the same way as Eq.~\eqref{eq:gauge_trans}.
We will show that there exists two noise models satisfying all the assumptions that are related by a gauge transformation (thus indistinguishable) but yields different values for the function corresponding to the cut $V_1\cup V_2$.
Starting with a noise model $\mathcal N = \{\mathcal E^S,\mathcal E^M, \Lambda\}$, we first calculate the gauge transformed noise model $\mathcal N'$.
The SPAM noise channels are transformed as
\begin{equation}\label{eq:spam_update_2}
\mathcal E^{S'} = \mathcal M\mathcal E^S,\quad \mathcal E^{M'} = \mathcal E^M\mathcal M^{-1},
\end{equation}
which are still Pauli diagonal maps.
Using exactly the same argument as in the proof of Theorem~\ref{th:nogo}, by choosing $\eta$ to be sufficiently close to $1$, these transformed maps are guaranteed to be CPTP and satisfy Assumption~4.
Secondly, the single-qubit unitaries are transformed as $\mathcal U' = \mathcal M\mathcal U\mathcal M^{-1}$. Calculate the following inner product for any $P,Q\in{\sf P}^n$,
\begin{equation}\label{eq:gauge_trans_2}
\begin{aligned}
\Tr(P\cdot\mathcal U'(Q))&=\Tr(\mathcal M^\dagger(P)\cdot \mathcal U(\mathcal M^{-1}(Q)) )\\
&= \eta^{\bm 1_{V_1}[\mathrm{pt}(P)]}(\eta^{-1})^{\bm 1_{V_1}[\mathrm{pt}(Q)]}\Tr(P\cdot\mathcal U(Q)).
\end{aligned}
\end{equation}
Here $\bm 1_{V_1}$ is the indicator function of $V_1$.
We see that $\Tr(P\cdot\mathcal U'(Q))=\Tr(P\cdot\mathcal U(Q))$ if $\mathrm{pt}(P)=\mathrm{pt}(Q)$.
A crucial observation is that a product of single-qubit unitaries can never change the pattern of the input Pauli. More precisely, $\mathcal U(Q)$ is a linear combination of Pauli operators with the same pattern as $Q$. Therefore, if $\mathrm{pt}(P)\ne\mathrm{pt}(Q)$, we would have $\Tr(P\cdot\mathcal U'(Q))=\Tr(P\cdot\mathcal U(Q))=0$.
Combining the two cases, we conclude $\mathcal U'=\mathcal U$, \textit{i.e.}, the single-qubit unitaries are still noiseless in $\mathcal N'$.
Finally, the noisy Clifford gates are transformed as
\begin{equation}
\begin{aligned}
\widetilde{\mathcal G}'&= \mathcal M{\mathcal G}\Lambda_{\mathcal G}\mathcal M^{-1}\\
&= \mathcal G\mathcal G^{-1}\mathcal M{\mathcal G}\Lambda_{\mathcal G}\mathcal M^{-1}\\
&\mkern-1.2mu=\nobreak\mathrel{\mathop:} \mathcal G\Lambda_{\mathcal G}'
\end{aligned}
\end{equation}
where the transformed noise channel $\Lambda_{\mathcal G}'\mathrel{\mathop:}\nobreak\mkern-1.2mu= \mathcal G^{-1}\mathcal M{\mathcal G}\Lambda_{\mathcal G}\mathcal M^{-1}$ is a Pauli diagonal map. We now calculate its Pauli eigenvalues. For $P\in{\sf P}^n$,
\begin{equation}\label{eq:f_update_2}
\begin{aligned}
\Lambda_{\mathcal G}'(P) &= \mathcal G^{-1}\mathcal M{\mathcal G}\Lambda_{\mathcal G}\mathcal M^{-1}(P)\\
&=\eta^{\bm 1_{V_1}[\mathrm{pt}(\mathcal G(P))]}(\eta^{-1})^{\bm 1_{V_1}[\mathrm{pt}(P)]}\lambda^{\mathcal G}_P~P\\
&=\left\{
\begin{aligned}
\eta \lambda_P^{\mathcal G},\quad& \mathrm{pt}(P)\in V_1,~\mathrm{pt}(\mathcal G(P))\in V_2.\\
\eta^{-1} \lambda_P^{\mathcal G},\quad& \mathrm{pt}(P)\in V_2,~\mathrm{pt}(\mathcal G(P))\in V_1.\,\\
\lambda_P^{\mathcal G},\quad& \text{otherwise}.\\
\end{aligned}
\right.
\end{aligned}
\end{equation}
Again, Assumption 4 guarantees that $\Lambda_{\mathcal G}'$ is a CPTP map satisfying all of our noise assumptions as long as $\eta$ is sufficiently close to $1$. We omit the argument here as it is the same as in the previous proof.
Define $t_p \mathrel{\mathop:}\nobreak\mkern-1.2mu= \log \eta$ where $p$ denotes the cut $V_1\cup V_2$. The above gauge transformation of the log Pauli fidelity can be written as
\begin{equation}
\bm l' = \bm l + t_p \bm v_p
\end{equation}
where $\bm v_p$ is the cut vector of $V = V_1\cup V_2$ as defined in Eq.~\eqref{eq:cut}.
We have just defined a gauge transformation $\mathcal M_p$ for an arbitrary cut $p$.
Fix a basis of the cut space $B$ (where vectors in $B$ has the form in Eq.~\eqref{eq:cut}). For a generic element of the cut space $\bm v\in U(G)$, we can decompose it as $\bm v = \sum_{p\in B} t_p \bm v_p$ ($t_p\in\mathbb{R}$). We define the gauge transformation $\mathcal M_{\bm v}$ associated with $\bm v$ as a consecutive application of the gauge transformations $\{\mathcal M_p\}$ for each $p\in B$, each with parameter $t_p$. Here we assume that each $|t_p|$ is sufficiently small, as otherwise we can rescale the vector. This implies that $\mathcal M_{\bm v}$ is a valid gauge transformation.
The effect of such a transformation is
\begin{equation}
\bm l' = \bm l + \bm v.
\end{equation}
Now, Definition~\ref{de:learnability} implies that a learnable function $\bm f$ must remain unchanged under gauge transformations (as they result in indistinguishable noise models), which means that $\bm f\cdot \bm l' = \bm f\cdot \bm l$. Thus, for all $\bm f\in F_L$, and all $\bm v \in U(G)$, we must have
\begin{equation}
\bm f\cdot \bm v = \bm f\cdot \bm l' - \bm f\cdot \bm l = 0.
\end{equation}
That is, $F_L$ must be orthogonal to the cut space $U(G)$. According to Lemma~\ref{le:complement}, $Z(G)$ is the orthogonal complement of $U(G)$, so we conclude that $F_L\subseteq Z(G)$. This completes the second part of our proof.
\end{proof}
\subsection{Learnability under no-crosstalk assumption}\label{sec:no_crosstalk}
As we commented before, the way we define the gate noise captures a general form of crosstalk~\cite{sarovar2020detecting}. One may ask, if we further make a favorable assumption that gate noise has no crosstalk, would this make the learning of noise much easier.
To consider this rigorously, we introduce the following optional assumption. See Fig.~\ref{fig:crosstalk} for an illustration.
\begin{itemize}
\item \textbf{Assumption 5} (No crosstalk.) For any $\mathcal G\in\mathfrak G$ that acts non-trivially only on a $k$-qubit subspace, the associated Pauli noise channel also acts non-trivially only on that subspace. In other words, if $\mathcal G = \mathcal G'\otimes \mathcal I$, we have $\widetilde{\mathcal G} = \left( \mathcal G'\circ\Lambda_{\mathcal G} \right)\otimes \mathcal I$ where $\Lambda_{\mathcal G}$ is an $k$-qubit Pauli channel
depending only on $\mathcal G$ and the (ordered) subset of qubits on which $\mathcal G$ acts.
\end{itemize}
\begin{figure}
\caption{Illustration of the crosstalk model. (a) A $4$-qubit circuit consists of three ideal CNOT gates. (b) Full crosstalk. The noise channels are $4$-qubit and depends on the qubits the CNOT acts on. (c) No crosstalk. The noise channel only acts on a $2$-qubit subspace. It can still depend on the qubits the CNOT acts on.}
\label{fig:crosstalk}
\end{figure}
Assumption 5 reduces the number of independent parameters of a noise model.
One might expect certain unlearnable functions to become learnable with this assumption.
Here, we show that the simple criteria of learnablity given in Theorem~\ref{th:nogo} still hold even in this case, as stated in the following proposition.
\begin{proposition}\label{prop:nogo_nocross}
With Assumption 1-5, for any $k$-qubit Clifford gate $\mathcal G$ and Pauli operator $P_a$, the Pauli fidelity $\lambda_a^{\mathcal G}$ is unlearnable if and only if $\mathcal G$ changes the pattern of $P_a$, \textit{i.e.}, $\mathrm{pt}(\mathcal G(P_a))\ne \mathrm{pt}(P_a)$.
\end{proposition}
\begin{proof}
We just need to modify the proof of Theorem~\ref{th:nogo}.
For the ``only if'' part, restrict our attention to the $k$-qubit subsystem that $\mathcal G$ acts on, and do a cycle benchmarking protocol as in the original proof. We can easily conclude that $\lambda_a^{\mathcal G}$ is learnable if $\mathrm{pt}(P_a) = \mathrm{pt}(\mathcal G(P_a))$.
For the ``if'' part, construct the same gauge transformation map as in the original proof.
That is, for an index $i\in[n]$ such that $\mathrm{pt}(P_a)_i\ne \mathrm{pt}(\mathcal G(P_a))_i$, let $\mathcal M = \mathcal D_i\otimes\mathcal I_{[n]\backslash i}$ where $D_i$ is the single-qubit deplorizing channel on the $i$th qubit with some parameter $\eta$.
With the no-crosstalk assumption, a generic $k$-qubit noisy Clifford gate $\widetilde{\mathcal T}$ transforms as
\begin{equation}
\widetilde{\mathcal T}\otimes\mathcal I \mapsto \mathcal M\circ (\widetilde{\mathcal T}\otimes\mathcal I)\circ \mathcal M^{-1}.
\end{equation}
If $\mathcal T$ does not act on the $i$th qubit, $\mathcal M$ commutes with $\widetilde{\mathcal T}$ and the noisy Clifford gate remains unchanged.
If $\mathcal T$ acts non-trivially on the $i$th qubit,
\begin{equation}
\widetilde{\mathcal T}\otimes\mathcal I \mapsto (\mathcal D_i\circ\widetilde{\mathcal T}\circ\mathcal D_i^{-1})\otimes\mathcal I.
\end{equation}
This means the transformed noise channel acts non-trivially only on the $k$-qubit subsystem that $\mathcal G$ acts on, thus satisfies the no-crosstalk assumption.
The Pauli fidelities of the noise channel will be updated as Eq.~\eqref{eq:paulifidelityupdate}.
Following the same argument of the original proof, we conclude that $\lambda_{a}^{\mathcal G}$ is unlearnable if $\mathrm{pt}(P_a) \neq \mathrm{pt}(\mathcal G(P_a))$.
\end{proof}
It is also possible to generalize the graph theoretical characterization in Theorem~\ref{th:space} to the no-crosstalk case.
One challenge in this case is that, different edges in the pattern transfer graph no longer stand for independent variables.
For example, consider a $3$-qubit system and a CNOT on the first two qubits.
Since $\mathrm{CNOT}(XI) = XX$, we would have the following two edges in the pattern transfer graph
$$e_{XII,\mathrm{CNOT}\otimes\mathcal I} = (100,110),\quad e_{XIX,\mathrm{CNOT}\otimes\mathcal I} = (101,111).$$
However, with the no-crosstalk assumption, we have
\begin{equation}
\lambda_{XII}^{\mathrm{CNOT}\otimes \mathcal I} =
\lambda_{XIX}^{\mathrm{CNOT}\otimes \mathcal I} =
\lambda_{XI}^{\mathrm{CNOT}},
\end{equation}
which means the above two edges represent the same Pauli fidelity.
As a result, a gauge transformation (as defined in the proof of Theorem~\ref{th:space}) that changes $\lambda_{XII}$ and $\lambda_{XIX}$ differently is no longer a valid transformation.
In other word, a cut represents a valid gauge transformation only if it cuts through all the edges for the same Pauli fidelity simultaneously.
This could decrease the number of unlearnable degrees of freedom.
We leave the precise characterization of the learnable space with no-crosstalk assumptions as an open question.
It is also interesting to study the learnability under other practical assumptions about the Pauli noise model, such as the sparse Pauli-Lindbladian model~\cite{berg2022probabilistic} and the Markovian graph model~\cite{flammia2020efficient,harper2020efficient}.
\subsection{Learnability of Pauli error rates}
We have been focusing on the learnability of Pauli fidelities $\bm\lambda$. One may ask similar questions about Pauli error rates $\bm p$.
It turns out that, at least in the weak-noise regime (\textit{i.e.}, $\lambda_a$ close to $1$), the learnability of $\bm p$ is $\bm \lambda$ are highly related. To see this, note that
\begin{equation}
\begin{aligned}
p_a &= \frac{1}{4^n}\sum_{b}(-1)^\expval{a,b}\lambda_b\\
&\approx \frac{1}{4^n}\sum_{b}(-1)^\expval{a,b}(\log\lambda_b + 1)\\
&=\frac{1}{4^n}\sum_{b}(-1)^\expval{a,b} l_b + \delta_{a,\bm 0},
\end{aligned}
\end{equation}
which means that $p_a$ is approximately a linear function of the logarithmic Pauli fidelity $\bm l$.
Therefore, one can in principle use Theorem~\ref{th:space} to completely decide the learnability of any Pauli error rates (with weak-noise approximation).
Furthermore, since the Walsh-Hadamard transformation is invertible, different $p_a$ corresponds to linearly-independent function of $\bm l$.
This means that the number of linearly independent equations we can obtain about the Pauli error rates is the same as the learnable degrees of freedom of the Pauli fidelities.
In Table~\ref{tab:CNOT_full}, we list a basis for all the learnable Pauli fidelities/Pauli error rates. One can see that there is an exact correspondence between these two. We leave a fully general argument for future study.
\begin{table}[!htp]
\centering
\begin{tabular}{|c|c|}
\hline
Learnable log Pauli fidelities & $l_{II},l_{ZI},l_{IX},l_{ZX},
l_{XZ},l_{YY},l_{XY},l_{YZ},$
\\&$l_{IZ}+l_{ZZ},l_{IY}+l_{ZY},l_{IZ}+l_{ZY},l_{XI}+l_{XX},l_{YI}+l_{YX},l_{XI}+l_{YX}$ \\
\hline
Learnable Pauli error rates & $p_{II},p_{ZI},p_{IX},p_{ZX},
p_{XZ},p_{YY},p_{XY},p_{YZ},$
\\ (approximately)&$p_{IZ}+p_{ZZ},p_{IY}+p_{ZY},p_{IZ}+p_{ZY},p_{XI}+p_{XX},p_{YI}+p_{YX},p_{XI}+p_{YX}$ \\
\hline
\end{tabular}
\caption{A complete basis for the learnable linear functions of log Pauli fidelities and Pauli error rates (the latter is approximate) for a single CNOT gate.}
\label{tab:CNOT_full}
\end{table}
\section{Additional details about the numerical simulations}\label{sec:numerics}
In this section, we provide more details about the numerical simulations mentioned in the main text. The simulation is conducted using \texttt{qiskit}~\cite{Qiskit}, an open-source Python package for quantum computing. We simulate a two-qubit system where single-qubit Clifford gates are noiseless, and CNOT is subject to amplitude damping channels on both qubits. Note that amplitude damping is not Pauli noise, but we apply randomized compiling and will only estimate its Pauli diagonal part. We also note that, \texttt{qiskit} adds the noise channel \emph{after} gate by default, but our theory assume the noise to be \emph{before} gate. These two models can be easily converted between each other via
\begin{equation}
\mathcal G\circ\Lambda_{\mathcal G} = (\mathcal G\circ\Lambda_{\mathcal G}\circ\mathcal G^{\dagger})\circ\mathcal G = \Lambda_{\mathcal G}'\circ\mathcal G.
\end{equation}
If $\mathcal G$ is Clifford, $\Lambda_{\mathcal G}$ is a Pauli channel if and only if $\Lambda_{\mathcal G}'$ is a Pauli channel. In the following, we will be consistent with our theory and assume the noise to be before gate.
Besides, we let the measurement to have $0.3\%$ bit-flip rate on each qubit and the state-preparation to be noiseless.
Fig.~\ref{fig:main_sim_cbraw} shows the estimates collected using standard CB and interleaved CB (circuits shown in Fig.~1 of main text). Compared to the true values, we see that both simulations yields accurate predictions of the learnable Pauli fidelities.
\begin{figure}
\caption{Numerical estimates of Pauli fidelities of a CNOT gate via standard CB (left) and CB with interleaved gates (right), using circuits shown in Fig.~1 of main text. Each Pauli fidelity is fitted using seven different circuit depths $L=[2,2^2,...,2^7]$. For each depth $C=30$ random circuits and $200$ shots of measurements are used.
The red cross shows the true fidelities and the red dash line shows the average of true fidelities within any two-Pauli group.
}
\label{fig:main_sim_cbraw}
\end{figure}
Fig.~\ref{fig:main_sim_cbfeasible} (a) calculates the physically feasible region according to the estimates in terms of $\{\lambda_{XX},\lambda_{ZZ}\}$, using approaches discussed in the main text.
Due to the special structure of the twirled amplitude damping noise (no $Z$-error), the feasible region for $\lambda_{XX}$ is extremely narrow. To eliminate the effect of statistical error, we allow a smoothing parameter $\varepsilon$ in calculating the physical region, making the constraints to be $p_a\ge-\varepsilon$. Here $\varepsilon$ is chosen to be the largest standard deviation in estimating the learnable Pauli fidelities. In Fig.~\ref{fig:main_sim_cbfeasible} (b)(c) we see that the true fidelity indeed falls into the physical region and is actually close to the lower-left corner of the physical region.
\begin{figure}
\caption{Feasible region of the learned Pauli noise model, using data from Fig.~\ref{fig:main_sim_cbraw}
\label{fig:main_sim_cbfeasible}
\end{figure}
Fig.~\ref{fig:app_sim_intercept} shows the simulation results of intercept CB. We see that, we obtain an accurate estimate even for the unlearnable Pauli fidelities. Besides, the estimate lies inside the physically feasible region up to a standard deviation.
This shows that intercept CB should work well in resolving the unlearnability if we do have access to noiseless state-preparation (and the method is robust against measurement noise). Therefore, failure of this method in experiment implies a non-negligible state-preparation error, as discussed in the main text.
\begin{figure}
\caption{The learned Pauli noise model using intercept CB. The feasible region (blue bars) are taken from Fig.~\ref{fig:main_sim_cbfeasible}
\label{fig:app_sim_intercept}
\end{figure}
\section{Justification for the claim in Sec.~\ref{sec:space}}\label{sec:justification}
We claim in Sec.~\ref{sec:space} that any measurement probability generated in experiment can be expressed as a polynomial of Pauli fidelities, and that each term in the polynomial can be learned in a CB experiment. This is the motivation why we only care for a single monomial of Pauli fidelities. Here we justify this claim.
Consider the most general experimental design: prepare some initial state $\rho_0$, apply some quantum circuit $\mathcal C$, and conduct a POVM measurement $\{E_j\}_j$. Denote the noisy realization of these objects with a tilde. Because of noise, the probability of obtaining a certain measurement outcome $j$ is
\begin{equation}
\mathrm{Pr}(j) = \Tr\left( \widetilde{E}_j\widetilde{\mathcal C}(\widetilde{\rho}_0) \right) = \Tr\left( E_j\left(\Lambda^M\circ\widetilde{\mathcal C}\circ\Lambda^S\right)(\rho_0) \right) \equiv \Tr\left( E_j\rho' \right).
\end{equation}
Here $\Lambda^S,\Lambda^M$ are the noise channels for state preparation and measurement, respectively. The Pauli fidelity of them are denoted by $\lambda_a^S,\lambda_a^M$ for Pauli operator $a$, respectively. We define $\rho'\mathrel{\mathop:}\nobreak\mkern-1.2mu= (\Lambda^M\circ\widetilde{\mathcal C}\circ\Lambda^S)(\rho_0)$ which encodes all the information that can be extracted from a quantum measurements. We will obtain a general formula for $\rho'$.
First note that a general noisy quantum circuit $\widetilde{\mathcal C}$ satisfying our assumptions can be expressed as
\begin{equation}
\widetilde{\mathcal C} = C\od m \circ \widetilde{\mathcal G}_{{m}} \circ \cdots \circ C\od 1 \circ \widetilde{\mathcal G}_{{1}} \circ C\od 0,
\end{equation}
where ${\mathcal G}_{{j}}\in{\mathfrak G}$ is an $n$-qubit Clifford gate and $C\od j$ is the tensor product of single-qubit gates. A crucial property for single-qubit gates is that they never change the Pauli pattern. More rigorously, one have that
\begin{equation}
C\od{j}(P_a) = \sum_{b\sim \mathrm{pt}(a)}c_{b,a}\od{j}P_b,\quad\forall P_a\in{\sf P}^n,
\end{equation}
where $c_{b,a}\od{j}\in\mathbfb R$, and the summation is over all $P_b$ that have the same Pauli pattern as $P_a$.
\noindent Now consider the action of $\widetilde{\mathcal C}$ on an arbitrary Pauli operator $P_a$.
\begin{equation}
\begin{aligned}
\widetilde{\mathcal C}(P_a) &= (C\od m \circ \widetilde{\mathcal G}_{{m}} \circ \cdots \circ C\od 1 \circ \widetilde{\mathcal G}_{{1}} \circ C\od 0)(P_a)\\
&= (C\od m \circ \widetilde{\mathcal G}_{{m}} \circ \cdots \circ C\od 1 \circ \widetilde{\mathcal G}_{{1}}) \left(\sum_{b_0\sim \mathrm{pt}(a)} c_{b_0,a}\od{0} P_{b_0} \right)\\
&= (C\od m \circ \widetilde{\mathcal G}_{{m}} \circ \cdots \circ C\od 1) \left(\sum_{b_0\sim \mathrm{pt}(a)} c_{b_0,a}\od 0\lambda_{b_0}^{\mathcal G_{1}} P_{\mathcal G_{1}(b_0)} \right)\\
&= (C\od m \circ \widetilde{\mathcal G}_{{m}} \circ \cdots \circ C\od 2) \left(\sum_{\substack{
b_0\sim \mathrm{pt}(a),\\
b_1\sim \mathrm{pt}(\mathcal G_{1}(b_0))
}} c_{b_1,\mathcal G_{1}(b_0)}\od{1}c_{b_0,a}\od{0} \lambda_{b_1}^{\mathcal G_{2}}\lambda_{b_0}^{\mathcal G_{1}} P_{\mathcal G_{2}(b_1)} \right)\\
&= \cdots\\
&= \sum_{\substack{
b_0\sim \mathrm{pt}(a),\\
b_1\sim \mathrm{pt}(\mathcal G_{1}(b_0)),\\
\dots\\
b_m\sim \mathrm{pt}(\mathcal G_{m}(b_{m-1}))
}}c_{b_m,\mathcal G_{m}(b_{m-1})}\od{m}\cdots c_{b_1,\mathcal G_{1}(b_0)}\od{1} c_{b_0,a}\od{0} \lambda_{b_{m-1}}^{\mathcal G_{m}}\cdots\lambda_{b_1}^{\mathcal G_{2}}\lambda_{b_0}^{\mathcal G_{1}} P_{b_m}.
\end{aligned}
\end{equation}
For any initial state $\rho_0$, we can decompose it via Pauli operators as
\begin{equation}
\rho_0 = \frac{1}{2^n} I + \sum_{a\ne\bm 0}\alpha_aP_a.
\end{equation}
Going through the state preparation noise, the quantum circuit, and the measurement noise, the state evolves to
\begin{equation}
\begin{aligned}\label{eq:rho'2new}
\rho' &= (\Lambda^{M}\circ\widetilde{\mathcal C}\circ \Lambda^{S})(\frac{1}{2^n}I + \sum_{a\ne\bm 0}\alpha_aP_a)\\
&= \frac{1}{2^n}I +\sum_{a\ne\bm 0}\alpha_a\sum_{\substack{
b_0\sim \mathrm{pt}(a),\\
b_1\sim \mathrm{pt}(\mathcal G_{1}(b_0)),\\
\dots\\
b_m\sim \mathrm{pt}(\mathcal G_{m}(b_{m-1}))
}}c_{b_m,\mathcal G_{m}(b_{m-1})}\od{m}\cdots c_{b_1,\mathcal G_{1}(b_0)}\od{1} c_{b_0,a}\od{0} ~\lambda_{\mathrm{pt}(b_m)}^M\lambda_{b_{m-1}}^{\mathcal G_{m}}\cdots\lambda_{b_1}^{\mathcal G_{2}}\lambda_{b_0}^{\mathcal G_{1}}\lambda_{\mathrm{pt}(a)}^S P_{b_m}\\
&\equiv\frac{1}{2^n}I +\sum_{a\ne\bm 0}\alpha_a\sum_{\substack{
b_0\sim \mathrm{pt}(a),\\
b_1\sim \mathrm{pt}(\mathcal G_{1}(b_0)),\\
\dots\\
b_m\sim \mathrm{pt}(\mathcal G_{m}(b_{m-1}))
}}c_{b_m,\mathcal G_{m}(b_{m-1})}\od{m}\cdots c_{b_1,\mathcal G_{1}(b_0)}\od{1} c_{b_0,a}\od{0} ~\Gamma_{\bm b,a} P_{b_m}.
\end{aligned}
\end{equation}
Here we define $\Gamma_{\bm b,a}=\lambda_{\mathrm{pt}(b_m)}^M\lambda_{b_{m-1}}^{\mathcal G_{m}}\cdots\lambda_{b_1}^{\mathcal G_{2}}\lambda_{b_0}^{\mathcal G_{1}}\lambda_{\mathrm{pt}(a)}^S$, which is a monomial of Pauli fidelities.
The measurement outcome probability $\mathrm{Pr}(j)$ is a linear combination of such $\Gamma_{\bm b,a}$ plus some constant.
Moreover, each $\Gamma_{\bm b,a}$ of the above form can also be learned from a simple experiment, by choosing the initial state to be a $+1$ eigenstate of $P_a$, measurement operator to be $P_{b_m}$, and $C^{(j)}$ to be the product of single-qubit Clifford gates satisfying $C^{(j)}(\mathcal G_j({b_{j-1}})) = {b_{j}}$ (which is possible because $\mathrm{pt}(b_j)=\mathrm{pt}(\mathcal G_j(b_{j-1}))$).
Therefore, to completely characterize a noise model, we only need to extract the products of Pauli fidelities in the form of $\Gamma_{\bm b,a}$. This justifies our earlier claim.
\end{appendix}
\end{document}
|
\begin{equation}gin{document}
\title[On $C^{2}$ solution of the free-transport equation in a disk]{On $C^{2}$ solution of the free-transport equation in a disk}
\author[G. Ko]{Gyounghun Ko}
\address[GK]{Department of Mathematics, Pohang University of Science and Technology, South Korea}
\email{[email protected]}
\author[D. Lee]{Donghyun Lee}
\address[DL]{Department of Mathematics, Pohang University of Science and Technology, South Korea}
\email{[email protected]}
\begin{equation}gin{abstract}
The free transport operator of probability density function $f(t,x,v)$ is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condition. We obtain initial-boundary compatibility conditions for $C^{1}_{t,x,v}$ and $C^{2}_{t,x,v}$ regularity of the solution. We also provide regularity estimates. \\
\end{abstract}
\mathrm{d}ate{\today}
\keywords{}
\maketitle
\thispagestyle{empty}
\setcounter{tocdepth}{2}
\tableofcontents
\section{Introduction}
The free transport equation (or free transport operator) is one of the most important ones in a wide area of mathematics. When we consider a probability density function $f : \mathbb{R}_{+}\times \Omega \times\mathbb{R}^{d}\rightarrow \mathbb{R}_{+}$, the free transport equation is written by
\[
\partial_{t}f + v\cdotot\nablala_{x}f = 0.
\]
Above equation is very simple and has explicit solution $f(t,x,v) = f_0(x-vt, v)$ when initial data $f_0$ is smooth and spatial domain is $\mathbb{R}^{d}$ or $\mathbb{T}^{d}$. However, if we consider general boundary problems, it becomes very complicated. One of the most important and ideal boundary conditions in kinetic theory is the specular reflection boundary condition,
\begin{equation} \label{BC}
f(t,x,v) = f(t,x,R_{x}v), \quad R_{x} = I - 2n(x)\otimes n(x),\quad x\in \partial\Omega,
\end{equation}
where $n(x)$ is outward unit normal vector on the boundary $\partial\Omega$ when $\partial\Omega$ is smooth. \eqref{BC} is motivated by billiard model and we usually analyze the problem through characteristics:
\begin{equation}gin{equation} \label{XV heu}
\begin{equation}gin{split}
X(s;t,x,v) &:= \text{position of a particle at time $s$ which was at phase space $(t,x,v)$}, \\
V(s;t,x,v) &:= \text{velocity of a particle at time $s$ which was at phase space $(t,x,v)$}, \\
\end{split}
\end{equation}
where $X(s;t,x,v)$ and $V(s;t,x,v)$ satisfy the following Hamiltonian structure,
\begin{equation} \label{Ham}
\frac{d}{ds}X(s;t,x,v) = V(s;t,x,v),\quad \frac{d}{ds}V(s;t,x,v) = 0, \\
\end{equation}
under billiard-like reflection condition on the boundary. Explicit formulation of $(X(s;t,x,v), V(s;t,x,v))$ will be given right after Definition \e^{\frac 12}f{notation}.
Since $X(t;t,x,v) = x$, $V(t;t,x,v)=v$ by definition, we can easily guess the following solution,
\[
f(t,x,v) = f_{0}(X(0;t,x,v), V(0;t,x,v)),
\]
which is same as $f_0(x-vt, v)$ when $\Omega=\mathbb{R}^{d}$. However, unlike to whole space case, regularity of the solution $f(t,x,v)$ depends on the regularity of trajectory \eqref{XV heu}. More precisely, when $X(0;t,x,v)\in \partial\Omega$, differentiability of \eqref{XV heu} break down in general. This means that for any time $t>0$, there exist corresponding $(x_{*},v_{*}) \in \Omega\times \mathbb{R}^{d}$ such that $f(t, \cdotot, \cdotot)$ is not differentiable at the point. Or equivalently, for any $(x,v)\in \Omega\times \mathbb{R}^{d}$, there exists some corresponding time $t$ such that $f(\cdotot, x, v)$ is not differentiable at that time. \\
Now let us consider general kinetic model which has hyperbolic structure, such as hard sphere or general cut-off Boltzmann equations. (Of course, there are lots of other kinetic literature which consider various boundary condition problems.) Although the Boltzmann equation (or other general kinetic equations) is much more complicated than the free transport equation, the recent development of the Boltzmann (or kinetic) boundary problems shows the regularity issue of the problems very well. \\
\indent In $\mathbb{T}^{3}$ or $\mathbb{R}^{3}$, many results have been known using high order regularity function spaces. We refer to some classical works such as \cite{DV, StrainJAMS,GuoVPB,GuoVMB}. (We note that the apriori assumption of \cite{DV} also covers some boundary condition problems, including specular reflection \eqref{BC}.) More recently, in the case of non-cutoff Boltzmann equation (which has regularizing effect), it is known that the solution is $C^{
\infty}$ by \cite{CILS}.\\
\indent However, when it comes to general boundary condition problems, a way of getting sufficient high order regularity estimate is not known and low regularity approach has been widely used. By defining mild solution, low regularity $L^{\infty}$ solutions have been studied after \cite{Guo10}. In \cite{LY2004}, they studied the pointwise estimate for the Green function of the linearized Boltzmann equation in $\mathbb{R}$. They also obtained weighted $L^\infty$ decay of the Boltzmann equation using the Green function approach. In \cite{UY2006}, they constructed $L^2\cap L^\infty_{\begin{equation}ta}$ solution to the Boltzmann equation in the whole space using Duhamel's principle and the spectral theory. Recently, authors in \cite{LLWW2022} provide new analysis to derive $L^\infty$ estimate rather than using Green function and Duhamel's principle. In addition, there are lots of references \cite{CKLVPB,DHWY2017,DHWZ2019,DKL2020,DW2019,KimLee,KimLeeNonconvex,KLP2022,LY2017}, where the low regularity solutions were studied for the cut-off type Boltzmann equation. We also refer to recent results \cite{AMSY2020,DLSS2021,GHJO2020,KGH2020}, etc., which deal with low regularity solution of the kinetic equation whose collision operator has regularizing effect such as non-cutoff Boltzmann or Landau equation. In fact, however, there are only a few results known about regularity of the Boltzmann equation with boundary conditions. We refer to \cite{GKTT2016,GKTT2017,Kim2011,KimLee2021}. \\
\indent As briefly explained above, regularity issue of boundary condition problems is very fundamental problem. In fact, even without complicated collision type operators, regularity of the free transport equation with boundary conditions has not been studied thoroughly to the best of author's knowledge. \\
\subsection{Statements of main theorems}
In this paper, we study classical $C^{2}_{t,x,v}$ regularity of the free transport equation in a 2D disk,
\begin{equation} \label{eq}
\partial_{t}f + v\cdotot\nablala_{x}f = 0,\quad x\in \Omega:=\{ x\in\mathbb{R}^{2} \ : \ |x| < 1\},
\end{equation}
with the specular reflection boundary condition \eqref{BC}. Note that $n(x)=x \in \partial\Omega$, since we consider unit disk $\partial\Omega = \{x\in\mathbb{R}^{2} : |x|=1 \}$. Our aim is to find initial-boundary compatibility conditions of initial data $f_0$ for $C^{1}$ and $C^{2}$ regularity of the solution $f(t,x,v)$. We expect the solution will be mild solution $f(t,x,v)=f_0(X(0;t,x,v), V(0;t,x,v))$ surely (See Definition \e^{\frac 12}f{notation} for $X$ and $V$.) By performing derivative of $f$ in terms of $t,x,v$ directly (up to second order), we will find some conditions of $f_0$ which contains first and second derivative in both $x$ and $v$. (See \eqref{C1 cond}, \eqref{C2 cond34}, \eqref{C2 cond 1}, and \eqref{C2 cond 2}.) \\
\hide
\begin{equation}
f(t,x,v) = f(t,x,R_{x}v), \quad R_{x} = I - 2n(x)\otimes n(x),\quad x\in \partial\Omega
\end{equation}
where $n(x)$ is outward unit normal vector
\unhide
In general, for smooth bounded domain $\Omega$, we define
\begin{equation}gin{equation*}
\Omega = \{ x\in \mathbb{R}^{2} : \xi(x) < 0 \},\quad \partial\Omega = \{ x\in \mathbb{R}^{2} : \xi(x) = 0 \}.
\end{equation*}
In the case of unit disk, we may choose
\begin{equation}gin{equation*}
\xi(x) = \frac{1}{2}|x|^{2} - \frac{1}{2},
\end{equation*}
and hence
\begin{equation}gin{equation*}
\nablala\xi(x) = x,\quad
\nablala^{2}\xi(x) = I. \\
\end{equation*}
Now let us define some notation to precisely describe characteristics $X(s;t,x,v)$ and $V(s;t,x,v)$.
\begin{equation}gin{definition} \label{notation}
Considering \eqref{Ham}, we define basic notations
\begin{equation}gin{equation} \notag
\begin{equation}gin{split}
t_{\mathbf{b}}(x,v) &:= \sup \big\{ s \geq 0 : x - sv \in \Omegamega \big\} , \\
x_{\mathbf{b}}(x,v) &:= x - t_{\mathbf{b}}(x,v)v = X ( t- t_{\mathbf{b}}(t,x,v);t,x,v) \ \text{1st bouncing point backward in time}, \\
v_{\mathbf{b}}(x,v) &:= v = \lim_{s\rightarrowt_{\mathbf{b}}(t,x,v)}V ( t- s;t,x,v), \\
t^{k}(t,x,v) & := t^{k-1} - t_{\mathbf{b}} (x^{k-1}, v^{k-1}), \ \text{k-th bouncing time backward in time}, \ t^{1}(t,x,v) := t-t_{\mathbf{b}}(x,v),\\
x^{k}(x,v) &:= x^{k-1} - t_{\mathbf{b}}(x^{k-1}, v^{k-1}) v^{k-1} = X(t^{k}; t^{k-1}, x^{k-1}, v^{k-1}) \\
&= \text{k-th bouncing point backward in time},\quad x_{\mathbf{b}} := x^{1}, \\
v^{k} &=
R_{x^{k}} v^{k-1} = R_{x^{k}} \lim_{s\rightarrow t^{k}-} V(s; t^{k-1}, x^{k-1}, v^{k-1}), \\
\end{split}
\end{equation}
where $R_{x^{k}}$ is defined in \eqref{BC}. We set $(t^0,x^0,v^0)=(t,x,v)$ and define the specular characteristics as
\begin{equation}gin{equation}\label{XV}
\begin{equation}gin{split}
X(s;t,x,v) &= \sum_{k} \mathbf{1}_{s \in ( t^{k+1}, t^{k}]}
X(s;t^{k}, x^{k}, v^{k}), \\
V(s;t,x,v) &= \sum_{k} \mathbf{1}_{s \in ( t^{k+1}, t^{k}]}
V(s;t^{k}, x^{k}, v^{k}).
\end{split}
\end{equation}
\end{definition}
We also use $\gamma_{\partialm}$ and $\gamma_{0}$ notation to denote
\begin{equation}gin{equation} \notag
\begin{equation}gin{split}
\gamma_{+} &:= \{ (x,v)\in \partial\Omega\times \mathbb{R}^{2} : v\cdotot n(x) > 0 \}, \\
\gamma_{0} &:= \{ (x,v)\in \partial\Omega\times \mathbb{R}^{2} : v\cdotot n(x) = 0 \}, \\
\gamma_{-} &:= \{ (x,v)\in \partial\Omega\times \mathbb{R}^{2} : v\cdotot n(x) < 0 \}. \\
\end{split}
\end{equation}
Note that unit disk $\Omega$ is uniformly convex and its linear trajectory \eqref{XV} is well-defined if $x\in\Omega$ (see velocity lemma \cite{Guo10} for example). However, we want to investigate regularity up to boundary $\overline{\Omega}$, so we carefully exclude $\gamma_0$ from $\overline{\Omega}\times \mathbb{R}^{2}$ since we do not define characteristics starting from (backward in time) $\gamma_0$. Hence, using \eqref{XV} and \eqref{Ham}, it is natural to write \eqref{eq} as the following mild formulation,
\begin{equation} \label{solution}
f(t,x,v) = f_{0}(X(0;t,x,v), V(0;t,x,v)),\quad (x,v) \in \mathcal{I} := \{\overline{\Omega}\times \mathbb{R}^{2} \}\backslash \gamma_{0}. \\
\end{equation}
\\
\indent Meanwhile, to study regularity of \eqref{solution}, the following quantity is very important,
\begin{equation}gin{equation} \label{def A}
\begin{equation}gin{split}
A_{v, y}
&:= \left[\left((v\cdotot n(y))I+(n(y) \otimes v) \right)\left(I-\frac{v\otimes n(y)}{v\cdotot n(y)}\right)\right] ,\quad (y,v)\in \{\partial\Omega\times \mathbb{R}^2\} \backslash \gamma_0. \\
\end{split}
\end{equation}
Notice that $A_{v,y}$ is a matrix-valued function $A_{v,y}: \{\mathbb{R}^d\times\partialartial \Omega\}\backslash \gamma_0 \rightarrow \mathbb{R}^d\times \mathbb{R}^d $. ($d=2$ in this paper in particular) In fact, $A_{v, y}$ can be written as
\begin{equation}gin{equation*}
\begin{equation}gin{split}
A_{v, y}
&= \nablala_{y}\big( (v\cdotot n(y))n(y)\big) =\left( (v \cdotot n(y) ) \nablala_y n(y) + (n(y)\otimes v ) \nablala_y n(y)\right),\\
\end{split}
\end{equation*}
which is identical to \eqref{def A} by \eqref{normal}. \\
Throughout this paper, we denote the $v$-derivative of the $i$-th column of the matrix $A_{v,y}$ by $\nablala_v A^i_{v,y}$, where $A^i$ be the $i$-th column of a matrix $A$ for $1\leq i \leq d$. For fixed $i$, it means that
\begin{equation}gin{equation*}
\nablala_{v} A^i_{v,x^1} =\left. \left(\nablala_v A^i_{v,y}\right)\right|_{y=x^1}.
\end{equation*}
It is important to note that we carefully distinguish between $\nablala_v A^i_{v,x^1}$ and $\nablala_v (A^i_{v,x^1(x,v)})$. \\
\hide
\begin{equation}gin{proposition}
[Faa di Bruno formula] For higher order $n$-derivatives, the following formula would be useful.
\[
(f\circ H)^{(n)} = \sum_{\sum_{j=1}^{n}j m_{j}=n} \frac{n!}{m_{1}!\cdotots m_{n}!} \big( f^{(m_{1}+\cdotots+m_{n})}\circ H \big) \partialrod_{j=1}^{n} \begin{equation}ig( \frac{H^{(j)}}{j!} \begin{equation}ig)^{m_{j}}
\]
\end{proposition}
\unhide
\begin{equation}gin{remark}
Assume $f_{0}$ satisfies \eqref{BC}. If $f_{0} \in C^{1} _{x,v}( \overline{\Omega}\times \mathbb{R}^{2})$, then
\begin{equation} \label{C1_v trivial}
\nablala_{v}f_{0}(x,v) = \nablala_{v}f_{0}(x,R_{x}v)R_{x},\quad \forall x\in\partial\Omega,\quad \forall v\in\mathbb{R}^{2},
\end{equation}
also hold. Similalry, if $f_{0} \in C^{2}_{x,v}( \overline{\Omega}\times \mathbb{R}^{2})$, then
\begin{equation} \label{C2_v trivial}
\nablala_{vv}f_{0}(x,v) = R_{x}\nablala_{vv}f_{0}(x,R_{x}v)R_{x},\quad \forall x\in\partial\Omega,\quad \forall v\in\mathbb{R}^{2},
\end{equation}
also holds as well as \eqref{C1_v trivial}. \\
\end{remark}
\begin{equation}gin{theorem} [$C^{1}$ regularity] \label{thm 1}
Let $f_{0}$ be $C^{1}_{x,v}( \overline{\Omega}\times\mathbb{R}^{2})$ which satisfies \eqref{BC}. If initial data $f_{0}$ satisfies
\begin{equation} \label{C1 cond}
\begin{equation}ig[ \nablala_x f_0( x,v) + \nablala_v f_0(x, v) \frac{ (Qv)\otimes (Qv) }{v\cdotot n} \begin{equation}ig] R_{x}
=
\nablala_x f_0(x, R_{x}v)
+ \nablala_v f_0(x, R_{x}v) \frac{ (QR_{x}v)\otimes (QR_{x}v) }{R_{x}v\cdotot n},\quad (x,v)\in \gamma_{-},
\end{equation}
then $f(t,x,v)$ defined in \eqref{solution} is a unique $C^{1}_{t,x,v}(\mathbb{R}_{+}\times \mathcal{I})$ solution of \eqref{eq}. We also note that if \eqref{C1 cond} holds, then it also holds for $(x,v)\in \gamma_{+}$. Here, $Q$ is counterclockwise rotation by $\frac{\partiali}{2}$ in $\mathbb{R}^{2}$. Moreover, if the initial condition \eqref{C1 cond} does not hold, then $f(t,x,v)$ is not of class $C^1_{t,x,v}$ at time $t$ such that $t^k(t,x,v)=0$ for some $k$.
\begin{equation}gin{remark}[Example of initial data satisfying \eqref{C1 cond}] \label{example}
In \eqref{C1 cond}, we consider the following special case
\begin{equation}gin{equation} \label{specialcase}
\nablala_x f_0(x,v)R_x = \nablala_x f_0(x,R_xv) \quad \textrm{and} \quad \nablala_vf_0(x,v)\frac{(Qv)\otimes (Qv)}{v\cdotot n}R_x = \nablala_v f_0(x,R_xv) \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n},
\end{equation}
for $(x,v)\in \gamma_-$. Since $Q^TR_xQ=-R_x$ and $v\cdotot n=-R_xv\cdotot n$, we derive
\begin{equation}gin{equation*}
\nablala_v f_0(x,v) \cdotot (Qv) = \nablala_v f_0(x,R_xv)\cdotot (QR_xv),
\end{equation*}
from the second condition above. Here, $A^T$ means transpose of a matrix $A$. From \eqref{C1_v trivial}, we get
\begin{equation}gin{equation*}
\nablala_v f_0(x,R_xv)\cdotot (R_xQv) = \nablala_v f_0(x,R_xv) \cdotot (QR_xv),
\end{equation*}
which implies that $\nablala_vf_0(x,v)\cdotot (Qv) = \nablala_v f_0(x,R_xv)\cdotot (R_xQv) =0$ because $R_xQ= -QR_x$. It means that $\nablala_v f_0(x,v)$ is parallel to $v$. Then, $ f_0(x,v)$ is a radial function with respect to $v$. Since the second condition in \eqref{specialcase} also holds for $(x,v)\in \gamma_+$, we deduce that a direction of $\nablala_v f_0(x,v)$ is $v^T$ for $v\in \gamma_- \cup\gamma_+$. In other words,
\begin{equation}gin{equation*}
f_0(x,v)=G(x,\vert v \vert), \quad (x,v)\in \gamma_-\cup \gamma_+,
\end{equation*}
where $G$ is a real-valued $C^1_{x,v}$ function. Moreover, $f_0$ can be continuously extended to $\gamma_0$ to satisfy $f_0 \in C^1_{x,v}(\partialartial \Omega\times \mathbb{R}^2)$. From the first condition $\nablala_x f_0 (x,v)R_x = \nablala_x f_0(x,R_xv)$ in \eqref{specialcase}, we have
\begin{equation}gin{equation*}
\nablala_x G(x,\vert v \vert) R_x = \nablala_x G(x,\vert v \vert).
\end{equation*}
Thus, $\nablala_x G(x,\vert v \vert)$ is orthogonal to $n(x)=x$, which means that the directional derivative $\nablala_x f_0(x,v) \cdotot n(x)$ be 0 for $x\in \partialartial \Omega$. In conclusion, $f_0(x,v)=G(x,\vert v \vert)$ such that $\nablala_x f_0(x,v) \cdotot n(x)=0$ for all $(x,v)\in \partialartial \Omega\times \mathbb{R}^2$ whenever $f_0$ satisfies \eqref{specialcase} for $(x,v)\in \gamma_-$.
\end{remark}
\hide
\[
Q =
\begin{equation}gin{pmatrix}
\vert & \vert & \vert \\
\hat{x}\times \widehat{v\times x} & \hat{x} & \widehat{v\times x} \\
\vert & \vert & \vert
\end{pmatrix}
\begin{equation}gin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{equation}gin{pmatrix}
\vert & \vert & \vert \\
\hat{x}\times \widehat{v\times x} & \hat{x} & \widehat{v\times x} \\
\vert & \vert & \vert
\end{pmatrix}^{-1}.
\]
\unhide
\end{theorem}
\begin{equation}gin{theorem} [$C^{2}$ regularity] \label{thm 2}
Let $f_{0}$ be $C^{2}_{x,v}( \overline{\Omega}\times\mathbb{R}^{2})$ which satisfies \eqref{BC} and \eqref{C1 cond}. (The condition \eqref{C1 cond} was necessary to satisfy $f(t,x,v)\in C^1_{t,x,v}$ in Theorem \e^{\frac 12}f{thm 1}). If we assume
\begin{equation} \label{C2 cond34}
\nablala_{x}f_0(x, R_{x}v) \partialarallel (R_{x}v)^{T},\quad \nablala_{v}f_0(x, R_{x}v) \partialarallel (R_{x}v)^{T},
\end{equation}
and
\begin{equation}gin{eqnarray}
&&R_{x} \begin{equation}ig[ \nablala_{xv}f_{0}(x,v) + \nablala_{vv}f_{0}(x,v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \begin{equation}ig] R_{x}
= \nablala_{xv}f_{0}(x, R_xv) + \nablala_{vv}f_{0}(x, R_xv) \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n} \notag \\
&&\quad\hspace{7.5cm} +
R_{x}
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x , R_xv) \mathcal{J}_1
\\
\nablala_{v}f_{0}(x , R_xv) \mathcal{J}_2
\end{bmatrix}
R_{x},
\label{C2 cond 1} \\
&&R_{x}\begin{equation}ig[ \nablala_{xx}f_{0}(x,v) + \nablala_{vx}f_{0}(x, v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n} + \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \nablala_{xv}f_{0}(x, v) \begin{equation}ig] R_{x} \notag \\
&&\quad = \nablala_{xx}f_{0}(x, R_xv) + \nablala_{vx}f_{0}(x, R_xv)\frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n}
+ \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n} \nablala_{xv}f_{0}(x, R_xv) \notag \\
&&\quad \quad
-2R_x
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x, R_xv) \nablala_{v}A^{1}_{v,x}
\\
\nablala_{v}f_{0}(x, R_xv) \nablala_{v}A^{2}_{v,x}
\end{bmatrix} R_xA_{v,x}R_x
+
A_{v,x}\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x, R_xv) \mathcal{J}_1 \\
\nablala_{v}f_{0}(x, R_xv) \mathcal{J}_2
\end{bmatrix}R_x
\notag \\
&&\quad \quad
- 2 R_x
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x, R_xv) \mathcal{K}_1
\\
\nablala_{v}f_{0}(x, R_xv) \mathcal{K}_2
\end{bmatrix} R_x,
\label{C2 cond 2}
\end{eqnarray}
where $x=(x_1,x_2), \; v=(v_1,v_2)$, and
\begin{equation}gin{align*}
&\mathcal{J}_1:=\frac{1}{v\cdotot x} \begin{equation}gin{bmatrix}
-4v_2x_1x_2 & 4v_1x_1x_2 \\
-2v_2(x_2^2-x_1^2) & 2v_1(x_2^2-x_1^2)
\end{bmatrix}, \quad
\mathcal{J}_2:= \frac{1}{v\cdotot x}\begin{equation}gin{bmatrix}
-2v_2(x_2^2-x_1^2) & 2v_1(x_2^2-x_1^2)\\
4v_2x_1x_2 & -4v_1x_1x_2
\end{bmatrix},\\
&\mathcal{K}_1:=\begin{equation}gin{bmatrix}
\mathrm{d}frac{4v_1^2v_2^2x_1^3 +2v_1v_2^3(3x_1^2x_2-x_2^3)+ 2v_2^4(3x_1x_2^2+x_1^3)}{(v\cdotot x)^3} & \mathrm{d}frac{-4v_1^3v_2x_1^3-2v_1^2v_2^2(3x_1^2x_2-x_2^3)-2v_1v_2^3(3x_1x_2^2+x_1^3)}{(v\cdotot x)^3}\\
\mathrm{d}frac{4v_2^4x_2^3+2v_1v_2^3(3x_1x_2^2-x_1^3)+2v_1^2v_2^2(3x_1^2x_2+x_2^3)}{(v\cdotot x)^3} & \mathrm{d}frac{-4v_1v_2^3x_2^3-2v_1^2v_2^2(3x_1x_2^2-x_1^3)-2v_1^3v_2(3x_1^2x_2+x_2^3)}{(v\cdotot x)^3}
\end{bmatrix},\\
&\mathcal{K}_2 := \begin{equation}gin{bmatrix}
\mathrm{d}frac{-4v_1^3v_2x_1^3-2v_1v_2^3(3x_1x_2^2+x_1^3) -2v_1^2v_2^2(3x_1^2x_2-x_2^3)}{(v\cdotot x)^3} & \mathrm{d}frac{4v_1^4x_1^3 +2v_1^2v_2^2(3x_1x_2^2+x_1^3)+2v_1^3v_2 (3x_1^2x_2-x_2^3)}{(v \cdotot x)^3}\\
\mathrm{d}frac{-4v_1v_2^3x_2^3 -2v_1^3v_2(3x_1^2x_2+x_2^3) -2v_1^2v_2^2(3x_1x_2^2-x_1^3)}{(v\cdotot x)^3} & \mathrm{d}frac{4v_1^2 v_2^2 x_2^3 +2v_1^4(3x_1^2x_2+x_2^3)+2v_1^3v_2(3x_1x_2^2-x_1^3)}{(v \cdotot x)^3}
\end{bmatrix},
\end{align*}
for all $(x,v)\in \gamma_-$, then $f(t,x,v)$ defined in \eqref{solution} is a unique $C^{2}_{t,x,v}(\mathbb{R}_{+}\times \mathcal{I})$ solution of \eqref{eq}.
In this case, $f_0(x,v)=G(x,\vert v \vert)$ satisfying $\nablala_x f_0(x, v)=0$ for $x \in \partialartial \Omega$, where $G$ is a real-valued $C^2_{x,v}$ function. Additionally, $f(t,x,v)$ is not of class $C^2_{t,x,v}$ at time $t$ such that $t^k(t,x,v)=0$ for some $k$ if one of the initial conditions \eqref{C2 cond34}, \eqref{C2 cond 1}, and \eqref{C2 cond 2} for $(x,v)\in \gamma_-$ is not satisfied.
\end{theorem}
\begin{equation}gin{remark} (Higher regularity)
If we want higher regularity such as $C^{3}$ and $C^{4}$, we should assume additional initial-boundary compatibility conditions for those regularities as we assumed \eqref{C2 cond34}-\eqref{C2 cond 2} in Theorem \e^{\frac 12}f{thm 2} for $C^{2}$ as well as \eqref{C1 cond}. Although the computation for higher regularity is available in principle, we should carefully check whether the additional conditions for higher regularity make lower regularity conditions trivial or not. Here, the trivial condition for \eqref{C2 cond 2} means
\[
\nablala_{x,v}f_0(x,v) = 0,\quad \forall (x,v)\in\gamma_{-}.
\]
In fact, the answer is given in Section 1.2. Because of very nontrivial null structure of \eqref{1st order}, imposing \eqref{C2 cond34}-\eqref{C2 cond 2} does not make \eqref{C1 cond} trivial, fortunately. Once we find a new initial-boundary compatibility condition for $C^{3}$, for example, we also have to check
\begin{equation} \notag
\begin{equation}gin{split}
&\text{Do additional compatibility conditions for $C^{3}$ regularity make } \\
&\quad \text{\eqref{C1 cond} or \eqref{C2 cond34}-\eqref{C2 cond 2} trivial, e.g. $\nablala f_0 = \nablala^{2}f_0=0$ on $\gamma_{-}$?}
\end{split}
\end{equation}
Whenever we gain conditions for higher order regularity, initial-boundary compatibility conditions are stacked and they might make lower order compatibility conditions just trivial ones. It is a very interesting question, but they require very complicated geometric considerations and obtaining higher order condition itself will be also very painful. But, if we impose very strong (trivial) high order initial-boundary compatibility conditions
\[
\nablala_{x,v}^{i}f_0(x,v) = 0,\quad \forall(x,v)\in\gamma_{-},\quad 1\leq i \leq k,
\]
then we will get $C^{k}$ regularity of the solution.
\end{remark}
\begin{equation}gin{remark} (Necessary conditions for $C^{2}$ regularity)
In Theorem \e^{\frac 12}f{thm 2}, initial conditions \eqref{C2 cond 1} and \eqref{C2 cond 2} are sufficient conditions for $f \in C^2_{t,x,v}$. Although these contain non-symmetric complicated first-order terms, we can obtain simpler necessary conditions.
Observe that the null space of $\mathcal{J}_i, \mathcal{K}_i$ is spanned by $v$, i.e.,
\begin{equation}gin{equation} \label{null J,K}
\mathcal{J}_i v =0, \quad \mathcal{K}_i v =0, \quad i=1,2.
\end{equation}
Multiplying the reflection matrix $R_x$ on both sides in \eqref{C2 cond 1} and \eqref{C2 cond 2}, we get necessary conditions for $C^{2}$ solution,
\hide
\begin{equation}gin{align*}
&\nablala_{xv} f_0(x,v) +\nablala_{vv} f_0(x,v)\frac{ (Qv)\otimes (Qv)}{v\cdotot n} = R_x \left[\nablala_{xv}f_{0}(x, R_xv) + \nablala_{vv}f_{0}(x, R_xv) \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n} \right]R_x \\
&\hspace{6.5cm} +
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x , R_xv) \mathcal{J}_1
\\
\nablala_{v}f_{0}(x , R_xv) \mathcal{J}_2
\end{bmatrix},\\
&\nablala_{xx}f_{0}(x,v) + \nablala_{vx}f_{0}(x, v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n} + \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \nablala_{xv}f_{0}(x, v)\\
&\quad = R_x \left[ \nablala_{xx}f_{0}(x, R_xv) + \nablala_{vx}f_{0}(x, R_xv)\frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n}
+ \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n} \nablala_{xv}f_{0}(x, R_xv)\right]R_x \\
&\quad \quad -2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x, R_xv) \nablala_{v}A^{1}_{v,x}
\\
\nablala_{v}f_{0}(x, R_xv) \nablala_{v}A^{2}_{v,x}
\end{bmatrix} R_xA_{v,x}
+
R_xA_{v,x}\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x, R_xv) \mathcal{J}_1 \\
\nablala_{v}f_{0}(x, R_xv) \mathcal{J}_2
\end{bmatrix}
- 2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x, R_xv) \mathcal{K}_1
\\
\nablala_{v}f_{0}(x, R_xv) \mathcal{K}_2
\end{bmatrix} ,
\end{align*}
\unhide
\begin{equation}gin{equation} \label{C2 nec cond}
\begin{equation}gin{split}
&v^T \left[ \nablala_{xv} f_0(x,v) +\nablala_{vv} f_0(x,v)\frac{ (Qv)\otimes (Qv)}{v\cdotot n}\right] v=(R_xv)^T \left[\nablala_{xv}f_{0}(x, R_xv) + \nablala_{vv}f_{0}(x, R_xv) \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n} \right](R_xv),\\
&v^T \left[ \nablala_{xx}f_{0}(x,v) + \nablala_{vx}f_{0}(x, v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n} + \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \nablala_{xv}f_{0}(x, v)\right]v \\
&=(R_xv)^T \left[\nablala_{xx}f_{0}(x, R_xv) + \nablala_{vx}f_{0}(x, R_xv)\frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n}
+ \frac{(QR_xv)\otimes (QR_xv)}{R_xv\cdotot n} \nablala_{xv}f_{0}(x, R_xv)\right](R_xv),
\end{split}
\end{equation}
for all $(x,v) \in \gamma_-$, where we used $R_x^2 =I$, \eqref{null J,K}, and $A_{v,x}v=0$ in Lemma \e^{\frac 12}f{lem_RA}.
\end{remark}
\begin{equation}gin{remark}\label{extension C2 cond34}
Using \eqref{C1_v trivial} and \eqref{C2 cond34} yields that
\begin{equation}gin{equation} \label{f0 gamma+}
\nablala_v f_0(x,v) \partialarallel v^T,
\end{equation}
for all $(x,v) \in \gamma_-\cup \gamma_+$. From \eqref{f0 gamma+}, we have
\begin{equation}gin{equation*}
\nablala_v f_0(x,v) \frac{(Qv) \otimes (Qv)}{v\cdotot n} R_x = \nablala_v f_0(x,R_xv)\frac{(QR_xv)\otimes(QR_xv)}{R_x v\cdotot n}.
\end{equation*}
Thus, the condition \eqref{C1 cond} in Theorem \e^{\frac 12}f{thm 1} becomes
\begin{equation}gin{equation*}
\nablala_x f_0(x,v) R_x = \nablala_x f_0(x,R_xv).
\end{equation*}
Similarly, by \eqref{C2 cond34} and the above result, we have
\begin{equation}gin{equation*}
\nablala_x f_0(x,v) \partialarallel v^T,
\end{equation*}
for all $(x,v) \in \gamma_-\cup \gamma_+$. Hence, we conclude that \eqref{C2 cond34} can be extended to $\gamma_-\cup\gamma_+$ under conditions \eqref{C1 cond} and \eqref{C2 cond34} for $(x,v)\in\gamma_-$.
\end{remark}
\begin{equation}gin{remark}[Extension to 3D sphere]
By symmetry, Theorem \e^{\frac 12}f{thm 1} and \e^{\frac 12}f{thm 2} also hold for three dimensional sphere if the rotation operator $Q$ is properly redefined in the plane spanned by $\{x, v\}$ for $x\in \partial\Omega$, $x\nparallel v\neq 0$. \\
\end{remark}
\begin{equation}gin{theorem} [Regularity estimates] \label{thm 3}
The $C^{1}(\mathbb{R}_{+}\times \mathcal{I})$ and $C^{2}(\mathbb{R}_{+}\times \mathcal{I})$ solutions of Theorem \e^{\frac 12}f{thm 1} and \e^{\frac 12}f{thm 2} enjoy the following regularity estimates :
\begin{equation} \label{C1 bound}
\|f\|_{C^{1}_{t,x,v}} \lesssim \|f_0\|_{C^{1}} \frac{|v|}{|v\cdotot n(x_{\mathbf{b}})|^{2}} \langle v \rangle^{2}(1 + |v|t),
\end{equation}
\begin{equation} \label{C2 bound}
\|f\|_{C^{2}_{t,x,v}} \lesssim \|f_0\|_{C^{2}} \frac{|v|^{2}}{|v\cdotot n(x_{\mathbf{b}})|^{4}} \langle v \rangle^{4}(1 + |v|t)^{2},
\end{equation}
where $x_{\mathbf{b}} = x_{\mathbf{b}}(x,v)$ and $\langle v \rangle := 1 + |v|$.
\end{theorem}
\subsection{Brief sketch of proofs and some important remarks}
In this paper, our aim is to analyze regularity of mild form \eqref{solution} where characteristic $(X(0;t, x, v), V(0;t, x,v))$ is well-defined (by excluding $\gamma_0$). If backward in time position $X(0;t,x,v) \notin \partial\Omega$, the characteristic is also a smooth function and we expect that the regularity of \eqref{solution} will be the same as initial data $f_0$ by the chain rule. When $X(0;t,x,v) \in \partial\Omega$, however, the derivative via the chain rule does not work anymore because of discontinuous behavior of velocity $V(0;t,x,v)$. Depending on perturbed directions, we obtain different directional derivatives. In fact, we can split directions into two pieces: one gives bouncing and the other does not. See \eqref{R12_v} and \eqref{set R_vel} for $C^{1}_{v}$ for example. By matching these directional derivatives and performing some symmetrization, we obtain symmetrized initial-boundary compatibility condition \eqref{C1 cond}. Of course, \eqref{C1_v trivial} also holds, but \eqref{C1_v trivial} is gained by taking the $v$-derivative of \eqref{BC} directly. We note that both $C^{1}_{x}$ and $C^{1}_{v}$ conditions yield identical initial compatibility condition \eqref{C1 cond}, and the condition for $C^{1}_{t}$ is just a necessary condition for \eqref{C1 cond}. \\
The analysis becomes much more complicated when we study $C^{2}$ conditions. Nearly all of our analysis consist of precise equalities, instead of estimates. This makes our business much harder. First, let us consider four cases: $\nablala_{xx}, \nablala_{xv}, \nablala_{vx}, \nablala_{vv}$. These yield very complicated initial-boundary compatibility conditions and in particular they contain derivatives of each column of reflection operator $R_x$ or $\nablala_{x,v}((n(x)\otimes n(x))v)$. It is nearly impossible to give proper geometric interpretation for each term. See \eqref{xv star1} and \eqref{xv star2} for example. \\
Nevertheless, it is quite interesting that the four conditions from $\nablala_{xx}, \nablala_{xv}, \nablala_{vx}, \nablala_{vv}$ can be rearranged with respect to the order of time $t$. By matching all directional derivatives, we obtain \eqref{Cond2 1}--\eqref{Cond2 4} which contain both second-order terms and first-order terms. However, the conditions from $\nablala_{xx}, \nablala_{xv}, \nablala_{vx}, \nablala_{vv}$ must satisfy transpose compatibility condition
\begin{equation} \label{trans comp}
\nablala_{xv}^{T}=\nablala_{vx} \ \ \text{and} \ \ \nablala_{xx}^{T} = \nablala_{xx},
\end{equation}
since we hope the solution to be $C^{2}$. However, it is extremely hard to find any good geometric meaning or properties of some terms like
\begin{equation} \label{1st order}
\nablala_{x}(R^{i}_{x^{1}(x,v)}),\quad \nablala_{x}(A^{i}_{v, x^{1}(x,v)}),\quad \text{for}\quad i=1,2,
\end{equation}
in \eqref{Cond2 1}--\eqref{Cond2 4}. If they do not have any special structures, the only way to get compatibility \eqref{trans comp} is to impose $\nablala_{x,v}f_0(x, Rv) = 0$ for all $(x,v)\in \gamma_{-}$. Then $C^{1}$ compatibility condition \eqref{C1 cond} becomes just trivial. Fortunately, however, the matrices of \eqref{1st order} have a rank $1$ structure. {\bf More surprisingly, all the null spaces are spanned by velocity $v$!} That is, from Lemma \e^{\frac 12}f{d_RA} and Lemma \e^{\frac 12}f{dx_A},
\[
\nablala_{x}(R_{x^1(x,v)}^1)v =0, \quad \nablala_{x}(R_{x^1(x,v)}^2) v =0,\quad \nablala_x(-2A_{v,x^1(x,v)}^1)v =0, \quad \nablala_x (-2A_{v,x^1(x,v)}^2)v=0.
\]
From these interesting results, we can derive necessary conditions \eqref{C2 cond34} for transpose compatibility \eqref{trans comp}. By imposing \eqref{C2 cond34}, we derive $C^{2}$ conditions as in Theorem \e^{\frac 12}f{thm 2}, while keeping $C^{1}$ condition \eqref{C1 cond} nontrivial. We note that all the conditions that include $\partial_{t}$ are repetitions of \eqref{Cond2 1}--\eqref{Cond2 4}. \\
In the last section, we study $C^{1}$ and $C^{2}$ regularity estimates of the solution \eqref{solution}. Essentially the regularity estimates of the solution come from the regularity estimates of characteristic $(X(0;t,x,v), V(0;t,x,v))$. For $C^{1}$ of $(X(0;t,x,v), V(0;t,x,v))$, we obtain Lemma \e^{\frac 12}f{est der X,V}. Note that we can find some cancellation that gives no singular bound for $\nablala_{v}X(0;t,x,v)$ which was found in \cite{GKTT2017} for general 3D convex domains. Growth in time need not to be exponential, but it is just linear in time $t$. The second derivative of characteristic is much more complicated and nearly impossible to try to find any cancellation, because of too many terms and combinations that appear. Instead, by studying the most singular terms only, we obtain rough bounds in Lemma \e^{\frac 12}f{2nd est der X,V}.
\section{Preliminaries}
Now, let us recall standard matrix notations which will be used in this paper. \\
\begin{equation}gin{definition}
When we perform matrix multiplications throughout this paper, we basically treat a n-dimensional vector $v$ as a {\it column} vector
\[
v = \begin{equation}gin{pmatrix}
v_{1} \\ \vdots \\ v_{n}
\end{pmatrix}.
\]
For about gradient of a smooth scalar function $a(x)$, however, we treat n-dimensional vector $\nablala a$ as a {\it row} vector,
\[
\nablala a(x) := (\partial_{x_{1}} a, \partial_{x_{2}} a, \cdotots, \partial_{x_{n}} a).
\]
For a smooth vector function $v : \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ with $v(x)= \begin{equation}gin{pmatrix}
v_{1}(x) \\ \vdots \\ v_{m}(x)
\end{pmatrix}$, we define $\nablala_{x}v(x)$ as $m\times n$ matrix,
\[
\nablala_{x}v := \begin{equation}gin{pmatrix}
\partial_{1} v_{1} & \cdotots & \partial_{n} v_{1} \\
\partial_{1} v_{2} & \cdotots & \partial_{n} v_{2} \\
\vdots & \vdots & \vdots \\
\partial_{1} v_{m} & \cdotots & \partial_{n} v_{m} \\
\end{pmatrix}_{m\times n}
=
\begin{equation}gin{pmatrix}
& \nablala_{x} v_{1} &\\
&\vdots& \\
&\nablala_{x} v_{m}& \\
\end{pmatrix}_{m\times n} .
\]
We use $\otimes$ to denote tensor product
\begin{equation}gin{equation*}
a\otimes b := \begin{equation}gin{pmatrix}
a_{1} \\ \vdots \\ a_{m}
\end{pmatrix}
\begin{equation}gin{pmatrix}
b_{1} & \cdotots & b_{n}
\end{pmatrix}. \\
\end{equation*}
\end{definition}
\begin{equation}gin{lemma}\label{matrix notation}
(1) (Product rule) For scalar function $a(x)$ and vector function $v(x)$,
\[
\nablala (a(x)v(x)) = a(x)\nablala v(x) + v\otimes \nablala a(x).
\]
(2) (Chain rule) For vector functions $v(x)$ and $w(x)$,
\[
\nablala (v(w(x))) = \nablala v(w(x)) \nablala w(x).
\]
(3) (Product rule) For vector functions,
\[
\nablala(v(x)\cdotot w(x)) = v(x)\nablala w(x) + w(x)\nablala v(x).
\]
(4) For matrix $d\times d$ matrix $A(x)$ and $d\times 1$ vector $v(x)$,
\begin{equation}gin{equation} \label{d_matrix}
\begin{equation}gin{split}
\nablala_{x} (A(x)v(x))
&= A(x)\nablala v(x)
+
\begin{equation}gin{pmatrix}
v(x)\nablala A^{1}(x) \\
\vdots \\
v(x)\nablala A^{d}(x)
\end{pmatrix} \\
&= A(x)\nablala v(x)
+ \sum_{k=1}^{d}
\partial_{k}A(x) E_{k},
\end{split}
\end{equation}
where $A^{i}(x)$ is $i$-th row of $A(x)$ and $E_{k}$ is $d\times d$ matrix whose $k$th column is $v$ and others are zero. (Here $\partial_{k}A(x)$ means elementwise $x_{k}$-derivative of $A(x)$.) Moreover, if $A = A(\theta(x))$ for some smooth $\theta:\Omega\rightarrow \mathbb{R}$, \\
\begin{equation} \label{d_matrix_theta}
\nablala_{x} (A(\theta)v(x))
= A(\theta)\nablala v(x)
+ \partial_{\theta}A(\theta)v \otimes \nablala_{x}\theta.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Only \eqref{d_matrix_theta} needs some explanation. When $A=A(\theta(x))$,
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nablala_{x} (A(\theta)v(x))
&= A(\theta)\nablala v(x)
+ \sum_{k=1}^{d}
\partial_{k}A(\theta) E_{k}
=
A(\theta)\nablala v(x)
+ \sum_{k=1}^{d}
\partial_{\theta}A(\theta) \partial_{k}\theta(x)E_{k} \\
&= A(\theta)\nablala v(x)
+ \partial_{\theta}A(\theta)
\begin{equation}gin{pmatrix}
& & \\
\partial_{1}\theta(x) v & \cdotots & \partial_{d}\theta(x)v \\
& & \\
\end{pmatrix} \\
&= A(\theta)\nablala v(x)
+ \partial_{\theta}A(\theta)v \otimes \nablala_{x}\theta(x).
\end{split}
\end{equation*}
\end{proof}
\begin{equation}gin{lemma} \label{nabla xv b}
We have the following computations where $x_{\mathbf{b}} = x_{\mathbf{b}}(x,v)$ and $t_{\mathbf{b}}=t_{\mathbf{b}}(x,v)$. \\
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nablala_{x}t_{\mathbf{b}} &= \frac{n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})} , \\
\nablala_{v}t_{\mathbf{b}} &= -t_{\mathbf{b}}\nablala_{x}t_{\mathbf{b}} = -t_{\mathbf{b}}\frac{n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})} , \\
\nablala_{x}x_{\mathbf{b}} &= I - \frac{v\otimes n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})}, \\
\nablala_{v}x_{\mathbf{b}} &= -t_{\mathbf{b}}\begin{equation}ig(I - \frac{v\otimes n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})} \begin{equation}ig). \\
\end{split}
\end{equation*}
\end{lemma}
\begin{equation}gin{proof}
Remind the definition of $x_{\mathbf{b}}$ and $t_{\mathbf{b}}$
\begin{equation}gin{equation*}
x_{\mathbf{b}}=x-t_{\mathbf{b}} v, \quad t_{\mathbf{b}}=\sup\{ s \; \vert \; x-sv \in \Omega\}.
\end{equation*}
Since $\xi(x) =0$ for $x \in \partial\Omega$, we have $\xi(x_{\mathbf{b}}) = \xi(x-t_{\mathbf{b}} v)=0$. Taking the $x\mbox{-}$derivative $\nablala_x$, we get
\begin{equation}gin{align*}
\nablala_x(\xi(x_{\mathbf{b}}))&= (\nablala\xi)(x_{\mathbf{b}}) -[ (\nablala \xi)(x_{\mathbf{b}}) \cdotot v]\nablala_x t_{\mathbf{b}} \\
&= 0,
\end{align*}
where the first equality comes from product rule in Lemma \e^{\frac 12}f{matrix notation}. Thus, we can derive
\begin{equation}gin{equation*}
\nablala_x t_{\mathbf{b}} = \frac{( \nablala \xi)(x_{\mathbf{b}})}{[ (\nablala \xi)(x_{\mathbf{b}}) \cdotot v]} = \frac{ n(x_{\mathbf{b}})}{v \cdotot n(x_{\mathbf{b}}) }.
\end{equation*}
Similarly, taking the $v\mbox{-}$derivative $\nablala_v$ and product rule in Lemma \e^{\frac 12}f{matrix notation} yields
\begin{equation}gin{equation*}
\nablala_v(\xi(x_{\mathbf{b}})) = (\nablala \xi)(x_{\mathbf{b}})(-t_{\mathbf{b}} I - v \otimes \nablala_v t_{\mathbf{b}})= 0,
\end{equation*}
which implies $\nablala_v t_{\mathbf{b}} = - t_{\mathbf{b}} \frac{n(x_{\mathbf{b}})}{ v\cdotot n(x_{\mathbf{b}})}.$ It follows from the calculation of $\nablala_x t_{\mathbf{b}}$ and $\nablala_v t_{\mathbf{b}}$ above that
\begin{equation}gin{align*}
\nablala_x x_{\mathbf{b}} &= \nablala_x ( x- t_{\mathbf{b}} v) = I - v \otimes \nablala_x t_{\mathbf{b}} = I - \frac{v \otimes n(x_{\mathbf{b}})}{v \cdotot n (x_{\mathbf{b}})} \\
\nablala_v x_{\mathbf{b}} &= \nablala_v (x-t_{\mathbf{b}} v) = -t_{\mathbf{b}} I - v \otimes \nablala_v t_{\mathbf{b}} = -t_{\mathbf{b}} \left(I - \frac{ v \otimes n(x_{\mathbf{b}})}{ v \cdotot n(x_{\mathbf{b}})}\right).
\end{align*}
\end{proof}
\begin{equation}gin{lemma} \label{d_n} For $n(x_{\mathbf{b}}(x,v))$, we have the following derivative rules,
\begin{equation}gin{equation} \label{normal}
\nablala_x [n(x_{\mathbf{b}})] = I - \frac{v \otimes n(x_{\mathbf{b}})}{v \cdotot n (x_{\mathbf{b}})}, \quad \nablala_v [n(x_{\mathbf{b}})] = -t_{\mathbf{b}} \begin{equation}ig( I - \frac{v \otimes n(x_{\mathbf{b}})}{v \cdotot n (x_{\mathbf{b}})} \begin{equation}ig),
\end{equation}
where $x_{\mathbf{b}}=x_{\mathbf{b}}(x,v)$.
\end{lemma}
\begin{equation}gin{proof}
For $\nablala_x n(x_{\mathbf{b}})$, we apply the chain rule in Lemma \e^{\frac 12}f{matrix notation} to $(\nablala \xi)(x_{\mathbf{b}})$ and $\frac{1}{\vert (\nablala \xi)(x_{\mathbf{b}})\vert }$ respectively. Because $\nablala \xi (x) \neq 0 $ at the boundary $x \in \partial\Omega$ in a circle, it is possible to apply the chain rule to $\frac{1}{\vert (\nablala \xi)(x_{\mathbf{b}})\vert }$. Taking $x\mbox{-}$derivative $\nablala_x$, one obtains
\begin{equation}gin{align*}
\nablala_x [(\nablala \xi)(x_{\mathbf{b}})] &= (\nablala ^2 \xi)(x_{\mathbf{b}})\nablala_x x_{\mathbf{b}}, \\
\nablala_x \left[ \frac{1}{ \vert (\nablala \xi)(x_{\mathbf{b}}) \vert} \right] & =- \frac{(\nablala\xi)(x_{\mathbf{b}}) (\nablala^2\xi)(x_{\mathbf{b}}) \nablala_xx_{\mathbf{b}}}{\vert (\nablala \xi)(x_{\mathbf{b}}) \vert^3}.
\end{align*}
Hence,
\begin{equation}gin{align*}
\nablala_x [n(x_{\mathbf{b}})] = \nablala_x \left[ \frac{ (\nablala \xi)(x_{\mathbf{b}})}{ \vert (\nablala \xi)(x_{\mathbf{b}}) \vert } \right] &= \frac{1}{ \vert (\nablala \xi)(x_{\mathbf{b}}) \vert } \nablala_x [ (\nablala \xi)(x_{\mathbf{b}}) ] + (\nablala \xi)(x_{\mathbf{b}}) \otimes \nablala_x \left [ \frac{1}{\vert (\nablala \xi)(x_{\mathbf{b}}) \vert} \right] \\
& = \frac{1}{ \vert (\nablala \xi)(x_{\mathbf{b}}) \vert }(\nablala ^2 \xi)(x_{\mathbf{b}})\nablala_x x_{\mathbf{b}} - \nablala \xi(x_{\mathbf{b}}) \otimes \frac{(\nablala\xi)(x_{\mathbf{b}}) (\nablala^2\xi)(x_{\mathbf{b}}) \nablala_xx_{\mathbf{b}}}{\vert (\nablala \xi)(x_{\mathbf{b}}) \vert^3}\\
&= \frac{1}{ \vert (\nablala \xi)(x_{\mathbf{b}}) \vert }\begin{equation}ig( I - n(x_{\mathbf{b}}) \otimes n(x_{\mathbf{b}})\begin{equation}ig) (\nablala^2 \xi)(x_{\mathbf{b}}) \nablala_x x_{\mathbf{b}}.
\end{align*}
Since $|\nablala\xi(x_{\mathbf{b}})| =1 $ and $\nablala^{2}\xi = I_{2}$, we deduce
\begin{equation}gin{align*}
\nablala_x [n(x_{\mathbf{b}})] &= \begin{equation}ig( I - n(x_{\mathbf{b}})\otimes n(x_{\mathbf{b}}) \begin{equation}ig) \begin{equation}ig( I - \frac{v \otimes n(x_{\mathbf{b}})}{v \cdotot n (x_{\mathbf{b}})} \begin{equation}ig) \\
&= I - \frac{v \otimes n(x_{\mathbf{b}})}{v \cdotot n (x_{\mathbf{b}})} - n(x_{\mathbf{b}})\otimes n(x_{\mathbf{b}}) + n(x_{\mathbf{b}})\otimes n(x_{\mathbf{b}}) \\
&= I - \frac{v \otimes n(x_{\mathbf{b}})}{v \cdotot n (x_{\mathbf{b}})}.
\end{align*}
The case for $\nablala_v [n(x_{\mathbf{b}})]$ is nearly same with extra term $-t_{\mathbf{b}}$ which comes from Lemma \e^{\frac 12}f{nabla xv b}. \\
\end{proof}
\hide
\begin{equation}gin{lemma}
For fixed $x\in\Omega$, we can classify direction $\mathbb{S}^{2}$ into three parts,
\begin{equation}gin{equation*}
\begin{equation}gin{split}
R_{0} &:= \{ \hat{r}\in \mathbb{S}^{2} : \nablala_{x}t_{\mathbf{b}}(x,v)\} \\
R_{1} &:= \{ \hat{r}\in \mathbb{S}^{2} : \} \\
R_{2} &:= \{ \hat{r}\in \mathbb{S}^{2} : \}
\end{split}
\end{equation*}
\end{lemma}
\begin{equation}gin{proof}
(i) From Proposition \eqref{nabla xv b}
\[
\frac{\partial}{\partial\varepsilon}t_{\mathbf{b}}(x+\varepsilon\hat{r}, v)\vert_{\varepsilon=0} = \nablala_{x}t_{\mathbf{b}}(x,v)\cdotot\hat{r} = \frac{\hat{r}\cdotot n(x_{\mathbf{b}}(x,v))}{v\cdotot n(x_{\mathbf{b}}(x,v))}
\]
\end{proof}
\unhide
\section{Initial-boundary compatibility condition for $C^{1}_{t,x,v}$}
\begin{equation}gin{lemma} \label{lem_RA}
Recall definition \eqref{def A} of the matrix $A_{v,x}$. We have the following identities, for $(x,v) \in \{\partial\Omega \times \mathbb{R}^d\} \backslash \gamma_0$,
\begin{equation}gin{equation} \label{RA}
\begin{equation}gin{split}
R_xA_{v,x} &= \frac{1}{v\cdotot n(x)} Q(v\otimes v)Q^{T} = \frac{1}{v\cdotot n(x)} (Qv)\otimes (Qv), \\
A_{v,x} R_x &= \frac{1}{v\cdotot n(x)} R_xQ(v\otimes v)Q^{T}R_x = -\frac{1}{R_xv\cdotot n(x)} (QR_xv)\otimes (QR_xv), \\
\end{split}
\end{equation}
\begin{equation}gin{equation} \label{A2}
\begin{equation}gin{split}
A^{2}_{v,x} &= \frac{1}{(v\cdotot n(x))^{2}} (QR_xv\otimes QR_xv)(Qv\otimes Qv),
\end{split}
\end{equation}
\begin{equation}gin{equation} \label{Av=0}
A_{v,x}v =0,
\end{equation}
where $Q := Q_{\frac{\partiali}{2}}$ is counterclockwise rotation by angle $\frac{\partiali}{2}$.
\end{lemma}
\begin{equation}gin{proof}
We compute
\begin{equation}gin{equation*}
\begin{equation}gin{split}
R_xA_{v,x}R_x &:= \left[\left((v\cdotot n(x))I - (n(x) \otimes v) \right)\left(I + \frac{v\otimes n(x)}{v\cdotot n(x)}\right)\right] \\
&= \big(Qv \otimes Qn(x)\big)\left(I + \frac{v\otimes n(x)}{v\cdotot n(x)}\right).
\end{split}
\end{equation*}
Now let us define $\tau(x)= Q_{-\frac{\partiali}{2}}n(x)$ as tangential vector at $x\in\partial\Omega$. ($n$ as y-axis and $\tau$ as x-axis) Then,
\begin{equation}gin{equation*}
\begin{equation}gin{split}
R_xA_{v,x}R_x &:= Qv \otimes \begin{equation}ig( -\tau - \frac{v\cdotot\tau}{v\cdotot n(x)}n(x) \begin{equation}ig) \\
&= -\frac{1}{v\cdotot n(x)} Qv\otimes \begin{equation}ig( (v\cdotot n(x))\tau + (v\cdotot\tau)n(x) \begin{equation}ig) \\
&= -\frac{1}{v\cdotot n(x)} Qv\otimes \big( R_xQ^{T}v \big) \\
&= \frac{1}{v\cdotot n(x)} Qv\otimes \big( R_xQv \big) \\
&= \frac{1}{v\cdotot n(x)} Q(v\otimes v)Q^{T}R_x, \\
\end{split}
\end{equation*}
and we get \eqref{RA} using $R_xQ=-R_xQ^T$, because
\[
Q^{T}R_xQ = I - 2Q^{T}(n(x)\otimes n(x))Q = I - 2\tau\otimes\tau = -R_x. \\
\]
\eqref{A2} is simply obatined by \eqref{RA}. By definition of $A_{v,x}$ in \eqref{def A}, one obtains that
\begin{equation}gin{align*}
A_{v,x}v = \left[\left((v\cdotot n(x))I+(n(x) \otimes v) \right)\left(I-\frac{v\otimes n(x)}{v\cdotot n(x)}\right)\right]v=\left((v\cdotot n(x))I+(n(x)\otimes v))\right)(v-v)=0.
\end{align*}
\end{proof}
Now, throughout this section, we study $C^{1}_{t,x,v}(\mathbb{R}_{+}\times \Omega\times \mathbb{R}^{2})$ of $f(t,x,v)$ of \eqref{solution} when
\begin{equation} \label{t1 zero}
0 = t^{1}(t,x,v) \ \text{or equivalently} \ t = t_{\mathbf{b}}(x,v).
\end{equation}
\subsection{$C^{1}_{v}$ condition of $f$}
Since we assume \eqref{t1 zero}, $X(0;t,x,v) = x^{1}(x,v) = x_{\mathbf{b}}(x,v) \in \partialartial \Omega$. To derive compatibility condition for $C^{1}_{v}$ of $f(t,x,v)$, we consider $v$-perturbation and use the following notation for perturbed trajectory:
\begin{equation}gin{equation} \label{XV epsilon v}
X^{\varepsilonsilon}(0) := X(0;t,x,v+\varepsilonsilon \hat{r}) , \quad V^{\varepsilonsilon}(0):=V(0;t,x,v+\varepsilonsilon \hat{r} ),
\end{equation}
where $\hat{r}\in\mathbb{R}^{2}$ is a unit-vector. As $\varepsilonsilon \rightarrow 0$, we simply get
\begin{equation}gin{equation*}
\lim_{\varepsilonsilon \rightarrow 0} X(0;t,x,v+\varepsilonsilon \hat{r}) = x^{1}(x,v) = x_{\mathbf{b}}(x,v),
\end{equation*}
from continuity of $X(0;t,x,v)$ in $v$. However, $V(0;t,x,v)$ is not continuous in $v$ because of \eqref{BC}. Explicitly, from Lemma \e^{\frac 12}f{nabla xv b},
\begin{equation} \label{R12_v}
\frac{\partial}{\partial\varepsilon}t_{\mathbf{b}}(x, v+\varepsilon\hat{r})\vert_{\varepsilon=0} = \nablala_{v}t_{\mathbf{b}}(x,v)\cdotot\hat{r} = -t_{\mathbf{b}}\frac{\hat{r}\cdotot n(x_{\mathbf{b}}(x,v))}{v\cdotot n(x_{\mathbf{b}}(x,v))},\quad \text{where}\quad v\cdotot n(x_{\mathbf{b}}(x,v)) < 0.
\end{equation}
So we define, for fixed $(x,v)$, $v\neq 0$,
\begin{equation}gin{equation} \label{set R_vel}
\begin{equation}gin{split}
R_{vel, 1} &:= \{ \hat{r}\in \mathbb{S}^{2} : \hat{r}\cdotot n(x_{\mathbf{b}}(x,v)) < 0 \}, \\
R_{vel, 2} &:= \{ \hat{r}\in \mathbb{S}^{2} : \hat{r}\cdotot n(x_{\mathbf{b}}(x,v)) \geq 0 \}. \\
\end{split}
\end{equation}
Then from \eqref{R12_v}, $\nablala_{v}t_{\mathbf{b}}(x,v)\cdotot\hat{r} > 0$ when $\hat{r}\in R_{vel, 1}$ and $\nablala_{v}t_{\mathbf{b}}(x,v)\cdotot\hat{r} \leq 0$ when $\hat{r}\in R_{vel, 2}$. Therefore, for two unit vectors $\hat{r}_1\in R_{vel, 1}$ and $\hat{r}_2\in R_{vel, 2}$, by continuity argument,
\begin{equation}gin{equation*}
\lim_{\varepsilonsilon \rightarrow 0+} V(0;t,x,v+\varepsilonsilon \hat{r}_1) = v, \quad \lim_{\varepsilonsilon \rightarrow 0+} V (0;t,x,v+\varepsilonsilon \hat{r}_2) = v^1=R_{x^{1}}v. \\
\end{equation*}
We consider directional derivatives with respect to $\hat{r}_1$ and $\hat{r}_2$. If $f$ belongs to the $C^1_v$ class, $\nablala_v f(t,x,v)$ exists and directional derivatives of $f$ with respect to $\hat{r}_1,\hat{r}_2$ will be $\nablala_v f(t,x,v) \hat{r}_1,\;\nablala_v f(t,x,v) \hat{r}_2$. Using \eqref{BC}, we have $f_{0}(x^{1}, v) = f_{0}(x^{1}, v^{1})$ and hence
\begin{equation}gin{align*}
\nablala_v f(t,x,v) \hat{r}_1 &= \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( f(t,x,v+\varepsilonsilon \hat{r}_1) - f(t,x,v) \right )\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\left( f_0(X(0;t,x,v+\varepsilonsilon \hat{r}_1),V(0;t,x,v+\varepsilonsilon \hat{r}_1)) - f_0(X(0;t,x,v),V(0;t,x,v)) \right)\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( f_0(X^{\varepsilonsilon}(0), V^{\varepsilonsilon}(0))- f_0 (X^{\varepsilonsilon}(0),v)+f_0(X^{\varepsilonsilon}(0),v) -f_0(X(0),v) \right) \\
&=\nablala_x f_0(X(0),v) \cdotot \lim_{s\rightarrow 0+} \nablala_v X(s) \hat{r}_1+ \nablala_v f_0(X(0),v) \lim_{s \rightarrow 0+} \nablala_v V(s)\hat{r}_1, \\
\nablala_v f(t,x,v) \hat{r}_2 &= \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( f(t,x,v+\varepsilonsilon \hat{r}_2) - f(t,x,v) \right )\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\left( f_0(X(0;t,x,v+\varepsilonsilon \hat{r}_2),V(0;t,x,v+\varepsilonsilon \hat{r}_2)) - f_0(X(0;t,x,v),V(0;t,x,v)) \right)\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( f_0(X^{\varepsilonsilon}(0), V^{\varepsilonsilon}(0))- f_0 (X^{\varepsilonsilon}(0),v^{1})+f_0(X^{\varepsilonsilon}(0),v^{1}) -f_0(X(0),v^{1}) \right) \\
&=\nablala_x f_0(X(0),v^{1}) \cdotot \lim_{s\rightarrow 0-} \nablala_v X(s) \hat{r}_2+ \nablala_v f_0(X(0),v^{1}) \lim_{s \rightarrow 0-} \nablala_v V(s)\hat{r}_2,
\end{align*}
which implies
\begin{equation}gin{eqnarray}
&& \nablala_v f(t,x,v) =\nablala_x f_0(X(0),v) \lim_{s\rightarrow 0+} \nablala_v X(s)+ \nablala_v f_0(X(0),v) \lim_{s \rightarrow 0+} \nablala_v V(s), \label{case12 r1}\\
&& \nablala_v f(t,x,v) =\nablala_x f_0(X(0),v^{1}) \lim_{s\rightarrow 0-} \nablala_v X(s)+ \nablala_v f_0(X(0),v^{1}) \lim_{s \rightarrow 0-} \nablala_v V(s). \label{case12 r2}
\end{eqnarray}
\noindent Since
\begin{equation} \label{nabla XV_v+}
\lim_{s\rightarrow 0+} \nablala_v X(s)= \lim_{s\rightarrow 0+}\nablala_{v}(x-v(t-s)) = -t I_{2\times 2}, \quad \lim_{s \rightarrow 0+} \nablala_v V(s) = \lim_{s \rightarrow 0+} \nablala_v v = I_{2\times 2},
\end{equation}
$\nablala_v f(t,x,v)$ of \eqref{case12 r1} becomes
\begin{equation}gin{equation} \label{c_1}
\nablala_v f(t,x,v) = -t \nablala_x f_0(X(0),v) + \nablala_v f_0(X(0),v). \\
\end{equation}
For \eqref{case12 r2}, using the product rule in Lemma \e^{\frac 12}f{matrix notation} and \eqref{normal} in Lemma \e^{\frac 12}f{d_n}, we have
\begin{equation}gin{equation} \label{nabla XV_v-}
\begin{equation}gin{split}
\lim_{s\rightarrow 0-} \nablala_v X(s)& = \lim_{s\rightarrow 0-} \nablala_v (x^1 - (t^1+s)v^1)= \lim_{s\rightarrow 0-} \nablala_v x^1 + v^{1}\otimes\nablala_{v}t_{\mathbf{b}} \\
&= -t\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right) - t\frac{v^1 \otimes n(x^1)}{ v \cdotot n(x^1)} \\
&= -t \begin{equation}ig( I -\frac{1}{v\cdotot n(x^1)} \big( 2(v\cdotot n(x^{1}))n(x^{1}) \big)\otimes n(x^{1}) \begin{equation}ig) = -tR_{x^{1}}, \\
\lim_{s \rightarrow 0-} \nablala_v V(s) &= \lim_{s \rightarrow 0-} \nablala_v (R_{x^{1}}v) \\
&=\lim_{s \rightarrow 0-} \left( I- 2(v \cdotot n(x^1) ) \nablala_v n(x^1) -2 n(x^1)\otimes n(x^1) -2 (n(x^1)\otimes v ) \nablala_v n(x^1)\right)\\
&=R_{x^{1}} +2t(v\cdotot n(x^1)) \left( I - \frac{ v \otimes n(x^1)}{ v \cdotot n(x^1)} \right) +2t (n(x^1) \otimes v)\left( I - \frac{ v \otimes n(x^1)}{ v \cdotot n(x^1)} \right) \\
&= R_{x^{1}} + 2t A_{v, x^{1}},
\end{split}
\end{equation}
where $A_{v, x^{1}}$ is defined in \eqref{def A}. Hence, using \eqref{nabla XV_v-}, $\nablala_v f(t,x,v)$ in \eqref{case12 r2} becomes
\begin{equation}gin{align} \label{c_2}
\begin{equation}gin{split}
\nablala_v f(t,x,v) &= -t \nablala_x f_0(X(0),R_{x^{1}}v) R_{x^{1}} \\
&\quad +\nablala_v f_0(X(0),R_{x^{1}}v)R_{x^{1}} + t\nablala_v f_0(X(0),R_{x^{1}}v)\left[2\left((v\cdotot n(x^1))I+(n(x^1) \otimes v) \right)\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right] \\
&= -t \nablala_x f_0(x^{1},R_{x^1}v) R_{x^{1}} + \nablala_v f_0(x^{1},R_{x^{1}}v) (R_{x^{1}} + 2tA_{v, x^{1}}),
\end{split}
\end{align}
where we used $v\cdotot n (x^1) = - v^1 \cdotot n(x^1)$. Meanwhile, taking $\nablala_{v}$ to specular reflection \eqref{BC} directly, we get
\begin{equation} \label{comp_v}
\nablala_{v}f_{0}(x,v) =\nablala_{v}f_{0}(x,R_{x}v)R_{x}, \quad \forall x\in\partial\Omega.
\end{equation}
Comparing \eqref{c_1}, \eqref{c_2}, and \eqref{comp_v}, we deduce
\begin{equation}gin{align} \label{c_v}
\begin{equation}gin{split}
\nablala_x f_0( x^{1},v) &= \nablala_x f_0(x^{1},R_{x^1}v) R_{x^{1}} - 2\nablala_v f_0(x^{1}, R_{x^{1}}v)A_{v,x^{1}},\quad (x^{1}, v)\in \gamma_{-}.
\end{split}
\end{align}
\subsection{$C^{1}_{x}$ condition of $f$}
Recall we assumed \eqref{t1 zero}. Similar to previous subsection, we define $x$-perturbed trajectory,
\begin{equation}gin{equation} \label{XV epsilon x}
X^{\varepsilonsilon}(0) :=X(0;t,x+\varepsilonsilon \hat{r}, v ), \quad V^{\varepsilonsilon}(0) := V(0;t,x+\varepsilonsilon \hat{r}, v),
\end{equation}
where $\hat{r}\in\mathbb{R}^{2}$ is a unit-vector. As $\varepsilonsilon \rightarrow 0$, we simply get
\begin{equation}gin{equation*}
\lim_{\varepsilonsilon \rightarrow 0} X(0;t,x+\varepsilonsilon\hat{r},v) = x^{1}(x,v).
\end{equation*}
Similar to previous subsection, using Lemma \e^{\frac 12}f{nabla xv b},
\begin{equation} \label{R12_x}
\frac{\partial}{\partial\varepsilon}t_{\mathbf{b}}(x+\varepsilon\hat{r}, v)\big\vert_{\varepsilon=0} = \nablala_{x}t_{\mathbf{b}}(x,v)\cdotot\hat{r} = \frac{\hat{r}\cdotot n(x_{\mathbf{b}}(x,v))}{v\cdotot n(x_{\mathbf{b}}(x,v))},\quad \text{where}\quad v\cdotot n(x_{\mathbf{b}}(x,v)) < 0.
\end{equation}
So we define, for fixed $(x,v)$, $v\neq 0$,
\begin{equation}gin{equation} \label{set R_sp}
\begin{equation}gin{split}
R_{sp, 1} &:= \{ \hat{r}\in \mathbb{S}^{2} : \hat{r}\cdotot n(x_{\mathbf{b}}(x,v)) > 0 \}, \\
R_{sp, 2} &:= \{ \hat{r}\in \mathbb{S}^{2} : \hat{r}\cdotot n(x_{\mathbf{b}}(x,v)) \leq 0 \}. \\
\end{split}
\end{equation}
Then from \eqref{R12_x}, $\nablala_{x}t_{\mathbf{b}}(x,v)\cdotot\hat{r} > 0$ when $\hat{r}\in R_{sp, 1}$ and $\nablala_{x}t_{\mathbf{b}}(x,v)\cdotot\hat{r} \leq 0$ when $\hat{r}\in R_{sp, 2}$. Therefore, for two unit vectors $\hat{r}_1\in R_{sp, 1}$ and $\hat{r}_2\in R_{sp, 2}$, by continuity argument,
\begin{equation}gin{equation*}
\lim_{\varepsilonsilon \rightarrow 0+} V(0;t,x,v+\varepsilonsilon \hat{r}_1) = v, \quad \lim_{\varepsilonsilon \rightarrow 0+} V (0;t,x,v+\varepsilonsilon \hat{r}_2) = v^1=R_{x^{1}}v.
\end{equation*}
Using similar arguments in previous subsection, we obtain
\begin{equation}gin{eqnarray}
&& \nablala_x f(t,x,v) \hat{r}_{1} =\nablala_x f_0(X(0),v) \lim_{s\rightarrow 0+} \nablala_x X(s)\hat{r}_{1} + \nablala_v f_0(X(0),v) \lim_{s \rightarrow 0+} \nablala_x V(s)\hat{r}_{1}, \label{case12 r1 x}\\
&& \nablala_x f(t,x,v)\hat{r}_{2} =\nablala_x f_0(X(0),Rv) \lim_{s\rightarrow 0-} \nablala_x X(s) \hat{r}_{2} + \nablala_v f_0(X(0),Rv) \lim_{s \rightarrow 0-} \nablala_x V(s) \hat{r}_{2}. \label{case12 r2 x}
\end{eqnarray}
Since
\begin{equation} \label{nabla XV_x+}
\lim_{s\rightarrow 0+} \nablala_x X(s) = I_{2\times 2},\quad \lim _{s \rightarrow 0+} \nablala_x V(s)=0_{2 \times 2},
\end{equation}
$\nablala_{x}f(t,x,v)$ of \eqref{case12 r1 x} becomes
\begin{equation}gin{equation} \label{c_3}
\nablala_x f(t,x,v) = \nablala_x f_0(X(0),v).
\end{equation}
For $\nablala_{x}f(t,x,v)$ of \eqref{case12 r2 x}, we apply the product rule in Lemma \e^{\frac 12}f{matrix notation} and \eqref{normal} in Lemma \e^{\frac 12}f{d_n} to get
\begin{equation}gin{equation} \label{nabla XV_x-}
\begin{equation}gin{split}
\lim_{s\rightarrow 0-} \nablala_x X(s)& = \lim_{s\rightarrow 0-} \nablala_x (x^1 - (t^1+s)v^1)=\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right) + \frac{v^1 \otimes n(x^1)}{ v \cdotot n(x^1)} = R_{x^{1}},\\
\lim_{s \rightarrow 0-} \nablala_x V(s) &= \lim_{s \rightarrow 0-} \nablala_x (R_{x^{1}}v) \\
&=\lim_{s \rightarrow 0-} \left( - 2(v \cdotot n(x^1) ) \nablala_x n(x^1) -2 (n(x^1)\otimes v ) \nablala_x n(x^1)\right)\\
&=-2(v\cdotot n(x^1)) \left( I - \frac{ v \otimes n(x^1)}{ v \cdotot n(x^1)} \right) -2 (n(x^1) \otimes v)\left( I - \frac{ v \otimes n(x^1)}{ v \cdotot n(x^1)} \right) \\
&= -2 A_{v,x^{1}}.
\end{split}
\end{equation}
Hence, using \eqref{nabla XV_x-}, $\nablala_x f(t,x,v)$ in \eqref{case12 r2 x} becomes
\begin{equation}gin{align} \label{c_4}
\begin{equation}gin{split}
\nablala_x f(t,x,v) &= \nablala_x f_0(X(0), R_{x^{1}}v) R_{x^{1}} - 2\nablala_v f_0(X(0),R_{x^{1}}v) A_{v,x^{1}}.
\end{split}
\end{align}
Combining \eqref{c_3} and \eqref{c_4},
\begin{equation}gin{align}\label{c_x}
\begin{equation}gin{split}
\nablala_x f_0( x^{1},v) &= \nablala_x f_0(x^{1},R_{x^{1}}v) R_{x^{1}} - 2\nablala_v f_0(x^{1}, R_{x^{1}}v) A_{v,x^{1}},\quad (x^{1}, v)\in \gamma_{-},
\end{split}
\end{align}
which is identical to \eqref{c_v}. \\
\hide
which exactly coincides with \eqref{c_v}. We rewrite compatibility condition as
\begin{equation}
\begin{equation}gin{split}
\nablala_x f_0( x^{1},v) &= \nablala_x f_0(x^{1},R_{x^{1}}v) R_{x^{1}}
- \nablala_v f_0(x^{1}, R_{x^{1}}v)\left[2\left((v\cdotot n(x^1))I+(n(x^1) \otimes v) \right)\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right],\quad (x^{1},v) \in \gamma_{-}.
\end{split}
\end{equation}
\unhide
\subsection{$C^{1}_{t}$ condition of $f$}
To check the $C^1_t$ condition, we define
\begin{equation}gin{align}\label{Perb_t}
X^\varepsilonsilon(0):=X(0;t+\varepsilonsilon, x,v), \quad V^\varepsilonsilon(0):= V(0;t+\varepsilonsilon,x,v).
\end{align}
More specifically,
\begin{equation}gin{align*}
X^\varepsilonsilon(0)=x^1-(t^1+\varepsilonsilon) R_{x^{1}}v, \quad V^\varepsilonsilon(0)= R_{x^{1}}v,\quad \varepsilonsilon > 0,
\end{align*}
and
\begin{equation}gin{align*}
X^\varepsilonsilon(0)=x-(t+\varepsilonsilon)v,\quad V^\varepsilonsilon(0)=v,\quad \varepsilonsilon < 0.
\end{align*}
Thus, the case ($\varepsilonsilon>0$) describes the situation after bounce (backward in time) and the case ($\varepsilonsilon<0$) describes the situation just before bounce (backward in time). Then, for $\varepsilonsilon>0$,
\begin{equation}gin{align*}
f_t(t,x,v)&= \lim_{\varepsilonsilon\rightarrow0+}\frac{f(t+\varepsilonsilon,x,v)-f(t,x,v)}{\varepsilonsilon} \\
&=\lim_{\varepsilonsilon\rightarrow 0+} \frac{f_0(X^\varepsilonsilon(0),V^\varepsilonsilon(0))-f_0(X(0),V(0))}{\varepsilonsilon}\\
&=\lim_{\varepsilonsilon\rightarrow 0+} \frac{f_0(X^\varepsilonsilon(0),R_{x^{1}}v)-f_0(X(0),R_{x^{1}}v)}{\varepsilonsilon}\\
&=\nablala_x f_0(x^1,R_{x^{1}}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{X^\varepsilonsilon(0)-X(0)}{\varepsilonsilon}\\
&=-\nablala_x f_0(x^1,R_{x^{1}}v)R_{x^{1}}v.
\end{align*}
We only consider the situation just before collision and then
\begin{equation}gin{align*}
f_t(t,x,v)&= \lim_{\varepsilonsilon\rightarrow0-}\frac{f(t+\varepsilonsilon,x,v)-f(t,x,v)}{\varepsilonsilon} \\
&= \lim_{\varepsilonsilon\rightarrow0-}\frac{f_0(X^\varepsilonsilon(0),v)-f_0(X(0),v)}{\varepsilonsilon}\\
&= \nablala_xf_0(x^1,v) \lim_{\varepsilonsilon\rightarrow0-} \frac{X^\varepsilonsilon(0)-X(0)}{\varepsilonsilon}\\
&=-\nablala_x f_0(x^1,v)v.
\end{align*}
Thus, we derive a $C^1_t$ condition
\begin{equation}gin{equation} \label{c_t}
\nablala_x f_0(x^1,v)v = \nablala_x f_0(x^1,R_{x^{1}}v)R_{x^{1}}v,\quad (x^{1}, v)\in \gamma_{-}.
\end{equation}
Actually, \eqref{c_t} is just particular case of \eqref{c_v}, because of \eqref{Av=0}.
\hide
{\color{blue}
\begin{equation}gin{remark} \label{trivial case}
Let us consider trivial case : $f(t,x,v) = f_{0}(v)$, spatially independent case. Since specular reflection holds for all $x\in \partial\Omega$, $f_{0}$ should be radial function, $f_{0}(v) = f_{0}(|v|)$. \eqref{Cond} also holds for this case, because vector $\nablala_{v}f_{0}(x^{1},Rv)$ has $Rv$ direction and
\begin{equation}
\begin{equation}gin{split}
&\underbrace{(Rv)}_{\text{row vector}}\left((v\cdotot n(x^1))I+(n(x^1) \otimes v) \right)\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right) \\
&= \left((v\cdotot n(x^1)) (Rv) + (Rv\cdotot n(x^1)) v \right)\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right) \\
&= (v\cdotot n(x^1)) Rv - (v\cdotot n(x^{1}))v - (Rv\cdotot v)n(x^{1}) + |v|^{2}n(x^{1}) \\
&= (v\cdotot n(x^1)) \big(v - 2(v\cdotot n(x^{1}))n(x^{1}) \big)- (v\cdotot n(x^{1}))v - \big(|v|^{2} - 2|v\cdotot n(x^{1})|^{2}\big) n(x^{1}) + |v|^{2}n(x^{1}) \\
&= 0.
\end{split}
\end{equation}
\end{remark}
}
\unhide
\subsection{Proof of Theorem \e^{\frac 12}f{thm 1}}
\begin{equation}gin{proof} [Proof of Theorem \e^{\frac 12}f{thm 1}]
If $0 \neq t^{k}$ for any $k\in \mathbb{N}$, then $X(0;t,x,v)$ and $V(0;t,x,v)$ are both smooth function of $(t,x,v)$. By chain rule and $f_0\in C^{1}_{x,v}$, $f(t,x,v)$ of \eqref{solution} is also $C^{1}_{t,x,v}$. \\
Now let us assume $ 0 = t^{k}(t,x,v)$ for some $k\in \mathbb{N}$. From discontinuous property of $V(0;t,x,v)$, we consider the following two cases:
\begin{equation}gin{align*}
& \quad \lim_{s\rightarrow 0+} \nablala_v V(s) \textcolor{blue}{ ( \text{or} \ \nablala_vX(s))} = \underbrace{ \lim_{s\rightarrow 0+} \frac{\partialartial V(s)\textcolor{blue}{( \text{or} \ \partialartial X(s))}}{\partialartial(t^{k-1},x^{k-1}, v^{k-1})} } \frac{\partialartial(t^{k-1},x^{k-1}, v^{k-1})}{\partialartial v},\\
& \quad \lim_{s\rightarrow 0-} \nablala_v V(s) \textcolor{blue}{( \text{or} \ \nablala_vX(s))}= \underbrace{ \lim_{s\rightarrow 0-} \frac{\partialartial V(s)\textcolor{blue}{( \text{or} \ \partialartial X(s))}}{\partialartial(t^{k},x^{k}, v^{k})}\frac{\partialartial(t^{k},x^k,v^k)}{\partialartial(t^{k-1},x^{k-1},v^{k-1})} } \frac{\partialartial(t^{k-1},x^{k-1}, v^{k-1})}{\partialartial v}.
\end{align*}
First, we note that the factor $\mathrm{d}isplaystyle \frac{\partialartial(t^{k-1},x^{k-1}, v^{k-1})}{\partialartial v}$ which is common for both of above is smooth. From Lemma \e^{\frac 12}f{nabla xv b}, $t^{1}(t,x,v) = t - t_{\mathbf{b}}(x,v)$, $x^{1}(x,v) = x - t_{\mathbf{b}}(x,v) v$, and $v^{1}(x,v) = R_{x_{\mathbf{b}}(x,v)}v$ are all smooth functions of $(x,v)$ if $(t^{1}, x^{1}, v^{1})$ is nongrazing at $x^{1}$. Now, let us consider the mapping
\[
(t^{1}, x^{1}, v^{1}) \mapsto (t^{2}, x^{2}, v^{2})
\]
which is smooth by
\[
t^{2} = t^{1} - t_{\mathbf{b}}(x^{1}, v^{1}),\quad x^{2} = x^{1} - v^{1}t_{\mathbf{b}}(x^{1}, v^{1}),\quad v^{2} = R_{x^{2}}v^{1}.
\]
(Note that the derivative of $t_{\mathbf{b}}$ on $\partial\Omega \times \mathbb{R}^{3}_{v}$ can be performed by its local parametrization.)
By the chain rule, we easily derive that $(t^{k}, x^{k}, v^{k})$ is smooth in $(x, v)$. For explicit computation and their Jacobian, we refer to \cite{KimLee}.
Now, it suffices to compare above two underbraced terms only. {\bf It means that no generality is lost by setting $k=1$.} \\
Initial-boundary compatibility conditions for $C^{1}_{t,x,v}$ were obtained in \eqref{c_v}, \eqref{c_x}, and \eqref{c_t}. Since compatibility conditions \eqref{c_x} and \eqref{c_t} are covered by \eqref{c_v}, $f(t,x,v)\in C^{1}_{t,x,v}$ once \eqref{c_v} holds. To change \eqref{c_v} into more symmetric presentation \eqref{C1 cond}, we apply \eqref{comp_v} and multiply invertible matrix $R_{x^{1}}$ on both sides from the right to obtain
\begin{equation}gin{equation*}
\begin{equation}gin{split}
&\big( \nablala_x f_0(x^{1},v) + \nablala_v f_0(x^{1}, v) R_{x^1}A_{v,x^{1}} \big) R_{x^1} = \nablala_x f_0(x^1, R_{x^1}v)
- \nablala_v f_0(x^1, R_{x^1}v) A_{v,x^{1}}R_{x^1}. \\
\end{split}
\end{equation*}
This yields
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\begin{equation}ig[ \nablala_x f_0( x,v) + \nablala_v f_0(x, v) \frac{ (Qv)\otimes (Qv) }{v\cdotot n(x)} \begin{equation}ig]R_x
&=
\nablala_x f_0(x, R_xv)
+ \nablala_v f_0(x, R_xv) \frac{ (QR_xv)\otimes (QR_xv) }{R_xv\cdotot n(x)},
\end{split}
\end{equation*}
by \eqref{RA}. \\
Now we claim that compatibility condition \eqref{c_v} also holds for $(x^{1}, v)\in \gamma_{+}$.
By multiplying $R_{x^{1}}$ both sides and using $R^2_{x^1} = I$, $R_{x^1}n(x^1) = -n(x^1)$, and \eqref{comp_v}, we obtain
\begin{equation} \label{C1 gamma+}
\begin{equation}gin{split}
\nablala_x f_0( x^{1}, R_{x^1}v) &= \nablala_x f_0(x^{1}, v) R_{x^{1}}
+ 2 \nablala_v f_0(x^{1}, v) \underbrace{ R_{x^1}\left[\left((v\cdotot n(x^1))I+(n(x^1) \otimes v) \right)\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right] R_{x^1} }.
\end{split}
\end{equation}
Since $R_{x^1}=R^{T}_{x^1}$ (transpose), the underbraced term is written as
\begin{equation}
\begin{equation}gin{split}
&R_{x^{1}} A_{v,x^{1}} R_{x^{1}} \\
&= R_{x^{1}} \left[\left((v\cdotot n(x^1))I+(n(x^1) \otimes v) \right)\left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right] R_{x^{1}} \\
&= -(R_{x^1}v\cdotot n(x^{1}))I - R_{x^1}v\otimes R_{x^1}n(x^{1}) + R_{x^1}n(x^{1})\otimes R_{x^1}v - \frac{R_{x^1}n(x^{1})\otimes R_{x^1}n(x^{1})}{v\cdotot n(x^{1})}|R_{x^1}v|^{2} \\
&= - \left[\left((R_{x^1}v\cdotot n(x^1))I+(n(x^1) \otimes R_{x^1}v) \right)\left(I-\frac{R_{x^1}v\otimes n(x^1)}{R_{x^1}v\cdotot n(x^1)}\right)\right] \\
&= -A_{R_{x^1}v,x^{1}},
\end{split}
\end{equation}
and hence \eqref{C1 gamma+} is identical to \eqref{c_v} when $(x^{1},v)\in \gamma_{+}$.
Finally, we will prove that $f(t,x,v)$ is not of class $C^1_{t,x,v}$ at time $t$ such that $t^k(t,x,v)=0$ for some $k$ if \eqref{C1 cond} does not hold. As we used the chain rule, we set $t^1(t,x,v)=0$. Thus, it suffices to prove that $f(t,x,v)$ is not of class $C^1_{t,x,v}$ at time $t$ which satisfies $t^1(t,x,v)=0$ if \eqref{C1 cond} is not satisfied for $(X(0;t,x,v),v)\in \gamma_-$. Remind directional derivatives with respect to $\hat{r}_1$ and $\hat{r}_2$ to get $f \in C^1_{t,x,v}(\mathbb{R}_+\times \mathcal{I})$. In $C^1_v$ case, we deduced two conditions \eqref{case12 r1} and \eqref{case12 r2} from directional derivatives. However, if initial data $f_0$ does not satisfy the condition \eqref{C1 cond} at $(X(0;t,x,v),v)\in \gamma_-$, two conditions cannot coincide. It means that $f(t,x,v)$ is not $C^1_v$ at $t$ such that $t^1(t,x,v)=0$. Similar to $C^1_{t,x}$ cases, we get the same result.
\end{proof}
\section{Initial-boundary compatibility condition for $C^{2}_{t,x,v}$}
As mentioned in the beginning of the previous section, we treat the problem \eqref{eq} as 2D problem in a unit disk $\{x\in\mathbb{R}^{2} : |x| < 1 \}$.
And, throughout this section, we use the following notation to interchange column and row for notational convenience,
\[
\begin{equation}gin{pmatrix}
a \\ b
\end{pmatrix}
\stackrel{c\leftrightarrow r}{=}
\begin{equation}gin{pmatrix}
a & b
\end{pmatrix}
,\quad
\begin{equation}gin{pmatrix}
a & b
\end{pmatrix}
\stackrel{r\leftrightarrow c}{=}
\begin{equation}gin{pmatrix}
a \\ b
\end{pmatrix}. \\
\]
Similar to previous section, we assume \eqref{t1 zero}, i.e., $0=t^{1}(t,x,v)$. We also assume $f_0$ satisfies specular reflection \eqref{BC} and $C^{1}_{t,x,v}$ compatibility condition \eqref{c_v} (or \eqref{C1 cond}) in this section. \\
\subsection{Condition for $\nablala_{xv}$}
Similar to previous section, we split perturbed direction into \eqref{set R_sp}. We also note that $\nablala_{v}f(t,x,v)$ can be written as \eqref{c_1} or \eqref{c_2}, which are identical by assuming \eqref{c_v}. First, using \eqref{c_1}, $\hat{r}_{1}$ of \eqref{set R_sp}, and notation \eqref{XV epsilon x}
\begin{equation}gin{equation} \label{nabla_xv f case1}
\begin{equation}gin{split}
&\nablala_{xv} f(t,x,v) \hat{r}_1 \stackrel{c\leftrightarrow r}{=} \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( \nablala_{v}f(t,x+\varepsilonsilon \hat{r}_1,v) - \nablala_{v}f(t,x,v) \right ) \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\begin{equation}ig( \nablala_{v}\big[ f_0(X(0;t,x+\varepsilonsilon \hat{r}_1,v),V(0;t,x+\varepsilonsilon \hat{r}_1,v)) \big] - \big( -t \nablala_x f_0(X(0),v) + \nablala_v f_0(X(0),v) \big)\begin{equation}ig) \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}V^{\varepsilon}(0) \\
&\quad\quad\quad\quad - \big( -t \nablala_x f_0(X(0),v) + \nablala_v f_0(X(0),v) \big) \begin{equation}ig\} \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ -t\big[ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) - \nablala_{x}f_{0}(X(0), v) \big]
+
\big[ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) - \nablala_{v}f_{0}(X(0), v) \big] \begin{equation}ig\} \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ -t\big[ \nablala_{x}f_{0}(X^{\varepsilon}(0), v) - \nablala_{x}f_{0}(X(0), v) \big]
+
\big[ \nablala_{v}f_{0}(X^{\varepsilon}(0), v) - \nablala_{v}f_{0}(X(0), v) \big] \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \nablala_{xx}f_{0}(X(0),v)\lim_{s\rightarrow 0+} \nablala_{x}X(s) (-t )\hat{r}_1 + \nablala_{xv}f_{0}(X(0),v)\lim_{s\rightarrow 0+}\nablala_{x}X(s) \hat{r}_1 \\
&= \begin{equation}ig( \nablala_{xx}f_{0}(x^{1},v) (-t ) + \nablala_{xv}f_{0}(x^{1},v) \begin{equation}ig) \hat{r}_1,
\end{split}
\end{equation}
where we have used \eqref{nabla XV_v+}, \eqref{nabla XV_x+}, $\nablala_{v}X^{\varepsilon}(0)=-t I_{2}$., and $\nablala_{v}V^{\varepsilon}(0)= I_{2}$. Similarly, using \eqref{c_2} and $\hat{r}_{2}$ of \eqref{set R_sp},
\begin{equation}gin{equation} \notag
\begin{equation}gin{split}
&\nablala_{xv} f(t,x,v) \hat{r}_2 \stackrel{c\leftrightarrow r}{=} \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( \nablala_{v}f(t,x+\varepsilonsilon \hat{r}_2,v) - \nablala_{v}f(t,x,v) \right )\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}V^{\varepsilon}(0) \\
&\quad\quad\quad - \big( -t \nablala_x f_0(X(0),R_{x^1}v) R_{x^{1}} + \nablala_v f_0(X(0),R_{x^1}v) (R_{x^1} + 2tA_{v, x^{1}})\big) \begin{equation}ig\} \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{\nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}X^{\varepsilon}(0) + t \nablala_x f_0(X(0),R_{x^1}v) R_{x^{1}} \begin{equation}ig\} \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}V^{\varepsilon}(0) - \nablala_v f_0(X(0),R_{x^1}v) (R_{x^1} + 2tA_{v, x^{1}}) \begin{equation}ig\} \\
&:= I_{xv,1} + I_{xv,2} .
\end{split}
\end{equation}
Using \eqref{nabla XV_v-} and \eqref{nabla XV_x-},
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{xv,1} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{\nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}X^{\varepsilon}(0) - \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s\rightarrow 0-}\nablala_{v}X(s) \\
&\quad\quad \quad\quad\quad + \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s\rightarrow 0-} \nablala_{v}X(s) + t \nablala_x f_0(X(0),R_{x^1}v) R_{x^{1}} \begin{equation}ig\} ,\quad\quad \lim_{s\rightarrow 0-} \nablala_{v}X(s) = -tR_{x^{1}}, \\
&= \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}X^{\varepsilon}(0) - \lim_{s\rightarrow 0-} \nablala_{v}X(s) \begin{equation}ig)
+ \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\begin{equation}ig( \nablala_{x}f_{0}(X^{\varepsilon}(0),V^{\varepsilon}(0)) - \nablala_{x}f_{0}(X(0), R_{x^1}v) \begin{equation}ig) (-tR_{x^1}) \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}X^{\varepsilon}(0) - \lim_{s\rightarrow 0-} \nablala_{v}X(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad + (-tR_{x^1}) \begin{equation}ig( \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) \lim_{s\rightarrow 0-}\nablala_{x}X(s) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)\lim_{s\rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig) \hat{r}_{2} \\
&=
\underbrace{ \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}X(0;t, x+\varepsilonsilon \hat{r}_{2}, v) - \lim_{s\rightarrow 0-} \nablala_{v}X(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{:=(*)_{xv,1}\hat{r}_{2} } \\
&\quad
+ (-tR_{x^1})\big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^1} + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \big] \hat{r}_{2}, \\
\end{split}
\end{equation*}
and
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{xv,2} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}V^{\varepsilon}(0) - \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s\rightarrow 0-}\nablala_{v}V(s) \\
&\quad\quad\quad\quad + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) (R_{x^{1}} + 2tA_{v,x^{1}}) - \nablala_v f_0(X(0),R_{x^1}v) (R_{x^1} + 2tA_{v, x^{1}})\begin{equation}ig\} \\
&= \nablala_{v}f_{0}(x^{1}, R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}V^{\varepsilon}(0) - \lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \\
&\quad\quad\quad\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) - \nablala_{v}f_{0}(x^{1}, R_{x^1}v) \begin{equation}ig) (R_{x^{1}} + 2tA_{v,x^{1}}) \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}V^{\varepsilon}(0) - \lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad\quad\quad\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}})\begin{equation}ig( \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) \lim_{s\rightarrow 0-} \nablala_{x}X(s) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)\lim_{s\rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig) \hat{r}_{2} \\
&=
\underbrace{ \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}V(0;t,x+\varepsilonsilon \hat{r}_{2}, v) - \lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{:=(*)_{xv,2} \hat{r}_{2}} \\
&\quad\quad\quad\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v, x^{1}}) \big] \hat{r}_{2}. \\
\end{split}
\end{equation*}
Now we compute two underbraced $(*)_{xv,1}$ and $(*)_{xv,2}$ \\
\begin{equation}gin{equation} \label{xv star1}
\begin{equation}gin{split}
(*)_{xv,1} \hat{r}_{2} &= \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}X^{\varepsilon}(0) - \lim_{s\rightarrow 0-} \nablala_{v}X(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&=
\begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{s \rightarrow 0-}\nablala_{x}(\partial_{v_{1}}X(s)) \hat{r}_{2}, \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{s \rightarrow 0-}\nablala_{x}(\partial_{v_{2}}X(s)) \hat{r}_{2} \begin{equation}ig]^{T} \\
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{s \rightarrow 0-}\nablala_{x}(\partial_{v_{1}}X(s))
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{s \rightarrow 0-}\nablala_{x}(\partial_{v_{2}}X(s))
\end{bmatrix}
\hat{r}_{2} \\
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-t R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-t R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}. \\
\end{split}
\end{equation}
Similarly,
\begin{equation}gin{equation} \label{xv star2}
\begin{equation}gin{split}
(*)_{xv,2} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \lim_{s \rightarrow 0-}\nablala_{x}(\partial_{v_{1}}V(s))
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \lim_{s \rightarrow 0-}\nablala_{x}(\partial_{v_{2}}V(s))
\end{bmatrix}
\hat{r}_{2} \\
&= \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1 + 2t A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2 + 2t A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}, \\
\end{split}
\end{equation}
where $A^i$ means $i$th column of matrix $A$. Therefore,
\begin{equation}gin{equation} \label{nabla_xv f case2}
\begin{equation}gin{split}
\nablala_{xv}f(t,x,v)
&= \underline{(*)_{xv,1}}_{\eqref{xv star1}} + \underline{(*)_{xv,2}}_{\eqref{xv star2}} \\
&\quad + (-tR_{x^1})\big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^1} + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \big] \\
&\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v, x^{1}}) \big]. \\
\end{split}
\end{equation}
From \eqref{nabla_xv f case1} and \eqref{nabla_xv f case2}, we get the following compatibility condition
\begin{equation}gin{equation} \label{xv comp}
\begin{equation}gin{split}
&(-t)\nablala_{xx}f_{0}(x^{1},v) + \nablala_{xv}f_{0}(x^{1},v) \\
&= \underline{(*)_{xv,1}}_{\eqref{xv star1}} + \underline{(*)_{xv,2}}_{\eqref{xv star2}} \\
&\quad + (-tR_{x^1}) \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^1} + (-tR_{x^1})\nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \\
&\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}}
+ (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v, x^{1}}) .
\end{split}
\end{equation}
\subsection{Condition for $\nablala_{vv}$}
We split perturbed direction into \eqref{set R_vel}. $\nablala_{v}f(t,x,v)$ can be written as \eqref{c_1} or \eqref{c_2}. Using \eqref{c_1}, $\hat{r}_{1}$ of \eqref{set R_vel}, and notation \eqref{XV epsilon v}, \\
\begin{equation}gin{equation} \label{nabla_vv f case1}
\begin{equation}gin{split}
&\nablala_{vv} f(t,x,v) \hat{r}_1 \\
&\stackrel{c\leftrightarrow r}{=} \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ -t\big[ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) - \nablala_{x}f_{0}(X(0), v) \big]
+
\big[ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) - \nablala_{v}f_{0}(X(0), v) \big] \begin{equation}ig\} \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ -t\big[ \nablala_{x}f_{0}(X^{\varepsilon}(0), v+\varepsilonsilon \hat{r}_{1} ) - \nablala_{x}f_{0}(X(0), v) \big]
+
\big[ \nablala_{v}f_{0}(X^{\varepsilon}(0), v+\varepsilonsilon \hat{r}_{1} ) - \nablala_{v}f_{0}(X(0), v) \big] \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ -t \nablala_{xx}f_{0}(X(0),v)\lim_{s\rightarrow 0+} \nablala_{v}X(s) -t \nablala_{vx}f_{0}(X(0),v)\lim_{s\rightarrow 0+} \nablala_{v}V(s) \\
&\quad + \nablala_{xv}f_{0}(X(0),v)\lim_{s\rightarrow 0+}\nablala_{v}X(s) + \nablala_{vv}f_{0}(X(0),v)\lim_{s\rightarrow 0+}\nablala_{v}V(s) \begin{equation}ig] \hat{r}_1 \\
&= \begin{equation}ig[ (-t ) \nablala_{xx}f_{0}(x^{1},v) (-t) + (-t )\nablala_{vx}f_{0}(x^{1},v) + \nablala_{xv}f_{0}(x^{1},v) (-t ) + \nablala_{vv}f_{0}(x^{1},v) \begin{equation}ig] \hat{r}_1,
\end{split}
\end{equation}
where we have used \eqref{nabla XV_v+} and note that we have $\nablala_{v}X^{\varepsilon}(0)=-t I_{2}$ and $\nablala_{v}V^{\varepsilon}(0)= I_{2}$ for $v+\varepsilon\hat{r}_1$ case also. \\
Similarly, using \eqref{c_2}, $\hat{r}_{2}$ of \eqref{set R_vel}, and notation \eqref{XV epsilon v},
\begin{equation}gin{equation*}
\begin{equation}gin{split}
&\nablala_{vv} f(t,x,v) \hat{r}_{2} \\
&\stackrel{c\leftrightarrow r}{=} \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{\nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}X^{\varepsilon}(0) + t \nablala_x f_0(X(0),R_{x^1}v) R_{x^{1}} \begin{equation}ig\} \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}V^{\varepsilon}(0) - \nablala_v f_0(X(0),R_{x^1}v) (R_{x^1} + 2tA_{v, x^{1}}) \begin{equation}ig\} \\
&:= I_{vv,1} + I_{vv,2},
\end{split}
\end{equation*}
and each $I_{vv,1},I_{vv,2}$ are estimated by
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{vv,1} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{\nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}X^{\varepsilon}(0) - \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s\rightarrow 0-}\nablala_{v}X(s) \\
&\quad\quad \quad\quad\quad + \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s\rightarrow 0-} \nablala_{v}X(s) + t \nablala_x f_0(X(0),R_{x^1}v) R_{x^{1}} \begin{equation}ig\} ,\quad \lim_{s\rightarrow 0-} \nablala_{v}X(s) = -tR_{x^{1}}, \\
&\stackrel{r \leftrightarrow c}{= } \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}X^{\varepsilon}(0) - \lim_{s\rightarrow 0-} \nablala_{v}X(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad + (-tR_{x^1}) \begin{equation}ig( \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) \lim_{s\rightarrow 0-}\nablala_{v}X(s) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)\lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \hat{r}_{2} \\
&=
\underbrace{ \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}X(0; t, x, v+\varepsilonsilon \hat{r}_{2}) - \lim_{s\rightarrow 0-} \nablala_{v}X(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{(*)_{vv,1}\hat{r}_{2} } \\
&\quad + (-tR_{x^1}) \big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^1}) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (R_{x^1} + 2tA_{v,x^{1}}) \big] \hat{r}_{2}, \\
\end{split}
\end{equation*}
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{vv,2} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{v}V^{\varepsilon}(0) - \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s\rightarrow 0-}\nablala_{v}V(s) \\
&\quad\quad\quad\quad + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) (R_{x^{1}} + 2tA_{v,x^{1}}) - \nablala_v f_0(X(0),R_{x^1}v) (R_{x^1} + 2tA_{v, x^{1}}) \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}V^{\varepsilon}(0) - \lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad\quad\quad\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \begin{equation}ig( \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) \lim_{s\rightarrow 0-} \nablala_{v}X(s) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)\lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \hat{r}_{2} \\
&=
\underbrace{ \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{v}V(0; t, x, v+\varepsilonsilon \hat{r}_{2}) - \lim_{s\rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{(*)_{vv,2} \hat{r}_{2} } \\
&\quad\quad\quad\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^1}) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (R_{x^{1}} + 2tA_{v,x^{1}}) \big] \hat{r}_{2}. \\
\end{split}
\end{equation*}
Similar to \eqref{xv star1} and \eqref{xv star2}, using Lemma \e^{\frac 12}f{nabla xv b},
\begin{equation}gin{equation} \label{vv star1}
\begin{equation}gin{split}
(*)_{vv,1} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(-t R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(-t R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}
=
t^{2}
\begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}, \\
\end{split}
\end{equation}
\begin{equation}gin{equation} \label{vv star2}
\begin{equation}gin{split}
(*)_{vv,2} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(R_{x^{1}(x,v)}^1 + 2t A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(R_{x^{1}(x,v)}^2 + 2t A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2} \\
&=
-t
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}
-
2t^{2}
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}
\\
&\quad +
2t
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{1}_{v,x^{1}}
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{2}_{v,x^{1}}
\end{bmatrix}
\hat{r}_{2}.
\\
\end{split}
\end{equation}
Hence, we get
\begin{equation}gin{equation} \label{nabla_vv f case2}
\begin{equation}gin{split}
&\nablala_{vv} f(t,x,v) \\
&= \underline{(*)_{vv,1}}_{\eqref{vv star1}} + \underline{(*)_{vv,2}}_{\eqref{vv star2}} \\
&\quad + (-tR_{x^1}) \big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^1}) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (R_{x^1} + 2tA_{v,x^{1}}) \big] \\
&\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^1}) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (R_{x^{1}} + 2tA_{v,x^{1}}) \big]. \\
\end{split}
\end{equation}
Then from \eqref{nabla_vv f case1} and \eqref{nabla_vv f case2} we get the following compatibility condition.
\begin{equation}gin{equation} \label{vv comp}
\begin{equation}gin{split}
&(-t ) \nablala_{xx}f_{0}(x^{1},v) (-t) + (-t )\nablala_{vx}f_{0}(x^{1},v) + \nablala_{xv}f_{0}(x^{1},v) (-t ) + \nablala_{vv}f_{0}(x^{1},v) \\
&= \underline{(*)_{vv,1}}_{\eqref{vv star1}} + \underline{(*)_{vv,2}}_{\eqref{vv star2}} \\
&\quad + (-tR_{x^1}) \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^1}) + (-tR_{x^1})\nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (R_{x^1} + 2tA_{v,x^{1}}) \\
&\quad + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^1}) + (R_{x^{1}} + 2tA^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (R_{x^{1}} + 2tA_{v,x^{1}}). \\
\end{split}
\end{equation}
\subsection{Condition for $\nablala_{xx}$}
We split perturbed direction into \eqref{set R_sp}. $\nablala_{x}f(t,x,v)$ can be written as \eqref{c_3} or \eqref{c_4}, which are identical due to \eqref{c_v}. Using \eqref{c_3}, $\hat{r}_{1}$ of \eqref{set R_sp}, and notation \eqref{XV epsilon x}, \\
\begin{equation}gin{equation} \label{nabla_xx f case1}
\begin{equation}gin{split}
&\nablala_{xx} f(t,x,v) \hat{r}_1 \stackrel{c \leftrightarrow r}{=} \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( \nablala_{x}f(t,x+\varepsilonsilon \hat{r}_1,v) - \nablala_{x}f(t,x,v) \right ) \\
&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\begin{equation}ig( \nablala_{x}\big[ f_0(X(0;t,x+\varepsilonsilon \hat{r}_1,v),V(0;t,x+\varepsilonsilon \hat{r}_1,v)) \big] - \nablala_{x}f_{0}(X(0),v) \begin{equation}ig) \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \underbrace{\nablala_{x}V^{\varepsilon}(0)}_{=0} - \nablala_{x}f_{0}(X(0),v) \begin{equation}ig\} \\
&\stackrel{r \leftrightarrow c}{=} \nablala_{xx}f_{0}(x^{1},v) \lim_{s \rightarrow 0+}\nablala_{x}X(s) \hat{r}_1
= \nablala_{xx}f_{0}(x^{1},v) \hat{r}_1,
\end{split}
\end{equation}
where we have used \eqref{nabla XV_v+}, \eqref{nabla XV_x+}, $\nablala_{x}X^{\varepsilon}(0) = I_{2}$, and $\nablala_{x}V^{\varepsilon}(0)= 0$. Similarly, using \eqref{c_4}, $\hat{r}_{2}$ of \eqref{set R_sp}, and notation \eqref{XV epsilon x}, \\
\begin{equation}gin{equation} \notag
\begin{equation}gin{split}
&\nablala_{xx} f(t,x,v) \hat{r}_2 \stackrel{c \leftrightarrow r}{=} \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( \nablala_{x}f(t,x+\varepsilonsilon \hat{r}_2,v) - \nablala_{x}f(t,x,v) \right )\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) \\
&\quad\quad\quad - \big( \nablala_x f_0(X(0), R_{x^1}v) R_{x^{1}} - 2\nablala_v f_0(X(0),R_{x^1}v) A_{v,x^{1}} \big) \begin{equation}ig\} \\
&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) - \nablala_x f_0(X(0), R_{x^1}v) R_{x^{1}} \begin{equation}ig\} \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) + 2\nablala_v f_0(X(0),R_{x^1}v) A_{v,x^{1}} \begin{equation}ig\} \\
&:= I_{xx,1} + I_{xx,2} ,
\end{split}
\end{equation}
where
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{xx,1} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) - \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s \rightarrow 0-}\nablala_{x}X(s) \\
&\quad + \begin{equation}ig( \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) R_{x^{1}} - \nablala_x f_0(X(0), R_{x^1}v) R_{x^{1}} \begin{equation}ig) \begin{equation}ig\} \\
&\stackrel{r \leftrightarrow c}{=}
\underbrace{ \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}X(0; t, x+\varepsilonsilon\hat{r}_{2}, v) - \lim_{s \rightarrow 0-}\nablala_{x}X(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{(*)_{xx,1}\hat{r}_{2}} \\
&\quad + R_{x^{1}} \big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \big] \hat{r}_{2},
\end{split}
\end{equation*}
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{xx,2} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) - \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0))\lim_{s \rightarrow 0-}\nablala_{x}V(s) \\
&\quad - 2\nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) A_{v,x^{1}} + 2\nablala_v f_0(X(0),R_{x^1}v) A_{v,x^{1}} \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}V^{\varepsilon}(0) - \lim_{s \rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad + (- 2A^{T}_{v,x^{1}}) \begin{equation}ig\{ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v)\lim_{s \rightarrow 0-}\nablala_{x}X(s) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)\lim_{s \rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig\} \hat{r}_{2} \\
&= \underbrace{ \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}V(0; t, x+\varepsilonsilon\hat{r}_{2}, v) - \lim_{s \rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{(*)_{xx,2}\hat{r}_{2}} \\
&\quad + (- 2A^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)(-2 A_{v,x^{1}}) \big] \hat{r}_{2}. \\
\end{split}
\end{equation*}
Similar to \eqref{xv star1} and \eqref{xv star2},
\begin{equation}gin{equation} \label{xx star1}
\begin{equation}gin{split}
(*)_{xx,1} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}, \\
\end{split}
\end{equation}
\begin{equation}gin{equation} \label{xx star2}
\begin{equation}gin{split}
(*)_{xx,2} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(- 2 A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(- 2 A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}. \\
\end{split}
\end{equation}
Hence,
\begin{equation}gin{equation} \label{nabla_xx f case2}
\begin{equation}gin{split}
&\nablala_{xx} f(t,x,v) \\
&= \underline{(*)_{xx,1}}_{\eqref{xx star1}} + \underline{(*)_{xx,2}}_{\eqref{xx star2}} \\
&\quad + R_{x^{1}} \big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \big] \\
&\quad + (- 2A^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)(-2 A_{v,x^{1}}) \big].
\end{split}
\end{equation}
Then, from \eqref{nabla_xx f case1} and \eqref{nabla_xx f case2}, we get the following compatibility condition
\begin{equation}gin{equation} \label{xx comp}
\begin{equation}gin{split}
&\nablala_{xx}f_{0}(x^{1},v) \\
&= \underline{(*)_{xx,1}}_{\eqref{xx star1}} + \underline{(*)_{xx,2}}_{\eqref{xx star2}} \\
&\quad + R_{x^{1}} \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + R_{x^{1}} \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \\
&\quad + (- 2A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + (- 2A^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)(-2 A_{v,x^{1}}). \\
\end{split}
\end{equation}
\subsection{Condition for $\nablala_{vx}$} We split perturbed direction into \eqref{set R_vel}. $\nablala_{x}f(t,x,v)$ can be written as \eqref{c_3} or \eqref{c_4}. Using \eqref{c_3}, $\hat{r}_{1}$ of \eqref{set R_vel}, and notation \eqref{XV epsilon v}, \\
\begin{equation}gin{equation} \label{nabla_vx f case1}
\begin{equation}gin{split}
&\nablala_{vx} f(t,x,v) \hat{r}_1 \stackrel{c\leftrightarrow r}{=} \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( \nablala_{x}f(t,x,v+\varepsilonsilon \hat{r}_1) - \nablala_{x}f(t,x,v) \right ) \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\begin{equation}ig( \nablala_{x}\big[ f_0(X(0;t,x ,v+\varepsilonsilon \hat{r}_1),V(0;t,x ,v+\varepsilonsilon \hat{r}_1)) \big] - \nablala_{x}f_{0}(X(0),v) \begin{equation}ig) \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) - \nablala_{x}f_{0}(X(0),v) \begin{equation}ig\} \\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), v+\varepsilonsilon \hat{r}_{1} ) \nablala_{x}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), v+\varepsilonsilon \hat{r}_{1} ) \underbrace{ \nablala_{x}V^{\varepsilon}(0) }_{=0} - \nablala_{x}f_{0}(X(0),v) \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \nablala_{xx}f_{0}(x^{1}, v)\lim_{s \rightarrow 0+}\nablala_{v}X(s)\hat{r}_{1} + \nablala_{vx}f_{0}(x^{1}, v) \lim_{s \rightarrow 0+}\nablala_{v}V(s)\hat{r}_{1} \\
&= \big( \nablala_{xx}f_{0}(x^{1}, v)(-t) + \nablala_{vx}f_{0}(x^{1}, v) \big) \hat{r}_{1} , \\
\end{split}
\end{equation}
where we have used \eqref{nabla XV_v+}, \eqref{nabla XV_x+}, $\nablala_{x}X^{\varepsilon}(0) = I_{2}$, and $\nablala_{x}V^{\varepsilon}(0)= 0$. Similalry, using \eqref{c_4}, $\hat{r}_{2}$ of \eqref{set R_vel}, and notation \eqref{XV epsilon v}, \\
\begin{equation}gin{equation} \notag
\begin{equation}gin{split}
&\nablala_{vx} f(t,x,v) \hat{r}_2 \stackrel{c\leftrightarrow r}{=} \lim _{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon}\left ( \nablala_{x}f(t,x,v+\varepsilonsilon \hat{r}_2) - \nablala_{x}f(t,x,v) \right )\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) + \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) \\
&\quad\quad\quad - \big( \nablala_x f_0(X(0), R_{x^1}v) R_{x^{1}} - 2\nablala_v f_0(X(0),R_{x^1}v) A_{v,x^{1}} \big) \begin{equation}ig\} \\
&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) - \nablala_x f_0(X(0), R_{x^1}v) R_{x^{1}} \begin{equation}ig\} \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) + 2\nablala_v f_0(X(0),R_{x^1}v) A_{v,x^{1}} \begin{equation}ig\} \\
&:= I_{vx,1} + I_{vx,2},
\end{split}
\end{equation}
where
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{vx,1} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}X^{\varepsilon}(0) - \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \lim_{s \rightarrow 0-}\nablala_{x}X(s) \\
&\quad + \begin{equation}ig( \nablala_{x}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) R_{x^{1}} - \nablala_x f_0(X(0), R_{x^1}v) R_{x^{1}} \begin{equation}ig) \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}X^{\varepsilon}(0) - \lim_{s \rightarrow 0-}\nablala_{x}X(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad + R_{x^{1}} \begin{equation}ig\{ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v)\lim_{s \rightarrow 0-}\nablala_{v}X(s) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)\lim_{s \rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig\} \hat{r}_{2} \\
&= \underbrace{ \begin{equation}ig[ \nablala_{x}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}X(0; t, x, v+\varepsilonsilon \hat{r}_{2}) - \lim_{s \rightarrow 0-}\nablala_{x}X(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{(*)_{vx,1}
\hat{r}_{2} } \\
&\quad + R_{x^{1}} \big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v)(-tR_{x^1}) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (R_{x^{1}} + 2tA_{v,x^{1}}) \big] \hat{r}_{2},
\end{split}
\end{equation*}
\begin{equation}gin{equation*}
\begin{equation}gin{split}
I_{vx,2} &:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig\{ \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) \nablala_{x}V^{\varepsilon}(0) - \nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0))\lim_{s \rightarrow 0-}\nablala_{x}V(s) \\
&\quad - 2\nablala_{v}f_{0}(X^{\varepsilon}(0), V^{\varepsilon}(0)) A_{v,x^{1}} + 2\nablala_v f_0(X(0),R_{x^1}v) A_{v,x^{1}} \begin{equation}ig\} \\
&\stackrel{r\leftrightarrow c}{=} \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}V^{\varepsilon}(0) - \lim_{s \rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig) \begin{equation}ig]^{T} \\
&\quad + (-2A^{T}_{v,x^{1}}) \begin{equation}ig\{ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v)\lim_{s \rightarrow 0-}\nablala_{v}X(s) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v)\lim_{s \rightarrow 0-}\nablala_{v}V(s) \begin{equation}ig\} \hat{r}_{2} \\
&= \underbrace{ \begin{equation}ig[ \nablala_{v}f_{0}(x^{1}, R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \begin{equation}ig( \nablala_{x}V(0; t, x, v+\varepsilonsilon \hat{r}_{2}) - \lim_{s \rightarrow 0-}\nablala_{x}V(s) \begin{equation}ig) \begin{equation}ig]^{T} }_{(*)_{vx,2}\hat{r}_{2}} \\
&\quad + (-2A^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^{1}}) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) ( R_{x^{1}} + 2tA_{v,x^{1}}) \big] \hat{r}_{2}.
\end{split}
\end{equation*}
Similar to \eqref{xv star1} and \eqref{xv star2},
\begin{equation}gin{equation} \label{vx star1}
\begin{equation}gin{split}
(*)_{vx,1} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}
= -t
\begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2},
\\
\end{split}
\end{equation}
\begin{equation}gin{equation} \label{vx star2}
\begin{equation}gin{split}
(*)_{vx,2} \hat{r}_{2}
&= \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(- 2 A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}(- 2 A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2} \\
&=
2t
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\hat{r}_{2}
-2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{1}_{v,x^{1}}
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{2}_{v,x^{1}}
\end{bmatrix}
\hat{r}_{2}.
\\
\end{split}
\end{equation}
Hence,
\begin{equation}gin{equation} \label{nabla_vx f case2}
\begin{equation}gin{split}
&\nablala_{vx} f(t,x,v) \\
&= \underline{(*)_{vx,1}}_{\eqref{vx star1}} + \underline{(*)_{vx,2}}_{\eqref{vx star2}} \\
&\quad + R_{x^{1}} \big[ \nablala_{xx}f_{0}(x^{1}, R_{x^1}v)(-tR_{x^1}) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (R_{x^{1}} + 2tA_{v,x^{1}}) \big] \\
&\quad + (-2A^{T}_{v,x^{1}}) \big[ \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^{1}}) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) ( R_{x^{1}} + 2tA_{v,x^{1}}) \big].
\end{split}
\end{equation}
Then from \eqref{nabla_vx f case1} and \eqref{nabla_vx f case2} we get the following compatibility condition
\begin{equation}gin{equation} \label{vx comp}
\begin{equation}gin{split}
&\nablala_{xx}f_{0}(x^{1}, v)(-t) + \nablala_{vx}f_{0}(x^{1}, v) \\
&= \underline{(*)_{vx,1}}_{\eqref{vx star1}} + \underline{(*)_{vx,2}}_{\eqref{vx star2}} \\
&\quad + R_{x^{1}} \nablala_{xx}f_{0}(x^{1}, R_{x^1}v)(-tR_{x^1}) + R_{x^{1}} \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) (R_{x^{1}} + 2tA_{v,x^{1}}) \\
&\quad + (-2A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) (-tR_{x^{1}}) + (-2A^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) ( R_{x^{1}} + 2tA_{v,x^{1}}). \\
\end{split}
\end{equation}
\subsection{Compatibility conditions for transpose : $\nablala_{xv}^{T} = \nablala_{vx}$ and $\nablala_{xx}^{T} = \nablala_{xx}$}
First, we claim that \eqref{xv comp}, \eqref{vv comp}, \eqref{xx comp}, and \eqref{vx comp} imply the following four conditions for $(x^{1}, v)\in \gamma_{-}$
\begin{equation}gin{eqnarray}
\nablala_{xv}f_{0}(x^{1},v)
&=& R_{x^{1}}\nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + R_{x^{1}}\nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v, x^{1}}) \notag \\
&&\quad + \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1) \\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} ,\quad x^{1}=x^{1}(x,v), \label{Cond2 1} \\
\nablala_{xx}f_{0}(x^{1},v)
&=& R_{x^{1}} \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + R_{x^{1}} \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)(-2A_{v,x^{1}}) \notag \\
&&\quad + (-2A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + (-2A^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \notag \\
&&\quad + \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
- 2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^2)
\end{bmatrix},
\label{Cond2 2} \\
\nablala_{vv}f_{0}(x^{1},v)
&=& R_{x^{1}}\nablala_{vv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}}, \label{Cond2 3} \\
\nablala_{vx}f_{0}(x^{1},v)
&=& R_{x^{1}}\nablala_{vx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + (-2A^{T}_{v,x^{1}})\nablala_{vv}f_{0}(x^{1}, R_{x^1}v)R_{x^{1}} \notag \\
&&\quad -2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{1}_{v,x^{1}}
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{2}_{v,x^{1}}
\end{bmatrix}. \label{Cond2 4}
\end{eqnarray}
\eqref{xx comp} is just identical to \eqref{Cond2 2}. Then applying \eqref{xx comp} to \eqref{vx comp} and \eqref{xv comp}, we obtain \eqref{Cond2 1} and \eqref{Cond2 4}, respectively. Finally, applying \eqref{Cond2 1}, \eqref{Cond2 2}, and \eqref{Cond2 4} to \eqref{vv comp}, we obtain \eqref{Cond2 3} which is true by taking $\nablala_{v}^{2}$ to \eqref{BC} directly. \\
From \eqref{Cond2 1}--\eqref{Cond2 4}, we must check conditions to guarantee necessary conditions, $\nablala_{xv}^{T} = \nablala_{vx}$ and $\nablala_{xx}^{T} = \nablala_{xx}$. \\
\subsubsection{$\nablala_{xv}^T=\nablala_{vx}$}
From \eqref{Cond2 1} and \eqref{Cond2 4}, we need
\begin{equation} \label{T invariant}
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}^{T}
=
-2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{1}_{v,x^{1}}
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{2}_{v,x^{1}}
\end{bmatrix}.
\end{equation}
To check \eqref{T invariant}, we explicitly compute $\nablala_x(R_{x^1(x,v)}^1),\nablala_x(R_{x^1(x,v)}^2),\nablala_v(-2A^1_{v,x^1}),$ and $\nablala_v(-2A_{v,x^1}^2)$ in the following Lemma.
\begin{equation}gin{lemma} \label{d_RA}
Recall reflection operator $R_{x^1}$ in \eqref{BC} and $A_{v,x^1}$ in \eqref{def A},
\begin{equation}gin{equation*}
A_{v,x^1} := \left[ \left((v\cdotot n(x^1))I +(n(x^1)\otimes v)\right) \left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right].
\end{equation*}
We write that $A^i$ is the $i$th column of matrix $A$ and $\nablala_vA^i_{v,y}$ be the $v$-derivative of $A_{v,y}^i$ for $1\leq i \leq 2$ and $(v,y) \in \mathbb{R}^2 \times \partialartial \Omega$. Then,
\begin{equation}gin{align*}
&\nablala_x (R_{x^1(x,v)}^1) = \begin{equation}gin{bmatrix}
\mathrm{d}frac{-4v_2n_1n_2}{v\cdotot n(x^1)} & \mathrm{d}frac{4v_1n_1n_2}{v\cdotot n(x^1)} \\
\mathrm{d}frac{-2v_2(n_2^2-n_1^2)}{v\cdotot n(x^1)} & \mathrm{d}frac{2v_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}
\end{bmatrix}, \quad
\nablala_x (R_{x^1(x,v)}^2)= \begin{equation}gin{bmatrix}
\mathrm{d}frac{-2v_2(n_2^2-n_1^2)}{v\cdotot n(x^1)} & \mathrm{d}frac{2v_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}\\
\mathrm{d}frac{4v_2n_1n_2}{v\cdotot n(x^1)} & \mathrm{d}frac{-4v_1n_1n_2}{v\cdotot n(x^1)}
\end{bmatrix},\\
&\nablala_v(-2A_{v,x^1}^1)= \begin{equation}gin{bmatrix}
-\mathrm{d}frac{2v_2^2n_1}{(v\cdotot n(x^1))^2} & -2n_2-\mathrm{d}frac{2v_1^2n_1^2n_2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{4v_1v_2n_1^3}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{2v_2^2n_1^2n_2}{(v\cdotot n(x^1))^2} \\
-\mathrm{d}frac{2v_2^2n_2}{(v\cdotot n(x^1))^2} & 2n_1 -\mathrm{d}frac{2v_1^2 n_1n_2^2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{4v_1v_2 n_1^2n_2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{2v_2^2 n_1n_2^2}{(v\cdotot n(x^1))^2}
\end{bmatrix},\\
&\nablala_v(-2A_{v,x^1}^2)= \begin{equation}gin{bmatrix}
2n_2+\mathrm{d}frac{2v_1^2n_1^2n_2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{4v_1v_2n_1n_2^2}{(v\cdotot n(x^1))^2} - \mathrm{d}frac{2v_2^2n_1^2n_2}{(v\cdotot n(x^1))^2} & -\mathrm{d}frac{2v_1^2n_1}{(v\cdotot n(x^1))^2} \\
-2n_1 -\mathrm{d}frac{2v_2^2 n_1n_2^2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{4v_1v_2 n_2^3}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{2v_1^2 n_1n_2^2}{(v\cdotot n(x^1))^2} & -\mathrm{d}frac{2v_1^2n_2}{(v\cdotot n(x^1))^2}
\end{bmatrix},
\end{align*}
where $v_i$ be the $i$th component of $v$. We denote the $i$th component $n_i(x,v)$ of $n(x^1)$ as $n_i$, that is, $n_i$ depends on $x,v$. Moreover, the following identity holds that
\begin{equation}gin{equation}\label{prop d_R}
\nablala_{x}(R_{x^1(x,v)}^1)v =0, \quad \nablala_{x}(R_{x^1(x,v)}^2) v =0.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Recall the definition of the reflection matrix $R_{x^1}$ and $-2A_{v,x^1}$:
\begin{equation}gin{align*}
R_{x^1}&=I-2n(x^1)\otimes n(x^1) = \begin{equation}gin{bmatrix}
1-2n_1^2 & -2n_1n_2 \\
-2n_1n_2 & 1-2n_2^2
\end{bmatrix},\\
-2A_{v,x^1}&= -2 \left[ \left((v\cdotot n(x^1))I +(n(x^1)\otimes v)\right) \left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right]\\
&=\begin{equation}gin{bmatrix}
-2v_2n_2 -\mathrm{d}frac{2v_1v_2n_1n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2 n_1^2}{ v\cdotot n(x^1)} & 2v_1n_2 + \mathrm{d}frac{2v_1^2n_1n_2}{v\cdotot n(x^1)} -\mathrm{d}frac{2v_1v_2n_1^2}{ v\cdotot n(x^1)} \\
2v_2n_1 -\mathrm{d}frac{2v_1v_2n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2n_1n_2}{v\cdotot n(x^1)} & -2v_1n_1 -\mathrm{d}frac{2v_1v_2n_1n_2}{v\cdotot n(x^1)} + \mathrm{d}frac{2v_1^2n_2^2}{v \cdotot n(x^1)}
\end{bmatrix}.
\end{align*}
To find $\nablala_x (R_{x^1(x,v)}^1),\nablala_x (R_{x^1(x,v)}^2)$, we use \eqref{normal} in Lemma \e^{\frac 12}f{d_n}:
\begin{equation}gin{equation} \label{comp_dn}
\nablala_x [n(x^1(x,v))] = I-\frac{v\otimes n(x^1)}{ v\cdotot n(x^1)}= \begin{equation}gin{bmatrix}
\mathrm{d}frac{v_2n_2}{v\cdotot n(x^1)} & -\mathrm{d}frac{v_1n_2}{v\cdotot n(x^1)} \\
-\mathrm{d}frac{v_2n_1}{v \cdotot n(x^1)} & \mathrm{d}frac{v_1n_1}{v\cdotot n(x^1)}
\end{bmatrix}.
\end{equation}
Firstly, we directly calculate $\nablala_x (R_{x^1(x,v)}^1)$ and $\nablala_x(R_{x^1(x,v)}^2)$ using \eqref{comp_dn}:
\begin{equation}gin{align*}
\nablala_x (R_{x^1(x,v)}^1) = \nablala_x \begin{equation}gin{bmatrix}
1-2n_1^2 \\ -2n_1n_2
\end{bmatrix}=
\begin{equation}gin{bmatrix}
\mathrm{d}frac{-4v_2n_1n_2}{v\cdotot n(x^1)} & \mathrm{d}frac{4v_1n_1n_2}{v\cdotot n(x^1)} \\
\mathrm{d}frac{-2v_2(n_2^2-n_1^2)}{v\cdotot n(x^1)} & \mathrm{d}frac{2v_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}
\end{bmatrix},\\
\nablala_x (R_{x^1(x,v)}^2) = \nablala_x \begin{equation}gin{bmatrix}
-2n_1n_2 \\ 1-2n_2^2
\end{bmatrix}=
\begin{equation}gin{bmatrix}
\mathrm{d}frac{-2v_2(n_2^2-n_1^2)}{v\cdotot n(x^1)} & \mathrm{d}frac{2v_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}\\
\mathrm{d}frac{4v_2n_1n_2}{v\cdotot n(x^1)} & \mathrm{d}frac{-4v_1n_1n_2}{v\cdotot n(x^1)}
\end{bmatrix}.
\end{align*}
Next, we calculate the $v$-derivative of $[-2A_{v,x^1}^1]$:
\begin{equation}gin{align*}
(\nablala_v (-2A_{v,x^1}^1))_{(1,1)} &= -\frac{2v_2n_1n_2 (v \cdotot n(x^1))-2v_1v_2n_1^2n_2}{(v\cdotot n(x^1))^2}-\frac{2v_2^2n_1^3}{(v\cdotot n(x^1))^2}=-\frac{2v_2^2n_1}{(v\cdotot n(x^1))^2}, \\
(\nablala_v (-2A_{v,x^1}^1))_{(1,2)} &=-2n_2-\frac{2v_1n_1n_2(v\cdotot n(x^1))-2v_1v_2n_1n_2^2}{(v\cdotot n(x^1))^2} +\frac{4v_2n_1^2 (v\cdotot n(x^1)) -2v_2^2n_1^2n_2}{(v\cdotot n(x^1))^2}\\
&= -2n_2-\mathrm{d}frac{2v_1^2n_1^2n_2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{4v_1v_2n_1^3}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{2v_2^2n_1^2n_2}{(v\cdotot n(x^1))^2},\\
(\nablala_v (-2A_{v,x^1}^1))_{(2,1)}&=-\frac{2v_2n_2^2 (v\cdotot n(x^1)) -2v_1v_2n_1n_2^2}{(v\cdotot n(x^1))^2}-\frac{2v_2^2n_1^2n_2}{(v\cdotot n(x^1))^2}=-\frac{2v_2^2n_2}{(v\cdotot n(x^1))^2},\\
(\nablala_v (-2A_{v,x^1}^1))_{(2,2)}&=2n_1-\frac{2v_1n_2^2(v \cdotot n(x^1))-2v_1v_2n_2^3}{(v\cdotot n(x^1))^2}+\frac{4v_2n_1n_2 (v\cdotot n(x^1)) -2v_2^2n_1n_2^2}{(v\cdotot n(x^1))^2}\\
&= 2n_1 -\mathrm{d}frac{2v_1^2 n_1n_2^2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{4v_1v_2 n_1^2n_2}{(v\cdotot n(x^1))^2} + \mathrm{d}frac{2v_2^2 n_1n_2^2}{(v\cdotot n(x^1))^2}.
\end{align*}
Similarly, we deduce the $v$-derivative of $[-2A_{v,x^1}^2]$. We derived $\nablala_x(R_{x^1(x,v)}^1)$ and $\nablala_x(R_{x^1(x,v)}^2)$, and then \eqref{prop d_R} follows from direct calculation that
\begin{equation}gin{align*}
\nablala_{x}(R_{x^1(x,v)}^1) v = \begin{equation}gin{bmatrix}
\mathrm{d}frac{-4v_2n_1n_2}{v\cdotot n(x^1)} & \mathrm{d}frac{4v_1n_1n_2}{v\cdotot n(x^1)} \\
\mathrm{d}frac{-2v_2(n_2^2-n_1^2)}{v\cdotot n(x^1)} & \mathrm{d}frac{2v_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}
\end{bmatrix}\begin{equation}gin{bmatrix}
v_1 \\ v_2
\end{bmatrix} = 0, \\
\nablala_x(R_{x^1(x,v)}^2) v =\begin{equation}gin{bmatrix}
\mathrm{d}frac{-2v_2(n_2^2-n_1^2)}{v\cdotot n(x^1)} & \mathrm{d}frac{2v_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}\\
\mathrm{d}frac{4v_2n_1n_2}{v\cdotot n(x^1)} & \mathrm{d}frac{-4v_1n_1n_2}{v\cdotot n(x^1)}
\end{bmatrix} \begin{equation}gin{bmatrix}
v_1 \\ v_2
\end{bmatrix}=0.
\end{align*}
\end{proof}
Back to the point, we find the condition of $\nablala_v f_0(x^1,Rv)$ satisfying \eqref{T invariant}. Since
\begin{equation}gin{align*}
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, Rv) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, Rv) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}^{T}=\begin{equation}gin{bmatrix}
\nablala_vf_0(x^1,Rv) \mathrm{d}frac{\partialartial}{\partialartial x_1} (R_{x^1(x,v)}^1) & \nablala_vf_0(x^1,Rv) \mathrm{d}frac{\partialartial}{\partialartial x_1} (R_{x^1(x,v)}^2)\\
\nablala_vf_0(x^1,Rv) \mathrm{d}frac{\partialartial}{\partialartial x_2} (R_{x^1(x,v)}^1) & \nablala_vf_0(x^1,Rv) \mathrm{d}frac{\partialartial}{\partialartial x_2} (R_{x^1(x,v)}^2)
\end{bmatrix}, \\
-2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, Rv) \nablala_{v}A^{1}_{v,x^{1}}
\\
\nablala_{v}f_{0}(x^{1}, Rv) \nablala_{v}A^{2}_{v,x^{1}} .
\end{bmatrix}=\begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \mathrm{d}frac{\partialartial}{\partialartial v_1}(-2A^1_{v,x^1}) & \nablala_v f_0(x^1,R_{x^1}v) \mathrm{d}frac{\partialartial}{\partialartial v_2}(-2A^1_{v,x^1})\\
\nablala_v f_0(x^1,R_{x^1}v) \mathrm{d}frac{\partialartial}{\partialartial v_1}(-2A^2_{v,x^1}) & \nablala_v f_0(x^1,R_{x^1}v) \mathrm{d}frac{\partialartial}{\partialartial v_2}(-2A^2_{v,x^1})
\end{bmatrix},
\end{align*}
it suffices to find the condition of $\nablala_v f_0(x^1,R_{x^1}v)$ such that
\begin{equation}gin{align*}
\nablala_vf_0(x^1,R_{x^1}v) \left( \mathrm{d}frac{\partialartial}{\partialartial x_1} (R_{x^1(x,v)}^1) -\mathrm{d}frac{\partialartial}{\partialartial v_1} (-2A_{v,x^1}^1)\right) =0, \quad
\nablala_vf_0(x^1,R_{x^1}v) \left( \mathrm{d}frac{\partialartial}{\partialartial x_2} (R_{x^1(x,v)}^1) -\mathrm{d}frac{\partialartial}{\partialartial v_1} (-2A_{v,x^1}^2)\right) =0,\\
\nablala_vf_0(x^1,R_{x^1}v) \left( \mathrm{d}frac{\partialartial}{\partialartial x_1} (R_{x^1(x,v)}^2) -\mathrm{d}frac{\partialartial}{\partialartial v_2} (-2A_{v,x^1}^1)\right) =0,\quad
\nablala_vf_0(x^1,R_{x^1}v) \left( \mathrm{d}frac{\partialartial}{\partialartial x_2} (R_{x^1(x,v)}^2) -\mathrm{d}frac{\partialartial}{\partialartial v_2} (-2A_{v,x^1}^2)\right) =0.
\end{align*}
We denote column vectors
\begin{equation}gin{align*}
K_1 := \mathrm{d}frac{\partialartial}{\partialartial x_1} (R_{x^1(x,v)}^1) -\mathrm{d}frac{\partialartial}{\partialartial v_1} (-2A_{v,x^1}^1), \quad K_2:= \mathrm{d}frac{\partialartial}{\partialartial x_2} (R_{x^1(x,v)}^1) -\mathrm{d}frac{\partialartial}{\partialartial v_1} (-2A_{v,x^1}^2),\\
K_3 := \mathrm{d}frac{\partialartial}{\partialartial x_1} (R_{x^1(x,v)}^2) -\mathrm{d}frac{\partialartial}{\partialartial v_2} (-2A_{v,x^1}^1), \quad K_4:=\mathrm{d}frac{\partialartial}{\partialartial x_2} (R_{x^1(x,v)}^2) -\mathrm{d}frac{\partialartial}{\partialartial v_2} (-2A_{v,x^1}^2).
\end{align*}
To determine whether $\nablala_v f_0(x^1,R_{x^1}v)$ is a nonzero vector or not for \eqref{T invariant}, we need to calculate the following determinant
\begin{equation}gin{align*}
\mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_i & K_j \\ \vert & \vert \end{bmatrix}, \quad 1\leq i < j \leq 4.
\end{align*}
If every determinant has a value of zero, then $\nablala_v f_0(x^1,R_{x^1}v)$ satisfying \eqref{T invariant} is not the zero vector. We now show that every determinant is $0$ and $\nablala_v f_0(x^1,R_{x^1}v)$ is parallel to a particular direction to satisfy \eqref{T invariant}. Using Lemma \e^{\frac 12}f{d_RA} and $\vert n(x^1) \vert = n_1^2 +n_2^2=1$,\\
\textrm{(Case 1)} $(K_1 \leftrightarrow K_4) $
\begin{equation}gin{align*}
\mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_1 & K_4 \\ \vert & \vert \end{bmatrix}&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2 \mathrm{d}et
\begin{equation}gin{bmatrix}
2v_2n_1n_2 -\mathrm{d}frac{v_2^2n_1}{v\cdotot n(x^1)} & v_1 (n_1^2-n_2^2)-\mathrm{d}frac{v_1^2n_1}{v\cdotot n(x^1)}\\
v_2(n_2^2-n_1^2)-\mathrm{d}frac{v_2^2n_2}{v\cdotot n(x^1)} & 2v_1n_1n_2 -\mathrm{d}frac{v_1^2n_2}{v\cdotot n(x^1)}
\end{bmatrix}\\
&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2\left[\left( 4v_1v_2n_1^2n_2^2-\frac{2v_1^2v_2n_1n_2^2}{v\cdotot n(x^1)} -\frac{2v_1v_2^2n_1^2n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{v_1^2v_2^2n_1n_2}{(v\cdotot n(x^1))^2}\right) \right. \\
&\quad \left. -\left(-v_1v_2(n_2^2-n_1^2)^2-\frac{v_1v_2^2n_2(n_1^2-n_2^2)}{v\cdotot n(x^1)} -\frac{v_1^2v_2n_1(n_2^2-n_1^2)}{v\cdotot n(x^1)}+\frac{v_1^2v_2^2n_1n_2}{(v\cdotot n(x^1))^2}\right) \right]\\
\hide
&=\left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2\left[\left( 4v_1v_2n_1^2n_2^2 +v_1v_2(n_2^2-n_1^2)^2\right)+\left(-\frac{2v_1^2v_2n_1n_2^2}{v\cdotot n(x^1)}+\frac{v_1^2v_2n_1(n_2^2-n_1^2)}{v \cdotot n(x^1)}\right) \right. \\
&\quad \left.+\left(\frac{v_1v_2^2n_2(n_1^2-n_2^2)}{v\cdotot n(x^1)} -\frac{2v_1v_2^2n_1^2n_2}{v\cdotot n(x^1)} \right) \right]\\
\unhide
&=\left(\frac{-2}{v\cdotot n(x^1)}\right)^2\left( v_1v_2 -\frac{v_1^2v_2 n_1}{ v\cdotot n(x^1)} -\frac{v_1v_2^2n_2}{v\cdotot n(x^1)}\right)\\
&=0,
\end{align*}
\textrm{(Case 2)} $(K_1 \leftrightarrow K_2)$
\begin{equation}gin{align*}
\mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_1 & K_2 \\ \vert & \vert \end{bmatrix}&=\left(\frac{-2}{v\cdotot n(x^1)}\right)^2 \mathrm{d}et
\begin{equation}gin{bmatrix}
2v_2n_1n_2 -\mathrm{d}frac{v_2^2n_1}{v\cdotot n(x^1)} & -v_1n_1n_2+v_2n_2^2 -\mathrm{d}frac{(v_2^2-v_1^2)n_1^2n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_1v_2n_1n_2^2}{v\cdotot n(x^1)} \\
v_2(n_2^2-n_1^2)-\mathrm{d}frac{v_2^2n_2}{v\cdotot n(x^1)} & -v_1n_2^2 -v_2n_1n_2 -\mathrm{d}frac{(v_2^2-v_1^2)n_1n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_1v_2n_2^3}{v \cdotot n(x^1)}
\end{bmatrix}\\
&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2 \left[ \left(-2v_1v_2 n_1 n_2^3 -2v_2^2 n_1^2 n_2^2 -\mathrm{d}frac{2(v_2^2-v_1^2)v_2n_1^2n_2^3}{v\cdotot n(x^1)}+\mathrm{d}frac{4v_1v_2^2n_1n_2^4}{ v \cdotot n(x^1)} \right. \right. \\
&\quad+ \left. \left.\mathrm{d}frac{v_1v_2^2n_1n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{v_2^3n_1^2n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{(v_2^2-v_1^2)v_2^2n_1^2n_2^2}{(v \cdotot n(x^1))^2} -\mathrm{d}frac{2v_1v_2^3n_1n_2^3}{(v\cdotot n(x^1))^2} \right) \right. \\
&\quad \left. -\left(-v_1v_2n_1n_2(n_2^2-n_1^2)+v_2^2n_2^2(n_2^2-n_1^2)-\mathrm{d}frac{(v_2^2-v_1^2)v_2n_1^2n_2(n_2^2-n_1^2)}{v\cdotot n(x^1)}+\mathrm{d}frac{2v_1v_2^2n_1n_2^2(n_2^2-n_1^2)}{v\cdotot n(x^1)} \right. \right.\\
&\quad \left. \left. +\mathrm{d}frac{v_1v_2^2n_1n_2^2}{v\cdotot n(x^1)} -\mathrm{d}frac{v_2^3n_2^3}{v\cdotot n(x^1)} +\mathrm{d}frac{(v_2^2-v_1^2)v_2^2n_1^2n_2^2}{(v\cdotot n(x^1))^2}-\mathrm{d}frac{2v_1v_2^3n_1n_2^3}{(v\cdotot n(x^1))^2} \right) \right] \\
\hide
&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2 \left[ \left(-v_1v_2n_1n_2-v_2^2n_2^2\right)+\left(-\frac{(v_2^2-v_1^2)v_2n_1^2n_2}{v\cdotot n(x^1)}+\frac{2v_1v_2^2n_1n_2^2}{v\cdotot n(x^1)}+\frac{v_2^3n_2}{v \cdotot n(x^1)} \right) \right]\\
\unhide
&=\left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2\left[ -\frac{v_2^3n_2^3}{v\cdotot n(x^1)} -\frac{v_2^3n_1^2n_2}{v\cdotot n(x^1)} +\frac{v_2^3n_2}{v\cdotot n(x^1)}\right]\\
&=0,
\end{align*}
\textrm{(Case 3)} $(K_1 \leftrightarrow K_3)$
\begin{equation}gin{align*}
\mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_1 & K_3 \\ \vert & \vert \end{bmatrix}&=\left(\frac{-2}{v\cdotot n(x^1)}\right)^2 \mathrm{d}et
\begin{equation}gin{bmatrix}
2v_2n_1n_2 -\mathrm{d}frac{v_2^2n_1}{v\cdotot n(x^1)} &-v_2n_1^2 -v_1n_1n_2-\mathrm{d}frac{(v_1^2-v_2^2)n_1^2n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_1v_2n_1^3}{v\cdotot n(x^1)}\\
v_2(n_2^2-n_1^2)-\mathrm{d}frac{v_2^2n_2}{v\cdotot n(x^1)} & v_1n_1^2 -v_2n_1n_2 -\mathrm{d}frac{(v_1^2-v_2^2)n_1n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_1v_2n_1^2n_2}{v\cdotot n(x^1)}
\end{bmatrix}\\
&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2 \left[ \left(2v_1v_2 n_1^3 n_2 -2v_2^2 n_1^2 n_2^2 -\mathrm{d}frac{2(v_1^2-v_2^2)v_2n_1^2n_2^3}{v\cdotot n(x^1)}+\mathrm{d}frac{4v_1v_2^2n_1^3n_2^2}{ v \cdotot n(x^1)} \right. \right. \\
&\quad- \left. \left.\mathrm{d}frac{v_1v_2^2n_1^3}{v\cdotot n(x^1)} +\mathrm{d}frac{v_2^3n_1^2n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{(v_1^2-v_2^2)v_2^2n_1^2n_2^2}{(v \cdotot n(x^1))^2} -\mathrm{d}frac{2v_1v_2^3n_1^3n_2}{(v\cdotot n(x^1))^2} \right) \right. \\
&\quad \left. -\left(-v_2^2n_1^2(n_2^2-n_1^2)-v_1v_2n_1n_2(n_2^2-n_1^2)-\mathrm{d}frac{(v_1^2-v_2^2)v_2n_1^2n_2(n_2^2-n_1^2)}{v\cdotot n(x^1)}+\mathrm{d}frac{2v_1v_2^2n_1^3(n_2^2-n_1^2)}{v\cdotot n(x^1)} \right. \right.\\
&\quad \left. \left. +\mathrm{d}frac{v_2^3n_1^2n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{v_1v_2^2n_1n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{(v_1^2-v_2^2)v_2^2n_1^2n_2^2}{(v\cdotot n(x^1))^2}-\mathrm{d}frac{2v_1v_2^3n_1^3n_2}{(v\cdotot n(x^1))^2} \right) \right]\\
\hide
&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2 \left[ \left(v_1v_2n_1n_2-v_2^2n_1^2\right)+\left(-\frac{(v_1^2-v_2^2)v_2n_1^2n_2}{v\cdotot n(x^1)} +\frac{2v_1v_2^2n_1^3}{v \cdotot n(x^1)} -\frac{v_1v_2^2n_1}{v\cdotot n(x^1)}\right)\right]\\
\unhide
&= \left(\mathrm{d}frac{-2}{v\cdotot n(x^1)}\right)^2\left[ \frac{v_1v_2^2n_1^3}{v\cdotot n(x^1)} +\frac{v_1v_2^2n_1n_2^2}{v \cdotot n(x^1)} -\frac{v_1v_2^2n_1}{v\cdotot n(x^1)}\right]\\
&=0.
\end{align*}
Moreover, from (Case 1) and (Case 2), we deduce
\begin{equation}gin{align*}
\mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_2 & K_4 \\ \vert & \vert \end{bmatrix}=0.
\end{align*}
Likewise, it holds that
\begin{equation}gin{align*}
\mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_2 & K_3 \\ \vert & \vert \end{bmatrix}=0, \quad \mathrm{d}et \begin{equation}gin{bmatrix} \vert & \vert \\ K_3 & K_4 \\ \vert & \vert \end{bmatrix}=0.
\end{align*}
Therefore, it means that we can find a nonzero vector $\nablala_v f_0(x^1,R_{x^1}v)$ satisfying \eqref{T invariant}. Since
\begin{equation}gin{align*}
\nablala_v f_0(x^1,R_{x^1}v) \begin{equation}gin{bmatrix} \vert \\ K_1 \\ \vert \end{bmatrix} = 0,
\end{align*}
$\nablala _v f_0 (x^1,R_{x^1}v)$ is orthogonal to the column vector $K_1$. More specifically, $\nablala_v f_0(x^1,R_{x^1}v)^T$ has the following direction
\begin{equation}gin{align*}
\frac{-2}{v\cdotot n(x^1)} \begin{equation}gin{bmatrix}
-v_2(n_2^2-n_1^2) + \mathrm{d}frac{v_2^2 n_2}{v\cdotot n(x^1)} \\
2v_2n_1n_2-\mathrm{d}frac{v_2^2n_1}{v\cdotot n(x^1)}
\end{bmatrix}&=\frac{-2}{(v\cdotot n(x^1))^2}
\begin{equation}gin{bmatrix}
-v_1v_2n_1(n_2^2-n_1^2) +2v_2^2n_1^2n_2\\
2v_1v_2n_1^2n_2 +v_2^2n_1(n_2^2-n_1^2)
\end{bmatrix}\\
&=\frac{2v_2n_1}{(v\cdotot n(x^1))^2} \begin{equation}gin{bmatrix}
n_2^2-n_1^2 & -2n_1n_2\\
-2n_1n_2 & n_1^2 -n_2^2
\end{bmatrix} \begin{equation}gin{bmatrix}
v_1 \\ v_2
\end{bmatrix}= \frac{2v_2n_1}{(v\cdotot n(x^1))^2} R_{x^1}v.
\end{align*}
Consequently, for \eqref{T invariant}, we get the following condition
\begin{equation}gin{align} \label{Cond3}
\nablala _v f_0(x,R_xv) \partialarallel (R_xv)^T,
\end{align}
for any $x \in \partialartial \Omega$. \\
\subsubsection{$\nablala_{xx}^T =\nablala_{xx}$} From \eqref{Cond2 2}, we need
\begin{equation}gin{equation*}
\begin{equation}gin{split}
&\left(\begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} +
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix}\right)^T
\\
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} +
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix}.
\end{split}
\end{equation*}
Thus, it suffices to check that
\begin{equation}gin{align*}
&\nablala_x f_0(x^1,R_{x^1}v) \frac{\partialartial}{\partialartial x_2}(R_{x^1(x,v)}^1) +\nablala_v f_0(x^1,R_{x^1}v) \frac{\partialartial}{\partialartial x_2} (-2A_{v,x^1(x,v)}^1)\\
&= \nablala_x f_0(x^1,R_{x^1}v) \frac{\partialartial}{\partialartial x_1}(R_{x^1(x,v)}^2) +\nablala_v f_0(x^1,R_{x^1}v) \frac{\partialartial}{\partialartial x_1} (-2A_{v,x^1(x,v)}^2).
\end{align*}
In other words, we have to find the condition of $\nablala_x f_0 (x^1,R_{x^1}v)$ to satisfy
\begin{equation}gin{align}\label{xx_sym2}
\nablala_xf_0(x^1,R_{x^1}v) \left[\frac{\partialartial}{\partialartial x_2}(R_{x^1(x,v)}^1)-\frac{\partialartial}{\partialartial x_1}(R_{x^1(x,v)}^2) \right] = \nablala_v f_0(x^1,R_{x^1}v) \left[ \frac{\partialartial}{\partialartial x_1} (-2A_{v,x^1(x,v)}^2)-\frac{\partialartial}{\partialartial x_2} (-2A_{v,x^1(x,v)}^1)\right].
\end{align}
Since we computed $\nablala_x (R_{x^1(x,v)}^1), \nablala_x (R_{x^1(x,v)}^2)$ in Lemma \e^{\frac 12}f{d_RA}, we represent $\nablala_x (-2A_{v,x^1(x,v)}^1)$ and $\nablala_x (-2A_{v,x^1(x,v)}^2)$ by components.
\begin{equation}gin{lemma} \label{dx_A} Recall the matrix $A_{v,x}$ defined in \eqref{def A}, and then
\begin{equation}gin{equation*}
A_{v,x^1} = \left[ \left((v\cdotot n(x^1))I +(n(x^1)\otimes v)\right) \left(I-\frac{v\otimes n(x^1)}{v\cdotot n(x^1)}\right)\right].
\end{equation*}
If we write that $A^i$ is the $i$th column of matrix $A$, then
\begin{equation}gin{align*}
&\nablala_x(-2A_{v,x^1(x,v)}^1) \\
&= \begin{equation}gin{bmatrix}
\mathrm{d}frac{4v_1^2v_2^2n_1^3 +2v_1v_2^3(3n_1^2n_2-n_2^3)+ 2v_2^4(3n_1n_2^2+n_1^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{-4v_1^3v_2n_1^3-2v_1^2v_2^2(3n_1^2n_2-n_2^3)-2v_1v_2^3(3n_1n_2^2+n_1^3)}{(v\cdotot n(x^1))^3}\\
\mathrm{d}frac{4v_2^4n_2^3+2v_1v_2^3(3n_1n_2^2-n_1^3)+2v_1^2v_2^2(3n_1^2n_2+n_2^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{-4v_1v_2^3n_2^3-2v_1^2v_2^2(3n_1n_2^2-n_1^3)-2v_1^3v_2(3n_1^2n_2+n_2^3)}{(v\cdotot n(x^1))^3}
\end{bmatrix},\\
&\nablala_x(-2A_{v,x^1(x,v)}^2)\\
&= \begin{equation}gin{bmatrix}
\mathrm{d}frac{-4v_1^3v_2n_1^3-2v_1v_2^3(3n_1n_2^2+n_1^3) -2v_1^2v_2^2(3n_1^2n_2-n_2^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{4v_1^4n_1^3 +2v_1^2v_2^2(3n_1n_2^2+n_1^3)+2v_1^3v_2 (3n_1^2n_2-n_2^3)}{(v \cdotot n(x^1))^3}\\
\mathrm{d}frac{-4v_1v_2^3n_2^3 -2v_1^3v_2(3n_1^2n_2+n_2^3) -2v_1^2v_2^2(3n_1n_2^2-n_1^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{4v_1^2 v_2^2 n_2^3 +2v_1^4(3n_1^2n_2+n_2^3)+2v_1^3v_2(3n_1n_2^2-n_1^3)}{(v \cdotot n(x^1))^3}
\end{bmatrix},
\end{align*}
where $v_i$ be the $i$th component of $v$. We denote the $i$th component $n_i(x,v)$ of $n(x^1)$ as $n_i$, that is, $n_i$ depends on $x,v$. Furthermore, it holds that
\begin{equation}gin{equation} \label{prop d_A}
\nablala_x(-2A_{v,x^1(x,v)}^1)v =0, \quad \nablala_x (-2A_{v,x^1(x,v)}^2)v=0.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
We write the matrix $-2A_{v,x^1}$ by components:
\begin{equation}gin{align*}
-2A_{v,x^1}=\begin{equation}gin{bmatrix}
-2v_2n_2 -\mathrm{d}frac{2v_1v_2n_1n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2 n_1^2}{ v\cdotot n(x^1)} & 2v_1n_2 + \mathrm{d}frac{2v_1^2n_1n_2}{v\cdotot n(x^1)} -\mathrm{d}frac{2v_1v_2n_1^2}{ v\cdotot n(x^1)} \\
2v_2n_1 -\mathrm{d}frac{2v_1v_2n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2n_1n_2}{v\cdotot n(x^1)} & -2v_1n_1 -\mathrm{d}frac{2v_1v_2n_1n_2}{v\cdotot n(x^1)} + \mathrm{d}frac{2v_1^2n_2^2}{v \cdotot n(x^1)}
\end{bmatrix}.
\end{align*}
For $\nablala_x(-2A_{v,x^1(x,v)}^1)$, we firstly take a derivative of $(1,1)$ component of $-2A_{v,x^1}$ with respect to $x_1$
\begin{equation}gin{align*}
&\frac{\partialartial}{\partialartial x_1} \left(-2v_2n_2 -\mathrm{d}frac{2v_1v_2n_1n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2 n_1^2}{ v\cdotot n(x^1)} \right)\\
&=-2v_2 \frac{\partialartial n_2}{\partialartial x_1} + \mathrm{d}frac{\left (-2v_1v_2 n_2\frac{\partialartial n_1}{\partialartial x_1} - 2 v_1v_2n_1 \frac{\partialartial n_2}{\partialartial x_1}\right)(v\cdotot n(x^1))+2v_1v_2n_1n_2\left(v_1\frac{\partialartial n_1}{\partialartial x_1}+v_2\frac{\partialartial n_2}{\partialartial x_1}\right)}{(v\cdotot n(x^1))^2}\\
& \quad+ \mathrm{d}frac{\left(4v_2^2n_1\frac{\partialartial n_1}{\partialartial x_1}\right)(v\cdotot n(x^1))-2v_2^2n_1^2\left(v_1\frac{\partialartial n_1}{\partialartial x_1}+v_2\frac{\partialartial n_2}{\partialartial x_1}\right)}{(v \cdotot n(x^1))^2}\\
\hide
&=\frac{4v_1v_2^2n_1^2-2v_1v_2^2n_2^2+6v_2^3n_1n_2}{(v\cdotot n(x^1))^2} +\frac{2v_1^2v_2^2n_1n_2^2-4v_1v_2^3n_1^2n_2+2v_2^4n_1^3}{(v\cdotot n(x^1))^3}\\
\unhide
&=\mathrm{d}frac{4v_1^2v_2^2n_1^3 +2v_1v_2^3(3n_1^2n_2-n_2^3)+ 2v_2^4(3n_1n_2^2+n_1^3)}{(v\cdotot n(x^1))^3},
\end{align*}
where we used \eqref{normal} in Lemma \e^{\frac 12}f{d_n}. Similarly, using \eqref{normal} in Lemma \e^{\frac 12}f{d_n}, we get
\begin{equation}gin{align*}
&\frac{\partialartial}{\partialartial x_2} \left(-2v_2n_2 -\mathrm{d}frac{2v_1v_2n_1n_2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2 n_1^2}{ v\cdotot n(x^1)} \right)\\
&=-2v_2 \frac{\partialartial n_2}{\partialartial x_2} + \mathrm{d}frac{\left (-2v_1v_2 n_2\frac{\partialartial n_1}{\partialartial x_2} - 2 v_1v_2n_1 \frac{\partialartial n_2}{\partialartial x_2}\right)(v\cdotot n(x^1))+2v_1v_2n_1n_2\left(v_1\frac{\partialartial n_1}{\partialartial x_2}+v_2\frac{\partialartial n_2}{\partialartial x_2}\right)}{(v\cdotot n(x^1))^2}\\
& \quad+ \frac{\left(4v_2^2n_1\frac{\partialartial n_1}{\partialartial x_2}\right)(v\cdotot n(x^1))-2v_2^2n_1^2\left(v_1\frac{\partialartial n_1}{\partialartial x_2}+v_2\frac{\partialartial n_2}{\partialartial x_2}\right)}{(v \cdotot n(x^1))^2}\\
\hide
&=\frac{-4v_1^2v_2n_1^2-6v_1v_2^2n_1n_2+2v_1^2v_2n_2^2}{(v \cdotot n(x^1))^2} + \frac{4v_1^2v_2^2n_1^2n_2-2v_1^3v_2n_1n_2^2-2v_1v_2^3n_1^3}{(v\cdotot n(x^1))^3} \\
\unhide
&= \mathrm{d}frac{-4v_1^3v_2n_1^3-2v_1^2v_2^2(3n_1^2n_2-n_2^3)-2v_1v_2^3(3n_1n_2^2+n_1^3)}{(v\cdotot n(x^1))^3},\\
&\frac{\partialartial}{\partialartial x_1} \left(2v_2n_1 -\mathrm{d}frac{2v_1v_2n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2n_1n_2}{v\cdotot n(x^1)}\right)\\
&=2v_2\frac{\partialartial n_1}{\partialartial x_1} - \frac{\left(4v_1v_2n_2\frac{\partialartial n_2}{\partialartial x_1}\right)(v\cdotot n(x^1))-2v_1v_2n_2^2\left(v_1\frac{\partialartial n_1}{\partialartial x_1}+v_2\frac{\partialartial n_2}{\partialartial x_1}\right) }{(v \cdotot n(x^1))^2}\\
&\quad+ \frac{\left( 2v_2^2n_2\frac{\partialartial n_1}{\partialartial x_1} +2v_2^2n_1\frac{\partialartial n_2}{\partialartial x_1}\right) (v\cdotot n(x^1))-2v_2^2n_1n_2\left( v_1 \frac{\partialartial n_1}{\partialartial x_1}+v_2 \frac{\partialartial n_2}{\partialartial x_1}\right)}{(v\cdotot n(x^1))^2}\\
\hide
&=\frac{6v_1v_2^2n_1n_2+4v_2^3n_2^2-2v_2^3n_1^2}{(v\cdotot n(x^1))^2} +\frac{2v_1^2v_2^2n_2^3-4v_1v_2^3n_1n_2^2+2v_2^4n_1^2n_2}{(v\cdotot n(x^1))^3}\\
\unhide
&=\mathrm{d}frac{4v_2^4n_2^3+2v_1v_2^3(3n_1n_2^2-n_1^3)+2v_1^2v_2^2(3n_1^2n_2+n_2^3)}{(v\cdotot n(x^1))^3},\\
&\frac{\partialartial}{\partialartial x_2} \left(2v_2n_1 -\mathrm{d}frac{2v_1v_2n_2^2}{v\cdotot n(x^1)} +\mathrm{d}frac{2v_2^2n_1n_2}{v\cdotot n(x^1)}\right)\\
&=2v_2\frac{\partialartial n_1}{\partialartial x_2} - \frac{\left(4v_1v_2n_2\frac{\partialartial n_2}{\partialartial x_2}\right)(v\cdotot n(x^1))-2v_1v_2n_2^2\left(v_1\frac{\partialartial n_1}{\partialartial x_2}+v_2\frac{\partialartial n_2}{\partialartial x_2}\right) }{(v \cdotot n(x^1))^2}\\
&\quad+ \frac{\left( 2v_2^2n_2\frac{\partialartial n_1}{\partialartial x_2} +2v_2^2n_1\frac{\partialartial n_2}{\partialartial x_2}\right) (v\cdotot n(x^1))-2v_2^2n_1n_2\left( v_1 \frac{\partialartial n_1}{\partialartial x_2}+v_2 \frac{\partialartial n_2}{\partialartial x_2}\right)}{(v\cdotot n(x^1))^2}\\
\hide
&\quad -\frac{2v_1v_2^2n_2^2}{(v\cdotot n(x^1))^2} +\frac{2v_1v_2^2n_1^2}{(v\cdotot n(x^1))^2} +\frac{2v_1^2v_2^2n_1n_2^2}{(v\cdotot n(x^1))^3} -\frac{2v_1v_2^3n_1^2n_2}{(v\cdotot n(x^1))^3}\\
\unhide
&=\mathrm{d}frac{-4v_1v_2^3n_2^3-2v_1^2v_2^2(3n_1n_2^2-n_1^3)-2v_1^3v_2(3n_1^2n_2+n_2^3)}{(v\cdotot n(x^1))^3}.
\end{align*}
Thus, we derived $\nablala_x(-2A_{v,x^1(x,v)}^1)$. Similar to $\nablala_x(-2A_{v,x^1(x,v)}^1)$, we can obtain $\nablala_x(-2A_{v,x^1(x,v)}^2)$, and the details are omitted. By the $\nablala_x(-2A_{v,x^1(x,v)}^1)$ and $\nablala_x(-2A_{v,x^1(x,v)}^2)$ formula above, direct calculation gives \eqref{prop d_A}:
\begin{equation}gin{footnotesize}
\begin{equation}gin{align*}
&\nablala_x(-2A_{v,x^1(x,v)}^1) v \\
&= \begin{equation}gin{bmatrix}
\mathrm{d}frac{4v_1^2v_2^2n_1^3 +2v_1v_2^3(3n_1^2n_2-n_2^3)+ 2v_2^4(3n_1n_2^2+n_1^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{-4v_1^3v_2n_1^3-2v_1^2v_2^2(3n_1^2n_2-n_2^3)-2v_1v_2^3(3n_1n_2^2+n_1^3)}{(v\cdotot n(x^1))^3}\\
\mathrm{d}frac{4v_2^4n_2^3+2v_1v_2^3(3n_1n_2^2-n_1^3)+2v_1^2v_2^2(3n_1^2n_2+n_2^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{-4v_1v_2^3n_2^3-2v_1^2v_2^2(3n_1n_2^2-n_1^3)-2v_1^3v_2(3n_1^2n_2+n_2^3)}{(v\cdotot n(x^1))^3}
\end{bmatrix} \begin{equation}gin{bmatrix}
v_1 \\ v_2
\end{bmatrix}=0,\\
&\nablala_x(-2A_{v,x^1(x,v)}^2) v \\
&=\begin{equation}gin{bmatrix}
\mathrm{d}frac{-4v_1^3v_2n_1^3-2v_1v_2^3(3n_1n_2^2+n_1^3) -2v_1^2v_2^2(3n_1^2n_2-n_2^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{4v_1^4n_1^3 +2v_1^2v_2^2(3n_1n_2^2+n_1^3)+2v_1^3v_2 (3n_1^2n_2-n_2^3)}{(v \cdotot n(x^1))^3}\\
\mathrm{d}frac{-4v_1v_2^3n_2^3 -2v_1^3v_2(3n_1^2n_2+n_2^3) -2v_1^2v_2^2(3n_1n_2^2-n_1^3)}{(v\cdotot n(x^1))^3} & \mathrm{d}frac{4v_1^2 v_2^2 n_2^3 +2v_1^4(3n_1^2n_2+n_2^3)+2v_1^3v_2(3n_1n_2^2-n_1^3)}{(v \cdotot n(x^1))^3}
\end{bmatrix} \begin{equation}gin{bmatrix}
v_1 \\ v_2
\end{bmatrix}=0.
\end{align*}
\end{footnotesize}
\end{proof}
Now, back to our consideration \eqref{xx_sym2}. By Lemma \e^{\frac 12}f{dx_A}, we have
\begin{equation}gin{align*}
\frac{\partialartial}{\partialartial x_2} (-2A_{v,x^1(x,v)}^1)= \frac{\partialartial}{\partialartial x_1}(-2A_{v,x^1(x,v)}^2),
\end{align*}
which implies that
\begin{equation}gin{align*}
\nablala_xf_0(x^1,R_{x^1}v) \left[\frac{\partialartial}{\partialartial x_2}(R_{x^1(x,v)}^1)-\frac{\partialartial}{\partialartial x_1}(R_{x^1(x,v)}^2) \right]=\frac{2}{v\cdotot n(x^1)}\nablala_xf_0(x^1,R_{x^1}v) \begin{equation}gin{bmatrix}
2v_1n_1n_2 +v_2(n_2^2-n_1^2) \\
v_1(n_2^2-n_1^2)-2v_2n_1n_2
\end{bmatrix}=0.
\end{align*}
It means that $\nablala_x f_0(x^1,R_{x^1}v)$ is orthogonal to $\frac{\partialartial}{\partialartial x_2}(R_{x^1(x,v)}^1)-\frac{\partialartial}{\partialartial x_1}(R_{x^1(x,v)}^2)$ and $\nablala_xf_0(x^1,R_{x^1}v)^T$ has the following direction
\begin{equation}gin{align*}
\begin{equation}gin{bmatrix}
-v_1(n_2^2-n_1^2)+2v_2n_1n_2 \\ 2v_1n_1n_2+v_2(n_2^2-n_1^2)
\end{bmatrix}=-\begin{equation}gin{bmatrix}
n_2^2-n_1^2 & -2n_1n_2 \\ -2n_1n_2 & n_1^2-n_2^2
\end{bmatrix}\begin{equation}gin{bmatrix}
v_1 \\ v_2
\end{bmatrix}=-R_{x^1}v.
\end{align*}
To hold $\nablala_{xx} f_0(x^1,R_{x^1}v)^T = \nablala_{xx} f_0 (x^1,R_{x^1}v)$, the following condition
\begin{equation}gin{align} \label{Cond4}
\nablala_xf_0(x,R_xv) \partialarallel (R_xv)^T,
\end{align}
must be satisfied for $x \in \partialartial \Omega$. \\
\subsection{Conditions including $\partial_{t}$}
In this subsection, we find conditions for $\partial_{tt}, \partial_{t}\nablala_{x}, \partial_{t}\nablala_{v}, \nablala_{x}\partial_{t}, \nablala_{v}\partial_{t}$. In the last subsubsection, we show that all these $\partial_{t}$ including compatibility conditions are covered by \eqref{Cond2 1}--\eqref{Cond2 4}, \eqref{Cond3}, and \eqref{Cond4}. \\
\subsubsection{$\partialartial_{tt}$} Using the same perturbation \eqref{Perb_t} in $C^1_t$ compatibility condition, we derive $C^2_t$ compatibility condition. For $\varepsilonsilon>0$,
\begin{equation}gin{align*}
\partialartial_t(f(t+\varepsilonsilon,x,v)-f(t,x,v))&= \partialartial_t (f_0(X^\varepsilonsilon(0),R_{x^1}v)-f_0(X(0),R_{x^1}v))\\
&=\left( \nablala_x f_0(X^\varepsilonsilon(0),R_{x^1}v)-\nablala_xf_0(X(0),R_{x^1}v)\right) (-R_{x^1}v) \\
&=(-R_{x^1}v)^T \left (\nablala_x f_0(X^\varepsilonsilon(0),R_{x^1}v) -\nablala_xf_0(X(0),R_{x^1}v) \right)^T,
\end{align*}
which implies
\begin{equation}gin{align*}
f_{tt}(t,x,v) &= \lim_{\varepsilonsilon \rightarrow 0+}\frac{ \partialartial_t f(t+\varepsilonsilon,x,v)-\partialartial_t f(t,x,v)}{\varepsilonsilon}\\
&=(-R_{x^1}v)^T \nablala_{xx} f_0(x^1,R_{x^1}v) \lim_{\varepsilonsilon\rightarrow 0+}\frac{ X^\varepsilonsilon(0)-X(0)}{\varepsilonsilon}\\
&=(-R_{x^1}v)^T \nablala_{xx} f_0(x^1,R_{x^1}v) (-R_{x^1}v).
\end{align*}
On the other hand, for $\varepsilonsilon<0$, it holds that
\begin{equation}gin{align*}
\partialartial_t(f(t+\varepsilonsilon,x,v)-f(t,x,v))= \partialartial_t (f_0(X^\varepsilonsilon(0),v)-f_0(X(0),v))&=\left( \nablala_x f_0(X^\varepsilonsilon(0),v)-\nablala_xf_0(X(0),v)\right) (-v)\\
&=(-v)^T \left( \nablala_x f_0(X^\varepsilonsilon(0),v)-\nablala_xf_0(X(0),v)\right)^T.
\end{align*}
Thus, we have
\begin{equation}gin{align*}
f_{tt}(t,x,v) &= \lim_{\varepsilonsilon \rightarrow 0-}\frac{ \partialartial_t f(t+\varepsilonsilon,x,v)-\partialartial_t f(t,x,v)}{\varepsilonsilon}\\
&=(-v)^T \nablala_{xx} f_0(x^1,v) \lim_{\varepsilonsilon\rightarrow 0-}\frac{ X^\varepsilonsilon(0)-X(0)}{\varepsilonsilon}\\
&=(-v)^T \nablala_{xx} f_0(x^1,v) (-v).
\end{align*}
To sum up, the condition
\begin{equation}gin{align} \label{time cond}
v^T \nablala_{xx}f_0(x^1,v)v = (R_{x^1}v)^T \nablala_{xx}f_0(x^1,R_{x^1}v)(R_{x^1}v),
\end{align}
must be satisfied to $f \in C^2_t$. \\
\subsubsection{$C^2_{t,x}$} We firstly use the perturbation \eqref{Perb_t} for $\varepsilonsilon <0$. From \eqref{c_3}, it holds that
\begin{equation}gin{equation} \label{nabla_tx f case1}
\begin{equation}gin{split}
\partialartial_t [\nablala_xf(t,x,v)]&= \lim_{\varepsilonsilon \rightarrow 0-} \frac{ \nablala_x f(t+\varepsilonsilon,x,v) - \nablala_xf(t,x,v)}{\varepsilonsilon}\\
&=\lim_{\varepsilonsilon \rightarrow 0-} \frac{1}{\varepsilonsilon} \left( \nablala_x \left[ f_0(X(0;t+\varepsilonsilon,x,v),V(0;t+\varepsilonsilon,x,v))\right]-\nablala_xf_0(X(0),v)\right)\\
&=\lim_{\varepsilonsilon \rightarrow 0-} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0(X^\varepsilonsilon(0),v)-\nablala_x f_0(X(0),v)\right)\\
&=-v^T \nablala_{xx} f_0(x^1,v),
\end{split}
\end{equation}
where we used $\nablala_x X^{\varepsilonsilon}(0) = I_2$ and $\nablala_x V^{\varepsilonsilon}(0)=0$. On the other hand, for $\varepsilonsilon>0$,
\begin{equation}gin{align*}
X^{\varepsilonsilon}(0):= X(0;t+\varepsilonsilon,x,v)=X(0;t,x-\varepsilonsilon v, v), \quad V^{\varepsilonsilon}(0):=V(0;t+\varepsilonsilon,x,v)=R_{x^1}v.
\end{align*}
Similar to previous case $\nablala_{xx}$, using \eqref{nabla XV_x-} and \eqref{c_4},
\begin{equation}gin{align*}
\partialartial_t [\nablala_xf(t,x,v)]&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{ \nablala_x f(t+\varepsilonsilon,x,v) - \nablala_xf(t,x,v)}{\varepsilonsilon}\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x \left[ f_0(X(0;t+\varepsilonsilon,x,v),V(0;t+\varepsilonsilon,x,v))\right] \right. \\
&\left.\quad - \left(\nablala_x f_0(X(0),R_{x^1}v)R_{x^1} -2\nablala_v f_0(X(0),R_{x^1}v)A_{v,x^1} \right) \right)\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x X^{\varepsilonsilon}(0) +\nablala_v f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0)) \nablala_x V^{\varepsilonsilon}(0)\right. \\
&\quad \left. -\left( \nablala_x f_0(X(0),R_{x^1}v)R_{x^1} -2\nablala_v f_0(X(0),R_{x^1}v)A_{v,x^1}\right) \right)\\
&=\lim _{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0 (X^{\varepsilonsilon}(0),R_{x^1}v) \nablala_x X^{\varepsilonsilon}(0) -\nablala_x f_0(X(0),R_{x^1}v)R_{x^1}\right) \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_v f_0(X^{\varepsilonsilon}(0),R_{x^1}v) \nablala_xV^{\varepsilonsilon}(0) +2\nablala_v f_0(X(0),R_{x^1}v)A_{v,x^1}\right) \\
&:= I_{tx,1}+I_{tx,2},
\end{align*}
where
\begin{equation}gin{align*}
I_{tx,1}&:=\lim_{\varepsilonsilon \rightarrow 0+}\frac{1}{\varepsilonsilon} \left(\nablala_x f_0 (X^{\varepsilonsilon}(0),R_{x^1}v) \nablala_x X^{\varepsilonsilon}(0) - \nablala_xf_0(X^\varepsilonsilon(0),R_{x^1}v) \lim_{s\rightarrow 0-}\nablala_x X(s) \right. \\
&\quad \left. +\nablala_x f_0(X^\varepsilonsilon(0),R_{x^1}v) \lim_{s\rightarrow 0-} \nablala_x X(s) -\nablala_x f_0(X(0),R_{x^1}v)R_{x^1}\right)\\
&\stackrel{r\leftrightarrow c}{=} \left[\nablala_xf_0(x^1,R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x X^{\varepsilonsilon}(0)-\lim_{s \rightarrow 0-} \nablala_x X(s) \right)\right]^T\\
&\quad + R_{x^1}\nablala_{xx}f_0(x^1,R_{x^1}v)(-R_{x^1}v)\\
&=\begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v)\nablala_x(R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix} (-v)+R_{x^1}\nablala_{xx} f_0(x^1,R_{x^1}v) (-R_{x^1}v), \\
I_{tx,2}&:= \lim_{\varepsilonsilon \rightarrow 0+}\frac{1}{\varepsilonsilon} \left(\nablala_v f_0 (X^{\varepsilonsilon}(0),R_{x^1}v) \nablala_x V^{\varepsilonsilon}(0) - \nablala_vf_0(X^\varepsilonsilon(0),R_{x^1}v) \lim_{s\rightarrow 0-}\nablala_x V(s) \right. \\
&\quad \left. +\nablala_v f_0(X^\varepsilonsilon(0),R_{x^1}v) \lim_{s\rightarrow 0-} \nablala_x V(s) -2\nablala_v f_0(X(0),R_{x^1}v)A_{v,x^1}\right)\\
&\stackrel{r\leftrightarrow c}{=} \left[\nablala_vf_0(x^1,R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x V^{\varepsilonsilon}(0)-\lim_{s \rightarrow 0-} \nablala_x V(s) \right)\right]^T\\
&\quad +(-2A^T_{v,x^1})\nablala_{xv} f_0(x^1,R_{x^1}v)\lim_{\varepsilonsilon \rightarrow 0+} \frac{ X^{\varepsilonsilon}(0)-X(0)}{\varepsilonsilon}\\
&= \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x(-2A_{v,x^1(x,v)}^2)
\end{bmatrix} (-v)+ (-2A^T_{v,x^1}) \nablala_{xv}f_0(x^1,R_{x^1}v) (-R_{x^1}v).
\end{align*}
Thus,
\begin{equation}gin{equation}\label{nabla_tx f case2}
\begin{equation}gin{split}
\partialartial_t [\nablala_x f(t,x,v)] &= (-v)^T \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix}^T+(-v)^T \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x(-2A_{v,x^1(x,v)}^2)
\end{bmatrix}^T\\
& \quad +(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T) R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(-2A_{v,x^1}).
\end{split}
\end{equation}
From \eqref{nabla_tx f case1} and \eqref{nabla_tx f case2}, we have the following condition
\begin{equation}gin{equation} \label{tx comp}
\begin{equation}gin{split}
(-v^T) \nablala_{xx}f_0(x^1,v) &= (-v)^T \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix}^T+(-v)^T \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x(-2A_{v,x^1(x,v)}^2)
\end{bmatrix}^T\\
& \quad +(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T) R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v) (-2A_{v,x^1}).
\end{split}
\end{equation}
\subsubsection{$C^2_{t,v}$} Similar to $C^2_{t,x}$, we use \eqref{c_1} and the perturbation \eqref{Perb_t} for $\varepsilonsilon<0$ to obtain
\begin{equation}gin{equation}\label{nabla_tv f case1}
\begin{equation}gin{split}
\partialartial_{t}[\nablala_{v}f(t,x,v)] &= \lim_{\varepsilonsilon \rightarrow 0-} \frac{ \nablala_v f(t+\varepsilonsilon,x,v) -\nablala_v f(t,x,v)}{ \varepsilonsilon}\\
&= \lim_{\varepsilonsilon \rightarrow 0-} \frac{1}{\varepsilonsilon} \left( \nablala_v \left[ f_0(X(0;t+\varepsilonsilon,x,v),V(0;t+\varepsilonsilon,x,v))\right]-(-t\nablala_x f_0(X(0),v)+\nablala_vf_0(X(0),v)) \right)\\
&= \lim_{\varepsilonsilon \rightarrow 0-} \frac{1}{\varepsilonsilon} \left( -(t+\varepsilonsilon) \nablala_x f_0(X^{\varepsilonsilon}(0),v) +\nablala_v f_0(X^{\varepsilonsilon}(0),v) +t\nablala_x f_0(X(0),v) -\nablala_v f_0(X(0),v) \right)\\
&=-\nablala_x f_0(x^1,v) -t(-v^T) \nablala_{xx}f_0(x^1,v) + (-v^T) \nablala_{vx}f_0(x^1,v),
\end{split}
\end{equation}
where we have used $\nablala_v X^{\varepsilonsilon}(0) = -(t+\varepsilonsilon) I_2, \nablala_v V^{\varepsilonsilon}(0) = I_2$.
For $\varepsilonsilon>0$, the perturbation \eqref{Perb_t} becomes
\begin{equation}gin{equation*}
X^{\varepsilonsilon}(0):=X(0;t+\varepsilonsilon,x,v) =X(0;t,x-\varepsilonsilon v,v) =x^1 -(t^1+\varepsilonsilon)R_{x^1}v, \quad V^{\varepsilonsilon}(0):=V(0;t+\varepsilonsilon,x,v)=R_{x^1}v.
\end{equation*}
By product rule, Lemma \e^{\frac 12}f{nabla xv b} and Lemma \e^{\frac 12}f{d_n}, one obtains that
\begin{equation}gin{align*}
\nablala_v [X^{\varepsilonsilon}(0)]&=\nablala_v [x^1-(t^1+\varepsilonsilon)R_{x^1}v] =-t\left(I-\frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -R_{x^1}v \otimes \nablala_v t^1 -\varepsilonsilon \nablala_v (R_{x^1}v)\\
&=-t \left(I-\frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right)-t R_{x^1}v \otimes \frac{n(x^1}{v\cdotot n(x^1)} -\varepsilonsilon (R_x^1 + 2t A_{v,x^1})\\
&= -tR_{x^1} -\varepsilonsilon (R_{x^1}+2tA_{v,x^1}), \\
\nablala_v[V^{\varepsilonsilon}(0)]&= \nablala_v [R_{x^1}v] = R_{x^1}+2tA_{v,x^1}.
\end{align*}
Through the $v$-derivative of $X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0)$ above and \eqref{c_2},
\begin{equation}gin{equation} \label{nabla_tv f case2}
\begin{equation}gin{split}
\partialartial_t[\nablala_{v} f(t,x,v)]&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{\nablala_v f(t+\varepsilonsilon,x,v) -\nablala_v f(t,x,v)}{\varepsilonsilon}\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} (\nablala_v [f_0(X(0;t+\varepsilonsilon,x,v),V(0;t+\varepsilonsilon,x,v)]\\
&\quad -(-t\nablala_x f_0(X(0),R_{x^1}v)R_{x^1}+ \nablala_v f_0(X(0),R_{x^1}v)(R_{x^1}+2tA_{v,x^1})))\\
&=-\nablala_x f_0(x^1,R_{x^1}v)\left(R_{x^1}+2tA_{v,x^1}\right) -t \left[ \lim_{\varepsilonsilon\rightarrow 0+} \frac{1}{\varepsilonsilon} \left(\nablala_xf_0(X^{\varepsilonsilon}(0),R_{x^1}v) -\nablala_xf_0(X^{\varepsilonsilon}(0),R_{x^1}v)\right)\right] R_{x^1} \\
&\quad + \left [\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon}\left( \nablala_v f_0(X^{\varepsilonsilon}(0),R_{x^1}v) -\nablala_v f_0(X(0),R_{x^1}v) \right)\right]\left(R_{x^1}+2tA_{v,x^1}\right)\\
&\stackrel{r\leftrightarrow c}{=} -\left(R_{x^1}+2tA_{v,x^1}\right)^T\nablala_x f_0(x^1,R_{x^1}v)^T -tR_{x^1} \nablala_{xx}f_0(x^1,R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{X^{\varepsilonsilon}(0)-X(0)}{\varepsilonsilon} \\
&\quad + \left(R_{x^1}+2tA_{v,x^1}\right)^T \nablala_{xv} f_0(x^1,R_{x^1}v) \lim_{\varepsilonsilon \rightarrow 0+} \frac{X^{\varepsilonsilon}(0)-X(0)}{\varepsilonsilon}\\
&\stackrel{c\leftrightarrow r}{=} -\nablala_xf_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1}) -t(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} \\
&\quad +(-v^T) R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1}).
\end{split}
\end{equation}
Summing \eqref{nabla_tv f case1} and \eqref{nabla_tv f case2} yields that
\begin{equation}gin{equation} \label{tv comp}
\begin{equation}gin{split}
&-\nablala_x f_0(x^1,v) -t(-v^T) \nablala_{xx}f_0(x^1,v) + (-v^T) \nablala_{vx}f_0(x^1,v)\\
&= -\nablala_xf_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1})-t(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} \\
&\quad +(-v^T) R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1}).
\end{split}
\end{equation}
\subsubsection{$C^2_{x,t}$} Similar to the $\nablala_{xv}$ case, using the same perturbation $\hat{r}_1$ of \eqref{set R_sp} and \eqref{c_3}, we have
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nablala_x[\partialartial_t f(t,x,v)]\hat{r}_1&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{ \partialartial_t f(t,x+\varepsilonsilon \hat{r}_1,v)- \partialartial_t f(t,x,v)}{\varepsilonsilon} \\
&= \lim_{\varepsilonsilon\rightarrow 0+} \left(\frac{ \nablala_x f(t,x+\varepsilonsilon \hat{r}_1,v) - \nablala_x f(t,x,v)}{\varepsilonsilon}\right)(-v)\\
&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left(\nablala_x [f_0(X(0;t,x+\varepsilonsilon \hat{r}_1,v),V(0;t,x+\varepsilonsilon \hat{r}_1,v))] -\nablala_x f_0(X(0),v)\right)(-v)\\
&= (-v^T) \nablala_{xx} f_0(x^1,v) \hat{r}_1,
\end{split}
\end{equation*}
where we have used $\nablala_x X^{\varepsilonsilon}(0)=I_2, \nablala_x V^{\varepsilonsilon}(0)=0$. Next, for $\hat{r}_2$ of \eqref{set R_sp}, using \eqref{Av=0} in Lemma \e^{\frac 12}f{lem_RA}, \eqref{c_4},\eqref{xx star1}, and \eqref{xx star2} gives
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nablala_x[\partialartial_t f(t,x,v)] \hat{r}_2 &= \lim_{\varepsilonsilon \rightarrow 0+} \frac{ \partialartial_t f(t,x+\varepsilonsilon \hat{r}_2,v)- \partialartial_t f(t,x,v)}{\varepsilonsilon}\\
&= \lim_{\varepsilonsilon\rightarrow 0+} \left(\frac{ \nablala_x f(t,x+\varepsilonsilon \hat{r}_2,v) - \nablala_x f(t,x,v)}{\varepsilonsilon}\right)(-v)\\
&= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} (\nablala_x [f_0(X(0;t,x+\varepsilonsilon \hat{r}_2,v),V(0;t,x+\varepsilonsilon \hat{r}_2,v))] \\
&\quad - (\nablala_x f_0(X(0),R_{x^1}v)R_{x^1} -2 \nablala_v f_0(X(0),R_{x^1}v) A_{v,x^1}) )(-v)\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_ xX^{\varepsilonsilon}(0) - \nablala_x f_0(X(0),R_{x^1}v)R_{x^1}\right) (-v) \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_v f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x V^{\varepsilonsilon}(0) -\nablala_v f_0(X(0),R_{x^1}v)(-2A_{v,x^1}) \right) (-v) \\
&:=I_{xt,1}+I_{xt,2},
\end{split}
\end{equation*}
where
\begin{equation}gin{align*}
I_{xt,1}&=\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_ xX^{\varepsilonsilon}(0) - \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0)) \lim_{s \rightarrow 0-} \nablala_x X(s) \right. \\
&\quad + \left. \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\lim_{s\rightarrow 0-} \nablala_x X(s) - \nablala_x f_0(X(0),R_{x^1}v)R_{x^1}\right)(-v)\\
&= (-v^T) \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1) \\
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix} \hat{r}_2\\
&\quad +(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}\hat{r}_2 +(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(-2A_{v,x^1})\hat{r}_2,\\
I_{xt,2} &= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_v f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x V^{\varepsilonsilon}(0) -\nablala_v f_0(X^\varepsilonsilon(0),V^{\varepsilonsilon}(0))\lim_{s\rightarrow 0-} \nablala_x V(s) \right. \\
&\quad + \left. \nablala_v f_0(X^\varepsilonsilon(0),V^{\varepsilonsilon}(0)) \lim_{s \rightarrow 0-} \nablala_x V(s) -\nablala_v f_0(X(0),R_{x^1}v)(-2A_{v,x^1})\right)(-v) \\
&=(-v^T) \begin{equation}gin{bmatrix}
\nablala_vf_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\hat{r}_2 \\
& \quad +(-v^T)(-2A^T_{v,x^1}) \left( \nablala_{xv} f_0(x^1,R_{x^1}v) R_{x^1} +\nablala_{vv} f_0(x^1,R_{x^1}v) (-2A_{v,x^1}) \right) \hat{r}_2,\\
&=(-v^T) \begin{equation}gin{bmatrix}
\nablala_vf_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\hat{r}_2.
\end{align*}
To sum up the above, we get the following condition:
\begin{equation}gin{equation} \label{xt comp}
\begin{equation}gin{split}
(-v^T) \nablala_{xx} f_0(x^1,v)&= (-v^T) \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1)\\
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)\end{bmatrix}
+(-v^T)
\begin{equation}gin{bmatrix}
\nablala_vf_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\\
&\quad +(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(-2A_{v,x^1}).
\end{split}
\end{equation}
\subsubsection{$C^2_{v,t}$} Using the perturbation $\hat{r}_1$ of \eqref{set R_sp} and \eqref{c_3},
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nablala_v [\partialartial_t f(t,x,v)] \hat{r}_1 &=\lim_{\varepsilonsilon\rightarrow 0+} \frac{ \partialartial_t f(t,x,v+\varepsilonsilon \hat{r}_1) -\partialartial_t f(t,x,v)}{\varepsilonsilon}\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \left(\frac{\nablala_x f(t,x,v+\varepsilonsilon \hat{r}_1) (-(v+\varepsilonsilon \hat{r}_1)) -\nablala_x f(t,x,v) (-v)}{\varepsilonsilon} \right) \\
&=-\lim_{\varepsilonsilon \rightarrow 0+} \nablala_x [f_0(X(0;t,x,v+\varepsilonsilon \hat{r}_1), V(0;t,x,v+\varepsilonsilon \hat{r}_1))]\hat{r}_1 \\
&+\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} ( \nablala_x [f_0(X(0;t,x,v+\varepsilonsilon \hat{r}_1), V(0;t,x,v+\varepsilonsilon \hat{r}_1))]-\nablala_x f_0(X(0),v))(-v)\\
&=-\nablala_x f_0(X(0),v) \hat{r}_1 +\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} (\nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))-\nablala_x f_0(X(0),v))(-v)\\
&=-\nablala_x f_0(x^1,v)\hat{r}_1 +(-v^T) \nablala_{xx} f_0(x^1,v) (-t\hat{r}_1) +(-v^T) \nablala_{vx} f_0(x^1,v) \hat{r}_1,
\end{split}
\end{equation*}
where $X^{\varepsilonsilon}(0):=X(0;t,x,v+\varepsilonsilon \hat{r}_1) = x-t(v+\varepsilonsilon \hat{r}_1), V^{\varepsilonsilon}(0):=V(0;t,x,v+\varepsilonsilon \hat{r}_1) =v+\varepsilonsilon \hat{r}_1$. Similar to the case $\nablala_{vx}$, for the perturbation $\hat{r}_2$ of \eqref{set R_sp}, using \eqref{nabla XV_x-}, \eqref{c_4} and \eqref{Av=0} in Lemma \e^{\frac 12}f{lem_RA} yields:
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\nablala_v[\partialartial_t f(t,x,v)] \hat{r}_2 &= \lim_{\varepsilonsilon \rightarrow 0+} \frac{\partialartial_t f(t,x,v+\varepsilonsilon \hat{r}_2) -\partialartial_t f(t,x,v)}{\varepsilonsilon}\\
&=\lim_{\varepsilonsilon \rightarrow 0+} \left( \frac{ \nablala_x f(t,x,v+\varepsilonsilon \hat{r}_2)(-(v+\varepsilonsilon \hat{r}_2))-\nablala_x f(t,x,v)(-v)}{\varepsilonsilon}\right)\\
&=-\lim_{\varepsilonsilon \rightarrow 0+} \nablala_x [f_0(X(0;t,x,v+\varepsilonsilon \hat{r}_2),V(0;t,x,v+\varepsilonsilon \hat{r}_2))]\hat{r}_2 \\
&\quad +\lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x [f_0(X(0;t,x,v+\varepsilonsilon \hat{r}_2), V(0;t,x,v+\varepsilonsilon \hat{r}_2))] \right. \\
&\quad - \left. \left(\nablala_x f_0(x^1,R_{x^1}v)R_{x^1}-2\nablala_v f_0(x^1,R_{x^1}v)A_{v,x^1} \right)\right)(-v)\\
&=-\left(\nablala_x f_0(x^1,R_{x^1}v) R_{x^1} +\nablala_v f_0(x^1,R_{x^1}v)(-2A_{v,x^1})\right) \hat{r}_2 \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x X^{\varepsilonsilon}(0) -\nablala_x f_0(X(0),R_{x^1}v)R_{x^1}\right)(-v) \\
&\quad + \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_v f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x V^{\varepsilonsilon}(0)-\nablala_v f_0(X(0),R_{x^1}v)(-2A_{v,x^1})\right)(-v)\\
&:=-\left(\nablala_x f_0(x^1,R_{x^1}v) R_{x^1} +\nablala_v f_0(x^1,R_{x^1}v)(-2A_{v,x^1})\right) \hat{r}_2 + I_{vt,1}+I_{vt,2},
\end{split}
\end{equation*}
where
\begin{equation}gin{align*}
I_{vt,1}&:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x X^{\varepsilonsilon}(0)-\nablala_x f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\lim_{s\rightarrow 0-} \nablala_x X(s) \right. \\
&\quad + \left. \nablala_x f_0(X^\varepsilonsilon(0),V^\varepsilonsilon(0))\lim_{s \rightarrow 0-} \nablala_x X(s) - \nablala_xf_0(X(0),R_{x^1}v)R_{x^1}\right) (-v) \\
&=(-v^T) \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_v (R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_v (R_{x^1(x,v)}^2)
\end{bmatrix}\hat{r}_2 \\
&\quad + (-v^T)R_{x^1} \left( \nablala_{xx} f_0(x^1,R_{x^1}v) (-tR_{x^1}) +\nablala_{vx} f_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1})\right)\hat{r}_2,\\
I_{vt,2}&:= \lim_{\varepsilonsilon \rightarrow 0+} \frac{1}{\varepsilonsilon} \left( \nablala_v f_0(X^{\varepsilonsilon}(0),V^{\varepsilonsilon}(0))\nablala_x V^{\varepsilonsilon}(0)-\nablala_v f_0(X^\varepsilonsilon(0),V^\varepsilonsilon(0))\lim_{s\rightarrow 0-} \nablala_xV(s) \right. \\
& \quad \left.+ \nablala_v f_0(X^\varepsilonsilon(0),V^\varepsilonsilon(0))\lim_{s\rightarrow 0-} \nablala_xV(s) -\nablala_v f_0(X(0),R_{x^1}v)(-2A_{v,x^1}) \right)(-v)\\
&=(-v^T)\begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\hat{r}_2\\
&\quad +(-v^T)(-2A^T_{v,x^1}) \left( \nablala_{xv} f_0(x^1,R_{x^1}v)(-tR_{x^1}) +\nablala_{vv} f_0(x^1,R_{x^1}v)(R_{x^1}+2tA_{v,x^1}) \right) \hat{r}_2\\
&=(-v^T)\begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\hat{r}_2.
\end{align*}
Thus, we have the following compatibility condition:
\begin{equation}gin{equation} \label{vt comp}
\begin{equation}gin{split}
&-\nablala_x f_0(x^1,v) +tv^T\nablala_{xx} f_0(x^1,v) +(-v^T) \nablala_{vx} f_0(x^1,v) \\
&=- \left(\nablala_x f_0(x^1,R_{x^1}v) R_{x^1} +\nablala_v f_0(x^1,R_{x^1}v)(-2A_{v,x^1})\right)\\
&\quad +(-v^T) \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_v (R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_v (R_{x^1(x,v)}^2)
\end{bmatrix} + (-v^T) \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\\
&\quad +tv^T R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} +(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(R_{x^1}+2tA_{v,x^1}).
\end{split}
\end{equation}
\subsubsection{Derive $C^2_{tt},C^2_{tx}, C^2_{tv},C^2_{xt},C^2_{vt}$ compatibility conditions from \eqref{Cond2 1}--\eqref{Cond2 4},\eqref{Cond3} and \eqref{Cond4}} So far, we have derived \eqref{Cond2 1}--\eqref{Cond2 4} to satisfy $f\in C^2_{xv},C^2_{xx},C^2_{vx},C^2_{vv}$. In \eqref{Cond2 1}--\eqref{Cond2 4}, since $\nablala_{xv} f_0(x^1,v)$ is the same as $\nablala_{vx} f_0(x^1,v)^T$, we need to assume \eqref{Cond3}. Similarly, we obtained \eqref{Cond3} because $\nablala_{xx} f_0(x^1,v)$ is a symmetric matrix. In this subsection, we will show that the compatibility conditions $C^2_{tt}$ \eqref{time cond}, $C^2_{tx}$ \eqref{tx comp}, $C^2_{tv}$ \eqref{tv comp}, $C^2_{xt}$ \eqref{xt comp}, and $C^2_{vt}$ \eqref{vt comp} are induced under \eqref{Cond2 1}--\eqref{Cond2 4},\eqref{Cond3}, and \eqref{Cond4}. Firstly, we consider $C^2_{tt}$ compatibility condition. Using \eqref{Av=0} in Lemma \e^{\frac 12}f{lem_RA}, \eqref{prop d_R}, and \eqref{prop d_A}, one has
\begin{equation}gin{equation*}
\begin{equation}gin{split}
v^T \nablala_{xx}f_0(x^1,v) v &= v^T \begin{equation}igg(R_{x^{1}} \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + R_{x^{1}} \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)(-2A_{v,x^{1}}) \\
&\quad + (-2A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + (-2A^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \\
&\quad + \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
- 2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\begin{equation}igg) v\\
&=v^TR_{x^1}\nablala_{xx}f_0(x^1,R_{x^1}v)R_{x^1} v +v^T \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} v \\
&\quad + v^T \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix}v\\
&= (R_{x^1}v)^T \nablala_{xx} f_0(x^1,R_{x^1}v) (R_{x^1}v).
\end{split}
\end{equation*}
In \eqref{tx comp}, the left-hand side is
\begin{equation}gin{align*}
(-v^T) \nablala_{xx} f_0(x^1,v) &= (-v^T) \begin{equation}igg(R_{x^{1}} \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + R_{x^{1}} \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)(-2A_{v,x^{1}}) \\
&\quad + (-2A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} + (-2A^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \\
&\quad + \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix}
- 2
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(A_{v,x^{1}(x,v)}^2)
\end{bmatrix}
\begin{equation}igg)\\
&= (-v^T) R_{x^1}\nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} + (-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1}) \\
&\quad + (-v^T) \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} +(-v^T) \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix},
\end{align*}
where we have used \eqref{Av=0}. When we assume \eqref{Cond4}, it holds that $\nablala_{xx}f_0(x^1,v)$ is a symmetric matrix. In other words,
\begin{equation}gin{align*}
&\left(\begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} +
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix}\right)^T \\
&= \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} +
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix},
\end{align*}
which implies that
\begin{equation}gin{equation} \label{vRA prop}
\begin{equation}gin{split}
&(-v^T) \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} +(-v^T) \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix}\\
&=\left( \left( \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^1)
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(R_{x^{1}(x,v)}^2)
\end{bmatrix} + \begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^1)
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}(-2A_{v,x^{1}(x,v)}^2)
\end{bmatrix}\right)(-v)\right)^T=0,
\end{split}
\end{equation}
due to \eqref{prop d_R} and \eqref{prop d_A}. Therefore, the left-hand side in \eqref{tx comp} becomes
\begin{equation}gin{equation} \label{tx comp left}
(-v^T) \nablala_{xx} f_0(x^1,v) = (-v^T) R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} + (-v^T) \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1}).
\end{equation}
Using \eqref{vRA prop}, the right-hand side in \eqref{tx comp} is
\begin{equation}gin{equation}\label{tx comp right}
\begin{equation}gin{split}
&(-v)^T \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix}^T+(-v)^T \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x(-2A_{v,x^1(x,v)}^2)
\end{bmatrix}^T\\
& \quad +(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T) R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v) (-2A_{v,x^1})\\
& =(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T) R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v) (-2A_{v,x^1}).\\
\end{split}
\end{equation}
From \eqref{tx comp left} and \eqref{tx comp right}, we derive \eqref{tx comp} under the assumption \eqref{Cond2 1}--\eqref{Cond2 4},\eqref{Cond3}, and \eqref{Cond4}. For the left-hand side in \eqref{tv comp}, we use \eqref{Av=0}, the $C^1$ compatibility condition \eqref{c_x}, \eqref{Cond2 1}--\eqref{Cond2 4}, and \eqref{vRA prop}:
\begin{equation}gin{align*}
&-\nablala_x f_0(x^1,v) +tv^T \nablala_{xx} f_0(x^1,v) + (-v^T) \nablala_{vx} f_0(x^1,v) \\
&= -\nablala_x f_0(x^1,R_{x^1}v)R_{x^1} - \nablala_v f_0(x^1,R_{x^1}v) (-2A_{v,x^1}) \\
&\quad +tv^TR_{x^1}\nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1} +tv^T R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1})\\
&\quad +(-v^T)R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v)R_{x^1} +(-v)^T \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v)\nablala_v(-2A_{v,x^1}^1)\\ \nablala_v f_0(x^1,R_{x^1}v)\nablala_v(-2A_{v,x^1}^2)
\end{bmatrix}.
\end{align*}
Since $\nablala_{xv}f_0(X(0),v)^T = \nablala_{vx} f_0(X(0),v)$ under \eqref{Cond3}, it holds that
\begin{equation}gin{equation} \label{RA prop}
\begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v)\nablala_v(-2A_{v,x^1}^1)\\ \nablala_v f_0(x^1,R_{x^1}v)\nablala_v(-2A_{v,x^1}^2)
\end{bmatrix}^T = \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1)\\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix}.
\end{equation}
Since \eqref{RA} in Lemma \e^{\frac 12}f{lem_RA}, \eqref{prop d_R}, \eqref{Cond3}, and the formula \eqref{RA prop} above, it follows that
\begin{equation}gin{equation} \label{A prop}
\begin{equation}gin{split}
&\nablala_v f_0(x^1,R_{x^1}v) (-2A_{v,x^1}) = C(R_{x^1}v)^T (-2A_{v,x^1})=-\frac{2C}{v\cdotot n(x^1)} v^T (Qv) \otimes (Qv) =0, \\
&(-v)^T\begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v)\nablala_v(-2A_{v,x^1}^1)\\ \nablala_v f_0(x^1,R_{x^1}v)\nablala_v(-2A_{v,x^1}^2)
\end{bmatrix}= \left( \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)
\end{bmatrix} (-v) \right)^T=0,
\end{split}
\end{equation}
where $C$ is an arbitrary constant. And then, one obtains that
\begin{equation}gin{equation} \label{tv comp left}
\begin{equation}gin{split}
&-\nablala_x f_0(x^1,v) +tv^T \nablala_{xx} f_0(x^1,v) + (-v^T) \nablala_{vx} f_0(x^1,v) \\
&= -\nablala_x f_0(x^1,R_{x^1}v)R_{x^1} +tv^TR_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1} +tv^T R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1})\\
&\quad +(-v^T)R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v)R_{x^1}.
\end{split}
\end{equation}
By \eqref{Av=0} and \eqref{Cond4}, the right-hand side in \eqref{tv comp} is
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& -\nablala_xf_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1})-t(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} \\
&\quad +(-v^T) R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (R_{x^1}+2tA_{v,x^1}) \\
&=-\nablala_x f_0(x^1,R_{x^1}v) R_{x^1} -2Ct (R_{x^1}v)^TA_{v,x^1} +tv^T R_{x^1} \nablala_{xx}f_0(x^1,R_{x^1}v) R_{x^1} \\
&\quad +tv^T R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1}) + (-v^T) R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) R_{x^1}\\
& = -\nablala_x f_0(x^1,R_{x^1}v)R_{x^1} +tv^TR_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1} +tv^T R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1})\\
&\quad +(-v^T)R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v)R_{x^1},
\end{split}
\end{equation*}
where $C$ is an arbitrary constant. Thus, the left-hand side in \eqref{tv comp} is the same as the right-hand side in \eqref{tv comp} under \eqref{Cond2 1}--\eqref{Cond2 4}, \eqref{Cond3}, and \eqref{Cond4}. The left-hand side in \eqref{xt comp} is as follows:
\begin{equation}gin{equation*}
(-v^T) \nablala_{xx}f_0(x^1,v) = (-v^T) R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} +(-v^T) \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1}),
\end{equation*}
by \eqref{tx comp left}. Using \eqref{vRA prop}, the right-hand side in \eqref{xt comp} can be further computed by
\begin{equation}gin{equation*}
\begin{equation}gin{split}
&(-v^T) \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^1) \\
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x (R_{x^1(x,v)}^2)\end{bmatrix}
+(-v^T)
\begin{equation}gin{bmatrix}
\nablala_vf_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\\
&\quad +(-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(-2A_{v,x^1})\\
&= (-v^T)R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1}+(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(-2A_{v,x^1}).
\end{split}
\end{equation*}
Hence, the \eqref{xt comp} condition can be deduced by \eqref{Cond2 1}--\eqref{Cond2 4},\eqref{Cond3}, and \eqref{Cond4}. Finally, the \eqref{vt comp} condition is the last remaining case. The left-hand side in \eqref{vt comp} comes from \eqref{tv comp left}:
\begin{equation}gin{align*}
&-\nablala_x f_0(x^1,v) +tv^T \nablala_{xx} f_0(x^1,v) + (-v^T) \nablala_{vx} f_0(x^1,v) \\
&= -\nablala_x f_0(x^1,R_{x^1}v)R_{x^1} +tv^TR_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v)R_{x^1} +tv^T R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) (-2A_{v,x^1})\\
&\quad +(-v^T)R_{x^1} \nablala_{vx}f_0(x^1,R_{x^1}v)R_{x^1}.
\end{align*}
Since \eqref{Av=0} in Lemma \e^{\frac 12}f{lem_RA}, \eqref{vRA prop}, \eqref{A prop}, and
\begin{equation}gin{align*}
\quad \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_v(R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_v(R_{x^1(x,v)}^2)
\end{bmatrix}&= (-t)\begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x(R_{x^1(x,v)}^1) \\\nablala_x f_0(x^1,R_{x^1}v) \nablala_x(R_{x^1(x,v)}^2)
\end{bmatrix},\\
\begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}&=(-t) \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}+ \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1}^1)\\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1}^2)
\end{bmatrix},
\end{align*}
the right-hand side in \eqref{vt comp} can be simplified as
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& - \left(\nablala_x f_0(x^1,R_{x^1}v) R_{x^1} +\nablala_v f_0(x^1,R_{x^1}v)(-2A_{v,x^1})\right)\\
&\quad +(-v^T) \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_v (R_{x^1(x,v)}^1) \\ \nablala_x f_0(x^1,R_{x^1}v) \nablala_v (R_{x^1(x,v)}^2)
\end{bmatrix} + (-v^T) \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1(x,v)}^2)
\end{bmatrix}\\
&\quad + tv^T R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} +(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(R_{x^1}+2tA_{v,x^1}) \\
&=-\nablala_x f_0(x^1,R_{x^1}v)R_{x^1} +tv^T \begin{equation}gin{bmatrix}
\nablala_x f_0(x^1,R_{x^1}v) \nablala_x(R_{x^1(x,v)}^1) \\\nablala_x f_0(x^1,R_{x^1}v) \nablala_x(R_{x^1(x,v)}^2)
\end{bmatrix}+tv^T \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_x (-2A_{v,x^1(x,v)}^2)
\end{bmatrix} \\
&\quad +(-v^T) \begin{equation}gin{bmatrix}
\nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1}^1) \\ \nablala_v f_0(x^1,R_{x^1}v) \nablala_v (-2A_{v,x^1}^2)
\end{bmatrix}
+ tv^T R_{x^1} \nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1} \\
&\quad +(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v)(R_{x^1}+2tA_{v,x^1})\\
&= -\nablala_x f_0(x^1,R_{x^1}v)R_{x^1}+tv^T R_{x^1}\nablala_{xx} f_0(x^1,R_{x^1}v) R_{x^1}+tv^T R_{x^1}\nablala_{vx} f_0(x^1,R_{x^1}v)(-2A_{v,x^1}) \\
&\quad +(-v^T)R_{x^1} \nablala_{vx} f_0(x^1,R_{x^1}v) R_{x^1}.
\end{split}
\end{equation*}
Hence, the \eqref{vt comp} condition can be obtained under \eqref{Cond2 1}--\eqref{Cond2 4},\eqref{Cond3}, and \eqref{Cond4}. \\
\hide
\subsubsection{Symmetric presentation of \eqref{Cond} under assumption \eqref{if}}
Above can be more simplified. Note that the 3rd condition of \eqref{Cond2 1}--\eqref{Cond2 4} is just obvious by specular reflection BC (taking $\nablala_{v}$ twice). From 1st and 4th condition, we have
\begin{equation}gin{equation}
\begin{equation}gin{split}
(-2A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}} &= (-2A^{T}_{v,x^{1}}) R_{x^{1}} \nablala_{xv}f_{0}(x^{1}, v) - (-2A^{T}_{v,x^{1}}) \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) (-2A_{v,x^{1}}) \\
R_{x^{1}} \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)(-2A_{v,x^{1}}) &= \nablala_{vx}f_{0}(x^{1}, v)R_{x^{1}}(-2A_{v,x^{1}}) - (-2A^{T}_{v,x^{1}})\nablala_{vv}f_{0}(x^{1}, R_{x^1}v)(-2A_{v,x^{1}}).
\end{split}
\end{equation}
Plugging into 2nd condition, \eqref{Cond2 2} is rewritten as
\begin{equation}gin{equation} \label{re 2}
\begin{equation}gin{split}
& \nablala_{xx}f_{0}(x^{1},v) + \nablala_{vx}f_{0}(x^{1}, v)R_{x^{1}}A_{v,x^{1}} + (R_{x^{1}}A_{v,x^{1}})^{T} \nablala_{xv}f_{0}(x^{1}, v) \\
&= R_{x^{1}}\nablala_{xx}f_{0}(x^{1}, R_{x^1}v)R_{x^{1}} + R_{x^{1}}\nablala_{vx}f_{0}(x^{1}, R_{x^1}v)(-A_{v,x^{1}} ) \\
&\quad + (-A^{T}_{v,x^{1}}) \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) R_{x^{1}}
+
{\color{blue}
\begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}[R_{x^{1}}]_{1}
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}[R_{x^{1}}]_{2}
\end{bmatrix}
}
\end{split}
\end{equation}
\\
{\bf Conclusion}
From \eqref{Cond2 3} and Lemma \e^{\frac 12}f{lem_RA}, \\
the \eqref{Cond2 1} can be written as symmetric form \\
\begin{equation}gin{equation} \label{sym Cond2_1}
\begin{equation}gin{split}
R_{x^{1}} \begin{equation}ig[ \nablala_{xv}f_{0}(x^{1},v) + \nablala_{vv}f_{0}(x^{1},v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \begin{equation}ig] R_{x^{1}}
&= \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) + \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) \frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)} .
\end{split}
\end{equation}
Similarly, 4th one give
\begin{equation}gin{equation} \label{sym Cond2_2}
\begin{equation}gin{split}
R_{x^{1}} \begin{equation}ig[ \nablala_{vx}f_{0}(x^{1},v) + \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \nablala_{vv}f_{0}(x^{1}, v) \begin{equation}ig] R_{x^{1}}
&= \nablala_{vx}f_{0}(x^{1}, R_{x^1}v) + \frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)} \nablala_{vv}f_{0}(x^{1}, R_{x^1}v) .
\end{split}
\end{equation}
\eqref{re 2} condition (\eqref{Cond2 2}) yields
\begin{equation}gin{equation} \label{sym Cond2_3}
\begin{equation}gin{split}
&R_{x^{1}}\begin{equation}ig[ \nablala_{xx}f_{0}(x^{1},v) + \nablala_{vx}f_{0}(x^{1}, v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n} + \frac{ (Qv)\otimes (Qv)}{v\cdotot n} \nablala_{xv}f_{0}(x^{1}, v) \begin{equation}ig] R_{x^{1}} \\
&= \nablala_{xx}f_{0}(x^{1}, R_{x^1}v) + \nablala_{vx}f_{0}(x^{1}, R_{x^1}v)\frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)}
+ \frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)} \nablala_{xv}f_{0}(x^{1}, R_{x^1}v) \\
&\quad +
{\color{blue}
\underbrace{
R_{x^{1}}
\begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}[R_{x^{1}}]_{1}
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_{x}[R_{x^{1}}]_{2}
\end{bmatrix}
R_{x^{1}}
}_{(?)}
}
\end{split}
\end{equation}
\unhide
\subsection{Proof of Theorem \e^{\frac 12}f{thm 2}}
\begin{equation}gin{proof} [Proof of Theorem \e^{\frac 12}f{thm 2}]
By the same argument of the proof of Theorem \e^{\frac 12}f{thm 1}, it suffices to set $k=1$. Through this section, we have shown that \eqref{Cond2 1}--\eqref{Cond2 4}, \eqref{Cond3}, and \eqref{Cond4} yield $C^{2}_{t,x,v}$ regularity of $f(t,x,v)$ of \eqref{solution}. However, \eqref{Cond2 3} is just an obvious consequence of \eqref{BC} and \eqref{Cond2 4} is identical to \eqref{Cond2 1} since we assume \eqref{C2 cond34} which is the same as \eqref{Cond3} and \eqref{Cond4}. So, we omit \eqref{Cond2 3} and \eqref{Cond2 4} in the statement.
In Remark \e^{\frac 12}f{extension C2 cond34}, under \eqref{C2 cond34}, we derived that
\begin{equation}gin{equation*}
\nablala_x f_0(x,v)R_x = \nablala_x f_0(x,R_xv) \quad \textrm{and} \quad \nablala_v f_0(x,v) \frac{(Qv) \otimes (Qv)}{v\cdotot n} R_x = \nablala_v f_0(x,R_xv)\frac{(QR_xv)\otimes(QR_xv)}{R_x v\cdotot n},
\end{equation*}
for all $(x,v) \in \gamma_- \cup \gamma_+$. In Remark \e^{\frac 12}f{example}, we showed that
\begin{equation}gin{equation*}
f_0(x,v)=G(x,\vert v \vert), \quad (x,v) \in \partialartial \Omega \times \mathbb{R}^2,
\end{equation*}
where $G$ is a $C^1_{x,v}$ function. Notice that the function $G$ must be $C^2_{x,v}$ to be $f_0 \in C^2_{x,v}(\bar\Omega\times \mathbb{R}^2)$ in Theorem \e^{\frac 12}f{thm 2}. Since $f_0(x,v)=G(x,\vert v \vert)$ be a radial function with respect to $v$ and $\nablala_x f_0(x,v) \partialarallel v^T$ for all $(x,v)\in \gamma_-\cup \gamma_+$, $\nablala_x f_0(x,v)$ must be $0$ on $\partialartial \Omega$.
Now let us change \eqref{Cond2 1} and \eqref{Cond2 2} into symmetric forms. First, we multiply \eqref{Cond2 1} by $R_{x^1}$ from both left and right. Then applying \eqref{Cond2 3} and \eqref{RA}, we obtain
\begin{equation}gin{align*}
R_{x^1} \begin{equation}ig[ \nablala_{xv}f_{0}(x^1,v) + \nablala_{vv}f_{0}(x^1,v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n(x^1)} \begin{equation}ig] R_{x^1}
&= \nablala_{xv}f_{0}(x^1, R_{x^1}v) + \nablala_{vv}f_{0}(x^1, R_{x^1}v) \frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)} \notag \\
&\quad+
R_{x^1}
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^1 , R_{x^1}v) \nablala_x(R^1_{x^1(x,v)})
\\
\nablala_{v}f_{0}(x^1, R_{x^1}v) \nablala_x(R^2_{x^1(x,v)})
\end{bmatrix}
R_{x^1}.
\end{align*}
Also, plugging the above into \eqref{Cond2 2} and using \eqref{RA} again, we obtain
\begin{equation}gin{align*}
&R_{x^1}\begin{equation}ig[ \nablala_{xx}f_{0}(x^1,v) + \nablala_{vx}f_{0}(x^1, v) \frac{ (Qv)\otimes (Qv)}{v\cdotot n(x^1)} + \frac{ (Qv)\otimes (Qv)}{v\cdotot n(x^1)} \nablala_{xv}f_{0}(x^1, v) \begin{equation}ig] R_{x^1} \\
&= \nablala_{xx}f_{0}(x^1, R_{x^1}v) + \nablala_{vx}f_{0}(x^1, R_{x^1}v)\frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)}
+ \frac{(QR_{x^1}v)\otimes (QR_{x^1}v)}{R_{x^1}v\cdotot n(x^1)} \nablala_{xv}f_{0}(x^1, R_{x^1}v) \\
&\quad -2R_x^1
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{1}_{v,x^1}
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_{v}A^{2}_{v,x^1}
\end{bmatrix} R_{x^1}A_{v,x^1}R_{x^1}
+
A_{v,x^1}\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_x(R^1_{x^1(x,v)}) \\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_x(R^2_{x^1(x,v)})
\end{bmatrix}R_{x^1} \\
&\quad
+ R_{x^1} \begin{equation}gin{bmatrix}
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_x (R^1_{x^1(x,v)})
\\
\nablala_{x}f_{0}(x^{1}, R_{x^1}v) \nablala_x(R^2_{x^1(x,v)})
\end{bmatrix} R_{x^1}
- 2 R_{x^1}
\begin{equation}gin{bmatrix}
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_x(A^1_{v,x^1(x,v)})
\\
\nablala_{v}f_{0}(x^{1}, R_{x^1}v) \nablala_x(A^2_{v,x^1(x,v)})
\end{bmatrix} R_{x^1}.
\end{align*}
By Lemma \e^{\frac 12}f{d_RA} and Lemma \e^{\frac 12}f{dx_A}, $\nablala_x(R^1_{x^1(x,v)}), \nablala_x(R^2_{x^1(x,v)}), \nablala_x(A^1_{v,x^1(x,v)})$, and $\nablala_v(A^2_{v,x^1(x,v)})$ depend only on $n(x^1)$ and $v$. We rewrite $x^1$ as $x$ for $(x,v) \in \gamma_-$ because $n(x^1)=x^1$. Since $\nablala_x f_0(x,v)=0$ for $x\in \partialartial \Omega$, we obtain \eqref{C2 cond 1} and \eqref{C2 cond 2}.
Lastly, we will prove that $f(t,x,v)$ is not of class $C^2_{t,x,v}$ at time $t$ such that $t^k(t,x,v)=0$ for some $k$ if one of these conditions \eqref{C2 cond34}, \eqref{C2 cond 1}, and \eqref{C2 cond 2} for $(x,v)\in \gamma_-$ does not hold. Similar to the proof of Theorem \e^{\frac 12}f{thm 1}, it suffices to set $k=1$ and prove that $f(t,x,v)$ is not a class of $C^2_{t,x,v}$ at time $t$ satisfying $t^1(t,x,v)=0$.
Let $t^*$ be time $t$ such that $t^1(t,x,v)=0$. Remind that the condition \eqref{C2 cond34} was necessary to satisfy $\nablala_{xv}^Tf_0(x,v)=\nablala_{vx}f_0(x,v)$ and $\nablala_{xx}^T f_0(x,v) = \nablala_{xx}f_0(x,v)$ for $x\in \partialartial \Omega$. In other words, $\nablala_{xv}f_0(x,v)^T \neq \nablala_{vx} f_0(x,v)$ and $\nablala_{xx}^Tf_0(x,v)\neq \nablala_{xx}f_0(x,v)$ without \eqref{C2 cond34}. In $\nablala_{xv}$ and $\nablala_{vx}$ with direction $\hat{r}_1$, we derived \eqref{nabla_xv f case1} and \eqref{nabla_vx f case1} at $t^*$:
\begin{equation}gin{equation*}
\nablala_{xv}f(t,x,v) = (-t) \nablala_{xx} f_0(x^1,v) + \nablala_{xv} f_0(x^1,v), \quad \nablala_{vx}f(t,x,v)=(-t) \nablala_{xx}f_0(x^1,v)+\nablala_{vx}f_0(x^1,v).
\end{equation*}
Thus, if \eqref{C2 cond34} is not provided, $\nablala_{xv}^T f(t,x,v) \neq \nablala_{vx}f(t,x,v)$. This implies that $f(t,x,v)$ is not $C^2_{t,x,v}$ at time $t^*$. Next, we do not assume \eqref{C2 cond 1} for $(x,v)\in \gamma_-$. The condition \eqref{C2 cond 1} is derived from $\nablala_{xv}$ compatibility condition \eqref{Cond2 1}. Therefore, directional derivatives \eqref{nabla_xv f case1} and \eqref{nabla_xv f case2} with respect to $\hat{r}_1$ and $\hat{r}_2$ are not the same. It means that $f(t,x,v)$ is not $C^2_{t,x,v}$ at time $t^*$.
Finally, we assume that \eqref{C2 cond 2} does not hold for $(x,v)\in \gamma_-$.
The condition \eqref{C2 cond 2} comes from $\nablala_{xx},\nablala_{xv}$ and $\nablala_{vx}$ compatibility conditions \eqref{Cond2 1}, \eqref{Cond2 2}, and \eqref{Cond2 4}. One may assume without loss of generality that the initial data $f_0$ satisfies \eqref{C2 cond34} and \eqref{C2 cond 1}. Then, only $\nablala_{xx}$ compatibility condition \eqref{Cond2 2} is not satisfied. Similar to the above, directional derivatives $\nablala_{xx}$ with respect to $\hat{r}_1$ and $\hat{r}_2$ are not the same. Then, $f(t,x,v)$ is not $C^2_{t,x,v}$ at time $t^*$ without \eqref{C2 cond 2}. This finishes the proof.
\end{proof}
\section{Regularity estimate of $f$}
\subsection{First order estimates of characteristics} Using Definition \e^{\frac 12}f{notation},
\begin{equation}gin{equation*}
V(0;t,x,v) = R_{\ell} R_{\ell-1} \cdotots R_{2} R_{1} v, \quad \text{for some $\ell$ such that}\quad t^{\ell+1} < 0 \leq t^{\ell},
\end{equation*}
where
\[
R_{j} = I - 2 n(x^{j})\otimes n(x^{j}).
\]
For above $\ell$,
\begin{equation}gin{equation*}
X(0;t,x,v) = x^{\ell} - v^{\ell}t^{\ell},
\end{equation*}
where inductively,
\[
x^{k} = x^{k-1} - v^{k-1}(t^{k-1} - t^{k}),\quad 2\leq k \leq \ell,
\]
and
\[
x^{1} = x - v(t-t^{1}) = x - vt_{\mathbf{b}}.
\]
Or using rotational symmetry, we can also express
\[
x^{\ell} = Q_{\theta}^{\ell-1}x^{1},
\]
where $Q_{\theta}$ is operator (matrix) which means rotation(on the boundary of the disk) by $\theta$. $\theta$ is uniquely determined by its first (backward in time) bounce angle $v\cdotot n(x_{\mathbf{b}})$. \\
\begin{equation}gin{lemma} \label{der theta}
Here, $\theta$ is the angle at which $v$ is rotated to $v^1$. Moreover, $\theta>0$ is the same as the angle of rotation from $x^{k}$ to $x^{k+1}$ for $k=1,2,\cdotots,l-1$. Then, derivatives of $\theta$ with respect to $x$ and $v$ are
\begin{equation}gin{equation} \label{d_theta}
\nablala_x \theta =-\frac{2}{\sin \frac{\theta}{2}} Q_{-\frac{\theta}{2}}n(x^{1}), \quad \nablala_v \theta = 2\left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} - \frac{1}{\vert v \vert}\right) Q_{-\frac{\theta}{2}}n(x^{1}),
\end{equation}
provided $n(x^1)\cdotot v \neq0$.
\end{lemma}
\begin{equation}gin{proof}
From the definition of $\theta$,
\begin{equation}gin{equation} \label{theta}
\cos \left( \frac{\partiali}{2} - \frac{\theta}{2} \right) = \sin \left( \frac{\theta}{2} \right) = - \left[ n(x^1) \cdotot \frac{v}{\vert v \vert} \right].
\end{equation}
Thus, taking $\nablala_x$ yields
\begin{equation}gin{align*}
\frac{1}{2} \cos \frac{\theta}{2} \nablala_x \theta= -\frac{v}{\vert v \vert} \nablala_x \left( n(x^1)\right)=-\frac{v}{\vert v \vert} \left( I - \frac{v \otimes n(x^1)}{v \cdotot n(x^1)} \right)=-\frac{v}{\vert v \vert}+ \frac{\vert v \vert}{v\cdotot n(x^1)} n(x^1),
\end{align*}
where we used the product rule in Lemma \e^{\frac 12}f{matrix notation} and \eqref{normal} in Lemma \e^{\frac 12}f{d_n}. Note that rotating an angle $\partialhi=\frac{\partiali}{2}-\frac{\theta}{2}>0$ on a normal vector $n(x^1)$ gives the vector $- \frac{v}{\vert v \vert}$. In other words, it holds that
\begin{equation}gin{equation} \label{v_n}
-\frac{v}{\vert v \vert} = Q_{\partialhi} n (x^1),
\end{equation}
where $Q_{\partialhi}= \begin{equation}gin{bmatrix} \cos \partialhi & -\sin \partialhi \\ \sin \partialhi & \cos \partialhi \end{bmatrix} =\begin{equation}gin{bmatrix} \sin \frac{\theta}{2} & -\cos \frac{\theta}{2} \\ \cos \frac{\theta}{2} & \sin \frac{\theta}{2} \end{bmatrix}$. Thus,
\begin{equation}gin{align*}
\nablala_x \theta &= \frac{2}{\cos \frac{\theta}{2}} \left( Q_{\partialhi} - \frac{1}{\sin \frac{\theta}{2}} I \right) n(x^1)=\frac{2}{\cos \frac{\theta}{2}\sin \frac{\theta}{2}} \begin{equation}gin{bmatrix} \sin^2 \frac{\theta}{2} -1 & -\cos \frac{\theta}{2} \sin \frac{\theta}{2} \\ \sin\frac{\theta}{2}\cos \frac{\theta}{2}& \sin^2 \frac{\theta}{2} -1\end{bmatrix}n(x^1) \\
&= -\frac{2}{\sin \frac{\theta}{2}} \begin{equation}gin{bmatrix} \cos \frac{\theta}{2} & \sin \frac{\theta}{2} \\ -\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix}n(x^1) = -\frac{2}{\sin \frac{\theta}{2}} Q_{-\frac{\theta}{2}}n(x^{1}).
\end{align*}
Similarly, taking the derivative $\nablala_v$ of both sides in \eqref{theta}:
\begin{equation}gin{align*}
\frac{1}{2} \cos \frac{\theta}{2} \nablala_v \theta=-\frac{v}{\vert v \vert} \nablala_v \left( n(x^1)\right) -n(x^1) \left( \frac{1}{\vert v \vert} I- \frac{v \otimes v}{\vert v \vert^3} \right)&=t_{\mathbf{b}}\frac{v}{\vert v \vert} \begin{equation}ig(I - \frac{v\otimes n(x^1)}{v\cdotot n(x^1)} \begin{equation}ig)-n(x^1) \left( \frac{1}{\vert v \vert} I- \frac{v \otimes v}{\vert v \vert^3} \right)\\
&=t_{\mathbf{b}} \frac{v}{\vert v \vert} -t_{\mathbf{b}} \frac{ \vert v \vert}{v\cdotot n(x^1)} n (x^1) - \frac{1}{\vert v \vert} n(x^1) + \frac{v \cdotot n(x^1)}{\vert v \vert^2} \frac{v}{\vert v \vert},
\end{align*}
where we used the product rule in Lemma \e^{\frac 12}f{matrix notation} and \eqref{normal} in Lemma \e^{\frac 12}f{d_n}. From \eqref{v_n},
\begin{equation}gin{align*}
\nablala_v \theta &= \frac{2}{\cos \frac{\theta}{2} \sin \frac{\theta}{2}} \left( -t_{\mathbf{b}}\sin\frac{\theta}{2} \left[Q_{\partialhi}-\frac{1}{\sin \frac{\theta}{2}}I \right]n(x^1)+\frac{\sin^2\frac{\theta}{2}}{\vert v \vert} \left[ Q_{\partialhi} - \frac{1}{\sin \frac{\theta}{2}} I\right]n(x^1) \right)\\
&=\frac{2 t_{\mathbf{b}}}{\sin \frac{\theta}{2}} \begin{equation}gin{bmatrix} \cos \frac{\theta}{2}& \sin \frac{\theta}{2} \\ -\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix}n(x^1) -\frac{2}{\vert v \vert} \begin{equation}gin{bmatrix} \cos \frac{\theta}{2} & \sin\frac{\theta}{2} \\ -\sin\frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix} n(x^1)\\
&=2\left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} - \frac{1}{\vert v \vert}\right) \begin{equation}gin{bmatrix} \cos \frac{\theta}{2} & \sin\frac{\theta}{2} \\ -\sin\frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix} n(x^1)
= 2\left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} - \frac{1}{\vert v \vert}\right) Q_{-\frac{\theta}{2}}n(x^{1}).
\end{align*}
\end{proof}
\begin{equation}gin{lemma} \label{X,V}
Let $(t,x,v) \in \mathbb{R}_+\times \Omega\times \mathbb{R}^2$. The specular characteristics $X(0;t,x,v)$ and $V(0;t,x,v)$ are defined in Definition \e^{\frac 12}f{notation}. Whenever $n(x^1)\cdotot v\neq0$, we have derivatives of the characteristics $X(0;t,x,v)$ and $V(0;t,x,v)$:
\begin{equation}gin{align} \label{n_x,v}
\begin{equation}gin{split}
\nablala_x X(0;t,x,v) &= Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) +t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_x \theta \right) - \frac{1}{\vert v \vert \sin \frac{\theta}{2}}Q_\theta^l \left(v \otimes n(x^1)\right) \\
&\quad -\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta -\partiali} \left(n(x^1) \otimes \nablala_x \theta \right), \\
\nablala_v X(0;t,x,v)&=-t_{\mathbf{b}} Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -t^l Q_\theta ^l+t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_v \theta \right)+\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} Q_{\theta}^l \left(v \otimes n(x^1)\right) \\
&\quad - \frac{2(l-1)\sin\frac{\theta}{2}}{\vert v \vert^3} Q_\theta^l \left(v \otimes v\right) -\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta - \partiali}\left(n(x^1) \otimes \nablala_v \theta\right), \\
\nablala_x V(0;t,x,v)&= -lQ_{l\theta-\frac{\partiali}{2}} \left( v\otimes \nablala_x \theta \right),\\
\nablala_v V(0;t,x,v)&= Q_{\theta} ^l -lQ_{l\theta-\frac{\partiali}{2}} \left( v\otimes \nablala_v \theta \right),
\end{split}
\end{align}
where $\theta$ is the angle given in Lemma \e^{\frac 12}f{der theta}, $t_{\mathbf{b}}$ is the backward exit time defined in Definition \e^{\frac 12}f{notation}, $l$ is the bouncing number, and $Q_\theta$ is a rotation matrix by $\theta$.
\end{lemma}
\begin{equation}gin{proof}
Recall
\begin{equation}gin{align*}
X(0;t,x,v) = x^l - v^l t^l , \quad V(0;t,x,v) = v^l.
\end{align*}
Using the rotation matrix $Q_\theta$, $x^l$ and $v^l$ can be expressed by
\begin{equation}gin{align} \label{x,v_l}
x^l = Q_{\theta}^{l-1} x^1, \quad v^l = Q_{\theta}^l v.
\end{align}
By the chain rule,
\begin{equation}gin{align*}
\frac{\partialartial{(X(0;t,x,v),V(0;t,x,v)})}{\partialartial{(x,v)}}= \frac{\partialartial{(X(0;t,x,v),V(0;t,x,v))}}{\partialartial{(t^l,x^l,v^l)}} \frac{\partialartial(t^l,x^l,v^l)}{\partialartial(x,v)}= \begin{equation}gin{bmatrix} -v^l & I & -t^l I \\ \textbf{0}_{2\times 1} & \textbf{0}_{2 \times 2} & I\end{bmatrix} \begin{equation}gin{bmatrix} \nablala_x t^l & \nablala_v t^l \\ \nablala_x x^l & \nablala_v x^l \\ \nablala_x v^l & \nablala_v v^l \end{bmatrix},
\end{align*}
where $I$ is a $2\times 2$ identity matrix. For the derivative of $X(0;t,x,v),V(0;t,x,v)$, it is necessary to find the derivative of $t^l,x^l,$ and $v^l$. Using the expression \eqref{x,v_l} and \eqref{d_matrix} in Lemma \e^{\frac 12}f{matrix notation}, we derive
\begin{equation}gin{align*}
\nablala_x x^l &= \nablala_x \left[ Q_\theta ^{l-1} x^1 \right]=Q_{\theta}^{l-1} \nablala_x x^1 -(l-1)\left(\begin{equation}gin{bmatrix} \sin(l-1)\theta & \cos(l-1)\theta \\ - \cos (l-1)\theta & \sin(l-1)\theta \end{bmatrix} x^1\right) \otimes \nablala_x \theta\\
&\hspace{.3cm} \qquad \qquad \qquad =Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -(l-1)\left(\begin{equation}gin{bmatrix} \sin(l-1)\theta & \cos(l-1)\theta \\ - \cos (l-1)\theta & \sin(l-1)\theta \end{bmatrix} x^1\right) \otimes \nablala_x \theta,\\
\nablala_v x^l &= \nablala_v \left[ Q_\theta ^{l-1} x^1 \right]=Q_{\theta}^{l-1} \nablala_v x^1 -(l-1)\left(\begin{equation}gin{bmatrix} \sin(l-1)\theta & \cos(l-1)\theta \\ - \cos (l-1)\theta & \sin(l-1)\theta \end{bmatrix} x^1\right) \otimes \nablala_v \theta\\
&\hspace{.3cm} \qquad \qquad \qquad =-t_{\mathbf{b}} Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -(l-1)\left(\begin{equation}gin{bmatrix} \sin(l-1)\theta & \cos(l-1)\theta \\ - \cos (l-1)\theta & \sin(l-1)\theta \end{bmatrix} x^1\right) \otimes \nablala_v \theta,\\
\nablala_x v^l &= \nablala_x \left[ Q_\theta^l v \right]= -l \left( \begin{equation}gin{bmatrix} \sin l \theta & \cos l \theta \\ -\cos l \theta & \sin l \theta \end{bmatrix} v\right) \otimes \nablala_x \theta, \\
\nablala_v v^l &= \nablala_v \left [ Q_\theta^l v \right]= Q_{\theta} ^l -l \left( \begin{equation}gin{bmatrix} \sin l \theta & \cos l \theta \\ -\cos l \theta & \sin l \theta \end{bmatrix} v\right) \otimes \nablala_v \theta.
\end{align*}
For the derivative of $t^l$, we rewrite $t^l$ as
\begin{equation}gin{align*}
t^l = t-(t-t^1) - \sum_{k=1}^{l-1}(t^k-t^{k+1})= t- t_{\mathbf{b}} -\sum_{k=1}^{l-1} (t^k-t^{k+1}).
\end{align*}
Since $\mathrm{d}isplaystyle t^k-t^{k+1}=\frac{2\sin\frac{\theta}{2}}{\vert v \vert}$ for all $k=1,2,\mathrm{d}ots,l-1$, it holds that
\begin{equation}gin{align} \label{t_ell}
t^l = t-t_{\mathbf{b}}- \frac{2(l-1)\sin \frac{\theta}{2}}{\vert v \vert}, \quad l-1 = \frac{\vert v \vert}{ 2 \sin \frac{\theta}{2}} \left(t-t_{\mathbf{b}} -t^l\right).
\end{align}
Taking the derivative of $t^l$ with respect to $x,v$
\begin{equation}gin{align} \label{nabla x,v t_ell}
\begin{equation}gin{split}
\nablala_x t^l &= -\nablala_x t_{\mathbf{b}} -\frac{(l-1) \cos \frac{\theta}{2}}{\vert v \vert} \nablala_x \theta=-\frac{n(x^1)}{v \cdotot n(x^1)}-\frac{(l-1) \cos \frac{\theta}{2}}{\vert v \vert} \nablala_x \theta=\frac{1}{\vert v \vert \sin\frac{\theta}{2}} n(x^1) -\frac{(l-1) \cos \frac{\theta}{2}}{\vert v \vert} \nablala_x \theta, \\
\nablala_v t^l &= -\nablala_v t_{\mathbf{b}} + \frac{2(l-1)\sin \frac{\theta}{2}}{\vert v \vert^3} v -\frac{(l-1) \cos \frac{\theta}{2}}{\vert v \vert} \nablala_v \theta=t_{\mathbf{b}} \frac{n(x^1)}{v \cdotot n(x^1)} +\frac{2(l-1)\sin \frac{\theta}{2}}{\vert v \vert^3} v -\frac{(l-1) \cos \frac{\theta}{2}}{\vert v \vert} \nablala_v \theta\\
&\hspace{7.1cm}=-\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} n(x^1) +\frac{2(l-1)\sin \frac{\theta}{2}}{\vert v \vert^3} v -\frac{(l-1) \cos \frac{\theta}{2}}{\vert v \vert} \nablala_v \theta.
\end{split}
\end{align}
Also note that, from \eqref{v_n} and \eqref{t_ell}, we have
\begin{equation}gin{equation} \label{cancel}
\begin{equation}gin{split}
&-(l-1)Q_{(l-1)\theta-\frac{\partiali}{2}} \left(x^1 \otimes \nablala \theta\right) +\frac{(l-1)\cos\frac{\theta}{2}}{\vert v \vert} Q_{\theta}^l \left(v \otimes \nablala \theta\right) \\
&= -(l-1)\left(Q_{(l-1)\theta -\frac{\partiali}{2}} +\cos \frac{\theta}{2} Q_{l\theta}Q_{\frac{\partiali}{2}-\frac{\theta}{2}}\right) \left(n(x^1) \otimes \nablala \theta \right) \\
&= - (l-1) Q_{(l-1)\theta -\frac{\partiali}{2}} \begin{equation}gin{bmatrix} \sin^2 \frac{\theta}{2} & \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\ -\sin \frac{\theta}{2} \cos \frac{\theta}{2} & \sin^2 \frac{\theta}{2} \end{bmatrix} \left(n(x^1) \otimes \nablala \theta \right) \\
&= -\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta -\partiali} \left(n(x^1) \otimes \nablala \theta \right).
\end{split}
\end{equation}
Hence, using \eqref{cancel} and $x^{1}=n(x^{1})$,
\begin{equation}gin{align*}
\begin{equation}gin{split}
\nablala_x X(0;t,x,v) &= \nablala_x x^l - t^l \nablala_x v^l -v^l \otimes \nablala_x t^l\\
&= Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -(l-1)\left(\begin{equation}gin{bmatrix} \sin(l-1)\theta & \cos(l-1)\theta \\ - \cos (l-1)\theta & \sin(l-1)\theta \end{bmatrix} x^1\right) \otimes \nablala_x \theta \\
& \quad +t^l l \left( \begin{equation}gin{bmatrix} \sin l \theta & \cos l \theta \\ -\cos l \theta & \sin l \theta \end{bmatrix} v\right) \otimes \nablala_x \theta-\frac{1}{\vert v \vert \sin \frac{\theta}{2}}Q_\theta^l \left(v \otimes n(x^1)\right) +\frac{(l-1)\cos\frac{\theta}{2}}{\vert v \vert} Q_{\theta}^l \left(v \otimes \nablala_x \theta\right) \\
&=Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) +t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_x \theta \right) - \frac{1}{\vert v \vert \sin \frac{\theta}{2}}Q_\theta^l \left(v \otimes n(x^1)\right) \\
&\quad -(l-1)Q_{(l-1)\theta-\frac{\partiali}{2}} \left(x^1 \otimes \nablala_x \theta\right) +\frac{(l-1)\cos\frac{\theta}{2}}{\vert v \vert} Q_{\theta}^l \left(v \otimes \nablala_x \theta\right) \\
&=Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) +t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_x \theta \right) - \frac{1}{\vert v \vert \sin \frac{\theta}{2}}Q_\theta^l \left(v \otimes n(x^1)\right) \\
&\quad -\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta -\partiali} \left(n(x^1) \otimes \nablala_x \theta \right), \\
\nablala_v X(0;t,x,v)&=\nablala_v x^l -t^l \nablala_v v^l-v^l \otimes \nablala_v t^l\\
&=-t_{\mathbf{b}} Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -(l-1)\left(\begin{equation}gin{bmatrix} \sin(l-1)\theta & \cos(l-1)\theta \\ - \cos (l-1)\theta & \sin(l-1)\theta \end{bmatrix} x^1\right) \otimes \nablala_v \theta \\
&\quad -t^l Q_\theta ^l+t^l l \left( \begin{equation}gin{bmatrix} \sin l \theta & \cos l \theta \\ -\cos l \theta & \sin l \theta \end{bmatrix} v\right) \otimes \nablala_v \theta\\
&\quad +\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} Q_{\theta}^l \left(v \otimes n(x^1)\right) - \frac{2(l-1)\sin\frac{\theta}{2}}{\vert v \vert^3} Q_\theta^l \left(v \otimes v\right)+ \frac{(l-1)\cos \frac{\theta}{2}}{\vert v \vert} Q_\theta^l\left( v \otimes \nablala_v \theta \right),\\
&= -t_{\mathbf{b}} Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) -t^l Q_\theta ^l+t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_v \theta \right)
+\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} Q_{\theta}^l \left(v \otimes n(x^1)\right)\\
&\quad - \frac{2(l-1)\sin\frac{\theta}{2}}{\vert v \vert^3} Q_\theta^l \left(v \otimes v\right) -(l-1)Q_{(l-1)\theta-\frac{\partiali}{2}}\left(x^1 \otimes \nablala_v \theta \right) + \frac{(l-1)\cos \frac{\theta}{2}}{\vert v \vert} Q_\theta^l\left( v \otimes \nablala_v \theta \right)\\
&=-t_{\mathbf{b}} Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x^1)}{v\cdotot n(x^1)} \right) - t^l Q_\theta ^l+t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_v \theta \right)+\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} Q_{\theta}^l \left(v \otimes n(x^1)\right) \\
&\quad - \frac{2(l-1)\sin\frac{\theta}{2}}{\vert v \vert^3} Q_\theta^l \left(v \otimes v\right)
-\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta - \partiali}\left(n(x^1) \otimes \nablala_v \theta\right) ,\\
\nablala_x V(0;t,x,v)&=\nablala_x v^l = -l \left( \begin{equation}gin{bmatrix} \sin l \theta & \cos l \theta \\ -\cos l \theta & \sin l \theta \end{bmatrix} v\right) \otimes \nablala_x \theta=-lQ_{l\theta-\frac{\partiali}{2}} \left( v\otimes \nablala_x \theta \right),\\
\nablala_v V(0;t,x,v)&= \nablala_v v^l = Q_{\theta} ^l -l \left( \begin{equation}gin{bmatrix} \sin l \theta & \cos l \theta \\ -\cos l \theta & \sin l \theta \end{bmatrix} v\right) \otimes \nablala_v \theta=Q_{\theta}^l -l Q_{l \theta-\frac{\partiali}{2}} \left( v\otimes \nablala_v \theta \right).
\end{split}
\end{align*}
\end{proof}
\begin{equation}gin{lemma}
The exit backward time $t_{\mathbf{b}}$ and the $l$-th bouncing backward time $t^l$ are defined in Definition \e^{\frac 12}f{notation}. Then, it holds that
\begin{equation}gin{align}\label{tb esti}
t_{\mathbf{b}} \leq \frac{2\sin \frac{\theta}{2}}{ \vert v \vert}, \quad t^l \leq \frac{2\sin \frac{\theta}{2}}{ \vert v \vert}.
\end{align}
\end{lemma}
\begin{equation}gin{proof}
Note that
\begin{equation}gin{align*}
t_{\mathbf{b}} = t-t^1= \frac{\vert x - x^1\vert}{ \vert v \vert }, \quad t^l = \frac{\vert x^l - X(0;t,x,v) \vert}{\vert v^l \vert}.
\end{align*}
Whenever $\theta$ is the angle at which $v$ is rotated to $v^1$, one obtains that
\begin{equation}gin{align*}
\vert x-x^1 \vert\leq 2 \sin \frac{\theta}{2}, \quad \vert x^l - X(0;t,x,v)\vert \leq 2 \sin \frac{\theta}{2}.
\end{align*}
From the above inequalities and $\vert v^l \vert = \vert v \vert $, we obtain
\begin{equation}gin{align*}
t_{\mathbf{b}} \leq \frac{2\sin \frac{\theta}{2}}{ \vert v \vert}, \quad t^l \leq \frac{2\sin \frac{\theta}{2}}{ \vert v \vert}.
\end{align*}
\end{proof}
\begin{equation}gin{lemma} \label{est der X,V}
Under the same assumption in Lemma \e^{\frac 12}f{X,V}, we have estimates of derivatives for the characteristics $X(0;t,x,v)$ and $V(0;t,x,v)$
\begin{equation}gin{align*}
\begin{equation}gin{split}
\vert \nablala_x X(0;t,x,v) \vert &\lesssim \frac{\vert v \vert} { \vert v \cdotot n(x_{\mathbf{b}}) \vert}\left( 1 + \vert v \vert t\right),\\
\vert \nablala_v X(0;t,x,v) \vert &\lesssim \frac{1} { \vert v \vert}\left( 1 + \vert v \vert t \right), \\
\vert \nablala_x V(0;t,x,v) \vert & \lesssim \frac{\vert v \vert^3}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert^2} \left( 1+ \vert v \vert t \right), \\
\vert \nablala_v V(0;t,x,v) \vert & \lesssim \frac{\vert v \vert}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert} \left( 1+ \vert v \vert t \right),
\end{split}
\end{align*}
where $n(x_{\mathbf{b}})$ is outward unit normal vector at $x_{\mathbf{b}} = x-t_{\mathbf{b}} v \in \partial\Omega$. \\
\end{lemma}
\begin{equation}gin{remark}
First-order derivatives of characteristics $(X,V)$ for general 3D convex domain were obtained in \cite{GKTT2017}. Lemma \e^{\frac 12}f{est der X,V} is simple version in 2D disk and its singular orders coincide with the results of \cite{GKTT2017}. \\
\end{remark}
\begin{equation}gin{proof}
By \eqref{n_x,v} in Lemma \e^{\frac 12}f{X,V}, we have
\begin{equation}gin{align*}
\nablala_x X(0;t,x,v) &= Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})} \right)-\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta -\partiali} \left(n(x_{\mathbf{b}}) \otimes \nablala_x \theta \right)\\
&\quad +t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_x \theta \right) - \frac{1}{\vert v \vert \sin \frac{\theta}{2}}Q_\theta^l \left(v \otimes n(x_{\mathbf{b}})\right).
\end{align*}
We define a matrix norm by
\begin{equation}gin{equation*}
\vert A \vert = \max _{i,j} a_{i,j},
\end{equation*}
where $a_{i,j}$ is the $(i,j)$ component of the matrix $A$. Then, we can easily check that
\begin{equation}gin{equation*}
\vert a \otimes b \vert \leq \vert a \vert \vert b \vert,
\end{equation*}
for any $a,b \in \mathbb{R}^n$. To find upper bound of $\nablala_x X(0;t,x,v)$, we only need to consider $\nablala_x \theta$ and $t^l \times l$. By \eqref{d_theta},\eqref{t_ell}, and \eqref{tb esti},
\begin{equation}gin{align} \label{e_1}
\vert \nablala_x \theta \vert =\left \vert \frac{2}{\sin \frac{\theta}{2}} Q_{-\frac{\theta}{2}} n(x_{\mathbf{b}}) \right \vert \leq \frac{2 \vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}}) \vert}, \quad t^l \times l \leq \frac{2 \sin\frac{\theta}{2}}{\vert v \vert}\times \left(\frac{\vert v \vert}{2\sin \frac{\theta}{2}}t +1 \right) \leq t+\frac{2}{\vert v \vert}.
\end{align}
Using the above inequalities, we derive that
\begin{equation}gin{align*}
\vert \nablala_x X(0;t,x,v) \vert &\leq 1+ \frac{ \vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}}) \vert} + \frac{ \vert v \vert t}{2} \vert \nablala_x \theta \vert + t^l l \vert v \vert \vert \nablala_x \theta \vert + \frac{1}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert} \vert v \vert \\
&\leq 1+\frac{ \vert v \vert}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert } + \frac{ \vert v \vert^2}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert} t + \frac{ 2\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert } \left( t + \frac{2}{\vert v \vert} \right) +\frac{\vert v \vert}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert}\\
&\lesssim \frac{\vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}}) \vert} \left( 1+ \vert v \vert t\right).
\end{align*}
\end{proof}
Recall the derivative $\nablala_v X(0;t,x,v)$ in Lemma \e^{\frac 12}f{X,V}.
\begin{equation}gin{align*}
\nablala_v X(0;t,x,v)&=-t_{\mathbf{b}} Q_{\theta}^{l-1}\left( I - \frac{v \otimes n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})} \right)-\frac{\vert v \vert(t-t_{\mathbf{b}}-t^l)}{2}Q_{(l-\frac{1}{2})\theta - \partiali}\left(n(x_{\mathbf{b}}) \otimes \nablala_v \theta\right) \\
&\quad -t^l Q_\theta ^l+t^l l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_v \theta \right)+\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} Q_{\theta}^l \left(v \otimes n(x_{\mathbf{b}})\right)- \frac{2(l-1)\sin\frac{\theta}{2}}{\vert v \vert^3} Q_\theta^l \left(v \otimes v\right).
\end{align*}
Similarly, to estimate $\nablala_v X(0;t,x,v)$, we need to estimate $\nablala_v \theta$. From \eqref{d_theta} and \eqref{tb esti}, we directly compute
\begin{equation}gin{align} \label{e_2}
\vert \nablala_v \theta \vert = 2 \left \vert \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}}- \frac{1}{\vert v \vert} \right) Q_{-\frac{\theta}{2}}n(x_{\mathbf{b}}) \right \vert \leq \frac{6}{\vert v \vert}.
\end{align}
Thus,
\begin{equation}gin{align*}
\vert \nablala_v X(0;t,x,v) \vert &\leq t_{\mathbf{b}} \left( 1+ \frac{\vert v \vert}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert} \right) +\frac{\vert v \vert t}{2} \vert \nablala_v \theta \vert + t^l + \left( t^l \times l \right) \vert v \vert \vert \nablala_v \theta \vert + \frac{ t_{\mathbf{b}}}{ \vert v \vert \sin \frac{\theta}{2}} \vert v \vert + \frac{2(l-1) \sin\frac{\theta}{2}}{\vert v \vert^3} \vert v \vert^2 \\
&\leq \frac{2\sin \frac{\theta}{2}}{\vert v \vert} \left( 1+ \frac{\vert v \vert}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert} \right)+\frac{\vert v \vert t}{2} \times \frac{6}{\vert v \vert}+\frac{2\sin \frac{\theta}{2}}{\vert v \vert} + 6\left(t +\frac{2}{\vert v \vert} \right)\\
&\quad + \frac{2\sin \frac{\theta}{2}}{\vert v \vert} \frac{1}{\vert v \vert \sin \frac{\theta}{2}} \vert v \vert + \frac{(t-t_{\mathbf{b}}-t^l)}{\vert v \vert^2} \vert v \vert^2\\
&\lesssim \frac{1}{\vert v \vert} \left (1+ \vert v \vert t \right),
\end{align*}
where we used \eqref{tb esti} and \eqref{e_1}. For $\nablala_{x,v} V(0;t,x,v)$, using \eqref{n_x,v}, \eqref{t_ell}, \eqref{e_1}, and \eqref{e_2} gives
\begin{equation}gin{align*}
\vert \nablala_x V(0;t,x,v) \vert &= \left \vert -l Q_{l\theta-\frac{\partiali}{2}} \left( v \otimes \nablala_x \theta \right) \right \vert \leq \left(\frac{\vert v \vert}{\vert 2\sin \frac{\theta}{2}\vert} t +1\right) \vert v \vert \vert \nablala_x \theta \vert \lesssim \frac{ \vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^2}(1+ \vert v \vert t), \\
\vert \nablala_v V(0;t,x,v) \vert&= \left \vert Q_{\theta}^l - lQ_{l\theta -\frac{\partiali}{2}} (v \otimes \nablala_v \theta) \right \vert \leq 1+ \left(\frac{\vert v \vert}{\vert 2 \sin \frac{\theta}{2}\vert} t +1\right) \vert v \vert \vert \nablala_v \theta \vert\lesssim \frac{\vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}}) \vert} (1+ \vert v \vert t).
\end{align*}
\subsection{Second-order estimates of characteristics}
\begin{equation}gin{lemma}
$n(x_{\mathbf{b}})$ is outward unit normal vector at $x_{\mathbf{b}}\in \partialartial \Omega$. For $(x_{\mathbf{b}},v) \notin \gamma_0$, it follows that
\begin{equation}gin{equation} \label{est der n}
\vert \nablala_x [n(x_{\mathbf{b}})] \vert \lesssim \frac{\vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}}) \vert}, \quad \vert \nablala_v [n(x_{\mathbf{b}})] \vert \lesssim \frac{1}{\vert v \vert}.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
We denote the components of $v$ and $n(x_{\mathbf{b}})$ by $(v_1,v_2)$ and $(n_1,n_2)$. By \eqref{normal} in Lemma \e^{\frac 12}f{d_n} and \eqref{tb esti}, we have
\begin{equation}gin{align*}
\nablala_x[n(x_{\mathbf{b}})] &= I- \frac{v\otimes n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})} = \frac{1}{v\cdotot n(x_{\mathbf{b}})}\begin{equation}gin{bmatrix}
v_2n_2 & -v_1n_2 \\
-v_2n_1 & v_1n_1
\end{bmatrix},\\
\nablala_v [n(x_{\mathbf{b}})]&=-t_{\mathbf{b}}\left(I- \frac{v\otimes n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})}\right)=\frac{-t_{\mathbf{b}}}{v\cdotot n(x_{\mathbf{b}})} \begin{equation}gin{bmatrix}
v_2n_2 & -v_1n_2 \\
-v_2n_1 & v_1n_1
\end{bmatrix},
\end{align*}
which is further bounded by
\begin{equation}gin{equation*}
\vert \nablala_x [n(x_{\mathbf{b}})] \vert \lesssim \frac{\vert v \vert }{\vert v \cdotot n(x_{\mathbf{b}})\vert}, \quad \vert \nablala_v [n(x_{\mathbf{b}})] \vert \lesssim \frac{1}{\vert v \vert}.
\end{equation*}
\end{proof}
\begin{equation}gin{lemma}
The exit backward time $t_{\mathbf{b}}$ and the $l$-th bouncing backward time $t^l$ are defined in Definition \e^{\frac 12}f{notation}. Then, we have the following estimates
\begin{equation}gin{equation} \label{est der t_ell}
\begin{equation}gin{split}
&\vert \nablala_x t^1 \vert \lesssim \frac{1}{\vert v \vert \vert \sin\frac{\theta}{2} \vert}, \quad \vert \nablala_v t^1 \vert \lesssim \frac{1}{\vert v \vert^2},\\
&\vert \nablala_x t^l \vert \lesssim \frac{1}{\vert v \vert \sin^2\frac{\theta}{2}}(1+\vert v\vert t), \quad \vert \nablala_v t^l \vert \lesssim \frac{1}{\vert v \vert^2 \vert \sin \frac{\theta}{2} \vert }(1+\vert v \vert t),
\end{split}
\end{equation}
whenever $v\cdotot n(x_{\mathbf{b}}) \neq 0$.
\end{lemma}
\begin{equation}gin{proof}
Since $t^1 = t-t_b$, it follows from Lemma \e^{\frac 12}f{nabla xv b} that
\begin{equation}gin{align*}
\nablala_x t^1 = -\nablala_x t_{\mathbf{b}} = -\frac{n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})}, \quad \nablala_v t^1 = -\nablala_v t_{\mathbf{b}} = t_{\mathbf{b}} \frac{n(x_{\mathbf{b}})}{v\cdotot n(x_{\mathbf{b}})}.
\end{align*}
Using the above and \eqref{tb esti} implies that
\begin{equation}gin{align*}
\vert \nablala_x t^1 \vert \lesssim \frac{1}{\vert v \vert \left \vert \sin \frac{\theta}{2}\right \vert}, \quad \vert \nablala_v t^1 \vert \lesssim \frac{1}{\vert v \vert^2}.
\end{align*}
By \eqref{nabla x,v t_ell} in the proof of Lemma \e^{\frac 12}f{X,V}, we have
\begin{equation}gin{align*}
\nablala_x t^l &= \frac{1}{\vert v \vert \sin \frac{\theta}{2}} n(x_{\mathbf{b}}) - \frac{(l-1)\cos \frac{\theta}{2}}{\vert v \vert} \nablala_x \theta, \\
\nablala_v t^l &= -\frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} n(x_{\mathbf{b}}) +\frac{2(l-1)\sin \frac{\theta}{2}}{\vert v \vert^3} v -\frac{(l-1)\cos \frac{\theta}{2}}{\vert v \vert} \nablala_v \theta.
\end{align*}
By \eqref{t_ell} in the proof of Lemma \e^{\frac 12}f{X,V}, the bouncing number $l$ can be bounded by
\begin{equation}gin{equation} \label{ell est}
l= 1+ \frac{\vert v \vert}{2\sin \frac{\theta}{2}} (t-t_{\mathbf{b}}-t^l) \leq 1+ \frac{\vert v \vert }{2\left \vert \sin \frac{\theta}{2}\right \vert} t \lesssim \frac{1}{\left \vert \sin\frac{\theta}{2}\right \vert} (1+\vert v \vert t).
\end{equation}
Then, from \eqref{tb esti}, \eqref{e_1},\eqref{e_2}, and \eqref{ell est}, one obtains that
\begin{equation}gin{align*}
\vert \nablala_x t^l \vert &\lesssim \frac{1}{\vert v \vert \vert \sin \frac{\theta}{2}\vert} + \frac{1}{\vert v \vert \sin^2 \frac{\theta}{2}}(1+\vert v \vert t)\lesssim \frac{1}{\vert v \vert \sin^2 \frac{\theta}{2}} (1+\vert v \vert t), \\
\vert\nablala_v t^l \vert &\lesssim \frac{1}{\vert v \vert^2}+\frac{1}{\vert v \vert^2}(1+\vert v \vert t) + \frac{1}{\vert v \vert^2}(1+\vert v \vert t) \lesssim \frac{1}{\vert v \vert^2} (1+\vert v \vert t).
\end{align*}
\end{proof}
\begin{equation}gin{lemma} \label{2nd est der X,V}
The characteristics $X(0;t,x,v)$ and $V(0;t,x,v)$ are defined in Definition \e^{\frac 12}f{notation}. Under the same assumption in Lemma \e^{\frac 12}f{X,V}, we have estimates for the second derivatives of characteristics
\begin{equation}gin{equation*}
\begin{equation}gin{split}
&\vert \nablala_{xx} X(0;t,x,v) \vert \lesssim \frac{\vert v \vert^4}{\vert v \cdotot n(x_{\mathbf{b}})\vert^4}(1+\vert v \vert^2 t^2), \quad \vert \nablala_{vx} X(0;t,x,v) \vert \lesssim \frac{\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3}(1+\vert v \vert^2 t^2), \\
&\vert \nablala_{xv} X(0;t,x,v) \vert \lesssim \frac{\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3}(1+\vert v \vert^2 t^2), \quad \vert \nablala_{vv}X(0;t,x,v) \vert \lesssim \frac{1}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^2}(1+\vert v \vert^2 t^2),\\
&\vert \nablala_{xx} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert^5}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^4} (1+\vert v \vert^2t^2), \quad \vert \nablala_{vx} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}})\vert^3}(1+\vert v \vert^2 t^2),\\
&\vert \nablala_{xv} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}})\vert^3}(1+\vert v \vert^2 t^2), \quad \vert \nablala_{vv} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}})\vert^2} (1+\vert v \vert^2 t^2),
\end{split}
\end{equation*}
where $\vert \nablala_{xx,xv,vv} X(0;t,x,v) (or \; V(0;t,x,v))\vert $ is given by $\sup_{i,j} \vert \nablala_{ij}X(0;t,x,v) (or \; \nablala_{ij}V(0;t,x,v))\vert$ for $i,j \in \{x_1,x_2,v_1,v_2\}$.
\end{lemma}
\begin{equation}gin{proof}
We denote the components of $v$ and $n(x_{\mathbf{b}})$ by $(v_1,v_2)$ and $(n_1,n_2)$. To estimate $\vert \nablala_{xx} X(0;t,x,v)\vert$, we need to determine which component in the matrix $\nablala_x X(0;t,x,v)$ has the highest singularity $\frac{1}{\sin \frac{\theta}{2}}$ and travel length $(1+\vert v \vert t)$ order when we take the derivative with respect to $x$. In estimates \eqref{e_1},\eqref{est der n},\eqref{est der t_ell}, and \eqref{ell est}, we already checked singularity and travel length order for some terms. Considering these estimates, we get the highest singularity and travel length order in the $x$-derivative of the (1,1) component of the matrix $\nablala_x X(0;t,x,v)$. Hence, we only consider the (1,1) component among components in the matrix $\nablala_x X(0;t,x,v)$. In fact, from Lemma \e^{\frac 12}f{X,V}, the (1,1) component $[\nablala_x X(0;t,x,v)]_{(1,1)}$ of the matrix $\nablala_x X(0;t,x,v)$ is
\begin{equation}gin{align*}
&[\nablala_x X(0;t,x,v)]_{(1,1)}\\
&= \cos((l-1)\theta) \frac{v_2n_2}{v\cdotot n(x_{\mathbf{b}})} + \sin((l-1)\theta) \frac{v_2 n_1}{v\cdotot n(x_{\mathbf{b}})}\\
&\quad +\frac{\vert v \vert(t^1-t^l)}{\sin \frac{\theta}{2}}\left(-n_1^2 \cos ((l-\frac{1}{2})\theta) \cos \frac{\theta}{2} -n_1n_2 \cos ((l-\frac{1}{2})\theta) \sin \frac{\theta}{2}+n_1n_2 \sin((l-\frac{1}{2})\theta)\cos \frac{\theta}{2} \right. \\
&\left. \qquad \qquad \qquad \qquad +n_2^2 \sin ((l-\frac{1}{2})\theta) \sin \frac{\theta}{2} \right)\\
&\quad-\frac{2t^l l}{\sin \frac{\theta}{2}} \left( v_1n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2n_1 \cos l\theta \cos \frac{\theta}{2} +v_2n_2 \cos l\theta \sin \frac{\theta}{2}\right)\\
&\quad -\frac{1}{\vert v \vert \sin \frac{\theta}{2}}\left( v_1 n_1 \cos l\theta -v_2n_1 \sin l\theta\right)\\
&\lesssim \frac{\vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}}) \vert} + \frac{1}{\vert \sin \frac{\theta}{2} \vert} (1+\vert v \vert t) +\frac{1}{\vert \sin \frac{\theta}{2}\vert} \lesssim \frac{\vert v \vert }{\vert v \cdotot n(x_{\mathbf{b}}) \vert}(1+\vert v \vert t) ,
\end{align*}
where the first inequality comes from \eqref{tb esti}, \eqref{ell est}, and
\begin{equation}gin{equation*}
t^1-t^l= \frac{2(l-1)\sin\frac{\theta}{2}}{\vert v \vert} \lesssim \frac{1}{\vert v \vert}(1+\vert v \vert t).
\end{equation*}
Similarly, the $(1,1)$ components of matrices $\nablala_v X(0;t,x,v), \nablala_x V(0;t,x,v)$, and $\nablala_v V(0;t,x,v)$ satisfy inequalities in Lemma \e^{\frac 12}f{est der X,V}. Similar as estimate $\vert \nablala_{xx} X(0;t,x,v)\vert$, we only consider $(1,1)$ components of derivative matrices for $X(0;t,x,v)$ and $V(0;t,x,v)$ to get estimates. When we differentiate $[\nablala_x X(0;t,x,v)]_{(1,1)}$ with respect to $x$, the terms containing $\frac{t^l l}{\sin \frac{\theta}{2}}$ are main terms that increase the singularity $\frac{1}{\sin \frac{\theta}{2}}$ and travel length $(1+\vert v \vert t)$ order. $\frac{t^l l}{\sin \frac{\theta}{2}}$ has a singularity order 1 and travel length order 1 because
\begin{equation}gin{equation*}
\left \vert \frac{t^l l}{\sin \frac{\theta}{2}}\right \vert \lesssim \frac{1}{\vert \sin \frac{\theta}{2}\vert } \times \frac{\vert \sin \frac{\theta}{2}\vert }{\vert v \vert} \times \frac{1}{\vert \sin \frac{\theta}{2}\vert}(1+\vert v \vert t)=\frac{ \vert v \vert}{\vert v \cdotot n(x_{\mathbf{b}})\vert}(1+\vert v \vert t),
\end{equation*}
where we have used \eqref{tb esti} and \eqref{ell est}. On the other hand, if we take of the term $\frac{t^l l}{\sin \frac{\theta}{2}}$ with respect to $x$, the singularity and travel length order become $4$ and $2$ respectively:
\begin{equation}gin{align*}
\left \vert \nablala_x \left(\frac{t^l l}{\sin \frac{\theta}{2}}\right)\right \vert =\left \vert \frac{l}{\sin \frac{\theta}{2}} \nablala_x t^l -\frac{t^l l\cos\frac{\theta}{2}}{2\sin^2 \frac{\theta}{2}} \nablala_x \theta\right \vert &\lesssim \frac{1}{\vert v \vert \sin^4 \frac{\theta}{2}}(1+\vert v \vert^2t^2) + \frac{1}{\vert v \vert \vert \sin^3 \frac{\theta}{2}\vert}(1+\vert v \vert t)\\
&\lesssim \frac{\vert v \vert^3}{\vert v\cdotot n(x_{\mathbf{b}})\vert^4} (1+\vert v \vert ^2 t^2),
\end{align*}
where \eqref{tb esti}, \eqref{e_1}, \eqref{est der t_ell}, and \eqref{ell est} have been used. Hence, it suffices to estimate the following terms in $[\nablala_x X(0;t,x,v)]_{(1,1)}$
\begin{equation}gin{equation*}
-\frac{2t^l l}{\sin \frac{\theta}{2}} \left( v_1n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2n_1 \cos l\theta \cos \frac{\theta}{2} +v_2n_2 \cos l\theta \sin \frac{\theta}{2}\right):=I_{1},
\end{equation*}
to obtain estimate for $\vert \nablala_{xx} X(0;t,x,v) \vert$. Taking the $x$-derivative to the above terms, one obtains
\begin{equation}gin{align*}
\nablala_x I_{1} &= \left ( \frac{-2l\nablala_x t^l}{\sin \frac{\theta}{2}} +\frac{2t^l l\cos\frac{\theta}{2}\nablala_x \theta}{2\sin^2 \frac{\theta}{2}} \right )\begin{equation}ig( v_1n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2n_1 \cos l\theta \cos \frac{\theta}{2} +v_2n_2 \cos l\theta \sin \frac{\theta}{2}\begin{equation}ig)\\
&\quad -\frac{2t^l l}{\sin \frac{\theta}{2}} \left( v_1 \sin l \theta \cos \frac{\theta}{2}\nablala_x n_1 +lv_1 n_1 \cos l\theta \cos \frac{\theta}{2}\nablala_x \theta -\frac{1}{2} v_1 n_1 \sin l\theta \sin \frac{\theta}{2} \nablala_x \theta \right. \\
&\qquad \qquad \quad \left. + v_1 \sin l \theta \sin \frac{\theta}{2}\nablala_x n_2 +lv_1 n_2 \cos l\theta \sin \frac{\theta}{2}\nablala_x \theta +\frac{1}{2} v_1 n_2 \sin l\theta \cos \frac{\theta}{2} \nablala_x \theta \right. \\
&\qquad \qquad \quad \left. + v_2 \cos l\theta \cos \frac{\theta}{2}\nablala_x n_1 -lv_2 n_1 \sin l \theta \cos \frac{\theta}{2} \nablala_x \theta -\frac{1}{2} v_2n_1\cos l\theta \sin \frac{\theta}{2}\nablala_x \theta \right. \\
&\qquad \qquad \quad \left. + v_2 \cos l\theta \sin \frac{\theta}{2}\nablala_x n_2 -lv_2 n_2 \sin l\theta \sin \frac{\theta}{2} \nablala_x \theta + \frac{1}{2} v_2 n_2 \cos l \theta \cos \frac{\theta}{2}\nablala_x \theta \right).
\end{align*}
Using \eqref{tb esti},\eqref{e_1},\eqref{est der n},\eqref{est der t_ell}, and \eqref{ell est}, one can further bound the above as
\begin{equation}gin{align*}
\vert \nablala_x I_{1}\vert &\lesssim \frac{\vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}})\vert^4} (1+\vert v \vert^2t^2) \times \vert v \vert + \frac{1}{\vert v \cdotot n(x_{\mathbf{b}})\vert}(1+\vert v \vert t) \times \left( \vert v \vert \vert \nablala_x n(x_{\mathbf{b}}) \vert + l\vert v \vert\vert \nablala_x \theta \vert\right )\\
&\lesssim \frac{\vert v \vert^4}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^4} (1+\vert v \vert^2 t^2).
\end{align*}
Therefore, we get
\begin{equation}gin{align*}
\vert \nablala_{xx} X(0;t,x,v) \vert \lesssim \frac{\vert v \vert^4}{\vert v\cdotot n(x_{\mathbf{b}}) \vert^4} (1+\vert v \vert^2t^2).
\end{align*}
For estimate of $\vert \nablala_{vx} X(0;t;x,v)\vert$, similar to the case $\vert \nablala_{xx} X(0;t,x,v) \vert$, we only consider terms $I_1$. By taking the $v$-derivative to $I_1$, we obtain
\begin{equation}gin{align*}
\nablala_v I_1&=\left ( \frac{-2l \nablala_v t^l}{\sin \frac{\theta}{2}} +\frac{2t^l l\cos\frac{\theta}{2}\nablala_v \theta}{2\sin^2 \frac{\theta}{2}} \right )\begin{equation}ig( v_1n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2n_1 \cos l\theta \cos \frac{\theta}{2} +v_2n_2 \cos l\theta \sin \frac{\theta}{2}\begin{equation}ig)\\
&\quad -\frac{2t^l l}{\sin \frac{\theta}{2}} \left( n_1 \sin l \theta \cos \frac{\theta}{2}\nablala_v v_1 + v_1 \sin l \theta \cos \frac{\theta}{2}\nablala_v n_1 +lv_1 n_1 \cos l\theta \cos \frac{\theta}{2}\nablala_v \theta -\frac{1}{2} v_1 n_1 \sin l\theta \sin \frac{\theta}{2} \nablala_v \theta \right. \\
&\qquad \qquad \quad \left. + n_2\sin l\theta \sin \frac{\theta}{2} \nablala_v v_1+ v_1\sin l \theta \sin \frac{\theta}{2} \nablala_v n_2 +lv_1 n_2 \cos l\theta \sin \frac{\theta}{2}\nablala_v \theta +\frac{1}{2} v_1 n_2 \sin l\theta \cos \frac{\theta}{2} \nablala_v \theta \right. \\
&\qquad \qquad \quad \left. + n_1 \cos l\theta \cos \frac{\theta}{2} \nablala_v v_2+v_2\cos l\theta \cos \frac{\theta}{2} \nablala_v n_1 -lv_2 n_1 \sin l \theta \cos \frac{\theta}{2} \nablala_v \theta -\frac{1}{2} v_2n_1\cos l\theta \sin \frac{\theta}{2}\nablala_v \theta \right. \\
&\qquad \qquad \quad \left. +n_2\cos l \theta \sin \frac{\theta}{2} \nablala_v v_2+ v_2\cos l\theta \sin \frac{\theta}{2} \nablala_v n_2 -lv_2 n_2 \sin l\theta \sin \frac{\theta}{2} \nablala_v \theta + \frac{1}{2} v_2 n_2 \cos l \theta \cos \frac{\theta}{2}\nablala_v \theta \right).
\end{align*}
Using \eqref{tb esti},\eqref{e_2},\eqref{est der n},\eqref{est der t_ell}, and \eqref{ell est} yields that
\begin{equation}gin{align*}
\vert \nablala_v I_1 \vert &\lesssim \left( \frac{1}{\vert v \vert^2 \vert \sin^3 \frac{\theta}{2}\vert} (1+\vert v \vert^2 t^2)+\frac{1}{\vert v \vert^2 \vert \sin ^2 \frac{\theta}{2}\vert}(1+\vert v \vert t) \right)\times \vert v \vert+\frac{1}{\vert v \cdotot n(x_{\mathbf{b}}) \vert} ( 1+ \vert v \vert \vert \nablala_v n(x_{\mathbf{b}}) \vert +l\vert v \vert \vert \nablala_v \theta\vert)\\
&\lesssim \frac{\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+\vert v \vert ^2 t^2).
\end{align*}
Hence, one obtains that
\begin{equation}gin{align*}
\vert \nablala_{vx} X(0;t,x,v) \vert \lesssim \frac{\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+\vert v \vert ^2 t^2).
\end{align*}
By Lemma \e^{\frac 12}f{X,V}, we write the $(1,1)$ component of $\nablala_v X(0;t,x,v)$:
\begin{equation}gin{align*}
&[\nablala_v X(0;t,x,v)]_{(1,1)}\\
&=-t_{\mathbf{b}} \left(\cos (l-1)\theta \frac{v_2n_2}{v\cdotot n(x_{\mathbf{b}})} +\sin (l-1)\theta \frac{v_2n_1}{v \cdotot n(x_{\mathbf{b}})} \right) -t^l \cos l\theta \\
&\quad +2lt^l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right) \left( v_1 n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2} +v_2 n_2 \cos l\theta \sin \frac{\theta}{2}\right) \\
&\quad + \frac{t_{\mathbf{b}}}{\vert v \vert \sin \frac{\theta}{2}} (v_1n_1\cos l\theta -v_2 n_1 \sin l \theta) -\frac{2(l-1)\sin \frac{\theta}{2}}{\vert v \vert^3}(v_1^2\cos l\theta - v_1v_2 \sin l \theta) \\
&\quad -\vert v \vert (t^1-t^l) \left(\frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}}-\frac{1}{\vert v \vert}\right) \left(-n_1^2 \cos (l-\frac{1}{2})\theta \cos \frac{\theta}{2} +n_1n_2 \sin (l-1)\theta +n_2^2 \sin (l-\frac{1}{2})\theta \sin \frac{\theta}{2}\right).
\end{align*}
Similar to $\nablala_x X(0;t,x,v)$, main terms in $\nablala_v X(0;t,x,v)$ are
\begin{equation}gin{align*}
2lt^l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right) \left( v_1 n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2} +v_2 n_2 \cos l\theta \sin \frac{\theta}{2}\right):=I_2.
\end{align*}
As we take derivative to $\nablala_v X(0;t,x,v)$ with respect to $x$ and $v$, $I_2$ mainly contributes to increase singularity and travel length order. Thus, we only differentiate terms $I_2$ to get estimate for $\vert \nablala_{xv} X(0;t,x,v) \vert $ and $\vert \nablala_{vv} X(0;t,x,v)\vert$. Firstly, taking $x$ derivative to $I_2$ gives
\begin{equation}gin{align*}
\nablala_x I_2 &= 2l \nablala_x t^l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right) \left( v_1 n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2} +v_2 n_2 \cos l\theta \sin \frac{\theta}{2}\right)\\
&\quad + 2lt^l \left( \frac{\nablala_x t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{t_{\mathbf{b}}\cos \frac{\theta}{2}\nablala_x \theta}{2\sin^2 \frac{\theta}{2}} \right)\begin{equation}ig( v_1 n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2} +v_2 n_2 \cos l\theta \sin \frac{\theta}{2}\begin{equation}ig)\\
&\quad +2lt^l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right)\left( v_1 \sin l\theta \cos \frac{\theta}{2} \nablala_xn_1+lv_1 n_1 \cos l\theta \cos \frac{\theta}{2} \nablala_x \theta -\frac{1}{2}v_1 n_1 \sin l \theta \sin \frac{\theta}{2} \nablala_x \theta \right. \\
&\qquad \qquad \qquad \qquad \qquad \quad +\left. v_1\sin l \theta \sin \frac{\theta}{2} \nablala_x n_2+lv_1n_2\cos l \theta \sin \frac{\theta}{2} \nablala_x \theta +\frac{1}{2} v_1 n_2 \sin l\theta \cos \frac{\theta}{2} \nablala_x \theta \right.\\
&\qquad \qquad \qquad \qquad \qquad \quad +\left. v_2\cos l\theta \cos \frac{\theta}{2}\nablala_x n_1 -lv_2n_1 \sin l \theta \cos \frac{\theta}{2} \nablala_x \theta -\frac{1}{2} v_2 n_1 \cos l\theta \sin \frac{\theta}{2} \nablala_x \theta \right. \\
&\qquad \qquad \qquad \qquad \qquad \quad +\left. v_2 \cos l\theta \sin \frac{\theta}{2} \nablala_x n_2 -lv_2n_2 \sin l \theta \sin \frac{\theta}{2} \nablala_x \theta +\frac{1}{2} v_2n_2 \cos l\theta \cos \frac{\theta}{2} \nablala_x \theta \right).
\end{align*}
Hence, it follows from \eqref{tb esti},\eqref{e_1},\eqref{est der n},\eqref{est der t_ell}, and \eqref{ell est} that
\begin{equation}gin{align*}
\vert \nablala_x I_2 \vert &\lesssim \frac{1}{\vert v \vert \vert \sin ^3 \frac{\theta}{2} \vert} (1+\vert v \vert^2 t^2) \times \frac{1}{\vert v \vert} \times \vert v \vert + \frac{1}{\vert v\vert}(1+\vert v \vert t) \times\frac{1}{\vert v \vert \sin^2 \frac{\theta}{2}} \times \vert v \vert \\
&\quad + \frac{1}{\vert v \vert}(1+\vert v \vert t)\times \frac{1}{\vert v \vert} \times \frac{\vert v \vert}{\sin ^2\frac{\theta}{2}} (1+\vert v \vert t)\\
&\lesssim \frac{\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+\vert v \vert^2 t^2),
\end{align*}
which yields $\vert \nablala_{xv} X(0;t,x,v) \vert$ estimate
\begin{equation}gin{align*}
\vert \nablala_{xv} X(0;t,x,v) \lesssim \frac{\vert v \vert^2}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+\vert v \vert^2 t^2).
\end{align*}
Similarly, we consider $\nablala_v I_2$:
\begin{equation}gin{align*}
\nablala_v I_2 &= 2l \nablala_v t^l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right) \left( v_1 n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2} +v_2 n_2 \cos l\theta \sin \frac{\theta}{2}\right)\\
&\quad + 2lt^l \left( \frac{\nablala_v t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{t_{\mathbf{b}}\cos \frac{\theta}{2}}{2\sin^2 \frac{\theta}{2}} \nablala_v \theta+\frac{v}{\vert v \vert^3}\right)\left( v_1 n_1 \sin l\theta \cos \frac{\theta}{2} +v_1 n_2 \sin l\theta \sin \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2} \right.\\
&\left. \qquad \hspace{5.7cm} +v_2 n_2 \cos l\theta \sin \frac{\theta}{2}\right)\\
&\quad +2lt^l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right)\left( n_1 \sin l\theta \cos \frac{\theta}{2} \nablala_v v_1+v_1 \sin l\theta \cos \frac{\theta}{2} \nablala_vn_1+lv_1 n_1 \cos l\theta \cos \frac{\theta}{2} \nablala_v \theta \right.\\
&\qquad \qquad \qquad \qquad \qquad \quad\left. -\frac{1}{2}v_1 n_1 \sin l \theta \sin \frac{\theta}{2} \nablala_v \theta
+n_2 \sin l \theta \sin \frac{\theta}{2} \nablala_v v_1 +v_1\sin l \theta \sin \frac{\theta}{2} \nablala_v n_2 \right. \\
&\qquad \qquad \qquad \qquad \qquad \quad\left. +lv_1n_2\cos l \theta \sin \frac{\theta}{2} \nablala_v\theta +\frac{1}{2} v_1 n_2 \sin l\theta \cos \frac{\theta}{2} \nablala_v \theta +n_1\cos l\theta \cos \frac{\theta}{2} \nablala_v v_2 \right.\\
&\qquad \qquad \qquad \qquad \qquad \quad +\left. v_2\cos l\theta \cos \frac{\theta}{2}\nablala_v n_1 -lv_2n_1 \sin l \theta \cos \frac{\theta}{2} \nablala_v \theta -\frac{1}{2} v_2 n_1 \cos l\theta \sin \frac{\theta}{2} \nablala_v \theta \right. \\
&\qquad \qquad \qquad \qquad \qquad \quad +\left. n_2 \cos l\theta \sin \frac{\theta}{2} \nablala_v v_2+v_2 \cos l\theta \sin \frac{\theta}{2} \nablala_v n_2 -lv_2n_2 \sin l \theta \sin \frac{\theta}{2} \nablala_v \theta \right.\\
&\qquad \qquad \qquad \qquad \qquad \quad \left.+\frac{1}{2} v_2n_2 \cos l\theta \cos \frac{\theta}{2} \nablala_v \theta \right).
\end{align*}
By \eqref{tb esti},\eqref{e_2},\eqref{est der n},\eqref{est der t_ell}, and \eqref{ell est}, the above can be further bounded by
\begin{equation}gin{align*}
\vert \nablala_v I_2 \vert &\lesssim \frac{1}{\vert v \vert^2 \sin^2 \frac{\theta}{2}} (1+\vert v \vert^2 t^2) \times \frac{1}{\vert v \vert} \times \vert v \vert + \frac{1}{\vert v \vert}(1+\vert v \vert t)\times \frac{1}{\vert v \vert^2 \vert \sin \frac{\theta}{2}\vert} \times \vert v \vert +\frac{1}{\vert v \vert}(1+\vert v \vert t)\times \frac{1}{\vert v \vert \vert \sin \frac{\theta}{2} \vert} (1+\vert v \vert t)\\
&\lesssim \frac{1}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^2} (1+\vert v \vert^2 t^2) .
\end{align*}
Hence, $\vert \nablala_{vv} X(0;t,x,v)\vert$ is bounded by
\begin{equation}gin{align*}
\vert \nablala_{vv} X(0;t,x,v) \vert \lesssim \frac{1}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^2} (1+\vert v \vert^2 t^2).
\end{align*}
To get estimate for $\vert \nablala_{xx} V(0;t,x,v)\vert$ and $\vert \nablala_{vx} V(0;t,x,v)\vert$, we now consider $[\nablala_{x} V(0;t,x,v)]_{(1,1)}$:
\begin{equation}gin{align*}
[\nablala_x V(0;t,x,v)]_{(1,1)} = \frac{2l}{\sin \frac{\theta}{2}} \left(v_1n_1 \sin l \theta \cos \frac{\theta}{2} +v_1n_2 \sin l \theta \sin \frac{\theta}{2} +v_2n_1 \cos l\theta \cos \frac{\theta}{2} + v_2n_2 \cos l\theta \sin \frac{\theta}{2}\right),
\end{align*}
by Lemma \e^{\frac 12}f{X,V}. In $[\nablala_x V(0;t,x,v)]_{(1,1)}$, the main terms are
\begin{equation}gin{align*}
\frac{2l}{\sin \frac{\theta}{2}} v_1n_1 \sin l \theta \cos \frac{\theta}{2} \quad \textrm{and} \quad \frac{2l}{\sin \frac{\theta}{2}} v_2 n_1 \cos l\theta \cos \frac{\theta}{2},
\end{align*}
because these terms have the highest singularity order in $[\nablala_x V(0;t,x,v)]_{(1,1)}$. Thus, for $\vert \nablala_{xx}V(0;t,x,v) \vert$, we now take the $x$-derivative for main terms:
\begin{equation}gin{align*}
&\nablala_x \left(\frac{2l}{\sin \frac{\theta}{2}}\left(v_1n_1\sin l \theta \cos \frac{\theta}{2} + v_2n_1 \cos l \theta \cos \frac{\theta}{2}\right)\right)\\
&= -\frac{l\cos \frac{\theta}{2}}{\sin ^2 \frac{\theta}{2}}\nablala_x \theta \left(v_1n_1\sin l \theta \cos \frac{\theta}{2} + v_2n_1 \cos l \theta \cos \frac{\theta}{2}\right)\\
&\quad + \frac{2l}{\sin \frac{\theta}{2}} \left(v_1 \sin l \theta \cos \frac{\theta}{2} \nablala_x n_1 +lv_1 n_1 \cos l\theta \cos \frac{\theta}{2} \nablala_x \theta -\frac{1}{2} v_1 n_1 \sin l\theta \sin \frac{\theta}{2} \nablala_x \theta \right. \\
&\qquad \qquad \quad + \left. v_2\cos l \theta \cos \frac{\theta}{2} \nablala_x n_1 -lv_2 n_1 \sin l\theta \cos \frac{\theta}{2} \nablala_x \theta -\frac{1}{2} v_2n_1 \cos l\theta \sin \frac{\theta}{2} \nablala_x \theta \right)\\
&:=I_3.
\end{align*}
By \eqref{e_1},\eqref{est der n}, and \eqref{ell est}, $I_3$ can be further bounded by
\begin{equation}gin{align*}
\vert I_3 \vert \lesssim \frac{\vert v \vert}{\sin^4 \frac{\theta}{2}}(1+\vert v \vert t) + \frac{\vert v \vert }{ \sin^4\frac{\theta}{2}}(1+\vert v \vert^2 t^2)\lesssim \frac{\vert v \vert^5}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^4}(1+\vert v \vert^2 t^2),
\end{align*}
which implies that
\begin{equation}gin{align*}
\vert \nablala_{xx} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert^5}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^4} (1+\vert v \vert^2t^2).
\end{align*}
Similarly, we firstly take the $v$-derivative for main terms in $[\nablala_x V(0;t,x,v)]_{(1,1)}$ and then estimate $v$-derivatives. Then, we deduce
\begin{equation}gin{align*}
\vert \nablala_{vx} V(0;t,x,v) \vert \lesssim \frac{ \vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+ \vert v \vert^2 t^2),
\end{align*}
where we have used \eqref{e_2},\eqref{est der n}, and \eqref{ell est}. Lastly, it remains to estimate $\vert \nablala_{xv}V(0;t,x,v)\vert$ and $\vert \nablala_{vv} V(0;t,x,v)\vert$. Let us consider the $(1,1)$ component of $\nablala_v V(0;t,x,v)$:
\begin{equation}gin{align*}
[\nablala_v V(0;t,x,v)]_{(1,1)}&=\cos l\theta -2l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right) \left(v_1n_1 \sin l \theta \cos \frac{\theta}{2} +v_1 n_2 \sin l \theta \sin \frac{\theta}{2} \right.\\
&\left. \qquad \qquad \qquad \qquad \qquad \qquad \quad + v_2 n_1 \cos l\theta \cos \frac{\theta}{2} +v_2n_2 \cos l\theta \sin \frac{\theta}{2}\right),
\end{align*}
by Lemma \e^{\frac 12}f{X,V}. Similar to previous cases, main terms in $[\nablala_v V(0;t,x,v)]_{(1,1)}$ are
\begin{equation}gin{align*}
-2l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right)\left(v_1n_1 \sin l \theta \cos \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2}\right):=I_4,
\end{align*}
by the same reason. Taking the $x$-derivative for $I_4$, we get
\begin{equation}gin{align*}
\nablala_x I_4 &= -2l \left( \frac{\nablala_x t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{t_{\mathbf{b}} \cos \frac{\theta}{2}}{2\sin^2\frac{\theta}{2}} \nablala_x \theta\right)\left(v_1n_1 \sin l \theta \cos \frac{\theta}{2} +v_2 n_1 \cos l\theta \cos \frac{\theta}{2}\right)\\
&\quad -2l \left( \frac{t_{\mathbf{b}}}{\sin \frac{\theta}{2}} -\frac{1}{\vert v \vert}\right)\left(v_1 \sin l \theta \cos \frac{\theta}{2} \nablala_x n_1 + lv_1n_1 \cos l\theta \cos \frac{\theta}{2}\nablala_x \theta -\frac{1}{2} v_1n_1 \sin l \theta \sin \frac{\theta}{2} \nablala_x \theta \right. \\
& \qquad \qquad \qquad \qquad \qquad +\left. v_2 \cos l \theta \cos \frac{\theta}{2} \nablala_x n_1 -l v_2n_1 \sin l \theta \cos \frac{\theta}{2} \nablala_x \theta -\frac{1}{2} v_2 n_1 \cos l\theta \sin \frac{\theta}{2} \nablala_x \theta\right).
\end{align*}
Using \eqref{tb esti},\eqref{e_1},\eqref{est der n},\eqref{est der t_ell}, and \eqref{ell est}, one obtains that
\begin{equation}gin{align*}
\vert \nablala_x I_4 \vert \lesssim \frac{1}{\vert \sin\frac{\theta}{2}\vert } (1+\vert v \vert t)\times \frac{1}{\vert v \vert \sin^2 \frac{\theta}{2}} \times \vert v \vert +\frac{1}{\vert \sin \frac{\theta}{2} \vert}(1+\vert v \vert t) \times \frac{1}{\vert v \vert} \times \frac{\vert v \vert}{ \sin^2 \frac{\theta}{2}}(1+\vert v \vert t)\lesssim \frac{\vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+\vert v \vert^2t^2).
\end{align*}
Hence, we get estimate for $\vert\nablala_{xv}V(0;t,x,v)\vert$
\begin{equation}gin{align*}
\vert \nablala_{xv} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert^3}{\vert v \cdotot n(x_{\mathbf{b}}) \vert^3} (1+\vert v \vert^2 t^2).
\end{align*}
Similarly, we take the $v$-derivative to main terms $I_4$ and estimate $\nablala_v I_4$ to get $\vert\nablala_{vv} V(0;t,x,v)\vert$. From \eqref{tb esti},\eqref{e_2},\eqref{est der n}, \eqref{est der t_ell}, and \eqref{ell est}, we derive
\begin{equation}gin{align*}
\vert \nablala_{vv} V(0;t,x,v) \vert \lesssim \frac{\vert v \vert}{ \vert v \cdotot n(x_{\mathbf{b}}) \vert^2} (1+\vert v \vert^2 t^2).
\end{align*}
\end{proof}
\subsection{Proof of Theorem \e^{\frac 12}f{thm 3}}
\begin{equation}gin{proof} [Proof of Theorem \e^{\frac 12}f{thm 3}]
\textit{Step 1} First, we prove $C^{1}$ estimate. Note that it is easy to derive
\begin{equation} \label{dt XV}
\partial_{t}X(0;t,x,v) = -v^{k},\quad \partial_{tt}X(0;t,x,v) = 0,\quad \partial_{t}V(0;t,x,v) = 0, \quad \partial_{tt}V(0;t,x,v) = 0,
\end{equation}
where we assumed $t^{k+1} < 0 < t^{k}$ for some integer $k$. For $i\in\{t,x,v\}$,
\begin{equation} \label{chain}
\nablala_{i}f(t,x,v) = \nablala_{x}f_0 \nablala_{i}X(0;t,x,v) + \nablala_{v}f_0 \nablala_{i} V(0;t,x,v).
\end{equation}
Hence using Lemma \e^{\frac 12}f{est der X,V} and \eqref{dt XV}, we obtain
\begin{equation}gin{equation*}
\begin{equation}gin{split}
|\partial_{t}f| &\lesssim \|f_0\|_{C^{1}}|v|, \\
|\nablala_{x}f| &\lesssim \|f_0\|_{C^{1}} \frac{|v|^{2}}{|v\cdotot n(x_{\mathbf{b}})|^{2}} \langle v \rangle (1 + |v|t), \\
|\nablala_{v}f| &\lesssim \|f_0\|_{C^{1}} \frac{1}{|v\cdotot n(x_{\mathbf{b}})|} \langle v \rangle (1 + |v|t), \\
\end{split}
\end{equation*}
where $x_{\mathbf{b}} = x_{\mathbf{b}}(x,v)$ and $\langle v \rangle := 1 + |v|$. So we obtain \eqref{C1 bound}. \\
\textit{Step 2} Now we compute second-order estimate. For $\nablala_{xx}f$, from \eqref{chain}, Lemma\e^{\frac 12}f{est der X,V}, and Lemma \e^{\frac 12}f{2nd est der X,V}, we obtain
\begin{equation}gin{equation*}
\begin{equation}gin{split}
|\nablala_{xx}f| &= |\nablala_{x} \big( \nablala_{x}f_0 \nablala_{x}X(0;t,x,v) + \nablala_{v}f_0 \nablala_{x} V(0;t,x,v) \big)| \\
&\lesssim \|f_0\|_{C^{1}} \big( |\nablala_{xx}X(0) | + |\nablala_{xx}V(0)| \big)
+ \|f_0\|_{C^{2}} \big( |\nablala_{x}X(0)| + |\nablala_{x}V(0)| \big)^{2} \\
&\lesssim \|f_0\|_{C^{2}} \frac{|v|^{4}}{|v\cdotot n(x_{\mathbf{b}})|^{4}} \langle v \rangle^{2} (1 + |v|t)^{2},
\end{split}
\end{equation*}
\begin{equation}gin{equation*}
\begin{equation}gin{split}
|\nablala_{vx}f| &= |\nablala_{v} \big( \nablala_{x}f_0 \nablala_{x}X(0;t,x,v) + \nablala_{v}f_0 \nablala_{x} V(0;t,x,v) \big)| \\
&\lesssim \|f_0\|_{C^{1}} \big( |\nablala_{vx}X(0) | + |\nablala_{vx}V(0)| \big)
+ \|f_0\|_{C^{2}} \big( |\nablala_{x}X(0)| + |\nablala_{x}V(0)| \big)\big( |\nablala_{v}X(0)| + |\nablala_{v}V(0)| \big) \\
&\lesssim \|f_0\|_{C^{2}} \frac{|v|^{2}}{|v\cdotot n(x_{\mathbf{b}})|^{3}} \langle v \rangle^{2} (1 + |v|t)^{2},
\end{split}
\end{equation*}
and
\begin{equation}gin{equation*}
\begin{equation}gin{split}
|\nablala_{vv}f| &= |\nablala_{v} \big( \nablala_{x}f_0 \nablala_{v}X(0;t,x,v) + \nablala_{v}f_0 \nablala_{v} V(0;t,x,v) \big)| \\
&\lesssim \|f_0\|_{C^{1}} \big( |\nablala_{vv}X(0) | + |\nablala_{vv}V(0)| \big)
+ \|f_0\|_{C^{2}} \big( |\nablala_{v}X(0)| + |\nablala_{v}V(0)| \big)^{2} \\
&\lesssim \|f_0\|_{C^{2}} \frac{1}{|v\cdotot n(x_{\mathbf{b}})|^{2}} \langle v \rangle^{2} (1 + |v|t)^{2},
\end{split}
\end{equation*}
where $|\nablala_{xx, vx, vv}X|$ means $\sup_{i,j,k}|\nablala_{ij}X_{k}(0;t,x,v)|$ for $i,j \in\{ x_{1}, x_{2}, v_{1}, v_{2}\}$ and $k \in \{1,2\}$. (Also similar for $\nablala_{ij}V$.) Combining above three estimates, we obtain \eqref{C2 bound}. Second derivative estimates which contain at least one $\partial_{t}$ also yield the same upper bound from \eqref{dt XV}. We omit the details. \\
\end{proof}
\noindent{\bf Acknowledgments.}
The authors thank Haitao Wang for suggestion and fruitful discussion. Their research is supported by the National Research Foundation of Korea(NRF) grant funded by the Korean government(MSIT)(No. NRF-2019R1C1C1010915). DL is also supported by the POSCO Science Fellowship of POSCO TJ Park Foundation. The authors sincerely appreciate the anonymous referees for their valuable comments and suggestions on the paper.
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|
\begin{document}
\title{Mixed Spatial and Temporal Decompositions
for Large Scale Multistage Stochastic
Optimization Problems}
\begin{abstract}
We consider multistage stochastic optimization problems
involving multiple units. Each unit is a (small) control system.
Static constraints couple units at each stage.
We present a mix of spatial and temporal decompositions
to tackle such large scale problems.
More precisely, we obtain theoretical bounds and policies
by means of two methods, depending whether the coupling constraints
are handled by prices or by resources.
We study both centralized and decentralized information structures.
We report the results of numerical experiments on the
management of urban microgrids. It appears that decomposition methods are
much faster and give better results than the standard Stochastic Dual
Dynamic Programming method, both in terms of bounds and of policy performance.
\end{abstract}
\section{Introduction}
Multistage stochastic optimization problems are, by essence,
complex because their solutions are indexed both by stages
(time) and by uncertainties (scenarios).
Another layer of complexity can come from spatial structure.
The large scale nature of such problems makes decomposition methods
appealing (we refer to
\citep{ruszczynski1997decomposition,carpentier2017decomposition}
for a broad description of decomposition methods in stochastic optimization problems).
We sketch decomposition methods along three dimensions:
\emph{temporal decomposition} methods like Dynamic Programming
break the multistage problem into a sequence of interconnected static
subproblems \citep{bellman57,bertsekas1995dynamic};
\emph{scenario decomposition} methods split large scale stochastic
optimization problems scenario by scenario, yielding deterministic
subproblems \citep{rockafellar1991scenarios,watson2011progressive,kim2018algorithmic};
\emph{spatial decomposition} methods break possible spatial
couplings in a global problem to obtain local decoupled subproblems
\cite{cohen80}.
These decomposition schemes have been applied in many fields,
and especially in energy management: Dynamic Programming methods have been
used for example in dam management \citep{shapiro2012final},
and scenario decomposition has been successfully applied to
the resolution of unit-commitment problems \citep{bacaud2001bundle},
among others.
Recent developments have mixed spatial decomposition
methods with Dynamic Programming to solve
large scale multistage stochastic optimization problems.
This work led to the introduction of the Dual Approximate
Dynamic Programming (DADP) algorithm, which was first
applied to unit-commitment problems with a single central
coupling constraint linking different stocks
\citep{barty2010decomposition}, and later applied to dams
management problems~\citep{carpentier2018stochastic}.
This article moves
one step further by considering altogether two types
of decompositions (by prices and by resources)
when dealing with general coupling
constraints among units.
General coupling constraints often arise from flows
conservation on a graph,
and our motivation indeed comes from district microgrid
management, where buildings (units) consume,
produce and store energy and are interconnected through
a network.
The paper is organized as follows.
In Sect.~\ref{chap:nodal}, we introduce a generic stochastic
multistage problem with different subsystems linked together
via a set of static coupling constraints.
We present price and resource decomposition schemes, that make use of
so-called admissible coordination processes. We show how to
bound the global Bellman functions above by a sum of local
resource-decomposed value functions, and below by a sum of
local price-decomposed value functions.
In Sect.~\ref{sec:genericdecomposition}, we study the special
case of deterministic coordination processes. First, we show
that the local price and resource decomposed value functions
satisfy recursive Dynamic Programming equations. Second, we
outline how to improve the bounds. Third, we show how to use the decomposed Bellman
functions to devise admissible policies for the global problem.
Finally, we provide an analysis of the decentralized information
structure, that is, when the controls of a given subsystem only
depend on the past observations of the noise in that same subsystem.
In Sect.~\ref{chap:district:numerics}, we present numerical
results for the optimal management of different microgrids
of increasing size and complexity.
We compare the two decomposition algorithms with
(state of the art) Stochastic Dual Dynamic Programming (SDDP) algorithm.
The analysis of case studies consisting of district
microgrids coupling up to 48 buildings together enlightens
that decomposition methods give better results in terms of cost
performance, and achieve up to a four times speedup in terms of computational
time.
\section{Upper and Lower Bounds by Spatial Decomposition}
\label{chap:nodal}
We focus in \S\ref{sec:nodal:generic} on a generic decomposable
optimization problem and present price and resource decomposition
schemes. In~\S\ref{sec:nodal:globalpb}, we apply these two
methods to a multistage stochastic optimization problem, by decomposing
a global static coupling constraint by means of so-called price
and resource coordination processes. For such problems,
we define the notions of centralized and decentralized information structures.
\subsection{Bounds for an Optimization Problem under Coupling Constraints
via Decomposition}
\label{sec:nodal:generic}
In~\S\ref{sec:nodal:genericproblem},
we introduce a generic optimization problem with coupled local units.
In~\S\ref{subsec:nodal:bounds}, we show how to bound its optimal value by
decomposition.
\subsubsection{Global Optimization Problem Formulation}
\label{sec:nodal:genericproblem}
Let~$\NODES$ be a finite set, representing local units~\( \node \in \NODES \)
(we use the letter~$\NODES$ as units can be seen as nodes on a graph).
Let~$\sequence{\mathcal{Z}^\node}{\node \in \NODES}$ be a family of sets and
$J^\node: \mathcal{Z}^\node \rightarrow \OpenIntervalClosed{-\infty}{+\infty}$, $\node \in \NODES$,
be local criteria, one for each unit, taking values in the extended reals
\( \OpenIntervalClosed{-\infty}{+\infty} \) ($+\infty$ included to allow for possible constraints).
Let $\sequence{\mathcal{R}^\node}{\node \in \NODES}$, be a family of vector spaces
and $\vartheta^\node: \mathcal{Z}^\node \rightarrow \mathcal{R}^\node$, $\node \in \NODES$,
be mappings that model local constraints.
From these \emph{local} data, we formulate a \emph{global}
minimization problem under constraints. We define the product
set~$\mathcal{Z} = \prod_{\node\in \NODES} \mathcal{Z}^\node $ and the product
space~$\mathcal{R}=\prod_{\node\in \NODES} \mathcal{R}^\node$.
Finally, we introduce a subset $S \subset \mathcal{R}$
that captures the coupling constraints between the $N$~units.
Using the notation \( z=\sequence{z^\node}{\node \in \NODES} \),
we define the \emph{global optimization} problem as
\begin{subequations}
\label{eq:gen:genpb}
\begin{equation}
V{\sharp} = \inf_{z \in \mathcal{Z}} \;
\sum_{\node \in \NODES} J^\node(z^\node) \eqfinv
\end{equation}
under the \emph{global coupling constraint}
\begin{equation}
\label{eq:gen:coupling}
\ba{\vartheta^\node(z^\node)}_{\node \in \NODES} \in -S \eqfinp
\end{equation}
\end{subequations}
The set~$S$ is called the \emph{primal admissible set},
and an element~$\sequence{r^\node}{\node \in \NODES}\in-S$
is called an \emph{admissible resource vector}.
We note that, without Constraint~\eqref{eq:gen:coupling},
Problem~\eqref{eq:gen:genpb} would decompose into $|\NODES|$ independent
subproblems in a straightforward manner.
We moreover assume that, for $\node \in \NODES$, the space~$\mathcal{R}^\node$
(resources) is paired with a space~$\mathcal{P}^\node$ (prices) by bilinear forms
$\bscal{\cdot}{\cdot}\; : \; \mathcal{P}^\node \times \mathcal{R}^\node \to \va{R}R$
(duality pairings).
We define the product space~$\mathcal{P}= \prod_{\node \in \NODES} \mathcal{P}^\node$,
so that~$\mathcal{R}$ and~$\mathcal{P}$ are paired by the duality pairing
$\bscal{p}{r} = \sum_{\node\in \NODES} \bscal{p^\node}{r^\node}$
(see \cite{rockafellar1974conjugate} for further details; a typical
example of paired spaces is a Hilbert space and itself).
\subsubsection{Upper and Lower Bounds from Price and Resource Value Functions}
\label{subsec:nodal:bounds}
Consider the global optimization problem~\eqref{eq:gen:genpb}.
For each~$\node \in \NODES$, we introduce \emph{local price value
functions} $\boldsymbol{U}nderline V^\node : \mathcal{P}^\node \to \ClosedIntervalOpen{-\infty}{+\infty}$
defined by
\begin{equation}
\label{eq:nodal:genericdualvf}
\boldsymbol{U}nderline V^\node\nc{p^\node}
= \inf_{z^\node \in \mathcal{Z}^\node} \; J^\node(z^\node) +
\bscal{p^\node}{\vartheta^\node(z^\node)}
\eqfinv
\end{equation}
where we have supposed that \( V^\node\nc{p^\node} < +\infty \),
and \emph{local resource value functions}
$\overline V^\node: \mathcal{R}^\node \to \OpenIntervalClosed{-\infty}{+\infty}$
defined by
\begin{equation}
\label{eq:nodal:genericprimalvf}
\overline V^\node\nc{r^\node}
= \inf_{z^\node\in \mathcal{Z}^\node} \; J^\node(z^\node)
\quad \text{s.t.}\ \quad \vartheta^\node(z^\node) = r^\node
\eqfinv
\end{equation}
where we have supposed that \( \overline V^\node\nc{r^\node} > -\infty \),
We denote by $S^\text{s.t.}ar \subset \mathcal{P}$ the dual cone
associated with the constraint set~$S$:
\begin{equation}
\label{eq:nodal:dualcone}
S^\text{s.t.}ar =
\ba{p \in \mathcal{P} \; |\;
\bscal{p}{r} \geq 0 \eqsepv
\forall r \in S} \eqfinp
\end{equation}
The cone~$S^\text{s.t.}ar$ is called the \emph{dual admissible set},
and an element~$\sequence{p^\node}{\node \in \NODES}\inS^\text{s.t.}ar$
is called an \emph{admissible price vector}.
We now establish lower and upper bounds for Problem~\eqref{eq:gen:genpb},
and show how they can be computed in a decomposed way, that is, unit by unit.
\begin{proposition}
\label{prop:nodal:valuefunctionsbounds}
For any admissible price vector
$p = \sequence{p^\node}{\node \in \NODES} \in S^\text{s.t.}ar$
and for any admissible resource vector
$r =\sequence{r^\node}{\node \in \NODES} \in -S$,
we have the following lower and upper decomposed estimates
of the global minimum $V{\sharp}$ of Problem~\eqref{eq:gen:genpb}:
\begin{equation}
\label{eq:nodal:generic:bounds}
\sum_{\node \in \NODES} \boldsymbol{U}nderline V^\node\nc{p^\node}
\; \leq \; V{\sharp} \; \leq \;
\sum_{\node \in \NODES} \overline V^\node\nc{r^\node}
\eqfinp
\end{equation}
\end{proposition}
\begin{proof}
Because we have supposed that \( V^\node\nc{p^\node} < +\infty \),
the left hand side of Equation~\eqref{eq:nodal:generic:bounds} belongs
to~$\ClosedIntervalOpen{-\infty}{+\infty}$. In the same way, the right
hand side belongs to~$\OpenIntervalClosed{-\infty}{+\infty}$.
For a given $p = \sequence{p^\node}{\node\in \NODES} \in S^\text{s.t.}ar$,
we have
\begin{align*}
\sum_{\node \in \NODES} \boldsymbol{U}nderline V^\node\nc{p^\node}
&= \sum_{\node \in \NODES} \inf_{z^\node \in \mathcal{Z}^\node} \;
J^\node(z^\node) + \bscal{p^\node}{\vartheta^\node(z^\node)}
\eqfinv \\
&= \inf_{z \in \mathcal{Z}} \; {\sum_{\node \in \NODES} J^\node(z^\node)} +
\bscal{p}{\sequence{\vartheta^\node(z^\node)}{\node \in \NODES}}
\eqfinv
\tag{since \( z=\sequence{z^\node}{\node \in \NODES} \)}
\\
&\le \inf_{z \in \mathcal{Z}} \; {\sum_{\node \in \NODES} J^\node(z^\node)} +
\bscal{p}{\sequence{\vartheta^\node(z^\node)}{\node \in \NODES}}
\eqfinv \\
&\hphantom{\le \inf_{z \in \mathcal{Z}}} \text{s.t.}\
{\sequence{\vartheta^\node(z^\node)}{\node \in \NODES}} \in - S
\tag{minimizing on a smaller set} \\
&\le \inf_{z \in \mathcal{Z}} \; {\sum_{\node \in \NODES} J^\node(z^\node) +0}
\tag{as $p\in S^\text{s.t.}ar$ and by definition~\eqref{eq:nodal:dualcone} of~$S^\text{s.t.}ar$}
\\
&\hphantom{\le \inf_{z \in \mathcal{Z}}} \text{s.t.}\
{\sequence{\vartheta^\node(z^\node)}{\node \in \NODES}} \in - S
\eqfinv
\end{align*}
which gives the lower bound inequality.
The upper bound is easily obtained,
as the optimal value $V{\sharp}$ of Problem~\eqref{eq:gen:genpb} is given by
$\inf_{\tilder \in -S} \sum_{\node \in \NODES} \overline V^\node\nc{\tilder^\node}
\leq \sum_{\node \in \NODES} \overline V^\node\nc{r^\node}$ for any $r \in -S$.
\end{proof}
\subsection{The Special Case of Multistage Stochastic Optimization Problems
\label{sec:nodal:globalpb}}
Now, we turn to the case where Problem~\eqref{eq:gen:genpb}
corresponds to a multistage stochastic optimization problem
elaborated from local data (local states, local controls, and
local noises), with global coupling constraints at each time step.
We use the notation \( \ic{r,s}=\na{r,r+1,\ldots,s-1,s} \) for two integers
$r \leq s$, and we consider a time span \( \ic{0,T} \) where $T
\in \NN^\text{s.t.}ar$ is a finite horizon.
\subsubsection{Local Data for Local Stochastic Control Problems}
\label{subsec:nodal:generic:localdata}
We detail the \emph{local} data describing
each unit. Let
$\ba{\XX_t^\node}_{t\in\ic{0,T}}$,
$\ba{\UU_t^\node}_{t\in\ic{0,T-1}}$
and $\ba{\WW_t^\node}_{t\in\ic{1,T}}$
be sequences of measurable spaces for each unit $\node \in \NODES$.
We consider two other sequences of measurable vector spaces
$\ba{\mathcal{R}_t^\node}_{t\in\ic{0,T-1}}$ and
$\ba{\mathcal{P}_t^\node}_{t\in\ic{0,T-1}}$
such that for all~$t$, $\mathcal{R}_t^\node$ and
$\mathcal{P}_t^\node$ are paired spaces,
equipped with a bilinear form~$\pscal{\cdot}{\cdot}$.
We also introduce, for all~$\node \in \NODES$ and for
all~$t \in \ic{0, T-1}$,
\begin{itemize}
\item
measurable \emph{local dynamics}
$g_t^\node : \XX_t^\node \times \UU_t^\node \times \WW_{t+1}^\node \to \XX_{t+1}^\node$,
\item
measurable \emph{local coupling functions}
$\Theta_t^\node: \XX_t^\node \times \UU_t^\node \to \mathcal{R}_t^\node$,
\item
measurable \emph{local instantaneous costs}
$L_t^\node: \XX_t^\node \times \UU_t^\node \times \WW_{t+1}^\node \rightarrow
\OpenIntervalClosed{-\infty}{+\infty}$,
\end{itemize}
and a measurable \emph{local final cost}
$K^\node: \XX^\node_T \rightarrow \OpenIntervalClosed{-\infty}{+\infty}$.
We incorporate possible local constraints (for instance constraints
coupling the control with the state) directly in the instantaneous
costs~$L_t^\node$ and the final cost~$K^\node$, since they are extended real
valued functions which can possibly take the value $+\infty$.
From local data given above, we define the global state,
control, noise, resource and price spaces at time~$t$ as
\begin{equation*}
\XX_t = \prod_{\node \in \NODES} \XX_t^\node , \;\:
\UU_t = \prod_{\node \in \NODES} \UU_t^\node , \;\:
\WW_t = \prod_{\node \in \NODES} \WW_t^\node , \;\:
\mathcal{R}_t = \prod_{\node \in \NODES} \mathcal{R}_t^\node , \;\:
\mathcal{P}_t = \prod_{\node \in \NODES} \mathcal{P}_t^\node
\eqfinp
\end{equation*}
We suppose given a \emph{global constraint set}
$S_t \subset \mathcal{R}_t$ \emph{at time~$t$}.
We define the global resource and price spaces~$\mathcal{R}$
and~$\mathcal{P}$, and the global constraint set~$S \subset \mathcal{R}$, as
\begin{equation}
\mathcal{R} = \prod_{t=0}^{T-1} \mathcal{R}_t
\eqsepv
\mathcal{P} = \prod_{t=0}^{T-1} \mathcal{P}_t
\eqsepv
S = \prod_{t=0}^{T-1} S_t
\subset \mathcal{R}
\eqfinv
\label{eq:nodal:global_constraint_set}
\end{equation}
and we denote by $S^\text{s.t.}ar \subset \mathcal{P}$ the dual cone of $S$
(see Equation~\eqref{eq:nodal:dualcone}).
\subsubsection{Centralized and Decentralized Information Structures}
\label{subsec:nodal:generic:globaldata}
We introduce a probability space $(\Omega, \mathcal{F}, \PP)$. For every unit
$\node \in\NODES$, we introduce \emph{local exogenous noise processes}
$\boldsymbol{W}^\node = \na{\boldsymbol{W}_t^\node}_{t\in \ic{1, T}}$, where each
$\boldsymbol{W}_t^\node:\Omega\to\WW_t^\node$ is a random variable.\footnote{Random variables
are denoted using bold letters.} We denote by
\begin{equation}
\label{eq:nodal:generic:globalnoise}
\boldsymbol{W}=(\boldsymbol{W}_1,\cdots,\boldsymbol{W}_T)
\quad\text{ where }\quad
\boldsymbol{W}_t = \sequence{\boldsymbol{W}_t^\node}{\node \in \NODES}
\end{equation}
the \emph{global noise process}.
\begin{subequations}
We consider two \emph{information structures}
\cite[Chap.~3]{carpentier2015stochastic}:
\begin{itemize}
\item
the \emph{centralized} information structure, represented
by the filtration~$\mathcal{F} = \np{\mathcal{F}_t}_{t \in \ic{0,T}}$,
associated with the global noise process~$\boldsymbol{W}$ in~\eqref{eq:nodal:generic:globalnoise},
where
\begin{equation}
\label{eq:nodal:globalinfo}
\mathcal{F}_t = \sigma(\boldsymbol{W}_1, \cdots, \boldsymbol{W}_t)
= \sigma\bp{ \sequence{\boldsymbol{W}_1^\node}{\node \in \NODES}, \cdots,
\sequence{\boldsymbol{W}_t^\node}{\node \in \NODES} }
\end{equation}
is the $\sigma$-field generated by all noises up to time~$t \in \ic{0,T}$,
with the convention~$\mathcal{F}_0 = \{\emptyset,\Omega\}$,
\item
the \emph{decentralized} information structure, represented
by the family \( \sequence{\mathcal{F}^\node}{\node \in \NODES} \)
of filtrations $\mathcal{F}^\node = \np{\mathcal{F}_t^\node}_{t \in \ic{0,T}}$,
where, for any unit $\node \in \NODES $ and any time $t \in \ic{0,T}$,
\begin{equation}
\label{eq:nodal:localinfo}
\mathcal{F}_t^\node = \sigma(\boldsymbol{W}_1^\node, \cdots, \boldsymbol{W}_t^\node)
\subset \mathcal{F}_t = \bigvee_{\node' \in \NODES} \mathcal{F}_t^{\node'}
\eqfinv
\end{equation}
with~$\mathcal{F}_0^\node = \{\emptyset,\Omega\}$.
The \emph{local} $\sigma$-field~$\mathcal{F}_t^\node$ captures
the information provided by the uncertainties up to time~$t$,
\emph{but only in unit~$\node$}.
\end{itemize}
\label{eq:nodal:infos}
\end{subequations}
In the sequel, for a given filtration~$\mathcal{G}$ and a given measurable
space~$\YY$, we denote by ${\mathbb L}^0(\Omega, \mathcal{G}, \PP ; \YY)$ the space
of~\emph{$\mathcal{G}$-adapted processes taking values in the space~$\YY$}.
\subsubsection{Global Stochastic Control Problem}
We denote by $\boldsymbol{X}_t = \sequence{\boldsymbol{X}_t^\node}{\node\in \NODES}$
and $\boldsymbol{U}_t = \sequence{\boldsymbol{U}_t^\node}{\node\in \NODES}$
families of random variables (each of them with values in~$\XX_t^\node$
and in $\UU_t^\node$).
The stochastic processes $\boldsymbol{X} = (\boldsymbol{X}_0,\cdots,\boldsymbol{X}_T)$
and~$\boldsymbol{U}= (\boldsymbol{U}_0,\cdots,\boldsymbol{U}_{T-1}) $ are called
\emph{global state} and \emph{global control} processes.
The stochastic processes $\boldsymbol{X}^\node = (\boldsymbol{X}_0^\node,\cdots,\boldsymbol{X}_T^\node)$
and $\boldsymbol{U}^\node = (\boldsymbol{U}_0^\node,\cdots,\boldsymbol{U}_{T-1}^\node)$ are called
\emph{local state} and \emph{local control processes}.
With the data detailed in \S\ref{subsec:nodal:generic:localdata}
and \S\ref{subsec:nodal:generic:globaldata}, we formulate
a family of optimization problems as follows.
At each time $t \in \ic{0, T}$, the \emph{global value function}
$V_t : \prod_{\node \in \NODES} \XX_{t}^{\node} \rightarrow [-\infty,+\infty]$
is defined, for all
$\sequence{x_t^\node}{\node \in \NODES} \in \prod_{\node \in \NODES}
\XX_{t}^{\node}$, by
(with the convention~$V_T=\sum_{\node \in \NODES}K^\node$)
\begin{subequations}
\label{eq:nodal:vf}
\begin{align}
V_t\bp{\sequence{x_t^\node}{\node \in \NODES}} = \inf_{\boldsymbol{X}, \boldsymbol{U}} \;
& \EE \bgc{\sum_{\node \in \NODES} \sum_{s=t}^{T-1}
L^\node_s(\va X_s^\node, \va U_s^\node, \va W^\node_{s+1}) +
K^\node(\boldsymbol{X}_T^\node)} \eqfinv
\label{eq:nodal:expected_value}
\\
\text{s.t.}
&\; \boldsymbol{X}_t^\node = x_t^\node \text{\, and \,} \va{f}ORALLTIMES{s}{t}{T\!-\!1}
\eqfinv \nonumber
\\
&\boldsymbol{X}_{s+1}^\node = {g}_s^\node(\boldsymbol{X}^\node_s, \boldsymbol{U}_s^\node, \boldsymbol{W}_{s+1}^\node)
\eqsepv \boldsymbol{X}_t^\node = x_t^\node
\eqfinv
\label{eq:nodal:dynamic}
\\
&\sigma(\boldsymbol{U}_s^\node) \subset \mathcal{G}_s^\node
\eqfinv
\label{eq:nodal:measurability}
\\
&
\ba{\Theta_s^\node(\boldsymbol{X}_s^\node, \boldsymbol{U}_s^\node)}_{\node \in \NODES} \in -S_s
\eqfinp
\label{eq:nodal:couplingcons}
\end{align}
\end{subequations}
In the global value function~\eqref{eq:nodal:vf}, the expected
value is taken \boldsymbol{W}rt\ (with respect to) the global uncertainty
process~$(\boldsymbol{W}_{t+1}, \cdots, \boldsymbol{W}_T)$.
We assume that measurability and integrability assumptions hold true,
so that the expected value in~\eqref{eq:nodal:expected_value} is well
defined. Constraints~\eqref{eq:nodal:measurability} --- where
$\sigma(\boldsymbol{U}_s^\node)$ is the $\sigma$-field generated by the random
variable~$\boldsymbol{U}_s^\node$ --- express the fact that each decision
$\boldsymbol{U}_s^\node$ is $\mathcal{G}_s^\node$-measurable, that is, measurable
either \boldsymbol{W}rt\ the global information~$\mathcal{F}_s$ (centralized information
structure) available at time~$s$ (see~Equation~\eqref{eq:nodal:globalinfo})
or \boldsymbol{W}rt\ the local information $\mathcal{F}_s^\node$ (decentralized
information structure) available at time~$s$ for unit~$\node$
(see~Equation~\eqref{eq:nodal:localinfo}), as detailed
in~\S\ref{subsec:nodal:generic:globaldata}.
Finally, Constraints~\eqref{eq:nodal:couplingcons} express
the global coupling constraint at time~$s$ between all units
and have to be understood in the $\PP$-almost sure sense.
We are mostly interested in the \emph{global optimization problem}~$
V_0\np{x_0}$,
where $x_0 = \sequence{x_0^\node}{\node \in \NODES} \in \XX_0$ is the initial
state, that is, Problem~\eqref{eq:nodal:vf} for~$t=0$.
\subsubsection{Local Price and Resource Value Functions}
\label{sec:nodal:localvaluefunctions}
As in \S\ref{subsec:nodal:bounds}, we define
local price and local resource value functions for the
global multistage stochastic optimization problems~\eqref{eq:nodal:vf}.
For this purpose, we introduce a duality pairing between stochastic processes.
For each $\node\in\NODES$, we consider subspaces
\( \boldsymbol{W}idetilde{{\mathbb L}}(\Omega,\mathcal{F},\PP ;\mathcal{R}^\node)
\subset {\mathbb L}^0(\Omega, \mathcal{F}, \PP ; \mathcal{R}^\node) \)
and
\( \boldsymbol{W}idetilde{{\mathbb L}}^{\text{s.t.}ar}(\Omega,\mathcal{F},\PP ;\mathcal{P}^\node)
\subset {\mathbb L}^0(\Omega, \mathcal{F}, \PP ; \mathcal{P}^\node) \) such that
the duality product terms
$\EE\bc{\sum_{t=0}^{T-1}\pscal{\va{p}_t^\node}{\Theta_t^\node(\boldsymbol{X}_t^\node,\boldsymbol{U}_t^\node)}}$
in Equation~\eqref{eq:nodal:priceproblem-t}
are well defined (like in the case of square integrable
random variables, when
$\Theta_t^\node(\boldsymbol{X}_t^\node, \boldsymbol{U}_t^\node) \in {\mathbb L}^2(\Omega,\mathcal{F}_t, \PP ; \va{R}R^d)$
and $\va{p}_t^\node \in {\mathbb L}^2(\Omega,\mathcal{F}_t, \PP ; \va{R}R^d)$).
Let $\node \in \NODES$ be a local unit, and
$\va{p}^\node = (\va{p}_0^\node, \cdots, \va{p}_{T-1}^\node)
\in \boldsymbol{W}idetilde{{\mathbb L}}^{\text{s.t.}ar}(\Omega,\mathcal{F},\PP ;\mathcal{P}^\node)$) be
a \emph{local price process} --- hence, adapted to the global
filtration $\mathcal{F}$ in \eqref{eq:nodal:globalinfo} generated by
the global noises (note that we do not assume that it is adapted
to the local filtration $\mathcal{F}^\node$ in \eqref{eq:nodal:localinfo}
generated by the local noises).
When specialized to the context of Problems~\eqref{eq:nodal:vf},
Equation~\eqref{eq:nodal:genericdualvf} gives,
at each time $t \in \ic{0, T}$,
what we call \emph{local price value functions}
$\boldsymbol{U}nderline V^\node_t\nc{\va{p}^\node} : \XX_{t}^{\node} \rightarrow
\ClosedIntervalOpen{-\infty}{+\infty}$
defined, for all $x_t^\node \in \XX_t^\node$, by
(with the convention~$\boldsymbol{U}nderline V^\node_T\nc{\va{p}^\node}=K^\node$)
\begin{align}
\boldsymbol{U}nderline V^\node_t\nc{\va{p}^\node}(x_t^\node) = \inf_{\boldsymbol{X}^\node, \boldsymbol{U}^\node} \;
& \EE \bigg[\sum_{s=t}^{T-1}
\Big(L^\node_s(\va X_s^\node,\va U_s^\node,\va W^\node_{s+1})
\nonumber \\
& \hspace{1.5cm} + \pscal{\va{p}_s^\node}{\Theta_s^\node(\boldsymbol{X}_s^\node, \boldsymbol{U}_s^\node)}\Big)
+ K^\node(\boldsymbol{X}_T^\node)\bigg] \eqfinv
\label{eq:nodal:priceproblem-t}
\\
\text{s.t.}
& \; \boldsymbol{X}_t^\node = x_t^\node \text{\, and \,} \va{f}ORALLTIMES{s}{t}{T\!-\!1},
\eqref{eq:nodal:dynamic}, \eqref{eq:nodal:measurability}.
\nonumber
\end{align}
We suppose that
\( \boldsymbol{U}nderline V^\node_t\nc{\va{p}^\node}(x_t^\node) < +\infty \)
in~\eqref{eq:nodal:priceproblem-t}.
We define the \emph{global price value function}
$\boldsymbol{U}nderline V_t\nc{\va{p}^\node} : \XX_{t} \rightarrow \ClosedIntervalOpen{-\infty}{+\infty}$
at time~$t\in\ic{0,T}$ as
the sum of the corresponding local price value functions, that is,
using the notation \( x_t=\sequence{x_t^\node}{\node \in \NODES} \),
\begin{equation}
\label{eq:global:priceproblem}
\boldsymbol{U}nderline V_t\nc{\va{p}}(x_t) = \sum_{\node \in \NODES}
\boldsymbol{U}nderline V^\node_t\nc{\va{p}^\node}(x_t^\node)
\eqsepv \forall x_t \in \XX_{t}
\eqfinp
\end{equation}
In the same vein, let
$\va{r}^\node =(\va{r}_0^\node,\cdots,\va{r}_{T-1}^\node)
\in \boldsymbol{W}idetilde{{\mathbb L}}(\Omega,\mathcal{F},\PP ; \mathcal{R}^\node)$
be a \emph{local resource process}.
Equation~\eqref{eq:nodal:genericprimalvf} gives,
at each time $t \in \ic{0, T}$,
what we call \emph{local resource value function},
$\overline V^\node_t\nc{\va{r}^\node} : \XX_{t}^{\node} \rightarrow \OpenIntervalClosed{-\infty}{+\infty}$
defined, for all $x_t^\node \in \XX_t^\node$, by
(with the convention~$\overline{V}^\node_T\nc{\va{r}^\node}=K^\node$)
\begin{subequations}
\label{eq:nodal:quantproblem-t}
\begin{align}
& \overline{V}^\node_t\nc{\va{r}^\node}(x_t^\node) =
\inf_{\boldsymbol{X}^\node, \boldsymbol{U}^\node} \;
\EE \bgc{ \sum_{s=t}^{T-1}
L^\node_s(\va X_s^\node, \va U_s^\node, \va W^\node_{s+1}) +
K^\node(\boldsymbol{X}_T^\node)} \eqfinv \\
\text{s.t.}
& \, \boldsymbol{X}_t^\node = x_t^\node \text{\, and \,} \va{f}ORALLTIMES{s}{t}{T\!-\!1},
\eqref{eq:nodal:dynamic}, \eqref{eq:nodal:measurability}
\text{\, and \,} \Theta_s^\node(\boldsymbol{X}_s^\node, \boldsymbol{U}_s^\node) = \va{r}_s^\node \eqfinp
\end{align}
\end{subequations}
We suppose that
\( \overline{V}^\node_t\nc{\va{r}^\node}(x_t^\node) > -\infty \)
in~\eqref{eq:nodal:quantproblem-t}.
We define the \emph{global resource value function}
\( \overline V_t\nc{\va{r}} : \XX_{t} \rightarrow \OpenIntervalClosed{-\infty}{+\infty} \)
at time $t \in \ic{0, T}$ as the sum
of the local resource value functions, that is,
\begin{equation}
\label{eq:global:quantproblem}
\overline V_t\nc{\va{r}}(x_t) = \sum_{\node \in \NODES}
\overline V^\node_t\nc{\va{r}^\node}(x_t^\node)
\eqsepv \forall x_t \in \XX_{t}
\eqfinp
\end{equation}
We call the global processes
$\va{p} \in \boldsymbol{W}idetilde{{\mathbb L}}^{\text{s.t.}ar}(\Omega, \mathcal{F}, \PP ; \mathcal{P})$
and
$\va{r} \in \boldsymbol{W}idetilde{{\mathbb L}}(\Omega,\mathcal{F},\PP ;\mathcal{R})$
respectively
\emph{price coordination processes}
and
\emph{ressource coordination processes}.
\subsubsection{Global Upper and Lower Bounds}
\label{subsec:nodal:globalprocess}
Applying Proposition~\ref{prop:nodal:valuefunctionsbounds}
to the local price value functions~\eqref{eq:nodal:priceproblem-t}
and resource value functions~\eqref{eq:nodal:quantproblem-t}
makes it possible to bound the values of the global problems~\eqref{eq:nodal:vf}.
For this purpose, we first define the notion of \emph{admissible}
price and resource coordination processes.
\begin{subequations}
We introduce the primal admissible set~$SSTO$ of stochastic processes
associated with the almost sure constraints~\eqref{eq:nodal:couplingcons}:
\begin{multline}
\label{eq:nodal:primaladmissibleset}
SSTO = \Big\{\va y = (\va y_0,\cdots,\va y_{T-1})
\in \boldsymbol{W}idetilde{{\mathbb L}}(\Omega,\mathcal{F},\PP;\mathcal{R})
\\
\;\; \text{\text{s.t.}} \;\;
\va y_t \in S_t \;\; \PP\text{-}\as \eqsepv
\va{f}ORALLTIMES{t}{0}{T\!-\!1}\Big\} \eqfinp
\end{multline}
Then, the dual admissible cone of~$SSTO$ is
\begin{multline}
\label{eq:nodal:dualadmissibleset}
SSTO^\text{s.t.}ar = \Big\{\va z = (\va z_0, \cdots, \va z_{T-1})
\in \boldsymbol{W}idetilde{{\mathbb L}}^{\text{s.t.}ar}(\Omega,\mathcal{F},\PP;\mathcal{P}) \\
\text{\text{s.t.}} \;\;
\EE \bc{\pscal{\va y_t}{\va z_t}} \geq 0 \eqsepv
\forall \: \va y \in SSTO \eqsepv
\va{f}ORALLTIMES{t}{0}{T\!-\!1}\Big\} \eqfinp
\end{multline}
\label{eq:nodal:admissiblesets}
\end{subequations}
We say that $\va{p} \in \boldsymbol{W}idetilde{{\mathbb L}}^{\text{s.t.}ar}(\Omega,\mathcal{F},\PP;\mathcal{P})$
is an \emph{admissible price coordination process}
if
$\va{p}\inSSTO^\text{s.t.}ar$, and that
$\va{r} \in \boldsymbol{W}idetilde{{\mathbb L}}(\Omega,\mathcal{F},\PP ;\mathcal{R})$
is an \emph{admissible resource coordination process}
if $\va{r} \in -SSTO$.
By considering admissible coordination processes, we will now
bound up and down the global value functions~\eqref{eq:nodal:vf} with
the local value functions~\eqref{eq:nodal:priceproblem-t}
and~\eqref{eq:nodal:quantproblem-t}.
\begin{proposition}
\label{prop:nodal:stochasticvaluefuncbounds}
Let $\va{p}=\sequence{\va{p}^\node}{\node \in \NODES}\inSSTO^\text{s.t.}ar$
be an admissible price coordination process,
and let $\va{r}=\sequence{\va{r}^\node}{\node\in \NODES} \in -SSTO$
be an admissible resource coordination process.
Then, for all~$t\in\ic{0,T}$ and
for all $x_t = \sequence{x_t^\node}{\node \in \NODES} \in \XX_t$,
we have the inequalities
\begin{equation}
\label{eq:nodal:stochasticvaluefuncbounds}
\sum_{\node \in \NODES} \boldsymbol{U}nderline{V}_t^\node\nc{\va{p}^\node}(x_t^\node) \leq
V_t(x_t) \leq
\sum_{\node \in \NODES} \overline{V}_t^\node\nc{\va{r}^\node}(x_t^\node) \eqfinp
\end{equation}
\end{proposition}
\begin{proof}
For~$t=0$, the proof of the following proposition is a direct
application of Proposition~\ref{prop:nodal:valuefunctionsbounds}
to Problem~\eqref{eq:nodal:vf}.
For~$t \in \ic{1,T\!-\!1}$,
from the definitions~\eqref{eq:nodal:admissiblesets}
of~$SSTO$ and~$SSTO^\text{s.t.}ar$, the assumption that
$(\va{r}_0, \cdots, \va{r}_{T-1})$
(resp.~$(\va{p}_0, \cdots, \va{p}_{T-1})$)
is an admissible process implies that
the reduced process $(\va{r}_t, \cdots, \va{r}_{T-1})$
(resp.~$(\va{p}_t, \cdots, \va{p}_{T-1})$) is also admissible
on the reduced time interval~$\ic{t,T-1}$, hence the result
by applying Proposition~\ref{prop:nodal:valuefunctionsbounds}.
\end{proof}
\section{Decomposition of Local Value Functions by Dynamic Programming}
\label{sec:genericdecomposition}
In~\S\ref{subsec:nodal:globalprocess}, we have obtained
upper and lower bounds of optimization problems by
spatial decomposition. We now give conditions under which
\emph{spatial decomposition} schemes can be made
\emph{compatible with temporal decomposition},
thus yielding a mix of spatial and temporal decompositions.
In~\S\ref{subsec:nodal:decomposedDPdeterministic}, we show
that the local price value functions~\eqref{eq:nodal:priceproblem-t}
and the local resource value functions~\eqref{eq:nodal:quantproblem-t}
can be computed by Dynamic Programming,
when price and resource processes are deterministic.
In~\S\ref{subsec:nodal:processdesign}, we sketch how to obtain
tighter bounds by appropriately choosing the deterministic price
and resource processes.
In~\S\ref{subsec:nodal:admissiblepolicy}, we show
how to use local price and resource value functions as
surrogates for the global Bellman value functions,
and then produce global admissible policies.
In~\S\ref{subsec:nodal:decentralizedinformation}, we analyze
the case of a \emph{decentralized information structure}.
In the sequel, we make the following key assumption.
\begin{assumption}
\label{hyp:independent}
The global uncertainty process $\np{\boldsymbol{W}_1, \cdots, \boldsymbol{W}_T}$
in \eqref{eq:nodal:generic:globalnoise}
consists of stagewise independent random variables.
\end{assumption}
In the case where $\mathcal{G}_t^\node=\mathcal{F}_t$ for all~$t\in\ic{0,T}$
and all~$\node \in \NODES$ (centralized information structure in~\S\ref{subsec:nodal:generic:globaldata}),
under Assumption~\ref{hyp:independent},
the global value functions~\eqref{eq:nodal:vf} satisfy
the Dynamic Programming equations \cite{carpentier2015stochastic}
\begin{subequations}
\label{eq:globaldp}
\begin{align}
V_T(x_T)
& = \sum_{\node \in \NODES} K^{\node}(x^{\node}_T)
\quad \text{ and, for \( t=T\!-\!1, \ldots, 0 \),}
\\
V_t(x_t)
& = \inf_{u_t \in \UU_t} \EE
\bgc{\sum_{\node \in \NODES} L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}^{\node}_{t+1}) +
V_{t+1}\bp{\sequence{\boldsymbol{X}_{t+1}^\node}{\node \in \NODES}}}
\\
& \hphantom{u_t \in \UU_t} \text{s.t.}\ \;
\boldsymbol{X}_{t+1}^{\node} = {g}_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}^{\node}_{t+1}) \eqfinv \\
& \hphantom{u_t \in \UU_t \text{s.t.}\ } \;
\ba{\Theta_t^\node(x_t^\node, u_t^\node)}_{\node \in \NODES} \in -S_t \eqfinp
\end{align}
\end{subequations}
In the case where $\mathcal{G}_t^{\node}=\mathcal{F}_t^{\node}$ for all~$t\in\ic{0,T}$
and all~$\node \in \NODES$ (decentralized information structure
in~\S\ref{subsec:nodal:generic:globaldata}), the common assumptions
under which the global value functions~\eqref{eq:nodal:vf} satisfy
Dynamic Programming equations are not met.
\subsection{Decomposed Value Functions by Deterministic Coordination Processes}
\label{subsec:nodal:decomposedDPdeterministic}
We prove now that, for deterministic coordination processes,
the local problems~\eqref{eq:nodal:priceproblem-t} and
\eqref{eq:nodal:quantproblem-t} satisfy local Dynamic Programming
equations.
We first study the local price value function~\eqref{eq:nodal:priceproblem-t}.
\begin{proposition}
\label{prop:nodal:dppriceconstant}
Let $p^{\node}= (p_0^{\node}, \cdots, p_{T-1}^{\node})
\in \mathcal{P}^{\node}$ be a deterministic price process.
Then, be it for the centralized or the decentralized information
structure (see \S\ref{subsec:nodal:generic:globaldata}),
the local price value functions~\eqref{eq:nodal:priceproblem-t}
satisfy the following recursive Dynamic Programming equations
\begin{subequations}
\label{eq:localdp}
\begin{align}
\boldsymbol{U}nderline{V}_T^{\node}\nc{p^{\node}}(x_T^{\node}) =
& \; K^{\node}(x_T^{\node})
\quad \text{and, for \( t=T\!-\!1, \ldots, 0 \),}
\\
\boldsymbol{U}nderline{V}_t^{\node}\nc{p^{\node}}(x_t^{\node}) =
& \inf_{u_t^\node \in \UU_t^{\node}} \;
\EE \Big[ L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\pscal{p_t^{\node}}{\Theta_t^{\node}(x_t^{\node}, u_t^{\node})}
\\
& \hspace{3.5cm} + \boldsymbol{U}nderline{V}_{t+1}^{\node}\nc{p^{\node}}
\bp{{g}_t^{\node}(x_t^{\node}, u_t^{\node},
\boldsymbol{W}_{t+1}^{\node})} \Big]
\eqfinp
\nonumber
\end{align}
\end{subequations}
\end{proposition}
\begin{proof}
Let $p^{\node} = (p_0^{\node},\cdots,p_{T-1}^{\node})\in \mathcal{P}^{\node}$
be a deterministic price vector. Then, the price value
function~\eqref{eq:nodal:priceproblem-t} has the following expression:
\begin{align}
\boldsymbol{U}nderline V^{\node}_0\nc{p^{\node}}(x_0^{\node}) = \inf_{\boldsymbol{X}^{\node}, \boldsymbol{U}^{\node}}
& \EE \bigg[\sum_{t=0}^{T-1}
L^{\node}_t(\va X_t^{\node}, \va U_t^{\node}, \va W^{\node}_{t+1}) \nonumber
\\
& \hspace{2.0cm} + \pscal{p_t^{\node}}{\Theta_t^{\node}(\boldsymbol{X}_t^{\node},\boldsymbol{U}_t^{\node})}
+ K^{\node}(\boldsymbol{X}_T^{\node}) \bigg]
\eqfinv
\label{eq:nodal:priceproblem-deter}
\\
\text{s.t.}
& \;\boldsymbol{X}_0^{\node} = x_0^{\node}\text{\, and \,}
\va{f}ORALLTIMES{s}{0}{T\!-\!1},
\eqref{eq:nodal:dynamic}, \eqref{eq:nodal:measurability} \nonumber
\eqfinp
\end{align}
In the case where~$\mathcal{G}_t^{\node}=\mathcal{F}_t$, and as
Assumption~\ref{hyp:independent} holds true, the optimal value
of Problem~\eqref{eq:nodal:priceproblem-deter} can be obtained
by the recursive Dynamic Programming equations~\eqref{eq:localdp}.
Consider now the case~$\mathcal{G}_t^{\node}=\mathcal{F}_t^{\node}$. Since the local
value function and local dynamics in~\eqref{eq:nodal:priceproblem-deter}
only depend on the local noise process~$\boldsymbol{W}^{\node}$, there is no loss of
optimality to replace the constraint $\sigma(\boldsymbol{U}_t^{\node}) \subset \mathcal{F}_t$
by $\sigma(\boldsymbol{U}_t^{\node}) \subset \mathcal{F}_t^{\node}$.
Moreover, Assumption~\ref{hyp:independent} implies that the local
uncertainty process $(\boldsymbol{W}_{1}^{\node},\dots,\boldsymbol{W}_{T}^{\node})$ consists
of stagewise independent random variables, so that the solution of
Problem~\eqref{eq:nodal:priceproblem-deter} can be obtained
by the recursive Dynamic Programming equations~\eqref{eq:localdp}
when replacing the \emph{global} $\sigma$-field~$\mathcal{F}_t$
by the \emph{local} $\sigma$-field~$\mathcal{F}_t^{\node}$
(see Equation~\eqref{eq:nodal:infos}).
\end{proof}
A similar result holds true for the local resource value
functions~\eqref{eq:nodal:quantproblem-t} as stated now in Proposition~\ref{prop:nodal:dpquantconstant}
whose proof is left to the reader.
\begin{proposition}
\label{prop:nodal:dpquantconstant}
Let $r^{\node}= (r_0^{\node}, \cdots, r_{T-1}^{\node})
\in \mathcal{R}^{\node}$ be a deterministic resource process.
Then, be it for the centralized or the decentralized information structure
in~\S\ref{subsec:nodal:generic:globaldata},
the local resource value functions~\eqref{eq:nodal:quantproblem-t}
satisfy the following recursive Dynamic Programming equations
\begin{subequations}
\begin{align}
\overline{V}_T^{\node}\nc{r^{\node}}(x_T^{\node}) =
& K^{\node}(x_T^{\node})
\quad \text{and, for \( t=T\!-\!1, \ldots, 0 \),}
\\
\overline{V}_t^{\node}\nc{r^{\node}}(x_t^{\node}) =
& \inf_{u_t^\node \in \UU_t^{\node}} \EE \Bc{L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node})
+ \overline{V}_{t+1}^{\node}\nc{r^{\node}}
\bp{{g}_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node})}}
\eqfinv
\nonumber
\\
& \text{s.t.}\ \;\; \Theta_t^{\node}(x_t^{\node}, u_t^{\node}) = r_t^{\node}
\eqfinp
\end{align}
\label{eq:nodal:localdpquant}
\end{subequations}
\end{proposition}
\subsection{Computing Upper and Lower Bounds, and Decomposed Value Functions}
\label{subsec:nodal:processdesign}
In the context of a deterministic \emph{admissible} price coordination process
$p^{\node} = (p_0^{\node},\cdots,p_{T-1}^{\node})\in S^\text{s.t.}ar$
and resource process
$r^{\node}= (r_0^{\node}, \cdots, r_{T-1}^{\node}) \in S$,
where~$S$ is defined in~\eqref{eq:nodal:global_constraint_set}, the double inequality
\eqref{eq:nodal:stochasticvaluefuncbounds}
in Proposition~\ref{prop:nodal:stochasticvaluefuncbounds} becomes
\begin{equation}
\label{eq:nodal:boundsvaluefunctiondeterministic}
\sum_{\node \in \NODES} \boldsymbol{U}nderline{V}_t^{\node}\nc{p^{\node}}(x_t^{\node})
\leq V_t(x_t)\leq
\sum_{\node \in \NODES} \overline{V}_t^{\node}\nc{r^{\node}}(x_t^{\node})
\eqfinp
\end{equation}
\begin{itemize}
\item
Both in the lower bound and the upper bound of~$V_t$
in~\eqref{eq:nodal:boundsvaluefunctiondeterministic},
the sum over units~$\node\in\NODES$ materializes the spatial decomposition
for the computation of the bounds. For each of the bounds, this
decomposition leads to independent optimization subproblems
that can be processed in parallel.
\item
For a given unit~$\node\in\NODES$,
the computation of the local value functions~$\boldsymbol{U}nderline{V}_t^{\node}\nc{p^{\node}}$
and~$\overline{V}_t^{\node}\nc{r^{\node}}$ for $t \in \ic{0,T}$
can be performed by Dynamic Programming
as stated in Propositions
\ref{prop:nodal:dppriceconstant} and~\ref{prop:nodal:dpquantconstant}.
The corresponding loop in backward time materializes the temporal
decomposition, processed sequentially.
\end{itemize}
Now, we suppose given an initial state
$x_0 = \sequence{x_0^\node}{\node \in \NODES} \in \XX_0$
and we sketch how, by suitably choosing the admissible coordination processes,
we can improve
the upper and lower bounds~\eqref{eq:nodal:boundsvaluefunctiondeterministic}
for~$V_0\np{x_0}$, that is, the optimal value of Problem~\eqref{eq:nodal:vf} for~$t=0$.
By Propositions \ref{prop:nodal:stochasticvaluefuncbounds}
and~\ref{prop:nodal:dppriceconstant}, for any deterministic
$p=(p_0,\cdots,p_{T-1}) \in S^\text{s.t.}ar$,
we have
\( \sum_{\node \in \NODES} \boldsymbol{U}nderline{V}_0^{\node}\nc{p^{\node}}(x_0^{\node})
\; \leq \; V_0(x_0) \).
As a consequence, solving the following optimization problem
\begin{equation}
\sup_{p \in S^\text{s.t.}ar} \sum_{\node \in \NODES}
\boldsymbol{U}nderline V^{\node}_0\nc{p^{\node}}(x_0^{\node})
\label{eq:nodal:relaxedconstraintdual}
\end{equation}
gives the greatest possible lower bound in the class
of deterministic admissible price coordination processes.
We can maximize Problem~\eqref{eq:nodal:relaxedconstraintdual}
\boldsymbol{W}rt~$p$ using a gradient-like ascent algorithm.
Updating $p$ requires the computation of the gradient of
$\sum_{\node \in \NODES} \boldsymbol{U}nderline{V}_0^{\node}\nc{p^{\node}}(x_0^{\node})$,
obtained when computing the price value functions.
The standard update formula corresponding to the gradient algorithm
(Uzawa algorithm) can be replaced by more sophisticated methods (Quasi-Newton).
By Propositions \ref{prop:nodal:stochasticvaluefuncbounds}
and~\ref{prop:nodal:dpquantconstant}, for any
deterministic $r=(r_0,\cdots,r_{T-1}) \in -S$,
we have
\( V_0\bp{\sequence{x_0^\node}{\node \in \NODES}} \; \leq \;
\sum_{\node \in \NODES} \overline{V}_0^{\node}\nc{r^{\node}}(x_0^{\node})
\).
As a consequence, solving the following optimization problem
\begin{equation}
\label{eq:nodal:overconstraint}
\inf_{r\in -S} \sum_{\node \in \NODES}
\overline{V}_0^{\node}\nc{r^{\node}}(x_0^{\node})
\end{equation}
gives the lowest possible upper bound in the set
of deterministic admissible resource coordination processes.
Again, we can minimize Problem~\eqref{eq:nodal:overconstraint}
\boldsymbol{W}rt~$r$ using a gradient-like algorithm.
Updating $r$ requires the computation of the gradient of
$\sum_{\node \in \NODES} \overline{V}_0^{\node}\nc{r^{\node}}(x_0^{\node})$, obtained
when computing the resource value functions.
Again, the standard update formula corresponding to the gradient
algorithm can be replaced by more sophisticated methods.
At the end of the procedure, we have obtained
a deterministic admissible price coordination process
$p=(p_0,\cdots,p_{T-1}) \in S^\text{s.t.}ar$
and a deterministic admissible resource coordination process
$r=(r_0,\cdots,r_{T-1}) \in -S$
such that~$V_0\np{x_0}$,
the optimal value of Problem~\eqref{eq:nodal:vf} for~$t=0$,
is tightly bounded above and below like
in~\eqref{eq:nodal:boundsvaluefunctiondeterministic}
for~$t=0$. We have also obtained the solutions
$\na{\boldsymbol{U}nderline{V}_t^{\node}\nc{p}}_{t\in \ic{0, T}}$
and
$\na{\overline{V}_t^{\node}\nc{r}}_{t\in \ic{0, T}}$
of the recursive Dynamic Programming Equations~\eqref{eq:localdp} and~\eqref{eq:nodal:localdpquant}
associated with these coordination processes.
\subsection{Devising Policies}
\label{subsec:nodal:admissiblepolicy}
Now that we have decomposed value functions,
we show how to devise policies.
By \emph{policy}, we mean a sequence
\( \gamma = \ba{\gamma_{t}}_{t\in \ic{0, T-1}} \) where,
for any \( t\in\ic{0,T{-}1} \),
each $\gamma_t$ is a \emph{state feedback}, that is,
a measurable mapping \( \gamma_t : \XX_t\to\UU_t \).
Here, we suppose that we have at our disposal pre-computed \emph{local}
value functions $\na{\boldsymbol{U}nderline V_t^{\node}}_{t\in \ic{0, T}}$
and $\na{\overline V_t^{\node}}_{t\in \ic{0, T}}$ solving
Equations~\eqref{eq:localdp} for the price value functions
and Equations~\eqref{eq:nodal:localdpquant} for the resource
value functions.
For instance, one could use the functions
$\na{\boldsymbol{U}nderline{V}_t^{\node}\nc{p}}_{t\in \ic{0, T}}$
and
$\na{\overline{V}_t^{\node}\nc{r}}_{t\in \ic{0, T}}$
obtained at the end of~\S\ref{subsec:nodal:processdesign}.
Using the sum of these local value functions
as a surrogate for a global Bellman value function,
we propose two \emph{global} policies as follows
(supposing that the $\argmin$ are not empty and that the resulting expressions
provide measurable mappings \cite{bertsekas-shreve:1996}):
\noindent
1) a \emph{global price policy}
\( \boldsymbol{U}nderline \gamma =
\ba{\boldsymbol{U}nderline\gamma_{t}}_{t\in \ic{0, T-1}} \)
with, for any \( t\in\ic{0,T{-}1} \), the feedback
$\boldsymbol{U}nderline \gamma_t:\XX_t\to\UU_t$ defined
for all $x_t = \sequence{x_t^\node}{\node \in \NODES} \in \XX_t$ by
\begin{align}
\boldsymbol{U}nderline\gamma_t(x_t) \in \argmin_{\sequence{u_t^\node}{\node \in \NODES}}
& \; \EE\bgc{\sum_{\node \in \NODES} L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\boldsymbol{U}nderline V_{t+1}^{\node}\bp{g_t^{\node}(x_t^{\node}, u_t^{\node},
\boldsymbol{W}_{t+1}^{\node})}} \eqsepv
\nonumber \\
\text{s.t.}\
& \; \ba{\Theta_t^\node(x_t^\node, u_t^\node)}_{\node \in \NODES}
\in -S_t
\eqfinv
\label{eq:nodal:globalpricepolicy}
\end{align}
2) a \emph{global resource policy}
\( \overline \gamma =
\ba{\overline\gamma_{t}}_{t\in \ic{0, T-1}} \)
with, for any \( t \in \ic{0,T{-}1} \), the feedback
$\overline \gamma_t: \XX_t \to \UU_t$ defined
for all $x_t = \sequence{x_t^\node}{\node \in \NODES} \in \XX_t$ by
\begin{align}
\overline\gamma_t(x_t) \in \argmin_{\sequence{u_t^\node}{\node \in \NODES}}
& \; \EE\bgc{\sum_{\node \in \NODES} L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\overline V_{t+1}^{\node}\bp{g_t^{\node}(x_t^{\node}, u_t^{\node},
\boldsymbol{W}_{t+1}^{\node})}}
\eqfinv
\nonumber \\
\text{s.t.}\
& \; \ba{\Theta_t^\node(x_t^\node, u_t^\node)}_{\node \in \NODES}
\in -S_t
\eqfinp
\label{eq:nodal:globalresourcepolicy}
\end{align}
Given a policy \( \gamma = \ba{\gamma_{t}}_{t\in \ic{0, T-1}} \)
and any time $t \in \ic{0, T}$, the expected cost of policy
$\gamma$ starting from state~$x_t$ at time~$t$ is equal to
\begin{align}
V_t^\gamma(x_t) =
& \; \EE\bgc{\sum_{\node \in \NODES} \sum_{s=t}^{T-1}
L_s^{\node}(\boldsymbol{X}_s^{\node},\gamma_s^{\node}(\boldsymbol{X}_s),\boldsymbol{W}_{t+1}^{\node}) +
K^{\node}(\boldsymbol{X}_T^{\node})}
\eqfinv
\label{eq:nodal:costpolicy}
\\
\text{s.t.}\
& \va{f}ORALLTIMES{s}{t}{T\!-\!1} \eqsepv
\boldsymbol{X}_{s+1}^{\node} = {g}_s^{\node}(\boldsymbol{X}^{\node}_s, \gamma_s^{\node}(\boldsymbol{X}_s), \boldsymbol{W}_{s+1}^{\node})
\eqsepv \boldsymbol{X}_t^{\node} = x_t^{\node}
\eqfinp
\nonumber
\end{align}
We provide several bounds hereafter.
\begin{proposition}
\label{prop:nodal:boundresourcepolicy}
Let $t \in \ic{0, T}$ and $x_t = \sequence{x_t^\node}{\node \in \NODES} \in \XX_t$
be a given state. Then, we have
\begin{subequations}
\begin{align}
\sum_{\node \in \NODES} \boldsymbol{U}nderline{V}_t^{\node}(x_t^{\node}) \leq V_t(x_t)
& \leq
V_t^{\overline \gamma}(x_t)
\leq \sum_{\node \in \NODES} \overline{V}_t^{\node}(x_t^{\node})
\eqfinv
\label{eq:nodal:boundresourcepolicy_a}
\\
V_t(x_t)
& \leq \inf
\ba{V_t^{\boldsymbol{U}nderline \gamma}(x_t),V_t^{\overline \gamma}(x_t)}
\eqfinp
\end{align}
\label{eq:nodal:boundresourcepolicy}
\end{subequations}
\end{proposition}
\begin{proof}
We prove the right hand side inequality
in~\eqref{eq:nodal:boundresourcepolicy_a} by backward induction.
At time $t = T$, the result is straightforward as
$\overline{V}_t^{\node} = K^{\node}$ for all $\node \in \NODES$.
Let $t \in \ic{0, T-1}$ such that the right hand side inequality
in~\eqref{eq:nodal:boundresourcepolicy_a} holds true at time $t+1$.
Then, for all $x_t \in \XX_t$, Equation \eqref{eq:nodal:costpolicy}
can be rewritten
\begin{equation*}
V_t^{\overline \gamma}(x_t) =
\EE\bgc{\sum_{\node \in \NODES}\bp{ L_t^{\node}(x_t^{\node}, \overline \gamma_t^{\node}(x_t),
\boldsymbol{W}_{t+1}^{\node})} + V_{t+1}^{\overline \gamma}(\boldsymbol{X}_{t+1}) } \eqfinv
\end{equation*}
Using the induction assumption, we deduce that
\begin{align*}
V_t^{\overline \gamma}(x_t)
&\leq
\EE\bgc{\sum_{\node \in \NODES} L_t^{\node}(x_t^{\node}, \overline \gamma_t^{\node}(x_t),
\boldsymbol{W}_{t+1}^{\node}) + \overline{V}_{t+1}^{\node}(\boldsymbol{X}_{t+1}^{\node}) } \eqfinp\\
\intertext{From the very definition~\eqref{eq:nodal:globalresourcepolicy}
of the global resource policy, $\overline{\gamma}$ , we obtain}
V_t^{\overline \gamma}(x_t)
& \leq \inf_{\sequence{u_t^\node}{\node \in \NODES}} \;
\EE \bgc{\sum_{\node \in \NODES} L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\overline{V}_{t+1}^{\node}(\boldsymbol{X}_{t+1}^{\node})} \eqfinv \\
& \hspace{1.0cm} \text{s.t.}\ \;
\ba{\Theta_t^\node(x_t^\node, u_t^\node)}_{\node \in \NODES} \in -S_t \eqfinp
\end{align*}
Introducing a deterministic admissible resource
process $\sequence{r_t^\node}{\node \in \NODES} \in -S_t$
and restraining the constraint with it reinforces
the inequality, thus giving
\begin{subequations}
\label{eq:nodal:proof:temp1}
\begin{align}
V_t^{\overline \gamma}(x_t)
& \leq \inf_{{\sequence{u_t^\node}{\node \in \NODES}}} \;
\EE \bgc{\sum_{\node \in \NODES} L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\overline{V}_{t+1}^{\node}(\boldsymbol{X}_{t+1}^{\node}) } \\
& \hspace{1.0cm} \text{s.t.}\ \;
\Theta_t^\node(x_t^\node, u_t^\node) = r_t^\node \eqsepv \forall {\node \in \NODES}
\eqfinv
\end{align}
\end{subequations}
so that
\begin{equation*}
V_t^{\overline \gamma}(x_t) \leq \sum_{\node \in \NODES}
\Bp{\inf_{u_t^{\node}}
\EE\bc{L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\overline{V}_{t+1}^{\node}(\boldsymbol{X}_{t+1}^{\node})}
\;\; \text{s.t.} \;\; \Theta_t^{\node}(x_t^{\node},u_t^{\node}) = r_t^{\node}}
\end{equation*}
as we do not have any coupling left in \eqref{eq:nodal:proof:temp1}.
By Equation~\eqref{eq:nodal:localdpquant}, we deduce that
\( V_t^{\overline \gamma}(x_t)\leq \sum_{\node \in \NODES}
\overline{V}_t^{\node}(x_t^{\node}) \),
hence the result at time~$t$.
Furthermore, for any admissible policy $\gamma$,
we have $V_t(x_t) \leq V_t^{\gamma}(x_t)$ as the global Bellman
function gives the minimal cost starting at any point $x_t \in \XX_t$.
We therefore obtain all the other inequalities
in~\eqref{eq:nodal:boundresourcepolicy}.
\end{proof}
\subsection{Analysis of the Decentralized Information Structure}
\label{subsec:nodal:decentralizedinformation}
An interesting consequence of Propositions
\ref{prop:nodal:dppriceconstant} and~\ref{prop:nodal:dpquantconstant}
is that the local price and resource value functions
$\boldsymbol{U}nderline{V}_t^{\node}\nc{p^{\node}}$ in~\eqref{eq:global:priceproblem}
and~$\overline{V}_t^{\node}\nc{r^{\node}}$
in~\eqref{eq:global:quantproblem}
remain the same when choosing either the centralized information
structure or the decentralized one in~\S\ref{subsec:nodal:generic:globaldata}.
By contrast, the global value functions~$V_t$ in~\eqref{eq:nodal:vf}
depend on that choice. Let us denote by~$V^{\mathrm{C}}_t$
(resp. $V^{\mathrm{D}}_t$) the value functions~\eqref{eq:nodal:vf}
in the centralized (resp. decentralized) case
where \( \sigma(\boldsymbol{U}_s^\node) \subset \mathcal{F}_s^\node \)
(resp. \( \sigma(\boldsymbol{U}_s^\node) \subset \mathcal{F}_s \)). Since the admissible
set induced by the constraint~\eqref{eq:nodal:measurability}
in the centralized case is larger than the one in the decentralized
case (because $\mathcal{F}_t^{\node} \subset \mathcal{F}_t$ by \eqref{eq:nodal:localinfo}),
we deduce that the lower bound is tighter for the centralized problem,
and the upper bound tighter for the decentralized problem:
for all $x_t = \sequence{x_t^\node}{\node \in \NODES} \in \XX_t$,
\begin{equation}
\label{eq:boundsCandD}
\sum_{\node \in \NODES} \boldsymbol{U}nderline{V}_t^{\node}\nc{p^{\node}}(x_t^{\node})
\leq V^{\mathrm{C}}_t(x_t)
\leq V^{\mathrm{D}}_t(x_t)
\leq \sum_{\node \in \NODES} \overline{V}_t^{\node}\nc{r^{\node}}(x_t^{\node})
\eqfinp
\end{equation}
Now, we show that, in some specific cases (but often encountered in practical
applications), the best upper bound
in~\eqref{eq:boundsCandD} is equal to the optimal value~$V^{\mathrm{D}}_t(x_t)$ of the decentralized problem.
\begin{proposition}
\label{prop:upperequalD}
If, for all~$t \in \ic{0,T-1}$, we have the equivalence
\begin{equation}
\label{eq:upperequalD-ass}
\begin{split}
\ba{\Theta_t^\node(\boldsymbol{X}_t^\node, \boldsymbol{U}_t^\node)}_{\node \in \NODES}
\in -S_t
\iff \\
\bp{\exists \sequence{r_t^\node}{\node \in \NODES}\in {-}S_t \eqsepv
\Theta_t^{\node}(\boldsymbol{X}_t^{\node}, \boldsymbol{U}_t^{\node})=r_t^{\node}
\quad \forall \node \in \NODES}
\eqfinv
\end{split}
\end{equation}
then the optimal value
$V^{\mathrm{D}}_0(x_0)$ of the decentralized problem ---
that is,
given by~\eqref{eq:nodal:vf} where \( \sigma(\boldsymbol{U}_s^\node) \subset \mathcal{F}_s^\node \)
in~\eqref{eq:nodal:measurability} --- satisfies
\begin{equation}
\label{eq:upperequalD-prop}
V_0^{\mathrm{D}}(x_0) =
\inf_{r \in -S} \;
\sum_{\node \in \NODES} \overline{V}_0^{\node}\nc{r^{\node}}(x_0^{\node}) \eqfinp
\end{equation}
\end{proposition}
\begin{proof}
Using Assumption~\eqref{eq:upperequalD-ass},
Problem~\eqref{eq:nodal:vf} for~$t=0$
can be written as
\begin{align*}
V_0^{\mathrm{D}}(x_0) =
& \inf_{r\in -S}
\Bgp{\sum_{\node \in \NODES} \inf_{\boldsymbol{X}^{\node}, \boldsymbol{U}^{\node}} \;
\EE\bgc{ \sum_{t=0}^{T-1}
L^{\node}_t(\va X_t^{\node}, \va U_t^{\node}, \va W^{\node}_{t+1}) +
K^{\node}(\boldsymbol{X}_T^{\node})}} \eqfinv \\
& \text{s.t.}\ \: \boldsymbol{X}_0^{\node} = x_0^{\node} \text{\, and \,}
\va{f}ORALLTIMES{s}{0}{T\!-\!1},
\eqref{eq:nodal:dynamic}, \eqref{eq:nodal:measurability},
\Theta_s^{\node}(\boldsymbol{X}_s^{\node}, \boldsymbol{U}_s^{\node}) = r_s^{\node}
\eqfinv
\nonumber\\
=
& \inf_{r\in -S} \;
\sum_{\node \in \NODES} \overline{V}_0^{\node}\nc{r^{\node}}(x_0^{\node}) \eqfinv
\end{align*}
the last equality arising from the definition
of~$\overline{V}_0^{\node}\nc{r^{\node}}$ in~\eqref{eq:nodal:quantproblem-t}
for $t=0$.
\end{proof}
As an application of the previous Proposition~\ref{prop:upperequalD},
we consider the case
of a decentralized information structure with an additional
\emph{independence assumption in space} (whereas
Assumption~\ref{hyp:independent} is an independence assumption \emph{in time}).
\begin{corollary}
We consider the case of a decentralized information structure
with the following two additional assumptions:
\begin{itemize}
\item
the random processes $\boldsymbol{W}^\node$, for $\node \in \NODES$, are independent,
\item
the coupling constraints~\eqref{eq:nodal:couplingcons}
are of the form
\( \sum_{\node \in \NODES}\Theta_t^{\node}(\boldsymbol{X}_t^{\node}, \boldsymbol{U}_t^{\node}) = 0 \).
\end{itemize}
Then, the assumptions of Proposition~\ref{prop:upperequalD}
are satisfied, so that Equality~\eqref{eq:upperequalD-prop} holds true.
\label{cor:upperequalD}
\end{corollary}
\begin{proof}
From the dynamic constraint~\eqref{eq:nodal:dynamic} and from
the measurability constraint~\eqref{eq:nodal:measurability},
we have that each term~$\Theta_t^{\node}(\boldsymbol{X}_t^{\node}, \boldsymbol{U}_t^{\node})$ is
$\mathcal{F}_t^{\node}$-measurable in the decentralized information structure case.
Since the random processes $\boldsymbol{W}^\node$, for $\node \in \NODES$, are independent,
so are the $\sigma$-fields~$\mathcal{F}_t^{\node}$, for $\node \in \NODES$, from which we
deduce that the random variables
$\Theta_t^{\node}(\boldsymbol{X}_t^{\node}, \boldsymbol{U}_t^{\node})$ are independent.
Now, these random variables sum up to zero.
But it is well-known that, if a sum of independent random variables
is zero, then every random variable in the sum is constant (deterministic).
Hence, each random variable $\Theta_t^{\node}(\boldsymbol{X}_t^{\node}, \boldsymbol{U}_t^{\node})$ is constant.
By introducing their constant values~$\sequence{r_t^\node}{\node \in \NODES}$,
the constraints~\eqref{eq:nodal:couplingcons} are written equivalently
$\Theta_t^{\node}(\boldsymbol{X}_t^{\node},\boldsymbol{U}_t^{\node}) - r_t^{\node} = 0$,
$\forall \: \node \in\NODES$,
and $\sum_{\node\in\NODES} r_t^{\node} = 0$.
We conclude with Proposition~\ref{prop:upperequalD}.
\end{proof}
\begin{remark}
\label{rem:decentralizedpolicy}
In the case of a decentralized information structure~\eqref{eq:nodal:localinfo},
it seems difficult to produce Bellman-based online policies.
Indeed, neither the global price policy in~\eqref{eq:nodal:globalpricepolicy}
nor the global resource policy in~\eqref{eq:nodal:globalresourcepolicy}
are implementable since both policies require the knowledge of the global
state $\sequence{x_t^\node}{\node \in \NODES}$ for each unit~$\node$, which is incompatible
with the information constraint~\eqref{eq:nodal:localinfo}.
Nevertheless, one can use the results given by resource decomposition
to compute a local state feedback as follows.
For a given deterministic admissible resource
process~$r \in -S$, solving at time~$t$ and for each~$\node \in\NODES$
the subproblem
\begin{align*}
\overline\gamma_t^{\node}(x_t^{\node}) \in \argmin_{u_t^{\node}}
& \; \EE\Bc{ L_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) +
\overline V_{t+1}^{\node}\bp{ g_t^{\node}(x_t^{\node}, u_t^{\node}, \boldsymbol{W}_{t+1}^{\node}) }} \eqfinv \\
\text{s.t.}\
& \Theta_t^{\node}(x_t^{\node}, u_t^{\node}) = r_t^{\node}
\end{align*}
generates a local state feedback
\( \overline\gamma_t^{\node} : \XX_t^{\node} \to\UU_t \) which is both
compatible with the decentralized information
structure~\eqref{eq:nodal:localinfo}
and such that the policy \( \overline \gamma =
\ba{\overline\gamma_{t}}_{t\in \ic{0, T-1}} \) is admissible
as it satisfies the global coupling
constraint~\eqref{eq:nodal:couplingcons}
between all units because $r \in -S$,
where $S$ is defined in~\eqref{eq:nodal:global_constraint_set}.
By contrast, replicating this procedure with a deterministic admissible
price process would produce a policy which would not satisfy the global
coupling constraint~\eqref{eq:nodal:couplingcons}.
\end{remark}
\section{Application to Microgrids Optimal Management}
\label{chap:district:numerics}
We illustrate the effectiveness of the two decomposition schemes
introduced in Sect.~\ref{sec:genericdecomposition}
by presenting numerical results.
In~\S\ref{Description_of_the_problems}, we describe an application
in the optimal management of urban microgrids.
In~\S\ref{ssec:nodalalgorithms}, we detail how we implement
algorithms to obtain bounds and policies.
In~\S\ref{Numerical_results}, we
illustrate the performance of the decomposition methods
with numerical results.
\subsection{Description of the Problems}
\label{Description_of_the_problems}
The energy management problem
and the structure of the microgrids
come from case studies
provided by the urban Energy Transition Institute
Efficacity\footnote{Established in 2014 with the French government support,
Efficacity aims to develop and implement innovative
solutions to build and manage energy-efficient
cities.}.
For more details on microgrid modeling and on the formulation
of associated optimization problems, the reader is referred
to the PhD thesis~\cite{thesepacaud}.
We represent a district microgrid by a directed graph
$(\mathfrak{N}, \mathfrak{A})$, with $\mathfrak{N}$ the set of nodes and $\mathfrak{A}$
the set of arcs. Each node of the graph corresponds to a building.
The buildings exchange energy through the edges of the graph,
hence coupling the different nodes of the graph by static
constraints (Kirchhoff law).
We manage the microgrids over a given day
in summer, with decisions taken every 15mn, so that $T = 96$.
Each building has its own electrical and domestic hot water demand profiles,
and possibly its own solar panel production.
At node~$\node$, we consider a random variable~$\va\boldsymbol{W}_t^\node$,
with values in $\WW_t^\node=\va{R}R^2$, representing the following couple
of uncertainties:
the local electricity demand minus the production of the solar panel;
the domestic hot water demand.
We also suppose given a corresponding finite probability distribution on the set~$\WW_t^\node$.
Each building is equipped with an electrical hot water tank;
some buildings have solar panels and some others have batteries.
We view batteries and electrical hot water tanks as energy stocks
so that, depending on the presence of battery inside the building,
we introduce a state~$\boldsymbol{X}_t^\node$ at node~$\node$ with dimension~2 or~1
(energy stored inside the water tank and energy stored in the battery),
and the same with the control $\boldsymbol{U}_t^\node$ at node~$\node$
(power used to heat the tank and power exchanged with the battery).
Each node of a graph is modelled as a local control system
in which the cost function corresponds to import electricity from the external
grid. Summing the costs and taking the expectation
(supposing that the $\np{\boldsymbol{W}_1, \cdots, \boldsymbol{W}_T}$
are stagewise independent random variables),
we obtain a global optimization problem of the form~\eqref{eq:nodal:vf}.
We consider six different problems with growing sizes.
Table~\ref{tab:numeric:pbsize} displays the different dimensions
considered.
\begin{table}[!ht]
\centering
{\normalsize
\begin{tabular}{|c|ccccc|}
\hline
Problem & $\card{\mathfrak{N}}$ & $\card{\mathfrak{A}}$ & $\mathbf{dim(\XX_t)}$ & $dim(\WW_t)$ & $supp(\boldsymbol{W}_t)$ \\
\hline
\hline
\textrm{3-nodes} & 3 & 3 & \textbf{4} & 6 & $10^3$ \\
\textrm{6-nodes} & 6 & 7 & \textbf{8} & 12 & $10^6$ \\
\textrm{12-nodes} & 12 & 16 & \textbf{16} & 24 & $10^{12}$ \\
\textrm{24-nodes} & 24 & 33 & \textbf{32} & 48 & $10^{24}$ \\
\textrm{48-nodes} & 48 & 69 & \textbf{64} & 96 & $10^{48}$ \\
\hline
\end{tabular}
}
\caption{Microgrid management problems with growing dimensions}
\label{tab:numeric:pbsize}
\end{table}
As an example, the 12-nodes problem consists of twelve buildings;
four buildings are equipped with a battery, and four other
buildings are equipped with solar panels.
The devices are dispatched so that a building equipped with a solar
panel is connected to at least one building with a battery.
\subsection{Computing Bounds, Decomposed Value Functions and Devising Policies}
\label{ssec:nodalalgorithms}
We apply the two decomposition algorithms, introduced
in~\S\ref{subsec:nodal:processdesign} and
in~\S\ref{subsec:nodal:admissiblepolicy},
to each problem as described in Table~\ref{tab:numeric:pbsize}.
We will term \emph{Dual Approximate Dynamic Programming}
(DADP) the price decomposition algorithm
and \emph{Primal Approximate Dynamic Programming} (PADP)
the resource decomposition algorithm described in
\S\ref{subsec:nodal:processdesign} and
in~\S\ref{subsec:nodal:admissiblepolicy}.
We compare DADP and PADP
with the well-known Stochastic Dual Dynamic Programming (SDDP)
algorithm (see~\cite{girardeau2014convergence} and references
inside) applied to the global problem.
In this part, we suppose given an initial state
$x_0 = \sequence{x_0^\node}{\node \in \NODES} \in \XX_0$.
Regarding the SDDP algorithm, it is not implementable in
a straightforward manner since the cardinality of the global noise
support becomes huge with the number~$\card{\mathfrak{N}}$ of nodes
(see Table~\ref{tab:numeric:pbsize}), so that the exact computation
of an expectation \boldsymbol{W}rt\ the global uncertainty
$\boldsymbol{W}_t= \sequence{\boldsymbol{W}_t^\node}{\node\in \mathfrak{N}}$ is out of reach.
To overcome this issue, we have resampled the probability distribution
of the global noise~$\sequence{\boldsymbol{W}_t^\node}{\node\in \mathfrak{N}}$ at
each time~$t$ by using the $k$-means clustering method
(see \cite{rujeerapaiboon2018scenario}).
Thanks to the convexity properties of the problem, the optimal quantization
yields a new optimization problem
whose optimal value is a lower bound for the optimal value
of the original problem (see \cite{lohndorfmodeling} for details).
Thus, the exact lower bound given by SDDP with resampling remains
a lower bound for the exact lower bound given by SDDP without resampling,
which itself is, by construction, a lower bound for the original problem.
Regarding DADP and PADP, we use a quasi-Newton algorithm
to perform the maximization \boldsymbol{W}rt\ $p$ in~\eqref{eq:nodal:relaxedconstraintdual}
and the minimization \boldsymbol{W}rt\ $r$ in~\eqref{eq:nodal:overconstraint}.
More precisely, the quasi-Newton algorithm is performed using Ipopt 3.12
(see~\citep{wachter2006implementation}). The algorithm stops either
when a stopping criterion is fulfilled or when no descent direction
is found.
Each algorithm (SDDP, DADP, PADP) returns a sequence
of global value functions indexed by time.
Indeed, SDDP produces approximate global value functions,
and, for DADP (resp. PADP), we sum the local price value functions
(resp. the local resource value functions) obtained as
solutions of the recursive Dynamic Programming
equations~\eqref{eq:localdp} (resp.~\eqref{eq:nodal:localdpquant}),
for the deterministic admissible price coordination process
$p=(p_0,\cdots,p_{T-1}) \in S^\text{s.t.}ar$
(resp. the deterministic admissible resource coordination process
$r=(r_0,\cdots,r_{T-1}) \in -S$)
obtained at the end of~\S\ref{subsec:nodal:processdesign}
for an initial state
$x_0 = \sequence{x_0^\node}{\node \in \NODES} \in \XX_0$.
As explained in~\S\ref{subsec:nodal:admissiblepolicy},
these global value functions yield policies.
Thus, we have three policies (SDDP, DADP, PADP)
that we can compare.
As the policies are admissible,
the three expected values of the associated costs are
\emph{upper bounds} of the optimal value of the global optimization problem.
\subsection{Numerical Results}
\label{Numerical_results}
We compare the three algorithms (SDDP, DADP, PADP)
regarding their execution time
in~\S\ref{Computation_of_the_Bellman_value_functions},
the quality of their theoretical bounds
in~\S\ref{Quality_of_the_theoretical_bounds},
and the performance of their policies in simulation
in~\S\ref{Policy_simulation_results}.
\subsubsection{CPU Execution Time}
\label{Computation_of_the_Bellman_value_functions}
Table~\ref{tab:district:numeric:optres} details CPU execution time
and number of iterations before reaching stopping criterion
for the three algorithms.
\begin{table}[!ht]
\centering
{\normalsize
\begin{tabular}{|l|ccccc|}
\hline
Problem & \textrm{3-nodes} \hspace{-0.2cm}
& \textrm{6-nodes} \hspace{-0.2cm}
& \textrm{12-nodes} \hspace{-0.2cm}
& \textrm{24-nodes} \hspace{-0.2cm}
& \textrm{48-nodes} \hspace{-0.2cm} \\
\hline
dim($\XX_t$) & 4 & 8 & 16
& 32 & 64 \\
\hline
\hline
SDDP CPU time & 1' & 3' & 10'
& 79' & 453' \\
SDDP iterations & 30 & 100 & 180
& 500 & 1500 \\
\hline
\hline
DADP CPU time & 6' & 14' & 29'
& 41' & 128' \\
DADP iterations & 27 & 34 & 30
& 19 & 29 \\
\hline
\hline
PADP CPU time & 3' & 7' & 22'
& 49' & 91' \\
PADP iterations & 11 & 12 & 20
& 19 & 20 \\
\hline
\end{tabular}
}
\caption{Comparison of CPU time and number of iterations for SDDP, DADP and PADP}
\label{tab:district:numeric:optres}
\end{table}
For a small-scale problem like \textrm{3-nodes} (second column
of Table~\ref{tab:district:numeric:optres}), SDDP is faster
than DADP and PADP. However, for the 48-nodes problem (last
column of Table~\ref{tab:district:numeric:optres}),
\emph{DADP and PADP} are \emph{more than three times faster}
than SDDP.
Figure~\ref{fig:nodal:cputime} depicts how much CPU
time take the different algorithms with respect to the state dimension.
For this case study, we observe
that the \emph{CPU time grows almost linearly} \boldsymbol{W}rt\ the dimension of the state
for DADP and PADP, whereas it grows exponentially for SDDP.
Otherwise stated, decomposition methods scale better than SDDP
in terms of CPU time for large microgrids instances.
\begin{figure}
\caption{CPU time for the three algorithms as a function
of the state dimension}
\label{fig:nodal:cputime}
\end{figure}
\subsubsection{Quality of the Theoretical Bounds}
\label{Quality_of_the_theoretical_bounds}
In Table~\ref{tab:district:numeric:upperlower},
we give the lower and upper bounds (of the optimal
cost~$V_0(x_0)$ of the global optimization problem)
achieved by the three algorithms (SDDP, DADP, PADP).
We recall that SDDP returns a lower bound of the optimal
cost~$V_0(x_0)$, both by nature and also because we used
a suitable resampling of the global uncertainty distribution
instead of the original distribution itself (see the discussion
in~\S\ref{ssec:nodalalgorithms}).
DADP and PADP lower and upper bounds are given by
Equation~\eqref{eq:nodal:relaxedconstraintdual}
and Equation~\eqref{eq:nodal:overconstraint} respectively.
In Table~\ref{tab:district:numeric:upperlower}, we observe that
\begin{itemize}
\item SDDP's and DADP's lower bounds are close to each other,
\item for problems with more than 12 nodes, DADP's lower
bound is up to 2.6\% better than SDDP's lower bound,
\item the gap between PADP's upper bound and
the two lower bounds is rather large.
\end{itemize}
\begin{table}[!ht]
\centering
{\normalsize
\begin{tabular}{|l|ccccc|}
\hline
Problem & \textrm{3-nodes} & \textrm{6-nodes} & \textrm{12-nodes} & \textrm{24-nodes} & \textrm{48-nodes} \\
\hline
\hline
SDDP LB & 225.2 & 455.9 & 889.7 & 1752.8 & 3310.3 \\
\hline
DADP LB & 213.7 & 447.3 & 896.7 & 1787.0 & 3396.4 \\
\hline
PADP UB & 252.1 & 528.5 & 1052.3 & 2100.7 & 4016.6 \\
\hline
\end{tabular}
}
\caption{Upper and lower bounds (of the optimal
cost~$V_0(x_0)$ of the global optimization problem) given by SDDP, DADP and PADP}
\label{tab:district:numeric:upperlower}
\end{table}
To sum up, DADP achieves a slightly better lower bound than SDDP,
with much less CPU time (and a parallel version of DADP would
give even better performance in terms of CPU time).
\subsubsection{Policy Simulation Performances}
\label{Policy_simulation_results}
In Table~\ref{tab:district:numeric:simulation},
we give the performances of the policies yielded by
the three algorithms.
The SDDP, DADP and PADP values are obtained by Monte Carlo simulation of the
corresponding policies on $5,000$ scenarios. The notation
$\pm$ corresponds to the 95\% confidence interval for the
numerical evaluation of the expected costs. We use
the value obtained by the SDDP policy as a reference,
a positive gap meaning that the corresponding policy
makes better than the SDDP policy.
All these values are \emph{statistical} upper bounds of the optimal
cost~$V_0(x_0)$ of the global optimization problem.
\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{|l|ccccc|}
\hline
Network & \textrm{3-nodes} & \textrm{6-nodes} & \textrm{12-nodes} &
\textrm{24-nodes} & \textrm{48-nodes} \\
\hline
\hline
SDDP value & 226 $\pm$ 0.6 & 471 $\pm$ 0.8 & 936 $\pm$ 1.1 & 1859 $\pm$ 1.6 & 3550 $\pm$ 2.3 \\
\hline
\hline
DADP value & 228 $\pm$ 0.6 & 464 $\pm$ 0.8 & 923 $\pm$ 1.2 & 1839 $\pm$ 1.6 & 3490 $\pm$ 2.3 \\
Gap & - 0.8 \% & + 1.5 \% & +1.4\% & +1.1\% & +1.7\% \\
\hline
\hline
PADP value & 229 $\pm$ 0.6 & 471 $\pm$ 0.8 & 931 $\pm$ 1.1 & 1856 $\pm$ 1.6 & 3508 $\pm$ 2.2 \\
Gap & -1.3\% & 0.0\% & +0.5\% & +0.2\% & +1.2\% \\
\hline
\end{tabular}
}
\caption{Simulation costs (Monte Carlo) for policies induced by
SDDP, DADP and PADP}
\label{tab:district:numeric:simulation}
\end{table}
We make the following observations:
\begin{itemize}
\item
for problems with more than 6 nodes,
both the DADP policy and the PADP policy beat
the SDDP policy,
\item
the DADP policy gives better results than the PADP policy,
\item
comparing with the last line of Table
\ref{tab:district:numeric:upperlower}, the statistical
upper bounds
are much closer to SDDP and DADP lower bounds than PADP's
exact upper bound.
\end{itemize}
For this last observation, our interpretation is as follows:
the PADP algorithm is penalized because, as the resource coordination
process is deterministic, it imposes constant
importation flows for every possible realization of
the uncertainties (see also the interpretation of PADP
in the case of a decentralized information structure
in~\S\ref{subsec:nodal:decentralizedinformation}).
\section{Conclusions}
We have considered multistage stochastic optimization problems
involving multiple units coupled by spatial static constraints.
We have presented a formalism for joint
temporal and spatial decomposition.
We have provided two fully parallelizable algorithms
that yield theoretical bounds, value functions
and admissible policies.
We have stressed the key role played by information structures in the
performance of the decomposition schemes.
We have tested these algorithms on the management of
several district microgrids. Numerical results have showed the effectiveness
of the approach: the price decomposition algorithm beats
the reference SDDP algorithm for large-scale problems with
more than 12~nodes, both in terms of theoretical bounds and
policy performance, and in terms of computation time. On problems
with up to 48~nodes (corresponding to 64~state variables), we have
observed that their performance scales well as the dimension of the state
grew: SDDP is affected by the well-known curse of dimensionality,
whereas decomposition-based methods are not.
Possible extensions are the following.
In~\S\ref{subsec:nodal:processdesign} and
in~\S\ref{subsec:nodal:admissiblepolicy},
we have presented a serial version of the decomposition algorithms,
but we believe that leveraging their parallel nature could decrease
further their computation time.
In~\S\ref{subsec:nodal:decomposedDPdeterministic},
we have only considered deterministic price and resource
coordination processes. Using larger search sets
for the coordination variables, e.g. considering
Markovian coordination processes, would make it
possible to improve the performance of the algorithms
(see \cite[Chap.~7]{thesepacaud} for further details).
However, one would need to analyze how to obtain a good
trade-off between accuracy and numerical performance.
\end{document}
|
\begin{document}
\title{Binomial transforms of the modified $k$-Fibonacci-like sequence}
\date{}
\author{Youngwoo Kwon\\
Department of mathematics, Korea University, Seoul, Republic of Korea\\
\href{mailto:[email protected]}{\tt [email protected]}\\
}
\maketitle
\begin{abstract}
This study applies the binomial, $k$-binomial, rising $k$-binomial and falling $k$-binomial transforms to the modified $k$-Fibonacci-like sequence. Also, the Binet formulas and generating functions of the above mentioned four transforms are newly found by the recurrence relations.
\end{abstract}
\section{Introduction}
The Fibonacci sequence $\left(F_{n}\right)_{n\geq0}$ is defined by the recurrence relation
\begin{align*}
F_{n+1}=&F_{n}+F_{n-1} \text{~for~} n\ge1
\end{align*}
with the initial conditions $F_{0}=0$ and $F_{1}=1$.
Many authors have studied the Fibonacci sequence, some of whom introduced new sequences related to it as well as proving many identities for them.
In particular, Falc$\acute{\rm{o}}$n and Plaza \cite{FP02} introduced the $k$-Fibonacci sequence.
\begin{definition}[\cite{FP02}]
For any positive real number $k$, the $k$-Fibonacci sequence $\left( F_{k,n} \right)_{n\geq0}$ is defined by recurrence relation
$$F_{k,n+1} = k F_{k,n} + F_{k,n-1} ~\text{for}~n\ge1$$
with the initial conditions $F_{k,0} =0$ and $F_{k,1} =1$.
\end{definition}
Also, Kwon \cite{YK} introduced the modified $k$-Fibonacci-like sequence.
\begin{definition}[\cite{YK}]
For any positive real number $k$, the modified $k$-Fibonacci-like sequence $\left(M_{k,n}\right)_{n\geq0}$ is defined by the recurrence relation
$$M_{k,n+1} = k M_{k,n} + M_{k,n-1} ~\text{for}~n\ge1 $$
with the initial conditions $M_{k,0} = M_{k,1} = 2$.
\end{definition}
The first few modified $k$-Fibonacci-like numbers are as follows:
\begin{align*}
M_{k,2}=& 2k+2,\\
M_{k,3}=& 2k^{2}+2k+2,\\
M_{k,4}=& 2k^{3}+2k^{2}+4k+2,\\
M_{k,5}=& 2k^{4}+2k^{3} +6k^{2}+4k+2.
\end{align*}
Kwon \cite{YK} studied the following identities between the $k$-Fibonacci sequence and the modified $k$-Fibonacci-like sequence.
$$M_{k,n} = 2 \left( F_{k,n} + F_{k,n-1}\right) \text{ and } F_{k,n} = \frac{1}{2}\sum_{i=0}^{n-1} M_{k,n-i}(-1)^{i}$$
Spivey and Steil \cite{SS} introduced various binomial transforms.
\begin{enumerate}
\item[(1)] The binomial transform $B$ of the integer sequence $A=\left\{a_{0}, a_1 , a_2 , \ldots \right\}$, which is denoted by $B(A)=\left\{b_{n}\right\}$ and defined by
$$b_{n} = \sum_{i=0}^{n}\binom{n}{i} a_{i}.$$
\item[(2)] The $k$-binomial transform $W$ of the integer sequence $A=\left\{a_{0}, a_1 , a_2 , \ldots \right\}$, which is denoted by $W(A)=\left\{w_{n}\right\}$ and defined by
$$w_{n} = \sum_{i=0}^{n}\binom{n}{i} k^{n} a_{i}.$$
\item[(3)] The rising $k$-binomial transform $R$ of the integer sequence $A=\left\{a_{0}, a_1 , a_2 , \ldots \right\}$, which is denoted by $B(A)=\left\{r_{n}\right\}$ and defined by
$$r_{n} = \sum_{i=0}^{n}\binom{n}{i} k^{i} a_{i} .$$
\item[(4)] The falling $k$-binomial transform $F$ of the integer sequence $A=\left\{a_{0}, a_1 , a_2 , \ldots \right\}$, which is denoted by $F(A)=\left\{f_{n}\right\}$ and defined by
$$f_{n} = \sum_{i=0}^{n}\binom{n}{i} k^{n-i} a_{i} .$$
\end{enumerate}
Other latest research \cite{BJS, FP03, YT02} also examined the various binomial transforms for several special sequences. These transforms are interesting and meaningful as they introduced several new approaches.
Based on those preceding studies, this study applies the four binomial transforms namely, binomial, $k$-binomial, rising $k$-binomial and falling $k$-binomial transforms to the modified $k$-Fibonacci-like sequence. This study also proves their properties.
\section{The binomial transform of the modified \texorpdfstring{$k$}{Lg}-Fibonacci-like sequence}
The binomial transform of the modified $k$-Fibonacci-like sequence $\left(M_{k,n}\right)_{n\geq0}$ is denoted by $B_{k}=\left(b_{k,n}\right)_{n \geq0}$ where
$$b_{k,n} = \sum_{i=0}^{n}\binom{n}{i}M_{k,i}.$$
The only binomial transforms of the modified $k$-Fibonacci-like sequences indexed in OEIS \cite{S} are as follows:
\begin{align*}
B_{1} =&\left\{2, 4, 10, 26, 68, 178, \ldots \right\} : A052995-\{0\} \text{ or } A055819-\{1\}\\
B_{2} =&\left\{2, 4, 12, 40, 136, 464, \ldots \right\} : A056236\\
B_{3} =&\left\{2, 4, 14, 58, 248, 1066, \ldots \right\} \\
B_{4} =&\left\{2, 4, 16, 80, 416, 2176, \ldots \right\} \\
B_{5} =&\left\{2, 4, 18, 106, 652, 4034, \ldots \right\}
\end{align*}
\begin{lemma}\label{binomial_T}
The binomial transform of the modified $k$-Fibonacci-like sequence satisfies the relation
$$b_{k,n+1} - b_{k,n} = \sum_{i=0}^{n}\binom{n}{i} M_{k,i+1}.$$
\end{lemma}
\begin{proof}
Note that $\binom{n}{0}=1$ and $\binom{n+1}{i} = \binom{n}{i}+\binom{n}{i-1}$.
The difference of the two consecutive binomial transforms is the following:
\begin{align*}
b_{k,n+1} -b_{k,n}&= \sum_{i=0}^{n+1} \binom{n+1}{i} M_{k,i}-\sum_{i=0}^{n}\binom{n}{i}M_{k,i}\\
&=\sum_{i=1}^{n}\left[\binom{n+1}{i}-\binom{n}{i} \right]M_{k,i} + M_{k,n+1} \\
&=\sum_{i=1}^{n}\binom{n}{i-1}M_{k,i}+M_{k,n+1}\\
&=\sum_{i=0}^{n-1}\binom{n}{i}M_{k,i+1} + \binom{n}{n}M_{k,n+1}=\sum_{i=0}^{n}\binom{n}{i}M_{k,i+1}
\end{align*}
\end{proof}
Note that $b_{k,n+1} = \sum_{i=0}^{n}\binom{n}{i}\left(M_{k,i}+M_{k,i+1}\right)$.
\begin{theorem}\label{binomial_Ta}
The binomial transform of the modified $k$-Fibonacci-like sequence $B_{k}=\left(b_{k,n}\right)_{n\geq0}$ satisfies the recurrence relation
$$b_{k,n+1} = (k+2)b_{k,n} - k b_{k,n-1} ~\text{for}~n\ge1$$
with the initial conditions $b_{k,0}=2$, $b_{k,1} = 4$.
\end{theorem}
\begin{proof}
By Lemma \ref{binomial_T}, since $b_{k,n+1} = \sum_{i=0}^{n}\binom{n}{i}\left(M_{k,i}+M_{k,i+1}\right)$, then we have
\begin{align*}
b_{k,n+1}=&M_{k,0} +M_{k,1} + \sum_{i=1}^{n}\binom{n}{i}\left(M_{k,i} + M_{k,i+1}\right)\\
=&M_{k,0}+M_{k,1}+\sum_{i=1}^{n}\binom{n}{i}\left(M_{k,i}+kM_{k,i}+M_{k,i-1}\right)\\
=&\left[(k+1)M_{k,0}+(k+1)\sum_{i=1}^{n}\binom{n}{i}M_{k,i}\right]\\
&+\sum_{i=1}^{n}\binom{n}{i}M_{k,i-1} + M_{k,1}-kM_{k,0}\\
=&(k+1)\sum_{i=0}^{n}\binom{n}{i}M_{k,i}+\sum_{i=1}^{n}\binom{n}{i}M_{k,i-1}+M_{k,1}-kM_{k,0}\\
=&(k+1)b_{k,n}+\sum_{i=1}^{n}\binom{n}{i}M_{k,i-1} +2-2k.
\end{align*}
On the other hand, in the case of $\binom{n-1}{n}=0$, we can obtain the following:
\begin{align*}
b_{k,n} &=kb_{k,n-1} + \sum_{i=1}^{n}\binom{n}{i}M_{k,i-1}+2-2k.
\end{align*}
Based on the above two identities, this study draws the below formulas.
$$b_{k,n+1} - (k+1)b_{k,n} = b_{k,n} - k b_{k,n-1},$$
and so
$$b_{k,n+1} = (k+2)b_{k,n} - k b_{k,n-1}.$$
\end{proof}
Binet's formulas are well known in the Fibonacci number theory. In this study, Binet's formula for the binomial transform of the modified $k$-Fibonacci-like sequence is suggested as the following:
\begin{theorem}\label{binet_T}
Binet's formula for the binomial transform of the modified $k$-Fibonacci-like sequence is given by
$$b_{k,n}= 4\frac{r_{1}^{n}-r_{2}^{n}}{r_{1}-r_{2}}- 2k\frac{r_{1}^{n-1}-r_{2}^{n-1}}{r_{1}-r_{2}},$$
where $r_{1}$ and $r_{2}$ are the roots of the characteristic equation $x^{2}-(k+2)x+k=0$, and $r_{1}>r_{2}$.
\end{theorem}
\begin{proof}
The characteristic polynomial equation of $b_{k,n+1} = (k+2)b_{k,n} - k b_{k,n-1}$ is $x^{2} - (k+2)x + k =0$, whose solution are $r_{1}$ and $r_{2}$ with $r_{1}>r_{2}$. The general term of the binomial transform may be expressed in the form, $b_{k,n}=C_{1}r_{1}^{n} + C_{2}r_{2}^{n}$ for some coefficients $C_{1}$ and $C_{2}$.
\begin{enumerate}
\item[(1)] $b_{k,0} = C_{1} + C_{2} = 2$
\item[(2)] $b_{k,1}=C_{1}r_{1} + C_{2}r_{2} = 4$
\end{enumerate}
Then
$$C_{1} = \frac{4-2r_{2}}{r_{1}-r_{2}} \text{ and } C_{2}=\frac{2r_{1} -4}{r_{1}-r_{2}}.$$
Therefore,
$$b_{k,n} = \frac{4-2r_{2}}{r_{1}-r_{2}} r_{1}^{n} + \frac{2r_{1} -4}{r_{1}-r_{2}} r_{2}^{n} =4 \frac{r_{1}^{n}-r_{2}^{n}}{r_{1}-r_{2}} -2k\frac{r_{1}^{n-1}-r_{2}^{n-1}}{r_{1}-r_{2}}. $$
\end{proof}
The binomial transform $B_{k}$ can be seen as the coefficients of the power series which is called the generating function. Therefore, if $b_{k}(x)$ is the generating function, then we can write
$$b_{k}(x)=\sum_{i=0}^{\infty} b_{k,i}x^{i} = b_{k,0}+b_{k,1}x+b_{k,2}x^{2}+\cdots.$$
And then,
\begin{align*}
(k+2)x b_{k}(x)=&(k+2)b_{k,0}x+(k+2)b_{k,1}x^2 + (k+2)b_{k,2}x^{3}+\cdots,\\
kx^{2}b_{k}(x)=&k b_{k,0}x^{2} + k b_{k,1}x^{3} + k b_{k,2} x^{4} + \cdots.
\end{align*}
Since $b_{k,n+1} - (k+2)b_{k,n} + k b_{k,n-1} = 0$, $b_{k,0}=2$, and $b_{k,1}=4$, then we have
\begin{align*}
&(1-(k+2)x+kx^{2}) b_{k} (x)\\
=& b_{k,0} + (b_{k,1} - (k+2)b_{k,0})x + (b_{k,2} - (k+2)b_{k,1} + kb_{k,0} )x^{2} + \cdots\\
=& b_{k,0} + (b_{k,1}-(k+2)b_{k,0})x\\
=&2+(4-(k+2)2)x = 2-2kx.
\end{align*}
Hence, the generating function for the binomial transform of the modified $k$-Fibonacci-like sequence $\left(b_{k,n}\right)_{n\geq0}$ is
$$b_{k}(x) = \frac{2(1-2kx)}{1-(k+2)x+kx^{2}}.$$
\section{The \texorpdfstring{$k$}{Lg}-binomial transform of the modified \texorpdfstring{$k$}{Lg}-Fibonacci-like sequence }
The $k$-binomial transform of the modified $k$-Fibonacci-like sequence $\left(M_{k,n}\right)_{n\geq0}$ is denoted by $W_{k}=\left(w_{k,n}\right)_{n\geq0}$ where
\begin{displaymath}
w_{k,n} =
\begin{cases}
\sum_{i=0}^{n}\binom{n}{i} k^{n}M_{k,i}, & \text{ for } k\ne 0 \text{ or } n\ne 0; \\
0, & \text{ if } k=0 \text{ and } n=0.
\end{cases}
\end{displaymath}
The first $k$-binomial transforms are as follows:
\begin{align*}
W_{1}=&\left\{2, 4, 10, 26, 68, 178, \ldots \right\} : A052995-\{0\} \text{ or } A055819-\{1\}\\
W_{2}=&\left\{2, 8, 96, 320, 1088, 3712, \ldots \right\}\\
W_{3}=&\left\{2, 12, 378, 1566, 6696, 28782, \ldots \right\}\\
W_{4}=&\left\{2, 16, 1024, 5120, 26624, \ldots \right\}\\
W_{5}=&\left\{2, 20, 2250, 13250, 81500, \ldots \right\}
\end{align*}
Note that the $1$-binomial transform $W_{1}$ coincides with the binomial transform $B_{1}$.
Note that
$$w_{k,n} = \sum_{i=0}^{n}\binom{n}{i}k^{n} M_{k,i} = k^{n} \sum_{i=0}^{n}\binom{n}{i} M_{k,i}= k^{n}b_{k,n},$$
$$\text{and so } w_{k,n+1} = k^{n+1}\sum_{i=0}^{n}\binom{n}{i}\left(M_{k,i} + M_{k,i+1}\right)$$
from Lemma \ref{binomial_T}
\begin{theorem}
The $k$-binomial transform of the modified $k$-Fibonacci-like sequence $W_{k}=\left(w_{k,n}\right)_{n\geq0}$ satisfies the recurrence relation
$$w_{k,n+1} =k(k+2)w_{k,n} - k^{3} w_{k,n-1} ~\text{for}~n\ge1$$
with the initial conditions $w_{k,0} = 2$, $w_{k,1} = 4k$.
\end{theorem}
\begin{proof}
By Theorem \ref{binomial_Ta}, we can easily obtain the following:
\begin{align*}
w_{k,n+1}&=k^{n+1} b_{k,n+1}\\
&=k^{n+1} \left[ (k+2)b_{k,n}-k b_{k,n-1}\right]\\
&=k^{n+1}(k+2)b_{k,n} - k^{n+2} b_{k,n-1}\\
&=k(k+2)w_{k,n} - k^{3} w_{k,n-1}
\end{align*}
\end{proof}
Similarly, Binet's formula for the $k$-binomial transform of the modified $k$-Fibonacci-like sequence is the following:
\begin{theorem}
Binet's formula for the $k$-binomial transform of the modified $k$-Fibonacci-like sequence is given by
$$w_{k,n}= 4\frac{s_{1}^{n}-s_{2}^{n}}{s_{1}-s_{2}}- 2k\frac{s_{1}^{n-1}-s_{2}^{n-1}}{s_{1}-s_{2}},$$
where $s_{1}$ and $s_{2}$ are the roots of the characteristic equation $x^{2}-k(k+2)x+k^{3}=0$, and $s_{1}>s_{2}$.
\end{theorem}
\begin{proof}
The proof is same as that of the binomial transform, which is in Theorem \ref{binet_T}.
\end{proof}
Similarly, the generating function for the $k$-binomial transform of the modified $k$-Fibonacci-like sequence is
$$w_{k}(x) = \frac{2(1-k^{2}x)}{1-k(k+2)x+k^{3}x^{2}}.$$
\section{The rising \texorpdfstring{$k$}{Lg}-binomial transform of the modified \texorpdfstring{$k$}{Lg}-Fibonacci-like sequence }
The rising $k$-binomial transform of the modified $k$-Fibonacci-like sequence $\left(M_{k,n}\right)_{n\geq0}$ is denoted by $R_{k}=\left(r_{k,n}\right)_{n\geq0}$ where
\begin{displaymath}
r_{k,n} = \begin{cases}
\sum_{i=0}^{n}\binom{n}{i} k^{i}M_{k,i}, & \text{ for } k\ne 0 \text{ or } n\ne 0; \\
0, & \text{ if } k=0 \text{ and } n=0.
\end{cases}
\end{displaymath}
The first rising $k$-binomial transforms are as follows:
\begin{align*}
R_{1}=&\left\{2, 4, 10, 26, 68, 178, \ldots \right\} : A052995-\{0\} \text{ or } A055819-\{1\}\\
R_{2}=&\left\{2, 6, 34, 198, 1154, 6726, \ldots \right\}\\
R_{3}=&\left\{2, 8, 86, 938, 10232, \ldots \right\}\\
R_{4}=&\left\{2, 10, 178, 3194, 57314, \ldots \right\}\\
R_{5}=&\left\{2, 12, 322, 8682, 234092, \ldots \right\}
\end{align*}
\begin{lemma}\label{rbinomial_T}
For any integer $n\ge0$ and $k\ne0$,
$$r_{k,n} = \sum_{i=0}^{n}\binom{n}{i}k^{i} M_{k,i} = M_{k,2n}.$$
\end{lemma}
\begin{proof}
This identity coincides with Theorem 4.10 in \cite{YK}.
\end{proof}
\begin{theorem}\label{rbinomial_Ta}
The rising $k$-binomial transform of the modified $k$-Fibonacci-like sequence $R_{k}=\left(r_{k,n}\right)_{n\geq0}$ satisfies the recurrence relation
$$r_{k,n+1} = (k^{2}+2)r_{k,n} - r_{k,n-1} ~\text{for}~n\ge1$$
with the initial conditions $r_{k,0} = 2$, $r_{k,1} = 2k+2$.
\end{theorem}
\begin{proof}
From the definition of the modified $k$-Fibonacci-like sequence, we obtain
\begin{align*}
M_{k,2n+2}=&kM_{k,2n+1}+M_{k,2n}\\
=&k\left(kM_{k,2n}+M_{k,2n-1}\right)+M_{k,2n}\\
=&(k^{2}+1)M_{k,2n}+kM_{k,2n-1}\\
=&(k^{2}+1)M_{k,2n}+M_{k,2n}-M_{k,2n-2}\\
=&(k^{2}+2)M_{k,2n}-M_{k,2n-2}.
\end{align*}
By Lemma \ref{rbinomial_T}, since $r_{k,n}=M_{k,2n}$, then we have
$$r_{k,n+1}=(k^{2}+2)r_{k,n}-r_{k,n-1}.$$
\end{proof}
Similarly, Binet's formula for the rising $k$-binomial transform of the modified $k$-Fibonacci-like sequence is the following:
\begin{theorem}
Binet's formula for the rising $k$-binomial transform of the modified $k$-Fibonacci-like sequence is given by
$$r_{k,n}= (2k+2)\frac{t_{1}^{n}-t_{2}^{n}}{t_{1}-t_{2}}- 2\frac{t_{1}^{n-1}-t_{2}^{n-1}}{t_{1}-t_{2}},$$
where $t_{1}$ and $t_{2}$ are the roots of the characteristic equation $x^{2}-(k^{2}+2)x+1=0$, and $t_{1}>t_{2}$.
\end{theorem}
\begin{proof}
The proof is same as that of the binomial transform, which is in Theorem \ref{binet_T}.
\end{proof}
Similarly, the generating function for the rising $k$-binomial transform of the modified $k$-Fibonacci-like sequence is
$$
r_{k}(x) = \frac{2-(2k^2 - 2k +2)x}{1-(k^{2}+2)x+x^{2}}.
$$
\section{The falling \texorpdfstring{$k$}{Lg}-binomial transform of the modified \texorpdfstring{$k$}{Lg}-Fibonacci-like sequence }
The falling $k$-binomial transform of the modified $k$-Fibonacci-like sequence $\left(M_{k,n}\right)_{n\geq0}$ is denoted by $F_{k}=\left(f_{k,n}\right)_{n\geq0}$ where
\begin{displaymath}
f_{k,n} =\begin{cases}
\sum_{i=0}^{n}\binom{n}{i} k^{n-i}M_{k,i}, & \text{ for } k\ne 0 \text{ or } n\ne 0; \\
0, & \text{ if } k=0 \text{ and } n=0.
\end{cases}
\end{displaymath}
The first falling $k$-binomial transforms are as follows:
\begin{align*}
F_{1}=&\left\{2, 4, 10, 26, 68, 178, \ldots \right\} : A052995-\{0\} \text{ or } A055819-\{1\}\\
F_{2}=&\left\{2, 6, 22, 90, 386, 1686, \ldots \right\}\\
F_{3}=&\left\{2, 8, 38, 206, 1208, 7370, \ldots \right\}\\
F_{4}=&\left\{2, 10, 58, 386, 2834, 22042 \ldots \right\}\\
F_{5}=&\left\{2, 12, 82, 642, 5612, 52722 \ldots \right\}
\end{align*}
\begin{lemma}\label{fbinomial_T}
The falling $k$-binomial transform of the modified $k$-Fibonacci-like sequence satisfies the relation
$$f_{k,n+1}-kf_{k,n} = \sum_{i=0}^{n}\binom{n}{i} k^{n-i}M_{k,i+1}.$$
\end{lemma}
\begin{proof}
The proof is similar to the proof of Lemma \ref{binomial_T}. And, we obtain
\begin{align*}
f_{k,n+1} - kf_{k,n}&= \sum_{i=0}^{n+1} \binom{n+1}{i}k^{n+1-i} M_{k,i}-\sum_{i=0}^{n}\binom{n}{i} k^{n+1-i}M_{k,i}\\
&=\sum_{i=1}^{n}\left[\binom{n+1}{i}-\binom{n}{i} \right]k^{n+1-i}M_{k,i} + M_{k,n+1} \\
&=\sum_{i=1}^{n}\binom{n}{i-1}k^{n+1-i}M_{k,i}+ M_{k,n+1}\\
&=\sum_{i=0}^{n-1}\binom{n}{i}k^{n-i}M_{k,i+1} + \binom{n}{n}M_{k,n+1}=\sum_{i=0}^{n}\binom{n}{i}k^{n-i}M_{k,i+1}.
\end{align*}
\end{proof}
Note that $f_{k,n+1} = \sum_{i=0}^{n}\binom{n}{i}\left( k^{n+1-i}M_{k,i}+ k^{n-i}M_{k,i+1}\right)$.
\begin{theorem}\label{fbinomial_Ta}
The falling $k$-binomial transform of the modified $k$-Fibonacci-like sequence $F_{k}=\left(f_{k,n}\right)_{n\geq0}$ satisfies the recurrence relation
$$f_{k,n+1} = 3kf_{k,n} - (2k^{2}-1)f_{k,n-1} ~\text{for}~n\ge1$$
with the initial conditions $f_{k,0} = 2$, $f_{k,1} = 2k+2$.
\end{theorem}
\begin{proof}
By Lemma \ref{fbinomial_T}, since $f_{k,n+1} = \sum_{i=0}^{n}\binom{n}{i}\left(k^{n+1-i}M_{k,i}+k^{n-i}M_{k,i+1}\right)$, then we have
\begin{align*}
f_{k,n+1}=&\sum_{i=0}^{n}\binom{n}{i}k^{n-i}\left(k M_{k,i}+M_{k,i+1}\right)\\
=&\sum_{i=1}^{n}\binom{n}{i}k^{n-i}\left(2kM_{k,i}+M_{k,i-1}\right)+k^{n}\left(k M_{k,0}+M_{k,1}\right)\\
=&2k\sum_{i=1}^{n}\binom{n}{i}k^{n-i}M_{k,i} +\sum_{i=1}^{n}\binom{n}{i}k^{n-i}M_{k,i-1} + k^{n}\left(kM_{k,0}+M_{k,1}\right)\\
=&2k\sum_{i=0}^{n}\binom{n}{i}k^{n-i}M_{k,i} +\sum_{i=1}^{n}\binom{n}{i}k^{n-i}M_{k,i-1}\\
& + k^{n}\left(kM_{k,0}+M_{k,1}-2kM_{k,0}\right)\\
=&2kf_{k,n} +\sum_{i=1}^{n}\binom{n}{i}k^{n-i}M_{k,i-1} + k^{n}\left(M_{k,1}-kM_{k,0}\right).
\end{align*}
On the other hand, in the case of $\binom{n-1}{n}=0$, we can obtain the following:
\begin{align*}
kf_{k,n} =&2k^2f_{k,n-1}+\sum_{i=1}^{n-1}\binom{n-1}{i}k^{n-i}M_{k,i-1}+k^{n}\left(M_{k,1}-kM_{k,0}\right)\\
=&2k^2f_{k,n-1}-\left[f_{k,n-1}-\sum_{i=0}^{n-1}\binom{n-1}{i}k^{n-1-i}M_{k,i}\right]\\
&+\sum_{i=0}^{n-2}\binom{n-1}{i+1}k^{n-1-i}M_{k,i}+k^{n}\left(M_{k,1}-kM_{k,0}\right)\\
=&\left(2k^{2}-1\right)f_{k,n-1}+\sum_{i=0}^{n-1}\left[\binom{n-1}{i}+\binom{n-1}{i+1}\right]k^{n-1-i}M_{k,i}\\
&+k^{n}\left(M_{k,1}-kM_{k,0}\right)\\
=&\left(2k^{2}-1\right)f_{k,n-1}+\sum_{i=0}^{n-1}\binom{n}{i+1}k^{n-1-i}M_{k,i}+k^{n}\left(M_{k,1}-kM_{k,0}\right)\\
=&\left(2k^{2}-1\right)f_{k,n-1}+\sum_{i=1}^{n}\binom{n}{i}k^{n-i}M_{k,i-1}+k^{n}\left(M_{k,1}-kM_{k,0}\right).
\end{align*}
Based on the above two identities, this study draws the below formulas.
$$f_{k,n+1}-2kf_{k,n}=kf_{k,n}-\left(2k^{2}-1\right)f_{k,n-1},$$
and so
$$f_{k,n+1} = 3kf_{k,n} - (2k^{2}-1) f_{k,n-1}.$$
\end{proof}
Similarly, Binet's formula for the falling $k$-binomial transform of the modified $k$-Fibonacci-like sequence is the following:
\begin{theorem}
Binet's formula for the falling $k$-binomial transform of the modified $k$-Fibonacci-like sequence is given by
$$f_{k,n}= (2k+2)\frac{u_{1}^{n}-u_{2}^{n}}{u_{1}-u_{2}}- 2\frac{u_{1}^{n-1}-u_{2}^{n-1}}{u_{1}-u_{2}},$$
where $u_{1}$ and $u_{2}$ are the roots of the characteristic equation $x^{2}-3kx+(2k^{2}-1)=0$, and $u_{1}>u_{2}$.
\end{theorem}
\begin{proof}
The proof is same as that of the binomial transform, which is in Theorem \ref{binet_T}.
\end{proof}
Similarly, the generating function for the falling $k$-binomial transform of the modified $k$-Fibonacci-like sequence is
$$
f_{k}(x) = \frac{2+(2-4k)x}{1-3kx+(2k^{2}-1)x^{2}}.
$$
\section{Conclusion}
This paper applies the four transforms- the binomial, $k$-binomial, rising $k$-binomial and falling $k$-binomial transforms- to the modified $k$-Fibonacci-like sequence. Although most of the results are rather similar to those of the previous sequences, this study is still meaningful as they introduce several new approaches and methods to derive the formulas. This study, furthermore, examines Binet's formulas and generating functions of the four transforms.
\end{document}
|
\begin{document}
\title{A note on counting flows in signed graphs}
\begin{abstract}
Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that
for every abelian group $\Gamma$ of order $n$, the number of nowhere-zero $\Gamma$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected
orientations), the situation is more subtle. For a finite group~$\Gamma$, let $\epsilon_2(\Gamma)$ be the largest integer $d$ so that $\Gamma$ has a
subgroup isomorphic to~$\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the
number of nowhere-zero $\Gamma$-flows in $G$ for every abelian group~$\Gamma$ with $\epsilon_2(\Gamma) = d$ and $|\Gamma| = 2^d n$. Beck and
Zaslavsky~\cite{BZ06} had previously established the special case of this result when $d=0$ (i.e., when $\Gamma$ has odd order).
\end{abstract}
\section{Introduction}
Throughout the paper we permit graphs to have both multiple edges and loops. Let $G$ be a graph equipped with an orientation of its edges and let $\Gamma$ be an abelian
group written additively. We say that a function $\phi : E(G) \rightarrow \Gamma$ is a $\Gamma$-\emph{flow} if it satisfies the following equation (Kirchhoff's law) for
every vertex $v \in V(G)$.
\[ \sum_{e \in \delta^+(v) } \phi(e) - \sum_{e \in \delta^-(v)} \phi(e) = 0, \]
where $\delta^+(v)$ ($\delta^-(v))$ denote the set of edges directed away from (toward) the vertex $v$.
We say that $\phi$ is \emph{nowhere-zero} if $0 \not\in \phi(E(G))$.
If $\phi$ is a $\Gamma$-flow and we switch the direction of an edge $e$ of $G$, we may obtain a
new flow by replacing~$\phi(e)$ by its additive inverse. Note that this does not affect the property of being nowhere-zero
. So, in particular, whenever some
orientation of $G$ has a nowhere-zero $\Gamma$-flow, the same will be true for every orientation. More generally, the number of nowhere-zero $\Gamma$-flows
in two different orientations of $G$ will always be equal, and we denote this important quantity by $\Phi(G,\Gamma)$.
Tutte~\cite{Tutte54} introduced the concept of a nowhere-zero $\Gamma$-flow and proved the following key theorem about counting them.
\begin{theorem}[Tutte~\cite{Tutte54}]
\label{tutte}
Let $G$ be a graph.
\begin{enumerate}
\item If $\Gamma $and $\Gamma'$ are abelian groups with $|\Gamma| = |\Gamma'|$, then $\Phi(G,\Gamma) = \Phi(G,\Gamma')$.
\item There exists a polynomial $f$ so that $\Phi(G,\Gamma) = f(n)$ for every abelian group~$\Gamma$ with $|\Gamma| = n$.
\end{enumerate}
\end{theorem}
Our interest in this paper is in counting nowhere-zero $\Gamma$-flows in signed graphs, so we proceed with an introduction to this setting. A \emph{signature} of a graph $G$ is a function $\sigma :
E(G) \rightarrow \{-1,1\}$. We say that a subgraph $H$ is \emph{positive} if $\prod_{e \in E(H)} \sigma(e) = 1$ and \emph{negative} if this product is $-1$, in particular we call
an edge $e$ \emph{positive} (\emph{negative}) if the graph $e$ induces is positive (negative). We say that two signatures $\sigma$ and $\sigma'$ are \emph{equivalent} if
the symmetric difference of the negative edges of $\sigma$ and the negative edges of $\sigma'$ is an edge-cut of $G$. Let us note that two signatures are equivalent if
and only if they give rise to the same set of negative cycles; this instructive exercise was observed by Zaslavsky~\cite{Zaslavsky}. Observe that if $\sigma$ is a signature and $C$ is an
edge-cut of $G$, then we may form a new signature $\sigma'$ equivalent to $\sigma$ by the following rule:
\[ \sigma'(e) = \left\{ \begin{array}{cl}
\sigma(e) & \mbox{if $e \not\in C$} \\
- \sigma(e) & \mbox{if $e \in C$.}
\end{array} \right. \]
So, in particular, for any signature $\sigma$ and a non-loop edge $e$, there is a signature $\sigma'$ equivalent to $\sigma$ with $\sigma'(e) = 1$. We define a
\emph{signed graph} to consist of a graph~$G$ together with a signature $\sigma_G$. As suggested by our terminology, we will only be interested in properties of signed
graphs which are invariant under changing to an equivalent signature.
Following Bouchet~\cite{Bouchet} we now introduce a notion of a half-edge so as to orient a signed graph. For
every graph $G$ we let $H(G)$ be a set of \emph{half edges} obtained from the set of edges $E(G)$ as follows. Each edge $e=uv$ contains two distinct half edges $h$ and $h'$ incident with $u$ and $v$, respectively. Note that if $u=v$, $e$ is a loop containing two half-edges both incident with $u$.
For a half-edge $h \in H(G)$, we let $e_h$~denote the edge of~$G$ that contains~$h$.
To orient
a signed graph $G$ we will equip each half edge with an arrow and direct it either toward or away from its incident vertex. Formally, we define an \emph{orientation} of a
signed graph $G$ to be a function $\tau : H(G) \rightarrow \{-1,1\}$ with the property that for every edge $e$ containing the half edges $h,h'$ we have
\[ \tau(h) \tau(h') = - \sigma_G(e). \]
We think of a half edge $h$ with $\tau(h) = 1$ ($\tau(h) = -1$) to be directed toward (away from) its endpoint. Note that in the case when $\sigma_G$ is identically 1, both arrows on every half edge are oriented consistently, and this aligns with the usual notion of orientation of an (ordinary) graph.
\begin{figure}
\caption{Orientations of edges in a signed graph}
\end{figure}
We define a $\Gamma$-\emph{flow} in such an orientation of a signed graph $G$ to be a function $\phi : E(G) \rightarrow \Gamma$ which obeys the following rule at every vertex $v$
\[ \sum_{ \{h \in H(G) \mid h \sim v \} } \tau(h) \phi(e_h). \]
As before, we call $\phi$ \emph{nowhere-zero} if $0 \not\in \phi(E(G))$. Note that in the case when $\sigma_{G}$ is identically 1, this notion agrees with our earlier notion of a (nowhere-zero) flow in an orientation
of a graph. Also note that, as before, we may obtain a new flow by reversing the orientation of an edge $e$ (i.e., by changing the sign of $\tau(h)$ for both half edges
contained in $e$) and then replacing $\phi(e)$ by its additive inverse. This new flow is nowhere-zero if and only if the original flow had this
property. In light of this, we may now define $\Phi(G,\Gamma)$ to be the number of nowhere-zero $\Gamma$-flows in some (and thus every) orientation
of the signed graph~$G$.
As we remarked, we are only interested in properties of signed graphs which are invariant under changing to an equivalent signature, and this is indeed the case for $\Phi(G,\Gamma)$. To see this, suppose that $\tau$ is an orientation of the signed graph $G$ and that $\phi$ is a nowhere-zero $\Gamma$-flow for this orientation. Assume that the signature $\sigma'_G$ is obtained from $\sigma_G$ by flipping the sign of every edge in the edge-cut $\delta(X)$ (here $X \subseteq V(G)$ and $\delta(X)$ is the set of edges with exactly one end in $X$). Modify the orientation $\tau$ to obtain a new orientation $\tau'$ by switching the sign of $h$ for every half edge incident with a vertex of $X$. It is straightforward to verify that $\tau'$ is now an orientation of the signed graph given by $G$ and $\sigma_G'$, and $\phi$ is still a $\Gamma$-flow for this new oriented signed graph.
Beck and Zaslavsky~\cite{BZ06} considered the problem of counting nowhere-zero flows in signed graphs and proved the following analogue of Tutte's Theorem~\ref{tutte} for
groups of odd order.
\begin{theorem}[Beck and Zaslavsky~\cite{BZ06}]
Let $G$ be a signed graph.
\begin{enumerate}
\item If $\Gamma,\Gamma'$ are abelian groups and $|\Gamma| = |\Gamma'|$ is odd, then $\Phi(G,\Gamma) = \Phi(G,\Gamma')$.
\item There exists a polynomial $f$ so that for every odd integer $n$, every abelian group $\Gamma$ with $|\Gamma|=n$ satisfies
$f(n) = \Phi(G,\Gamma)$.
\end{enumerate}
\end{theorem}
The purpose of this note is to extend the above theorem to allow for groups of even order by incorporating another parameter. For any finite group $\Gamma$ we define
$\epsilon_2(\Gamma)$ to be the largest integer $d$ so that $\Gamma$ contains a subgroup isomorphic to $\mathbb{Z}_2^d$ (here $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$).
\begin{theorem}
\label{maingroup}
Let $G$ be a signed graph and let $d \ge 0$.
\begin{enumerate}
\item If $\Gamma$ and $\Gamma'$ are abelian groups with $|\Gamma| = |\Gamma'|$ and $\epsilon_2(\Gamma) = \epsilon_2(\Gamma')$, then $\Phi(G,\Gamma) = \Phi(G,\Gamma')$.
\item For every nonnegative integer $d$, there exists a polynomial $f_d$ so that $\Phi(G,\Gamma) = f_d(n)$ for every abelian group $\Gamma$ with $\epsilon(\Gamma) = d$
and $|\Gamma| = 2^dn$.
\end{enumerate}
\end{theorem}
The proof of the above theorem is a straightforward adaptation of Tutte's original method, so it may seem surprising it was not proved earlier. The cause of this may be
some confusion over whether or not it was already done. The paper by Beck and Zaslavsky~\cite{BZ06} includes a footnote with the following comment: ``Counting of flows in groups of
even order has been completely resolved by Cameron et al.''. This refers to an interesting paper of Cameron, Jackson, and Rudd~\cite{CJR} which concerns problems such as counting
the number of orbits of nowhere-zero flows under a group action. However, the methods developed in this paper only apply to counting nowhere-zero flows in (ordinary) graphs
for the reason that the incidence matrix of an oriented graph is totally unimodular. Since the corresponding incidence matrices of oriented signed graphs are generally
not totally unimodular (and not equivalent to such matrices under elementary row and column operations), our result does not follow from Cameron et al.
Before giving the proof of our theorem, let us pause to make one further comment about nowhere-zero flows in signed graphs which consist of a single loop edge~$e$. For a
loop edge $e$ with signature $1$ we may obtain a nowhere-zero flow by assigning any nonzero value $x$ to the edge $e$. So, two groups $\Gamma$ and $\Gamma'$ will have
the same number of nowhere-zero flows for this graph if and only if $|\Gamma| = |\Gamma'|$. If, on the other hand, our graph consists of a single loop edge $e$ which is
negative, then the number of nowhere-zero $\Gamma$-flows in this graph will be precisely the number of nonzero group elements $y$ for which $2y = 0$ (i.e., the number of
elements of order~2). All elements of order~2 form (together with the zero element) a subgroup isomorphic to~$\mathbb{Z}_2^{\epsilon_2(\Gamma)}$, thus
this number is precisely $2^{{\epsilon}_2(\Gamma)} - 1$. So, in order for two groups $\Gamma$ and $\Gamma'$ to have the same number of
nowhere-zero flows on this graph, they
must satisfy $\epsilon_2(\Gamma) = \epsilon_2(\Gamma')$. By our main theorem, two groups $\Gamma$ and $\Gamma'$ will satisfy $\Phi(G,\Gamma) = \Phi(G,\Gamma')$ for every signed graph $G$ if and only if this holds for every one edge graph. This statement is in precise analogy with the situation for flows in ordinary graphs.
\begin{figure}
\caption{Two graphs that determine $\Phi(G,\Gamma)$ for every other graph~$G$.}
\end{figure}
We close the introduction by mentioning related results about the number of integer flows.
Tutte~\cite{Tutte49} defined a nowhere-zero $n$-flow to be a $\mathbb{Z}$-valued flow that only uses values~$k$
with $0 < |k| < n$. Surprisingly, a graph has a nowhere-zero $n$-flow if and only if it has
a nowhere-zero $\mathbb{Z}_n$-flow. Let us use $\Phi(G,n)$ to denote the number of nowhere-zero $n$-flows on~$G$.
While $\Phi(G,n)$ and $\Phi(G,\mathbb{Z}_n)$ are either both zero or both nonzero, the actual values differ.
An analogical statement to the second part of Theorem~\ref{tutte} is again true, by a result of
Kochol~\cite{Kochol}; that is, $\Phi(G,n)$ is a polynomial in~$n$. His result has already been extended for bidirected graphs.
Beck and Zaslavsky~\cite{BZ06} prove that for a signed graph~$G$, $\Phi (G,n)$~is a quasipolynomial of period 1 or 2; that is,
there are polynomials~$p_0$ and~$p_1$ such that $\Phi(G,n)$ is equal to~$p_0(n)$ for even~$n$ and to~$p_1(n)$ for odd~$n$.
Both the Kochol's and the Beck and Zaslavsky's result is proved by an illustrative application of Ehrhart's theorem~\cite{Ehrhart, Sam}.
\section{The proof}
\label{sec:group}
The proof of our main theorem requires the following lemma about counting certain solutions to an equation in an abelian group.
\begin{lemma}
\label{abeliancount}
Let $\Gamma$ be an abelian group with $\epsilon_2(\Gamma) = d$ and $|\Gamma| = 2^d n$. Then the number of solutions to $2x_1 + \dots + 2x_t = 0$ with
$x_1, \ldots, x_t \in \Gamma \setminus \{ 0 \}$ is given by the formula
\[
\sum_{s=0}^{t} (2^d)^s (2^d-1)^{t-s} {t \choose s} \sum_{i=1}^{s-1} (-1)^{i-1} (n-1)^{s-i} \,.
\]
\end{lemma}
\begin{proof}
We claim that for every abelian group of order $m$, the number of solutions to $x_1 + \dots + x_t = 0$ with $x_1, \ldots, x_t \neq 0$ is given by the formula
\[
\sum_{i=1}^{t-1} (-1)^{i-1} (m-1)^{t-i} \,.
\]
We prove this by induction on $t$. The base case $t=1$ holds trivially. For the inductive step, we may assume $t \ge 2$. The total number of solutions to the given
equation for which $x_1, \ldots, x_{t-1}$ are nonzero, but $x_{t}$ is permitted to have any value is exactly $(m-1)^{t-1}$ since we may choose the nonzero terms $x_1,
\ldots, x_{t-1}$ arbitrarily and then set $x_{t} = - \sum_{i=1}^{t-1} x_i$ to obtain a solution. By induction, there are exactly $\sum_{i=1}^{t-2} (-1)^{i-1}
(m-1)^{t-1-i}$ of these solutions for which $x_{t} = 0$. We conclude that the number of solutions with all variables nonzero is
\[ (m-1)^{t-1} - \sum_{i=1}^{t-2} (-1)^{i-1} (m-1)^{t-1-i} = \sum_{i=1}^{t-1} (-1)^{i-1} (m-1)^{t-i} \]
as claimed.
Now, to prove the lemma, we consider the group homomorphism $\psi : \Gamma \rightarrow \Gamma$ given by the rule $\psi (x) = x+x$. Note that the kernel of $\psi$,
denoted $ker(\psi)$, is isomorphic to $\mathbb{Z}_2^d$. Now $x_1, \ldots, x_t$ satisfy $2x_1 + \dots + 2x_t = 0$ if and only if
$\psi(x_1), \ldots, \psi(x_t)$ satisfy $\psi(x_1) + \dots + \psi(x_t) = 0$.
So, to count the number of solutions to $2x_1 + \dots + 2x_t = 0$ in $\Gamma$ with all variables nonzero, we may count all possible solutions
to $y_1 + \dots + y_t = 0$ within the group $\psi(\Gamma)$ and then, for each such solution, count the number of nonzero sequences $x_1, \ldots, x_t$ in $\Gamma$ with $
\psi(x_i) = y_i$. For every $y_i \in \psi(\Gamma)$, the pre-image $\psi^{-1}(y_i)$ is a coset of $ker(\psi)$. So the number of nonzero elements $x_i$ with $\psi(x_i) =
y_i$ will equal $2^d$ if $y_i \neq 0$ and $2^d - 1$ if $y_i = 0$. Now we will combine this with the claim proved above. For every $0 \le s \le t$, the number of
solutions to $y_1 + \dots + y_t=0$ in the group $\psi(\Gamma)$ with exactly $s$ nonzero terms is given by
\[
{t \choose s} \sum_{i=1}^{s-1} (-1)^{i-1} (n-1)^{s-i} \,.
\]
Each such solution will be the image of exactly $(2^d)^s (2^d-1)^{t-s}$ nonzero sequences $x_1, \ldots, x_t \in \Gamma$. Summing over all $s$ gives the desired formula.
\end{proof}
We also require the usual contraction-deletion formula for counting nowhere-zero flows.
\begin{observation}
\label{contdelobs}
Let $G$ be an oriented signed graph and let $e \in E(G)$ satisfy $\sigma_G(e) = 1$.
\begin{enumerate}
\item If $e$ is a loop edge, then $\Phi(G,\Gamma) = (|\Gamma|-1) \Phi({G \setminus e},\Gamma)$.
\item If $e$ is not a loop edge, then $\Phi(G,\Gamma) = \Phi({G/e},\Gamma) - \Phi({G \setminus e},\Gamma)$.
\end{enumerate}
\end{observation}
\begin{proof} The first part follows from the observation that every nowhere-zero flow in $G$ is obtained from a nowhere-zero flow in $G \setminus e$ by choosing an arbitrary nonzero value for $e$. The second part follows from the usual contraction-deletion formula for flows. Suppose $\phi$ is a nowhere-zero flow in $G / e$, and return to the original graph $G$ by uncontracting $e$. It follows from elementary considerations that there is a unique value $\phi(e)$ we can assign to $e$ so that $\phi$ is a flow. It follows that $\Phi(G/e,\Gamma)$ is precisely the number of $\Gamma$-flows in $G$ for which all edges except possibly $e$ are nonzero. This latter count is exactly $\Phi(G,\Gamma) + \Phi(G \setminus e,\Gamma)$ and this completes the proof.
\end{proof}
Equipped with these lemmas, we are ready to prove our main theorem about counting group-valued flows.
\begin{proof}[Proof of Theorem~\ref{maingroup}]
For the first part, we proceed by induction on $|E(G)|$. Our base cases will consist of one vertex graphs $G$ for which every edge has signature $-1$. In this case we
may orient $G$ so that every half-edge is directed toward its endpoint. If the edges are $e_1, \ldots, e_t$, then to find a nowhere-zero flow we need to assign each
edge~$e_i$ a nonzero value~$x_i$ so that $2x_1 + \dots + 2x_t = 0$.
By Lemma~\ref{abeliancount}, the number of ways to do this is the same for $\Gamma$ and $\Gamma'$.
For the inductive step, we may assume $G$ is connected, as otherwise the result follows by applying induction to each component. If $G$ has a loop edge $e$ with
$\sigma_G(e) = 1$, then the result follows from the previous lemma and induction on $G \setminus e$. Otherwise $G$ must have a non-loop edge $e$. By possibly switching
to an equivalent signature, we may assume that $\sigma_G(e) = 1$. Now our result follows from the previous lemma and induction on $G \setminus e$ and $G / e$.
The second part of the theorem follows by a very similar argument. In the base case when $G$ is a one vertex graph in which every edge has signature $-1$, the desired
polynomial is given by Lemma~\ref{abeliancount}. For the inductive step, we may assume $G$ is connected, as otherwise the result follows by applying induction to each
component and taking the product of these polynomials. If we are not in the base case, then $G$ must either have a loop edge with signature $1$ or a non-loop edge $e$
which we may assume has signature $1$. In either case, Observation~\ref{contdelobs} and induction yield the desired result.
\end{proof}
\end{document}
|
\begin{document}
\title{Correlations Between Quantumness and Learning Performance in Reservoir Computing with a Single Oscillator}
\author{Arsalan~Motamedi}
\email{[email protected]}
\affiliation{Institute for Quantum Computing, Department of Physics \& Astronomy University of Waterloo, Waterloo, ON, N2L 3G1, Canada}
\author{Hadi~Zadeh-Haghighi}
\email{[email protected]}
\affiliation{Department of Physics and Astronomy, Institute for Quantum Science and Technology, Quantum Alberta, and Hotchkiss Brain Institute, University of Calgary, Calgary, AB T2N 1N4, Canada}
\author{Christoph Simon}
\email{[email protected]}
\affiliation{Department of Physics and Astronomy, Institute for Quantum Science and Technology, Quantum Alberta, and Hotchkiss Brain Institute, University of Calgary, Calgary, AB T2N 1N4, Canada}
\date{\today}
\maketitle
\section{Introduction}
The theory of quantum information processing has been thriving over the past few decades, offering various advantages, including efficient algorithms for breaking Rivest–Shamir–Adleman (RSA) encryption, exponential query complexity speed-ups, improvement of sensors and advances in metrology, and the introduction of secure communication protocols \cite{shor1999polynomial, MacQuarrie_2020,Harrow_2009, nielsen2002quantum, bennett2020quantum, degen2017quantum, simon2017towards, rivest1983cryptographic}. Nevertheless, the challenge of error correction and fault-tolerant quantum computing is still the biggest obstacle to the realization of a quantum computer. Despite threshold theorems giving the hope of fault-tolerant computation on quantum hardware \cite{aharonov1997fault, knill1998resilient, kitaev2003fault, shor1996fault}, a successful realization of such methods is only recently accomplished on intermediate-size quantum computers \cite{acharya2022suppressing}, and the implementation of a large scale quantum computer is yet to be achieved. Moreover, today's quantum hardware contain only a few tens of qubits. Hence we are in the noisy intermediate-scale quantum (NISQ) era, and it is of interest to know what tasks could be performed by such limited noisy devices that are hard to do with classical computers \cite{temme2017error, bharti2022noisy, kandala2019error, preskill2018quantum}.
In the past few years, and on the classical computing side, neuromorphic (brain-inspired) computing has shown promising results \cite{farquhar2006field, hopfield1982neural, schmidhuber2015deep, goodfellow2020generative}, most notably the celebrated artificial neural networks used in machine learning. Neuromorphic computing uses a network of neurons to access a vast class of parametrized non-linear functions. Despite being very successful in accuracy, these models are hard to train due to the need to optimize many parameters. Another obstacle in the training of such models is the vanishing gradient problem \cite{pascanu2013difficulty, basodi2020gradient}.
A subfield of brain-inspired computing, derived from recurrent neural networks, is reservoir computing, where the learning is to be performed only at the readout. Notably, this simplification - optimizing over a smaller set of parameters - allows for circumventing the problem of barren plateaus encountered in the training of recurrent neural networks. Despite such simplifications, reservoir computing still shows remarkable performance \cite{maass2002real,jaeger2004harnessing,tanaka2019recent, rohm2018multiplexed, nature1, nakajima2018reservoir}. Reservoir computing methods are often applied to temporal categorization, regression, and also time series prediction \cite{schrauwen2007overview, mammedov2022weather}. Notably, there have been successful efforts on the physical implementation of (classical) reservoir computing \cite{tanaka2019recent, kan2022physical, nakajima2018reservoir}.
More recently, the usefulness of quantum computing in the field of machine learning has been studied \cite{biamonte2017quantum, QML, schuld2015introduction}. In addition to that, there are novel attempts to introduce an appropriate quantum counterpart for classical neuromorphic (in particular reservoir) computing. There have been different reservoir models considered, which could mostly be categorized as spin-based or oscillator-based \cite{fujii2021quantum,PhysRevResearch.3.013077, luchnikov2019simulation, martinez2020information, nokkala2021gaussian} (corresponding to finite and infinite dimensional Hilbert spaces).
On the quantum reservoir computing front, there have been efforts such as \cite{PhysRevResearch.3.013077}, where one single Kerr oscillator is exploited for fundamental signal processing tasks. The approach used in \cite{nokkala2021gaussian} for quantum reservoir computing introduces non-linearity through the encoding of input signals in Gaussian states. Their approach has been proven to be universal, meaning that it can accurately approximate fading memory functions with arbitrary precision. \cite{pfeffer2022quantum} predicts time series using a spin-based model. \cite{ghosh2021quantum} exploits a network of interacting quantum reservoirs for tasks like quantum state preparation and tomography. \cite{vintskevich2022computing} proposes heuristic approaches for optimized coupling of two quantum reservoirs. An analysis of the effect of quantumness is performed in \cite{PhysRevResearch.3.013077}, where the authors consider dimensionality as a quantum resource. The effects of quantumness have been studied more concretely in \cite{gotting2023exploring}, where they consider an Ising model as their reservoir and show that the dimension of the phase space used in the computation is linked to the system's entanglement. Also, \cite{pfeffer2022quantum} demonstrates that quantum entanglement might enhance reservoir computing. Specifically, they show that a quantum model with a few strongly entangled qubits could perform as well as a classical reservoir with thousands of perceptions, and moreover, performance declines when the reservoir is decomposed into separable subsets of qubits.
In this work, we explore how well a single quantum non-linear oscillator performs time series predictions. In particular, we are focused on the prediction of the Mackey-Glass (MG) time series \cite{mackey1977oscillation}, which is often used as a benchmark task in reservoir computing. We then investigate the role of quantumness in the quantum learning model. We use two quantumness measures, the Wigner negativity and the Lee-Jeong measure. The latter was originally introduced as a measure for macroscopicity, but here we demonstrate that it is a non-classicality measure as well. Using our approaches, we observe that quantumness correlates with learning performance, and that it does so more strongly than dimensionality.
The paper is organized as follows. In \cref{sec:rc} we introduce the reservoir computing method used in this work. \cref{sec:pts} shows the performance of the method. \cref{sec:q} analyzes the effect of quantumness measures on performance. Finally, \cref{sec:discussion} provides a discussion of the findings.
\begin{figure}
\caption{A schematic representation of the computation model, either classical or quantum. In the learning process, we find the proper $A$ that is to predict the sample set $G$, based on the outputs of the reservoir. The dynamics of the reservoir is controlled via sample set $F$.}
\label{fig:Schem}
\end{figure}
\begin{figure*}
\caption{Performance of the trained quantum model.}
\label{fig:MG1}
\label{fig:MG2}
\label{fig:1}
\end{figure*}
\section{Reservoir Computing}\label{sec:rc}
In this work, we use the approach introduced by Govia et al. \cite{PhysRevResearch.3.013077} to feed the input signal to the reservoir by manipulating the Hamiltonian. We describe the state of our quantum and classical systems using $\hat\rho$ and $a$, respectively. Let us consider a time interval of length $\Delta t$ that has been discretized with $N$ equidistant points $t_1< \cdots< t_N$. We can expand this set by considering $M$ future values $t_{N+1}< \cdots< t_{N+M}$, which are also equidistantly distributed. It is worth noting that the interval $t_N - t_1$ is equal to $\Delta t$. Our objective is to estimate a set of future values of the function $f$, which is denoted by $G = \left(f(t_{i})\right)_{i=N+1}^M$, given the recent observations $F=\left(f(t_i)\right)_{i=1}^N$. To this end, we evolve our system so that $\hat\rho(t_j)$ (respectively, $a(t_j)$) depends on $f(t_1), \cdots, f(t_j)$. We then obtain observations $s(t_i) = \langle\hat{O} \hat\rho(t_i)\rangle$ for an observable $\hat O$ (respectively, $s(t_i) = \left(h \circ a\right){t_i}$ for a function $h$). Finally, we perform a linear regression on $\left( s(t_i)\right)_{i=1}^N$ to predict $G$. In what follows we elaborate on the system's evolution in both classical and quantum cases.
For the classical reservoir, we consider the following evolution
\begin{align}\label{eq:classicEv}
\dot{a} = -iK(1+2\, |a|^2)a - \frac{\kappa}{2} a - i\alpha f(t)
\end{align}
with $K$, $\kappa$, and $\alpha$ being the reservoir's natural frequency, dissipation rate, and the amplifier of the input signal, respectively. We let $s(t) = \tanh\left(\Re{a(t)}\right)$. The quantum counterpart of the evolution described by \cref{eq:classicEv} is the following Markovian dynamics
\cite{PhysRevResearch.3.013077,gardiner2004quantum}
\begin{equation}
\begin{split}
\frac{d}{dt}\hat\rho(t) &= -i [\hat{H}(t), \hat\rho(t)] + \kappa\, \mathcal{D}_a(\hat\rho)\\
\text{where }\, \hat{H}(t) &= K\, \hat{N}^2 + \alpha \, f(t)\, \hat{X}, \\ \text{and }\, \mathcal{D}_a(\hat\rho) &= \hat{a} \hat\rho \hat{a}^\dagger - \{\hat{N}, \hat\rho\}/2. \label{eq:ev-2}
\end{split}
\end{equation}
Note that we use the properly scaled parameters, such that the uncertainty principle becomes $\Delta x \, \Delta p \geq 1/2$. The parameters $(\alpha, \kappa, K)$ are the same as above (in \eqref{eq:classicEv}). The operators $\hat X$, $\hat a$, and $\hat N$ represent the position, annihilation, and the number operator respectively. We let $s(t) = \tr ( \hat\rho(t) \,\tanh \hat X)$. Utilizing the non-linear quantum evolution \eqref{eq:ev-2} as our quantum reservoir, we perform the learning of the MG time series.
Let us now provide details on the process of linear regression. Our objective is to find the predictor $A$ that satisfies the relationship
$
G \approx A s
$
(note that we think of $G$ and $s$ as column vectors). To this end, we conduct the experiment $T$ times and collect the resulting column vectors as ${ \vec s_1, \vec s_2, \cdots, \vec s_T}$. We then define a matrix $\mathbf{S}$ as the concatenation of these column vectors
\begin{align}
\mathbf{S} := \begin{pmatrix}
s_1 |\, s_2 |\, \cdots | s_T
\end{pmatrix}
\end{align}
Similarly, we define the matrix $\mathbf{G}$ as
\begin{align}
\mathbf G := \begin{pmatrix}
G_1 | G_2|\, \cdots \, | G_T
\end{pmatrix}.
\end{align}
Finally, we choose $A$ by applying Tikhonov regularization \cite{shalev2014understanding}, which results in the following choice of $A$
\begin{align}\label{eq:W}
A = \mathbf G \mathbf{S}^T (\mathbf{S}\mathbf{S}^T + \gamma \mathbb{I})^{-1}
\end{align}
with $\gamma$ and $\mathbb I$ being a regularization parameter, and the identity matrix, respectively. One should note that $\mathbf G$, and $\mathbf S$ written in \eqref{eq:W} correspond to $T$ training samples that we take. The matrix $A$ evaluated above is then used for the prediction of the test data. \cref{fig:Schem} provides a schematic representation of the reservoir training.
Overall, to predict time series, the reservoir is initially trained to determine $A$ by using equation \eqref{eq:W}. After training, a set of initial values outside of the training data is inputted into the oscillator. The oscillator uses $A$ to predict future values, which are then used as initial values for further predictions.
\section{Results}\label{sec:res}
This section investigates the performance of a single Kerr non-linear oscillator trained on the MG chaotic time series, as well as the effect of different quantumness measures on the performance of the reservoir. Specifically, in Section \cref{sec:pts}, we discuss how well the non-linear oscillator could learn when trained on chaotic time series, while in Section \cref{sec:q}, we examine the impact of various quantumness measures. Lastly, we outline further investigations, including the effects of noise.
To simulate quantum dynamics in the Fock space, we truncate every operator in the number basis, making them $d_{\text{t}}$-dimensional. Notably, the simulation results in this work use a dimension $d_{\text{t}}\geq 20$, which is sufficiently large as most of the states considered have a significant overlap with the subspace spanned by the number states $\ket n$ for $n\leq10$ (For instance, we use the coherent state $\ket\alpha$ with $\alpha=1+\iota$, the overlap of which with the first $20$ Fock states is larger than $1-6.5\times 10^{-15}$).
\subsection{Learning Time Series}\label{sec:pts}
In what follows we report the results obtained by training our single non-linear oscillator.
Here, we consider the prediction the chaotic MG series. MG is formally defined as the solution to the following delay differential equation
\begin{align}
\dot{x}(t) = \beta \frac{x(t-\tau)}{1+x(t-\tau)^n} - \gamma x(t).
\end{align}
We use the parameters $\beta = 0.2$, $\gamma = 0.1$, $n=10$, and $\tau = 17$ throughout this work. The performance of the trained reservoir on the test data is presented in \cref{fig:MG1}. \cref{fig:MG2} shows the delayed embedding of the predicted MG, which is compared to the actual diagram. One can readily observe that this model is successful in learning MG.
\subsection{Quantumness}\label{sec:q}
In this section, we introduce our quantumness measure and study the effect of quantumness on that basis on the accuracy of the learning model.
\begin{figure}
\caption{
Average quantumness ($Q$) during the evolution is shown to be decreasing as $\kappa$ (the photon loss rate) increases. The states used as initial states are the mixed state (labeled as `mix') being proportional to $\ket{\alpha}
\label{fig:Wigs}
\end{figure}
\subsubsection{Quantumness Measures}\label{sec:quant}
Our main goal here is to determine if there is a correlation between the quantumness of the system and the accuracy of the learning process. An affirmative answer would be a quantum advantage for this computational model. To this end, we need to quantify the quantumness of a state in Fock space. We point out that there has been extensive research done on the quantification of quantumness \cite{ groisman2007quantumness, ollivier2001quantum, takahashi1986wigner}.
Furthermore, there has been a line of research in the study of the macroscopicity of quantum states and their effective size \cite{frowis2018macroscopic, nimmrichter2013macroscopicity, leggett1985quantum, leggett2016note}. One such measure in the Fock space is defined by
Lee and Jeong \cite{lee2011quantification} as follows:
\begin{equation}
\begin{split}
I(\hat\rho) := \pi \bigg( &\int_{x,p} \big(\partial_xW(x,p)\big)^2\\ &+\big(\partial_pW(x,p)\big)^2 - 2W(x,p)^2 \bigg)
\end{split}
\end{equation}
where $x,p$ are dimensionless space and momentum components (in such scale, the uncertainty principle becomes $\Delta x\, \Delta p \geq 1/2$). This formulation can also be found in \cite{I}. The following identities are pointed out in \cite{lee2011quantification}:
\begin{itemize}
\item $I(\ket{\alpha} \bra{\alpha}) = 0$, for any coherent state $\ket\alpha$.
\item $\forall n \in \mathbb{N}: I(\sum_{i=0}^{n-1} \frac{1}{n}\, \ket{i}\bra{i}) = 0$, where $\ket{i}$ are the Fock states (i.e., the eigen-vectors of the number operator).
\end{itemize}
It is worth mentioning that this measure can obtain negative values \cite{I}. Intuitively, one could think of $I(\hat\rho)$ as the fineness of the Wigner function associated with $\hat\rho$. The aforementioned results may suggest that the positiveness of $I$ indicates non-classicality, as the coherent state and a diagonal density matrix in the Fock basis are considered as classical states. In the following theorem, we prove that if $I(\hat\rho) >0$, then $\rho$ cannot be written as a mixture of coherent states, hence being non-classical.
\begin{thm}\label{thm}
If $\hat\rho$ is a mixture of coherent states, then $I(\hat\rho)\leq 0$.
\end{thm}
The proof is provided in \cref{PfOfThm}.
One should also note that $I$ is computable in a much shorter time as it reformulated as (\cite{lee2011quantification})
\begin{align}
I(\hat\rho) = \tr\left(\hat\rho \mathcal{D}_a(\hat\rho)\right).
\end{align}
On the other hand, the computation of Wigner negativity with the current algorithms is costly, as it requires the computation of the entire Wigner function.
We hence use the following quantumness measure $Q$
\begin{align}
Q(\hat\rho) = \begin{cases}
I(\hat\rho) \, &\text{if } I(\hat\rho) >0,\\
0 & \text{o.w.}
\end{cases}
\end{align}
We make this choice as we do not want our quantumness measure to obtain negative values. We observe that this measure is consistent with some intuitions regarding the quantumness of reservoir computing, which have been previously used in \cite{PhysRevResearch.3.013077}. In particular, the intuition that by increasing $\kappa$ we should reach a classical limit, which is illustrated in \cref{fig:Wigs}. Finally, we point out the fact that the quantumness of the state changes during evolution. This is indeed observed in \cref{fig:QuantKappa}.
\begin{figure}
\caption{Quantumness and Wigner plots of the reservoir's state evolution. The oscillator parameters are $(\alpha, \kappa, K) = (1.2, 0.1, 0.05)$.}
\label{fig:QuantKappa}
\end{figure}
\subsubsection{Example States}
To study the quantumness effect, here, we used different initial states. For instance, we considered training the reservoir initialized at different states including the cat state (i.e., normalized $\ket{\alpha} + \ket{-\alpha}$), the corresponding mixed state (i.e., normalized $\ket{\alpha}\bra{\alpha} + \ket{-\alpha}\bra{-\alpha}$), the coherent state $\ket \alpha$, and the number state $\ket n$. We further added the data obtained from the training of a classical model. We changed the training data sizes to better compare different states and models. The result is presented in \cref{fig:trainingCurve}. Despite not showing a significant advantage for states with high quantumness, this diagram reveals that the quantum model for the reservoir outperforms the classical model in the task of MG prediction.
\begin{figure}
\caption{Training curves for both classical and quantum models. Different initial states for the quantum model have been considered. Those are the mixed state (labeled as `mix') being proportional to $\ket{\alpha}
\label{fig:trainingCurve}
\end{figure}
\subsubsection{Random States}
Since the distinction between our examples in \cref{fig:trainingCurve} does not demonstrate a clear correlation between the quantumness and the performance, we pick random states and scrutinize the correlations between the training accuracy and quantumness measures. Interestingly, \cref{fig:hist} shows a clear correspondence between the quantumness and the performance; providing us with a piece of evidence that the quantumness does indeed help achieve more precise predictions. For the initial random states, we fix a dimension $d$, then pick a state according to the Haar random measure on $\mathcal H(\mathbb C^d)$ \cite{haar1933massbegriff}.
To this end, we use \cite[Proposition 7.17]{watrous2018theory}. In particular, we consider the set of $2d$ independent and identically distributed (i.i.d.) standard Gaussian random variables $X_1, Y_1, \cdots, X_d, Y_d$ to construct the state
\begin{align}\label{eq:Gaussian}
\ket \psi = \frac{\left(X_1 + \iota Y_1, \cdots, X_d + \iota Y_d\right)}{\sqrt{\sum_{i} X_i^2 + Y_i^2}}
\end{align}
which is a Haar random state in $\mathcal H(\mathbb C^d)$. To more elaborate, according to \cite[Proposition 7.17]{watrous2018theory}, the distribution of the states generated by \eqref{eq:Gaussian}, are invariant under the action of the unitaries acting on $\mathcal{H}(\mathbb C^d)$. We repeated this random state generation for $d=4$ to $d=10$, collecting 20 samples for each dimension, and truncated the Fock space by considering the subspace spanned by the first $25$ number states (i.e., $d_{\text{t}}=25$) to obtain the results presented in \cref{fig:MainA}. We emphasize that the same training protocol is applied to all such states in every dimension.
\begin{figure*}
\caption{Training accuracy for the task of Mackey-Glass prediction, using $140$ random states ($20$ datapoints for each $d = 4, \cdots, 10$). We elaborate on the random selection process at the end of section \ref{sec:q}
\label{fig:MainA}
\label{fig:hist}
\label{fig:Main}
\end{figure*}
The correlation coefficients of the data show a relationship between the quantumness and the training accuracy. We further investigated the effects of quantumness using the t-tests. Our hypothesis test (\cref{fig:hist})
resulted in a $p$-value of $2.1\times 10^{-4}$. As a reminder, the $p$-value in our case of study is an estimation of a lower bound on the probability of observing the obtained results under the hypothesis that the quantumness has no positive effect on the training accuracy. Thus, such a small $p$-value confirms the efficacy of quantumness on the training outcome. We further investigate the relation of dimensionality with the error. We note that increasing the dimension used for the oscillator could be understood as increasing the complexity and the power of the model. From \cref{fig:MainA} we observe that although dimensionality correlates well with quantumness, it does not correlate much with accuracy. This suggests that quantumness is a more effective factor than complexity.
A similar analysis is presented in \cref{app:Wigner}, where we consider the Wigner negativity as our quantumness measure. Other than that, we also investigate the robustness of the model by introducing a variety of noises to the evolution. Our results, presented in \cref{app:noise}, suggest that the model can tolerate a considerable amount of noise.
\section{Discussion}\label{sec:discussion}
In this study, we focused on evaluating the effectiveness of a quantum non-linear oscillator in making time-series predictions and examining how quantumness impacts the quantum learning model. We utilize two quantumness metrics, namely the Wigner negativity and the Lee-Jeong measure \cite{lee2011quantification}. The former is a widely accepted measure, while the latter was initially introduced to measure macroscopicity and in this work is shown to be a quantumness measure as well. Through our methodologies, we discovered that quantumness has a stronger correlation with performance than the dimensionality of the reservoir's state. Overall, our findings contribute to a deeper understanding of the role of quantumness in continuous-variable reservoir computing and highlight its potential for enhancing the performance of this computational model.
Our work raises a number of important questions. Firstly, we aim to determine what specific structures within a reservoir computing model will lead to quantum speed-ups. Additionally, one can investigate the impact of quantumness on a network of oscillators in future research. Notably, when dealing with a network of continuous variable oscillators, entanglement as a measure of quantumness could also be examined. It is worth mentioning that our method has potential for implementation on actual quantum hardware, and may even be feasible with current limited devices due to the strong Kerr non-linearity present in models for a transmon superconducting qubit \cite{bertet2012circuit}.
\section{Code Availability}
The codes used for the generation of the plots of this manuscript are publicly available at \href{https://github.com/arsalan-motamedi/QRC}{https://github.com/arsalan-motamedi/QRC}.
\section{Contributions}
All authors contributed extensively to the presented work. H.Z-H. and C.S. conceived the original ideas and supervised the project. A.M. performed analytical studies and numerical simulations and generated different versions of the manuscript. H.Z-H. and C.S. verified the calculations, provided detailed feedback on the manuscript, and applied many insightful updates.
\begin{appendices}
\crefalias{section}{appendix}
\section{Comparisons with Wigner negativity}\label{app:Wigner}
In this appendix, we examine the quantumness of the system by Wigner negativity. We recall that the Wigner negativity is the volume under the $W=0$ plane for a Wigner function, and is often considered as a quantumness measure in the literature \cite{hudson1974wigner, kenfack2004negativity}. The computation of Wigner negativity is more costly than the Lee-Jeong measure introduced in \cref{sec:quant}. Hence, we only compute it for the initial state. The result of this study is presented in \cref{fig:WigNeg}. As we observe, there is a strong correlation between the two quantumness measures. Furthermore, the Wigner negativity correlates well with the test error of the experiment.
\begin{figure}
\caption{The effects of Wigner negativity on the training performance. As observed, we get a correlation between Wigner negativity and test error, which again highlights the effect of quantumness. Moreover, we observe that there is a strong correlation between the quantumness measures considered in this work.}
\label{fig:WigNeg}
\end{figure}
\section{Noise}\label{app:noise}
Noise is an inevitable factor in quantum devices, and it is of profound importance for a quantum computing approach to be robust to noise. We show the robustness of this approach by considering different noise models as explained below.
\begin{figure}
\caption{Noisy reservoir learning MG. The MG training process is performed when there are a variety of noises applied to the reservoir.}
\label{fig:noisepred}
\label{fig:noisephase}
\label{fig:NoisyMG}
\end{figure}
\textit{Dephasing, Pumping, and the White noise.} We introduce a dephasing error by considering the Lindbladian operators $L_n = \lambda \ket{n}\bra{n}$. As observed in the experiments, this will not affect the results as much. Furthermore, the incoherent pumping error is simulated by considering the Lindbladian corresponding to $\lambda a^\dagger$. On top of those, we add white noise to the input of the reservoir. This is performed via changing the equations of evolution \eqref{eq:ev-2} through the substitution $f(t) \rightarrow f(t) + \lambda' n(t)$. Here, $n(t)$ is the white noise of unit power, and $\lambda'$ controls its strength. \cref{fig:NoisyMG} shows the performance of the reservoir's output under both incoherent pumping, dephasing error, and white noise (see \cref{eq:ev-2}). We have set $(K,\kappa, \alpha, \lambda, \lambda')= (0.05, 0.15, 1.2, 0.05, 0.02)$.
It is worth mentioning that noise in the context of reservoir computing is shown to be useful in certain cases
\cite{noise, fry2023optimizing}.
\section{Proof of Theorem \ref{thm}}\label{PfOfThm}
Recall the Gaussian integrals, which we will use at the end
\begin{equation}\label{eq:GI}
\begin{split}
\int_{\xi\in\mathbb{R}} e^{-\frac{\xi^2}{a^2}} \, d\xi &= \sqrt{\pi\, a^2},\\
\int_{\xi\in\mathbb{R}} \xi^2 \, e^{-\frac{\xi^2}{a^2}}\, d\xi &= \frac{a^2}{2}\, \sqrt{\pi\, a^2}.
\end{split}
\end{equation}
Note that for any coherent state $\ket{\alpha}$, one has
\begin{align*}
W_{\alpha} = \frac{1}{\pi}\, e^{- \big[ (x-\text{Re}(\alpha))^2 + (p-\text{Im}(\alpha))^2 \big]}
\end{align*}
Let us consider a set of $K$ coherent states, namely $\{\rho_{i} = \ket{\alpha_i}\bra{\alpha_i}: i=1,2,\cdots,K \}$, and define $x_i : = \text{Re}(\alpha_i)$, $p_i := \text{Im}(\alpha_i)$. Also let $(q_i)_{i\in [K]}$ be a probability distribution over $K$ objects. We can then consider the mixture of coherent states as
\begin{align*}
\rho = \sum_{i=1}^K q_i\, \ket{\alpha_i}\bra{\alpha_i}
\end{align*}
since the Wigner function is linear with respect to the density matrix, one has
\begin{align*}
W_{\rho}(x,p) = \sum_{i=1}^K q_i\, W_{\alpha_i}(x,p)
\end{align*}
Hence, we get
\begin{align*}
\frac{1}{\pi}\, I(\rho) &= \int_{x,p} \big(\sum_{i\in[K]} \, q_i\, \partial_xW_i(x,p)\big)^2 \notag \\
& \quad \quad \quad + \big(\sum_{i\in[K]} \, q_i\, \partial_pW_i(x,p)\big)^2 \notag \\
& \quad \quad \quad - 2 \big(\sum_{i\in[K]} \, q_i\, W_i(x,p)\big)^2 \\
&= \sum_{i,j\in[k]} \, q_i\, q_j\, \int_{x,p}\bigg( \partial_{x} W_i\, \partial_{x} W_j\notag \\
&\qquad \quad+ \partial_{p} W_i\, \partial_{p} W_j - 2W_i\, W_j \bigg)
\end{align*}
note that the terms in the summation above with $i=j$ could be rewritten as $q_i^2\, I(\rho_i) = 0$ since any cohrent state $\rho_i$ has zero quantumness i.e., $I(\rho_i)=0$. Furthermore, \cref{claim1} below guarantees that for any choice of $i,j$ the expression in the parenthesis is non-positive, and hence, the proof is complete.
\begin{claim}\label{claim1}
For any two coherent states, say $\ket{\alpha_0}$ and $\ket{\alpha_1}$, both of the following inequalities hold
\begin{equation}
\begin{split}
\int_{x,p} \bigg(\partial_{x} W_0\, \partial_{x} W_1 - W_0\, W_1\bigg) &\leq 0\\
\int_{x,p} \bigg(\partial_{p} W_0\, \partial_{p} W_1 - W_0\, W_1\bigg) &\leq 0
\end{split}
\end{equation}
\end{claim}
\begin{proof}
One has
\begin{align*}
W_0 = \frac{1}{\pi}\, e^{- \big[ (x-x_0)^2 + (p-p_0)^2 \big]}
\end{align*}
hence
\begin{equation*}
\begin{split}
\partial_x W_0 &= -2 \frac{(x-x_0)}{\pi}\, e^{- \big[ (x-x_0)^2 + (p-p_0)^2 \big]},\\ \partial_p W_0 &= -2 \frac{(p-p_0)}{\pi}\, e^{- \big[ (x-x_0)^2 + (p-p_0)^2 \big]}
\end{split}
\end{equation*}
and similar expressions for $W_1$ and its derivatives. Let us now prove the first inequality. Define
\begin{align*}
\mathcal A := \int_{x,p} \bigg(\partial_{x} W_0\, \partial_{x} W_1 - W_0\, W_1\bigg)
\end{align*}
then, by direct substitution one gets
\begin{align*}
\mathcal A = \frac{1}{\pi^2}\, &\int_{x,p}\, \big( 4(x-x_0)(x-x_1) - 1 \big) \notag \\
& \times e^{- \big[ (x-x_0)^2 + (x-x_1)^2 + (p-p_0)^2 + (p-p_1)^2 \big] }
\end{align*}
we may now use the elementary identities
$
(x-x_0)(x-x_1) =\left(x - \frac{x_0+x_1}{2}\right)^2 - \left(\frac{x_0 - x_1}{2} \right)^2$ and $
(x-x_0)^2 + (x-x_1)^2 =\left(x - \frac{x_0+x_1}{2}\right)^2 + \left(\frac{x_0 - x_1}{2} \right)^2
$
and further, letting $\Delta x := x_0 - x_1$, $\Delta p = p_0-p_1$, and $\overline{x} := \frac{x_0 + x_1}{2}$, and $\overline{p}= \frac{p_0+p_1}{2}$ to conclude
\begin{align*}
\mathcal A &= \frac{e^{-\frac{1}{2} ( \Delta x^2 + \Delta p^2) }}{\pi^2}\, \notag \\
& \quad \quad \times \int_{x,p} \big[ 4\big(x - \overline{x}\big)^2 - (\Delta x)^2 - 1 \big] \\
&\quad \qquad e^{ -2(x - \overline{x})^2 -2(p - \overline{p})^2 }\\
&= -\frac{e^{-\frac{1}{2} ( \Delta x^2 + \Delta p^2)}}{2\pi}\, (\Delta x)^2 \leq 0
\end{align*}
(for the last equality, we use the Gaussian integrals in \cref{eq:GI}). A similar argument gives the second inequality of the claim.
\end{proof}
\end{appendices}
\pagebreak
\widetext
\begin{center}
\textbf{\large Supplementary Material for: Correlations Between Quantumness and Learning Performance in Reservoir Computing with a Single Oscillator}
\end{center}
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\section{R\"ossler Attractor}\label{sec:chaos}
This section provides the performance in the training of R\"ossler attractor. The R\"ossler attractor \cite{rossler1976equation, rossler1979equation} is a three-dimensional motion following the dynamics
\begin{equation}
\begin{cases}
\frac{dx}{dt} &= -y-z\\
\frac{dy}{dt} &= x + ay\\
\frac{dz}{dt} &= b+ z(x-c)
\end{cases}
\end{equation}
where $(a,b,c)\in\mathbb R^3$ are constant parameters, which we set to $(0.2,0.2,5.7)$ in our experiment. \cref{fig:Ros1} and \cref{fig:Ros2} show the model's learning results for this chaotic time series. We highlight that each component of the oscillator is learned independently from the others.
\begin{figure}
\caption{R\"ossler attractor training with a quantum oscillator.}
\label{fig:Ros1}
\label{fig:Ros2}
\label{fig:Rossler}
\end{figure}
\section{More on Noise}
In this section, we investigate the effect of adding white noise on the input to the reservoir. This error is introduced by the change equations of evolution through the substitution $f(t) \rightarrow f(t) + \lambda n(t)$. Here, $n(t)$ is the white noise of unit power, and $\lambda$ controls its strength. Firstly, we consider a white noise applied to the input to the reservoir. \cref{fig:ArbF} shows the outcome of learning noisy period functions. Despite significant signal distortion caused by noise, the oscillator demonstrates the ability to learn the underlying periodic functions. We further investigated the effect of training sawtooth signal with different noise levels, which resulted in \cref{fig:stn}. A similar experiment, this time with the MG, resulted in a training error of $0.053$. Noting that the training error in the noiseless case results in an error of $0.047$, we conclude that the model is robust to this noise model for a variety of prediction tasks. We made the choice of parameters $\alpha = 0.1$, $\kappa = K = 0.05$ in obtaining the results.
\begin{figure}
\caption{Test error of training noisy sawtooth function. The initial states are the cat state and its classical mixture i.e., the normalized $\ket{\alpha}
\label{fig:stn}
\end{figure}
\begin{figure}
\caption{Learning noisy periodic functions. The input to the reservoir is contaminated by white noise. However, the reservoir is still able to learn the input signal.}
\label{fig:ArbF}
\end{figure}
\section{Evolution Animations}
Animations showing the evolution of the Wigner function throughout the process are prepared and made available online at \href{https://github.com/arsalan-motamedi/QRC/tree/main/EvolutionAnimations}{https://github.com/arsalan-motamedi/QRC/tree/main/EvolutionAnimations}.
\end{document}
|
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{document}
\title{Well-rounded zeta-function of planar arithmetic lattices}
\operatorname{Aut}hor{Lenny Fukshansky}\thanks{The author was partially supported by a grant from the Simons Foundation (\#208969 to Lenny Fukshansky) and by the NSA Young Investigator Grant \#1210223.}
\address{Department of Mathematics, 850 Columbia Avenue, Claremont McKenna College, Claremont, CA 91711}
\email{[email protected]}
{\mathcal S}ubjclass[2010]{11H06, 11H55, 11M41, 11E45}
\keywords{arithmetic lattices, integral lattices, well-rounded lattices, Dirichlet series, zeta-functions}
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{abstract}
We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving upon a result of \cite{kuehnlein}. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence properties of a zeta-function of similarity classes of such lattices, building on the results of \cite{fletcher_jones}.
\end{abstract}
\maketitle
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{\mathcal S}ection{Introduction}
\label{intro}
Let ${\mathcal L}ambda = A{\mathbb Z}^2 {\mathcal S}ubset {\mathbb R}^2$ be a lattice of full rank in the plane, where $A=({{{\boldsymbol 1}_Ldsymbol 0}ldsymbol a}_1 {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol a}_2)$ is a basis matrix. The corresponding norm form is defined as
$$Q_A({{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x}) = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x}^t A^t A {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x}.$$
We say that ${\mathcal L}ambda$ is {\it arithmetic} if the entries of the matrix $A^tA$ generate a 1-dimensional ${\mathbb Q}$-vector subspace of ${\mathbb R}$. This property is easily seen to be independent of the choice of a basis. We define $\operatorname{det}({\mathcal L}ambda)$ to be $|\operatorname{det}(A)|$, again independent of the basis choice, and (squared) {\it minimum} or {\it minimal norm}
$$|{\mathcal L}ambda| = \min \{ \|{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x}\|^2 : {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x} \in {\mathcal L}ambda {\mathcal S}etminus \{{{\boldsymbol 1}_Ldsymbol 0}\} \} = \min \{ Q_A({{{\boldsymbol 1}_Ldsymbol 0}ldsymbol w}y) : {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol w}y \in {\mathbb Z}^2 {\mathcal S}etminus \{{{\boldsymbol 1}_Ldsymbol 0}\} \},$$
where $\|\ \|$ stands for the usual Euclidean norm. Then each ${{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x} \in {\mathcal L}ambda$ such that $\|{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x}\|^2 = |{\mathcal L}ambda|$ is called a {\it minimal vector}, and the set of minimal vectors of ${\mathcal L}ambda$ is denoted by $S({\mathcal L}ambda)$. A planar lattice ${\mathcal L}ambda$ is called {\it well-rounded} (abbreviated WR) if the set $S({\mathcal L}ambda)$ contains a basis for ${\mathcal L}ambda$; we will refer to such a basis as a {\it minimal basis} for~${\mathcal L}ambda$.
While in this note we focus on the planar case, the notion of WR lattices is defined in every dimension: a full-rank lattice in ${\mathbb R}^N$ is WR if it contains $N$ linearly independent minimal vectors -- the fact that these form a basis for the lattice is a low-dimensional phenomenon, only valid for $N \leq 4$. WR lattices are important in discrete optimization, in particular in the investigation of sphere packing, sphere covering, and kissing number problems \cite{martinet}, as well as in coding theory \cite{esm}. Properties of WR lattices have also been investigated in \cite{mcmullen} in connection with Minkowski's conjecture and in \cite{lf:robins} in connection with the linear Diophantine problem of Frobenius. Furthermore, WR lattices are used in cohomology computations of $\operatorname{SL}_N({\mathbb Z})$ and its subgroups \cite{ash}. These considerations motivate the study of distribution properties of WR lattices. Distribution of WR lattices in the plane has been studied in \cite{wr1}, \cite{wr2}, \cite{wr3}, \cite{fletcher_jones}, \cite{kuehnlein}. In particular, these papers investigate various aspects of distribution properties of WR sublattices of a fixed planar lattice.
An important equivalence relation on lattices is geometric similarity: two lattices ${\mathcal L}ambda_1, {\mathcal L}ambda_2 {\mathcal S}ubset {\mathbb R}^2$ are called {\it similar}, denoted ${\mathcal L}ambda_1 {\mathcal S}im {\mathcal L}ambda_2$, if there exists $\alpha \in {\mathbb R}_{>0}$ and $U \in O_2({\mathbb R})$ such that ${\mathcal L}ambda_2 = \alpha U {\mathcal L}ambda_1$. It is easy to see that similar lattices have the same algebraic structure, i.e., for every sublattice $\Gamma_1$ of a fixed index in ${\mathcal L}ambda_1$ there is a sublattice $\Gamma_2$ of the same index in ${\mathcal L}ambda_2$ so that $\Gamma_1 {\mathcal S}im \Gamma_2$. A WR lattice can only be similar to another WR lattice, so it makes sense to speak of WR similarity classes of lattices. In \cite{kuehnlein} it has been proved that a planar lattice contains infinitely many non-similar WR sublattices if and only if it contains one. This is always the case for arithmetic planar lattices. If the lattice in question is not arithmetic, it may still have infinitely many non-similar WR sublattices depending on the value of a certain invariant described in \cite{kuehnlein}. In any case, it appears that non-arithmetic planar lattices contain fewer WR sublattices than arithmetic ones in the sense which we discuss below.
Given an infinite finitely generated group $G$, it is a much-studied problem to determine the asymptotic growth of $\# \left\{ H \leq G : \left| G:H \right| \leq N \right\}$, the number of subgroups of index no greater than $N$, as $N \to \infty$ (see \cite{lubot}). One approach that has been used by different authors with great success entails looking at the analytic properties of the corresponding Dirichlet-series generating function ${\mathcal S}um_{H \leq G} \left| G:H \right|^{-s}$ and then using some Tauberian theorem to deduce information about the rate of growth of partial sums of its coefficients (see \cite{sautoy}, as well as Chapter~15 of \cite{lubot}). In case $G$ is a free abelian group of rank 2, i.e. a planar lattice, this Dirichlet series allows to count sublattices of finite index, and is a particular instance of the Solomon zeta-function (see \cite{reiner}, \cite{solomon}). We will use a similar approach while restricting to just WR sublattices, which is a more delicate arithmetic problem.
Fix a planar lattice ${\mathcal O}mega$, and define the {\it zeta-function of WR sublattices} of ${\mathcal O}mega$ to be
$$\zeta_{\operatorname{WR}}({\mathcal O}mega,s) = {\mathcal S}um_{\operatorname{WR} {\mathcal L}ambda {\mathcal S}ubseteq {\mathcal O}mega} \frac{1}{\left| {\mathcal O}mega : {\mathcal L}ambda \right|^s} = {\mathcal S}um_{n=1}^{\infty} \frac{\# \{\operatorname{WR} {\mathcal L}ambda {\mathcal S}ubseteq {\mathcal O}mega : \left| {\mathcal O}mega : {\mathcal L}ambda \right| = n\}}{n^s}$$
for $s \in {\mathbb C}$. The rate of growth of coefficients of this function can be conveyed by studying its abscissa of convergence and behavior of the function near it. For brevity of notation, we will say that an arbitrary Dirichlet series $f(s) = {\mathcal S}um_{n=1}^{\infty} a_n n^{-s}$ has an {\it abscissa of convergence} with a {\it real pole of order $\mu$} at $s=\rho$ if $f(s)$ is absolutely convergent for ${\mathcal R}e(s) > \rho$, and for $s \in {\mathbb R}$
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{lim_def}
\lim_{s \to \rho^+} (s-\rho)^{\mu} {\mathcal S}um_{n=1}^{\infty} \frac{a_n}{n^s}
\end{equation}
exists and is nonzero. Notice that this notion does not imply existence of analytic continuation for $f(s)$, but is merely a statement about the rate of growth of the coefficients of $f(s)$, which is precisely what we require. For instance, in \cite{wr1} and \cite{wr2} it has been established that $\zeta_{\operatorname{WR}}({\mathbb Z}^2,s)$ has abscissa of convergence with a real pole of order 2 at $s=1$. Furthermore, it has been shown in \cite{kuehnlein} that if ${\mathcal O}mega$ is a non-arithmetic planar lattice containing WR sublattices, then $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ has abscissa of convergence with a real pole of order 1 at $s=1$ (in fact, Lemma~3.3 of~\cite{kuehnlein} combined with Theorem~4 on p.~158 of~\cite{lang} imply the existence of analytic continuation of $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ in this situation to ${\mathcal R}e(s) > 1-{\varepsilon}$ for some ${\varepsilon} > 0$ with a pole of order 1 at $s=1$). It is natural to expect that the situation for any arithmetic lattice is the same as it is for ${\mathbb Z}^2$; in fact, another result of \cite{kuehnlein} states that for any arithmetic lattice ${\mathcal O}mega$, $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ has abscissa of convergence at $s=1$, and it is conjectured that it has a pole of order 2 at $s=1$. The main goal of the present paper is to prove the following result in this direction.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{thm} \label{main} Let ${\mathcal O}mega$ be a planar arithmetic lattice. Then $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ has abscissa of convergence with a real pole of order 2 at $s=1$ in the sense of \eqref{lim_def} above. Moreover,
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{growth_bnd_1}
\# \{\operatorname{WR}\ {\mathcal L}ambda {\mathcal S}ubseteq {\mathcal O}mega : \left| {\mathcal O}mega : {\mathcal L}ambda \right| \leq N\} = O(N \log N)
\end{equation}
as $N \to \infty$.
\end{thm}
{\mathcal S}mallskip
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{rem} \label{WR_growth} To compare, Theorem~4.20 of \cite{sautoy} combined with Lemma~3.3 (and the Corollary following it) of \cite{kuehnlein} imply that if ${\mathcal O}mega$ is a non-arithmetic planar lattice containing WR sublattices, then the right hand side of \eqref{growth_bnd_1} is equal to $O(N)$. It should be pointed out that by writing that a function of $N$ is equal to $O(N \log N)$ (respectively, $O(N)$) we mean here that it is asymptotically bounded from above and below by nonzero multiples of $N \log N$ (respectively,~$N$). On the other hand, it is a well known fact (outlined, for example, on p. 793 of \cite{sautoy}) that for any planar lattice~${\mathcal O}mega$,
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{growth_bnd_2}
\# \{ {\mathcal L}ambda {\mathcal S}ubseteq {\mathcal O}mega : \left| {\mathcal O}mega : {\mathcal L}ambda \right| \leq N\} {\mathcal S}im \left( \boldsymbol\varphii^2/12 \right) N^2
\end{equation}
as $N \to \infty$.
\end{rem}
The organization of this paper is as follows. In Section~\ref{IWR}, we start by reducing the problem to integral WR (abbreviated IWR) lattices in Lemma~\ref{reduce}: a planar lattice ${\mathcal L}ambda = A{\mathbb Z}^2$ is called {\it integral} if the coefficient matrix $A^t A$ of its quadratic form $Q_A$ has integer entries (this definition does not depend on the choice of a basis). We then introduce zeta-functions of similarity classes of planar IWR lattices, objects of independent interest, and study their convergence properties in Theorem~\ref{IWR_zeta}. Our arguments build on the parameterization of planar IWR lattices obtained in \cite{fletcher_jones}. In Section~\ref{IWR_subl} we continue using this parameterization to obtain an explicit description of IWR sublattices of a fixed planar IWR lattice which are similar to another fixed IWR lattice (Theorem~\ref{two_IWR_1}), and use it to determine convergence properties of the Dirichlet series generating function of all such sublattices (Lemma~\ref{zeta_two}). Finally, in Lemma~\ref{zeta_two_1} we decompose $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ for a fixed IWR planar lattice ${\mathcal O}mega$ into a sum over similarity classes of sublattices and observe that this sum can be represented as a product of the two different types of Dirichlet series that we investigated above; hence the result of Theorem~\ref{main} follows by Lemma~\ref{reduce}.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}gskip
{\mathcal S}ection{Integral WR lattices in the plane}
\label{IWR}
Integral lattices are central objects in arithmetic theory of quadratic forms and in lattice theory. IWR lattices have recently been studied in \cite{fletcher_jones}. The significance of IWR planar lattices for our purposes is reflected in the following reduction lemma.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{lem} \label{reduce} Let ${\mathcal O}mega$ be an arithmetic planar lattice. Then there exists some IWR planar lattice ${\mathcal L}ambda$ such that $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ has the same abscissa of convergence with pole of the same order as $\zeta_{\operatorname{WR}}({\mathcal L}ambda,s)$.
\end{lem}
\boldsymbol\varphiroof Lemma 2.1 of \cite{kuehnlein} guarantees that ${\mathcal O}mega$ has a WR sublattice, call it ${\mathcal O}mega'$; naturally, ${\mathcal O}mega'$ must also be arithmetic. Let $A$ be a basis matrix for ${\mathcal O}mega'$, then entries of $A^tA$ span a 1-dimensional vector space over ${\mathbb Q}$, meaning that there exists $\alpha \in {\mathbb R}_{>0}$ such that the matrix $\alpha A^tA$ is integral. Then the lattice ${\mathcal L}ambda := {\mathcal S}qrt{\alpha} A {\mathbb Z}^2$ is integral and is similar to ${\mathcal O}mega'$, hence it is also WR. Since ${\mathcal L}ambda$ is just a scalar multiple of ${\mathcal O}mega'$, it is clear that $\zeta_{\operatorname{WR}}({\mathcal L}ambda,s)$ has the same abscissa of convergence with pole of the same order as $\zeta_{\operatorname{WR}}({\mathcal O}mega',s)$, which is the same as that of $\zeta_{\operatorname{WR}}({\mathcal O}mega,s)$ by Lemma~3.2 of~\cite{kuehnlein}.
\endproof
Moreover, it is easy to see that these properties of zeta-function of WR sublattices are preserved under similarity.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{lem} \label{sim_red} Assume that ${\mathcal L}ambda_1, {\mathcal L}ambda_2$ are two planar lattices such that ${\mathcal L}ambda_1 {\mathcal S}im {\mathcal L}ambda_2$. Then $\zeta_{\operatorname{WR}}({\mathcal L}ambda_1,s) = \zeta_{\operatorname{WR}}({\mathcal L}ambda_2,s)$.
\end{lem}
\boldsymbol\varphiroof Similar lattices have the same numbers of WR sublattices of the same indices. The statement of the lemma follows immediately.
\endproof
Lemmas~\ref{reduce} and~\ref{sim_red} imply that we can focus our attention on similarity classes of IWR lattices to prove Theorem~\ref{main}. Integrality is not preserved under similarity, however a WR similarity class may or may not contain integral lattices. WR similarity classes containing integral lattices, we will call them IWR similarity classes, have been studied in \cite{fletcher_jones} -- these are precisely the WR similarity classes containing arithmetic lattices. Let us write $\left< {\mathcal L}ambda \right>$ for the similarity class of the lattice ${\mathcal L}ambda$, then a result of \cite{fletcher_jones} states that the set of IWR similarity classes is
$${\mathcal I}WR = \left\{ \left< \Gamma_D(p,q) \right> : \Gamma_D(p,q) = \frac{1}{{\mathcal S}qrt{q}} {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q & p \\ 0 & r{\mathcal S}qrt{D} \end{pmatrix} {\mathbb Z}^2 \right\},$$
where $(p,r,q,D)$ are all positive integer 4-tuples satisfying
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{prqD}
p^2+Dr^2=q^2,\ \gcd(p,q)=1,\ \frac{p}{q} \leq \frac{1}{2}, \text{ and } D \text{ squarefree}.
\end{equation}
It is also discussed in \cite{fletcher_jones} that $\Gamma_D(p,q)$ is a {\it minimal} integral lattice with respect to norm in its similarity class. In particular, every integral lattice ${\mathcal L}ambda \in \left< \Gamma_D(p,q) \right>$ is of the form ${\mathcal L}ambda = {\mathcal S}qrt{k}\ U \Gamma_D(p,q)$ for some $k \in {\mathbb Z}_{>0}$, $U \in O_2({\mathbb R})$, and so
$$|{\mathcal L}ambda| \geq |\Gamma_D(p,q)| = q.$$
The set ${\mathcal I}WR$ can be represented as
$${\mathcal I}WR = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}gsqcup_{D \in {\mathbb Z}_{>0} \text{ squarefree}} {\mathcal I}WR(D),$$
where for each fixed positive squarefree integer $D$, ${\mathcal I}WR(D) := \left\{ \left< \Gamma_D(p,q) \right> \right\}$ is the set of IWR similarity classes of {\it type} $D$.
Let us define the {\it minimum} and {\it determinant zeta-functions} of IWR similarity classes of type $D$ in the plane:
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{z^m_IWR}
\zeta^m_{{\mathcal I}WR(D)}(s) = {\mathcal S}um_{\left< \Gamma_D(p,q) \right> \in {\mathcal I}WR(D)} \frac{1}{|\Gamma_D(p,q)|^s} = {\mathcal S}um_{\left< \Gamma_D(p,q) \right> \in {\mathcal I}WR(D)} \frac{1}{q^s},
\end{equation}
and
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{z^d_IWR}
\zeta^d_{{\mathcal I}WR(D)}(s) = {\mathcal S}um_{\left< \Gamma_D(p,q) \right> \in {\mathcal I}WR(D)} \frac{1}{\operatorname{det} \Gamma_D(p,q)^s} = \frac{1}{D^{s/2}} {\mathcal S}um_{\left< \Gamma_D(p,q) \right> \in {\mathcal I}WR(D)} \frac{1}{r^s},
\end{equation}
where $s \in {\mathbb C}$. Since
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{rq}
\frac{{\mathcal S}qrt{3}}{2} \times \frac{1}{{\mathcal S}qrt{D}} \times q \leq r \leq \frac{1}{{\mathcal S}qrt{D}} \times q,
\end{equation}
we have
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{zeta_md_ineq}
\zeta^m_{{\mathcal I}WR(D)}(s) \leq \zeta^d_{{\mathcal I}WR(D)}(s) \leq \left( \frac{2}{{\mathcal S}qrt{3}} \right)^s \zeta^m_{{\mathcal I}WR(D)}(s)
\end{equation}
for all real $s$, and so $\zeta^m_{{\mathcal I}WR(D)}(s)$ and $\zeta^d_{{\mathcal I}WR(D)}(s)$ have the same convergence properties. We can establish the following result.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{thm} \label{IWR_zeta} For every real value of $s > 1$,
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{zeta_bnd}
\frac{1}{\left( 2{\mathcal S}qrt{3D} \right)^s} \frac{\zeta(2s-1)}{\zeta(2s)} \leq \zeta^d_{{\mathcal I}WR(D)}(s) \leq \left( \frac{2}{{\mathcal S}qrt{3}} \right)^s \zeta^m_{{\mathcal I}WR(D)}(s) \leq \left( \frac{4D}{{\mathcal S}qrt{3}} \right)^s \zeta_{{\mathbb Q}({\mathcal S}qrt{-D})}(s),
\end{equation}
where $\zeta(s)$ is the Riemann zeta-function and $\zeta_{{\mathbb Q}({\mathcal S}qrt{-D})}(s)$ is the Dedekind zeta-function of the imaginary quadratic number field ${\mathbb Q}({\mathcal S}qrt{-D})$. Hence the Dirichlet series $\zeta^d_{{\mathcal I}WR(D)}(s)$ and $\zeta^m_{{\mathcal I}WR(D)}(s)$ are absolutely convergent for ${\mathcal R}e(s) > 1$, and for $s \in {\mathbb R}$ the limits
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{lim_md}
\lim_{s \to 1^+} (s-1) \zeta^d_{{\mathcal I}WR(D)}(s),\ \lim_{s \to 1^+} (s-1) \zeta^m_{{\mathcal I}WR(D)}(s)
\end{equation}
exist and are nonzero. Moreover, the $N$-th partial sums of coefficients of these Dirichlet series are equal to $O(N)$ as $N \to \infty$.
\end{thm}
\boldsymbol\varphiroof Let $D$ be a fixed positive squarefree integer. Lemma~1.3 of \cite{fletcher_jones} guarantees that $p,r,q \in {\mathbb Z}_{>0}$ satisfy \eqref{prqD} if and only if
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{mn_par}
p = \frac{| m^2-Dn^2 |}{2^e \gcd(m,D)},\ r = \frac{2mn}{2^e \gcd(m,D)},\ q = \frac{m^2 + Dn^2}{2^e \gcd(m,D)},
\end{equation}
for some $m,n \in {\mathbb Z}$ with $\gcd(m,n)=1$ and ${\mathcal S}qrt{\frac{D}{3}} \leq \frac{m}{n} \leq {\mathcal S}qrt{3D}$, where
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{e_def}
e = \left\{ {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{array}{ll}
0 & \mbox{if either $2 \mid D$, or $2 \mid (D+1), mn$} \\
1 & \mbox{otherwise.}
\end{array}
\right.
\end{equation}
Then
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{z_IWR_mn}
\zeta^m_{{\mathcal I}WR(D)}(s) = {\mathcal S}um_{{\mathcal S}ubstack{m,n \in {\mathbb Z}_{>0},\ \gcd(m,n)=1 \\ {\mathcal S}qrt{\frac{D}{3}} \leq \frac{m}{n} \leq {\mathcal S}qrt{3D}}} \left( \frac{2^e \gcd(m,D)}{m^2 + Dn^2} \right)^s,
\end{equation}
and so for each real $s > 1$,
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{eqnarray}
\label{z_IWR_mn_up}
\zeta^m_{{\mathcal I}WR(D)}(s) & \leq & (2D)^s {\mathcal S}um_{{\mathcal S}ubstack{m,n \in {\mathbb Z} {\mathcal S}etminus \{0\} \\ {\mathcal S}qrt{\frac{D}{3}} \leq \frac{m}{n} \leq {\mathcal S}qrt{3D}}} \frac{1}{ \left( m^2 + Dn^2 \right)^s} \nonumber \\
& \leq & \left( 2D \right)^s {\mathcal S}um_{m,n \in {\mathbb Z} {\mathcal S}etminus \{0\}} \frac{1}{ \left( m^2 + Dn^2 \right)^s} = \left( 2D \right)^s \zeta_{{\mathbb Q}({\mathcal S}qrt{-D})}(s).
\end{eqnarray}
Now, the Dedekind zeta-function of a number field converges absolutely for ${\mathcal R}e(s) > 1$ and has a simple pole at $s=1$.
On the other hand, for all real $s >1$,
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{eqnarray}
\label{z_IWR_mn_low}
\zeta^d_{{\mathcal I}WR(D)}(s) & \geq & {\mathcal S}um_{{\mathcal S}ubstack{m,n \in {\mathbb Z} {\mathcal S}etminus \{0\},\ \gcd(m,n)=1 \\ {\mathcal S}qrt{\frac{D}{3}} \leq \frac{m}{n} \leq {\mathcal S}qrt{3D}}} \frac{1}{\left( 2mn \right)^s} \nonumber \\
& \geq & \frac{1}{\left( 2{\mathcal S}qrt{3D} \right)^s} {\mathcal S}um_{n=1}^{\infty} \frac{a_n}{n^{2s}},
\end{eqnarray}
where $a_n$ is the cardinality of the set
$$S_n = \left\{ m \in {\mathbb Z}_{>0} : n{\mathcal S}qrt{\frac{D}{3}} \leq m \leq n {\mathcal S}qrt{3D},\ \gcd(m,n)=1 \right\}.$$
We will now produce a lower bound on $a_n$ for every $n \geq 1$. For each $m \in S_n$, let $s_n(m) = m \operatorname{mod} n$, then
$$a_n = |S_n| \geq \left| \left\{ s_n(m) : m \in S_n \right\} \right|.$$
Notice that
$${\mathcal S}qrt{3D} - {\mathcal S}qrt{D/3} = {\mathcal S}qrt{D} ({\mathcal S}qrt{3} - 1/{\mathcal S}qrt{3}) > 1$$
for each $D$, and hence
$$\left\{ s_n(m) : m \in S_n \right\} = \left\{ k \in {\mathbb Z} : 1 \leq k < n, \gcd(k,n) =1 \right\},$$
meaning that $a_n \geq \varphi(n)$, the Euler $\varphi$-function of $n$. Therefore
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{zeta_phi}
\zeta^d_{{\mathcal I}WR(D)}(s) \geq \frac{1}{\left( 2{\mathcal S}qrt{3D} \right)^s} {\mathcal S}um_{n=1}^{\infty} \frac{\varphi(n)}{n^{2s}} = \frac{1}{\left( 2{\mathcal S}qrt{3D} \right)^s} \frac{\zeta(2s-1)}{\zeta(2s)}
\end{equation}
for all real $s > 1$ by Theorem 288 of \cite{hardy}. The right hand side of \eqref{zeta_phi} converges absolutely for ${\mathcal R}e(s) > 1$ and has a simple pole at $s=1$. The inequality \eqref{zeta_bnd} now follows upon combining \eqref{zeta_md_ineq} with \eqref{z_IWR_mn_up} and~\eqref{zeta_phi}.
Since each Dirichlet series can be written in the form ${\mathcal S}um_{n=1} b_n n^{-s}$ for some coefficient sequence $\{b_n\}_{n=1}^{\infty}$, we will refer to ${\mathcal S}um_{n=1}^N b_n$ as its $N$-th partial sum of coefficients. Now Theorem~4.20 of \cite{sautoy} guarantees that the $N$-th partial sums of coefficients of Dirichlet series $\frac{1}{\left( 2{\mathcal S}qrt{3D} \right)^s} \frac{\zeta(2s-1)}{\zeta(2s)}$ and $\left( \frac{4D}{{\mathcal S}qrt{3}} \right)^s \zeta_{{\mathbb Q}({\mathcal S}qrt{-D})}(s)$ are equal to $O(N)$ as $N \to \infty$. Inequality \eqref{zeta_bnd} implies that the same must be true about $N$-th partial sums of coefficients of Dirichlet series $\zeta^m_{{\mathcal I}WR(D)}(s)$ and $\zeta^d_{{\mathcal I}WR(D)}(s)$, and that $\zeta^m_{{\mathcal I}WR(D)}(s)$ and $\zeta^d_{{\mathcal I}WR(D)}(s)$ are absolutely convergent for ${\mathcal R}e(s) > 1$ with limits in \eqref{lim_md} existing and nonzero for $s \in {\mathbb R}$. This finishes the proof of the theorem.
\endproof
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{rem} \label{ht_zeta} There is a connection between the zeta-function $\zeta^m_{{\mathcal I}WR(D)}(s)$ and the height zeta-function of the corresponding Pell-type rational conic. One can define a height function on points ${{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x} = (x_1, x_2,x_3) \in {\mathbb Z}^3$ as
$$H({{{\boldsymbol 1}_Ldsymbol 0}ldsymbol x}) = \frac{1}{\gcd(x_1,x_2,x_3)} \max_{1 \leq i \leq 3} |x_i|.$$
It is easy to see that $H$ is in fact projectively defined, and hence induces a function on a rational projective space. Let $D$ be a fixed positive squarefree integer, then the set of all integral points $(p,r,q)$ satisfying
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{int_con}
p^2+Dr^2=q^2,\ \gcd(p,r,q)=1,\ q > 0
\end{equation}
is precisely the set of all distinct representatives of projective rational points on the Pell-type conic
$$X_D({\mathbb Q}) = \{ [x,y,z] \in {\mathbb P}({\mathbb Q}^3) : x^2+Dy^2=z^2 \}.$$
For each point $[x,y,z] \in X_D({\mathbb Q})$ there is a unique $(p,r,q)$ satisfying \eqref{int_con}, and
$$H([x,y,z]) = H(p,r,q)=q.$$
Hence the height zeta-function of $X_D({\mathbb Q})$ is
$${\mathcal S}um_{[x,y,z] \in X_D({\mathbb Q})} \frac{1}{H([x,y,z])^s} = {\mathcal S}um_{(p,r,q) \text{ as in \eqref{int_con}}} \frac{1}{q^s},$$
where $s \in {\mathbb C}$.
\end{rem}
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}gskip
{\mathcal S}ection{IWR sublattices of IWR lattices}
\label{IWR_subl}
In this section we further investigate distribution properties of planar IWR lattices and prove Theorem~\ref{main}. Theorem~1.3 of \cite{fletcher_jones} guarantees that every IWR lattice of type $D$ contains IWR sublattices belonging to every similarity class of this type, and none others. Hence $\zeta^m_{{\mathcal I}WR(D)}(s)$ and $\zeta^d_{{\mathcal I}WR(D)}(s)$ are zeta-functions of minimal lattices over similarity classes of IWR sublattices of any IWR lattice of type $D$ in the plane. It will be convenient to define
$${\mathcal O}mega_D(p,q) = {\mathcal S}qrt{q}\ \Gamma_D(p,q) = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q & p \\ 0 & r{\mathcal S}qrt{D} \end{pmatrix} {\mathbb Z}^2$$
for each $(p,r,q,D)$ satisfying \eqref{prqD}. Then for a fixed choice of $D,p_0,q_0$ the lattice ${\mathcal O}mega_D(p_0,q_0)$ contains IWR sublattices similar to each ${\mathcal O}mega_D(p,q)$. We will now describe explicitly how these sublattices look like. We start with a simple example of such lattices.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{lem} \label{two_IWR} Let $(p,r,q,D)$ and $(p_0,r_0,q_0,D)$ satisfy \eqref{prqD}. Let
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{kmn}
k=m^2+Dn^2
\end{equation}
for some $m,n \in {\mathbb Z}$, not both zero, and let
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{U}
U = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} \frac{m}{{\mathcal S}qrt{k}} & -\frac{n{\mathcal S}qrt{D}}{{\mathcal S}qrt{k}} \\ \frac{n{\mathcal S}qrt{D}}{{\mathcal S}qrt{k}} & \frac{m}{{\mathcal S}qrt{k}} \end{pmatrix}.
\end{equation}
Then $U$ is a real orthogonal matrix such that the lattice
$${\mathcal L}ambda = {\mathcal S}qrt{k}\ r_0q_0 U {\mathcal O}mega_D(p,q)$$
is an IWR sublattice of ${\mathcal O}mega_D(p_0,q_0)$ similar to ${\mathcal O}mega_D(p,q)$ with
$$\left| {\mathcal O}mega_D(p_0,q_0) : {\mathcal L}ambda \right| = r_0q_0rqk.$$
\end{lem}
\boldsymbol\varphiroof As indicated in the proof of Theorem~1.3 of \cite{fletcher_jones},
$${{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q_0 & p_0 \\ 0 & r_0{\mathcal S}qrt{D} \end{pmatrix} {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} r_0q & r_0p-rp_0 \\ 0 & rq_0 \end{pmatrix} = r_0q_0 {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q & p \\ 0 & r{\mathcal S}qrt{D} \end{pmatrix},$$
and so $r_0q_0 {\mathcal O}mega_D(p,q)$ is a sublattice of ${\mathcal O}mega_D(p_0,q_0)$ of index $rq$. Now notice that
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{eqnarray*}
{\mathcal L}ambda & = & {\mathcal S}qrt{k} r_0q_0 {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} \frac{m}{{\mathcal S}qrt{k}} & -\frac{n{\mathcal S}qrt{D}}{{\mathcal S}qrt{k}} \\ \frac{n{\mathcal S}qrt{D}}{{\mathcal S}qrt{k}} & \frac{m}{{\mathcal S}qrt{k}} \end{pmatrix} {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q & p \\ 0 & r{\mathcal S}qrt{D} \end{pmatrix} {\mathbb Z}^2 \\
& = & {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q_0 & p_0 \\ 0 & r_0{\mathcal S}qrt{D} \end{pmatrix} {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} mr_0q - np_0q & m(r_0p-p_0r)-n(p_0p+Dr_0r) \\ nq_0q & mrq_0+npq_0 \end{pmatrix} {\mathbb Z}^2,
\end{eqnarray*}
and hence is a sublattice of ${\mathcal O}mega_D(p_0,q_0)$ of index $r_0q_0rqk$.
\endproof
Lemma~\ref{two_IWR} demonstrates some examples of sublattices of ${\mathcal O}mega_D(p_0,q_0)$ similar to ${\mathcal O}mega_D(p,q)$. We will now describe all such sublattices.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{thm} \label{two_IWR_1} A sublattice ${\mathcal L}ambda$ of ${\mathcal O}mega_D(p_0,q_0)$ is similar to ${\mathcal O}mega_D(p,q)$ as above if and only if
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{subl}
{\mathcal L}ambda = {\mathcal S}qrt{Q_{p_0,q_0,p,q}(m,n)}\ U \Gamma_D(p,q),
\end{equation}
for some $m,n \in {\mathbb Z}$, not both zero, where $Q_{p_0,q_0,p,q}(m,n)$ is a positive definite binary quadratic form, given by \eqref{Q} below, and $U$ is a real orthogonal matrix as in \eqref{U_orth} with the angle $t$ satisfying \eqref{sin_cos}, where $x,y$ are as in \eqref{cong_sol_1} or \eqref{cong_sol_2}. In this case,
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{index}
\left| {\mathcal O}mega_D(p_0,q_0) : {\mathcal L}ambda \right| = \frac{rQ_{p_0,q_0,p,q}(m,n)}{r_0q_0}.
\end{equation}
\end{thm}
\boldsymbol\varphiroof By Theorem~1.1 of \cite{fletcher_jones}, ${\mathcal L}ambda {\mathcal S}im {\mathcal O}mega_D(p,q)$ if and only if
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{L1}
{\mathcal L}ambda = {\mathcal S}qrt{\frac{k}{q}}\ U {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q & p \\ 0 & r{\mathcal S}qrt{D} \end{pmatrix} {\mathbb Z}^2
\end{equation}
for some positive integer $k$ and a real orthogonal matrix
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{U_orth}
U = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} \cos t & -{\mathcal S}in t \\ {\mathcal S}in t & \cos t \end{pmatrix} \text{ or } {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} \cos t & {\mathcal S}in t \\ {\mathcal S}in t & -\cos t \end{pmatrix}
\end{equation}
for some value of the angle $t$. On the other hand, ${\mathcal L}ambda {\mathcal S}ubset {\mathcal O}mega_D(p_0,q_0)$ if and only if
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{L2}
{\mathcal L}ambda = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q_0 & p_0 \\ 0 & r_0{\mathcal S}qrt{D} \end{pmatrix} C {\mathbb Z}^2,
\end{equation}
where $C$ is an integer matrix. Therefore ${\mathcal L}ambda$ as in \eqref{L1} is a sublattice of ${\mathcal O}mega_D(p_0,q_0)$ if and only if it is of the form \eqref{L2} with
$$C = \alpha\ {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q(r_0{\mathcal S}qrt{D}\cos t - p_0{\mathcal S}in t) & (r_0p-rp_0){\mathcal S}qrt{D}\cos t - (pp_0+rr_0D){\mathcal S}in t \\ qq_0{\mathcal S}in t & q_0p{\mathcal S}in t + q_0r{\mathcal S}qrt{D}\cos t \end{pmatrix}$$
or
$$C = \alpha\ {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q(r_0{\mathcal S}qrt{D}\cos t - p_0{\mathcal S}in t) & (r_0p+rp_0){\mathcal S}qrt{D}\cos t - (pp_0-rr_0D){\mathcal S}in t \\ qq_0{\mathcal S}in t & q_0p{\mathcal S}in t - q_0r{\mathcal S}qrt{D}\cos t \end{pmatrix}$$
where $\alpha = \frac{{\mathcal S}qrt{k}}{q_0r_0 {\mathcal S}qrt{qD}}$. These conditions imply that we must have
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{sin_cos}
\cos t = \frac{xp_0+yq_0}{{\mathcal S}qrt{qk}},\ {\mathcal S}in t = \frac{xr_0{\mathcal S}qrt{D}}{{\mathcal S}qrt{qk}}
\end{equation}
for some integers $x,y$ satisfying one of the following two systems of congruences:
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{cong_1}
\left. {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{array}{ll}
q_0rx + (p_0r - r_0p)y \equiv 0 (\operatorname{mod} qr_0) \\
(p_0r + r_0p)x + q_0ry \equiv 0 (\operatorname{mod} qr_0)
\end{array}
\right\},
\end{equation}
or
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{cong_2}
\left. {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{array}{ll}
q_0rx + (p_0r + r_0p)y \equiv 0 (\operatorname{mod} qr_0) \\
(p_0r - r_0p)x + q_0ry \equiv 0 (\operatorname{mod} qr_0)
\end{array}
\right\}.
\end{equation}
First assume \eqref{cong_1} is satisfied. Notice that
$$\operatorname{det} {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{pmatrix} q_0r & p_0r-r_0p \\ p_0r+r_0p & q_0r \end{pmatrix} = (qr_0)^2 \equiv 0 (\operatorname{mod} qr_0),$$
which means that a pair $(x,y)$ solves the system \eqref{cong_1} if and only if it solves one of these two congruences. Hence it is enough to solve the first congruence of \eqref{cong_1}. Define $d_1 = \gcd(q_0r,qr_0)$ and $d_2 = \gcd(d_1,p_0r - r_0p)$, and let $a,b \in {\mathbb Z}$ be such that
$$aq_0r+bqr_0=d_1.$$
It now easily follows that the set of all possible solutions to \eqref{cong_1} is
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{cong_sol_1}
(x,y) = \left\{ \left( \frac{a(r_0p-p_0r)n}{d_2} + \frac{qr_0m}{d_1},\ \frac{d_1n}{d_2} \right) : n,m \in {\mathbb Z} \right\}.
\end{equation}
Combining \eqref{sin_cos} with \eqref{cong_sol_1}, we see that
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{qk1}
qk = \left( \frac{\left( ap_0(r_0p-p_0r) + d_1q_0 \right)n}{d_2} + \frac{qp_0r_0m}{d_1} \right)^2 + Dr_0^2 \left( \frac{a(r_0p-p_0r)n}{d_2} + \frac{qr_0m}{d_1} \right)^2.
\end{equation}
Then the right hand side of \eqref{qk1} is a positive definite integral binary quadratic form in the variables $m,n$:
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{eqnarray}
\label{Q1}
Q^1_{p_0,q_0,p,q}(m,n) & = & \left\{ \frac{q_0^2 q^2 r_0^2}{d_1^2} \right\} \times m^2 \nonumber \\
& + & \left\{ \frac{ a^2 (r_0p-p_0r)^2 q_0^2 + 2ad_1p_0q_0(r_0p-p_0r) + d_1^2q_0^2}{d_2^2} \right\} \times n^2 \nonumber \\
& + & \left\{ \frac{2a(r_0p-p_0r)qq_0^2r_0 + 2d_1qq_0r_0p_0}{d_1d_2} \right\} \times mn.
\end{eqnarray}
One can observe that all three coefficients of $Q^1_{p_0,q_0,p,q}(m,n)$ are divisible by $q$. Then define
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{Q}
Q_{p_0,q_0,p,q}(m,n) = \frac{1}{q} Q^1_{p_0,q_0,p,q}(m,n),
\end{equation}
which is again a positive definite integral binary quadratic form.
Now notice that the system of congruences in \eqref{cong_2} is the same as the one in \eqref{cong_1} with the order of equations reversed and the variables $x$ and $y$ reversed. Hence the solution set for \eqref{cong_2} is
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{cong_sol_2}
(x,y) = \left\{ \left( \frac{d_1n}{d_2},\ \frac{a(r_0p-p_0r)n}{d_2} + \frac{qr_0m}{d_1} \right) : n,m \in {\mathbb Z} \right\}.
\end{equation}
Combining \eqref{sin_cos} with \eqref{cong_sol_2}, we see that if \eqref{cong_2} is satisfied, then
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{qk2}
qk = \left( \frac{\left( aq_0(r_0p-p_0r)+d_1p_0 \right)n}{d_2} + \frac{qq_0r_0m}{d_1} \right)^2 + \frac{Dr_0^2d_1^2n^2}{d_2^2}.
\end{equation}
Then the right hand side of \eqref{qk2} is precisely $Q^1_{p_0,q_0,p,q}(n,m)$.
In either case, we have
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{k_value}
k = \frac{1}{q} Q^1_{p_0,q_0,p,q}(m,n) = Q_{p_0,q_0,p,q}(m,n)
\end{equation}
for some $m,n \in {\mathbb Z}$, not both zero. Then \eqref{subl} follows upon combining \eqref{L1} with \eqref{k_value}. Now we notice that
$$\left| {\mathcal O}mega_D(p_0,q_0) : {\mathcal L}ambda \right| = \frac{\operatorname{det} {\mathcal L}ambda}{\operatorname{det} {\mathcal O}mega_D(p_0,q_0)},$$
and so \eqref{index} follows from \eqref{subl}. This completes the proof of the lemma.
\endproof
Now define $S_D(p_0,q_0)$ to be the set of all IWR sublattices of ${\mathcal O}mega_D(p_0,q_0)$, and $S_D(p_0,q_0,p,q)$ to be the set of all IWR sublattices of ${\mathcal O}mega_D(p_0,q_0)$ which are similar to ${\mathcal O}mega_D(p,q)$. Then
$$S_D(p_0,q_0) = {{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}gsqcup S_D(p_0,q_0,p,q).$$
Define
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{Z_D_1}
Z_{D,p_0,q_0,p,q}(s) = {\mathcal S}um_{{\mathcal L}ambda \in S_D(p_0,q_0,p,q)} \frac{1}{\left| {\mathcal O}mega_D(p_0,q_0) : {\mathcal L}ambda \right|^s}
\end{equation}
and
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{Z_D_2}
Z_{D,p_0,q_0}(s) = {\mathcal S}um_{{\mathcal L}ambda \in S_D(p_0,q_0)} \frac{1}{\left| {\mathcal O}mega_D(p_0,q_0) : {\mathcal L}ambda \right|^s} = {\mathcal S}um_{(p,q) \text{ as in \eqref{prqD}}} Z_{D,p_0,q_0,p,q}(s)
\end{equation}
for $s \in {\mathbb C}$.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{lem} \label{zeta_two} For every squarefree positive integer $D$ and integer triples $(p_0,r_0,q_0)$ and $(p,r,q)$ satisfying \eqref{prqD}, the Dirichlet series $Z_{D,p_0,q_0,p,q}(s)$ is absolutely convergent for ${\mathcal R}e(s) > 1$. Moreover, it has analytic continuation to all of ${\mathbb C}$ except for a simple pole at $s=1$.
\end{lem}
\boldsymbol\varphiroof By Theorem~\ref{two_IWR_1},
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{eps1}
Z_{D,p_0,q_0,p,q}(s) = \left( \frac{r_0q_0}{r} \right)^s {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s},
\end{equation}
where the sum on the right hand side of \eqref{eps1} is the Epstein zeta-function of the positive definite integral binary quadratic form $Q_{p_0,q_0,p,q}(m,n)$; it is known to converge absolutely for ${\mathcal R}e(s) > 1$ and has analytic continuation to all of ${\mathbb C}$ except for a simple pole at $s=1$ (this is a classical result, which can be found for instance in Chapter~5, \S 5 of \cite{koecher}; in fact, the authors of \cite{koecher} indicate that the existence of a simple pole at $s=1$ goes as far back as the work of Kronecker, 1889). The lemma follows.
\endproof
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{lem} \label{zeta_two_1} For every squarefree positive integer $D$ and integer triple $(p_0,r_0,q_0)$ satisfying \eqref{prqD}, the Dirichlet series $Z_{D,p_0,q_0}(s)$ is absolutely convergent for ${\mathcal R}e(s) > 1$ and for $s \in {\mathbb R}$ the limit
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{lim_ZD}
\lim_{s \to 1^+} (s-1)^2 Z_{D,p_0,q_0}(s)
\end{equation}
exists and is nonzero. Moreover, if we write $Z_{D,p_0,q_0}(s) = {\mathcal S}um_{n=1}^{\infty} b_n n^{-s}$, then the $N$-th partial sum of coefficients of $Z_{D,p_0,q_0}(s)$ is
$${\mathcal S}um_{n=1}^N b_n = O(N \log N)$$
as $N \to \infty$.
\end{lem}
\boldsymbol\varphiroof Combining \eqref{Z_D_2}, \eqref{eps1}, and \eqref{rq} we obtain for every real $s>0$
$$Z_{D,p_0,q_0}(s) = (r_0q_0)^s {\mathcal S}um_{(p,r,q) \text{ as in \eqref{prqD}}} \left( \frac{1}{r^s} {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s} \right).$$
Combining this observation with Theorem~\ref{IWR_zeta} implies that
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{eqnarray}
\label{eps2}
&\ & \left( \frac{r_0q_0}{2{\mathcal S}qrt{3}} \right)^s \frac{\zeta(2s-1)}{\zeta(2s)} \inf_{(p,r,q) \text{ as in \eqref{prqD}}} {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s} \\
& \leq & \left( r_0q_0 {\mathcal S}qrt{D} \right)^s \zeta^d_{{\mathcal I}WR(D)}(s) \inf_{(p,r,q) \text{ as in \eqref{prqD}}} {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s} \nonumber \\
& \leq & Z_{D,p_0,q_0}(s) \nonumber \\
&\leq & \left( r_0q_0 {\mathcal S}qrt{D} \right)^s \zeta^d_{{\mathcal I}WR(D)}(s) {\mathcal S}up_{(p,r,q) \text{ as in \eqref{prqD}}} {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s} \nonumber \\
&\leq & \left( \frac{4r_0q_0D^{\frac{3}{2}}}{{\mathcal S}qrt{3}} \right)^s \zeta_{{\mathbb Q}({\mathcal S}qrt{-D})}(s) {\mathcal S}up_{(p,r,q) \text{ as in \eqref{prqD}}} {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s}. \nonumber
\end{eqnarray}
Theorem~\ref{IWR_zeta} and Lemma~\ref{zeta_two} now imply that for each $(p,r,q)$ as in \eqref{prqD} the Dirichlet series
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{Dir1}
\left( \frac{r_0q_0}{2{\mathcal S}qrt{3}} \right)^s \frac{\zeta(2s-1)}{\zeta(2s)} {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s}
\end{equation}
and
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol e}gin{equation}
\label{Dir2}
\left( \frac{4r_0q_0D^{\frac{3}{2}}}{{\mathcal S}qrt{3}} \right)^s \zeta_{{\mathbb Q}({\mathcal S}qrt{-D})}(s) {\mathcal S}um_{(m,n) \in {\mathbb Z}^2 {\mathcal S}etminus \{ {{\boldsymbol 1}_Ldsymbol 0} \}} \frac{1}{Q_{p_0,q_0,p,q}(m,n)^s}
\end{equation}
are absolutely convergent for ${\mathcal R}e(s) > 1$ and have analytic continuation to the half-plane ${\mathcal R}e(s) > 0$ except for a pole of order 2 at $s=1$. Then Theorem~4.20 of \cite{sautoy} implies that the $N$-th partial sums of coefficients of all the Dirichlet series as in \eqref{Dir1} and \eqref{Dir2} must be equal to $O(N \log N)$. Then \eqref{eps2} implies that the $N$-th partial sum of coefficients of $Z_{D,p_0,q_0}(s)$ is also $O(N \log N)$, and $Z_{D,p_0,q_0}(s)$ is absolutely convergent for ${\mathcal R}e(s) > 1$, where the limit of~\eqref{lim_ZD} exist and is nonzero for $s \in {\mathbb R}$.
\endproof
\boldsymbol\varphiroof[Proof of Theorem~\ref{main}] The theorem now follows upon combining Lemmas~\ref{reduce},~\ref{sim_red}, and~\ref{zeta_two_1}.
\endproof
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}gskip
{\bf Acknowledgment.} I would like to thank Stefan K\"uhnlein for providing me with a copy of his manuscript \cite{kuehnlein}, as well as for discussing the problem with me, and for his useful remarks about this paper. I would also like to thank Michael D. O'Neill and Bogdan Petrenko for their helpful comments on this paper. Finally, I would like to thank the referee for the suggestions and corrections which improved the quality of this paper.
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}gskip
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}bliographystyle{plain}
{{{\boldsymbol 1}_Ldsymbol 0}ldsymbol i}bliography{iwr_zeta}
\end{document}
|
\begin{document}
\begin{frontmatter}
\title{Common fixed
points and endpoints of multi-valued generalized weak contraction
mappings}
\author{Congdian Cheng}
\address{College of Mathematics and Systems
Science, Shenyang Normal University,
Shenyang, 110034, China}
\begin{abstract}
Let $(X, d)$ be a complete metric space, and let $S, T :
X\rightarrow
CB(X)$ be a duality of multi-valued generalized
weak contraction mappings or a duality of generalized $\varphi$-weak
contraction mappings. We discuss the common fixed points and
endpoints of the two kinds of multi-valued weak mappings. Our
results extend and improve some results given by Daffer and Kaneko
(1995), Rouhani and Moradi (2010), and Moradi and Khojasteh (2011).
\end{abstract}
\begin{keyword} multi-valued mapping\sep weak contraction\sep
common fixed point\sep common endpoint\sep Hausdorff metric
\MSC 47H10 \sep 54C60
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{1}Let $(X, d)$ be a metric space and $CB(X)$ denote the class
of closed and bounded subsets of X. Also let $S, T : X\rightarrow
2^X$ be a multi-valued mapping. A point x is called a fixed point of
$T$ if $x\in Tx$. Define $Fix(T) = \{x\in X : x \in Tx\}$. An
element $x \in X$ is said to be an endpoint (or stationary point) of
a multi-valued mapping $T$ if $Tx = \{x\}$. We denote the set of all
endpoints of $T$ by $End(T)$.
A bivariate mapping $\phi: X\times X\rightarrow[0, +\infty)$ is
called compactly positive if $\inf\{\phi(x,y): a\leq d(x,y)\leq
b\}>0$ for each finite interval $[a, b]\subseteq (0, +\infty)$. A
mapping $T : X \rightarrow CB(X)$ is called weakly contractive if
there exists a compactly positive mapping $\phi$
such that
$$H(Tx, Ty)
\leq d(x, y)-\phi(x, y)$$ for each $x, y \in X$, where
$$H(A, B) :=\max\{ \sup\limits_{x\in B}
d(x, A), \sup\limits_{x\in A} d(x, B)\},$$ denoting the Hausdorff
metric on $CB(X)$. (see [1].)
A mapping $T : X \rightarrow CB(X)$ is called an generalized
$\varphi$-weak contraction if there exists a map $\varphi: [0,
+\infty)\rightarrow[0, +\infty)$ with $\varphi(0)=0$ and
$\varphi(t)<t$ for all $t>0$ such that
$$H(Sx, Ty)
\leq \varphi(N(x, y))$$
for all $x, y \in X$, where
$$N(x, y)
:= \max \{d(x, y), d(x, Tx), d(y, Ty), \frac{d(x, Ty) + d(y,
Tx)}{2}\}. $$
Two mappings $S, T : X \rightarrow CB(X)$ ($S, T : X \rightarrow X$)
are called a duality of generalized weak contractions if there
exists a bivariate mapping $\alpha: X\times X\rightarrow[0, 1)$ such
that
$$H(Sx, Ty)
\leq \alpha(x, y)M(x, y)$$ for all $x, y \in X$ (or equivalently, if
there exists a bivariate mapping $\phi: [0, +\infty)\rightarrow[0,
+\infty)$ with $\phi(0)=0$ and $\phi(t)>0$ for all $t>0$
such that
$$H(Sx, Ty)
\leq M(x, y)-\phi(x, y)$$ for each $x, y \in X$), where
$$\begin{array}{rcl}M(x, y): =
\max\{d(x,y), d(x, Sx), d(y, Ty), \frac{d(x, Ty)+d(y,
Sx)}{2}\}.\end{array}$$
Also, two mappings $S, T : X \rightarrow CB(X)$ are called a duality
of generalized $\varphi$-weak contractions if there exists a
bivariate mapping $\varphi: [0, +\infty)\rightarrow[0, +\infty)$
with $\varphi(0)=0$ and $\varphi(t)<t$ for all $t>0$ such that
$$H(Sx, Ty)
\leq \varphi(M(x, y))$$
for all $x, y \in X$ (or
equivalently, if there exists a bivariate mapping $\varphi: [0,
+\infty)\rightarrow[0, +\infty)$ with $\varphi(0)=0$ and
$\varphi(t)>0$ for all $t>0$ such that
$$H(Sx, Ty)
\leq M(x, y)-\varphi(M(x, y))$$ for all $x, y \in X$).
A mapping $T : X \rightarrow CB(X)$ has the approximate endpoint
property if $$\inf\limits_{x\in X}\sup\limits_{y\in Tx}d(x, y) =
0.$$
The fixed points for multi-valued contraction mappings have been
the subject of the research area on fixed points for more than forty
years, for example, see [1-5] and the references therein. The
investigation of endpoints of multi-valued mappings was made as
early as 30 years ago, and has received great attention in recent
years, see e.g. [5-10]. Among other studies, several important
results related closely to the present work are as follows.
First, in the following theorem, Nadler [2] (1969) extended the
Banach contraction principle to multi-valued mappings.\\
\noindent{\bf Theorem 1.1.} \textit{Let $(X, d)$ be a complete
metric space. Suppose that $T : X \rightarrow CB(X)$ is a
contraction mapping in the sense that for some $0\leq \alpha<1,
H(Tx, Ty) \leq \alpha d(x, y)$ for all $x, y \in X$. Then there
exists a point $x \in X$ such that $x \in Tx$.}\\
\noindent Then, Daffer and Kaneko [1] (1995) proved the next Theorem
1.2 and
Theorem 1.3.\\
\noindent{\bf Theorem 1.2} ([1, Theorem 3.3]) \textit{Let $(X, d)$
be a complete metric space. Suppose that $T : X \rightarrow CB(X)$
be such that $H(Tx, Ty) \leq \alpha N(x, y)$ for $0\leq\alpha<1$,
for all $x, y \in X$. If $x\rightarrow d(x, Tx)$ is lower
semicontinuous (l.s.c.), then there
exists a point $x_0\in X$ such that $x_0\in Tx_0$.}\\
\noindent{\bf Theorem 1.3} ([1, Theorem 2.3]). \textit{Let $(X, d)$
be a complete metric and $T : X \rightarrow CB(X)$ weakly
contractive. Assume that
$$\liminf
\limits_{ \beta\rightarrow 0}\frac{\lambda(\alpha, \beta)}{\beta} >
0 \hspace{3mm}(0<\alpha\leq\beta),$$ where $\lambda(\alpha,
\beta)=\inf\{\phi(x, y)|x, y\in X, \alpha\leq d(x, y)\leq\beta\}$
for each finite interval $[\alpha, \beta]\subset(0, \infty)$. Then
$T$
has a fixed point in $X$.}\\
\noindent Lately, Zhang and Song [3, Theorem 2.1] (2009) proved a
theorem on the existence of a common fixed point for a duality of
two single valued generalized $\varphi$-weak contraction mappings.
By extending two single valued mappings in the Theorem of Zhang
and Song [3] to two multi-valued mappings, and By extending one
multi-valued mapping in Theorem 1.2 to a duality of multi-valued
mappings, Rouhani and Moradi [4] (2010) proved the following
coincidence theorem, without assuming $x\longrightarrow d(x, Tx)$ or
$x\rightarrow d(x, Sx)$ to be l.s.c.\\
\noindent{\bf Theorem 1.4} ([4, Theorem 3.1]). \textit{Let $(X, d)$
be a complete metric space, and let $T, S : X \rightarrow CB(X)$ be
two multivalued mappings such that for all $x, y\in X, H(Tx,Sy) \leq
\alpha M(x, y)$, where $0 \leq \alpha <1$. Then there exists a point
$x \in X$ such that $x \in Tx$ and $x \in Sx$ (i.e., $T$ and $S$
have a common fixed point). Moreover, if either $T$ or $S$ is single
valued, then
this common fixed point is unique.}\\\\
Further they also proved the Theorem 1.5 below.\\
\noindent{\bf Theorem 1.5} ([4, Theorem 4.1]). \textit{Let $(X, d)$
be a complete metric space and let $T : X \rightarrow X$ and $S : X
\rightarrow CB(X)$ be two mappings such that for all $x, y \in X$,
$$H({Tx}, Sy)\leq M(x, y)-\varphi(M(x, y)),$$
where $\varphi: [0,
+\infty)\rightarrow[0, +\infty)$ is l.s.c. with $\varphi(0)=0$ and
$\varphi(t)>0$ for all $t>0$. Then there exists a unique point
$x \in X$ such that $Tx = x \in Sx$.}\\
\noindent Finally, for the endpoint of multi-valued mappings,
Amini-Harandi [6] (2010) proved Theorem 1.6 below.\\
\noindent{\bf
Theorem 1.6} ([6, Theorem 2.1]).\textit{Let $(X, d)$ be a complete
metric space and $T$ be a multi-valued mapping that satisfies
$$H(Tx, Ty)
\leq \varphi(d(x, y)),$$ for each $x, y \in X$ , where $\varphi :
[0,+\infty) \rightarrow [0,+\infty)$ is upper semicontinuous
(u.s.c.), $\varphi(t) < t$ for each $t
> 0$ and satisfies $\liminf \limits_{
t\rightarrow \infty }(t - \varphi(t)) > 0$. Then $T$ has a unique
endpoint if and only if T has the approximate endpoint property.}\\
\noindent Moradi and Khojasteh [7] (2011) extended the result of
Amini-Harandi
to the following theorem 1.7.\\
\noindent{\bf Theorem 1.7} ([7, Theorem 2.1]). \textit{Let $(X, d)$
be a complete metric space and $T$ be a multi-valued mapping that
satisfies
$$H(Tx, Ty)
\leq \varphi(N(x, y)), $$ for each $ x, y \in X$ , where $\varphi :
[0,+\infty) \rightarrow [0,+\infty)$ is u.s.c. with $\varphi(t) < t$
for all $t
> 0$ and $\liminf \limits_{
t\rightarrow \infty }(t - \varphi(t)) > 0$. Then $T$ has a unique
endpoint if and only if T has the approximate endpoint property.}\\
Motivated by the contributions stated above, the present work make a
further study on the common fixed point for a duality of generalized
weak ($\varphi$-weak) contractions, and also make a study on the
common endpoint for a duality of generalized weak ($\varphi$-weak)
contractions. Our contributions extend the results of Theorem 1.3,
Theorem 1.4, Theorem 1.5 and Theorem 1.7.
\section{Preliminaries}
\label{1} This section proposes several Lemmas for our posterior
discussions.\\
\noindent{\bf Lemma 2.1.} \textit{Let $(X, d)$ be a complete metric
space
and $S, T : X\rightarrow CB(X)$ are a duality of generalized weak
(or
$\varphi$-weak) contractions. Then $Fix(S)=Fix(T)$.}\\
\noindent{\bf Proof}. Let $x\in Fix(S)$. Then
$$\begin{array}{rcl}& &d(x, Tx)\leq H(Sx,Tx)\leq\alpha(x,x)M(x,x)\\
&=&\alpha(x,x)\max\{d(x,x), d(x,Sx), d(x,Tx), \frac{d(x,Tx)+d(x,Sx)
}{2}\}\\
&=&\alpha(x,x)d(x, Tx).\end{array}$$ Since $\alpha(x,x)<1$, this
implies $d(x,Tx)=0$.
That is, $x\in Fix(T)$. Hence, $Fix(S)=Fix(T)$.\\
\noindent{\bf Lemma 2.2.} \textit{Let $(X, d)$ be a complete metric
space, $\gamma\in[0,1)$,
and $x_{n}$ be a sequence of $X$ that satisfies
$$\begin{array}{rcl}d(x_n, x_{n+1}) \leq\gamma d(x_{n-1},
x_n)+\frac{1}{2^n}
\end{array}\eqno (2.1)$$
for all $n \in {\mathbb{N}}$ ($x_0 \in X$). Then $\{x_n\}$ is
convergent.}\\
\noindent{\bf Proof}. By (2.1), for each $n\in {\mathbb{N}}$,
$$\begin{array}{rcl} d(x_n, x_{n+1})&\leq &\gamma d(x_{n-1},
x_n)+\frac{1}{2^n}\\&\leq&\gamma[\gamma d(x_{n-2},
x_{n-1})+\frac{1}{2^{n-1}}]+\frac{1}{2^n}\\&=&\gamma^2 d(x_{n-2},
x_{n-1})+\frac{\gamma}{2^{n-1}}+\frac{1}{2^n}\\& &\cdots\\&\leq&
\gamma^n d(x_{0}, x_{1})+\frac{\gamma^{n-1}}{2^{1}}+\cdots
+\frac{\gamma^1}{2^{n-1}}+\frac{\gamma^0}{2^n}\\&\leq&\frac{M}{1-\gamma}
(\frac{\gamma^n}{2^0}+\frac{\gamma^{n-1}}{2^{1}}+\cdots
+\frac{\gamma^1}{2^{n-1}}+\frac{\gamma^0}{2^n}),
\end{array}$$
where $M=\max\{d(x_{0}, x_{1}),1\}$. Without loss of generality,
assume $M=1$. Then
$$\begin{array}{rcl} d(x_n, x_{n+1})\leq
\frac{\gamma^n}{2^0}+\frac{\gamma^{n-1}}{2^{1}}+\cdots
+\frac{\gamma^1}{2^{n-1}}+\frac{\gamma^0}{2^n}.
\end{array}\eqno
(2.2)$$ By (2.2), for any $n, m \in {\mathbb{N}}$, we have
$$\begin{array}{rcl}& & d(x_n, x_{n+m})\\&\leq& d(x_n, x_{n+1})+d(x_{n+1},
x_{n+2})+\cdots+d(x_{n+m-1},
x_{n+m})\\&\leq&\{[\frac{\gamma^n}{2^0}+\frac{\gamma^{n-1}}{2^1}
+\frac{\gamma^{n-2}}{2^2}+\cdots+\frac{\gamma^{0}}{2^n}]\\&
&+[\frac{\gamma^{n+1}}{2^0}+\frac{\gamma^{n}}{2^1}
+\frac{\gamma^{n-1}}{2^2}+\cdots+\frac{\gamma^{1}}{2^n}+\frac{\gamma^{0}}{2^{n+1}}]\\&
&\cdots\\& &+ [\frac{\gamma^{n+m-1}}{2^0}+\frac{\gamma^{n+m-2}}{2^1}
+\frac{\gamma^{n+m-3}}{2^2}+\cdots\\&
&+\frac{\gamma^{m-1}}{2^n}+\frac{\gamma^{m-2}}{2^{n+1}}+\cdots+
\frac{\gamma^{0}}{2^{n+m-1}}]\}\\&=&\{[\frac{1}{2^0}(\gamma^{n}+\gamma^{n+1}+
\cdots+\gamma^{n+m-1})+\\& &\frac{1}{2^1}(\gamma^{n-1}+\gamma^{n}+
\cdots+\gamma^{n+m-2})+\cdots\\&
&+\frac{1}{2^n}(\gamma^{0}+\gamma^{1}+
\cdots+\gamma^{m-1})]+[\frac{1}{2^{n+1}}(\gamma^{0}+\gamma^{1}+
\cdots+\gamma^{m-2})\\& &+\frac{1}{2^{n+2}}(\gamma^{0}+\gamma^{1}+
\cdots+\gamma^{m-3})\\&
&+\cdots+\frac{1}{2^{n+m-2}}(\gamma^{0}+\gamma^{1})+\frac{1}{2^{n+m-1}}(\gamma^{0})]\}
\\&=&\{[\frac{1}{2^0}(\frac{\gamma^{n}-\gamma^{n+m}}{1-\gamma})+
\frac{1}{2^1}(\frac{\gamma^{n-1}-\gamma^{n+m-1}}{1-\gamma})+\cdots+
\frac{1}{2^n}(\frac{\gamma^{0}-\gamma^{m-1}}{1-\gamma})]\\&
&+[\frac{1}{2^{n+1}}(\frac{\gamma^{0}-\gamma^{m-1}}{1-\gamma})+
\frac{1}{2^{n+2}}(\frac{\gamma^{0}-\gamma^{m-2}}{1-\gamma})+\cdots+
\frac{1}{2^{n+m-1}}(\frac{\gamma^{0}-\gamma^{1}}{1-\gamma})]\}
\\&<&\frac{1}{(1-\gamma)}[(\frac{\gamma^{n}}{2^0}+
\frac{\gamma^{n-1}}{2^1}+\cdots+ \frac{\gamma^{0}}{2^n})\\&
&+(\frac{1}{2^{n+1}}+ \frac{1}{2^{n+2}}+\cdots+
\frac{1}{2^{n+m-1}})]\\&=
&\frac{1}{(1-\gamma)}\{\gamma^{n}[\frac{1}{(2\gamma)^0}+
\frac{1}{(2\gamma)^1}+\cdots+
\frac{1}{(2\gamma)^n}]+\frac{\frac{1}{2^{n+1}}-\frac{1}{2^{n+m}}}{1-\frac{1}{2}}\}
\\&<&\frac{1}{(1-\gamma)}\{\gamma^{n}[\frac{1-\frac{1}{(2\gamma)^{n+1}}}{1-\frac{1}{(2\gamma)}}]
+\frac{1}{2^{n}}\}.\end{array}\eqno (2.3)$$ In terms of (2.3), if
$(2\gamma)>1$, then
$$\begin{array}{rcl}& & d(x_n, x_{n+m})\\&<& \frac{1}{(1-\gamma)}
\{\gamma^n[\frac{2\gamma}{2\gamma-1}]+\frac{1}{2^n}\}=\frac{1}{(1-\gamma)}\cdot
\frac{(2\gamma)^{n+1}+2\gamma-1}{(2\gamma-1)2^n}\\&<&\frac{2\gamma
}{(2\gamma-1)(1-\gamma)}\cdot (\gamma^n+\frac{1}{2^n})<\frac{4
\gamma^{n+1}}{(2\gamma-1)(1-\gamma)}.\end{array}\eqno (2.4)$$
Otherwise, $(2\gamma)<1$, we have
$$\begin{array}{rcl}& & d(x_n, x_{n+m})\\&=& \frac{1}{(1-\gamma)}
[\gamma^n\cdot\frac{1-(2\gamma)^{n+1}}{(2\gamma)^{n}-(2\gamma)^{n+1}}+\frac{1}{2^n}]<
\frac{1}{(1-\gamma)}
[\frac{\gamma^n}{(2\gamma)^{n}-(2\gamma)^{n+1}}+\frac{1}{2^n}]\\&=&\frac{1}{(1-\gamma)}
[\frac{1}{2^{n}(1-2\gamma)}+\frac{1}{2^n}]=\frac{1}{(1-\gamma)(1-2\gamma)2^{n-1}}.\end{array}\eqno
(2.5)$$
From (2.4) and (2.5), we can easily know that the sequence $\{x_n\}$
is a Cauchy sequence. So it is convergent. This ends the proof.
$\square$\\
\noindent{\bf Lemma 2.3.} \textit{Let $(X, d)$ be a complete metric
space
and $S, T : X\rightarrow CB(X)$ are a duality of generalized weak
contractions. Let also $\{x_{n}\}$ be a convergent sequence of $X$
that satisfies $x_{n+1}\in Sx_{n}$ for each even $n \in
{\mathbb{N}}$, $\lim\limits_{n\rightarrow\infty}x_{n}=x^\ast$ and
$\limsup\limits_{k\rightarrow\infty}\alpha(x_{2k}, x^\ast)<1$. Then
$x^\ast\in Fix(T)=Fix(S)$.}\\
\noindent{\bf Proof}. In terms of the conditions, for each even $n
\in {\mathbb{N}}$, we have
$$\begin{array}{rcl}d(x_{n+1}, Tx^\ast) &\leq& H(Sx_{n}, Tx^\ast)
\leq\alpha(x_{n}, x^\ast)M(x_{n},
x^\ast);
\end{array}\eqno
(2.6)$$
$$\begin{array}{rcl}& &M(x_{n}, x^\ast)\\&=&\max\{d(x_{n}, x^\ast), d(x_{n}, Sx_{n}),
d(x^\ast, Tx^\ast),\\& & \frac{d(x_{n}, Tx^\ast)+d(x^\ast,
Sx_{n})}{2}\}\\&\leq&\max\{d(x_{n}, x^\ast), d(x_{n}, x_{n+1}),
d(x^\ast, Tx^\ast),\\& & \frac{d(x_{n}, x^\ast)+d(x^\ast,
Tx^\ast)+d(x^\ast, x_{n})+d(x_{n},
Sx_{n})}{2}\}\\&\leq&\max\{d(x_{n}, x^\ast), d(x_{n}, x_{n+1}),
d(x^\ast, Tx^\ast),\\& &d(x_{n}, x^\ast)+ \frac{d(x^\ast,
Tx^\ast)+d(x_{n}, x_{n+1})}{2}\}.
\end{array}\eqno
(2.7)$$Note that $\lim\limits_{n\rightarrow\infty}x_{n}=x^\ast$.
Combining (2.6) and (2.7), we further obtain
$$\begin{array}{rcl}d(x^\ast, Tx^\ast)&\leq&[\limsup\limits_{k\rightarrow\infty}
\alpha(x_{2k}, x^\ast)]\limsup\limits_{k\rightarrow\infty}M(x_{2k},
x^\ast)
\\&\leq&[\limsup\limits_{k\rightarrow\infty}\alpha(x_{2k}, x^\ast)]d(x^\ast,
Tx^\ast).
\end{array}$$
Since $\limsup\limits_{k\rightarrow\infty}\alpha(x_{2k}, x^\ast)<1$, this
implies $d(x^\ast, Tx^\ast)=0$. That is $x^\ast\in Tx^\ast$. The
proof
completes. $\square$\\
\noindent{\bf Lemma 2.4.} \textit{Let $(X, d)$ be a complete metric
space
and $S, T : \rightarrow CB(X)$ are a duality of generalized weak
contractions. Then we have the the conclusions as follows.\\
(1) $End(S)=End(T) (\subseteq Fix(S)=Fix(T))$ and $|End(S)|\leq 1$.
Here $|End(S)|$ denotes the cardinal number of $End(S)$. (This
implies that $S$ and $T$ have an unique common endpoint, or have no
endpoint.)\\
(2) If $S$ and $T$ have common endpoint, then $\inf\limits_{x\in
X}[H(\{x\}, Sx)+H(\{x\}, Tx)]$=0, termed as the
approximate endpoint property of duality $S$ and $T$.\\
(3) If either $S$ or $T$ is single valued, then
$End(S)=End(T)=Fix(S)=Fix(T)$. (This implies that the fixed points
of $S$ and
$T$ must be endpoints.)}\\
\noindent{\bf Proof}. Let $x\in End(S)$. Then $x\in
Fix(S)=Fix(T)$ from Lemma 2.1. This implies $M(x,x)=0$. Therefore,
we have
$$\begin{array}{rcl}& &H(\{x\}, Tx)= H(Sx,Tx)\leq\alpha(x,x)M(x,x)=0.\end{array}$$
This means $Tx=\{x\}$. That is, $x\in End(T)$. Hence
$End(S)=End(T)$.
Let $x,y\in End(S)=End(T)$. Then $M(x,y)=d(x,y)$, further
$$d(x,y)=H(\{x\}, \{y\})=
H(Sx,Ty)\leq\alpha(x,y)M(x,y)=\alpha(x,y)d(x,y).$$ For
$\alpha(x,y)<1$, this implies $d(x,y)=0$. That is $x=y$. Hence
$|End(S)|\leq 1$.
We have proved (1). (2) is obvious. Next we further prove (3).
Suppose that one of $S$ and $T$ is single valued. Without loss of
generality, we assume $S$ is single valued. Then it is obvious that
$End(S)=Fix(S)$. So $End(T)=End(S)=Fix(S)=Fix(T)$. This ends the
proof. $\square$
\section{Fixed
point theory} \label{1} In the section, we focus on studying the
fixed point theory.
We are now in a position to prove our first theorem, which extends
Theorem 2.3 of Daffer and Kaneko [1] by generalizing one mapping
$T$ to two mappings $S$ and $T$, and by improving the other
conditions, which also extends Theorem 3.1 of Rouhani and Moradi [4]
by replacing the constant contraction factor $\alpha$ with an
general
$\alpha(x, y)$.\\
\noindent{\bf Theorem 3.1.} \textit{Let $(X, d)$ be a complete
metric space
and $S, T : \rightarrow CB(X)$ are a duality of generalized weak
contractions that satisfies
$$\sup\{\alpha(x_{2k-2},
x_{2k-1}),\alpha(x_{2k},
x_{2k-1})|k\in{\mathbb{N}}\}<1 \eqno (3.1)$$
for any sequence $\{x_n\}$ of $X$ with $\{d(x_{n}, x_{n+1})\}$ to
be monotone decreasing, and $\alpha$ is
u.s.c. (or
$\limsup\limits_{n\rightarrow\infty}\alpha(x_{n}, x^\ast)<1$ if
$\lim\limits_{n\rightarrow\infty}x_{n}=x^\ast$).
Then $Fix(S)=Fix(T)\neq\emptyset$.}\\
\noindent{\bf Proof.} (1) By Lemma 2.1, $Fix(S)=Fix(T)$. To complete
the proof, what we need is only to prove
$Fix(S)=Fix(T)\neq\emptyset$. Arguing by contradiction, we assume
$Fix(S)=Fix(T)=\emptyset$.
\noindent(2) Let $x_0 \in X$. Then $d(x_0,Sx_0)>0$. It is obvious
that we can choose a $x_1 \in Sx_0$ such that $0<d(x_0,
x_1)<d(x_0,Sx_0)+1$, and $d(x_1,Tx_1)>0$.
Let $\varepsilon_1=\min\{\frac{1}{2}, [1-\alpha(x_0, x_1)]d(x_0,
x_1)\}$. Then there exists a $x_2 \in Tx_1$ such that $0<d(x_1,
x_2)<d(x_1,Tx_1)+\varepsilon_1$. Let
$\varepsilon_2=\min\{\frac{1}{2^2}, [1-\alpha(x_2, x_1)]d(x_1,
x_2)\}$. Then there exists a $x_3 \in Sx_2$ such that $0<d(x_2,
x_3)<d(x_2,Sx_2)+\varepsilon_2$. Inductively, we have the general
fact as follows.
For each $k\in {\mathbb{N}}$, let
$$\varepsilon_{2k-1}=\min\{\frac{1}{2^{2k-1}}, [1-\alpha(x_{2k-2},
x_{2k-1})]d(x_{2k-2}, x_{2k-1})\}.$$ Then there exists a $x_{2k} \in
Tx_{2k-1}$ such that
$$0<d(x_{2k-1},
x_{2k})<d(x_{2k-1},Tx_{2k-1})+\varepsilon_{2k-1}\leq
d(x_{2k-1},Tx_{2k-1})+\frac{1}{2^{2k-1}}.$$ Let also
$$\varepsilon_{2k}=\min\{\frac{1}{2^{2k}}, [1-\alpha(x_{2k},
x_{2k-1})]d(x_{2k-1}, x_{2k})\}.$$ Then there exists a $x_{2k+1} \in
Sx_{2k}$ such that $$0<d(x_{2k},
x_{2k+1})<d(x_{2k},Sx_{2k})+\varepsilon_{2k}\leq
d(x_{2k},Sx_{2k})+\frac{1}{2^{2k}}.$$
\noindent(3) For the sequence $\{x_{n}\}$ constructed above,
$\forall n \in {\mathbb{N}}$, when $n$ is odd, we have
$$[1-\alpha(x_{n-1}, x_{n})]d(x_{n-1}, x_{n})\geq
\varepsilon_{n}.\eqno (3.2)$$ Further,
$$\begin{array}{rcl}d(x_n, x_{n+1})&< &
d(x_n, Tx_{n})+\varepsilon_{n} \leq H(Sx_{n-1},
Tx_{n})+\varepsilon_{n} \\&\leq&\alpha(x_{n-1}, x_n)M(x_{n-1},
x_n)+\varepsilon_{n}\\&\leq&\alpha(x_{n-1}, x_n)M(x_{n-1},
x_n)+\frac{1}{2^n},
\end{array}\eqno (3.3)$$
$$\begin{array}{rcl} & &
M(x_{n-1}, x_n)\\
& \leq& \max\{d(x_{n-1}, x_n), d(x_{n-1}, Sx_{n-1}), d(x_n,
Tx_n),\\& & \frac{d(x_{n-1}, Tx_n) + d(x_{n}, Sx_{n-1})}{2}\}\\&\leq
& \max\{d(x_{n-1}, x_n), d(x_{n-1}, x_{n}), d(x_n, x_{n+1}),\\& &
\frac{d(x_{n-1}, x_n) +d(x_{n}, Tx_{n})}{2}\} \\& \leq&
\max\{d(x_{n-1}, x_n), d(x_n, x_{n+1}), \frac{d(x_{n-1}, x_n)
+d(x_{n}, x_{n+1})}{2}\}\\ & =& \max\{d(x_{n-1}, x_n), d(x_n,
x_{n+1})\}.
\end{array}\eqno (3.4)$$
If $\max\{d(x_{n-1}, x_n), d(x_n, x_{n+1})\}=d(x_n, x_{n+1})$, i.e.
$d(x_{n-1}, x_n)\leq d(x_n, x_{n+1})$, then
$$[1- \alpha(x_{n-1}, x_n)]d(x_{n-1}, x_n)\leq[1- \alpha(x_{n-1},
x_n)]d(x_{n}, x_{n+1}),\eqno (3.5)$$ and from (3.3) and (3.4), we
obtain
$$\begin{array}{rcl}& &d(x_n, x_{n+1})<
\alpha(x_{n-1}, x_n)d(x_n, x_{n+1})+\varepsilon_{n}\\&\Rightarrow&
[1- \alpha(x_{n-1}, x_n)]d(x_{n}, x_{n+1})<\varepsilon_{n}.
\end{array}\eqno (3.6)$$
Combing (3.5) and (3.6), we obtain $[1- \alpha(x_{n-1},
x_n)]d(x_{n-1}, x_n)<\varepsilon_{n}$. This contradicts (3.2).So
$\max\{d(x_{n-1}, x_n), d(x_n, x_{n+1})\}\neq d(x_{n-1}, x_n)$.This
yields to $\max\{d(x_{n-1}, x_n), d(x_n, x_{n+1})\}=d(x_{n-1}, x_n)$
and $d(x_n, x_{n+1})<d(x_{n-1}, x_n)$. Also, from (3.3) and (3.4),
we obtain
$$\begin{array}{rcl}d(x_n, x_{n+1})&< &
\alpha(x_{n-1}, x_n)d(x_{n-1}, x_n)+\frac{1}{2^n}.
\end{array}\eqno (3.7)$$
When $n$ is even, we have
$$[1-\alpha(x_{n}, x_{n-1})]d(x_{n-1}, x_{n})\geq
\varepsilon_{n},\eqno (3.8)$$
$$\begin{array}{rcl}& &d(x_n, x_{n+1})=d( x_{n+1},x_n)\\&< &
d(Sx_n, x_{n})+\varepsilon_{n} \leq H(Sx_{n},
Tx_{n-1})+\varepsilon_{n} \\&\leq&\alpha( x_n,x_{n-1})M(x_{n},
x_{n-1})+\varepsilon_{n}\\&\leq&\alpha( x_n,x_{n-1})M(x_{n},
x_{n-1})+\frac{1}{2^n},
\end{array}\eqno (3.9)$$
$$\begin{array}{rcl} & &
M(x_{n}, x_{n-1})\\
& \leq& \max\{d(x_{n-1}, x_n), d(x_{n}, Sx_{n}), d(x_{n-1},
Tx_{n-1}),\\& & \frac{d(x_{n}, Tx_{n-1}) + d(x_{n-1},
Sx_{n})}{2}\}\\&\leq & \max\{d(x_{n-1}, x_n), d(x_n, x_{n+1}),
d(x_{n-1}, x_{n}),\\& & \frac{d(x_{n-1}, x_n) +d(x_{n},
Sx_{n})}{2}\}
\\& \leq& \max\{d(x_{n-1}, x_n), d(x_n, x_{n+1}), \frac{d(x_{n-1},
x_n) +d(x_{n}, x_{n+1})}{2}\}\\ & =& \max\{d(x_{n-1}, x_n), d(x_n,
x_{n+1})\}.
\end{array}\eqno (3.10)$$
From $(3.8)$, (3.9) and (3.10), in the same way as used above, we
can also obtain $d(x_n, x_{n+1})<d(x_{n-1}, x_n)$ and
$$\begin{array}{rcl}d(x_n, x_{n+1})&< &
\alpha(x_n, x_{n-1})d(x_{n-1}, x_n)+\frac{1}{2^n}.
\end{array}\eqno (3.11)$$
(4)
From (3), it is obvious that the sequence $\{d(x_{n}, x_{n+1})\}$ is
monotone decreasing. Hence (3.1) holds. So, there exists a
$\gamma<1$ such that $$\max\{\alpha(x_{2k-2},
x_{2k-1}),\alpha(x_{2k},
x_{2k-1})\}<\gamma$$ for all $k\in {\mathbb{N}}$. Therefore
using (3.7) and $(3.11)$ we can obtain (2.1). Thus $\{x_{n}\}$ is
convergent from Lemma 2.2.
Finally, let $\lim\limits_{n\rightarrow\infty}x_{n}=x^\ast$. Then,
since $\alpha$ is u.s.c. we have
$\limsup\limits_{n\rightarrow\infty}\alpha(x_{n},
x^\ast)\leq\alpha(x^\ast, x^\ast)<1$. Note that the approach we
produce the sequence $\{x_{n}\}$. By Lemma 2.3, $x^\ast\in Tx^\ast$.
This contradicts $Fix(T)=\emptyset$. So
$Fix(S)=Fix(T)\neq\emptyset$. The proof completes. $\square$
As an application we propose a proof of Theorem 1.3 from Theorem
3.1
as follows.
\noindent{\bf Proof of Theorem 1.3}. let
$$\begin{array}{rcl}\alpha(x, y)=
\left\{\begin{array}{l}1-\frac{\phi(x, y)}{d(x, y)}, d(x, y)\neq
0;\\0, d(x, y)=0,\end{array}\right.
\end{array}$$
and $S=T$. Then $S$ and $T$ are a duality of generalized weak
contractions. Let also $\{x_{n}\}$ be a sequence of $X$ with
$\{d(x_{n},x_{n+1})\}$ to be monotone decreasing. And assume
$\lim\limits_{n\rightarrow \infty}d(x_{n},x_{n+1})=r$.
If $r>0$, then $\lambda(r,d(x_{1},x_{2}))>0$ for $\phi$ is compactly
positive. On the other hand,
$\phi(x_{n},x_{n+1})\geq\lambda(r,d(x_{1},x_{2}))$ since
$r<d(x_{n},x_{n+1})\leq d(x_{1},x_{2})$. So we have
$$\alpha(x_{n},x_{n+1})=1-\frac{\phi(x_{n},x_{n+1})}{d(x_{n},x_{n+1})}\leq
1-\frac{\lambda(r,d(x_{1},x_{2}))}{d(x_{1},x_{2})}<1.$$ With the
same argument, $\alpha(x_{n+1},x_{n})\leq
1-\frac{\lambda(r,d(x_{1},x_{2}))}{d(x_{1},x_{2})}$. Hence (3.1)
holds. If $r=0$, then
$$\begin{array}{rcl}\limsup\limits_{n\rightarrow
\infty}\alpha(x_{n},x_{n+1})&=&\limsup\limits_{n\rightarrow
\infty}[1-\frac{\phi(x_{n},x_{n+1})}{d(x_{n},x_{n+1})}]\\&\leq&
\limsup\limits_{n\rightarrow
\infty}[1-\frac{\lambda(d(x_{n-1},x_{n}),
d(x_{n},x_{n+1}))}{d(x_{n},x_{n+1})}]\\&\leq&
1-\liminf\limits_{n\rightarrow
\infty}\frac{\lambda(d(x_{n-1},x_{n}),
d(x_{n},x_{n+1}))}{d(x_{n},x_{n+1})}\\&\leq&1-\liminf\limits_{\beta\rightarrow
\infty}\frac{\lambda(\alpha, \beta)}{\beta}<1.\end{array}$$ With the
same argument, $\limsup\limits_{n\rightarrow
\infty}\alpha(x_{n+1},x_{n})<1$. Hence (3.1) holds.
Let $\lim\limits_{n\rightarrow \infty}x_{n}=x^\ast$. Without loss of
generality, assume $d(x_{n}, x^\ast)\neq 0$ for all $
n\in{\mathbb{N}}$. Then
$$\begin{array}{rcl}& &\limsup\limits_{n\rightarrow\infty}\alpha(x_{n}, x^\ast)
=\limsup\limits_{n\rightarrow\infty}[1-\frac{\phi(x_{n},
x^\ast)}{d(x_{n}, x^\ast)}]\leq\\& &
1-\liminf\limits_{n\rightarrow\infty}\frac{\phi(x_{n},
x^\ast)}{d(x_{n}, x^\ast)}\leq 1-\liminf\limits_{\beta\rightarrow
0}\frac{\lambda(\alpha, \beta)}{\beta}<1.\end{array}$$
Combing the results above, by Theorem 3.1, $T$ has fixed point.
This ends the proof.
$\square$\\
\noindent{\bf Theorem 3.2.} \textit{Let $(X, d)$ be a complete
metric space
and $S, T : \rightarrow CB(X)$ are a duality of generalized $\varphi$-weak
contractions that satisfies $\varphi$ is u.s.c. and
$$\limsup\limits_{t\rightarrow 0}\frac{\varphi(t)}{t}<1.\eqno (3.13)$$
Then $Fix(S)=Fix(T)\neq\emptyset$.}\\
\noindent{\bf Proof}. For any $(x, y)\in X\times X$, put
$$\begin{array}{rcl}\alpha(x, y)=
\left\{\begin{array}{l}\frac{\varphi(M(x, y))}{M(x, y)}, M(x, y)\neq
0;\\0, M(x, y)=0.\end{array}\right.
\end{array}$$
Then it can be easily verify that $H(Sx, Ty)\leq\alpha(x, y)M(x,
y)$. That is, $S, T : \rightarrow CB(X)$ are a duality of
generalized weak contractions with the $\alpha(x, y)$. Note that the
conditions $\alpha$ is u.s.c. and (3.1) are used only in the step
(4) of the proof of Theorem 3.1. We can easily know that the steps
(1), (2) and (3) can be used to prove Theorem 3.2. So the proof can
be accomplished by proposing the step $(4)'$ below.
\noindent $(4)'$ $\forall n\in{\mathbb{N}}$, assume first that $n$
is odd. Note that $0<d(x_{n-1}, x_{n})\leq M(x_{n-1}, x_{n})$ and
$$\max\{d(x_{n-1}, x_{n}), d(x_n, x_{n+1})\}=d(x_{n-1}, x_{n}).$$ By
(3.4), we have $M(x_{n-1}, x_{n})=d(x_{n-1}, x_{n})>0$. This leads
to
$$\alpha(x_{n-1}, x_{n})=\frac{\varphi(d(x_{n-1}, x_{n}))}{d(x_{n-1},
x_{n})}.\eqno (3.14)$$ Further, from (3.7), we obtain $$d(x_{n},
x_{n+1})\leq \varphi(d(x_{n-1}, x_{n}))+\frac{1}{2^n}.\eqno (3.15)$$
When $n$ is even, with the same argument, we have (3.15) and
$$\alpha(x_{n}, x_{n-1})=\frac{\varphi(d(x_{n-1},
x_{n}))}{d(x_{n-1}, x_{n})}.\eqno (3.16).$$ Since the sequence
$\{d(x_{n}, x_{n+1})\}$ is monotone decreasing and bounded below, it
is convergent. Let $\lim\limits_{n\rightarrow\infty}d(x_n,
x_{n+1})=r$. For $\varphi$ is u.s.c. using (3.15) we have $r\leq
\varphi(r)$. This implies $r=0$ because $\varphi(t)<t$ for all
$t>0$. Therefore, according to (3.14) and (3.16), we respectively
have
$$\limsup\limits_{k\rightarrow\infty}\alpha(x_{2k-2}, x_{2k-1})=
\limsup\limits_{k\rightarrow\infty}\frac{\varphi(x_{2k-2},
x_{2k-1})}{d(x_{2k-2}, x_{2k-1})}\leq \limsup\limits_{t\rightarrow
0}\frac{\varphi(t)}{t}<1.$$
$$\limsup\limits_{k\rightarrow\infty}\alpha(x_{2k}, x_{2k-1})=
\limsup\limits_{k\rightarrow\infty}\frac{\varphi(x_{2k},
x_{2k-1})}{d(x_{2k}, x_{2k-1})}\leq \limsup\limits_{t\rightarrow
0}\frac{\varphi(t)}{t}<1.$$Hence (3.1) holds.
Using
(3.7) and $(3.11)$ we obtain (2.1). Thus $x_{n}$ is convergent from Lemma 2.2.
Finally, let $\lim\limits_{n\rightarrow\infty}x_{n}=x^\ast$. Then
for each even $n$, we have (2.7). This reduces to
$\limsup\limits_{k\rightarrow\infty}M(x_{2k}, x^\ast)\leq d(x^\ast,
Tx^\ast)$. So there exists a positive number $b$ such that
$M(x_{2k}, x^\ast)\leq b$. For $\varphi$ is u.s.c. and
$\limsup\limits_{t\rightarrow 0}\frac{\varphi(t)}{t}<1$,
$\sup\{\frac{\varphi(t)}{t}|t\in(0, b]\}<1$. Therefore,
$\limsup\limits_{n\rightarrow\infty}\alpha(x_{n},
x^\ast)=\limsup\limits_{n\rightarrow\infty}\frac{\varphi(M(x_{n},
x^\ast))}{M(x_{n}, x^\ast)}\leq\sup\{\frac{\varphi(t)}{t}|t\in(0,
b]\}<1$. By Lemma 2.3, $x^\ast\in Tx^\ast$. This contradicts
$Fix(T)=\emptyset$. So $Fix(S)=Fix(T)\neq\emptyset$. The proof ends.
$\square$
Theorem 3.2 extends Theorem 4.1 of Rouhani and Moradi [4] by
allowing both two mappings $S$ and $T$ to be multi-valued. However,
we add the condition (3.1). Whether Theorem 3.2 holds or not without
the condition (3.1) is a topic for us to further pursue.
\section{Endpoint theory} \label{1}Now we turn to address the endpoint theory.
In terms of Theorem 3.1 (Theorem 3.2) and Lemma 2.4, we can
immediately get the next corollary.\\
{\bf corollary 3.2.} \textit{Under the conditions of Theorem 3.1 (
Theorem 3.2), if either $S$ or $T$ is single valued, then there
exists a unique common fixed point for $S$ and $T$, which is also
a unique common endpoint of theirs.}\\
For $S$ and $T$ are all multi-valued, we have Theorem 4.1 and
Theorem 4.2 below.\\
{\bf Theorem 4.1.} \textit{Let $(X, d)$ be a complete metric space
and $S, T : X\rightarrow CB(X)$ are a duality of generalized weak
contractions that satisfies $\alpha$ is u.s.c. and
$$\limsup\limits_{n, m\rightarrow\infty}\alpha(x_n,x_m)<1\eqno (4.1)$$
if $\lim\limits_{n,
m\rightarrow\infty}d(x_n,x_m)[1-\alpha(x_n,x_m)]=0$. Then $S$ and
$T$ have a unique common endpoint if they have the approximate
endpoint property.}\\
\noindent{\bf Proof}. Suppose that $S$ and $T$ have the approximate
endpoint property. Then there exists a sequence $\{x_n\}$ such that
$$\lim\limits_{n\rightarrow \infty}[H(\{x_n\}, Sx_n)+ H(\{x_n\},
Tx_n)]= 0.$$
For all $m, n \in {\mathbb{N}}$, we have
$$\begin{array}{rcl}& & M(x_n, x_m)\\ &=& \max\{d(x_n, x_m), d(x_n, Sx_n), d(x_m,
Tx_m), \\& &\frac{d(x_n, Tx_m) + d(x_m, Sx_n)}{ 2 }\}\\& \leq &
\max\{d(x_n, x_m), H(\{x_n\}, Sx_n), H(\{x_m\}, Tx_m),\\& &
\frac{d(x_n, x_m)+H(\{x_m\}, Tx_m) +d(x_n, x_m)+ H(\{x_n\},
Sx_n)}{2}\} \\&\leq & d(x_n, x_m) + H(\{x_n\}, Sx_n) + H(\{x_m\},
Tx_m). \end{array}\eqno (4.2)$$ Note that $d(x_n, x_m)\leq
H(\{x_n\}, Sx_n) +H(Sx_n, Tx_m)+ H(\{x_m\}, Tx_m)$. From (4.2), we
further have
$$\begin{array}{rcl}& & M(x_n, x_m)\\&\leq &d(x_n, x_m)
-H(\{x_n\}, Sx_n) - H(\{x_m\}, Tx_m)\\& & + 2H(\{x_n\}, Sx_n) +
2H(\{x_m\}, Tx_m)\\ &\leq & H(Tx_n, Tx_m) + 2H(\{x_n\}, Sx_n) +
2H(\{x_m\}, Tx_m). \end{array}\eqno (4.3)$$ This reduces to
$$\begin{array}{rcl}& & M(x_n, x_m)\\&\leq & \alpha(x_n,
x_m) M(x_n, x_m) + 2H(\{x_n\}, Sx_n) + 2H(\{x_m\},
Tx_m).\end{array}\eqno (4.4)$$ Note that $d(x_n, x_m)\leq M(x_n,
x_m)$. Using (4.4) we obtain
$$\begin{array}{rcl}& & d(x_n, x_m)[1-\alpha(x_n,
x_m)]\\&\leq & M(x_n, x_m)[1-\alpha(x_n, x_m)]\\&\leq & 2H(\{x_n\},
Sx_n) + 2H(\{x_m\}, Tx_m)\\&\Rightarrow &\lim\limits_{
n,m\rightarrow \infty}d(x_n, x_m)[1-\alpha(x_n, x_m)]=
0.\end{array}\eqno (4.5)$$ For (4.5), we have (4.1). Using also
(4.4), we obtain
$$\limsup\limits_{
n,m\rightarrow \infty}M(x_n, x_m)\leq[\limsup\limits_{
n,m\rightarrow \infty}\alpha(x_n, x_m)]\limsup\limits_{
n,m\rightarrow \infty}M(x_n, x_m).$$ By (4.1), this yields to
$\limsup\limits_{ n,m\rightarrow \infty}M(x_n, x_m)=0$. Hence
$\limsup\limits_{ n,m\rightarrow \infty}d(x_n, x_m)=0$, i.e.
$\{x_n\}$ is a Cauchy sequence.
Let $\lim\limits_{ n\rightarrow \infty}x_n=x^\ast$. For all $ n \in
{\mathbb{N}}$, we have
$$\begin{array}{rcl}& &H(\{x_n\}, Tx^\ast)- H(\{x_n\}, Sx_n)\\&\leq& H(Sx_n, Tx^\ast)
\leq \alpha(x_n, x^\ast)M(x_n, x^\ast)\\& =&\alpha(x_n,
x^\ast)\max\{d(x_n, x^\ast), d(x_n, Sx_n), d(x^\ast, Tx^\ast),\\&
&\frac{d(x_n, Tx^\ast)+d(x^\ast, Sx_n)}{2}\}\\ &\leq&\alpha(x_n,
x^\ast)\max\{d(x_n, x^\ast), d(x_n, Sx_n), d(x^\ast, Tx^\ast),\\&
&\frac{d(x_n, x^\ast)+d(x^\ast, Tx^\ast)+d(x^\ast, x_n)+d(x_n,
Sx_n)}{2}\}.
\end{array}$$Noting also $\alpha$ is u.s.c. we obtain
$$H(\{x^\ast\}, Tx^\ast)\leq\alpha(x^\ast,
x^\ast)d(x^\ast, Tx^\ast)\leq\alpha(x^\ast, x^\ast)H(\{x^\ast\},
Tx^\ast).$$ For $\alpha(x^\ast, x^\ast)<1 $, we conclude that
$H({x^\ast}, Tx^\ast)=0$. This means $Tx^\ast = \{x^\ast\}$.
Finally, the uniqueness of the endpoint is concluded from Lemma
2.4. $\square$
The following Theorem 4.2 is our final result, which extends the
Theorem 2.1 of Moradi and Khojasteh [7] to the case where both two
mappings are multi-valued.\\
{\bf Theorem 4.2.} \textit{Let $(X, d)$ be a complete metric space
and $S, T : \rightarrow CB(X)$ are a duality of $\varphi$-generalized weak
contractions that satisfies $\varphi$ is s.u.c. and
$$\liminf\limits_{t\rightarrow \infty}[t-\varphi(t)]>0.\eqno (4.6)$$
Then $S$ and $T$ have a unique common endpoint if they have the
approximate endpoint property.}\\
\noindent{\bf Proof}. Suppose that $S$ and $T$ have the approximate
endpoint property. Then there exists a sequence $\{x_n\}$ such that
$$\lim\limits_{n\rightarrow \infty}[H(\{x_n\}, Sx_n)+ H(\{x_n\},
Tx_n)]= 0,$$as well as (4.2) and (4.3) hold.
By (4.3) we have
$$\begin{array}{rcl}& & M(x_n, x_m)\\&\leq & \varphi( M(x_n, x_m)) + 2H(\{x_n\}, Sx_n) + 2H(\{x_m\},
Tx_m).\end{array}\eqno (4.7)$$ If $\limsup\limits_{ n,m\rightarrow
\infty}M(x_n, x_m)=+\infty$, then
$$\begin{array}{rcl}\liminf\limits_{
t\rightarrow \infty}[t-\varphi(t)]\leq\liminf\limits_{
n,m\rightarrow \infty}[M(x_n, x_m)-\varphi(M(x_n, x_m))]\leq
0.\end{array}$$ This contradicts (4.6). So $\limsup\limits_{
n,m\rightarrow \infty}M(x_n, x_m)<+\infty$. Noting also $\varphi(t)$
is u.s.c. and using (4.7), we obtain
$$\limsup\limits_{
n,m\rightarrow \infty}M(x_n, x_m)\leq\limsup\limits_{ n,m\rightarrow
\infty}\varphi(M(x_n, x_m))\leq\varphi(\limsup\limits_{
n,m\rightarrow \infty}M(x_n, x_m)).$$ Note that $\limsup\limits_{
n,m\rightarrow \infty}M(x_n, x_m)<+\infty$ and $\varphi(t)<t$ for
all $t>0$. This implies
$\limsup\limits_{ n,m\rightarrow
\infty}M(x_n, x_m)=0$. Thus $\{x_n\}$ is Cauchy sequence.
Let $\lim\limits_{ n\rightarrow \infty}x_n=x^\ast$. For all $ n \in
{\mathbb{N}}$, we have
$$\begin{array}{rcl}& &H(\{x_n\}, Tx^\ast)- H(\{x_n\}, Sx_n)\\&\leq& H(Sx_n, Tx^\ast)
\leq \varphi(M(x_n, x^\ast)).
\end{array}$$ This reduces to
$$H(\{x^\ast\}, Tx^\ast)\leq \limsup\limits_{ n,m\rightarrow \infty}\varphi(M(x_n,
x^\ast))\leq\varphi(\limsup\limits_{ n,m\rightarrow \infty}M(x_n,
x^\ast)). \eqno(4.8)$$On the other hand,
$$\begin{array}{rcl}M(x_n, x^\ast) &\leq&\max\{d(x_n,
x^\ast), d(x_n, Sx_n), d(x^\ast, Tx^\ast),\\& &\frac{d(x_n,
x^\ast)+d(x^\ast, Tx^\ast)+d(x^\ast, x_n)+d(x_n, Sx_n)}{2}\}.
\end{array}\eqno(4.9)$$
If $H(\{x^\ast\}, Tx^\ast)\neq 0$, from (4.8) and (4.9), we have
$$\begin{array}{rcl}& &H(\{x^\ast\}, Tx^\ast)<\limsup\limits_{ n,m\rightarrow \infty}M(x_n,
x_m)\leq d(x^\ast, Tx^\ast)\leq H(\{x^\ast\}, Tx^\ast).\end{array}$$
This contradiction shows $H(\{x^\ast\}, Tx^\ast)=0$. That is,
$Tx^\ast = \{x^\ast\}$. Finally, the uniqueness of the endpoint is
concluded from Lemma 2.4. $\square$
\noindent{\bf Remark 4.3.} By taking $S=T$, we
can immediately obtain the Theorem 2.1 of Moradi and Khojasteh [7]
from Theorem 4.2 and Lemma
2.4.\\
{\bf Acknowledgements}
The author cordially thank the anonymous referees for their valuable
comments which lead to the improvement of this paper.
\end{document}
|
\begin{document}
\begin{center}
{\Large \bf{On Warped Product Gradient Yamabe Soliton}}
\textbf{\bf{Tokura, W. I.$^1$, Adriano, L. R.$^2$}, Pina R. S.$^3$.}\\
\textbf{\footnotesize \textit{ Instituto de Matemática e Estatística-UFG}}\\
\textbf{\footnotesize \textit{$^1$email:
[email protected]}} \textbf{\footnotesize
\textit{$^2$email: [email protected]}} \textbf{\footnotesize
\textit{$^3$email: [email protected]}}
{\Large}
\end{center}
\begin{abstract}
In this paper, we provide a necessary and sufficient conditions for
the warped product $M=B\times_{f}F$ to be a gradient Yamabe soliton
when the base is conformal to an $n$-dimensional pseudo-Euclidean
space, which are invariant under the action of an
$(n-1)$-dimensional translation group, and the fiber $F$ is
scalar-constant. As application, we obtain solutions in steady case
with fiber scalar-flat. Besides, on the warped product we consider
the potential function as separable variables and obtain some
characterization of the base and the fiber.
\end{abstract}
\section{Introduction and main statements}
A \textit{Yamabe soliton} is a pseudo-Riemannian manifold $(M,g)$
admitting a vector field \linebreak$X\in \mathfrak{X}(M)$ such that
\begin{equation}\label{eq:01}(S_{g}-\rho)g=\frac{1}{2}\mathfrak{L}_{X}g,
\end{equation}
where $S_{g}$ denotes the scalar curvature of $M$, $\rho$ is a real
number and $\mathfrak{L}_{X}g$ denotes the Lie derivative of the
metric $g$ with respect to $X$. We say that $(M^{n},g)$ is
shrinking, steady or expanding, if $\rho>0$, $\rho=0$ , $\rho<0$,
respectively. When $X=\nabla h$ for some smooth function $h\in
\mathbf{C^{\infty}}(M)$, we say that $(M^{n},g,\nabla h)$ is an
\textit{gradient Yamabe soliton} with potential function $h$. In
this case the equation \eqref{eq:01} turns out
\begin{equation}\label{eq8}(S_{g}-\rho)g=Hess(h),
\end{equation}
where $Hess(\phi)$ denote the hessian of $\phi$. When $\phi$ is constant, we
call it a \textit{trivial Yamabe soliton}.
Yamabe solitons are self-similar solutions for the Yamabe flow
$$
\frac{\partial}{\partial t}g(t)=-R_{g(t)}g(t),
$$
and are important to
understand the geometric flow since they can appear as singularity
models. It has been known that every compact gradient Yamabe soliton
is of constant scalar curvature, hence, trivial since $f$ is
harmonic, see \cite{Cho}, \cite{Dask}, \cite{Hsu}. For the
non-compact case many interesting results is obtained in \cite{Cao}, \cite{Cat}, \cite{Ma}, \cite{Ma1}, \cite{Wu}.
As pointed in \cite{Cal} and \cite{Lopes}, it is important to
emphasize here that although the Yamabe flow are wellposed in the
Riemannian setting, they do not necessarily exist in the
semi-Riemannian case, where even the existence of short-time
solutions is not guaranteed in general due to the lack of
parabolicity. However, the existence of self-similar solutions of
the flow is equivalent to the existence of Yamabe solitons as in
\eqref{eq:01}. Semi-Riemannian Yamabe solitons have been intensively
studied, showing many differences with respect to the Riemannian
case, see for instance \cite{Bat} and \cite{Cal}.
Brozos-Vázquez et al. in \cite{Bro}, obtain a local characterization
of pseudo Riemannian ma\-ni\-fold endowed with gradient Yamabe soliton
metric, its results establish that if a pseudo Riemannian gradient
Yamabe soliton $(M,g)$, with potential function $h$ and such that
\linebreak$|\nabla h|\neq0$ is locally isometric to warped product of
unidimensional base and scalar-constant fiber. In the Riemannian
context a global structure result was given in \cite{Cao}.
In \cite{Dask} Daskalopoulos and Sesum investigated gradient Yamabe
soliton and proved that all complete locally conformally flat
gradient Yamabe solitons with positive sectional curvature are
rotationally symmetric. Proceeding in the same locally conformally
flat context, Neto and Tenenblat in \cite{Bene} consider the study of
pseudo Riemannian manifold
$(\mathbb{R}^{n},\frac{1}{\varphi^{}2}g_{0})$, where $g_{0}$ is the
canonical pseudo metric, and obtain a necessary and sufficient
condition to this manifold be a gradient Yamabe soliton. In the
search for invariant solutions they consider the invariant action of
an $(n-1)$-dimensional translation group and exhibit a complete
solution in steady case.
Recently, Pina and De Sousa in \cite{Pina}, consider the study of
gradient Ricci solitons on warped product structure
$M=B^{n}\times_{f}F^{m}$, where the base is conformal to an $n$-dimensional
pseudo-Euclidean space, invariant under the action of an
$(n-1)$-dimensional translation group, the fiber chosen to be an
Einstein manifold and potential function $h$ depending only on the
base and they give a necessary an sufficient condiction for $M$ to be a
gradient Ricci soliton.
As far as we know, there are no results for gradient Yamabe solitons
related to its potential function in the warped products of two
Riemannian manifolds of arbitrary dimensions. Thus, in this paper we
consider the study of gradient Yamabe solitons with warped product
structure, where we choose the fiber with dimension greater than
$1$. Initially we provide a sufficient condiction for the potential
function on warped product depends only on the base.
\begin{proposition}\label{eq3}Let $M=B\times_{f}F$ be a warped product
manifold with metric $\tilde{g}$. If the metric
$\tilde{g}=g_{B}\oplus f^{2}g_{F}$ is a gradient Yamabe soliton with
potential function $h:M\rightarrow \mathbb{R}$ and there exist a
pair of orthogonal vectors $(X_i,X_{j})$ of the base $B$, such that
$Hess_{g_{B}}(f)(X_{i},X_{j})\neq 0$, then the potential function
$h$ depends only on the base.
\end{proposition}
\begin{observation}
The previous proposition extend for Yamabe solitons the result obtained in \cite{Kim} where the authors studied warped product gradient Ricci solitons with one-dimensional base.
\end{observation}
Motivated by natural extension of Ricci solitons given by Rigoli,
Pigola and Setti in \cite{Pigola} the authors Barbosa and Ribeiro in
\cite{Barbosa}, define the concept of \textit{Almost Yamabe soliton}
allowing the constant $\rho$ in definition of Yamabe soliton
\eqref{eq:01} to be a differentiable function on $M$. The following
example was obtained by Barbosa and Ribeiro in \cite{Barbosa}, where
the manifold is endowed with a warped product metric.
\begin{example}Let $M^{n+1}=\mathbb{R}\times_{\cosh t}\mathbb{S}^{n}$
with metric $g=dt^{2}+\cosh^{2} t g_{0}$, where $g_{0}$ is the
canonical metric of $\mathbb{S}^{n}$. Taking $(M^{n+1},g,\nabla h,\rho)$, where $h(t,x)=\sinh t$ and \newline$\rho(t,x)=\sinh t+n$. A
straightforward computation gives that
$M^{n+1}=\mathbb{R}\times_{\cosh t}\mathbb{S}^{n}$ is a noncompact almost gradient Yamabe soliton.
\end{example}
In what follows, inspired by Proposition \ref{eq3}, we consider a
warped product gradient Yamabe soliton $M=B\times_{f}F$ with
potential function $h$ splitting of the form
\begin{equation}
\label{h12}
h(x,y)=h_{1}(x)+h_{2}(y),\ \mbox{where}\ h_{1}\in\mathcal{C}^{\infty}(B)\
\mbox{and}\ h_{2}\in\mathcal{C}^{\infty}(F),
\end{equation}
and get the following characterization theorem.
\begin{theorem}
\label{theo1.1}Let $M=B\times_{f}F$ be a warped product manifold with metric $\tilde{g}=g_{B}\oplus
f^{2}g_{F}$, and gradient Yamabe soliton structure with potential
function $h:B\times F\rightarrow\mathbb{R}$ given by \eqref{h12}, then one of the following
cases occurs
\begin{description}
\item[(a)]$M$ is the Riemannian product between a \textbf{trivial gradiente Yamabe soliton} and a \textbf{gradient Yamabe soliton}.
\item[(b)]$M$ is the Riemannian product between two \textbf{gradient Yamabe solitons}.
\item[(c)]$M$ is the warped product between a \textbf{Almost gradient Yamabe solitons} and a \textbf{trivial gradiente Yamabe soliton}.
\end{description}
\end{theorem}
This characterization theorem shows us that if we take the potential
function depending only on the base then the fiber $F$ is of
constant scalar curvature. In what follows we will take a warped
product gradient Yamabe soliton with potential function of the form
$h(x,y)=h_{1}(x)+constant$, the base conformal to an $n$-dimensional
pseudo-Euclidean space, and the fiber chosen to be an
scalar-constant space. More precisely, let $(\mathbb{R}^{n},g)$ be
the pseudo-Euclidean space, $n\geq3$ with coordinates
$x=(x_{1},\dots,x_{n})$ and $g_{ij}=\delta_{ij}\epsilon_{i}$ and let
$M=(\mathbb{R}^{n},\bar{g})\times_{f}F^{m}$ be a warped product
where $\bar{g}=\frac{1}{\varphi^{2}}g$, $F$ a semi-Riemannian
scalar-constant manifold with curvature $\lambda_{F}$, $m\geq1$,
$f$,$\varphi$, $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$, smooth
functions, and $f$ is a positive function. Then we obtain necessary
and sufficient conditions for the warped product metric
$g_{B}\oplus f^{2}g_{F}$ to be a gradient Yamabe soliton.
\begin{theorem}\label{eq:02}Let $(\mathbb{R}^{n},g)$ be a
pseudo-Euclidean space, $n\geq3$ with coordinates \newline$x=(x_{1},\dots,x_{n})$ and $g_{ij}=\delta_{ij}\epsilon_{i}$, and
let $M=(\mathbb{R}^{n},\bar{g})\times_{f}F^{m}$ be a warped product
where $\bar{g}=\frac{1}{\varphi^{2}}g$, $F$ a semi-Riemannian
scalar-constant manifold with curvature $\lambda_{F}$, $m\geq1$,
$f$,$\varphi$, $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$, smooth
functions, and $f$ is a positive function. Then the warped product
metric $\tilde{g}$ is a gradient Yamabe soliton with potential
function $h$ if, and only if, the functions $f, \varphi, h$
satisfy
\begin{equation}\label{eq:19}h_{,x_{i}x_{j}}+\frac{\varphi_{,x_{j}}}{\varphi}h_{,x_{i}}+\frac{\varphi_{,x_{i}}}{\varphi}h_{,x_{j}}=0\hspace{1cm}
i\neq j,
\end{equation}
\begin{equation}\label{eq:20}
\begin{split}
\Big{[}(n-1)\left(2\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}x_{k}}-n\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}^{2}\right)+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}\left(\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}x_{k}}-(n-2)\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}f_{,x_{k}}\right)+\\
-\frac{m(m-1)}{f^2}\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}}^{2}-\rho\Big{]}\frac{\varepsilon_{i}}{\varphi^{2}}=h_{,x_{i}x_{i}}+2\frac{\varphi_{,x_{i}}}{\varphi}h_{,x_{i}}-\varepsilon_{i}\sum_{k}\varepsilon_{k}\frac{\varphi_{,x_{k}}}{\varphi}h_{,x_{k}}\hspace{1cm}i=j,
\end{split}
\end{equation}
\begin{equation}\label{eq:21}
\begin{split}
(n-1)\left(2\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}x_{k}}-n\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}^{2}\right)+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}\left(\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}x_{k}}-(n-2)\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}f_{,x_{k}}\right)+\\
-\frac{m(m-1)}{f^2}\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}}^{2}-\rho=\frac{\varphi^{2}}{f}\sum_{k}\varepsilon_{k}f_{,x_{k}}h_{,x_{k}}.
\end{split}
\end{equation}
\end{theorem}
In order to obtain solutions for equations in Theorem \ref{eq:02},
we consider $f$, $\varphi$ and $h$ invariant under the action of an
$(n-1)$-dimensional translation group, and
$\xi=\sum_{i=1}^{n}\alpha_{i}x_{i}, \alpha_{i}\in\mathbb{R}$, be a
basic invariant for the $(n-1)$-dimensional translation group, then
we obtain
\begin{theorem}\label{eq4}Let $(\mathbb{R}^{n},g)$ be a pseudo-Euclidean space, $n\geq3$ with coordinates \newline$x=(x_{1},\dots,x_{n})$, $g_{ij}=\delta_{ij}\epsilon_{i}$ and
let $M=(\mathbb{R}^{n},\bar{g})\times_{f}F^{m}$ be a warped product
where $\bar{g}=\frac{1}{\varphi^{2}}g$, $F$ a semi-Riemannian
scalar-constant manifold with curvature $\lambda_{F}$, $m\geq1$,
$f$,$\varphi$, $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$, smooth
functions and $f>0$. Consider the functions $f(\xi)$,
$\varphi(\xi)$ and $h(\xi)$, where \newline$\xi=\sum_{k=1}^{n}\alpha_{k}x_{k},\alpha_{k}\in\mathbb{R}$ and
$\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}$ or
$\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=0$. Then the warped
product metric $\tilde{g}$ is a gradient Yamabe soliton with
potential function $h$ if, and only if, $f$, $h$ and $\varphi$,
satisfy
\begin{equation}\label{eq:09}
h''+2\frac{\varphi'h'}{\varphi}=0
\end{equation}
\begin{equation}\label{eq1}
\begin{split}
\varepsilon_{k_{0}}[(n-1)(2\varphi\varphi''-n(\varphi')^{2})-2\frac{m}{f}(\varphi^{2}f''-(n-2)\varphi\varphi'f')-&\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}+
\varphi'h'\varphi]\\&=\rho-\frac{\lambda_{F}}{f^{2}}
\end{split}
\end{equation}
\begin{equation}\label{eq2}
\begin{split}
\varepsilon_{k_{0}}[(n-1)(2\varphi\varphi''-n(\varphi')^{2})-2\frac{m}{f}(\varphi^{2}f''-(n-2)\varphi\varphi'f')-&\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}-
\frac{\varphi^{2}}{f}f'h']\\&=\rho-\frac{\lambda_{F}}{f^{2}}
\end{split}
\end{equation}
when
$\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}$.
And
\begin{equation}\label{eq:10}
h''+2\frac{\varphi'h'}{\varphi}=0
\end{equation}
\begin{equation}\label{eq10}
\rho-\frac{\lambda_{F}}{f^{2}}=0
\end{equation}
when $\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=0.$
\end{theorem}
It is interesting to know how geometry of the fiber manifold $F$
affects the geometry of the warped product
$M=(\mathbb{R}^{n},\bar{g})\times_{f}F^{m}$. He in \cite{He0} has
shown that any complete steady gradient Yamabe soliton on
$\mathbb{R}\times_{f}F$ is necessarily isometric to the Riemannian
product with constant $f$ and $F$ being of zero scalar curvature.
Moreover, he showed that there is no complete steady gradient Yamabe
soliton on $\mathbb{R}\times_{f}F^{n}$ with $n\geq2$ and $F$
positive scalar constant manifold.
As consequence of Theorem \ref{eq4} in the context of lightlike
vector invariance and scalar-constant fiber, we prove that if $F$
has positive scalar constant curvature then there is no shirinking
or steady gradient Yamabe soliton
$M=(\mathbb{R}^{n},\bar{g})\times_{f}F^{m}$ and when $F$ has
negative constant scalar curvature, there is no expanding or steady
gradient Yamabe soliton \newline$M=(\mathbb{R}^{n},\bar{g})\times_{f}F^{m}$,
this is translated into the following corollary.
\begin{cor}\label{cor1.7}In the context of Theorem \ref{eq4}, if $X=\sum_{k}\alpha_{k}\frac{\partial}{\partial_{k}}$ is a lightlike vector, assume that $\lambda_{F}>0$, then there is no expanding or
steady gradient Yamabe soliton with warped metric $\tilde{g}$ and
potential function $h$. Similarly, if we assume that $\lambda_{F}<0$, then there is no shrinking or
steady gradient Yamabe soliton with warped metric $\tilde{g}$ and
potential function $h$.
\end{cor}
Now, by equations \eqref{eq:09} and \eqref{eq:10} in Theorem \ref{eq4} we easily see that a necessary condition for
$M=(\mathbb{R}^{n},\overline{g})\times_{f}F^{m}$ be a gradient
Yamabe soliton with invariant solution $f(\xi)$,
$\varphi(\xi)$ and $h(\xi)$, where
$\xi=\sum_{k=1}^{n}\alpha_{k}x_{k},\alpha_{k}\in\mathbb{R}$ is that
$h$ is a monotone function. That is,
\begin{equation}h'(\xi)=\frac{\alpha}{\varphi^{2}(\xi)},\nonumber
\end{equation}
for some $\alpha\in\mathbb{R}$.
We provide solutions for ODE in Theorem \ref{eq4} in two cases:
$h'=0$ and $h'\neq0$, with metric $\tilde{g}=g_{B}\oplus f^{2}g_{F}$
be a steady gradient Yamabe soliton, i.e. $\rho=0$, and $F$ is a
scalar-flat pseudo-Riemannian manifold.
\begin{theorem}\label{eq5}In the context of Theorem \ref{eq4}, if $\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}\neq 0$ and $F$ scalar-flat fiber, then the warped product metric $\tilde{g}$ is a steady gradient
Yamabe soliton with potential function $h$ and $h'\neq0$ if, and
only if, $f$, $h$ and $\varphi$, satisfies
\begin{equation}\label{eqa}f(\xi)=\frac{e^{c}}{\varphi(\xi)},
\end{equation}
\begin{equation}\label{eqb}h(\xi)=\alpha\int\frac{1}{\varphi^{2}(\xi)}d\xi,
\end{equation}
\begin{equation}\label{eqc}(n+m-1)(n+m+2)\int\frac{\varphi d\varphi}{\alpha-\frac{2\beta(n+m-1)(n+m+2)}{n+m-2}\varphi^{\frac{n+m}{2}+1}}=\xi+\nu,\hspace{0,2cm}c\in\mathbb{R}
\end{equation}
where $c$, $\nu$, $\alpha$, $\beta\in \mathbb{R}$ with
$\alpha\neq0$.
\end{theorem}
\begin{theorem}\label{eq7} In the context of Theorem \ref{eq4}, if $\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}\neq 0$ and $F$ scalar-flat fiber, then, given a smooth function $\varphi>0$, the warped product metric
$\tilde{g}$ is a steady gradient Yamabe soliton with potential
function $h$ and $h'=0$ if, and only if, $f$ and $h$, satisfies
\begin{equation}\label{eq23}h(\xi)=\text{constant},
\end{equation}
\begin{equation}\label{eq22}f=\varphi^{\frac{n-2}{m+1}}e^{\int z_{p}d\xi}\left(\int e^{-(m+1)\int z_{p}d\xi}d\xi+\frac{2}{m+1}C\right)^{\frac{2}{m+1}}
\end{equation}
for an appropriate function $z_{p}$.
\end{theorem}
In the null case $\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=0$ we
obtain
\begin{theorem}\label{eq6} In the context of Theorem \ref{eq4}, if $\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=0$ and $F$ scalar-flat fiber, then, given smooth
functions $\varphi(\xi)$ and $f(\xi)$,the warped product metric
$\tilde{g}$ is a steady gradient Yamabe soliton with potential
function $h$ if, and only if
\begin{equation}h(\xi)=\alpha\int\frac{1}{\varphi^{2}(\xi)}d\xi.\nonumber
\end{equation}
\end{theorem}
\begin{observation}As pointed in \cite{Chenn}, see Theorem 3.6, a necessary and
sufficient condition to warped product $B\times_{f}F$ be a
conformally flat is that the function $f$ defines a global conformal
deformation such that $(B,\frac{1}{f^{2}}g_{B})$ is a space of
constant curvature $c$ and $F$ has constant curvature $-c$. With
this observation, we see that the solutions of theorems \ref{eq5},
\ref{eq7} and \ref{eq6} defines a non locally conformally flat
metric if the warping function $f$ is not constant.
\end{observation}
\begin{observation}As we can see in the proof of Theorem \ref{eq4}, if $\rho$ is a function defined only on the
base, then we can easily extend Theorem \ref{eq4} into context of
almost gradient Yamabe solitons. In the particular case of lightlike
vectors there are infinitely many solutions, that is, given
$\varphi$ and $f$
\begin{equation}\rho(\xi)=\frac{\lambda_{F}}{f(\xi)^{2}}\nonumber
\end{equation}
\begin{equation}h(\xi)=\alpha\int\frac{1}{\varphi^{2}(\xi)}d\xi\nonumber
\end{equation}
provide a family of almost gradient Yamabe soliton with warped
product structure.
\end{observation}
Before proving our main results, we present some examples
illustrating the above theorems.
\begin{example}In Theorem \ref{eq5}, consider $\beta=0$, then we
have
\begin{equation}f(\xi)=\frac{e^{c}\sqrt{(n-1)(n+m+2)}}{\sqrt{2\alpha(\xi+\nu)}},\nonumber
\end{equation}
\begin{equation}h(\xi)=\frac{\alpha}{2}(n-1)(n+m+2)\ln|\xi+\nu|\nonumber
\end{equation}
\begin{equation}\varphi(\xi)=\sqrt{\frac{2\alpha(\xi+\nu)}{(n-1)(n+m+2)}}\nonumber
\end{equation}
where $\alpha(\xi+\nu)>0$ and $c\in\mathbb{R}$. Thus, the metric
$\tilde{g}=\frac{1}{\varphi^{2}}g_{0}\oplus f^{2}g_{F}$ is a steady
gradient Yamabe soliton defined in the semi-space of Euclidean space
$\mathbb{R}^{n}$ with potential function $h$.
\end{example}
\begin{example}In Theorem \ref{eq7} consider the warped product $M=(R^{2},\bar{g})\times_{f}F^{3}$. Given the function $\varphi(\xi)=e^{\frac{3\xi^{2}}{4}}$
we have that
\begin{equation}\bar{g}=e^{-\frac{3\xi^{2}}{2}}g_0, \hspace{0,3cm} h(\xi)=\text{constant},\hspace{0,3cm}f(\xi)=e^{\frac{\xi}{2}}\nonumber
\end{equation}
defines a steady gradient Yamabe soliton in warped metric.
\end{example}
\begin{example}In Theorem \ref{eq6} consider the Lorentzian space $(\mathbb{R}^{n},g)$ with coordinates $(x_{1},\dots,x_{n})$
and signature $\epsilon_{1}=-1$, $\epsilon_{k}=1$ for all $k\geq2$,
and $F^{m}$ a complete scalar flat manifold. Let $\xi=x_{1}+x_{2}$
and choose $\varphi(\xi)=\frac{1}{1+\xi^{2}}$. Then, for
$\alpha\neq0$
\begin{equation}\bar{g}=(1+\xi^{2})^{2}g, \hspace{0,3cm} h(\xi)=\alpha\left(\xi+\frac{2}{3}\xi^{3}+\frac{1}{5}\xi^{5}\right),\hspace{0,3cm}f\in\mathcal{C}^{\infty}\nonumber
\end{equation}
defines a steady gradient Yamabe soliton
$(\mathbb{R}^{n},\bar{g})\times_{f}(F^{m},g_{flat})$ with potential
function $h$ and warping function $f$. Observe that, since the
conformal function $\varphi$ is bounded, we have that $\bar{g}$ is
complete, and consequently
$\tilde{g}=\frac{1}{\varphi^{2}}g_{0}\oplus f^{2}g_{F}$ is complete.
\end{example}
\begin{section}{Proofs of the Main Results}
\begin{myproof}[\textbf{Proof of Proposition 1.1}]Let $M=B\times_{f}F$ be a gradient Yamabe soliton with potential function $h:M\rightarrow\mathbb{R}$,
then by equation \eqref{eq8}, we obtain
\begin{equation}\label{eq9}(S_{\tilde{g}}-\rho)\tilde{g}=Hess(h)
\end{equation}
Now, It is well known that on warped metric $\tilde{g}$ the scalar
curvature of base $B$, fiber $F$ and $M$ is related by(see chapter 7
of \cite{oneil})
\begin{equation}S_{\tilde{g}}=S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle grad_{B}f,grad_{B}f\rangle}{f^{2}}\label{eq:04}
\end{equation}
where $\Delta^{B}$ denote the laplacian defined on $B$. Then
considering $X_{1},X_{2},\dots,X_{n}\in \mathcal{L}(B)$ and
$Y_{1},Y_{2},\dots,Y_{m}\in \mathcal{L}(F)$, where $\mathcal{L}(B)$
and $\mathcal{L}(F)$ are respectively the space of lifts of vector
fiels on $B$ and $F$ to $B\times F$, we obtain substituting equation
\eqref{eq:04} in \eqref{eq9} that
\begin{equation*}
\begin{cases}
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}g_{B}(X_{i},X_{j})=Hess_{\tilde{g}}h(X_{i},X_{j})&(i)\\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}\tilde{g}(X_{i},Y_{j})=Hess_{\tilde{g}}h(X_{i},Y_{j})&(ii) \\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}f^{2}g_{F}(Y_{i},Y_{j})=Hess_{\tilde{g}}h(Y_{i},Y_{j}).&(iii)
\end{cases}
\end{equation*}
Thus, using the fact $\tilde{g}(X_{i},Y_{j})=0$, we obtain by expression $(ii)$ that
\begin{equation}Hess(h)(X_{i},Y_{j})=0.\nonumber
\end{equation}
Now, using Lemma 2.1 of \cite{He}, we obtain
\begin{equation}\label{eqe}h(x,y)=z(x)+f(x)v(y)
\end{equation}
where $z:B\rightarrow\mathbb{R}$ and $v:F\rightarrow\mathbb{R}$.
Then since
$$Hess_{\tilde{g}}h(X_{i},X_{j})=Hess_{g_{B}}h(X_{i},X_{j}),$$
we have by expression $(i)$ that
\begin{equation}\label{eqd}\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}g_{B}(X_{i},X_{j})=Hess_{g_{B}}h(X_{i},X_{j}).
\end{equation}
We have from \eqref{eqe} that the right size of \eqref{eqd} is given by
\begin{equation}\label{eqf}Hess_{g_{B}}z+vHess_{g_{B}}f.
\end{equation}
By hypothesis, there are two ortogonal vectors fields $X_{i}$, $X_{j}$ such that $Hess_{g_{B}}f(X_{i},X_{j})\neq0$. Then combining this fact with equations \eqref{eqd} and \eqref{eqf}, we obtain
\begin{equation}v=-\frac{Hess_{g_{B}}z(X_{i},X_{j})}{Hess_{g_{B}}f(X_{i},X_{j})}.
\end{equation}
This show that $v(y)$ is constant, and then by expression \eqref{eqe} we have that $h$ depends only on the base.
\end{myproof}
\begin{myproof}[\textbf{Proof of Theorem \ref{theo1.1}}]Let $M=B\times_{f}F$ be a warped product with gradient Yamabe soliton structure and potential function
$h(x,y)=h_{1}(x)+h_{2}(y)$. In the same way as in the proof of Proposition \ref{eq3},
for $X_{1},X_{2},\dots,X_{n}\in \mathcal{L}(B)$ and
$Y_{1},Y_{2},\dots,Y_{m}\in \mathcal{L}(F)$ we obtain
\begin{equation}Hess(h)(X_{i},Y_{j})=0.\nonumber
\end{equation}
As we know, the connection of warped product is particularly simple, that is, for \newline$X\in \mathcal{L}(B)$ and $Y\in \mathcal{L}(F)$, we have
$$\nabla_{X}Y=\nabla_{Y}X=\frac{X(f)}{f}Y.$$
Thus,
\begin{equation}Hess(h)(X_{i},Y_{j})=X_{i}(Y_{j}(h))-(\nabla_{X_{i}}Y_{j})h=X_{i}(Y_{j}(h))-\frac{X_{i}(f)}{f}Y_{j}=0.\nonumber
\end{equation}
Establishing the notation $h_{,x_{i}}=X_{i}(h)$,
$h_{,x_{i}x_{j}}=X_{j}(X_{i}(h))$, we have that
\begin{equation}h_{,y_{j}x_{i}}-\frac{f_{,x_{i}}}{f}h_{,y_{j}}=0-\frac{f_{,x_{i}}}{f}(h_{2})_{,y_{j}}=0\hspace{0,4cm}\forall i,j.\nonumber
\end{equation}
Then, $f$ is constant or $h(x,y)=h_{1}(x)+constant$. We separate the
proof in three cases:
$Case (I):$ ($f$ is constant and $h(x,y)=h_{1}(x)+constant$). In
this case, $M=B\times_{f}F$ is a Riemannian product and we have
\begin{equation*}
\begin{cases}
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\rho\Big{)}g_{B}(X_{i},X_{j})=Hess_{g_{B}}h_{1}(X_{i},X_{j})&(i)\\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\rho\Big{)}\tilde{g}(X_{i},Y_{j})=Hess_{\tilde{g}}h(X_{i},Y_{j})=0&(ii) \\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\rho\Big{)}f^{2}g_{F}(Y_{i},Y_{j})=Hess_{g_{F}}h_{2}(Y_{i},Y_{j})+f\nabla f(h_{1})g_{F}(Y_{i},Y_{j})&(iii)
\end{cases}
\end{equation*}
where in equation $(iii)$ we use Proposition 35 of \cite{oneil} and
the Hessian definition to get
\begin{eqnarray}\label{eq13}
Hess_{\tilde{g}}h(Y_{i},Y_{j})& = &
Y_{i}(Y_{j}(h))-(\nabla_{Y_{i}}Y_{j})^{M}h\\
&=& Y_{i}(Y_{j}(h))-(\mathcal{H}(\nabla_{Y_{i}}Y_{j})+\mathcal{V}(\nabla_{Y_{i}}Y_{j}))(h)\nonumber\\
&=& Y_{i}(Y_{j}(h))+\frac{\langle Y_{i},
Y_{j}\rangle}{f}grad_{\tilde{g}}f(h)-\nabla_{Y_{i}}^{F}Y_{j}(h)\nonumber\\
&=& Y_{i}(Y_{j}(h))+fg_{F}(Y_{i},
Y_{j})grad_{\tilde{g}}f(h)-\nabla_{Y_{i}}^{F}Y_{j}(h)\nonumber\\
&=&Hess_{g_{F}}h_{2}(Y_{i},Y_{j})+f\nabla
f(h_{1})g_{F}(Y_{i},Y_{j}).\nonumber
\end{eqnarray}
Since $S_{g_{F}}$ is constant on $B$, we have from $(i)$ that $B$
is a gradient Yamabe soliton of the form $(B,g_{B},\nabla
h_{1},-\frac{S_{g_{F}}}{f^{2}}+\rho)$. Furthermore, since
$h(x,y)=h_{1}(x)+cte$ we have by $(iii)$ that $F$ is a trivial
gradient Yamabe soliton of the form $(F,g_{F},\nabla
0,f^{2}\rho-f^{2}S_{g_{B}})$. This proves the item $(a)$.
$Case (II):$ ($f$ is constant and $h(x,y)=h_{1}(x)+h_{2}$, $h_{2}$
not necessarily constant). In this case, $M=B\times_{f}F$ is a
Riemannian product and we have
\begin{equation*}
\begin{cases}
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\rho\Big{)}g_{B}(X_{i},X_{j})=Hess_{g_{B}}h_{1}(X_{i},X_{j})&(i)\\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\rho\Big{)}\tilde{g}(X_{i},Y_{j})=Hess_{\tilde{g}}h(X_{i},Y_{j})=0&(ii) \\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\rho\Big{)}f^{2}g_{F}(Y_{i},Y_{j})=Hess_{g_{F}}h_{2}(Y_{i},Y_{j})+f\nabla f(h_{1})g_{F}(Y_{i},Y_{j}).&(iii)
\end{cases}
\end{equation*}
Since $S_{g_{F}}$ is constant on $B$, we have that $B$ is a gradient
Yamabe soliton of the form $(B,g_{B},\nabla
h_{1},-\frac{S_{g_{F}}}{f^{2}}+\rho)$. Furthermore, by equation
$(iii)$ we have that $F$ is a gradient Yamabe soliton of the form
$(F,g_{F},\nabla h_{2},f^{2}\rho-f^{2}S_{g_{B}})$. This proves the
item $(b)$.
$Case (III):$ ($f$ is non constant and $h(x,y)=h_{1}(x)+constant$).
In this case we have:
\begin{equation*}
\begin{cases}
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}g_{B}(X_{i},X_{j})=Hess_{g_{B}}h_{1}(X_{i},X_{j})&(i)\\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}\tilde{g}(X_{i},Y_{j})=Hess_{\tilde{g}}h(X_{i},Y_{j})=0&(ii) \\
\Big{(}S_{g_{B}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho\Big{)}f^{2}g_{F}(Y_{i},Y_{j})=f\nabla
f(h)g_{F}(Y_{i},Y_{j})&(iii)
\end{cases}
\end{equation*}
Since $f>0$, by equation $(iii)$ we have that
\begin{equation} (S_{g_{F}}-\psi) g_{F}(Y_{i},Y_{j})=0
\end{equation}
where $\psi=-f^{2}S_{g_{B}}+2fd\Delta^{B}f+d(d-1)\langle
grad_{B}f,grad_{B}f\rangle+f^{2}\rho+f\nabla f(h)$.
Now, since $\psi$ depend only on $B$, we have that $\psi$ is
constant on $F$, then by equation $(iii)$, we have that $F$ is a
trivial gradient Yamabe soliton. Furthermore, by equation $(i)$ we
have that $(B,g_{B})$ is a gradient almost Yamabe soliton of the
form
$$
(B,g_{B},\nabla
h_{1},-[\frac{S_{g_{F}}}{f^{2}}-\frac{2d}{f}\Delta^{B}f-d(d-1)\frac{\langle
grad_{B}f,grad_{B}f\rangle}{f^{2}}-\rho] ).
$$
This proves the item $(c)$.
\end{myproof}
\begin{myproof}[\textbf{Proof of Theorem \ref{eq:02}}]
Let $M$ be a warped product with gradient Yamabe soliton structure
and potential function $h$, that is,
\begin{equation}(S_{\tilde{g}}-\rho)\tilde{g}=Hess_{\tilde{g}}(h).\label{eq:200}
\end{equation}
By the same arguments used in proof of Proposition \ref{eq3}, for
$X_{1},X_{2},\dots,X_{n}\in \mathcal{L}(B)$ and
$Y_{1},Y_{2},\dots,Y_{m}\in \mathcal{L}(F)$ we obtain
\begin{equation*}
\begin{cases}
\Big{(}S_{\overline{g}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2m}{f}\Delta_{\overline{g}}f-m(m-1)\frac{\langle
grad_{\overline{g}}f,grad_{\overline{g}}f\rangle}{f^{2}}-\rho\Big{)}\overline{g}(X_{i},X_{j})=Hess_{\tilde{g}}h(X_{i},X_{j})&(i)\\
\Big{(}S_{\overline{g}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2m}{f}\Delta_{\overline{g}}f-m(m-1)\frac{\langle
grad_{\overline{g}}f,grad_{\overline{g}}f\rangle}{f^{2}}-\rho\Big{)}\tilde{g}(X_{i},Y_{j})=Hess_{\tilde{g}}h(X_{i},Y_{j})=0&(ii) \\
\Big{(}S_{\overline{g}}+\frac{S_{g_{F}}}{f^{2}}-\frac{2m}{f}\Delta_{\overline{g}}f-m(m-1)\frac{\langle
grad_{\overline{g}}f,grad_{\overline{g}}f\rangle}{f^{2}}-\rho\Big{)}f^{2}g_{F}(Y_{i},Y_{j})=Hess_{\tilde{g}}h(Y_{i},Y_{j}).&(iii)
\end{cases}
\end{equation*}
It is well known that for the conformal metric
$\bar{g}=\frac{1}{\varphi^{2}}g_{0}$, the Christofel symbol is given
by
\begin{equation}\bar{\Gamma}_{ij}^{k}=0,\ \bar{\Gamma}_{ij}^{i}=-\frac{\varphi_{,x_{j}}}{\varphi},\ \bar{\Gamma}_{ii}^{k}=\epsilon_{i}\epsilon_{k}\frac{\varphi_{,x_{k}}}{\varphi}\;\ \mbox{and}\;\ \bar{\Gamma}_{ii}^{i}=-\frac{\varphi_{,x_{i}}}{\varphi}.\nonumber
\end{equation}
Then, we obtain by Hessian definiton that
\begin{equation}\label{eq:22}
\begin{cases}
Hess_{\overline{g}}(h)_{ij}=h_{,x_{i}x_{j}}+\frac{\varphi_{,x_{i}}h_{,x_{j}}}{\varphi}+\frac{\varphi_{,x_{j}}h_{,x_{i}}}{\varphi}& i\neq j\\
Hess_{\overline{g}}(h)_{ii}=h_{,x_{i}x_{i}}+2\frac{\varphi_{,x_{i}}h_{,x_{i}}}{\varphi}-\varepsilon_{i}\sum_{k=1}^{n}\varepsilon_{k}\frac{\varphi_{,x_{k}}}{\varphi}h_{,x_{k}}&i=j.
\end{cases}
\end{equation}
The Ricci curvature is given by
$$Ric_{\overline{g}}=\frac{1}{\varphi^{2}}\Big{\{}(n-2)\varphi
Hess_{g}(\varphi)+[\varphi\Delta_{g}\varphi-(n-1)|\nabla_{g}\varphi|^{2}]g\Big{\}}$$
and then we easily see that the scalar curvature on conformal metric
is given by
\begin{equation}\label{eq12}S_{\overline{g}}=(n-1)(2\varphi\Delta_{g}\varphi-n|\nabla_{g}\varphi|^{2})=(n-1)(2\varphi\sum_{k=1}^{n}\varepsilon_{k}\varphi_{,x_{k}x_{k}}-n\sum_{k=1}^{n}\varepsilon_{k}\varphi_{,x_{k}}^{2}).
\end{equation}
Since $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$, we obtain
\begin{equation}\label{eq:24}
Hess_{\tilde{g}}h(X_{i},X_{j})=Hess_{\overline{g}}h(X_{i},X_{j}),
\hspace{0,4cm} \forall i,j.
\end{equation}
On the other hand
\begin{equation}\label{eq:23}
\begin{cases}
S_{F}g_{F}=\lambda_{F}g_{F}\\
\tilde{g}(Y_{i},Y_{j})=f^{2}g_{F}(Y_{i},Y_{j})\\
\Delta_{\overline{g}}f=\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}x_{k}}-(n-2)\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}f_{,x_{k}}\\
\tilde{g}(grad_{\overline{g}}f,grad_{\overline{g}}f)=\overline{g}(grad_{\overline{g}}f,grad_{\overline{g}}f)=\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}}^{2}.
\end{cases}
\end{equation}
Now, substituting the second equation of $\eqref{eq:22}$, the
equations of \eqref{eq:23} and equation \eqref{eq12} in $(i)$, we
have
\begin{equation*}
\begin{split}
\Big{[}(n-1)(2\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}x_{k}}-n\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}^{2})+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}(\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}x_{k}}-(n-2)\varphi\sum_{k}\varepsilon_{k}\varphi_{,x_{k}}f_{,x_{k}})+\\
-\frac{m(m-1)}{f^2}\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}}^{2}-\rho\Big{]}\frac{\varepsilon_{i}}{\varphi^{2}}=h_{,x_{i}x_{i}}+2\frac{\varphi_{,x_{i}}}{\varphi}h_{,x_{i}}-\varepsilon_{i}\sum_{k}\varepsilon_{k}\frac{\varphi_{,x_{k}}}{\varphi}h_{,x_{k}}
\end{split}
\end{equation*}
which is the equation \eqref{eq:20}.
Analogously, substituting the first equation of \eqref{eq:22} and
equation \eqref{eq:23} in $(i)$, we obtain
\begin{equation}h_{,x_{i}x_{j}}+\frac{\varphi_{,x_{j}}}{\varphi}h_{,x_{i}}+\frac{\varphi_{,x_{i}}}{\varphi}h_{,x_{j}}=0
\end{equation}
which is the equation \eqref{eq:19}.
in the similar way that equation \eqref{eq13}, we have that
\begin{eqnarray}\label{eq:25}
Hess_{\tilde{g}}h(Y_{i},Y_{j})& = &
Y_{i}(Y_{j}(h))-(\nabla_{Y_{i}}Y_{j})^{M}h\nonumber\\
&=&
Hess_{gF}h(Y_{i},Y_{j})+fg_{F}(Y_{i},
Y_{j})grad_{\tilde{g}}f(h)\nonumber\\
&=& fg_{F}(Y_{i},Y_{j})grad_{\tilde{g}}f(h)\nonumber\\
&=&f\varphi^{2}\sum_{k}\varepsilon_{k}f_{,x_{k}}h_{,x_{k}}g_{F}(Y_{i},Y_{j}).
\end{eqnarray}
Then, substituting equation \eqref{eq:25}, \eqref{eq:23} and
equation \eqref{eq12} in $(iii)$, we obtain equation \eqref{eq:21}.
A direct calculation shows us the converse implication. This
concludes the proof of Theorem \ref{eq:02}.
\end{myproof}
\begin{myproof}[\textbf{Proof of Theorem \ref{eq4}}]Since we are assuming that $\varphi(\xi)$, $h(\xi)$
and $f(\xi)$ are functions of $\xi$, where
$\xi=\sum_{k}\alpha_{k}x_{k}$, $\alpha_{k}\in\mathbb{R}^{n}$ and
$\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}$ or
$\sum_{k=1}^{n}\varepsilon_{k}\alpha_{k}^{2}=0$, then we have
\begin{equation*}\varphi_{,x_{i}}=\varphi'\alpha_{i};\hspace{0,1cm}
\varphi_{,x_{i}x_{j}}=\varphi''\alpha_{i}\alpha_{j};\hspace{0,1cm}f{,x_{i}}=f'\alpha_{i};\hspace{0,1cm}
f_{,x_{i}x_{j}}=f''\alpha_{i}\alpha_{j};\hspace{0,1cm}
h_{,x_{i}}=h''\alpha_{i};\hspace{0,1cm}
h_{,x_{i}x_{j}}=h''\alpha_{i}\alpha_{j}.
\end{equation*}
Substituting these expressions into \eqref{eq:19} of Theorem \ref{eq:02}, we obtain
\begin{equation}\label{eq:33}\Big{(}h''+2\frac{h'\varphi'}{\varphi}\Big{)}\alpha_{i}\alpha_{j}=0,
\hspace{0,4cm} \forall i\neq j.
\end{equation}
Similarly, considering equations \eqref{eq:20} and \eqref{eq:21} of Theorem \ref{eq:02},
we obtain
\begin{equation}\label{eq:30}
\begin{split}
\Big{[}(n-1)(2\varphi\varphi''\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-n(\varphi')^{2}\sum_{k}\varepsilon_{k}\alpha_{k}^{2})+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}(\varphi^{2}f''\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-(n-2)\varphi\varphi'f'\sum_{k}\varepsilon_{k}\alpha_{k}^{2})+\\
-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-\rho\Big{]}\frac{\varepsilon_{i}}{\varphi^{2}}=h''\alpha_{i}^{2}+2\alpha_{i}^{2}\frac{\varphi'}{\varphi}h'-\varepsilon_{i}h'\frac{\varphi'}{\varphi}\sum_{k}\varepsilon_{k}\alpha_{k}^{2}
\end{split}
\end{equation}
for $i\in\{1,2,\dots,n\}$, and
\begin{equation}\label{eq:31}
\begin{split}
(n-1)(2\varphi\varphi''\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-n(\varphi')^{2}\sum_{k}\varepsilon_{k}\alpha_{k}^{2})+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}(\varphi^{2}f''\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-(n-2)\varphi\varphi'f'\sum_{k}\varepsilon_{k}\alpha_{k}^{2})+\\
-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-\rho=\frac{\varphi^{2}}{f}f'h'\sum_{k}\varepsilon_{k}\alpha_{k}^{2}.
\end{split}
\end{equation}
If there exist $i,j$, $i\neq j$ such that $\alpha_{i}\alpha_{j}\neq
0$, then we get by equation \eqref{eq:33} that
\begin{equation}\label{eq:34}\left(h''+2\frac{h'\varphi'}{\varphi}\right)=0.
\end{equation}
It follows from \eqref{eq:34} that the equation \eqref{eq:30} is
summed to
\begin{equation}
\begin{split}
\Big{[}(n-1)(2\varphi\varphi''\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-n(\varphi')^{2}\sum_{k}\varepsilon_{k}\alpha_{k}^{2})+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}(\varphi^{2}f''\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-(n-2)\varphi\varphi'f'\sum_{k}\varepsilon_{k}\alpha_{k}^{2})+\\
-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}\sum_{k}\varepsilon_{k}\alpha_{k}^{2}-\rho\Big{]}\frac{\varepsilon_{i}}{\varphi^{2}}=-\varepsilon_{i}h'\frac{\varphi'}{\varphi}\sum_{k}\varepsilon_{k}\alpha_{k}^{2}.
\end{split}
\end{equation}
Thus, isolating the term
$\sum_{k}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}$, we
obtain the equation \eqref{eq1}.
In the same way, isolating the term
$\sum_{k}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}$ in
\eqref{eq:31} we obtain the equation \eqref{eq2}.
Thus, if
$\sum_{k}\varepsilon_{k}\alpha_{k}^{2}=\varepsilon_{k_{0}}$, then we
obtain equations \eqref{eq:09}, \eqref{eq1} and \eqref{eq2}. In the
case \newline$\sum_{k}\varepsilon_{k}\alpha_{k}^{2}=0$, we easily see that
the equation \eqref{eq:09} is summed to
\begin{equation}
\begin{cases}
h''+2\frac{\varphi'h'}{\varphi}=0 \\
\rho-\frac{\lambda_{F}}{f^{2}}=0.\\
\end{cases}
\end{equation}
Now, we need to consider the case $\alpha_{k_{0}}=1$ and
$\alpha_{k}=0$ $\forall k\neq k_{0}$. In this case, equation
\eqref{eq:33} is trivially satisfied, and since equation
\eqref{eq:31} does not depend on the index $i$, we have that
equation \eqref{eq:31} is equivalent to equation $\eqref{eq2}$.
Finally, we need to show the validity of equation \eqref{eq:09} and
$\eqref{eq1}$. Observe that taking $i=k_{0}$, that is,
$\alpha_{k_{0}}=1$, in \eqref{eq:30}, we get
\begin{equation}
\begin{split}
\Big{[}(n-1)(2\varphi\varphi''\varepsilon_{k_{0}}-n(\varphi')^{2}\varepsilon_{k_{0}})+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}(\varphi^{2}f''\varepsilon_{k_{0}}-(n-2)\varphi\varphi'f'\varepsilon_{k_{0}})+\\
-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}\varepsilon_{k_{0}}-\rho\Big{]}\frac{\varepsilon_{k_{0}}}{\varphi^{2}}=h''+2\frac{\varphi'}{\varphi}h'-h'\frac{\varphi'}{\varphi}=h''+\frac{\varphi'}{\varphi}h'
\end{split}
\end{equation}
and for $i\neq k_{0}$, that is, $\alpha_{i}=0$, we have
\begin{equation}
\begin{split}
\Big{[}(n-1)(2\varphi\varphi''\varepsilon_{k_{0}}-n(\varphi')^{2}\varepsilon_{k_{0}})+\frac{\lambda_{F}}{f^{2}}-\frac{2m}{f}(\varphi^{2}f''\varepsilon_{k_{0}}-(n-2)\varphi\varphi'f'\varepsilon_{k_{0}})+\\
-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}\varepsilon_{k_{0}}-\rho\Big{]}\frac{\varepsilon_{i}}{\varphi^{2}}=-\varepsilon_{i}\varepsilon_{k_{0}}\frac{\varphi'}{\varphi}h'.
\end{split}
\end{equation}
However, this equations are equivalent to equations \eqref{eq:09}
and $\eqref{eq1}$. This complete the proof of Theorem \ref{eq4}.
\end{myproof}
\begin{myproof}[\textbf{Proof of Corollary \ref{cor1.7}}]By Theorem \ref{eq4}, we have that $M$ is a gradient
Yamabe soliton with potential function $h$ if, and only if,
\begin{equation}
\begin{cases}
h''+2\frac{\varphi'h'}{\varphi}=0 \\
\rho-\frac{\lambda_{F}}{f^{2}}=0.\\
\end{cases}
\end{equation}
Thus, we have that $\lambda_{F}$ and $\rho$ always
have the same signal. Therefore, there is no gradient Yamabe soliton
$M$ expanding/shrinking with fiber trivial gradient Yamabe soliton
shrinking/expanding.
\end{myproof}
\begin{myproof}[\textbf{Proof of Theorem \ref{eq5}}]Since $\lambda_{F}=\rho=0$ we have by equation \eqref{eq1} and \eqref{eq2}
of Theorem \ref{eq4} that
\begin{equation}\varphi'h'\varphi=-\frac{\varphi^{2}}{f}f'h'\nonumber
\end{equation}
and by condition $h'\neq0$, we obtain
\begin{equation}\label{eq14}\frac{\varphi'}{\varphi}=-\frac{f'}{f}.
\end{equation}
Integrating this equation we have
$$f(\xi)=\frac{e^{c}}{\varphi(\xi)}$$
for some $c\in\mathbb{R}$, which is equation \eqref{eqa} of Theorem \ref{eq5}.
Integrating the equation \eqref{eq:09}, we have that
\begin{equation}\label{eq19}h'(\xi)=\frac{\alpha}{\varphi^{2}(\xi)}
\end{equation}
for some $\alpha\neq0$, and
$$h(\xi)=\alpha\int\frac{1}{\varphi^{2}(\xi)}d\xi$$
which is equation \eqref{eqb} of Theorem \ref{eq5}.
Substituting equation \eqref{eq19} into \eqref{eq1}
we have
\begin{equation}\label{eq15}(n-1)(2\varphi\varphi''-n(\varphi')^{2})-2\frac{m}{f}(\varphi^{2}f''-(n-2)\varphi\varphi'f')-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}+\alpha\frac{\varphi'}{\varphi}=0.
\end{equation}
Inserting equation \eqref{eq14} into \eqref{eq15} we obtain
\begin{equation}\label{eq16}\varphi\varphi''-\frac{(n+m)}{2}(\varphi')^{2}+\frac{\alpha}{2(n+m-1)}\frac{\varphi'}{\varphi}=0.
\end{equation}
Consider $\varphi(\xi)^{1-\frac{n+m}{2}}=\omega(\xi)$, then
\begin{equation}\omega'(\xi)=(1-\frac{n+m}{2})\varphi^{-\frac{n+m}{2}}\varphi',\hspace{0,5cm}\omega''(\xi)=(1-\frac{m+n}{2})\left(\varphi^{-\frac{n+m}{2}-1}(\varphi\varphi''-\frac{(n+m)}{2}(\varphi')^{2})\right)
\end{equation}
and we obtain that the differential equation \eqref{eq16} is
equivalent to
\begin{equation}\label{eq17}\omega''(\xi)+\frac{\alpha}{2(n+m-1)}\omega'(\xi)\omega(\xi)^{\frac{4}{n+m-2}}=0.
\end{equation}
Integrating equation \eqref{eq17} we have
$$\omega'(\xi)+\frac{\alpha(n+m-2)}{2(n+m+2)(n+m-1)}\omega(\xi)^{\frac{n+m+2}{n+m-2}}=\beta,\hspace{0,3cm}\beta\in\mathbb{R}.
$$
Thus,
\begin{equation}-\int\frac{1}{\frac{\alpha(n+m-2)}{2(n+m+2)(n+m-1)}\omega(\xi)^{\frac{n+m+2}{n+m-2}}-\beta}d\omega=\xi+\nu\nonumber
\end{equation}
and then
\begin{equation}(n+m-1)(n+m+2)\int\frac{\varphi d\varphi}{\alpha-\frac{2\beta(n+m-1)(n+m+2)}{n+m-2}\varphi^{\frac{n+m}{2}+1}}=\xi+\nu\nonumber
\end{equation}
which is equation \eqref{eqc} of Theorem \ref{eq5}. Then we prove the
necessary condition. Now a direct calculation shows us the converse
implication. This concludes the proof of Theorem \ref{eq5}.
\end{myproof}
\begin{myproof}[\textbf{Proof of Theorem \ref{eq7}}]Since $h'=0$ and $\lambda_{F}=\rho=0$ we have by equation \eqref{eq1} and \eqref{eq2} of Theorem \ref{eq4} that
\begin{equation}(n-1)(2\varphi\varphi''-n(\varphi')^{2})-2\frac{m}{f}(\varphi^{2}f''-(n-2)\varphi\varphi'f')-\frac{m(m-1)}{f^2}\varphi^{2}(f')^{2}=0\nonumber
\end{equation}
which is equivalent to
\begin{equation}\left(\frac{f'}{f}-\frac{(n-2)}{(m+1)}\frac{\varphi'}{\varphi}\right)^{2}+\frac{2}{m+1}\left(\frac{f'}{f}-\frac{(n-2)}{(m+1)}\frac{\varphi'}{\varphi}\right)^{'}+\frac{(n+m-1)}{m(m+1)^{2}}\left(n(\frac{\varphi'}{\varphi})^{2}-2\frac{\varphi''}{\varphi}\right)=0.\nonumber
\end{equation}
Consider
$z=\frac{f'}{f}-\frac{(n-2)}{(m+1)}\frac{\varphi'}{\varphi}$, then
\begin{equation}\label{eq20}z^{2}+\frac{2}{m+1}z'+\frac{(n+m-1)}{m(m+1)^{2}}\left(n(\frac{\varphi'}{\varphi})^{2}-2\frac{\varphi''}{\varphi}\right)=0.
\end{equation}
Now, recall that the Ricatti differential equation is a differential
equation of the form
\begin{equation}\label{eq21}z(\xi)'=p(\xi)+q(\xi)z(\xi)+r(\xi)z(\xi)^{2}
\end{equation}
where $p$, $q$ and $r$ are smooth functions on $\mathbb{R}$, and by
Picard theorem we have that the solutions of \eqref{eq21} is given
by
$$z(\xi)=z_{p}(\xi)+\frac{e^{\int P(\xi)d\xi}d\xi}{-\int r(\xi)e^{\int P(\xi)d\xi}d\xi+c}$$
where $P(\xi)=q(\xi)+2z_{p}(\xi)r(\xi)$, $z_{p}(\xi)$ is a
particular solution of \eqref{eq21} and $c$ is a constant.
Observe that \eqref{eq20} is a Ricatti differential equation with
\begin{equation}q(\xi)=0,\hspace{0,2cm} r(\xi)=-\frac{m+1}{2}\hspace{0,2cm} \text{and}\hspace{0,2cm}
p(\xi)=-\frac{(n+m-1)}{2m(m+1)}\left(n(\frac{\varphi'}{\varphi})^{2}-2\frac{\varphi''}{\varphi}\right)
\end{equation}
Then we obtain
$$\frac{f'(\xi)}{f(\xi)}=\frac{(n-2)}{(m+1)}\frac{\varphi'(\xi)}{\varphi(\xi)}+z_{p}(\xi)+\frac{e^{-(m+1)\int z_{p}(\xi)d\xi}}{\frac{m+1}{2}\int e^{-(m+1)\int z_{p}(\xi)d\xi}d\xi+c}$$
and thus,
$$f=\varphi^{\frac{n-2}{m+1}}e^{\int z_{p}d\xi}\left(\int e^{-(m+1)\int z_{p}d\xi}d\xi+\frac{2}{m+1}C\right)^{\frac{2}{m+1}}$$
where $z_{p}(\xi)$ is a particular solution of \eqref{eq20}. This
expression is equation \eqref{eq22} of Theorem \ref{eq:02}.
Now, since $h'=0$, we have that $h(\xi)=constant$, which is equation
\eqref{eq23} of theorem 1.5. Then we prove the necessary condition.
Now a direct calculation shows us the converse implication. This
concludes the proof of Theorem \ref{eq:02}.
\end{myproof}
\begin{myproof}[\textbf{Proof of Theorem \ref{eq6}}]In this case, since $\lambda_{F}=\rho=0$, we have
by differential equation \eqref{eq:10} and \eqref{eq10} that
$$h(\xi)=\alpha\int\frac{1}{\varphi^{2}(\xi)}d\xi$$
for some $\alpha\neq0$ and $f$, $\varphi$ are arbitrary.
Then we prove the necessary condition. Now a direct calculation
shows us the converse implication. This concludes the proof of
Theorem \ref{eq6}.
\end{myproof}
\end{section}
\vskip0.8cm
\noindent
{Willian Isao Tokura} (e-mail: [email protected])\\[2pt]
Instituto de Matem\'atica e Estat\'istica\\
Universidade Federal de Goi\'as\\
74001-900-Goi\^ania-GO\\
Brazil\\
\noindent{Levi Adriano } (e-mail: [email protected])\\[2pt]
Instituto de Matem\'atica e Estat\'istica\\
Universidade Federal de Goi\'as\\
74001-900-Goi\^ania-GO\\
Brazil\\
\noindent{Romildo da Silva Pina} (e-mail: [email protected])\\[2pt]
Instituto de Matem\'atica e Estat\'istica\\
Universidade Federal de Goi\'as\\
74001-900-Goi\^ania-GO\\
Brazil
\end{document}
|
\begin{document}
\title{Entanglement and nonlocality in multi-particle systems }
\author{M. D. Reid, Q. Y. He and P. D. Drummond}
\affiliation{ARC Centre of Excellence for Quantum-Atom Optics, Centre for Atom
Optics and Ultrafast Spectroscopy, Swinburne University of Technology,
Melbourne 3122, Australia}
\begin{abstract}
\textbf{Entanglement, the Einstein-Podolsky-Rosen (EPR) paradox and
Bell's failure of local-hidden-variable (LHV) theories are three historically
famous forms of {}``quantum nonlocality''. We give experimental
criteria for these three forms of nonlocality in multi-particle systems,
with the aim of better understanding the transition from microscopic
to macroscopic nonlocality. We examine the nonlocality of $N$ separated
spin $J$ systems. First, we obtain multipartite Bell inequalities
that address the correlation between spin values measured at each
site, and then we review spin squeezing inequalities that address
the degree of reduction in the variance of collective spins. The latter
have been particularly useful as a tool for investigating entanglement
in Bose-Einstein condensates (BEC). We present solutions for two topical
quantum states: multi-qubit Greenberger-Horne-Zeilinger (GHZ) states,
and the ground state of a two-well BEC. }
\textbf{Keywords:} entanglement, quantum nonlocality, multi-particle,
two-well Bose-Einstein condensates (BEC)
\textbf{PACS numbers:} 03.65.Ta, 42.50.St, 03.65.Ud, 03.75.Gg
\end{abstract}
\maketitle
\section{\textbf{Introduction}}
Nonlocality in quantum mechanics has been extensively experimentally
investigated. Results to date support the quantum prediction, first
presented by Bell, that quantum theory is inconsistent with a combination
of premises now generally called {}``local realism'' \cite{Bell,CHSH}.
However, the extent that quantum mechanics is inconsistent with local
realism at a more mesoscopic or macroscopic level is still not well
understood. Schr\"odinger presented the case that loss of realism
macroscopically would be a concern, and raised the question of how
to link the loss of local realism with macroscopic superposition states
\cite{Schrodinger-1,Schrodinger-2,Schrodinger-3,legg}.
The advent of entangled Bose-Einstein condensate (BEC) states leads
to new possibilities for testing mesoscopic and macroscopic quantum
mechanics. With this in mind, the objective of this article is to
give an overview of a body of work that explores nonlocality in multi-particle
or multi-site systems.
Three types of nonlocality are reviewed: \emph{entanglement }\cite{Schrodinger-1},
the \emph{Einstein-Podolsky-Rosen (EPR) paradox} \cite{epr}, and\emph{
Bell's nonlocality} \cite{Bell,CHSH}. Examples of criteria to demonstrate
each of these nonlocalities is presented, first for multi-site {}``qubits''
(many spin $1/2$ particles) and then for multi-site {}``qudits''
(many systems of higher dimensionality such as high spin particles).
The criteria presented in this paper are useful for detecting the
nonlocality of the $N$-qubit (or $N$-qudit) Greenberger-Horne-Zeilinger
(GHZ) states \cite{ghz,mermin90}. These states are extreme superpositions
that were shown by GHZ to demonstrate a very striking {}``all or
nothing'' type of nonlocality. This nonlocality can manifest as a
violation of a Bell inequality, and at first glance these violations,
because they increase exponentially with $N$, appear to indicate
a more extreme nonlocality as the size $N$ of the system increases
\cite{mermin bellghz}.
We point out, however, that the detection of \emph{genuine} $N$-body
nonlocality, as first discussed by Svetlichny \cite{genuine,collspinmol},
requires much higher thresholds. Genuine $n$-party nonlocality (e.g.
genuine entanglement) requires that the nonlocality is shared among
\emph{all} $N$ parties, or particles. The violations in this case
do not increase with $N$, and the detection over many sites is very
sensitive to loss and inefficiencies.
Finally, we review and outline how to detect entanglement \cite{collspinmol}
and the EPR paradox using collective spin measurements. This approach
has recently been employed to establish a genuine entanglement of
many particles in a BEC \cite{exp multi,treutnature}.
\section{Three Famous types of nonlocality}
The earliest studies of nonlocality concerned bipartite systems. Einstein-Podolsky-Rosen
(EPR) \cite{epr} began the debate about quantum nonlocality, by pointing
out that for some quantum states there exists an inconsistency between
the premises we now call {}``local realism'' and the completeness
of quantum mechanics.
\emph{Local realism} (LR) may be summarized as follows. EPR argued
\cite{epr,epr rev } first for {}``locality'', by claiming that
there could be no {}``action-at-a-distance''. A measurement made
at one location cannot instantaneously affect the outcomes of measurements
made at another distant location. EPR also argued for {}``reality'',
which they considered in the following context. Suppose one can predict
with certainty the result of a measurement made on a system, without
disturbing that system. Realism implies that this prediction is possible,
only because the outcome for that measurement was a predetermined
property of the system. EPR called this predetermined property an
{}``\emph{element of reality}'', though most often the element of
reality is interpreted as a {}``\emph{hidden variable}''. The essence
of EPR's local realism assumption is that results of measurements
made on a system at one location come about because of predetermined
properties of that system, and because of their local interactions
with the measurement apparatus, not because of measurements that are
made simultaneously at a distant locations.
\subsection{EPR paradox}
EPR argued that for states such as the spin $1/2$ singlet state
\begin{equation}
|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_{A}|\downarrow\rangle_{B}-|\downarrow\rangle_{A}|\uparrow\rangle_{B}\right)\label{eq:spinbohm}
\end{equation}
there arises an inconsistency of the LR premises with the quantum
predictions. Here, we define $|\uparrow\rangle_{A/B}$ and $|\downarrow\rangle_{A/B}$
as the spin {}``up'' and {}``down'' eigenstates of $J_{A/B}^{Z}$
for a system at location $A/B$. For the state (\ref{eq:spinbohm}),
the prediction of the spin component $J_{A}^{Z}$ can be made by measurement
of the component $J_{B}^{Z}$ at $B$. From quantum theory, the two
measurements are perfectly anticorrelated. According to EPR's Local
Realism premise (as explained above), there must exist an {}``element
of reality'' to describe the predetermined nature of the spin at
$A$. We let this element of reality be symbolized by the variable
$\lambda_{z}$, and we note that $\lambda_{z}$ assumes the values
$\pm1/2$ (\ref{eq:spinbohm}).
Calculation shows that there is a similar prediction of a perfect
anti-correlation for the other spin component pairs. Therefore, according
to LR, each of the spin components $J_{A}^{Y}$ and $J_{A}^{X}$ can
also be represented by an element of reality, which we denote $\lambda_{x}$
and $\lambda_{y}$ respectively. A moment's thought tells us that
the if there is a state for which all three spins are completely and
precisely predetermined in this way, then this {}``state'' cannot
be a quantum state. Such a {}``state'' is generally called a {}``local
hidden variable (LHV) state'', and the set of three variables are
{}``hidden'', since they are not part of standard quantum theory.
Hence, EPR argued, quantum mechanics is incomplete.
Since perfect anticorrelation is experimentally impossible, an operational
criterion for an EPR paradox can be formulated as follows. Consider
two observables $X$ and $P$, with commutators like position and
momentum. The Heisenberg Uncertainty Principle is $\Delta X\Delta P\geq1$,
where $\Delta X$ and $\Delta P$ are the variances of the outcomes
of measurements for $X$ and $P$ respectively. The EPR paradox criterion
is \cite{reidepr}
\begin{equation}
\Delta_{inf}X\Delta_{inf}P<1,\label{eq:eprcrit}
\end{equation}
where $\Delta_{inf}X\equiv V(X|O_{B})$ is the {}``variance of inference''
i.e. the variance of $X$ conditional on the measurement of an observable
$O_{B}$ at a distant location $B$. The $\Delta_{inf}P\equiv V(P|Q_{B})$
is defined similarly where $Q_{B}$ is a second observable for measurement
at $B$. This criterion reflects that the combined uncertainty of
inference is reduced below the Heisenberg limit. Of course, the reduced
uncertainty applies over an ensemble of measurements, where only one
of the conjugate measurements is made at a time. This criterion is
also applicable to optical quadrature observables, where it has been
experimentally violated, although without causal separation. With
spin commutators, other types of uncertainty principle can be used
to obtain analogous inferred uncertainty limits.
The demonstration of an EPR paradox through the measurement of correlations
satisfying Eq. (\ref{eq:eprcrit}) is a proof that local realism is
inconsistent with the completeness of quantum mechanics (QM). Logically,
one must: discard local realism, the completeness of QM, or both.
However, it does not indicate which alternative is correct.
\subsection{Schr\"odinger's Entanglement}
Schr\"odinger \cite{Schrodinger-1,Schrodinger-2,Schrodinger-3} noted
that the state (\ref{eq:spinbohm}) is a special sort of state, which
he called an an \emph{entangled} state. An entangled state is one
which cannot be factorized: for a pure state, we say there is entanglement
between $A$ and $B$ if we cannot write the composite state $|\psi\rangle$
(that describes all measurements at the two locations) in the form
$|\psi\rangle=|\psi\rangle_{A}|\psi\rangle_{B}$, where $|\psi\rangle_{A/B}$
is a state for the system at $A/B$ only.
For mixed states, there is said to be entanglement when the density
operator for the composite system cannot be written as a mixture of
factorizable states \cite{peres}. A mixture of factorizable states
is said to be a \emph{separable} state, which where there are just
two sites, is written as
\begin{equation}
\rho=\sum_{R}P_{R}\rho_{A}^{R}\rho_{B}^{R}.\label{eq:sep2}
\end{equation}
If the density operator cannot be written as (\ref{eq:sep2}), then
the mixed system possesses \emph{entanglement} (between $A$ and $B$).
More generally, for $N$ sites, full separability implies
\begin{equation}
\rho=\sum_{R}P_{R}\rho_{1}^{R}...\rho_{N}^{R}.\label{eq:sepN}
\end{equation}
If the density operator cannot be expressed in the fully separable
form (\ref{eq:sepN}), there is entanglement between at least two
of the sites.
We consider measurements $\hat{X}_{k}$, with associated outcomes
$X_{k}$, that can be performed on the $k$-th system ($k=1,...,N$).
For a separable state (\ref{eq:sepN}), it follows that the joint
probability for outcomes is expressible as
\begin{equation}
P(X_{1},...,X_{N})=\int_{\lambda}P(\lambda)P_{Q}(X_{1}|\lambda)...P_{Q}(X_{N}|\lambda)d\lambda\,,\label{eq:sepent}
\end{equation}
where we have replaced for convenience of notation the index $R$
by $\lambda$, and used a continuous summation symbolically, rather
than a discrete one, so that $P(\lambda)\equiv P_{R}$. The subscript
$Q$ represents {}``quantum'', because there exists the quantum
density operator $\rho_{k}^{\lambda}\equiv\rho_{k}^{R}$ for which
$P(X_{k}|\lambda)\equiv\langle X_{k}|\rho_{k}^{\lambda}|X_{k}\rangle$.
In this case, we write $P(X_{k}|\lambda)\equiv P_{Q}(X_{k}|\lambda)$,
where the subscript $Q$ reminds us that this is a quantum probability
distribution. The model (\ref{eq:sepent}) implies (\ref{eq:sep2})
\cite{wisesteer,wisesteer2}, and has been studied in Ref. \cite{eric steer},
in which it is referred to as a \emph{quantum separable model }(QS).
We can test nonlocality when each system $k$ is spatially separated.
We will see from the next section that LR implies the form (\ref{eq:sepent}),
but without the subscripts {}``Q'', that is, without the underlying
local states designated by $\lambda$ necessarily being quantum states.
If the quantum separable QS model can be shown to fail where each
$k$ is spatially separated, one can only have consistency with Local
Realism if there exist underlying local states that are \emph{non-quantum.
}This is an EPR paradox, since it is an argument to complete quantum
mechanics, based on a requirement that LR be valid.
The EPR paradox necessarily requires entanglement \cite{epr rev ,mallon}.
The reason for this is that for separable states (\ref{eq:sep2}-\ref{eq:sepent}),
the uncertainty relation that applies to each of the states $|\psi\rangle_{A}$
and $|\psi\rangle_{B}$ will imply a minimum level of local uncertainty,
which means that the noncommuting observables cannot be sufficiently
correlated to obtain an EPR paradox. In other words, the entangled
state (\ref{eq:spinbohm}) can possess a greater correlation than
possible for (\ref{eq:sep2}).
Schr\"odinger also pointed to two paradoxes \cite{Schrodinger-1,Schrodinger-2,Schrodinger-3}
in relation to the EPR paper. These gedanken-experiments strengthen
the apparent need for the existence of EPR {}``elements of reality'',
in situations involving macroscopic systems, or spatially separated
ones. The first is famously known as the Schr\"odinger's cat paradox,
and emphasizes the importance of EPR's {}``elements of reality''
at a \emph{macroscopic} level. Reality applied to the state of a cat
would imply a cat to be either dead or alive, prior to any measurement
that might be made to determine its {}``state of living or death''.
We can define an {}``element of reality'' $\lambda_{cat}$ , to
represent that the cat is \emph{predetermined} to be dead (in which
case $\lambda_{cat}=-1$) or alive (in which case $\lambda_{cat}=+1$).
Thus, the observer looking inside a box, to make a measurement that
gives the outcome {}``dead'' or {}``alive'', is simply uncovering
the value of $\lambda_{cat}$. Schr\"odinger's point was that the
element of reality specification is not present in the quantum description
$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|dead\rangle+|alive\rangle\right)$
of a superposition of two macroscopically distinguishable states.
The second paradox raised by Schr\"odinger concerns the apparent
{}``action at-a-distance'' that seems to occur for the EPR entangled
state. Unless one identifies an element of reality for the outcome
$A$, it would seem to be the action of the measurement of $B$ that
immediately enables prediction of the outcome for the measurement
at $A$. Schr\"odinger thus introduced the notion of {}``\emph{steering}''.
While all these paradoxes require entanglement, we emphasize that
entanglement \emph{per se} is a relatively common situation in quantum
mechanics. It is necessary for a quantum paradox, but does not by
itself demonstrate any paradox.
\subsection{Bell's nonlocality: failure of local hidden variables (LHV)}
EPR claimed as a solution to their EPR paradox that hidden variables
consistent with local realism would exist to further specify the quantum
state. It is the famous work of Bell that proved the impossibility
of finding such a theory. This narrows down the two alternatives possible
from a demonstration of the EPR paradox, and shows that local realism
itself is invalid.
Specifically, Bell considered the predictions of a Local Hidden Variable
(LHV) theory, to show that they would be different to the predictions
of the spin-half EPR state (\ref{eq:spinbohm}). Following Bell \cite{Bell,CHSH},
we have a \emph{local hidden variable model} (LHV) if the joint probability
for outcomes of simultaneous measurements performed on the $N$ spatially
separated systems is given by
\begin{equation}
P(X_{1},...,X_{N})=\int_{\lambda}P(\lambda)P(X_{1}|\lambda)...P(X_{N}|\lambda)d\lambda\,.\label{eq:bell}
\end{equation}
Here $\lambda$ are the {}``local hidden variables'' and $P(X_{k}|\lambda)$
is the probability of $X_{k}$ given the values of $\lambda$, with
$P(\lambda)$ being the probability distribution for $\lambda$. The
factorization in the integrand is Bell's locality assumption, that
$P(X_{k}|\lambda)$ depends on the parameters $\lambda$, and the
measurement choice made at $k$ only. The hidden variables $\lambda$
describe a\emph{ }local state\emph{ }for each site, in that the probability
distribution $P(X_{k}|\lambda)$ for the measurement at $k$ is given
as a function of the $\lambda$. The form of (\ref{eq:bell}) is formally
similar to (\ref{eq:sepN}) except in the latter there is the additional
requirement that the local states are quantum states. If (\ref{eq:bell})
fails, then we have\emph{ }proved\emph{ }a\emph{ }failure of all LHV
theories\emph{, }which we refer to as a \emph{Bell violation }or\emph{
Bell nonlocality} \cite{eric steer}\emph{. }
The famous Bell-Clauser-Horne-Shimony-Holt (CHSH) inequalities follow
from the LHV model, in the $N=2$ case. Bell considered measurements
of the spin components $J_{A}^{\theta}=\cos\theta J_{A}^{X}+\sin\theta J_{A}^{Y}$
and $J_{B}^{\theta}=\cos\phi J_{B}^{X}+\sin\phi J_{B}^{Y}$. He then
defined the spin product $E(\theta,\phi)=\langle J_{A}^{\theta}J_{B}^{\phi}\rangle$
and showed that for the LHV model, there is always the constraint
\begin{equation}
B=E(\theta,\phi)-E(\theta,\phi')+E(\theta',\phi)+E(\theta',\phi')\leq2.\label{eq:bellchsh}
\end{equation}
The quantum prediction for an entangled Bell state (\ref{eq:spinbohm})
is $E(\theta,\phi)=\cos(\theta-\phi)$ and the inequality is violated
for the choice of angles
\begin{equation}
\theta=0,\theta'=\pi/2,\phi=\pi/4,\phi'=3\pi/4\label{eq:angles}
\end{equation}
for which the quantum prediction becomes $B=2\sqrt{2}$. Tsirelson's
theorem proves the value of $B=2\sqrt{2}$ to be the maximum violation
possible for any quantum state \cite{tsirel}. We note that experimental
inefficiencies mean that violation of the CHSH inequalities for causally
separated detectors is difficult, and has so far always required additional
assumptions in the interpretation of experimental data.
\subsection{Steering as a special nonlocality }
Recently, Wiseman et al (WJD) \cite{wisesteer,wisesteer2} have constructed
a hybrid separability model, called the Local Hidden State Model (LHS),
the violation of which is confirmation of Schr\"odinger's {}``steering''
(Figure 1). The bipartite local hidden state model (LHS) assumes
\begin{equation}
P(X_{A},X_{B})=\int_{\lambda}P(\lambda)P(X_{A}|\lambda)P_{Q}(X_{B}|\lambda)d\lambda\,.\label{eq:bell-1}
\end{equation}
Thus, for one site $A$ which we call {}``Alice'', we assume a local
hidden variable (LHV) state, but at the second site $B$, which we
call {}``Bob'', we assume a local quantum state (LQS). The violation
of this model occurs iff there is a {}``\emph{steering}'' of Bob's
state by Alice \cite{steerexp}.
WJD pointed out the association of steering with the EPR paradox \cite{wisesteer}.
The EPR criterion is also a criterion for steering, as defined by
the violation of the LHS model. An analysis of the EPR argument when
generalized to allow for imperfect correlation and arbitrary measurements
reveals that violation of the LHS model occurs iff there is an EPR
paradox \cite{eric steer,epr rev }. As a consequence, the violation
of the LHS model is referred to as demonstration of a type of nonlocality
called {}``\emph{EPR steering}'' \cite{eric steer}. EPR steering
confirms the incompatibility of local realism with the \emph{completeness}
of quantum mechanics, just as with the approach of EPR in their original
paper \cite{epr}.
The notion of steering can be generalized to consider $N$ sites,
or observers \cite{ericmulti}. The multipartite LHS model is (Figure
1)
\begin{multline}
P(X_{1},...,X_{N})=\\
\int d\lambda P(\lambda)\prod_{j=1}^{T}P_{Q}(X_{j}|\lambda)\prod_{j=T+1}^{N}P(X_{j}|\lambda),\label{eq:LHS_model_multipartite}
\end{multline}
where here we have $T$ quantum states, and $N-T$ LHV local states.
We use the symbol $T$ to represent the quantum states, since these
are the{}``trusted sites'' in the picture put forward by WJD \cite{wisesteer}.
This refers to an application of this generalized steering to a type
of quantum cryptography in which an encrypted secret is being shared
between sites. At some of the sites, the equipment and the observers
are trusted, while at other sites this is not the case.
In this picture, which is an application of the LHS model, an observer
$C$ wishes to establish entanglement between two observers Alice
and Bob. The violation of the QS model is sufficient to do this, provided
each of the two observers Alice and Bob can be trusted to report the
values for their local measurements. It is conceivable however that
they report instead statistics that can give a violation of LHS model,
so it seems as if there is entanglement when there is not. WJD point
out the extra security present if instead there is the stronger requirement
of violation of the LHV model, in which the untrusted observers are
identified with a LHV state.
Cavalcanti et al \cite{ericmulti} have considered the multipartite
model (\ref{eq:LHS_model_multipartite}), and shown that violation
of (\ref{eq:LHS_model_multipartite}) where $T=1$ is sufficient to
imply an \emph{EPR steering} paradox exists between at least two of
the sites. Violation where $T=0$ is proof of Bell's nonlocality,
and violation where $T=N$ is a confirmation of entanglement (quantum
inseparability).
\subsection{Hierarchy of nonlocality }
WJD established formally the concept of a hierarchy of nonlocality
\cite{wisesteer,wisesteer2}. Werner \cite{werner} showed that some
classes of entangled state can be described by Local Hidden Variable
theories and hence cannot exhibit a Bell nonlocality. WJD showed that
not all entangled states are {}``steerable'' states, defined as
those that can exhibit EPR steering. Similarly, they also showed that
not all EPR steerable states exhibit Bell nonlocality. However, we
see from the definitions that all EPR steering states must be entangled,
and all Bell-nonlocal states ( defined as those exhibiting Bell nonlocality)
must be EPR steering states. Thus, the Bell-nonlocal states are a
strict subset of EPR steering states, which are a strict subset of
entangled states, and a hierarchy of nonlocality is established.
\begin{figure}
\caption{\emph{The LHS model:}
\end{figure}
\section{Multiparticle Nonlocality}
Experiments that have been performed on many microscopic systems support
quantum mechanics. Those that test Bell's theorem \cite{Bell,CHSH},
or the equivalent, are the most useful, since they directly refute
the assumption of local realism. While these experiments still require
additional assumptions, it is generally expected that improved technology
will close the remaining loopholes.
There remains however the very important question of whether reality
will hold macroscopically. Quantum mechanics predicts the possibility
of superpositions of two macroscopically distinguishable states \cite{legg},
like a cat in a superposition of dead and alive states. Despite the
apparent paradox, there is increasing evidence for the existence of
mesoscopic and macroscopic quantum superpositions.
As with microscopic systems, there is a need to verify the loss of
reality for macroscopic superpositions in an objective sense, by following
Bell's example and comparing the predictions of quantum mechanics
with those based on premises of local realism. The first steps toward
this have been taken, through theoretical studies of nonlocality for
multi-particle systems. Two limits have been rather extensively examined.
The first is that of bipartite qudits. The second is multipartite
qubits. Surprisingly, while it may have been thought that the violation
of LR would diminish or vanish at a critical number of particles,
failure of local realism has been shown possible according to quantum
mechanics, for arbitrarily large numbers of particles. The third possibility
of multipartite qudits has not been treated in as much detail.
\subsection{Bipartite qudits}
The simplest mesoscopic extension of the Bell case (\ref{eq:spinbohm})
is to consider bipartite qudits: two sites of higher dimensionality.
The maximally entangled state in this case is
\begin{equation}
|\psi\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|jj\rangle,\label{eq:maxentqudit}
\end{equation}
where $|jj\rangle$ is abbreviation for $|j\rangle_{A}|j\rangle_{B}$,
and $d$ is the dimensionality of the systems at $A$ and $B$. In
this case at each site $A$ and $B$ the possible outcomes are $j=0,...,d-1$.
This system can be realized by two spin $J$ systems, for which the
outcomes are $x$ given by $-J,-J+1,...,J-1,J$, so that $d=2J+1$,
and $j$ of Eq. (\ref{eq:maxentqudit}) is $j\equiv x+J$ where $x$
is the outcome of spin. It can also be realized by multi-particle
systems.
It was shown initially by Mermin, Garg and Drummond, and Peres and
others \cite{highd,multibell,drumspinbell,peresspin,gsisspin,franmrspin}
that quantum systems could violate local realism for large $d$. The
approach was to use the classic Bell inequalities derived for binary
outcomes.
Later, Kaszlikowski et al \cite{high D K} showed that for maximally
entangled states (\ref{eq:maxentqudit}), the strength of violation
actually becomes stronger for increasing $d$. A new set of Bell inequalities
for bipartite qudits was presented by Collins et al (CGLMP) et al
\cite{collins high d,fuqdit} and it was shown subsequently by Acin
et al \cite{acin} that greater violations can be obtained with non-maximally
entangled states, and that the violations increase with $d$. Chen
et al \cite{chen} have shown that the violation of CGLMP inequalities
increases as $d\rightarrow\infty$ to a limiting value.
We wish to address the question of how the entanglement and EPR steering
nonlocalities increase with $d$. Since Bell nonlocality implies both
EPR steering and entanglement, these nonlocalities also increase with
$d.$ However, since there are distinct nested classes of nonlocality,
the violation could well be greater, for an appropriate set of measures
of the nonlocalities, and this problem is not completely solved for
the CGLMP approach. We later investigate alternative criteria that
show differing levels of violation for the different classes of nonlocality.
\subsection{Multipartite qubits: MABK Bell inequalities}
The next mesoscopic - macroscopic scenario that we will consider is
that of many distinct single particles $-$ the multi-site qubit system.
The interest here began with the Greenberger-Horne-Zeilinger (GHZ)
argument \cite{ghz}, which revealed a more extreme {}``all-or-nothing''
form of nonlocality for the case of three and four spin $1/2$ particle
(three or four qubits), prepared in a so-called GHZ state. The N qubit
GHZ state is written
\begin{equation}
|\Psi\rangle_{GHZ}=\frac{1}{\sqrt{2}}\{|0\rangle^{\otimes N}+|1\rangle^{\otimes N}\},\label{eq:ghz-1}
\end{equation}
where $|0\rangle$ and $|1\rangle$ in this case are spin up/ down
eigenstates. Mermin then showed that for this extreme superposition,
there corresponded a greater violation of LR, in the sense that the
new {}``Mermin'' Bell inequalities were violated by an amount that
increased exponentially with $N$ \cite{mermin bellghz}. These new
multipartite Bell inequalities of Mermin were later generalized by
Ardehali, Belinski and Klyshko, to give a set of MABK Bell inequalities
\cite{ard,bkmabk}.
The MABK inequalities define moments like $\langle J_{A}^{+}J_{B}^{+}J_{C}^{-}\rangle$,
where $J^{\pm}=J^{X}\pm iJ^{Y}$ and $J^{X}$, $J^{Y}$, $J^{Z}$,
$J^{2}$ are the standard quantum spin operators. In the MABK case
of qubits, Pauli operators are used, so that the spin outcomes $\pm1/2$
are normalized to $\pm1$. The $J^{X/Y}$ are redefined accordingly.
The moments are defined generally by
\begin{equation}
\prod_{N}=\langle\Pi_{k=1}^{N}J_{k}^{s_{k}}\rangle\label{eq:prodNmabk}
\end{equation}
where $s_{k}=\pm1$ and $J^{s_{1}}\equiv J^{+}$ and $J^{s_{-1}}\equiv J^{-}$.
A LHV theory expresses such moments as the integral of a complex number
product:
\begin{equation}
\prod_{N}=\int d\lambda P(\lambda)\Pi_{N,\lambda}\label{eq:prodNmabkhidden}
\end{equation}
where $\Pi_{N,\lambda}=\Pi_{k=1}^{N}\langle J_{k}^{s_{k}}\rangle_{\lambda}$
and $\langle J_{k}^{\pm}\rangle_{\lambda}=\langle J_{k}^{X}\rangle\pm i\langle J_{k}^{Y}\rangle$
where $\langle J_{k}^{X/Y}\rangle_{\lambda}$ is the expected value
of outcome for measurement $J^{X/Y}$ made at site $k$ given the
local hidden state $\lambda$. The $\Pi_{N,\lambda}$ is a complex
number product, which Mermin \cite{mermin bellghz} showed has the
following extremal values: for $N$ odd, a magnitude $2^{N/2}$ at
angle $\pi/4$ to real axis; for $N$ even, magnitude $2^{N/2}$ aligned
along the real or imaginary axis. With this algebraic constraint,
LR will imply the following inequalities, for odd $N$:
\begin{equation}
Re\prod_{N},\, Im\prod_{N}\leq2^{(N-1)/2}.\label{eq:mabkodd}
\end{equation}
For even $N$, the inequality $ $$Re\prod_{N},\, Im\prod_{N}\leq2^{N/2}$
will hold. However, it is also true, for even $N$, that \cite{ard}
\begin{equation}
Re\prod_{N}+Im\prod_{N}\leq2^{N/2}.\label{eq:mabkeven}
\end{equation}
The Eqns (\ref{eq:mabkodd}-\ref{eq:mabkeven}) are the MABK Bell
inequalities \cite{bkmabk}. Maximum violation of these inequalities
is obtained for the $N$-qubit Greenberger-Horne-Zeilinger (GHZ) state
(\ref{eq:ghz-1}) \cite{wernerwolf}. For optimal angle choice, a
maximum value
\begin{equation}
\langle\mathrm{Re}\Pi_{N}\rangle,\mathrm{\langle Im}\Pi_{N}\rangle=2^{N-1}\label{eq:qm1}
\end{equation}
can be reached for the left -side of (\ref{eq:mabkodd}), while for
a different optimal angle choice, the maximum value
\begin{equation}
\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle=2^{N-1/2}\label{eq:qm2}
\end{equation}
can be reached for the left-side of (\ref{eq:mabkeven}). MABK Bell
inequalities became famous for the prediction of exponential gain
in violation as the number of particles (sites), $N$, increases.
The size of violation is most easily measured as the ratio of left-side
to right-side of the inequalities (\ref{eq:mabkodd},\ref{eq:mabkeven}),
seen to be $2^{(N-1)/2}$ for the MABK inequalities. Werner and Wolf
\cite{wernerwolf} showed the quantum prediction to be maximum for
two-setting inequalities.
\subsection{MABK-type EPR steering and entanglement inequalities for multipartite
qubits}
Recently, MABK-type inequalities have been derived for EPR steering
and entanglement \cite{ericmulti}. Entanglement is a failure of quantum
separability, where each of the local states in (\ref{eq:LHS_model_multipartite})
are quantum states ($T=N$). EPR steering occurs when there is failure
of the LHS model with $T=1$. To summarize the approach of Ref. \cite{ericmulti},
we note the statistics of each \emph{quantum }state must satisfy a
quantum uncertainty relation
\begin{equation}
\Delta^{2}J^{X}+\Delta^{2}J^{Y}\geq1.\label{eq:hup-1}
\end{equation}
As a consequence, for every \emph{quantum} local state $\lambda$,
\begin{equation}
\langle J^{X}\rangle^{2}+\langle J^{Y}\rangle^{2}\leq1,\label{eq:hupconseqquantum}
\end{equation}
which implies the complex number product can have arbitrary phase,
leading to the new nonlocality inequalities, which apply for all $N$,
even or odd, and $T>0$:
\begin{eqnarray}
\langle\mathrm{Re}\Pi_{N}\rangle,\langle\mathrm{Im}\Pi_{N}\rangle & \leq & 2^{(N-T)/2},\label{eq:merminsteer}\\
\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle & \leq & 2^{(N-T+1)/2}.\label{eq:merminsteerstat-2}
\end{eqnarray}
For $T=N$, these inequalities if violated will imply entanglement,
as shown by Roy \cite{roy}. If violated for $T=1$, there is EPR
steering. As pointed out in \cite{ericmulti}, the exponential gain
factor of the violation with the number of particles $N$ increases
for increasing $T$: the strength of violation as measured by left
to right side ratio is $2^{(N+T-2)/2}$, but for both inequalities
(\ref{eq:merminsteer}-\ref{eq:merminsteerstat-2}).
\subsection{CFRD Multipartite qudit Bell, EPR steering and entanglement inequalities}
We now summarize an alternative approach to nonlocality inequalities,
developed by Cavalcanti, Foster, Reid and Drummond (CFRD) \cite{cfrd,vogelcfrd,acincfrd,cfrd he func,cfrdhepra}.
These hold for any operators, and are not restricted to spin-half
or qubits. We shall apply this approach to the case of a hierarchy
of inequalities, with some quantum and some classical hidden variable
states. Consider
\begin{eqnarray}
|\prod_{N}| & \leq & \int d\lambda P(\lambda)\Pi_{k=1}^{N}|\langle J_{k}^{s_{k}}\rangle_{\lambda}|\label{eq:prodNmabkhidden-1}\\
& = & \int d\lambda P(\lambda)\Pi_{k=1}^{N}\{\langle J_{k}^{X}\rangle_{\lambda}^{2}+\langle J_{k}^{Y}\rangle_{\lambda}^{2}\}^{1/2}.
\end{eqnarray}
We can see that for any LHV, because the variance is always positive,
one can derive an inequality for any operator
\begin{equation}
\langle J_{k}^{X}\rangle_{\lambda}^{2}+\langle J_{k}^{Y}\rangle_{\lambda}^{2}\leq\langle(J_{k}^{X})^{2}\rangle_{\lambda}+\langle(J_{k}^{Y})^{2}\rangle_{\lambda}\label{eq:LHVspinvar}
\end{equation}
but then for a quantum state in view of the uncertainty relation (\ref{eq:hup-1}),
it is the case that for qubits (spin-1/2)
\begin{equation}
\langle J_{k}^{X}\rangle_{\lambda}^{2}+\langle J_{k}^{Y}\rangle_{\lambda}^{2}\leq\langle(J_{k}^{X})^{2}\rangle_{\lambda}+\langle(J_{k}^{Y})^{2}\rangle_{\lambda}-1.\label{eq:lqsvarhur}
\end{equation}
For the particular case of qubits, the outcomes are $\pm1$ so that
simplification occurs, to give final bounds based on local realism
that are identical to (\ref{eq:merminsteer}-\ref{eq:merminsteerstat-2}).
We note that at $T=0$, there is also a CFRD Bell inequality, but
it is weaker than that of MABK, in the sense that the violation is
not as strong as is not predicted for $N=2$. Since this approach
holds for any operator, we now can generalize to arbitrary spin.
The expression (\ref{eq:LHVspinvar}-\ref{eq:lqsvarhur}) also holds
for arbitrary spin, for which case we revert to the usual spin outcomes
(rather than the Pauli spin outcomes of $\pm1$). The LHV result for
arbitrary spin is constrained by (\ref{eq:LHVspinvar}). The quantum
result however requires a more careful uncertainty relation that is
relevant to higher spins. In fact, for systems of fixed dimensionality
$d$, or fixed spin $J$, the {}``qudits'', the following uncertainty
relation holds
\begin{equation}
\Delta^{2}J^{X}+\Delta^{2}J^{Y}\geq C_{J},\label{eq:cj}
\end{equation}
where the $C_{J}$ has been derived and presented in Ref. \cite{cj}.
The use of the more general result (\ref{eq:cj}) gives the following
higher-spin (qudit) nonlocality inequalities derived in Ref. \cite{higherspin steerq}:
\begin{multline}
|\langle\prod_{k=1}^{N}J_{k}^{s_{k}}\rangle|^{2}\leq\int d\lambda P(\lambda)\prod_{k=1}^{N}|\langle J_{k}^{s_{k}}\rangle_{\lambda}|^{2}\\
\leq\left\langle \prod_{k=1}^{T}(J_{k}^{X})^{2}+(J_{k}^{Y})^{2}-C_{k})\prod_{k=T+1}^{N}(J_{k}^{X})^{2}+(J_{k}^{Y})^{2}\right\rangle .\label{eq:ineqcom}
\end{multline}
Thus:
\begin{enumerate}
\item Entanglement is verified if ($T=N$)
\begin{eqnarray}
|\langle\prod_{k=1}^{N}J_{k}^{s_{k}}\rangle|^{2} & > & \langle\prod_{k=1}^{N}[(J_{k})^{2}-(J_{k}^{Z})^{2}-C_{J}]\rangle.\label{eq:spinjent}
\end{eqnarray}
\item An EPR-steering nonlocality is verified if ($T=1$)
\begin{eqnarray}
|\langle\prod_{k=1}^{N}J_{k}^{s_{k}}\rangle|^{2} & > & \langle[(J_{1})^{2}-(J_{1}^{Z})^{2}-C_{J}]\nonumber \\
& & \ \ \times\prod_{k=2}^{N}[(J_{k}^{X})^{2}+(J_{k}^{Y})^{2}]\rangle.\label{eq:spinjsteer}
\end{eqnarray}
\item Bell inequality ($T=0$). The criterion to detect failure of the LHV
theories is
\begin{eqnarray}
|\langle\prod_{k=1}^{N}J_{k}^{s_{k}}\rangle|^{2} & > & \langle\prod_{k=0}^{N}[(J_{k}^{X})^{2}+(J_{k}^{Y})^{2}]\rangle.
\end{eqnarray}
\end{enumerate}
These criteria will be called the {}``$C_{J}"$ CFRD nonlocality
criteria, and allow investigation of nonlocality in multisite qudits,
where the spin $J$ is fixed.
We investigate predictions for quantum states that are maximally entangled,
or not so\textcolor{black}{, according to measures of entanglement
that are justified for pure states. Maximally-entangled, highly correlated
states for a fixed spin $J$ are written
\begin{eqnarray}
|\Psi\rangle_{max} & = & \frac{1}{\sqrt{d}}\sum_{m=-J}^{J}|m\rangle_{1}|m\rangle_{2}....|m\rangle_{N}\nonumber \\
& = & \frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|j\rangle_{1}|j\rangle_{2}...|j\rangle_{N},\label{eq:maxentst}
\end{eqnarray}
where $|m\rangle_{k}\equiv|J,m\rangle_{k}$ is the eigenstate of $J_{k}^{2}$
and $J_{k}^{Z}$ (eigenvalue $m$ for $J_{k}^{Z}$), defined at site
$k$, and the dimensionality is $d=2J+1$. This state is the extension
of (\ref{eq:maxentqudit}) for multiple sites. We follow \cite{acin}
however and consider more generally the non-maximally entangled but
highly correlated spin states of form
\begin{eqnarray}
|\psi\rangle_{non} & = & \frac{1}{\sqrt{n}}[r_{-J}|J,-J\rangle^{\otimes N}+r_{-J+1}|J,-J+1\rangle^{\otimes N}\nonumber \\
& & \ \ \ \ +...+r_{J}|J,+J\rangle^{\otimes N}],\label{eq:nonmaximally state}
\end{eqnarray}
where $|J,m\rangle^{\otimes N}=\Pi_{k=1}^{N}|J,m\rangle_{k}$, $n={\displaystyle \sum_{m=-J}^{J}}r_{m}^{2}$.
Here we will restrict to the case of real parameters symmetrically
distributed around $m=0$. The amplitude $r_{m}$ can be selected
to optimize the nonlocality result. It is known for example, with
$N$ sites and a spin-$1$ system that the optimized state:
\begin{eqnarray}
|\psi\rangle & = & \frac{1}{\sqrt{r^{2}+2}}(|1,-1\rangle^{\otimes N}+r|1,0\rangle^{\otimes N}\nonumber \\
& & \ \ \ \ +|1,+1\rangle^{\otimes N}),\label{eq:staterspin1}
\end{eqnarray}
will give improved violation over the maximally entangled state (for
which the amplitudes are uniform) for some Bell inequalities \cite{acin}. }
\textcolor{black}{With the optimization described above, we summarize
}the results explained in Ref. \cite{higherspin steerq} that a growth
of the violation of the nonlocality inequalities for increasing number
$N$ of spin sites is maintained with arbitrary $d$. This is shown
in Figure 2 for qudits $d=2$ and $d=3$ (spin $J=1/2$ and $J=1$),
but means for higher $d$ that one can obtain in principle a violation
of inequalities for arbitrary $d$ by increasing $N$. Thus, quantum
mechanics predicts that at least for some states, increasing contradiction
with separable theories is possible, as the number of sites increases,
even where one has at each site a system of high spin. These results
are consistent with those obtained by other authors \cite{Cabello,multisitequdit,multisitequditson}.
\begin{figure}
\caption{\emph{Showing nonlocality to be possible for large numbers $N$ of
spin systems.}
\end{figure}
\section{Genuine Multiparticle Nonlocality: qubit example}
Svetlichny \cite{genuine} addressed the following question. How many
particles are \emph{genuinely} entangled? The above nonlocality inequalities
can fail if separability/ locality fails between a single \emph{pair}
of sites. To prove \emph{all} $N$ sites are entangled, or that the
Bell nonlocality is shared between \emph{all} $N$ sites, is a more
challenging task, and one that relates more closely to the question
of multi-particle quantum mechanics.
To detect genuine nonlocality, one needs to construct different criteria.
For example where $N=3$, to show genuine tripartite entanglement,
we need to exclude that the statistics can be described by bipartite
entanglement i.e., by the models
\begin{equation}
\rho=\sum_{R}P_{R}\rho_{AB}^{R}\rho_{C}^{R},\,\rho=\sum_{R}P_{R}\rho_{A}^{R}\rho_{BC}^{R},\,\rho=\sum_{R}P_{R}\rho_{B}^{R}\rho_{AC}^{R},\label{eq:genentmodel}
\end{equation}
where $\rho_{IJ}^{R}$ can be \emph{any} density operator for composite
system $I$ and $J$. These models can fail \emph{only} if there is
genuine tripartite entanglement. Thus, to show there is a genuine
tripartite Bell nonlocality, one needs to falsify all models encompassing
bipartite Bell nonlocality, i.e..
\begin{equation}
P(x_{\theta},x_{\phi},x_{\vartheta})=\int d\lambda P(\lambda)P_{AB}(x_{\theta},x_{\phi}|\lambda)P_{C}(x_{\vartheta}|\lambda)\label{eq:gennolocmodel}
\end{equation}
and the permutations. In the expansion (\ref{eq:gennolocmodel}),
locality is not assumed between $A$ and $B$, but is assumed between
a composite system $AB$, and $C$. This model allows bipartite entanglement
between $A$ and $B$, but not tripartite entanglement. To test genuine
nonlocality or entanglement, it is therefore useful to consider hybrid
local-nonlocal models. What is a condition for genuine $N$ partite
entanglement?
Consider again the $N$-qubit system. A recent analysis \cite{ericmulti}
follows Svetlichny \cite{genuine} and Collins \emph{et al. }(CGPRS)
\cite{collspinmol}, to consider a hybrid local-nonlocal model in
which Bell nonlocality \emph{can} exist, but only if shared among
$k=N-1$ or fewer parties. Separability must then be retained between
any two groups $A$ and $B$ of $ $$k$ and $N-k$ parties respectively,
if $k>N/2$ , and one can write:
\begin{equation}
\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle=\int_{\lambda}d\lambda P(\lambda)\langle\prod_{j=1}^{k}F_{j}^{s_{j}}\rangle_{A,\lambda}\langle\prod_{j=k+1}^{N}F_{j}^{s_{j}}\rangle_{B,\lambda}.\label{eqn:sepave-1}
\end{equation}
Violation of all such {}``$k$ - nonlocality'' models then implies
the nonlocality to be genuinely {}``$k+1$ partite''. We summarize
Ref. \cite{ericmulti} who use (\ref{eqn:sepave-1}) to consider
consequences of the hybrid model (\ref{eqn:sepave-1}) for the three
different types of nonlocality. Multiplying out $\prod_{j=1}^{N}F_{j}^{s_{j}}=\mathrm{Re}\Pi_{N}+i\mathrm{Im}\Pi_{N}$
reveals recursive relations $\mathrm{Re}\Pi_{N}=\mathrm{Re}\Pi_{N-1}\sigma_{x}^{N}-\mathrm{Im}\Pi_{N-1}\sigma_{y}^{N}$,
$\mathrm{Im}\Pi_{N}=\mathrm{Re}\Pi_{N-1}\sigma_{y}^{N}+\mathrm{Im}\Pi_{N-1}\sigma_{x}^{N}$
which imply algebraic constraints that must hold for all theories
\cite{mermin bellghz}
\begin{eqnarray}
\langle\mathrm{Re}\Pi_{N}\rangle,\langle\mathrm{Im}\Pi_{N}\rangle & \leq & 2^{N-1},\label{eq:alg1}\\
\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle & \leq & 2^{N}.\label{eq:alg2}
\end{eqnarray}
These recursive relations and the CHSH lemma summarized by Ardehali
\cite{ard} gives the Svetlichny-CGPRS inequality \cite{genuine,collspinmol}
\[
\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle\leq2^{N-1}
\]
the violation of which confirms genuine $N$ partite Bell-nonlocality.
The quantum prediction maximizes at (\ref{eq:qm2}) to predict violation
by a \emph{constant }amount ($S_{N}=\sqrt{2}$) \cite{ghose-1,ghose2}.
In order to investigate the other nonlocalities, for example the genuine
multipartite steering, the authors of Ref. \cite{ericmulti} suggest
the hybrid approach of \emph{quantizing} $B$, the group of $N-k$
qubits, but not group $A$. In this case, the extremal points of the
hidden variable product $\langle\Pi_{k}^{A}\rangle_{\lambda}=\langle\prod_{j=1}^{k}F_{j}^{s_{j}}\rangle_{A,\lambda}$
of $A$ is constrained only by the \emph{algebraic} limit (\ref{eq:alg1}),
whereas the product $\langle\Pi_{N-k}^{B}\rangle_{\lambda}\equiv\langle\prod_{j=k+1}^{N}F_{j}^{s_{j}}\rangle_{\lambda}$
for group $B$ is constrained by the \emph{quantum} result (\ref{eq:qm2}).
We note that a criterion for genuine $N$-qubit entanglement is obtained
by constraining \emph{both} $A$ and $B$ to be quantum, leading to
the condition
\begin{equation}
\langle\mathrm{Re}\Pi_{N}\rangle,\langle\mathrm{Im}\Pi_{N}\rangle\leq2^{N-2}
\end{equation}
(as derived in Ref. \cite{tothguhne}), and $\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle\leq2^{N-3/2}$.
These are violated by (\ref{eq:qm1}-\ref{eq:qm2}) to confirm genuine
$N$qubit entanglement ($S_{N}=2$).
In short, genuine $N$ particle nonlocality can be confirmed using
MABK Bell inequalities for $N$ qubits, but a higher threshold is
required. The threshold is reached by the quantum prediction of the
GHZ states, but the higher bound implies the level of violation is
\emph{no longer} exponentially increasing with $N$. As a related
consequence, the higher threshold also implies a much higher bound
for efficiency, which makes multi-particle nonlocality difficult to
detect for increasingly larger systems.
\begin{figure}
\caption{\emph{Genuine nonlocality and entanglement:}
\end{figure}
\section{Investigating Entanglement using Collective measurements: spin squeezing
inequalities}
While detection of individual qubits could be fulfilled in many systems,
the demonstration of a large multi-particle nonlocality would likely
require exceptional detection efficiencies if one is to detect a \emph{genuine}
multi-particle nonlocality for large $N$. We thus review and outline
a complementary approach, which is the measurement of the \emph{collective
}spin of a system.
\subsection{Spin squeezing entanglement criterion}
Consider $N$ identical spin-$J$ particles (Figure 3). One defines
the collective spin operator
\begin{equation}
J^{X}=\sum_{k=1}^{N}J_{k}^{X}\label{eq:spincoll}
\end{equation}
and similarly a $J^{Y}$ and $J^{Z}$. Entanglement between the spin
$J$ particles can be inferred via measurements of these collective
operators. The concept of spin squeezing was pioneered by Kitagawa
and Ueda \cite{spinsq-1kueda}, and Wineland et al \cite{wineland e}.
To investigate entanglement, we note that for each particle, or quantum
site $k$, the Heisenberg uncertainty relation holds
\begin{equation}
\Delta J_{k}^{X}\Delta J_{k}^{Y}\geq|\langle J_{k}^{Z}\rangle|/2.\label{eq:hup}
\end{equation}
If the system is fully separable (no entanglement) then
\begin{equation}
\rho=\sum_{R}P_{R}\rho_{1}^{R}...\rho_{k}^{R}...\rho_{N}^{R}.\label{eq:fulsep}
\end{equation}
For a mixture, the variance is greater than the average of the variances
of the components, which for a product state is the sum of the individual
variances \cite{hofman}. Thus, separability implies
\begin{equation}
\Delta^{2}J^{X}\geq\sum_{R}P_{R}\sum_{k=1}^{N}\Delta^{2}J_{k}^{X}.\label{eq:varmin}
\end{equation}
The next point to note is that for a fixed dimensionality spin- $J$
system, there is a constraint on the \emph{minimum }value for the
variance of spin. The constraint on the minimum arises because of
the constraint on the \emph{maximum} variance, which for fixed spin
$J$ must be bounded by
\begin{equation}
\Delta^{2}J^{Y}\leq J^{2}.\label{eq:varmax}
\end{equation}
This implies, by the uncertainty relation, the lower bound on the
minimum variance for a spin J system
\begin{equation}
\Delta^{2}J^{X}\geq\langle J^{Z}\rangle^{2}/4J^{2}.\label{eq:hupspinsq}
\end{equation}
Then we can prove, using (\ref{eq:varmin}) to get the first line,
\begin{eqnarray}
\Delta^{2}J^{X} & \geq & \frac{1}{4J^{2}}\sum_{k=1}^{N}\sum_{R}P_{R}\langle J_{k}^{Z}\rangle_{R}^{2}\nonumber \\
& \geq & \frac{1}{4J^{2}}\sum_{k=1}^{N}|\sum_{R}P_{R}\langle J_{k}^{Z}\rangle_{R}|^{2}\nonumber \\
& = & \frac{1}{4J^{2}}\sum_{k=1}^{N}|\langle J_{k}^{Z}\rangle|^{2}\label{eq:proofspinsq}
\end{eqnarray}
and the Cauchy Schwarz inequality to get the second to last line (use
$(\sum x^{2})(\sum y^{2})\geq|\sum xy|^{2}$ where $x=\sqrt{P_{R}}$
and $y=\sqrt{P_{R}}\langle J_{k}^{Z}\rangle_{R}$). We can rewrite
and use the Cauchy-Schwarz inequality again (this time, $x=1/\sqrt{N}$
and $y=\langle J_{k}^{Z\rangle}\rangle/\sqrt{N}$), to obtain
\begin{eqnarray}
\Delta^{2}J^{X} & = & \frac{N}{4J^{2}}\sum_{k=1}^{N}\frac{1}{N}|\langle J_{k}^{Z}\rangle|^{2}\nonumber \\
& \geq & \frac{N}{4J^{2}}|\sum_{k=1}^{N}\frac{1}{N}\langle J_{k}^{Z}\rangle|^{2}\nonumber \\
& = & \frac{1}{4NJ^{2}}|\langle J^{Z}\rangle|^{2}.\label{eq:proofspinsq-1}
\end{eqnarray}
We can express the result as
\begin{equation}
y=x^{2}/4J,\label{eq:analyquad}
\end{equation}
where $y=\Delta^{2}J^{X}/J$ and $x=|\langle J^{Z}\rangle|/J$. For
$J=1/2$, we obtain the result that for a fully separable state,
\begin{equation}
\Delta^{2}J^{X}\geq|\langle J^{Z}\rangle|^{2}/N\label{eq:spinsqentcrit}
\end{equation}
($y=x^{2}/2$). This result for spin $1/2$ was first derived by Sorenson
et al \cite{sorespinsqzoller}, and is referred to as the {}``spin
squeezing criterion'' to detect entanglement. Failure of (\ref{eq:spinsqentcrit})
reflects a reduction in variance (hence {}``squeezing''), and is
confirmation that there is entanglement between at least two particles
(sites). The criterion is often expressed in terms of the parameter
defined by Wineland et al \cite{wineland e}, that is useful measure
of interferometric enhancement, as
\begin{equation}
\xi=\frac{\sqrt{N}\Delta J^{X}}{|\langle J^{Z}\rangle|}<1.\label{eq:spinsqe}
\end{equation}
\begin{figure}
\caption{Detecting entanglement within the atoms of a two-component BEC using
the spin squeezing criterion (\ref{eq:spinsqe}
\end{figure}
The spin squeezing criterion has been used to investigate entanglement
within a group of atoms in a BEC by Esteve et al, Gross et al and
Riedel et al \cite{Germany-spin&entanglement,exp multi,treutnature}.
In fact, spin squeezing is predicted for the ground state of the following
two-mode Hamiltonian
\begin{equation}
H=\kappa(a^{\dagger}b+ab^{\dagger})+\frac{g}{2}[a^{\dagger}a^{\dagger}aa+b^{\dagger}b^{\dagger}bb],\label{hamgs}
\end{equation}
which is a good model for a two-component BEC. Here $\kappa$ denotes
the conversion rate between the two components, and $g$ is a self
interaction term. More details on one method of solution of this Hamiltonian
and some other possible entanglement criteria are given in Ref. \cite{eprbec he}.
To summarize, collective spin operators can be defined in the Schwinger
representation: \textcolor{black}{
\begin{eqnarray}
J^{Z} & = & (a^{\dagger}a-b^{\dagger}b)/2,\nonumber \\
J^{X} & = & (a^{\dagger}b+ab^{\dagger})/2,\nonumber \\
J^{Y} & = & (a^{\dagger}b-ab^{\dagger})/(2i),\nonumber \\
J^{2} & = & \hat{N}(\hat{N}+2)/4,\nonumber \\
\hat{N} & = & a^{\dagger}a+b^{\dagger}b.\label{eq:schwinger}
\end{eqnarray}
}The system is viewed as $N$ atoms, each with two-levels (components)
available to it. For each atom, the spin is defined in terms of boson
operators $J_{i}^{Z}=(a_{i}^{\dagger}a_{i}-b_{i}^{\dagger}b_{i})/2$
where the total number for each atom is $N_{i}=1$, and the outcomes
for $a_{i}^{\dagger}a_{i}$ and $b_{i}^{\dagger}b_{i}$ are $0$ and
$1$. The collective spin defined as $J^{Z}=\sum_{i}J_{i}^{Z}=\sum_{i}(a_{i}^{\dagger}a_{i}-b_{i}^{\dagger}b_{i})/2$
can then be re-expressed in terms of the total occupation number sums
of each level. Figure 4 shows predictions for the variance of the
collective spins $J^{Z}$ or $J^{Y}$, where the mean spin is aligned
along direction $J^{X}$, as a function of ratio $Ng/\kappa$, for
a fixed number of atoms $N$ and a fixed intercomponent coupling.
In nonlinear regimes, indicated by $g\neq0$, we see $\xi<1$ is predicted,
which is sufficient to detect entanglement. Heisenberg relations imply
$\xi\geq1/\sqrt{N}$.
Sorenson and Molmer \cite{soremol} have evaluated the exact minimum
variance of the spin squeezing for a fixed $J$. Their result for
$J=1/2$ agrees with (\ref{eq:hupspinsq}) and also (\ref{eq:spinsqentcrit}),
but for $J\geq1$ there is a tighter lower bound for the minimum variance,
which can be expressed as
\begin{equation}
\Delta^{2}J^{X}/J\geq F_{J}(\langle J^{Z}\rangle/J),\label{eq:min functsm}
\end{equation}
where the functions $F_{J}$ are given in Ref. \cite{soremol}.
The above criteria hold for particles that are effectively indistinguishable.
It is usually of most interest to detect entanglement when the particles
involved are distinguishable, or, even better, causally separated.
We ask how to detect entanglement between spatially-separated or at
least distinguishable groups of spin $J$. We examined this question
in Section III, and considered criteria that were useful for superposition
states with mean zero spin amplitude.
Another method put forward by Sorenson and Molmer (SM) is as follows.
The separability assumption (\ref{eq:fulsep}) will imply \cite{soremol},
\begin{equation}
\Delta^{2}J^{Z}\geq NJF_{J}(\langle J^{X}\rangle/NJ),\label{eq:smolvar}
\end{equation}
where we have for convenience exchanged the notation of $X$ and $Z$
directions (compared to (\ref{eq:min functsm})). The expression applies
when considering $N$ states $\rho_{k}^{R}$ which have a fixed spin
$J$, and could be useful where the mean spin is nonzero.
\subsection{Depth of entanglement and genuine entanglement}
We note from (\ref{eq:min functsm}) that the minimum variance (maximum
spin squeezing) reduces as $J$ increases. Sorenson and Molmer (SM)
showed how this feature can be used to demonstrate that a minimum
number of particles or sites are genuinely entangled \cite{soremol}.
If
\begin{equation}
\Delta^{2}J^{Z}/NJ<F_{J_{0}}(\langle J^{X}\rangle/NJ),\label{eq:geentpart}
\end{equation}
then we must have $J>J_{0}$ and so a minimum number $N_{0}$ of particles
(where the maximum spin for a block of $N$ atoms is $J=N/2$, we
must have $N_{0}=2J_{0}$ are involved, to allow the higher spin value.
It will be useful to summarize the proof of this result giving some
detail as follows. Consider a system with the density matrix
\begin{eqnarray}
\rho & = & \sum_{R}P_{R}\rho^{R}\nonumber \\
\nonumber \\
& = & \sum_{R}P_{R}\prod_{i=1}^{N_{R}}\rho_{i}^{R}.\label{eq:rho-1}
\end{eqnarray}
We will consider for the sake of simplicity \emph{that the overall
system has a fixed number of atoms $N_{T}$ and a fixed total spin
$J_{tot}$.} The density operator (\ref{eq:rho-1}) describes a system
in a mixture of states $\rho_{R}$, with probability $P_{R}$. For
each possibility $R$, there are $N_{R}$ blocks each with $N_{R,i}$
atoms and a total spin $J_{R,i}$ (note that $J_{R,i}\leq N_{R,i}/2$)
(Figure 5).
We note that if the maximum number of atoms in each block is $N_{0}$
then the \emph{maximum} spin for the block is $J_{0}=N_{0}/2$. Also,
if the total number of atoms is fixed, at $N_{T}$, then $N_{T}=\sum_{i=1}^{N_{R}}N_{R,i}$,
which implies that each $\rho_{i}^{R}$ has a definite number $N_{R,i}$
, meaning it cannot be a superposition of state of different numbers.
Similarly, for a product state the total spin must be the sum of the
individual spins (as readily verified on checking Clebsch-Gordan coefficients),
which implies that if the total spin is fixed, then each $\rho_{i}^{R}$
has a fixed spin (that is, cannot be in a superposition state of different
spins).
\begin{figure}
\caption{\emph{Genuine multi-particle entanglement:}
\end{figure}
Using again that the variance of the mixture cannot be less than the
average of its components, and that the variance of the product state
$\rho^{R}$ is the sum of the variances $(\Delta^{2}J^{Z})_{R,i}$
of each factor state $\rho_{i}^{R}$, we apply (\ref{eq:min functsm}),
that the variance has a lower bound determined by the spin $J_{R,i}$.
Thus we can write:
\begin{eqnarray}
\Delta^{2}J^{Z} & \geq & \sum_{R}P_{R}\sum_{i=1}^{N_{R}}(\Delta^{2}J^{Z})_{R,i}\nonumber \\
& \geq & \sum_{R}P_{R}\sum_{i=1}^{N_{R}}J_{R,i}F_{J_{R,i}}(|\langle J^{X}\rangle|_{R,i}/J_{R,i}).
\end{eqnarray}
\textcolor{black}{Now we can use the fact that the curves $F_{J}$
are nested to form a decreasing sequence at each value of their domain,
as $J$ increases, as explained by Sorenson and Molmer. We then apply
the steps based on the SM proof (lines (6) -(8) of their paper), which
uses convexity of the functions $F_{J}$. We cannot exclude that the
total spin of a block can be zero, $J_{R,i}=0$, for which $(\Delta^{2}J^{Z})_{R,i}\geq0$,
but such blocks do not contribute to the summation and can be formally
excluded. We define the total spin $\sum_{i=1}^{N_{R}}J_{R,i}=J_{tot}^{R}$
for each $\rho^{R}$ but note that for fixed total spin this is equal
to $J_{tot}$, and we also note that $J_{tot}^{R}\leq J_{0}$. In
the later steps below, we define the total spin as $J_{tot}=\sum_{R}P_{R}J_{tot}^{R}$
and the collective spin operator $J^{Z}$.}
\begin{eqnarray}
\Delta^{2}J^{Z} & \geq & \sum_{R}P_{R}\sum_{i=1}^{N_{R}}J_{R,i}F_{J_{0}}(\langle J^{X}\rangle_{R,i}/J_{R,i})\nonumber \\
& = & \sum_{R}P_{R}J_{tot}^{R}\sum_{i=1}^{N_{R}}\frac{J_{R,i}}{J_{tot}^{R}}F_{J_{0}}(\langle J^{X}\rangle_{R,i}/J_{R,i})\nonumber \\
& \geq & \sum_{R}P_{R}J_{tot}^{R}F_{J_{0}}(\frac{\sum_{i=1}^{N_{R}}\langle J^{X}\rangle_{R,i}}{J_{tot}^{R}})\nonumber \\
& = & J_{tot}\sum_{R}P_{R}\frac{J_{tot}^{R}}{J_{tot}}F_{J_{0}}(\frac{\sum_{i=1}^{N_{R}}\langle J^{X}\rangle_{R,i}}{J_{tot}^{R}})\nonumber \\
& \geq & J_{tot}F_{J_{0}}(\sum_{R}P_{R}\frac{1}{J_{tot}}\sum_{i=1}^{N_{R}}\langle J^{X}\rangle{}_{R,i})\nonumber \\
& \geq & J_{tot}F_{J_{0}}(\frac{1}{J_{tot}}\sum_{R}P_{R}\sum_{i=1}^{N_{R}}\langle J^{X}\rangle{}_{R,i})\nonumber \\
& = & J_{tot}F_{J_{0}}(\langle J^{X}\rangle/J_{tot}).\label{eq:proof2-1}
\end{eqnarray}
The total spin $J_{tot}$ is maximum at $J_{tot}=N/2$ where $N$
is total number of atoms over both systems, but is assumed measurable.
Thus, if the maximum number of atoms in each block does not exceed
$N_{0}$, then the inequality (\ref{eq:proof2-1}) must always hold.
The violation of (\ref{eq:proof2-1}) is a demonstration of a group
of atoms that are genuinely entangled \cite{soremol}.
\begin{figure}
\caption{Detecting multi-particle entanglement in ground state of a two-component
BEC, as modeled by ({\ref{hamgs}
\end{figure}
The predictions of the model (\ref{hamgs}) are given in Figure 6,
for a range of values of $N$ (the total number of atoms). \textcolor{black}{In
each case, there is a constant total spin, $J\equiv J_{tot}$, given
by $\langle(J^{X})^{2}+(J^{Y})^{2}+(J^{Z})^{2}\rangle=J(J+1)$ where
$J=N/2$. We keep $N$ and the interwell coupling $\kappa$ fixed,
and note that the variance of $J^{Z}$ decreases with increasing $g$,
while the variance in $J^{X}$ increases. Evaluation of the normalized
quantities of the SM inequality (\ref{eq:proof2-1}) are given in
the second plot of Figure 6. Comparing with the functions $F_{J_{0}}$
reveals the prediction of a full $N$ particle entanglement, where
$N$ is an integer value. }
We note this treatment does not itself test nonlocality, or even the
quantum separability models (\ref{eq:sepN}-\ref{eq:sepent}) because
measurements are not taken at distinct locations. However, it can
reveal, \emph{within a quantum framework}, an underlying entanglement,
of the type that could give nonlocality if the individual spins could
be measured at different locations. The great advantage however of
the collective criteria is the reduced sensitivity to efficiency,
since it is no longer necessary to measure the spin at each site.
The depth of spin squeezing has been used recently and reported at
this conference to infer blocks of entangled atoms in BEC condensates
\cite{exp multi,treutnature}.
To test nonlocality between sites, the criteria need will involve
measurements made at the different spatial locations. How to detect
entanglement between two-modes using spin operators \cite{hillzub,schvogent,spinsq,toth2009,spinsqkorb},
and how to detect a true Einstein-Podolsky-Rosen (EPR) entanglement
\cite{epr rev ,reidepr,eprbohmparadox,sumuncerduan,spinprodg,Kdechoum,cavalreiduncer}
in BEC \cite{murraybecepr,eprbecbar,eprbec he} are the topics of
much current interest.
\subsection{EPR steering nonlocality with atoms }
An interesting question is whether one derive criteria, involving
collective operators, to determine whether there are stronger underlying
nonlocalities. How can we infer whether the one group of atoms $A$
can {}``steer'' a second group $B$, as shown in schematic form
in Figure 7? This would confirm an EPR paradox between the two groups,
that the correlations imply inconsistency between Local Realism (LR)
and the completeness of quantum mechanics. This is an interesting
task since very little experimental work has been done on confirming
EPR paradoxes between even single atoms. Steering paradoxes between
groups of atoms raise even more fundamental questions about mesoscopic
quantum mechanics.
\begin{figure}
\caption{Is {}
\end{figure}
As an example, we thus consider the following. EPR steering is demonstrated
between $N$ sites when the LHS model (\ref{eq:bell-1}) fails with
$T=1$ fails. The system (which we will call $B$) at the one site
corresponding to $T=1$ is described by a local quantum state LQS,
which means it is constrained by the uncertainty principle. All other
groups are described by a Local Hidden Variable Theory (LHV), and
thus are constrained to have only a non-negative variance. For this
first group (only) there is the SM minimum variance (implied by quantum
mechanics):
\begin{equation}
\Delta^{2}J_{B}^{X}\geq J_{B}F_{J}(\langle J^{Z}\rangle/J_{B})\label{eq:smolvar-1}
\end{equation}
Hence, with this assumption, we follow the approach of Section V.
B, to write (where we assume the maximum spin of the steered group
$B$ is $J_{0}$)
\begin{eqnarray}
\Delta^{2}J^{X} & \geq & \sum_{R}P_{R}\{J_{R,B}F_{J_{0}}(\langle J^{Z}\rangle_{R,B}/J_{R,B})\}\nonumber \\
& = & \sum_{R}P_{R}J_{tot,}^{R}\frac{J_{R,B}}{J_{tot}^{R}}F_{J_{0}}(\langle J^{Z}\rangle_{R,B}/J_{R,B})\nonumber \\
& \geq & \sum_{R}P_{R}J_{tot}^{R}F_{J_{0}}(\frac{\langle J^{Z}\rangle_{R,B}}{J_{tot}^{R}})\nonumber \\
& = & J_{tot}\sum_{R}P_{R}\frac{J_{tot}^{R}}{J_{tot}}F_{J_{0}}(\frac{\sum_{i=1}^{N_{R}}\langle J^{Z}\rangle_{R,i}}{J_{tot}^{R}})\nonumber \\
& \geq & J_{tot}F_{J_{0}}(\sum_{R}P_{R}\frac{1}{J_{tot}}\langle J^{Z}\rangle{}_{R,B})\nonumber \\
& \geq & J_{tot}F_{J_{0}}(\frac{1}{J_{tot}}\sum_{R}P_{R}\langle J^{Z}\rangle{}_{R,B})\nonumber \\
& = & J_{tot}F_{J_{0}}(\langle J_{B}^{Z}\rangle/J_{tot})\label{eq:proof2-1-1}
\end{eqnarray}
If the inequality is violated, a {}``steering'' between the two
groups is confirmed possible: group $A$ {}``steers'' group $B$.
In this case, the spins of spatially separated systems $B$ would
need to be measured, and potential such {}``EPR'' systems have been
proposed, with a view to this sort of experiment in the future.
\section{Conclusion}
We have examined a strategy for testing multi-particle nonlocality,
by first defining three distinct levels of nonlocality: (1) entanglement,
(2) EPR paradox/ steering, or (3) failure of local hidden variable
(LHV) theories (which we call Bell's nonlocality). We next focused
on two types of earlier studies that yielded information about nonlocality
in systems of more than two particles.
The first study originated with Greenberger, Horne and Zeilinger (GHZ)
and considers $N$ spatially separated in $1/2$ particles, on which
individual spin measurements are made. The study revealed that nonlocality
involving $N$ spatially separated (spin $1/2$) particles can be
more extreme. Mermin showed that the deviation of the quantum prediction
from the classical LHV boundaries can grow exponentially with $N$
for this scenario. Here we have summarized some recent results by
us that reveal similar features for entanglement and EPR steering
nonlocalities. Inequalities are presented that enable detection of
these nonlocalities in this multipartite scenario, for certain correlated
quantum states. The results are also applicable to $N$ spin $J$
particles (or systems), and thus reveal nonlocality can survive for
$N$ systems even where these systems have a higher dimensionality.
We then examined the meaning of {}``multi-particle nonlocality'',
in the sense originated by Svetlichny, that there is an {}``$N-$body''
nonlocality, necessarily shared among \emph{all} $N$systems. For
example, three-particle entanglement is defined as an entanglement
that cannot be modeled using two-particle entangled or separable states
only. Such entanglement, generalized to $N$ parties, is called genuine
$N$ partite entanglement. We present some recent inequalities that
detect such genuine nonlocality for the GHZ/ Mermin scenario of $N$
spin $1/2$ particles, and show a higher threshold is required that
will imply a much greater sensitivity to inefficiencies $\eta$. In
other words, the depth of violation of the Bell or nonlocality inequalities
determines the level of \emph{genuine} multi-particle nonlocality.
This led to the final focus of the paper, which examined criteria
that employ collective spin measurements.\textcolor{red}{{} }\textcolor{black}{For
example,}\textcolor{red}{{} }the spin squeezing entanglement criterion
of Sorenson et al enables entanglement to be confirmed between $N$
spin $1/2$ particles, based on a reduction in the overall variance
({}``squeezing'') of a single collective spin component. The criterion
works because of the finite dimensionality of the spin Hilbert space,
which means only higher spin systems $-$ as can be formed from entangled
spin $1/2$ states $-$ can have larger variances in one spin component,
and hence smaller variances in the other. As shown by Sorenson and
Molmer, even greater squeezing of the spin variances will imply larger
entanglement, between more particles. Hence the depth of spin squeezing,
as with the depth of Bell violations in the GHZ Mermin example above,
will imply genuine entanglement between a minimum number of particles.
This result has recently been used to detect experimental multi-particle
entanglement in BEC systems. We present a model of the ground state
BEC for the two component system, calculating the extent of such multi-particle
squeezing.
We make the final point that, while collective spin measurements are
useful in detecting multi-particle entanglement and overcoming problems
that are encountered with detection inefficiencies, the method does
not address tests of nonlocality unless the measured systems can be
at least in principle spatially separated. This provides motivation
for studies of entanglement and EPR steering between groups of atoms
in spatially distinct environments.
\begin{acknowledgments}
\end{acknowledgments}
We wish to thank the Humboldt Foundation, Heidelberg University, and
the Australian Research Council for funding via AQUAO COE and Discovery
grants, as well as useful discussions with Markus Oberthaler, Philip
Treutlein, and Andrei Sidorov.
\end{document}
|
\begin{document}
\title{Faithfulness and learning hypergraphs from discrete distributions}
\author{Anna Klimova \\
{\small{Institute of Science and Technology, Austria} }\\
{\small \texttt{[email protected]}}\\
{}\\
\and
Caroline Uhler \\
{\small{Institute of Science and Technology, Austria} }\\
{\small \texttt{[email protected]}}\\
{}\\
\and Tam\'{a}s Rudas \\
{\small{E\"{o}tv\"{o}s Lor\'{a}nd University, Budapest, Hungary}}\\
{\small \texttt{[email protected]}}\\
}
\date{}
\maketitle
\begin{abstract}
In this paper, we study the concepts of faithfulness and strong-faithfulness for discrete distributions. In the discrete setting, graphs are not sufficient for describing the association structure. So we consider hypergraphs instead, and introduce the concept of parametric (strong-) faithfulness with respect to a hypergraph. Assuming strong-faithfulness, we build uniformly consistent parameter estimators and corresponding procedures for a hypergraph search. The strength of association in a discrete distribution can be quantified with various measures, leading to different concepts of strong-faithfulness. We explore these by computing lower and upper bounds for the proportions of distributions that do not satisfy strong-faithfulness.
\end{abstract}
\begin{keywords}
contingency tables, directed acyclic graphs, hierarchical log-linear models, hypergraphs, (strong-) faithfulness
\end{keywords}
\section{Introduction}\label{intro}
A graphical model is a set of probability distributions whose association structure can be identified with a graph. Given a graph, the Markov property entails a set of conditional independence relations that are fulfilled by distributions in the model. Distributions in the model that obey no further conditional independence relations are called \emph{faithful to the graph}. For each undirected graphical model, as well as for each directed acyclic graph (DAG) model, there is a distribution that is faithful to the graph \perp\!\!\!\perptep*[cf.][]{SpirtesBook}. Moreover, the Lebesgue measure of the set of parameters corresponding to distributions that are unfaithful to a graphical model is zero; this result was proven by \perp\!\!\!\perpte*{SpirtesBook} for the case of multivariate normal distributions, by \perp\!\!\!\perpte{MeekFaith} for discrete distributions on multi-way contingency tables, and by \perp\!\!\!\perpte*{Pena2009} for arbitrary sample spaces and dominating measures. It is also well-known, that a DAG model may include distributions that are unfaithful to it but are not Markov to any nested DAG. This kind of unfaithfulness may occur due to path cancellation and can arise both in the discrete and in the multivariate normal settings \perp\!\!\!\perptep[cf.][]{ZhangSp2008, UhlerRaskutti2013}.
In the discrete case, the non-existence of a graph to which a distribution is faithful is related to the presence of higher than first order interactions in this distribution. Graph learning algorithms \perp\!\!\!\perptep[cf.][]{SpirtesBook}, which do not recognize the presence of higher order interactions, may produce a graph which does not reveal the true association structure \perp\!\!\!\perptep[cf.][]{StudenyBook}. In order to avoid such errors, graph learning algorithms usually assume the existence of a DAG to which the distribution is faithful. Since the Lebesgue measure of the set of parameters corresponding to the distributions that are unfaithful to the underlying graph is zero, the faithfulness assumption is not considered to be restrictive in the context of graphical search. While graph search procedures assuming faithfulness are pointwise consistent, they are not uniformly consistent and thus cannot simultaneously control Type I and Type II errors with a finite sample size \perp\!\!\!\perptep*{RSSWassUnifCons}. To ensure existence of a uniformly consistent learning procedure, strong-faithfulness of a distribution to the underlying DAG is needed \perp\!\!\!\perptep{ZhangSpirtesLambdaFaith}. \perp\!\!\!\perpte*{UhlerFaithGeometry} analyzed the Gaussian setting and showed that the strong-faithfulness assumption may, in fact, be very restrictive and the corresponding proportions of distributions which do not satisfy strong-faithfulness may become very large as the number of nodes grows.
The concepts of faithfulness and strong-faithfulness were originally introduced in the causal search framework, where they are linked to identifiability of causal effects. However, as we show in this paper using the discrete setting, these concepts are also important for identifiability of more general parameters of association. In Section~\ref{sectionGraphFaith}, we define the concept of a model class being closed under a faithfulness relation: for each positive distribution, there exists a model in such a class to which it is faithful. By giving examples of distributions that are not faithful to any directed or undirected graphical model, we show that these model classes are not closed under the faithfulness relation which is based on the corresponding Markov property. Further, we introduce the concept of parametric faithfulness of a distribution to a hypergraph (instead of a graph). This concept seems more adequate for categorical data, where hypergraphs can be used to represent hierarchical log-linear models. Indeed, we show that the class of models associated with hypergraphs is closed under a parametric faithfulness relation.
In Section~\ref{sectionStrongFaith}, we describe two major difficulties with the concept of strong-faithfulness in the discrete case. First, in contrast to role of correlations in the multivariate normal case, there is no single standard measure of the strength of association in a joint distribution. Therefore, depending on the measure of association, different variants of strong-faithfulness may be considered. Second, the proportion of strong-faithful distributions depends on the parameterization used and can only be computed if the parameter space has finite volume. We explore the consequences of different parameterizations and measures of association for the case of the $2 \times 2$ contingency table. We define parametric strong-faithfulness with respect to a hypergraph under a parameterization based on the log-linear interaction parameters. Assuming strong-faithfulness, we show that the maximum likelihood estimators of the interaction parameters associated with the hyperedges are uniformly consistent. As a result, we give a set of conditions, under which Type I and Type II errors can be controlled with a finite sample size. We also discuss the uniform consistency of model selection procedures for a hypergraph search, for example, using the approaches described by \perp\!\!\!\perpte{EdwardsBook, EdwardsNote}.
In Section~\ref{SecProp}, we estimate the proportion of distributions that do not satisfy the parametric strong-faithfulness assumption with respect to a given hypergraph. We give an exact formulation of these proportions, under a parameterization based on conditional probabilities, for hypergraphs whose hyperedges form a decomposable set. The association structure of such distributions may be discovered incorrectly during a hypergraph learning procedure. Finally, we define the concept of projected strong-faithfulness, which applies to distributions which do not belong to the hypergraph, and estimate the proportions of projected strong-faithful distributions for several hypergraphs for the $2 \times 2 \times 2$ contingency table.
In Section \ref{SecConcl}, we conclude the paper with a brief discussion of our results and their implications.
\section{Graphical and parametric faithfulness} \label{sectionGraphFaith}
In this section, we first review the concept of faithfulness with respect to a graph. We then introduce parametric faithfulness with respect to a hypergraph and show that this is a more relevant concept for categorical data.
\subsection{Faithfulness with respect to a graph}
Let $\mathcal{V}_1, \dots, \mathcal{V}_K$ be random variables taking values in $\mathcal{I} = \mathcal{I}_1 \times \dots \times \mathcal{I}_K $, a Cartesian product of finite sets. $\mathcal{I}$ describes a $K$-way contingency table and a vector $\boldsymbol i = (i_1,\dots,i_K) \in \mathcal{I}$ forms a cell. A subset $M \subseteq \{1,\dots ,K\}$ specifies a marginal of the joint distribution of $\mathcal{V}_1, \dots, \mathcal{V}_K$, and $M=\emptyset$ is the empty marginal. For $M=(k_1,\dots k_t)$, the set $\mathcal{I}_M = \mathcal{I}_{k_1} \times \dots \times \mathcal{I}_{k_t}$ is a \emph{marginal table}, and the canonical projection $\boldsymbol i_M$ of the cell $\boldsymbol i$ onto the set $\mathcal{I}_M$ is a \emph{marginal cell}. We parameterize the population distribution by cell probabilities $\boldsymbol p =(p_{\boldsymbol i})_{\boldsymbol i\in \mathcal{I}}$, where $p_{\boldsymbol i} \in (0,1)$ and $\sum_{\boldsymbol i \in \mathcal{I} }p_{\boldsymbol i} = 1$, and denote by $\mathcal{P}$ the set of all distributions on $\mathcal{I}$. A subset of $\mathcal{P}$ is called a \emph{model}. For simplicity of exposition, we assume that $\mathcal{V}_1, \dots, \mathcal{V}_K$ are binary, $\mathcal{I}$ is treated as a sequence of cells ordered lexicographically, and
a distribution $P\in \mathcal{P}$ is addressed by its parameter, $\boldsymbol p$.
A \emph{graphical model} is a set of probability distributions, whose association structure can be identified with a graph with vertices $V = \{1, \dots, K\}$, where each vertex $i$ is associated with a random variable $\mathcal{V}_i$. In the following, we will identify each vertex with its associated random variable. The absence of an edge between two vertices means that the corresponding random variables satisfy some (conditional) independence relation. A detailed description of graphical models for discrete as well as for multivariate normal distributions can be found in \perp\!\!\!\perpte{EdwardsBook}, among others. In the sequel, we only consider undirected graphical models and DAG models.
A graphical model identified with an undirected graph (also called a \emph{graphical log-linear model} in the discrete setting) is a set of probability distributions on $V$ that satisfy the \emph{local undirected Markov property}: Every node is conditionally independent of its non-neighbors given its neighbors. In the discrete case, such models are a sublcass of hierarchical log-linear models. A graphical model identified with a directed acyclic graph, a DAG model, is a set of probability distributions on $V$ that satisfy the \emph{directed Markov property}: Every node is conditionally independent of its non-descendants given its parents. A distribution that satisfies the Markov property with respect to a graph is called \emph{Markov} to it.
A distribution which is Markov to a graph, is said to be \emph{faithful} to it if all conditional independencies in this distribution can be derived from the graph. The faithfulness relation can be thought of as a decision rule that classifies a distribution $\boldsymbol p$ in a model $\mathcal{M}$ as faithful or unfaithful to it:
$$\mathbb{F}(\boldsymbol p, \mathcal{M}) = \left\{\begin{array}{ll} 1 & \mbox{ if } \boldsymbol p \mbox{ is faithful to } \, \mathcal{M},\\
0 & \mbox{ otherwise}.\\
\end{array}\right.$$
\begin{definition}
A class $\frak{C}$ of models on $\mathcal{P}$, where $\frak{C}$ is partially ordered with respect to inclusion, is said to be \emph{closed} under the faithfulness relation indicated by $\mathbb{F}$, if for every non-empty $\mathcal{M} \in \frak{C}$ and for every $\boldsymbol p \in \mathcal{M}$ such that $\mathbb{F}(\boldsymbol p , \mathcal{M}) = 0$, there exists an $\mathcal{M}' \in \frak{C}$ with $\mathcal{M}' \subset \mathcal{M}$ and $\mathbb{F}(\boldsymbol p , \mathcal{M}') = 1$.
\end{definition}
This definition implies that a class $\frak{C}$ is closed under the faithfulness relation indicated by $\mathbb{F}$ if and only if for every $\boldsymbol p \in \mathcal{P}$ there exists an $\mathcal{M} \in \frak{C}$, such that $\mathbb{F}(\boldsymbol p, \mathcal{M}) = 1$. Graphical log-linear models and DAG models are specified by a list of conditional independence relations which, in turn, comprise other conditional independencies. Thus, these model classes have a natural partial order implied by the conditional independence relation. We now show that these classes are not closed under the corresponding faithfulness relations.
\begin{proposition}
\label{prop_undirected}
The class of graphical log-linear models is not closed under the faithfulness relation defined by the local undirected Markov property.
\end{proposition}
The following example is given as a proof.
\begin{example}\label{FourVarUnfaith}
Let $V=\{A, B, C, D\}$ and consider the log-linear model $[ABC][ABD]$ \perp\!\!\!\perptep[cf.][]{Agresti2002}. This is the model of conditional independence of $C$ and $D$ given $A$ and $B$. All distributions in this model are Markov to the graph in Figure \ref{CiDgivABgraph}. Consider the distribution parameterized by
\begin{eqnarray*}
\boldsymbol p &=& (0.022, 0.062, 0.063, 0.103, 0.103, 0.063, 0.062, 0.022, \\
&&0.103, 0.063, 0.062, 0.022,0.022,0.062,0.063,0.103)',
\end{eqnarray*}
where the cell probabilities are ordered lexicographically. In this distribution, the conditional odds ratios ($\mathcal{COR}$) of $C$ and $D$ given the levels of $A$ and $B$ are equal to $1$:
\begin{equation*}
\mathcal{COR}(CD\mid A=i, B=j) = \frac{p_{ij00}p_{ij11}}{p_{ij01}p_{ij10}} = 1, \,\, \mbox{for all } i, j \in \{0, 1\}.
\end{equation*}
Hence, the distribution is in the model. The $(A,B)$-marginal of this distribution is uniform:
$$
\begin{array}{c|cc}
& {B}=0 & {B}=1 \\
\hline
{A} = 0& 1/4 & 1/4\\
{A} = 1& 1/4 & 1/4\\
\end{array},
$$
and thus $A \perp\!\!\!\perp B$. So the distribution is unfaithful to the graph in Figure \ref{CiDgivABgraph}. In addition, since the conditional odds ratios of $A$, $B$ and $C$ given $D$ and of $A$, $B$, and $D$ given $C$ are not equal to $1$:
\begin{eqnarray*}
\mathcal{COR}(ABC\mid D=0) &=& \frac{p_{0000}p_{1100}p_{0110}p_{1010}}{p_{0100}p_{1000}p_{0010}p_{1110}} \approx 0.04418483,\\
\mathcal{COR}(ABC\mid D=1) &=& \frac{p_{0001}p_{1101}p_{0111}p_{1011}}{p_{0101}p_{1001}p_{0011}p_{1111}} \approx 0.04418483,\\
\mathcal{COR}(ABD\mid C=0) &=& \frac{p_{0000}p_{1100}p_{0101}p_{1001}}{p_{0100}p_{1000}p_{0001}p_{1101}} \approx 0.04710518,\\
\mathcal{COR}(ABD\mid C=1) &=& \frac{p_{0010}p_{1110}p_{0111}p_{1011}}{p_{0110}p_{1010}p_{0011}p_{1111}} \approx 0.04710518,
\end{eqnarray*}
the distribution cannot be Markov to any nested undirected graph.
\qed
\end{example}
The situation described in Example \ref{FourVarUnfaith} is distinctive to discrete distributions. In the Gaussian setting, marginal independence of more than two variables implies their joint independence. Thus, a multivariate normal distribution whose components are pairwise independent is Markov and faithful to a graph with no edges. But in the discrete case, a joint distribution of pairwise independent variables may have a non-trivial structure of higher than first order interactions. Next, we prove that also the class of DAG models is not closed with respect to the faithfulness relation.
\begin{proposition}
The class of DAG models is not closed under the faithfulness relation defined by the directed Markov property.
\end{proposition}
As a proof, two examples are given. The second example only pertains to the discrete case.
\begin{figure}
\caption{$C \perp\!\!\!\perp D \mid A,B$.}
\label{CiDgivABgraph}
\caption{$A \perp\!\!\!\perp C \mid B, \quad B \perp\!\!\!\perp D \mid A,C$.}
\label{DAGcycle}
\end{figure}
\begin{example}\label{normalDAGunfaith}
Let $V=\{A, B, C, D\}$ and consider the model specified by two conditional independence relations: $A \perp\!\!\!\perp C \mid B$ and $B \perp\!\!\!\perp D \mid A,C$. Any distribution in this model is Markov to the DAG in Figure \ref{DAGcycle}. For example, the distribution parameterized by
\begin{eqnarray*}
\boldsymbol p &=& (0.006, 0.006, 0.0288, 0.0192, 0.06, 0.06, 0.072, 0.048, 0.0056, \\
&&0.0504, 0.187148, 0.0368516, 0.021, 0.189, 0.175452, 0.0345484)',
\end{eqnarray*}
is in the model. However, this distribution also satisfies the additional independence relation $A \perp\!\!\!\perp D$. This independence relation is not reflected in the graph. Thus the distribution is unfaithful to the graph in Figure \ref{DAGcycle}.
Next, we show that there is no DAG that fulfills all three (conditional) independence relations $A \perp\!\!\!\perp D$, $A \perp\!\!\!\perp C\mid B$, $B \perp\!\!\!\perp D\mid A,C$. If such a DAG existed, then its skeleton would have three edges: $AB$, $BC$, $CD$. In order to satisfy faithfulness, $A \perp\!\!\!\perp D$ requires that $A\to B\leftarrow C$ or $B\to C\leftarrow D$. However, $A \perp\!\!\!\perp C\mid B$ is unfaithful to $A\to B\leftarrow C$ and $B \perp\!\!\!\perp D\mid A,C$ is unfaithful to $B\to C\leftarrow D$.
\qed
\end{example}
\begin{remark}
\label{rem_Gaussian}
One can also construct an instance of Example \ref{normalDAGunfaith} using multivariate normal distributions by choosing the partial correlations in such a way that the causal effect associated with the edge $A\to D$ cancels with the causal effect associated with the path $A\to B\to C\to D$ (see Figure \ref{DAGcycle}). This shows that also Gaussian DAG models are not closed under the faithfulness relation defined by the directed Markov property.
\end{remark}
The next example illustrates a situation that occurs only in the discrete case. To construct this example, we will use the fact that, in contrary to the Gaussian case, a discrete distribution with pairwise independent random variables can have non-vanishing interactions of higher than the first order.
\begin{example}\label{ExampleIntro}
Let $V=\{A, B, C\}$. Consider the distribution parameterized by
\begin{equation}\label{distr18}
\boldsymbol p = (1/8-\delta,1/8+\delta, 1/8+\delta,1/8-\delta, 1/8+\delta,1/8-\delta, 1/8-\delta,1/8+\delta)',
\end{equation}
where $\delta \in (-1/8, 1/8)$. Its marginals are uniform resulting in pairwise independence: $A \perp\!\!\!\perp B$, $A \perp\!\!\!\perp C$, and $B \perp\!\!\!\perp C$. The second order odds ratio of this distribution,
\begin{equation*}
\frac{p_{000}p_{011}p_{101}p_{110}}{p_{001}p_{010}p_{100}p_{111}} = \left(\frac{1/8-\delta}{1/8+\delta}\right)^{4},
\end{equation*}
does not vanish, implying that $A$, $B$, and $C$ are not jointly independent. The distribution belongs to the graphical log-linear model that can be identified with the graph shown in row 1 of Table \ref{allgraphsABC}. Further, since each pairwise independence holds, the distribution is Markov to the DAGs shown in rows 2, 3, and 4 of Table \ref{allgraphsABC}. However, the distribution is not faithful to these DAGs and it is not Markov to any of the nested DAGs (rows 5, 6, 7 and 8). \qed
\end{example}
The association structure of a distribution that is unfaithful to every model in a given class can be considered within a larger model class. We have described examples of discrete distributions for which there is no undirected graphical model or DAG model to which they are faithful. Graphical models (directed and undirected) for discrete distributions are a subclass of hierarchical marginal log-linear models \perp\!\!\!\perptep*{RudasBergsma, RudasBN2006} and can be considered within this larger class. We revisit Example \ref{normalDAGunfaith} to motivate the introduction of \emph{parametric faithfulness}, a generalization of the concept of faithfulness that can be applied to the class of hierarchical marginal log-linear models. In the following, we show that under this natural generalization of faithfulness, we can find a model in the class of hierarchical marginal log-linear models to which the distribution described in Example \ref{normalDAGunfaith} is faithful.
\textbf{Example \ref{normalDAGunfaith}} (revisited):
A marginal log-linear parameterization \perp\!\!\!\perptep{RudasBergsma} for the DAG in Figure~\ref{DAGcycle} can be derived from the set of marginals $$\mathcal{M} = \{(A, D), (A, B, C), (A, B, C, D)\}.$$ The corresponding parameters are:
\begin{eqnarray}\label{MargEx}
\lambda_{\emptyset}^{AD}, \, \, \lambda_{A*}^{AD}, \, \, \lambda_{*D}^{AD}, \, \, \lambda_{AD}^{AD}, \, \, \lambda_{*B*}^{ABC}, \, \, \lambda_{**C}^{ABC}, \, \, \lambda_{*BC}^{ABC}, \, \, \lambda_{AB*}^{ABC}, \, \, \lambda_{A*C}^{ABC}, \, \, \lambda_{ABC}^{ABC}, \nonumber \\
\\
\lambda_{*B*D}^{ABCD}, \, \, \lambda_{**CD}^{ABCD}, \, \, \lambda_{AB*D}^{ABCD}, \, \, \lambda_{A*CD}^{ABCD}, \, \lambda_{*BCD}^{ABCD}, \, \, \lambda_{ABCD}^{ABCD}. \nonumber
\end{eqnarray}
The conditional independencies $A \perp\!\!\!\perp C \mid B$ and $B \perp\!\!\!\perp D \mid A,C$ are obtained by taking
\begin{eqnarray}\label{MargEx2}
\lambda_{A*C}^{ABC} = 0, \,\, \lambda_{ABC}^{ABC} = 0, \,\, \lambda_{*B*D}^{ABCD} = 0, \,\, \lambda_{AB*D}^{ABCD} = 0, \,\, \lambda_{*BCD}^{ABCD} = 0, \,\, \lambda_{ABCD}^{ABCD} = 0.
\end{eqnarray}
Any distribution that is Markov to the DAG in Figure \ref{DAGcycle} can be parameterized by the remaining marginal log-linear parameters.
The faithfulness relation in the class of marginal log-linear models can be defined as a relationship between the parameters of a distribution and the parameters of a model that contains the distribution. A distribution which also satisfies the marginal independence $A \perp\!\!\!\perp D$, has $\lambda_{AD}^{AD} = 0$ and thus belongs to a nested marginal log-linear model, to which it is faithful in the parametric sense. \qed
This example motivates taking a parametric approach (instead of a graphical approach) to faithfulness. In the next section we introduce the concept of parametric faithfulness for discrete distributions more formally.
\subsection{Parametric faithfulness}
\label{sec_par_faith}
Let $\mathcal{P}$ denote the full exponential family of distributions. We choose a mixed parameterization $(\boldsymbol \mu, \boldsymbol \nu)$ of this family, where $\boldsymbol \mu$ denotes the vector of mean value parameters and $\boldsymbol \nu$ the vector of canonical parameters \perp\!\!\!\perptep[cf.][]{Barndorff1978}. Let $\frak{C}$ be a class of partially ordered exponential families that are obtained by setting some of the components of $\boldsymbol \mu$ and/or some of the components of $\boldsymbol \nu$ to zero, and let $\mathcal{M} \in \frak{C}$. Assume that $\mathcal{M}$ is parameterized by $(\boldsymbol \mu_{\mathcal{M}}, \boldsymbol \nu_{\mathcal{M}})$, where $\boldsymbol \mu_{\mathcal{M}} \subseteq \boldsymbol \mu$, $\boldsymbol \nu_{\mathcal{M}} \subseteq \boldsymbol \nu$, $\boldsymbol \mu \setminus \boldsymbol \mu_{\mathcal{M}} = \boldsymbol 0$, and $\boldsymbol \nu \setminus \boldsymbol \nu_{\mathcal{M}} = \boldsymbol 0$. We define faithfulness as a relationship between the parameters of a distribution and the parameters of a model containing the distribution under consideration.
\begin{definition} \label{FaithNormalDef}
A distribution $\boldsymbol p \in \mathcal{M}$ parameterized by $(\boldsymbol \mu_{\mathcal{M}}(\boldsymbol p), \boldsymbol \nu_{\mathcal{M}}(\boldsymbol p))$, satisfies the \emph{parametric faithfulness relation with respect to} $\mathcal{M}$ if none of the components of $\boldsymbol \mu_{\mathcal{M}}(\boldsymbol p)$ or $\boldsymbol \nu_{\mathcal{M}}(\boldsymbol p)$ vanish.
\end{definition}
The class of discrete exponential families, where the canonical parameters are the interactions of the variables in $V$ of order up to $K-1$, corresponds to the class of hierarchical log-linear models on $V$. More precisely, let $\mathcal{M}=\{M_1, \dots,M_T\}$ be a set of incomparable subsets of $V$. Then the hierarchical log-linear model generated by $\mathcal{M}$ is the set of distributions in $\mathcal{P}$ that satisfy
\begin{equation}\label{LLMdef}
\mbox{log } p_{\boldsymbol i} = \sum_{M \subseteq V: M \subseteq M_j \in \mathcal{M}} \gamma_M(\boldsymbol i_M),
\end{equation}
where $\gamma_{M'}(\boldsymbol i_{M'}) = 0$ implies $\gamma_{M''} (\boldsymbol i_{M''}) = 0$ for any $M''\supseteq M'$, and $\gamma_{M}$ are called the \emph{interaction parameters} (interactions for short). Their identifiability is assumed in the sequel.
The set $\mathcal{M}$ partitions the power set of $V$ into a descending class, consisting of subsets of $M_1, \dots, M_T$, and a complementary ascending class. The partition induces a mixed parameterization of $\mathcal{P}$ with the canonical parameters equal to the conditional odds ratios (or their logarithms) of the subsets in the ascending class, given the remaining variables, and the mean value parameters equal to the marginal distributions of the subsets in the descending class. Under this parameterization, the canonical parameters of the distributions in the model generated by $\mathcal{M}$ are equal to $1$ (or zero) and the distributions are parameterized by the mean value parameters \perp\!\!\!\perptep{RudasSAGE}. The structure of the highest order interactions of the distributions in the hierarchical log-linear model generated by $\mathcal{M}=\{M_1, \dots,M_T\}$ is described next. In the sequel, $\bar{M}_t= V \setminus M_t$.
\begin{lemma}\label{interactions}
There exists a parameterization of $\mathcal{P}$ under which, for every $t = 1, \dots, T$, the interaction parameter $\gamma_t$ is equal to the logarithm of the conditional odds ratio of $M_t$ given $\bar{M}_t = \boldsymbol i_{\bar{M}_t}$, and is invariant of the choice of $\boldsymbol i_{\bar{M}_t}$.
\end{lemma}
\begin{proof}
There exists a marginal log-linear parameterization of $\mathcal{P}$ under which for every $t = 1, \dots, T$, the interaction parameter $\gamma_{t}$, corresponding to the generating marginal $M_t$, is the average log conditional odds ratio of $M_t$ conditioned on and averaged over $\bar{M}_t$ \perp\!\!\!\perptep[cf.][]{RudasBN2006}:
$$\gamma_{M_t} = \frac{1}{|\mathcal{I}_{\bar{M}_t}|}\sum_{\boldsymbol i_{\bar{M}_t}}\mbox{log } \mathcal{COR}(M_t\mid \bar{M}_t = \boldsymbol i_{\bar{M}_t}).$$ Since $M_{t}$ is a maximal interaction, $\mathcal{COR}(M'\mid \bar{M}' = \boldsymbol i_{\bar{M}'}) = 1,$ for any $M' \supsetneq M_t$.
Further, it can be shown by induction on the elements of the ascending class of $M_1, \dots, M_T$, that
$$\mathcal{COR}(M'\mid\bar{M}' = \boldsymbol i_{\bar{M}'}) = \frac{ \mathcal{COR}(M_t \mid (M'\setminus M_t)\cup\bar{M}' =(\boldsymbol i_{M'\setminus M_t}, \boldsymbol i_{\bar{M}'}))}{\mathcal{COR}(M_t \mid (M'\setminus M_t)\cup\bar{M}' =(\boldsymbol j_{M'\setminus M_t}, \boldsymbol i_{\bar{M}'}))} = \frac{\mathcal{COR}(M_t \mid \bar{M}_t = \boldsymbol i_{\bar{M}_t})}{\mathcal{COR}(M_t \mid \bar{M}_t = \boldsymbol j_{\bar{M}_t})},$$
and thus,
$$\mbox{log } \mathcal{COR}(M_t\mid \bar{M}_t = \boldsymbol i_{\bar{M}_t}) = \mbox{log } \mathcal{COR}(M_t\mid \bar{M}_t = \boldsymbol j_{\bar{M}_t}),$$
for any $\boldsymbol i_{\bar{M}_t}$ and $\boldsymbol j_{\bar{M}_t}$. Hence,
$$\gamma_{M_t} = \mbox{log } \mathcal{COR}(M_t\mid \bar{M}_t = \boldsymbol i_{\bar{M}_t}),$$
for any $\boldsymbol i_{\bar{M}_t}$.
\end{proof}
The association structure of a discrete distribution in a hierarchical log-linear model generated by $\mathcal{M}$ can be described with a hypergraph, $\mathcal{H}= \mathcal{H}(\mathcal{M})$ with vertices $V=\{1, \dots, K\}$ and hyperedges equal to the generating marginals, or, equivalently, to the maximum non-vanishing interactions in $\mathcal{M}$. Faithfulness to a hypergraph is naturally defined as follows:
\begin{definition}\label{HypergrFaithDef}
A distribution is \emph{faithful to a hypergraph} $\mathcal{H}$ if the non-vanishing maximal interactions of this distribution coincide with hyperedges of $\mathcal{H}$.
\end{definition}
This definition implies that a distribution in the log-linear model generated by $\mathcal{M}=\{M_1,\dots M_T\}$ is faithful to the hypergraph with hyperedges $M_1,\dots M_T$ if, for all $t \in \{1,\dots, T\}$, none of the conditional odds ratios of $M_t$ given the variables in $\bar{M}_t = V \setminus M_t$ is equal to $1$. In the following result, we show that the class of hypergraphs is closed under the parametric faithfulness relation.
\begin{theorem} \label{pIII}
The class of hypergraphs in $\mathcal{P}$ is closed under the faithfulness relation specified by Definition \ref{HypergrFaithDef}.
\end{theorem}
\begin{proof}
Let $P \in \mathcal{P}$. In the following, we show that there exists a hypergraph to which $P$ is faithful. First, derive the ascending class, $\mathcal{A}$, of subsets of $V$ such that the log conditional odds ratios of the elements of $\mathcal{A}$ given the remaining variables vanish on $P$. Next, find the maximal (with respect to inclusion) elements, $M_1, \dots, M_T$, of the complement of $\mathcal{A}$. Then, by construction, $P$ is faithful to the hypergraph with hyperedges $M_1, \dots,M_T$.
\end{proof}
\begin{remark}
This paper is solely concerned with discrete distributions. However, it is worth pointing out that Definition \ref{FaithNormalDef} makes sense for exponential families in general. In particular, multivariate normal distributions can be described using an exponential family whose canonical parameters correspond to pairwise interactions between the random variables in $V$. We mentioned in Remark \ref{rem_Gaussian} that there are examples of distributions in Gaussian DAG models that are not Markov to any nested DAG, and hence the class of Gaussian DAG models is not closed under the faithfulness relation. However, the class of multivariate normal exponential families is closed under the parametric faithfulness relation. This is the case since setting an additional canonical parameter to zero leads to a nested exponential family.
\end{remark}
\section{Parametric Strong-Faithfulness}\label{sectionStrongFaith}
In order to test statistical hypotheses when working with data, a stronger version of faithfulness is needed. In this section, we generalize the notion of parametric faithfulness to parametric strong-faithfulness and discuss difficulties arising with this concept in the discrete setting.
\begin{table}[b!]
\centering
\caption{Selected parameterizations, measures of association, and strong-faithfulness conditions for the $2 \times 2$ contingency table.}
\begin{tabular}{l|p{20mm}|p{26mm}|p{59mm}}
\hline
& & & \\
Parametrization & Parameter space& Variation independence& Association function; $\lambda$-strong-faithfulness condition \\
\hline
& & &\\
Cell probabilities: & & & \\
$p_{00}, p_{01},$ & \multicolumn{1}{c|}{Simplex} & \multicolumn{1}{c|}{No} & $\phi_1 = \left|\mbox{log} \left(\frac{p_{00}p_{11}}{p_{01}p_{10}} \right)\right| > \lambda$ \\ [5pt]
$p_{10}, p_{11}$& \multicolumn{1}{c|}{$\Delta_3$} & \\
& & & $\phi_2 = \left|\frac{p_{00}p_{11} - p_{01}p_{10}}{p_{00}p_{11} + p_{01}p_{10}}\right| > \lambda$ \\
& & &\\
\hline
& & &\\
Conditional probabilities: & & &\\ [5pt]
$\theta_1 = \mathbb{P}(A=0)$, & \multicolumn{1}{c|}{$(0,1)^3$} & \multicolumn{1}{c|}{Yes} & $\phi_3 = |\theta_2 - \theta_3| > \lambda$ \\
$\theta_2 = \mathbb{P}(B=0\mid A=0)$, & & \\
$\theta_3 = \mathbb{P}(B=0\mid A=1)$ & & \\
\hline
\end{tabular}
\label{StrFtwoway}
\end{table}
\subsection{Strong-faithfulness in the discrete setting}
\label{subsec_strong_faith}
A distribution in a model is faithful to it if the model fully describes the conditional independence structure in this distribution. It is further called \emph{strong-faithful} if the conditional dependencies present in the distribution are strong enough. The concept of strong-faithfulness, originally defined by \perp\!\!\!\perpte{ZhangSpirtesLambdaFaith}, is usually applied to multivariate normal distributions: For a given $\lambda > 0$, a multivariate normal distribution in a DAG model is $\lambda$\emph{-strong-faithful} with respect to this DAG if all non-zero partial correlations are bounded away from zero by $\lambda$. A formal definition of strong-faithfulness in the discrete case has not been proposed, although some analogies were used. For example, \perp\!\!\!\perpte*{Zuk} made use of the assumption that the conditional probabilities in a Bayesian network are bounded between $\lambda$ and $1 - \lambda$. This can be seen as a form of strong-faithfulness.
In the discrete setting, one problem is that many variants of strong-faithfulness relations can be considered. Whether a distribution is $\lambda$-strong-faithful to a model, depends on the choice of parameterization and the measure of association. This is illustrated in the following example for two binary random variables.
\begin{example}\label{22param}
Let $V = \{A, B\}$ and consider the saturated model $[AB]$, which allows for interaction between $A$ and $B$. A distribution in which this interaction vanishes is unfaithful to $[AB]$ and belongs to the model of independence, $A \perp\!\!\!\perp B$. A distribution in which the association between $A$ and $B$ is strong enough is called strong-faithful to $[AB]$. While in the multivariate normal setting the partial correlations are a standard measure of association, in the discrete setting there are many viable choices of association measures, see \perp\!\!\!\perpte{GoodmanKruskal74}. Table \ref{StrFtwoway} illustrates different possible definitions of strong-faithfulness based on three different measures of association, the log odds ratio, $\phi_1$, Yule's coefficient of association, $\phi_2$, and the absolute difference between the conditional probabilities, $\phi_3$. In all three cases, the parameter space has finite volume. So it is possible to estimate the proportion (relative volume) of distributions that do not satisfy the $\lambda$-strong-faithfulness relation with respect to $[AB]$. Figure \ref{2by2Prop} shows that this proportion varies considerably depending on the chosen parameterization and association measure.
\qed
\end{example}
\begin{figure}
\caption{The proportions of distributions that are not $\lambda$-strong-faithful to the model $[AB]$ with respect to different association measures, see Example \ref{22param}
\label{2by2Prop}
\end{figure}
The proportion of distributions in a model that do not satisfy the strong-faithfulness relation with respect to this model is of importance for model selection procedures, which are often based on the strong-faithfulness assumption. Lemma \ref{interactions} justifies the use of $\phi_1$ to define strong-faithfulness in the discrete case. In the following, we propose the concept of strong-faithfulness to a hypergraph and, assuming strong-faithfulness, prove existence of uniformly consistent estimators of the hypergraph parameters.
\subsection{Strong-faithfulness with respect to a hypergraph}
Let $\mathcal{H}$ be the hypergraph generated by a set of marginals $\mathcal{M}= \{M_1, \dots, M_T\}$. For $\boldsymbol p \in \mathcal{H}$ let $\boldsymbol \gamma(\boldsymbol p) = (\gamma_1(\boldsymbol p), \dots, \gamma_T(\boldsymbol p))$ denote the set of interaction parameters of $\boldsymbol p$ corresponding to the hyperedges of $\mathcal{H}$.
\begin{definition}
For $\lambda >0$, a distribution $\boldsymbol p \in \mathcal{H}$ is $\lambda$\emph{-strong-faithful} to $\mathcal{H}$ if
\begin{equation}\label{tube}
\operatorname{min} \{|\gamma_1(\boldsymbol p)|, \dots, |\gamma_T(\boldsymbol p)|\} > \lambda.
\end{equation}
\end{definition}
As described in Section \ref{subsec_strong_faith}, one can, in principle, use different measures of association to define strong-faithfulness. The advantage of the definition given here is that it generalizes the original definition of strong-faithfulness given by \perp\!\!\!\perpte{ZhangSpirtesLambdaFaith}. For a hypergraph generated by two-way marginals the interactions $\boldsymbol \gamma(\boldsymbol p)$ are analogous to partial correlations of a multivariate normal distribution \perp\!\!\!\perptep[cf.][]{WermuthAnalogies}. Therefore, the definition of strong-faithfulness to a hypergraph proposed here is consistent with the original definition of strong-faithfulness of a multivariate normal distribution with respect to a DAG given by \perp\!\!\!\perpte{ZhangSpirtesLambdaFaith}. In addition, as we will show in Section \ref{sec_hypergraph_search}, strong-faithfulness with respect to a hypergraph allows to build uniformly consistent algorithms for learning hypergraphs.
In the following example, we illustrate the concept of strong-faithfulness with respect to a hypergraph for distributions on the $2 \times 2 \times 2$ contingency table.
\begin{example}\label{222marginal}
Let $V = \{A, B, C\}$. A distribution of $V$ can be parameterized by
\begin{equation*}
\mbox{log } \boldsymbol p = \mathbf{M} \boldsymbol \gamma,
\end{equation*}
where
\begin{equation*}
\mathbf{M}=\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
\end{array}\right),
\end{equation*}
and
$$\boldsymbol \gamma = (\gamma^{\emptyset}, \gamma^{A}_{1}, \gamma^{B}_{1}, \gamma^{C}_{1} , \gamma^{AB}_{11}, \gamma^{AC}_{11}, \gamma^{BC}_{11}, \gamma^{ABC}_{111})$$
are the interaction parameters corresponding to the marginal distributions indicated in the superscript. The matrix $\mathbf{M}$ is of full rank, and it can easily be shown that
\begin{align*}
&\gamma^{\emptyset} = \mbox{log } p_{000}, \hspace{14mm}
\gamma^{A}_{1} = \mbox{log } \frac{p_{100}}{p_{000}} ,\\
&\gamma^{B}_{1} = \mbox{log } \frac{p_{010}}{p_{000}}, \hspace{13mm}
\gamma^{C}_{1} = \mbox{log } \frac{p_{001}}{p_{000}},\\
&\gamma^{AB}_{11} = \mbox{log } \frac{p_{000}p_{110}}{p_{010}p_{100}}, \quad
\gamma^{AC}_{11} = \mbox{log } \frac{p_{000}p_{101}}{p_{001}p_{100}}, \\
&\gamma^{BC}_{11} = \mbox{log } \frac{p_{000}p_{011}}{p_{001}p_{010}}, \quad \gamma^{ABC}_{111} =
\mbox{log } \frac{p_{001}p_{010}p_{100}p_{111}}{p_{000}p_{011}p_{101}p_{110}}.
\end{align*}
\noindent In the following table we give the $\lambda$-strong-faithfulness conditions for several hypergraph models:
\begin{equation*}
\begin{tabular}{l| l}
{Hyperedges} & \multicolumn{1}{c}{Strong-faithfulness constraints} \\
\hline
& \\
$\{ABC\}$ & $|\gamma^{ABC}_{111}| > \lambda$ \\ [5pt]
$\{AB\}, \{AC\}, \{BC\}$ & $\operatorname{min} \{|\gamma^{AB}_{11}|, |\gamma^{AC}_{11}|, |\gamma^{BC}_{11}|\} > \lambda$ \\ [5pt]
$\{AC\}, \{BC\}$ & $\operatorname{min} \{|\gamma^{AC}_{11}|, |\gamma^{BC}_{11}|\} > \lambda$ \\ [5pt]
$\{A\}, \{BC\}$ & $\operatorname{min} \{|\gamma^{A}_{1}|, |\gamma^{BC}_{11}|\} > \lambda$ \\ [5pt]
$\{A\}, \{B\}, \{C\}$ & $\operatorname{min} \{|\gamma^{A}_{1}|, |\gamma^{B}_{1}|, |\gamma^{C}_{1}| \} > \lambda$ \\ [5pt]
\end{tabular}
\end{equation*}
\end{example}
\subsection{Hypergraph search}
\label{sec_hypergraph_search}
In this section, we discuss how to construct hypothesis tests, when the association measure is based on the interaction parameters $\boldsymbol{\gamma}(\boldsymbol p)$, and how to perform a hypergraph search based on these hypothesis tests.
Let $\mathcal{H}$ be a hypergraph generated by the marginals $M_1,\dots , M_T$ and let $\gamma_1,\dots ,\gamma_T$ be the corresponding interaction parameters. We denote by $\mathcal{H}_{\lambda}$ the set of distributions that are $\lambda$-strong-faithful to the hypergraph $\mathcal{H}$, i.e.,
$$\mathcal{H}_{\lambda} = \{\boldsymbol p \in \mathcal{H}: \,\, \operatorname{min}\{|\gamma_1(\boldsymbol p)|, \dots, |\gamma_T(\boldsymbol p)|\} > \lambda\},$$
and define
$$\mathcal{H}_{\lambda, \delta} = \mathcal{H}_{\lambda} \cap \{ \boldsymbol p \in \mathcal{P}: \,\, p_{\boldsymbol i} \in [\delta, 1), \, \sum_{\boldsymbol i \in \mathcal{I}} p_{\boldsymbol i} = 1\},$$
where $\delta > 0$ is small enough so $\mathcal{H}_{\lambda, \delta}$ is not empty.
If $M_t$, for $t \in \{1, \dots, T\}$, is an interaction of order $h_t$, then the conditional odds ratio of $M_t$ given the variables in $\bar{M}_t$ is the ratio of the product of some $2^{h_t}$ cell probabilities and the product of a disjoint set of $2^{h_t}$ cell probabilities. Since $p_{\boldsymbol i} \in [\delta, 1)$ for all $\boldsymbol i \in \mathcal{I}$, the interaction parameter $\,\,|\gamma_t(\boldsymbol p)| \leq 2^{h_t}\mbox{log } ((1-\delta)/\delta)$, and, therefore,
$$|\gamma_t(\boldsymbol p)| \leq C(\delta), \quad \mbox{for } t=1, \dots, T,$$
where $C(\delta) = 2^{{\operatorname{max}}\{h_1, \dots, h_T\}} \mbox{log } ((1-\delta)/\delta)$. Here, $C(\delta)$ is an upper bound on the interaction parameters (it plays the same role as the constant $M$ in Assumption (A4) of \perp\!\!\!\perpte{KalischBullm} for the Gaussian setting).
\begin{theorem}
Let $\mathbf Y$ have a multinomial distribution with parameters $N$ and $\boldsymbol p$. Assume that, under the log-linear model corresponding to $\mathcal{H}$, the maximum likelihood estimates of the interaction parameters
$$\hat{\boldsymbol \gamma}^{(N)}(\boldsymbol p)=(\hat{\gamma}_1^{(N)}(\boldsymbol p), \dots, \hat{\gamma}_T^{(N)}(\boldsymbol p))=({\gamma}_1^{(N)}(\hat{\boldsymbol p}), \dots, {\gamma}_T^{(N)}(\hat{\boldsymbol p}))$$
exist and are unique. Then, $\hat{\boldsymbol \gamma}^{(N)}(\boldsymbol p)$ is a uniformly, over $\mathcal{H}_{\lambda, \delta}$, consistent estimator of ${\boldsymbol \gamma}(\boldsymbol p)$.
\end{theorem}
\begin{proof}
For $t \in \{1, \dots, T\}$, $\,\gamma_t(\boldsymbol p) = \boldsymbol c_{t}' \operatorname{log } \boldsymbol p$, where $\boldsymbol c_t$ is a vector in $\mathbb{Z}^{|\mathcal{I}|}$ whose components are comprised of equal number, $2^{h_t}$, of $1$'s and $-1$'s, and some $0$'s. By Theorem 14.6-4 in \perp\!\!\!\perpte{BFH}, as $N \to \infty$, $\gamma_t(\boldsymbol p)$ is asymptotically normal with mean zero and variance
\begin{equation}\label{asVar}
{\textrm{var}}(\hat{\gamma}_t) = \frac{1}{N}\boldsymbol c'_t \textrm{diag}^{-1}(\boldsymbol p)\boldsymbol c_t.
\end{equation}
For every $\boldsymbol p \in \mathcal{H}_{\lambda, \delta}$,
\begin{equation}\label{VarBound}
{\textrm{var}}(\hat{\gamma}_t) \leq \frac{\boldsymbol c_t'\boldsymbol c_t}{N\delta},
\end{equation}
and thus, using the Chebyshev inequality,
\begin{eqnarray}\label{ConsBound}
\mathbb{P}\left(|\hat{\gamma}_t^{(N)}(\boldsymbol p) - \gamma_t(\boldsymbol p)| < \epsilon\right) &=& \mathbb{P}\left(|Z|< \frac{\sqrt{N} \epsilon}{\sqrt{\boldsymbol c_{t}'\textrm{diag}^{-1}(\boldsymbol p)\boldsymbol c_{t}}}\right)
\geq \mathbb{P}\left(|Z| < \epsilon\sqrt{\frac{N\delta}{\boldsymbol c_t'\boldsymbol c_t}}\right) \nonumber \\
\nonumber \\
\nonumber \\
&\geq& 1 - \frac{\boldsymbol c_t'\boldsymbol c_t}{N\delta\epsilon^2} \geq 1 - \frac{\underset{t =1, \dots, T}{\operatorname{max}}(\boldsymbol c_t'\boldsymbol c_t)}{N\delta\epsilon^2}, \quad \forall \epsilon > 0,
\end{eqnarray}
where $Z$ is a random variable with a standard normal distribution. Therefore, $\mathbb{P}(|\hat{\gamma}_t^{(N)}(\boldsymbol p) - {\gamma_t}(\boldsymbol p)| < \epsilon) \to 1$ for every $t = 1, \dots,T$, uniformly over $\boldsymbol p \in \mathcal{H}_{\lambda, \delta}$. Since the lower bound in (\ref{ConsBound}) does not depend on $t$, the proof is complete.
\end{proof}
We now address the question of how to select the threshold $\lambda$ in a hypergraph learning procedure. We fix a $t \in \{1, \dots, T\}$ and consider testing the ``one-hyperedge'' hypothesis $H_{0t}: \, \gamma_t = 0$ versus $H_{1t}: \, \gamma_t \neq 0$ under a significance level $\alpha$. Let $\hat{\boldsymbol p}$ be the observed distribution and let $\hat{\gamma}_t = \gamma_t(\hat{\boldsymbol p}) = \boldsymbol c_t \mbox{log }\hat{\boldsymbol p}$ denote the corresponding interaction parameter. By Slutsky's Theorem,
$$\sqrt{N} \frac{\hat{\gamma}_t}{\sqrt{\boldsymbol c'_t \textrm{diag}^{-1}(\hat{\boldsymbol p})\boldsymbol c_t}} \to N(0, 1), \, \mbox{ as } N \to \infty,$$
and thus we reject the null hypothesis if
\begin{equation}\label{TestSt}
\frac{|\hat{\gamma}_t|}{\sqrt{ \frac{1}{N}\boldsymbol c'_t \textrm{diag}^{-1}(\hat{\boldsymbol p})\boldsymbol c_t}} > z_{1-\alpha/2},
\end{equation}
where $z_{1-\alpha/2} = \Phi^{-1}(1-\alpha/2)$ is the corresponding quantile of the standard normal distribution. With such a procedure, the probability of wrongly rejecting the null hypothesis does not exceed $\alpha$.
\begin{theorem}\label{powerTh}
Let $\epsilon \in (0,1/2)$ and set
\begin{equation}\label{LambdaVar}
\lambda^*_N = \frac{z_{1-\alpha/2}}{N^{1/2-\epsilon}}{\underset{t =1, \dots, T}{\operatorname{min}}\sqrt{\boldsymbol c_t'\boldsymbol c_t}}.
\end{equation}
For the distributions that are $\lambda^*_N$-strong-faithful to $\mathcal{H}$, the
power of the one-hyperedge test approaches $1$ as $N \to \infty$.
\end{theorem}
\begin{proof}
The asymptotic variance of $\hat{\gamma}_t$ is bounded below by
$${\textrm{var}}(\hat{\gamma}_t) = \frac{1}{N}\boldsymbol c'_t \textrm{diag}^{-1}(\hat{\boldsymbol p})\boldsymbol c_t \geq \frac{1}{N} \boldsymbol c_t'\boldsymbol c_t.$$
Since for an $h_t$-order interaction, the vector $\boldsymbol c_t$ has $2^{h_t}$ components equal to $1$, $2^{h_t}$ components equal to $-1$, and the remaining components equal to zero, we have $\sqrt{\boldsymbol c_t'\boldsymbol c_t} = 2^{(h_t + 1)/2}$. The distributions that are $\lambda^*_N$-strong-faithful to $\mathcal{H}$ satisfy (\ref{TestSt}) for all $t = 1, \dots, T$.
For these distributions the power of the one-hyperedge test is bounded below:
$$\Phi\left(\frac{|\hat{\gamma}_t|}{\sqrt{ \frac{1}{N}\boldsymbol c'_t \textrm{diag}^{-1}(\hat{\boldsymbol p})\boldsymbol c_t}} - z_{1-\alpha/2}\right) \geq \Phi\left(z_{1-\alpha/2}(\frac{\sqrt{\boldsymbol c_t'\boldsymbol c_t}N^{-1/2+\epsilon}}{\sqrt{ \frac{1}{N}\boldsymbol c'_t \textrm{diag}^{-1}(\hat{\boldsymbol p})\boldsymbol c_t}} - 1)\right) \textrm{ for all } t\in\{1,\dots T\}$$
and approaches $1$ as $N \to \infty$.
\end{proof}
Examples of $\lambda^*_N$ computed for hyperedges of different sizes are shown in Table \ref{lambdasTest}. In this paper, we do not investigate any multiple comparison issues arising with testing several one-hyperedge hypotheses at the same time.
\begin{table}
\centering
\caption{Possible threshold values for the parameter $\lambda$.}
\label{lambdasTest}
\begin{tabular}{l|c|l}
\hline
& & \\
Hyperedge & \multicolumn{1}{c|}{The order of the odds ratio} & \multicolumn{1}{c}{$\lambda^*_N$} \\
& & \\
\hline
& & \\
$[AB]$ & $h = 1$ & $\lambda^*_N = \frac{z_{1-\alpha/2}}{N^{1/4}} \cdot 2$\\
& & \\
$[ABC]$ & $h=2$ & $\lambda^*_N = \frac{z_{1-\alpha/2}}{N^{1/4}} \cdot 2^{3/2}$ \\
& & \\
$[ABCD]$ & $h=3$ & $\lambda^*_N = \frac{z_{1-\alpha/2}}{N^{1/4}}\cdot 4$\\
& & \\
$[ABCDE]$ & $h=4$ & $\lambda^*_N = \frac{z_{1-\alpha/2}}{N^{1/4}}\cdot 2^{5/2}$\\
\hline
\end{tabular}
\end{table}
For learning a hypergraph, any model selection procedure for hierarchical log-linear models can be applied. A review of such procedures can be found in \perp\!\!\!\perpte{EdwardsBook}. Backward selection which starts from the saturated model and, using the edge removal mechanism described by \perp\!\!\!\perpte {EdwardsNote}, goes through a sequence of nested hypergraphs, is a polynomial time algorithm that is appropriate for high dimensions. Uniform consistency of the maximum likelihood estimates for the maximal interactions of the distributions in $\mathcal{H}_{\lambda, \delta}$ entails that backward selection is a uniformly consistent procedure and the hypergraph will be determined correctly.
\section{Proportions of strong-unfaithful distributions}\label{SecProp}
As shown in the previous section, strong-faithfulness ensures the existence of uniformly consistent tests for developing methods for learning the underlying hypergraph. If the parameter space has finite volume it is possible to estimate the proportion (relative volume) of the distributions that are not $\lambda$-strong-faithful to a model of interest and thus whose association structure may be discovered incorrectly. \perp\!\!\!\perpte{UhlerFaithGeometry} analyzed the proportion of distributions that are not $\lambda$-strong-faithful to a DAG in the Gaussian setting. Partial correlations define varieties and strong-unfaithful distributions correspond to the parameters that lie in a tube around these varieties. So the relative volume of these tubes corresponds to the proportion of distributions that don't satisfy the strong-faithfulness assumption, and lower bounds on these volumes were given for different classes of DAGs. The following example illustrates how one can estimate such volumes in the discrete case.
\textbf{Example \ref{22param}} (revisited):
Consider a hierarchical log-linear parameterization of the distributions on the $2\times 2$ contingency table:
\begin{align}\label{22corner}
\operatorname{log} p_{00} &=\gamma^{\emptyset}, \nonumber \\
\operatorname{log} p_{01} &=\gamma^{\emptyset}+ \gamma^{B}_{1},\\
\operatorname{log} p_{10} &= \gamma^{\emptyset} + \gamma^{A}_{1}, \nonumber \\
\operatorname{log} p_{11} &= \gamma^{\emptyset} + \gamma^{A}_{1}+\gamma^{B}_{1}+ \gamma^{AB}_{11}. \nonumber
\end{align}
The interaction parameter, $\gamma_{11}^{AB}$, which was denoted by $\phi_1$ in Example \ref{22param} and in the corresponding Table \ref{StrFtwoway} and Figure \ref{2by2Prop}, can be expressed in terms of conditional probabilities $\theta_1 = \mathbb{P}(A = 0)$, $\theta_2 = \mathbb{P}(B=0\mid A=0)$, and $\theta_3 = \mathbb{P}(B=0\mid A=1)$:
\begin{equation*}
\gamma^{AB}_{11} =\mbox{log } \left(\frac{p_{00}p_{11}}{p_{01}p_{10}}\right)= \mbox{log } \frac{\theta_2(1-\theta_3)}{(1-\theta_2)\theta_3} = \mbox{log } \frac{\theta_2}{1-\theta_2} - \mbox{log } \frac{\theta_3}{1-\theta_3}.
\end{equation*}
Let
\begin{eqnarray*}
\mathcal{H}_{\lambda} &=&\left \{(\theta_1, \theta_2, \theta_3) \in (0,1)^3: \,\, \left|\mbox{log } \frac{\theta_2}{1-\theta_2} - \mbox{log } \frac{\theta_3}{1-\theta_3}\right| > \lambda \right\}.
\end{eqnarray*}
The volume of its complement, $\bar{\mathcal{H}}_{\lambda}$, is equal to:
\begin{eqnarray}
\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) &=& \mathrm{vol} \left \{(\theta_1, \theta_2, \theta_3) \in (0,1)^3: \,\, e^{-\lambda} < \frac{\theta_2}{1-\theta_2} \cdot \frac{1-\theta_3}{\theta_3} < e^{\lambda} \right \}\nonumber\\
&=& \int_{0}^1 d \theta_2 \left(
\frac{\theta_2}{\theta_2(1-e^{-\lambda}) + e^{-\lambda}} -
\frac{\theta_2}{\theta_2(1-e^{\lambda}) + e^{\lambda}}
\right) \nonumber\\
{}\nonumber\\
&=& \frac{e^{2\lambda} - 2\lambda e^{\lambda} - 1}{(1-e^{\lambda})^2},\label{eq_gamma2}
\end{eqnarray}
where the integral was computed by substitution. The parameter space $(0,1)^3$ has a unit volume. Hence, the relative proportion of distributions that are not $\lambda$-strong-faithful to $[AB]$ is equal to $\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) $. For small $\lambda$ this proportion is approximately $\frac{\lambda}{3}$, which is consistent with the simulation results for $\phi_1$ in Figure \ref{2by2Prop}.\qed
\begin{theorem}\label{BoundMax}
Let $\mathcal{H}$ be a hypergraph whose hyperedges $M_1, \dots, M_T$ are interactions of order $h_1, \dots, h_T$ respectively, and let
$\bar{\mathcal{H}}_{\lambda}$ be the set of distributions that are not $\lambda$-strong-faithful to $\mathcal{H}$. Then,
$$\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) \geq \underset{t \in \{1, \dots, T\}}{\operatorname{max}} \mathrm{vol} \{\boldsymbol p \in \mathcal{H}: \,\, |\gamma_{t}(\boldsymbol p)| < \lambda\} \geq \underset{t \in \{1, \dots, T\}}{\operatorname{max}}\left(\frac{e^{2\mu} - 2\mu e^{\mu} - 1}{(1-e^{\mu})^2}\right)^{2^{h_t-1}},$$
where $\mu=\lambda/2^{h_t-1}$
\end{theorem}
\begin{proof}
By Lemma \ref{interactions}, the parameter $\gamma_{t}$ for $t \in \{1, \dots, T\}$ is the log conditional odds ratio of $M_t$, given the variables in $\bar{M}_t$. We can express $\gamma_t$ using a corresponding set of variation independent conditional probabilities:
\begin{eqnarray*}
\gamma_{t} = \mbox{log } \left(\frac{\theta_{1}}{1-\theta_{1}} \cdots \frac{\theta_{2^{h_t-1}}}
{1-\theta_{2^{h_t-1}}}\cdot
\frac{1-\theta_{2^{h_t-1}+1}}{\theta_{2^{h_t-1}+1}} \cdots \frac{1-\theta_{2^{h_t}}}{ \theta_{2^{h_t}}}\right).\end{eqnarray*}
Therefore,
\begin{eqnarray*}
&&\mathrm{vol} \{\boldsymbol p \in \mathcal{H}: \,\, |\gamma_{t}(\boldsymbol p)| < \lambda\} = \mathrm{vol} \left\{\boldsymbol \theta \in (0,1)^m: \,\, |\gamma_{t}| < \lambda \right\}\\
&& \\
&=&\mathrm{vol}\left\{(\theta_1, \dots, \theta_{2^{h_t}})\!\in\!(0,1)^{2^{h_t}}\!:\, \left| \mbox{log }\!\!\left(\!\frac{\theta_{1}}{1-\theta_{1}} \cdots \frac{\theta_{2^{h_t-1}}}
{1-\theta_{2^{h_t-1}}}\cdot
\frac{1-\theta_{2^{h_t-1}+1}}{\theta_{2^{h_t-1}+1}} \cdots \frac{1-\theta_{2^{h_t}}}{ \theta_{2^{h_t}}}\!\right)\!\right| < \lambda \right\} \\
&\geq&
\left(\mathrm{vol}\,\left\{(\zeta_1, \zeta_2)\in(0,1)^2:\,\, \left| \mbox{log } \frac{\zeta_1}{1-\zeta_1} - \mbox{log } \frac{\zeta_2}{1-\zeta_2}\right| < \frac{\lambda}{2^{h_t-1}}\right\} \right)^{2^{h_t-1}} \\
&=& \left(\frac{e^{2\mu} - 2\mu e^{\mu} - 1}{(1-e^{\mu})^2}\right)^{2^{h_t-1}},
\end{eqnarray*}
where $\mu = {\lambda}/{2^{h_t-1}}$, and for the last equation we used (\ref{eq_gamma2}). Since $\bar{\mathcal{H}}_{\lambda} = \{\boldsymbol p \in \mathcal{H}: \,\, |\gamma_t(\boldsymbol p)| < \lambda \mbox{ for at least one } t \in \{1, \dots, T\}\}$,
$$\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) \geq \mathrm{vol} \{\boldsymbol p \in \mathcal{H}: \,\, |\gamma_t(\boldsymbol p)| < \lambda \mbox{ for all } t \in \{1, \dots, T\}\},$$
and thus
$$\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) \geq \underset{t \in \{1, \dots, T\}}{\operatorname{max}} \left(\frac{e^{2\mu} - 2\mu e^{\mu} - 1}{(1-e^{\mu})^2}\right)^{2^{h_t-1}}, \quad \mbox{for } \mu = {\lambda}/{2^{h_t-1}}.$$
\end{proof}
As we will see in the following result, for hypergraphs whose hyperedges are variation independent we can in fact give an exact formulation of the proportion of distributions that don't satisfy strong-faithfulness.
\begin{theorem} \label{conjecture}
Let $\mathcal{H}$ be the hypergraph generated by marginals $M_1, \dots, M_T$. Assume that there exists a parameterization of $\mathcal{H}$ under which the interaction parameters corresponding to $M_1, \dots, M_T$ are variation independent and, further, that the parameter space has finite volume. Then, the proportion of distributions that are not $\lambda$-strong-faithful to $\mathcal{H}$ is
\begin{equation}\label{VolumeDecomp}
\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) = 1 - (1-\boldsymbol \nu_{M_1}) \cdots (1-\boldsymbol \nu_{M_T}),
\end{equation}
where $\boldsymbol \nu_{M_t}$, for $t = 1, \dots, T$, is the proportion of distributions that are not $\lambda$-strong-faithful to $M_t$.
\end{theorem}
\begin{proof}
Consider a parameterization of $\mathcal{H}$ under which the maximal interaction parameters corresponding to $M_1, \dots, M_T$ are variation independent. For $t \in \{1, \dots, T\}$, let $\boldsymbol \nu_{M_t}$ denote the proportion of distributions that do not satisfy $\lambda$-strong-faithfulness with respect to $M_t$. Since the joint range of the variation independent parameters is equal to the Cartesian product of the individual ranges, the proportion of distributions that are not $\lambda$-strong-faithful to $\mathcal{H}$ is equal to
$1 - (1-\boldsymbol \nu_{M_1}) \cdots (1-\boldsymbol \nu_{M_T}).$
\end{proof}
The maximal interaction parameters in decomposable log-linear models \perp\!\!\!\perptep{Haberman} and in ordered decomposable marginal log-linear models \perp\!\!\!\perptep{RudasBergsma} are variation independent, and Theorem \ref{conjecture} applies. Let $\mathcal{H}$ be the hypergraph generated by a decomposable sequence of marginals $M_1, \dots, M_T$. The interaction parameter associated with a hyperedge $M_t$ can be expressed using a variation independent set of conditional probabilities. Under this parameterization, the proportion of distributions that do not satisfy $\lambda$-strong-faithfulness with respect to the hyperedge $M_t$ is equal to
\begin{eqnarray}\label{nu}
\boldsymbol \nu_{h_t} = \mathrm{vol}\!\left\{\!(\theta_1, \dots, \theta_{2^{h_t}})\!:\, \left|\log \!\left(\frac{\theta_{1}}{1-\theta_{1}} \cdots \frac{\theta_{2^{h_t-1}}}
{1-\theta_{2^{h_t-1}}}\cdot
\frac{1-\theta_{2^{h_t-1}+1}}{\theta_{2^{h_t-1}+1}} \cdots \frac{1-\theta_{2^{h_t}}}{ \theta_{2^{h_t}}}\right)\!\right| < \lambda \right\},
\end{eqnarray}
where $h_t$ denotes the order of interaction $M_t$. The proportion of distributions that are not $\lambda$-strong-faithful to $\mathcal{H}$ is calculated using (\ref{VolumeDecomp}).
Figure \ref{Chains} shows values of $\boldsymbol \nu_{h}$ as functions of $h$ and $\lambda$. The concrete computations involved in Theorem \ref{conjecture} are illustrated in Example \ref{ChainExample}. We next analyze hypergraphs with a special ``chain'' structure and show that in this case Equation (\ref{VolumeDecomp}) simplifies.
\begin{definition}
A hypergraph $\mathcal{H}$ is called a chain of order $h$ if the generating sequence of marginals
$\{M_1, \dots M_T\}$, where $\cup_{t = 1}^T M_t = V$, is decomposable and all of the hyperedges correspond to $h$-th order interactions of the joint distribution.
\end{definition}
For example, the hypergraph generated by $\{A,B\}$, $\{B,C\}$, $\{C,D\}$, $\{D,E\}$ is a chain of order $1$ of length $4$, and the hypergraph generated by $\{A,B,C\}$ and $\{A,B,D\}$ is a chain of order 2 of length 2.
\begin{corollary} \label{ChainH}
Let a hypergraph $\mathcal{H}$ be a chain of order $h$ of length $L$. Then the proportion of distributions that are not $\lambda$-strong-faithful to $\mathcal{H}$ is equal to
$$1 - (1-\boldsymbol \nu_h)^L,$$
where
\begin{eqnarray}\label{nuH}
\boldsymbol \nu_{h} = \mathrm{vol}\left\{(\theta_1, \dots, \theta_{2^{h}}):\,\, \left|\log \left(\frac{\theta_{1}}{1-\theta_{1}} \cdots \frac{\theta_{2^{h-1}}}
{1-\theta_{2^{h-1}}}\cdot
\frac{1-\theta_{2^{h-1}+1}}{\theta_{2^{h-1}+1}} \cdots \frac{1-\theta_{2^{h}}}{ \theta_{2^{h}}}\right)\right| < \lambda \right\}.
\end{eqnarray}
\end{corollary}
For a chain of order $1$, the proportion of distributions that are not $\lambda$-strong-faithful to $\mathcal{H}$ is especially simple:
$$\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) = 1 - (1-\boldsymbol \nu_1)^T,$$
where
\begin{eqnarray}\label{nu1}
\boldsymbol \nu_1 = \frac{e^{2\lambda} - 2\lambda e^{\lambda} - 1}{(1-e^{\lambda})^2}.
\end{eqnarray}
The proportions for chains of several orders were estimated using Monte-Carlo method and are displayed in Figure \ref{ChainsProportions}.
\begin{figure}
\caption{Proportions of distributions that do not satisfy strong-faithfulness with respect to a single hyperedge. See Equation (\ref{nu}
\label{Chains}
\caption{Proportions of distributions that are not $\lambda$-strong-faithful to a first order chain. See Corollary \ref{ChainH}
\label{ChainsProportions}
\end{figure}
\begin{example} \label{ChainExample}
We demonstrate the volume computation using the chain $[AB][BC][CD]$ of order 1.
The maximal interaction parameters corresponding to the hyperedges are:
\begin{eqnarray*}
\gamma_1 &=& \mbox{log } \mathcal{COR}(AB \mid CD) = \mbox{log } \frac{p_{00kl}p_{11kl}}{p_{01kl}p_{10kl}}, \\
\gamma_2 &=& \mbox{log } \mathcal{COR}(BC \mid AD) = \mbox{log } \frac{p_{i00l}p_{i11l}}{p_{i01l}p_{i10l}}, \\
\gamma_3 &=& \mbox{log } \mathcal{COR}(CD \mid AB) = \mbox{log } \frac{p_{ij00}p_{ij11}}{p_{ij01}p_{ij10}},
\end{eqnarray*}
where $i, j, k, l \in \{0, 1\}$ are fixed categories of $A$, $B$, $C$, $D$ respectively. The chain can be described by two conditional independence relations: $A \perp\!\!\!\perp C \mid B$ and $AB \perp\!\!\!\perp D \mid C$. Thus the distributions in a chain model can be parameterized using the conditional probabilities:
\begin{align*}
&\theta_{0} = \mathbb{P}(B = 0), \\
&\theta_{10} = \mathbb{P}(A=0 \mid B = 0), \quad \theta_{11} = \mathbb{P}(A = 0 \mid B = 1), \\
&\theta_{20} = \mathbb{P}(C = 0 \mid B = 0), \quad \theta_{21} = \mathbb{P}(C = 0 \mid B = 1), \\
&\theta_{30} = \mathbb{P}(D = 0 \mid C = 0), \quad \theta_{31} = \mathbb{P}(D = 0 \mid C = 1).
\end{align*}
The parameters $\boldsymbol \theta$ are variation independent, and, for $t = 1, 2, 3$,
$$\gamma_t = \mbox{log } \left(\frac{\theta_{t0}}{1-\theta_{t0}} \cdot \frac{1-\theta_{t1}}{\theta_{t1}}\right).$$
Let
\begin{eqnarray*}
\boldsymbol \nu_1 &=& \mathrm{vol}\left \{(\theta_1, \theta_2) \in (0,1)^2: \,\, \left|\mbox{log } \frac{\theta_1}{1-\theta_1} - \mbox{log } \frac{\theta_2}{1-\theta_2} \right| < \lambda \right\} =\frac{e^{2\lambda} - 2\lambda e^{\lambda} - 1}{(1-e^{\lambda})^2}.
\end{eqnarray*}
Using the binomial formula, we obtain that
$\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) = 1 - (1 - \boldsymbol \nu_1)^3$. \qed
\end{example}
\textbf{Example \ref{FourVarUnfaith}} (revisited):
The maximal non-vanishing interactions of a distribution that is faithful to a hypergraph $\mathcal{H}$ with hyperedges $\{A,B,C\}$ and $\{A,B,D\}$ can be described using the interaction parameters equal to the logarithm of the second order conditional odds ratios of $ABC$ given $D$ and of $ABD$, given $C$:
\begin{eqnarray*}
\gamma^{ABC}_0 &=& \mbox{log } \mathcal{COR}(ABC\mid D=0),\\
\gamma^{ABC}_1 &=& \mbox{log } \mathcal{COR}(ABC\mid D=1),\\
\gamma^{ABD}_0 &=& \mbox{log } \mathcal{COR}(ABD\mid C=0),\\
\gamma^{ABD}_1 &=& \mbox{log } \mathcal{COR}(ABD\mid C=1).
\end{eqnarray*}
Using conditional probabilities,
\begin{eqnarray*}
\theta_1 &=& \mathbb{P}(C=0\mid A=0, B=0), \quad \theta_2 = \mathbb{P}(C=0\mid A=0, B=1), \\
\theta_3 &=& \mathbb{P}(C=0\mid A=1, B=0), \quad \theta_4 = \mathbb{P}(C=0\mid A=1, B=1), \\
\theta_5 &=& \mathbb{P}(D=0\mid A=0, B=0), \quad \theta_6 = \mathbb{P}(D=0\mid A=0, B=1), \\
\theta_7 &=& \mathbb{P}(D=0\mid A=1, B=0), \quad \theta_8 = \mathbb{P}(D=0\mid A=1, B=1),
\end{eqnarray*}
one obtains
\begin{eqnarray*}
\gamma^{ABC}_0 &=& \gamma^{ABC}_1 = \mbox{log } \frac{\theta_1}{1-\theta_1} + \mbox{log } \frac{\theta_4}{1-\theta_4} - \mbox{log } \frac{\theta_2}{1-\theta_2} - \mbox{log } \frac{\theta_3}{1-\theta_3}, \\
\gamma^{ABD}_0 &=& \gamma^{ABD}_1 = \mbox{log } \frac{\theta_5}{1-\theta_5} + \mbox{log } \frac{\theta_8}{1-\theta_8} - \mbox{log } \frac{\theta_6}{1-\theta_6} - \mbox{log } \frac{\theta_7}{1-\theta_7}.
\end{eqnarray*}
A distribution is not $\lambda$-strong-faithful to the hypergraph $\mathcal{H}$ if at least one of the following inequalities holds:
$$|\gamma^{ABC}_0| < \lambda, \mbox{ or } |\gamma^{ABD}_0| < \lambda.$$
Hence, the proportion of distributions that are not $\lambda$-strong-faithful to the hypergraph $\mathcal{H}$ is equal to $1 - (1 - \boldsymbol \nu_2)^2$,
where \begin{eqnarray*}
\boldsymbol \nu_2 &=& \mathrm{vol}\left \{(\theta_1, \dots, \theta_4) \in (0,1)^4: \, \left|\mbox{log } \frac{\theta_1}{1-\theta_1} + \mbox{log } \frac{\theta_4}{1-\theta_4} - \mbox{log } \frac{\theta_2}{1-\theta_2} - \mbox{log } \frac{\theta_3}{1-\theta_3} \right| < \lambda \right\}.
\end{eqnarray*}
We were not able to find a closed-form expression for $\boldsymbol \nu_2$. It can be shown that $\boldsymbol \nu_2$ is bounded above by $\boldsymbol \nu_1$ and thus the volume of distributions that are not $\lambda$-strong-faithful to the hypergraph $\mathcal{H}$ is bounded above by the volume computed for the chain of the same length of order $1$, that is,
$\mathrm{vol}(\bar{\mathcal{H}}_{\lambda}) \leq 1 - (1-\boldsymbol \nu_{1})^2.$
\qed
\begin{remark}
The concept of strong-faithfulness can be extended to distributions that do not belong to a given hypergraph model. Let $\boldsymbol p, \boldsymbol q \in \mathcal{P}$, and let $\rho$ be a divergence function. The distance from a distribution $\boldsymbol p$ to a hypergraph $\mathcal{H}$ can be defined as
\begin{equation}\label{distModel}
\rho(\boldsymbol p, \mathcal{H}) = \underset{\boldsymbol q \in \mathcal{H}}{\mbox{min }} \rho(\boldsymbol p, \boldsymbol q).
\end{equation}
In particular, $\rho(\boldsymbol p, \mathcal{H}) = 0$ if and only if $\boldsymbol p \in \mathcal{H}$. We denote by $\boldsymbol p_{\mathcal{H,\rho}}$ the projection of $\boldsymbol p$ onto the hypergraph $\mathcal{H}$, i.e.:
$$\boldsymbol p_{\mathcal{H}, \rho} = \underset{\boldsymbol q \in \mathcal{H}}{\mbox{argmin }} \rho(\boldsymbol p, \boldsymbol q),$$
and call a distribution $\boldsymbol p$ \emph{projected-$\lambda$-strong-faithful to} $\mathcal{H}$ with respect to $\rho$ (for $\lambda >0$) if $\boldsymbol p_{\mathcal H, \rho}$ is $\lambda$-strong-faithful to $\mathcal{H}$. The concept of projected-strong-faithfulness is relevant in various estimation procedures.
\end{remark}
We end by illustrating the concept of projected-strong-faithfulness by estimating the proportions of projected-$\lambda$-strong-faithful distributions for several hypergraph models on the $2 \times 2 \times 2$ contingency table.
\begin{figure}
\caption{Proportions of distributions that are not projected-$\lambda$-strong-faithful computed for several hypergraphs on the $2 \times 2 \times 2$ contingency table.}
\label{C1}
\end{figure}
\textbf{Example \ref{222marginal}} (revisited):
To determine the distance from a distribution $\boldsymbol p$ to a hypergraph $\mathcal{H}$ we use the likelihood function under the corresponding log-linear model. Relative frequencies of distributions that do not satisfy the projected-$\lambda$-strong-faithfulness relation for different hypergraphs and different values of $\lambda$ are displayed in Figure \ref{C1}.
\qed
\section{Conclusion}\label{SecConcl}
We demonstrated that the association structure of discrete data can be very complex, and some distributions are not faithful to any undirected graphical model or any DAG. Thus, the attractive simplicity of graphical models may be misleading. In Section \ref{sectionGraphFaith}, we proposed the concept of parametric faithfulness, which can be applied to any exponential family, including those which cannot be specified using Markov properties.
We considered the class of hypergraphs which can be identified with hierarchical log-linear models. We showed that for any distribution there exists a hypergraph to which it is parametrically faithful and suggested to conduct the search in this class. As the class also contains graphical models, if a model structure which can be described by a graph is appropriate, it will be discovered (see the consistency result in Section \ref{sectionStrongFaith}).
Our work is relevant for the popular causal search algorithms, referred to in Section \ref{intro}, which assume (strong-) faithfulness. The findings described in Sections \ref{sectionStrongFaith} and \ref{SecProp} imply that, depending on the quantitative expression for association and on the choice of the cut-off parameter $\lambda$, to define strong-faithfulness, the resulting model selection procedures may yield different results for the same data.
\begin{table}
\centering
\label{allgraphsABC}
\caption{Some graphical models on three nodes.}
\begin{tabular}{m{7mm}m{32mm}m{65mm}m{35mm}}
\hline
& & & \\
& \multicolumn{1}{c}{Log-Linear Model} & \multicolumn{1}{c}{Conditional Independence} & \multicolumn{1}{c}{Graph} \\ [2ex]
\hline
1 & \multicolumn{1}{c}{[ABC]} & \multicolumn{1}{c}{None} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{ABCpic.pdf}\end{minipage}}
\\ [4ex]
\hline
2 & \multicolumn{1}{c}{[A][B]} & \multicolumn{1}{c}{$A \perp\!\!\!\perp B$} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{AindB.pdf}\end{minipage}} \\ [4ex]
\hline
3 & \multicolumn{1}{c}{[A][C]} & \multicolumn{1}{c}{$A \perp\!\!\!\perp C$} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{AindC.pdf}\end{minipage}} \\ [4ex]
\hline
4 & \multicolumn{1}{c}{[B][C]} & \multicolumn{1}{c}{$B \perp\!\!\!\perp C$} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{BindC.pdf}\end{minipage}} \\ [4ex]
\hline
5 & \multicolumn{1}{c}{[AB][C]} & \multicolumn{1}{c}{$AB \perp\!\!\!\perp C$} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{CindAB.pdf}\end{minipage}} \\ [4ex]
\hline
6 & \multicolumn{1}{c}{[AC][B]} & \multicolumn{1}{c}{$AC \perp\!\!\!\perp B$} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{BindAC.pdf}\end{minipage}} \\ [4ex]
\hline
7 & \multicolumn{1}{c}{[A][BC]} & \multicolumn{1}{c}{$A \perp\!\!\!\perp BC$} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{AindBC.pdf}\end{minipage}} \\ [4ex]
\hline
8 & \multicolumn{1}{c}{[A][B][C]} & \multicolumn{1}{c}{\begin{tabular}{lll} $A \perp\!\!\!\perp B$, & $A \perp\!\!\!\perp C$, & $B \perp\!\!\!\perp C$, \\
$A \perp\!\!\!\perp B | C$, & $A \perp\!\!\!\perp C | B$, & $B \perp\!\!\!\perp C | A$ \end{tabular}} & \multicolumn{1}{c}{\begin{minipage}{.2\textwidth}\includegraphics[scale=0.5]{A_B_Cpic.pdf}\end{minipage}} \\ [4ex]
\hline
\end{tabular}
\end{table}
\end{document}
|
\begin{document}
\begin{abstract}
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heAbstract
\end{abstract}
\maketitle
\else
\begin{document}
\journalname{Mathematical Programming}
\title{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heTitle\thanks{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heFunding.}}
\titlerunning{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heShortTitle}
\author{
Puya Latafat\and
Andreas Themelis\and
Panagiotis Patrinos
}
\authorrunning{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heShortAuthor}
\institute{
P. Latafat
\at
Tel.: +32 (0)16 374408\\
\email{[email protected]}
\and
A. Themelis
\at
Tel.: +32 (0)16 374573\\
\email{[email protected]}
\and
P. Patrinos
\at
Tel.: +32 (0)16 374445\\
\email{[email protected]}
\and
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heAddressKU.
}
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract}
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heAbstract
\keywords{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heKeywords}
\subclass{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}heSubjclass}
\end{abstract}
\fi
\section{Introduction}\label{sec:Introduction}
This paper addresses block-coordinate (BC) proximal gradient methods for problems of the form
\begin{equation}\label{eq:P}
\minimize_{\bm x=(x_1,\dots,x_N)\in\R^{\sum_in_i}}
\@ifstar\@@P\@Phi(\bm x)
{}\coloneqq{}
F(\bm x)
{}+{}
G(\bm x),
\quad\text{where}\quad
\textstyle
F(\bm x)\coloneqq\tfrac1N\sum_{i=1}^N f_i(x_i),
\end{equation}
in the following setting.
\begin{ass}[problem setting]\label{ass:basic}
In problem \eqref{eq:P} the following hold:
\begin{enumeratass}
\item\label{ass:f}
function \(f_i\) is \(L_{f_i}\)-smooth (Lipschitz differentiable with modulus \(L_{f_i}\)), \(i\in[N]\);
\item\label{ass:g}
function \(G\) is proper and lower semicontinuous (lsc);
\item\label{ass:phi}
a solution exists: \(\argmin\@ifstar\@@P\@Phi\neq\emptyset\).
\end{enumeratass}
\end{ass}
Unlike typical cases analyzed in the literature where \(G\) is separable \cite{tseng2001convergence,tseng2009coordinate,nesterov2012efficiency,beck2013convergence,bolte2014proximal,richtarik2014iteration,lin2015accelerated,chouzenoux2016block,hong2017iteration,xu2017globally}, we here consider the complementary case where it is only the smooth term \(F\) that is assumed to be separable. The main challenge in analyzing convergence of BC schemes for \eqref{eq:P} especially in the nonconvex setting is the fact that even in expectation the cost does not necessarily decrease along the trajectories.
Instead, we demonstrate that the forward-backward envelope (FBE) \cite{patrinos2013proximal,themelis2018forward} is a suitable Lyapunov function for such problems.
Several BC-type algorithms that allow for a nonseparable nonsmooth term have been considered in the literature, however, all in convex settings.
In \cite{tseng2008block,tseng2010coordinate} a class of convex composite problems is studied that involves a linear constraint as the nonsmooth nonseparable term.
A BC algorithm with a Gauss-Southwell-type rule is proposed and the convergence is established using the cost as Lyapunov function by exploiting linearity of the constraint to ensure feasibility. A refined analysis in \cite{necoara2013random,necoara2014random} extends this to a random coordinate selection strategy.
Another approach in the convex case is to consider randomized BC updates applied to general averaged operators. Although this approach can allow for fully nonseparable problems, usually separable nonsmooth functions are considered in the literature.
The convergence analysis of such methods relies on establishing quasi-Fej\'er monotonicity \cite{iutzeler2013asynchronous,combettes2015stochastic,pesquet2015class,bianchi2016coordinate,peng2016arock,latafat2019new}.
In a primal-dual setting in \cite{fercoq2019coordinate} a combination of Bregman and Euclidean distance is employed as Lyapunov function.
In \cite{hanzely2018sega} a BC algorithm is proposed for strongly convex algorithms that involves coordinate updates for the gradient followed by a full proximal step, and the distance from the (unique) solution is used as Lyapunov function.
The analysis and the Lyapunov functions in all of the above mentioned works rely heavily on convexity and are not suitable for nonconvex settings.
Thanks to the nonconvexity and nonseparability of \(G\), many machine learning problems can be formulated as in \eqref{eq:P}, a primary example being constrained and/or regularized finite sum problems \cite{bertsekas2011incremental,shalevshwartz2013stochastic,defazio2014finito,defazio2014saga,mairal2015incremental,reddi2016proximal,reddi2016stochastic,schmidt2017minimizing}
\begin{equation}\label{eq:FSP}
\textstyle
\minimize_{x\in\R^n}
\varphi(x)
{}\coloneqq{}
\tfrac1N\sum_{i=1}^N f_i(x)
{}+{}
g(x),
\end{equation}
where \(\func{f_i}{\R^n}{\R}\) are smooth functions and \(\func{g}{\R^n}{\Rinf}\) is possibly nonsmooth, and everything here can be nonconvex.
In fact, one way to cast \eqref{eq:FSP} into the form of problem \eqref{eq:P} is by setting
\begin{equation}\label{eq:FINITOG}
\textstyle
G(\bm x)
{}\coloneqq{}
\tfrac1N\sum_{i=1}^Ng(x_i)
{}+{}
\indicator_C(\bm x),
\end{equation}
where
\(
C
{}\coloneqq{}
\set{\bm x\in\R^{nN}}[x_1=x_2=\dots=x_N]
\)
is the consensus set, and \(\indicator_C\) is the indicator function of set \(C\), namely
\(
\indicator_C(\bm x)=0
\)
for \(\bm x\in C\) and \(\infty\) otherwise.
Since the nonsmooth term \(g\) is allowed to be nonconvex, formulation \eqref{eq:FSP} can account for nonconvex constraints such as rank constraints or zero norm balls, and nonconvex regularizers such as \(\ell^p\) with \(p\in[0,1)\), \cite{hou2012complexity}.
Another prominent example in distributed applications is the \emph{``sharing''} problem \cite{boyd2011distributed}:
\begin{equation}\label{eq:SP}
\minimize_{\bm x\in\R^{nN}}\@ifstar\@@P\@Phi(\bm x)
{}\coloneqq{}
\textstyle
\tfrac1N\sum_{i=1}^Nf_i(x_i)
{}+{}
g\Bigl(\sum_{i=1}^Nx_i\Bigr)
.
\end{equation}
where \(\func{f_i}{\R^n}{\R}\) are smooth functions and \(\func{g}{\R^n}{\Rinf}\) is nonsmooth, and all are possibly nonconvex. The sharing problem is cast as in \eqref{eq:P} by setting \(G\coloneqq g \circ A\), where \(A\coloneqq[\I_n~\dots~\I_n]\in\R^{n\times nN}\) (\(I_r\) denotes the \(r\times r\) identity matrix).
\subsection{The main block-coordinate algorithm}\label{sec:BC}
While gradient evaluations are the building blocks of smooth minimization, a fundamental tool to deal with a nonsmooth lsc term \(\func{\psi}{\R^r}{\Rinf}\) is its \DEF{\(V\)-proximal mapping}
\begin{equation}\label{eq:prox}
\prox_\psi^V(x)
{}\coloneqq{}
\argmin_{w\in\R^r}\set{
\psi(w)
{}+{}
\tfrac12\|w-x\|^2_V
},
\end{equation}
where \(V\) is a symmetric and positive definite matrix and \(\|{}\cdot{}\|_V\) indicates the norm induced by the scalar product \((x,y)\mapsto\innprod{x}{Vy}\).
It is common to take \(V=t^{-1}\I_r\) as a multiple of the \(r\times r\) identity matrix \(\I_r\), in which case the notation \(\prox_{t\psi}\) is typically used and \(t\) is referred to as a stepsize.
While this operator enjoys nice regularity properties when \(g\) is convex, such as (single valuedness and) Lipschitz continuity, for nonconvex \(g\) it may fail to be a well-defined function and rather has to be intended as a point-to-set mapping \(\ffunc{\prox_\psi^V}{\R^r}{\R^r}\).
Nevertheless, the value function associated to the minimization problem in the definition \eqref{eq:prox}, namely the \emph{Moreau envelope}
\begin{equation}\label{eq:Moreau}
\psi^V(x)
{}\coloneqq{}
\min_{w\in\R^r}\set{
\psi(w)
{}+{}
\tfrac12\|w-x\|^2_V
},
\end{equation}
is a well-defined real-valued function, in fact locally Lipschitz continuous, that lower bounds \(\psi\) and shares with \(\psi\) infima and minimizers.
The proximal mapping is available in closed form for many useful functions, many of which are widely used regularizers in machine learning; for instance, the proximal mapping of the \(\ell^0\) and \(\ell^1\) regularizers amount to hard and soft thresholding operators.
In many applications the cost to be minimized is structured as the sum of a smooth term \(h\) and a proximable (\ie with easily computable proximal mapping) term \(\psi\).
In these cases, the \emph{proximal gradient method} \cite{fukushima1981generalized,attouch2013convergence} constitutes a cornerstone iterative method that interleaves gradient descent steps on the smooth function and proximal operations on the nonsmooth function, resulting in iterations of the form
\(
x^+
{}\in{}
\prox_{\gamma\psi}(x-\gamma\nabla h(x))
\)
for some suitable stepsize $\gamma$.
Our proposed scheme to address problem \eqref{eq:P} is a BC variant of the proximal gradient method, in the sense that only some coordinates are updated according to the proximal gradient rule, while the others are left unchanged.
This concept is synopsized in \Cref{alg:BC}, which constitutes the general algorithm addressed in this paper.
\begin{algorithm}
\caption{General forward-backward block-coordinate scheme}
\label{alg:BC}
\begin{algorithmic}[1]
\Require
\(\bm x^0\in\R^{\sum_in_i}\),~
\(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\),
{\small \(i\in[N]\)}
\Statex
\(
\Gamma=\blockdiag(\gamma_1\I_{n_1},\dots,\gamma_N\I_{n_N})
\),~
\(k=0\)
\item[{\sc Repeat} until convergence]
\State
\(
\bm z^k
{}\in{}
\prox_G^{\Gamma^{-1}}\bigl(
\bm x^k-\Gamma\nabla F(\bm x^k)
\bigr)
\)
\State\label{state:BC:sampling}
select a set of indices \(I^{k+1}\subseteq[N]\)
\State
update~~
\(x_i^{k+1}= z_i^k\)
~for \(i\in I^{k+1}\)
~~and~~
\(x_i^{k+1}= x_i^k\)
~for \(i\notin I^{k+1}\),~
\(k\gets k+1\)
\item[{\sc Return}]
\(\bm z^k\)
\end{algorithmic}
\end{algorithm}
Although seemingly wasteful, in many cases one can efficiently compute individual blocks without the need of full operations.
In fact BC \Cref{alg:BC} bridges the gap between a BC framework and a class of incremental methods where a global computation typically involving the full gradient is carried out incrementally via performing computations only for a subset of coordinates.
Two such broad applications, problems \eqref{eq:FSP} and \eqref{eq:SP}, are discussed in the dedicated \Cref{sec:Finito,sec:Sharing}, where among other things we will show that \Cref{alg:BC} leads to the well known Finito/MISO algorithm \cite{defazio2014finito,mairal2015incremental}.
\subsection{Contribution}
\begin{enumerate}[
leftmargin=0pt,
labelwidth=7pt,
itemindent=\labelwidth+\labelsep,
label=\rlap{{\bf\arabic*)}}\hspace*{\labelwidth},
]
\item
To the best of our knowledge this is the first analysis of BC schemes with a nonseparable nonsmooth term and in the fully nonconvex setting.
While the original cost \(\@ifstar\@@P\@Phi\) cannot serve as a Lyapunov function, we show that the forward-backward envelope (FBE) \cite{patrinos2013proximal,themelis2018forward} decreases surely, not only in expectation (\Cref{thm:sure}).
\item
This allows for a quite general convergence analysis for different sampling criteria.
This paper in particular covers randomized strategies (\Cref{sec:random}) where at each iteration one or more coordinates are sampled with possibly time-varying probabilities, as well as essentially cyclic (and in particular cyclic and shuffled) strategies in case the nonsmooth term is convex (\Cref{sec:cyclic}).
\item
We exploit the Kurdyka-\L ojasiewicz (KL) property to show global (as opposed to subsequential) and linear convergence when the sampling is essentially cyclic and the nonsmooth function is convex, without imposing convexity requirements on the smooth functions (\Cref{thm:cyclic:global}).
\ifaccel
\item
When \(G\) is convex and \(F\) is twice continuously differentiable, the FBE is continuously differentiable.
If, additionally, \(F\) is (strongly) convex and quadratic, then the FBE is (strongly) convex and has Lipschitz-continuous gradient.
Owing to these favorable properties, we propose a new BC Nesterov-type acceleration algorithm for minimizing the sum of a block-separable convex quadratic plus a nonsmooth convex function, whose analysis directly follows from existing work on smooth BC minimization \cite{allen2016even}.
\fi
\item
As immediate byproducts of our analysis we obtain
{\bf (a)} an incremental algorithm for the sharing problem \cite{boyd2011distributed} that to the best of our knowledge is novel (\Cref{sec:Sharing}), and
{\bf (b)} the Finito/MISO algorithm \cite{defazio2014finito,mairal2015incremental} leading to a much simpler and more general analysis than available in the literature with new convergence results both for randomized sampling strategies in the fully nonconvex setting and for essentially cyclic samplings when the nonsmooth term is convex (\Cref{sec:Finito}).
\end{enumerate}
\subsection{Organization}
The rest of the paper is organized as follows.
The core of the paper lies in the convergence analysis of \Cref{alg:BC} detailed in \Cref{sec:convergence}: \Cref{sec:FBE} introduces the FBE, fundamental tool of our methodology and lists some of its properties whose proofs are detailed in the dedicated \Cref{sec:proofs:FBE}, followed by other ancillary results documented in \Cref{sec:auxiliary}.
The algorithmic analysis begins in \Cref{sec:sure} with a collection of facts that hold independently of the chosen sampling strategy, and later specializes to randomized and essentially cyclic samplings in the dedicated \Cref{sec:random,sec:cyclic}.
\Cref{sec:Finito,sec:Sharing} discuss two particular instances of the investigated algorithmic framework, namely (a generalization of) the Finito/MISO algorithm for finite sum minimization and an incremental scheme for the sharing problem, both for fully nonconvex and nonsmooth formulations.
Convergence results are immediately inferred from those of the more general BC \Cref{alg:BC}.
\Cref{sec:Conclusions} concludes the paper.
\section{Convergence analysis}\label{sec:convergence}
We begin by observing that \Cref{ass:basic} is enough to guarantee the well definedness of the forward-backward operator in \Cref{alg:BC}, which for notational convenience will be henceforth denoted as \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\).
Namely, \(\ffunc{\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}}{\R^{\sum_in_i}}{\R^{\sum_in_i}}\) is the point-to-set mapping
\begin{align*}
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)
{}\coloneqq{} &
\prox_G^{\Gamma^{-1}}\left(\Fw{\bm x}\right)
\\
\numberthis\label{eq:T}
{}={} &
\argmin_{\bm w\in\R^{\sum_in_i}}\set{
F(\bm x)+\innprod{\nabla F(\bm x)}{\bm w-\bm x}
{}+{}
G(\bm w)
{}+{}
\tfrac12\|\bm w-\bm x\|_{\Gamma^{-1}}^2
}.
\end{align*}
\begin{lem}\label{thm:osc}
Suppose that \Cref{ass:basic} holds, and let \(\Gamma\coloneqq\blockdiag(\gamma_1\I_{n_1},\dots,\gamma_N\I_{n_N})\) with \(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\), \(i\in[N]\).
Then \(\prox_G^{\Gamma^{-1}}\) and \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}\) are locally bounded, outer semicontinuous (osc), nonempty- and compact-valued mappings.
\begin{proof}
See \Cref{proof:thm:osc}.
\end{proof}
\end{lem}
\subsection{The forward-backward envelope}\label{sec:FBE}
The fundamental challenge in the analysis of \eqref{eq:P} is the fact that, without separability of \(G\), descent on the cost function cannot be established even in expectation.
Instead, we show that the \emph{forward-backward envelope} (FBE) \cite{patrinos2013proximal,themelis2018forward} can be used as Lyapunov function.
This subsection formally introduces the FBE, here generalized to account for a matrix-valued stepsize parameter \(\Gamma\), and lists some of its basic properties needed for the convergence analysis of \Cref{alg:BC}.
Although easy adaptations of the similar results in \cite{patrinos2013proximal,themelis2018forward,themelis2019acceleration}, for the sake of self-inclusiveness the proofs are detailed in the dedicated \Cref{sec:proofs:FBE}.
\begin{subequations}
\begin{defin}[forward-backward envelope]\label{def:FBE}
In problem \eqref{eq:P}, let \(f_i\) be differentiable functions, \(i\in[N]\), and for \(\gamma_1,\dots,\gamma_N>0\) let
\(
\Gamma=\blockdiag(\gamma_1\I_{n_1},\dots,\gamma_N\I_{n_N})
\).
The forward-backward envelope (FBE) associated to \eqref{eq:P} with stepsize \(\Gamma\) is the function
\(
\func{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}}{\R^{\sum_in_i}}{[-\infty,\infty)}
\)
defined as
\begin{equation}
\label{eq:FBE}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}\coloneqq{}
\inf_{\bm w\in\R^{\sum_in_i}}\set{
F(\bm x)+\innprod{\nabla F(\bm x)}{\bm w-\bm x}
{}+{}
G(\bm w)
{}+{}
\tfrac12\|\bm w-\bm x\|_{\Gamma^{-1}}^2
}.
\end{equation}
\end{defin}
\Cref{def:FBE} highlights an important symmetry between the Moreau envelope and the FBE: similarly to the relation between the Moreau envelope \eqref{eq:Moreau} and the proximal mapping \eqref{eq:prox}, the FBE \eqref{eq:FBE} is the value function associated with the proximal gradient mapping \eqref{eq:T}.
By replacing any minimizer \(\bm z\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\) in the right-hand side of \eqref{eq:FBE} one obtains yet another interesting interpretation of the FBE in terms of the \(\Gamma^{-1}\)-augmented Lagrangian associated to \eqref{eq:P}
\begin{align}
\nonumber
\LL(\bm x,\bm z,\bm y)
{}\coloneqq{} &
F(\bm x)+G(\bm z)+\innprod{\bm y}{\bm x-\bm z}
{}+{}
\tfrac12\|\bm x-\bm z\|_{\Gamma^{-1}}^2,
\shortintertext{namely,}
\label{eq:FBEz}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={} &
F(\bm x)+\innprod{\nabla F(\bm x)}{\bm z-\bm x}
{}+{}
G(\bm z)
{}+{}
\tfrac12\|\bm z-\bm x\|_{\Gamma^{-1}}^2
\\
{}={} &
\LL(\bm x,\bm z,-\nabla F(\bm x)).
\shortintertext{
Lastly, by rearranging the terms it can easily be seen that
}
\label{eq:FBEMoreau}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={} &
F(\bm x)
{}-{}
\tfrac12\|\nabla F(\bm x)\|_\Gamma^2
{}+{}
G^{\Gamma^{-1}}(\Fw{\bm x}),
\end{align}
hence in particular that the FBE inherits regularity properties of \(G^{\Gamma^{-1}}\) and \(\nabla F\), some of which are summarized in the next result.
\end{subequations}
\begin{lem}[FBE: fundamental inequalities]\label{thm:FBEineq}
Suppose that \Cref{ass:basic} is satisfied and let \(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\), \(i\in[N]\).
Then, the FBE \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is a (real-valued and) locally Lipschitz-continuous function.
Moreover, the following hold for any \(\bm x\in\R^{\sum_in_i}\):
\begin{enumerate}
\item\label{thm:leq}
\(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)\leq\@ifstar\@@P\@Phi(\bm x)\).
\item\label{thm:geq}
\(
\tfrac12\|\bm z-\bm x\|^2_{\Gamma^{-1}-\Lambda_F}
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)-\@ifstar\@@P\@Phi(\bm z)
{}\leq{}
\tfrac12\|\bm z-\bm x\|^2_{\Gamma^{-1}+\Lambda_F}
\)
for any \(\bm z\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\), where
\(
\Lambda_F
{}\coloneqq{}
\tfrac1N
\blockdiag\bigl(L_{f_1}\I_{n_1},\dots, L_{f_n}\I_{n_N}\bigr)
\).
\item\label{thm:strconcost}
If in addition each $f_i$ is $\mu_{f_i}$-strongly convex and $G$ is convex, then for every \(\bm x\in\R^{\sum_in_i}\)
\[
\tfrac12\|\bm z-\bm x^\star\|_{\mu_F}^2
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)-\min\@ifstar\@@P\@Phi
\]
where \(\bm x^\star\coloneqq\argmin\@ifstar\@@P\@Phi\),
\(
\mu_F
{}\coloneqq{}
\frac1N\blockdiag\bigl(\mu_{f_1}\I_{n_1},\dots,\mu_{f_N}\I_{n_N}\bigr)
\),
and \(\bm z=\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\).
\end{enumerate}
\begin{proof}
See \Cref{proof:thm:FBEineq}.
\end{proof}
\end{lem}
Another key property that the FBE shares with the Moreau envelope is that minimizing the extended-real valued function \(\@ifstar\@@P\@Phi\) is equivalent to minimizing the continuous function \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\).
Moreover, the former is level bounded iff so is the latter.
This fact will be particularly useful for the analysis of \Cref{alg:BC}, as it will be shown in \Cref{thm:sure} that the FBE (surely) decreases along its iterates.
As a consequence, despite the fact that the same does not hold for \(\@ifstar\@@P\@Phi\) (in fact, iterates may even be infeasible), coercivity of \(\@ifstar\@@P\@Phi\) is enough to guarantee boundedness of \(\seq{\bm x^k}\) and \(\seq{\bm z^k}\).
\begin{lem}[FBE: minimization equivalence]\label{thm:FBEmin}
Suppose that \Cref{ass:basic} is satisfied and that \(\gamma_i\in(0,\nicefrac{N}{L_i})\), \(i\in[N]\).
Then the following hold:
\begin{enumerate}
\item\label{thm:min}
\(\min\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}=\min\@ifstar\@@P\@Phi\);
\item\label{thm:argmin}
\(\argmin\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}=\argmin\@ifstar\@@P\@Phi\);
\item\label{thm:LB}
\(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is level bounded iff so is \(\@ifstar\@@P\@Phi\).
\end{enumerate}
\begin{proof}
See \Cref{proof:thm:FBEmin}.
\end{proof}
\end{lem}
We remark that the kinship of \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) and \(\@ifstar\@@P\@Phi\) extends also to local minimality; the interested reader is referred to \cite[Th. 3.6]{themelis2018proximal} for details.
\subsection{A sure descent lemma}\label{sec:sure}
We now proceed to the theoretical analysis of \Cref{alg:BC}.
Clearly, some assumptions on the index selection criterion are needed in order to establish reasonable convergence results, for little can be guaranteed if, for instance, one of the indices is never selected.
Nevertheless, for the sake of a general analysis it is instrumental to first investigate which properties hold independently of such criteria.
After listing some of these facts in \Cref{thm:sure}, in \Cref{sec:random,sec:cyclic} we will specialize the results to randomized and (essentially) cyclic sampling strategies.
\begin{lem}[sure descent]\label{thm:sure}
Suppose that \Cref{ass:basic} is satisfied.
Then, the following hold for the iterates generated by \Cref{alg:BC}:
\begin{enumerate}
\item\label{thm:Igeq}
\(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\sum_{i\in I^{k+1}}\tfrac{\xi_i}{2\gamma_i}\|z_i^k-x_i^k\|^2
\),
where \(\xi_i\coloneqq\frac{N-\gamma_iL_{f_i}}{N}\), \(i\in[N]\), are strictly positive;
\item\label{thm:decrease}
\(\seq{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)}\) monotonically decreases to a finite value \(\@ifstar\@@P\@Phi_\star\geq\min\@ifstar\@@P\@Phi\);
\item\label{thm:omega}
\(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is constant (and equals \(\@ifstar\@@P\@Phi_\star\) as above) on the set of accumulation points of \(\seq{\bm x^k}\);
\item\label{thm:xdiff}
the sequence \(\seq{\|\bm x^{k+1}-\bm x^k\|^2}\) has finite sum (and in particular vanishes);
\item\label{thm:bounded}
if \(\@ifstar\@@P\@Phi\) is coercive, then \(\seq{\bm x^k}\) and \(\seq{\bm z^k}\) are bounded.
\end{enumerate}
\begin{proof}
\begin{proofitemize}
\item\ref{thm:Igeq}~
To ease notation, let
\(
\Lambda_F
{}\coloneqq{} \tfrac1N
\blockdiag\bigl(L_{f_1}\I_{n_1},\dots, L_{f_n}\I_{n_N}\bigr)
\)
and for \(\bm w\in\R^{\sum_in_i}\) let \(w_I\in\R^{\sum_{i\in I}n_i}\) denote the slice \((w_i)_{i\in I}\), and let \(\Lambda_{F_I},\Gamma_I\in\R^{\sum_{i\in I}n_i\times\sum_{i\in I}n_i}\) be defined accordingly.
Start by observing that, since \(\bm z^{k+1}\in\prox_G^{\Gamma^{-1}}(\Fw{\bm x^{k+1}})\), from the proximal inequality on $G$ it follows that
\begin{align*}
G(\bm z^{k+1})-G(\bm z^k)
{}\leq{} &
\tfrac12\|\bm z^k-\bm x^{k+1}+\Gamma\nabla F(\bm x^{k+1})\|_{\Gamma^{-1}}^2
{}-{}
\tfrac12\|\bm z^{k+1}-\bm x^{k+1}+\Gamma\nabla F(\bm x^{k+1})\|_{\Gamma^{-1}}^2
\\
={} &
\numberthis\label{eq:proxIneqFBS}
\tfrac12\|\bm z^k-\bm x^{k+1}\|_{\Gamma^{-1}}^2
{}-{}
\tfrac12\|\bm z^{k+1}-\bm x^{k+1}\|_{\Gamma^{-1}}^2
{}+{}
\innprod{\nabla F(\bm x^{k+1})}{\bm z^k-\bm z^{k+1}}.
\end{align*}
We have
\ifarxiv\else
\bgroup\mathtight[0.5]
\fi
\begin{align*}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})-\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}={} &
{\color{red}
F(\bm x^{k+1})
}
{}+{}
\innprod{\nabla F(\bm x^{k+1})}{\bm z^{k+1}-\bm x^{k+1}}
{\color{blue}
{}+{}
G(\bm z^{k+1})
}
{}+{}
\tfrac12\|\bm z^{k+1}-\bm x^{k+1}\|_{\Gamma^{-1}}^2
\\
&
{}-{}
\left(
{\color{red}
F(\bm x^k)+\innprod{\nabla F(\bm x^k)}{\bm z^k-\bm x^k}
}
{\color{blue}
{}+{}
G(\bm z^k)
}
{}+{}
\tfrac12\|\bm z^k-\bm x^k\|_{\Gamma^{-1}}^2
\right)
\shortintertext{
{\color{red}
apply the upper bound in \eqref{eq:Lip} with \(\bm w=\bm x^{k+1}\)
}
and
{\color{blue}
the proximal inequality \eqref{eq:proxIneqFBS}
}
}
{}\leq{} &
{\color{red}
\innprod{\nabla F(\bm x^k)}{\bm x^{k+1}-\bm z^k}
{}+{}
\tfrac12\|\bm x^{k+1}-\bm x^k\|_{\Lambda_F}^2
}
{}+{}
\innprod{\nabla F(\bm x^{k+1})}{{\color{blue}\bm z^k}-\bm x^{k+1}}
\\
&
{}-{}
\tfrac12\|\bm z^k-\bm x^k\|_{\Gamma^{-1}}^2
{\color{blue}
{}+{}
\tfrac12\|\bm z^k-\bm x^{k+1}\|_{\Gamma^{-1}}^2
}.
\end{align*}
\ifarxiv\else
\egroup
\fi
To conclude, notice that the \(\ell\)-th block of \(\nabla F(\bm x^k)-\nabla F(\bm x^{k+1})\) is zero for \(\ell\notin I\), and that the \(\ell\)-th block of \(\bm x^{k+1}-\bm z^k\) is zero if \(\ell\in I\).
Hence, the scalar product vanishes.
For similar reasons, one has
\(
\|
\bm z^k-\bm x^{k+1}
\|^2_{\Gamma^{-1}}
{}-{}
\|\bm z^k-\bm x^k\|_{\Gamma^{-1}}^2
{}={}
{}-{}
\|z_I^k-x_I^k\|_{\Gamma_I^{-1}}^2
\)
and
\(
\|\bm x^{k+1}-\bm x^k\|_{\Lambda_F}^2
{}={}
\|z_I^k-x_I^k\|_{\Lambda_{F_I}}^2
\),
yielding the claimed expression.
\item\ref{thm:decrease}~
Monotonic decrease of \(\seq{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)}\) is a direct consequence of assert \ref{thm:Igeq}.
This ensures that the sequence converges to some value \(\@ifstar\@@P\@Phi_\star\), bounded below by \(\min\@ifstar\@@P\@Phi\) in light of \Cref{thm:min}.
\item\ref{thm:omega}~
Directly follows from assert \ref{thm:decrease} together with the continuity of \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\), see \Cref{thm:FBEineq}.
\item\ref{thm:xdiff}~
Denoting
\(
\xi_{\rm min}
{}\coloneqq{}
\min_{i\in[N]}\set{
\xi_i
}
\)
which is a strictly positive constant, it follows from assert \ref{thm:Igeq} that for each \(k\in\N\) it holds that
\begin{align*}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})-\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}\leq{} &
{}-{}
\sum_{\mathclap{i\in I^{k+1}}}{
\tfrac{\xi_i}{2\gamma_i}\|z_i^k-x_i^k\|^2
}
\\
{}\leq{} &
{}-{}
\tfrac{\xi_{\rm min}}{2}
\sum_{i\in I^{k+1}}{
\gamma_i^{-1}\|z_i^k-x_i^k\|^2
}
\\
\numberthis\label{eq:SDx}
{}={} &
{}-{}
\tfrac{\xi_{\rm min}}{2}
\|\bm x^{k+1}-\bm x^k\|_{\Gamma^{-1}}^2.
\end{align*}
By summing for \(k\in\N\) and using the positive definiteness of \(\Gamma^{-1}\) together with the fact that \(\min\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}=\min\@ifstar\@@P\@Phi>\infty\) as ensured by \Cref{thm:min} and \Cref{ass:phi}, we obtain that
\(
\sum_{k\in\N}\|\bm x^{k+1}-\bm x^k\|^2
{}<{}
\infty
\).
\item\ref{thm:bounded}~
It follows from assert \ref{thm:decrease} that the entire sequence \(\seq{\bm x^k}\) is contained in the sublevel set \(\set{\bm w}[\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm w)\leq\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^0)]\), which is bounded provided that \(\@ifstar\@@P\@Phi\) is coercive as shown in \Cref{thm:LB}.
In turn, boundedness of \(\seq{\bm z^k}\) then follows from local boundedness of \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}\), cf. \Cref{thm:osc}.
\qedhere
\end{proofitemize}
\end{proof}
\end{lem}
\subsection{Randomized sampling}\label{sec:random}
In this section we provide convergence results for \Cref{alg:BC} where the index selection criterion complies with the following requirement.
\begin{ass}[randomized sampling requirements]\label{ass:random}
There exist \(p_1,\dots,p_N>0\) such that, at any iteration and independently of the past, each $i\in[N]$ is sampled with probability at least $p_i$.
\end{ass}
Our notion of randomization is general enough to allow for time-varying probabilities and mini-batch selections.
The role of parameters \(p_i\) in \Cref{ass:random} is to prevent that an index is sampled with arbitrarily small probability.
In more rigorous terms,
\(
\@ifstar\@@P\@P*{i\in I^{k+1}}
{}\geq{}
p_i
\)
shall hold for all \(i\in[N]\), where \(\@ifstar\@@P\@P{}\) represents the probability conditional to the knowledge at iteration \(k\).
Notice that we do not require the \(p_i\)'s to sum up to one, as multiple index selections are allowed, similar to the setting of \cite{bianchi2016coordinate,latafat2019new} in the convex case.
Due to the possible nonconvexity of problem \eqref{eq:P}, unless additional assumptions are made not much can be said about convergence of the iterates to a unique point.
Nevertheless, the following result shows that any accumulation point \(\bm x^\star\) of sequences $\seq{\bm x^k}$ and $\seq{\bm z^k}$ generated by \Cref{alg:BC} is a stationary point, in the sense that it satisfies the necessary condition for minimality
\(
0\in\hat\partial\@ifstar\@@P\@Phi(\bm x^\star)
\),
where \(\hat\partial\) denotes the (regular) nonconvex subdifferential, see \cite[Th. 10.1]{rockafellar2011variational}.
\begin{thm}[randomized sampling: subsequential convergence]\label{thm:random:subseq}
Suppose that \Cref{ass:basic,ass:random} are satisfied.
Then, the following hold almost surely for the iterates generated by \Cref{alg:BC}:
\begin{enumerate}
\item\label{thm:res}
the sequence \(\seq{\|\bm x^k-\bm z^k\|^2}\) has finite sum (and in particular vanishes);
\item\label{thm:decreasez}
the sequence \(\seq{\@ifstar\@@P\@Phi(\bm z^k)}\) converges to \(\@ifstar\@@P\@Phi_\star\) as in \Cref{thm:decrease};
\item\label{thm:cluster}
\(\seq{\bm x^k}\) and \(\seq{\bm z^k}\) have same cluster points, all stationary and on which \(\@ifstar\@@P\@Phi\) and \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) equal \(\@ifstar\@@P\@Phi_\star\).
\end{enumerate}
\begin{proof}
In what follows, \(\@ifstar\@@E\@E{}\) denotes the expectation conditional to the knowledge at iteration \(k\).
\begin{proofitemize}
\item\ref{thm:res}~
Let
\(
\xi_i\coloneqq\frac{N-\gamma_iL_{f_i}}{N}>0
\),
\(i\in[N]\), be as in \Cref{thm:Igeq}.
We have
\begin{align*}
\@ifstar\@@E\@E{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})}
{}\overrel*[\leq]{\ref{thm:Igeq}}{} &
\@ifstar\@@E\@E{
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\sum_{i\in I^{k+1}}{
\tfrac{\xi_i}{2\gamma_i}\|z_i^k-x_i^k\|^2
}
}
\\
{}={} &
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\sum_{I\in\Omega}{
\@ifstar\@@P\@P{\mathcal I^{k+1}=I}
\sum_{i\in I}{
\tfrac{\xi_i}{2\gamma_i}\|z_i^k-x_i^k\|^2
}
}
\\
{}={} &
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\sum_{i=1}^N{
\sum_{I\in\Omega,I\ni i}{
\@ifstar\@@P\@P{\mathcal I^{k+1}=I}
\tfrac{\xi_i}{2\gamma_i}\|z_i^k-x_i^k\|^2
}
}
\\
\numberthis\label{eq:EFBE+}
{}\leq{} &
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\sum_{i=1}^N{
\tfrac{p_i\xi_i}{2\gamma_i}\|z_i^k-x_i^k\|^2
},
\end{align*}
where $\Omega\subseteq 2^{[N]}$ is the sample space ($2^{[N]}$ denotes the power set of $[N]$).
Therefore,
\begin{equation}\label{eq:ExSD}
\@ifstar\@@E\@E{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})}
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\tfrac\sigma2
\|\bm x^k-\bm z^k\|_{\Gamma^{-1}}^2
\quad\text{where }
\sigma
{}\coloneqq{}
\min_{i=1\dots N}{
p_i\xi_i
}
{}>{}
0.
\end{equation}
The claim follows from the Robbins-Siegmund supermartingale theorem, see \eg \cite{robbins1985convergence} or \cite[Prop. 2]{bertsekas2011incremental}.
\item\ref{thm:decreasez}~
Observe that
\(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\|\bm z^k-\bm x^k\|^2_{\Gamma^{-1}+\Lambda_F}
{}\leq{}
\@ifstar\@@P\@Phi(\bm z^k)
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\|\bm z^k-\bm x^k\|^2_{\Gamma^{-1}-\Lambda_F}
\)
holds (surely) for \(k\in\N\) in light of \Cref{thm:geq}.
The claim then follows by invoking \Cref{thm:decrease} and assert \ref{thm:res}.
\item\ref{thm:cluster}~
In the rest of the proof, for conciseness the ``almost sure'' nature of the results will be implied without mention.
It follows from assert \ref{thm:res} that a subsequence \(\seq{\bm x^k}[k\in K]\) converges to some point \(\bm x^\star\) iff so does the subsequence \(\seq{\bm z^k}[k\in K]\).
Since \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^k)\ni\bm z^k\) and both \(\bm x^k\) and \(\bm z^k\) converge to \(\bm x^\star\) as \(K\ni k\to\infty\), the inclusion \(0\in\hat\partial\@ifstar\@@P\@Phi(\bm x^\star)\) follows from \Cref{thm:critical}.
Since the full sequences \(\seq{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)}\) and \(\seq{\@ifstar\@@P\@Phi(\bm z^k)}\) converge to the same value \(\@ifstar\@@P\@Phi_\star\) (cf. \Cref{thm:decrease} and assert \ref{thm:decreasez}), due to continuity of \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) (\Cref{thm:FBEineq}) it holds that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^\star)=\@ifstar\@@P\@Phi_\star\), and in turn the bounds in \Cref{thm:geq} together with assert \ref{thm:res} ensure that \(\@ifstar\@@P\@Phi(\bm x^\star)=\@ifstar\@@P\@Phi_\star\) too.
\qedhere
\end{proofitemize}
\end{proof}
\end{thm}
When \(G\) is convex and \(F\) is strongly convex (that is, each of the functions \(f_i\) is strongly convex), the FBE decreases \(Q\)-linearly in expectation along the iterates generated by the randomized BC-\Cref{alg:BC}.
\begin{thm}[randomized sampling: linear convergence under strong convexity]\label{thm:random:linear}
Additionally to \Cref{ass:basic,ass:random}, suppose that \(G\) is convex and that each \(f_i\) is \(\mu_{f_i}\)-strongly convex.
Then, for all \(k\) the following hold for the iterates generated by \Cref{alg:BC}:
\begin{subequations}\label{subeq:random:linear}
\begin{align}
\label{eq:random:Qlinear}
\@ifstar\@@E\@E{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})-\min\@ifstar\@@P\@Phi}
{}\leq{} &
(1-c)
\bigl(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\min\@ifstar\@@P\@Phi\bigr)
\\
\@ifstar\@@E\@E[]{\@ifstar\@@P\@Phi(\bm z^k)-\min\@ifstar\@@P\@Phi}
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^0)-\min\@ifstar\@@P\@Phi\bigr)(1-c)^k
\\
\tfrac12\@ifstar\@@E\@E[]{\|\bm z^k-\bm x^\star\|^2_{\mu_F}}
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^0)-\min\@ifstar\@@P\@Phi\bigr)(1-c)^k
\end{align}
\end{subequations}
where \(\bm x^\star\coloneqq\argmin\@ifstar\@@P\@Phi\),
\(
\mu_F
{}\coloneqq{} \tfrac1N
\blockdiag\bigl(\mu_{f_1}\I_{n_1},\dots\mu_{f_n}\I_{n_N}\bigr)
\),
and denoting \(\xi_i=\frac{N-\gamma_iL_{f_i}}{N}\), \(i\in[N]\),
\begin{equation}\label{eq:cwc}
c
{}={}
\min_{i\in[N]}{
\set{\tfrac{\xi_ip_i}{\gamma_i}}
{}\bigg/{}
\max_{i\in[N]}\set{\tfrac{N-\gamma_i\mu_{f_i}}{\gamma_i^2\mu_{f_i}}}
}.
\end{equation}
Moreover, by setting the stepsizes \(\gamma_i\) and minimum sampling probabilities \(p_i\) as
\begin{equation}\label{eq:gammaLinear}
\gamma_i
{}={}
\tfrac{N}{\mu_{f_i}}
\left(1-\sqrt{1-1/\kappa_{i}}\right)
\quad\text{and}\quad
p_i
{}={}
\frac{
\left(\sqrt{\kappa_i}+\sqrt{\kappa_i-1}\right)^2
}{
\sum_{j=1}^N\left(\sqrt{\kappa_j}+\sqrt{\kappa_j-1}\right)^2
}
\end{equation}
with
\(
\kappa_i\coloneqq\frac{L_{f_i}}{\mu_{f_i}}
\),
\(i\in[N]\), then the constant \(c\) in \eqref{subeq:random:linear} can be tightened to
\begin{equation}\label{eq:cbc}
c
{}={}
\tfrac{1}{
\sum_{i=1}^N\left(
\sqrt{\kappa_i}+\sqrt{\kappa_i-1}
\right)^2
}.
\end{equation}
\begin{proof}
Since \(\bm z^k\) is a minimizer in \eqref{eq:FBE}, the necessary stationarity condition reads
\(
\Gamma^{-1}(\bm x^k-\bm z^k)-\nabla F(\bm x^k)
{}\in{}
\partial G(\bm z^k)
\).
Convexity of \(G\) then implies
\[
G(\bm x^\star)
{}\geq{}
G(\bm z^k)
{}+{}
\innprod{\Gamma^{-1}(\bm x^k-\bm z^k)-\nabla F(\bm x^k)}{\bm x^\star-\bm z^k},
\]
whereas from strong convexity of \(F\) we have
\[
F(\bm x^\star)
{}\geq{}
F(\bm x^k)
{}+{}
\innprod{\nabla F(\bm x^k)}{\bm x^\star-\bm x^k}
{}+{}
\tfrac12\|\bm x^k-\bm x^\star\|^2_{\mu_F}.
\]
By combining these inequalities into \eqref{eq:FBEz}, and denoting \(\@ifstar\@@P\@Phi_\star\coloneqq\min\@ifstar\@@P\@Phi=\min\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) (cf. \Cref{thm:min}), we have
\begin{align*}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star
{}\leq{} &
\tfrac12\|\bm z^k-\bm x^k\|^2_{\Gamma^{-1}}
{}-{}
\tfrac12\|\bm x^\star-\bm x^k\|^2_{\mu_F}
{}+{}
\innprod{\Gamma^{-1}(\bm z^k-\bm x^k)}{\bm x^\star-\bm z^k}
\\
{}={} &
\tfrac12\|\bm z^k-\bm x^k\|_{\Gamma^{-1}-\mu_F}^2
{}+{}
\innprod{(\Gamma^{-1}-\mu_F)(\bm z^k-\bm x^k)}{\bm x^\star-\bm z^k}
{}-{}
\tfrac12\|\bm x^\star-\bm z^k\|_{\mu_F}^2.
\end{align*}
Next, by using the inequality
\(
\innprod{\bm a}{\bm b}
{}\leq{}
\tfrac12\|\bm a\|_{\mu_F}^2
{}+{}
\tfrac12\|\bm b\|^2_{\mu_F^{-1}}
\)
to cancel out the last term, we obtain
\begin{align*}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star
{}\leq{} &
\tfrac12\|\bm z^k-\bm x^k\|_{\Gamma^{-1}-\mu_F}^2
{}+{}
\tfrac12\|(\Gamma^{-1}-\mu_F)(\bm x^k-\bm z^k)\|_{\mu_F^{-1}}^2
\\
{}={} &
\tfrac12\|\bm z^k-\bm x^k\|_{\Gamma^{-2}\mu_F^{-1}(\I-\Gamma\mu_F)}^2,
\numberthis\label{eq:QUB}
\end{align*}
where the last identity uses the fact that the matrices are diagonal.
Combined with \eqref{eq:EFBE+} the claimed \(Q\)-linear convergence \eqref{eq:random:Qlinear} with factor \(c\) as in \eqref{eq:cwc} is obtained.
The $R$-linear rates in terms of the cost function and distance from the solution are obtained by repeated application of \eqref{eq:random:Qlinear} after taking (unconditional) expectation from both sides and using \Cref{thm:FBEineq}.
To obtain the tighter estimate \eqref{eq:cbc}, observe that \eqref{eq:EFBE+} with the choice
\[
\textstyle
p_i
{}\coloneqq{}
\tfrac{1}{\gamma_i\mu_{f_i}}
\tfrac{N-\gamma_i\mu_{f_i}}{N-\gamma_iL_{f_i}}
\left(
\sum_j{
\tfrac{1}{\gamma_j\mu_{f_j}}
\tfrac{N-\gamma_j\mu_{f_j}}{N-\gamma_jL_{f_j}}
}
\right)^{-1},
\]
which equals the one in \eqref{eq:gammaLinear} with $\gamma_i$ as prescribed, yields
\begin{align*}
\@ifstar\@@E\@E{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{k+1})-\@ifstar\@@P\@Phi_\star}
{}\leq{} &
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star
{}-{}
\left(
\textstyle
2N\sum_j{
\tfrac{1}{\gamma_j\mu_j}
\tfrac{N-\gamma_j\mu_j}{N-\gamma_jL_j}
}
\right)^{-1}
\sum_{i=1}^N{
\tfrac{N-\gamma_i\mu_{f_i}}{\gamma_i^2\mu_{f_i}}
\|z_i^k-x_i^k\|^2
}
\\
{}={} &
\textstyle
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star
{}-{}
\left(
2N\sum_j{
\tfrac{1}{\gamma_j\mu_j}
\tfrac{N-\gamma_j\mu_j}{N-\gamma_jL_j}
}
\right)^{-1}
\|\bm z^k-\bm x^k\|_{\Gamma^{-1}\mu_F^{-1}(\Gamma^{-1}-\mu_F)}^2.
\end{align*}
The assert now follows by combining this with \eqref{eq:QUB} and replacing the values of \(\gamma_i\) as proposed in \eqref{eq:gammaLinear}.
\end{proof}
\end{thm}
Notice that as \(\kappa_i\)'s approach \(1\) the linear rate tends to \(1-\nicefrac1N\).
\subsection{Cyclic, shuffled and essentially cyclic samplings}\label{sec:cyclic}
In this section we analyze the convergence of the BC-\Cref{alg:BC} when a cyclic, shuffled cyclic or (more generally) an essentially cyclic sampling \cite{tseng1987relaxation,tseng2001convergence,hong2017iteration,chow2017cyclic,xu2017globally} is used.
As formalized in the following standing assumption, an additional convexity requirement for the nonsmooth term \(G\) is needed.
\begin{ass}[essentially cyclic sampling requirements]\label{ass:cyclic}
In problem \eqref{eq:P}, function \(G\) is convex.
Moreover, there exists $T\geq 1$ such that in \Cref{alg:BC} each index is selected at least once within any interval of $T$ iterations.
\end{ass}
Note that having \(T<N\) is possible because of our general sampling strategy where sets of indices can be sampled within the same iteration.
For instance, \(T=1\) corresponds to \(I^{k+1}=[N]\) for all \(k\), in which case \Cref{alg:BC} would reduce to a (full) proximal gradient scheme.
Two notable special cases of single index selection rules are the cyclic and shuffled cyclic sampling strategies.
\begin{itemize}[
leftmargin=*,
label={},
itemindent=0cm,
labelsep=0pt,
partopsep=0pt,
parsep=0pt,
listparindent=0pt,
topsep=0pt,
]
\item{\sc Shuffled cyclic sampling:} corresponds to setting
\begin{equation}\label{eq:ShufCyclicRule}
I^{k+1}=\set{\pi_{\lfloor\nicefrac kN\rfloor}\bigl(\mod(k,N)+1\bigr)}\quad \text{for all}\quad k\in\N,
\end{equation}
where $\pi_0,\pi_1,\dots$ are permutations of the set of indices $[N]$ (chosen randomly or deterministically).
\item{\sc Cyclic sampling:} corresponds to the case \eqref{eq:ShufCyclicRule} with $\pi_{\lfloor\nicefrac kN\rfloor}=\id$, \ie,
\begin{equation}\label{eq:cyclicRule}
I^{k+1}=\set{\mod(k,N)+1}\quad \text{for all}\quad k\in\N.
\end{equation}
\end{itemize}
We remark that in practice it has been observed that an effective sampling technique is to use random shuffling after each cycle \cite[\S2]{bertsekas2015convex}.
Consistently with the deterministic nature of the essentially cyclic sampling, all results of the previous section hold surely, as opposed to almost surely.
\begin{thm}[essentially cyclic sampling: subsequential convergence]\label{thm:cyclic:subseq}
Suppose that \Cref{ass:basic,ass:cyclic} are satisfied.
Then, all the asserts of \Cref{thm:random:subseq} hold surely.
\begin{proof}
We first establish an important descent inequality for \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) after every $T$ iterations, cf. \eqref{eq:Essential_cyclic_descent}.
Convexity of \(G\), entailing $\prox_{G}^{\Gamma^{-1}}$ being Lipschitz continuous (cf. \Cref{thm:FNE}), allows the employment of techniques similar to those in \cite[Lemma 3.3]{beck2013convergence}.
Since all indices are updated at least once every \(T\) iterations, one has that
\begin{equation}\label{eq:ki}
\ki
{}\coloneqq{}
\min\set{t\in[T]}[
\text{\(i\) is sampled at iteration \(T\nu+t-1\)}
]
\end{equation}
is well defined for each index \(i\in[N]\) and \(\nu\in\N\).
Since \(i\) is sampled at iteration \(T\nu+\ki-1\) and \(x_i^{T\nu}=x_i^{T\nu+1}=\dots=x_i^{T\nu+\ki-1}\) by definition of \(\ki\), it holds that
\begin{align*}
x_i^{T\nu+\ki}
{}={} &
x_i^{T\nu+\ki-1}
{}+{}
\trans{U_i}\,\left(
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\ki-1})
{}-{}
\bm x^{T\nu+\ki-1}
\right)
\\
\numberthis\label{eq:equiiter_ki}
{}={} &
x_i^{T\nu}
{}+{}
\trans{U_i}\,\left(
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\ki-1})
{}-{}
\bm x^{T\nu+\ki-1}
\right),
\end{align*}
\ifaccel\else
where $U_i\in \R^{(\sum_jn_j)\times n_i}$ denotes the $i$-th block column of the identity matrix so that for a vector $v\in \R^{n_i}$
\begin{equation}\label{eq:U}
U_iv
{}={}
\trans{(0,\dots, 0, \!\overbracket{\,v\,}^{\mathclap{i\text{-th}}}\!, 0, \dots, 0)}.
\end{equation}
\fi
For all $t\in[T]$ the following holds
\begin{align*}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T(\nu+1)})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})
{}={} &
\sum_{\tau=1}^T\left(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\tau})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\tau-1})
\right)
\\
{}\leq{} &
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+t})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+t-1})
\\
{}\leq{} &
-\tfrac{\xi_{\rm min}}{2}
\|\bm x^{T\nu+t}-\bm x^{T\nu+t-1}\|_{\Gamma^{-1}}^2,
\numberthis\label{eq:descent_esscyc}
\end{align*}
where $\xi_i\coloneqq \tfrac{N-\gamma_{i}L_{f_{i}}}{N}$ as in \Cref{thm:Igeq}, $\xi_{\rm min}\coloneqq \min_{i\in[N]}\set{\xi_i}$, and the two inequalities follow from \Cref{thm:Igeq}.
Moreover, using triangular inequality for $i\in[N]$ yields
\begin{align*}
\|\bm x^{T\nu+\ki-1}-\bm x^{T\nu}\|_{\Gamma^{-1}}
{}\leq{} &
\sum_{\tau=1}^{\ki-1}\|\bm x^{T\nu+\tau}-\bm x^{T\nu+\tau-1}\|_{\Gamma^{-1}}
\\
\numberthis\label{eq:new25}
{}\leq{} &
\tfrac{T}{\sqrt{\xi_{\rm min}\nicefrac{}{2}}}
\left(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T(\nu+1)})
\right)^{\nicefrac12},
\end{align*}
where the second inequality follows from \eqref{eq:descent_esscyc} together with the fact that \(\ki\leq T\).
For all \(i\in[N]\), from the triangular inequality and the \(L_{\bf T}\)-Lipschitz continuity of \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}\) (\Cref{thm:TLip}) we have
\begin{align*}
\gamma_i^{-\nicefrac12}
\|\trans{U_i}\,(\bm x^{T\nu}-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu}))\|
{}\leq{} &
\gamma_i^{-\nicefrac12}
\|\trans{U_i}\,\bigl(\bm x^{T\nu}-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\ki-1})\bigr)\|
\\
&
{}+{}
\gamma_i^{-\nicefrac12}
\|\trans{U_i}\,\bigl(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\ki-1})-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})\bigr)\|
\\
{}\leq{} &
\gamma_i^{-\nicefrac12}
\|x_i^{T\nu+\ki-1}-x_i^{T\nu+\ki}\|
\\
&
{}+{}
\|\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu+\ki-1})-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})\|_{\Gamma^{-1}}
\\
{}\leq{} &
\|\bm x^{T\nu+\ki-1}-\bm x^{T\nu+\ki}\|_{\Gamma^{-1}}
{}+{}
L_{\bf T}\|\bm x^{T\nu+\ki-1}-\bm x^{T\nu}\|_{\Gamma^{-1}}
\\
\numberthis\label{eq:sqrtbound}
{}\overrel[\leq]{\eqref{eq:descent_esscyc},~\eqref{eq:new25}}{} &
\tfrac{1+TL_{\bf T}}{\sqrt{\xi_{\rm min}/2}}
\left(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T(\nu+1)})
\right)^{\nicefrac12}.
\end{align*}
By squaring and summing over \(i\in[N]\), we obtain
\begin{equation}\label{eq:Essential_cyclic_descent}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T(\nu+1)})-\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})
{}\leq{}
-\tfrac{\xi_{\rm min}}{2N(1+TL_{\bf T})^2}
\|\bm z^{T\nu}-\bm x^{T\nu}\|^{2}_{\Gamma^{-1}}.
\end{equation}
By telescoping the inequality and using the fact that \(\min\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}=\min\@ifstar\@@P\@Phi\)
\ifarxiv
by
\else
shown in
\fi
\Cref{thm:min}, we obtain that
\(
\seq{\|\bm z^{T\nu}-\bm x^{T\nu}\|^2_{\Gamma^{-1}}}[\nu\in\N]
\)
has finite sum, and in particular vanishes.
Clearly, by suitably shifting, for every \(t\in[T]\) the same can be said for the sequence
\(
\seq{\|\bm z^{T\nu+t}-\bm x^{T\nu+t}\|^2_{\Gamma^{-1}}}[\nu\in\N]
\).
The whole sequence
\(
\seq{\|\bm z^k-\bm x^k\|^2}
\)
is thus summable, and we may now infer the claim as done in the proof of \Cref{thm:random:subseq}.
\end{proof}
\end{thm}
In the next theorem explicit linear convergence rates are derived under the additional strong convexity assumption for the smooth functions.
The cyclic and shuffled cyclic cases are treated separately, as tighter bounds can be obtained by leveraging the fact that within cycles of \(N\) iterations every index is updated exactly once.
\begin{thm}[essentially cyclic sampling: linear convergence under strong convexity]\label{thm:cyclic:linear}
Additionally to \Cref{ass:basic,ass:cyclic}, suppose that each function \(f_i\) is \(\mu_{f_i}\)-strongly convex.
Then, denoting
\(
\delta
{}\coloneqq{}
\min_{i\in[N]}\set{
\tfrac{\gamma_i\mu_{f_i}}{N}
}
\)
and
\(
\Delta
{}\coloneqq{}
\max_{i\in[N]}\set{
\tfrac{\gamma_iL_{f_i}}{N}
}
\),
for all \(\nu\in\N\) the following hold for the iterates generated by \Cref{alg:BC}:
\begin{subequations}\label{subeq:cyclic:linear}
\begin{align}
\label{eq:cyclic:Qlinear}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T(\nu+1)})-\min\@ifstar\@@P\@Phi
{}\leq{} &
(1-c)
\bigl(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})-\min\@ifstar\@@P\@Phi\bigr)
\\ \label{eq:cyclic:Rlinear1}
\@ifstar\@@P\@Phi(\bm z^{T\nu})-\min\@ifstar\@@P\@Phi
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^0)-\min\@ifstar\@@P\@Phi\bigr)(1-c)^\nu
\\ \label{eq:cyclic:Rlinear2}
\tfrac12\|\bm z^{T\nu}-\bm x^\star\|^2_{\mu_F}
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^0)-\min\@ifstar\@@P\@Phi\bigr)(1-c)^\nu
\end{align}
\end{subequations}
where \(\bm x^\star\coloneqq\argmin\@ifstar\@@P\@Phi\),
\(
\mu_F
{}\coloneqq{} \tfrac1N
\blockdiag\bigl(\mu_{f_1}\I_{n_1},\dots\mu_{f_n}\I_{n_N}\bigr)
\),
and
\begin{equation}\label{eq:cyclic:cwc}
c
{}={}
\frac{
\delta(1-\Delta)
}{
N\bigl(1+T(1-\delta)\bigr)^2
(1-\delta)
}.
\end{equation}
In the case of shuffled cyclic \eqref{eq:ShufCyclicRule} or cyclic \eqref{eq:cyclicRule} sampling, the inequalities can be tightened by
replacing \(T\) with \(N\) and with
\begin{equation}\label{eq:linearShuffledCyclic}
c
{}={}
\frac{\delta(1-\Delta)}{N\left(2-\delta\right)^{2}\left(1-\delta\right)}.
\end{equation}
\begin{comment}
the following tighter bound holds
\begin{equation}\label{eq:linearShuffledCyclic}
{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N(\nu+1)})-\min\@ifstar\@@P\@Phi}
{}\leq{}
\left(1-c\right)
\left(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N\nu})-\min\@ifstar\@@P\@Phi\right),
\quad \text{where} \quad
c
{}={}
\frac{\delta(1-\Delta)}{N\left(2-\delta\right)^{2}\left(1-\delta\right)}.
\end{equation}
\end{comment}
\begin{proof}
\begin{proofitemize}
\item\emph{The general essentially cyclic case.}~
Since \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}\) is \(L_{\bf T}\)-Lipschitz continuous with \(L_{\bf T}=1-\delta\) as shown in \Cref{thm:contractive}, inequality \eqref{eq:Essential_cyclic_descent} becomes
\[
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T(\nu+1)})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})
{}\leq{}
-\tfrac{1-\Delta}{2N(1+T(1-\delta))^2}
\|\bm z^{T\nu}-\bm x^{T\nu}\|^2_{\Gamma^{-1}}.
\]
Moreover, it follows from \eqref{eq:QUB} that
\begin{equation}\label{eq:strongLB}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{T\nu})-\@ifstar\@@P\@Phi_\star
{}\leq{}
\tfrac12
(\delta^{-1}-1)
\|\bm z^{T\nu}-\bm x^{T\nu}\|_{\Gamma^{-1}}^2.
\end{equation}
By combining the two inequalities the claimed \(Q\)-linear convergence \eqref{eq:cyclic:Qlinear} with factor \(c\) as in \eqref{eq:cyclic:cwc} is obtained.
In turn, the $R$-linear rates \eqref{eq:cyclic:Rlinear1} and \eqref{eq:cyclic:Rlinear2} follow from \Cref{thm:FBEineq}.
\item\emph{The shuffled cyclic case.}~
Let us now suppose that the sampling strategy follows a shuffled rule as in \eqref{eq:ShufCyclicRule} with permutations \(\pi_0,\pi_1,\dots\) (hence in the cyclic case $\pi_\nu=\id$ for all $\nu\in\N$).
Let $U_i$ be as in \eqref{eq:U} and $\xi_{\rm min}$ as in the proof of \Cref{thm:cyclic:subseq}. Observe that \(\ki=\pi_\nu^{-1}(i)\leq N\) for \(\ki\) as defined in \eqref{eq:ki}.
For all $t\in[N]$
\begin{align*}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N(\nu+1)}) -\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N\nu})
{}\leq{} &
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N\nu+t-1}) -\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N\nu})
\\
{}\leq{} &
-\tfrac{\xi_{\rm min}}{2}\sum_{\tau=1}^{t-1} \|\bm x^{N\nu+\tau}-\bm x^{N\nu+\tau-1}\|^2_{\Gamma^{-1}}
\\
\numberthis\label{eq:tighterNoTri}
{}={} &
-\tfrac{\xi_{\rm min}}{2}\|\bm x^{N\nu+t-1}-\bm x^{N\nu}\|^2_{\Gamma^{-1}},
\end{align*}
where the equality follows from the fact that at every iteration a different coordinate is updated (and that $\Gamma$ is diagonal), and the inequalities from \Cref{thm:Igeq}. Similarly, \eqref{eq:descent_esscyc} holds with $T$ replaced by \(N\) (despite the fact that \(T\) is not necessarily \(N\), but is rather bounded as \(T\leq 2N-1\)).
By using \eqref{eq:tighterNoTri} in place of \eqref{eq:new25}, inequality \eqref{eq:sqrtbound} is tightened as follows
\[
\gamma_i^{-\nicefrac12}
\|\trans{U_i}(\bm x^{N\nu}-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^{N\nu}))\|
{}\leq{}
\tfrac{1+L_{\bf T}}{\sqrt{\xi_{\rm min}/2}}
\left(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N\nu})
{}-{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N(\nu+1)})
\right)^{\nicefrac12}.
\]
By squaring and summing for \(i\in[N]\) we obtain
\begin{equation}\label{eq:cyclic_descent}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N(\nu+1)})-\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{N\nu})
{}\leq{}
-\tfrac{\xi_{\rm min}}{2N(1+L_{\bf T})^2}
\|\bm z^{N\nu}-\bm x^{N\nu}\|^2_{\Gamma^{-1}}
{}={}
-\tfrac{1-\Delta}{2N(1+L_{\bf T})^2}
\|\bm z^{N\nu}-\bm x^{N\nu}\|^2_{\Gamma^{-1}},
\end{equation}
where \(L_{\bf T}=1-\delta\) as discussed above.
By combining this and \eqref{eq:strongLB} (with \(T\) replaced by \(N\)) the improved coefficient \eqref{eq:linearShuffledCyclic} is obtained.
\qedhere
\end{proofitemize}
\end{proof}
\end{thm}
Note that if one sets $\gamma_i = \alpha N/L_{f_i}$ for some $\alpha\in(0,1)$, then $\delta = \alpha\min_{i\in[N]} \set{\nicefrac{\mu_{f_i}}{L_{f_i}}}$ and $\Delta=\alpha$.
With this selection, as the condition number approaches $1$ the rate in \eqref{eq:linearShuffledCyclic} tends to $1-\frac{\alpha}{N\left(2-\alpha\right)^{2}}$.
\begin{comment}
{\color{red}Similarly to the argument in the randomized case, the $R$-linear rate
\[
\|\bm x^{N\nu}-\bm x^\star\|^2_M
{}\leq{}
4(1-c)^k
\left(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^0)-\min\@ifstar\@@P\@Phi\right)
\]
for the (shuffled) cyclic case with $M$ as in \Cref{thm:strconcost} is obtained.}
\end{comment}
\subsection{Global and linear convergence with KL inequality}
The convergence analyses of the randomized and essentially cyclic cases both rely on a descent property on the FBE that quantifies the progress in the minization of \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) in terms of the squared forward-backward residual \(\|\bm x-\bm z\|^2\).
A subtle but important difference, however, is that the inequality \eqref{eq:ExSD} in the former case involves a conditional expectation, whereas \eqref{eq:Essential_cyclic_descent} in the latter does not.
The \emph{sure} descent property occurring for essentially cyclic sampling strategies is the key for establishing global (as opposed to subsequential) convergence based on the Kurdyka-\L ojasiewicz (KL) property \cite{lojasiewicz1963propriete,lojasiewicz1993geometrie,kurdyka1998gradients}.
A similar result is achieved in \cite{xu2017globally}, which however considers the complementary case to problem \eqref{eq:P} where the nonsmooth function \(G\) is assumed to be separable, and thus the cost function itself can serve as Lyapunov function.
\begin{defin}[KL property with exponent \(\theta\)]\label{def:KL}
A proper lsc function \(\func{h}{\R^n}{\Rinf}\) is said to have the \DEF{Kurdyka-{\L}ojasiewicz} (KL) property with exponent \(\theta\in(0,1)\) at \(\bar w\in\dom h\) if there exist \(\varepsilon,\eta,\varrho>0\) such that
\[
\psi'(h(w)-h(\bar w))\dist(0,\partial h(w))\geq 1
\]
holds for all \(w\) such that \(\|w-\bar w\|<\varepsilon\) and \(h(\bar w)<h(w)<h(\bar w)+\eta\), where \(\psi(s)\coloneqq\varrho s^{1-\theta}\).
We say that \(h\) satisfies the KL property with exponent \(\theta\) (without mention of \(\bar w\)) if it satisfies the KL property with exponent \(\theta\) at any \(\bar w\in\dom\partial h\).
\end{defin}
Semialgebraic functions comprise a wide class of functions that enjoy this property \cite{bolte2007clarke,bolte2007lojasiewicz}, which has been extensively exploited to provide convergence rates of optimization algorithms \cite{attouch2009convergence,attouch2010proximal,attouch2013convergence,bolte2014proximal,frankel2015splitting,ochs2014ipiano,li2016douglas,xu2013block}.
Based on this, in the next result we provide sufficient conditions ensuring global and \(R\)-linear convergence of \Cref{alg:BC} with essentially cyclic sampling.
\begin{thm}[essentially cyclic sampling: global and linear convergence]\label{thm:cyclic:global}
Additionally to \Cref{ass:basic,ass:cyclic}, suppose that \(\@ifstar\@@P\@Phi\) has the KL property with exponent \(\theta\in(0,1)\) (as is the case when \(f_i\) and \(G\) are semialgebraic), and is coercive.
Then, any sequences \(\seq{\bm x^k}\) and \(\seq{\bm z^k}\) generated by \Cref{alg:BC} converge to (the same) stationary point \(\bm x^\star\).
Moreover, if \(\theta\leq\nicefrac12\) then \(\seq{\|\bm z^k-\bm x^k\|}\), \(\seq{\bm x^k}\) and \(\seq{\bm z^k}\) converge at $R$-linear rate.
\begin{proof}
Let \(\seq{\bm x^k}\) and \(\seq{\bm z^k}\) be sequences generated by \Cref{alg:BC} with essentially cyclic sampling, and let \(\@ifstar\@@P\@Phi_\star\) be the limit of the sequence \(\seq{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)}\) as in \Cref{thm:decrease}.
To avoid trivialities, we may assume that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)\gneqq\@ifstar\@@P\@Phi_\star\) for all \(k\), for otherwise the sequence \(\seq{\bm x^k}\) is asymptotically constant, and thus so is $\seq{\bm z^k}$.
Let \(\Omega\) be the set of accumulation points of \(\seq{\bm x^k}\), which is compact and such that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\equiv\@ifstar\@@P\@Phi_\star\) on \(\Omega\), as ensured by \Cref{thm:cyclic:subseq}.
It follows from \Cref{thm:loja} and \cite[Lem. 1(ii)]{attouch2009convergence} that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) enjoys a \emph{uniform} KL property on \(\Omega\); in particular,
\(
\psi'(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star)\dist(0,\partial\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k))
{}\geq{}
1
\)
holds for all \(k\) large enough such that \(\bm x^k\) is sufficiently close to \(\Omega\) and \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)\) is sufficiently close to \(\@ifstar\@@P\@Phi_\star\), where \(\psi(s)=\varrho s^{1-\theta'}\) for some \(\varrho>0\) and \(\theta'=\max\set{\theta,\nicefrac12}\).
Combined with \Cref{thm:subdiffdist}, for all \(k\) large enough we thus have
\begin{equation}\label{eq:KL}
\psi'(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star)
{}\geq{}
\frac{c}{\|\bm x^k-\bm z^k\|_{\Gamma^{-1}}},
\end{equation}
where
\(
c
{}\coloneqq{}
\frac{
N\min_i\set{\sqrt{\gamma_i}}
}{
N+\max_i\set{\gamma_iL_{f_i}}
}
{}>{}
0
\).
Let
\(
\Delta_k\coloneqq\psi(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)-\@ifstar\@@P\@Phi_\star)
\).
By combining \eqref{eq:KL} and \eqref{eq:Essential_cyclic_descent} we have that there exists a constant \(c'>0\) such that
\begin{equation}\label{eq:KLinequality}
\Delta_{(\nu+1)T}
{}-{}
\Delta_{\nu T}
{}\leq{}
\psi'(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{\nu T})-\@ifstar\@@P\@Phi_\star)
\left(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{(\nu+1)T})-\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{\nu T})\right)
{}\leq{}
-c'
\|\bm x^{\nu T}-\bm z^{\nu T}\|_{\Gamma^{-1}}
\end{equation}
holds for all $\nu\in\N$ large enough (the first inequality uses concavity of \(\psi\)).
By summing over \(\nu\) (sure) summability of the sequence \(\seq{\|\bm x^{\nu T}-\bm z^{\nu T}\|}[\nu\in\N]\) is obtained.
By suitably shifting, for every \(t\in[T]\) the same can be said for the sequence
\(
\seq{\|\bm z^{T\nu+t}-\bm x^{T\nu+t}\|}[\nu\in\N]
\),
and since \(T\) is finite we conclude that the whole sequence
\(
\seq{\|\bm z^k-\bm x^k\|}
\)
is summable.
Since \(\|\bm x^{k+1}-\bm x^k\|\leq\|\bm z^k-\bm x^k\|\) we conclude that \(\seq{\bm x^k}\) has finite length and is thus convergent (to a single point), and consequently so is \(\seq{\bm z^k}\).
\begin{comment}
Consider the \(\Gamma^{-1}\)-augmented Lagrangian defined in \eqref{eq:AugLagrangian} and let
\(
\mathcal{L}_k
{}\coloneqq{}
\LL(\bm x^k,\bm z^k,-\nabla F(\bm x^k))
\)
and similarly
\(
\partial\mathcal{L}_k
{}\coloneqq{}
\partial\LL(\bm x^k,\bm z^k,-\nabla F(\bm x^k))
\).
Note that
\(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}={}
\mathcal{L}_k
\);
to avoid trivialities, we may thus assume that \(\mathcal{L}_k\gneqq\@ifstar\@@P\@Phi_\star\) for all \(k\), for otherwise the sequence \(\seq{\bm x^k}\) is asymptotically constant, cf. \eqref{eq:SDx}, and thus so is $(\bm x^k,\bm z^k,-\nabla F(\bm x^k))$.
Let \(\Omega\) be the set of accumulation points of \(\seq{\bm x^k}\), which is compact and such that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\equiv\@ifstar\@@P\@Phi_\star\) on \(\Omega\) for some \(\@ifstar\@@P\@Phi_\star\in\R\), as ensured by \Cref{thm:cluster}.
Then, since \(\|\bm x^k-\bm z^k\|\to0\) and \(\nabla F\) is continuous,
\(
\Omega'
{}\coloneqq{}
\set{\bigl(\bm x,\bm x,-\nabla F(\bm x)\bigr)}[\bm x\in\Omega]
\)
is the set of cluster points of $(\bm x^k,\bm z^k,-\nabla F(\bm x^k))$, which is also
compact and on which \(\LL\) is constantly equal to \(\@ifstar\@@P\@Phi_\star\).
Since \(F\) and \(G\) are semialgebraic, known properties of semialgebraic functions (see \eg \cite[\S8.3.1]{ioffe2017variational}) ensure that \(\LL\) is semialgebraic, and as such it possesses the KL property on \(\Omega'\), see \cite[Thm. 3 and Lem. 6]{bolte2014proximal}: there exists a continuous increasing concave function
\(
\func{\psi}{[0,\varepsilon)}{[0,\infty)}
\)
(for some \(\varepsilon>0\)) which is differentiable on \((0,\varepsilon)\) and with \(\psi(0)=0\), such that
\(
\psi'(\mathcal{L}_k-\@ifstar\@@P\@Phi_\star)\dist(0,\partial\mathcal{L}_k)
{}\geq{}
1
\)
for all \(k\) large enough such that $(\bm x^k,\bm z^k,-\nabla F(\bm x^k))$ is sufficiently close to \(\Omega'\) and \(\mathcal{L}_k\) is sufficiently close to \(\@ifstar\@@P\@Phi_\star\).
The optimality condition for \(\bm z\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\) reads
\begin{equation}\label{eq:proxsubgrad}
\Gamma^{-1}(\bm x-\bm z)-\nabla F(\bm x)
{}\in{}
\partial G(\bm z),
\end{equation}
hence
\(
\partial\mathcal{L}_k
{}\ni{}
\bigl(
\Gamma^{-1}(\bm x^k-\bm z^k),~
0,~
\bm x^k-\bm z^k
\bigr)
\),
which implies that
\[
\textstyle
\dist(0,\partial\mathcal{L}_k)
{}\leq{}
\sqrt{\gamma_{\rm min}^{-1}+\gamma_{\rm max}}\,
\|\bm x^k-\bm z^k\|_{\Gamma^{-1}},
\]
where \(\gamma_{\rm min}\coloneqq\min_{i\in[N]}\set{\gamma_i}\) and \(\gamma_{\rm max}\coloneqq\max_{i\in[N]}\set{\gamma_i}\).
Combined with the KL inequality, we obtain
\begin{equation}\label{eq:KL}
\textstyle
\psi'(\mathcal L_k-\@ifstar\@@P\@Phi_\star)
{}\geq{}
\frac{1}{
\sqrt{\gamma_{\rm min}^{-1}+\gamma_{\rm max}}\,
\|\bm x^k-\bm z^k\|_{\Gamma^{-1}}
}.
\end{equation}
Denote
\(
\Delta_k\coloneqq\psi(\mathcal{L}_k-\@ifstar\@@P\@Phi_\star)
\), \(
\sigma'
{}={}
\tfrac{\xi_{\rm min}}{N(1+TL_{\bf T})^2}
\)
and let \(\xi_{\rm min}\) and \(L_{\bf T}\) be as in the proof of \Cref{thm:cyclic:subseq}.
We have for all $\nu\in\N$
\begin{equation}\label{eq:KLinequality}
\Delta_{(\nu+1)T}
{}-{}
\Delta_{\nu T}
{}\leq{}
\psi'(\mathcal{L}_{\nu T}-\@ifstar\@@P\@Phi_\star)
\left(\mathcal{L}_{(\nu+1)T}-\mathcal{L}_{\nu T}\right)
{}\overrel*[\leq]{\eqref{eq:Essential_cyclic_descent},\,\eqref{eq:KL}}{}
-\tfrac{\sigma'}{2\sqrt{\gamma_{\rm min}^{-1}+\gamma_{\rm max}}}
\|\bm x^{\nu T}-\bm z^{\nu T}\|_{\Gamma^{-1}},
\end{equation}
where the first inequality uses concavity of \(\psi\).
By summing over \(\nu\in\N\) (sure) summability of the sequence \(\seq{\|\bm x^{\nu T}-\bm z^{\nu T}\|}[\nu\in\N]\) is obtained.
By suitably shifting, for every \(t\in[T]\) the same can be said for the sequence
\(
\seq{\|\bm z^{T\nu+t}-\bm x^{T\nu+t}\|}[\nu\in\N]
\),
and since \(T\) is finite we conclude that the whole sequence
\(
\seq{\|\bm z^k-\bm x^k\|}
\)
is summable.
Since \(\|\bm x^{k+1}-\bm x^k\|\leq\|\bm z^k-\bm x^k\|\) we conclude that \(\seq{\bm x^k}\) has finite length and is thus convergent (to a single point), and consequently so is \(\seq{\bm z^k}\).
Suppose now that \(\theta\leq\nicefrac12\), so that \(\psi(s)=\varrho\sqrt s\).
Then,
\[
\|\bm x^{\nu T}-\bm z^{\nu T}\|_{\Gamma^{-1}}
{}\overrel[\geq]{\eqref{eq:KL}}{}
\tfrac{2c}{\varrho}
\sqrt{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{\nu T})-\@ifstar\@@P\@Phi_\star}
{}={}
\tfrac{2c}{\varrho^2}
\psi(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^{\nu T})-\@ifstar\@@P\@Phi_\star)
{}={}
\tfrac{2c}{\varrho^2}
\Delta_{\nu T}.
\]
Combined with \eqref{eq:KLinequality} it follows that $\seq{\Delta_{\nu T}}[\nu\in\N]$ conveges $Q$-linearly.
By rearranging \eqref{eq:KLinequality} as
\[
c'\|\bm x^{\nu T}-\bm z^{\nu T}\|_{\Gamma^{-1}}
{}\leq{}
\Delta_{\nu T}
{}-{}
\Delta_{(\nu+1)T}
{}\leq{}
\Delta_{\nu T},
\]
\(R\)-linear convergence of \(\seq{\|\bm x^{\nu T}-\bm z^{\nu T}\|}[\nu\in\N]\) follows.
By suitably shifting, for every \(t\in[T]\) the same can be said for the sequence
\(
\seq{\|\bm z^{T\nu+t}-\bm x^{T\nu+t}\|}[\nu\in\N]
\),
and since \(T\) is finite we conclude that the whole sequence
\(
\seq{\|\bm z^k-\bm x^k\|}
\)
converges $R$-linearly. On the other hand, since \(\|\bm x^{k+1}-\bm x^k\|\leq\|\bm z^k-\bm x^k\|\), also \(\seq{\|\bm x^{k+1}-\bm x^k\|}\) converges \(R\)-linearly, hence so does \(\seq{\bm x^k}\). By combining the two, we conclude that also $\seq{\bm z^k}$ converges $R$-linearly.
\begin{comment}
As in the proof of \Cref{thm:cyclic:global} for \(k\) large enough inequality \eqref{eq:KL} holds, that is,
\[
\textstyle
\rho(1-\theta)(\mathcal L_{\nu T}-\@ifstar\@@P\@Phi_\star)^{-\theta}
{}\geq{}
\frac{1}{
\sqrt{\gamma_{\rm min}^{-1}+\gamma_{\rm max}}\,
\|\bm x^k-\bm z^k\|_{\Gamma^{-1}}
}
\]
owing to the fact that \(\psi(s)=\rho s^{1-\theta}\).
Therefore,
\begin{align*}
\Delta_{\nu T}
{}={}
\rho(\mathcal L_{\nu T}-\@ifstar\@@P\@Phi_\star)^{1-\theta}
{}\leq{} &
\rho
\left(
\rho(1-\theta)\sqrt{\gamma_{{\rm min}}^{-1}+\gamma_{{\rm max}}}\,\|{\bm x}^{\nu T}-{\bm z}^{\nu T}\|_{\Gamma^{-1}}
\right)^{\frac{1-\theta}{\theta}}
\\
{}\leq{} &
\rho^2(1-\theta)\sqrt{\gamma_{{\rm min}}^{-1}+\gamma_{{\rm max}}}\,\|{\bm x}^{\nu T}-{\bm z}^{\nu T}\|_{\Gamma^{-1}},
\end{align*}
where in the second inequality we used the fact $(1-\theta)/\theta\geq 1$ and that the base of the exponent is smaller than one for \(k\) large enough since $\|{\bm x}^{\nu T}-{\bm z}^{\nu T}\|_{\Gamma^{-1}}$ converges to zero.
Combined with \eqref{eq:KLinequality} it follows that $\seq{\Delta_{\nu T}}[\nu\in\N]$ conveges $Q$-linearly.
By rearranging \eqref{eq:KLinequality} as
\[
\tfrac{\sigma'}{2\sqrt{\gamma_{\rm min}^{-1}+\gamma_{\rm max}}}
\|\bm x^{\nu T}-\bm z^{\nu T}\|_{\Gamma^{-1}}
{}\leq{}
\Delta_{\nu T}
{}-{}
\Delta_{(\nu+1)T}
{}\leq{}
\Delta_{\nu T}
\]
\(R\)-linear convergence of \(\seq{\|\bm x^k-\bm z^k\|}\) follows.
By suitably shifting, for every \(t\in[T]\) the same can be said for the sequence
\(
\seq{\|\bm z^{T\nu+t}-\bm x^{T\nu+t}\|}[\nu\in\N]
\),
and since \(T\) is finite we conclude that the whole sequence
\(
\seq{\|\bm z^k-\bm x^k\|}
\)
converges $R$-linearly. On the other hand since $\|\bm x^{k+1}-\bm x^k\|\leq \|\bm z^k-\bm x^k\|$, $\seq{\|\bm x^{k+1}-\bm x^k\|}$ also converges $R$-linearly, hence so does $\seq{\bm x^k}$.
\end{comment}
\end{proof}
\end{thm}
\section{Nonconvex finite sum problems: the Finito/MISO algorithm}\label{sec:Finito}
As mentioned in \Cref{sec:Introduction}, if \(G\) is of the form \eqref{eq:FINITOG} then problem \eqref{eq:P} reduces to the finite sum minimization presented in \eqref{eq:FSP}.
Most importantly, the proximal mapping of the original nonsmooth function \(G\) can be easily expressed in terms of that of the small function \(g\) in the reduced finite sum reformulation, as shown in the next lemma.
\begin{lem}
Given \(\gamma_i>0\), \(i\in[N]\), let
\(
\Gamma
{}\coloneqq{}
\blockdiag(\gamma_1I_n,\dots,\gamma_NI_n)
\)
and
\(
\hat\gamma
{}\coloneqq{}
\bigl(\sum_{i=1}^N\gamma_i^{-1}\bigr)^{-1}
\).
Then, for \(G\) as in \eqref{eq:FINITOG} and any \(\bm u\in\R^{Nn}\)
\[
\textstyle
\prox_G^{\Gamma^{-1}}(\bm u)
{}={}
\set{(\hat v,\dots,\hat v)}[
\hat v
{}\in{}
\prox_{\hat\gamma g}(\hat u)
]
\quad\text{where}\quad
\hat u
{}\coloneqq{}
\hat\gamma
\sum_{i=1}^N\gamma_i^{-1}u_i.
\]
\begin{proof}
Observe first that for every \(w\in\R^n\) one has
\begin{align*}
\textstyle
\sum_i\gamma_i^{-1}\|w-u_i\|^2
{}={} &
\textstyle
\sum_i\gamma_i^{-1}\|\hat u-u_i\|^2
{}+{}
\sum_i\gamma_i^{-1}\|w-\hat u\|^2
{}+{}
\smashoverbrace{
\textstyle
2\sum_i\gamma_i^{-1}\innprod{\hat u-u_i}{w-\hat u}
}{
=0
}
\\
\numberthis\label{eq:mean}
{}={} &
\textstyle
\sum_i\gamma_i^{-1}\|\hat u-u_i\|^2
{}+{}
\hat\gamma^{-1}\|w-\hat u\|^2.
\end{align*}
Next, observe that since \(\dom G\subseteq C\) (the consensus set),
\begin{align*}
\prox_G^{\Gamma^{-1}}(\bm u)
{}={} &
\argmin_{\bm w\in\R^{Nn}}\set{\textstyle
G(\bm w)+\sum_{i=1}^N\tfrac{1}{2\gamma_i}\|w_i-u_i\|^2
}
\\
{}={} &
\argmin_{\bm w\in\R^{Nn}}\set{\textstyle
G(\bm w)+\sum_{i=1}^N\tfrac{1}{2\gamma_i}\|w_i-u_i\|^2
}[
w_1=\dots=w_N
]
\\
{}={} &
\argmin_{(w,\dots,w)}\set{\textstyle
g(w)+\sum_{i=1}^N\tfrac{1}{2\gamma_i}\|w-u_i\|^2
}
\\
{}\overrel*{\eqref{eq:mean}}{} &
\argmin_{(w,\dots,w)}\set{\textstyle
g(w)
{}+{}
\tfrac{1}{2\hat\gamma}\|w-\hat u\|^2
}
{}={}
\set{(\hat v,\dots,\hat v)}[
\hat v\in\prox_{\hat\gamma g}(\hat u)
]
\end{align*}
as claimed.
\end{proof}
\end{lem}
If all stepsizes are set to a same value \(\gamma\), so that \(\Gamma=\gamma\I_{Nn}\), then the forward-backward step reduces to
\begin{align*}
\bm z
{}\in{}
\prox_G^{\Gamma^{-1}}(\bm x-\Gamma\nabla F(\bm x))
\quad\Leftrightarrow\quad &
\bm z=(\bar z,\dots,\bar z),
\\
\numberthis\label{eq:FBFinito}
&
\bar z
{}\in{}
\prox_{\gamma g\nicefrac{}N}\left(
\textstyle
\tfrac1N\sum_{j=1}^N\bigl(
x_j-\tfrac\gamma N\nabla f_j(x_j)
\bigr)
\right).
\end{align*}
The argument of \(\prox_{\gamma g\nicefrac{}{N}}\) is the (unweighted) average of the forward operator.
By applying \Cref{alg:BC} with \eqref{eq:FBFinito}, Finito/MISO \cite{defazio2014finito,mairal2015incremental} is recovered.
Differently from the existing convergence analyses, ours covers fully nonconvex and nonsmooth problems, more general sampling strategies and the possibility to select different stepsizes \(\gamma_i\) for each block, which can have a significant impact on the performance compared to the case where all stepsizes are equal.
Moreover, to the best of our knowledge this is the first work that shows global convergence and linear rates even when the smooth functions are nonconvex.
The resulting scheme is presented in \Cref{alg:Finito}.
We remark that the consensus formulation to recover Finito/MISO (although from a different umbrella algorithm) was also observed in \cite{davis2016smart} in the convex case.
Moreover, the Finito/MISO algorithm with cyclic sampling is also studied in \cite{mokhtari2018surpassing} when \(g\equiv0\) and $f_i$ are strongly convex functions; consistently with \Cref{ass:cyclic}, our analysis covers the more general essentially cyclic sampling even in the presence of a nonsmooth convex term $g$ and allowing the smooth functions $f_i$ to be nonconvex.
Randomized Finito/MISO with $g\equiv 0$ is also studied in the recent work \cite{qian2019miso}; although their analysis is limited to a single stepsize, in the convex case it is allowed to be larger than our worst-case stepsize \(\min_i\gamma_i\).
\begin{algorithm}
\caption{Nonconvex proximal Finito/MISO for problem \eqref{eq:FSP}
}
\label{alg:Finito}
\begin{algorithmic}[1]
\item[{\sc Require}]
\(
x^{\rm init}\in\R^n
\),~
\(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\),
{\small \(i\in[N]\)}
\Statex
\(
\hat\gamma
{}\coloneqq{}
\bigl(\sum_{i=1}^N\gamma_i^{-1}\bigr)^{-1}
\),~~
\(
s_i
{}={}
x^{\rm init}-\frac{\gamma_i}{N}\nabla f_i(x^{\rm init})
\)~
\(i\in[N]\),~~
\(
\hat s
{}={}
{\hat\gamma}\sum_{i=1}^N\gamma_i^{-1}s_i
\)
\item[{\sc Repeat} until convergence]
\State
select a set of indices \(I\subseteq[N]\)
\State
\(
z
{}\in{}
\prox_{\hat\gamma g}(\hat s)
\)
\For{ \(i\in I\) }
\State
\(
v
{}\gets{}
z-\frac{\gamma_i}{N}\nabla f_i(z)
\)
\State
update~~
\(
\hat s
{}\gets{}
\hat s+\tfrac{\hat\gamma}{\gamma_i}(v-s_i)
\)
~~and~~
\(
s_i\gets v
\)
\@ifstar\@@E\@EndFor
\item[{\sc Return} $z$ ]
\end{algorithmic}
\end{algorithm}
The convergence results from \Cref{sec:convergence} are immediately translated to this setting by noting that the bold variable ${\bm z}^k$ corresponds to $(z^k,\dots,z^k)$.
Therefore, $\@ifstar\@@P\@Phi({\bm z^k})= \varphi(z^k)$ where $\varphi$ is the cost function for the finite sum problem.
\begin{cor}[subsequential convergence of \Cref{alg:Finito}]\label{thm:Finito:convergence}
In the finite sum problem \eqref{eq:FSP} suppose that \(\argmin\varphi\) is nonempty, \(g\) is proper and lsc, and each \(f_i\) is \(L_{f_i}\)-Lipschitz differentiable, \(i\in[N]\).
Then, the following hold almost surely (resp. surely) for the sequence $\seq{z^k}$ generated by \Cref{alg:Finito} with randomized sampling strategy as in \Cref{ass:random} (resp. with any essentially cyclic sampling strategy and $g$ convex as required in \Cref{ass:cyclic}):
\begin{enumerate}
\item
the sequence \(\seq{\varphi(z^k)}\) converges to a finite value \(\varphi_\star\leq\varphi(x^{\rm init})\);
\item
all cluster points of the sequence \(\seq{z^k}\) are stationary and on which \(\varphi\) equals \(\varphi_\star\).
\end{enumerate}
If, additionally, \(\varphi\) is coercive, then the following also hold:
\begin{enumerate}[resume]
\item
\(\seq{z^k}\) is bounded (in fact, this holds surely for arbitrary sampling criteria).
\end{enumerate}
\end{cor}
\begin{cor}[linear convergence of \Cref{alg:Finito} under strong convexity]
Additionally to the assumptions of \Cref{thm:Finito:convergence}, suppose that \(g\) is convex and that each \(f_i\) is \(\mu_{f_i}\)-strongly convex.
The following hold for the iterates generated by \Cref{alg:Finito}:
\begin{itemize}[leftmargin=*,label={},itemindent=-0.5cm,labelsep=0pt,partopsep=0pt,parsep=0pt,listparindent=0pt,topsep=0pt]
\item{\sc Randomized sampling:}
under \Cref{ass:random},
\begin{align*}
\@ifstar\@@E\@E[]{\varphi(z^k)-\min\varphi}
{}\leq{} &
(\varphi(x^{\rm init})-\min\varphi)
(1-c)^k
\\
\tfrac12\@ifstar\@@E\@E[]{\|z^k-x^\star\|^2}
{}\leq{} &
\frac{
N(\varphi(x^{\rm init})-\min\varphi)
}{
\sum_i\mu_{f_i}
}
(1-c)^k
\end{align*}
holds for all \(k\in\N\), where \(c\) is as in \eqref{eq:cwc} and \(x^\star\coloneqq\argmin\varphi\).
If the stepsizes \(\gamma_i\) and the sampling probabilities \(p_i\) are set as in \Cref{thm:random:linear}, then the tighter constant \(c\) as in \eqref{eq:cbc} is obtained.
\item{\sc Shuffled cyclic or cyclic sampling:}
under either sampling strategy \eqref{eq:ShufCyclicRule} or \eqref{eq:cyclicRule},
\begin{align*}
\varphi(z^{\nu N})-\min\varphi
{}\leq{} &
(\varphi(x^{\rm init})-\min\varphi)
(1-c)^\nu
\\
\tfrac12\@ifstar\@@E\@E[]{\|z^{\nu N}-x^\star\|^2}
{}\leq{} &
\frac{
N(\varphi(x^{\rm init})-\min\varphi)
}{
\sum_i\mu_{f_i}
}
(1-c)^\nu
\end{align*}
holds surely for all \(\nu\in\N\), where \(c\) is as in \eqref{eq:linearShuffledCyclic}.
\end{itemize}
\end{cor}
The next result follows from \Cref{thm:cyclic:global} once the needed properties of \(\@ifstar\@@P\@Phi\) as in the umbrella formulation \eqref{eq:P} are shown to hold.
\begin{cor}[global convergence of \Cref{alg:Finito}]\label{thm:Finito:global}
In the finite sum problem \eqref{eq:FSP}, suppose that \(\varphi\) has the KL property with exponent \(\theta\in(0,1)\) (as is the case when \(f_i\) and \(g\) are semialgebraic) and coercive, \(g\) is proper convex and lsc, and each \(f_i\) is \(L_{f_i}\)-Lipschitz differentiable, \(i\in[N]\).
Then the sequence \(\seq{z^k}\) generated by \Cref{alg:Finito} with any essentially cyclic sampling strategy as in \Cref{ass:cyclic} converges surely to a stationary point for \(\varphi\).
Moreover, if \(\theta\leq\nicefrac12\) then it converges at \(R\)-linear rate.
\begin{proof}
Function \(\@ifstar\@@P\@Phi=F+G\) be as in \eqref{eq:FINITOG} clearly is coercive and satisfies \Cref{ass:basic}.
In order to invoke \Cref{thm:cyclic:global} is suffices to show that there exists a constant \(c>0\) such that
\begin{equation}\label{eq:Dist}
\dist(0,\partial\@ifstar\@@P\@Phi(\bm x))
{}\geq{}
c\dist(0,\partial\varphi(x))
\quad
\text{for all \(x\in\R^n\) and \(\bm x=(x,\dots,x)\),}
\end{equation}
as this will ensure that \(\@ifstar\@@P\@Phi\) enjoys the KL property at \(\bm x^\star=(x^\star,\dots,x^\star)\) with same desingularizing function (up to a positive scaling).
Notice that for \(x\in\R^n\) and \(\bm x=(x,\dots,x)\), one has
\(
\bm v\in\partial G(\bm x)
\)
iff
\(
\frac1N\sum_{i=1}^Nv_i
{}\in{}
\partial g(x)
\).
Since
\(
\partial\@ifstar\@@P\@Phi(\bm x)
{}={}
\tfrac1N\mathop\times_{i=1}^N\nabla f_i(x_i)+\partial G(\bm x)
\)
and
\(
\partial\varphi(x)
{}={}
\tfrac1N\sum_{i=1}^N\nabla f_i(x)
{}+{}
\partial g(x)
\),
see \cite[Ex. 8.8(c) and Prop. 10.5]{rockafellar2011variational}, for \(x\in\R^n\) and denoting \(\bm x=(x,\dots,x)\) we have
\begin{align*}
\dist(0,\partial\varphi(x))
{}\leq{} &
\inf_{\bm v\in\partial G(\bm x)}{
\left\|\textstyle
\tfrac1N\sum_{i=1}^N\nabla f_i(x)
{}+{}
\tfrac1N\sum_{i=1}^Nv_i
\right\|
}
\\
{}\leq{} &
\tfrac1N\inf_{\bm v\in\partial G(\bm x)}{
\textstyle
\sum_{i=1}^N\|\nabla f_i(x)+v_i\|
}
{}={}
\tfrac1N\inf_{\bm u\in\partial\@ifstar\@@P\@Phi(\bm x)}{
\newnorm{\bm u}
},
\end{align*}
where \(\newnorm{{}\cdot{}}\) is the norm in \(\R^{Nn}\) given by
\(
\newnorm{\bm w}=\sum_{i=1}^N\|w_i\|
\).
Inequality \eqref{eq:Dist} then follows by observing that
\(
\inf_{\bm u\in\partial\@ifstar\@@P\@Phi(\bm x)}{
\newnorm{\bm u}
}
\)
is the distance of \(0\) from \(\partial\@ifstar\@@P\@Phi(\bm x)\) in the norm \(\newnorm{{}\cdot{}}\), hence that \(\newnorm{{}\cdot{}}\leq c'\|{}\cdot{}\|\) for some \(c'>0\).
\end{proof}
\end{cor}
\section{Nonconvex sharing problem}\label{sec:Sharing}
In this section we consider the sharing problem \eqref{eq:SP}.
As discussed in \Cref{sec:Introduction}, \eqref{eq:SP} fits into the problem framework \eqref{eq:P} by simply letting \(G\coloneqq g \circ A\), where
\(A\coloneqq[\I_n~\dots~\I_n]\in\R^{n\times nN}\).
By arguing as in \cite[Th. 6.15]{beck2017first} it can be shown that, when \(A\) has full row rank, the proximal mapping of $G=g\circ A$ is given by
\begin{equation}\label{eq:sharingprox}
\prox_G^{\Gamma^{-1}}(\bm u)
{}={}
\bm u+\Gamma\trans A(A\Gamma\trans A\,)^{-1}\left(\prox^{(A\Gamma\trans A\,)^{-1}}_g\left(A\bm u\right)-A\bm u\right).
\end{equation}
Since \(A\Gamma\trans A=\sum_{i=1}^N\gamma_i\) for the sharing problem \eqref{eq:SP},
\begin{align*}
\bm v
{}\in{}
\prox_G^{\Gamma^{-1}}(\bm u)
~~\Leftrightarrow~~ &
\bm v
{}={}
(u_1+\gamma_1w,\dots,u_N+\gamma_Nw)
\\
&
\textstyle
w
{}\in{}
\tilde\gamma^{-1}\left(\prox_{\tilde{\gamma}g}(\tilde u)-\tilde u\right),
~~
\tilde\gamma\coloneqq\sum_{i=1}^N\gamma_i,
~~
\tilde u\coloneqq \sum_{i=1}^Nu_i.
\end{align*}
Consequently general BC \Cref{alg:BC} when applied to the sharing problem \eqref{eq:SP} reduces to \Cref{alg:Sharing}.
\begin{algorithm}
\caption{Block-coordinate method for nonconvex sharing problem \eqref{eq:SP}}
\label{alg:Sharing}
\begin{algorithmic}[1]
\item[{\sc Require}]
\(
x_i^{\rm init}\in\R^{n}
\),~
\(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\),
{\small \(i\in [N]\)}
\Statex
\(
\tilde\gamma
{}\coloneqq{}
\sum_{i=1}^N\gamma_i
\),~~
\(
s_i
{}={}
x_i^{\rm init}-\frac{\gamma_i}{N}\nabla f_i(x_i^{\rm init})
\)~
\(i\in [N]\),~~\(
\tilde s
{}={}
\sum_{i=1}^N s_i
\)
\item[{\sc Repeat} until convergence]
\State
select a set of indices \(I\subseteq[N]\)
\State $w \gets \tilde{\gamma}^{-1}(\prox_{\tilde{\gamma}g}(\tilde s)-\tilde s)$
\For{ \(i\in I\) }
\State
\(
v_i {}\gets{} s_i + \gamma_i w - \tfrac{\gamma_i}{N} \nabla f_i(s_i + \gamma_i w )
\)
\State
update~~
\(
\tilde s
{}\gets{}
\tilde s+(v_i-s_i)
\)
~~and~~
\(
s_i \gets v_i
\)
\@ifstar\@@E\@EndFor
\item[{\sc Return}]
$\bm z=(s_1 + \gamma_1 w ,\dots,s_N + \gamma_N w)$ with $w\in\tilde{\gamma}^{-1}(\prox_{\tilde{\gamma}g}(\tilde s)-\tilde s)$
\end{algorithmic}
\end{algorithm}
\begin{rem}[generalized sharing constraint]
Another notable instance of $G=g\circ A$ well suited for the BC framework of \Cref{alg:BC} is when \(g=\indicator_{\set0}\) and \(A=[A_1~\dots~A_N]\), \(A_i\in\R^{n\times n_i}\) such that $A$ is full rank.
This models the generalized sharing problem
\[
\minimize_{\bm x\in\R^{\sum_in_i}}{\textstyle
\tfrac1N\sum_{i=1}^Nf_i(x_i)
}
\quad\stt{}\textstyle
\sum_{i=1}^NA_ix_i=0.
\]
In this case \eqref{eq:sharingprox} simplifies to
\[
\left(\prox_G^{\Gamma^{-1}}(\bm u)\right)_i
{}={}
u_i-\gamma_i\trans{A_i}\mathcal A^{-1}\sum_{i=1}^NA_iu_i,
\]
where $\mathcal A\coloneqq A\Gamma\trans A$ can be factored offline and \(\sum_{i=1}^NA_ix_i\) can be updated in an incremental fashion in the same spirit of \Cref{alg:Sharing}.
\end{rem}
The convergence results for \Cref{alg:Sharing} summarized below fall as special cases of those in \Cref{sec:convergence}.
\begin{cor}[convergence of \Cref{alg:Sharing}]\label{thm:sharing:convergence}
In the sharing problem \eqref{eq:SP}, suppose that \(\argmin\@ifstar\@@P\@Phi\) is nonempty, \(g\) is proper and lsc, and each \(f_i\) is \(L_{f_i}\)-Lipschitz differentiable, \(i\in[N]\). Consider the sequences $\seq{w^k}$ and $\seq{\bm s^k}$ generated by \Cref{alg:Sharing} and let $\seq{\bm z^k}=\seq{s_1^k + \gamma_1 w^k ,\dots,s_N^k + \gamma_N w^k}$.
Then, the following hold almost surely (resp. surely) with randomized sampling strategy as in \Cref{ass:random} (resp. with any essentially cyclic sampling strategy and $g$ convex as required in \Cref{ass:cyclic}):
\begin{enumerate}
\item
the sequence \(\seq{\@ifstar\@@P\@Phi(\bm z^k)}\) converges to a finite value \(\@ifstar\@@P\@Phi_\star\leq\@ifstar\@@P\@Phi(\bm x^{\rm init})\);
\item
all cluster points of the sequence \(\seq{\bm z^k}\) are stationary and on which \(\@ifstar\@@P\@Phi\) equals \(\@ifstar\@@P\@Phi_\star\).
\end{enumerate}
If, additionally, \(\@ifstar\@@P\@Phi\) is coercive, then the following also hold:
\begin{enumerate}[resume]
\item
\(\seq{\bm z^k}\) is bounded (in fact, this holds surely for arbitrary sampling criteria).
\end{enumerate}
\end{cor}
\begin{cor}[linear convergence of \Cref{alg:Sharing} under strong convexity]\label{cor:RLinSharing}
Additionally to the assumptions of \Cref{thm:sharing:convergence}, suppose that \(g\) is convex and that each \(f_i\) is \(\mu_{f_i}\)-strongly convex.
The following hold:
\begin{itemize}[leftmargin=*,label={},itemindent=-0.5cm,labelsep=0pt,partopsep=0pt,parsep=0pt,listparindent=0pt,topsep=0pt]
\item{\sc Randomized sampling:}
under \Cref{ass:random},
\begin{align*}
\@ifstar\@@E\@E[]{\@ifstar\@@P\@Phi(\bm z^k)-\min\@ifstar\@@P\@Phi}
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^{\rm init})-\min\@ifstar\@@P\@Phi\bigr)(1-c)^k
\\
\tfrac12\@ifstar\@@E\@E[]{\|\bm z^k-\bm x^\star\|^2_{\mu_F}}
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^{\rm init})-\min\@ifstar\@@P\@Phi\bigr)(1-c)^k
\end{align*}
holds for all \(k\in\N\), where \(\bm x^\star\coloneqq\argmin\@ifstar\@@P\@Phi\),
\(
\mu_F
{}\coloneqq{} \tfrac1N
\blockdiag\bigl(\mu_{f_1}\I_{n_1},\dots\mu_{f_n}\I_{n_N}\bigr)
\),
and \(c\) is as in \eqref{eq:cwc}.
If the stepsizes \(\gamma_i\) and the sampling probabilities \(p_i\) are set as in \Cref{thm:random:linear}, then the tighter constant \(c\) as in \eqref{eq:cbc} is obtained.
\item{\sc Shuffled cyclic or cyclic sampling:}
under either sampling strategy \eqref{eq:ShufCyclicRule} or \eqref{eq:cyclicRule},
\begin{align*}
\@ifstar\@@P\@Phi(\bm z^{N\nu})-\min\@ifstar\@@P\@Phi
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^{\rm init})-\min\@ifstar\@@P\@Phi\bigr)(1-c)^\nu
\\
\tfrac12\|\bm z^{N\nu}-\bm x^\star\|^2_{\mu_F}
{}\leq{} &
\bigl(\@ifstar\@@P\@Phi(\bm x^{\rm init})-\min\@ifstar\@@P\@Phi\bigr)(1-c)^\nu
\end{align*}
holds surely for all \(\nu\in\N\), where \(c\) is as in \eqref{eq:linearShuffledCyclic}.
\end{itemize}
\end{cor}
We conclude with an immediate consequence of \Cref{thm:cyclic:global} that shows that (strong) convexity is in fact not necessary for global or linear convergence to hold.
\begin{cor}[global and linear convergence of \Cref{alg:Sharing}]\label{thm:Sharing:global}
In problem \eqref{eq:SP}, suppose that \(\@ifstar\@@P\@Phi\) has the KL property with exponent \(\theta\in(0,1)\) (as is the case when \(g\) and \(f_i\) are semialgebraic) and is coercive, \(g\) is proper convex lsc, and each \(f_i\) is \(L_{f_i}\)-Lipschitz differentiable, \(i\in[N]\).
Then the sequence $\seq{\bm z^k}$ as defined in \Cref{thm:sharing:convergence} with any essentially cyclic sampling strategy as in \Cref{ass:cyclic} converges surely to a stationary point for \(\@ifstar\@@P\@Phi\).
Moreover, if \(\theta\leq\nicefrac12\) it converges with \(R\)-linear rate.
\end{cor}
\ifaccel
\section{Accelerated block-coordinate proximal gradient}
The work \cite{allen2016even} introduced a coordinate descent method for smooth convex minimization, in which each coordinate is randomly sampled according to an ad hoc probability distribution that provably leads to a remarkable speed up with respect to uniform sampling strategies.
The unified analysis of BC-algorithms and the analytical tool introduced in this paper, the forward backward envelope function, allow the extention of this approach to nonsmooth convex minimization of the form \eqref{eq:P},
where functions \(f_i\) are convex quadratic and \(G\) is convex but possibly nonsmooth:
\begin{ass}[requirements for the fast BC-\Cref{alg:Fast}]\label{ass:Fast}
In problem \eqref{eq:P}, \(\func{G}{\R^{\sum_in_i}}{\Rinf}\) is proper convex and lsc, and
\(f_i(x_i)\coloneqq\tfrac12\trans{x_i}H_ix_i+\trans{q_i}x_i\) is convex quadratic, with \(L_{f_i}\coloneqq\lambda_{\rm max}(H_i)\) and \(\mu_{f_i}\coloneqq\lambda_{\rm min}(H_i)\geq0\),
\(i\in[N]\).
\end{ass}
Let $U_i\in \R^{{\sum_{i=1}^N n_i}\times n_i}$ denote the $i$-th block column of the identity matrix so that for a vector $v\in \R^{n_i}$
\begin{equation}\label{eq:U}
U_iv= (0,\dots, 0, \!\overbracket{\,v\,}^{\mathclap{i\text{-th}}}\!, 0, \dots, 0).
\end{equation}
The accelerated BC scheme based on \cite{allen2016even} (for both strongly convex and convex cases) is given in \Cref{alg:Fast}.
Similarly to the approach of \cite{patrinos2014douglas} where an accelerated Douglas-Rachford algorithm is proposed, in order to derive \Cref{alg:Fast} we consider the scaled problem \( \minimize_{\tilde{\bm x}} \@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x})\) where $\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C \coloneqq \@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\circ Q^{-1/2}$, and $Q$ is the symmetric positive definite matrix
\begin{equation} \label{eq:QQ}
Q
{}\coloneqq{}
\blockdiag(Q_1,\dots,Q_N)\succ0
\quad \text{with } Q_i
{}\coloneqq{}
\gamma_i^{-1}\I-\tfrac{1}{N}H_i\in\R^{n_i\times n_i},~i\in[N].
\end{equation}
As detailed in \Cref{thm:convex}, whenever \Cref{ass:Fast} is satisfied $\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C$ is a convex Lipschitz-differentiable function, and its gradient is given by $\nabla \@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x}) = Q^{1/2}(\bm x-\prox_G^{\Gamma^{-1}}(\bm x-\Gamma\nabla F(\bm x)))$ where $\bm x=Q^{-1/2}\tilde{\bm x}$.
Note that, based on \Cref{thm:convex}, \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C\) is \(1\)-smooth along the \(i\)-th block (in the notation of \cite{allen2016even}, \(L_i=1\), $S_\alpha=N$, and \(p_i=\nicefrac1N\)).
Hence the parameters of the algorithm simplify substantially resulting in uniform sampling.
Moreover, when functions \(f_i\) are \(\mu_{f_i}\)-strongly convex, by \Cref{thm:convex} \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C\) is $\sigma$-strongly convex with $\sigma= \frac{1}{N} \min_{i\in [N]} \{\gamma_i\mu_{f_i}\}$.
\Cref{alg:Fast} is obtained by applying the fast BC to this problem and scaling the variables by $Q^{-1/2}$.
Specifically, the update rule as in \cite{allen2016even} reads
\[
\begin{cases}[ r @{{}={}} l ]
\tilde{\bm x}^+ & \tau\tilde{\bm w}+(1-\tau)\tilde{\bm y}
\\
\tilde{\bm y}^+ & \tilde{\bm x}-U_i\trans{U_i}\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x}^+)
{}={}
\tilde{\bm x}-U_iQ_i^{\nicefrac12}(x_i^+-z_i^+)
\\
\tilde{\bm w}^+ & \tfrac{1}{1+\eta\sigma}(\tilde{\bm w}+\eta\sigma\tilde{\bm x}^+-N\eta U_i\trans{U_i}\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x}^+))
{}={}
\tfrac{1}{1+\eta\sigma}(\tilde{\bm w}+\eta\sigma\tilde{\bm x}^+-N\eta U_iQ_i^{\nicefrac12}(x_i^+-z_i^+)),
\end{cases}
\]
where \(\bm z^+=\prox_G^{\Gamma^{-1}}(\bm x^+-\Gamma\nabla F(\bm x^+))\).
Since \(Q^{-\nicefrac12}U_iQ_i^{\nicefrac12}=U_i\), premultiplying by \(Q^{-\nicefrac12}\) yields
\[
\begin{cases}[ r @{{}={}} l ]
\bm x^+ & \tau\bm z+(1-\tau)\bm y
\\
\bm z^+ & \prox_G^{\Gamma^{-1}}(\bm x^+-\Gamma\nabla F(\bm x^+))
\\
\bm y^+ & \bm x+U_i(z_i^+-x_i^+)
\\
\bm w^+ & \tfrac{1}{1+\eta\sigma}(\bm w+\eta\sigma\bm x^++N\eta U_i(z_i^+-x_i^+)).
\end{cases}
\]
For computational efficiency, vectors $\Gamma\nabla F(\bm x^k)$ and $\Gamma\nabla F(\bm w^k)$ are stored in variables $\bm r^k$ and $\bm v^k$ and updated recursively using the fact that gradients are affine, in such a way that each iteration requires only the evaluation of the sampled gradient (see \Cref{state:d}).
For similar reasons, in \Cref{alg:Fast} the iterates start with the $\bm y$-update rather than the $\bm x$-update as in \cite{allen2016even}.
Moreover, in the same spirit of \Cref{alg:BC} this accelerated variant can be implemented efficiently whenever the individual blocks of \(\bm z^+\) can be computed efficiently, similarly to the cases discussed in \Cref{sec:Finito,sec:Sharing}.
\begin{algorithm}
\caption{Accelerated block-coordinate proximal gradient for problem \eqref{eq:P} under \Cref{ass:Fast}}
\label{alg:Fast}
\begin{algorithmic}[1]
\item[{\sc Require}]
\(\bm x^0\in\R^{\sum_in_i}\),~
\(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\),~$i\in[N]$,
\(
\sigma
{}\coloneqq{}
\frac{1}{N} \min_{i\in [N]} \{\gamma_i\mu_{f_i}\}
\)
\State\label{state:sigma_beta}
{\bf if} \(\sigma=0\),~~{\bf then}~~
\(\eta = \nicefrac{1}{N^2}\)
~~{\bf otherwise}~~
set
\(
\tau
{}={}
\frac{2}{1+\sqrt{1+\nicefrac{4N^2}{\sigma}}}
\),
\(
\eta
{}={}
\frac{1}{\tau N^2}
\)~~
{\bf end if}
\State
\(
\bm w^0
{}={}
\bm x^0
\),~
\(
(\bm v^0,\bm r^0)
{}={}
(\Gamma\nabla F(\bm x^0),\Gamma\nabla F(\bm x^0))
\),~
\(
\bm z^{0}
{}={}
\prox_G^{\Gamma^{-1}}\bigl(\bm x^{0}-\bm r^{0}\bigr)
\)
\def\myVar#1{
\fillwidthof[l]{\bm x^{k+1}}{#1}
}
\For{ \(k=0,1,\dots\) }
\State
sample \(i\in[N]\) uniformly
\State\label{state:d}
\(
\myVar{\bm y^{k+1}}
{}\gets{}
\bm x^{k}+U_{i}\bigl(z_{i}^{k}-x_{i}^{k}\bigr)
\),\quad
\(
d
{}\gets{}
\tfrac{\gamma_i}{N}\nabla f_i(z_i^{k})-r_i^{k}
\)
\State
\(
\myVar{\bm v^{k+1}}
{}={}
\frac{1}{1+\eta\sigma}\Bigl(\bm v^{k}+\eta\sigma\bm r^{k}+N{\eta} U_id\Bigr)
\),\quad
\(
\bm w^{k+1}
{}={}
\frac{1}{1+\eta\sigma}\Bigl(\bm w^{k}+\eta\sigma\bm x^{k}+{N\eta}U_{i}\bigl(z_{i}^{k}-x_{i}^{k}\bigr)\Bigr)
\)
\State{\bf if}~~\(\sigma=0\),~~{\bf then}~~
\(
\eta
{}\gets{}
\frac{k+3}{2N^2}
\),~~
\(
\tau
{}\gets{}
\frac{2}{k+3}
\);
~~{\bf end if}
\State\label{state:FBEgrad}
\(
\myVar{\bm x^{k+1}}
{}={}
\tau \bm w^{k+1}+(1-\tau)\bm y^{k+1}
\),\quad
\(
\bm r^{k+1}
{}={}
\tau \bm v^{k+1}+(1-\tau)(\bm r^{k}+U_i d)
\)
\State
\(
\myVar{\bm z^{k+1}}
{}={}
\prox_G^{\Gamma^{-1}}\bigl(\bm x^{k+1}-\bm r^{k+1}\bigr)
\)
\@ifstar\@@E\@EndFor{}
\end{algorithmic}
\end{algorithm}
The convergence rate results follow directly from those of \cite{allen2016even} with parameters \(L_i=1\) and $S_\alpha=N$ as described above.
\begin{thm}[convergence rates of \Cref{alg:Fast}]
Suppose that \Cref{ass:basic,ass:Fast} are satisfied.
Then, the iterates generated by \Cref{alg:Fast} satisfy
\[
\mathbb{E}\left[{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm y^k)-\min \@ifstar\@@P\@Phi}\right]
{}\leq{}
\frac{2N^2\|\bm x^0 - \bm x^\star\|^2_Q}{(k+1)^2},
\]
where $Q$ is as in \eqref{eq:QQ}.
Moreover, in the strongly convex case ($\sigma= \frac{1}{N} \min_{i\in [N]} \{\gamma_i\mu_{f_i}\}>0$)
\[
\mathbb{E}\left[{\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm y^k)-\min \@ifstar\@@P\@Phi}\right]
{}\leq{}
O(1)
(1-c)^k\left(
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^0)-\min \@ifstar\@@P\@Phi
\right)
\quad\text{where}\quad
\textstyle
c
{}={}
\left(
\frac12
{}+{}
\sqrt{
\frac14
{}+{}
\frac{N^2}{\sigma}
}
\right)^{-1}.
\]
\end{thm}
Note that in the strongly convex case it follows from \Cref{thm:strconcost} that the distance from the solution decreases \(R\)-linearly as
\[
\@ifstar\@@E\@E[]{ \|\bm y^{k}-\bm x^\star\|^2_M}
{}\leq{}
O(1)\left(1-c\right)^k
\left(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^0)-\min\@ifstar\@@P\@Phi\right),
\]
where $M$ is as in \Cref{thm:strconcost}.
\fi
\section{Conclusions}\label{sec:Conclusions}
We presented a general block-coordinate forward-backward algorithm for minimizing the sum of a separable smooth and a nonseparable nonsmooth function, both allowed to be nonconvex.
The framework is general enough to encompass regularized finite sum minimization and sharing problems, and leads to (a generalization of) the Finito/MISO algorithm \cite{defazio2014finito,mairal2015incremental} with new convergence results and with another novel incremental-type algorithm.
The forward-backward envelope is shown to be a particularly suitable Lyapunov function for establishing convergence: additionally to enjoying favorable continuity properties, \emph{sure} descent (as opposed to in expectation) occurs along the iterates.
Possible future developments include extending the framework to account for a nonseparable smooth term, for instance by ``quantifying the strength of coupling'' between blocks of variables as in \cite[\S7.5]{bertsekas1989parallel}.
\ifarxiv
\fi
\begin{appendix}
\section{The key tool: the forward-backward envelope}\label{sec:appendix}
This appendix contains some proofs and auxiliary results omitted in the main body.
We begin by observing that, since \(F\) and \(-F\) are 1-smooth in the metric induced by
\(
\Lambda_F\coloneqq\tfrac1N\blockdiag(L_{f_1}\I_{n_1},\dots,L_{f_N}\I_{n_N})
\),
one has
\begin{equation}\label{eq:Lip}
F(\bm x)+\innprod{\nabla F(\bm x)}{\bm w-\bm x}
{}-{}
\tfrac12\|\bm w-\bm x\|_{\Lambda_F}^2
{}\leq{}
F(\bm w)
{}\leq{}
F(\bm x)+\innprod{\nabla F(\bm x)}{\bm w-\bm x}
{}+{}
\tfrac12\|\bm w-\bm x\|_{\Lambda_F}^2
\end{equation}
for all \(\bm x,\bm w\in\R^{\sum_in_i}\), see \cite[Prop. A.24]{bertsekas2016nonlinear}.
Let us denote
\[
\M(\bm w,\bm x)
{}\coloneqq{}
F(\bm x)+\innprod{\nabla F(\bm x)}{\bm w-\bm x}
{}+{}
G(\bm w)
{}+{}
\tfrac12\|\bm w-\bm x\|_{\Gamma^{-1}}^2
\]
the quantity being minimized (with respect to \(\bm w\)) in the definition \eqref{eq:FBE} of the FBE.
It follows from \eqref{eq:Lip} that
\begin{equation}\label{eq:bounds}
\@ifstar\@@P\@Phi(\bm w)
{}+{}
\tfrac12\|\bm w-\bm x\|^2_{\Gamma^{-1}-\Lambda_F}
{}\leq{}
\M(\bm w,\bm x)
{}\leq{}
\@ifstar\@@P\@Phi(\bm w)
{}+{}
\tfrac12\|\bm w-\bm x\|^2_{\Gamma^{-1}+\Lambda_F}
\end{equation}
holds for all \(\bm x,\bm w\in\R^{\sum_in_i}\).
In particular, \(\M\) is a \emph{majorizing model} for \(\@ifstar\@@P\@Phi\), in the sense that \(\M(\bm x,\bm x)=\@ifstar\@@P\@Phi(\bm x)\) and \(\M(\bm w,\bm x)\geq\@ifstar\@@P\@Phi(\bm w)\) for all \(\bm x,\bm w\in\R^{\sum_in_i}\).
In fact, as explained in \Cref{sec:FBE}, while a \(\Gamma\)-forward-backward step \(\bm z\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\) amounts to evaluating a minimizer of \(\M({}\cdot{},\bm x)\), the FBE is defined instead as the minimization value, namely \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)=\M(\bm z,\bm x)\) where \(\bm z\) is any element of \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\).
\subsection{Proofs of \texorpdfstring{\Cref{sec:FBE}}{\S\ref*{sec:FBE}}}\label{sec:proofs:FBE}
\begin{appendixproof}{thm:osc}
For \(\bm x^\star\in\argmin\@ifstar\@@P\@Phi\) it follows from \eqref{eq:Lip} that
\[
\min\@ifstar\@@P\@Phi
{}\leq{}
F(\bm x)
{}+{}
G(\bm x)
{}\leq{}
G(\bm x)
{}+{}
F(\bm x^\star)
{}+{}
\innprod{\nabla F(\bm x^\star)}{\bm x-\bm x^\star}
{}+{}
\tfrac12\|\bm x^\star-\bm x\|_{\Lambda_F}^2.
\]
Therefore, \(G\) is lower bounded by a quadratic function with quadratic term \(-\tfrac12\|{}\cdot{}\|_{\Lambda_F}^2\), and thus is prox-bounded in the sense of \cite[Def. 1.23]{rockafellar2011variational}.
The claim then follows from \cite[Th. 1.25 and Ex. 5.23(b)]{rockafellar2011variational} and the continuity of the forward mapping \(\Fw{}\).
\end{appendixproof}
\begin{appendixproof}{thm:FBEineq}
Local Lipschitz continuity
\ifarxiv
of the FBE
\fi
follows from \eqref{eq:FBEMoreau} in light of \Cref{thm:osc} and \cite[Ex. 10.32]{rockafellar2011variational}.
\begin{proofitemize}
\item\ref{thm:leq}~
Follows by replacing \(\bm w=\bm x\) in \eqref{eq:FBE}.
\item\ref{thm:geq}~
Directly follows from \eqref{eq:bounds} and the identity \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)=\M(\bm z,\bm x)\) for \(\bm z\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\).
\item\ref{thm:strconcost}~
By strong convexity, denoting \(\@ifstar\@@P\@Phi_\star\coloneqq\min\@ifstar\@@P\@Phi\), we have
\[
\@ifstar\@@P\@Phi_\star
{}\leq{}
\@ifstar\@@P\@Phi(\bm z)-\tfrac12\|\bm z-\bm x^\star\|_{\mu_F}^2
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}-{}
\tfrac12\|\bm z-\bm x^\star\|_{\mu_F}^2
\]
where the second inequality follows from \Cref{thm:geq}.
\qedhere
\end{proofitemize}
\end{appendixproof}
\begin{appendixproof}{thm:FBEmin}
\begin{proofitemize}
\item\ref{thm:min} and \ref{thm:argmin}~
It follows from \Cref{thm:leq} that \(\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\leq\min\@ifstar\@@P\@Phi\).
Conversely, let \(\seq{\bm x^k}\) be such that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)\to\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) as \(k\to\infty\), and for each \(k\) let \(\bm z^k\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^k)\).
It then follows from \Cref{thm:leq,thm:geq} that
\[
\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}
{}\leq{}
\min\@ifstar\@@P\@Phi
{}\leq{}
\liminf_{k\to\infty}\@ifstar\@@P\@Phi(\bm z^k)
{}\leq{}
\liminf_{k\to\infty}\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}={}
\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}},
\]
hence \(\min\@ifstar\@@P\@Phi=\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\).
Suppose now that \(\bm x\in\argmin\@ifstar\@@P\@Phi\) (which exists by \Cref{ass:basic}); then it follows from \Cref{thm:geq} that \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)=\set{\bm x}\) (for otherwise another element would belong to a lower level set of \(\@ifstar\@@P\@Phi\)).
Combining with \Cref{thm:leq} with \(\bm z=\bm x\) we then have
\[
\min\@ifstar\@@P\@Phi
{}={}
\@ifstar\@@P\@Phi(\bm z)
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}\leq{}
\@ifstar\@@P\@Phi(\bm x)
{}={}
\min\@ifstar\@@P\@Phi.
\]
Since \(\min\@ifstar\@@P\@Phi=\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\), we conclude that \(\bm x\in\argmin\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\), and that in particular \(\inf\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}=\min\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\).
Conversely, suppose \(\bm x\in\argmin\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) and let \(\bm z\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\).
By combining \Cref{thm:leq,thm:geq} we have that \(\bm z=\bm x\), that is, that \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)=\set{\bm x}\).
It then follows from \Cref{thm:geq} and assert \ref{thm:min} that
\[
\@ifstar\@@P\@Phi(\bm x)
{}={}
\@ifstar\@@P\@Phi(\bm z)
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={}
\min\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}
{}={}
\min\@ifstar\@@P\@Phi,
\]
hence \(\bm x\in\argmin\@ifstar\@@P\@Phi\).
\item\ref{thm:LB}~
Due to \Cref{thm:leq}, if \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is level bounded clearly so is \(\@ifstar\@@P\@Phi\).
Conversely, suppose that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is not level bounded.
Then, there exist \(\alpha\in\R\) and \(\seq{\bm x^k}\subseteq\lev_{\leq\alpha}\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) such that \(\|\bm x^k\|\to\infty\) as \(k\to\infty\).
Let \(\lambda=\min_i\set{\gamma_i^{-1}-L_{f_i}N^{-1}}>0\), and for each \(k\in\N\) let \(\bm z^k\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x^k)\).
It then follows from \Cref{thm:geq} that
\[
\min\@ifstar\@@P\@Phi
{}\leq{}
\@ifstar\@@P\@Phi(\bm z^k)
{}\leq{}
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x^k)
{}-{}
\tfrac\lambda2\|\bm x^k-\bm z^k\|^2
{}\leq{}
\alpha
{}-{}
\tfrac\lambda2\|\bm x^k-\bm z^k\|^2,
\]
hence \(\seq{\bm z^k}\subseteq\lev_{\leq\alpha}\@ifstar\@@P\@Phi\) and
\(
\|\bm x^k-\bm z^k\|^2
{}\leq{}
\tfrac2\lambda(\alpha-\min\@ifstar\@@P\@Phi)
\).
Consequently, also the sequence \(\seq{\bm z^k}\subseteq\lev_{\leq\alpha}\@ifstar\@@P\@Phi\) is unbounded, proving that \(\@ifstar\@@P\@Phi\) is not level bounded.
\qedhere
\end{proofitemize}
\end{appendixproof}
\subsection{Further results}\label{sec:auxiliary}
This section contains a list of auxiliary results invoked in the main proofs of \Cref{sec:convergence}.
\begin{lem}\label{thm:critical}
Suppose that \Cref{ass:basic} holds, and let two sequences \(\seq{\bm u^k}\) and \(\seq{\bm v^k}\) satisfy \(\bm v^k\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm u^k)\) for all \(k\) and be such that both converge to a point \(\bm u^\star\) as \(k\to\infty\).
Then, \(\bm u^\star\in\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm u^\star)\), and in particular \(0\in\hat\partial\@ifstar\@@P\@Phi(\bm u^\star)\).
\begin{proof}
Since \(\nabla F\) is continuous, it holds that \(\Fw{\bm u^k}\to\Fw{\bm u^\star}\) as \(k\to\infty\).
From outer semicontinuity of \(\prox_G^{\Gamma^{-1}}\) \cite[Ex. 5.23(b)]{rockafellar2011variational} it then follows that
\[
\bm u^\star
{}={}
\lim_{k\to\infty}
\bm v^k
{}\in{}
\limsup_{k\to\infty}
\prox_G^{\Gamma^{-1}}(\Fw{\bm u^k})
{}\subseteq{}
\prox_G^{\Gamma^{-1}}(\Fw{\bm u^\star})
{}={}
\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm u^\star),
\]
where the limit superior is meant in the Painlevé-Kuratowski sense, cf. \cite[Def. 4.1]{rockafellar2011variational}.
The optimality conditions defining \(\prox_G^{\Gamma^{-1}}\) \cite[Th. 10.1]{rockafellar2011variational} then read
\begin{align*}
0
{}\in{} &
\hat\partial\left(
G+\tfrac12\|{}\cdot{}-(\Fw{\bm u^\star})\|_{\Gamma^{-1}}^2
\right)(\bm u^\star)
{}={}
\hat\partial G(\bm u^\star)
{}+{}
\Gamma^{-1}\left(
\bm u^\star - (\Fw{\bm u^\star})
\right)
\\
{}={} &
\hat\partial G(\bm u^\star)
{}+{}
\nabla F(\bm u^\star)
{}={}
\hat\partial\@ifstar\@@P\@Phi(\bm u^\star),
\end{align*}
where the first and last equalities follow from \cite[Ex. 8.8(c)]{rockafellar2011variational}.
\end{proof}
\end{lem}
\begin{lem}
Suppose that \Cref{ass:basic} holds and that function \(G\) is convex.
Then, the following hold:
\begin{enumerate}
\item\label{thm:FNE}
\(\prox_G^{\Gamma^{-1}}\) is (single-valued and) firmly nonexpansive (FNE) in the metric $\|{}\cdot{}\|_{\Gamma^{-1}}$; namely,
\[
\|
\prox_G^{\Gamma^{-1}}(\bm u)
{}-{}
\prox_G^{\Gamma^{-1}}(\bm v)
\|_{\Gamma^{-1}}^2
{}\leq{}
\innprod{
\prox_G^{\Gamma^{-1}}(\bm u)
{}-{}
\prox_G^{\Gamma^{-1}}(\bm v)
}{
\Gamma^{-1}(\bm u-\bm v)
}
{}\leq{}
\|
\bm u
{}-{}
\bm v
\|_{\Gamma^{-1}}^2
\quad\forall\bm u,\bm v;
\]
\item\label{thm:MoreauGrad}
the Moreau envelope \(G^{\Gamma^{-1}}\) is differentiable with \(\nabla G^{\Gamma^{-1}}=\Gamma^{-1}(\id-\prox_G^{\Gamma^{-1}})\);
\item\label{thm:subdiffdist}
for every \(\bm x\in\R^{\sum_in_i}\) it holds that
\(
\dist(0,\partial\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x))
{}\leq{}
\tfrac{
N+\max_i\set{\gamma_iL_{f_i}}
}{
N\min_i\set{\sqrt{\gamma_i}}
}
\|\bm x-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\|_{\Gamma^{-1}}
\);
\item\label{thm:TLip}\label{thm:contractive}
\(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}\) is \(L_{\bf T}\)-Lipschitz continuous in the metric $\|{}\cdot{}\|_{\Gamma^{-1}}$ for some \(L_{\bf T}\geq0\); if in addition \(f_i\) is \(\mu_{f_i}\)-strongly convex, \(i\in[N]\), then \(L_{\bf T}\leq 1-\delta\) for \(\delta=\frac1N\min_{i\in[N]}\set{\gamma_i\mu_{f_i}}\).
\end{enumerate}
\begin{proof}
\begin{proofitemize}
\item\ref{thm:FNE} and \ref{thm:MoreauGrad}~
See \cite[Prop.s 12.28 and 12.30]{bauschke2017convex}.
\item\ref{thm:subdiffdist}
Let \(D\subseteq\R^{\sum_in_i}\) be the set of points at which \(\nabla F\) is differentiable.
From the chain rule of differentiation applied to the expression \eqref{eq:FBEMoreau} and using assert \ref{thm:MoreauGrad}, we have that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is differentiable on \(D\) with gradient
\[
\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={}
\bigl[
\I-\Gamma\nabla^2F(\bm x)
\bigr]
\Gamma^{-1}
\bigl[
\bm x-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)
\bigr]
\quad
\forall\bm x\in D.
\]
Since \(D\) is dense in \(\R^{\sum_in_i}\) owing to Lipschitz continuity of \(\nabla F\), we may invoke \cite[Th. 9.61]{rockafellar2011variational} to infer that \(\partial\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)\) is nonempty for every \(\bm x\in\R^{\sum_in_i}\) and
\[
\partial\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}\supseteq{}
\partial_B\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={}
\bigl[
\I-\Gamma\partial_B\nabla F(\bm x)
\bigr]
\Gamma^{-1}
\bigl[
\bm x-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)
\bigr]
{}={}
\bigl[
\Gamma^{-1}-\partial_B\nabla F(\bm x)
\bigr]
\bigl[
\bm x-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)
\bigr],
\]
where \(\partial_B\) denotes the (set-valued) Bouligand differential \cite[\S7.1]{facchinei2003finite}.
The claim now follows by observing that
\(
\partial_B\nabla F(\bm x)
{}={}
\tfrac1N\blockdiag(\partial_B\nabla f_1(x_1),\dots,\partial_B\nabla f_N(x_N))
\)
and that each element of \(\partial_B\nabla f_i(x_i)\) has norm bounded by \(L_{f_i}\).
\item\ref{thm:TLip}~
Lipschitz continuity follows from assert \ref{thm:FNE} together with the fact that Lipschitz continuity is preserved by composition.
Suppose now that \(f_i\) is \(\mu_{f_i}\)-strongly convex, \(i\in[N]\).
By \cite[Thm 2.1.12]{nesterov2013introductory} for all $x_i,y_i\in\R^{n_i}$
\begin{equation}\label{eq:smoothStrcvx}
\langle\nabla f_i(x_i)-\nabla f_i(y_i),x_i-y_i\rangle\geq\tfrac{\mu_{f_i}L_{f_i}}{\mu_{f_i}+L_{f_i}}\|x_i-y_i\|^2+\tfrac1{\mu_{f_i}+L_{f_i}}\|\nabla f_i(x_i)-\nabla f_i(y_i)\|^2.
\end{equation}
For the forward operator we have
\begin{align*}
&
\|
(\id-\tfrac{\gamma_i}{N}\nabla f_i)(x_i)
{}-{}
(\id-\tfrac{\gamma_i}{N}\nabla f_i)(y_i)
\|^2
\\
{}={} &
\|x_i-y_i\|^2
{}+{}
\tfrac{\gamma_i^2}{N^2}
\|\nabla f_i(x_i)-\nabla f_i(y_i)\|^2
{}-{}
\tfrac{2\gamma_i}{N}
\innprod{x_i-y_i}{\nabla f_i(x_i)-\nabla f_i(y_i)}
\\
\overrel[\leq]{\eqref{eq:smoothStrcvx}}{} &
\Bigl(
1-\tfrac{\gamma_i^2\mu_{f_i}L_{f_i}}{N^2}
\Bigr)
\|x_i-y_i\|^2
{}-{}
\tfrac{\gamma_i}{N}
\Bigl(
2-\tfrac{\gamma_i}{N}(\mu_{f_i}+L_{f_i})
\Bigr)
\innprod{\nabla f_i(x_i)-\nabla f_i(y_i)}{x_i-y_i}
\\
{}\leq{} &
\left(1-\tfrac{\gamma_i^2\mu_{f_i}L_{f_i}}{N^2}\right)
\|x_i-y_i\|^2
{}-{}
\tfrac{\gamma_i\mu_{f_i}}{N}
\left(2-\tfrac{\gamma_i}{N}(\mu_{f_i}+L_{f_i})\right)
\|x_i-y_i\|^2
\\
{}={} &
\left(1-\tfrac{\gamma_i\mu_{f_i}}{N}\right)^2
\|x_i-y_i\|^2,
\end{align*}
where strong convexity and the fact that $\gamma_i<\nicefrac{N}{L_{f_i}}\leq\nicefrac{2N}{(\mu_{f_i}+L_{f_i})}$ was used in the second inequality.
Multiplying by $\gamma_i^{-1}$ and summing over $i$ shows that \(\id-\Gamma\nabla F\) is \((1-\delta)\)-contractive in the metric \(\|{}\cdot{}\|_{\Gamma^{-1}}\), and so is \(\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}=\prox_G^{\Gamma^{-1}}\circ(\Fw{})\) as it follows from assert \ref{thm:FNE}.
\qedhere
\end{proofitemize}
\end{proof}
\end{lem}
The next result recaps an important property that the FBE inherits from the cost function \(\@ifstar\@@P\@Phi\) that is instrumental for establishing global convergence and asymptotic linear rates for the BC-\Cref{alg:BC}.
The result falls as special case of \cite[Th. 5.2]{yu2019deducing} after observing that
\[
\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={}
\inf_{\bm w}\set{
\@ifstar\@@P\@Phi(\bm w)
{}+{}
D_H(\bm w,\bm x)
},
\]
where
\(
D_H(\bm w,\bm x)
{}={}
H(\bm w)-H(\bm x)-\innprod{\nabla H(\bm x)}{\bm w-\bm x}
\)
is the Bregman distance with kernel \(H=\tfrac12\|{}\cdot{}\|_{\Gamma^{-1}}^2-F\).
\begin{lem}[{\cite[Th. 5.2]{yu2019deducing}}]\label{thm:loja}
Suppose that \Cref{ass:basic} holds and for \(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\), \(i\in[N]\), let \(\Gamma=\blockdiag(\gamma_1\I_{n_1},\dots,\gamma_N\I_{n_N})\).
If \(\@ifstar\@@P\@Phi\) has the KL property with exponent \(\theta\in(0,1)\) (as is the case when \(f_i\) and \(G\) are semialgebraic), then so does \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) with exponent
\(
\max\set{\nicefrac12,\theta}
\).
\end{lem}
\ifaccel
\begin{lem}[FBE: convexity and block-smoothness]\label{thm:convex}
Suppose that \Cref{ass:basic,ass:Fast} are satisfied, and consider the notation introduced therein.
Let \(\gamma_i\in(0,\nicefrac{N}{L_{f_i}})\) be fixed.
Define
\(
Q_i
{}\coloneqq{}
\gamma_i^{-1}\I-\tfrac{1}{N}H_i\in\R^{n_i\times n_i}
\),
\(
Q
{}\coloneqq{}
\blockdiag(Q_1,\dots,Q_N)
\),
and
\(
H
{}\coloneqq{}
\tfrac1N\blockdiag(H_1,\dots,H_N)
\).
Then, $\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C \coloneqq\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}} \circ Q^{-1/2}$ is convex and smooth with $\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C (\tilde{\bm x}) = Q^{1/2}(\bm x-\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x))$ where $\bm x=Q^{-1/2}\tilde{\bm x}$.
In fact, for any \(\tilde{\bm x},\tilde{\bm x}'\in\R^{\sum_in_i}\) it holds that
\begin{equation}\label{eq:FBEComposedsmooth}
0
{}\leq{}
\innprod{\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x}')-\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x})}{\tilde{\bm x}'-\tilde{\bm x}}
{}\leq{}
\|\tilde{\bm x}'-\tilde{\bm x}\|^2.
\end{equation}
In particular, function $\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C$ is $1$-smooth along each block $i\in[N]$. If, additionally, all functions \(f_i\) are strongly convex, then
$\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C$ is \(\sigma\)-strongly convex with $\sigma\coloneqq \tfrac{1}{N}\min_{i\in [N]}\left\{\gamma_i\mu_{f_i}\right\}$.
\begin{proof}
Since $\gamma_i<N/L_{f_i}$, $Q$ is positive definite.
We begin by showing that for any \(\bm x,\bm x'\in\R^{\sum_in_i}\) it holds that
\begin{equation}\label{eq:FBEsmooth}
0\leq \|\bm x'-\bm x\|^2_{Q}- \|Q(\bm x'-\bm x)\|^2_{\Gamma}
{}\leq{}
\innprod{\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x')-\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)}{\bm x'-\bm x}
{}\leq{}
\|\bm x'-\bm x\|^2_Q.
\end{equation}
It follows from \Cref{thm:MoreauGrad}, the chain rule of differentiation applied to \eqref{eq:FBEMoreau}, and the twice continuous differentiability of \(F\) that \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is continuously differentiable with
\(
\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)
{}={}
Q(\bm x-\bm z)
\).
For \(\bm z^x\coloneqq\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm x)\) and \(\bm z^{x'}\coloneqq\@ifstar\operatorname T_\gamma^{\text{\sc fb}}\operatorname T_\Gamma^{\text{\sc fb}}(\bm {x'})\) it holds that
\begin{equation}\label{eq:innprodGrad}
\innprod{\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm {x'})-\nabla\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}(\bm x)}{\bm {x'}-\bm x}
{}={}
\innprod{
Q(\bm {x'}-\bm z^{x'}-\bm x+\bm z^x)
}{
\bm {x'}-\bm x
}
{}={}
\|\bm {x'}-\bm x\|^2_Q
{}-{}
\innprod{
\bm z^{x'}-\bm z^x
}{
Q(\bm {x'}-\bm x)
}.
\end{equation}
In order to bound the last scalar product, observe that
\[
0
{}\leq{}
\innprod{
\Gamma^{-1}(\bm z^{x'}-\bm z^x)
}{
(\bm {x'}-\Gamma\nabla F(\bm {x'}))
{}-{}
(\bm x-\Gamma\nabla F(\bm x))
}
{}\leq{}
\bigl\|
(\bm {x'}-\Gamma\nabla F(\bm {x'}))
{}-{}
(\bm x-\Gamma\nabla F(\bm x))
\bigr\|_{\Gamma^{-1}}^2,
\]
as it follows from \Cref{thm:FNE}.
Since \(\id-\Gamma\nabla F=\Gamma Q{}\cdot{} - \Gamma\bm q\) (with $\bm q\coloneqq(\tfrac1Nq_1,\dots,\tfrac1Nq_N)$), the above inequality simplifies to
\[
0
{}\leq{}
\innprod{
\bm z^{x'}-\bm z^x
}{
Q(\bm {x'}-\bm x)
}
{}\leq{}
\|\Gamma Q(\bm {x'}-\bm x)\|_{\Gamma^{-1}}^2,
\]
which combined with \eqref{eq:innprodGrad} results in the claimed \eqref{eq:FBEsmooth}.
If additionally \(\mu_{f_i}>0\) for all \(i\), then \(\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}\) is \(1\)-strongly convex in the metric \(\|{}\cdot{}\|^2_{Q - Q\Gamma Q}\) (by observing that \(Q-Q\Gamma Q\succ 0\)). The result in \eqref{eq:FBEComposedsmooth} follows by using \eqref{eq:FBEsmooth} with the change of variables $\bm x=Q^{-1/2}\tilde{\bm x}$, $\bm x'=Q^{-1/2}\tilde{\bm x}'$ and noting that $\nabla \@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C(\tilde{\bm x}) = Q^{-1/2}\nabla \@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}} (\bm x)$.
Since $\Gamma$ is block-wise a multiple of identity it commutes with any block-diagonal matrix. Therefore, when $f_i$ are strongly convex, using the lower bound in \eqref{eq:FBEsmooth} and the above change of variable we obtain that $\@ifstar\@ifstar\@@P\@Phi_\gamma^{\text{\sc fb}}\@ifstar\@@P\@Phi_\Gamma^{\text{\sc fb}}C$ is strongly convex in the metric \(\|{}\cdot{}\|^2_{\I - \Gamma Q}\). The result follows by noting that $\I - \Gamma Q= \Gamma H$.
\end{proof}
\end{lem}
\fi
\end{appendix}
\ifarxiv
\else
\phantomsection
\addcontentsline{toc}{section}{References}
\fi
\end{document}
|
\begin{document}
\title{On diagram groups over Fibonacci-like semigroup presentations and their generalizations}
\author{
V. S. Guba\thanks{This work is partially supported by the Russian Foundation
for Basic Research, project no. 19-01-00591 A.}\\
Vologda State University,\\
15 Lenin Street,\\
Vologda\\
Russia\\
160600\\
E-mail: guba{@}uni-vologda.ac.ru}
\date{}
\maketitle
\begin{abstract}
We answer the question by Matt Brin on the structure of diagram groups over semigroup
presentation ${\mathcal P}=\langle\,ngle a,b,c\mid a=bc,b=ca,c=ab\,\ranglengle$. In the talk on Oberwolfach workshop,
Brin conjectured that the diagram group over $\mathcal P$ with base $a$ is isomorphic to the generalized
Thompson's group $F_9$. We confirm this conjecture and consider some generalizations of this fact.
\end{abstract}
\section{Introduction}
\langle\,bel{backgr}
In this background Section we recall the concept of diagram groups and introduce some terminology. The contents of the present
Section is essentially known. Some defininions and examples from here repeat the ones from \cite{Gu04}. Detailed information about
diagram groups can be found in \cite{GbS}.
First of all, let us recall the concept of a semigroup diagram and
introduce some notation. To do this, we consider the following example.
Let ${\mathcal P}=\langle\, a,b\mid aba=b,bab=a\,\rangle$ be the semigroup presentation. (In the next Section we will work with it.)
It is easy to see by the following algebraic calculation
$$
a^5=a(bab)a(bab)a=(aba)(bab)(aba)=bab=a
$$
that the words $a^5$ and $a$ are equal modulo ${\mathcal P}$. The same can be seen from the following picture
\begin{center}
\begin{picture}(90.00,37.00)
\put(00.00,23.00){\circle*{1.00}}
\put(10.00,23.00){\circle*{1.00}}
\put(20.00,23.00){\circle*{1.00}}
\put(30.00,23.00){\circle*{1.00}}
\put(30.00,23.00){\circle*{1.00}}
\put(40.00,23.00){\circle*{1.00}}
\put(50.00,23.00){\circle*{1.00}}
\put(60.00,23.00){\circle*{1.00}}
\put(60.00,23.00){\circle*{1.00}}
\put(70.00,23.00){\circle*{1.00}}
\put(80.00,23.00){\circle*{1.00}}
\put(90.00,23.00){\circle*{1.00}}
\put(00.00,23.00){\line(1,0){90.00}}
\bezier{152}(10.00,23.00)(25.00,35.00)(40.00,23.00)
\bezier{240}(50.00,23.00)(80.00,23.00)(50.00,23.00)
\bezier{164}(50.00,23.00)(65.00,37.00)(80.00,23.00)
\bezier{240}(0.00,23.00)(30.00,23.00)(0.00,23.00)
\bezier{156}(0.00,23.00)(17.00,11.00)(30.00,23.00)
\bezier{164}(30.00,23.00)(44.00,9.00)(60.00,23.00)
\bezier{164}(60.00,23.00)(74.00,9.00)(90.00,23.00)
\put(5.00,25.00){\makebox(0,0)[cc]{$a$}}
\put(24.00,32.00){\makebox(0,0)[cc]{$a$}}
\put(45.00,25.00){\makebox(0,0)[cc]{$a$}}
\put(65.00,32.00){\makebox(0,0)[cc]{$a$}}
\put(84.00,25.00){\makebox(0,0)[cc]{$a$}}
\put(23.00,16.00){\makebox(0,0)[cc]{$b$}}
\put(44.00,13.00){\makebox(0,0)[cc]{$a$}}
\put(65.00,16.00){\makebox(0,0)[cc]{$b$}}
\put(15.00,20.00){\makebox(0,0)[cc]{$b$}}
\put(24.00,25.00){\makebox(0,0)[cc]{$a$}}
\put(35.00,21.00){\makebox(0,0)[cc]{$b$}}
\put(53.00,21.00){\makebox(0,0)[cc]{$b$}}
\put(66.00,25.00){\makebox(0,0)[cc]{$a$}}
\put(74.00,21.00){\makebox(0,0)[cc]{$b$}}
\bezier{520}(0.00,23.00)(45.00,-24.00)(90.00,23.00)
\put(44.00,2.00){\makebox(0,0)[cc]{$a$}}
\end{picture}
\end{center}
This is a {\em diagram\/} $\Delta$ over the semigroup presentation ${\mathcal P}$.
It is a plane graph with $10$ vertices, $15$ (geometric) edges and $6$ faces
or {\em cells\/}. Each cell corresponds to an elementary transformation of a
word, that is, a transformation of the form
$p\cdot u\cdot q\to p\cdot v\cdot q$, where $p$, $q$ are words (possibly,
empty), $u=v$ or $v=u$ belongs to the set of defining relations. The
diagram $\Delta$ has the leftmost vertex denoted by $\iota(\Delta)$ and
the rightmost vertex denoted by $\tau(\Delta)$. It also has the {\em top
path\/} $\mathop{\mbox{\bf top}}(\Delta)$ and the {\em bottom path\/} $\mathop{\mbox{\bf bot}}(\Delta)$ from
$\iota(\Delta)$ to $\tau(\Delta)$. Each cell $\pi$ of a diagram can be
regarded as a diagram itself. The above functions $\iota$, $\tau$, $\mathop{\mbox{\bf top}}$,
$\mathop{\mbox{\bf bot}}$ can be applied to $\pi$ as well. We do not distinguish isotopic
diagrams.
We say that $\Delta$ is a $(w_1,w_2)$-diagram whenever the label of its
top path is $w_1$ and the label of its bottom path is $w_2$. In our example,
we deal with an $(a^5,a)$-diagram. If we have two diagrams such that the
bottom path of the first of them has the same label as the top path of the
second, then we can naturally {\em concatenate\/} these diagrams by
identifying the bottom path of the first diagram with the top path of the
second diagram. The result of the concatenation of a $(w_1,w_2)$-diagram and
a $(w_2,w_3)$-diagram obviously is a $(w_1,w_3)$-diagram. We use the sign
$\circ$ for the operation of concatenation. For any diagram $\Delta$ over
${\mathcal P}$ one can consider its {\em mirror image\/} $\Delta^{-1}$. A diagram may
have {\em dipoles\/}, that is, subdiagrams of the form $\pi\circ\pi^{-1}$,
where $\pi$ is a single cell. To {\em cancel\/} (or {\em reduce\/}) the
dipole means to remove the common boundary of $\pi$ and $\pi^{-1}$ identifying $\mathop{\mbox{\bf top}}(\pi)$ with $\mathop{\mbox{\bf bot}}(\pi^{-1})$.
In any diagram, we can cancel all its dipoles, step by step. The result does not depend on the order of
cancellations. A diagram is {\em irreducible\/} whenever it has no dipoles.
The operation of cancelling dipoles has an inverse operation called the
{\em insertion\/} of a dipole. These operations induce an equivalence
relation on the set of diagrams (two diagrams are {\em equivalent\/} whenever
one can go from one of them to the other by a finite sequence of
cancelling/inserting dipoles). Each equivalence class contains exactly one
irreducible diagram.
For any nonempty word $w$, the set of all $(w,w)$-diagrams forms a monoid
with the identity element $\varepsilon(w)$ (the diagram with no cells). The
operation $\circ$ naturally induces some operation on the set of equivalence
classes of diagrams. This operation is called a {\em product\/} and
equivalent diagrams are called {\em equal\/}. (The sign $\equiv$ will be
used to denote that two diagrams are isotopic.) So the set of all
equivalence classes of $(w,w)$-diagrams forms a group that is called the
{\em diagram group\/} over ${\mathcal P}$ with {\em base\/} $w$. We denote this
group by ${\mathcal D}({\mathcal P},w)$. We can think of this group as of the set
of all irreducible $(w,w)$-diagrams. The group operation is the
concatenation with cancelling all dipoles in the result. An inverse
element of a diagram is its mirror image. We also need one more natural
operation on the set of diagrams. By the {\em sum\/} of two diagrams we
mean the diagram obtained by identifying the rightmost vertex of the
first summand with the leftmost vertex of the second summand. This
operation is also associative. The sum of diagrams $\Delta_1$,
$\Delta_2$ is denoted by $\Delta_1+\Delta_2$.
Now let us recall some information about generalized Thompson's groups $F_r$.
This family was introduced by K. S. Brown in \cite{Bro}. Additional facts about these groups
can be found in \cite{BCS,Stein}.
The family of generalized Thompson's groups can be defined as follows.
The group $F_r$ is the group of all piecewise linear self homeomorphisms
of the unit interval $[0,1]$ that are orientation preserving (that is,
send $0$ to zero and $1$ to $1$) with all slopes integer powers of $r$
and such that their singularities (breakpoints of the derivative) belong
to $\mathbb Z[\,\frac1r\,]$. The group $F_r$ admits a presentation given by
\be{presfp}
\langle\, x_0,x_1,x_2,\ldots\mid x_jx_i=x_ix_{j+r-1}\ (i<j)\,\rangle.
\end{equation}
\noindent
This presentation is infinite, but a close examination shows that the
group is actually finitely generated, since $x_0$, $x_1$, \dots, $x_{p-1}$
are sufficient to generate it. In fact, the group is finitely presented; see \cite{Bro}.
The finite presentation is awkward, and it is not used much. The symmetric
and simple nature of the infinite presentation makes it much more adequate
for almost all purposes.
One way in which the infinite presentation is very useful is in the construction of the normal forms.
A word given in the generators $x_i$ and their inverses, can have its generators moved around
according to the relators, and the result is the following well-known statement:
\begin{thm}
\langle\,bel{stnf}
An element in $F_r$ always admits an expression of the form
$$
x_{i_1}x_{i_2}\cdots x_{i_m}x_{j_n}^{-1}\cdots x_{j_2}^{-1}x_{j_1}^{-1},
$$
where
$$
i_1\le i_2\le\cdots\le i_m,\ j_1\le j_2\le\cdots\le j_n.
$$
\end{thm}
In general, this expression is not unique, but for every element there is
a unique word of this type which satisfies certain technical condition.
This unique word is called the {\em standard normal form\/} for the
element of $F_r$.
The case $r=2$ corresponds to famous R. Thompson's group $F=F_2$.
It is known \cite{GbS} that groups $F_r$ are diagram groups over the
semigroup presentation ${\mathcal P}_r=\langle\, x\mid x=x^r\,\rangle$ with base $x$ (note that
for any base $x^k$, where $k\ge1$, we get an isomorphic group).
Now let us compare the diagram representation of $F$ with the
representation of its elements by piecewise-linear homeomorphisms of
the closed unit interval $[0,1]$. Let $\Delta$ be an $(x^p,x^q)$-diagram
over ${\mathcal P}$. We will show how to assign to it a piecewise-linear function
from $[0,p]$ onto $[0,q]$. Each positive edge of $\Delta$ is homeomorphic
to the unit interval $[0,1]$. So we assign a coordinate to each point of
this edge (the leftmost end of an edge has coordinate $0$, the rightmost one
has coordinate $1$). Let $\pi$ be an $(x,x^r)$-cell of $\Delta$. Let us map
$\mathop{\mbox{\bf top}}(\pi)$ onto $\mathop{\mbox{\bf bot}}(\pi)$ linearly, that it, the point on the edge
$\mathop{\mbox{\bf top}}(\pi)$ with coordinate $t\in[0,1]$ is taken to the point on
$\mathop{\mbox{\bf bot}}(\pi)$ with coordinate $rt$ (the bottom path of $\pi$ has length
$r$ so it is naturally homeomorphic to $[0,r]$). The same thing can be
done for an $(x^r,x)$-cell of $\Delta$. Thus for any cell $\pi$ of $\Delta$
we have a natural mapping $T_\pi$ from $\mathop{\mbox{\bf top}}(\pi)$ onto $\mathop{\mbox{\bf bot}}(\pi)$ (we
call it a {\em transition map\/}). Now let $t$ be any number in $[0,p]$. We
consider the point $o$ on $\mathop{\mbox{\bf top}}(\Delta)$ that has coordinate $t$. If $o$ is
not a point of $\mathop{\mbox{\bf bot}}(\Delta)$, then it is an internal point on the top path
of some cell. Thus we can apply the corresponding transition map to $o$. We
repeat this operation until we get a point $o'$ on the path $\mathop{\mbox{\bf bot}}(\Delta)$.
The coordinate of this point is a number in $[0,q]$. Hence we have a function
$f_\Delta\colon[0,p]\to[0,q]$ induced by $\Delta$. It is easy to see this
will be a piecewise-linear function. When we concatenate diagrams, this
corresponds to the composition of the PL functions induced by these diagrams.
For groups $F_r$, which are the diagram group ${\mathcal D}({\mathcal P}_r,x)$, we have the
homomorphism from it to $PLF[0,1]$. It is known this is an monomorphism.
The following elementary fact was essentially used several times in \cite{GuSa99,Gu00}
and some other papers.
\begin{lm}
\langle\,bel{longpath}
Let ${\mathcal P}=\langle\, X\mid{\mathcal R}\,\rangle$ be a semigroup presentation. Suppose that all
defining relations of ${\mathcal P}$ have the form $a=A$, where $a\in X$ and $A$ is
a word of length at least $2$. Also assume that all letters in the
left-hand sides of the defining relations are different. Then any
irreducible diagram $\Delta$ over ${\mathcal P}$ is the concatenation of the form
$\Delta_1\circ\Delta_2^{-1}$, where the top path of each cell of both
$\Delta_1$, $\Delta_2$ has length $1$. The longest positive path in $\Delta$
from $\iota(\Delta)$ to $\tau(\Delta)$ coincides with the bottom path of
$\Delta_1$ and the top path of $\Delta_2^{-1}$.
\end{lm}
Note that $\langle\, x\mid x=x^r\,\rangle$ obviously satisfies the conditions of the
Lemma. The same concerns the presentation $\langle\, a,b\mid a=bab, b=aba\,\rangle$,
which was considered in the beginning of this Section. Let us recall the
idea of the proof. Let $p$ be the longest positive path in $\Delta$ from
$\iota(\Delta)$ to $\tau(\Delta)$. It cuts $\Delta$ into two parts. It
suffices to prove that all cells in the ``upper" part correspond to the
defining relations of the form $a=A$, where $a$ is a letter, and none of
them corresponds to $A=a$. Assume the contrary. Suppose that there is a
cell $\pi$ in the upper part of $\Delta$ with the top label $A$ and the
bottom label $a$. The bottom path of $\pi$ cannot be a subpath in $p$ since
$p$ is chosen the longest. So the bottom edge of $\pi$ belongs to the top
path of some cell $\pi'$. The diagram $\Delta$ has no dipoles. All letters
in the left-hand sides of the defining relations are different. So the top
path of $\pi'$ cannot have length $1$. This means that we have found a new
cell in the upper part of $\Delta$ that also corresponds to the defining
relation of the form $A=a$. Applying the same argument to $\pi'$, we get a
process that never terminates. This is impossible since the cells that appear
during the process cannot repeat. This completes the proof.
\section{Main Results}
\langle\,bel{flp}
Let $a_1$, $a_2$, ... , $a_n$ be a finite alphabet. By definition, $a_{n+1}=a_1$, $a_{n+2}=a_2$. Consider the
following semigroup presentation
\be{fs}
{\mathcal P}_n=\langle\,ngle a_1,\ldots,a_n\mid a_i=a_{i+1}a_{i+2}\ (1\le i\le n)\,\ranglengle.
\end{equation}
The semigroup presented by ${\mathcal P}_n$ is called {\em Fibonacci semigroup\/}. One can ask what are the
diagram groups $G_n=\mathcal D(\mathcal P_n,a_1)$. The case $n=1$ is trivial, it gives the diagram group over
$\langle\,ngle x\mid x=xx\,\ranglengle$ so it is Thompson's group $F$. For $n=2$ one has the presentation $\langle\,ngle a,b\mid
a=ba,b=ab\,\ranglengle$. It was shown in \cite{GoSa17} that $G_2$ (the so called Jones' subgroup) is isomorphic to
$F_3$.
In his talk on an Oberwolfach worksop, Matt Brin asked about the group $G_3$, see \cite[Question 73]{BBN}. He conjectured
that this diagram group is isomorphic to $F_9$. Notice that $\mathcal P_3$ can be written as $\langle\,ngle a,b,c\mid
a=bc,b=ca,c=ab\,\ranglengle$. This presentation is not {\em complete\/}. This means that for the Thue system $ab\to
c$, $bc\to a$, $ca\to b$ there are no unique normal forms. For complete semigroup presentations, there exists a
technique of their calculation from \cite{GbS}. Sometimes it is possible to consider a completion, but here it
has a complicated form. Indeed, the semigroup given by $\mathcal P_3$ is the quaternion group $Q_8$. So this way
of description looks very unclear.
Here we present a purely geometric way to find the diagram group. First of all, let us mention that one can
avoid generator $c$ replacing it by $ab$. In generators $a$, $b$ the presentaion becomes $\langle\,ngle a,b\mid
a=bab,b=aba\,\ranglengle$. It was considered as an example in the beginning of the Introduction. The semigroup
given by it is the same as above.
There is a fact from \cite[Section 4]{GuSa05} that ordinary Tietze transformations of semigroup presentations
lead to the same diagram groups. (This can also be shown directly.) So we have one more generalization of the
class of semigroup presentations under consideration.
Let $a_1$, $a_2$, ... , $a_n$ be a finite alphabet as above and let $r\ge2$ be an integer. For any $j$ from $1$
to $r$ we set $a_{n+j}=a_j$. Now for every $i$ from $1$ to $n$ we consider a relation of the form
$a_i=a_{i+1}\ldots a_{i+r}$. By $\mathcal P_{nr}$ we denote a semigroup presentation given by these relations:
\be{pnr}
{\mathcal P}_{nr}=\langle\,ngle a_1,\ldots,a_n\mid a_i=a_{i+1}\ldots a_{i+r}\ (1\le i\le n)\,\ranglengle.
\end{equation}
This class of presentations was introduced by Johnson in \cite{Jo74} in order to generalize the concept of a
Fibonacci group. Since $\mathcal P_{nr}$ is also a semigroup presentation, one can introduce the corresponding
semigroups as well. For $r=2$ we have the above Fibonacci-like presentations. Now we can consider diagram groups
$G_{nr}$ defined as $\mathcal D(\mathcal P_{nr},a_1)$. The group we are interested in is $G_{32}\cong G_{23}$.
We confirm Brin's conjecture about it.
\begin{thm}
\langle\,bel{f9}
The diagram group with base $a$ over semigroup presentation $\langle\,ngle a,b,c\mid a=bc,b=ca,c=ab\,\ranglengle$ is
isomorphic to generalized Thompson's group $F_9$.
\end{thm}
{\bf Proof.}\
We consider this group as a diagram group over $\mathcal P_{23}=\langle\,ngle a,b\mid a=bab,b=aba\,\ranglengle$. It is known
that the groups $F_r$ have no proper non-Abelian homomorphic images. So it suffices to construct a homomorphism
from $F_9$ to the diagram group $G=G_{23}$ showing it is surjective. Therefore, this will give us an
isomorphism.
The group $F_9$ will be considered as the diagram group over $\langle\,ngle x\mid x=x^9\,\ranglengle$ with base $x$. A
diagram over this presentation is a plane graph composed from cycles of even length. By induction on the number
of cells it is easy to show that the graph is bipartite. So we can give colours to its vertices. Let the initial
vertex of a diagram $\Delta$ gets the colour 1. Then the other vertices get their colours uniquely.
Now we relabel the diagram: if a positive edge goes from a vertex of colour 1 to the vertex of color 2, then we
give it label $a$. Otherwise it has label $b$. As a result, we get a diagram denoted by $\Delta'$. Each
cell $x=x^9$ becomes a cell of one of the two forms: $a=a(ba)^4$ or $b=b(ab)^4$. The same for inverse cells.
We have the following derivation over $\mathcal P_{23}$: $a=bab=(aba)(bab)(aba)=a(ba)^4$, and similarly for the
other equality. Semigroup diagrams for these equalities consist of $4$ cells. They will be called {\em basic\/}. We fill the cells
of the above form by basic diagrams. This gives us the diagram $\Delta''$ over $\mathcal P_{23}$.
The rule $\Delta\mapsto\Delta''$ induces a homomorphism of groupoids of diagrams. (Notice that cancelling a
dipole in a diagram $\Delta$ over $x=x^9$ leads to cancelling $4$ dipoles in $\Delta''$ so the mapping we have
defined preserves equivalence of diagrams.) In particular, we have a homomorphism from $F_9$ as the diagram group
over $x=x^9$ with base $x$ to $G$ as the diagram over $\mathcal P_{23}$ with base $a$.
Now let $\Psi$ be a reduced $(a,a)$-diagram over $\mathcal P_{23}$. We would like to find a preimage of it in
$F_9$. According to Lemma~\ref{longpath}, we decompose $\Psi$ as $\Psi_1\circ\Psi_2^{-1}$ where $\Psi_1$,
$\Psi_2$ are positive diagrams. It holds that $\mathop{\mbox{\bf bot}}{\Psi_1}=\mathop{\mbox{\bf top}}{\Psi_2^{-1}=p}$, where $p$ is the longest
positive path in $\Psi$ from $\iota(\Psi)$ to $\tau(\Psi)$.
Now we will change $\Psi=\Psi_1\circ\Psi_2^{-1}$ and the path $p$ step by step inserting some dipoles. The
current situation will always have the same notation. Suppose that the first edge of $p$ has label $b$. In this
case we replace the subdiagram $\varepsilon(b)$ that consists of one edge by a dipole of the two cells
$(b=aba)\circ(aba=b)$. The new longest path in the diagram we obtain will be still denoted by $p$.
Now look and the subwords of the form $aa$ or $bb$ of the label of $p$. Choose the leftmost of them. If it is
$aa$ then we replace the second edge labelled by $a$ by the dipole $(a=bab)\circ(bab=b)$. If it is $bb$ then
we also replace the second edge of it by the dipole $(b=aba)\circ(aba=b)$.
After a finite number of steps, the label of the longest path $p$ becomes $abab\ldots$\ . The last letter in it
will have label $a$. This follows from parity arguments and the fact that the terminal vertex of $\Psi$ has
colour $2$. Now we have $\Psi=\Psi_1\circ\Psi_2^{-1}$ where $\Psi_1$, $\Psi_2$ are positive $(a,a(ba)^m)$-diagrams
for some $m$. It suffices to show that each diagram with this property belongs to the image of our mapping
$\Delta\mapsto\Delta''$. This means that every positive $(a,a(ba)^m)$-diagram over $\mathcal P_{23}$ can be
composed from basic diagrams. Also we claim a symmetric statement: every positive $(b,b(ab)^m)$-diagram over $\mathcal P_{23}$
can be composed from basic diagrams.
Let $\Phi$ be one of these diagrams. We proceed by induction on the number of cells in it. If there are no cells
($m=0$) then we can nothing to prove. Otherwise let us define the {\em depth\/} of an edge in the diagram. The
top edge will have depth $0$ by definition. All other edges belong to the bottom path of a cell $\pi$. If its top edge has depth $d$,
then we assign depth $d+1$ to our edge. The only important thing for us is whether $d$ is even or odd. So we talk about even and odd edges.
Now we remark the following.
1) Let $e_1$, ... , $e_s$ be all edges coming out of a vertex, read from top to bottom. Then labels of them always change from
$a$ to $b$ and vice versa, and the same for parity of their depth. The same for edges that come into a vertex.
2) If two consecutive edges have the same label, then they have different partity. Otherwise, if the labels are $ab$ or $ba$,
the parity is the same.
The first part is clear. As for the second one, let us consider only one case of the edges labelled by $ab$. Let $e$ be the highest
edge that ends at $v$ (the vertex between $a$ and $b$) and let $f$ be the highest edge that starts at $v$. It is easy to see that $ef$ is a
part of the bottom path of a cell. Therefore, $e$ and $f$ have different labels and the same depth. Now everything follows from 1).
The cases $ba$, $aa$, $bb$ are similar.
Now we look again at the path $p$ (the bottom of $\Phi$). Its first label is $a$, so the first edge is even. Therefore,
all edges of $p$ are even according to 2) since $p$ has label $abab...a$. If $m > 0$ then $\Phi$ has a top cell $a=bab$ with the bottom
path $e_1e_2e_3$. Deleting the top cell gives us a sum of 3 diagrams: $(e_1,p_1)+(e_2,p_2)+(e_3,p_3)$, where $p=p_1p_2p_3$. Each edge
of $p_i$ has odd parity in the $i$-th sumand. Therefore, $e_i$ does not belong to $p_i$. So there exists a top cell in each of the
summands. Together with the cell $a=bab$ we have deleted, they form a basic diagram.
Removing three cells with top edges $e_i$ ($i=1,2,3)$, we get a sum of $9$ positive diagrams. Now all edges of $p$ have even depth so the
inductive assumption can be applied to these summands. This completes the proof.
So this answers Brin's question, and now we look at some generalizations. The next Fibonacci-like presentation
in the series is $\mathcal P_{42}$. Its relations are $a=bc$, $b=cd$, $c=da$, $d=ab$. Applying Tietze
transformations, we rewrite the presentation as $\langle\,ngle a,b\mid a=baba,b=abaab\,\ranglengle$, where $d\to ab$, $c\to da\to
aba$. In the second relation $b=abaab$ we replace its third occurrence of $a$ to the right-hand side by
$(ba)^2$. This gives us a Tietze-equivalent presentation $\mathcal P=\langle\,ngle a,b\mid a=(ba)^2,b=(ab)^4$. The diagram
group over $\mathcal P$ with base $a$ is the same as the one over $\mathcal P_{42}$ according to general facts from
\cite{GuSa05}.
\begin{thm}
\langle\,bel{f11}
The diagram group with base $a$ over semigroup presentation $\langle\,ngle a,b,c\mid a=bc,b=cd,c=da,d=ab\,\ranglengle$ is
isomorphic to generalized Thompson's group $F_{11}$.
\end{thm}
{\bf Proof.}\ The idea of the proof is similar to the one for Theorem~\ref{f9}. We will work with presentation $\mathcal P=\langle\,ngle a,b\mid a=(ba)^2,b=(ab)^4$
instead of $\mathcal P_{42}$. Our aim is to construct a homomorphism from $F_{11}$ to $G=\mathcal D(\mathcal
P,a)$. Notice that we have no longer a symmetry between $a$ and $b$. The group $F_{11}$ will be the diagram
group with base $x$ over $x=x^{11}$, as usual. Any diagram $\Delta$ over it is still a bipartite graph since
$11$ is odd. So each vertex gets a colour $1$ or $2$ and each edge will have a label $a$ or $b$ by the same
rules as above. This new diagram over $a=a(ba)^5$, $b=b(ab)^5$ will be denoted by $\Delta'$.
Both relations can be derived from $\mathcal P$. Indeed, $a=baba$, and then we replace the first occurrence of
$b$ to the right-hand side by $(ab)^4$. Thus we have a diagram of two cells over $\mathcal P$ for $a=a(ba)^5$. As for the second
equality, we take $b=ababab$ and replace the first $a$ by $(ba)^2$. This gives a two-cell diagram over $\mathcal P$ for
$b=b(ab)^5$. These two diagrams over $\mathcal P$ will be called basic. Replacing the cells of $\Delta'$ by
basic diagrams lead to the diagram $\Delta''$. In a standard way, the mapping $\Delta\mapsto\Delta''$ induces
the homomorphism of the groupoids of diagrams, and therefore we have a group homomorphism from $F_{11}$ to $G$.
Our aim is to establish its surjectivity.
Now let $\Psi$ be a reduced diagram over $\mathcal P$. As in the proof of the previous theorem, we let
$\Psi=\Psi_1\circ\Psi_2^{-1}$ where $p$ is the common part of the two pieces. We are going to insert certain
dipoles to $\Psi$ in such a way that the label of $p$ will have the form $abab\ldots$\ .
Suppose that the label of $p$ starts with $b$. Then we insert a dipole of the form $(b=(ab)^4)\circ((ab)^4=b)$
instead of the first edge of $p$. The new path is still denoted by $p$. If its label has an occurrence of $aa$
or $bb$ then we take the leftmost of them. In case it is $aa$, we replace the second edge by the dipole
$(a=(ba)^2)\circ((ba)^2=a)$. In case it is $bb$, the second edge is replaced by a dipole from the beginning of this
paragraph. So in a finite number of steps, we get a decomposition into a product of two diagrams, positive
and negative. It suffices to take a positive $(a,abab\ldots)$-diagram $\Phi$ showing that it is in the image
of the mapping $\Delta\mapsto\Delta''$.
Now we are proving that any positive $(a,abab\ldots)$-diagram over $\mathcal P$ can be composed from basic
diagrams together with an additional statement for a $(b,baba\ldots)$-diagram over $\mathcal P$. We prove
both facts simultaneously by induction on the number of cells in a diagram $\Phi$ with this property.
If $\Phi$ has no cells, there is nothing to prove. Let $\Phi$ have $a$ as a top label. Notice that the defining
relations of $\mathcal P$ always preserve the last letter of a word. So $a$ cannot be equal modulo this
presentation to a word that ends with $b$. Hence $\Phi$ is an $(a,(ab)^ma)$-diagram for some $m\ge1$. The top
cell of $\Phi$ has the form $a=(ba)^2$. Since the bottom path $p$ starts with $a$, the first letter $b$ of the
word $(ba)^2$ must correspond to the top path of a cell $b=(ab)^4$. These two cells form a basic diagram. So we
can cut it off. The rest will be a diagram with top path $a(ba)^5$ and bottom path $p$ labelled by $(ab)^ma$.
All vertices of a positive diagram belong to its bottom path. So it decomposes into a sum of diagrams for which
the top label of each of them is $a$ or $b$. If it is $a$, then the bottom label of a summand ends with $a$.
The length of the bottom path is odd so the bottom label has the form $(ab)^ka$ for some $k\ge0$. If the top
label of a summand is $b$, the same argument shows that the bottom label is of the form $(ba)^kb$. Thus all the
summands satisfy the inductive assumption (they have fewer cells than $\Phi$). Therefore they can be decomposed
into basic diagrams.
Now let $\Phi$ have $b$ as a top label. The top cell now is $b=(ab)^4$. The bottom path $p$ now starts with $b$.
Thus the first letter $a$ of $(ab)^4$ is the top path of a cell $a=(ba)^2$. The two cells together form a basic
diagram. We cut it off, and then repeat the same arguments as in the previous paragraph.
The image of the homomorphism is not Abelian. As above, we use the fact that generalized Thompson's groups $F_r$ have
no proper non-Abelian homomorphic images. Thus we have an isomorphism $F_{11}\cong G_{42}$.
The proof is complete.
Notice that the Fibonacci {\bf group} presented by $\mathcal P$ is a cyclic group $\mathbb Z_5$. The semigroup
with the same presentation is also finite, it has $10$ elements. However, for $n\ge5$ the Fibonacci semigroups presented by (\ref{fs})
turn out to be infinite. This makes unclear the structure of diagram groups $G_{n2}$ for that case (it is even possible that the groups
may be trivial). As for the generalization into another direction, we are able to describe completely the diagram groups over (\ref{pnr})
for the case $n=2$.
\begin{thm}
\langle\,bel{johnn}
Let $s$ be a positive integer.
The diagram group with base $a$ over $\langle\,ngle a,b\mid a=b(ab)^s,b=a(ba)^s\,\ranglengle$ is isomorphic to
generalized Thompson's group $F_{(2s+1)^2}$.
The diagram group with base $a$ over $\langle\,ngle a,b\mid a=(ba)^s,b=(ab)^s\,\ranglengle$ is isomorphic to
generalized Thompson's group $F_{4s-1}$.
So the group $G_{2r}=\mathcal D(\mathcal P_{2r},a)$ is isomorphic to $F_{r^2}$ for odd $r$ and $F_{2r-1}$ for
even $r$, where $\mathcal P_{2r}=\langle\,ngle a,b\mid a=ba\ldots,b=ab\ldots\,\ranglengle$ with the right-hand sides of the
defining relations of length $r\ge2$.
\end{thm}
{\bf Proof.}\ The case of odd $r=2s+1$ has the same proof as in Theorem~\ref{f9}. Basic diagrams here consist of $r+1$
cells. They correspond to the derivation $a=b(ab)^s$ with further replacements of all the $r$ letters of the
right-hand side according to the defining relators, and similarly for $b=a(ba)^s$ (we have a total symmetry here).
The bottom label of basic diagrams have length $r^2$. The proof goes without any changes for the general case.
Now let $r=2s$ be even. The construction of basic diagrams here is simpler. They consist of two cells only.
There is some similarity here to the construction from the proof of Theorem~\ref{f11}. Namely, we take the
cell $a=(ba)^s$ and replace the first letter in the right-hand side by $(ab)^s$. As a result, we get an
$(a,(ab)^{2s-1}a)$-diagram of two cells. We call it basic as well as the $(b,(ba)^{2s-1}b)$-diagram of two cells.
The bottom paths here have length $4s-1=2r-1$ so we are able to construct a homomorphism from $F_{2r-1}$ to the
diagram group and then show it is an isomorphism. The construction here is slightly easier than the one from
the proof of Theorem~\ref{f11} because of symmetry. This completes the proof.
\end{document}
|
\begin{document}
\ifJOC
\TITLE{SOS-SDP: an Exact Solver for Minimum Sum-of-Squares Clustering}
\ARTICLEAUTHORS{
\AUTHOR{Veronica Piccialli, Antonio M.~Sudoso}
\AFF{University of Rome Tor Vergata,
\EMAIL{\href{mailto:[email protected]}{[email protected]}},
ORCiD: 0000-0002-3357-9608,
\EMAIL{\href{mailto:[email protected]}{[email protected]}},
ORCiD: 0000-0002-2936-9931, \URL{}}
\AUTHOR{Angelika Wiegele}
\AFF{Universität Klagenfurt, \EMAIL{\href{mailto:[email protected]}{[email protected]}},
ORCiD: 0000-0003-1670-7951}
}
\else
\title{SOS-SDP: an Exact Solver for Minimum Sum-of-Squares Clustering}
\date{\today}
\author{Veronica Piccialli, Antonio M.~Sudoso, Angelika Wiegele}
\fi
\ifJOC
\ABSTRACT{
The minimum sum-of-squares clustering problem (MSSC) consists of partitioning $n$ observations into $k$ clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure where valid inequalities are iteratively added to the Peng-Wei SDP relaxation. The upper bound is computed with the constrained version of $k$-means where the initial centroids are extracted from the solution of the SDP relaxation. In the branch-and-bound procedure, we incorporate instance-level must-link and cannot-link constraints to express knowledge about which data points should or should not be grouped together. We manage to reduce the size of the problem at each level preserving the structure of the SDP problem itself. The obtained results show that the approach allows to successfully solve for the first time real-world instances up to 4000 data points.
}
\maketitle
\else
\maketitle
\begin{abstract}
The minimum sum-of-squares clustering problem (MSSC) consists of partitioning $n$ observations into $k$ clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure where valid inequalities are iteratively added to the Peng-Wei SDP relaxation. The upper bound is computed with the constrained version of $k$-means where the initial centroids are extracted from the solution of the SDP relaxation. In the branch-and-bound procedure, we incorporate instance-level must-link and cannot-link constraints to express knowledge about which data points should or should not be grouped together. We manage to reduce the size of the problem at each level preserving the structure of the SDP problem itself. The obtained results show that the approach allows to successfully solve for the first time real-world instances up to 4000 data points.
\end{abstract}
\fi
\section{Introduction}\label{sec:intro}
Clustering is the task of partitioning a set
of objects into homogeneous and/or well-separated groups, called clusters. Cluster analysis is the discipline that studies methods and algorithms for clustering objects according to a suitable similarity measure. It belongs to unsupervised learning since it does not use class labels. Two main clustering approaches exist: hierarchical clustering, which assumes a tree structure in the data and builds nested clusters, and partitional clustering. Partitional clustering generates all the clusters at the same time without assuming a nested structure. Among partitional clustering, the minimum sum-of-squares clustering problem (MSSC) or sum-of-squares (SOS) clustering, is one of the most popular and well studied. MSSC asks to partition $n$ given data points into $k$ clusters so that the sum of the Euclidean distances from each data point to the cluster centroid is minimized.
The MSSC commonly arises in a wide range of disciplines and applications, as for example image segmentation \citep{dhanachandra2015image, shi2000normalized}, credit risk evaluation \citep{CARUSO2021100850}, biology \citep{jiang2004cluster}, customer segmentation \citep{syakur2018integration}, document clustering \citep{mahdavi2009harmony}, and as a technique for the missing values imputation \citep{zhang2006clustering}.
The MSSC can be stated as follows for fixed $k$:
\begin{subequations}
\label{eq:MSSC}
\begin{align}
\min~ & \sum_{i=1}^n \sum_{j=1}^k x_{ij}\|p_i - c_j\|^2 \\
\textrm{s.t.}~ & \sum_{j=1}^k x_{ij} = 1,\quad \forall i \in \{1,\dots,n\} \label{eq:MSSCa}\\
& \sum_{i=1}^n x_{ij} \ge 1, \quad \forall j \in \{1,\dots,k\}\label{eq:MSSCb}\\
& x_{ij} \in \{0,1\}, \quad \forall i \in \{1,\dots,n\}\; \forall j \in \{1,\dots,k\} \\
& c_j \in \mathbb{R}^d, \quad \forall j \in \{1, \dots, k \}.
\end{align}
\end{subequations}
Here, $p_i \in \mathbb{R}^d$, where $d$ is the number of features, $i \in \{1,\dots,n\}$, are the data points, and the centers of the $k$ clusters are at the (unknown) points $c_j$, $j\in \{1,\dots,k\}$.
For convenience, we sometimes collect all the data points $p_i$ as rows in a matrix $W_p$.
The binary decision variable $x_{ij}$ expresses whether data point $i$ is assigned to cluster $j$ or not.
Constraints~\eqref{eq:MSSCa} make sure that each point is assigned to a cluster, and constraints~\eqref{eq:MSSCb} guarantee that none of the $k$ clusters is empty.
Setting the gradient of the objective function with respect to $c$ to zero yields
\begin{equation*}
\sum_{i=1}^n x_{ij}(c^r_j - p^r_i) = 0,\quad \forall j \in \{1,\dots,k\}\; \forall r \in \{1,\dots,d\}
\end{equation*}
and we obtain the formula for the point in the center of each cluster
\begin{equation*}
c^r_j = \frac{\sum_{i=1}^n x_{ij} p^r_i}{\sum_{i=1}^n x_{ij}}, \quad \forall j \in \{1,\dots,k\}\; \forall r \in \{1,\dots,d\}.
\end{equation*}
Replacing the formula for $c$ in~\eqref{eq:MSSC}, we get
\begin{subequations}
\label{eq:MSSC2}
\begin{align}
\min~ & \sum_{i=1}^n \sum_{j=1}^k x_{ij}\Big\|p_i - \frac{\sum_{l=1}^n x_{lj} p_l}{\sum_{l=1}^n x_{lj}}\Big\|^2 \\
\textrm{s.t.}~ & \sum_{j=1}^k x_{ij} = 1,\quad \forall i \in \{1,\dots,n\}\\
& \sum_{i=1}^n x_{ij} \ge 1, \quad \forall j \in \{1,\dots,k\}\\
& x_{ij} \in \{0,1\}, \quad \forall i \in \{1,\dots,n\}\; \forall j \in \{1,\dots,k\}.
\end{align}
\end{subequations}
\subsection{Literature Review}
The MSSC is known to be NP-hard in $\mathbb{R}^2$ for general values of $k$ \citep{mahajan2012planar}, and in higher dimension even for $k=2$ \citep{aloise2009np}. The one-dimensional case is proven to be solvable in polynomial time. In particular, \cite{wang2011ckmeans} proposed an $O(kn^2)$ time and $O(kn)$ space dynamic programming algorithm for solving this special case. Because of MSSC's computational complexity, heuristic approaches and approximate algorithms are usually preferred over exact methods.
The most popular heuristic for solving MSSC is $k$-means \citep{macqueen1967some, lloyd1982least},
that alternates the centroid initialization with the assignments of points until centroids do not move anymore. The main disadvantage of $k$-means is that it produces locally optimal solutions that can be far from the global minimum, and it is extremely sensitive to the initial assignment of centroids. For this reason, a lot of research has been dedicated to finding efficient initialization for $k$-means (see for example \cite{arthur2006k,improvedkmeans2018,franti2019much} and references therein). However, an efficient initialization may not be enough in some instances, so that different strategies have been implemented in order to improve the exploration capability of the algorithm. A variety of heuristics and metaheuristics have been proposed, following the standard metaheuristic framework, e.g., simulated annealing \citep{lee2021simulated}, tabu search \citep{ALSULTAN19951443}, variable neighborhood search \citep{HANSEN2001405,Orlov2018}, iterated local search \citep{likas2003global}, evolutionary algorithms \citep{MAULIK20001455,SARKAR1997975}). In the work of \cite{tao2014new,BAGIROV201612,KARMITSA2017367,KARMITSA2018245}, DC (Difference of Convex functions) programming is used to define efficient heuristic algorithms for clustering large datasets.
The algorithm $k$-means has also been used as a local search subroutine in different algorithms, as in the population-based metaheuristic developed in \cite{gribel2019hg} and in the differential evolution scheme proposed in \cite{Schoen2021}.
Recently, thanks to the enhancements in computers' computational power and to the progress in mathematical programming, the exact resolution of MSSC has become way more achievable. In this direction, mathematical programming algorithms based on branch-and-bound and column generation have produced guaranteed globally optimal solutions for small and medium scale instances. Due to the NP-hardness of the MSSC, the computational time of globally optimal algorithms quickly increases with the size of the problem. However, besides the importance of finding optimal solutions for some clustering applications, certified optimal solutions remain extremely valuable as a benchmark tool since they can be used for evaluating, improving, and developing heuristics and approximate methods.
Compared to the huge number of papers proposing heuristics and approximate methods for the MSSC problem, the number of articles proposing exact algorithms is much smaller.
One of the earliest attempts was the integer programming formulation proposed by \citet{rao1971cluster}, which requires the cluster sizes to be fixed in advance and is limited to small instances. A first branch-and-bound algorithm was proposed by \citet{koontz1975branch} and extended by \citet{diehr1985evaluation}.
The idea is to use partial clustering solutions on a subset $S$ of the main dataset $D$ to determine improved bounds and clusters on the entire sample by a branch-and-bound search. The key observation is that the optimal objective function value of the MSSC on $D$ is greater or equal than the optimal objective function value of the MSSC on $S$ plus the optimal objective function value of the MSSC on $D - S$.
This approach was later improved by \citet{brusco2006repetitive},
who developed a repetitive-branch-and-bound algorithm (RBBA). After a proper reordering of the entities in $D$, RBBA solves a sequence of subproblems of increasing size with the branch-and-bound technique. While performing a branch-and-bound for a certain subproblem, Brusco's algorithm exploits the optimal solutions found for the previous subproblems which provide tighter bounds compared to the ones used by \cite{koontz1975branch} and \cite{diehr1985evaluation}. RBBA provided optimal solutions for well separated synthetic datasets with up to 240 objects. Poorly separated problems with no inherent cluster structure were optimally solved for up to 60 objects.
\citet{sherali2005global} proposed a different branch-and-bound algorithm where tight lower bounds are determined by using the reformulation-linearization-technique (RLT), see \citet{sherali1998reformulation}. The authors claim that this algorithm allows for the exact resolution of problems of size up to 1000 entities, but those results seem to be hard to reproduce. The computing times in an attempted replication by \citet{aloise2011evaluating} were already high for real datasets with about 20 objects.
A column generation algorithm for MSSC was proposed by \citet{du1999interior}. The master problem is solved by an interior point method, whereas the auxiliary problem of finding a column with negative reduced cost is expressed as a hyperbolic program with binary variables. Variable-neighborhood-search heuristics are used to find a good initial solution
and to accelerate the resolution of the auxiliary problem. This approach has been considered a successful one, since it solved for the first time medium size benchmark instances (i.e., instances with 100--200 entities), including the popular Iris dataset, which encounters 150 entities. However, the bottleneck of the algorithm lies in the resolution of the auxiliary problem, and more precisely, in the unconstrained quadratic 0-1 optimization problem. Later this algorithm was further improved by \citet{aloise2012improved} who define a different geometric-based approach for solving the auxiliary problem. In particular, the solution of the auxiliary problem is achieved by solving a certain number of convex quadratic problems. If the points to be clustered are in the plane, the maximum number of convex problems to solve is polynomially bounded. When the points are not in the plane, in order to solve the auxiliary problems the cliques in a certain graph (induced by the current solution of the master problem) have to be found. The algorithm is more efficient when the graph is sparse, and the graph becomes sparser when the number of clusters $k$ increases. Therefore, the algorithm proposed in \citet{aloise2012improved} is particularly efficient in the plane and when $k$ is large. Their method was able to provide exact solutions for large scale problems, including one instance of 2300 entities when the ratio between $n$ and $k$ is small.
Recently, \citet{peng2007approximating} by using matrix arguments proved the equivalence between the MSSC formulation and a model called 0-1 semidefinite programming (SDP), in which the eigenvalues of the matrix variable are binary. Using this result, \citet{aloise2009branch} proposed a branch-and-cut algorithm for MSSC where lower bounds are obtained from the linear programming relaxation of the 0-1 SDP model. This algorithm manages to obtain exact solutions for datasets up to 200 entities with computing times comparable with those obtained by the column generation method proposed by \citet{du1999interior}.
Constant-factor approximation algorithms have also been developed in the literature, both for fixed number of clusters $k$ and for fixed dimension $d$ \citep{kanungo2004local}. Among these methods, \citet{peng2007approximating} proposed a rounding procedure to extract a feasible solution of the original MSSC
from the approximate solution of the relaxed SDP problem. More in detail, they use
the Principal Component Analysis (PCA) to reduce the dimension of the dataset and then perform clustering on the projected PCA space. They showed that this algorithm can provide a 2-approximate solution to the MSSC.
More recently, \citet{prasad2018improved} proposed a new approximation algorithm that utilizes an improved copositive conic reformulation of the MSSC. Starting from this reformulation, the authors derived a hierarchy of accurate SDP relaxations obtained by replacing the completely positive cone with progressively tighter semidefinite outer approximations. Their SDP relaxations provide better lower bounds than the Peng-Wei one but do not scale well when the size of the problem increases.
\subsection*{Main results and outline}
The main contributions of this paper are the following:
\begin{description}
\item[(i)] we define the first SDP based branch-and-bound algorithm for MSSC, and we use a cutting-plane procedure for strengthening the bound, following a recent strand of research \citep{demeijer2021sdpbased};
\item[(ii)] we define a shrinking procedure that allows reducing the size of the problem when introducing must link constraints;
\item[(iii)] we exploit the SDP solution for a smart initialization of the constrained version of $k$-means that yields high quality upper bounds;
\item[(iv)] for the first time, we manage to find the exact solution for instances of size up to $n=4000$.
\end{description}
This paper is structured as follows. In Section~\ref{sec:bound} we introduce equivalent formulations for the MSSC and derive relaxations based on semidefinite programming (SDP).
In Section~\ref{sec:branching} we analyze the SDP problems that arise at each node within the branch-and-bound tree and discuss the selection of the branching variable.
In Section~\ref{sec:bab} the details about the bound computation are discussed, including a post-processing procedure that produces a ``safe'' bound from an SDP that is solved to medium precision only.
Section~\ref{sec:heuristic} gives all the details on the heuristic used to generate feasible clusterings.
The details of our implementation and exhaustive numerical results are presented in Section~\ref{sec:numericalresults}.
Finally, Section~\ref{sec:conclusion} concludes the paper.
\subsection*{Notation}
Let ${\mathcal S}^n$ denote the set of all $n\times n$ real symmetric matrices. We denote by $M\succeq 0$ that matrix $M$ is positive semidefinite and let ${\mathcal S}_+^n$ be the set of all positive semidefinite matrices of order
$n\times n$. We denote by $\inprod{\cdot}{\cdot}$ the
trace inner product. That is, for any
$M, N \in \mathbf{R}^{n\times n}$, we define $\inprod{M}{N}:= \textrm{trace} (M^\top N )$. Its associated norm is the Frobenius norm, denoted by $\| M\|_F := \sqrt{\textrm{trace} (M^\top M )}$. We define the linear map $\mathcal{A}: {\mathcal S}^n \rightarrow \mathbb{R}^{m_1}$ as $(\mathcal{A}(X))_i = \inprod{A_i}{X}$, where $A_i \in {\mathcal S}^n$, $i=1,\dots,m_1$, and the linear map $\mathcal{B}: {\mathcal S}^n \rightarrow \mathbb{R}^{m_2}$ as $(\mathcal{B}(X))_i = \inprod{B_i}{X}$, where $B_i \in {\mathcal S}^n$, $i=1,\dots,m_2$.
We define by $e_n$ the vector of all ones of length $n$. We omit the subscript in case the dimension is clear from the context.
We denote by $E_{i}$ the symmetric matrix such that $\inprod{E_{i}}{Z}$ is the sum of row~$i$ of $Z$.
\section{A Lower Bound based on Semidefinite Programming}\label{sec:bound}
We briefly remind the Peng-Wei SDP relaxation to Problem~\eqref{eq:MSSC2} that will be the basis of the bounding procedures within our exact algorithm.
Consider matrix $W$ where the entries are the inner products of the data points, i.e., $W_{ij} = p_i^\top p_j$ for $i,j \in \{1,\dots,n\}$. Furthermore, collect the binary decision variables $x_{ij}$ from~\eqref{eq:MSSC2} in the $n\times k$ matrix $X$ and define matrix $Z$ as
\begin{equation*}
Z = X(X^\top X)^{-1}X^\top.
\end{equation*}
\citet{peng2007approximating} introduced a different but equivalent formulation for the MSSC, yielding the following
optimization problem:
\begin{subequations}
\label{eq:PengSDP}
\begin{align}
\min~ & \inprod{-W}{Z} \\
\textrm{s.t.}~ & Ze = e\\
& \textrm{tr}(Z) = k\\
& Z \ge 0, \ Z^2 = Z, \ Z = Z^\top.
\end{align}
\end{subequations}
We can convert Problem~\eqref{eq:PengSDP} into a rank constrained optimization problem. In fact we can replace the constraints $Z^2 = Z$ and $Z = Z^\top$ with a rank constraint and a positive semidefiniteness constraint on $Z$, yielding the following problem:
\begin{subequations}
\label{eq:RankSDP}
\begin{align}
\min~ & \inprod{-W}{Z} \\
\textrm{s.t.}~ & Ze = e\\
& \textrm{tr}(Z) = k\\
& Z \ge 0, \ Z \in {\mathcal S}^np\\
& \textrm{rank}(Z) = k.
\end{align}
\end{subequations}
In order to prove the equivalence of Problems~\eqref{eq:PengSDP} and~\eqref{eq:RankSDP}, we need the definition of an idempotent matrix and its characterization in terms of eigenvalues given by Lemma~\ref{lemma:ideig}.
\begin{definition}
A symmetric matrix $Z$ is idempotent if $Z^2 = ZZ = Z$.
\end{definition}
\begin{lemma}\label{lemma:ideig}
A symmetric matrix $Z$ is idempotent if and only if all its eigenvalues are either 0 or 1.
\end{lemma}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Let $Z$ be idempotent, $\lambda$ be an eigenvalue and $v$ a
corresponding eigenvector then $\lambda v = Zv = ZZv = \lambda Zv = \lambda^2 v$.
Since $v \neq 0$ we find $\lambda - \lambda^2 = \lambda (1 - \lambda) = 0$ so either $\lambda = 0$ or $\lambda = 1$.
To prove the other direction, consider the eigenvalue decomposition of $Z$,
$Z = P \Lambda P^\top$,
where $\Lambda$ is a diagonal matrix having the eigenvalues $0$ and $1$ on the diagonal, and $P$ is orthogonal. Then, since $\Lambda^2 = \Lambda$, we get
\begin{equation*}
Z^2 = P \Lambda P^\top P \Lambda P^\top = P \Lambda^2 P^\top = P \Lambda P^\top = Z.
\end{equation*}
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
\begin{theorem}
Problems~\eqref{eq:PengSDP} and~\eqref{eq:RankSDP} are equivalent.
\end{theorem}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Let $Z$ be a feasible solution of Problem \eqref{eq:PengSDP}. We first show that $Z^2 = Z$ and $Z = Z^T$ imply $Z \in {\mathcal S}^np$. In fact, for all $v$ we have:
\begin{equation*}
v^\top Z v = v^\top Z^2 v = v^\top Z Z v = v^\top Z (v^\top Z^\top)^\top = (v^\top Z) (v^\top Z)^\top = \| v^\top Z \|_2^2 \geq 0.
\end{equation*}
Since $Z$ is symmetric idempotent, the number of eigenvalues equal to 1 is $\textrm{tr}(Z) = \textrm{rank}(Z) = k$.
To prove the other direction, let $Z$ be a feasible solution of Problem \eqref{eq:RankSDP}. If $\textrm{rank}(Z) = k$, then $Z$ has $n-k$ eigenvalues equal to 0. Furthermore, let $\lambda_1 \geq \lambda_2 \geq \ldots > \lambda_n\ge 0$ be the eigenvalues of Z, then
\begin{equation*}
\textrm{tr}(Z) = \sum_{i=1}^{n} \lambda_i = \sum_{i=1}^{k} \lambda_i + \sum_{i=k+1}^{n} \lambda_i = \sum_{i=1}^{k} \lambda_i = k.
\end{equation*}
Constraints $Z\succeq 0$, $Z\ge 0$ and $Ze=e$ imply that the eigenvalues of $Z$ are bounded by one (see, e.g., Lemma~\ref{lem:eigboundZ}). Hence, the trace constraint is satisfied if and only if the positive eigenvalues are all equal to~1. This shows that $\lambda(Z) \in \{0, 1\}$ and therefore $Z$ is symmetric idempotent.
\ifJOC
\par
\Halmos
\endproof
\else
\end{proof}
\fi
By dropping the non-convex rank constraint from Problem~\eqref{eq:RankSDP}, we obtain the SDP relaxation which is the convex optimization problem
\begin{subequations}
\label{eq:SDP}
\begin{align}
\min~ & \inprod{-W}{Z} \\
\textrm{s.t.}~ & Ze = e\\
& \textrm{tr}(Z) = k\\
& Z \ge 0, \ Z \in {\mathcal S}^np
\end{align}
\end{subequations}
\subsection{Strengthening the Bound through Inequalities}
The SDP relaxation~\eqref{eq:SDP} can be tightened by adding valid inequalities and solving the resulting SDP in a cutting-plane fashion. In this section, we present the class of inequalities we use for strengthening the bound. For each class, we describe the separation routine used.
We consider three different sets of inequalities:
\begin{description}
\item[Pair inequalities.] In any feasible solution of~\eqref{eq:RankSDP}, it holds that \begin{equation}\label{eq:pairs}
Z_{ij}\le Z_{ii},\quad Z_{ij}\le Z_{jj}\quad \forall i,j \in \{1,\dots,n\}, i\not=j.
\end{equation}
This set of $n(n-1)$ inequalities were used by \citet{peng2005new} and in the branch-and-cut proposed by \citet{aloise2009branch}.
\item[Triangle Inequalities.]
The triangle inequalities are based on the observation that if points $i$ and $j$ are in the same cluster and points $j$ and $h$ are in the same cluster, then points $i$ and $h$ necessarily must be in the same cluster. The resulting $3\binom{n}{3}$ inequalities are:
\begin{equation}\label{eq:triangle}
Z_{ij}+Z_{ih}\le Z_{ii}+Z_{jh}\quad \forall i,j,h \in \{1,\dots,n\}, i,j,h ~\mathrm{distinct}.
\end{equation}
These inequalities were already introduced by \citet{peng2005new}, and used also by \citet{aloise2009branch}.
\item[Clique Inequalities.] If the number of clusters is $k$, for any subset $Q$ of $k+1$ points at least two points have to be in the same cluster (meaning that at least one $Z_{ij}$ needs to be positive and equal to $Z_{ii}$ for all $(i,j)\in Q$). This can be enforced by the following inequalities:
\begin{equation}\label{eq:clique}
\sum_{(i,j)\in Q,i<j}Z_{ij}\ge \frac{1}{n-k+1} \quad\forall Q\subset\{1,\ldots,n\},\,|Q|=k+1.
\end{equation}
These $\binom{n}{k+1}$ inequalities are similar to the clique inequalities for the $k$-partitioning problem \citep{chopra1993partition}, the difference lies in the right hand side, that in that case is equal to~1, whereas here we use the smallest possible value that an element on the diagonal of $Z$ can hold.
\end{description}
Pair and triangle inequalities are known to be valid for Problem~\eqref{eq:PengSDP}, see \cite{peng2005new} and \cite{de2020ratio}.
It remains to show that also the clique inequalities are valid.
\begin{lemma}
The clique inequalities~\eqref{eq:clique} are valid for Problem~\eqref{eq:PengSDP}.
\end{lemma}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
The left hand side of \eqref{eq:clique} has $\binom{k+1}{2}$ terms, and we know that $Z_{ii}\ge\frac{1}{n-k+1}$, since the cardinality of a cluster can be at most $n-k+1$.
Given that the number of clusters is $k$, for any set of $k+1$ points at least two points have to be in the same cluster, say points $i$ and $j$. Then, for any feasible clustering $Z$, at least the element $Z_{ij}$ in the left hand side of \eqref{eq:clique} needs to be different from zero, therefore equal to $Z_{ii}$, and hence \eqref{eq:clique} must hold.
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
\section{Branching: Subproblems within a Branch-and-Bound Algorithm and Variable Selection}\label{sec:branching}
Our final goal is to develop a branch-and-bound scheme to solve the MSSC to optimality using relaxation~\eqref{eq:SDP} strengthened by some of the inequalities~\eqref{eq:pairs}--\eqref{eq:clique}. In this section we examine the problems that arise after branching. To keep the presentation simple and since everything carries over in a straightforward way, we omit in this section the inclusion of inequalities~\eqref{eq:pairs}--\eqref{eq:clique}.
The branching decisions are as follows. Given a pair $(i,j)$,
\begin{itemize}
\item points $p_i$ and $p_j$ should be in different clusters, i.e., they \textit{cannot link} or
\item points $p_i$ and $p_j$ should be in the same cluster, i.e., they \textit{must link}.
\end{itemize}
By adding constraints due to the branching decisions, the problem changes. However, the structure of the SDP remains similar.
In this section we describe the subproblems to be solved at each node in the branch-and-bound tree. Each such SDP is of the form
\begin{subequations}
\label{eq:SDPbab}
\begin{align}
\min~ & \inprod{-\mathcal{T}^{\ell} W (\mathcal{T}^{\ell})^\top}{Z^{\ell}} \\
\textrm{s.t.}~ & Z^{\ell} e^{\ell} = e \\
& \inprod{\textrm{Diag}(e^{\ell})}{Z^{\ell}} = k\\
& Z^\ell_{ij} = 0 \quad (i,j) \in \textrm{CL}\\
&Z^{\ell} \ge 0, \ Z^{\ell} \in \mathcal{S}^+_{n-\ell}
\end{align}
\end{subequations}
where $\textrm{CL}$ (cannot link) is the set of pairs that must be in different clusters and matrix $\mathcal{T}^{\ell}$ and vector $e^\ell$ encode the branching decisions that ask data points to be in the same cluster (i.e., they must link). We describe this in detail in the subsequent sections.
\subsection{Branching Decisons}
In case we want to have $i$ and $j$ in different clusters, we add the constraint $Z_{ij} = 0$ to the SDP, i.e., we add the pair $(i,j)$ to the set $\textrm{CL}$.
In the other case, i.e., when the decison is to have $i$ and $j$ in the same cluster, we proceed as follows.
Assume at the current node we have $n$ points and we decide that on this branch the two points $p_i$ and $p_j$ have to be in the same cluster. We can reduce the size of $W_p$ (the matrix having data points $p_i$ as rows) by substituting row $i$ by $p_{i}+p_j$ and omitting row $j$. To formalize this procedure, we introduce the following notation.
Let $b(r) = (i,j)$, $i<j$, be the branching pair in branching decision at level $r$ and $b(1),\dots,b(\ell)$ a sequence of consecutive branching decisions. Furthermore, let $g(r)=(\underline{i}, \underline{j})$ be the corresponding global indices.
Define $\mathcal{T}^{\ell} \in \{0,1\}^{(n-\ell) \times n}$ as
\[
\mathcal{T}^{\ell} = T^{b(\ell)} T^{b(\ell-1)} \dots T^{b(1)}
\]
where the $(n-r)\times (n-r+1)$ matrix $T^{b(r)}$ for branching decision $b(r)=(i,j)$ is defined by
\[
T^{b(r)}_{s,\cdot} = \left\{ \begin{array}{ll}
u_s & \textrm{if}~ 1 \le s < i ~\textrm{and}~ i+1\le s\le j\\
u_i + u_j & \textrm{if}~ s=i\\
u_{s+1} & \textrm{if}~ j<s\le n-r
\end{array}\right.
\]
with $u_s$ being the unit vector of size $(n-r+1)$.
Furthermore, we define $T^{b(0)} = I_n$.
Note that $T^{b(r)}\cdot M$ builds a matrix of size $(n-r)\times (n-r+1)$ by adding rows $i$ and $j$ of $M$ and putting the result into row $i$ while row $j$ is removed and all other rows remain the same.
We also define the vector $e^\ell \in \mathbb{R}^{n-\ell}$ as
\[
e^\ell = \mathcal{T}^\ell e
\]
where $e$ is the vector of all ones of length $n$.
\begin{remark}\label{rem:ell}
Note that in $(e^\ell)$ the number of points that have been fixed to belong to the same cluster along the branching decisions $b(1),\dots,b(\ell)$ are given.
Furthermore, $\mathcal{T}^\ell(\mathcal{T}^{\ell})^\top = \textrm{Diag}(e^\ell)$.
\ifJOC
$\triangle$
\fi
\end{remark}
We now show that this shrinking operation corresponds to the must-link branching decisions. Consider the following two semidefinite programs.
\begin{subequations}
\label{eq:SDPell}
\begin{align}
\min~ & -\inprod{\mathcal{T}^{\ell} W (\mathcal{T}^{\ell})^\top}{Z^{\ell}} \\
\textrm{s.t.}~ & Z^{\ell} e^{\ell} = e_{n-\ell} \label{eq:SDPellb}\\
& \inprod{\mathcal{T}^{\ell}(\mathcal{T}^{\ell})^\top}{Z^{\ell}} = k \label{eq:SDPellk}\\
&Z^{\ell} \ge 0, Z^{\ell} \in \mathcal{S}^+_{n-\ell}
\end{align}
\end{subequations}
and
\begin{subequations}
\label{eq:SDPbranch}
\begin{align}
\min~ & -\inprod{W}{Z} \\
\textrm{s.t.}~ & Ze = e\\
& \inprod{I}{Z} = k\\
& Z_{i\cdot} = Z_{j\cdot} \quad \forall \{i,j\} \in g(l), ~ l \in \{1,\dots, \ell\} \label{eq:SDbranch-rows}\\
&Z \ge 0, Z \in {\mathcal S}^np
\end{align}
\end{subequations}
\begin{theorem}
Problems~\eqref{eq:SDPell} and~\eqref{eq:SDPbranch} are equivalent.
\end{theorem}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Let $Z^{\ell}$ be a feasible solution of Problem~\eqref{eq:SDPell}. Define $Z = (\mathcal{T}^{\ell})^\top Z^{\ell} \mathcal{T}^{\ell}$. This is equivalent to expanding the matrix by replicating the rows according to branching decisions. Therefore, \eqref{eq:SDbranch-rows} holds by construction.
Clearly, $Z \ge 0$ and $Z\in {\mathcal S}^np$ hold as well. Moreover, we have that
\[\inprod{I}{Z} = \inprod{I}{(\mathcal{T}^{\ell})^\top Z^{\ell} \mathcal{T}^{\ell}} = \inprod{\mathcal{T}^{\ell}(\mathcal{T}^{\ell})^\top}{Z^{\ell}} = k
\]
and
\[
Ze = (\mathcal{T}^{\ell})^\top Z^{\ell} \mathcal{T}^{\ell} e = (\mathcal{T}^{\ell})^\top Z^{\ell} e^{\ell} =
(\mathcal{T}^{\ell})^\top e_{n-\ell} = e_n.
\]
Furthermore,
\[ \inprod{W}{Z} = \inprod{W}{(\mathcal{T}^{\ell})^\top Z^{\ell} \mathcal{T}^{\ell}}
= \inprod{(\mathcal{T}^{\ell})W(\mathcal{T}^{\ell})^\top}{Z^{\ell}}
\]
and thus $Z$ is a feasible solution of Problem~\eqref{eq:SDPbranch} and the values of the objective functions coincide.
We next prove that any feasible solution of Problem~\eqref{eq:SDPbranch} can be transformed into a feasible solution of Problem~\eqref{eq:SDPell} with the same objective function value. In order to do so, we define the matrix
\[ \mathcal{D}^\ell = \textrm{Diag}(1/e^\ell)\]
where $1/e^\ell$ denotes the vector that takes the inverse elementwise. It is straightforward to check that
\[
\mathcal{D}^\ell \mathcal{T}^\ell (\mathcal{T}^\ell)^\top \mathcal{D}^\ell = \mathcal{D}^\ell.
\]
Assume that $Z$ is a feasible solution of Problem~\eqref{eq:SDPbranch}
and set $Z^{\ell} = \mathcal{D}^\ell \mathcal{T}^{\ell} Z (\mathcal{T}^{\ell})^\top \mathcal{D}^\ell$. If $Z$ is nonnegative and positive semidefinite, then so is $Z^\ell$.
Furthermore, we can derive
\begin{align*}
\inprod{\mathcal{T}^{\ell}(\mathcal{T}^{\ell})^\top}{Z^{\ell}} & =\inprod{\mathcal{T}^{\ell}(\mathcal{T}^{\ell})^\top}{\mathcal{D}^\ell \mathcal{T}^{\ell} Z (\mathcal{T}^\ell)^\top \mathcal{D}^\ell}\\
&= \inprod{\mathcal{D}^\ell\mathcal{T}^{\ell}(\mathcal{T}^{\ell})^\top \mathcal{D}^\ell}{\mathcal{T}^\ell Z (\mathcal{T}^\ell)^\top }\\
&= \inprod{\mathcal{D}^\ell}{ \mathcal{T}^\ell Z (\mathcal{T}^\ell)^\top }
= \sum_{l=1}^{n-\ell} \frac{1}{e^\ell_l} \sum_{j \in g(l)} \sum_{i\in g(l)} Z_{ij}\\
&\textrm{s.t.}ackrel{(*)}{=} \sum_{l=1}^{n-\ell} \frac{1}{e^\ell_l} \sum_{j \in g(l)} \sum_{i\in g(l)} Z_{ii}
= \sum_{l=1}^{n-\ell} \frac{1}{e^\ell_l} e^\ell_l \sum_{i\in g(l)} Z_{ii}\\
&= \sum_{l=1}^{n-\ell} \sum_{i\in g(l)} Z_{ii}= \sum_{i=1}^n Z_{ii}= k.
\end{align*}
Note that the equality~$(*)$ holds since $Z_{i,j} = Z_{r,s}$ for any $i,j,r,s \in g(l)$. This ensures that constraint~\eqref{eq:SDPellk} holds for $Z^\ell$.
To prove~\eqref{eq:SDPellb} consider the equations
\begin{align*}
Z^\ell e^\ell & = \mathcal{D}^\ell \mathcal{T}^{\ell} Z (\mathcal{T}^{\ell})^\top \mathcal{D}^\ell e^\ell
= \mathcal{D}^\ell \mathcal{T}^{\ell} Z (\mathcal{T}^{\ell})^\top e_{n-\ell} \\
&= \mathcal{D}^\ell \mathcal{T}^{\ell} Z e
= \mathcal{D}^\ell \mathcal{T}^{\ell} e = \mathcal{D}^\ell e^\ell = e_{n-\ell}.
\end{align*}
It remains to show that the objective function values coincide.
\begin{align*}
\inprod{\mathcal{T}^{\ell} W (\mathcal{T}^{\ell})^\top}{Z^{\ell}}
&=
\inprod{\mathcal{T}^{\ell} W (\mathcal{T}^{\ell})^\top}{\mathcal{D}^\ell \mathcal{T}^{\ell} Z (\mathcal{T}^{\ell})^\top \mathcal{D}^\ell}\\
&=
\inprod{ W }{(\mathcal{T}^{\ell})^\top\mathcal{D}^\ell \mathcal{T}^{\ell} Z (\mathcal{T}^{\ell})^\top \mathcal{D}^\ell\mathcal{T}^{\ell}}\\
&=
\inprod{ W }{Z}.\\
\end{align*}
As for the last equation, note that pre- and postmultiplying $Z$ by $(\mathcal{T}^{\ell})^\top\mathcal{D}^\ell \mathcal{T}^{\ell}$ ``averages'' over the respective rows of matrix $Z$. Since these respective rows are identical due to~\eqref{eq:SDbranch-rows}, the last equation holds.
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
\begin{remark}
The addition of constraints $Z_{ij}=0$ for datapoints $i,j$ that should not belong to the same cluster also goes through in the above equivalence. However, to keep the presentation simple we did not include it in the statement of the theorem above.
\ifJOC
$\triangle$
\fi
\end{remark}
\begin{remark}
It is straightforward to include the additional constraints~\eqref{eq:pairs}, \eqref{eq:triangle}, and~\eqref{eq:clique} in the subproblems, i.e., in case of shrinking the problem, the constraints are still valid. Again, to keep notation simple, we omitted these constraints in the presentation above.
Further discussions on including these inequalities are in Section~\ref{sec:boundcomp}.
\ifJOC
$\triangle$
\fi
\end{remark}
\subsection{Variable Selection for Branching}\label{sec:variableselection}
In a matrix $Z$ corresponding to a clustering, for each pair $(i,j)$ either $Z_{ij}=0$ or $Z_{ii} = Z_{ij}$. \citet{peng2005new} propose a simple branching scheme. Suppose that for the optimal solution of the SDP relaxation there are indices $i$ and $j$ such that $Z_{ij}(Z_{ii}-Z_{ij}) \neq 0$ then one can produce a cannot-link branch with $Z_{ij} = 0$ and a must-link branch with $Z_{ii} = Z_{ij}$. Regarding the variable selection the idea is to choose indices $i$ and $j$ such that in both branches we expect a significant improvement of the lower bound. In~\cite{peng2005new} the branching pair is chosen as the \[\argmax_{i,j} \{ \min \{Z_{ij},Z_{ii}-Z_{ij}\} \}.\]
Here we propose a variable selection strategy that is coherent with the way we generate the cannot-link and the must-link subproblems. In fact, we observe that in a matrix $Z$ corresponding to a clustering, for each pair $(i, j)$ either $Z_{ij} = 0$ or $Z_{i\cdot} = Z_{j\cdot}$. This motivates the following strategy to select a pair of data points to branch on
\[\argmax_{i,j} \{ \min \{Z_{ij}, \|Z_{i\cdot} - Z_{j\cdot}\|_2^2 \} \}.\]
In case this maximizer gives a value close to zero, say $10^{-5}$, the SDP solution corresponds to a feasible clustering.
\subsubsection*{Variable selection on the shrunk problem}
The strategy for the variable selection still carries over on the shrunk problems.
Since $Z$ is obtained from $Z^\ell$ only by repeating rows and columns, every pair $(Z^\ell_{ij}, Z^\ell_{ii})$ appears also in $Z$ and vice versa.
Moreover, within the already merged points, by construction $Z^\ell_{ii}=Z^\ell_{ij}$ and hence this can never be a branching candidate again.
\section{Branch-and-Bound Algorithm}\label{sec:bab}
We now put the bound computation (see Sections~\ref{sec:bound} and~\ref{sec:branching}) together with our way of branching (see Section~\ref{sec:variableselection}) to form our algorithm \texttt{SOS-SDP}. The final ingredient, a heuristic for providing upper bounds, is described in Section~\ref{sec:heuristic}.
\subsection{The Bound Computation}\label{sec:boundcomp}
In order to obtain a strong lower bound, we solve the SDP relaxation~\eqref{eq:SDP} strengthened by the inequalities given in Section~\ref{sec:bound}.
The enumeration of all pair and triangle inequalities is computationally intractable even for medium size instances. Therefore we use a similar separation routine for both types of inequalities:
\begin{enumerate}
\item Generate randomly up to $t$ inequalities violated by at least $\varepsilon_{\mathrm{viol}}$
\item Sort the $t$ inequalities by decreasing violation
\item Add to the current bounding problem the $p\ll t$ most violated ones.
\end{enumerate}
As for the clique inequalities, we use the heuristic separation routine described in \cite{ghaddar2011branch} for the minimum $k$-partition problem, that returns at most $n$ valid clique inequalities. More in detail, at each cutting-plane iteration, these cuts are determined by finding $n$ subsets $Q$ with a greedy principle. For each point $i \in S = \{1, \dots, n\}$, $Q$ is initialized as $Q = \{i\}$. Then, until the cardinality of $Q$ does not reach the size $k+1$, $Q$ is updated as $Q = Q \cup \{\argmin_{j \in S \setminus Q} \sum_{q \in Q} Z_{qj} \}$.
We denote by $\mathcal{A}(Z^\ell) = b$ the equations from the must-link and cannot-link constraints and by $l \leq \mathcal{B}(Z^\ell) \leq u$ the inequalities representing the cutting planes.
The cutting-plane procedure performed at each node is outlined in Algorithm~\ref{alg:cpproc}.
We stop the procedure when we reach the maximum number of iterations $cp_\textrm{max}$. Another stopping criterion is based on the relative variation of the bound between two consecutive iterations. If the variation is lower than a tolerance $\varepsilon_{\textrm{cp}}$, the cutting-pane method terminates, and we branch.
At each node, we use a cuts inheritance procedure to quickly retrieve several effective inequalities from the parent node and save a significant number of cutting-plane iterations during the bound computation of the children. More in detail, the inequalities that were included in the parent node during the last cutting-plane iteration are passed to its children and included in their problem from the beginning.
While inheriting inequalities in the $(i,j)$ must link child, the shrinking procedure must be taken into account, updating the indices in the inequalities involved and deleting inequalities involving both points $i$ and $j$.
In addition to the cuts inheritance, we use a cuts management procedure. A standard cutting-plane algorithm expects the valid inequalities not to be touched after having been included. The efficiency of state-of-the-art SDP solvers considerably deteriorates as we add these cuts, especially when solving large scale instances in terms of $n$. For this reason, after solving the current SDP, we remove the constraints that are not active at the optimum. Of course, inactive constraints may become active again in the subsequent cutting-plane iteration, and this operation could prevent the lower bounds from increasing monotonically; however, empirical results show that this situation happens rarely, and in this case, we decide to stop the cutting-plane procedure and we branch. From the practical standpoint, we notice that removing inactive constraints makes a huge difference since it keeps the SDP problem to a computationally tractable size. The result is that each cutting-plane iteration is more lightweight in comparison to the standard version, and this significantly impacts the overall efficiency of our branch-and-bound algorithm. Our strategy turns out to be more efficient than adding cuts only at the root node and inheriting them in the children. Indeed, if we add cuts only at the root node, the number of nodes in the tree increases since the bound does not improve as much as by repeating the separation routine in each node. Even though the single node is faster since only one SDP is solved, the overall computational time increases.
\begin{algorithm}
\SetKw{Init}{Initialization:}
\SetKw{Or}{or}
\SetKw{Stop}{stop;}
\KwData{A subproblem defined through the current set of equalities $\mathcal{A}(Z^\ell) = b$, and inequalities $l \leq \mathcal{B}(Z^\ell) \leq u$, the current global upper bound $\varphi$, the maximum number of cutting-plane iterations $cp_{\max}$, the cutting-plane tolerance $\varepsilon_{\mathrm{cp}}$, the cuts violation tolerance $\varepsilon_{\mathrm{viol}}$, and the cuts removal tolerance $\varepsilon_{\mathrm{act}}$.}
\KwResult{A lower bound $\hat{\delta}^\ell$ on the optimal value of the subproblem}
\Init $i \leftarrow 1$, $\delta_0 \leftarrow -\infty$\
\mathbb{R}epeat{no violated inequalities found}{
solve the current SDP relaxation:
\begin{equation*}
\hat{\delta}_i^\ell = \min \big\{ \inprod{-\mathcal{T}^{\ell} W (\mathcal{T}^{\ell})^\top}{Z^{\ell}} \colon \mathcal{A}(Z^\ell) = b, \ l \leq \mathcal{B}(Z^\ell) \leq u, \ Z^\ell \ge 0, \ Z^\ell \in {\mathcal S}^nml_+ \big\}
\end{equation*}
and let $\hat{Z}_i^\ell$ be the optimizer\;
\If{$\hat{\delta}_i^\ell \geq \varphi$}{
\Stop the node can be pruned\;
}
\If{$i \geq cp_{\max}$ \Or $\frac{| \hat{\delta}_i^\ell - \hat{\delta}_{i-1}^\ell |}{\hat{\delta}_{i-1}} \leq \varepsilon_{\mathrm{cp}}$}{\Stop return the lower bound $\hat{\delta}_i^\ell$ and branch\;}
remove inactive inequalities with tolerance $\varepsilon_{\mathrm{act}}$ by updating $(\mathcal{B}(\cdot), l, u)$\;
apply the separation routines for pair, triangle and clique inequalities with tolerance $\varepsilon_{\mathrm{viol}}$ and add them to $(\mathcal{B}(\cdot), l, u)$\;
\eIf{no violated inequalities found}{\Stop return the lower bound $\hat{\delta}_i^\ell$ and branch\;}
{add the inequalities by updating $(\mathcal{B}(\cdot), l, u)$\;
set $i \leftarrow i + 1$\;}
}
\caption{The node processing loop in the branch-and-cut algorithm}
\label{alg:cpproc}
\end{algorithm}
\subsection{Post-processing Using Error Bounds}
Using the optimal solution of the SDP relaxation whithin a branch-and-bound framework requires the computation of ``safe'' bounds. Such safe bounds are obtained by solving the SDP to high precision, which, however, is out of reach when using first-order methods.
In order to obtain a safe bound, we run a post-processing procedure where
we use a method to obtain rigorous lower bounds on the
optimal value of our SDP relaxation introduced by \citet{JaChayKeil2007}.
Before describing our post-processing, we state a result bounding the eigenvalues of any feasible solution of~\eqref{eq:SDP}.
\begin{lemma}\label{lem:eigboundZ}
Let $Z\succeq 0$ and $Z\ge 0$. Furthermore, let $Ze=e$. Then the eigenvalues of $Z$ are bounded by one.
\end{lemma}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Let $\lambda$ be an eigenvalue of $Z$ with eigenvector $v$, i.e., $Zv = \lambda v$. This implies
\begin{equation*}
\lambda |v_i| = | \sum_{j=1}^n z_{ij}v_j| \le \max_{1\le j\le n} |v_j| \sum_{i=1}^n z_{ij} = \max_{1\le j\le n} |v_j| \mbox{ for all } i\in \{1,\dots,n\}
\end{equation*}
by nonnegativity of $Z$ and since the row sums of $Z$ are one.
Therefore, the inequality
\begin{equation*}
\lambda \le \frac{\max_{1\le j\le n} |v_j|}{|v_i|}
\end{equation*}
holds for all $i\in \{1,\dots,n\}$, and in particular for $i \in \argmax_{1\le j\le n} |v_j|$ which proves $\lambda \le 1$.
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
We now restate Lemma~3.1 from~\cite{JaChayKeil2007} in our context.
\begin{lemma}\label{lem:jansson}
Let $S$, $Z$ be symmetric matrices that satisfy
$0 \leq \lambda_{\min}(Z)$ and $\lambda_{\max}(Z) \leq \bar{z}$
for some $\bar{z} \in \mathbb{R}$.
Then the inequality
\begin{equation*}
\left\langle S,Z\right\rangle \geq \bar{z}\sum_{i \colon \lambda_i(S) <0}\lambda_i(S)
\end{equation*}
holds.
\end{lemma}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Let $S$ have the eigenvalue decomposition
$S=Q\Lambda Q^\top$ where $QQ^\top=I$ and $\Lambda=\textrm{Diag}(\lambda(S))$.
Then
\begin{equation*}
\left\langle S,Z\right\rangle = \inprod{Q\Lambda Q^\top}{Z} = \inprod{\Lambda}{Q^\top Z Q}
= \sum_{i=1}^n \lambda_i(S)Q_{\cdot,i}^\top Z Q_{\cdot,i}
\end{equation*}
where $Q_{\cdot,i}$ is column $i$ of matrix $Q$.
Because of the bounds on the eigenvalues of $Z$, we have $0 \leq Q_{\cdot,i}^\top Z Q_{\cdot,i} \leq \bar{z}.$ Therefore
$\left\langle S,Z\right\rangle \geq \bar{z}\sum_{i \colon \lambda_i <0}\lambda_i(S)$.
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
\begin{theorem}\label{thm:errorbound}
Consider the SDP~\eqref{eq:SDP} together with equations $\mathcal{A}(Z)=b$ (e.g., from cannot-link constraints) and inequalities $l \le \mathcal{B}(Z) \le u$ (representing cutting planes)
with optimal objective function value $p^*$.
Denote the dual variables by $(\tilde{y},\tilde{u},\tilde{v},\tilde{w},\tilde{P})$, with $\tilde{y}\in \mathbb{R}^{n+1}$, $\tilde{u}$, $\tilde{v}$, $\tilde{w}$ being vectors of appropriate size, $\tilde{P}\in {\mathcal S}^n$, $\tilde{P} \ge 0$ and set $\tilde{S} = -W - \sum_{i=1}^n \tilde{y}_iE_i - \tilde{y}_{n+1}I - \mathcal{A}^\top(\tilde{u}) + \mathcal{B}^\top(\tilde{v}) - \mathcal{B}^\top(\tilde{w}) - \tilde{P}$. Then
\begin{equation*}
p^* \ge \sum_{i=1}^n\tilde{y}_i + k\tilde{y}_{n+1} + b^\top \tilde{u} - l^\top \tilde{v} + u^\top\tilde{w} + \bar{z} \sum_{i\colon \lambda_i(\tilde{S}) < 0} \lambda_i(\tilde{S}).
\end{equation*}
\end{theorem}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Let $Z^*$ be an optimal solution of~\eqref{eq:SDP} with the additional constraints $\mathcal{A}(Z) = b$ and $l \le \mathcal{B}(Z) \le u$ and $(\tilde{y},\tilde{z},\tilde{u},\tilde{v},\tilde{w},\tilde{P})$ dual feasible. Then
\begin{align*}
\inprod{-W}{Z^*} &- ( \sum_{i=1}^n \tilde{y}_i + k\tilde{y}_{n+1} + b^\top \tilde{u} - l^\top \tilde{v} + u^\top\tilde{w}) \\
& = \inprod{-W}{Z^*} - \sum_{i=1}^n \tilde{y}_i\inprod{E_i}{Z^*} - \tilde{z}\inprod{I}{Z^*} - \inprod{\mathcal{A}(Z^*)}{\tilde{u}} + \inprod{\mathcal{B}(Z^*)}{\tilde{v}} - \inprod{\mathcal{B}(Z^*)}{\tilde{w}} \\
&= \inprod{-W - \sum_{i=1}^n \tilde{y}_iE_i - \tilde{y}_{n+1}I - \mathcal{A}^\top(\tilde{u}) + \mathcal{B}^\top(\tilde{v}) - \mathcal{B}^\top(\tilde{w})}{Z^*} \\
& = \inprod{\tilde{P} + \tilde{S}}{Z^*} = \inprod{\tilde{P}}{Z^*} + \inprod{\tilde{S}}{Z^*}.
\end{align*}
We have $\tilde{P}\ge 0$, $Z^* \ge 0$. Furthermore, the eigenvalues of $Z^*$ are nonnegative and bounded by one (Lemma~\ref{lem:eigboundZ}). Using this and Lemma~\ref{lem:jansson}, we obtain
\begin{align*}
p^* = \inprod{-W}{Z^*} &\ge \sum_{i=1}^n \tilde{y}_i + k\tilde{y}_{n+1} + b^\top \tilde{u} - l^\top \tilde{v} + u^\top\tilde{w} + \inprod{\tilde{S}}{Z^*} \\
& \ge \sum_{i=1}^n \tilde{y}_i + k\tilde{y}_{n+1} + b^\top \tilde{u} - l^\top \tilde{v} + u^\top\tilde{w} + \sum_{i\colon \lambda_i(\tilde{S}) < 0} \lambda_i(\tilde{S}).
\end{align*}
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
Before stating the result used in the branch-and-bound tree after merging data points, we introduce the following notation. Let $E_i^\ell$ be the symmetric matrix such that $\inprod{E_i^\ell}{Z^\ell} = (Z^\ell e^\ell)_i$.
\begin{corollary}
Consider the SDP~\eqref{eq:SDPell} together with equations $\mathcal{A}(Z^\ell)=b$ (e.g., from cannot-link constraints) and inequalities $l \le \mathcal{B}(Z^\ell) \le u$ (representing cutting planes) with optimal objective function value $p^*$.
Let $\tilde{y} \in \mathbb{R}^{n-\ell+1}$, $\tilde{u}$, $\tilde{v}$, $\tilde{w}$ being vectors of appropriate size, $\tilde{P}\in {\mathcal S}^nml$, $\tilde{P} \ge 0$ and set
$\tilde{S} = -W^\ell - \sum_{i=1}^{n-\ell} \tilde{y}_i E^\ell_i + \tilde{y}_{n+\ell+1}\textrm{Diag}(e^\ell) - \mathcal{A}^\top(\tilde{u}) + \mathcal{B}^\top(\tilde{v}) - \mathcal{B}^\top(\tilde{w}) - \tilde{P}$.
Then
\begin{equation*}
p^* \ge \sum_{i=1}^{n-\ell} \tilde{y}_i + k\tilde{y}_{n-\ell+1} + b^\top \tilde{u} - l^\top \tilde{v} + u^\top\tilde{w} + \sum_{i\colon \lambda_i(\tilde{S}) < 0} \lambda_i(\tilde{S}).
\end{equation*}
\end{corollary}
\ifJOC
\proof{Proof.}
\else
\begin{proof}
\fi
Constraint~\eqref{eq:SDPellb} implies that the row-sum of any row in $Z^\ell$ is bounded by one since
\begin{equation*}
\sum_{j=1}^{n-\ell} z^\ell_{ij} \le \sum_{j=1}^{n-\ell} z^\ell_{ij}e^\ell_j = 1 \quad \mbox{for all } i \in \{1,\dots, n-\ell\}.
\end{equation*}
Hence using the same arguments as in Lemma~\ref{lem:eigboundZ} we can bound the eigenvalues by one and apply Theorem~\ref{thm:errorbound}.
\ifJOC
\Halmos
\endproof
\else
\end{proof}
\fi
\section{Heuristic}\label{sec:heuristic}
The most popular heuristic for solving MSSC is $k$-means \citep{macqueen1967some, lloyd1982least}. It can be viewed as a greedy algorithm.
During each update step, all the data points are assigned to their nearest centers. Afterwards, the cluster centers are repositioned by calculating the mean of the assigned observations to the respective centroids. The update process is performed until the centroids are no longer updated and therefore all observations remain at the assigned clusters.
In this paper, we use COP $k$-means \citep{wagstaff2001constrained}, a constrained version of $k$-means that aims at finding high quality clusters using prior knowledge. COP $k$-means is a constrained clustering algorithm that belongs to a class of semi-supervised machine learning algorithms. Constrained clustering incorporates a set of must-link and cannot-link constraints that define a relationship between two data instances: a must-link constraint (ML) is used to specify that the two points in the must-link relation should be in the same cluster, whereas a cannot-link constraint (CL) is used to specify that the two points in the cannot-link relation should not be in the same cluster. These sets of constraints, which are naturally available as branching decisions while visiting the branch-and-bound tree, represent the prior knowledge on the problem for which $k$-means will attempt to find clusters that satisfy the specified ML and CL constraints. The algorithm returns an empty partition if no such clustering exists which satisfies the constraints. COP $k$-means is described in Algorithm~\ref{alg:copkmeans}.
\begin{algorithm}
\caption{COP $k$-means}
\label{alg:copkmeans}
\FuncSty{K-MEANS(}\ArgSty{dataset $\mathcal{D}$, initial cluster centers $m_1, \dots, m_k$, must-link constraints ML $\subseteq \mathcal{D} \times \mathcal{D}$, cannot-link constraints CL $\subseteq \mathcal{D} \times \mathcal{D}$}\FuncSty{)}
\mathbb{R}epeat{convergence}{
\ForEach{data point $s_i \in \mathcal{D}$}{
$j \leftarrow \argmin \big\{ \|s_i - m_j\|^2 \colon j \in \{1,\dots,k\} \And $ \\ \hspace{3cm}$\texttt{VIOLATE\_CONSTRAINT}(s_i, C_j, \textrm{ML}, \textrm{CL}) \mathrm{~is~ false}\big\}$\;
\eIf{$j<\infty$}{
assign $s_i$ to $C_j$\;}{
\KwRet empty partition\;}}
\ForEach{cluster $C_j$}{
$m_j \leftarrow $ mean of the data points $s_i$ assigned to $C_j$\;}
}
\KwRet $C_1, \dots, C_k$
\FuncSty{VIOLATE\_CONSTRAINTS(}\ArgSty{data point $s_i$, cluster $C_j$, must-link constraints $\textrm{ML} \subseteq \mathcal{D} \times \mathcal{D}$, cannot-link constraints $\textrm{CL} \subseteq \mathcal{D} \times \mathcal{D}$}\FuncSty{)}
\ForEach{$(s_i, s_h) \in \textrm{ML}$}{
\lIf{$s_h \notin C_j$}{
\KwRet true}}
\ForEach{$(s_i, s_h) \in \textrm{CL}$}{
\lIf{$s_h \in C_j$}{
\KwRet true}}
\KwRet false\;
\end{algorithm}
Like other local solvers for non-convex optimization
problems, $k$-means (both in the unconstrained and constrained version) is very sensitive to the choice of the initial centroids, therefore, it often converges to a local minimum rather than the global minimum of the MSSC objective. To overcome this drawback, the algorithm is initialized with several different starting points, choosing then the clustering with the lowest objective function \citep{franti2019much}.
In the literature, several initialization algorithms have been proposed to prevent $k$-means to get stuck in a low quality local minimum. The most popular strategy for initializing $k$-means is $k$-means++ \citep{arthur2006k}. The basic idea behind this approach is to spread out the $k$ initial cluster centers to avoid the poor clustering that can be found by the standard $k$-means algorithm with random initialization. More in detail, in $k$-means++, the first cluster center is randomly chosen from the data points. Then, each subsequent cluster center is chosen from the remaining data points with probability proportional to its squared distance from the already chosen cluster centers.
We aim to exploit the information available in the solution of the SDP relaxation in order to extract a centroid initialization for COP $k$-means.
In the literature, theoretical properties of the Peng-Wei relaxation have been studied under specific stochastic models. A feasible clustering can be derived by the solution of the SDP relaxation~(\ref{eq:SDP}) by a rounding step. Sometimes, the rounding step is unnecessary because the SDP relaxation finds a solution that is feasible for the original MSSC. This phenomenon is known in the literature as exact recovery or tightness of the relaxation. Recovery guarantees have been established under a model called the subgaussian mixtures model, whose special cases include the stochastic ball model and Gaussian mixture model \citep{awasthi2015relax, iguchi2017probably, mixon2017clustering, li2020birds}. Under this distributional setting, cluster recovery is guaranteed with high probability whenever the distances between the clusters are sufficiently large. However, the generative assumption may not be satisfied by real data, and this implies that in general a rounding procedure is needed, and if possible also a bound improvement. Instead of building a rounding procedure, we decide to derive a ``smart'' initialization for the constrained $k$-means based on the solution of our bounding problem.
Here, we build the initialization exploiting the matrix $Z_{SDP}$ solution of the current bounding problem. The idea is that if the relaxation were tight, then $Z_{SDP}$ would be a clustering feasible for the rank constrained SDP \eqref{eq:RankSDP}, and hence would allow to easily recover the centroids. If the relaxation is not tight, the closest rank-$k$ approximation is built and it is used to recover the centroids. More in detail, let $Z$ be a feasible solution of the rank constrained SDP~\eqref{eq:RankSDP}. It is straightforward~\citep{mixon2017clustering} to see that $Z$ can be written as the sum of $k$ rank-one matrices:
\begin{equation}
\label{eq:clustering_matrix}
Z = \sum_{j=1}^{k} \frac{1}{|C_j|} \mathbbm{1}_{C_j} \mathbbm{1}^\top_{C_j},
\end{equation}
where $\mathbbm{1}_{C_j} \in \{0,1\}^n$ is the indicator vector of the $j$-th cluster, i.e., the $i$-th component of $\mathbbm{1}_{C_j}$ is 1 if the data point $p_i \in C_j$ and 0 otherwise.
If we post-multiply $Z$ by the data matrix $W_p \in \mathbf{R}^{n\times d}$ whose $i$-th row is the data point $p_i$, we obtain a matrix $M = ZW_p$ with a well defined structure. In fact, from equation (\ref{eq:clustering_matrix}) it follows that, for each $j \in \{1, \dots, k\}$, $M$ contains $|C_j|$ rows equal to the centroid of the data points assigned to $C_j$. If the SDP relaxation is tight, the different rows of $M$ are equal to the optimal centroids. In this case, it is natural to use the convex relaxation directly to obtain the underlying ground truth solution without the need for a rounding step. In practice, the optimizer of the SDP relaxation may not always be a clustering matrix, i.e., a low-rank solution as described by equation~(\ref{eq:clustering_matrix}).
The idea now is to build the rank-$k$ approximation $\hat{Z}$ which is obtained exploiting the following result.
\begin{proposition} \citep{eckart1936approximation}
Let $X$ be a positive semidefinite matrix with the eigenvalues $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n\ge 0$ and the corresponding eigenvectors $v_1, v_2,\ldots,v_n$. If $X$ has rank $r$, for any $k < r$, the best rank $k$ approximation of $X$, for both the Frobenius and the spectral norms is given by
\begin{equation}\label{eq:rankkapp}
\hat{X} = \sum_{i=i}^k \lambda_i v_i v_i^\top,
\end{equation}
which is the truncated eigenvalue decomposition of $X$.
\end{proposition}
Then, we compute the approximate centroid matrix $M=\hat{Z}W_p$.
In order to derive the $k$ centroids, the unconstrained $k$-means is applied to the rows of matrix $M$. Finally, the obtained centroids are used in order to initialize the algorithm COP $k$-means, which is run just once. The procedure is summarized in Algorithm~\ref{alg:sdpinit}.
\begin{algorithm}
\caption{SDP-based initialization of $k$-means}
\label{alg:sdpinit}
\FuncSty{SDP-INIT(}\ArgSty{dataset $\mathcal{D}$, number of clusters $k$, must-link constraints ML $\subseteq \mathcal{D} \times \mathcal{D}$, cannot-link constraints CL $\subseteq \mathcal{D} \times \mathcal{D}$}\FuncSty{)}
solve the SDP relaxation and obtain the optimizer $Z_{SDP}$\;
find the best rank $k$ approximation of $Z_{SDP}$ and obtain $\hat{Z}$ by \eqref{eq:rankkapp}\;
compute $M = \hat{Z}W_p$\;
cluster the rows of $M$ with unconstrained $k$-means to get the centroids $m_1, \dots, m_k$\;
use $m_1, \dots, m_k$ as the starting point of constrained $k$-means\;
\end{algorithm}
The intuition is that the better the SDP solution, the better the initialization, and hence the produced clustering. In order to confirm this intuition, we show the behavior of the heuristic on a synthetic example with 150 points in 2 dimensions. We denote by circles the points in $W_p$, by crosses the rows of matrix $M$ produced at Step 3, by diamonds the centroids obtained by clustering the rows of $M$ at Step 4 of the Algorithm~\ref{alg:sdpinit}.
In Figure~\ref{fig:heuristic3} we assume $k=3$ and apply our heuristic on different solutions of the SDPs generated during our bounding procedure: in Figure~\ref{fig:heuristic3}~(a) we use as $Z_{SDP}$ the solution obtained by solving problem \eqref{eq:SDP}, and we can see that there is some gap (the upper and lower bounds are displayed on top of each figure) and that matrix $M$ has many different rows. In Figures~\ref{fig:heuristic3}~(b), (c), and (d) we consider as $Z_{SDP}$ the solution of the SDP obtained by performing respectively 1, 2 and 3 iterations of adding cutting-planes, i.e., solving problem~(\ref{eq:SDP}) with some additional constraints~(\ref{eq:pairs})--(\ref{eq:clique}). It is clear how the rows of $M$ converge to three different centroids that, in this case, correspond to the optimal solution (the gap here is zero).
The use of \texttt{SDP-INIT} as a standalone initialization procedure could be expensive since it needs to solve a certain number of SDP problems and to perform an eigenvalue decomposition on the solution that gives the best lower bound. However, when embedded in our branch-and-bound, the extra cost of running \texttt{SDP-INIT} is only the computation of the spectral decomposition of the SDP solution providing the lower bound at the node, which is negligible with respect to the bound computation.
The effectiveness of the proposed heuristic algorithm is confirmed by the numerical results presented in Section~\ref{sec:heurnr}.
\begin{comment}
\begin{figure}
\caption{An instance with 150~points and $k = 2$.}
\label{fig:heuristic2}
\end{figure}
\end{comment}
\begin{figure}
\caption{An instance with 150~points and $k = 3$.}
\label{fig:heuristic3}
\end{figure}
\begin{comment}
\begin{figure}
\caption{$k = 4$.}
\label{fig:heuristic4}
\end{figure}
\end{comment}
\begin{comment}
\begin{figure}
\caption{$k = 5$.}
\label{fig:heuristic5}
\end{figure}
\end{comment}
\section{Numerical Results}\label{sec:numericalresults}
In this section we describe the implementation details and we show the numerical results of \texttt{SOS-SDP} on synthetic and real-world datasets.
\subsection{Details on the Implementation}
\texttt{SOS-SDP} is implemented in C++ and we use as internal subroutine for computing the bound SDPNAL+ \citep{sdpnalplus, zhao-sun-toh-2010}, which is implemented in MATLAB. SDPNAL+ is called using the MATLAB Engine API that enables running MATLAB code from C++ programs. We note that solvers based on interior point methods are not practical when solving instances with such a large number of constraints. We run our experiments on a machine with Intel(R) Xeon(R) 8124M CPU @ 3.00GHz with 16 cores, 64 GB of RAM, and Ubuntu Server 20.04. The C++ Armadillo library \citep{sanderson2016armadillo} is extensively used to handle matrices and linear algebra operations efficiently.
\texttt{SOS-SDP} can be efficiently executed in a multi-thread environment. In order to guarantee an easy and highly configurable parallelization, we use the thread pool pattern. This pattern allows controlling the number of threads the branch-and-bound is creating and saving resources by reusing threads for processing different nodes of the tree.
We adopt the same branch-and-bound configuration for each instance. In particular, we visit the tree with the best-first search strategy. When the problem at a given level is divided into the \emph{must-link} and the \emph{cannot-link} sub-problems, each node is submitted to the thread pool and run in parallel with the other threads of the pool.
Each thread of the branch-and-bound algorithm runs in a separate MATLAB session. Furthermore, since numerical algebra and linear functions are multi-threaded in MATLAB, these functions automatically execute on multiple computational threads in a single MATLAB session. To balance resource allocations for multiple MATLAB sessions and use all the available cores of the machine, we set a maximum number of computational threads allowed in each session.
\paragraph{Branch-and-bound setting}
On all the numerical tests, we adopt the following parameters setting.
As for the pair and triangle inequalities, we randomly separate at most 100000 valid cuts, we sort them in decreasing order with respect to the violation, and we select the first 5\% of violated ones, yielding at most 5000 pairs and at most 5000 triangles added in each cutting-plane iteration. Since effective inequalities are inherited from the parent to its children, at the root node the maximum number of cutting-plane iterations is set to $cp_{\mathrm{max}} = 50$, whereas for the children this number is set to 30.
The tolerance for checking the violation of the cuts is set to $\varepsilon_{\mathrm{viol}} = 10^{-4}$, whereas the tolerance for identifying the active inequalities is set to $\varepsilon_{\mathrm{act}} = 10^{-6}$. Finally, we set the accuracy tolerance of SDPNAL+ to $10^{-5}$.
As for the parallel setting, we use different configurations depending on the size of the instances since the solver requires a higher number of threads to efficiently solve large size problems.
For small instances ($n < 500$) we create a pool of 16 threads, each of them running on a session with a single component thread. For medium instances ($500 \leq n < 1000$) we use a pool of 8 threads, each of them running on a session with 2 component threads. For ($1000 \leq n < 1500$) we use a pool of 4 threads, each of them runs on a session with 4 component threads. Finally, for large scale instances ($n \geq 1500$) we use a pool of 2 threads, each of them running on a session with 8 component threads. In all cases, the MATLAB session for the computation at the root node uses all the available cores. The source code is available at \url{https://github.com/INFORMSJoC/2021.0096} \citep{SOS-SDP2021}.
\subsection{Benchmark Instances}
In order to test extensively the efficiency of \texttt{SOS-SDP} we use both artificial datasets that are built in such a way to be compliant with the MSSC assumptions and real-world datasets.
\paragraph{Artificial Instances}
Due to the minimization of the sum of squared Euclidean distances, an algorithm that solves the MSSC finds spherically distributed clusters around the centers. In order to show the effectiveness of our algorithm on instances compliant with the MSSC assumptions, we generate very large scale Gaussian datasets in the plane $(d=2)$ with varying number of data points $n \in \{2000, 2500, 3000 \}$, number of clusters $k \in \{10, 15\}$ and degree of overlap. More in detail, we sample $n$ points from a mixture of $k$ Gaussian distributions $\mathcal{N}(\mu_j, \Sigma_j)$ with equal mixing proportions, mean $\mu_j$ and shared spherical covariance matrix $\Sigma_j = \sigma I$, where $\sigma \in \{0.5, 1.0\}$ is the standard deviation. The cluster centers $\mu_j$ are sampled from a uniform distribution in the interval $[-\frac{n}{1000}-k, \frac{n}{1000}+k]$. We use the following notation to name the instances: $\{n\}\_\{k\}\_\{\sigma\}$. Note that in this case, we know in advance the correct number of clusters, so we only solve the instances for that value of~$k$.
\paragraph{Real-world Datasets}
We use a set of 34 real-world datasets coming from different domains, with a number of entities $n$ ranging between $75$ and $4177$, and with a number of features $d$ ranging between $2$ and $20531$. The datasets' characteristics are reported in Table~\ref{tab:datasets}.
\begin{table}
\begin{center}
\begin{tabular}{lcc}
\toprule
Dataset & $n$ & $d$ \\
\midrule
Ruspini & 75 & 2 \\
Voice & 126 & 310 \\
Iris & 150 & 4 \\
Wine & 178 & 13 \\
Gr202 & 202 & 2 \\
Seeds & 210 & 7 \\
Glass & 214 & 9 \\
CatsDogs & 328 & 14773 \\
Accent & 329 & 12 \\
Ecoli & 336 & 7 \\
RealEstate & 414 & 5 \\
Wholesale & 440 & 11 \\
ECG5000 & 500 & 140 \\
Hungarian & 522 & 20 \\
Wdbc & 569 & 30 \\
Control & 600 & 60 \\
Heartbeat & 606 & 3053 \\
\bottomrule
\end{tabular}\qquad
\begin{tabular}{lcc}
\toprule
Dataset & $n$ & $d$ \\
\midrule
Strawberry & 613 & 235 \\
Energy & 768 & 16 \\
Gene & 801 & 20531 \\
SalesWeekly & 810 & 106 \\
Vehicle & 846 & 18 \\
Arcene & 900 & 10000 \\
Wafer & 1000 & 152 \\
Power & 1096 & 24 \\
Phishing & 1353 & 9 \\
Aspirin & 1500 & 63 \\
Car & 1727 & 11 \\
Wifi & 2000 & 7 \\
Ethanol & 2000 & 27 \\
Mallat & 2400 & 1024 \\
Advertising & 3279 & 1558 \\
Rice & 3810 & 7 \\
Abalone & 4177 & 10 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Characteristics of the real world datasets. They all can be downloaded at the UCI \citep{uci}, UCR \citep{UCRArchive2018} and sGDML \citep{chmiela2019sgdml} websites.}
\label{tab:datasets}
\end{table}
\subsection{Branch-and-Bound Results on Artificial instances}
In Table \ref{tab:res_art} we report the dataset name according to the notation $\{n\}\_\{k\}\_\{\sigma\}$, the optimal objective function $f_\textrm{opt}$, the number of cutting-plane iterations at the root (cp), the number of cuts added in the last cutting-plane iteration at the root ($\textrm{cuts}_\textrm{cp}$), the gap at the root ($gap_0$) when problem \eqref{eq:SDPbab} is solved without adding valid inequalities, in brackets the gap at the end of the cutting-plane procedure at the root node ($gap_{cp}$),
the number of nodes of the branch-and-bound tree (N), and the wall clock time in seconds (time).
\begin{tabularx}{\textwidth}{lcccccccc}
\toprule
Dataset & $f_\textrm{opt}$ & cp & $\textrm{cuts}_\textrm{cp}$ & $gap_0$ $(gap_{cp})$ & N & time \\
\midrule
2000\_10\_0.5 & 955.668 & 0 & 0 & 0.000039 (0.000039) & 1 & 848.88 \\
2000\_10\_1.0 & 3601.310 & 3 & 10999 & 0.006171 (0.003578) & 3 & 8794.17 \\
2000\_15\_0.5 & 955.800 & 1 & 6177 & 0.001556 (0.000009) & 1 & 1155.06 \\
2000\_15\_1.0 & 3658.730 & 3 & 11035 & 0.006192 (0.002059) & 3 & 8351.91 \\
2500\_10\_0.5 & 1199.080 & 1 & 5249 & 0.000184 (0.000083) & 1 & 2859.30 \\
2500\_10\_1.0 & 4522.350 & 12 & 11539 & 0.008008 (0.000553) & 1 & 20495.43 \\
2500\_15\_0.5 & 1194.550 & 0 & 0 & 0.000699 (0.000699) & 1 & 1049.76 \\
2500\_15\_1.0 & 4574.360 & 6 & 10146 & 0.005311 (0.000971) & 1 & 10245.69 \\
3000\_10\_0.5 & 1446.480 & 0 & 0 & 0.000067 (0.000067) & 1 & 2220.21 \\
3000\_10\_1.0 & 5512.370 & 9 & 10769 & 0.004601 (0.000606) & 1 & 27781.38 \\
3000\_15\_0.5 & 1439.940 & 0 & 0 & 0.000433 (0.000433) & 1 & 2003.94 \\
3000\_15\_1.0 & 5537.200 & 10 & 15608 & 0.006245 (0.001205) & 3 & 38330.01 \\
\bottomrule
\caption{Results for the artificial datasets.}
\label{tab:res_art}
\end{tabularx}
As we increase $\sigma$, the cluster separation decreases, and the degree of overlap increases (see Figure \ref{fig:art}). In this scenario, the SDP relaxation is not tight anymore and the global minimum is certified by our specialized branch-and-bound algorithm.
For $\sigma = 0.5$ each problem is solved at the root with zero (i.e., the SDP relaxation is tight) or with at most one cutting-plane iteration. As we decrease the cluster separation by increasing $\sigma$ the problem becomes harder since some clusters overlap and the cluster boundaries are less clear. In this case, more cutting-plane iterations are needed (up to a maximum of 12 iterations). In any case, we need at most 3 nodes for solving these instances, and this confirms that, if the generative assumption is met, the cutting-plane procedure at the root node is the main ingredient for success. In the next section, we show how the behavior changes in real world instances, where we do not have information on the data distribution and on the correct value of $k$. In this case, the overall branch-and-bound algorithm becomes fundamental in order to solve the problems.
\begin{figure}
\caption{Artificial instances for $n=2000$ and $d=2$.}
\label{fig:art}
\end{figure}
\subsection{Branch-and-Bound Results on Real World Datasets}
The MSSC requires the user to specify the number of clusters $k$ to generate. Determining the right $k$ for a data set is a different issue from the process of solving the clustering problem. This is still an open problem since, depending on the chosen distance measure, one value of $k$ may be better than another one. Hence, choosing $k$ is often based on assumptions on the application, prior knowledge of the properties of the dataset, and practical experience. In the literature, clustering validity indices in conjunction with the $k$-means algorithm are commonly used to determine the ``right'' number of clusters. Most of these methods minimize or maximize an external validity index by running a clustering algorithm (for example $k$-means) several times for different values of $k$.
We recall that the basic idea behind the MSSC is to define clusters such that the total within-cluster sum of squares is minimized. This objective function measures the compactness of the clustering and we want it to be as small as possible. The ``elbow method'' is probably the most popular method for determining the number of clusters. It requires running the $k$-means algorithm with an increasing number of clusters. The suggested $k$ can be determined by looking at the MSSC objective as a function of $k$ and by finding the inflection point. The location of the inflection point (knee) in the plot is generally considered as an indicator of the appropriate number of clusters. The drawback of this method is that the identification of the knee could not be obvious. Hence, different validity indices have been proposed in the literature to identify the suitable number of clusters or to check whether a given dataset exhibits some kind of a structure that can be captured by a clustering algorithm for a given $k$. All these indices are computed aposteriori given the clustering produced for different values of $k$. In addition to the elbow method, we use three cluster validity measures that are compliant with the assumptions of the MSSC: namely the Silhouette index \citep{rousseeuw1987silhouettes}, the Calinski–Harabasz (CH) index \citep{calinski1974dendrite} and Davies–Bouldin (DB) index \citep{davies1979cluster}.
The Silhouette index determines how well each object lies within its cluster and is given by the average Silhouette coefficient over all the data points. The Silhouette coefficient is defined for each data point and is composed of two scores: the mean distance between a sample and all other points in the same class and the mean distance between a sample and all other points in the next nearest cluster. The CH index is the ratio of the sum of between-clusters dispersion and within-cluster dispersion for all clusters. The DB index is defined as the average similarity between each cluster and its most similar one.
The Silhouette index and the CH index are higher when the clusters are dense and well separated, which relates to the standard concept of clustering, whereas for the DB index lower values indicate a better partition.
Since the exact resolution of the MSCC problem could be expensive and time consuming from the computational point of view, one may be interested in finding the global solution for a specified or restricted number of clusters. In practice, one can run the $k$-means algorithm for different values of $k$ and then use the exact algorithm to find and certify the global optimum for the $k$ suitable for the application of interest.
Hence, we choose to run our algorithm on a large number of datasets, and for each dataset, we run it only for the suggested number of clusters obtained with the help of the criteria mentioned above.
Whenever there is some ambiguity, i.e., the different criteria suggest different values of $k$, we run our algorithm for all the suggested values. With this criterion, we end up solving $54$ clustering instances with different size $n$, different dimension $d$, and different values of $k$.
In Table \ref{tab:res_dataset} we report:
\begin{itemize}
\item the dataset name
\item the number of clusters ($k$)
\item the optimal objective function ($f_\textrm{opt}$). We add a $(*)$ whenever the optimum we certify is not found by $k$-means at the root node
\item the number of cutting-plane iterations at root (cp)
\item the number of inequalities of the last SDP problem solved at the root in the cutting-plane procedure ($\textrm{cuts}_\textrm{cp}$)
\item the gap at the root ($gap_0$) when problem \eqref{eq:SDPbab} is solved without adding valid inequalities, and in brackets the gap at the end of the cutting-plane procedure at the root node ($gap_{cp}$)
\item the number of nodes (N) of the branch-and-bound tree
\item the wall clock time in seconds (time).
\end{itemize}
Small and medium scale instances ($n < 1000$) are considered solved when the relative gap tolerance
is less or equal than $10^{-4}$, whereas for large scale instances ($n \geq 1000$) the branch-and-bound algorithm is stopped when the tolerance is less or equal than $10^{-3}$, which we feel is an adequate tolerance for large scale real-world applications. The gap measures the difference between the best upper and lower bounds and it is calculated as $(UB - LB) / UB$.
The numerical results show that our method is able to solve successfully all the instances up to a size of $n=4177$ entities.
\begin{longtable}{lcccccccc}
\toprule
Dataset & $k$ & $f_\textrm{opt}$ & cp & $\textrm{cuts}_\textrm{cp}$ & $gap_0$ $(gap_{cp})$ & N & time \\
\midrule
Ruspini & 4 & 1.28811e+04 & 0 & 0 & 2.23e-04 (2.23e-04) & 1 & 2.55 \\
Voice & 2 & 1.13277e+22 & 2 & 7593 & 5.40e-02 (1.66e-06) & 1 & 14.45 \\
Voice & 9 & 5.74324e+20* & 4 & 6115 & 1.07e-01 (6.45e-04) & 3 & 128.35 \\
Iris & 2 & 1.52348e+02 & 2 & 7701 & 1.10e-02 (2.19e-06) & 1 & 17 \\
Iris & 3 & 7.88514e+01 & 4 & 7136 & 4.23e-02 (1.18e-04) & 5 & 83.3 \\
Iris & 4 & 5.72285e+01 & 4 & 7262 & 4.28e-02 (4.20e-04) & 3 & 104.55 \\
Wine & 2 & 4.54375e+06 & 3 & 8162 & 3.45e-02 (2.69e-07) & 1 & 53.55 \\
Wine & 7 & 4.12138e+05* & 4 & 5759 & 5.81e-02 (1.03e-04) & 3 & 87.55 \\
Gr202 & 6 & 6.76488e+03 & 6 & 6607 & 6.72e-02 (8.53e-04) & 17 & 298.35 \\
Seeds & 2 & 1.01161e+03* & 9 & 10186 & 4.31e-02 (4.77e-04) & 29 & 957.1 \\
Seeds & 3 & 5.87319e+02 & 4 & 6620 & 2.67e-02 (1.26e-05) & 1 & 68.85 \\
Glass & 3 & 1.14341e+02 & 5 & 6799 & 4.68e-02 (1.64e-04) & 3 & 193.8 \\
Glass & 6 & 7.29647e+01* & 7 & 3014 & 5.45e-02 (4.36e-04) & 5 & 198.9 \\
CatsDogs & 2 & 1.14099e+05 & 1 & 5368 & 1.83e-03 (2.23e-09) & 1 & 108.8 \\
Accent & 2 & 3.28685e+04 & 0 & 0 & 6.55e-06 (6.55e-06) & 1 & 11.05 \\
Accent & 6 & 1.84360e+04* & 8 & 4523 & 2.94e-02 (2.08e-05) & 1 & 244.8 \\
Ecoli & 3 & 2.32610e+01 & 4 & 10101 & 7.71e-03 (1.89e-04) & 3 & 181.9 \\
RealEstate & 3 & 5.50785e+07 & 3 & 6236 & 1.59e-02 (3.51e-05) & 1 & 104.55 \\
RealEstate & 5 & 2.18711e+07 & 5 & 8006 & 6.82e-02 (2.64e-05) & 1 & 258.4 \\
Wholesale & 5 & 2.04735e+03 & 6 & 7668 & 6.43e-02 (2.06e-05) & 1 & 421.6 \\
Wholesale & 6 & 1.73496e+03* & 10 & 11161 & 6.32e-02 (7.06e-04) & 3 & 1782.45 \\
ECG5000 & 2 & 1.61359e+04 & 3 & 9312 & 1.02e-03 (7.49e-05) & 1 & 119 \\
ECG5000 & 5 & 1.15458e+04 & 25 & 6289 & 4.93e-02 (1.01e-04) & 3 & 2524.5 \\
Hungarian & 2 & 8.80283e+06 & 7 & 11265 & 1.03e-02 (1.31e-05) & 1 & 551.65 \\
Wdbc & 2 & 7.79431e+07 & 5 & 8645 & 3.21e-02 (2.10e-05) & 1 & 436.05 \\
Wdbc & 5 & 2.05352e+07* & 23 & 10662 & 7.45e-02 (5.27e-04) & 15 & 2436.95 \\
Control & 3 & 1.23438e+06 & 6 & 12381 & 2.80e-03 (1.26e-04) & 9 & 895.9 \\
Heartbeat & 2 & 2.79391e+04 & 0 & 0 & 8.15e-06 (8.15e-06) & 1 & 66.3 \\
Strawberry & 2 & 2.79363e+03 & 15 & 23776 & 5.44e-02 (4.02e-04) & 37 & 5250.45 \\
Energy & 2 & 9.64123e+03 & 0 & 0 & 9.86e-09 (9.86e-09) & 1 & 18.7 \\
Energy & 12 & 4.87456e+03 & 0 & 0 & 4.03e-07 (4.03e-07) & 1 & 29.75 \\
Gene & 5 & 1.78019e+07* & 2 & 15589 & 1.83e-03 (1.30e-04) & 3 & 3851.35 \\
Gene & 6 & 1.70738e+07 & 5 & 14620 & 3.82e-03 (2.08e-04) & 11 & 9896.55 \\
SalesWeekly & 2 & 1.44942e+06* & 6 & 8508 & 2.50e-02 (1.33e-03) & 9 & 2341.75 \\
SalesWeekly & 3 & 7.09183e+05* & 4 & 9096 & 1.03e-03 (9.44e-05) & 1 & 262.65 \\
SalesWeekly & 5 & 5.20938e+05* & 4 & 11811 & 1.67e-03 (1.12e-04) & 5 & 1045.5 \\
Vehicle & 2 & 7.29088e+06 & 5 & 10395 & 7.68e-03 (3.72e-04) & 11 & 1842.8 \\
Arcene & 2 & 3.48490e+10 & 3 & 36100 & 2.59e-03 (1.26e-04) & 3 & 1369.35 \\
Arcene & 3 & 2.02369e+10 & 0 & 0 & 3.50e-06 (3.50e-06) & 1 & 758.2 \\
Arcene & 5 & 1.69096e+10* & 7 & 8327 & 7.57e-03 (1.55e-04) & 27 & 6885 \\
Wafer & 2 & 6.19539e+04 & 3 & 7254 & 7.82e-04 (1.00e-04) & 1 & 379.1 \\
Wafer & 4 & 4.42751e+04 & 22 & 16957 & 1.97e-02 (8.76e-04) & 1 & 6756.65 \\
Power & 2 & 3.22063e+03 & 3 & 11350 & 1.05e-02 (2.89e-03) & 3 & 3381.3 \\
Phishing & 9 & 3.15888e+03* & 46 & 12459 & 2.48e-02 (7.00e-04) & 1 & 18866.6 \\
Aspirin & 3 & 1.27669e+04 & 2 & 10000 & 4.39e-03 (3.02e-03) & 9 & 3779.1 \\
Car & 4 & 5.61600e+03 & 23 & 38582 & 1.61e-03 (1.02e-05) & 1 & 5989.95 \\
Ethanol & 2 & 7.26854e+03 & 0 & 0 & 5.33e-08 (5.33e-08) & 1 & 310.25 \\
Wifi & 5 & 2.04311e+05 & 7 & 20886 & 1.13e-02 (2.18e-03) & 7 & 22754.5 \\
Mallat & 3 & 9.08648e+04 & 5 & 17092 & 3.61e-03 (9.59e-04) & 1 & 5970.4 \\
Mallat & 4 & 7.45227e+04 & 6 & 15305 & 6.80e-03 (4.49e-03) & 5 & 26344.9 \\
Advertising & 2 & 5.00383e+06* & 1 & 12533 & 1.53e-03 (2.16e-05) & 1 & 6465.1 \\
Advertising & 8 & 4.54497e+06* & 4 & 19948 & 2.98e-03 (1.08e-04) & 1 & 25114.1 \\
Rice & 2 & 1.39251e+04 & 24 & 7258 & 1.43e-02 (7.14e-03) & 5 & 103710.2 \\
Abalone & 3 & 1.00507e+03 & 0 & 0 & 3.14e-04 (3.14e-04) & 1 & 9428.2 \\
\bottomrule
\caption{Results for the real world datasets}
\label{tab:res_dataset}
\end{longtable}
To the best of our knowledge, the exact algorithm proposed in \cite{aloise2012improved} represents the actual state-of-the-art. Indeed it is the only algorithm able to exactly solve instances of size larger than 1000, satisfying one of the following strong assumptions (due to the geometrical approach involved): either the instance is on the plane ($d=2$) or the required number of clusters is large with respect to the number of points. Indeed they were able to solve a TSP instance with $d=2$ of size $n=2392$ for numbers of clusters ranging from $k=2$ to $k=10$, and for large number of clusters ($k$ between 100 and 400), and an instance of size $n=2310$ with $d=19$ but only for large number of clusters ($k$ between 230 and 500). Our algorithm has orthogonal capabilities in some sense to the one proposed in \cite{aloise2012improved}, since is not influenced by the number of features (we solve problems with thousands of features, which would be completely out of reach for the algorithm in~\cite{aloise2012improved}). Indeed, in the SDP formulation, the number of features is hidden in the matrix $W$, which is computed only once, so that it does not influence the computational cost of the algorithm. On the other hand, it is well known that the difficulty (and the gap) of the SDP relaxation \eqref{eq:SDPbab} increases when the boundary of the clusters are confused, and this phenomenon becomes more frequent when the number of clusters is high with respect to the number of points, and far away from the correct $k$ for the MSSC objective function. The strength of our bounding procedure is confirmed by 28 problems out of 54 solved at the root. Among these 28 problems, only 8 are tight, in the sense that problem \eqref{eq:SDPbab} without inequalities produces the optimal solution.
The efficiency of \texttt{SOS-SDP} comes from the combination of the cutting-plane procedure that allows us to close a significant amount of the gap even when the bound without inequalities is not tight, and the heuristic that when the SDP solution is good allows us to find the optimal solution. Note that in 15 out of 34 instances, our algorithm certifies the optimality of a solution that $k$-means at the root could not find.
Overall, the number of nodes of the branch-and-bound tree is always smaller than 40, but the computational cost of the single node may be high due to the high number of cutting-plane iterations. The values of cuts$_{\mathrm{cp}}$ confirm that the removal of inactive inequalities is effective, and allows to keep the number of inequalities moderate so that the SDP at each cutting-plane iteration is computationally tractable.
\subsection{Numerical Results of \texttt{SDP-INIT} }\label{sec:heurnr}
In order to test the efficiency of our initialization of constrained $k$-means, we report the behaviour at the root node on a subset of real-world datasets. We selected the most popular on the UCI website with size in the range of 150--569. To have more difficult instances, we run the heuristic for all the values of $k$ in the range from $2$ to $10$. Note that for $k$ far from the values suggested by the validation indices, the optimal solution may be constituted by overlapped and confused clusters that are more difficult to find for any heuristic.
In Table~\ref{tab:heuristic}, we report the results obtained by our heuristic, compared with 50 runs of $k$-means initialized with $k$-means++ and with random initialization.
In each table, we report:
\begin{itemize}
\item the lower bound obtained by solving the basic SDP relaxation ($LB_0$), and the corresponding heuristic solution ($UB_0$)
\item the lower bound obtained after performing $CP$ cutting-plane iterations $LB_{CP}$ and the corresponding heuristic solution ($UB_{CP}$)
\item the solution produced by $k$-means after 50 runs initialized with $k$-means++ ($UB_{++}$)
\item the solution produced by $k$-means after 50 runs randomly initialized ($UB_{RAND}$)
\end{itemize}
We highlight the best solution in boldface. The results show that the solution $UB_{CP}$ is always the best, apart from 1 case. Note that in many cases, the solution $UB_{0}$ is fairly competitive both in terms of bound quality and computational effort since it requires the solution of exactly one SDP.
\begin{table}
\begin{center}
\footnotesize
\begin{tabular}{cccccccc}
\toprule
$K$ & $CP$ & $LB_{0}$ & $LB_{CP}$ & $UB_{0}$ & $UB_{CP}$ & $UB_{++}$ & $UB_{RAND}$ \\
\midrule
\multicolumn{8}{l}{Iris dataset}\\
\midrule
2 & 2 & 1.50679e+02 & 1.52348e+02 & \fontseries{b}\selectfont{1.52348e+02} & \fontseries{b}\selectfont{1.52348e+02} & \fontseries{b}\selectfont{1.52348e+02} & \fontseries{b}\selectfont{1.52348e+02} \\
3 & 4 & 7.55144e+01 & 7.88421e+01 & 7.88557e+01 & \fontseries{b}\selectfont{7.88514e+01} & 7.88518e+01 & 7.88527e+01 \\
4 & 6 & 5.47766e+01 & 5.72281e+01 & \fontseries{b}\selectfont{5.72285e+01} & \fontseries{b}\selectfont{5.72285e+01} & 5.72560e+01 & 5.72560e+01 \\
5 & 3 & 4.38467e+01 & 4.64369e+01 & 4.64612e+01 & \fontseries{b}\selectfont{4.64462e+01} & \fontseries{b}\selectfont{4.64462e+01} & 4.64612e+01 \\
6 & 4 & 3.67110e+01 & 3.90175e+01 & 3.90660e+01 & \fontseries{b}\selectfont{3.90400e+01} & 3.90660e+01 & \fontseries{b}\selectfont{3.90400e+00} \\
7 & 6 & 3.18467e+01 & 3.42788e+01 & 3.43058e+01 & \fontseries{b}\selectfont{3.42982e+01} & 3.44090e+01 & 3.43859e+01 \\
8 & 3 & 2.88697e+01 & 2.99660e+01 & \fontseries{b}\selectfont{2.99904e+01} & \fontseries{b}\selectfont{2.99904e+01} & \fontseries{b}\selectfont{2.99904e+01} & 3.04762e+01 \\
9 & 3 & 2.64849e+01 & 2.77836e+01 & 2.79408e+01 & \fontseries{b}\selectfont{2.77861e+01} & 2.78921e+01 & 2.83071e+01 \\
10 & 4 & 2.44186e+01 & 2.58329e+01 & 2.62712e+01 & \fontseries{b}\selectfont{2.58341e+01} & 2.59644e+01 & 2.65776e+01 \\
\midrule
\multicolumn{8}{l}{Glass dataset}\\
\midrule
2 & 6 & 1.35499e+02 & 1.36525e+02 & 1.36537e+02 & \fontseries{b}\selectfont{1.36528e+02} & \fontseries{b}\selectfont{1.36528e+02} & 1.36537e+02 \\
3 & 5 & 1.08991e+02 & 1.14320e+02 & \fontseries{b}\selectfont{1.14341e+02} & \fontseries{b}\selectfont{1.14341e+02} & \fontseries{b}\selectfont{1.14341e+02} & \fontseries{b}\selectfont{1.14341e+02} \\
4 & 5 & 9.14749e+01 & 9.47742e+01 & 9.48402e+01 & \fontseries{b}\selectfont{9.47899e+01} & 9.48402e+01 & 9.48402e+01 \\
5 & 6 & 7.87104e+01 & 8.34045e+01 & 8.40062e+01 & \fontseries{b}\selectfont{8.35054e+01} & 8.42973e+01 & 8.40502e+01 \\
6 & 8 & 6.89918e+01 & 7.29430e+01 & \fontseries{b}\selectfont{7.29647e+01} & \fontseries{b}\selectfont{7.29647e+01} & 7.37947e+01 & 7.43696e+01 \\
7 & 8 & 6.19552e+01 & 6.47908e+01 & 6.53398e+01 & \fontseries{b}\selectfont{6.47973e+01} & 7.08087e+01 & 6.66828e+01 \\
8 & 6 & 5.61534e+01 & 5.85654e+01 & 5.87606e+01 & \fontseries{b}\selectfont{5.85699e+01} & 5.90119e+01 & 6.08941e+01 \\
9 & 10 & 5.12932e+01 & 5.37277e+01 & 5.41810e+01 & \fontseries{b}\selectfont{5.37580e+01} & 5.55979e+01 & 5.61847e+01 \\
10 & 4 & 4.70718e+01 & 4.93411e+01 & 4.97866e+01 & \fontseries{b}\selectfont{4.97382e+01} & 5.15837e+01 & 5.25047e+01 \\
\midrule
\multicolumn{8}{l}{Wholesale dataset}\\
\midrule
2 & 2 & 3.48221e+03 & 3.48656e+03 & \fontseries{b}\selectfont{3.48657e+03} & \fontseries{b}\selectfont{3.48657e+03} & \fontseries{b}\selectfont{3.48657e+03} & \fontseries{b}\selectfont{3.48657e+03} \\
3 & 5 & 2.85705e+03 & 2.91234e+03 & \fontseries{b}\selectfont{2.91252e+03} & \fontseries{b}\selectfont{2.91252e+03} &\fontseries{b}\selectfont{2.91252e+03} & \fontseries{b}\selectfont{2.91254e+03} \\
4 & 9 & 2.33207e+03 & 2.46555e+03 & \fontseries{b}\selectfont{2.46558e+03} & \fontseries{b}\selectfont{2.46558e+03} & \fontseries{b}\selectfont{2.46558e+03} & \fontseries{b}\selectfont{2.46558e+03} \\
5 & 7 & 1.91575e+03 & 2.04735e+03 & 2.04741e+03 & \fontseries{b}\selectfont{2.04735e+03} & 2.04891e+03 & 2.04891e+03 \\
6 & 10 & 1.63098e+03 & 1.73382e+03 & 1.74322e+03 & \fontseries{b}\selectfont{1.73496e+03} & 1.74096e+03 & 1.75359e+03 \\
7 & 12 & 1.44236e+03 & 1.52350e+03 & 1.52551e+03 & \fontseries{b}\selectfont{1.52383e+03} & 1.52693e+03 & 1.53677e+03 \\
8 & 11 & 1.28695e+03 & 1.36289e+03 & 1.36949e+03 & \fontseries{b}\selectfont{1.36290e+03} & 1.36621e+03 & 1.39735e+03 \\
9 & 10 & 1.14692e+03 & 1.21928e+03 & 1.22008e+03 & \fontseries{b}\selectfont{1.21978e+03} & \fontseries{b}\selectfont{1.21978e+03} & 1.26105e+03 \\
10 & 6 & 1.03078e+03 & 1.07843e+03 & 1.08010e+03 & \fontseries{b}\selectfont{1.07843e+03} & 1.13670e+03 & 1.21282e+03 \\
\midrule
\multicolumn{8}{l}{Wdbc dataset}\\
\midrule
2 & 6 & 7.54429e+07 & 7.79415e+07 & \fontseries{b}\selectfont{7.79431e+07} & \fontseries{b}\selectfont{7.79431e+07} & \fontseries{b}\selectfont{7.79431e+07} & \fontseries{b}\selectfont{7.79431e+07} \\
3 & 27 & 4.14673e+07 & 4.72612e+07 & 4.74219e+07 & \fontseries{b}\selectfont{4.72648e+07} & \fontseries{b}\selectfont{4.72648e+07} & 4.74999e+07 \\
4 & 22 & 2.62662e+07 & 2.91013e+07 & 2.92269e+07 & \fontseries{b}\selectfont{2.92265e+07} & \fontseries{b}\selectfont{2.92265e+07} & \fontseries{b}\selectfont{2.92265e+07} \\
5 & 20 & 1.90062e+07 & 2.05248e+07 & 2.05806e+07 & \fontseries{b}\selectfont{2.05352e+07} & \fontseries{b}\selectfont{2.05352e+07} & 2.06727e+07 \\
6 & 6 & 1.47880e+07 & 1.55897e+07 & 1.69771e+07 & 1.69343e+07 & \fontseries{b}\selectfont{1.66461e+07} & 1.71215e+07 \\
7 & 22 & 1.20747e+07 & 1.31868e+07 & 1.32742e+07 & \fontseries{b}\selectfont{1.32470e+07} & 1.32655e+07 & 1.33533e+07 \\
8 & 8 & 1.02027e+07 & 1.07390e+07 & 1.12114e+07 & \fontseries{b}\selectfont{1.12064e+07} & 1.12441e+07 & 1.15090e+07 \\
9 & 3 & 8.83658e+06 & 9.09983e+06 & \fontseries{b}\selectfont{9.43290e+06} & \fontseries{b}\selectfont{9.43290e+06} & 9.47386e+06 & 1.05951e+07 \\
10 & 1 & 7.72013e+06 & 7.72013e+06 & \fontseries{b}\selectfont{8.37902e+06} & \fontseries{b}\selectfont{8.37902e+06} & 8.54589e+06 & 9.83225e+06 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Heuristic performance for selected datasets. Best upper bounds found are typeset in boldface.}\label{tab:heuristic}
\end{table}
\section{Conclusions}\label{sec:conclusion}
We developed an exact solution algorithm for the minimum sum-of-squares clustering problem (MSSC) using tools from semidefinite programming.
We use a semidefinite relaxation that exploits three types of valid inequalities in a cutting plane fashion to generate tight lower bounds for the MSSC.
Besides these lower bounds, the semidefinite relaxation also provides a primal solution that can be used for generating data to initialize constrained $k$-means, which is known to be sensitive concerning the starting point. Numerical experiments undoubtedly demonstrate the advantage of using this initialization procedure.
We implemented a branch-and-bound algorithm using the ingredients described above. Our way of branching allows us to decrease the size of the problem while going down the branch-and-bound tree. Notably, the shrinking procedure preserves the structure of the problem which is beneficial for our routine computing the bounds in each node of the branch-and-bound tree.
Our code is parallelized in two ways: the nodes in the branch-and-bound tree are evaluated in parallel and the bound computation within a node is executed in a multi-threaded MATLAB environment.
The numerical results impressively exhibit the efficiency of our algorithm: we can solve real-world instances up to 4000~data points. To the best of our knowledge, no other exact solution methods can handle generic instances of that size.
Moreover, the dimension of the data points does not influence the performance of our algorithm, we solve instances with more than 20\;000 features.
Our algorithm can be extended to deal with certain constrained versions of sum-of-squares clustering like those with diameter constraints, split constraints, density constraints, or capacity constraints \citep{davidson2005clustering, duong:2017}. This is left for future work. Also, kernel-based clustering is a promising extension that we plan to consider \citep{dhillon2004kernel}.
Finally, we have ideas in mind on how to use our algorithm in a heuristic fashion for obtaining high quality solutions for huge graphs.
\ifJOC
\ACKNOWLEDGMENT{
Parts of this project were carried out during a research stay of the third author at the University Tor Vergata, funded by the University of Rome Tor Vergata Visiting Professor grant 2018.
Furthermore, this project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant
agreement MINOA No~764759.
We thank Kim-Chuan Toh for bringing our attention to the work of \citet{JaChayKeil2007} and
for providing an implementation of the method therein.}
\else
{\small
\vspace*{1ex}\noindent
Veronica Piccialli,
\href{mailto:[email protected]}{\url{[email protected]}},
University of Rome Tor Vergata,
via del Politechnico, 00133 Rome, Italy
ORCiD: 0000-0002-3357-9608
\vspace*{1ex}\noindent
Antonio M. Sudoso,
\href{mailto:[email protected]}{\url{[email protected]}},
University of Rome Tor Vergata,
via del Politecnico, 1, 00133 Rome, Italy
ORCiD: 0000-0002-2936-9931
\vspace*{1ex}\noindent
Angelika Wiegele,
\href{mailto:[email protected]}{\url{[email protected]}},
Alpen-Adria-Universität Klagenfurt,
Universitätsstraße 65--67, 9020 Klagenfurt, Austria,
ORCiD: 0000-0003-1670-7951
}
\fi
\end{document}
|
\begin{document}
\begin{abstract}
Given a multivariate complex polynomial ${p\in\mathbb{C}[z_1,\ldots,z_n]}$, the imaginary projection $\mathcal{I}(p)$ of $p$ is defined as the projection of the variety $\mathcal{V}(p)$ onto its imaginary part. We focus on studying the imaginary projection of complex polynomials and we state explicit results for certain families of them with arbitrarily large degree or dimension. Then, we restrict to complex conic sections and give a full characterization of their imaginary projections, which generalizes a classification for the case of real conics. That is, given a bivariate complex polynomial $p\in\C[z_1,z_2]$ of total degree two, we describe the number and the boundedness of the components in the complement of $\mathcal{I}(p)$ as well as their boundary curves and the spectrahedral structure of the components. We further show a realizability result for strictly convex complement components which is in sharp contrast to the case of real polynomials.
\end{abstract}
\maketitle
\section{Introduction\label{se:intro}}
Given a polynomial $p\in\C[\z]:=\C[z_1, \ldots ,z_n]$, the imaginary projection $\I(p)$ as introduced in \cite{jtw-2019} is the projection of the variety $\mathcal{V}(p)\subseteq\C^n$ onto its imaginary part, that is,
\begin{equation}
\label{eq:imagproj1}
\ \I(p)= \ \left\{ \z_{\rm im} = ((z_1)_{\rm im}, \ldots, (z_n)_{\rm im}) \ : \ \mathbf{z} \in \mathcal{V}(p)\right\} \ \subseteq \ \R^n,
\end{equation}
where $(\cdot)_{\rm im}$ is the imaginary part of a complex number.
Recently, there has been wide-spread research interest in mathematical branches which are directly connected to the imaginary projection of polynomials.
As a primary motivation, the imaginary projection provides a
comprehensive geometric view for notions of \emph{stability of
polynomials} and generalizations thereof. A polynomial $p\in\C[\z]$ is called \textit{stable}, if $p(\z)=0$ implies $(z_j)_{\rm im}\leq 0$ for some $j\in[n]$.
In terms of the imaginary projection $\I(p)$, we can express the stability of $p$ as
the condition $\I(p)\cap\R^n_{>0}=\emptyset$. Stable polynomials
have applications in many branches of mathematics
including combinatorics (\cite{braenden-hpp} and see \cite{brown-wagner-2020}
for the connection of the imaginary projection to combinatorics),
differential equations \cite{borcea-braenden-2010},
optimization \cite{straszak-vishnoi-2017},
probability theory \cite{bbl-2009}, and
applied algebraic geometry \cite{volcic-2019}.
Further application areas include
theoretical computer science \cite{mss-interlacing1, mss-interlacing2},
statistical physics \cite{borcea-braenden-leeyang1}, and control theory \cite{ms-2000}, see also the surveys \cite{pemantle-2012} and
\cite{wagner-2010}.
Recently, various generalizations and variations of the stability
notion have been studied, such as stability with respect to
a polyball \cite{gkv-2016,gkv-2017},
conic stability \cite{dgt-conic-pos-map-2019,joergens-theobald-conic},
Lorentzian polynomials \cite{braenden-huh-2020}, or positively
hyperbolic varieties \cite{rvy-2021}. Exemplarily, regarding the
conic stability,
a polynomial $p\in\C[\z]$ is called \textit{$K$-stable} for a proper cone $K\subset \R^n$ if $p(\z)\neq 0$, whenever $\z_{\rm im}\in \inter K$,
where $\inter$ is the interior. In terms of
the imaginary projection, this condition can be equivalently expressed as
$\I(p)\cap \inter K=\emptyset$.
Another motivation comes from the close connection of the imaginary
projection to
hyperbolic polynomials and hyperbolicity
cones \cite{garding-59}.
As shown in \cite{joergens-theobald-hyperbolicity}, in case of a
real \emph{homogeneous} polynomial $p$,
the components of the complement
$\I(p)^\mathsf{c}$ coincide with the hyperbolicity cones
of $p$.
These concepts play a central role in hyperbolic programming,
see \cite{gueler-97,naldi-plaumann-2018,
nesterov-tuncel-2016,saunderson-2019}.
A prominent open question in this research direction is the generalized
Lax conjecture, which claims that every hyperbolicity cone is
spectrahedral, see \cite{vinnikov-2012}.
Representing
convex sets by spectrahedra is not only motivated by the general
Lax conjecture, but also by the question of effective handling convex
semialgebraic sets (see, for example, \cite{bpt-2013,kpv-2015}).
Recently, the conjecture that every convex semialgebraic set would be
the linear projection of a spectrahedron, the
``Helton-Nie conjecture'', has
been disproven by Scheiderer \cite{scheiderer-spectrahedral-shadows}.
Moreover, the imaginary projection closely relates to and complements
the notions of \emph{amoebas}, as introduced by Gel'fand, Kapranov
and Zelevinsky \cite{gkz-1994}, and \emph{coamoebas}.
The amoeba $\mathcal{A}(p)$
of a polynomial $p$
is defined as
{\small $\,{\!\mathcal{A}(p)\! := \!\{(\ln|z_1|, \ldots,\ln|z_n|) \!:\!
\mathbf{z} \in\! \mathcal{V}(p) \cap (\C^*)^n \}}$},
so it considers the logarithm of the absolute value of a complex
number rather than its imaginary part. The coamoeba of a polynomial
deals with the phase of a complex number. Each of these three viewpoints
of a complex variety gives a set in a real space with the characteristic
property that the complement of the closure consists of finitely
many \emph{convex} connected components. See
\cite{forsgard-johansson-2015}, \cite{gkz-1994}
and \cite{jtw-2019}
for the convexity properties of amoebas, coamoebas,
and imaginary projections, respectively. Due to their convexity
phenomenon, these structures provide natural classes in recent
developments of convex algebraic geometry.
For amoebas, an exact upper bound on the number of components in
the complement is known \cite{gkz-1994}.
For the coamoeba of a polynomial $p$, it has been conjectured that
there are at most $n!\vol\New(p)$ connected components in the complement,
where $\vol$ denotes the volume and $\New(p)$ the Newton polytope
of $p$, see \cite{forsgard-johansson-2015}
for more background as well as a proof for the special case
$n=2$.
For imaginary projections, a tight upper bound is known in the homogeneous
case \cite{joergens-theobald-hyperbolicity}, but for the non-homogeneous
case there only exists a lower bound \cite{jtw-2019}.
Currently, no efficient method is known to calculate the imaginary
projection for a general real or complex polynomial. For
some families of polynomials, the imaginary projection has been
explicitly characterized,
including complex linear polynomials and real quadratic polynomials,
see \cite{jtw-2019} and \cite[Proposition 3.2]{joergens-theobald-conic}.
However, since imaginary projections for non-linear
complex polynomials exhibit new structural phenomena compared
to the real case,
even the characterization of the imaginary projection of complex
conics had remained elusive so far.
Our primary goal is to reveal fundamental and surprising differences between imaginary projections of \emph{real polynomials} and \emph{complex polynomials}.
In fixed degree and dimension, for a polynomial $p$ with non-real coefficients, the algebraic degree of the {\it boundary} of the imaginary projection $\partial\I(p):= \overline{\I(p)}\cap\overline{\I(p)^\mathsf{c}}$
can be higher than the case of real coefficients. Here $(.)^\mathsf{c}$ and $\overline{(.)}$ are the complement and Euclidean closure, respectively.
These incidences already begin when the degree and dimension are both two. However, the contrast is not only concerning the boundary degrees, but also the arrangements and the strict convexity of the components in $\I(p)^\mathsf{c}$.
We start with structural results which serve to work out the differences
between the case of real and complex coefficients. Our first result is a sufficient
criterion on the roots of the \emph{initial form}
of an arbitrarily large degree non-real
bivariate complex polynomial to have the real plane as its imaginary projection,
see Theorem~\ref{th:EvenDeg} and Corollary~\ref{co:wholeplane2}.
Next, we characterize the imaginary projections of $n$-dimensional
multivariate complex quadratics with hyperbolic initial form,
see Theorem~\ref{th:ndimhyperbol1} and Corollary~\ref{co:ndimhyperbol2}.
In the two-dimensional case,
although by generalizing from real to complex conics, the bounds on the number of bounded and unbounded components in the complement of the imaginary projections remain unchanged,
the
possible arrangements of these components, strictness of their convexity, and the algebraic degrees of their boundaries strongly differ. See Corollaries \ref{co:alg-degrees} and
\ref{cor:oneUnbdd}.
For conic sections with real coefficients, it was shown
by J\"orgens, Theobald, and de~Wolff \cite{jtw-2019} that the boundary
$\partial\I(p)$ consists of pieces which are algebraic
curves of degree at most two. In sharp contrast to this,
for complex polynomials, the boundary may not be algebraic and the degree of its irreducible pieces can go up to 8.
For example, despite the simple expression of the polynomial $p = z_1^2+{\rm i}z_2^2+z_2$, an exact description
of
$\I(p)$ is
\begin{equation}
\label{eq:example1}
\begin{array}{r@{\hspace*{0.5ex}}l}
\mathcal{I}(p) \ = \ & \{ y \in \R^2 \, : \,
-64 y_1^8-128 y_1^4 y_2^4-64 y_2^8
+256 y_1^4 y_2^3+256 y_2^7-272 y_1^4 y_2^2
\\ [1ex]
& \, -400 y_2^6+144 y_1^4 y_2+304 y_2^5-27 y_1^4-112 y_2^4+16 y_2^3 \le 0\}
\setminus \{(0,1/2)\},
\end{array}
\end{equation}
and the describing polynomial in~\eqref{eq:example1} is irreducible over $\C$.
In this example, the set ${\I(p)}^{\mathsf{c}}$ consists of a single convex connected and bounded component.
Any polynomial vanishing on the boundary will also vanish on the single
point $(0,1/2)$ which is not part of the boundary $\partial \I(p)$.
Thus, $\partial \I(p)$ is not algebraic.
See Figure~\ref{fi:example1} for an illustration and we return to this
example in Section \ref{se:higherdegree} and at the end of Section~\ref{se:non-hyperbolic}.
\begin{figure}
\caption{{\small(A)}
\label{fi:example1}
\end{figure}
Since the topology of the imaginary projection in $\R^n$ is invariant under the action of $G_n:=\C^n\rtimes \GL_n(\R)$, that is the semi-direct product of $\GL_n(\R)$ and complex translations,
the problem to understand the imaginary projections
naturally leads to a polynomial classification problem.
As starting point, recall that
under the action of the affine group
$\text{Aff}(\C^2)$, there are precisely five orbits for complex conics, with the following representatives:
\[
\begin{matrix}
z_1^2 \text{ (one line)},&&&
z_1^2+1 \text{ (two parallel lines)},&&&
z_1^2-z_2 \text{ (parabola)},
\end{matrix}
\]
\[
\begin{matrix}
z_1^2+z_2^2 \text{ (two crossing lines)},&&&
z_1^2+z_2^2-1 \text{ (circle)}.
\end{matrix}
\]
However, the arrangement of the components in $\I(p)^\mathsf{c}$
is not invariant under the action of $\text{Aff}(\C^2)$, but only under its restriction to $G_2$. There are several other related classifications of complex conic sections. Newstead \cite{Newstead} has classified the set of projective complex conics under real linear transformations. However, out of a projective setting his method becomes ineffective as it is based on the arrangements of four intersection points between a conic and its conjugate.
On the other hand, by considering the real part and the
imaginary part of a complex conic $p$, under
the action of $G_2$ the classification of conic sections has some relations
to the problem of
classifying pairs of real conics.
Systematic
classifications of this kind
are mostly done in the projective setting and are well understood. See
\cite{briand-2007,levy-1964,petitjean-2010,uhlig-1976}. However,
those classifications rely
on the invariance of the number and multiplicity of real intersection points between the two real conics.
The drawback here is that under complex translations on $p$, these numbers are not invariant anymore, except at infinity.
To capture the invariance under $G_2$, we
develop a novel classification based on the initial forms of complex conics. This classification is adapted to the imaginary projection
and it is rather fine but coarse enough to allow handling the inherent
algebraic degree of 8 in the boundary description of the imaginary projection.
Finally, we show that non-real complex conics can significantly improve a realization result on the complement of the imaginary projections. In \cite{joergens-theobald-hyperbolicity}, for any given integer $k\ge 1$, they present a polynomial $p$ of degree $d=4\lceil \frac{k}{4}\rceil+2$ as a product of real conics, such that $\I(p)^\mathsf{c}$ has at least $k$ components that are strictly convex and bounded.
Using non-real conics, we furnish a degree $d/2+1$ polynomial having exactly $k$ components with these properties. See Theorem~\ref{th:StrictlyConvexComplex} and Question \ref{ques:deg}.
The paper is structured as follows. Section \ref{se:prelim} provides our notation and the necessary background on the imaginary projection of polynomials and contains
the classification of the imaginary projection for the case of real conics. Section~\ref{se:higherdegree}
deals with complex plane curves and provides a highlighting example where the complex versus real coefficients make a remarkable difference in the complexity of the imaginary projection.
Moreover, we determine a family of arbitrarily large degree non-real plane curves with a full-space imaginary projection, based on the arrangements of roots of the initial form.
In Section~\ref{se:QuadraticsWithHyperbolicInit}, we set the degree to be two and let the dimension grow and we classify the imaginary projections of complex quadratics with hyperbolic initial form.
In Sections~\ref{se:mainclassification}
and~\ref{se:non-hyperbolic}, we restrict the degree and dimension both to be two and we provide a full classification of the imaginary projections for affine complex conics based on their initial forms. Moreover, we determine in which classes the components in the complement of the imaginary projection have a spectrahedral description and also state them explicitly.
Section~\ref{se:mainclassification}
contains our main classification theorems and the corollaries differentiating the cases of complex and real coefficients. The part where the initial form is hyperbolic is already covered in \ref{se:QuadraticsWithHyperbolicInit}. Each subsection of Section~\ref{se:non-hyperbolic} treats one of the remaining classes and explains their spectrahedral structure. In particular, we show that the only class where the components in the complement are not necessarily spectrahedral is the case where the initial form has two distinct non-real roots in $\P^1_\C$ such that they do not form a complex conjugate pair. In Section~\ref{se:convex}, we
prove a realization result for strictly convex complement components, which highlights another contrast between the imaginary projections of complex and real polynomials.
Section~\ref{se:outlook} gives some open questions.
\section{Preliminaries and background}\label{se:prelim}
For a set $S\subseteq\R^n$, we denote by
$\overline{S}$ the topological closure of $S$ with respect to the Euclidean topology on $\R^n$ and by $S^{\mathsf{c}}$
the complement of $S$ in $\R^n$. The \textit{algebraic degree} of $S$ is the degree of its closure with respect to the
Zariski topology. The set of non-negative and the set of strictly positive real numbers are
abbreviated by $\R_{\ge 0}$ and $\R_{>0}$ throughout the text.
Moreover,
bold letters will denote $n$-dimensional vectors.
By $\P^n$ and $\P^n_{\R}$, we denote the $n$-dimensional complex and real
projective spaces, respectively.
For a polynomial $p \in \C[\mathbf{z}]$, the imaginary
projection $\mathcal{I}(p)$ is defined in~\eqref{eq:imagproj1} and its boundary $ \overline{\I(p)}\cap\overline{\I(p)^\mathsf{c}}$
is denote by $\partial\I(p)$.
\begin{theorem}\cite{jtw-2019}
Let $p\in\C[\z]$ be a complex polynomial. The set $\overline{\I(p)}^\mathsf{c}$ consists
of a finite number of convex connected components.
\end{theorem}
We denote by $a_{\rm{re}}$ and $a_{\rm{im}}$ the real and the imaginary parts of a complex number $a\in\C$, i.e., $a$ is written in the form $a_{\rm re}+{\rm i}a_{\rm im}$, such that $a_{\rm re},a_{\rm im}\in\R$. Let $p\in\C[\z]$ be a complex polynomial. After substituting
$z_j = x_j+{\rm i}y_j$
for all $1\le j\le n$, the complex polynomial can be written in the form
\[p(\z) =p_{\rm re}(\x,\y)+{\rm i}p_{\rm im}(\x,\y),\]
such that $p_{\rm re},p_{\rm im}\in\R[\x,\y]$. We call the real polynomials $p_{\rm re}$ and $p_{\rm im}$, the \textit{real part} and the \textit{imaginary part} of $p$, respectively. Thus, finding $\I(p)$ is equivalent to determining the values of $\y$ for which the real polynomial system
\begin{equation}\label{PolySystem}
p_{\rm re}\,(\x,\y)=0 \; \text{ and } \;
p_{\rm im}(\x,\y)=0
\end{equation}
has real solutions for $\x$.
\begin{definition}\label{def:complexConic}
Let $p\in\C[z_1,z_2]$ be a quadratic polynomial, i.e., $p = a z_1^2 + b z_1 z_2 + c z_2^2 + d z_1 + e z_2 +f$ such that $a,b,c,d,e,f\in\C$. We say that $p$ is the defining polynomial of a complex conic, or shortly, a \textit{complex conic} if its total degree equals two, i.e., at least one of the coefficients $a,b$, or $c$ is non-zero. A complex conic
$p$ is called a \textit{real conic} if all coefficients of $p$ are real.
\end{definition}
The following lemma from \cite{jtw-2019} shows how real linear transformations and complex translations act on the imaginary projection. These are the key ingredients for computing the imaginary projection of every class of conic sections.
\begin{lemma}\label{le:group-actions-improj}
Let $p\in\C[\z]$ and $A\in\R^{n\times n}$ be an invertible matrix. Then \[{\I(p(A\z))=A^{-1}\I(p(\z)).}\]
Moreover, a real translation $\z\mapsto \z+\aaa, \ \aaa\in\R^n$ does not change the imaginary projection. An imaginary translation $\z\mapsto \z+{\rm i}\aaa, \ \aaa\in\R^n$ shifts the imaginary projection into the direction $-\aaa$.
\end{lemma}
By the previous lemma, to classify the imaginary projection of polynomials we consider their orbits under the action of the group $G_n:=\C^n\rtimes \text{GL}_n(\R)$, given by real linear transformations and complex translations. Further let
$\text{Aff}(\K^n):=\K^n\rtimes\text{GL}_n(\K)$ be the general affine group for $\K=\R$ or $\K=\C$. The real dimensions of these groups are
\[
\begin{matrix}
\dim_\R(\text{Aff}(\C^n))=2\dim_\R(\text{Aff}(\R^n))=2(n^2+n),&&\dim_\R(G_n)=n^2+2n.
\end{matrix}
\]
Up to the action of $G_2$, a real conic $p\in\R[z_1,z_2]$ is equivalent to a conic given by one of the following polynomials.
\setlength{\columnsep}{0pt}
\begin{multicols}{2}
\begin{itemize}
\item[($i$)] $z_1^2+z_2^2-1$ (ellipse),
\item[($ii$)] $z_1^2-z_2^2-1$ (hyperbola),
\item[($iii$)] $z_1^2+z_2$ (parabola),
\item [($iv$)]$z_1^2+z_2^2+1$ (empty set),
\item[($v$)] $z_1^2-z_2^2$ (pair of crossing lines),
\item[($vi$)] $z_1^2-1$ (parallel lines/one line $z_1^2$),
\item [($vii$)]$z_1^2+z_2^2$ (isolated point),
\item [($viii$)]$z_1^2+1$ (empty set).
\end{itemize}
\end{multicols}
In \cite{jtw-2019}, a full classification of the imaginary projection for real quadratics was shown. In particular, the following theorem is the classification for real conics. For illustrations of the cases, see
Figure~\ref{fi:real-classification}. The theorem that comes after provides the imaginary projection of some families of real quadratics. Furthermore, they state the subsequent question as an open problem.
\begin{theorem}\label{th:RealConicChar}
Let $p\in\R[z_1,z_2]$ be a real conic. For the normal forms
(i)--(viii) from above, the imaginary projections $\I(p)\subseteq\R^2$
are as follows.
\hspace{-3mm} \begin{multicols}{2}
\begin{itemize}
\item[($i$)] $\I(p) =\R^2$,
\item[($ii$)] $\I(p) = \{-1\le y_1^2-y_2^2<0\}\cup\{\mathbf{0}\}$,
\item[($iii$)] $\I(p) =\R^2\setminus\{(0,y_2):y_2\neq 0\}$,
\item [($iv$)]$\I(p) =\{\y\in\R^2:y_1^2+y_2^2-1\ge 0\}$,
\item[($v$)] $\I(p) =\{\y\in\R^2:y_1^2=y_2^2\}$,
\item[($vi$)] $\I(p) =\{\y\in\R^2:y_1=0\}$,
\item [($vii$)]$\I(p) =\R^2$,
\item [($viii$)]$\I(p) =\{\y\in\R^2:y_1=\pm 1\}$.
\end{itemize}
\end{multicols}
\end{theorem}
\begin{theorem}\label{th:real-Quad}
Let $p\in\C[z_1,\dots,z_n]$ be $p = \sum_{i=1}^{n-1}z_i^2-z_{n}^2+k$ for $k\in\{\pm 1\}$. Then
\[
\I(p) = \begin{cases}
\left\{\y \in \R^n \ : \ y_n^2<\sum_{i=1}^{n-1} y_i^2 \right\}
\cup \{\mathbf{0}\} & \text{if } k =1,
\\
\left\{\y \in \R^n \ : \ y_{n}^{2}-\sum_{i=1}^{n-1} y_i^2 \le 1 \right\}
& \text{if } k =-1. \\
\end{cases}
\]
\end{theorem}
\begin{figure}
\caption{The imaginary projections of the real conic sections and their complements are colored in gray and blue, respectively. The cases $(i)$ and $(vii)$ are skipped, as their
imaginary projection is the whole plane.}
\label{fi:real-classification}
\end{figure}
The following question, which is true for real quadratics $p\in\C[\z]$,
was asked in \cite[Open problem 3.4]{jtw-2019}.
In Section~\ref{subs:real-non-real},
we show that it is not true in general even for complex conics.
\begin{question}\label{que:open-close}
Let $p\in\C[\z]$ be a polynomial. Is $\I(p)$ open if and only if $\I(p)=\R^n$?
\end{question}
We use the {\it initial form} of $p$ abbreviated by $\init(p)(\z)=p^h(\z,0)$ , where $p^h$ is the homogenization of $p$. The initial form consists of the terms of $p$ with the maximal total degree. Furthermore, a complex polynomial $p \in \C[\z]$
is called \emph{hyperbolic} w.r.t. $\e\in\R^n$ if the univariate polynomial $t\mapsto p(\x+t\e)$ is real-rooted. Note that any hyperbolic polynomial is a, possibly complex, multiple of a real polynomial.
Finally, a \textit{spectrahedron} is a set of the form
\[
\{ \x \in \R^n \ : \ A_0 + \sum_{j=1}^n A_j x_j \succeq 0\},
\]
where $A_1, \ldots, A_n$ are real symmetric matrices of size $d$. Here, ``$\succeq 0$'' denotes the positive semidefiniteness
of a matrix. We also speak of a spectrahedral set if the set is given by positive
definite conditions, i.e., by strict conditions.
\section{Imaginary projections of complex plane curves\label{se:higherdegree}}
In this section, we determine the imaginary projection of some families of arbitrarily high degree complex plane curves.
Our point of departure is the characterization of real conics in
Theorem \ref{th:RealConicChar}. In the following example, which is an affine version of case ($B_{+}$) in Newstead's classification \cite{Newstead},
we show that by allowing non-real coefficients the imaginary projection of a complex conic can significantly change in terms of the algebraic degree of its boundary. See Corollary \ref{co:alg-degrees}.
\begin{remark}\label{re:quarticRoots}
Recall that the discriminant of a univariate polynomial
$p(z) = \sum_{j=0}^n a_j z^j$ is given
by $\Disc(p) = (-1)^{\frac{1}{2}n(n-1)}\frac{1}{a_n} \Res(p,p')$,
where $\Res$ denotes the resultant. For a quartic, having negative discriminant implies the existence of a real root. However, a positive discriminant can correspond to either four
real roots or none. Let
{\small
\[
P = 8 a_2 a_4 - 3 a_3^2, \,
R = a_3^{3}+8a_1a_4^{2}-4a_4a_3a_2,\,
D = 64a_4^{3}a_0-16a_4^{2}a_2^{2}+16a_4a_3^{2}a_2-16a_4^{2}a_3a_1-3a_3^{4}.
\]}
If $\Disc(p)>0$, then $p=0$ has four real roots if $P < 0$ and $D<0$, and no real roots otherwise. Finally, if the discriminant is zero, the only conditions under which there is no real solution is having $D=R=0$ and $P>0$
(see, e.g., \cite[Theorem 9.13 (vii)]{janson-2011}).
\end{remark}
\begin{example}\label{ex:caseB}
Let $p=z_1^2+{\rm i}z_2^2+z_2$. For simplifying the calculations, we use the translation $z_2\mapsto z_2+{\rm{i}}/2$ to eliminate the linear term.
This turns the equation $p=0$ into
$
q := z_1^2+{\rm i}z_2^2+{\rm{i}}/4=0.
$
Building the real polynomial system as introduced in (\ref{PolySystem}) implies
\[q_{\mathrm{re}} = x_1^2-2x_2y_2-y_1^2 = 0 \; \text{ and } \;
q_{\mathrm{im}} = 4x_2^2 +8x_1y_1 - 4y_2^2 + 1= 0.
\]
First assume $y_1\neq 0$. Substituting $x_1$ from $q_{\mathrm{im}}=0$ into $q_{\mathrm{re}}=0$ gives\[
16x_2^4 + (-32y_2^2 + 8)x_2^2 - 128y_1^2y_2x_2 - 64y_1^4 + 16y_2^4 - 8y_2^2 + 1 = 0.
\]
We calculate the discriminant of the above equation with respect to $x_2$. By the previous remark, there is a real solution for $x_2$ if the discriminant is negative, i.e.,
\[
-64y_1^8 - 128y_1^4y_2^4 - 64y_2^8 - 80y_1^4y_2^2 + 48y_2^6 + y_1^4 - 12y_2^4 + y_2^2< 0.
\]
Now we need to check the conditions where the discriminant is zero or positive. To show the positive discriminant implies no real solution for $x_2$,
we rewrite the condition with the substitution $u = y_1^4$:
\[
\Delta:=-64u^2 + (-128y_2^4 - 80y_2^2 + 1)u - 64y_2^8 + 48y_2^6 - 12y_2^4 + y_2^2>0.
\]
It is a quadratic polynomial in $u$ with negative leading coefficient. It can only be positive between the two roots for $u$ in $\Delta=0$. Those are
\[
-y_2^4 - \frac{5}{8}y_2^2 + \frac{1}{128} \pm \frac{\sqrt{32768y_2^6 + 3072y_2^4 + 96y_2^2 + 1}}{128}.
\]
To obtain $\Delta >0$, we need to have a solution $u>0$, i.e., we need to have either $-y_2^4 - \frac{5}{8}y_2^2 + \frac{1}{128}\ge 0$ or otherwise
{\small
\[
\left(-y_2^4 - \frac{5}{8}y_2^2 + \frac{1}{128}\right)^2>\frac{32768y_2^6 + 3072y_2^4 + 96y_2^2 + 1}{128^2}.
\] }
The first inequality implies $y_2^2\le \frac{3\sqrt{3}-5}{16}$ and after simplifications the second inequality implies $y_2^2< 1/4$. The polynomial $P$ from the previous remark for the quartic polynomials evaluates to $
4(1-4y_2^2),$ which is positive for $y_2^2< 1/4$. Therefore, for $\Delta>0$, there is no real solution for $x_2$. It remains now to consider the case $\Delta=0$. Since $y_1\neq 0$, to have $R=-262144y_2y_1^2=0$ we need $y_2=0$. Substituting $y_2=0$ in $D=0$ implies $-4096y_1^4 - 960=0$, which is a contradiction. Therefore, if $y_1\neq 0$, the imaginary projection of $q$ consists of points $\y \in\R^2$ for which $\Delta\le 0$.
Now assume $y_1=0$. From $q_{\mathrm{im}}=0$ we can observe that
$\mathbf{0} \not\in \mathcal{I}(q)$. Thus, assume $y_2\neq 0$. Solving $q_{\mathrm{re}}=0$ for $x_2$ and substituting in $q_{\mathrm{im}}=0$ implies
$x_1^4-y_2^2(4y_2^2 - 1)=0.$
This equation has a real solution if and only if $-y_2^2(4y_2^2 - 1)\le0$. Substituting $y_1=0$ in $\Delta$ allows to write $\Delta$ in terms of $y_2$, which gives
$\Delta_{y_2} = -y_2^2(4y_2^2 - 1)^3.$
Therefore, the imaginary projection on the $y_2$-axis is $\{(0,y_2)\in\R^2 :\Delta_{y_2}\le 0\}\setminus\{(0,0)\}$. Thus,
\[
\mathcal{I}(q)=\{\y \in\R^2:-64y_1^8 - 128y_1^4y_2^4 - 64y_2^8 - 80y_1^4y_2^2 + 48y_2^6 + y_1^4 - 12y_2^4 + y_2^2\le 0\}\setminus\{\mathbf0\}.
\]
The irreducibility of the polynomial above over $\C$ can
be verified for example using {\sc Maple}. For the original polynomial $p$, this gives the inequality description for
$\mathcal{I}(p)$ stated in~\eqref{eq:example1} in the Introduction.
\end{example}
Even in the case of real polynomials, extending the case of real conics by letting the degree or the number of variables be greater than two dramatically increases the difficulty of characterizing the imaginary projection. Let us see one such example of a cubic plane curve, i.e., where we have two unknowns and the total degree is three.
\begin{example}
Let $p\in\R[\z] = \R[z_1,z_2]$ be of the form $p = z_1^3 + z_2^3 - 1$. The similar attempt as before to calculate the imaginary projection $\I(p)$ is to separate the real and the imaginary parts of $p$ according to \eqref{PolySystem},
\[
p_{\rm re} = x_1^3 - 3x_1y_1^2 + x_2^3 - 3x_2y_2^2 - 1=0 \;
\text{ and } \;
p_{\rm im} = 3x_1^2y_1 + 3x_2^2y_2 - y_1^3 - y_2^3 =0.
\]
Despite the simplicity of the polynomial $p$, one cannot use the previous techniques to find the values of $\y \in\R^2$ such that the above system has real solutions for $\x$. The reason is that both $x_1$ and $x_2$ appear in higher degree than one in both equations. The resultant with respect to one of $x_1$ or $x_2$ is a univariate polynomial of degree six in the other, where we lack the exact tools to specify the reality of the roots.
\end{example}
In the following theorem, we show that the imaginary projection of a generic complex plane curve of odd degree is
the whole plane.
\begin{thm}\label{th:EvenDeg}
Let $p\in\C[z_1,z_2]$ be a complex bivariate polynomial of total degree $d$ such that its initial form has no real roots in $\P^1$. If $d$ is odd then the imaginary projection $\mathcal{I}(p)$ is $\R^2$.
As a consequence, the imaginary projection of a generic
complex bivariate polynomial of odd total degree is $\R^2$.
\end{thm}
\begin{proof}
Since the initial form has no real roots, it can be written in the form
\[
\init(p) = \prod_{j=1}^{d}(z_1-\alpha_j z_2),
\]
where $\alpha_j\notin\R$ for $1\le j \le d$.
Substitute $z_j=x_j+ {\rm i}y_j$ for $j=1,2$ in $p$ and form the polynomial system $p_{\rm re} = p_{\rm im} = 0$ as introduced in (\ref{PolySystem}). For any fixed $\y \in\R^2$, both equations are of total degree $d$ in $x_1$ and $x_2$. Denote by $p_{\mathrm{re}}^h$ and $p_{\mathrm{im}}^h$, the homogenization of these two polynomials by a new variable $x_3$.
Since both, $p_{\mathrm{re}}^h$ and $p_{\mathrm{im}}^h$, have odd degree, the number of complex intersection points (counted with multiplicities) is odd while the number of non-real intersection points (counted with multiplicities) is even. Thus, there is a real intersection point
in $\P^2_{\R}$.
We claim that this intersection point lies in the affine piece where $x_3 = 1$. This implies that for any given $\y \in\R^2$, there exist $x_1,x_2\in\R$ for which $p_{\mathrm{re}}=p_{\mathrm{im}}=0$ and therefore completes the proof.
To prove our claim, we show that the two curves defined by $p_{\mathrm{re}}^h=0$ and $p_{\mathrm{im}}^h=0$ do not intersect at infinity, i.e., their intersection point has $x_3\neq 0$. Let us assume that they intersect at infinity and set $x_3 = 0$ in $p_{\mathrm{re}}^h$ and $p_{\mathrm{im}}^h$. This substitution turns the complex polynomial $p_{\mathrm{re}}^h+ {\rm i}p_{\mathrm{im}}^h$ into
\[
q:=\prod_{j=1}^{d}(x_1-\alpha_j x_2).
\]
Thus, for the two projective curves to intersect at infinity we need to have $q=0$. Since $\alpha_j\notin\R$ for $1\le j \le d$, the only real solution for $x_1$ and $x_2$ is zero. This is a contradiction.
\end{proof}
\begin{cor}
\label{co:wholeplane2}
Let $p\in\C[z_1,z_2]$ be a complex bivariate polynomial. The imaginary projection $\I(p)$ is $\R^2$ if $p$ has a factor $q$ such that the total degree of $q$ is odd and its initial form has no real roots in $\P^1$.
\end{cor}
\begin{proof}
Since for $p_1,p_2 \in \C[\z]$, we have $\I(p_1 \cdot p_2)
= \I(p_1) \cup \I(p_2)$, we claim that if there is a factor $q$ in $p$ whose imaginary projection is $\R^2$, then $\I(p)=\R^2$. The result now follows from the previous theorem.
\end{proof}
In the following section, instead of the dimension we set the degree to be two and characterize the imaginary projection for a
certain family of quadratic hypersurfaces.
\section{Complex quadratics with hyperbolic initial form\label{se:QuadraticsWithHyperbolicInit}}
As we have seen in Example \ref{ex:caseB}, the methods used to compute the imaginary projection of real quadratics is not always useful for complex ones. However, for a certain family, namely the quadratics with hyperbolic initial form, one can build up on the methods for the real case. To classify the imaginary projections of any family of polynomials, Lemma \ref{le:group-actions-improj} suggests bringing them to their proper normal forms.
\begin{lemma}\label{le:hyp-Init-quadratic-forms}
Under the action of $G_n$, any quadratic polynomial $p\in\C[z_1, \dots, z_n]$ with hyperbolic initial form can be transformed to one of the following normal forms:
\begin{enumerate}
\item $z_1^2+\alpha z_2+ r z_3+\gamma$,\\
\item $\sum_{i=1}^{j}z_i^2 - z_{j+1}^2+\alpha z_{j+2}+r z_{j+3}+\gamma \qquad\text{for some } j=1,\dots,n-1$,
\end{enumerate}
\noindent such that terms containing $z_k$ do not appear for $k>n$, and $\alpha,r,\gamma\in\C$.
\end{lemma}
\begin{proof}
The initial form $\init(p)$ is a hyperbolic polynomial of degree two. That is, after a real linear transformation it can be either $z_1^2$ or of the form $\z'^TM\z'$ such that $\z' = (z_1,\dots, z_{j+1})$ for some $1\le j\le n-1$ and $M$ is a square matrix of size $j+1$ with signature $(j,1)$. See \cite{garding-59}. This explains the initial forms in (1) and (2).
Any term $\lambda z_j$ for some $1\le j \le n$, such that $z_j$ appears in our transformed initial forms, cancels out by one of the translations $z_j\mapsto z_j\pm \frac{\lambda}{2}$ without changing the initial form. Finally, we show that the number of linear terms in the rest of the variables is at most two. Consider the complex linear form $\sum_{j=1}^{n}\lambda_j z_j$.
For $1\le j \le n$, let $\lambda_j= r_j + {\rm i} s_j$ such that $r_j,s_j\in\R$. We can now write the sum as
$
(\sum_{j=1}^{n}r_j z_j)+{\rm i}(\sum_{j=1}^{n}s_j z_j)
$. If in the real part at least one of the $r_j$, say, $r_1$, is non-zero, then a
sequence of linear transformations $z_1\mapsto z_1-\frac{r_j}{r_1}z_j$ for $j=2,\dots,n$, cancels out $\sum_{j=2}^{n}r_j z_j$. Similarly, the complex part reduces to only one term.
\end{proof}
We first focus on the case where $n=2$. In this case, we explicitly express the unbounded spectrahedral components forming $\I(p)^\mathsf{c}$. The following subsection covers part of the proof of Theorem \ref{th:complex-classification1}.
\subsection{Complex conics with hyperbolic initial form}\label{subs:Conic-Hyp-Init}
To match them with our classification of conics in Theorem~\ref{th:conic-classification1}, we do a real linear transformation in the case (2) and write them as
\[
\begin{matrix}
\text{(1a.1)} \,p=z_1^2+\gamma,&&&&\text{(1a.2)}\,p=z_1^2+\gamma z_2\,\,\,\,\,\,\gamma\neq 0,&&&&\text{(1b)} \,p=z_1z_2+\gamma,
\end{matrix}
\]
\noindent for some $\gamma\in\C$.
To find $\I(p)$ for each normal form, we compute the resultant of the two real polynomials, as introduced in (\ref{PolySystem}), with respect to $x_i$ to have a univariate polynomial in $x_j$, where $i,j\in\{1,2\}$, and $i\neq j$. Then we use the discriminantal conditions on the univariate polynomials to argue about the real roots.
First consider the normal form (1a.1). If $\gamma_{\mathrm{im}}=0$, then we have the real conics of the cases $(vi)$ and $(viii)$ in Theorem \ref{th:RealConicChar}.
The two real polynomials $
p_{\mathrm{re}} = x_1^2-y_1^2+\gamma_{\mathrm{re}} =0
\; \text{ and } \;
p_{\mathrm{im}} = 2 x_1 y_1+\gamma_{\mathrm{im}} =0
$ form the system (\ref{PolySystem}) here.
From $\gamma_{\mathrm{im}}\neq 0$, we need to have $y_1\neq 0$. Now substituting $x_1 = \frac{-\gamma_{\mathrm{im}}}{2 y_1}$ from $p_{\mathrm{im}}=0$ into $p_{\mathrm{re}}=0$ and solving for $y_1^2$ implies
$
y_1^2 = \frac{1}{2}\left(\gamma_{\mathrm{re}}+\sqrt{\gamma_{\mathrm{re}}^2+\gamma_{\mathrm{im}}^2}\right).
$
Therefore,
\begin{equation}
\tag{\text{1a.1}}
\mathcal{I}(p) =
\begin{cases}
\text{A unique line} &\text{if } \gamma\in\R_{\le 0},\\
\text{Two parallel lines} &\text{otherwise}.
\end{cases}
\end{equation}
Clearly, the closures of the components in the complement are spectrahedra.
Now consider (1a.2) which is a generalization of the parabola case $(iii)$ in Theorem~\ref{th:RealConicChar}, where $\gamma=1$.
Similarly to the previous case, we build the corresponding polynomial system as (\ref{PolySystem}). The discriminantal condition after substituting $x_2$ from $p_{\mathrm{im}} = 0$ into $p_{\mathrm{re}} = 0$ implies that there exists a real $x_1$
if and only if ${4|\gamma|^2(y_1^2+\gamma_{\mathrm{im}}y_2)\ge 0}$.
Hence, $\I(p)^\mathsf{c}$ consists of $\y\in\R^2$ such that
$y_1^2+\gamma_{\mathrm{im}}y_2< 0$.
This inequality specifies the open subset of $\R^2$ bounded by the parabola $y_1^2+\gamma_{\mathrm{im}}y_2= 0$ and containing its focus.
Therefore,
\begin{equation}
\tag{\text{1a.2}}
\mathcal{I}(p) =
\begin{cases}
\R^2\setminus \{(0,y_2):y_2\neq 0\} &\text{if } \gamma\in\R, \\
\{\y \in\R^2 : y_1^2+\gamma_{\mathrm{im}}y_2\ge 0\}&\text{otherwise.}
\end{cases}
\end{equation}
Notice that this incidence of $\I(p)^\mathsf{c}$ consisting of one unbounded component does not occur for real conics. See Corollary \ref{cor:oneUnbdd}. Further, $\I(p)^\mathsf{c}$ for $\gamma\notin\R$ is given by the unbounded spectrahedral set defined by
\begin{equation*}
\left(\begin{matrix}
1 & y_1 \\
y_1 & -\gamma_{\mathrm{im}}y_2
\end{matrix}\right)\succ 0.
\end{equation*}
For the last case (1b) from the corresponding real polynomial system $p_{\mathrm{re}} =p_{\mathrm{im}}=0$, one can simply check that $\gamma=0$ implies $\mathcal{I}(p)=\{\y\in\R^2:y_1y_2=0\}$.
Now let $\gamma\neq 0$ and first assume $y_1y_2\neq 0$.
After the substitution of $x_2$ from $p_{\mathrm{im}}=0$ to $p_{\mathrm{re}}=0$, the discriminantal condition on the quadratic univariate polynomial to have a real $x_1$ implies
\[
\gamma_{\mathrm{re}} - |\gamma| \le 2 y_1 y_2 \le \gamma_{\mathrm{re}}+ |\gamma|.
\]
If $\gamma\in\R\setminus\{0\}$, then $\mathbf{0}$ is the only point with $y_1y_2=0$ that is included in $\I(p)$. If $\gamma\notin\R$, then the union of the two axes except the origin is included in $\I(p)$. Thus,
\begin{equation}
\tag{\text{1b}}
\I(p)=
\begin{cases}
\text{The union of the two axes $y_1$ and $y_2$} & \text{if } \gamma = 0,
\\
\left\{\y \in \R^2 \ : \ 0 < y_1 y_2 \le \gamma \right\}
\cup \{\mathbf{0}\} & \text{if } \gamma \in \R_{>0}, \\
\left\{\y \in \R^2 \ : \ \gamma \le y_1 y_2 < 0 \right\}
\cup \{\mathbf{0}\} & \text{if } \gamma \in \R\setminus\R_{\ge0}, \\
\left\{\y \in \R^2 \ : \
\frac{1}{2}(\gamma_{\mathrm{re}} - |\gamma|) \le y_1 y_2 \le \frac{1}{2}( \gamma_{\mathrm{re}} + |\gamma| ) \right\}
\setminus \{\mathbf{0}\} & \text{if } \gamma \not\in \R.
\end{cases}
\end{equation}
\begin{corollary}\label{co:spectra-double-real-root}
Let $p\in\C[z_1,z_2]$ be a complex conic with hyperbolic initial form. The complement $\I(p)^\mathsf{c}$
of the imaginary projection consists of only unbounded spectrahedral components.
\end{corollary}
\begin{proof}
We saw this already for the cases (1a.1) and (1a.2). Therefore, we only prove the statement for (1b). There are four
unbounded components, namely in each quadrant one, and
no bounded component in $\mathcal{I}(p)^\mathsf{c}$.
The closures of the four unbounded components after setting
\[w = \sqrt{\frac{1}{2}( |\gamma|+\gamma_{\mathrm{re}})}\,\quad
\text{and}\quad u = \sqrt{\frac{1}{2}( |\gamma| - \gamma_{\mathrm{re}})}\,\]
have
the following representations as spectrahedra.
In the quadrants $y_1y_2\geq 0$, they are expressed by ${y_1 y_2 - \frac{1}{2}(\gamma_{\mathrm{re}} + |\gamma|) \ge 0}$, or equivalently, $S_1(y_1,y_2)\succeq 0$ and ${ S_2(y_1,y_2)\succeq 0}$, where
\[
S_1(y_1,y_2)=\left( \begin{array}{ccc}
y_1 && w \\
w & &y_2\\
\end{array} \right),
\quad
S_2(y_1,y_2)=\left( \begin{array}{cc}
-y_1 & w \\
w & -y_2 \\
\end{array} \right).
\]
In the quadrants with $y_1y_2\leq 0$, they are expressed by ${y_1 y_2 - \frac{1}{2} (\gamma_{\mathrm{re}} - |\gamma|) \le 0}$, or equivalently, $S_3(y_1,y_2)\succeq 0$ and $S_4(y_1,y_2)\succeq 0$, where
\[
S_3(y_1,y_2)=\left( \begin{array}{cc}
y_1 & u \\
u & - y_2 \\
\end{array} \right),
\quad
S_4(y_1,y_2)=\left( \begin{array}{ccc}
- y_1 && u \\
u & &y_2
\end{array} \right).
\] \end{proof}
Given a conic $q$, an explicit description of the components of $\I(q)^\mathsf{c}$ can be derived by using those of its normal form $p$ and applying on $\y$ the inverse operations turning $q$ to $p$.
We close this subsection by providing two examples for the cases (1a.2) and (1b) and their corresponding spectrahedral components.
\begin{example}\label{ex:(1a.2)}
Let $q(z_1,z_2)=z_1^2+2z_1z_2+z_2^2+2{\rm i}z_2+1$. By applying the transformation $A$ and the translation $\w$ given by
\[
A:=\begin{pmatrix}
1 & -1 \\
0 & 1
\end{pmatrix} \quad \text{and} \quad \w:=\begin{pmatrix}
0 \\ \rm{i/2}
\end{pmatrix},
\]
the conic $q$ is transformed to its normal form $p=z_1^2+2{\rm i}z_2$. Thus, we have
{\small
\[\I(p)^\mathsf{c}=\left\{ y \in\R^2:\begin{pmatrix}
1 & y_1 \\
y_1 & -2y_2
\end{pmatrix}\succ 0\right\}
\ \text{and} \ \
\mathcal{I}(q)^\mathsf{c}=\left\{ y \in\R^2:\begin{pmatrix}
1 & y_1+y_2 \\
y_1+y_2 & -2y_2+1
\end{pmatrix}\succ 0\right\},
\] } \\
such that $\I(q)^\mathsf{c}$ is obtained by the inverse transformations for $\y$ in $\I(p)^\mathsf{c}$.
Figure \ref{fig:improjComplex} (1a) illustrates $\I(q)^\mathsf{c}$.
\end{example}
\begin{example}\label{ex:1b}
Let $q(z_1,z_2)=z_1^2-z_2^2+2{\rm i}$. Applying $A=\frac{1}{2}\left(\begin{matrix}
-1 & -1 \\
-1 & 1
\end{matrix}\right)$ transfers the conic $q$ into $p = z_1z_2+2{\rm i}=0$. The value of both $u$ and $w$ introduced in the proof of Corollary \ref{co:spectra-double-real-root} is 1. By applying $A^{-1}$ to $\y$, the matrices $S_1,\ldots,S_4$ transform to
{\small
\begin{align*}
T_1(y_1,y_2)=& \begin{pmatrix}
-y_1-y_2 & 1 \\
1 & -y_1+y_2
\end{pmatrix}, \ \qquad T_2(y_1,y_2)= \ \left(\begin{matrix}
y_1+y_2 && 1 \\
1 && y_1-y_2
\end{matrix}\right), \\ T_3(y_1,y_2)=& \ \begin{pmatrix}
-y_1-y_2 &1 &\!\!\\
1& y_1-y_2&\!\!
\end{pmatrix}, \ \qquad T_4(y_1,y_2)= \ \begin{pmatrix}
y_1+y_2 &1\, \\
1& -y_1+y_2\,
\end{pmatrix}.
\end{align*} }
Thus, the complement of the imaginary projection as shown in Figure~\ref{fig:dist-real} is given by \[
\overline{\I(q)^\mathsf{c}}=\bigcup_{j=1}^4\left\{ \y \in\R^2:T_j(y_1,y_2)\succeq 0\right\}.
\]
\begin{figure}
\caption{The first four pictures represent $T_j(y_1,y_2)\succeq 0$ for $1\le j\le4$, and the last one shows their union, which gives $\I(q)^\mathsf{c}
\label{fig:dist-real}
\end{figure}
\end{example}
In the example above all four components are strictly convex, which can not occur in the case of real conics. This provides a key ingredient in the proof of Theorem \ref{th:StrictlyConvexComplex}.
\subsection{Higher dimensional complex quadratics}
We now let the dimension to be at least three and we use the normal forms provided in Lemma \ref{le:hyp-Init-quadratic-forms} to show the following classification of the imaginary projection.
To avoid redundancy, for each quadratic polynomial we set $n$ to be the largest index of $z$ appearing in its normal form. Since we have already covered the case of conics, we need to consider $n\ge 3$.
\begin{theorem}
\label{th:ndimhyperbol1}
Let $n \ge 3$ and $p\in\C[z_1,\dots,z_n]$ be a quadratic polynomial with hyperbolic initial form. Up to the action of $G_n$, the imaginary projection $\I(p)$ is either $\R^n$, $\R^n\setminus\{(0,\dots,0,y_n)\in\R^n : y_n\neq 0\}$, or otherwise we can write $p$ as $p= \sum_{i=1}^{n-1}z_i^2 - z_{n}^2+\gamma$ for some $\gamma\in\C$ such that $|\gamma|=1$ and we get
\begin{equation*}
\mathcal{I}(p)=
\begin{cases}
\left\{\y \in \R^n \ : \ y_n^2<\sum_{i=1}^{n-1} y_i^2 \right\}
\cup \{\mathbf{0}\} & \text{if } \gamma =1,
\\
\left\{\y \in \R^n \ : \ y_{n}^{2}-\sum_{i=1}^{n-1} y_i^2 \le 1 \right\}
& \text{if } \gamma =-1, \\
\left\{\y \in \R^n \ : \
y_{n}^{2}-\sum_{i=1}^{n-1} y_i^2 \le\frac{1}{2}(1-\gamma_{\rm re}) \right\}
\setminus \{\mathbf{0}\} & \text{if } \gamma \not\in \R.
\end{cases}
\end{equation*}
\end{theorem}
\begin{proof}
By real scaling and complex translations, any of the forms in Lemma \ref{le:hyp-Init-quadratic-forms} drops into one of the following cases:
\[
\begin{matrix}
\text{(a)}\, \alpha=r=\gamma=0,&&&
\text{(b)}\, \alpha=1,\,\, \text{and}\,\, r=\gamma=0,&&&
\text{(c)}\,\alpha\notin\R,\,\, \text{and} \,\,r,\gamma=0,
\end{matrix}
\]
\[
\begin{matrix}
\text{(d)}\, \alpha\notin\R,\, r=1,\,\, \text{and} \,\,\gamma=0,&&&
\text{(e)}\,\alpha=r=0, \,\,\text{and} \,\,\gamma\neq 0.
\end{matrix}
\]
For the normal form (1) all cases but (d) drop into the conic sections discussed previously. Case (d) is similar for both normal forms (1) and (2). Thus we focus on (2).
The imaginary projection for the cases (a) and (b) are known from the real classification and they are $\R^n$ and $ \R^n\setminus\{(0,\dots,0,y_n)\in\R^n : y_n\neq 0\}$, respectively. See \cite[Theorem 5.4]{jtw-2019}.
In case (c) after building the system (\ref{PolySystem}) and considering two cases, based on whether the real part of $\alpha$ is zero or not, one can then check that $\mathcal{I}(p) = \R^n$ as follows. We have
\[
\begin{array}{rcl}
p_{\rm re} &=&\sum_{i=1}^{n-2}x_i^2-x_{n-1}^{2}-\sum_{i=1}^{n-2}y_i^2+y_{n-1}^{2}+\alpha_{\rm re}x_n-\alpha_{\rm im}y_n,\\
p_{\rm im} &=&2\sum_{i=1}^{n-2}x_iy_i-2 x_{n-1} y_{n-1}+\alpha_{\rm im} x_{n}+\alpha_{\rm re} y_{n}.
\end{array}
\]
First assume $\alpha_{\rm re} = 0$. For any $\y\in\R^n$, the equation $p_{\rm re} = 0$ has solutions $(x_1,\dots,x_{n-1})\in\R^{n-1}$. By substituting any of those solutions in $p_{\rm im} = 0$ we can solve it for $x_n$ and get a real solution. Now let $\alpha_{\rm re}\neq 0$. In this case, we substitute $x_n$ from the second equation into the first. For any $\y\in\R^n$, we get $\sum_{i=1}^{n-2}(x_i-r_i)^2-(x_{n-1}-r_{n-1})^{2} = r_n$ for some $r_1,\dots,r_{n}\in\R$ and therefore, there always exists a real solution $(x_1,\dots,x_{n-1})\in\R^{n-1}$.
Similarly, in the case (d), for any $\y\in\R^n$, there exists a real solution $(x_1,\dots,x_{n-1})\in\R^{n-1}$ for $p_{\rm im} = 0$ and for any $\y\in\R^n$ and any $(x_1,\dots,x_{n-1})\in\R^{n-1}$, there exists a real $x_n$ for $p_{\rm re}=0$. Thus $\I(p) = \R^n$ in this case, too.
Now we focus on case (e).
Let $p= \sum_{i=1}^{n-1}z_i^2 - z_{n}^2+\gamma$ for some $\gamma\in\C \setminus \{0\}$. Building the real system (\ref{PolySystem}) for $p$ yields
\[
\begin{matrix}
p_{\rm re} &=&\sum_{i=1}^{n-1}x_i^2-x_{n}^{2}-\sum_{i=1}^{n-1}y_i^2+y_{n}^{2}+\gamma_{\rm re},&&
p_{\rm im} &=&2\sum_{i=1}^{n-1}x_iy_i-2 x_{n} y_{n}+\gamma_{\rm im}.
\end{matrix}
\]
We can assume $|\gamma| = 1$. Note that $\{\mathbf{0}\}\in\mathcal{I}(p)$ if and only if $\gamma\in\R$. We can thus exclude the origin in the following calculations. Moreover, Theorem \ref{th:real-Quad} shows the cases where
$\gamma = \pm 1$. Thus, we need to consider the case $\gamma\notin\R$.
Let $T$ be an orthogonal transformation on $\R^{n-1}$. Invariance of the polynomials $\sum_{j=1}^{n-1}{y}_j^2$ and $\sum_{j=1}^{n-1}x_jy_j$ under
the mapping $(x,y) \mapsto (T(x),T(y))$ implies
\begin{center}
$(y_1,y_2,\dots,y_{n}) \in \mathcal{I}(p)\qquad$ if and only if $\qquad (y'_1,\dots,y'_{n-1},y_n) \in \mathcal{I}(p)$,
\end{center}
where $(y'_1,\dots,y'_{n-1}) = T(y_1,\dots,y_{n-1})$.
For a given $\y\in\I(p)$, let $T$ be a transformation with the property
$T(y_1,\dots,y_{n-1})=(\sqrt{\sum_{i=1}^{n-1}y_i^2},0,\dots, 0)$ and set $(x'_1,\dots,x'_{n-1})=T(x_1,\dots,x_{n-1})$.
We can now rewrite the simplified polynomial system as
\[
\begin{matrix}
p_{\rm re} &=&\sum_{i=1}^{n-1}{x'_{i}}^{2}-x_{n}^{2}-{y'_1}^{2}+y_{n}^{2}+\gamma_{\rm re}, &&&
p_{\rm im} &=& 2 x'_{1} y'_{1}-2 x_{n} y_{n}+\gamma_{\rm im}.
\end{matrix}
\]
First consider $y'_1 = 0$. This implies $y_n\neq 0$. Solving $p_{\rm im} =0$ for $x_n$ and substituting in $p_{\rm re} = 0$ implies
\[
4y_{n}^2(\sum_{i=1}^{n-1}{x'_i}^{2})=\left(\gamma_{\rm re}^{2}+\gamma_{\rm im}^{2}\right)-\left(2y_{n}^{2}+\gamma_{\rm re}^{}\right)^{2} = 1-\left(2y_{n}^{2}+\gamma_{\rm re}\right)^{2}.
\]
This has a real solution for $(x'_1,\dots,x'_{n-1})$ if and only if $y_n^2\le\frac{1-\gamma_{\rm re}}{2}$. Now assume $y'_1 \neq 0$. Observe that if ${y'_1}^2 = y_n^2$ then we always get a real solution. Thus assume $\frac{y_{n}^{2}}{{y'_1}^2}-1\neq 0$.
Solving $p_{\rm im} =0$ for $x'_1$ and substituting in $p_{\rm re} = 0$ implies
{\small\[
\left(\frac{y_{n}^{2}}{{y'_1}^2}-1\right) \left(x_{n}-\frac{\gamma_{\rm im} y_{n}}{2 {y'_1}^2 \left(\frac{y_{n}^{2}}{{y'_1}^2}-1\right)}\right)^{2}+\sum_{i=2}^{n-1}{x'_i}^{2}+ \frac{\left(y_{n}^{2}-{y'_1}^2\right)^{2}+\gamma_{\rm re}\left(y_{n}^{2}-{y'_1}^2\right)-\left(\frac{\gamma_{\rm im}^{}}{2}\right)^{2}}{ y_{n}^{2}- {y'_1}^2}=0.
\]}
If ${y'_1}^2 > y_n^2$, there always is a real solution and otherwise, it has a real solution if and only if $\left(y_{n}^{2}-{y'_1}^{2}\right)^{2}+\gamma_{\rm re}\left(y_{n}^{2}-{y'_1}^{2}\right)-\left(\frac{\gamma_{\rm im}^{}}{2}\right)^{2} \le 0$. That is, $y_{n}^{2}-{y'_1}^{2}\le\frac{1-\gamma_{\rm re}^{}}{2}$.
To get the imaginary projection of the original system, it is enough to do the inverse transformation $T^{-1}$. This completes the proof.
\end{proof}
\begin{cor}
\label{co:ndimhyperbol2}
Let $p\in\C[z_1,\dots,z_n]$ be a quadratic polynomial with hyperbolic initial form. Then
\begin{itemize}
\item[(1)] the complement $\mathcal{I}(p)^\mathsf{c}$ is either empty or it consists of
\subitem- one, two, three, or four unbounded components; or
\subitem- two unbounded components and a single point.
\item[(2)] the complement of the closure $\overline{\mathcal{I}(p)}^\mathsf{c}$ is either empty or unbounded.
\item[(3)] the algebraic degrees of the irreducible components in $\partial\I(p)$ are at most two.
\end{itemize}
\end{cor}
\section{The main classification of complex conics\label{se:mainclassification}}
In this section, we give a classification of the imaginary projection $\mathcal{I}(p)$ where $p\in\C[\z] = \C[z_1,z_2]$ is a complex conic as in Definition \ref{def:complexConic}. We state our topological classification in terms of the number and boundedness of the components in $\mathcal{I}(p)^\mathsf{c}$.
In particular, this implies that the number of bounded and unbounded components do not exceed one and four, respectively. Furthermore, $\mathcal{I}(p)^\mathsf{c}$ cannot contain both bounded and unbounded components for some complex conic $p$.
A main achievement of this section is to establish a suitable classification and normal forms of complex conics under the action of the group $G_2$.
There are infinitely many orbits on the set of complex conics under this action, since
the real dimension of $G_2$ is $8$ and the set of complex conics has real dimension $10$. Each of our normal forms corresponds to infinitely many orbits that share their topology of imaginary projection by Lemma \ref{le:group-actions-improj}.
As a consequence of the obstructions in the existing classifications of conics that we discussed in the Introduction, we developed our own classification of conic sections. It is based on the five distinct arrangement possibilities for the roots of the initial form in $\P^1$ that are grouped in two main cases, depending on whether the initial form of the complex conic is hyperbolic or not:
\setlength{\columnsep}{-10pt}
\begin{multicols}{2}
\begin{itemize}
\item[] \hspace{-0.82cm}\underline{Hyperbolic initial form}
\item[]
\item[(1a)] A double real root
\item [(1b)]Two distinct real roots
\item[] \hspace{-0.85cm}\underline{Non-hyperbolic initial form}
\hspace{-3mm} \item [(2a)]A double non-real root
\item [(2b)]One real and one non-real root
\item [(2c)]Two distinct non-real roots
\end{itemize}
\end{multicols}
\begin{thm}[\textbf{Topological Classification}]
\label{th:complex-classification1}
Let $p\in\C[z_1,z_2]$ be a complex conic. For the above five
cases, the set $\I(p)^\mathsf{c}$ is
\setlength{\columnsep}{-10pt}
\begin{multicols}{2}
\begin{itemize}
\item[]
\item[(1a)] the union of one, two, or three
\item[]unbounded components.
\item [(1b)] the union of four
\item[]unbounded components.
\item[]
\item [(2a)] empty.
\item[(2b)] empty, a single point,
\item[]or a line segment.
\item [(2c)]empty or one bounded component,
\item[]possibly open.
\end{itemize}
\end{multicols}
In particular, the components of $\mathcal{I}(p)^\mathsf{c}$ are spectrahedral in all
the first four classes. This is not true in general for the last class {\rm(2c)}.
\end{thm}
The following corollary relates the boundedness of the components in $\mathcal{I}(p)^\mathsf{c}$ to the hyperbolicity of the initial form $\init(p)$.
\begin{cor}\label{co:hyperbolicityBdd}
Let $p\in\C[z_1,z_2]$ be a complex conic. Then $\mathcal{I}(p)^\mathsf{c}$ consists of unbounded components if and only if the initial form of $p$ is hyperbolic. Otherwise, $\mathcal{I}(p)^\mathsf{c}$ is empty or consists of one bounded component.
Moreover, if there is a bounded component with
non-empty interior, then $\init(p)$ has two distinct non-real roots.
\end{cor}
Figure~\ref{fig:improjComplex} represents the types that do not appear for real coefficients. For instance, the middle picture, labeled as (2b), shows the case where $\I(p)^\mathsf{c}$ consists of a bounded component with empty interior. This can not occur if $p$ has only real coefficients.
The other two pictures are discussed in the next two corollaries. The following corollary compares the algebraic degrees of the irreducible components in the boundary $\partial\I(p)$. Its proof comes at the end of the next section.
\begin{figure}
\caption{The complements of the imaginary projections are colored in blue. The
pictures show cases in the classification of the imaginary
projection for complex conics which do not appear for real conics.
The orange line in the right figure represents a generic line intersecting
the boundary in two points, which is used to prove the
non-spectrahedrality of this example in Section~\ref{se:non-hyperbolic}
\label{fig:improjComplex}
\end{figure}
\begin{cor}
\label{co:alg-degrees}
Let $p\in\C[z_1,z_2]$ be a complex conic.
\begin{enumerate}
\item The boundary $\partial\mathcal{I}(p)$ may not be algebraic. The algebraic degree of any irreducible component in its Zariski closure is at most 8. The bound is tight. If $\mathcal{I}(p)^\mathsf{c}$ has no bounded
components, then $\partial\mathcal{I}(p)$ is algebraic and it consists of irreducible pieces of degree at most two.
\item If all coefficients are real, then $\partial\mathcal{I}(p)$ is algebraic and it consists of irreducible pieces of degree at most two.
\end{enumerate}
\end{cor}
Example \ref{ex:caseB}, that is shown in Figure \ref{fig:improjComplex} (2c), illustrates an instance where the above contrast appears. The next corollary compares the number and strict convexity of the unbounded components that occur in $\mathcal{I}(p)^\mathsf{c}$ when $p$ is a complex or a real conic.
\begin{cor}\label{cor:oneUnbdd}
Let $p\in\C[z_1,z_2]$ be a complex conic.
\begin{enumerate}
\item The number of unbounded components in $\mathcal{I}(p)^\mathsf{c}$ can be any integer $0\le k\le 4$ and up to 4 of them can be strictly convex.
\item If all coefficients are real, the number of unbounded components in $\mathcal{I}(p)^\mathsf{c}$ can be any integer $0\le k\le 4$ except for $k=1$ and up to 2 of them can be strictly convex.
\end{enumerate}
\end{cor}
The proof follows from Theorems \ref{th:RealConicChar} and \ref{th:complex-classification1}, together with Example \ref{ex:1b}. The highlighting difference in the previous corollary, i.e., when $\I(p)^\mathsf{c}$ has one unbounded component, appears in the first class (1a) where the initial form has a double real root. Example \ref{ex:(1a.2)} provides such an instance and is shown in Figure \ref{fig:improjComplex} (1a).
Theorem~\ref{th:complex-classification1} is only proven by the end of Section \ref{se:non-hyperbolic}. In the previous section, we discussed the case where $p$ has hyperbolic initial form in details.
It remains to consider the case where $\init(p)$ is not hyperbolic. As in Subsection \ref{subs:Conic-Hyp-Init}, we first need to compute proper normal forms and then by Lemma~\ref{le:group-actions-improj}, it suffices to compute the imaginary projections of those forms for each case.
\begin{thm}[\textbf{Normal Form Classification}]
\label{th:conic-classification1}
With respect to the group $G_2$, there are infinitely many orbits for the complex conic sections with the following representatives.
\setlength{\columnsep}{-20pt}
\begin{multicols}{2}
\begin{itemize}
\item[]
\item[]
\item[(1a)] $\begin{array}{l}
{\rm(1a.1)}\,\,p=z_1^2+\gamma
\\
{\rm(1a.2)}\,\ p=z_1^2+\gamma z_2
\end{array}$
\item[]
\item[(1b)] $p= z_1z_2+\gamma$
\item[]
\item[]
\item[(2a)] $\begin{array}{l}
{\rm(2a.1)}\,\,p=(z_1-{\rm i} z_2)^2 + \gamma
\\
{\rm(2a.2)}\,\ p = (z_1 - {\rm i}z_2)^2 + \gamma z_2
\end{array}$
\item[]
\item[(2b)] $p = z_2 (z_1 - \alpha z_2) + \gamma$
\item[(2c)] $\begin{array}{l}
{\rm(2c.1)}\,\,p = z_1^2+z_2^2+\gamma
\\
{\rm(2c.2)}\,\ p=(z_1 - {\rm i} z_2)(z_1 - \alpha z_2)+\gamma
\end{array}$
\end{itemize}
\end{multicols}
\noindent for some $\gamma,\alpha\in\C$ such that, to avoid overlapping, we assume $\gamma\neq 0$ in {\rm(1a.2)} and {\rm(2a.2)}, $\alpha\notin\R$ in {\rm(2b)} and {\rm(2c.2)}, and finally $\alpha\neq \pm {\rm i}$ in \rm{(2c.2)}.
\end{thm}
\begin{proof}
By applying a real linear transformation we first map the roots of $\init(p)$ to $(0:1)$ in (1a),
to $(1:0)$ and $(0:1)$ in (1b), to $({\rm i : 1})$ in (2a), to $(1:0)$ and $(\alpha,1)$ such that $\alpha\notin\R$ in (2b), to $(\pm{\rm i} : 1)$ in (2c.1), and to $({\rm i} : 1)$ and $(\alpha : 1)$ such that $\alpha\notin\R$ and $\alpha\neq\pm{\rm i}$ in (2c.2). Then, similar to the proof of Lemma \ref{le:hyp-Init-quadratic-forms}, by eliminating some linear terms or the constant by complex translations we arrive at the given normal forms for each case.
Since the arrangements of the two roots in $\P^1$ is invariant under the action of $G_2$, the given five cases lie in different orbits. Note that the orbits of the subcases in each case do not overlap. For the subcases of (1a), in (1a.2), $z_1$ and $z_2$ may be transformed to $az_1+bz_2+e$ and $cz_1+dz_2+f$ with $a,b,c,d\in\R$ and $e,f\in\C$. This leads to $(az_1+bz_2+e)^2+\gamma$. Since $z_2^2$ does not appear in the normal form of case (1a.2), we get $b=0$ and thus $z_2$ can not appear. Further $z_1^2+\gamma_1$ and $z_1^2+\gamma_2$ with $\gamma_1\neq \gamma_2$ belong to different orbits since the previous argument enforces $a=1,b=0,e=0$. The other cases are similar. Thus, for any of the eight normal forms, there are infinitely many orbits corresponding to each $\gamma\in\C$ (and $\alpha\in\C$ in some cases).
\end{proof}
\section{Complex conics with non-hyperbolic initial form\label{se:non-hyperbolic}}
We complete the proof of the Topological Classification
Theorem~\ref{th:complex-classification1} by treating the case where the complex
conic $p \in \C[\z] = \C[z_1,z_2]$ does not have a hyperbolic initial form.
In particular, we see that, as previously stated in Corollary \ref{co:hyperbolicityBdd}, if the initial form of $p$ is not hyperbolic, then $\mathcal{I}(p)^\mathsf{c}$ is empty or consists of one bounded component whose interior is non-empty only if $\init(p)$ has two distinct non-real roots in $\P^1$.
The overall steps in computing the imaginary projection of the cases with non-hyperbolic initial form are as follows. After building up the real polynomial system for the classes (2b) and (2c.1) of
Theorem~\ref{th:conic-classification1}
as in~\eqref{PolySystem}, we use the same techniques as in Subsection \ref{subs:Conic-Hyp-Init}. However, in the case (2a), by the nature of the polynomial system, we directly argue that the imaginary projection is $\R^2$. In the last case (2c.2), we do not explicitly represent the components of $\I(p)^\mathsf{c}$. Instead, in Theorem~\ref{OnebddComp}
we prove that it does not contain any unbounded components and the number of bounded components does not exceed one.
\subsection{A double non-real root (2a)} We show that in this case we have a full space imaginary projection.
First consider the normal form (2a.1). We have
\[ \begin{matrix}
p_{\mathrm{re}} & = &
x_1 ^2- x_2^2+ 2 y_2 x_1+ 2 y_1 x_2 +
\gamma_{\mathrm{re}}x_2-y_1^2 + y_2^2 -\gamma_{\mathrm{im}}y_2&=& 0, \\
p_{\mathrm{im}} & = &
- 2 x_1 x_2 +2 y_1x_1 - 2y_2 x_2 + \gamma_{\mathrm{im}}x_2 + 2 y_1 y_2 + \gamma_{\mathrm{re}}y_2&= &0.
\end{matrix}\]
We prove $\mathcal{I}(p) = \R^2$ by showing that for every given $\y \in \R^2$,
these two real conics in $\x=(x_1,x_2)$ have a real intersection point. For any fixed $\y \in \R^2$, the bivariate polynomial
$p_\mathrm{re}$ in $\x$
has the quadratic part $x_1^2-x_2^2$, and hence, the equation $p_{\mathrm{re}}=0$ defines
a real hyperbola in $\x$ with
asymptotes
$x_1=x_2+c_1$ and $x_1=-x_2+c_2$ for some constants $c_1,c_2 \in \R$; possibly the hyperbola
degenerates to
a union of these two lines.
The degree two part of the polynomial $p_{\mathrm{im}}$ is given by $-2x_1x_2$
and hence, the equation $p_{\mathrm{im}}=0$
defines a real hyperbola in $\x$ with asymptotes
$x_1= d_1$ and $x_2=d_2$ for some constants $d_1, d_2 \in \R$;
possibly the hyperbola may degenerate to a union of these two lines.
Since the two hyperbolas have a real intersection point, the
claim follows. The case (2a.2) is similar.
\subsection{One real and one non-real root (2b)}\label{subs:real-non-real}
This case gives the system of
equations
\[ \begin{matrix}
p_{\mathrm{re}} & = & - \alpha_{\mathrm{re}} x_2^2+ x_1 x_2 + 2 \alpha_{\mathrm{im}} y_2 x_2 + \alpha_{\mathrm{re}} y_2^2 - y_1 y_2
+ \gamma_{\mathrm{re}} & = & 0, \\
p_{\mathrm{im}} & = & - \alpha_{\mathrm{im}} x_2^2+ y_2 x_1+ y_1x_2 -2 \alpha_{\mathrm{re}}y_2 x_2
+ \alpha_{\mathrm{im}} y_2^2 + \gamma_{\mathrm{im}} & =& 0.
\end{matrix}
\]
First assume $y_2\neq 0$. By solving the second equation for $x_1$, substituting the solution into the first equation and clearing the denominator, we get a univariate cubic polynomial in $x_2$ with non-zero leading coefficient. Since real cubic polynomials always have a real root,
this shows that for $\y \in \R^2$ with $y_2 \neq 0$, there is a solution $\x \in \R^2$.
It remains to consider $y_2 = 0$.
In this case, the second equation has a real solution in $x_2$
whenever the corresponding discriminant $y_1^2 + 4 \alpha_{\mathrm{im}} \gamma_{\mathrm{im}}$ is non-negative, and if one of these solutions
is non-zero, the first equation then gives a real solution for $x_1$.
The special case that in the second equation both solutions for $x_2$ are
zero, can only occur for $y_1 = 0$ and $\gamma_{\mathrm{im}} = 0$. Then the first
equation has a real solution for $x_1$ if and only if $\gamma_{\mathrm{re}} = 0$.
Altogether, we obtain
\begin{equation*}
\tag{\text{2b}}
\mathcal{I}(p) \ = \ \begin{cases}
\R^2 & \text{ if } \gamma=0 \ \ \text{or} \ \ \alpha_{\mathrm{im}}\gamma_{\mathrm{im}} > 0, \\
\R^2 \setminus \{\mathbf{0}\} & \text{ if } \gamma\in\R\setminus\{0\},\\
\R^2 \setminus \{(y_1,0) \ : \ y_1^2 < -4 \alpha_{\mathrm{im}} \gamma_{\mathrm{im}}\} &
\text{ if } \alpha_{\mathrm{im}} \gamma_{\mathrm{im}} < 0.
\end{cases}
\end{equation*}
Note that when $\gamma\in\R\setminus\{0\}$ then $\I(p)$ is open but not $\R^2$. This answers Question \ref{que:open-close}.
See Figure \ref{fig:improjComplex} (2b) for the imaginary projection of $p = z_2(z_1-{\rm i} z_2)-{\rm i}$ from this class.
\subsection{Two distinct non-real roots (2c)}\label{subs:2complex}
First we show that in (2c.1), i.e., where the roots of the initial form are complex conjugate, the imaginary projection is one open bounded component. After forming the polynomial system (\ref{PolySystem}), the same methods as those in Subsection \ref{subs:Conic-Hyp-Init}, i.e., taking the resultant of the two polynomials $p_{\rm re}$ and $p_{\rm im}$ with respect to $x_2$ and checking the discriminantal conditions to have a real $x_1$, lead to the imaginary projection
\begin{equation*}\label{k-disc}
\tag{\text{2c.1}}
\I(p) = \Big\{\y\in\R^2 : y_1^2+y_2^2\ge \frac{1}{2}(\gamma_{\mathrm{re}}+\sqrt{\gamma_{\mathrm{re}}^2+\gamma_{\mathrm{im}}^2})\Big\}.
\end{equation*}
In particular, we have $\I(p) = \R^2$ if and only if $\gamma_{\mathrm{im}}=0$ and $\gamma_{\mathrm{re}}\le0$. Hence, in the case of two non-real conjugate roots,
$\mathcal{I}(p)^{\mathsf{c}}$ consists of either one or zero bounded component
and it is a spectrahedral set.
The subsequent lemma shows that for the case (2c) in general
$\mathcal{I}(p)^\mathsf{c}$ is either empty or consists of one
bounded component.
\begin{lemma}\label{OnebddComp}
Let $p = (z_1 - \alpha z_2) (z_1 - \beta z_2) + d z_1 + e z_2 + f$
with $\alpha, \beta \not \in \R$ and $d,e,f \in \C$.
Then
\begin{enumerate}
\item $\mathcal{I}(p)^{\mathsf{c}}$ has at most one bounded component.
\item $\mathcal{I}(p)^{\mathsf{c}}$ does not have unbounded components.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Assume that there are at least two bounded components in $\mathcal{I}(p)^{\mathsf{c}}$. By Lemma~\ref{le:group-actions-improj},
we can assume without loss of generality that the
$y_1$-axis intersects both components.
Solving $p=0$ for $z_1$ gives
{\small
\begin{equation}
\label{eq:one-component-branch1}
z_1 \ = \ \frac{\alpha + \beta}{2}z_2
- \frac{d}{2}
+ \sqrt[\C]{\left( \frac{\alpha-\beta}{2} \right)^2 z_2^2 - e z_2 -f } \, .
\end{equation}
}
By letting $z_2\in\R$
we obtain two continuous
branches $y_1^{(1)}(z_2)$ and $y_1^{(2)}(z_2)$
satisfying~\eqref{eq:one-component-branch1}.
Therefore, the set
$\mathcal{I}(p) \cap \{\y \in \R^2 \, : \, y_2 = 0\}$ has at most two connected components.
This is a contradiction to our assumption that the $y_1$-axis intersects the two bounded components in $\mathcal{I}(p)^\mathsf{c}$.
For (2), assume that there exists an unbounded component in the complement of $\mathcal{I}(p)$. The convexity implies that it must contain a ray. By Lemma~\ref{le:group-actions-improj},
we can assume without loss
of generality that the ray is the non-negative part of the $y_1$-axis.
Similarly to the proof of (1), we set $y_2 = 0$ and check
the imaginary projection on $y_1$-axis, using the two complex solutions
in~\eqref{eq:one-component-branch1}.
Since $\alpha \neq \beta$, we have
$D:=
\left( \frac{\alpha-\beta}{2} \right)^2
\neq 0$, where $D$ is the discriminant of $\init(p)$ with
$z_2$ substituted to 1. We consider two cases: $D \not\in \R_{>0}$ and $D \in \R_{>0}$.
In both cases we get into a contradiction to the assumption that the unbounded component contains the non-negative part of the $y_1$-axis.
First assume $D \not\in \R_{>0}$.
For $z_2 \to \pm \infty$,
the imaginary part of the radicand is dominated by the imaginary part of
the square root of $D$. Since $D \not\in \R_{>0}$ at least one of the two expressions
{\small
\begin{equation*}
\label{eq:dominating1}
\left(\frac{\alpha+\beta}{2}\right)_{\mathrm{im}} \pm
\sqrt{\frac{-D_{\mathrm{re}}+\sqrt{D_{\mathrm{re}}^2+D_{\mathrm{im}}^2}}{2}}\,\,
\end{equation*}}
is non-zero. Thus, letting $z_2\mapsto\pm\infty$, implies $y_1\mapsto+\infty$ in at least one of the branches.
Now assume $D \in \R_{>0}$. This implies $(\alpha - \beta)/2 \in \R$. Thus $(\alpha + \beta)/2 \notin \R$, since otherwise it contradicts with $\alpha,\beta\notin\R$. In this case, by letting $z_2$ grow to infinity, the dominating expression for $y_1$ is
$\frac{1}{2}(\alpha+\beta)_{\rm im}z_2.$
Therefore, $y_1$ converges to $+\infty$ in one of the two branches.
In both cases, for some $s>0$, the ray $\{(y_1,0)\in\R^2:y_1\ge s\}$ lies in the imaginary projection. This completes the proof.
\end{proof}
Before, in Example \ref{ex:caseB} we have shown that the defining polynomial of the imaginary projection can be irreducible of degree 8. The previous lemma enables us to
show that $\mathcal{I}(q)^\mathsf{c}$
has exactly one bounded component. Note that $\mathbf{0}\in\mathcal{I}(q)^\mathsf{c}$. Let $B_\epsilon$ be an open ball with center at the origin and radius $\epsilon$. By letting $y_1$ and $y_2$ converge to zero, the dominating part of $\Delta$ is $y_1^4+y_2^2$. Thus, for sufficiently small $\epsilon$, any non-zero point in $B_\epsilon$ has $\Delta>0$. Therefore, $\mathcal{I}(q)^\mathsf{c}$ contains an open ball around the origin.
Now the claim follows from Theorems \ref{OnebddComp}.
In this example, the imaginary projection is Euclidean closed, i.e., $\overline{\mathcal{I}(q)}=\mathcal{I}(q)$, however, its boundary is not Zariski closed.
We claim that the set $\mathcal{I}(q)^\mathsf{c}$ is not a spectrahedron.
By the characterization of Helton and Vinnikov \cite{helton-vinnikov-2007},
it suffices to show that $\overline{\mathcal{I}(q)}$
is not rigidly convex. That is, if $h$ is a defining polynomial
of minimal degree for the component $\mathcal{I}(q)^\mathsf{c}$, then we have to
show that a generic line $\ell$ through the interior of $\mathcal{I}(q)^\mathsf{c}$ does not
meet the variety $V:=\{\x \in \R^2 \, : \, h(\x) = 0\}$ in exactly $\deg(h)$
many real points, counting multiplicities.
However, this can be checked immediately. For example, the
line $y_1 = 1/3$ intersects the variety $V$ in exactly
two real points, and any sufficiently small perturbation of the line
preserves the number of real intersection points. See Figure \ref{fig:improjComplex} (2c).
This completes the proof of Theorem~\ref{th:complex-classification1}. We now prove Corollary \ref{co:alg-degrees} by showing that 8 is an upper bound.
\noindent
\textit{Proof of Corollary \ref{co:alg-degrees}}. For the first four classes we have precisely computed the boundaries $\partial\I(p)$ and they are algebraic with irreducible components of degree at most two. It remains to consider the case (2c), more precisely (2c.2), where
$p=(z_1 - {\rm i} z_2)(z_1 - \alpha z_2)+\gamma$
for some $\alpha,\gamma\in\C$, $\alpha\notin\R$, and $\alpha\neq {\rm \pm i}$.
Using Remark \ref{re:quarticRoots}, we show that the degrees of the irreducible components in the Zariski closure of $\partial\I(p)$ do not exceed $8$. This, together with Example \ref{ex:caseB}, completes the proof of (1). We separate the real and the imaginary parts as before.
{\small
\[
p_{\mathrm{re}} =x_{1}^{2}\!+(\!(\alpha_{\mathrm{im}}+1) y_{2}\!)-\alpha_{\mathrm{re}} x_{2}) x_{1}-\alpha_{\mathrm{im}} x_{2}^{2}+(\!(\!\alpha_{\mathrm{im}}+1) y_{1}-2 \alpha_{\mathrm{re}} y_{2}) x_{2}+\alpha_{\mathrm{re}} y_{2} y_{1}+\alpha_{\mathrm{im}} y_{2}^{2}-y_{1}^{2}\!+\gamma_{\mathrm{re}}=0,
\]\[
p_{\mathrm{im}} =\! ((\alpha_{\mathrm{im}}+1) x_{2}+\alpha_{\mathrm{re}} y_{2}-2 y_{1}) x_{1}-\alpha_{\mathrm{re}} x_{2}^{2}+(\alpha_{\mathrm{re}} y_{1}+2 \alpha_{\mathrm{im}} y_{2}) x_{2}+\alpha_{\mathrm{re}} y_{2}^{2}-(\alpha_{\mathrm{im}}+1) y_{1} y_{2}-\gamma_{\mathrm{im}}= 0.
\]
}
First we assume $(\alpha_{\mathrm{im}}+1) x_{2}+\alpha_{\mathrm{re}} y_{2}-2 y_{1}\neq0$. Solving $p_{\mathrm{im}} =0$ for $x_1$
and substituting in $p_{\mathrm{re}} = 0$ returns
{\small
\[
\Big( \alpha_{\mathrm{im}}(\alpha_{\mathrm{re}}^{2}+(\alpha_{\mathrm{im}}+1)^2) \Big) x_2^4
-\Big((\alpha_{1}^{2}+\alpha_{2}^{2}+6 \alpha_{2}+1) (-\alpha_{1} y_{2}+y_{1} (\alpha_{2}+1)) \Big) x_2^3
+\Big((\alpha_{1}^{2}+5 \alpha_{2}^{2}+14 \alpha_{2}+5) y_{1}^{2}
\]
\[
-y_{1} \alpha_{1} (\alpha_{1}^{2}+\alpha_{2}^{2}+14 \alpha_{2}+9) y_{2}+(4 \alpha_{1}^{2}+\alpha_{2} (\alpha_{1}^{2}+(\alpha_{2}-1)^2)) y_{2}^{2} +(k_{2} \alpha_{1}-2 k_{1} -k_{1} \alpha_{2})\alpha_{2}-k_{2} \alpha_{1}-k_{1}\Big) x_{2}^{2}
\]
\[
+\Big(8(-\alpha_{2}-1) y_{1}^{3}+8 \alpha_{1} (\alpha_{2}+2) y_{1}^{2} y_{2}-(\alpha_{2} (\alpha_{1}^{2}+\alpha_{2}^{2}-\alpha_{2}-1)+9\alpha_{1}^{2}+1) y_{1} y_{2}^{2}+\alpha_{1} (\alpha_{1}^{2}+(\alpha_{2}^{}-1)^{2}) y_{2}^{3}
\]
\[
+4 k_{1}(\alpha_{2}+1) y_{1}+((\alpha_{1}^{2}-(\alpha_{2}-1)^{2}) k_{2}-2 k_{1} \alpha_{1} (\alpha_{2}+1)) y_{2}\Big)x_2 + 4 y_{1}^{4}-8 \alpha_{1} y_{1}^{3} y_{2}+(5 \alpha_{1}^{2}+(\alpha_{2}-1)^{2}) y_{1}^{2} y_{2}^{2}
\]
\[
-\alpha_{1} (\alpha_{1}^{2}+(\alpha_{2}-1)^{2}) y_{1} y_{2}^{3}-4 k_{1} y_{1}^{2}+4\alpha_{1} k_{1} y_{1} y_{2}-\alpha_{1} (k_{1} \alpha_{1}+\alpha_{2} k_{2}-k_{2}) y_{2}^{2}-k_{2}^{2}.
\]
}
Since $\alpha\notin\R$, the leading coefficient is non-zero. Therefore, we have a quartic univariate polynomial in $x_2$. The relevant
polynomials for the decision of whether this polynomial has a real root for $x_2$ are $P,D$ and the discriminant $\Disc$ from Remark~\ref{re:quarticRoots}.
By computing these polynomials, we observe that $\Disc$ decomposes as $Q_1^2\cdot q$, where $Q_1$ is a quadratic polynomial and $q$ is
of degree $8$ in $\mathbf{y}$.
The total degrees of $P$ and $D$ are $2$ and $4$, respectively.
Now let us assume $(\alpha_{\mathrm{im}}+1) x_{2}+\alpha_{\mathrm{re}} y_{2}-2 y_{1} = 0$. If $\alpha_{\mathrm{im}}\neq -1$, then substituting $x_2 = \frac{-\alpha_{\mathrm{re}} y_{2}+2 y_{1}}{\alpha_{\mathrm{im}}+1}$ into $p_{\mathrm{im}}=0$ is the quadratic $Q_1$. Otherwise, the substitution $\alpha_{\mathrm{im}}= -1$ and $y_{1}=\frac{\alpha_{\mathrm{re}} y_{2}}{2}$ in $p_{\mathrm{re}}$ and $ p_{\mathrm{im}}$, and setting $s = 2p_{\mathrm{im}}-\alpha_{\mathrm{re}}p_{\mathrm{re}}$ simplifies the original system to
\[\begin{matrix}
p_{\mathrm{re}} &=& \alpha_{\mathrm{re}}^{2} y_{2}^{2}-4 \alpha_{\mathrm{re}} x_{1} x_{2}-8 \alpha_{\mathrm{re}} x_{2} y_{2}+4 x_{1}^{2}+4 x_{2}^{2}-4 y_{2}^{2}+4 \gamma_{\mathrm{re}}&=&0,\\
\\
s &=& 2 (2 \alpha_{\mathrm{re}}^{2} x_{1}+3 \alpha_{\mathrm{re}}^{2} y_{2}+4 y_{2})x_{2}-(\alpha_{\mathrm{re}}^{3} y_{2}^{2}+4 \alpha_{\mathrm{re}} x_{1}^{2}+4 \gamma_{\mathrm{re}} \alpha_{1}-4 \gamma_{\mathrm{im}})&=&0.
\end{matrix}
\]
If the coefficient of $x_2$ in $s$ is non-zero, then solving $s=0$ for $x_2$ and substituting in $p_{\mathrm{re}}=0$ results in a quartic polynomial in $x_1$ with non-zero leading coefficient. In this case, the polynomials Disc, P, and D from Remark \ref{re:quarticRoots} are all univariate in $y_2$. The decomposition of the discriminant in this case consists of the polynomial $q$ after the substitution $y_{1}=\frac{\alpha_{\mathrm{re}} y_{2}}{2}$ and the square of a quadratic polynomial $Q_2$. The total degrees of $P$ and $D$ are $2$ and $4$, respectively.
Otherwise, solving $2 \alpha_{\mathrm{re}}^{2} x_{1}+3 \alpha_{\mathrm{re}}^{2} y_{2}+4 y_{2}=0$ for $x_1$ and substituting in $s=0$, results in $Q_2$. In all the cases that we have discussed above, the degree of none of the irreducible factors appearing in the polynomials that could possibly form the $\partial\I(p)$ exceeds 8. Example \ref{ex:caseB} shows an example where this bound is reached. This completes the proof of (1). (2) follows from Theorem \ref{th:RealConicChar}.
$\Box$
We have precisely verified the imaginary projections for all the normal forms in Theorem \ref{th:conic-classification1} except for (2c.2) . In particular, we have shown that if $p$ is not of the class (2c.2), then $\I(p) = \R^2$ if and only if there exist some $\gamma,\alpha\in\C$, and $\alpha\notin\R$ such that $p$ can be transformed to one of the following normal forms.
\begin{equation}\label{list:ConicImprojR2}
\begin{cases}
(2a): (z_1 - {\rm i}z_2)^2 + \gamma z_2\quad\text{or}\quad (z_1-{\rm i} z_2)^2 + \gamma\\
(2b): z_2 (z_1 - \alpha z_2) + \gamma & \text{for}\quad \gamma=0 \,\,\,\text{or}\,\,\, \alpha_{\mathrm{im}}\gamma_{\mathrm{im}} < 0,\\
(2c.1): z_1^2+z_2^2+\gamma & \text{for}\quad \gamma_{\mathrm{im}}=0 \,\,\,\text{and}\,\,\, \gamma_{\mathrm{re}}\le0.
\end{cases}
\end{equation}
An example for a complex conic of class (2c.2) where the imaginary projection is $\R^2$ is $p = z_1^2 - 3{\rm i}z_1z_2-2z_2^2$. The reason is that for any given $(y_1,y_2)\in\R^2$, the polynomial $p$ vanishes on the point $(-y_2+{\rm i} y_1 , y_1+{\rm i} y_2)$. Answering the following question completes the verification of complex conics with a full-space imaginary projection.
\begin{question}
Let $p\in\C[z_1,z_2]$ be a complex conic of the form $p=(z_1 - {\rm i}z_2) (z_1 - \alpha z_2) + \gamma$ such that $\alpha\notin\R$ and $\alpha\neq\pm {\rm i}$. Under
which conditions on the coefficients $\gamma,\alpha\in\C$ does $\mathcal{I}(p)$
coincide with $\R^2$?
\end{question}
\section{convexity results}\label{se:convex}
For the case of complex plane conics, we have shown in Theorem \ref{OnebddComp} that there can be at most one bounded component in the complement of its imaginary projection. An example of such a conic is $z_1^2+z_2^2+1 = 0$, where the unique bounded component is the unit disc, which in particular is strictly convex. In the following theorem, we show that for any $k>0$, there exists a complex plane curve whose complement of the imaginary projection has exactly $k$ strictly convex bounded components.
For the case of real coefficients, only the lower bound of $k$ and no
exactness result is known
(see \cite[Theorem 1.3]{joergens-theobald-hyperbolicity}).
Allowing non-real coefficients lets us break the symmetry of the imaginary projection with respect to the origin and this enables us to fix the number of components exactly instead of giving a lower bound. Furthermore, using a non-real conic which has four strictly convex unbounded components, illustrated in Figure \ref{fig:dist-real}, notably drops the degree of the corresponding polynomial.
\begin{theorem}\label{th:StrictlyConvexComplex}
For any $k>0$ there exists a polynomial $p\in\C[z_1,z_2]$ of degree $2\lceil \frac{k}{4}\rceil+2$ such that $\mathcal{I}(p)^\mathsf{c}$ consists of exactly $k$ strictly convex bounded components.
\end{theorem}
\begin{proof}
Let $R^{\varphi}$ be the rotation map and $g:\C^2\rightarrow\C^2$ be defined as
\[
g(z_1,z_2) = z_1z_2+2{\rm i}.
\]
Note that the equation
\begin{equation}\label{eq:m=2}
\prod_{j=0}^{m-1}(g\circ R^{\pi j/2m})(z_1,z_2) =0
\end{equation}
where $m=\lceil \frac{k}{4}\rceil$ as before, has $4m$ unbounded components in the complement of its imaginary projection. We need to find a circle that intersects with $k$ of them and does not intersect with the rest $4m-k$ components. By symmetry of the construction of the equation above, the smallest distance between the origin $O$ and each component is the same for all the components. The following picture shows the case $m=2$.
\begin{figure}
\caption{The imaginary projection of (\ref{eq:m=2}
\end{figure}
Let $C$ be the boundary of the imaginary projection of
$
z_1^2 +z_2^2 +r^2
$
where $r = |OA_1|$. The center of $C$ is the origin and it passes through all $4m$ points $A_1,\dots,A_{4m}$ that minimize the distance from the origin to each component. A sufficiently small perturbation of the center and the diameter can result in a circle $C'$ with center $(a,b)$ and radius $s$ that only intersects the interiors of the first $k$ unbounded components.
Now define \[q:=(z_1-{\rm i}a)^2+(z_2-{\rm i}b)^2+s^2.\]
By Lemma \ref{le:group-actions-improj} and the fact that the imaginary projection of the multiplication of two polynomials is the union of their imaginary projections,
the polynomial
\[
p := q \cdot \prod_{j=0}^{m-1}(g\circ R^{\pi j/2m})(z_1,z_2),
\]
has exactly $k$ strictly convex bounded components in $\mathcal{I}(p)^\mathsf{c}$.
\end{proof}
Although, by generalizing from real to complex coefficients, we improved the degree of the desired polynomial from $d=4\lceil \frac{k}{4}\rceil+2$ to $d/2+1$, it is not the optimal degree.
For instance if $k=1$, the polynomial $z_1^2+z_2^2+1$ has the desired imaginary projection, while the degree is $2<4$. Thus, we can ask the following question.
\begin{question}\label{ques:deg}
For $k>0$, what is the smallest integer $d>0$ for which there exists a polynomial $p\in\C[z_1,z_2]$ of degree $d$ such that $\mathcal{I}(p)^\mathsf{c}$ consists of exactly $k$ strictly convex bounded components.
\end{question}
\section{Conclusion and open questions\label{se:outlook}}
We have classified the imaginary projections of complex conics and revealed
some phenomena for polynomials with complex coefficients in higher degrees and dimensions.
It seems widely open to come up with a classification of the imaginary projections
of bivariate cubic polynomials, even in the case of real coefficients. In particular, the maximum number of components in the complement of the imaginary projection for both complex and real polynomials of degree $d$ where $d\ge 3$ is currently unknown. We have shown that in degree two they coincide for real and complex conics, however, this may not be the case for cubic polynomials.
\subsection*{Acknowledgment.} We thank the anonymous referees for
their helpful comments.
\end{document}
|
\begin{document}
\title{Exact rate analysis for quantum repeaters with imperfect memories and \\entanglement swapping as soon as possible}
\author{Lars Kamin}
\email{[email protected]}
\author{Evgeny Shchukin}
\email{[email protected]}
\author{Frank Schmidt}
\email{[email protected]}
\author{Peter van Loock}
\email{[email protected]}
\affiliation{Johannes-Gutenberg University of Mainz, Institute of
Physics, Staudingerweg 7, 55128 Mainz, Germany}
\begin{abstract}
We present an exact rate analysis for a secret key that can be shared among two parties employing a linear quantum repeater chain. One of our main motivations is to address the question whether simply placing quantum memories along a quantum communication channel can be beneficial in a realistic setting. The underlying model assumes deterministic entanglement swapping of single-spin quantum memories and it excludes probabilistic entanglement distillation, and thus two-way classical communication, on higher nesting levels. Within this framework, we identify the essential properties of any optimal repeater scheme: entanglement distribution in parallel, entanglement swapping as soon and parallel quantum storage as little as possible. While these features are obvious or trivial for the simplest repeater with one middle station, for more stations they cannot always be combined. We propose an
optimal scheme including channel loss and memory dephasing, proving its optimality for the case of two stations and conjecturing it for the general case. In an even more realistic setting, we consider
additional tools and parameters such as memory cut-offs, multiplexing, initial state and swapping
gate fidelities, and finite link coupling efficiencies in order to identify potential
regimes in memory-assisted quantum key distribution beyond one middle station that exceed the rates of the smallest quantum repeaters as well as those obtainable in all-optical schemes unassisted by stationary memory qubits and two-way classical communication. Our analytical treatment enables us to determine simultaneous trade-offs between various parameters,
their scaling, and their
influence on the performance ordering among different types of protocols, comparing two-photon interference after dual-rail qubit
transmission with one-photon interference of single-rail qubits or, similarly, optical interference of coherent states.
We find that for experimental parameter values that are highly demanding but not impossible (up to 10s coherence time, about 80\% link coupling, and state or gate infidelities in the regime of 1-2\%), one secret bit can be shared per second over a total distance of 800km with repeater stations placed at every 100km -- a significant improvement over ideal point-to-point or realistic twin-field quantum key distribution at GHz clock rates.
\end{abstract}
\pacs{03.67.Mn, 03.65.Ud, 42.50.Dv}
\keywords{quantum repeaters, quantum memory}
\maketitle
\section{Introduction}\label{sec:Introduction}
Recent progress on quantum computers with tens of qubits led to
experimental demonstrations of quantum devices that are able to solve
specifically adapted problems which are not soluble in an efficient manner with
the help of classical computers alone. These devices are primarily based upon
solid-state (superconducting) systems \cite{Arute2019,Qiskit}, however, there are also photonics approaches \cite{Pan2020}.
While these schemes still have to be enhanced in terms of size, i.e. the number
of qubits (scalability), their error robustness and corresponding logical encoding
(fault tolerance), as well as their range of applicability (eventually reaching universality),
this progress represents a threat to common classical communication systems.
Eventually, this may compromise our current key distribution protocols.
Although there are recent developments in classical cryptography to address
the threat imposed by such quantum devices (``post-quantum cryptography''),
quantum mechanics also gives a possible solution to
this by means of quantum key distribution (QKD) \cite{NLRMP,PirRMP}.
Many QKD protocols have been proposed such as the most prominent, so-called BB84 scheme \cite{BB84}.
Indeed among the various quantum technologies that promise to enable their users to fulfil tasks impossible without quantum resources, quantum communication is special. Unlike quantum computers there are already commercially available quantum communication systems intended for costumers who wish to communicate in the classical, real world in a basically unconditionally secure fashion -- independent of mathematically unproven assumptions exploiting the concept of QKD. QKD systems are naturally realized for photonic systems using non-classical optical quantum states such as single-photon, weak \cite{Hwang2002, Lo2004} or even bright coherent states \cite{PirRMP}.
\subsection{Previous works and state of the art}
Current point-to-point QKD systems, directly connecting the sender (Alice) and the receiver (Bob) via an optical-fiber channel, are limited in distance due to the exponentially growing transmission loss along the channel. Typical maximal distances are 100-200km.
A very recent QKD variant, so-called twin-field (TF) QKD\cite{Lucamarini},
allows to push these limits farther (basically doubling the effective distance)
by placing an (untrusted) middle station between Alice and Bob. Remarkably, TF QKD achieves this loss scaling advantage in an all-optical fashion with no need for quantum storage at the middle station and at an, in principle, unlimited clock rate with no need for two-way classical communication. It further inherits the improved security features of measurement-device-independent (MDI) QKD schemes \cite{LoCurty,PirBraun}. However, the original TF QKD concept is not known to be further scalable beyond the effective distance doubling.
In classical communication, the distance problem is straightforwardly overcome by introducing repeater stations along the fiber channel (about every 50-100km) in order to reamplify (and typically reshape) the optical pulses.
On a fundamental level, the famous No-Cloning-theorem \cite{NoCloning,Dieks1982}, prohibits such solutions for quantum communication.
As a possible remedy, the concept of quantum repeaters has been developed \cite{BriegelDur,Dur1999,Hartmann2007}.
With the help of sufficiently short-range entanglement distributions,
quantum memories, entanglement distillation and swapping, in principle,
scalable long-distance, fiber-based quantum communication becomes possible,
including long-range QKD. The original quantum repeater proposals
assumed small-scale non-universal quantum computers at each repeater node
in order to perform the necessary gates for the entangled-pair manipulations,
and hence clearly appeared to be technologically less demanding than
a fully-fledged fault-tolerant and universal quantum computer.
Related to this, for QKD applications including those over large distances,
there are very powerful, classical post-processing techniques which
allow to relax the minimal requirements on the experimental states and gates.
Nonetheless, as a whole, these original quantum repeater systems would
still have high experimental requirements.
This led to some quantum repeater proposals specifically adapted
to certain matter memory systems and light-matter interfaces.
Probably the most prominent such proposal is the ``DLCZ'' quantum repeater \cite{DLCZ, Sangouard},
based upon atomic-ensemble nodes that no longer rely upon
the execution of difficult two-qubit entangling gates,
but instead only require linear-optical state manipulations
and photon detectors. Other schemes rely upon single emitters in solid-state repeater nodes, especially colour centers in diamond \cite{ChildressNV, Humphreys2017}. Alternative proposals employ optical coherent states and their cavity-QED interactions with single-spin-based quantum memory nodes \cite{HybridPRL}.
These proposals made a possible realization of a large-scale quantum repeater more likely, but as a complete implementation, they would still be fundamentally limited in their achievable (secret) key rates per second.
The reason for this is the need for two-way classical communication
on all, including the highest ``nesting'' levels in order to conduct entanglement distillation and confirm successful entanglement swappings when these are probabilistic.
Today this type of quantum repeater schemes are referred to as 1st-generation quantum repeaters. A memory-assisted QKD scheme was proposed in Ref. \cite{tf_repeater}, extending the TF concept to memory-based quantum repeaters.
In principle, this scheme achieves an effective distance doubling compared with standard quantum repeaters or, equivalently, it exhibits the standard loss scaling with about half as many
memory stations as in a standard quantum repeater (while the other half are all-optical
stations with beam splitter and photon detectors).
Apart from a certain level of memory assistance,
this repeater scheme also relies upon two-way classical communication
(between the nearest stations)
and hence can operate only at a limited clock rate determined
by the classical signalling time per segment.
Moreover, for its large-scale operation the scheme would
require an additional element for quantum error correction.
Alternative schemes circumventing the fundamental limitations
are the so-called 2nd- and 3rd-generation quantum repeaters
that exploit quantum error correction codes to suppress
the effect of gate and memory errors or channel loss, respectively \cite{JiangRvw}. A 3rd-generation quantum repeater no longer requires
quantum memories and two-way classical communication and so it can be,
in principle, realized in an all-optical fashion at a clock rate
only limited by the local error correction operations.
It is important to stress that all these quantum repeaters
are designed to allow for a genuine long-distance quantum state transfer.
In the QKD context, this means that the intermediate stations along
the repeater channel may be untrusted. If instead sufficiently many trusted stations can be
placed along the communication channel between Alice and Bob,
and the quantum signals can be converted into classical information
at each station (as a whole, effectively corresponding to classically connected, independent,
sufficiently short-range QKD links), large-scale QKD is already possible and being demonstrated \cite{ChinaDaily}.
Conceptually, this also applies to long-range links enabled by satellites \cite{Yin2017, Vallone2015}.
It is only the genuine quantum repeater that incorporates
two main features at the same time:
{\it long-distance scalability and long-distance privacy}.
From a practical point of view, it is expected that global quantum communication
systems will be a combination of both elements: genuine fiber-based quantum repeaters
over intermediate distances (thousands of km) and satellite-based quantum links
bridging even longer distances (tens of thousands of km; the earth's
circumference is about 40000km).
While such truly global quantum communication may eventually lead
to some form of a ``quantum internet'' \cite{WehnerHanson},
only the coherent long-distance quantum state transfer
as enabled by a genuine quantum repeater allows to
consider applications that go beyond long-range QKD.
In fact, the original quantum repeater proposals
were not specifically intended for or adapted to long-range QKD.
They can be used for any application that relies upon
the distribution of entangled states over large distances
including large-scale quantum networks.
Obvious applications are distributed quantum tasks
such as distributed quantum computing, coherently
connecting quantum computers which are spatially
far apart.
These ultimate long-distance quantum communication applications
will then impose much higher demands on the fault tolerance
of the experimental quantum states and gates.
In particular, QKD-specifc classical post-processing
will no longer be applicable.
In this work, we shall consider small to intermediate-scale quantum repeaters
that allow to do QKD or coherently connect quantum nodes
at a corresponding size and at a reasonably practical clock rate.
\subsection{This work}
In this work, we will focus on small-scale or medium-size quantum repeater systems
beyond a single middle station and without probabilistic entanglement distillation
on higher ``nesting levels". This class of quantum repeaters is of great interest
for at least two reasons.
(i) There are now first experiments
of memory-enhanced quantum communication basically demonstrating memory-assisted MDI QKD \cite{Lukin, Rempe}. Therefore the natural next step for the experimentalists will be to
connect such elementary modules to obtain larger repeater systems
with {\it two or more intermediate stations}, thus
bridging larger distances and, unlike memory-assisted MDI QKD, ultimately relying upon classical communication between the repeater stations \cite{White}.
These next near-term experiments will aim at a distance extension still independent
of additional and more complicated schemes such as entanglement distillation
on ``higher nesting levels''.
Restricting the entanglement manipulations to the level of the elementary
repeater segments will also help to avoid the use of long-distance two-way classical signalling
like in a fully scalable 1st-generation quantum repeater, and hence
allow for still limited but reasonable repeater clock rates.
In this regime, comparing (secret key) rates per second of the quantum repeaters with those
of an (ideal) point-to-point link or TF QKD scheme is in some way most fair and meaningful.
While the current experimental repeater demonstrations with a single repeater station \cite{Lukin,Rempe} would still suffer from too low clock rates and link coupling efficiencies before giving a practical repeater advantage, an urgent theoretical question
is whether, under practical realistic circumstances, it really helps to place memory stations along a quantum communication channel and execute memory-assisted QKD without extra active quantum error correction. In principle, placing a middle station between Alice and Bob allows to gain a repeater advantage per channel use \cite{NL,WehPar,White}.
Omitting the non-scalable all-optical TF approach,
is there a practical benefit also in terms of secret bits per second
when using a two-segment quantum repeater?
Moreover, and this is the focus of the present work,
is there even a further advantage when adding more stations beyond a single
middle station under realistic assumptions and with no extra
quantum error correction?
We will see that for up to eight repeater segments, covering distances up to around 800km, the quantum repeaters treated in this work, assuming experimental parameter values that are demanding but not impossible to achieve in practice, can exceed the performance limits
of the other schemes. For larger distances, the attainable absolute rates of point-to-point quantum communication become extremely small. However, for quantum repeaters, additional elements
of quantum error correction will be needed, as otherwise the final rates would vanish and no gain can be expected over point-to-point communication.
(ii) The second point refers to the theoretical treatment. Typically, the repeater rates can be calculated either numerically including many protocol variations and (experimental) degrees of freedom \cite{Coopmans} or approximately in certain regimes \cite{Sangouard} (there are also semi-analytical approaches, see Refs. \cite{Kuzmin2021,Kuzmin2019}).
If errors are neglected an exact and even optimized raw rate calculation is possible
even for non-unit (but constant) entanglement swapping probabilities
using the formalism of Markov chains and decision processes \cite{PvL,Shchukin2021} (see also Refs. \cite{Vinay2019,Khatri2019}).
This approach works well for repeaters up to about ten segments;
for too many repeater segments the resulting linear equation systems become intractable.
Nonetheless, for the smallest repeaters with only a single middle station,
it was shown how to calculate secret key rates even including
various experimental parameters, though partially also employing approximations for the raw rates \cite{NL,WehPar}.
In this work we will go beyond the case of a single middle station
and present exact calculations of {\it secret key rates obtainable with
realistic small and intermediate-scale quantum repeaters}.
The theoretical difficulty here is, even already when only channel loss and memory dephasing is considered, that for repeaters beyond a single middle station there are various distribution and swapping strategies and so it becomes non-trivial to determine the optimal ones.
The usual treatment in this case is based upon the so-called doubling strategy where for a repeater with a power-of-two number of segments only certain pairs of segments will be connected in order to double the distances at each repeater level. As a consequence, sometimes entanglement connections will be postponed even though neighboring pairs may be ready already, thus unnecessarily accumulating more memory dephasing errors. With regards to memory dephasing, the best strategy appears to be {\it to swap as soon as possible} and here we will show how this type of repeater strategy can be exactly and analytically treated. This element is the crucial step that enables us to propose optimal quantum repeater schemes.
On the hardware side, memory-based quantum repeaters require
sufficiently long-lived quantum memories and efficient, typically
light-matter-based interfaces converting flying into stationary qubits.
In the context of our theoretical treatment, the stationary qubits
are assumed to be represented by single spins in a suitable
solid-state quantum node such as colour (NV or SiV) centers in diamond,
usually separately treated as short-lived electronic
and long-lived nuclear spins \cite{LukinSiV, HansonNV}.
As for efficient quantum emitters and short-lived quantum memories
semiconductor quantum dots may be considered too \cite{White}.
Alternatively, various types of atom or ion qubits could be taken into account \cite{White}.
While all these different hardware platforms
have their own assets and disadvantages (e.g.
the required temperatures which range from room or modestly low temperatures
for atoms/ions/NV to cryogenic temperatures for NV/SiV/quantum dots),
and every one eventually requires a specifically adapted physical model,
to a certain extent the quantum repeater performance based
on these elements and assuming only a single repeater station
can be assessed (or at least qualitatively bounded from above) using a fairly simple
physical model that includes {\it three experimental parameters}:
the link coupling efficiency, the memory coherence time,
and the experimental clock rate \cite{White}.
In order to incorporate an appropriate experimental
memory coherence time into the model,
qubit dephasing errors can be considered where
the stationary qubit is never lost but subject to
random phase flips with a probability exponentially
growing with the storage time.
Already this rather simple model is theoretically
non-trivial, because it leads to two distinct impacts
on the final secret key rates.
On the one hand, a finite link coupling efficiency
(including all constant inefficiencies per segment from the
sources, detectors, and interfaces) and a segment-length-dependent
transmission efficiency affect the raw rate
of the qubit transmission (which, if expressed as rate per second,
also directly depends on the repeater clock rate).
Thereby, in logarithmic rate-versus-distance plots (like those frequently shown
later in this article), a finite link coupling leads to an offset
towards smaller rates at zero distance, while a finite channel transmission
results in a certain (negative) slope.
On the other hand, a finite memory coherence time influences the final Alice-Bob state fidelity
or QKD error rate
(which also indirectly depends on the repeater clock rate,
i.e. the time duration per entanglement distribution attempt
per segment, determining the possible number of distribution attempts
within a given memory coherence time).
This becomes manifest as an increase of the (negative) slope for growing distances,
moving from an initially repeater-like slope towards one corresponding to
a point-to-point transmission.
There are interesting concepts to suppress this latter effect
by introducing more sophisticated memory models such as memory buffers
or cut-offs. Especially a memory cut-off \cite{CollinsPrl}
has turned out to be useful without the need for additional experimental resources.
It means that a maximal storage time is
imposed at every memory node and any loaded stationary qubits waiting
for a longer duration will be reinitialized.
As a result, state fidelities can be kept high at the expense of
a decreasing raw rate due to the frequently occurring reinitializations
(which implies that a memory cut-off must neither be set too low nor too high).
Theoretically, including memory cut-offs into the rate analysis
significantly increases the complexity (becoming manifest
in e.g. quickly growing Markov-chain matrices) \cite{PvL}.
For small quantum repeaters, especially those with only one middle station,
a secret key rate analysis remains possible \cite{WehPar, White}.
For larger quantum repeaters, the effective rates may be calculated
via recursively obtained expressions \cite{Jiang}, via different kinds of approximations and assumptions \cite{Elkouss2021} or with the help of numerical simulations \cite{Coopmans}.
Nonetheless, in our treatment, we shall explicitly include
a memory cut-off in some protocols allowing us to extrapolate
its positive impact on other schemes.
We choose to incorporate random dephasing as the
dominating source of memory errors.
While memory dephasing is generally an error to be taken into account,
it is particularly important for those stationary qubits
encoded into single solid-state spins, e.g. for colour centers
or quantum dots \cite{White}.
We omit (time-dependent) memory decay (loss) which additionally becomes relevant
for atomic memories, either as collective spin modes
of atomic ensembles or in the form of an individual atom in
a cavity (generally, atoms and trapped ions may be subject to both dephasing and decay) \cite{DLCZ,HybridPRL,HybridNJP,Rempe}.
It turns out that the effect of memory dephasing can be accurately
included into the statistical repeater model, since the
total, accumulated dephasing in the final Alice-Bob density operator
follows a simple sum rule \cite{tf_repeater}.
Thus, the statistical averaging can be applied to the final state,
for which we derive a recursive formula that also includes
depolarizing errors from the initially distributed states and
from the imperfect Bell measurement gates in every
entanglement swapping operation. The main complication will be
to determine the correct dephasing variables for the different
swapping strategies and identify the optimal schemes. As a result, we extend the simple model of Ref.~\cite{White}
not only with regards to the repeater's size, but also to include additional experimental parameters:
{\it besides the above three parameters we then have
one or two extra parameters for the initially distributed states}
(taking into account initial dephasing or depolarization errors
depending on the protocol) {\it and one extra depolarization
parameter for the local gates and Bell measurements.}
Our analytical treatment enables us to identify
the scaling of the various parameters, their specific impact
onto the repeater performance (for QKD, affecting either the raw rate
or the error-dependent secret key fraction), and the
resulting trade-offs. Most apparent is the trade-off
for quantum repeaters with $n$ segments and $n-1$ intermediate memory stations
leading to an improved loss scaling with an $n$-times bigger
effective attenuation distance compared with a point-to-point link ($n=1$),
but a final state fidelity parameter
decreasing as the power of $2n-1$ (assuming equal gate and initial state
error rates).
We will then be able to consider repeater protocol variations
with an improved scaling of the basic loss and fidelity parameters.
Based upon the above-mentioned TF concept with coherent states
or basically replacing two-photon by one-photon interferences at the
beam splitter stations, these repeaters exhibit
a $2n$-times bigger effective attenuation distance
while keeping the $2n-1$ power scaling of the final state fidelity parameter
for $n-1$ memory stations. However, they are subject to some
extra intrinsic (dephasing) errors even when only channel loss is considered, which will turn out to be an essential complication that prevents to fully exploit the improved
scaling of the basic parameters in comparison with the standard repeater protocols that do not suffer from intrinsic dephasing.
Comparing different repeater protocols and incorporating
the optimized memory dephasing from our statistical model into them,
we find that for experimental parameter values that are highly demanding but not impossible
(up to 10s coherence time, 80\% link coupling, and state or gate infidelities in the regime of 1-2\%), one secret bit can be shared per second over a total distance of 800km.
This represents a significant improvement over ideal point-to-point or realistic TF QKD at GHz clock rates.
In particular, the repeaterless, point-to-point bound \cite{PLOB}, for e.g. 800km is $3 \times 10^{-16}$ bits per channel use
or $0.3 \mu$bits per second (at GHz clock rate).
We will see that, in order to clearly beat this with those reasonable experimental parameters from above, the number of repeater stations
must neither be too high nor too low, and so placing a station at every 100km will work well.
As mentioned before, our schemes are generally independent of the typically
used doubling strategies in quantum repeaters
(which are most suitable to incorporate entanglement distillation
in a systematic way and which are included as a special case in our sets of swapping strategies). Instead we will consider general memory-assisted entanglement distribution with possible QKD applications. Compatible with our analysis are also schemes that aim at an enhanced
initial state distribution efficiency or fidelity as, for example, in multiplexing-assisted or the above-mentioned 2nd-generation quantum repeaters.
In any case, the subsequent steps after the initial distributions in each repeater segment are simple entanglement swapping steps combined with quantum storage in single spins. For the entanglement swapping we assume unit success probability. This assumption is experimentally justified for systems where Bell measurements or, more generally, gates can be performed in a deterministic fashion, for instance, with atoms or ions or solid-state-based spin qubits \cite{White}.
For a linear quantum repeater chain, this system is still remarkably complex.
The assumption of deterministic entanglement swapping will allow us to calculate
the exact (secret key) rates in a quantum repeater up to eight segments.
We will distinguish schemes with sequential and parallel entanglement distributions and also consider different swapping strategies. Based on {\it two characteristic random variables},
the total repeater waiting time and the accumulated dephasing time of the final state, and their probability generating functions, we will be able to determine exact, optimized secret key rates.
In principle, this gives us access to the {\it full
statistics of this class of quantum repeaters.}
Optimality here refers to the minimal dephasing among all parallel-distribution (and hence maximal raw-rate)
schemes. For three segments and two intermediate stations, we show that the resulting secret key
rates are optimal among all schemes. For more segments and stations we conjecture this to hold too, however, there is the loophole that sequential-distribution schemes (generally exhibiting smaller raw rates) may accumulate less dephasing and as a result, in combination, lead to a higher secret key rate. We conclude that our treatment gives evidence for any optimal scheme to distribute entangled pairs in parallel, to swap as soon as possible, and to simultaneously store qubits as little as possible. However, here the first and the third property are not compatible, which leads to another trade-off between
high efficiencies (raw rates) and small state fidelities (high error rates) as commonly encountered for entanglement distribution and quantum repeaters.
The (partially or fully) sequential schemes have the advantage that parallel storage of qubits can be avoided to a certain (or even a full) extent. However, since the sequential schemes are overall slower, their total dephasing may still exceed that of the fastest repeater schemes with parallel storage.
For up to eight repeater segments, our optimal scheme, exhibiting the smallest total dephasing among all fast repeater schemes, also exhibits a smaller total dephasing than the fully sequential scheme.
The outline of this paper is as follows.
In Sec.~\ref{sec:QRwonemiddlestation} we will first review the known results
and existing approaches to analyze secret key rates for the smallest possible
quantum repeater based upon a single middle station, including calculations
of the repeater raw rate and physical error models to describe the evolution
of the relevant density operators. The methods for the statistical analysis
-- probability generating functions, and the figure of merit
to quantitatively assess the repeater performance -- a QKD secret key rate,
will be introduced in Sec.~\ref{sec:Methods}.
In Sec.~\ref{sec:Physical Modelling} we will then start introducing our
new, generalized treatment for quantum repeaters beyond a single middle station.
For this, we present two subsections on the two characteristic random variables
-- the waiting time and the dephasing time, which contain the entire statistical
information of the class of quantum repeaters considered in our work.
In order to be able to take into account optimal strategies
for the initial entanglement distribution and the subsequent entanglement swapping
in more complex quantum repeaters with two or more intermediate repeater stations,
we discuss in detail in various subsections sequential and parallel distribution
as well as optimal swapping schemes. Still in Sec.~\ref{sec:Physical Modelling},
we show how these optimizations can be applied to the statistics of
various quantum repeaters, explicitly calculating the probability generating functions
of the two basic random variables for two-, three-, four- and eight-segment
quantum repeaters. In particular, for the four- and eight-segment cases we will show
how and to what extent our optimized and exact treatment of the memory dephasing
will improve the relevant quantities of the final state density operators
as compared with the usually employed, canonical schemes such as ``doubling''.
The interesting case of a three-segment repeater and its optimization will be discussed in more detail in an appendix.
Finally, in Sec.~\ref{sec:Secret Key Rate} we will analyze the secret key rates of all proposed schemes and compare them for various repeater sizes with the ``PLOB" bound \cite{PLOB}.
For this, we will explicitly consider the extended set of experimental parameters
and insert experimentally meaningful values (representing current and future experimental
capabilities) for them.
A particular focus will be on the initial state and gate parameters
and their impact on the repeater performance.
We shall compare the performances of different schemes, discuss the possibility of including multiplexing, and examine what influence a memory cut-off and what (scaling) advantages the different types of encoding for the flying qubits can have.
For the latter, we discuss in more detail schemes based on the TF concept and, for the comparison between different schemes and encodings, the final secret key rates per second.
Sec.~\ref{sec:Conclusion} concludes the paper with a final summary of the results and their implications. Various additional technical details can be found in the appendices.
\section{Quantum repeaters with one middle station}\label{sec:QRwonemiddlestation}
A small quantum repeater composed of two segments and one middle station,
as schematically shown in Fig.~\ref{fig:2seg},
is pretty well understood and it is known how to obtain the secret key rates
in a QKD scheme assisted by a single memory station,
even including experimental imperfections \cite{NL,WehPar,White,tf_repeater},
including memory cutoffs \cite{WehPar,White,PvL, CollinsPrl},
and for general, probabilistic entanglement swappping \cite{PvL}.
First experimental demonstrations of memory-enhanced quantum communication
are also based on this simplest repeater setting \cite{Lukin}.
In such a small quantum repeater, there is only a single Bell measurement
on the spin memories at the central station, and so the entanglement swapping
``strategy'' is clear. Later we will briefly discuss the two-segment case
as a special case of our more general rate analysis treatment, easily
deriving the statistical properties of the two basic random repeater variables,
the total waiting and dephasing times, and obtaining the optimal scheme \cite{White, tf_repeater}.
\begin{figure}
\caption{A two-segment quantum repeater. Each segment has length $L_0$
and is characterized by a distribution success probability $p$,
a (geometrically distributed) random number of distribution attempts $N$
(with expectation value $\bar N = 1/p$), and a ``final'' two-qubit state $\hat\rho$ (subscripts denote segments or qubits at the nodes). ``Final'' here means that the, in general, imperfectly distributed states may be further subject to memory dephasing for a maximal number of $m$ time steps (distribution attempts). After an imperfect swapping operation $\mathcal S$ (error parameter $\mu$), the repeater end nodes share an entangled state over distance $2 L_0$.}
\label{fig:2seg}
\end{figure}
The smallest, two-segment quantum repeater also serves as a basic building block
for general, larger quantum repeaters.
In the scheme of Fig.~\ref{fig:2seg},
each segment distributes an entangled pair of (mostly) stationary qubits
by connecting its end nodes through flying qubits.
The goal is to share entanglement between the two qubits at the end nodes
of the whole repeater.
The specific entanglement distribution scheme in each segment depends
on the repeater protocol and it may involve memory nodes sending
or receiving photons \cite{White}.
In the notation of Fig.~\ref{fig:2seg}, from
an entangled state $\hat{\varrho}_{12}$ of qubits 1 and 2 and an entangled state $\hat{\varrho}_{34}$ of qubits 3 and
4, we create an entangled state $\hat{\varrho}_{14}$ of qubits 1 and 4. Here the states $\hat{\varrho}_{12}$ and
$\hat{\varrho}_{34}$ subject to the Bell measurement for the entanglement swapping operation are those quantum states present in the segments at the moment when the swapping is
performed. If, for example, segment 1 generates an entangled state earlier than segment 2, then $\hat{\varrho}_{12}$ enters the swapping step in the form of the initially, distributed state
(which is not necessarily a pure maximally entangled state) after it was subject to memory dephasing while waiting for segment 2.
Thus, our physical model includes state imperfections that originate from
the initial distribution as well as from the storage time,
as we shall discuss in detail below. In addition, we will include
an error parameter for the swapping gate itself.
\subsection{Raw rate}\label{sec:rawrate}
The entanglement distribution in an elementary segment is typically not a deterministic process and
several attempts are necessary to successfully share an
entangled pair of qubits among two neighboring stations.
If the probability of successful generation in each attempt is $p$, then the number of time steps
until success is a geometrically distributed random variable $N$ with success parameter $p$. We denote the failure probability as $q = 1 - p$.
The parameter $p$ is primarily given by the probability that a photonic qubit is successfully transmitted via a fiber channel of length $L_0$ connecting two stations, $\exp(-L_0/\unit[22]{km})$. It also includes local state preparation/detection, fiber coupling, frequency conversion, and memory ``write-in'' efficiencies.
The random variables for different segments (in
Fig.~\ref{fig:2seg} denoted as $N_1$ and $N_2$ for the first and the second segment, respectively) are independent and
identically distributed geometric random variables. Only when both segments have generated an entangled state, we perform a
swapping operation on the adjacent ends (nodes 2 and 3) of the segments and, when successful, we will be left with an entangled
state of qubits 1 and 4.
In general, the swapping operation is also non-deterministic, but here we consider only
the case of deterministic swapping. Under this simple assumption we can still cover
a large class of physically relevant and realistic repeater schemes and obtain exact and optimized rates for them. Moreover, especially for larger repeaters (still with no entanglement distillations), this assumption allows
to circumvent the need for classical communication times longer than the elementary time $\tau$
(as defined below) in order to confirm
successful entanglement swapping operations on ``higher'' repeater levels
beyond the initial distributions in each segment.
Physically, this assumption requires that in our schemes the Bell measurements
for entanglement swapping (including the memory ``read-out'' operations) can be performed deterministically. Nonetheless, the swapping operations can still be imperfect,
introducing errors in the states, as will be described below.
Due to the non-deterministic nature of the initial entanglement generation,
the whole process of
entanglement distribution is also non-deterministic and fully described by the number of attempts up to and including
the successful distribution (so, this number is always larger than zero). The real, wall-clock time needed for
entanglement generation or distribution can be obtained from the number of attempts by multiplying it with an
elementary time unit, typically $\tau = L_0/c_f$, where again $L_0$ is the length of the segment and $c_f = c/n_\mathrm{r}$ is the speed of light
in the optical fiber ($c$ is the speed of light in vacuum and $n_\mathrm{r}$ is the index of refraction of the fiber, and depending on the specific distribution protocol there may be an extra factor 2).
The elementary time unit is actually composed of the classical (and quantum) signalling time
per segment $\tau$ and the local processing time. However, for typical $L_0$ values
as considered here, the former largely dominates over the latter, and so we may neglect
the local times, as they would hardly change the final secret key rates \cite{White}.
If one of the two segments generates entanglement earlier than the other, then the created state must be kept in memory.
The exact technique employed to implement this quantum memory is irrelevant for our analysis. The simplest model assumes
that the state can be kept in memory for arbitrarily long. A useful assumption in the realistic setting with imperfect quantum memories is to set a certain limit of $m$ time
units on the memory storage time, thus restarting the creation process whenever this threshold is reached.
\subsection{Errors}
When the quantum repeater is employed for long-range QKD, errors will become manifest
in terms of a reduced secret key fraction, as introduced in the subsequent section.
In order to compute this secret key fraction, we need to know the finally distributed state (density operator) of the
complete repeater system, and for this we require a more detailed physical model.
We shall establish a relation between the finally distributed state as a
function of the initial states in each segment and various errors that appear in the process of entanglement
distribution. The physical model is rather common and has been used before in
several works, both analytical and numerical.
Especially, a two-segment quantum repeater can be treated analytically based on simple Pauli errors
representing memory dephasing and gate (Bell measurement) errors.
We address the effect of imperfect quantum storage at a memory node
via a dephasing model where the stored quantum state is waiting for an adjacent
segment to successfully generate or distribute entanglement.
This kind of memory error can be modelled by a one-qubit dephasing
channel,
\begin{equation}\label{eq:Gl}
\Gamma_\lambda(\hat{\varrho})=(1 - \lambda) \hat{\varrho} +\lambda Z \hat{\varrho} Z,
\end{equation}
where $Z$ is a qubit Pauli phase flip operator.
We assume that $0 \leqslant \lambda < 1/2$, and any such number can be represented as $\lambda = (1 - e^{-\alpha})/2$
for some $\alpha > 0$. We denote the map in Eq.~\eqref{eq:Gl} also as $\Gamma_\alpha$. To avoid confusion, throughout this work we use the following definition:
\begin{equation}\label{eq:Ga}
\Gamma_\alpha(\hat{\varrho}) = \frac{1 + e^{-\alpha}}{2} \hat{\varrho} +
\frac{1 - e^{-\alpha}}{2} Z \hat{\varrho} Z.
\end{equation}
The definition for a dephasing two-qubit channel is obtained from Eqs.~\eqref{eq:Gl}-\eqref{eq:Ga} by the replacement $Z
\to Z \otimes I$ if the dephasing acts on the first qubit and by $Z \to I \otimes Z$ if the dephasing acts on the second
qubit.
Errors may also occur when a Bell state measurement is performed. This kind of errors is modelled by a two-qubit depolarizing channel,
\begin{equation}
\tilde{\Gamma}_\mu(\hat{\varrho}) = \mu \hat{\varrho} + (1-\mu) \frac{\hat{\mathbb{1}}}{4}.
\end{equation}
We do not consider dark counts of the detectors,
since the optical propagation distances $L_0$ after which
a detection attempt takes place remain sufficiently small in
any quantum relay or repeater.
Thanks to recent technological developments typical dark count rates can be reduced
far below 1 dark count per second. In Ref.~\cite{Schuck2013} they were shown to be in the range of
\(\unit{mHz} \). Dark counts of such a low frequency have no significant impact on the secret key rate in our schemes.
Let us now apply this to the case of a two-segment quantum repeater.
The Bell measurement of qubits 2 and 3 produces from a pair of states $\hat{\varrho}_{12}$ and $\hat{\varrho}_{34}$ a
state $\hat{\varrho}_{14}$, see Fig.~\ref{fig:2seg}. The initial state $\hat{\varrho}_{1234} = \hat{\varrho}_{12}
\otimes \hat{\varrho}_{34}$ of all four qubits 1, 2, 3 and 4 is the product of the states of qubits 1, 2 and qubits 3,
4. After the measurement the state $\hat{\varrho}_{14}$ of qubits 1 and 4 becomes
\begin{equation}\label{eq:Srho}
\hat{\varrho}_{14} \equiv \mathcal{S}(\hat{\varrho}_{1234}) = \frac{\Tr_{23}(\hat{P}_{23}
\tilde{\Gamma}_{\mu, 23}(\hat{\varrho}_{1234}) \hat{P}_{23})}{\Tr(\hat{P}_{23}
\tilde{\Gamma}_{\mu, 23}(\hat{\varrho}_{1234}) \hat{P}_{23})},
\end{equation}
where $\mu$ describes the imperfection of the measurement and $\hat{P}_{23} = |\Psi^+\rangle_{23}\langle\Psi^+|$ is one of the four measurement operators
in the two-qubit Bell state basis of the central subsystem (qubits 2 and 3),
$\{ |\Phi^\pm\rangle_{23}\langle\Phi^\pm|,\,|\Psi^\pm\rangle_{23}\langle\Psi^\pm| \}$,
where $|\Phi^\pm\rangle = (|00\rangle \pm |11\rangle)/\sqrt{2},\,
|\Psi^\pm\rangle = (|10\rangle \pm |01\rangle)/\sqrt{2}$, for qubits
defined via the two $Z$ eigenstates $|0\rangle, \,|1\rangle$
(for any one of the other three Bell measurement outcomes, the analysis below
is similarly applicable).
In this case, Eq.~\eqref{eq:Srho}
reduces to
\begin{equation}\label{eq:rhod}
\hat{\varrho}_{14} \equiv \mathcal{S}(\hat{\varrho}_{1234}) =
\frac{_{23}\langle\Psi^+|\tilde{\Gamma}_{\mu, 23}(\hat{\varrho}_{1234})|\Psi^+\rangle_{23}}
{\Tr(_{23}\langle\Psi^+|\tilde{\Gamma}_{\mu, 23}(\hat{\varrho}_{1234})|\Psi^+\rangle_{23})}.
\end{equation}
A simple way to compute the right-hand side of this relation for an arbitrary density operator
$\hat{\varrho}_{1234}$ is given in App.~\ref{app:Trace Identities}.
In general, states of the form
\begin{equation}\label{eq:rho0}
\hat{\varrho}_0 = \tilde{\Gamma}_{\mu_0}\bigl(F_0\dyad{\Psi^+} + (1-F_0)\dyad{\Psi^-}\bigr)
\end{equation}
play an important role in the full theory presented below. It is easy to verify that
\begin{equation}
(I \otimes Z) \hat{\varrho}_0 (I \otimes Z) = (Z \otimes I) \hat{\varrho}_0 (Z \otimes I),
\end{equation}
so it does not matter whether $\Gamma_\alpha$ acts on the first or second qubit of $\hat{\varrho}_0$ and either
application we simply denote as $\Gamma_\alpha(\hat{\varrho}_0)$. An easily checkable relation is
\begin{equation}\label{eq:Ga2}
\Gamma_\alpha(\hat{\varrho}_0) = \tilde{\Gamma}_{\mu_0}\bigl(F\dyad{\Psi^+} + (1-F)\dyad{\Psi^-}\bigr),
\end{equation}
where the new parameter $F$ is expressed in terms of the original one, $F_0$, as
\begin{equation}\label{eq:Fprime}
F = \frac{1}{2}(2F_0-1) e^{-\alpha} + \frac{1}{2}.
\end{equation}
The initial fidelity parameter $F_0$ (describing an initial dephasing
of the distributed states) combined with the $\mu_0$-dependent initial
depolarization are both included in the initial $\hat \rho_0$ in Eq.~\eqref{eq:rho0},
because later this will allow for an elegant recursive state relation for
larger repeaters. It will also allow to switch between different initial
physical errors depending on the specific repeater realization.
In general, the maps in Eq.~\eqref{eq:Ga} satisfy the relation $\Gamma_\alpha \circ \Gamma_\beta = \Gamma_{\alpha + \beta}$. In
particular, we have $\Gamma_\alpha \circ \ldots \circ \Gamma_\alpha = \Gamma_{k\alpha}$, where $\Gamma_\alpha$ is used
$k$ times on the left-hand side. So, applying $\Gamma_\alpha$ to the state $\hat{\varrho}_0$ given by
Eq.~\eqref{eq:rho0} several times, we have to multiply $\alpha$ in Eq.~\eqref{eq:Fprime} by this number of times.
In a two-segment quantum repeater, if we start with the distributed states $\hat{\varrho}_{12}$ and $\hat{\varrho}_{34}$ of the special form
(similar to Eq.~\eqref{eq:rho0})
\begin{equation}\label{eq:Drho}
\begin{split}
\hat{\varrho}_{12} &= \tilde{\Gamma}_{\mu_1}\bigl(F_1|\Psi^+\rangle_{12}\langle\Psi^+| + (1 - F_1)|\Psi^-\rangle_{12}\langle\Psi^-|\bigr), \\
\hat{\varrho}_{34} &= \tilde{\Gamma}_{\mu_2}\bigl(F_2|\Psi^+\rangle_{34}\langle\Psi^+| + (1 - F_2)|\Psi^-\rangle_{34}\langle\Psi^-|\bigr), \\
\end{split}
\end{equation}
then the ``swapped'', finally distributed state $\hat{\varrho}_{14}$, given by Eq.~\eqref{eq:rhod}, is also of the same form,
\begin{equation}\label{eq:rho14}
\hat{\varrho}_{14} = \tilde{\Gamma}_{\mu_d}\bigl(F_d|\Psi^+\rangle_{14}\langle\Psi^+| + (1 - F_d)|\Psi^-\rangle_{14}\langle\Psi^-|\bigr),
\end{equation}
where $\mu_d = \mu \mu_1 \mu_2$ and $F_d$ reads as
\begin{equation}\label{eq:Fd}
F_d = \frac{1}{2}(2F_1 - 1)(2F_2 - 1) + \frac{1}{2}.
\end{equation}
We see that the form of the state is preserved by the total distribution procedure
of a two-segment repeater. The same conclusion will be applicable to larger
repeaters as well --- if all segments start in a state of the form given by Eq.~\eqref{eq:rho0}, then the finally distributed
state will also be of the same form.
For the two-segment repeater,
let us now assume that both segments generate the same state as in Eq.~\eqref{eq:rho0},
but not necessarily simultaneously, and so generally only after
some waiting time we perform the entanglement swapping and distribute entanglement over the two segments. If the first segment
generates entanglement after $N_1$ time units, and the second segment after $N_2$ time units, and we perform the entanglement swapping
after $N$ time units, with $N \geqslant N_1, N_2$, then the states $\hat{\varrho}_{12}$ and $\hat{\varrho}_{34}$ prior to swapping will be of
the form in Eq.~\eqref{eq:Drho} with $\mu_1 = \mu_2 = \mu_0$ and
\begin{equation}
\begin{split}
F_1 &= \frac{1}{2}(2F_0 - 1)e^{-(N-N_1)\alpha} + \frac{1}{2}, \\
F_2 &= \frac{1}{2}(2F_0 - 1)e^{-(N-N_2)\alpha} + \frac{1}{2}. \\
\end{split}
\end{equation}
The final, distributed state is then given by Eq.~\eqref{eq:rho14} where, according to Eq.~\eqref{eq:Fd}, the parameters are $\mu_d
= \mu \mu^2_0$ and
\begin{equation}
F_d = \frac{1}{2}(2F_0-1)^2 e^{-(2N-N_1-N_2)\alpha} + \frac{1}{2}.
\end{equation}
This distributed state is subject to less dephasing when we swap as early as possible, thus $N = \max(N_1, N_2)$, so the
integer term in front of $\alpha$ is equal to $2\max(N_1, N_2) - N_1 - N_2 = |N_1 - N_2|$. Extra factors depending on the number of spins subject to dephasing in one segment (in particular, a factor of 2 for one spin pair) can be absorbed into $\alpha$. The precise physical meaning of $\alpha$ will be discussed later when we calculate the memory-assisted secret key rates in a quantum repeater. Furthermore, here we omitted explicit factors depending on the number of memory qubits that are subject to dephasing in a single repeater segment (in our model this will be one or two spins).
\section{Methods and figure of merit}\label{sec:Methods}
Before we move to the more general case of more than two segments and more than
just one middle station, we need some general methods and tools from statistics.
This will enable us to derive an analytic, statistical model for larger quantum repeaters
beyond one middle station (the physical model remains basically the same as for the
small, elementary two-segment quantum repeater), where we calculate average values
or moments of two random variables: the total repeater waiting time $K_n$ and the total
(i.e., the totally accumulated) memory dephasing time $D_n$.
As a quantitative figure of merit, it is useful to consider the secret key rate
of QKD, as it combines in a single quantity the two typically competing effects in
a quantum repeater system: the speed at which quantum states can be distributed
over the entire communication distance
and the quality of the totally distributed quantum states.
These two effects are naturally related to the above-mentioned two
random variables.
For our purposes here, throughout we shall rely on asymptotic expressions for the secret key rate
omitting effects of finite key lengths.
Of course, alternatively, one could also treat the total state distribution efficiencies
and qualities (fidelities) separately and individually, and then also consider
quantum repeater applications beyond long-range QKD.
\subsection{Probability generating function}
The method of probability generating functions (PGFs) plays an important role in our treatment of statistical properties
of quantum repeaters. For any random variable $X$ taking integer non-negative values its PGF $G_{X}(t)$ is defined via
\begin{equation}
G_X(t) = \mathbf{E}[t^X] = \sum^{+\infty}_{k = 0} \mathbf{P}(X = k)t^k.
\end{equation}
The series on the right-hand side converges at least for all complex values of $t$ such that $|t| \leqslant 1$. The PGF
contains all statistical information about $X$, which can be easily extracted if an explicit expression for $G_X(t)$ is
known. For example, the average value of $X$, $\mathbf{E}[X] \equiv \overline{X}$, and its variance $\mathbf{V}[X]
\equiv \sigma^2_X = \mathbf{E}[(X - \overline{X})^2]$, are expressed as follows:
\begin{equation}\label{eq:PGF}
\begin{split}
\mathbf{E}(X) &= G'_X(1), \\
\mathbf{V}(X) &= G^{\prime\prime}_X(1) + G'_X(1) - G^{\prime 2}_X(1).
\end{split}
\end{equation}
For any $\alpha \geqslant 0$ the random variable $e^{-\alpha X}$ has a finite average value, which can be computed as
\begin{equation}\label{eq:PGF_2}
\mathbf{E}[e^{-\alpha X}] = G_X(e^{-\alpha}).
\end{equation}
Note that for this random variable, besides the mean or average value, any statistical moment can be easily obtained and the $k$th-moment simply becomes $\mathbf{E}[e^{-\alpha k X}] = G_X(e^{-k \alpha})$.
Two kinds of random variables appear in our model of quantum repeaters where one is related to the raw rate
and the other to the secret key fraction of QKD as introduced below. It is not always possible to get a compact expression for the PGF of these
random variables explicitly, but when it is, we use the equations above to obtain statistical properties of the
corresponding random variables.
\subsection{Secret key rate}\label{sec:skr}
The main figure of merit in our study is the quantum repeater secret key rate, which can be defined as the product of two quantities,
\begin{equation}
S = Rr,
\end{equation}
where $R$ is the raw rate and $r$ is the secret key fraction. The raw rate is simply the inverse average
waiting time,
\begin{equation}
R = \frac{1}{T},
\end{equation}
where $T = \mathbf{E}[K]$ is the average number of steps $K$ needed to successfully distribute one entangled qubit pair over the entire communication distance between Alice and Bob (giving an average time duration in seconds when multiplied with an appropriate time unit $\tau$). The secret key fraction of the BB84 QKD protocol \cite{BB84,PirRMP}, assuming one-way post-processing, is given by
\begin{equation}\label{eq:skf}
r = 1 - h(\overline{e_x}) - h(\overline{e_z}),
\end{equation}
where $e_x$ and $e_z$ are the quantum bit error rates (QBERs),
\begin{equation}\label{eq:exez}
\begin{split}
e_z &= \langle 00|\hat{\varrho}_n|00\rangle + \langle 11|\hat{\varrho}_n|11\rangle, \\
e_x &= \langle +-|\hat{\varrho}_n|+-\rangle + \langle -+|\hat{\varrho}_n|-+\rangle,
\end{split}
\end{equation}
and $h(p)$ is the binary entropy function,
\begin{equation}
h(p) = -p\log_2(p) - (1-p)\log_2(1-p).
\end{equation}
The QBERs $e_x$ and $e_z$ in Eq.~\eqref{eq:exez} are obtainable from the final, distributed state $\hat{\varrho}_n$ of an
$n$-segment quantum repeater, which in our case will depend on the dephasing random variable, and so we have to insert average values in Eq.~\eqref{eq:skf}
as indicated by the bars. We thus need a complete model of quantum repeaters to compute the statistical properties of the relevant
random variables associated with the number of steps to distribute entanglement or the density operator of the distributed state.
Given such a model,
the aim of our work is to compute and analyze secret key rates of quantum repeaters with an increasing size, up to eight segments, considering and optimizing different distribution and swapping schemes.
Besides the most common BB84 QKD protocol,
alternatively, we may also consider the six-state protocol
\cite{sixstate}
which would slightly improve the secret key rate.
Assuming again one-way post-processing, the secret key fraction $r$
of the six-state protocol is given by $1-H(\boldsymbol{\lambda})$
\cite[App. A]{RevModPhys.81.1301}
where $H(\cdot)$ is the Shannon entropy and the vector $\boldsymbol{\lambda}$ must contain the corresponding weights of the four Bell states in the final density operator $\hat{\varrho}_n$. Throughout this work all secret key rates
are calculated from their asymptotic expressions and hence effects
of finite key lengths are not included here. This simplifies the analytical treatment of a quantum repeater chain, which, as we will see, quickly becomes rather complex for a growing number of stations, involving many distinct choices and strategies for the entanglement manipulations. Moreover, our rate analysis shall also be useful to assess and compare the performances of different quantum repeaters in applications beyond QKD.
\section{Quantum repeaters beyond one middle station}\label{sec:Physical Modelling}
\begin{figure*}
\caption{``Doubling'' swapping scheme for a four-segment quantum repeater. This is the most common swapping strategy which allows to systematically include entanglement distillation at each repeater ``nesting level". Without extra distillation, however, ``doubling'' is never optimal: combined with fast, parallel distributions it exhibits increased parallel storage times and hence memory dephasing (while combined with sequential distributions the repeater waiting times become suboptimal). Memory cut-off parameters are omitted in the illustration.}
\label{fig:4segD}
\end{figure*}
\begin{figure*}
\caption{``Iterative'' swapping scheme for a four-segment quantum repeater. The swapping operations are performed step by step (here from left to right). Also this scheme, when executed with parallel distributions in each segment, leads to an increase of the total dephasing. However, if combined with sequential distributions, the accumulated dephasing times can be reduced (with always at most one spin or spin pair being subject to a long dephasing) at the expense of a growing repeater waiting time. Memory cut-off parameters are omitted in the illustration.}
\label{fig:4segI}
\end{figure*}
Larger repeaters with more than two segments and one middle station can now be modeled in a way similar to the two-segment case
discussed above.
However, the extended, more general case is also more complex and there are both different ways to perform the initial
entanglement distributions in all elementary segments and different ways to connect the successfully distributed
segments via entanglement swapping. For the initial distributions we make a distinction between
sequential and parallel schemes, where the former refers to a scheme in which, according to a predetermined order,
the distributions are attempted step by step starting from e.g. the first segment. In a parallel scheme,
the distributions are attempted simultaneously in all segments, which obviously leads to a smaller total
repeater waiting time than for the sequential distribution schemes. Nonetheless, since the sequential schemes
do make use of the quantum memories, they do already offer the repeater-like scaling advantage over
point-to-point quantum communication links. Even for a two-segment quantum repeater, we may choose
a sequential scheme, where we first only distribute e.g. the left segment and only once we succeeded there we attempt
to distribute the right segment. Experimentally, this can be of relevance for those realizations where only a single short-term
quantum memory is available at every station for the light-matter interface
and another quantum memory for the longer-term storage (e.g., respectively, an electronic and a nuclear spin in colour-center-based repeater nodes) \cite{WehnerNV,HansonNV}.
Theoretically and conceptually, there are at least two advantages of a (fully) sequential distribution approach \cite{tf_repeater}.
First, the two basic random variables of a quantum repeater are very simple and so the secret key rates are fairly easy to calculate. Second, always only at most one entangled qubit pair (or even only a single spin if e.g. Alice measures her qubit immediately) may be subject to memory dephasing during all distribution steps.
For the entanglement connections via entanglement swapping, the two-segment case is special,
as there is only one swapping to be performed at the end when pairs in both segments are available.
However, already with three segments and two repeater stations there is no unique swapping order anymore,
and we may either fix the order or ``dynamically'' choose where we swap as soon as swapping is possible
for two neighboring, successfully distributed segments. In a fixed scheme, two neighboring segments,
though ready, may have to wait before being connected.
Thus, the choice of the entanglement swapping scheme
has a significant impact on the totally accumulated dephasing time. In a worst-case scenario, we could wait until
all segments have been distributed and then do all the entanglement connections at the very end;
for deterministic entanglement swapping, like in our model, this would not affect the raw waiting times,
but it would lead to a maximal total dephasing.
In this case, a sequential distribution where entanglement swapping takes place immediately when a new, successfully bridged segment is available can lead to a higher secret key rate than a combination of parallel distribution and swapping at the end (where the rates of the latter scheme may still only be obtainable approximately) \cite{tf_repeater}.
The crucial innovation in our analytical treatment here is that we will be able to obtain the exact secret key rates for schemes that combine fast, parallel distributions with fast, immediate swappings (and hence a suppressed level of parallel storage). In other words, among all parallel-distribution schemes we will calculate the exact rates that are optimized with regards to the total repeater dephasing.
\subsection{Waiting times}
The average total waiting times in a quantum repeater or even the full statistics
of the waiting-time random variable can be, in principle, obtained
via the Markov chain formalism, even when the swapping is probabilistic \cite{PvL, Shchukin2021}.
More generally, the PGFs as introduced earlier contain the full statistical information,
and for deterministic swapping, we can obtain the PGF of $K_n$ through combinatorics.
In order to minimize the total waiting time, the distributions should occur in parallel.
However, there is no unique way to
perform the entanglement swapping, and so let us briefly consider this aspect in the context
of the waiting times. For example, for a four-segment repeater, two possible swapping strategies are shown in Figs.~\ref{fig:4segD} and
\ref{fig:4segI}. Both schemes are for a fixed swapping order, while we may distribute the
individual segments in parallel.
In the first scheme, typically referred to as ``doubling'', we swap the two halves of the repeater independently and only when both are ready, we
swap them too. In the second scheme, we swap the segments one after the other starting in one of the repeater's ends
(here the left end); we may refer to this scheme as ``iterative'' swapping.
Other schemes are possible, and the more segments the repeater has, the more possibilities for performing swappings
there are. The raw rate of a repeater is characterized by the number of steps, $K_n$, needed to successfully distribute
an entangled pair, and this random variable can be expressed in terms of the geometric random variables $N_i$ associated with each segment. For
example, for the swapping schemes shown in Figs.~\ref{fig:4segD} and \ref{fig:4segI}, when combined with parallel distributions, we have $K_4 = \max(N_1, N_2, N_3, N_4)$, so the two schemes have the same raw rate.
In general, the waiting times of all such schemes that distribute in parallel are of a similar form.
Those schemes that we later classify as ``optimal'' in terms of the whole secret key rate
are assumed to be parallel distribution schemes. Conversely, combining iterative swapping with sequential distribution can lead to a reduced accumulated dephasing time at the expense of an increased total repeater waiting time. We shall discuss the accumulated dephasing times next.
\subsection{Dephasing times}
In order to treat the total dephasing time in a quantum repeater with more than two segments,
we have to generalize the methods and the model that led to the result for the distributed state for two segments,
Eq.~\eqref{eq:rho14} and Eq.~\eqref{eq:Fd}, and the discussion below, to larger repeaters
with, in pinciple, an arbitrary number of segments $n$.
In fact, we did the two-segment derivations in such a way that
an $n$-segment extension is now straightforward.
We obtain the following expression for the final, distributed state in the general case:
\begin{equation}\label{eq:rhon}
\begin{split}
\hat{\varrho}_n = \tilde{\Gamma}_{\mu_n}\Biggl[&\frac{1
+(2F_0 - 1)^n e^{-\alpha D_n}}{2} \dyad{\Psi^+} \Biggr. \\
+\Biggl.&\frac{1 - (2F_0 - 1)^n e^{-\alpha D_n}}{2} \dyad{\Psi^-}\Biggr],
\end{split}
\end{equation}
where $\mu_n = \mu^{n-1} \mu^n_0$ and $D_n = D_n(N_1, \ldots, N_n)$ is a random variable describing the total number of
time units that contribute to the total dephasing in the final output state. For $n=2$, the expression for $D_2(N_1, N_2) = |N_1 - N_2|$ has been obtained before, for larger $n$ the value of $D_n$ now depends on the swapping scheme. As before, we omitted explicit factors depending on the number of memory qubits that are subject to dephasing in a single repeater segment (one or two spins in our model) which also depends on the application and the specific execution of the protocol. Such factors can always be absorbed into $\alpha$.
The precise physical meaning of $\alpha$ will be discussed later when we calculate the memory-assisted secret key rates in a quantum repeater.
The QBERs for the state in Eq.~\eqref{eq:rhon} are easy to compute,
\begin{equation}\label{eq:QBER}
\begin{split}
e_z &= \frac{1}{2}(1 - \mu^{n-1}\mu^n_0), \\
e_x &= \frac{1}{2}(1 - \mu^{n-1}\mu^n_0 (2F_0 - 1)^n e^{-\alpha D_n}).
\end{split}
\end{equation}
For one of the averages, we have $\overline{e_z} = e_z$, and in order to obtain the other average $\overline{e_x}$ we need to calculate the expectation value
$\mathbf{E}[e^{-\alpha D_n}]$. This average can be obtained with the help of Eq.~\eqref{eq:PGF_2} if we know the
PGF of $D_n$. Again, in principle, we can get the full statistics of $D_n$ (and functions of it) from this PGF. More specifically, according to Eq.~\eqref{eq:PGF_2}, for the random variable $e^{-\alpha D_n}$ we can easily obtain all statistical moments of order $k$, $\mathbf{E}[e^{-\alpha D_n k}]$. This may be useful for a rate analysis that includes keys of a finite length, though here in this work we shall focus on asymptotic keys.
The PGF of $D_n$, however, is generally harder to obtain than that of $K_n$. For example, the PGF of $D_n$ is not obtainable
via the absorption time of a Markov chain (unlike that of $K_n$, which is obtainable even when the entanglement swapping is probabilistic) \cite{PvL, Shchukin2021}.
Nonetheless, at least without considering the more complicated case including
a memory cut-off, we can calculate the relevant PGF of $D_n$ by analyzing all permutations of the basic variables
(there are also other, more elegant, but still not so efficient and well scalable methods to treat the statistics of $D_n$,
e.g. based on algebraic geometry).
We see that in order to compute the secret key rate of a quantum repeater we need to study the two integer-valued random variables
$K_n$ and $D_n$. The former describes the number of steps to successfully distribute entanglement and is responsible for
the repeater's raw rate. The latter describes the quality of the final state and strongly depends on the swapping
scheme. For example, for a four-segment repeater with a predetermined swapping order like the iterative scheme
in Fig.~\ref{fig:4segI}, we could actually also choose to adapt the initial entanglement distributions
to the swapping strategy and hence wait with every subsequent distribution step until the corresponding connection from the left
has been performed. Since this is no longer parallel distribution (it is ``sequential'' distribution), we would obtain an increased total waiting time.
However, the accumulated dephasing time may be reduced this way, as we discuss in the next subsection.
In general, we may also consider schemes with a memory cut-off, where we put a certain restriction of $m$ time units on the
maximum time a qubit can be kept in memory. So, in this case, we study four variables --- the total number of distribution steps and the total dephasing, both with and without cut-off. In order to maximize the secret key rate we need a scheme with small $\mathbf{E}[K_n]$ and large
$\mathbf{E}[e^{-\alpha D_n}]$. In the following subsections, we will introduce different schemes for performing
the entanglement swapping and, where
possible, compute the PGFs of the corresponding random variables. The PGF of $K_n$ is denoted as $G_n(t)$ and that of $D_n$ as
$\tilde{G}_n(t)$. For the corresponding quantities with cut-off $m$ we use the superscript $[m]$, e.g. $K^{[m]}_n$.
We will see and argue that there are three basic properties that a quantum repeater protocol (unassisted by additional
quantum error detection or correction) should exhibit: distribute the entangled states in each segment in parallel,
swap the initially distributed states as soon as possible, and avoid parallel storage of already distributed pairs
as much as possible. It is obvious that all these three ``rules'' cannot be fully obeyed at the same time.
In particular, parallel distribution will ultimately lead to some degree of parallel storage.
\subsection{Sequential distribution schemes}
In what we refer to as a sequential entanglement distribution scheme,
the initial, individual pairs are no longer distributed in parallel but
strictly sequentially according to a predetermined order.
If this order is chosen in a suitable way, it is possible that
at any time during the repeater protocol at most one entangled pair
is subject to dephasing (apart from small constant dephasing units
for single attempts), because once a new pair is available
an entanglement connection can be immediately performed and only then
another new segment starts distributing.
This may lead to a reduced accumulated dephasing time.
Moreover, from a secret key rate analysis point of view,
an appropriate sequential scheme can allow for a straightforward
calculation of the statistics of both random variables,
the total waiting and the accumulated dephasing times, even when a memory
cut-off is included.
Let us consider a simple, sequential distribution and swapping scheme where the
above discussion applies and the secret key rate can be computed exactly by means of elementary
combinatorics. In this scheme, we start by distributing entanglement in segment 1 (most left segment),
and only after a success we start to attempt distributions in segment 2.
As soon as we succeed there too, we immediately swap segments 1 and 2 and start to distribute entanglement in segment 3. As soon
as we succeed with the distribution in segment 3, we swap segment 3 with the first two, already connected segments, start distributing
in segment 4, and so on, repeating this process until entanglement has also been distributed in the most right segment
followed by a final entanglement swapping step.
This scheme, for $n=4$, is also illustrated by Fig.~\ref{fig:4segI}. The variables $K_n$ and $D_n$ for this scheme and general $n$
are thus defined as
\begin{equation}
K^{\mathrm{seq}}_n = N_1 + \ldots + N_n, \quad D^{\mathrm{seq}}_n = N_2 + \ldots + N_n.
\end{equation}
The PGFs of these random variables are just powers of the PGF of the geometric distribution:
\begin{equation}
G^{\mathrm{seq}}_n(t) = \left(\frac{pt}{1 - qt}\right)^n, \ \tilde{G}^{\mathrm{seq}}_n(t) = \left(\frac{pt}{1 - qt}\right)^{n-1}.
\end{equation}
In App.~\ref{app:SeqPGF} we derive the following expressions for the PGFs of the random variables with memory cut-off. We assume an accumulated, global cut-off where the total storage (dephasing) time across all segments must not exceed the value $m$. The
PGF of $K^{[m]}_n$ is given by
\begin{equation}\label{eq:Gmnt}
G^{[m]}_n(t) = \frac{p^n t^n \sum^{m-n+1}_{j=0}\binom{j+n-2}{n-2}q^j t^j}{1-qt-p\sum^{n-2}_{i=0}\binom{m}{i}p^i q^{m-i} t^{m+1}},
\end{equation}
and the PGF of $D^{[m]}_n$ becomes
\begin{equation}
\tilde{G}^{[m]}_n(t) = \frac{t^{n-1}\sum^{m-n+1}_{j=0} \binom{j+n-2}{n-2}q^j t^j}{\sum^{m-n+1}_{i=0}\binom{m}{i+n-1}p^i q^{m-n+1-i}}.
\end{equation}
Because it takes at least one time step for each segment to succeed, we have the inequalities $n \leqslant K^{[m]}_n$
and $n-1 \leqslant D^{[m]}_n \leqslant m$, which agree with the PGFs of these quantities presented above. Moreover, for
$m \to +\infty$ we have
\begin{equation}\label{eq:GGinf}
G^{[+\infty]}_n(t) = G^{\mathrm{seq}}_n(t), \quad \tilde{G}^{[+\infty]}_n(t) = \tilde{G}^{\mathrm{seq}}_n(t).
\end{equation}
These relations are easy to prove, just note that
\begin{equation}
\begin{split}
\sum^{m-n+1}_{i=0} &\binom{m}{i+n-1}p^i q^{m-n+1-i} \\
&= \frac{1}{p^{n-1}} \left[1 - \sum^{n-2}_{i=0}\binom{m}{i}p^i q^{m-i}\right].
\end{split}
\end{equation}
The binomial coefficient $\binom{m}{i}$ is polynomial in $m$ of $i$-th degree, and thus $\binom{m}{i} q^m \to 0$ when $m
\to +\infty$ for all $i = 0, \ldots, n-2$, which proves the relations of Eq.~\eqref{eq:GGinf}.
There are also variations of the above sequential cutoff scheme.
In the previous scheme we only abort a round when we already waited $m$ time units. Now consider the case where we already waited $m/2$ time units, but only a small number of segments succeeded. Hence, it is highly unlikely that we will succeed in all segments within the $m$ time steps. Therefore, it is better not to waste time and already abort the current round to start from scratch. A very simple strategy following this idea makes use of an individual (local) cutoff in each segment. However, it is beneficial to use a different cutoff in every segment; one should choose a smaller cutoff in the first segments and then increase the cutoff for later segments. The rationale behind this is that in the first segments we have not invested much effort and can discard rather aggressively, whereas later we should discard less aggressively since we already consumed lots of resources.
The advanced protocol is uniquely defined by a vector of cutoffs $\vec{m}=(m_1,\dots,m_{n-1})$ and the random variables $K_n$ and $D_n$ for this protocol and general $n$ are given by
\begin{equation}
K_{n}^{\mathrm{seq},\vec{m}}= \tilde{N}^{(m_{n-1})}+(T_{n-1}-1)m_{n-1}+\sum_{j=1}^{T_{n-1}} K_{n-1,j}^{\mathrm{seq},\vec{m}},
\end{equation}
where $K_1^{\mathrm{seq},\vec{m}}$ is geometrically distributed with parameter $p$, $\tilde{N}^{(m_{n-1})}$ follows a truncated geometric distribution with cutoff $m_{n-1}$, and $T_{n-1}$ is a geometric random variable with parameter $(1-q^{m_{n-1}})$ describing the number of starts of the protocol. For the dephasing we have
\begin{equation}
D_{n}^{\mathrm{seq},\vec{m}}=\tilde{N}^{m_1}+\ldots+\tilde{N}^{m_{n-1}}.
\end{equation}
The PGF of $K_{n}^{\mathrm{seq},\vec{m}}$ is calculated in App.~\ref{app:SeqPGF} and given recursively by
\begin{equation}
G^{[\vec{m}]}_n(t)=\tilde{G}_2^{[m_{n-1}]}(t)t^{-m_{n-1}}P^{(m_{n-1})}\left(G_{n-1}^{[\vec{m}]}(t)t^{m_{n-1}}\right),
\end{equation}
where $P^{(m)}(t)=\frac{(1-q^m)t}{1-q^m t}$ and $G^{[\vec{m}]}_1=G^{seq}_1$.
The PGF of $D_{n}^{\mathrm{seq},\vec{m}}$ is simply given by
\begin{equation}
\tilde{G}^{[\vec{m}]}_{n}(t)=\prod_{j=1}^{n-1} \tilde{G}^{[m_j]}_2(t)\,,
\end{equation}
since the sum of independent random variables translates to a product for PGFs.
As the state quality only depends on the total dephasing time, the best sequential protocol would count the total number of storage steps and would discard following a cutoff which is a function of the number of already succeeded segments, and one may also make use of the early aggressive discarding.
\subsection{Parallel distribution schemes}
A more efficient class of schemes is constructed when we do not wait for some segments to finish before we start others.
In these schemes we start all segments independently and distribute in parallel. It follows that for these schemes without cut-off we
have
\begin{equation}
K^{\mathrm{par}}_n = \max(N_1, \ldots, N_n),
\end{equation}
which means that all such schemes give the same raw rate. In App.~\ref{app:GKn} we derive the following expressions
for the PGF of $K_n$:
\begin{equation}\label{eq:GKn}
\begin{split}
G^{\mathrm{par}}_n(t) &= t\sum^n_{i = 1}(-1)^{i+1} \binom{n}{i}\frac{1-q^i}{1 - q^i t} \\
&= 1 + (1-t)\sum^n_{i=1} (-1)^i \binom{n}{i} \frac{1}{1-q^i t}.
\end{split}
\end{equation}
The two expressions are identical, since their difference reduces to $(1 - 1)^n = 0$. From the first expression it is clear that the values of $K_n$ start at 1, as it must be, because it takes at least one time unit to distribute
entanglement. In the other expression the necessary property of all PGFs becomes manifest, $G_n(1) = 1$. From the first relation
of Eqs.~\eqref{eq:PGF} we get the well-known expression for the average waiting time of a quantum repeater with parallel distribution and deterministic entanglement swapping (at any time when possible, e.g. at the very end)
\begin{equation}\label{eq:Knpar}
\overline{K^{\mathrm{par}}_n} = \frac{\mathrm{d}}{\mathrm{d}t}G^{\mathrm{par}}_n(t) \Big\vert_{t=1} = \sum^n_{i=1} (-1)^{i+1} \binom{n}{i} \frac{1}{1 - q^i},
\end{equation}
which has been obtained in Ref.~\cite{PhysRevA.83.012323} (but the full waiting time probability distribution has not). Importantly, however, all other relevant expressions, the total number of
distribution steps including memory cut-off as well as the finally distributed quantum state including memory imperfections, both for the model with and without memory cut-off, depend on the particular swapping
strategy chosen (e.g. unnecessarily postponing some or even all entanglement swapping steps until the very end maximizes the amount of parallel storage and hence the total dephasing in the final state). For this, there is a growing number of choices for larger repeaters, and in the following
we shall derive an optimal swapping scheme that results in a minimal total dephasing time
(while sharing the high raw rates, i.e. the minimal total waiting times, with all parallel distribution schemes).
\subsubsection{Optimal swapping scheme}\label{sec:optimalswapscheme}
Because all schemes (without cut-off) considered in this subsection have equal raw rates, the best secret key rate is
determined by the optimal scheme with regards to the secret key fraction.
In this subsection we shall present this scheme. In contrast to the
schemes presented in Figs.~\ref{fig:4segD} and \ref{fig:4segI}, which are fixed, the optimal swapping scheme is
dynamic. In a fixed scheme the order of swappings is fixed at the beginning and does not depend on the order in which
the segments become ready. For example, for the ``doubling'' scheme as shown in Fig.~\ref{fig:4segD} for $n=4$, we never swap
segments 2 and 3, even if they are ready and segments 1 and 4 are not. We always wait for segments 1 and 2 or segments 3
and 4 to become ready, swap these pairs, and then swap the larger segments to finish the entanglement distribution over
the whole repeater. In a dynamical scheme we do not follow a prescribed order and can swap the segments based on their
state. Of course, we can freely mix and match fixed and dynamic behaviours. For example, for $n=8$, we can first swap four
pairs of segments in a fixed way and then swap the four new, larger segments dynamically. We now show that the fully
dynamic scheme, where we always swap the segments that are ready, is the optimal one.
To prove this statement, we give two characterizations of this fully dynamic scheme. One is the straightforward
translation of the description to the definition, but this definition is not explicitly optimal. The other one is
optimal by construction, but it is not fully dynamic explicitly. We then show that the two constructions coincide, which
will demonstrate the validity of our statement.
Swapping an earliest pair of segments means that we choose an index $i$ for which $\max(N_i, N_{i+1})$ is minimal (there
can be several such indices), swap the pair of segments $i$ and $i+1$, and recursively apply this procedure to the other
segments (if there are several such pairs, choose one of them arbitrarily). If we denote the dephasing random variable
of this scheme as $\tilde{D}_n$, then its formal definition reads as
\begin{widetext}
\begin{equation}\label{eq:Dtilde}
\tilde{D}_n(N_1, \ldots, N_n) = |N_{i_0} - N_{i_0 + 1}|
+\tilde{D}_{n-1}(N_1, \ldots, N_{i_0-1}, \max(N_{i_0}, N_{i_0+1}), N_{i_0+2}, \ldots, N_n),
\end{equation}
\end{widetext}
where $i_0 = \argmin_i \max(N_i, N_{i+1})$. This definition is a greedy, locally optimal scheme, which optimizes only
one step. As it is known from algorithm theory, greedy algorithms do not always produce globally optimal results. By
doing only locally optimal steps, we may miss an opportunity for a much better reward in the future if we make a
non-optimal step now. Fortunately, in this case the greedy, locally optimal scheme expressed by Eq.~\eqref{eq:Dtilde}
does give the globally optimal result, as we show below.
In any scheme, the first step will be to swap a pair of neighbouring segments, let us say segments $i$ and $i+1$. We do
this at the time moment $\max(N_i, N_{i+1})$, and the contribution of these segments to the total dephasing is $|N_i -
N_{i+1}|$. After this swapping, we are left with $n-1$ new segments, one of which is the combination of two original
ones. Any initial segment $j$, where $j \not= i, i+1$, generates an entangled state after $N_j$ time units, and the
combined segment ``generates'' entanglement after $\max(N_i, N_{i+1})$ time units. If we swap these $n-1$ segments in
any way in $D_{n-1}$ time units, then the total swapping takes $D_n = |N_i - N_{i+1}| + D_{n-1}$ time units. To find the
minimal dephasing we simply take the minimum over $i = 1, \ldots, n-1$ of this expression, and recursively apply it for
the new segments. If we denote the dephasing random variable corresponding to this scheme as $D^\star_n$, then this
description translates into the following definition:
\begin{equation}\label{eq:Dstar}
\begin{split}
&D^\star_n(N_1, \ldots, N_n) = \min_{i = 1, \ldots, n-1}\Bigl[|N_i - N_{i+1}| \Bigr. \\
\Bigl.&+ D^\star_{n-1}(N_1, \ldots, N_{i-1}, \max(N_i, N_{i+1}), N_{i+2}, \ldots, N_n) \Bigr].
\end{split}
\end{equation}
The base case of this recursive definition is $D^\star_2(N_1, N_2) \equiv D_2(N_1, N_2) = |N_1 - N_2|$. This definition
by construction gives the globally minimal number of dephasing time units required to distribute long-distance entanglement if it takes $N_i$ time units for segment $i$ to generate entanglement.
We now have two quantities, the locally optimal one, given by Eq.~\eqref{eq:Dtilde}, and the globally optimal one, given
by Eq.~\eqref{eq:Dstar}. The former has semantics of swapping the earliest, but may not be globally optimal. The latter
is optimal by construction, but does not necessarily correspond to the swapping earliest strategy. It turns out that the
two quantities coincide, at least for all $n = 2, \ldots, 8$. A straightforward way to check this is to consider all
possible inequality relations between $N_i$. There are $n!$ such relations, which correspond to the permutations of
$N_i$ in the following inequality
\begin{equation}\label{eq:N1Nn}
N_1 \leqslant \ldots \leqslant N_n.
\end{equation}
For any given inequality relation between $N_i$ we can compute both quantities explicitly in terms of $N_i$. For
example, for the relation in Eq.~\eqref{eq:N1Nn} both quantities reduce to the same expression, $\tilde{D}_n = D^\star_n = N_n -
N_1$. For all other possible relations we have
\begin{equation}
\tilde{D}_n(N_1, \ldots, N_n) = D^\star_n(N_1, \ldots, N_n),
\end{equation}
for all $n = 2, \ldots, 8$. This can be easily verified with the help of a computer algebra system. Our conjecture is
that the statement is valid for all $n \geqslant 2$, but in this work we consider repeaters with up to eight segments only,
and for such $n$ we have verified this statement directly.
In contrast to the sequential scheme introduced earlier, there is no compact expression for the PGF of the optimal
scheme here. Each case will be considered separately in the next subsections. Where possible, we present explicit expressions
of the PGFs of the quantities in question. The main difficulty is encountered for those schemes with memory cut-off, and hence when including a cut-off, even for smaller repeaters (but $n>2$) we only consider the fully sequential scheme, for which we have got the exact expressions.
In the following subsections, we discuss quantum repeaters for $n=2$, $3$, $4$, and $8$ segments.
Although the case $n=2$ is rather well known and there is no set of different swapping strategies to choose from in this case, it will
be briefly reproduced based on the formalism introduced in this work.
The case $n=3$ is interesting, as it represents the simplest, nontrivial case beyond one middle station,
already requiring a choice regarding distribution and swapping strategies
(here, in the main text, the focus remains on schemes with an optimal dephasing for parallel distribution;
in App.~\ref{app:Optimality 3 segments}, we discuss the full secret key rate for $n=3$ including all possible distribution schemes).
Finally, the cases $n=4$ and $n=8$ are chosen, as they allow for a comparison with ``doubling''
(see Fig.~\ref{fig:4segD}). Larger quantum repeaters with $n>8$ become increasingly difficult to treat
(in terms of the optimized total dephasing).
We will later also see that for $n=8$, without additional methods of quantum error detection or correction,
the necessary experimental parameter values in our model become already highly demanding.
\subsubsection{Two-segment repeater}
This is the simplest kind of a quantum repeater. The PGF $G_2(t)$ of $K_2 = \max(N_1, N_2)$ is given by
Eq.~\eqref{eq:GKn} with $n=2$ and in this case reads as
\begin{equation}
G_2(t) = \frac{p^2 t (1 + qt)}{(1 - qt)(1 - q^2 t)}.
\end{equation}
As we noted before, there is only one choice for the dephasing variable, $D_2 = |N_1 - N_2|$ (parallel distribution). In Appendix~\ref{app:PGF Parallel schemes}, we derive
the following expression for the PGF of this variable:
\begin{equation}
\tilde{G}_2(t) = \frac{p^2}{1 - q^2} \frac{1 + q t}{1 - q t}.
\end{equation}
There we also show that the PGFs of the variables with cut-offs are
\begin{equation}
\begin{split}
G^{[m]}_2(t) &= \frac{p^2 t (1 + qt - 2(qt)^{m+1})}{(1 - qt)(1 - q^2 t - 2p (qt)^{m+1})}, \\
\tilde{G}^{[m]}_2(t) &= \frac{p}{1 + q - 2q^{m+1}} \frac{1 + qt - 2(qt)^{m+1}}{1 - qt}.
\end{split}
\end{equation}
It is obvious that we have the same consistency relations as for the sequential distribution scheme:
\begin{equation}
G^{[+\infty]}_2(t) = G_2(t), \quad \tilde{G}^{[+\infty]}_2(t) = \tilde{G}_2(t).
\end{equation}
\subsubsection{Three-segment repeater}\label{sssec:Par-distr: 3-segment repeater}
For three segments there are various ways how to distribute entanglement. One could use a fully sequential scheme, start at one end and distribute entanglement in concurrent segments. Alternatively, one could consider schemes where pairs of segments generate entanglement in parallel and the remaining segment goes last or, the other way around, it goes first. There are also combined distribution schemes with ``overlapping" parallel and sequential distributions. Finally, there are those schemes which attempt to generate entanglement in all segments at once and thereby use different swapping schemes. Among the latter here only the potentially optimal scheme is of interest, as it minimizes the accumulated dephasing, while having the same total waiting time as any other parallel distribution scheme.
However, it could still be the case that a scheme from the other, slower class of schemes performs better in terms of the full secret key rate. This is possible, because there is typically a trade-off between the raw rate and the dephasing or, more generally, the QBER. In particular, the fully sequential distribution scheme is interesting, since its total dephasing becomes minimal, as there is basically always only one segment waiting at every time step. On the other hand, for the fully parallel schemes the raw rate is optimal.
In App.~\ref{app:Optimality 3 segments} we present all possible schemes for $n=3$ and calculate the PGFs of their total waiting and dephasing times. Then we use these results to obtain the secret key rate for each scheme and to compare the different schemes. We also show in the appendix that the PGF of the optimal dephasing random variable, equivalently defined by Eqs.~\eqref{eq:Dtilde} and \eqref{eq:Dstar}, reads as
\begin{equation}
\tilde{G}^\star_3(t) = \frac{p^3}{1-q^3} \frac{1 + (q+2q^2)t - (2q^2+q^3)t^3 - q^4 t^4}{(1-qt)(1-q^2t)(1-qt^2)}.
\end{equation}
It turns out that with regards to the full secret key rate the parallel-distribution optimal-dephasing scheme is indeed optimal in all relevant regimes and especially in the limit of improving hardware parameters, which can be seen in Fig.~\ref{fig:Comparison_3_segments_non tau=0.1} and Fig.~\ref{fig:Comparison_3_segments_non tau=10} for two different memory coherence times. In the same section one can also find a more detailed discussion of the figure. In addition, aiming at the most general treatment of the $n=3$ case, we also consider the scenario where Alice and Bob measure their qubits immediately, thus suppressing their memory dephasing, and we apply this to all possible schemes. The comparison of these ``immediate-measurement" schemes is shown in Fig.~\ref{fig:Comparison_3_segments_immediate tau=0.1} and Fig.~\ref{fig:Comparison_3_segments_immediate tau=10}, again for two different coherence times. The conclusion remains the same: overall ``optimal" is optimal.
However, note that the option with immediate measurements for Alice and Bob only exists when they operate the quantum repeater for the purpose of long-range QKD. More advanced quantum repeater applications may require quantum storage for the qubits at each end (user) node. In any case, the memory qubits at each intermediate repeater node are (jointly) measured as soon as possible when the two adjacent segments are filled with an entangled pair (or even later, depending on the particular swapping strategy, but in App.~\ref{app:Optimality 3 segments} we only consider swap-as-soon-as-possible schemes that minimize the dephasing).
The above discussion leads us to the conclusion that there are three basic properties that a quantum repeater protocol (unassisted by additional quantum error detection or correction) should exhibit: distribute the entangled states in each segment in parallel, swap the initially distributed states as soon as possible, and avoid parallel storage of already distributed pairs as much as possible. It is obvious that all these three``rules" cannot be fully obeyed at the same time. However, our optimal scheme has the optimal balance with regards to these rules for three segments. We conjecture that this also holds true for larger $n>3$-segment repeaters.
\subsubsection{Four-segment repeater}\label{sssec:Par-distr: 4-segment repeater}
Of particular interest to us is the case of a four-segment repeater which is commonly operated via ``doubling". Here we are now able to discuss more general schemes, especially those that would always swap as soon as possible, unlike doubling where the second and third segments may not be immediately connected even when they are both ready. Overall there are many more schemes than in the previous $n=3$ case, and here for $n=4$ we focus on the parallel-distribution schemes. All these schemes (without cut-off) have identical $K_4 = \max(N_1, N_2, N_3, N_4)$, whose PGF is given by Eq.~\eqref{eq:GKn}
for $n=4$. The dephasing variable $D_4$ and its PGF become different for different schemes. One such scheme, the common ``doubling", is illustrated in
Fig.~\ref{fig:4segD}, where we first swap the pairs of segments 1, 2 and 3, 4 independently and then swap the two larger
segments. Note that the swappings will typically take place at different
moments in time - one pair of segments will usually swap earlier than the other. The state of the faster pair that goes into the final swapping operation is the state of these segments after their connection and at the moment when the final swapping is done, and so the state has been subject to a corresponding memory dephasing. For example, if the swapping of segments
1 and 2 is done first, the state of the distributed state over segments 1 and 2 just after the swapping is
$\hat{\varrho}_{14} = \mathcal{S}(\hat{\varrho}_{12} \otimes \hat{\varrho}_{34})$. If $k$ time units later segments 3
and 4 swap, producing the state $\hat{\varrho}_{58} = \mathcal{S}(\hat{\varrho}_{56} \otimes \hat{\varrho}_{78})$, the
former state becomes $\Gamma_{k\alpha}(\hat{\varrho}_{14})$, and the state distributed over the whole repeater is
\begin{equation}\label{eq:Sk}
\hat{\varrho}_{18} = \mathcal{S}(\Gamma_{k\alpha}(\mathcal{S}(\hat{\varrho}_{12} \otimes \hat{\varrho}_{34})) \otimes
\mathcal{S}(\hat{\varrho}_{56} \otimes \hat{\varrho}_{78})),
\end{equation}
instead of just $\hat{\varrho}_{18} = \mathcal{S}(\mathcal{S}(\hat{\varrho}_{12} \otimes \hat{\varrho}_{34}) \otimes
\mathcal{S}(\hat{\varrho}_{56} \otimes \hat{\varrho}_{78}))$. Again, as before, we omitted any extra factors that depend on the number of spins subject to dephasing in a single repeater segment. So, Fig.~\ref{fig:4segD} shows just a workflow of swapping
operations, while the exact expressions should be adjusted according to the respective time differences. The dephasing variable $D_4$ in
this doubling scheme is defined as follows:
\begin{equation}
\begin{split}\label{eq:doublingvariable}
D^{\mathrm{dbl}}_4 &= |N_1 - N_2| + |N_3 - N_4| \\
&+ |\max(N_1, N_2) - \max(N_3, N_4)|.
\end{split}
\end{equation}
The first two terms are due to the possible time difference for generating entangled states within each pair of
segments. The last term is due to the time difference between the pairs (e.g. the difference of the two maxima is $k$ time steps in
Eq.~\eqref{eq:Sk}). Note that this particular form of $D^{\mathrm{dbl}}_4$ is consistent with the commonly used "doubling" where the initial distributions happen in parallel, but the swapping strategy is fixed and sometimes disallows to swap as soon as possible. In Appendix~\ref{app:PGF Parallel schemes}, we derive the PGF of this random dephasing variable,
\begin{equation}
\tilde{G}^{\mathrm{dbl}}_4(t) = \frac{p^4}{1-q^4} \frac{P^{\mathrm{dbl}}_4(q, t)}{Q^{\mathrm{dbl}}_4(q, t)},
\end{equation}
where the numerator and denominator are given by
\begin{displaymath}
\begin{split}
P^{\mathrm{dbl}}_4(q, t) &= 1 + (q^2+3q^3)t + (3q+3q^2-q^5)t^2 \\
&- (q^3-q^5)t^3 + (q^3-3q^6-3q^7)t^4 \\
&- (3q^5+q^6)t^5 - q^8t^6, \\
Q^{\mathrm{dbl}}_4(q, t) &= (1-q^2t)(1-q^3t)(1-qt^2)(1-q^2t^2).
\end{split}
\end{displaymath}
The dephasing variable corresponding to the iterated scheme as shown in Fig.~\ref{fig:4segI} differs from that of the doubling
scheme. In the iterative scheme we first distribute entanglement over segments 1 and 2, then extend it over segment 3,
and finally over segment 4. Note that the figure can be understood to illustrate both sequential distribution and iterated swapping. In the
sequential distribution scheme, we would start to generate entanglement in each segment only when all previous segments (e.g. from left to right) have successfully generated entanglement. In the iterated swapping scheme, all segments may start simultaneously (parallel distribution), thus increasing the chances to swap sooner, but also the number of qubits potentially stored in parallel. The variable
$D^{\mathrm{itr}}_4$ for this scheme is
\begin{displaymath}
\begin{split}
D^{\mathrm{itr}}_4(N_1, N_2, N_3, N_4) &= |N_1 - N_2| + |\max(N_1, N_2) - N_3| \\
&+ |\max(N_1, N_2, N_3) - N_4|.
\end{split}
\end{displaymath}
The PGF of this random variable is rather large and reads as
\begin{equation}
\tilde{G}^{\mathrm{itr}}_4(t) = \frac{p^4}{1-q^4} \frac{P^{\mathrm{itr}}_4(q, t)}{Q^{\mathrm{itr}}_4(q, t)},
\end{equation}
where the numerator and denominator are given by
\begin{displaymath}
\begin{split}
P^{\mathrm{itr}}_4(q, t) &= 1+3q^3t+(4q^2-q^4-2q^5)t^2 \\
&+(q-q^2-3q^3-6q^4+2q^5+q^6)t^3\\
&+(-2q^2-5q^3+q^4+2q^5-q^6-3q^7)t^4\\
&+(-2q^2+4q^4-4q^6+2q^8)t^5\\
&+(3q^3+q^4-2q^5-q^6+5q^7+2q^8)t^6 \\
&+(-q^4-2q^5+6q^6+3q^7+q^8-q^9)t^7\\
&+(2q^5+q^6-4q^8)t^8-3q^7t^9-q^{10}t^{10}, \\
Q^{\mathrm{itr}}_4(q, t) &= (1-qt)(1-q^2t)(1-q^3t)(1-qt^2)\\
&\times (1-q^2t^2)(1-qt^3).
\end{split}
\end{displaymath}
We present an example for another, mixed swapping strategy
in App.~\ref{app:mixedstr}.
For the dephasing random variable $D^\star_4$, corresponding to the optimal swapping scheme given by
Eq.~\eqref{eq:Dstar} for $n=4$, we derive the following PGF:
\begin{equation}
\tilde{G}^\star_4(t) = \frac{p^4}{1-q^4} \frac{P^\star_4(q, t)}{Q^\star_4(q, t)},
\end{equation}
where the numerator and denominator read as
\begin{displaymath}
\begin{split}
P^\star_4(q, &t) = 1 + (q+2q^2+3q^3)t + (q+2q^2+q^4)t^2 \\
&-(3q^2+4q^3+4q^4)t^3 - (4q^5+4q^6+3q^7)t^4 \\
&+ (q^5+2q^7+q^8)t^5 + (3q^6+2q^7+q^8)t^6 + q^9t^7, \\
Q^\star_4(q, &t) = (1-qt)(1-q^2t)(1-q^3t)(1-qt^2)(1-q^2t^2).
\end{split}
\end{displaymath}
\subsubsection{Eight-segment repeater}\label{sssec:Par-distr: 8-segment repeater}
As before, again all parallel-distribution schemes (without cut-off) have identical total waiting times,
$K_8 = \max(N_1, \ldots, N_8)$, whose PGF is given by
Eq.~\eqref{eq:GKn} for $n=8$. For the dephasing variable there are many more possibilities now. We shall consider and compare five
different schemes -- the doubling and the optimal schemes, and three less important schemes, which nevertheless exhibit
an interesting behavior. The somewhat less important ones
are described and discussed in App.~\ref{app:mixedstr}.
The optimal dephasing $D^\star_8$ is defined equivalently by
Eqs.~\eqref{eq:Dtilde}-\eqref{eq:Dstar} for $n=8$ and the doubling dephasing $D^{\mathrm{dbl}}_8$ is defined recursively
as
\begin{equation}
\begin{split}
D^{\mathrm{dbl}}_8&(N_1, \ldots, N_8) = D^{\mathrm{dbl}}_4(N_1, \ldots, N_4) \\
&+ D^{\mathrm{dbl}}_4(N_5, \ldots, N_8) \\
&+ |\max(N_1, \ldots, N_4) - \max(N_5, \ldots, N_8)|,
\end{split}
\end{equation}
with $D^{\mathrm{dbl}}_4$ defined as in Eq.~\eqref{eq:doublingvariable}.
The comparison of the five different schemes can be found in App.~\ref{app:mixedstr}.
In this appendix, App.~\ref{app:mixedstr}, we present some figures
showing the ratios between the average dephasing
of the four sub-optimal schemes and the optimal scheme,
with and without exponentiation.
We can then compare the relative positions of the curves in Fig.~\ref{fig:Ee} with those of the curves of the ratios
\begin{equation}\label{eq:r3}
\frac{\mathbf{E}[D^{\mathrm{sch}}_8]}{\mathbf{E}[D^{\mathrm{opt}}_8]} =
\frac{\tilde{G}^{\mathrm{sch}\prime}_8(1)}{\tilde{G}^{\mathrm{opt}\prime}_8(1)},
\end{equation}
which are shown in Fig.~\ref{fig:Ea}. Looking at the two figures, we see that
\begin{equation}
\mathbf{E}[D^{\mathrm{dbl}}_8] > \mathbf{E}[D^{44}_8], \quad
\mathbf{E}[e^{-\alpha D^{\mathrm{dbl}}_8}] < \mathbf{E}[e^{-\alpha D^{44}_8}].
\end{equation}
This behavior is in full agreement with the properties of the exponential function: if $x > y \geqslant 0$ and $\alpha >
0$, then $e^{-\alpha x} < e^{-\alpha y}$. But for the other pair of schemes we have
\begin{equation}\label{eq:DD}
\mathbf{E}[D^{242}_8] > \mathbf{E}[D^{2222}_8], \quad
\mathbf{E}[e^{-\alpha D^{242}_8}] > \mathbf{E}[e^{-\alpha D^{2222}_8}].
\end{equation}
Nonetheless there is no contradiction here. This is a known property of nonlinear functions of random variables. This property can be
observed even in the simplest case of random variables $X$ and $Y$ each taking two values only. One can easily construct
an example such that $\mathbf{E}[X] > \mathbf{E}[Y]$ and $\mathbf{E}[e^{-\alpha X}] > \mathbf{E}[e^{-\alpha Y}]$. However,
the inequalities \eqref{eq:DD} show that it is not necessary to consider artificial constructions. This property can be
observed for simple and natural schemes.
The important conclusion is that the optimal scheme by construction minimizes $\mathbf{E}[D]$, but to have the highest fidelity of the
distributed state we need to maximize $\mathbf{E}[e^{-\alpha D}]$. For an ordinary nonnegative function $f(x)$ and a
positive parameter $\alpha > 0$ the minimum of $f(x)$ is the maximum of $e^{-\alpha f(x)}$ and vice versa, but for
random variables this is not necessarily true. Strictly speaking, in general, we know only the scheme that minimizes
$\mathbf{E}[D]$, but not the scheme that maximizes $\mathbf{E}[e^{-\alpha D}]$. The two schemes seem to be identical, but
there is no strict proof of this statement. We have to rely on evidence based on computing the properties of some schemes
explicitly and comparing them. For the examples for $n=8$ given in this section
and in the appendix, we see that dividing the exponentiated dephasing of
all other schemes by that of the optimal scheme gives a number smaller than one,
whereas the same ratios without exponentiation give a number greater than one.
Thus, minimal dephasing corresponds to minimal dephasing errors,
and the optimal dephasing scheme exhibits the smallest fraction of dephasing errors.
To summarize, our optimization of the secret key rates obtainable
with different distribution and swapping strategies is based
on three steps. First, we can rely upon the proof of the minimal dephasing variable for up to $n=8$ segments given in Sec.~\ref{sec:optimalswapscheme}
assuming parallel initial distributions
(it is already non-trivial to extend this proof to larger $n>8$).
Second, in order to compare the average dephasing errors in the final density operators, we need to consider the average dephasing exponentials for the different schemes. Finally, in order to assess the optimality of the secret key rate over all possible schemes, we have to take into account also those schemes where the initial distributions no longer occur in parallel which generally leads to smaller raw rates, but at the same time can result in a smaller dephasing by (partially) avoiding parallel storage. For the first non-trivial case beyond a single middle station, we have explicitly gone through all these three steps, namely for the case of a three-segment repeater with two intermediate stations (App.~\ref{app:Optimality 3 segments}), and found that ``optimal" is optimal. For larger repeaters beyond eight segments, $n>8$, we conjecture that our ``optimal" scheme also gives the best secret key rate. This includes conjecturing that our minimized dephasing is minimal also for $n>8$, that it minimizes the dephasing errors in the final density operator, and that overall the dephasing-optimized parallel-distribution approach is superior to any partially or fully sequential distribution scheme. Especially the last point cannot be taken for granted. In App.~\ref{app:8segmentsimmediate} we present some rate calculations for $n=8$ where, beyond a certain distance, ``optimal" can be beaten by a sequential scheme. However, there we allow for immediate measurements at an end node only for the sequential scheme (for which this is easy to include), but not for ``optimal"; a comparison which is slightly unfair and also only relevant for QKD applications. In the case of non-immediate-measurement schemes including potential beyond-QKD applications, ``optimal" remains optimal.
\section{Secret key rate analysis}\label{sec:Secret Key Rate}
A useful and practically relevant figure of merit for quantifying a quantum repeater's performance is its secret key rate in long-range QKD, which determines the amount of secret key generated in bits per channel use or second. As briefly reviewed in Sec.~\ref{sec:skr}, the secret key rate consists of two parts: the raw rate or yield and the secret key fraction. The former quantifies how long it takes to send a raw quantum bit or to (effectively) generate entanglement, independent of the quality of the final state; the latter then determines the average amount of secret key that can be extracted from a single raw bit, depending on the particular QKD protocol chosen and including the corresponding procedures for the classical post-processing.
Here we will focus on the asymptotic BB84 secret key rate $S=Rr=r/T$ with one-way post-processing. In the most general scenario of long-range memory-assisted QKD, i.e.
including a finite swapping probability $a$ and a memory cut-off parameter $m$,
this secret key rate is given by
\begin{equation}\label{eq:secret key rate}
S(p,a,m)=\frac{1-h(\overline{e_x}(p,a,m))-h(\overline{e_z}(p,a,m))}{T(p,a,m)},
\end{equation}
where \(h\) is the binary entropy function, \(T\) the average number of steps needed to successfully distribute long-distance entanglement, and \(e_x\), \(e_z\) are the QBERs of Eq.~\eqref{eq:QBER}.
The probability of successful entanglement generation in a single attempt in a single elementary segment is $p$, as introduced in Sec.~\ref{sec:rawrate}.
The denominator of $S$, $T = \mathbf{E}[K]$, is basically the total raw waiting time of the repeater which generally depends on $p$ and $a$ where $a$ is a finite success probability of the entanglement swapping using the same notation
as in Refs.~\cite{PvL,Shchukin2021} (where it was shown how to compute \cite{PvL} and optimize \cite{Shchukin2021}
$T=\mathbf{E}[K]$ for arbitrary $a$).
The dependency on the cut-off parameter $m$ means: the smaller $m$ becomes, the longer it takes to distribute an entangled state. The numerator of $S$, $r$, generally also depends on $p$, $a$, and $m$ through the QBERs.
Recall that we have to take the averages here, $\overline{e_z} = e_z$ and $\overline{e_x}$ obtainable via $\mathbf{E}[e^{-\alpha D_n}]$.
A smaller $m$ can lead to a higher state quality with a smaller total dephasing and thus to a larger secret key fraction $r$. It is generally hard to optimize $S$ over general $p$, $a$, and $m$. Our approach here is based on the simplifying (and experimentally still relevant) assumption $a=1$ (deterministic entanglement swapping) and the idea that the highest secret key rates will be obtainable with the fastest schemes (parallel distributions minimizing the total waiting time) and, among these, with those that swap entanglement as soon as possible (minimizing the total dephasing time, see Sec.~\ref{sec:optimalswapscheme}). While for a two-segment repeater the cases of deterministic and non-deterministic swapping can be treated similarly, for repeater chains with more than a single middle station ($n>2$) our results for optimizing distribution and swapping strategies only hold for the deterministic swapping case. Using the results of all previous sections the secret key rate can then be calculated. Therefore, in what follows we always have $a=1$.
The above secret key rate $S$ is expressed in terms of bits per channel use. For a rate per second, the average total number of distribution attempts $T$ must by multiplied with the duration of a single attempt in seconds, i.e. the elementary time unit $\tau = L_0/c_f$. Note that a single attempt or channel use is uniquely defined only for direct channel transmission in a point-to-point link, whereas the channel in a quantum repeater is used directly only between neighboring memory stations. Since our model always assumes that the interfaces at each station connect a single channel (to the left or to the right) with a single memory qubit (unit memory ``buffer"), those channel segments that belong to already successfully distributed pairs remain unused until new attempts in these segments will be started (e.g. when the memory cut-off has been exceeded or when a long-distance pair has been finally created). Nonetheless, at every attempt, we shall always count a full channel use over the entire distance despite the growing number of unused channel segments during memory-assisted long-distance entanglement distribution. Thus, strictly speaking, we underestimate the secret key rate per channel use and one could continue distributing pairs in all channel segments provided sufficient memory qubits are available.
The parameter values as given in Tab.~\ref{tab:constants} have been used to obtain the quantitative results discussed in this section.
Most parameters there have been introduced in the previous sections in the context of our physical model.
The resulting probability to distribute entanglement over one link in terms of the parameters of Tab.~\ref{tab:constants} now includes a
zero-distance link-coupling efficiency
\begin{equation}
p(L_0)=p_{\mathrm{link}}\cdot e^{-\frac{L_0}{L_{\mathrm{att}}}},
\end{equation}
with $p(0) = p_{\mathrm{link}}$ and where $p_{\mathrm{link}} = \eta_\mathrm{c} \cdot \eta_\mathrm{d} \cdot \eta_\mathrm{p}$
incorporates various efficiencies of the experimental hardware independent of the channel transmission itself, especially wavelength conversion, fiber coupling, preparation, and detector efficiencies.
\begin{table*}
\begin{tabular}{c|c|c|c}
Constant & Meaning & Current value & Improved value \\
\hline
\hline
$a$ & swapping probability & $1$ & $1$\\
$\tau_{\mathrm{coh}}$ & coherence time & $\unit[0.1]{s}$ & $\unit[10]{s}$ \\
$\mu$ & gate depolarisation (Bell measurement) & $0.97$ & $1$ \\
$\mu_0$ & initial state depolarisation & $0.97$ & $1$ \\
$F_0$ & initial state fidelity (dephasing) & $1$ & $1$ \\
$L_{\mathrm{att}}$ & attenuation length & $\unit[22]{km}$ & $\unit[22]{km}$ \\
$n_\mathrm{r}$ & index of refraction & $1.44$ & $1.44$ \\
$\eta_\mathrm{p}$ & preparation efficiency & * & * \\
$\eta_\mathrm{c}$ & \begin{tabular}{@{}c@{}}
photon-fibre coupling efficiency $\times$\\
wavelength conversion\\
\end{tabular} & * & *\\
$\eta_\mathrm{d}$ & detector efficiency & * & * \\
\hline
$p_{\mathrm{link}}:=\eta_\mathrm{c} \cdot \eta_\mathrm{d} \cdot \eta_\mathrm{p}$ & total efficiency & $0.05$ & $0.7$
\end{tabular}
\caption{Experimental parameter values used to calculate secret key rates. The star symbols * allow for various choices. The exact choices vary for each experimental platform. Some of the ``improved values" are the ideal values which allow to consider idealized, fundamental scenarios such as ``channel-loss-only" or ``channel-loss-and-memory-dephasing-only" (for which we may also set $p_{\mathrm{link}}=1$).}
\label{tab:constants}
\end{table*}
In the context of our statistical and physical model
the memory coherence time \(\tau_{\mathrm{coh}}\) in Tab.~\ref{tab:constants},
an experimentally determined parameter that describes the average speed of the memory dephasing, can be converted into a (dimensionless) effective coherence time in units of the repeater's elementary time unit, $\tau_{\mathrm{coh}}/\tau$. Equivalently, we can say that the (number of) dephasing time (steps) $D_n$ is to be multiplied with an elementary time $\tau$ before it can be divided by $\tau_{\mathrm{coh}}$ in $\mathbf{E}[e^{-D_n \tau/\tau_{\mathrm{coh}}}]$. In any case, we absorb both $\tau$ and $\tau_{\mathrm{coh}}$ in our dimensionless $\alpha$ dephasing parameter,
\begin{equation}
\alpha(L_0)=\frac{\tau}{\tau_{\mathrm{coh}}}=\frac{L_0}{c_f \tau_{\mathrm{coh}}}.
\end{equation}
Thus, $\alpha$ can be referred to as an inverse effective coherence time.
Note that in order to count the dephasing times appropriately in a specific protocol, we may have to add an extra factor of 2 (depending on the number of spins dephasing at each time step in a certain elementary or extended segment) and a constant dephasing term $\sim 2n$ that takes into account memory dephasing that occurs even when the first distribution attempt in a segment succeeds.
Any missing factors in the dephasing can be reinterpreted in terms of $\alpha$ or $\tau_{\mathrm{coh}}$, e.g. a missing factor of 2 corresponds to a coherence time twice as big.
In Tab.~\ref{tab:constants}, two sets of current and improved parameter values are listed, which specifically refer to $\tau_{\mathrm{coh}}$ and $p_{\mathrm{link}}$ for which we choose 0.1s or 10s and 0.05 or 0.7, respectively. The other state and gate fidelity parameters will be either set to unity or close to but below one
(in some of the following plots we will also treat them as a free parameter).
We will see that in memory-assisted QKD without additional quantum error detection or correction, the fidelity parameters must always be above a certain threshold value which (obviously) grows with the number of stations (and which generally depends on the particular QKD protocol and the classical post-processing method).
To compare the performance of each repeater protocol with a direct point-to-point link over the total distance $L$, we will use the PLOB bound \cite{PLOB}, which is given by
\begin{equation}
S^{\mathrm{PLOB}}(L)=-\log_2(1-e^{-\frac{L}{L_{\mathrm{att}}}}).
\end{equation}
It represents an upper bound on the number of secret bits that can be shared per channel use. For example, for $e^{-\frac{L}{L_{\mathrm{att}}}}=1/2$ corresponding to $L=15$km, we have $S^{\mathrm{PLOB}} = 1$, and so at most one secret bit can be distributed per channel use (per mode) independent of the optical encoding.
It will also be useful to consider an upper bound on the number of secret bits that can be shared with the help of a quantum repeater \cite{PLOB_QR},
\begin{equation}
S^{\mathrm{PLOB,QR}}(L_0)=-\log_2(1-e^{-\frac{L_0}{L_{\mathrm{att}}}}),
\end{equation}
corresponding to the PLOB bound for one segment (in the case of equal segment lengths $L_0$). For a point-to-point link, $n=1$ with $L=L_0$, we thus use the notation $S^{\mathrm{PLOB}}=S^{\mathrm{PLOB,QR}}$.
The rates we will focus on first in the following are to be understood as secret key rates per channel use. Later we shall also discuss secret key rates per second.
\subsection{Two-segment repeater}\label{sec:Two-Segment Repeater}
Let us start with the rates for the simplest case: a two-segment quantum repeater with one middle station. We shall only consider one scheme, the ``optimal" scheme, with and without a memory cut-off.
First, we address the question whether and when it is possible to overcome the PLOB bound with a two-segment repeater given the (current and improved) parameter values from Tab.~\ref{tab:constants}. We stick to \(F_0=1\) and, for illustrative clarity, we set \(\mu=\mu_0\) (while, first, $\mu$ is not fixed). Physically, this means that the repeater states when initially distributed in each segment and then manipulated at the middle station for the Bell measurement are subject to the same depolarizing error channels (and there is no extra initial dephasing). The cut-off parameter \(m\) is chosen most appropriately such that the final secret key rate is close to optimal over the entire range.
In Fig.~\ref{fig:Contour_2_segments} one can see various contour plots of the secret key rate. For convenience, we translated the error parameter \(\mu\) into a fidelity, $F = (3\mu + 1)/4$. The plots clearly indicate the minimal fidelity values below which the rates drop below the PLOB bound or even to zero rates, for different total repeater distances \(L\). The resulting contours are color-coded such that a particular color represents the secret key rate to be e.g. twice the rate of the PLOB bound. Thus, one can see that in certain parameter regimes it becomes impossible to beat the PLOB bound with a two-segment repeater. However, if both the memory coherence time $\tau_{\mathrm{coh}}$ and the link efficiency \(p_{\mathrm{link}}\) take on their improved values, it is possible to reach secret key rates as high as \(500\)-times the rate of the PLOB bound, and beyond, in a certain distance regime.
In Fig.~\ref{fig:SKR_2_segments}, we show the resulting secret key rates for the experimental parameters from Tab.~\ref{tab:constants}, for both the scheme with and without a memory cut-off. This time the error parameter \(\mu=\mu_0\) is fixed, and it either takes on its ``current" or its ``improved" (ideal) value. For comparison, as a reference, we also included the raw rates in each case. The loss scaling of the rates in all schemes is, as expected, proportional to $p_{\mathrm{link}} \,e^{-\frac{L}{2 L_{\mathrm{att}}}}=p_{\mathrm{link}}\sqrt{e^{-\frac{L}{L_{\mathrm{att}}}}}$ (corresponding to a linear decrease with distance due to the log scale representation). The effect of the different experimental parameter values is clearly visible. The choice of $p_{\mathrm{link}}=0.05$ or $p_{\mathrm{link}}=0.7$ determines the offset along the $y$-axis (rate axis) at zero distance. A higher $p_{\mathrm{link}}$ allows to cross the PLOB bound at a smaller distance. Note that the PLOB bound itself can arbitrarily exceed the value of one secret bit towards zero distance; in our schemes we always distribute qubits and so one secret bit per channel use is the maximum (and depending on the number of modes to encode the photonic qubits there could be extra factors, ``per mode"). The choice of $\tau_{\mathrm{coh}}=\unit[0.1]{s}$ or $\tau_{\mathrm{coh}}=\unit[10]{s}$ determines when (at which distance) the (negative) slope of the secret key rate increases such that the repeater switches from a $\sqrt{e^{-\frac{L}{L_{\mathrm{att}}}}}$ to a $e^{-\frac{L}{L_{\mathrm{att}}}}$ (PLOB-like) scaling, or even worse. This effect is an effect of the memory dephasing that occurs even when \(\mu=\mu_0=1\). If, in addition, \(\mu=\mu_0=0.97<1\), the secret key rates can drop abruptly down to zero, since then the QBERs have nonzero contributions both in $e_z$ and $e_x$, see Eq.~\eqref{eq:QBER}. Note that this effect happens also when either of the two parameters, $\mu$ or $\mu_0$, drop below one, i.e. when either the gates or the initial states become imperfect. Also note that non-unit $\mu$ or $\mu_0$ in addition lead to an increased $y$-axis offset which will become more apparent for larger repeaters with larger $n$.
However, a memory cut-off can significantly change the picture, and it can increase the achievable distance compared to the scheme without a cut-off (compare the solid yellow with the solid green curves in Fig.~\ref{fig:SKR_2_segments}). More specifically,
beyond distances when the rates of the no cut-off scheme drop dramatically, the cut-off scheme still scales proportional to the PLOB bound.
Note that for the scheme with cut-off, even the raw rates (dashed green curves) can switch from an $L/2$ to an $L$ scaling (like PLOB), because a finite cut-off value ``simulates" an imperfect memory in the raw rate (whose loss scaling resembles the scaling without a quantum memory, i.e. that of the PLOB bound, in the limit of $m=1$) \cite{CollinsPrl}.
Again, one can also see that with ``current" parameter values, see Fig.~\ref{fig:SKR_2_segments}(a), it is impossible to beat the PLOB bound
(here even when \(\mu=\mu_0=1\), see Fig.~\ref{fig:SKR_2_segments}(b)), but with improving values for the coherence time and the link efficiency, it becomes possible. This holds even when only one of the two parameters, $p_{\mathrm{link}}$ or $\tau_{\mathrm{coh}}$, is improved, as long as we can cross PLOB at a sufficiently small distance or maintain the repeater's slope for sufficiently long, respectively.
\begin{figure*}
\caption{Contour plots illustrating the minimal fidelity requirements to overcome the PLOB bound by a two-segment repeater for different parameter sets. In all contour plots, \(\mu = \mu_0\) and \(F_0=1\) has been used.}
\label{fig:Contour_2_segments}
\end{figure*}
\begin{figure*}
\caption{Rates (secret key or raw) for a two-segment repeater over distance \(L\) for different experimental parameters.}
\label{fig:SKR_2_segments}
\end{figure*}
In the next section we will turn to a four-segment repeater
(a three-segment repeater is discussed in great detail in App.~\ref{app:Optimality 3 segments}).
\subsection{Four-segment repeater}\label{sec:Four-Segment Repeater}
As we have seen in Sec.~\ref{sssec:Par-distr: 4-segment repeater}, there are various swapping strategies possible for a four-segment repeater in contrast to a simple two-segment repeater. Our conjecture is (see also App.~\ref{app:Optimality 3 segments} for the case $n=3$) that the ``optimal" scheme is optimal in the regimes of increasingly good hardware parameters. Thus, let us first again focus on the minimal fidelities to overcome the PLOB bound for this scheme, similar to our analysis for two segments, but now without cut-off only. The results are shown in Fig.~\ref{fig:Contour_4_segments}. It becomes apparent that now a much higher fidelity or equivalently \(\mu\) is needed, but in turn also much higher secret key rates, \(10^4\)-times the PLOB rate and beyond, are possible. Since we have $n=4$ now, non-unit $\mu$ values have a stronger impact on the QBERs, see Eq.~\eqref{eq:QBER}. At the same time, however, the loss scaling becomes proportional to $p_{\mathrm{link}} \,e^{-\frac{L}{4 L_{\mathrm{att}}}}=p_{\mathrm{link}}\sqrt[4]{e^{-\frac{L}{L_{\mathrm{att}}}}}$. Furthermore, note that a different scaling of the contours is observable. This effect is due to the lack of a memory cut-off.
\begin{figure*}
\caption{Contour plots illustrating the minimal fidelity requirements to overcome the PLOB bound by a four-segment repeater for different parameter sets. In all contour plots, \(\mu = \mu_0\) and \(F_0=1\) has been used.}
\label{fig:Contour_4_segments}
\end{figure*}
Next, we consider the secret key rates for a particular choice of the experimental parameters including $\mu = \mu_0$ according to Tab.~\ref{tab:constants}. Besides the ``optimal" scheme, now we also include the sequential and the doubling schemes in the rate analysis (sequential/iterative swapping together with sequential distributions and doubling with parallel distributions). In Fig.~\ref{fig:SKR_4_segments}, one can see the
PLOB bound and the secret key rates for the sequential scheme with and without a cut-off, for the doubling scheme and for the optimal scheme (both without a cut-off). In addition, again the raw rates are shown as a reference, and the corresponding three dashed curves are the raw rates for (equivalently) doubling and ``optimal", and for the sequential scheme with and without cut-off. Compared to the previous two-segment repeater, it is now easier to overcome the PLOB bound, but the crossing happens at longer distances, since the four-segment repeater starts with a lower rate at \(L=\unit[0]{km}\).
\begin{figure*}
\caption{Rates (secret key or raw) for a four-segment repeater over distance \(L\) for different experimental parameters.}
\label{fig:SKR_4_segments}
\end{figure*}
\subsection{Eight-segment repeater}\label{sec:Eight-Segment Repeater}
In comparison with the usual treatment of quantum repeaters via doubling the links at each repeater level, the next logical step is to consider an eight-segment repeater. For eight segments, there is an increasing number of possible distribution and swapping strategies, and for the swapping we have discussed this in more detail in Sec.~\ref{sssec:Par-distr: 8-segment repeater}. Here we will only consider the sequential, the doubling, and the optimal schemes (the former one with sequential distributions, the latter two with parallel distributions). Again, in Fig.~\ref{fig:Contour_8_segments}, we present limitations on the error parameter \(\mu\) to overcome the PLOB rate at different distances. The regions are color-coded as before. Compared to the limits observed for a two-segment repeater they exhibit a different behaviour now, but this is again due to the fact that we do not consider a cut-off scheme here. The requirements for the fidelity or \(\mu\) are higher, but this was expected, since the secret key fraction includes terms \(\propto \mu^{2n-1} \), again setting \(\mu_0=\mu\). Nevertheless, for sufficiently high fidelities, the attainable secret key rates are much higher than for any of the previously considered repeater schemes, becoming as high as \(10^8\)-times the rate of the PLOB bound, and beyond.
Finally, we have also evaluated the performance of an eight-segment repeater for our experimental parameter set. Now caution is required when these plots are compared directly with the previous ones, as we had to improve the ``current", non-unit value of \(\mu\) to \(\mu=0.99\). Without this fidelity adjustment, it would be impossible to achieve a non-zero secret key rate for an eight-segment repeater (see next section). The $\mu$-scaling with $n$ in the QBERs prohibits to scale up a realistic quantum repeater to arbitrarily large distances and $n$ values, as long as no extra elements for quantum error detection or correction are included. For example, in a 2nd-generation quantum repeater, the effective $\mu_0$ and $\mu$ values could be kept close to one, at the expense of extra resources for quantum error correction and a typically decreasing initial distribution efficiency $p$ (for instance, due to an extra step of entanglement distillation for the distributed, encoded memory qubits). In principle, our formalism could be also applied to such a more sophisticated scenario by considering the effective changes of $\mu$, $\mu_0$, and $p$ (and possibly $\alpha$ too).
Nevertheless, our plots presented in Fig.~\ref{fig:SKR_8_segments} show that an eight-segment quantum repeater in a memory-assisted QKD scheme is, in principle, already able to cover large distances by reaching usable rates up to
\(\unit[1000]{km}\) or even \(\unit[1200]{km}\), provided that \(\mu=0.99\) or \(\mu\rightarrow 1\), respectively. Apart from this, the behaviour of an eight-segment repeater is very similar to that of the previous four-segment repeater.
\subsection{Minimal $\mu$ values}
We have already seen that the secret key rate of memory-assisted QKD is highly sensitive to the depolarizing errors that we use to model the imperfect gates and the imperfect initial states in the quantum repeater. Here let us explicitly give some minimal values for the error parameter $\mu$ which at least have to be achieved in order to obtain a non-zero secret key fraction for QKD protocols restricted to one-way post-processing (see Tab.~\ref{tab_minimalmu}). More generally, in principle, much higher error rates can be tolerated by allowing for two-way post-processing in the QKD protocols \cite{twowayqkd}. However, in this work, we primarily utilize the secret key rate as a practical and useful quantitative figure of merit to assess a quantum repeater's performance. Nonetheless, the quantum repeater schemes that we consider may also be employed for other, more general quantum information and communication tasks. Thus, we decided not to include schemes with two-way post-processing, as this would certainly lead to a narrower specialization towards QKD applications. Clearly, in the context of long-range QKD, we believe that considering schemes with two-way post-processing will be very valuable, since potential, future large-scale quantum repeaters will be rather noisy and therefore protocols which still work for large error rates are very useful. Such a further optimization of our schemes with a special focus on long-range QKD is possible and we leave this option for future work.
It is easy to check that the concatenation of two depolarizing channels with parameters $\mu_1$ and $\mu_2$ is equivalent to a single depolarizing channel with parameter $\mu_1\mu_2$. Thus, for an $n$-segment repeater, we would expect a total depolarizing channel with parameter $\mu_n=\mu_0^n\mu^{n-1}$. We have carefully and systematically checked and confirmed this in the first part of the paper including other parameters too, such as constant initial and time-dependent memory dephasing.
For the BB84 and the six-state protocols, the amount of tolerable noise, such that a secret key can still be obtained with one-way post-processing, has been extensively studied. For BB84 the error threshold lies at $Q=11.0\%$ and for the six-state protocol it is $Q=12.6\%$ \cite[App. A]{RevModPhys.81.1301}. Since a maximally mixed state results in an error rate of $50\%$, this gives us a constraint on the minimal values of $\mu_n\geq1-2Q$.
More specifically, the BB84 secret key fraction of Eq.~\eqref{eq:skf} on which we focus here vanishes when the two QBERs both exceed $Q=11\%$. This is the case for $\mu_n < 1-2Q$ even when all other elements are perfect, i.e. even when there is no memory dephasing at all ($\alpha \rightarrow 0$). In this case, the two QBERs as described by Eq.~\eqref{eq:QBER} coincide (also assuming zero initial dephasing $F_0=0$) and neither includes a random variable. These two constant QBERs then express the sole faultiness of the repeater elements without any time-dependent quantum storage (i.e., only the initial states and the gates) which can suffice to prevent Alice and Bob from finally sharing a non-zero secret key.
\begin{table}[]
\begin{tabular}{l|l|l|l|l}
$n$ & \begin{tabular}{@{}c@{}}\quad $\mu_0=1$, \quad \\\quad BB84 \quad \end{tabular} & \begin{tabular}{@{}c@{}} \quad $\mu_0=\mu$, \quad \\ \quad BB84 \quad \end{tabular} & \begin{tabular}{@{}c@{}} \quad $\mu_0=1$, \quad \\ \quad 6-state \quad \end{tabular} & \begin{tabular}{@{}c@{}} \quad $\mu_0=\mu$, \quad \\ \quad 6-state \quad \end{tabular} \\\hline
2 & 0.780 & 0.920 & 0.748 & 0.908 \\\hline
4 & 0.920 & 0.965 & 0.908 & 0.959 \\\hline
8 & 0.965 & 0.984 & 0.959 & 0.981
\end{tabular}
\caption{Minimal values of $\mu$ required for a non-zero secret key rate in one-way post-processing protocols.}
\label{tab_minimalmu}
\end{table}
\begin{figure*}
\caption{Contour plots illustrating the minimal fidelity requirements to overcome the PLOB bound by an eight-segment repeater for different parameter sets. In all contour plots, \(\mu = \mu_0\) and \(F_0=1\) has been used.}
\label{fig:Contour_8_segments}
\end{figure*}
\begin{figure*}
\caption{Rates (secret key/raw) for an eight-segment repeater over distance \(L\) for different experimental parameters.}
\label{fig:SKR_8_segments}
\end{figure*}
\subsection{Comparisons}
\subsubsection{Sequential vs. doubling vs. optimal schemes}\label{sec:Comparison: Sequential vs. Doubling vs. Optimal scheme}
In the previous sections (together with the appendix) we have presented our results for the obtainable secret key rates of two-, three-, four- and eight-segment quantum repeaters based on various entanglement distribution and swapping strategies. While it is generally straightforward to include a memory cut-off for the case of two segments, for more than two segments, we have achieved this only for the fully sequential scheme.
This was depicted in green in the (non-contour) plots for four and eight segments.
The memory cut-off allows to maintain a scaling proportional to the PLOB bound even beyond the distance where the scheme without cut-off drops more quickly. As a consequence, the cut-off can significantly increase the achievable distance. However, it is hard to obtain an exact result for the secret key rate for the more complicated swapping strategies. Nonetheless, for larger distances, one could extrapolate the behaviour of the doubling and optimal schemes including a cut-off by simply continuing the curves with lines parallel to the PLOB bound after the drops. Alternatively, inferring from our plots, at larger distances one can rely on a continuation of the curves that behaves exactly like the sequential scheme with memory cut-off. Both approaches give us a fairly good picture of the behaviour of the doubling and optimal schemes including the cut-off.
Nevertheless, the optimal scheme outperforms all other schemes without a cut-off before each one drops completely. The doubling scheme achieves almost similar rates, although it starts earlier to decline. The secret key rates are similar thanks to the equivalent, high raw rates of the doubling and optimal schemes (both being based upon parallel entanglement distributions), and due to our general assumption of deterministic entanglement swapping with $a=1$ \cite{Shchukin2021} \footnote{for $a<1$, regimes exist where in terms of the raw rates ``doubling'' performs strictly worse than ``swap as soon as possible'' \cite{Shchukin2021}, similar to regimes here for the full secret key rates with $a=1$ when the dephasing becomes dominant.}.
Thus, for the doubling scheme one could additionally incorporate nested entanglement distillations in the usual, well-known way, which would allow to reduce the QBERs at the expense of the effective raw rates and with the need of extra physical resources.
While the differences between the doubling and optimal schemes may not be
so large for the repeater sizes mainly considered here ($n\leq 8$),
our exact statistical treatment enabled us to determine the optimal swapping scheme (optimizing the dephasing) and thus allows for a rigorous, quantitative comparison with the non-optimal doubling and possible other (including ``mixed") schemes.
The fully sequential scheme, based on sequential entanglement distributions, leads to the lowest raw rate. The longer total waiting times of this scheme also contribute to an increased accumulated dephasing. On the other hand, the dephasing of the fully sequential scheme remains limited, as only one segment is waiting at any time step. Thus, although theoretically the sequential scheme is the
easiest to calculate, experimentally it would typically result in the lowest secret key rate. Nonetheless, the fully sequential scheme is conceptually special and serves as a very useful reference for comparison with the other schemes.
\subsubsection{Two- vs. four- vs. eight-segment repeaters}\label{sec:Comparison: 2 vs. 4 vs. 8 segment repeaters}
In this section, let us finally address one of the main questions that motivates the exact secret key rate analysis that we have presented: is there an actual benefit of additional (memory) stations and repeater segments compared with schemes that work entirely without quantum memories (such as point-to-point links or twin-field QKD) or compared to schemes with a smaller number of memory stations? More specifically, is it useful to replace a simple two-segment repeater by a four- or eight-segment repeater in a realistic setting, i.e. even when the extra quantum memories are subject to additional preparation and operational errors and contribute to an increased accumulated memory dephasing?
In the preceding section with Tab.~\ref{tab_minimalmu} we saw that the sole faultiness of the memory qubit initial states and gates, even with no time- and distance-dependent memory dephasing, can make the secret key rate completely vanish, and this effect grows with the segment number $n$.
In the last section of the paper, we shall also look at schemes that minimize the actual number of memory stations by combining the twin-field QKD and repeater memory concepts, for instance, in a four-segment scheme with only one of the three intermediate stations being equipped with memory qubits.
Now here we only consider the ``optimal" scheme (generally and rigorously only without memory cut-off, as discussed before), since this ensures we always consider the highest possible secret key rates. By adding extra repeater stations the requirements on the initial state preparations and the Bell measurements become much higher, where the corresponding terms scale as \(\propto \mu^{n-1} \mu_0^n \) in the QBERs. We stress again that in order to achieve a non-zero secret key rate for the eight-segment repeater, we had to alter the non-ideal value of \(\mu\) of Tab.~\ref{tab:constants} to a sufficiently large value, \(\mu=0.99\), see also Tab.~\ref{tab_minimalmu}.
For a fair comparison, this value is then also used here to obtain the curves of the two- and four-segment repeaters.
\begin{figure*}
\caption{Comparison of secret key rates of the two-, four-, and eight-segment repeaters at total distances \(L\) for different experimental parameters.}
\label{fig:SKR_comparison}
\end{figure*}
The resulting secret key rates can be seen in Fig.~\ref{fig:SKR_comparison}. As one would expect, for example, the scaling
changes from \(\sqrt{e^{-\frac{L}{L_{\mathrm{att}}}}} \) to \(\sqrt[8]{e^{-\frac{L}{L_{\mathrm{att}}}}} \)
when the transition from a two-segment to an eight-segment repeater is considered.
However, the rate at \(L=\unit[0]{km} \) decreases when increasing the number of segments.
This effect occurs for the raw rates (and the secret key rates assuming $\mu=1$), but it becomes more apparent for $\mu=0.99$.
Still, at long distances, eight segments are superior to a smaller number of segments.
Therefore, acknowledging that the necessary $\mu$ requirements are extremely demanding but not entirely impossible to achieve in practice, we conclude that it is indeed beneficial to add repeater stations. In particular, the effect of the memory dephasing alone (besides channel loss), for possible coherence times like those in Tab.~\ref{tab:constants} and used throughout the plots, will not prevent the benefit of adding more stations.
Even when both $p_{\mathrm{link}}$ and $\tau_{\mathrm{coh}}$ take on their lowest of the two considered values as shown in Fig.~\ref{fig:SKR_comparison}(b), by placing seven memory stations along the channel it is in principle still possible to exceed the PLOB bound significantly. However, realistically, when $\mu<1$ like in Fig.~\ref{fig:SKR_comparison}(a), all secret key rates stay below the PLOB bound.
In this case it becomes crucial that either $p_{\mathrm{link}}$ (Fig.~\ref{fig:SKR_comparison}(c)) or $\tau_{\mathrm{coh}}$ (Fig.~\ref{fig:SKR_comparison}(e)) is sufficiently large such that the curves can cross PLOB at a sufficiently small distance (thanks to the small $y$-axis offset) or they can maintain their repeater loss scaling for sufficiently long distances, respectively. Recall that all rates shown and discussed here are per channel use.
Further it should be stressed here that we did not explicitly include time-dependent memory loss (assuming that the memory imperfections are dominated by the time-dependent memory dephasing), which can additionally jeopardise the benefits of adding more, in this case lossy memory stations \cite{PirEisert}. (If this loss is detectable it may lead to a non-deterministic entanglement swapping like in the ``DLCZ" quantum repeater, which is harder to accurately analyze and optimize even for a constant swapping probability \cite{Shchukin2021}; if the loss remains partially undetected at each station, it can lead to a reduced final state fidelity and thus an increased QBER.)
Let us discuss the comparison of repeaters with different segment numbers in a little more detail.
It is indeed quite subtle and for this we shall also take into account larger repeater systems, far beyond the $n=8$ case. For the general discussion, it is helpful to first consider the fully sequential scheme, as in this case we have access to all relevant (physical and statistical) quantities even for large repeaters, see Tab.~\ref{tab:seqperchanneluse}. If we only consider channel loss or, equivalently, if we only look at the raw rates, there is an optimal number of segments for a given total distance.
In Tab.~\ref{tab:seqperchanneluse}, among the possibilities considered there, this is $n=80$ for $L=800$km, and so we should put stations every $L_0=10$km.
If we include the memory dephasing (``channel-loss-and-memory-dephasing-only case"), we observe that not only the average (number of) waiting time (steps) $\mathbf{E}[K_n]$, but also the average (number of) dephasing time (steps) $\mathbf{E}[D_n]$ is minimized for $n=80$ when $L=800$km. In fact, these two averages, $n/p$ and $(n-1)/p$, respectively, become identical for larger $n$, and both grow in the two limits of many and very few segments, $L_0\rightarrow 0$ ($n\rightarrow \infty$) and $L_0\rightarrow L/2$ ($n\rightarrow 2$), respectively. However, when changing the segment length $L_0$, also the inverse effective coherence time $\alpha=L_0/(c_f \tau_{\mathrm{coh}})$ will change, where now $\alpha$ is simply maximal at $L_0=L/2$ and it steadily becomes smaller when $L_0\rightarrow 0$ at fixed $\tau_{\mathrm{coh}}$. Note that below a certain $L_0$ value the repeater's elementary time unit is no longer dominated by the classical communication times and instead the maximal local processing times must go into $\alpha$ which we refer to as $\alpha^{\mathrm{loc}}$. This effect implies that in order to maximize the effective coherence time $\tau_{\mathrm{coh}}/\tau$, one should simply use as many stations as possible, eventually approaching the limitation given by the local processing times at each station. For these we may typically assume
$\alpha^{\mathrm{loc}}_1=\tau/\tau_{\mathrm{coh}}={\rm MHz}^{-1}/0.1{\rm s}=0.00001$ and
$\alpha^{\mathrm{loc}}_2=\tau/\tau_{\mathrm{coh}}={\rm MHz}^{-1}/10{\rm s}=0.0000001$.
However, the first really relevant quantity to assess the effect of the memory dephasing is the effective average dephasing time $\alpha \mathbf{E}[D_n]$ that is related to the memory dephasing channel evolution. Interestingly, for the fully sequential scheme, this quantity, $\alpha \mathbf{E}[D_n]=(L/n)(n-1)/(c_f \tau_{\mathrm{coh}} p)$, converges for growing $n$ (small $L_0$) to $L/(c_f \tau_{\mathrm{coh}} p)$ with $p\rightarrow 1$. For example, in Tab.~\ref{tab:seqperchanneluse}, for $L=800$km, we have $L/(c_f \tau_{\mathrm{coh}} p)=0.0374$ for $\tau_{\mathrm{coh}}=0.1$s and $L/(c_f \tau_{\mathrm{coh}} p)=0.0004$ for $\tau_{\mathrm{coh}}=10$s. These limits are attainable for about $n=8000$ and for $n=800$, respectively. With $\tau_{\mathrm{coh}}=10$s the limit is also almost attainable for $n=80$, so again $L_0=10$km, and there is no further benefit by further increasing $n$. However, we also have $\alpha^{\mathrm{loc}}_1 \mathbf{E}[D_n]=0.00001\times (n-1)/p=0.0804$ for $n=8000$ and $\alpha^{\mathrm{loc}}_2 \mathbf{E}[D_n]=0.0000001\times (n-1)/p=0.0001$ for $n=800$.
\begin{table*}
\begin{tabular}{c|c|c|c|c|c|c|c}
$n$ & 1 & 2 & 4 & 8 & 80 & 800 & 8000 \\
\hline
\hline
$L_0$[km] & 800 & 400 & 200 & 100 & 10 & 1 & 0 \\
\hline
$\mathbf{E}[K_n]$ & $\sim 10^{16}$ & $\sim 10^{8}$ & 35497 & 754 & 126 & 837 & 8036 \\
\hline
$R$ & $\sim 10^{-16}$ & $\sim 10^{-8}$ & $\sim 10^{-5}$ & 0.0013 & 0.0079 & 0.0012 & 0.0001 \\
\hline
$\mathbf{E}[D_n]$ & - & $\sim 10^{8}$ & 26623 & 659 & 124 & 836 & 8035 \\
\hline
$\alpha_1$ & - & 0.0192 & 0.0096 & 0.0048 & 0.0005 & $\sim 10^{-5}$ & $\sim 10^{-6}$ \\
\hline
$\alpha_1 \mathbf{E}[D_n]$ & - & $\sim 10^{6}$ & 256 & 3.1674 & 0.0598 & 0.0402 & 0.0386 \\
\hline
$\alpha_2$ & - & 0.0002 & 0.0001 & $\sim 10^{-5}$ & $\sim 10^{-6}$ & $\sim 10^{-7}$ & $\sim 10^{-8}$ \\
\hline
$\alpha_2 \mathbf{E}[D_n]$ & - & 15131 & 2.5576 & 0.0317 & 0.0006 & 0.0004 & 0.0004 \\
\hline
$\mathbf{E}[e^{-\alpha_1 D_n}]$ & - & $\sim 10^{-7}$ & $\sim 10^{-6}$ & $0.0729$ & $0.9420$ & $0.9606$ & $0.9621$ \\
\hline
$\mathbf{E}[e^{-\alpha_2 D_n}]$ & - & $ \sim 10^{-6}$ & $0.1573$ & $0.9689$ & $0.9994$ & $0.9996$ & $0.9996$ \\
\hline
$r_1(\mu=1)$ & - & $\sim 10^{-13}$ & $\sim 10^{-12}$ & $0.0038$ & $0.8106$ & $0.8603$ & $0.8646$ \\
\hline
$r_2(\mu=1)$ & - & $\sim 10^{-9}$ & $0.0179$ & $0.8843$ & $0.9961$ & $0.9972$ & $0.9973$ \\
\hline
$r_1(\mu=0.99)$ & - & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
\hline
$r_2(\mu=0.99)$ & - & $0$ & $0$ & $0.2203$ & $0$ & $0$ & $0$ \\
\hline
$S_1(\mu=1)$ & - & $\sim 10^{-21}$ & $\sim 10^{-17}$ & $\sim 10^{-6}$ & $0.0064$ & $0.0010$ & $0.0001$ \\
\hline
$S_2(\mu=1)$ & - & $\sim 10^{-17}$ & $\sim 10^{-7}$ & $0.0012$ & $0.0079$ & $0.0012$ & $0.0001$ \\
\hline
$S_1(\mu=0.99)$ & - & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
\hline
$S_2(\mu=0.99)$ & - & $0$ & $0$ & $0.0003 $ & $0$ & $0$ & $0$ \\
\hline
$S^{\mathrm{PLOB,QR}}(L_0)$
& $\sim 10^{-16}$ & $\sim 10^{-8}$ & 0.0002 & 0.0154 & 1.4530 & 4.4921 & 7.7846
\end{tabular}
\caption{Overview of the relevant quantities for the {\it fully sequential scheme}: segment number $n$, segment length $L_0$[km], average (number of) waiting time (steps) $\mathbf{E}[K_n]$, raw rate $R$, average (number of) dephasing time (steps) $\mathbf{E}[D_n]$, inverse effective coherence time $\alpha_1=L_0/(c_f 0.1{\rm s})$, effective average dephasing time $\alpha_1 \mathbf{E}[D_n]$, inverse effective coherence time $\alpha_2=L_0/(c_f 10{\rm s})$, effective average dephasing time $\alpha_2 \mathbf{E}[D_n]$, average dephasing fractions $\mathbf{E}[e^{-\alpha_1 D_n}]$ and $\mathbf{E}[e^{-\alpha_2 D_n}]$, secret key fractions and rates, $r$ and $S$, for different $\mu=\mu_0$ (subscript corresponds to the choice of $\alpha_1$ or $\alpha_2$, $\mu=1$ is the channel-loss-and-memory-dephasing-only case), and the (repeater-assisted) capacity bound $S^{\mathrm{PLOB,QR}}(L_0)$.
We further assumed $p_{\mathrm{link}}=F_0=1$ for the link coupling efficiency and the initial state dephasing.}
\label{tab:seqperchanneluse}
\end{table*}
\begin{table*}
\begin{tabular}{c|c|c|c|c|c|c|c}
$n$ &1 & 2 & 4 & 8 & 80 & 800 & 8000\\
\hline
\hline
$L_0$[km] &800 & 400 & 200 & 100 & 10 & 1 & 0.1\\
\hline
$ \mathbf{E}[K_n] $ & $ \sim 10^{16} $ & $ \sim 10^{8} $ & $ 18487$ & $ 255$ & $ 5.4 $ & $ 2.9 $ & $ 2.2 $\\
\hline
$R $ & $ \sim 10^{-16} $ & $ \sim 10^{-8} $ & $ \sim 10^{-5} $ & $ 0.0039$ & $ 0.1841 $ & $ 0.3490 $ & $ 0.4646 $\\
\hline
$\mathbf{E}[D_n]$ & - & $ \sim 10^{8} $ & $ 22923$ & $ 488$ & $<124$ & $<836$ & $<8035$\\
\hline
$\alpha_1$ & - & $ 0.0192$ & $ 0.0096$ & $ 0.0048$ & $ 0.0005$ & $ \sim 10^{-5} $ & $ \sim 10^{-6} $\\
\hline
$\alpha_1 \mathbf{E}[D_n]$ &- & $ \sim 10^{6} $ & $ 220$ & $ 2.3484$ & $ <0.0582$ & $ <0.0391$ & $ <0.0376$ \\
\hline
$\alpha_2$ & - & $ 0.0002$ & $ 0.0001 $ & $ \sim 10^{-5} $ & $ \sim 10^{-6} $ & $ \sim 10^{-7} $ & $ \sim 10^{-8} $\\
\hline
$\alpha_2 \mathbf{E}[D_n]$ &- & $ 15131$ & $ 2.2022$ & $ 0.0235$ & $<0.0006$ & $<0.0004$ & $<0.0004$\\
\hline
$\mathbf{E}[e^{-\alpha_1 D_n}]$ &- &$ \sim 10^{-6} $ & $ \sim 10^{-5} $ & $ 0.1552$ & $ >0.9420$ & $ >0.9606$ & $ >0.9621$ \\
\hline
$\mathbf{E}[e^{-\alpha_2 D_n}]$ &- &$ \sim 10^{-4} $ & $ 0.2215$ & $ 0.9769$ & $>0.9994$ & $>0.9996$ & $>0.9996$ \\
\hline
$r_1(\mu=1)$ &- &$ \sim 10^{-13} $ & $ \sim 10^{-11} $ & $ 0.0174$ & $>0.8106$ & $>0.8603$ & $>0.8646$ \\
\hline
$r_2(\mu=1)$ &- &$ \sim 10^{-9} $ & $ 0.0357$ & $ 0.9090$ & $>0.9961$ & $>0.9972$ & $>0.9973$ \\
\hline
$r_1(\mu=0.99)$ &- &$ 0$ & $ 0$ & $ 0$ & $ 0$ & $ 0$ & $ 0$\\
\hline
$r_2(\mu=0.99)$ &- &$ 0$ & $ 0$ & $ 0.2323$ & $ 0$ & $ 0$ & $ 0$\\
\hline
$S_1(\mu=1)$ &- &$ \sim 10^{-21} $ & $ \sim 10^{-15} $ & $ 0.0001 $ & $>0.0064$ & $>0.0010$ & $>0.0001$ \\
\hline
$S_2(\mu=1)$ &- &$ \sim 10^{-17} $ & $ \sim 10^{-6} $ & $ 0.0036$ & $>0.0079$ & $>0.0012$ & $>0.0001$ \\
\hline
$S_1(\mu=0.99)$ &- &$ 0$ & $ 0$ & $ 0$ & $ 0$ & $ 0$ & $ 0$\\
\hline
$S_2(\mu=0.99)$ &- &$ 0$ & $ 0$ & $ 0.0009$ & $ 0$ & $ 0$ & $ 0$\\
\hline
$S^{\mathrm{PLOB,QR}}(L_0)$
& $\sim 10^{-16}$ & $\sim 10^{-8}$ & 0.0002 & 0.0154 & 1.4530 & 4.4921 & 7.7846
\end{tabular}
\caption{Overview of the relevant quantities for the {\it optimal scheme}: segment number $n$, segment length $L_0$[km], average (number of) waiting time (steps) $\mathbf{E}[K_n]$, raw rate $R$, average (number of) dephasing time (steps) $\mathbf{E}[D_n]$, inverse effective coherence time $\alpha_1=L_0/(c_f 0.1{\rm s})$, effective average dephasing time $\alpha_1 \mathbf{E}[D_n]$, inverse effective coherence time $\alpha_2=L_0/(c_f 10{\rm s})$, effective average dephasing time $\alpha_2 \mathbf{E}[D_n]$, average dephasing fractions $\mathbf{E}[e^{-\alpha_1 D_n}]$ and $\mathbf{E}[e^{-\alpha_2 D_n}]$, secret key fractions and rates, $r$ and $S$, for different $\mu=\mu_0$ (subscript corresponds to the choice of $\alpha_1$ or $\alpha_2$, $\mu=1$ is the channel-loss-and-memory-dephasing-only case), and the (repeater-assisted) capacity bound $S^{\mathrm{PLOB,QR}}(L_0)$. For the cases $n>8$, not all exact values are available and hence we inserted approximate values or (lower or upper) bounds.
We assumed $p_{\mathrm{link}}=F_0=1$ for the link coupling efficiency and the initial state dephasing.}
\label{tab:optperchanneluse}
\end{table*}
Next let us consider the relevant quantities for the optimal scheme as presented in Tab.~\ref{tab:optperchanneluse}. In this case we no longer have access
to all exact values for larger repeaters $n>8$. However, there is a distinction between the waiting times $K_n$ and the dephasing times $D_n$. For the total waiting times or the raw rates $R$ we can calculate the numbers for small and also for larger $n$ according to the exact analytical expression in Eq.~\eqref{eq:Knpar}.
There are also good approximations for both small $n$ (small $p$) and larger $n$ ($p$ closer to one) which may be easier to calculate \cite{PvL, Elkouss2021, Eisenberg}.
Importantly, unlike the case of the fully sequential scheme, the raw rate $R$ now grows with all $n$ (though slowly for larger $n$) thanks to the fast, parallel distributions in all segments together with the loss scaling that improves with $n$.
This behaviour even matches that of the repeater-assisted capacity bounds for increasing $n$, as given in the last row of Tab.~\ref{tab:optperchanneluse}.
However, recall that for our qubit-based quantum repeaters the raw rate can never exceed one secret bit per channel use, whereas $S^{\mathrm{PLOB,QR}}(L_0)$ can, for decreasing $L_0$.
For the average total dephasing we can calculate the exact values up to $n=8$. Comparing these values in Tabs.~\ref{tab:seqperchanneluse} and \ref{tab:optperchanneluse}, we see that the optimal scheme accumulates less dephasing than the fully sequential scheme when $n=4, 8$. The two competing effects in the fully sequential scheme, long total waiting time versus minimal number of simultaneously stored memory qubits per elementary time unit, overall result in a larger dephasing rate in comparison with our optimal scheme for $n\leq 8$. We extrapolate this relative behaviour to larger $n$ and therefore assume that the dephasing values of the fully sequential scheme may serve as upper bounds on those for the optimal scheme when $n>8$ in Tab.~\ref{tab:optperchanneluse}. We make the same assumption for the other dephasing-dependent quantities, in particular, the secret key fractions, for which the fully sequential values then serve as lower bounds. Looking at the entries of Tab.~\ref{tab:optperchanneluse} for the optimal scheme, as a final result, we conclude that while for $\mu=1$ (``channel-loss-and-memory-dephasing-only'' case) it may be best to choose as many segments as $n=80$ (i.e. stations are placed at every 10km), similar to what is best for the fully sequential scheme (Tab.~\ref{tab:seqperchanneluse}), for $\mu=0.99<1$ we must not go to segment numbers higher than $n=8$. In fact, for $\mu=0.99$, both for the sequential and the optimal schemes, effectively the only non-zero secret key rate is obtainable for $n=8$ and the larger of the two coherence times considered, with a factor-three enhancement for the optimal scheme over the sequential one. If $n>8$, the faulty states and gates make $S$ vanish, if $n<8$ the small raw rates and the high effective average dephasing times do not permit practically usable secret key rates. Note that the entire discussion here in the context of Tabs.~\ref{tab:seqperchanneluse} and \ref{tab:optperchanneluse} is for a total distance of $L=800$km. We may infer that an elementary segment length of $L_0 \sim 100$km is not only highly compatible with existing classical repeater and fiber network architectures, but also seems to offer a good balance between an improved memory-assisted loss scaling and an only limited addition of extra faulty elements. This conclusion here holds for our repeater setting based upon heralded loss-tolerant entanglement distribution, deterministic entanglement swapping, and a memory dephasing model. Similar elementary lengths have been used before for schemes with probabilistic entanglement swapping and memory loss \cite{DLCZ, Sangouard}.
For schemes with deterministic entanglement swapping, but a less loss-tolerant entanglement distribution mechanism, \cite{HybridPRL} smaller segment lengths may be preferable.
We will include such schemes, exhibiting an intrinsic channel-loss-dependent dephasing, into the discussion in a later section. Let us now consider a simple form of multiplexing in order to improve the repeater performance, provided sufficient extra resources are available.
\subsection{Multiplexing}
Operating $M$ repeater chains in parallel automatically leads to an enhancement of the overall rates by a factor of $M$. However, since in this case the corresponding number of channels grows as well by a factor of $M$, the rates per channel use remain unchanged. The situation becomes different though when the chains can ``interact" with each other. In particular, the loss scaling of heralded entanglement distributions can be improved, at least for small systems in an MDI QKD setting (even without the use of quantum memories but with the need for a nondestructive heralding) \cite{Azuma2015}.
For memory-based quantum repeaters, memory imperfections may be compensated via multiplexing techniques \cite{CollinsPrl,MunroNatPhot,LutPRA,RazLut}.
Experimentally, multiplexing can be realized through various degrees of freedom.
Apart from spatial multiplexing with additional memory qubits at each station that can be coupled to additional fiber channels, this can be forms of temporal or spectral multiplexing where a single fiber may be employed sequentially at a high clock rate \cite{Cody_Jones} or at the same time with multiple wavelengths, respectively.
In this section, we shall incorporate a simple form of multiplexing into our formalism and our repeater models and systems. We have seen that either high total efficiencies or sufficiently long coherence times are needed to achieve usable secret key rates at long distances. We will now see that multiplexing can be understood as a means to effectively enhance the memory coherence time. In the following we will describe in more detail which kind of multiplexing we consider and why it indeed effectively increases the coherence time.
The simplest way to include multiplexing in our repeater models is by using \(M\) memories simultaneously to generate entanglement. These memories can either be connected to the same fiber by a switch or they may each be coupled to their own fiber channel. For simplicity, we consider the switch to be perfect such that both approaches become equivalent (and where the additional channel uses take place either in time or in space). A lossy switch could be easily incorporated into our model by using an additional parameter which is included in \(p_{\mathrm{link}}\) (note that the loss from the switch is time-independent and so
always the same). A possible setup for a two-segment repeater with multiplexing is shown in Fig.~\ref{fig:figure-multiplexing}. Here all entanglement distribution attempts happen simultaneously. Since we have \(M\) replica of all memories and channels, this setup acts as if \( p \mapsto 1-(1-p)^M\), provided that memory qubits from different chains can talk to each other in the middle station so that we may again swap as soon as possible.
\begin{figure}
\caption{Multiplexing in a two-segment repeater.}
\label{fig:figure-multiplexing}
\end{figure}
For an $M$-multiplexing let us thus define the effective distribution probability $p_{\mathrm{eff}}=1-(1-p)^M$. For small $p$, only keeping linear terms, we have $p_{\mathrm{eff}}\approx M p$.
As the expected waiting time in a single segment is then given by $\frac{1}{Mp}$, we can already gain insight on the possibility that multiplexing increases the effective coherence time by a factor of $M$.
More specifically, for example, for the fully sequential scheme the expectation value of $D_n$ is $(n-1)/p$, thus the transition \( p \mapsto p_{\mathrm{eff}}\approx M p\) reduces the number of dephasing steps, on average, by a factor of $1/M$.
This is equivalent to an increase of the coherence time by a factor $M$.
In the following, let us be more precise and show what `small' $p$ really means in terms of the corresponding segment length $L_0$.
In fact, including multiplexing, the secret key rates in dependence of the repeater distance behave in a more complicated way and one can see that for small distances the rate is nearly constant and only for larger distances the rates behave as we would expect from the non-multiplexed schemes.
In the general, exact model using $p_{\mathrm{eff}}=1-(1-p)^M$, it becomes clear that the above-mentioned behaviour originates from this general expression for $p_{\mathrm{eff}}$.
In Fig.~\ref{fig:ruleofthumb}(a) one can see that $p_{\mathrm{eff}}$ can be divided in three regimes. In the first regime of small $L_0$, $p_{\mathrm{eff}}$ is a constant. In the second regime of large $L_0$, $p_{\mathrm{eff}}$ is a simple exponential decay, while in between it has a more complicated form interpolating both regimes.
In the first regime, the effective probability is nearly constant, because in our simple multiplexing protocol we only make use of a single `entanglement excitation' in each segment of the parallelized repeater chains, but for small $L_0$ we would typically have multiple excitations in each segment.
Thus, increasing $L_0$ decreases the number of excitations, but as we anyway only make use of a single one, this barely matters
(making use of more excitations and keeping the `residual entanglement' could potentially further enhance the rates \cite{RazaviProc};
however, here our focus is on a simple and clear interpretation of the impact of the multiplexing on the coherence time and the memory dephasing in our statistical model).
In the second regime of rather large $L_0$, the contributions of multiple excitations can be neglected and therefore the rates behave exactly like in the $M=1$ case.
Hence, this regime two is exactly that where we can increase the effective coherence time by a factor of $M$ with the help of multiplexing.
We can give a rough rule of thumb for the minimal length of $L_0$ when one may use the simple approximation of increasing the coherence time by a factor of $M$.
For this we assume $p=\exp(-\frac{L_0}{L_{\mathrm{att}}})$ \footnote{When considering $p_{\mathrm{link}}<1$ one can incorporate this as an additional length of $-\ln(p_{\mathrm{link}})L_{\mathrm{att}}$ regarding $L_0$.} and take the minimizing argument of $\frac{\partial^2\ln\left(p_{\mathrm{eff}}\right)}{\partial L_0^2}$ for a given $M$ in order to estimate the midpoint of the interpolating regime.
For general $M$ this expression can be nicely fitted to an expression of the form $c_1 \ln\left(c_2 M+c_3\right)+c_4$, as one can see in Fig.~\ref{fig:ruleofthumb}(b).
One should then consider $L_0$ to be slightly larger for the approximation to hold.
\begin{figure}
\caption{(a) $p_{\mathrm{eff}
\label{fig:ruleofthumb}
\end{figure}
\begin{figure*}
\caption{Rates (secret key/raw) of (a,b) two- and (c,d) four-segment repeaters using multiplexing $M=10$ at distances \(L\) for different experimental parameters. The rate of a repeater without multiplexing, but with the same coherence time is shown in orange, whereas the rate of a repeater using multiplexing is shown in red. Additionally, a repeater without multiplexing, but with an equivalent effective coherence time is presented in dashed black. All rates are expressed per channel use and hence include a division by $M$.}
\label{fig:SKR_multiplexing}
\end{figure*}
Let us give another, more rigorous derivation of the effective coherence time in the presence of multiplexing.
The coherence time primarily characterises the increasing decline of the secret key rate with distance. However, a massive drop actually happens when the secret key fraction \(r\) reaches zero, which is possible when $e_z>0$, i.e. when $\mu<1$ or $\mu_0<1$. Thus, let us determine the probability at which \(r=0\) holds with multiplexing and from that deduce an equivalent coherence time without multiplexing. Since the QBER \(e_z\) is constant ($e_z=\overline{e_z})$, we have to solve for the expectation value of \(\overline{e_x}\) such that
\begin{equation}
1-h(e_z) \overset{!}{=} h(\overline{e_x}).
\end{equation}
In order to find the probability \(p\) or equivalently the distance at which the drop happens, let us use the Taylor series of the binary entropy function at \(x=\frac{1}{2}\),
\begin{align}
h(x)= 1- \frac{1}{2 \ln(2)}\sum_{n=1}^{\infty} \frac{\left(1-2x\right)^{2n}}{n\left(2n-1\right)},\; \forall\; 0<x<1.
\end{align}
Then one finds for \(\overline{e_x}\) up to first order:
\begin{equation}
\overline{e_x}= \frac{1}{2} - \sqrt{\frac{\ln(2)h(e_z)}{2}},
\end{equation}
where only the negative root is possible, as \(0\leq e_x \leq \frac{1}{2}\). Inserting \(\overline{e_x}\) and solving for $\mathbf{E}[e^{-\alpha D_n}]$ gives
\begin{equation}
\mathbf{E}[e^{-\alpha D_n}] = \frac{\sqrt{2\ln(2)h(e_z)}}{\mu^{n-1} \mu_0^{n} \left(2 F_0-1\right)^n}.
\end{equation}
If $\mu=\mu_0=1$, including especially the channel-loss-and-memory-dephasing-only case (for which also $F_0=1$), we have $h(e_z)=0$ and so the requirement becomes $\mathbf{E}[e^{-\alpha D_n}]=0$, which is impossible.
However, as soon as $e_z>0$, i.e. $\mu<1$ or $\mu_0<1$, a sufficiently small non-zero (average) dephasing fraction $\mathbf{E}[e^{-\alpha D_n}]$ leads to a zero secret key fraction.
As we can always calculate this expectation value by our previously derived PGFs, we now have an accurate and systematic way to derive the probability $p$ (or the total distance $L=n L_0$) at which the drop takes place for given values of $n$, $\tau_{\mathrm{coh}}$, $\mu$, $\mu_0$, and $F_0$. Recall that the inverse effective coherence time $\alpha=L_0/(c_f \tau_{\mathrm{coh}})$ typically also depends on $L_0$.
On the other hand, we may use the above relation to determine an (inverse) effective coherence time by calculating the drop for a repeater with multiplexing and then the equivalent \(\alpha\), which would be needed to achieve the same distance without any multiplexing. From this \(\alpha\) one can recover the coherence time \(\tau_{\mathrm{coh}}\) and finds the approximate relation
\begin{equation}\label{eq:MPrelation}
\tau_{\mathrm{coh}} \mapsto M \cdot \tau_{\mathrm{coh}},
\end{equation}
when a multiplexing of \(M\) is used and the remaining setup is kept the same. Thus, one can achieve an \(M\)-times longer effective coherence time with the help of multiplexing.
In Fig.~\ref{fig:SKR_multiplexing}, we show the rates of two- and four-segment repeaters using a multiplexing of \(M=10\) in red. Note that because we use the SKR per channel use, the rates are obtained including a division by $M$. The rates of the same repeaters without multiplexing are presented in orange. Furthermore, a repeater without multiplexing, but with the equivalent `effective' coherence time of \(\tau_{\mathrm{eff}} = M \tau_{\mathrm{coh}}\) is shown in dashed black. One can see that for small distances, i.e large probabilities, the multiplexed repeater does not quite behave like its non-multiplexed counterpart with an effectively increased coherence time. A clear splitting between the red and black curves is visible. However, for larger distances, especially after crossing the PLOB bound, the multiplexed repeater behaves exactly the same as if simply memories with an effectively longer coherence time were used. For smaller link efficiencies, the splitting becomes much less pronounced, as can be seen in the plots on the right of Fig.~\ref{fig:SKR_multiplexing}. All this holds for both two and four segments, according to Fig.~\ref{fig:SKR_multiplexing}.
In particular, for small link efficiencies, the secret key rate of an equivalent repeater with \(\tau_{\mathrm{eff}} = M \tau_{\mathrm{coh}}\) is
almost indistinguishable from a repeater with multiplexing. This is in agreement with the above discussion on the occurrence of single versus multiple `entanglement excitations' in each segment where the latter are then highly suppressed even at short distances due to the small value of $p_{\mathrm{link}}$.
Thus, for practical purposes, in all our discussions, we may treat several cases equivalently, for instance, a repeater with $\tau_{\mathrm{coh}}=10$s and $M=1$ would be equivalent to a repeater with $\tau_{\mathrm{coh}}=1$s and $M=10$.
\subsection{Secret key rate per second}\label{sec:Secret Key Rate per Second}
In a real-world application, the important figure of merit is not the rate per channel use, it will be the rate per second. In particular, a memory-asissted QKD system or generally a memory-based quantum repeater, as typically based upon light-matter interactions and classical communication at least between neighboring stations, has a limited `clock rate'.
Classical communication is needed to declare successful
transmission of photons for the entanglement distribution. In general, also extra communication would be needed to signal any successful entanglement swapping, but as we assumed deterministic swapping no such communication is needed in our repeater models.
As we already discussed frequently throughout the paper, a repeater's performance generally depends on an elementary time unit $\tau$, which is contained in the inverse effective coherence time $\alpha=\tau/\tau_{\mathrm{coh}}$, where generally $\tau=\tau_{\mathrm{clock}}+L_0/c_f$ including the experimental local processing time $\tau_{\mathrm{clock}}$. We have mostly argued that in the relevant distance regimes, this quantity is dominated by the (quantum and classical) communication times between neighboring stations, thus $\tau = L_0/c_f$ and $\alpha=L_0/(c_f \tau_{\mathrm{coh}})$.
Already with segment
lengths above \(\unit[10]{km}\), one can neglect the local clock rates, since these are much higher than the rates given
by the transmission times.
An extra factor of two could be included in $\tau$ for some protocols due to the $L_0$-transmission of a photon entangled with a memory qubit and the classical answer (sent back over $L_0$) heralding its successful transmission. However, this would depend on the specific protocol and so we have chosen the simplest, minimal form $\tau = L_0/c_f$.
Only for very short segment lengths do we have $\alpha\approx \alpha^{\mathrm{loc}}=\tau_{\mathrm{clock}}/\tau_{\mathrm{coh}}={\rm MHz}^{-1}/\tau_{\mathrm{coh}}$ assuming experimental clock rates $\tau_{\mathrm{clock}}^{-1}$ typically of the order of MHz.
However, there are repeater schemes that are independent of additional classical communication and the decision to keep or reinitialize a memory state can be made at the memory station. These schemes may be referred to as ``node receives photons" (NRP) as opposed to the class of schemes with ``node sends photons" (NSP) \cite{White}.
An NRP protocol and application that circumvents the need of extra signal waiting times can be realized with two ``segments'' and a middle station in memory-assisted MDI QKD \cite{White}.
Such a scheme, when treating it as an elementary quantum repeater unit or module
many of which a large-scale repeater can be made of, may be referred to as a ``quantum repeater cell'', actually composed of two half-segments \cite[Fig. 6b]{White}.
In this case, even for large (half-)segment length $L_0$, we have $\alpha = \alpha^{\mathrm{loc}}=\tau_{\mathrm{clock}}/\tau_{\mathrm{coh}}$.
For completeness, we show the rates of such an NRP-based two-segment scheme
in the form of contour plots in App.~\ref{app:nrp}.
By circumventing the need for extra classical communication
and thus significantly reducing the effective memory dephasing,
the minimal state and gate fidelity values can even be kept constant
over large distance regimes.
However, as soon as the NRP concept is applied to repeaters beyond a single middle station effectively connecting complete repeater segments, \cite[Fig. 6a]{White}
the need for extra classical communication to initiate an entanglement swapping operation can no longer be entirely avoided
(though there are ideas to still partially benefit from the NRP concept) \cite{Cody_Jones}.
A quantum repeater cell can also be considered employing the NSP protocol \cite{NL} and one such cell (two half-segments) or the corresponding complete segment can then be used as an elementary quantum repeater unit \cite[Fig. 4]{White}.
For the NSP concept, the extra signal waiting time is generally required at every distribution attempt.
In any case or protocol, the repeater's elementary time unit $\tau$ determines the effective coherence time $\tau_{\mathrm{coh}}/\tau$ and as such, even when the rates per channel use are considered, it determines how many distribution attempts are possible within a given $\tau_{\mathrm{coh}}$ and hence how big the effective dephasing time $\alpha D_n$ becomes.
Compared with memory-assisted quantum communication schemes, a big asset of an all-optical point-to-point quantum communication link is that it can operate at a very high clock rate, typically of the order of GHz, only limited by the speed of Alice's laser (quantum state) source and Bob's (quantum state) detector.
For such a direct state transmission, no extra classical communication is required as for heralding the successful transfer of entangled photons between repeater links. Thus, the rate per second is simply given by the two local clock rates, especially the time it takes to generate the photonic qubit states
or any other quantum states in QKD based on different types of encoding
(however, thanks to the known linear bounds on the key distribution via a long and lossy point-to-point quantum communication channel \cite{PLOB, TGW}, it is clear that the rate scaling of qubit-based QKD cannot be beaten by any form of non-qubit encoding).
Other all-optical schemes such as MDI QKD or twin-field QKD, which are no longer point-to-point and do include a middle station between Alice and Bob, also benefit from such high clock rates. The remarkable feature of twin-field QKD is that it shares both advantages: the high clock rate with point-to-point quantum communication and the $L \rightarrow L/2$ loss scaling gain with memory-based two-segment quantum repeaters. In order to assess whether there is a real benefit of employing a two-segment quantum repeater or even adding extra repeater stations, we must eventually consider the rates per second and take into account the corresponding clock rates in all schemes. As a consequence, comparing clock rates of MHz with those of GHz (of memory-based versus all-optical quantum communication), there is a penalty of a factor of about 1000 from the start for the memory-based approach. In the regime where $\alpha\approx L_0/(c_f \tau_{\mathrm{coh}})$, this penalty even gets worse. In this case, when $\tau\approx L_0/c_f$, there are at least two disadvantages of $\tau$ growing with $L_0$: a reduced effective coherence time $\tau_{\mathrm{coh}}/\tau$ and a reduced raw rate per second $R/\tau$. Beating the PLOB bound for the rates per channel use is only a necessary criterion that a quantum repeater can be beneficial. In order to confirm a real benefit, we have to consider the secret key rates per second $S/\tau=r R/\tau$. Thus, even with perfect memories $\tau_{\mathrm{coh}}\rightarrow\infty$, the different $\tau$ values matter. The situation is similar to throwing two or more dices at once at a fast rate. To get all dices showing six eyes this may still be faster than throwing them very slowly while being allowed to only continue with the unsuccessful dices in each round. The final raw and secret key rates per second obtainable with our two most prominent and mostly discussed repeater schemes, the fully sequential scheme and the optimal scheme, are given in Tabs.~\ref{tab:seqpersecond} and \ref{tab:optpersecond}, respectively.
\begin{table*}
\begin{tabular}{c|c|c|c|c|c|c|c}
$n$ &1 & 2 & 4 & 8 & 80 & 800 & 8000\\
\hline
\hline
$L_0$[km] &800 & 400 & 200 & 100 & 10 & 1 & 0.1\\
\hline
$R /\tau $ & $\unit[\sim 10^{-14}]{Hz}$ & $\unit[\sim 10^{-6}]{Hz}$ & $\unit[0.0293]{Hz}$ & $\unit[2.8]{Hz}$ &
$\unit[165.2]{Hz}$ & $\unit[248.7]{Hz}$ & $\unit[259.1]{Hz}$\\
\hline
$S_1(\mu=1)/\tau $ &- &$\unit[\sim 10^{-18}]{Hz}$ & $\unit[\sim 10^{-14}]{Hz}$ & $\unit[0.0106]{Hz}$ & $\unit[133.9]{Hz}$ & $\unit[213.9]{Hz}$ & $\unit[224.0]{Hz}$\\
\hline
$S_2(\mu=1)/\tau $ &- &$\unit[\sim 10^{-14}]{Hz}$ & $\unit[0.0005]{Hz}$ & $\unit[2.4]{Hz}$ & $\unit[164.5]{Hz}$ & $\unit[248.0]{Hz}$ & $\unit[258.4]{Hz}$\\
\hline
$S_1(\mu=0.99)/\tau $ &- &$\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$\\
\hline
$S_2(\mu=0.99)/\tau $ &- &$\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0.6086]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$\\
\hline
$S^{\mathrm{PLOB,QR}}(L_0)/\tau $ &$\unit[\sim 10^{-7}]{Hz}$ & $\unit[18.3]{Hz}$ & $\unit[0.2]{MHz}$ & $\unit[15.5]{MHz}$ & $\unit[1.5]{GHz}$ & $\unit[4.5]{GHz}$ & $\unit[7.8]{GHz}$
\end{tabular}
\caption{Overview of the relevant quantities for the {\it fully sequential scheme} of Tab.~\ref{tab:seqperchanneluse} calculated per second (shown are only those entries that change, but again with segment number $n$, segment length $L_0$[km]): raw rate $R/\tau$, secret key rate $S/\tau$ for different $\mu=\mu_0$ (again subscript corresponds to the choice of $\alpha_1$ or $\alpha_2$, $\mu=1$ is the channel-loss-and-memory-dephasing-only case), and the (repeater-assisted) capacity bound per elementary time unit $S^{\mathrm{PLOB,QR}}(L_0)/\tau$ where we choose $\tau={\rm GHz}^{-1}$ for the cases $n=1,2$, i.e. the bounds, expressed per second, on all-optical point-to-point and twin-field QKD. Note that for realistic but still GHz-clock-rate twin-field QKD we rather have $S/\tau\sim1$Hz. In any of the other, memory-based scenarios, we choose $\tau=\tau_{\mathrm{clock}}+L_0/c_f$ with $\tau_{\mathrm{clock}}={\rm MHz}^{-1}$.
We again assumed $p_{\mathrm{link}}=F_0=1$ for the link coupling efficiency and the initial state dephasing.}
\label{tab:seqpersecond}
\end{table*}
\begin{table*}
\begin{tabular}{c|c|c|c|c|c|c|c}
$n$ &1 & 2 & 4 & 8 & 80 & 800 & 8000\\
\hline
\hline
$L_0$[km] &800 & 400 & 200 & 100 & 10 & 1 & 0.1\\
\hline
$R /\tau $ & $\unit[\sim 10^{-14}]{Hz}$ & $\unit[\sim
10^{-6}]{Hz}$ & $\unit[0.0563]{Hz}$ & $\unit[8.2]{Hz}$ & $\unit[3.8]{kHz}$ & $\unit[72.7]{kHz}$ & $\unit[967.2]{kHz}$\\
\hline
$S_1(\mu=1)/\tau $ &- &$\unit[\sim 10^{-18}]{Hz}$ & $\unit[\sim 10^{-12}]{Hz}$ & $\unit[0.1423]{Hz}$ & $>\unit[3.1]{kHz}$ & $>\unit[62.5]{kHz}$ & $>\unit[832.1]{kHz}$\\
\hline
$S_2(\mu=1)/\tau $ &- &$\unit[\sim 10^{-14}]{Hz}$ & $\unit[0.0020]{Hz}$ & $\unit[7.4]{Hz}$ & $>\unit[3.8]{kHz}$ & $>\unit[72.4]{kHz}$ & $>\unit[964.5]{kHz}$\\
\hline
$S_1(\mu=0.99)/\tau $ &- &$\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$\\
\hline
$S_2(\mu=0.99)/\tau $ &- &$\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[1.9]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$ & $\unit[0]{Hz}$\\
\hline
$S^{\mathrm{PLOB,QR}}(L_0)/\tau $ &$\unit[\sim 10^{-7}]{Hz}$ & $\unit[18.3]{Hz}$ & $\unit[0.2]{MHz}$ & $\unit[15.5]{MHz}$ & $\unit[1.5]{GHz}$ & $\unit[4.5]{GHz}$ & $\unit[7.8]{GHz}$
\end{tabular}
\caption{Overview of the relevant quantities for the {\it optimal scheme} of Tab.~\ref{tab:optperchanneluse} calculated per second (shown are only those entries that change, but again with segment number $n$, segment length $L_0$[km]): raw rate $R/\tau$, secret key rate $S/\tau$ for different $\mu=\mu_0$ (again subscript corresponds to the choice of $\alpha_1$ or $\alpha_2$, $\mu=1$ is the channel-loss-and-memory-dephasing-only case), and the (repeater-assisted) capacity bound per elementary time unit $S^{\mathrm{PLOB,QR}}(L_0)/\tau$ where we choose $\tau={\rm GHz}^{-1}$ for the cases $n=1,2$, i.e. the bounds, expressed per second, on all-optical point-to-point and twin-field QKD. Note that for realistic but still GHz-clock-rate twin-field QKD we rather have $S/\tau\sim1$Hz. In any of the other, memory-based scenarios, we choose $\tau=\tau_{\mathrm{clock}}+L_0/c_f$ with $\tau_{\mathrm{clock}}={\rm MHz}^{-1}$.
We again assumed $p_{\mathrm{link}}=F_0=1$ for the link coupling efficiency and the initial state dephasing.}
\label{tab:optpersecond}
\end{table*}
\subsection{Application and comparison of protocols}
Let us now consider various quantum repeater protocols
based on different types of the optical encoding
and calculate their corresponding secret key rates per second
using the methods developed in the preceding sections.
We shall look at (i) a kind of standard scheme employing
two-mode (dual-rail, DR) photonic qubits distributed through the optical-fiber channels (either emitted from a central source of entangled photon pairs
and written into the spin memory qubits or emitted from the repeater nodes
employing spin-photon entangled states and utilizing two-photon interference in the middle of each segment), \cite{White}
(ii) a scheme based upon spin-photon (spin-light-mode) entanglement and one-photon interference with an encoding similar to that
introduced by Cabrillo et al. \cite{cabrillo} effectively using one-mode (single-rail, SR) photonic qubits,
(iii) a scheme that extends the concepts of twin-field QKD with coherent states
to a specific variant of memory-assisted QKD, i.e. a kind of twin-field quantum repeater \cite{tf_repeater}.
We refer to scheme (ii) as the Cabrillo scheme and discuss it in more detail in App.~\ref{app:cabrillo}.
For all three schemes we consider a quantum repeater with $n=1,2,3,4,8$ segments
matching the size of the repeater systems that we have formally/theoretically treated in great detail in the first parts of this paper.
We always use the previously derived ``optimal'' quantum repeater protocol
that belongs to the fastest schemes and gives the smallest dephasing among all fast schemes.
The two schemes (ii) and (iii) share the potential benefit that
for quantum repeaters with $n$ segments and $n-1$ intermediate memory stations
(not counting the memories at Alice and Bob or assuming immediate measurements there) they lead to an improved loss scaling with a $2n$-times bigger
effective attenuation distance compared with a point-to-point link
(unlike the standard scheme (i) that only achieves an $n$-times bigger
effective attenuation distance), but a final state fidelity parameter
still decreasing as the power of $2n-1$ (assuming equal gate and initial state
error rates) like the standard scheme (i).
However, scheme (ii) has an intrinsic error
during the distribution step due to the initial two-photon terms
in combination with channel loss.
Similarly, scheme (iii) is more sensitive to channel loss
exhibiting an intrinsic loss-dependent dehasing error,
because the optical state is a phase-sensitive continuous-variable state \cite{HybridPRL}.
The two models of channel-loss-induced errors for schemes (ii) and (iii)
thus slightly differ, while the transmission loss scaling is identical.
As a consequence, for both (ii) and (iii), we have the constraint that
the excitation amplitudes (the weights of the non-vacuum terms)
must not become too large.
Despite the above-mentioned benefits compared with scheme (i) it will turn out that the intrinsic errors of schemes (ii) and (iii)
represent an essential complication that prevents to fully exploit the improved
scaling of the basic parameters in comparison with the standard repeater protocols.
For a fair comparison, assuming similar types of initial state imperfections in all three schemes, we set $\mu_0=1$ with $F_0=0.99,0.98$ and so replace the initial depolarizing error for scheme (i) by an initial dephasing error.
Thus, in the expressions of the QBERs as given by
Eq.~\eqref{eq:QBER}, the contribution of $\mu_0^n$ to the initial error scaling
from the analysis of the preceding sections (where $F_0=1)$ is now replaced by a corresponding scaling with $F_0<1$. The gate error scaling with $\mu^{n-1}$ remains unchanged in all schemes. Of course, our formalism also allows to focus on specific schemes including initial state errors with $\mu_0<1$.
In this case, the specific contributions of the different elements in each elementary repeater unit (segments, half-segments, ``cells'') \cite{White} to the link coupling efficiency $p_{\mathrm{link}}$ and the initial state error parameters $\mu_0$ or $F_0$ depend of the particular protocol \cite{White}.
For example, zooming in on an NSP segment, \cite{White} we have a squared contribution from the two spin-photon entangled states on the left and on the right, $\mu_{\mathrm{sp,ph}}^2$, and another possible gate error factor, $\mu_{\mathrm{OBM}}$, coming from the optical Bell measurement in the middle of the segment.
In this scenario, already in a single segment, we effectively have one imperfect entanglement swapping operation (acting on the two photons in the middle of the segment) connecting two initially distributed, depolarized entangled states (the two spin-photon states), to which our physical model directly applies replacing our initial $\mu_0$ for one segment according to $\mu_0 \rightarrow \mu_{\mathrm{sp,ph}}^2 \mu_{\mathrm{OBM}}$.
This overall initial distribution error will most likely be dominated by the
imperfect spin-photon states, assuming near-error-free (though probabilistic)
photonic Bell measurements, thus $\mu_0 \sim \mu_{\mathrm{sp,ph}}^2$.
In a full NRP segment, the memory write-in may be realized via quantum teleportation using a locally prepared spin-photon state and an optical Bell measurement on the photon that arrives from the fiber channel and the local photon. In this scenario, already in a single complete segment, we may effectively have three initial entangled states (two local spin-photon states on the left and on the right together with one distributed entangled photon pair emitted from a source in the middle of the segment) and two optical Bell measurements, \cite[Fig. 6a]{White} with our model resulting in a $\mu_0 \sim \mu_{\mathrm{ph,ph}} \mu_{\mathrm{sp,ph}}^2 \mu_{\mathrm{OBM}}^2$ scaling of the initial error parameter for one segment (i.e., similar to the effective final scaling of a three-segment repeater in our more abstract model,
with $\mu_0 \rightarrow \mu_{\mathrm{sp,ph}}$ and $\mu \rightarrow \mu_{\mathrm{OBM}}$, and setting for this simplifying analogy, quite unrealistically, $\mu_{\mathrm{sp,ph}}=\mu_{\mathrm{ph,ph}}$). Assuming near-error-free Bell measurements, and near-perfect (though possibly only probabilistically created) photon pairs, we would again arrive at an overall scaling of $\mu_0 \sim \mu_{\mathrm{sp,ph}}^2$ for the initial error parameter. In case of an entangled photon pair source that deterministically produces imperfect photon-photon states (such as a quantum dot source), we would have $\mu_0 \sim \mu_{\mathrm{ph,ph}}\mu_{\mathrm{sp,ph}}^2$ instead. There is also the option of a heralded memory write-in that no longer relies on the generation of local spin-photon states and optical Bell measurements \cite{Rempe}.
In this case, our physical model has to be slightly adapted to such a scenario and a decomposition of the different error channels, including an imperfect memory write-in operation, into one effective initial error channel should be considered.
Thus, zooming in on our general initial-state error parameters $\mu_0$ or $F_0$
for a specific implementation is straightforwardly possible, but it will eventually lead to even stronger fidelity requirements for the individual experimental components that contribute to $\mu_0$ or $F_0$.
The different contributions to the link coupling efficiencies $p_{\mathrm{link}}$
can be similarly decomposed into the different experimental elements,
also including some differences for the different types of quantum repeater units and protocols \cite{White}.
However, note that for our comparison in this section, especially assuming that two photonic states are combined in the middle of each segment (i.e. in a kind of NSP scenario), the two-photon interference of scheme (i) results in
a quadratic disadvantage not only for the channel transmission
but also in terms of the link coupling efficiency $p_{\mathrm{link}}$ in comparison
with the protocols based on one-photon interference
(schemes (ii) and (iii)), $p_{\mathrm{link,(i)}}=p_{\mathrm{link,(ii)}}^2=p_{\mathrm{link,(iii)}}^2$.
For this let us write in short $p_{\mathrm{link,DR}}=p_{\mathrm{link,TF}}^2$, given the similarity of schemes (ii) and (iii).
In Fig. \ref{fig:SKR_per_sec_tf} we compare the secret key rates for the
dual-rail scheme (i) (DR), the Cabrillo scheme (ii), and the twin-field repeater (iii) (TF).
The two twin-field-type schemes include a free parameter describing the number of excitations. More excitations lead to a higher transmission rate at the expense of a lower state quality. In the plots we optimize this parameter for each data point to obtain the maximal secret key rate.
Recall, for the DR scheme, we introduce a small dephasing via the parameter $F_0<1$ in order to avoid comparing perfect initial entangled states with noisy ones.
When comparing schemes (ii) and (iii) one can see that for $\mu\approx1$ (iii) performs better while for lower $\mu$ (ii) is the better performing scheme.
This is because the probability of an error is smaller for the Cabrillo scheme, but the error would affect both QBERs of the BB84 protocol, significantly reducing the secret key rate. For the TF scheme (iii) we have an effect on only one of the two error rates.
When $\mu$ gets smaller, all schemes have a non-vanishing error rate in both bases and therefore the lower error rate of the Cabrillo scheme is helpful.
Figure~\ref{fig:SKR_per_sec_tf} shows that, although the DR scheme has a scaling disadvantage in comparison to both other schemes, it is often highly competitive, since both twin-field-type schemes suffer from their low initial probabilities of success when only weak excitations can be used to avoid introducing too much noise from the loss channel. Considering a memory coherence time of 10 seconds, a gate error parameter $\mu\geq0.97$, and coupling efficiencies as $p_{\mathrm{link,TF}}=0.9$, one can already overcome the PLOB bound with only three memory stations using either the DR scheme (i) or the TF protocol (iii).
For this comparison in terms of secret bits per second, we assume a
source repetition rate of \unit[1]{GHz} for an ideal point-to-point link
as associated with the PLOB bound per channel use.
Note that we do not include an extra factor of $1/2$ for the final rates which would strictly be needed in the DR-based scheme in comparison with the PLOB bound for a single-mode loss channel. Here the parallel transmission of the two modes
for a DR qubit does not change the rates per second and this optical encoding does not cause an extra experimental resource overhead (in fact, it even simplifies the optical transmission circumventing the need for long-distance phase stabilization as for the TF-type schemes). Moreover, an optical point-to-point direct transmission would most likely be based on DR qubit transmission as well. The other, previously mentioned factor $2$ that occurs in front of the effective inverse coherence time $\alpha$ when the two spins of a two-qubit spin pair simultaneously dephase while waiting in one segment has now been included here for each segment
(i.e. a small improvement would be possible when Alice and Bob measure their spins immediately).
In Fig.~\ref{fig:SKR_per_sec_tf}, we always assume a coherence time $\tau_{\mathrm{coh}}=\unit[10]{s}$, $p_{\mathrm{link,TF}}=0.9$, and $M=1$.
Recall from our discussions of the possibility of multiplexing that
we may equivalently consider schemes for which, for instance,
$\tau_{\mathrm{coh}}=\unit[1]{s}$ and $M=10$ according to Eq.~\eqref{eq:MPrelation}.
The plots lead to the following observations.
The two TF-type schemes (ii) and (iii) more heavily rely upon
sufficiently good error parameters than the DR scheme (i).
In Figs.~\ref{fig:SKR_per_sec_tf}(a) and (b) for two different initial dephasing
fidelities (which is only relevant for DR), we see that only the TF scheme (iii)
performs as good as DR with a gate error as low as $\mu=0.999$.
In this case, for the given parameters, TF even allows to reach slightly larger distances compared with DR, both going well above $L=1200$ km
giving more than a hundredth of a secret bit per second at such distances.
Note that in order to achieve this, the TF scheme requires a loss scaling
with a $16$-times bigger
effective attenuation distance compared with a point-to-point link,
whereas the DR scheme only has to exhibit an $8$-times bigger
effective attenuation distance (``$n=8$ TF'' vs. ``$n=8$ DR'').
The number of memory stations is the same for both, namely seven (not counting those at Alice and Bob).
With increasing gate errors $\mu\leq 0.99$, as shown in Figs.~\ref{fig:SKR_per_sec_tf}(c)-(g), only the DR scheme allows to reach distances above or near $L=1000$km. If both error parameters, that for the gates, $\mu$, and that for the initial states, $F_0$, are no longer sufficiently good (both or in combination), also the DR scheme ceases to reach large distances and barely beats the PLOB bound (see Figs.~\ref{fig:SKR_per_sec_tf}(f) and (g)).
For the two TF-type schemes (ii) and (iii), we generally checked both types of detectors, on-off as well as photon-number-resolving
(Fig.14 shows the results for on-off detections),
and we did not see a significant difference in the logarithmic plots
of the secret key rates for both schemes. The reason is that for larger distances the two-photon events at either of the two detectors (detectable via PNRDs) get increasingly unlikely compared with one-photon detection events coming from the two-photon terms in combination with the loss of one photon during transmission
(causing errors which remain undetectable via PNRDs).
The practically most relevant situation is shown in Figs.~\ref{fig:SKR_per_sec_tf}(c)-(e). In particular, for the numbers chosen there, i.e. state and gate errors of the order of 1-2\%, the DR scheme reaches
a distance of $L=800$km with about one secret bit per second, and even beyond with a lower rate. The link coupling efficiency for this scenario, like in all others, is $p_{\mathrm{link,DR}}=p_{\mathrm{link,TF}}^2=0.81$; the coherence time is $\tau_{\mathrm{coh}}=\unit[10]{s}$. The number of segments is $n=8$ (``$n=8$ DR'', dotted yellow curve) corresponding to a memory station placed at every $L_0=100$km.
The result for this scheme is consistent with the results obtained for $S_2(\mu=0.99)$ and especially $S_2(\mu=0.99)/\tau$ in Tabs.~\ref{tab:optperchanneluse} and \ref{tab:optpersecond}, respectively, for $n=8$. However, note that for the values in Tabs.~\ref{tab:optperchanneluse} and \ref{tab:optpersecond} we chose $p_{\mathrm{link}}=F_0=1$ and $\mu=\mu_0$, slightly different from the parameter choice for Fig.~\ref{fig:SKR_per_sec_tf}(c)
where $\mu_0=1$ and $F_0=0.99$ playing the role of an imperfect state parameter instead of $\mu_0$ (in addition, we have $p_{\mathrm{link}}=0.81$ for DR, and also two spins dephasing at any time step included).
Reiterating the previous discussions in Secs.~\ref{sec:Comparison: 2 vs. 4 vs. 8 segment repeaters}, the choice of $L_0 \sim 100$km seems not only highly compatible with existing classical repeater and fiber network architectures, but also offers a good balance between an improved memory-assisted loss scaling and an only limited addition of extra faulty elements.
Here now we found, in particular, that the standard DR scheme (i) provides another good choice in order to really benefit from these well balanced parameters.
Finally, we also considered the six-state QKD protocol \cite{sixstate} instead of BB84, but this only improved the final rates marginally. In the case of $\mu=0.98$ and $\mu_0=1$, the rate could be, in principle, improved significantly for $n=8$, but for these parameters, in practice, it is easier to use BB84 and $n=4$ instead. When considering sufficiently good error parameter values like $\mu=0.99$, such that $n=8$ outperforms $n=4$, then again there is only a minimal improvement by employing the six-state QKD protocol.
\begin{figure*}
\caption{Secret key rates per second. We always assume a coherence time $\tau_{\mathrm{coh}
\label{fig:SKR_per_sec_tf}
\end{figure*}
\section{Conclusion}\label{sec:Conclusion}
We presented a statistical model based on two random variables
and their probability-generating functions (PGFs)
in order to describe, in principle, the full statistics
of the rates obtainable in a memory-based quantum repeater chain.
The physical repeater model assumes a heralded initial
entanglement distribution with a certain elementary probability
for each repeater segment
(including fiber channel transmission and all link coupling efficiencies),
deterministic entanglement swapping to connect the segments,
and single-spin quantum memories at each repeater station that are subject
to time-dependent memory dephasing.
No active quantum error correction is performed on any of the repeater
``levels'', while our model does not even rely upon the basic assumption of any nested repeater level structure.
The two basic statistical variables associated with this
physical repeater model are the total repeater waiting time and the total, accumulated dephasing time.
In the context of an application in long-range quantum cryptography, our model corresponds to a form of memory-assisted quantum key distribution, for which we calculated the (asymptotic, primarily BB84-type) secret key rates as a figure of merit to assess the repeater performance against known benchmarks and all-optical quantum communication schemes.
Apart from the theoretical complexity that grows with the size of the repeater
(i.e., the number of repeater segments), it was clear from the start that experimentally the memory-assisted schemes of our model cannot go arbitrarily far while still producing a non-zero secret key rate. One motivation and goal of our work was to quantify this intuition and to provide an answer to the question whether it is actually beneficial, in a real setting, to add faulty memory stations to a quantum communication line. Existing works had their focus on the smallest repeaters with only two segments and one middle station. So, the aim was to further explore these smallest repeaters and then extend them to repeaters of a larger scale, answering the above question.
Within this framework, we determined an optimal repeater scheme
that belongs to the class of the fastest schemes (minimizing the average total waiting time and hence maximizing the long-distance entanglement distribution ``raw rate'') and, in addition, minimizes the average accumulated memory dephasing within the class of the fastest schemes. We have achieved this optimization for medium-size quantum repeaters with up to eight segments.
In particular, for the minimal dephasing, this led us to a scheme to ``swap as soon as possible''. The technically most challenging element of our treatment is to determine an explicit analytical expression for the random dephasing variable of the fast schemes and its PGF.
In order to confirm the correspondence of the minimum of the dephasing variable with the minimal QKD quantum bit error rate (for the variable related to memory dephasing), we calculated the relevant expectation values and compared the optimal scheme with schemes based on other, different swapping strategies. More generally, our formalism enables one to also consider mixed strategies in which different types of entanglement distribution and swapping can be combined, including the traditionally used doubling strategy that allows to systematically incorporate methods for quantum error detection (entanglement distillation).
Our new results especially apply to quantum repeaters beyond one middle station for which an optimization of the distribution and swapping strategies is no longer obvious. For the special case of three repeater segments, assuming only channel loss and memory dephasing, we showed that our optimal scheme gives the highest secret key rate among not only all the fastest schemes but among all schemes including overall slower schemes that may still potentially lead to a smaller accumulated dephasing. We conjecture that our optimal scheme also gives the highest secret key rate for more than three segments under the same physical assumptions. A rigorous proof of this is non-trivial, because the number of distinct swapping and distribution strategies grows fast with the number of repeater segments. Moreover, in a long-range QKD application, some of the spin qubits may be measured immediately which is generally hard to include in the statistical analysis and the optimization for all possible schemes; for three segments though we did include this additional complexity of the protocols. Towards applications beyond QKD, this extra variation may no longer be relevant.
We identified three criteria that should be satisfied by an optimal repeater scheme: distribute entanglement in parallel as fast as possible, store entanglement in parallel as little as possible, and swap entanglement as soon as possible. It is not always possible to satisfy these conditions at the same time, and we discussed specific schemes that are particularly good or bad with regards to some of the criteria. For example, a fully sequential repeater scheme is particularly slow, but avoids parallel storage of many spin qubits. Nonetheless, since it is overall slow, the fully sequential scheme can still accumulate more dephasing. We presented a detailed analysis comparing such different repeater protocols and approaches.
With regards to a more realistic quantum repeater modelling, we considered additional tools and parameters such as memory cut-offs, multiplexing,
initial state and swapping gate fidelities in order to identify
potential regimes in memory-assisted quantum key distribution beyond one middle station where, exploiting our optimized swapping strategy, it becomes useful to add further memory stations along the communication line and connect them via two-qubit swapping operations. Importantly, we found that the initial state and gate fidelities must exceed certain minimal values (generally depending on the specific QKD protocol including post-processing), as otherwise the sole faultiness of the spin-qubit preparations and operations prevents to obtain a non-zero secret key rate even when no imperfect quantum storage (no memory dephasing) at all takes place and independent of the finite channel transmission.
This effect becomes stronger with an increasing number of repeater nodes, scaling with the power of $2n-1$ for the error parameters in the QKD secret key rate. Once this minimal state and gate fidelity criterion is fulfilled and when the other experimental imperfections are included too, especially the time-dependent memory dephasing, it is essential to consider the exact secret key rates obtainable in optimized repeater protocols in order to conclude whether a genuine quantum repeater advantage over direct transmission schemes is possible or not.
This is what our work aimed at and achieved based on the standard notion of asymptotic QKD figures of merit.
By quantifying the influence of (within our physical model) basically all relevant experimental parameters on the final long-range QKD rate, we were able to determine the scaling and trade-offs of these parameters and analytically calculate exact, optimal rates. A quantum repeater of $n=L/L_0$ segments is thereby characterized by the parameter set $(p,a,\alpha)$ where $p$ is the entanglement distribution probability per segment (including the $n$-dependent channel transmission and zero-distance link coupling efficiency per segment), $a$ is the entanglement swapping success probability, and $\alpha$ is the inverse effective memory coherence time which, in most protocols, depends on $n$ via the quantum and classical communication times per distribution attempt (we also considered small-scale two-segment protocols without this dependence and ideas exist to minimize the impact of the inevitable signal waiting times for the elementary units of larger repeaters in combination with high experimental source and processing clock rates \cite{Cody_Jones}).
In addition, we have introduced a set of initial state and gate parameters $(\mu_0/F_0, \mu)$ where $\mu_0$ and $F_0$ can be adapted to the specific protocols. Additional memory parameters can be collected as $(m,M,B)$ where $m$ is the memory cut-off (maximal time at which any spin qubit is stored), $M$ is the number of simultaneously employed memory qubits in a simple multiplexing scenario with $M$ repeater chains used in parallel, and $B$ is the ``memory buffer'' (the number of memory qubits per half station in a single repeater chain).
In our work, we focussed on schemes with $a=1$ and $B=1$. The use of $B>1$ memories at each station would allow to continue the optical quantum state transfer even in segments that already possess successfully distributed states
and to potentially replace the earlier distributed lower-quality pairs (subject to memory dephasing) by the later distributed pairs.
We also did not put the main emphasis on the use and optimization of $m$, though we did include this option in some schemes. We found that $M>1$ leads to an effective improvement of the memory coherence time by a factor of $M$.
In this setting, the three essential experimental parameters that have to be sufficiently good are the link coupling efficiency (via $p$), the memory coherence time (via $\alpha$), and the state/gate error parameter $\mu_0$/$\mu$. While the latter must not go below the above-mentioned limits, generally two of these three parameters should be sufficiently good as a rule of thumb in order to exceed the repeaterless bound and obtain practically meaningful rates. If this is the case, or even better, if all three are of high quality, memory-assisted quantum key distribution based on heralded entanglement distribution and swapping without additional quantum error correction or detection is possible to allow Alice and Bob to share a secret key at a rate orders of magnitude faster than in all-optical quantum state transmission schemes. For instance, for a total distance of 800km and experimental parameter values that are highly demanding but not impossible (up to 10s coherence time, about 80\% link coupling, and state or gate infidelities in the regime of 1-2\%), one secret bit can be shared per second with repeater stations placed at every 100km, providing the best balance
between a minimal number of extra faulty repeater elements and a sufficient number of repeater stations for an improved loss scaling.
{\it Acknowledgement:}
We thank the BMBF in Germany for support via Q.Link.X/QR.X and
the BMBF/EU for support via QuantERA/ShoQC.
\appendix
\section{Derivation of Eq.~\eqref{eq:GKn}}\label{app:GKn}
In this section we derive the PGF $G_n(t)$ of the random variable $K_n$ defined via
\begin{equation}
K_n = \max(N_1, \ldots, N_n),
\end{equation}
where $N_i$ are the geometrically distributed random variables with parameter $p$. We have
\begin{equation}\label{eq:app:GKn}
\begin{split}
G_n(t) &= \sum^{+\infty}_{k_1, \ldots, k_n = 1} p q^{k_1 - 1} \ldots p q^{k_n - 1} t^{\max(k_1, \ldots, k_n)} \\
&= p^n t F_n(q, t),
\end{split}
\end{equation}
where the function $F_n(x, t)$ is defined as
\begin{equation}
F_n(x, t) = \sum^{+\infty}_{k_1, \ldots, k_n = 0} x^{k_1 + \ldots + k_n} t^{\max(k_1, \ldots, k_n)}.
\end{equation}
The series on the right-hand side of this definition converges for all $|x|<1$ and $|t| \leqslant 1$, since we have
\begin{equation}
|F_n(x, t)| \leqslant \sum^{+\infty}_{k_1, \ldots, k_n = 0} |x|^{k_1 + \ldots + k_n} = \frac{1}{(1-|x|)^n}.
\end{equation}
The function $F_n(x, t)$ can be written in a compact form, having only a finite number of terms. We have
\begin{equation}
\begin{split}
&\frac{F_n(x, t)}{1-t} = \sum^{+\infty}_{k_1, \ldots, k_n = 0} \sum^{+\infty}_{k = \max(k_1, \ldots, k_n)}
x^{k_1 + \ldots + k_n} t^k \\
&= \sum^{+\infty}_{k = 0} t^k \sum^k_{k_1, \ldots, k_n = 0} x^{k_1 + \ldots + k_n}
= \sum^{+\infty}_{k = 0} t^k \left(\frac{1 - x^{k+1}}{1-x}\right)^n.
\end{split}
\end{equation}
Expanding the $n$-th power on the right-hand side and applying simple algebraic transformations, we obtain the following
compact expression:
\begin{equation}
F_n(x, t) = \frac{1 - t}{(1 - x)^n t} \sum^n_{i = 0} (-1)^i \binom{n}{i} \frac{1}{1 - x^i t}.
\end{equation}
From Eq.~\eqref{eq:app:GKn} we derive the following expression for the PGF of $K_n$:
\begin{equation}
\begin{split}
G_n(t) &= (1 - t) \sum^n_{i = 0} (-1)^i \binom{n}{i} \frac{1}{1 - q^i t} \\
&= 1 + (1-t)\sum^n_{i = 1} (-1)^i \binom{n}{i} \frac{1}{1 - q^i t},
\end{split}
\end{equation}
which is exactly the expression presented in the main text.
\section{Trace identities}\label{app:Trace Identities}
We have
\begin{equation}\label{eq:Trid}
\begin{split}
{}_{23}\langle&\Psi^+| \tilde{\Gamma}_{\mu, 23}(\hat{\varrho}_{1234})|\Psi^+\rangle_{23} \\
&= \mu \cdot {}_{23}\langle\Psi^+| \hat{\varrho}_{1234}|\Psi^+\rangle_{23}
+ \frac{1 - \mu}{4} \Tr_{23}(\hat{\varrho}_{1234}).
\end{split}
\end{equation}
Here we show how to compute the quantities on the right-hand side of this equality. A
simple way is to work with density matrices. We use the order of basis elements
induced by the tensor product. From the one-qubit basis $(|0\rangle, |1\rangle)^T$
we obtain the two-qubit basis
\begin{equation}\label{eq:B2}
\begin{pmatrix}
|0\rangle \\
|1\rangle
\end{pmatrix}
\otimes
\begin{pmatrix}
|0\rangle \\
|1\rangle
\end{pmatrix}
=
\begin{pmatrix}
|00\rangle \\
|01\rangle \\
|10\rangle \\
|11\rangle
\end{pmatrix}.
\end{equation}
Taking the tensor product once again, we obtain the ordering of four-qubit basis vectors $|0000\rangle$, $|0001\rangle$,
$|0010\rangle$, $|0011\rangle$, $|0100\rangle$, $|0101\rangle$, $|0110\rangle$, $|0111\rangle$, $|1000\rangle$,
$|1001\rangle$, $|1010\rangle$, $|1011\rangle$, $|1100\rangle$, $|1101\rangle$, $|1110\rangle$, $|1111\rangle$. If a
four-qubit state is described by a density operator $\hat{\varrho}_{1234}$ which has a $16 \times 16$
density matrix $\varrho$ in the standard basis ordered as described above, then two-qubit partial diagonal states have the
following matrices in the basis \eqref{eq:B2}:
\begin{equation}\label{eq:D23}
\begin{split}
{}_{23}\langle 00|\hat{\varrho}_{1234}|00\rangle_{23} &= \rho[1, 2, 9, 10] \\
{}_{23}\langle 01|\hat{\varrho}_{1234}|01\rangle_{23} &= \rho[3, 4, 11, 12] \\
{}_{23}\langle 10|\hat{\varrho}_{1234}|10\rangle_{23} &= \rho[5, 6, 13, 14] \\
{}_{23}\langle 11|\hat{\varrho}_{1234}|11\rangle_{23} &= \rho[7, 8, 15, 16],
\end{split}
\end{equation}
where $\varrho[I]$, $I$ being a set of 1-based indices, is the submatrix of $\varrho$ with row and column indices in
$I$. For the off-diagonal states we have
\begin{equation}\label{eq:O23}
\begin{split}
{}_{23}\langle 01|\hat{\varrho}_{1234}|10\rangle_{23} &= \rho[3, 4, 11, 12 | 5, 6, 13, 14] \\
{}_{23}\langle 10|\hat{\varrho}_{1234}|01\rangle_{23} &= \rho[5, 6, 13, 14 | 3, 4, 11, 12],
\end{split}
\end{equation}
where $\varrho[I|J]$ is the submatrix of $\varrho$ with row indices in $I$ and column indices in $J$.
The state of the form given by Eq.~\eqref{eq:Drho}
\begin{equation}
\hat{\varrho} = \tilde{\Gamma}_{\mu}\bigl(F|\Psi^+\rangle\langle\Psi^+| + (1 - F)|\Psi^-\rangle\langle\Psi^-|\bigr)
\end{equation}
has the following density matrix in the basis \eqref{eq:B2}:
\begin{equation}
\varrho =
\frac{1}{4}
\begin{pmatrix}
1 - \mu & 0 & 0 & 0 \\
0 & 1 + \mu & 2\mu(2F - 1) & 0 \\
0 & 2\mu(2F - 1) & 1 + \mu & 0 \\
0 & 0 & 0 & 1 - \mu
\end{pmatrix}.
\end{equation}
Taking the Kronecker product of two states of this form, Eq.~\eqref{eq:Trid} together with the relations
Eqs.~\eqref{eq:D23}-\eqref{eq:O23} lead to the final form of the distributed state given by Eq.~\eqref{eq:rho14}.
\section{Computing PGFs of the sequential scheme}\label{app:SeqPGF}
\begin{figure*}
\caption{A visualization of the entanglement distribution process with the sequential scheme for $n=4$.}
\label{fig:seq1}
\end{figure*}
In the sequential scheme the number of steps $K_n$ and the dephasing $D_n$ are given by
\begin{equation}
K_n = N_1 + \ldots + N_n, \quad D_n = N_2 + \ldots + N_n.
\end{equation}
Their PGFs are thus the $n$-th and $(n-1)$-th power of the single-segment PGF:
\begin{equation}
G_n(t) = \left(\frac{p t}{1 - q t}\right)^n, \quad \tilde{G}_n(t) = \left(\frac{p t}{1 - q t}\right)^{n-1}.
\end{equation}
In the case of cutoff, the process of entanglement distribution is visualized in Fig.~\ref{fig:seq1}. There are zero or
more failure parts, with number of steps generating function $B^{[m]}_n(t)$, and one and only one success part, with
generating function $A^{[m]}_n(t)$. The total PGF $G^{[m]}_n(t)$ of the number of steps $K^{[m]}_n$ is thus given by
\begin{equation}
G^{[m]}_n(t) = \frac{A^{[m]}_n(t)}{1 - B^{[m]}_n(t)}.
\end{equation}
We start with the derivation of the failure parts's PGF. The PGF of the top line is clearly
\begin{equation}
G_0(t) = \frac{pt}{1-qt}.
\end{equation}
Among the rest $n-1$ lines there are $i$ lines that succeed, where $0 \leqslant i \leqslant n-2$, so we have to put $i$ $p$'s into $m$ places and the rest $m-i$ places will be taken by $q$'s. We thus have
\begin{equation}
B^{[m]}_n(t) = G_0(t) \sum^{n-2}_{i=0} \binom{m}{i}p^iq^{m-i}t^m.
\end{equation}
For the success part's PGF we have
\begin{equation}
A^{[m]}_n(t) = G_0(t) \sum^m_{j=n-1} \binom{j-1}{n-2}p^{n-1}q^{j-n+1} t^j,
\end{equation}
since the length of the success part can vary from $n-1$ to $m$ (we need to put at least $n-1$ $p$'s there). The position of the last $p$ is fixed, so we need to place $n-2$ $p$'s into $j-1$ places and the rest $j-n+1$ will be taken by $q$'s. Making substitution $j \to j-n+1$, we arrive to the expression \eqref{eq:Gmnt} of the main text.
The random variable for the waiting time of the scheme involving multiple cutoffs is given by
\begin{equation}
K_{n}^{\mathrm{seq},\vec{m}}= \tilde{N}^{(m_{n-1})}-m_{n-1}+\sum_{j=1}^{T_{n-1}} \left(K_{n-1,j}+m_{n-1}\right)\,.
\end{equation}
Exploiting that sums of independent random variables correspond to products of their PGFs and using \cite[Satz 3.8]{Klenke2020} for the sum one immediately obtains the result in the main text.
\section{Computing dephasing PGFs for parallel schemes}\label{app:PGF Parallel schemes}
In this section we derive explicit expressions for the PGFs of the dephasing random variables $D_n$ for different
schemes considered in the main text. All these schemes have the same property --- if the order of $N_i$'s is known then
one can obtain an analytical expression for the corresponding random variable $D_n$ explicitly. Having an explicit
expression for $D_n$, we can compute a part of its PGF corresponding to a given order of arguments. Combining these
parts for all possible ordering of arguments, we get the expression for PGF of $D_n$.
More formally, the space $\Omega = \mathbb{N}^n$ of elementary events consists of all $n$-vectors $\vec{N} = (N_1,
\ldots, N_n)$ of positive integers. The components $N_i$ are independent identically distributed (i.i.d.) random
variables with geometric distribution with success probability $p$, so $N_i$ is the number of attempts (including the
last successful one) of the $i$-th segment to distribute entanglement. The failure probability we denote $q = 1-p$. To
every point $\vec{N} = (N_1, \ldots, N_n) \in \Omega$ we assign the probability
\begin{equation}
\mathbf{P}(\vec{N}) = pq^{N_1 - 1} \ldots pq^{N_n-1} = p^n q^{N_1 + \ldots + N_n - n}.
\end{equation}
The sum of these probabilities is obviously 1, so we have a valid probability space $(\Omega, \mathbf{P})$.
The PGF of every component $N_i$ is given by the following simple expression:
\begin{equation}
g_{N_i}(t) = \frac{pt}{1-qt}.
\end{equation}
To find PGFs of more complicated random variables involving several components, we appropriately partition $\Omega$,
compute the partial PGF on each part and then combine these partial results into the full expression. For every permutation
$\pi \in S_n$ we define a subset of $\Omega$ which is determined by the corresponding relations between $n$ arguments.
For $n=2$ we have two permutations $(12)$ and $(21)$ with corresponding relations $N_1 \leqslant N_2$ and $N_2 < N_1$.
For $n=3$ we have six permutations and six corresponding relations
\begin{equation}
\begin{split}
&N_1 \leqslant N_2 \leqslant N_3 \quad N_1 \leqslant N_3 < N_2 \quad N_2 < N_1 \leqslant N_3 \\
&N_2 \leqslant N_3 < N_1 \quad N_3 < N_1 \leqslant N_2 \quad N_3 < N_2 < N_1.
\end{split}
\end{equation}
To make all these subsets non-overlapping, we use strict inequality between an inversion and non-strict inequality in
other positions between numbers in permutations. We thus have the following decomposition:
\begin{equation}\label{eq:omega}
\Omega = \bigsqcup_{\pi \in S_n} \Omega_\pi,
\end{equation}
where $\Omega_\pi$ is the subset determined by the relations corresponding to $\pi$. For any point $\vec{N} \in
\Omega_\pi$ we can obtain an explicit expression for $D_n$ for any scheme. In Table~\ref{tbl:1} we show all possible
relations between four arguments and the expression corresponding to the optimal and doubling schemes in the case of
$n=4$. Expressions corresponding to different $\pi$ might be the same, as can be seen for the doubling scheme.
The PGF of $D_n$ is defined as
\begin{equation}
\tilde{G}_n(t) = \sum^{+\infty}_{d=0} \mathbf{P}(D_n=d) t^d = \sum_{\vec{N} \in \Omega} \mathbf{P}(\vec{N}) t^{D_n(\vec{N})}.
\end{equation}
Using the decomposition in Eq.~\eqref{eq:omega}, we introduce the partial PGFs via
\begin{equation}
\tilde{G}_n(\pi|t) = \sum_{\vec{N} \in \Omega_\pi} p^nq^{N_1 + \ldots + N_n - n} t^{D_n(N_1, \ldots, N_n)},
\end{equation}
where $D_n(N_1, \ldots, N_n)$ is given explicitly as an appropriate linear combination of $N_i$'s. The total PGF $\tilde{G}_n(t)$ is then just the sum of all of these partial PGFs:
\begin{equation}
\tilde{G}_n(t) = \sum_{\pi \in S_n} \tilde{G}_n(\pi|t).
\end{equation}
We demonstrate computing these sums by an example for $n=4$. We have the correspondence
\begin{equation}
\pi = (2134) \to N_2 < N_1 \leqslant N_3 \leqslant N_4
\end{equation}
and the explicit expressions
\begin{equation}
\begin{split}
D^\star_4(N_1, N_2, N_3, N_4) &= N_4 - N_2, \\
D^{\mathrm{dbl}}_4(N_1, N_2, N_3, N_4) &= 2N_4 - N_2 - N_3.
\end{split}
\end{equation}
For the partial PGFs we have
\begin{widetext}
\begin{equation}
\begin{split}
\tilde{G}^\star_4(\pi|t) &= \sum^{+\infty}_{N_2=1}\sum^{+\infty}_{N_1=N_2+1}\sum^{+\infty}_{N_3=N_1}\sum^{+\infty}_{N_4=N_3} p^4 q^{N_1+N_2+N_3+N_4-4} t^{N_4 - N_2} = \frac{p^4}{1-q^4}\frac{q^3 t}{(1-qt)(1-q^2t)(1-q^3t)}, \\
\tilde{G}^{\mathrm{dbl}}_4(\pi|t) &= \sum^{+\infty}_{N_2=1}\sum^{+\infty}_{N_1=N_2+1}\sum^{+\infty}_{N_3=N_1}\sum^{+\infty}_{N_4=N_3} p^4 q^{N_1+N_2+N_3+N_4-4} t^{2N_4 - N_2- N_3} = \frac{p^4}{1-q^4}\frac{q^3 t}{(1-q^2t)(1-q^3t)(1-qt^2)}.
\end{split}
\end{equation}
\end{widetext}
Summing up the expression for all $\pi \in S_4$, we obtain the expressions for $\tilde{G}^\star_4(t)$ and
$\tilde{G}^{\mathrm{dbl}}_4(t)$ presented in the main text. For completeness, we also give the optimal PGFs for $n=2$ and $n=3$:
\begin{displaymath}
\begin{split}
\tilde{G}^\star_2(t) &= \frac{p^2}{1-q^2} \frac{1+qt}{1-qt}, \\
\tilde{G}^\star_3(t) &= \frac{p^3}{1-q^3} \frac{1+(q+2q^2)t-(2q^2+q^3)t^3-q^4t^4}{(1-qt)(1-q^2t)(1-qt^2)}.
\end{split}
\end{displaymath}
The size of the expressions grows rather quickly with $n$, so we do not present them explicitly for $n>4$.
We see that obtaining $\tilde{G}_n(t)$ reduces to computing sums of many geometrical series, which is a rather trivial
task. The only nontrivial part of this algorithm is its superexponential $n!$-complexity. So, this algorithm is applicable
only for small $n$; we used it up to $n=8$, which is of practical relevance.
\begin{table}[ht]
\begin{tabular}{|l|l|l|}
\hline
\hfil Permutation & \hfil $D^\star_4(\vec{N})$ & \hfil $D^{\mathrm{dbl}}_4(\vec{N})$ \\
\hline
$N_1 \leqslant N_2 \leqslant N_3 \leqslant N_4$ & $N_4-N_1$ & $2N_4-N_1-N_3$ \\
$N_1 \leqslant N_2 \leqslant N_4 < N_3$ & $2N_3-N_1-N_4$ & $2N_3-N_1-N_4$ \\
$N_1 \leqslant N_3 < N_2 \leqslant N_4$ & $N_2+N_4-N_1-N_3$ & $2N_4-N_1-N_3$ \\
$N_1 \leqslant N_3 \leqslant N_4 < N_2$ & $2N_2-N_1-N_3$ & $2N_2-N_1-N_3$ \\
$N_1 \leqslant N_4 < N_2 \leqslant N_3$ & $2N_3-N_1-N_4$ & $2N_3-N_1-N_4$ \\
$N_1 \leqslant N_4 < N_3 < N_2$ & $2N_2-N_1-N_4$ & $2N_2-N_1-N_4$ \\
$N_2 < N_1 \leqslant N_3 \leqslant N_4$ & $N_4-N_2$ & $2N_4-N_2-N_3$ \\
$N_2 < N_1 \leqslant N_4 < N_3$ & $2N_3-N_2-N_4$ & $2N_3-N_2-N_4$ \\
$N_2 \leqslant N_3 < N_1 \leqslant N_4$ & $N_4-N_2$ & $2N_4-N_2-N_3$ \\
$N_2 \leqslant N_3 \leqslant N_4 < N_1$ & $N_1-N_2$ & $2N_1-N_2-N_3$ \\
$N_2 \leqslant N_4 < N_1 \leqslant N_3$ & $2N_3-N_2-N_4$ & $2N_3-N_2-N_4$ \\
$N_2 \leqslant N_4 < N_3 < N_1$ & $N_1+N_3-N_2-N_4$ & $2N_1-N_2-N_4$ \\
$N_3 < N_1 \leqslant N_2 \leqslant N_4$ & $N_2+N_4-N_1-N_3$ & $2N_4-N_1-N_3$ \\
$N_3 < N_1 \leqslant N_4 < N_2$ & $2N_2-N_1-N_3$ & $2N_2-N_1-N_3$ \\
$N_3 < N_2 < N_1 \leqslant N_4$ & $N_4-N_3$ & $2N_4-N_2-N_3$ \\
$N_3 < N_2 \leqslant N_4 < N_1$ & $N_1-N_3$ & $2N_1-N_2-N_3$ \\
$N_3 \leqslant N_4 < N_1 \leqslant N_2$ & $2N_2-N_1-N_3$ & $2N_2-N_1-N_3$ \\
$N_3 \leqslant N_4 < N_2 < N_1$ & $N_1-N_3$ & $2N_1-N_2-N_3$ \\
$N_4 < N_1 \leqslant N_2 \leqslant N_3$ & $2N_3-N_1-N_4$ & $2N_3-N_1-N_4$ \\
$N_4 < N_1 \leqslant N_3 < N_2$ & $2N_2-N_1-N_4$ & $2N_2-N_1-N_4$ \\
$N_4 < N_2 < N_1 \leqslant N_3$ & $2N_3-N_2-N_4$ & $2N_3-N_2-N_4$ \\
$N_4 < N_2 \leqslant N_3 < N_1$ & $N_1+N_3-N_2-N_4$ & $2N_1-N_2-N_4$ \\
$N_4 < N_3 < N_1 \leqslant N_2$ & $2N_2-N_1-N_4$ & $2N_2-N_1-N_4$ \\
$N_4 < N_3 < N_2 < N_1$ & $N_1-N_4$ & $2N_1-N_2-N_4$ \\
\hline
\end{tabular}
\caption{Explicit expressions for the optimal and doubling dephasing for all possible relations between arguments in the case of $n=4$.} \label{tbl:1}
\end{table}
\section{Optimality for three segments}\label{app:Optimality 3 segments}
Here we will compare all possible schemes for a 3-segment repeater, when swapping is applied as soon as possible. We will not consider any scheme, which swaps only at the end or delays the entanglement swapping, as this increases the dephasing even further. For each scheme we calculate the random variables for the waiting time and the dephasing. In case of the dephasing the probability generating function is most useful, whereas for the waiting time we will only state the expectation value.
Moreover, we will consider two different types of schemes. The first type, which we will indicate by ``imm'', describes schemes where Alice and Bob measure their qubits immediately. This scenario is especially useful in QKD applications. The second type of schemes we consider is indicated by a subscript ``non'' and describes schemes, where Alice and Bob do not measure immediately and these types of schemes are important in non-QKD applications. A possible case of usage for those schemes is transferring quantum information between quantum computers by exchanging entangled photons. Here Alice and Bob will not measure their qubits until they share entanglement between each other.
\subsection{Sequential schemes}
\begin{figure}
\caption{Sequential arrangements of entanglement generation in a three-segment repeater. The number in each segment corresponds to the moment when it starts.}
\label{fig:Sequential a}
\label{fig:Sequential b}
\label{fig:Sequential c}
\label{fig:Different schemes 3 segments seq}
\end{figure}
Let us start with sequential schemes, where entanglement generation only takes places in one segment after another. There are three possibilities. First, one starts generating entanglement in Alice's or Bob's segment and always connects adjacent segments after the previous one has finished successfully. Note that here entanglement swapping is performed as soon as possible. We will call this scheme ``sequential a'', see Fig.~\ref{fig:Sequential a}.
The second possibility is given by starting with the left or right segment, followed by the segment on the opposite side. Thus, no entanglement swapping is possible. Finally, the middle segment is connected. Let us call this scheme ``sequential b'', see Fig.~\ref{fig:Sequential b}.
The third possible arrangement is given by starting in the middle, continuing with the left or right segment and finishing of with the remaining segment on the opposing side, see Fig.~\ref{fig:Sequential c}. All other sequential arrangements for three segments are equivalent to those three schemes.
These three sequential schemes share the same waiting time, which is
\begin{equation}
K^{\mathrm{seq}}_3=N_1+ N_2 +N_3,
\end{equation}
and has the expectation value
\begin{equation}
\mathbf{E}[K^{\mathrm{seq}}_3]=\frac{3}{p}.
\end{equation}
Obviously, the dephasing of the schemes differs, and we also have to distinguish between schemes measuring immediately and non-immediately. At first, let us consider immediate schemes, as it will turn out the random variables of the non-immediate schemes are just scaled by a factor of two, although it might not be the random variable of the same scheme. We find
\begin{equation}
\begin{split}
D^{\mathrm{seq,a}}_{3,\mathrm{imm}} &= N_2 + N_3 ,\\
D^{\mathrm{seq,b}}_{3,\mathrm{imm}} &= 2 N_2 + N_3, \\
D^{\mathrm{seq,c}}_{3,\mathrm{imm}} &= 2 N_1 + N_3 .
\end{split}
\end{equation}
Since \(N_2\) and \(N_3\) are i.i.d., the probability generating function (PGF) of \(D^{\mathrm{seq,a}}_{3,\mathrm{imm}}\) is given by
\begin{equation}
\tilde{G}^{\mathrm{seq,a}}_{3, \mathrm{imm}}(t)=g_{N_2}(t) \cdot g_{N_3}(t) = \left( \frac{pt}{1-qt} \right)^2.
\end{equation}
Due to the general relation
\begin{equation}
g_{2X}(t)=\mathbf{E}[t^{2X}]=\mathbf{E}[(t^2)^X]=g_X(t^2)
\end{equation}
valid for any discrete random variable \(X\), we have
\begin{align}
\tilde{G}^{\mathrm{seq,b}}_{3,\mathrm{imm}}(t)= g_{N_2}(t^2) \cdot g_{N_3}(t) = \frac{p^2 t^3}{(1-qt)(1-qt^2)}.
\end{align}
The same holds true for the PGF of the immediate measurement scheme ``sequential c'', because
\(N_1\) and \(N_2\) are i.i.d.. Thus, its PGF is also given by
\begin{equation}
\tilde{G}^{\mathrm{seq,c}}_{3,\mathrm{imm}}(t)= \frac{p^2 t^3}{(1-qt)(1-qt^2)},
\end{equation}
which shows, that this scheme is actually equivalent to ``sequential b'' and will not be considered separately in the later comparison.
On the other hand, for non-immediate measurements we find the random variables to be
\begin{equation}
\begin{split}
D^{\mathrm{seq,a}}_{3,\mathrm{non}} &= 2 D^{\mathrm{seq,a}}_{3,\mathrm{imm}} = 2\left( N_2 + N_3 \right), \\
D^{\mathrm{seq,b}}_{3,\mathrm{non}} &= 2 D^{\mathrm{seq,b}}_{3,\mathrm{imm}} = 2\left( 2 N_2 + N_3 \right), \\
D^{\mathrm{seq,c}}_{3,\mathrm{non}} &= 2 D^{\mathrm{seq,a}}_{3,\mathrm{imm}} = 2\left( N_1 + N_3 \right).
\end{split}
\end{equation}
By using the same argument as before, we find the corresponding PGFs
\begin{equation}
\begin{split}
G^{\mathrm{seq,a}}_{3,\mathrm{non}}(t) &= G^{\mathrm{seq,a}}_{3,\mathrm{imm}}(t^2) ,\\
G^{\mathrm{seq,b}}_{3,\mathrm{non}}(t) &= G^{\mathrm{seq,b}}_{3,\mathrm{imm}}(t^2) , \\
G^{\mathrm{seq,c}}_{3,\mathrm{non}}(t) &= G^{\mathrm{seq,a}}_{3,\mathrm{imm}}(t^2).
\end{split}
\end{equation}
Again, the scheme ``sequential c'' is equivalent to another scheme, but now it is ``sequential a''. Therefore, the non-immediate version of ``sequential c'' will not be treated separately from ``sequential a''.
\subsection{Two segments simultaneously at the start}
\begin{figure}
\caption{Possible arrangements of entanglement generation in a three-segment repeater, when two segments start simultaneously. The number in each segment corresponds to the moment when it starts.}
\label{fig:two start a}
\label{fig:two start b}
\label{fig:Different schemes 3 segments sim start}
\end{figure}
When we generate entanglement in two segments simultaneously, we can do that by starting with these two segments or by finishing with these two. Here we will consider the case where one starts with them and we only have two different arrangements. However, we still have to distinguish between measuring immediately or not.
For the first scheme in consideration, the middle and the left (or equivalently right) segment start generating entanglement at once. They swap as soon as both are done and then the last segment starts generating entanglement, see Fig.~\ref{fig:two start a}. Let us call this scheme ``start a''.
The dephasing random variables in this case are
\begin{equation}
\begin{split}
D^{\mathrm{start,a}}_{3,\mathrm{imm}}&=
\begin{cases}
N_2-N_1+N_3 & N_1 \leq N_2\\
2\left(N_1-N_2\right)+N_3 & N_2 < N_1
\end{cases}, \\
D^{\mathrm{start,a}}_{3,\mathrm{non}}&=2|N_1-N_2|+2N_3.
\end{split}
\end{equation}
The PGF of \(D^{\mathrm{start,a}}_{3,\mathrm{non}}\) is obviously reads as
\begin{equation}
\tilde{G}^{\mathrm{start,a}}_{3,\mathrm{non}}(t)=\tilde{G}_{2}(t^2) \cdot g_{N_3}(t^2) = \frac{p^3t^2(1+qt^2)}{(1-q^2)(1-qt^2)^2}.
\end{equation}
For immediate measurements use the methods presented in the previous section and derive the PGF of \(D^{\mathrm{start,a}}_{3,\mathrm{imm}}\)
\begin{equation}
\tilde{G}^{\mathrm{start,a}}_{3,\mathrm{imm}}(t) = \frac{p^3 t (1-q^2 t^3)}{(1-q^2) (1-q t)^2 (1-q t^2)}.
\end{equation}
The second scheme is realised when we start with both the left and the right segment at once. As in the second sequential scheme there is no swapping possible, when both segments finished and one has to wait for the middle segment. We will call this scheme ``start b''. In pictures, it can be seen in Fig.~\ref{fig:two start b}. Here we have for the dephasing random variables
\begin{equation}
\begin{split}
D^{\mathrm{start,b}}_{3,\mathrm{imm}} &= |N_1-N_3|+2N_2, \\
D^{\mathrm{start,b}}_{3,\mathrm{non}} &= 2|N_1-N_3|+4N_2 = 2D^{\mathrm{start,b}}_{3,\mathrm{imm}}.
\end{split}
\end{equation}
We can simplify the calculation, by considering the immediate scheme first and using \(g_{2X}(t)=g_X(t^2)\). The PGF is given by
\begin{displaymath}
\tilde{G}^{\mathrm{start,b}}_{3,\mathrm{imm}}(t) = \tilde{G}_{2}(t) \cdot g_{2N_3}(t)
= \frac{p^3 t^2 (1+qt)}{(1-q^2)(1-qt)(1-qt^2)}.
\end{displaymath}
Hence, the PGF of the non-immediate version is simply
\begin{align}
\tilde{G}^{\mathrm{start,b}}_{3,\mathrm{non}}(t) = \tilde{G}^{\mathrm{start,b}}_{3,\mathrm{imm}}(t^2).
\end{align}
The waiting time is the same for both schemes in this subsection and amounts to
\begin{equation}
K^{\mathrm{simult.}}_3=\max(N_1,N_2)+N_3,
\end{equation}
with an expectation value of
\begin{equation}
\mathbf{E}[K^{\mathrm{simult.}}_3]=\frac{5-3p}{(2-p)p}.
\end{equation}
\subsection{Two segments simultaneously at the end}
\begin{figure}
\caption{Possible arrangements of entanglement generation in a three-segment repeater, when only one segment starts and the rest finishes simultaneously. The number in each segment corresponds to the moment when it starts.}
\label{fig:two end a}
\label{fig:two end b}
\label{fig:Different schemes 3 segments sim end}
\end{figure}
Finally, the last possible arrangement of two simultaneous segments is to start them in the last step. The waiting time stays the same as in the previous case, but again, there are two possibilities for the dephasing and two to perform measurements,i.e. immediate or non-immediate. The first scheme is realised, when we start with the segment in the middle and when it finishes, the left and right segment start generating entanglement simultaneously. We will call this scheme ``end a'' and it is shown schematically in Fig.~\ref{fig:two end a}. In this case the dephasing random variables are given by
\begin{equation}
\begin{split}
D^{\mathrm{end,a}}_{3,\mathrm{imm}} &= N_1 + N_3 , \\
D^{\mathrm{end,a}}_{3,\mathrm{non}} &=2\max(N_1,N_3),
\end{split}
\end{equation}
with the PGFs
\begin{equation}
\begin{split}
\tilde{G}^{\mathrm{end,a}}_{3,\mathrm{imm}}(t) &= \tilde{G}^{\mathrm{seq,a}}_{3, \mathrm{imm}}(t) = \left( \frac{pt}{1-qt} \right)^2, \\
\tilde{G}^{\mathrm{end,a}}_{3,\mathrm{non}}(t) &= G^{\mathrm{par}}_n(t^2) = \frac{p^2 t^2 (1+qt^2)}{(1-qt^2)(1-q^2t^2)}.
\end{split}
\end{equation}
The second possibility is to start with the left or right segment and after it finished generate entanglement
simultaneously in the remaining segments. The schemes and random variables are equivalent independent whether one starts
with the left or right segment. We will call this scheme ``end b'' and its schematic representation, when starting with
the left segment, is shown in Fig.~\ref{fig:two end b}. Similarly to the scheme ``start a'', the dephasing random
variables depended on the order of successful entanglement generation.
Let us consider the scheme where we do not measure immediately as an example. First, assume that we started with the
left segment and it finished successfully after \(N_1\) attempts. Then both the middle and the right segment start
generating entanglement simultaneously. If the middle segments succeeds first after \(N_2\) attempts, we can swap
immediately and again have only one segment waiting. Eventually, the right segment will succeed after \(N_3\) attempts,
and we can also swap it. In total the dephasing will equal \(D^{\mathrm{end,b}}_{3,\mathrm{non}}=2N_3\), because
\(2N_2\) cancels out. This is the optimal case of this scheme.
Alternatively, it could also happen that the right segment finishes first, and we have two segments waiting for the
middle to succeed. In this case, we have \(D^{\mathrm{end,b}}_{3,\mathrm{non}}=4N_2-2N_3\). Hence, in total the
dephasing is
\begin{equation}
D^{\mathrm{end,b}}_{3,\mathrm{non}}=
\begin{cases}
2N_3 & N_3 \geq N_2 \\
4N_2-2N_3 & N_3 < N_2
\end{cases}.
\end{equation}
A similar consideration yields the dephasing random variable of the immediate measurement scheme to be \begin{equation}
D^{\mathrm{end,b}}_{3,\mathrm{imm}}=
\begin{cases}
N_3 & N_3 \geq N_2 \\
2N_2-N_3 & N_3 < N_2
\end{cases}.
\end{equation}
As mentioned a few times so far, we can exploit that \(g_{2X}(t)=g_X(t^2)\), and thus we calculate the PGF of the immediate scheme first, which reads as
\begin{equation}
\tilde{G}^{\mathrm{end,b}}_{3,\mathrm{imm}}(t) = \frac{p^2 t \left(1-q^2 t^3\right)}{(1-qt) \left(1- q^2 t\right) \left(1-q t^2\right)}.
\end{equation}
Therefore, the PGF of of \(D^{\mathrm{end,b}}_{3,\mathrm{non}}\) is given by
\begin{equation}
\tilde{G}^{\mathrm{end,b}}_{3,\mathrm{non}}(t) = \tilde{G}^{\mathrm{end,b}}_{3,\mathrm{imm}}(t^2),
\end{equation}
and we have covered all possibles schemes of this subsection.
\subsection{Overlapping schemes}
Before, considering fully parallel schemes, we turn our attention to a mixture of the previous simultaneous schemes. We will call the schemes of this section overlapping schemes. The procedure is as follows, we start generating entanglement in two segments simultaneously and as soon as one of the two segments finishes, we start with the remaining one as well. Thus, the two processes of entanglement generation are overlapping, explaining the naming. In Fig.~\ref{fig:Different overlapping schemes} a schematic version of the overlapping schemes can be seen.
\begin{figure}
\caption{Possible arrangements of entanglement generation in a three-segment repeater, when two segments start simultaneously and the remaining segment starts as soon as one is successful. The number in each segment corresponds to the moment when it starts and the star indicates that this segment starts as soon as one of the others finished.}
\label{fig:Overlapping a}
\label{fig:Overlapping b}
\label{fig:Different overlapping schemes}
\end{figure}
There are two different possible arrangements presented in Fig.~\ref{fig:Overlapping a} and Fig.~\ref{fig:Overlapping
b}. In the former one the left (or equivalently the right) and the middle segment start from the beginning. This scheme
will be called ``overlapping, a''. The latter scheme starts with both outer segments and will be called ``overlapping,
b".
For the scheme ``overlapping, a'' we find with immediate measurements the dephasing random variable to be
\begin{equation}
D^{\mathrm{over,a}}_{3,\mathrm{imm}}=
\begin{cases}
N_3 &\Omega_1 \\
2(N_2-N_1)-N_3 & \Omega_2 \\
N_1-N_2+N_3 &\Omega_3 \\
\end{cases}
\end{equation}
where we have chosen the partition $\Omega = \mathbb{N}^3 = \Omega_1 \sqcup \Omega_2 \sqcup \Omega_3$ given by the following inequalities:
\begin{equation}
\begin{split}
\Omega_1 &= N_1 \leq N_2, N_2-N_1 \leq N_3, \\
\Omega_2 &= N_1 < N_2, N_2-N_1 > N_3, \\
\Omega_3 &= N_2< N_1.
\end{split}
\end{equation}
The dephasing varies depending on the order in which the segments finish, since one cannot swap or measure depending on
which segment is done first. Thus, we have three different cases. One can calculate the full PGF of the dephasing in a
similar way to the previous schemes and finds
\begin{align}
\tilde{G}^{\mathrm{over,a}}_{3,\mathrm{imm}}(t) = \frac{p^3 t (1 + q - 2 q^2 t - q t^2 + q^4 t^4)}
{ (1 - q^2) (1 -q t)^2 (1 - q^2 t) (1\! -q t^2)}.
\end{align}
For the non-immediate version of the scheme ``overlapping, a'', we do not have to take the measurements into account,
but this still does not result in more symmetries simplifying the expression. Hence, one has to consider all possible
orders separately and we find the dephasing to be
\begin{equation}
D^{\mathrm{over,a}}_{3,\mathrm{non}}=
\begin{cases}
2N_3 & \Omega_1 \\
2\left(2\left(N_2-N_1\right)-N_3\right) & \Omega_2 \\
2N_3 & \Omega_3 \\
2\left(N_1-N_2\right) & \Omega_4 \\
\end{cases}
\end{equation}
where the partition in this case is given by
\begin{equation}
\begin{split}
\Omega_1 &= N_1 \leq N_2, N_2-N_1 \leq N_3, \\
\Omega_2 &= N_1 < N_2, N_2-N_1 > N_3, \\
\Omega_3 &= N_2< N_1, N_1-N_2 \leq N_3, \\
\Omega_4 &= N_2< N_1, N_1-N_2 > N_3.
\end{split}
\end{equation}
The resulting PGF reads as
\begin{displaymath}
\tilde{G}^{\mathrm{over,a}}_{3,\mathrm{non}}(t) = \frac{p^3 t^2 (1 +2 q - q (1 + q) t^4 - q^3 t^6)}
{(1 - q^2) (1 - q t^2) (1 - q^2 t^2) (1 - q t^4)}.
\end{displaymath}
The other overlapping scheme possesses more symmetry, thus we find more compact expressions for the random variables. It mainly depends on the relative difference of steps between the outer segments. We find for the immediate and non-immediate scheme
\begin{equation}
\begin{split}
D^{\mathrm{over,b}}_{3,\mathrm{imm}} &=
\begin{cases}
2N_2 - \abs{N_1 - N_3} & \abs{N_1-N_3} < N_2 \\
\abs{N_1-N_3} & \abs{N_1-N_3} \geq N_2
\end{cases}, \\
D^{\mathrm{over,b}}_{3,\mathrm{non}} &=
\begin{cases}
4N_2 - 2\abs{N_1- N_3} & \abs{N_1-N_3} < N_2 \\
2\abs{N_1\!-\!N_3} \! & \abs{N_1-N_3} \geq N_2
\end{cases}.
\end{split}
\end{equation}
By case analysis we derive the PGFs
\begin{equation}
\begin{split}
\tilde{G}^{\mathrm{over,b}}_{3,\mathrm{imm}}(t) &= \frac{p^3 t (t + q (2 - t^2 (1 + q + q^2 t)))}{(1 - q^2) (1 - q t) (1 - q^2 t) (1 - q t^2)}, \\
\tilde{G}^{\mathrm{over,b}}_{3,\mathrm{non}}(t) &= \tilde{G}^{\mathrm{over,b}}_{3,\mathrm{imm}}(t^2).
\end{split}
\end{equation}
Finally, the only missing piece is the waiting time of the overlapping schemes and its expectation value. The random variable of the waiting time is
\begin{equation}
K^{\mathrm{over}}_3 = \min(N_1,N_2) + \max(|N_1-N_2|,N_3).
\end{equation}
Its expectation value is found to be
\begin{equation}
E[K^{\mathrm{over}}_3] = \frac{8 - 3p \left(3 - p \right)}{p\left(2 - p\right)^2}.
\end{equation}
\subsection{Parallel schemes}
Here we only consider the potentially optimal scheme, since all parallel schemes posses the same raw rate, but differ in dephasing. In the optimal scheme the dephasing is minimized, such that it has the best secret key rate of all schemes of this class.
\begin{table}[ht]
\begin{tabular}{|l|l|l|}
\hline
\hfil Domain & \hfil $D^\star_{3, \mathrm{non}}$ & \hfil $D^\star_{3, \mathrm{imm}}$ \\
\hline
$N_1 \leqslant N_2 \leqslant N_3$ & $2(N_3 - N_1)$ & $N_3 - N_1$ \\
$N_1 \leqslant N_3 < N_2$ & $2(2N_2 - N_1 - N_3)$ & $2N_2 - N_3 - N_1$ \\
$N_2 < N_1 \leqslant N_3$ & $2(N_3 - N_2)$ & $N_1 + N_3 - 2N_2$ \\
$N_2 \leqslant N_3 < N_1$ & $2(N_1 - N_2)$ & $N_1 + N_3 - 2N_2$ \\
$N_3 < N_1 \leqslant N_2$ & $2(2N_2 - N_1 - N_3)$ & $2N_2 - N_3 - N_1$ \\
$N_3 < N_2 < N_1$ & $2(N_1 - N_3)$ & $N_1 - N_3$ \\
\hline
\end{tabular}
\caption{The values of $D^\star_{3, \mathrm{non}}$ and $D^\star_{3, \mathrm{imm}}$ on the domains of the partition.}\label{tbl:D3nm}
\end{table}
The waiting time is \(K^{\mathrm{par}}_3=\max(N_1,N_2,N_3)\) and following \eqref{eq:Knpar} or Appendix~\ref{app:GKn} its expectation value is
\begin{equation}
E[K^{\mathrm{par}}_3]=\frac{1 + q \left(4 + 3 q \left(1 + q\right)\right)}{1 + q - q^3 - q^4}.
\end{equation}
The dephasing PGF can be computed with our partitioning approach. The six domains and the values of the dephasing variables in these domains are given in Table~\ref{tbl:D3nm}. The final result reads as
\begin{displaymath}
\begin{split}
\tilde{G}^\star_{3,\mathrm{non}}(t) &= \frac{p^3}{1-q^3} \frac{1+q(1+2q)t^2-q^2(2+q)t^6-q^4t^8}{(1-qt^2)(1-q^2t^2)(1-qt^4)} \\
\tilde{G}^\star_{3,\mathrm{imm}}(t) &= \frac{p^3}{1 - q^3} \frac{1+q^2t-2q^3t^2-2q^2t^3+q^3t^4+q^5t^5}{(1-qt)^2(1-q^2t)(1-qt^2)}
\end{split}
\end{displaymath}
\subsection{Comparisons}
Finally, as we have calculated all necessary statistical quantities we are able to compare the previously discussed schemes. Again as a remark, we only considered schemes here, which swap as soon as possible, as delaying the entanglement swapping increases the dephasing, which in turn decreases the SKR.
First, we consider the immediate measurement schemes. In Fig.~\ref{fig:Comparison_3_segments_immediate tau=0.1} ($\tau_{\mathrm{coh}}= \unit[0.1]{s}$) and Fig.~\ref{fig:Comparison_3_segments_immediate tau=10} ($\tau_{\mathrm{coh}}= \unit[10]{s}$), one can see a comparison of all immediate measurement schemes for a three-segment repeater using the previously discussed schemes. In both figures the SKR of the ``optimal'' scheme is represented in orange. As mentioned earlier, the scheme ``seq, c'' is equivalent to ``seq, b'' in this setting and thus not considered separately. For both coherence times the optimal schemes outperforms all other schemes. Especially for shorter distances, the optimal scheme performs clearly better than others. Only for longer distances, where the rate of any three-segment repeater drops, the schemes ``over, b'', ``over, a'' and ``end, b'' catch up, but do not surpass it. Typically, one would not use this regime of a repeater, as the rates are too low. Additionally, in the limit of increasing hardware resources, i.e. \( p_{\mathrm{link}} \rightarrow 1 ,\; \mu \rightarrow 1 , \; \mu_0 \rightarrow 1 \), the optimal scheme keeps performing the best. Thus, we conclude that the immediate measurement version of the optimal scheme is truly optimal for \(n \leq 3\).
Next, in Fig.~\ref{fig:Comparison_3_segments_non tau=0.1} ($\tau_{\mathrm{coh}}= \unit[0.1]{s}$) and Fig.~\ref{fig:Comparison_3_segments_non tau=10} ($\tau_{\mathrm{coh}}= \unit[10]{s}$) one can see the same comparison of different swapping schemes using non-immediate measurements. Again, the ``optimal" scheme is presented in orange. This time the sequential schemes ``seq, a'' and ``seq, c'' are equivalent and thus are not considered separately.
As one can see, the optimal scheme outperforms all other schemes in the ideal case when \(\mu=\mu_0=1\) for all choices of \(\tau_{\mathrm{coh}}\) and \(p_{\mathrm{link}}\). Furthermore, it also provides the highest secret key rate in the non-ideal case until close to the drop-off. The scheme ``end a'' surpasses it only at those distances either close to or after both start declining dramatically, thus increasing the achievable distance. As discussed before, one typically would not use the regime of an repeater. However, if the main goal is to achieve the longest achievable distance possible, then the scheme ``end a'' performs the best.
In the end, the optimal scheme provides the best secret key rate under most realistic use scenarios. Moreover, it is truly optimal in the limit of increasing hardware parameters, i.e. \( p_{\mathrm{link}} \rightarrow 1 ,\; \mu \rightarrow 1 , \; \mu_0 \rightarrow 1 \). Thus, it will be beneficial to use the ``optimal'' scheme as technology progresses and the hardware resources increase. Hence, our conclusion for non-immediate schemes is that the ``optimal" scheme is optimal under improving hardware parameters for \(n \leq 3\).
We conjecture that the same is true for both immediate and non-immediate measurement schemes for all \(n\geq 3\)-segment repeaters. This should be investigated in future research.
\begin{figure*}
\caption{Comparison of secret key rates of three-segment repeaters performing \emph{immediate}
\label{fig:Comparison_3_segments_immediate tau=0.1}
\end{figure*}
\begin{figure*}
\caption{Comparison of secret key rates of three-segment repeaters performing \emph{immediate}
\label{fig:Comparison_3_segments_immediate tau=10}
\end{figure*}
\begin{figure*}
\caption{Comparison of secret key rates of three-segment repeaters performing \emph{non-immediate}
\label{fig:Comparison_3_segments_non tau=0.1}
\end{figure*}
\begin{figure*}
\caption{Comparison of secret key rates of three-segment repeaters performing \emph{non-immediate}
\label{fig:Comparison_3_segments_non tau=10}
\end{figure*}
\section{Comparison of ``optimal" with fully sequential and Alice immediately measuring (n=8)}\label{app:8segmentsimmediate}
The fully sequential scheme, in which repeater segments are sequentially filled with entangled pairs from, for example, left to right is the overall slowest scheme leading to the smallest raw rates. However, a potential benefit is that parallel qubit storage can be almost entirely avoided. More specifically, when the first segment on the left is filled and waiting for the second segment to be filled too, the first segment waits for a random number of $N_2$ steps, whereas the second segment always only waits for one constant dephasing unit
(for each distribution attempt in the second segment). Thus, omitting the constant dephasing in each segment, the accumulated time-dependent random dephasing of the fully sequential scheme has only contributions from a single memory pair subject to memory dephasing at any elementary time step.
On average, this gives a total dephasing of $(n-1)/p$ which is the sum of the average waiting time in one segment for segments 2 through $n$, as discussed
in detail in the main text.
In a QKD application, Alice's qubit can be measured immediately
(and so can Bob's qubit at the very end when the entangled pair of the most right segment is being distributed). This way there is another factor of $1/2$ improvement possible for the effective dephasing, since at any elementary time step there is always only a single memory qubit dephasing instead of a qubit pair. In Fig.~\ref{fig:Comparison_8_segments_imm_vs_non_imm}, for eight repeater segments, we compare this fully sequential scheme and immediate measurements by Alice and Bob with the ``optimal'' scheme (parallel distribution and swap as soon as possible) where Alice and Bob store their qubits during the whole long-distance distribution procedure to do the BB84 measurements only at the very end. We see that a QKD protocol in which Alice and Bob measure their qubits immediately can be useful in order to go a bit farther. However, note that in the ``optimal'' scheme Alice and Bob may also measure their qubits immediately, resulting in higher rates but also requiring a more complicated rate analysis.
\begin{figure*}
\caption{Comparison of eight-segment repeaters for a total distance \(L\) and different experimental parameters. The ``optimal'' scheme (red) performing BB84 measurements at the end is compared with the fully sequential scheme (orange without memory cut-off, green with cut-off) performing immediate measurements on Alice's / Bob's sides.}
\label{fig:Comparison_8_segments_imm_vs_non_imm}
\end{figure*}
\section{Mixed strategies for distribution and swapping}\label{app:mixedstr}
In this appendix we shall illustrate that our formalism based on the calculation
of PGFs for the two basic random variables is so versatile that we can also obtain the rates for all kinds of mixed strategies. This applies to both the initial entanglement distributions and the entanglement swappings. In fact, for the case of three repeater segments ($n=3$), we have already explicitly calculated the secret key rates for all possible schemes with swapping as soon as possible, but with variations in the initial distribution strategies, see
App.~\ref{app:Optimality 3 segments}. This enabled us to consider schemes that are overall slower (exhibiting smaller raw rates), but can have a smaller accumulated dephasing. While swapping as soon as possible is optimal with regards to a minimal dephasing time, it may sometimes also be useful to consider a different swapping strategy. The most commonly considered swapping strategy is doubling which implies that it can sometimes happen that neighboring, ready segments will not be connected, as this would be inconsistent with a doubling of the covered repeater distances at each step. A conceptual argument for doubling could be that for a scalable (nested) repeater system one can incorporate entanglement distillation steps in a systematic way. A theoretical motivation to focus on doubling has been that rates are more easy to calculate -- a motivation that is rendered obsolete through the present work, at least for repeaters of size up to $n=8$. Nonetheless we shall give a few examples for mixed strategies for $n=4$ and $n=8$ segments.
For $n=4$ segments, in addition to those schemes discussed in the main text, let us consider another possibility where we distribute entanglement over the first three segments in the optimal way and then extend it
over the last segment. Note that this scheme is a variation of the swapping strategy, while the initial distributions still occur in parallel. As a consequence, it can happen that either segment 4 waits for the first three segments to accomplish their distributions and connections or the first three segments have to wait for segment 4. This part of the dephasing corresponds to the last term in the next equation below. The scheme serves as an illustration of the rich choice of possibilities for the swapping
strategies even when only $n=4$. We have
\begin{equation}\label{eq:D4-3-1}
\begin{split}
D^{31}_4(N_1, N_2, N_3, N_4) &= D^\star_3(N_1, N_2, N_3) \\
&+ |\max(N_1, N_2, N_3) - N_4|.
\end{split}
\end{equation}
The PGF of this random variable reads as
\begin{equation}
\tilde{G}^{31}_4(t) = \frac{p^4}{1-q^4} \frac{P^{31}_4(q, t)}{Q^{31}_4(q, t)},
\end{equation}
where the numerator and denominator are given by
\begin{displaymath}
\begin{split}
P^{31}_4(q, &t) = 1 + (q^2 + 3 q^3) t + (q + 3 q^2 - q^4 - q^5 ) t^2 \\
&+ (-2 q^2 - 4 q^3 - 4 q^4 + q^5 + q^6) t^3 \\
&+ (-q^2 - 3 q^3 - q^4 -3 q^6 - 3 q^7 ) t^4 \\
&+ (-2 q^2 - q^3 + 2 q^4 - 2 q^6 + q^7 + 2 q^8 ) t^5 \\
&+ (3 q^3 + 3 q^4 + q^6 + 3 q^7 + q^8 ) t^6 \\
&+ (-q^4 - q^5 + 4 q^6 + 4 q^7 + 2 q^8 ) t^7 \\
&+ (q^5 + q^6 - 3 q^8 - q^9 ) t^8 - (3 q^7 + q^8 ) t^9 - q^{10} t^{10}, \\
Q^{31}_4(q, &t) = (1-qt)(1-q^2t)(1-q^3t)(1-qt^2)\\
&\times (1-q^2t^2)(1-qt^3).
\end{split}
\end{displaymath}
If we take the derivatives (see Eq.~\eqref{eq:PGF}),
we can obtain the following relation,
\begin{equation}
\mathbf{E}[D^{\mathrm{dbl}}_4] = \mathbf{E}[D^{31}_4].
\end{equation}
This means that the two random variables have the same expectation values, even though their distributions are different. For
the secret key fraction we need the averages of the exponential of these variables, which essentially leads to the values of the
corresponding PGFs (see Eq.~\eqref{eq:PGF_2}). These do differ, as Fig.~\ref{fig:DD} illustrates. It shows the ratio
\begin{equation}\label{eq:r}
\frac{\mathbf{E}[e^{-\alpha D^{31}_4}]}{\mathbf{E}[e^{-\alpha D^{\mathrm{dbl}}_4}]} =
\frac{\tilde{G}^{31}_4(e^{-\alpha})}{\tilde{G}^{\mathrm{dbl}}_4(e^{-\alpha})},
\end{equation}
as a function of $\alpha$. The two random variables have the same average, but the average $\mathbf{E}[e^{-\alpha
D^{31}_4}]$ is larger than the other, so in the scheme corresponding to the random variable given by Eq.~\eqref{eq:D4-3-1}, the
distributed state has a higher fidelity than the final state in the doubling scheme.
\begin{figure}
\caption{The ratio given by Eq.~\eqref{eq:r}
\label{fig:DD}
\end{figure}
\begin{figure}
\caption{The ratio in Eq.~\eqref{eq:r2}
\label{fig:Ee}
\end{figure}
\begin{figure}
\caption{The ratio in Eq.~\eqref{eq:r3}
\label{fig:Ea}
\end{figure}
For the case $n=8$, among a large number of other possibilities to swap the segments, we consider the following three (in addition, the doubling and optimal schemes are discussed in the main text). The
first scheme is to swap the two halves of the repeater in the optimal way (for four segments) and then swap the two larger
segments. We loosely denote the dephasing variable of these scheme as $D^{44}_8$, whose definition reads as
\begin{equation}
\begin{split}
D^{44}_8&(N_1, \ldots, N_8) = D^\star_4(N_1, \ldots, N_4) \\
&+ D^\star_4(N_5, \ldots, N_8) \\
&+ |\max(N_1, \ldots, N_4) - \max(N_5, \ldots, N_8)|.
\end{split}
\end{equation}
Another possibility is to divide the repeater in four pairs, swap them and then swap the four larger segments optimally.
The expression for this dephasing variable $D^{2222}_8$ is a straightforward translation of this description:
\begin{equation}
\begin{split}
D^{2222}_8&(N_1, \ldots, N_8) = |N_1 - N_2| + \ldots + |N_7 - N_8| \\
&+ D^\star_4(\max(N_1, N_2), \ldots, \max(N_7, N_8)).
\end{split}
\end{equation}
Finally, we can divide the segments into three groups, consisting of two, four, and two segments. The middle group we
swap optimally (for four segments), and then we swap the three larger segments in the optimal way (for three segments). The
definition of the corresponding random variable $D^{242}_8$ reads as
\begin{equation}
\begin{split}
D^{242}_8&(N_1, \ldots, N_8) = |N_1 - N_2| + |N_7 - N_8| \\
&+ D^\star_4(N_3, \ldots, N_6) + D^\star_3(\max(N_1, N_2), \\
&\max(N_3, \ldots, N_6), \max(N_7, N_8)).
\end{split}
\end{equation}
The PGFs of all these variables have all the same form,
\begin{equation}
\frac{p^8}{1 - q^8} \frac{P(q, t)}{Q(q, t)},
\end{equation}
with appropriate polynomials $P(q, t)$ and $Q(q, t)$. The numerator polynomials $P(q, t)$ are quite large and contain
around one thousand terms, so we do not present them here.
We can compare the performances of different schemes by plotting the ratios
\begin{equation}\label{eq:r2}
\frac{\mathbf{E}[e^{-\alpha D^{\mathrm{sch}}_8}]}{\mathbf{E}[e^{-\alpha D^{\mathrm{opt}}_8}]} =
\frac{\tilde{G}^{\mathrm{sch}}_8(e^{-\alpha})}{\tilde{G}^{\mathrm{opt}}_8(e^{-\alpha})},
\end{equation}
similar to Eq.~\eqref{eq:r}, for $\mathrm{sch} = \mathrm{dbl}, 2222, 242, 44$. We see that among the five
schemes the doubling scheme is the worst with regards to dephasing, and the scheme 44 is the closest to the optimal scheme, see Fig.~\ref{fig:Ee}.
This means that the commonly used parallel-distribution doubling scheme, though fast in terms of $K_8$, is really inefficient in terms of dephasing $D_8$
by disallowing to swap when neighboring segments are ready
on all ``nesting'' levels \cite{Shchukin2021}.
\section{Two-Segment ``Node-Receives-Photon'' Repeaters}
\label{app:nrp}
Figure~\ref{fig:NRP_Contour_2_segments}
shows the BB84 rates in a two-segment quantum repeater
based on the NRP concept with one middle station
receiving optical quantum signals sent from
two outer stations at Alice and Bob.
By circumventing the need for extra classical communication
and thus significantly reducing the effective memory dephasing,
the minimal state and gate fidelity values can even be kept constant
over large distance regimes.
For the experimental clock rate we have chosen $\tau_{\mathrm{clock}}=\unit[10]{MHz}$,
limited by the local interaction and processing times
of the light-matter interface at the middle station.
\begin{figure*}
\caption{Contour plots illustrating the minimal fidelity requirements to overcome the PLOB bound by a two-segment NRP repeater for different parameter sets. In all contour plots, \(\mu = \mu_0\), \(\tau_{\mathrm{clock}
\label{fig:NRP_Contour_2_segments}
\end{figure*}
\section{Calculation for Cabrillo's scheme}\label{app:cabrillo}
First, we consider two entangled states of a single-rail qubit with a quantum memory ($\gamma\in\mathbb{R}$)
\begin{align}
\frac{1}{1+\gamma^2} &\left[ \ket{\uparrow,\uparrow,0,0} +\gamma \ket{\uparrow,\downarrow,0,1} \right. \nonumber \\
&\hspace{20pt} + \left. \gamma\ket{\downarrow,\uparrow,1,0}+\gamma^2\ket{\downarrow,\downarrow,1,1} \right].
\end{align}
After applying a lossy channel with transmission parameter $\eta=p_{\mathrm{link}}\exp(-\frac{L_0}{2L_{\mathrm{att}}})$ to both optical modes, we obtain the following state after introducing two additional environmental modes
\begin{widetext}
\begin{align}
\frac{1}{1+\gamma^2} &\Bigg[ \gamma^2 \ket{\downarrow,\downarrow} \otimes \left(\eta\ket{1,1,0,0} + \sqrt{\eta(1-\eta)} \left(\ket{1,0,0,1} + \ket{0,1,1,0}\right) + (1-\eta) \ket{0,0,1,1} \right) \nonumber\\
&\hspace{20pt} + \left. \gamma \ket{\uparrow,\downarrow} \otimes \left(\sqrt{\eta} \ket{0,1,0,0} + \sqrt{1-\eta} \ket{0,0,0,1} \right) \right. \nonumber\\
&\hspace{20pt} + \left. \gamma \ket{\downarrow,\uparrow} \otimes \left(\sqrt{\eta} \ket{1,0,0,0} + \sqrt{1-\eta} \ket{0.0,1,0} \right) \right.\\
&\hspace{20pt} + \ket{\uparrow,\uparrow,0,0,0,0} \Bigg]\nonumber
\end{align}
\end{widetext}
We apply a 50:50 beam splitter to the (non-environmental) optical mode and obtain the state
\begin{widetext}
\begin{align}
\frac{1}{1+\gamma^2} \Bigg[&\gamma^2 \ket{\downarrow,\downarrow} \otimes \sqrt{\frac{\eta(1-\eta)}{2}} \left(\ket{1,0,0,1}+\ket{0,1,0,1}+\ket{1,0,1,0}-\ket{0,1,1,0}\right) \nonumber\\
&\hspace{20pt} + \gamma^2 \ket{\downarrow,\downarrow} \otimes \frac{\eta}{2}\left(\ket{2,0,0,0}-\ket{0,2,0,0}\right) \nonumber \\
&\hspace{20pt} +\gamma^2 \ket{\downarrow,\downarrow} \otimes (1-\eta)\ket{0,0,1,1} \nonumber \\
&\hspace{20pt} \left. + \gamma \ket{\uparrow,\downarrow} \otimes \left(\sqrt{\frac{\eta}{2}} \left(\ket{1,0,0,0}-\ket{0,1,0,0}\right) + \sqrt{1-\eta} \ket{0,0,0,1} \right)\right.\\
&\hspace{20pt} \left. + \gamma \ket{\downarrow,\uparrow} \otimes \left(\sqrt{\frac{\eta}{2}} \left(\ket{1,0,0,0}+\ket{0,1,0,0}\right) + \sqrt{1-\eta}\ket{0,0,1,0}\right) \right.\nonumber\\
&\hspace{20pt} +\ket{\uparrow,\uparrow,0,0,0,0} \Bigg]\nonumber\,.
\end{align}
\end{widetext}
We can obtain entangled memory states by post-selecting single photon events at the detectors. If we detect a single photon at the first detector and no photon at the other, we obtain the following (unnormalized) 2-memory reduced density operator (see \cite[App. E]{tf_repeater})
\begin{align}
\frac{\gamma^2\eta}{(1+\gamma)^2}\left[\ket{\Psi^+}\bra{\Psi^+}+\gamma^2(1-\eta)\ket{\downarrow,\downarrow}\bra{\downarrow,\downarrow}\right].
\end{align}
When using simple on/off detectors instead of photon number resolving detectors (PNRD) two-photon events will also lead to a detection event. The two-memory state after a two-photon event is given by
\begin{align}
\frac{\gamma^4\eta^2}{4(1+\gamma^2)^2}\ket{\downarrow,\downarrow}\bra{\downarrow,\downarrow}\,.
\end{align}
Thus, the probability of a successful entanglement generation is given by $p_{\mathrm{PNRD}}=\frac{2\gamma^2\eta}{(1+\gamma^2)^2}(1+\gamma^2(1-\eta))$, when using PNRD, and $p_{\mathrm{on/off}}=\frac{2\gamma^2\eta}{(1+\gamma^2)^2}(1+\gamma^2(1-\frac{3}{4}\eta))$, when using on/off detectors. The factor 2 comes from the possibility to detect the photon at the other detector instead, although in this case the memory state differs by a single-qubit $Z$-operation. After a suitable twirling, we can find a one-qubit Pauli channel which maps the state $\ket{\Psi^+}\bra{\Psi^+}$ to the actual memory state, i.e. we can claim that the loss channel acting on the optical modes induces a Pauli channel on the memories. We can parametrize this Pauli channel by the tuple of error probabilities $(p_I,p_X,p_Y,p_Z)$ and for the case with PNRDs this tuple is given by
\begin{align}
\frac{1}{1+\gamma^2(1-\eta)}\left(1,\frac{\gamma^2}{2}(1-\eta),\frac{\gamma^2}{2}(1-\eta),0\right)
\end{align}
and for on/off detectors it is given by
\begin{align}
\frac{1}{1+\gamma^2(1-\frac{3}{4}\eta)}\left(1,\frac{\gamma^2}{2}(1-\frac{3}{4}\eta),\frac{\gamma^2}{2}(1-\frac{3}{4}\eta),0\right)\,.
\end{align}
When we consider an $n$-segment repeater, we have to consider a concatenation of $n$ such Pauli channels and we finally obtain the error rates
\begin{align}
e_x&=\frac{1}{2}\left(1-\mu^{n-1}\mu_0^{n}\frac{(2F_0-1)^n\mathbf{E}[e^{-\alpha D_n}]}{(1+\gamma^2(1-\eta))^n}\right),\\
e_z&=\frac{1}{2}\left(1-\mu^{n-1}\mu_0^{n}\left(\frac{1-\gamma^2(1-\eta)}{1+\gamma^2(1-\eta)}\right)^n\right)
\end{align}
in the case of PNRDs. When we consider on/off detectors, we can simply replace $\eta$ by $\frac{3}{4}\eta$ in the error rates.
\end{document}
|
\begin{document}
\frontmatter
\title{Interpolation synthesis for quadratic polynomial
inequalities and combination with \textit{EUF}}
\titlerunning{Hamiltonian Mechanics}
\author{\small Ting Gan\inst{1} \and Liyun Dai\inst{1} \and Bican Xia\inst{1} \and Naijun Zhan\inst{2} \and Deepak Kapur\inst{3}
\and Mingshuai Chen \inst{2}
}
\authorrunning{Ting Gan et al.}
\tocauthor{}
\institute{LMAM \& School of Mathematical Sciences, Peking University\\
\email{\{gant,dailiyun,xbc\}@pku.edu.cn},
\and
State Key Lab. of Computer Science, Institute of Software, CAS \\
\email{[email protected]}
\and
Department of Computer Science, University of New Mexico\\
\email{[email protected]}
}
\maketitle
\begin{abstract}
An algorithm for generating interpolants for formulas which are
conjunctions of quadratic polynomial
inequalities (both strict and nonstrict) is proposed. The
algorithm is based on a key observation that
quadratic polynomial inequalities can be linearized if they are
concave. A generalization of Motzkin's
transposition theorem is proved, which is used to generate
an interpolant between two mutually contradictory conjunctions
of polynomial inequalities, using semi-definite programming in time complexity
$\mathcal{O}(n^3+nm))$ with a given threshold, where $n$ is the number of variables
and $m$ is the number of inequalities.
Using the framework proposed by \cite{SSLMCS2008}
for combining
interpolants for a combination of quantifier-free theories which
have their own interpolation algorithms, a combination algorithm is
given for the combined theory of concave quadratic polynomial
inequalities and the equality theory over uninterpreted functions
symbols (\textit{EUF}). The proposed approach is applicable to all existing abstract domains like
\emph{octagon}, \emph{polyhedra}, \emph{ellipsoid} and so on, therefore it can be used to improve
the scalability of existing verification techniques for programs and hybrid systems.
In addition, we also discuss how to
extend our approach to formulas beyond concave quadratic polynomials
using Gr\"{o}bner basis.
\end{abstract}
\keywords{Program verification, Interpolant, Concave quadratic polynomials, Motzin's theorem, Semi-definite programming}.
\section{Introduction}
Interpolants have been popularized by McMillan \cite{mcmillan05} for automatically generating
invariants of programs. Since then, developing efficient algorithms for
generating interpolants for various theories has become an active area
of research; in
particular, methods have been developed for generating
interpolants for Presburger arithmetic (both for integers as well
as for rationals/reals), theory of equality over uninterpreted
symbols as well as their combination. Most of these methods
assume the availability of a refutation proof of $\alpha \land
\beta$ to generate a ``reverse" interpolant of $(\alpha, \beta)$;
calculi have been proposed to label an inference node in a
refutational proof depending upon whether symbols of formulas on
which the inference is applied are purely from $\alpha$ or
$\beta$. For propositional calculus, there already existed
methods for generating interpolants from resolution proofs
\cite{krajicek97,pudlak97} prior
to McMillan's work, which generate different interpolants from
those done by McMillan's method. This led D'Silva et al \cite{SPWK10} to study
strengths of various interpolants.
In Kapur, Majumdar and Zarba \cite{KMZ06}, an intimate connection between
interpolants and quantifier elimination was established. Using
this connection, existence of quantifier-free as well as
interpolants with quantifiers were shown for a variety of
theories over container data structures. A CEGAR based
approach was generalized for verification of programs over
container data structures using interpolants. Using this connection between
interpolant
generation and quantifier elimination, Kapur \cite{Kapur13}
has shown that interpolants form a lattice ordered using
implication, with the interpolant generated from $\alpha$ being
the bottom of such a lattice and the interpolant generated from
$\beta$ being the top of the lattice.
Nonlinear polynomials inequalities have been found useful to express
invariants for software involving sophisticated number theoretic
functions as well as hybrid systems; an interested reader may see
\cite{ZZKL12,ZZK13} where different controllers involving nonlinear polynomial
inequalities are discussed for some industrial applications.
We propose an algorithm to generate interpolants for
quadratic
polynomial inequalities (including strict inequalities).
Based on the insight that for analyzing the solution space
of concave quadratic polynomial (strict) inequalities, it
suffices to linearize them. We prove
a generalization of Motzkin's transposition theorem
to be applicable for quadratic polynomial inequalities (including
strict as well as nonstrict). Based on this result, we
prove the existence of interpolants for two mutually
contradictory conjunctions $\alpha, \beta$ of concave quadratic polynomial
inequalities and give an algorithm for computing an interpolant
using semi-definite programming.
The algorithm is recursive with the basis step of the algorithm
relying on an additional condition on concave quadratic
polynomials appearing in nonstrict inequalities
that any nonpositive constant combination of these polynomials is
never a nonzero sum of square polynomial (called $\mathbf{NSOSC}$).
In this case, an interpolant output by the algorithm is either a
strict inequality or a nonstrict inequality much like in the
linear case.
In case,
this condition is not satisfied by the nonstrict inequalities,
i.e., there is a nonpositive constant combinations of polynomials
appearing as nonstrict inequalities that is a negative of a sum
of squares,
then new mutually contradictory conjunctions
of concave quadratic polynomials in fewer variables are derived
from the input augmented with the equality relation deduced, and
the algorithm is recursively invoked on the smaller problem. The
output of this algorithm is in general an interpolant that is a
disjunction of conjunction of polynomial nonstrict or strict inequalities.
The $\mathbf{NSOSC}$ condition can be checked in polynomial time using
semi-definite programming.
We also show how separating terms $t^-, t^+$ can be constructed using common
symbols in $\alpha, \beta$ such that $\alpha \Rightarrow t^- \le x \le t^+$ and
$\beta \Rightarrow t^+ \le y \le t^-$, whenever $(\alpha \land
\beta) \Rightarrow x = y$. Similar to the construction for
interpolants, this construction has the same recursive structure with
concave quadratic polynomials satisfying NSOSC as the basis
step.
This result enables the use of the
framework proposed in \cite{RS10} based on
hierarchical theories and a combination method for generating
interpolants by Yorsh and Musuvathi, from combining equality
interpolating quantifier-free theories for generating
interpolants for the combined theory of quadratic polynomial
inequalities and theory of uninterpreted symbols.
Obviously, our results are significant in program verification as
all well-known abstract domains, e.g. \emph{octagon}, \emph{polyhedra}, \emph{ellipsoid} and so on,
which are widely used in the verification of programs and hybrid systems, are
\emph{quadratic} and \emph{concave}. In addition, we also discuss the possibility to
extend our results to general polynomial formulas by allowing polynomial equalities whose
polynomials may be neither \emph{concave} nor \emph{quadratic} using Gr\"{o}bner basis.
We develop a combination algorithm for generating
interpolants for the combination of concave quadratic polynomial
inequalities and uninterpreted function symbols.
In \cite{DXZ13}, Dai et al. gave an
algorithm for generating interpolants for conjunctions of
mutually contradictory nonlinear polynomial inequalities
based on the existence of a witness guaranteed by Stengle's
\textbf{Positivstellensatz} \cite{Stengle} that can be computed using
semi-definite programming.
Their algorithm is incomplete in general but if every variables ranges
over a bounded interval (called Archimedean condition), then
their algorithm is complete. A major limitation of their work is
that formulas $\alpha, \beta$ cannot have uncommon
variables\footnote{See however an expanded version of their paper
under preparation where they propose heuristics using program
analysis for eliminating uncommon variables.}.
However, they do not give any
combination algorithm for generating interpolants in the presence
of uninterpreted function symbols appearing in $\alpha, \beta$.
The paper is organized as follows. After discussing some
preliminaries in the next section,
Section 3 defines concave quadratic polynomials, their matrix
representation and their linearization. Section 4
presents the main contribution of the paper.
A generalization of Motzkin's transposition theorem for quadratic
polynomial inequalities is presented. Using this result, we
prove the existence of interpolants for two mutually
contradictory conjunctions $\alpha, \beta$ of concave quadratic polynomial
inequalities and give an algorithm (Algorithm 2) for computing an interpolant
using semi-definite programming.
Section 5 extends this algorithm to the combined theory of
concave quadratic inequalities and \textit{EUF} using the framework used
in \cite{SSLMCS2008,RS10}.
Implementation and experimental results using the proposed
algorithms are briefly reviewed in Section 6, and
we conclude and discus future work in Section 7.
\begin{comment}
{\color{red} \em I reformulated this part accordingly. Deepak, could you have a look and revise this paragraph?. }
The paper is organized as follows. After discussing some
preliminaries in the next section, \oomit{we review in Section 3, the
results for interpolant generation for nonlinear polynomial
inequalities; we discuss the key assumptions used in this
work. Section 4 discusses how many of these assumptions can be
relaxed and the results can be generalized.}
Section 3 is devoting to the main contribution of the paper, where we present
a generalization of Motzkin's transposition theorem for quadratic
polynomial inequalities. Using this result, we
prove the existence of interpolants for two mutually
contradictory conjunctions $\alpha, \beta$ of concave quadratic polynomial
inequalities and give an algorithm for computing an interpolant
using semi-definite programming in Section 4.
In Section 5, after
briefly reviewing the framework used in \cite{RS10} for generating
interpolants for the combined theory of linear inequalities and
equality over uninterpreted symbols, we show how the result of the
previous section can be used to develop an algorithm for
generating interpolants for the combined theory of quadratic
polynomial inequalities and equality over uninterpreted symbols.
Implementation and experimental results of our approach are provided in Section 6, and
we conclude this paper and discus future work in Section 7.
\end{comment}
\section{Preliminaries}
Let $\mathbb{N}$, $\mathbb{Q}$ and $\mathbb{R}$
be the set of natural, rational and real numbers, respectively.
Let $\mathbb{R}[x]$ be the polynomial ring over $\mathbb{R}$ with
variables $x=(x_1,\cdots,x_n)$. An atomic polynomial formula
$\varphi$ is of the form $ p(x) \diamond 0$, where $p(x) \in
\mathbb{R}[x]$, and
$\diamond$ can be any of $=, >, \ge, \neq$; without any
loss of generality, we can assume $\diamond$ to be any of $>, \ge$.
An arbitrary polynomial formula is
constructed from atomic ones with Boolean connectives and quantifications over real numbers.
Let $\mathbf{PT} ( \mathbb{R} )$ be a first-order theory of polynomials with
real coefficient, In this paper, we are focusing on
quantifier-free fragment of $\mathbf{PT}(\mathbb{R})$.
Later we discuss quantifier-free theory of equality of terms over
uninterpreted function symbols and its combination with
the quantifier-free fragment of $\mathbf{PT}(\mathbb{R})$. Let $\Sigma$ be a set
of (new) function symbols.
Let $\PT ( \RR )^{\Sigma}$ be the extension of
the quantifier-free theory with uninterpreted function symbols in $\Sigma$.
For convenience, we use $\bot$ to stand for \emph{false} and
$\top$ for \emph{true} in what follows.
\begin{definition}
A model $\mathcal{M}=(M,{f_{\mathcal{M}}})$ of $\PT ( \RR )^{\Sigma}$ consists of a model $M$ of
$\PT ( \RR )$ and a function $f_{\mathcal{M}}: \mathbb{R}^n \rightarrow \mathbb{R}$ for each $f\in \Sigma$ with arity $n$.
\end{definition}
\oomit{We have already defined the atomic formulas, thus, a formulas can be defined
easily refer to first-order logic. A formula is called closed, or a sentence, if it
has no free variables. A formula or a term is called ground if it has no occurrences
of variables.
Let $\mathcal{T}$ be a theory (here is $\PT ( \QQ )$, $\PT ( \RR )$,$\PT ( \QQ )^{\Sigma}$ or $\PT ( \RR )^{\Sigma}$), we define truth,
satisfiability and entail of a first-order formula in the standard way. }
\begin{definition}
Let $\phi$ and $\psi$ be formulas of a considered theory $\mathcal{T}$, then
\begin{itemize}
\item $\phi$ is \emph{valid} w.r.t. $\mathcal{T}$, written as $\models_{\mathcal{T}} \phi$, iff $\phi$ is
true in all models of $\mathcal{T}$;
\item $\phi$ \emph{entails} $\psi$ w.r.t. $\mathcal{T}$, written as $\phi \models_{\mathcal{T}} \psi$,
iff for any model of $\mathcal{T}$, if $\psi$ is true in the model, so is $\phi$;
\item $\phi$ is \emph{satisfiable} w.r.t. $\mathcal{T}$, iff there exists a model of $\mathcal{T}$
such that in which $\phi$ is true; otherwise \emph{unsatisfiable}.
\end{itemize}
\end{definition}
Note that $\phi$ is unsatisfiable iff $\phi \models_{\mathcal{T}} \bot$.
Craig showed that given two formulas $\phi$ and $\psi$ in a
first-order theory $\mathcal{T}$ such that
$\phi \models \psi$, there always exists an \emph{interpolant} $I$ over
the common symbols of $\phi$ and $\psi$ such that $\phi \models
I, I \models \psi$. In the verification literature, this
terminology has been abused following \cite{mcmillan05}, where an
\emph{reverse} interpolant $I$ over the common symbols of $\phi$ and
$\psi$ is defined for $\phi \wedge\psi \models
\bot$ as: $\phi \models I$ and $I \wedge \psi \models
\bot$.
\begin{definition}
Let $\phi$ and $\psi$ be two formulas in a theory $\mathcal{T}$ such that
$\phi \wedge \psi \models_{\mathcal{T}} \bot$. A formula $I$ said to be
a \emph{(reverse) interpolant} of $\phi$ and
$\psi$ if the following conditions hold:
\begin{enumerate}
\item[i] $\phi \models_{\mathcal{T}} I$;
\item[ii] $I \wedge \psi \models_{\mathcal{T}} \bot$; and
\item[iii] $I$ only contains common symbols and free variables shared by $\phi$ and
$\psi$.
\end{enumerate}
\end{definition}
If $\psi$ is closed, then $\phi \models_{\mathcal{T}} \psi$ iff
$\phi \wedge \neg \psi \models_{\mathcal{T}} \bot$. Thus, $I$ is an interpolant of
$\phi$ and $\psi$ iff $I$ is a reverse interpolant of $\phi$ and
$\neg \psi$.
In this paper, we just deal with reveres interpolant, and from now
on, we abuse interpolant and reverse interpolant.
\subsection{Motzkin's transposition theorem}
Motzkin's transposition theorem \cite{schrijver98} is one of the
fundamental results about linear inequalities; it also served as
a basis of the interpolant generation algorithm for the
quantifier-free
theory of linear inequalities in \cite{RS10}.
The theorem has several variants as well.
Below we give two of them.
\begin{theorem}[Motzkin's transposition theorem \cite{schrijver98}] \label{motzkin-theorem} Let $A$ and $B$ be matrices and let $\va$ and $\vb$ be
column vectors. Then there exists a vector $x$ with $Ax \ge \va$ and $Bx > \vb$, iff
\begin{align*}
&{\rm for ~all~ row~ vectors~} \yy,\zz \ge 0: \\
&~(i) {\rm ~if~} \yy A + \zz B =0 {\rm ~then~} \yy \va + \zz \vb \le 0;\\
&(ii) {\rm ~if~} \yy A + \zz B =0 {\rm~and~} \zz\neq 0 {\rm ~then~} \yy \va + \zz \vb < 0.
\end{align*}
\end{theorem}
\begin{corollary} \label{cor:linear}
Let $A \in \mathbb{R}^{r \times n}$ and $B \in \mathbb{R}^{s \times n}$ be matrices and $\va \in \mathbb{R}^r$ and
$\vb \in \mathbb{R}^s$ be column vectors. Denote by $A_i, i=1,\ldots,r$ the $i$th row of $A$ and by
$B_j, j=1,\ldots,s$ the $j$th row of $B$. Then there does not exist a vector $x$ with
$Ax \ge \va$ and $Bx > \vb$, iff there exist real numbers $\lambda_1,\ldots,\lambda_r \ge 0$
and $\eta_0,\eta_1,\ldots,\eta_s \ge0$ such that
\begin{align}
&\sum_{i=1}^{r} \lambda_i (A_i x - \alpha_i) + \sum_{j=1}^{s} \eta_j (B_j x -\beta_j) + \eta_0 \mathbf{Eq}uiv 0, \label{eq:corMoz1}\\
&\sum_{j=0}^{s} \eta_j > 0.\label{eq:corMoz2}
\end{align}
\end{corollary}
\begin{proof}
The ``if" part is obvious. Below we prove the ``only if" part.
By Theorem \ref{motzkin-theorem}, if
$Ax \ge \va$ and $Bx > \vb$ have no common solution, then
there exist two row vectors $\yy \in \mathbb{R}^r$ and $\zz \in \mathbb{R}^s$ with $\yy \ge 0$ and $\zz\ge 0$
such that
\[ (\yy A+\zz B=0 \wedge \yy \va+ \zz \vb > 0) \vee (\yy A+\zz B=0 \wedge \zz \neq 0 \wedge \yy \va+ \zz \vb \ge 0).\]
Let $\lambda_i=y_i, i=1,\ldots, r$, $\eta_j = z_j, j=1,\ldots, s$ and $\eta_0 = \yy \va+ \zz \vb$.
Then it is easy to check that Eqs. (\ref{eq:corMoz1}) and (\ref{eq:corMoz2}) hold. \qed
\end{proof}
\section{Concave quadratic polynomials and their linearization}
\oomit{As we know, the existing algorithms for interpolant generation fell mainly into two classes. One is
proof-based, they first require explicit construction of proofs, then an interpolant can be computed,
\cite{krajicek97,mcmillan05,pudlak97,KB11}. Another is constraint solving based, they first construct an
constrained system, then solve it, from which an interpolant can be computed, \cite{RS10,DXZ13}. The works are all deal
with propositional logic or linear inequalities over reals except \cite{DXZ13,KB11}, which can deal with
nonlinear case. Unfortunately, in \cite{DXZ13} the common variables, i.e. $(iii)$ in Definition \ref{crain:int}, can not be handled well; and \cite{KB11}, which is a variant of SMT solver based on interval arithmetic, is too much
rely on the interval arithmetic. Consider
the following example,
\begin{example} \label{exam:pre}
Let $f_1 = x_1, f_2 = x_2,f_3= -x_1^2-x_2^2 -2x_2-\zz^2, g_1= -x_1^2+2 x_1 - x_2^2 + 2 x_2 - \yy^2$. Two formulas $\phi:=(f_1 \ge 0) \wedge (f_2 \ge0) \wedge (g_1 >0)$,
$\psi := (f_3 \ge 0)$. $\phi \wedge \psi \models \bot$.
\end{example}
We want to generate an interpolant for
$\phi \wedge \psi \models \bot$. The existing algorithms can not
be used directly to obtain an interpolant, since this example is
nonlinear and not all the variables are common variables in
$\phi$ and $\psi$.
An algorithm may be exploited based on CAD, but the efficiency of
CAD is fatal weakness.
In this paper, we provide a complete and efficient algorithm to generate interpolant for a special nonlinear case
(CQ case, i.e. $\phi$ and $\psi$ are defined by the conjunction of a set of concave quadratic polynomials "$>0$" or "$\ge 0$"), which
contains Example \ref{exam:pre}. }
\begin{definition} [Concave Quadratic] \label{quad:concave}
A polynomial $f \in \mathbb{R}[x]$ is called {\em concave quadratic (CQ)}, if the following two conditions hold:
\begin{itemize}
\item[(i)] $f$ has total degree at most $2$, i.e., it has the form
$f = x^T A x + 2 \va^T x + a$, where $A$ is a real symmetric matrix, $\va$ is a column vector and $a \in \mathbb{R}$ is a constant;
\item[(ii)] the matrix $A$ is negative semi-definite, written as $A \preceq
0$.\footnote{$A$ being negative semi-definite has many equivalent
characterizations: for every vector $x$, $x ^T A x \le 0$;
every $k$th minor of $A$ $\le 0$ if $k$ is odd and $\ge 0$
otherwise; a Hermitian matrix whose eigenvalues are
nonpositive.}
\end{itemize}
\end{definition}
\begin{example}
Let $g_1= -x_1^2+2 x_1 - x_2^2 + 2 x_2 - y^2$, then it can be expressed as
\begin{align*}
g_1={\left( \begin{matrix}
&x_1\\
&x_2\\
&y
\end{matrix} \right)}^T
{\left( \begin{matrix}
&-1&0&0\\
&0&-1&0\\
&0&0&-1
\end{matrix} \right)}
{\left( \begin{matrix}
&x_1\\
&x_2\\
&y
\end{matrix} \right)} +2 {\left( \begin{matrix}
&1\\
&1\\
&0
\end{matrix} \right)}^T
{\left( \begin{matrix}
&x_1\\
&x_2\\
&y
\end{matrix} \right)}.
\end{align*}
The degree of $g_1$ is 2, and the corresponding $A={\left( \begin{matrix}
&-1&0&0\\
&0&-1&0\\
&0&0&-1
\end{matrix} \right)} \preceq 0$. Thus, $g_1$ is CQ.
\end{example}
It is easy to see that if $f \in \mathbb{R}[x]$ is linear, then $f$ is
CQ because its total degree is $1$ and the
corresponding $A$
is $0$ which is of course negative semi-definite.
A quadratic polynomial can also be represented as an inner
product of matrices (cf. \cite{laurent}), i.e.,
$ f(x)=\left<P,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right)\right >.$
\subsection{Linearization} \label{linearization}
Consider quadratic polynomials
$f_i$ and $g_j$ ($i=1,\ldots,r$,
$j=1,\ldots,s$),
\begin{align*}
f_i=x^T A_i x+2\va_i^Tx+a_i,\\
g_j=x^T B_j x+2\vb_j^Tx+b_j,
\end{align*}
where $A_i$, $B_j$ are symmetric $n\times n$ matrices,
$\va_i,\vb_j\in \mathbb{R}^n$, and $a_i,b_j\in \mathbb{R}$;
let
$P_i:=\left( \begin{matrix}
a_i & \va_i^T \\
\va_i & A_i
\end{matrix}
\right),~
Q_j:=\left( \begin{matrix}
b_j & \vb_j^T \\
\vb_j & B_j
\end{matrix}
\right)$
be
$(n+1)\times(n+1)$ matrices, then
\begin{align*}
f_i(x)=\left<P_i,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right)\right >,~~
g_j(x)=\left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right)\right >.
\end{align*}
\begin{comment}
The moment matrix $M_1(\yy)$ is of the form
$\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right)$
for some $x\in \mathbb{R}^n$ and some symmetric $n\times n $ matrix
$\XX$, and the localizing constraint $M_0(f_i\yy)$ and
$M_0(g_j\yy)$ read $\left <P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right)\right >$
and
$\left <Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right)\right >$.
Therefore, the moment relaxations of order $1$ can be reformulated as
\end{comment}
For CQ polynomials $f_i$s and $g_j$s in which each $A_i \preceq
0$, $B_j \preceq 0$, define
\begin{equation}
K=\{x \in \mathbb{R}^n \mid f_1(x) \ge0,\ldots,f_r(x)\ge0, g_1(x)>0,\ldots, g_s(x)>0 \}.
\label{eq:opt}
\end{equation}
Given a quadratic polynomial
$ f(x)=\left<P,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right)\right >$,
its \emph{linearization} is defined as
$f(x)=\left<P,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right)\right >$,
where
$\left( \begin{matrix}
1 & x^T\\
x & \XX
\end{matrix}
\right)\succeq 0$.
\
Let
\begin{align*}
\overline{\XX}=(&\XX_{(1,1)},\XX_{(2,1)},\XX_{(2,2)},\ldots,
\XX_{(k,1)}, \ldots,\XX_{(k,k)}, \ldots, \XX_{(n,1)}, \ldots,\XX_{(n,n)})
\end{align*}
be the vector variable
with $\frac{n(n+1)}{2}$ dimensions corresponding to the matrix $\XX$.
Since $\XX$ is a symmetric
matrix,
$\left<P,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >$
is a linear expression in $x,\overline{\XX}$.
Now, let
\begin{align}
&K_1 = \{x \mid \left( \begin{matrix}
1 & x^T\\
x & \XX
\end{matrix}
\right)\succeq 0 ,
\
\wedge_{i=1}^r \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \ge 0 , \nonumber \\
& \quad \quad \quad \quad
\wedge_{j=1}^s \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > > 0, \mbox{ for some } \XX \},\label{eq:mom1}
\end{align}
which is the set of all $x\in \mathbb{R}^n$ on linearizations of the above $f_i$s and $g_j$s.
\oomit{Thus,
\begin{aligned}
\left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \ge 0 ,
\
&\left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > > 0,
\end{aligned}
are respectively linearizations of quadratic nonstrict inequalities $f_i
\ge 0$'s and strict inequalities $g_j > 0$'s in which all
quadratic terms are abstracted by new variables.
A reader should note that the condition
\begin{aligned}
\left( \begin{matrix}
1 & x^T\\
x & \XX
\end{matrix}
\right)\succeq 0.
\end{aligned}
is critical.
}
\begin{comment}
Define
\begin{equation*}
K_2:=\left\{ x\in \mathbb{R}^n\mid \sum_{i=1}^r t_if_i(x)\ge 0 \mbox{ for all } t_i\ge 0 \mbox{ for which } \sum_{i=1}^r t_iA_i\preceq 0 \right\}.
\end{equation*}
Note that the definition of $K_2$ is independent on $g_j$.
\end{comment}
In \cite{fujie,laurent}, when $K$ and $K_1$ are defined only with $f_i$ without $g_j$, i.e., only with
non-strict inequalities, it is proved that $K=K_1$.
\oomit{The set $K$ is defined by a set of quadratic inequalities. The
set $K_1$ is defined by a positive semi-definite constraint and a
set of linear inequalities.}
By the following Theorem \ref{the:1},
we show that $K=K_1$ also holds even in
the presence of strict inequalities when $f_i$ and $g_j$ are CQ. So, when
$f_i$ and $g_j$ are CQ, the CQ
polynomial inequalities can be transformed equivalently to a set of
linear inequality constraints and a positive semi-definite
constraint.
\begin{theorem} \label{the:1}
Let $f_1,\ldots,f_r$ and $g_1,\ldots,g_s$ be CQ polynomials, $K$ and
$K_1$ as above, then $K=K_1$.
\end{theorem}
\begin{proof}
For any $x \in K$, let $\XX=x x^T$. Then it is easy to see that
$x,\XX$ satisfy (\ref{eq:mom1}). So $x \in K_1$, that is $K \subseteq K_1$.
Next, we prove $K_1 \subseteq K$.
Let $x \in K_1$, then there exists a symmetric $n\times n $ matrix $\XX$ satisfying
(\ref{eq:mom1}).
Because
$\left( \begin{matrix}
1 & x^T\\
x & \XX
\end{matrix}
\right)\succeq 0$,
we have $\XX - x x^T \succeq 0$.
Then by the last two conditions in (\ref{eq:mom1}), we have
\begin{align*}
f_i(x) &= \left< P_i,\left( \begin{matrix}
1 & x^T \\
x & x x^T
\end{matrix}
\right ) \right> =
\left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right> +
\left<P_i,\left( \begin{matrix}
0 & 0 \\
0 & x x^T - \XX
\end{matrix}
\right ) \right> \\
&=\left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right> +
\left<A_i , x x^T - \XX \right> \ge \left<A_i , x x^T - \XX \right>, \\[1mm]
g_j(x) &= \left< Q_j,\left( \begin{matrix}
1 & x^T \\
x & x x^T
\end{matrix}
\right ) \right> =
\left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right> +
\left<Q_j,\left( \begin{matrix}
0 & 0 \\
0 & x x^T - \XX
\end{matrix}
\right ) \right> \\
&=\left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right> +
\left<B_j , x x^T - \XX \right> > \left<B_j , x x^T - \XX \right>.
\end{align*}
Since $f_i$ and $g_j$ are all CQ, $A_i \preceq 0$ and $B_j \preceq 0$.
Moreover, $\XX -x x^T \succeq 0$, i.e.,
$x x^T -\XX \preceq 0$. Thus,
$\left<A_i , x x^T - \XX \right> \ge 0$ and
$\left<B_j , x x^T - \XX \right> \ge 0$.
Hence, we have $f_i(x) \ge 0$ and $g_j(x) > 0$, so $x \in K$,
that is
$K_1 \subseteq K$.
\qed
\end{proof}
\subsection{Motzkin's theorem in Matrix Form}
If
$\left<P,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >$
is seen as a linear expression in $x,\overline{\XX}$,
then Corollary \ref{cor:linear}
can be reformulated as:
\begin{corollary} \label{cor:matrix}
Let $x$ be a column vector variable of dimension $n$ and $\XX$ be a $n \times n$
symmetric matrix variable. Suppose $P_0,P_1,\ldots, P_r$ and $Q_1,\ldots, Q_s$ are
$(n+1) \times (n+1)$ symmetric matrices. Let
\begin{align*}
W \hat{=} \{ (x,\XX) \mid
\wedge_{i=1}^r \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \ge 0,
\wedge_{i=1}^s \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > > 0 \oomit{,
~for~
\begin{matrix}
i=0,1,\ldots,r, \\
j=1,\ldots,s.
\end{matrix} }
\},
\end{align*}
then $W=\emptyset$ iff there exist $\lambda_0, \lambda_1,\ldots,\lambda_r \ge 0$
and $\eta_0,\eta_1,\ldots,\eta_s \ge 0$ such that
\begin{align*}
&\sum_{i=0}^{r}\lambda_i \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > +
\sum_{j=1}^{s}\eta_j \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > + \eta_0 \mathbf{Eq}uiv 0, \mbox{ and }\\
&\eta_0 + \eta_1 + \ldots + \eta_s > 0.
\end{align*}
\end{corollary}
\section{Algorithm for generating interpolants for Concave
Quadratic Polynomial inequalities}
\begin{problem} \label{CQ-problem}
Given two formulas $\phi$ and $\psi$
on $n$ variables with $\phi \wedge \psi
\models \bot$,
where
\begin{eqnarray*}
\phi & = & f_1 \ge 0 \wedge \ldots \wedge f_{r_1} \ge 0 \wedge g_1 >0 \wedge \ldots \wedge g_{s_1} > 0, \\
\psi & = & f_{r_1+1} \ge 0 \wedge \ldots \wedge f_{r} \ge 0 \wedge g_{s_1+1} >0 \wedge \ldots \wedge g_{s} > 0,
\end{eqnarray*}
in which
$f_1, \ldots, f_{r}, g_1, \ldots, g_s$ are all CQ,
develop an algorithm
to generate a (reverse) Craig interpolant
$I$ for $\phi$ and $\psi$, on the common variables of
$\phi$ and $\psi$,
such that $\phi \models I$ and $I \wedge
\psi \models \bot$. For convenience, we partition the variables appearing in the
polynomials above into three disjoint subsets $x=(x_1,\ldots,x_d)$ to stand for
the common variables appearing in both $\phi$ and $\psi$, $\yy=(y_1,\ldots,y_u)$ to stand for the
variables appearing only in $\phi$ and $\zz=(z_1,\ldots,z_v)$
to stand for the variables appearing only in $\psi$, where $d+u+v=n$.
\end{problem}
Since linear inequalities are trivially concave quadratic
polynomials, our algorithm (Algorithm $\mathbf{IGFQC}$ in Section \ref{sec:alg}) can deal
with the linear case too. In fact, it is a generalization of the
algorithm for linear inequalities.
\oomit{Actually, since our main result (Theorem \ref{the:main}) is a generalization of Motzkin's theorem, our algorithm is essentially the same as other interpolant generation algorithms (e.g., \cite{RS10}) based on Motzkin's theorem when all the $f_i$ and $g_j$ are linear.
\begin{example}
For the formulas in Example \ref{exam:pre}, the input and output of our algorithm are
\begin{description}
\item[{\sf In:}] $\phi :f_1 = x_1, f_2 = x_2,g_1= -x_1^2+2 x_1 - x_2^2 + 2 x_2 - y^2$;\\
$\psi : f_3= -x_1^2-x_2^2 -2x_2-z^2$
\item[{\sf Out:}] $\frac{1}{2}x_1^2+\frac{1}{2}x_2^2+2x_2 > 0$
\end{description}
\end{example}
In the following sections, we come up with an condition $\mathbf{NSOSC}$ (Definition \ref{def:sosc}) and generalize the
Motzkin's transposition theorem (Theorem \ref{motzkin-theorem}) to Theorem \ref{the:main} for concave quadratic case when this condition hold in Section \ref{theory}.
When the condition $\mathbf{NSOSC}$ holds, we give a method to solve Problem 1 based on Theorem \ref{the:main} in Section \ref{sec:hold}. For the general case of concave quadratic ,
Problem 1 is solved in a recursive way in Section \ref{g:int}. }
The proposed algorithm is recursive: the base case is when no sum
of squares (SOS)
polynomial can be generated by a nonpositive constant combination of
nonstrict inequalities in $\phi \land \psi$.
When this condition is not satisfied, i.e., an SOS polynomial
can be generated by a nonpositive constant combination of
nonstrict inequalities in $\phi \land \psi$, then it is possible
to identify variables which can be eliminated by replacing them
by linear expressions in terms of other variables and thus
generate equisatisfiable problem with fewer variables on which
the algorithm can be recursively invoked.
\begin{lemma} \label{lemma:1}
Let $U \in \mathbb{R}^{(n+1) \times (n+1)}$ be a matrix.
If
$\left<U,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \le 0$
for any $x \in \mathbb{R}^n$ and symmetric matrix $\XX \in \mathbb{R}^{n\times n}$ with
$\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix} \right ) \succeq 0$
, then $U \preceq 0$.
\end{lemma}
\begin{proof}
Assume that $U \not \preceq 0$.
Then there exists a column vector $\yy=(y_0,y_1,\ldots,y_n)^T \in \mathbb{R}^{n+1}$ such that $c:=\yy^T U\yy=\left< U, \yy \yy^T \right>>0$. Denote $M= \yy \yy^T$, then $M \succeq 0$.
If $y_0 \neq 0$, then let $x = (\frac{y_1}{y_0},\ldots,\frac{y_n}{y_0})^T$, and $\XX= x x^T$.
Thus,
$\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) =
\left( \begin{matrix}
1 & x^T \\
x & x x^T
\end{matrix}
\right ) = \frac{1}{y_0^2} M \succeq $,
and
$\left<U,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right> = \left<U, \frac{1}{y_0^2} M \right> = \frac{c}{y_0^2} >0$,
which contradicts with
$\left<U,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \le 0$.
If $\yy_0= 0$, then $M_{(1,1)} = 0$.
Let $M{'}=\frac{|U_{(1,1)}|+1 }{c} M$, then $M{'} \succeq 0$. Further,
let $M{''} = M{'}+\left( \begin{matrix}
1 & 0& \cdots & 0 \\
0 & 0& \cdots & 0 \\
\vdots & \vdots &\ddots &\vdots \\
0 & 0& \cdots & 0
\end{matrix}
\right )$.
Then $M{''}\succeq 0$ and $M{''}_{(1,1)}=1$.
Let
$\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right )= M{''} $, then
\begin{align*}
\left<U,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >&=
\left<U, M{''} \right > =
\left<U,
M{'}+\left( \begin{matrix}
1 & 0& \cdots & 0 \\
0 & 0& \cdots & 0 \\
\vdots & \vdots &\ddots &\vdots \\
0 & 0& \cdots & 0
\end{matrix}
\right )
\right >\\
&=
\left<U, \frac{|U_{(1,1)}|+1 }{c} M
+\left( \begin{matrix}
1 & 0& \cdots & 0 \\
0 & 0& \cdots & 0 \\
\vdots & \vdots &\ddots &\vdots \\
0 & 0& \cdots & 0
\end{matrix}
\right )
\right > \\
&=
\frac{|U_{(1,1)}|+1 }{c}\left<U, M
\right > + U_{(1,1)}\\
&=|U_{(1,1)}|+1+ U_{(1,1)} >0,
\end{align*} which also contradicts with
$\left<U,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \le 0$.
Thus,
the assumption does not hold, that is $U\preceq 0$. \qed
\end{proof}
\begin{lemma} \label{lemma:2}
Let $\mathcal{A} = \{ \yy \in \mathbb{R}^m \mid A_i \yy-\va_i \ge 0, B_j \yy-\vb_j > 0,
~for~ i=1,\ldots,r, j=1,\ldots,
\}$ be a nonempty set
and $\mathcal{B} \subseteq \mathbb{R}^m$ be an nonempty convex closed set. If
$\mathcal{A} \cap \mathcal{B} = \emptyset$ and there does not exist a linear form $L(\yy)$
such that
\begin{align} \label{separ:1}
\forall \yy \in \mathcal{A}, L(\yy) >0, ~and~~
\forall \yy \in \mathcal{B}, L(\yy) \le 0,
\end{align}
then there is a linear form $L_0(\yy) \not\mathbf{Eq}uiv 0$
and $\delta_1, \ldots, \delta_r \ge 0$
such that
\begin{align}
L_0(\yy) = \sum_{i=1}^{r} \delta_i (A_i \yy - \alpha_i)~and~~
\forall \yy \in \mathcal{B}, L_0(\yy) \le0.
\end{align}
\end{lemma}
\begin{proof}
Since $\mathcal{A}$ is defined by a set of linear inequalities, $\mathcal{A}$ is a
convex set.
Using the separation theorem on disjoint convex sets,
cf. e.g. \cite{barvinok02},
there exists a linear
form $L_0(\yy) \not\mathbf{Eq}uiv 0$ such that
\begin{align}
\forall \yy \in \mathcal{A}, L_0(\yy) \ge 0, ~and~~
\forall \yy \in \mathcal{B}, L_0(\yy) \le0.
\end{align}
From (\ref{separ:1}) we have that
\begin{align} \label{y0}
\exists\yy_0 \in \mathcal{A}, ~~ L_0(\yy_0)=0.
\end{align}
Since
\begin{align}
\forall \yy \in \mathcal{A}, L_0(\yy) \ge 0,
\end{align}
then
\begin{align*}
&A_1 \yy -\alpha_1 \ge0\wedge \ldots \wedge A_r \yy -\alpha_r \ge0 \wedge \\
&B_1 \yy -\beta_1 > 0\wedge \ldots \wedge B_s \yy -\beta_s > 0 \wedge -L_0(\yy) >0
\end{align*}
has no solution w.r.t. $\yy$.
Using Corollary \ref{cor:linear}, there exist $\lambda_1,\ldots,\lambda_r \ge0$, $\eta_0, \ldots, \eta_s \ge0$ and $\eta \ge0$ such that
\begin{align}
&\sum_{i=1}^{r} \lambda_i (A_i\yy - \alpha_i) + \sum_{j=1}^{s} \eta_j (B_j \yy -\beta_j)+
\eta (-L_0(\yy)) + \eta_0 \mathbf{Eq}uiv 0, \label{eq:lem2} \\
&\sum_{j=0}^{s} \eta_j + \eta > 0. \label{ineq:lem2}
\end{align}
Applying $\yy_0$ in (\ref{y0}) to (\ref{eq:lem2}) and (\ref{ineq:lem2}), it follows
\begin{align*}
\eta_0=\eta_1=\ldots=\eta_s=0,~~ \eta >0.
\end{align*}
For $i=1,\ldots,r$, let $\delta_i=\frac{\lambda_i}{\eta} \ge 0$, then
\begin{align*}
L_0(\yy) = \sum_{i=1}^{r} \delta_i (A_i \yy -\alpha_i)~and~~
\forall \yy \in \mathcal{B}, L_0(\yy) \le0. ~~~ \qed
\end{align*}
\end{proof}
The lemma below asserts the existence of a strict linear inequality
separating $\mathcal{A}$ and $\mathcal{B}$ defined above,
for the case when any nonnegative constant combination of the linearization of $f_i$s is
positive.
\begin{lemma} \label{linear}
Let $\mathcal{A} = \{ \yy \in \mathbb{R}^m \mid A_i \yy-\va_i \ge 0, B_j \yy-\vb_j > 0,
~for~ i=1,\ldots,r, j=1,\ldots,
\}$ be a nonempty set
and $\mathcal{B} \subseteq \mathbb{R}^m$ be an nonempty convex closed set, $\mathcal{A} \cap \mathcal{B} = \emptyset$.
There exists a linear form $L(x,\overline{\XX})$ such that
\begin{align*}
\forall (x,\overline{\XX}) \in \mathcal{A}, L(x,\overline{\XX}) >0, ~and~~
\forall (x,\overline{\XX}) \in \mathcal{B}, L(x,\overline{\XX}) \le0,
\end{align*}
whenever there does not exist $\lambda_i \ge 0$, s.t.,
$ \sum_{i=1}^{r} \lambda_i P_i \preceq 0$.
\end{lemma}
\begin{proof}
Proof is by contradiction. \oomit{ i.e., there does not exists a linear form $L(x,\overline{\XX})$ such that
\begin{align*}
\forall (x,\overline{\XX}) \in \mathcal{A}, L(x,\overline{\XX}) >0, ~and~~
\forall (x,\overline{\XX}) \in \mathcal{B}, L(x,\overline{\XX}) \le0.
\end{align*}}
Given that $\mathcal{A}$ is defined by a set of linear inequalities
and $\mathcal{B}$ is a closed convex nonempty set,
by Lemma \ref{lemma:2},
there exist a linear form $L_0(x,\overline{\XX}) \not\mathbf{Eq}uiv 0$
and $\delta_1, \ldots, \delta_r \ge 0$
such that
\begin{align*}
L_0(x,\overline{\XX}) = \sum_{i=1}^{r} \delta_i
\left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >~and~~
\forall (x,\overline{\XX}) \in \mathcal{B}, L_0(x,\overline{\XX}) \le0.
\end{align*}
I.e. there exists an symmetrical matrix $\mathbf{L} \not\mathbf{Eq}uiv 0$ such that
\begin{align}
&\left<\mathbf{L},\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >
\mathbf{Eq}uiv \sum_{i=1}^{r} \delta_i
\left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >, \label{eq:mat}\\
& \forall (x,\overline{\XX}) \in \mathcal{B},
\left<\mathbf{L},\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \le0. \label{ineq:mat}
\end{align}
Applying Lemma \ref{lemma:1} to (\ref{ineq:mat}), it follows $\mathbf{L}
\preceq 0$. This implies that $\sum_{i=1}^{r} \delta_i P_i =\mathbf{L} \preceq 0$,
which is in contradiction to
the assumption that there does not exist $\lambda_i \ge 0$, s.t.,
$ \sum_{i=1}^{r} \lambda_i P_i \preceq 0$
\qed
\end{proof}
\begin{definition} \label{def:sosc}
For given formulas $\phi$ and $\psi$ as in Problem 1,
it satisfies the non-existence of an SOS condition ($\mathbf{NSOSC}$) iff
there do not exist $\delta_1\ge0, \ldots, \delta_r\ge 0$, such that
$-(\delta_1 f_1 + \ldots + \delta_r f_r)$ is a non-zero SOS.
\end{definition}
\oomit{The above condition implies that there is no \textcolor{blue}{nonnegative} constant combination
of nonstrict inequalities which is always {nonpositive}.
\textcolor{green}{If quadratic polynomials appearing in $\phi$ and $\psi$ are
linearized, then the above condition is equivalent to requiring
that every nonnegative linear combination of the linearization of
nonstrict inequalities in $\phi$ and $\psi$ is {negative.}}
The following theorem gives a method for generating an
interpolant for this case by considering linearization of the
problem and using Corollary \ref{cor:matrix}. In that sense, this
theorem is a generalization of Motzkin's theorem to CQ polynomial inequalities. }
The following theorem gives a method for generating an
interpolant when the condition $\mathbf{NSOSC}$\ holds by considering linearization of the
problem and using Corollary \ref{cor:matrix}. In that sense, this
theorem is a generalization of Motzkin's theorem to CQ polynomial inequalities.
The following separation lemma about a nonempty convex set $\mathcal{A}$ generated by
linear inequalities that is disjoint from another nonempty closed
convex set $\mathcal{B}$ states that if there is no strict linear
inequality that holds over $\mathcal{A}$ and does not hold on any
element in $\mathcal{B}$, then there is a hyperplane separating
$\mathcal{A}$ and $\mathcal{B}$, which is a nonnegative linear
combination of nonstrict inequalities.
\begin{theorem}
\label{the:main}
Let $f_1,\ldots,f_r,g_1,\ldots,g_s$ are CQ polynomials
and the $K$ is defined as in (\ref{eq:opt}) with $K=\emptyset$. If the condition $\mathbf{NSOSC}$ holds,
then there exist $\lambda_i\ge 0$ ($i=1,\cdots,r$), $\eta_j \ge 0$ ($j=0,1,\cdots,s$) and a quadratic SOS polynomial $h \in \mathbb{R}[x]$ such that
\begin{align}
&\sum_{i=1}^{r} \lambda_i f_i +\sum_{j=1}^{s} \eta_j g_j + \eta_0 + h \mathbf{Eq}uiv 0,\\
&\eta_0+\eta_1 + \ldots + \eta_s = 1.
\end{align}
\end{theorem}
The proof uses the fact that if $f_i$s satisfy the $\mathbf{NSOSC}$
condition, then the linearization of $f_i$s and $g_j$s can be
exploited to generate an interpolant expressed in terms of $x$.
The main issue is to decompose the result from the
linearized problem into two components giving an interpolant.
\begin{proof}
Recall from Section \ref{linearization} that
\begin{align*}
f_i = \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right ) \right > ,~~
g_j = \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right ) \right >.
\end{align*}
Let
\begin{equation}\label{eq:mom}
\begin{aligned}
&\mathcal{A}:=\{ (x,\overline{\XX}) \mid
\wedge_{i=1}^r \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > \ge 0,
\wedge_{j=1}^s \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > > 0
\oomit{~for~
\begin{matrix}
i=1,\ldots,r \\
j=1,\ldots,s
\end{matrix} }
\}, \\
&\mathcal{B}:=\{(x,\overline{\XX})\mid \left( \begin{matrix}
1 & x^T\\
x & \XX
\end{matrix}
\right)\succeq 0 \}, \\
\end{aligned}
\end{equation}
be linearizations of the CQ polynomials $f_i$s
and $g_j$s,
where \begin{align*}
\overline{\XX}=(&\XX_{(1,1)},\XX_{(2,1)},\XX_{(2,2)},\ldots,
\XX_{(k,1)}, \ldots,\XX_{(k,k)}, \ldots, \XX_{(n,1)}, \ldots,\XX_{(n,n)}).
\end{align*}
By Theorem \ref{the:1}, $\mathcal{A} \cap \mathcal{B} =K_1=K=\emptyset$.
Since $f_i$s satisfy the $\mathbf{NSOSC}$ condition, its linearization
satisfy the condition of Lemma \ref{linear}; thus
there exists a linear form $\mathcal{L}(x,\XX)=\left<L,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right>$ such that
\begin{align}
& \mathcal{L}(x,\XX) > 0, ~for~ (x,\XX) \in \mathcal{A}, \label{lin-sep1}\\
& \mathcal{L}(x,\XX) \le 0, ~for~ (x,\XX) \in \mathcal{B} \label{lin-sep2}.
\end{align}
Applying Lemma \ref{lemma:1}, it follows $L \preceq 0$.
Additionally, applying Lemma \ref{cor:matrix} to
(\ref{lin-sep1}) and denoting $-L$ by $P_0$,
there exist $\overline{\lambda_0}, \overline{\lambda_1},\ldots,\overline{\lambda_r} \ge 0$
and $\overline{\eta_0},\overline{\eta_1},\ldots,\overline{\eta_s} \ge 0$ such that
\begin{align*}
& \sum_{i=0}^{r}\overline{\lambda_i} \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > +
\sum_{j=1}^{s}\overline{\eta_j} \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > + \overline{\eta_0} \mathbf{Eq}uiv 0, \\
& \overline{\eta_0} + \overline{\eta_1} + \ldots + \overline{\eta_s} > 0.
\end{align*}
Let $\lambda_i=\frac{\overline{\lambda_i}}{\sum_{j=0}^s \overline{\eta_j}}$,
$\eta_j=\frac{\overline{\eta_j}}{\sum_{j=0}^s \overline{\eta_j}}$, then
\begin{align}
& \lambda_0
\left<-U,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right >+
\sum_{i=1}^{r}\lambda_i \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > +
\sum_{j=1}^{s}\eta_j \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & \XX
\end{matrix}
\right ) \right > + \eta_0 \mathbf{Eq}uiv 0, \label{lin-sep5}\\
& \eta_0 + \eta_1 + \ldots + \eta_s =1. \label{lin-sep6}
\end{align}
Since for any $x$ and symmetric matrix $\XX$, (\ref{lin-sep5}) holds, by setting $\XX=xx^T$,
\begin{align*}
\lambda_0
\left<-U,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right ) \right >+
\sum_{i=1}^{r}\lambda_i \left<P_i,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right ) \right > +
\sum_{j=1}^{s}\eta_j \left<Q_j,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right ) \right > + \eta_0 \mathbf{Eq}uiv 0,
\end{align*}
which means that
\begin{align*}
h+\sum_{i=1}^{r} \lambda_i f_i +\sum_{j=1}^{s} \eta_j g_j + \eta_0 \mathbf{Eq}uiv 0,
\end{align*}
where $h=\lambda_0 \left<-U,\left( \begin{matrix}
1 & x^T \\
x & xx^T
\end{matrix}
\right ) \right >$. Since $U\preceq 0$, $-U \succeq 0$. Hence $h$ is a quadratic SOS polynomial.
\qed
\end{proof}
\subsection{Base Case: Generating Interpolant when NSOSC is satisfied} \label{sec:hold}
Using the above theorem, it is possible to generate an
interpolant for $\phi$ and $\psi$ from the SOS polynomial $h$ obtained using
the theorem which can be split into two SOS polynomials in the
common variables of $\phi$ and $\psi$.
This is
proved in the following theorem using some lemma as follows.
\begin{lemma} \label{h:sep}
Given a quadratic SOS polynomial
$h(x,\yy,\zz) \in \mathbb{R}[x,\yy,\zz]$ on variables
$x=(x_1,\cdots,x_d) \in
\mathbb{R}^{d}$,$\yy=(y_1,\cdots,y_u)
\in \mathbb{R}^{u}$
and $\zz=(z_1,\cdots,z_v) \in \mathbb{R}^{v}$ such that the coefficients of
$y_i z_j$ ($i=1,\cdots,u,j=1,\cdots,v$) are all vanished when
expanding $h(x,\yy,\zz)$, there exist two quadratic
polynomial $h_1(x,\yy) \in \mathbb{R}[x,\yy]$ and
$h_2(x,\zz) \in \mathbb{R}[x,\zz]$ such that $h=h_1+h_2$,
moreover, $h_1$ and $h_2$ both are SOS.
\end{lemma}
\begin{proof}
Since $h(x,\yy,\zz)$ is a quadratic polynomial and the coefficients of $y_i z_j$ ($i=1,\cdots,u,j=1,\cdots,v$) are all vanished when expanding $h(x,\yy,\zz)$, we have
\begin{align*}
h(x,\yy_1,\cdots,\yy_u,\zz)=a_1 y_1^2 + b_1(x,y_2,\cdots,y_u) y_1 + c_1(x,y_2,\cdots,y_u,\zz),
\end{align*}
where $a_1 \in \mathbb{R}$, $b_1(x,y_2,\cdots,y_u) \in \mathbb{R}[x,y_2,\cdots,y_u]$ is a linear function and $c_1(x,y_2,\cdots,y_u,\zz)\in \mathbb{R}[x,y_2,\cdots,y_u,\zz]$ is a
quadratic polynomial.
Since $h(x,\yy,\zz)$ is an SOS polynomial, so
\begin{align*}
\forall (x,y_1,\cdots,y_u,\zz) \in \mathbb{R}^{d+u+v} ~~~ h(x,y_1,\cdots, y_u, \zz) \geq 0.
\end{align*}
Thus $a_1=0 \wedge b_1 \mathbf{Eq}uiv 0$ or $a_1>0$.
If $a_1=0 \wedge b_1 \mathbf{Eq}uiv 0$ then we denote
\begin{align*}
p_1(x,y_2,\cdots,y_u,\zz)=c_1(x,y_2,\cdots,y_u,\zz), ~~ q_1(x,y_1,\cdots,y_u)=0;
\end{align*}
otherwise, $a_1 >0$, and we denote
{\small \begin{align*}
p_1(x,y_2,\cdots,y_u,\zz)=h(x,-\frac{b_1}{2 a_1},y_2,\cdots,y_u,\zz),~~
q_1(x,y_1,\cdots,y_u)=a_1 (y_1 + \frac{b_1}{2 a_1})^2.
\end{align*} }
Then, it is easy to see $p_1(x,y_2,\cdots,y_u,\zz)$ is a quadratic polynomial satisfying
\begin{align*}
h(x,y_1,\cdots,y_u,\zz)= p_1(x,y_2,\cdots,y_u,\zz)+q_1(x,y_1,\cdots,y_u),
\end{align*}
and
\begin{align*}
\forall (x,y_2,\cdots,y_u,\zz) \in \mathbb{R}^{r+s-1+t} ~~~ p_1(x,y_2,\cdots, y_u, \zz) \geq 0,
\end{align*}
moreover, the coefficients of $y_i z_j$ ($i=2,\cdots,s,j=1,\cdots,t$) are all vanished when expanding $p_1(x,y_2,\cdots,y_u,\zz)$, and $q_1(x,y_1,\cdots,y_u)\in \mathbb{R}[x,\yy]$ is an SOS.
With the same reason, we can obtain $p_2(x,y_3,\cdots,y_u,\zz)$, $\cdots$, $p_u(x,\zz)$ and
$q_2(x,y_2, \cdots,y_u)$, $\cdots$, $q_s(x,y_u)$ such that
\begin{align*}
p_{i-1}(x,y_{i},\cdots,y_u,\zz) = p_{i}(x,y_{i+1},\cdots,y_u,\zz)+
q_i(x,y_{i},\cdots,y_u), \\[2mm]
\forall (x,y_{i+1},\cdots,y_u,\zz) \in \mathbb{R}^{d+u-i+v} ~ p_i(x,y_{i+1},\cdots, y_u, \zz) \geq 0,\\
q_i(x,y_{i},\cdots,y_u) {\rm~ is ~ a ~ SOS ~polynomial},
\end{align*}
for $i=2,\cdots,u$.
Therefore, let
\begin{align*}
h_1(x,\yy)=q_1(x,y_1,\cdots,y_u) + \cdots + q_s(x,y_u), ~~
h_2(x,\zz)=p_u(x,\zz),
\end{align*}
we have $h_1(x,\yy) \in \mathbb{R}[x,\yy]$ is an SOS and
$\forall (x,\zz) \in \mathbb{R}^{r+t} ~ h_2(x,\zz)= p_u(x,\zz) \geq 0$.
Hence, $h_2(x,\zz)$ is also an SOS, because that for the case of degree $2$, a polynomial is
positive semi-definite iff it is an SOS polynomial.
Thus $h_1(x,\yy) \in \mathbb{R}[x,\yy]$ and $h_2(x,\zz) \in \mathbb{R}[x,\zz]$ are both SOS,
moreover,
{\small \begin{align*}
h_1+h_2=q_1+\cdots +q_{u-1}+q_u +p_u
=q_1+\cdots+ q_{u-1} +p_{u-1}
=&\cdots
= q_1+p_1
=h. ~~ \qed
\end{align*} }
\end{proof}
The above proof of Lemma \ref{h:sep} gives a method to express
$h, h_1, h_2$ as sums of squares of linear expressions and a
nonnegative real number.
\begin{lemma}\label{lem:split}
Let $h, h_1, h_2$ be as in the statement of Lemma \ref{h:sep}. Then,
{\small \begin{align*}
\mathrm{(H)}:~ h(x,\yy,\zz)=&a_1 (y_1 -l_1(x,y_2,\ldots,y_u))^2 + \ldots + a_u (y_u -l_u(x))^2+\\
&a_{u+1} (z_1 -l_{u+1}(x,z_2,\ldots,z_v))^2 + \ldots + a_{u+v} (z_v -l_{u+v}(x))^2+\\
&a_{u+v+1}(x_1 - l_{u+v+1}(x_2,\ldots,x_d))^2 + \ldots + a_{u+v+d} (x_d - l_{u+v+d})^2 \\
&+a_{u+v+d+1},
\end{align*} }
where $a_i \ge 0$ and $l_j$ is a linear expression
in the corresponding variables, for $ i=1,\ldots,u+v+d+1$, $
j=1,\ldots,u+v+d$. Further,
{\small \begin{align*}
\mathrm{(H1)}:~ &h_1(x,\yy)=a_1 (y_1 -l_1(x,y_2,\ldots,y_u))^2 + \ldots + a_u (y_u -l_u(x))^2+\\
&\frac{a_{u+v+1}}{2}(x_1 - l_{u+v+1}(x_2,\ldots,x_d))^2 + \ldots + \frac{a_{u+v+d}}{2} (x_d - l_{u+v+d})^2 +\frac{a_{u+v+d+1}}{2}, \\[2mm]
\mathrm{(H2)}:~ &h_2(x,\zz)=a_{u+1} (z_1 -l_{u+1}(x,z_2,\ldots,z_v))^2 + \ldots + a_{u+v} (\zz_v -l_{u+v}(x))^2+\\
&\frac{a_{u+v+1}}{2}(x_1 - l_{u+v+1}(x_2,\ldots,x_d))^2 + \ldots + \frac{a_{u+v+d}}{2} (x_d - l_{u+v+d})^2+\frac{a_{u+v+d+1}}{2}.
\end{align*} }
\end{lemma}
\begin{theorem} \label{the:int}
Let $\phi$ and $\psi$ as defined in Problem 1 with $\phi\wedge\psi\models\bot$, which satisfy
$\mathbf{NSOSC}$.
Then there exist $\lambda_i\ge 0$ ($i=1,\cdots,r$), $\eta_j \ge 0$ ($j=0,1,\cdots,s$) and two quadratic SOS polynomial $h_1 \in \mathbb{R}[x,\yy]$ and $h_2 \in \mathbb{R}[x,\zz]$ such that
\begin{align}
& \sum_{i=1}^{r} \lambda_i f_i +\sum_{j=1}^{s} \eta_j g_j + \eta_0 + h_1+h_2 \mathbf{Eq}uiv 0, \label{con1:inte}\\
& \eta_0+\eta_1 + \ldots + \eta_s = 1.\label{con2:inte}
\end{align}
Moreover, if $\sum_{j=0}^{s_1} \eta_j > 0$, then $I >0$ is an interpolant,
otherwise $I \ge 0$ is an interpolant, where $I = \sum_{i=1}^{r_1}
\lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j + \eta_0 + h_1 \in \mathbb{R}[x]$.
\end{theorem}
\begin{proof}
From Theorem \ref{the:main}, there exist $\lambda_i\ge 0$ ($i=1,\cdots,r$), $\eta_j \ge 0$ ($j=0,1,\cdots,s$) and a quadratic SOS polynomial $h \in \mathbb{R}[x,\yy,\zz]$ such that
\begin{align}
& \sum_{i=1}^{r} \lambda_i f_i +\sum_{j=1}^{s} \eta_j g_j + \eta_0 + h \mathbf{Eq}uiv 0, \label{r1}\\
& \eta_0+\eta_1 + \ldots + \eta_s = 1. \label{r2}
\end{align}
Obviously, (\ref{r1}) is equivalent to the following formula
\begin{align*}
\sum_{i=1}^{r_1} \lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j +\eta_0+
\sum_{i=r_1+1}^{r} \lambda_i f_i +\sum_{j=s_1+1}^{s} \eta_j g_j+ h \mathbf{Eq}uiv 0,
\end{align*}
It's easy to see that
\begin{align*}
\sum_{i=1}^{r_1} \lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j +\eta_0 \in \mathbb{R}[x,\yy], ~~
\sum_{i=r_1+1}^{r} \lambda_i f_i +\sum_{j=s_1+1}^{s} \eta_j g_j \in \mathbb{R}[x,\zz].
\end{align*}
Thus, for any $1\le i \le u$, $1\le j \le v$, the term $\yy_i \zz_j$ does not appear in
\begin{align*}
\sum_{i=1}^{r_1} \lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j +\eta_0+
\sum_{i=r_1+1}^{r} \lambda_i f_i +\sum_{j=s_1+1}^{s} \eta_j g_j .
\end{align*}
Since all the conditions in Lemma \ref{h:sep} are satisfied, there exist two quadratic
SOS polynomial $h_1 \in \mathbb{R}[x,\yy]$ and $h_2 \in \mathbb{R}[x,\zz]$ such that
$h=h_1+h_2$.
Thus, we have
\begin{align*}
&\sum_{i=1}^{r_1} \lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j +\eta_0+h_1 \in \mathbb{R}[x,\yy],\\
&\sum_{i=r_1+1}^{r} \lambda_i f_i +\sum_{j=s_1+1}^{s} \eta_j g_j+h_2 \in \mathbb{R}[x,\zz],\\
&\sum_{i=1}^{r_1} \lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j +\eta_0+h_1+
\sum_{i=r_1+1}^{r} \lambda_i f_i +\sum_{j=s_1+1}^{s} \eta_j g_j+h_2 \mathbf{Eq}uiv 0
\end{align*}
Besides, as
\begin{align*}
I=\sum_{i=1}^{r_1} \lambda_i f_i +\sum_{j=1}^{s_1} \eta_j g_j +\eta_0+h_1
=-(\sum_{i=r_1+1}^{r} \lambda_i f_i +\sum_{j=s_1+1}^{s} \eta_j g_j+h_2),
\end{align*}
we have $I \in \mathbb{R}[x]$.
It is easy to see that
\begin{itemize}
\item if $\sum_{j=0}^{s_1} \eta_j > 0$ then
$\phi \models I >0$ and $\psi \wedge I>0 \models \bot$,
so $I > 0$ is an interpolation; and
\item if $\sum_{j=s_1+1}^{s} \eta_j > 0$ then
$\phi \models I \ge 0$ and $\psi \wedge I\ge 0 \models \bot$,
hence $I \ge 0$ is an interpolation. \qed
\end{itemize}
\oomit{ Since $\sum_{j=0}^s \eta_j =1$ implies either $ \sum_{j=0}^{s_1} \eta_j > 0$
or $\sum_{j=s_1+1}^{s} \eta_j > 0$.
\qed}
\end{proof}
\subsection{Computing Interpolant using Semi-Definite Programming} \label{sec:sdp}
\oomit{When the condition $\mathbf{NSOSC}$ hold, from Theorem \ref{the:int} we can see that, the problem of
interpolant generation can be reduced to the following constraint solving problem.
{\em Find:
\begin{center}
real numbers $\lambda_i\ge 0~(i=1,\cdots,r)$, $\eta_j \ge 0~(j=0,1,\cdots,s)$, and\\
two quadratic SOS polynomials $h_1 \in \mathbb{R}[x,\yy]$ and $h_2 \in \mathbb{R}[x,\zz]$,
\end{center}
such that
\begin{align*}
& \sum_{i=1}^{r} \lambda_i f_i +\sum_{j=1}^{s} \eta_j g_j + \eta_0 + h_1+h_2 \mathbf{Eq}uiv 0, \\
& \eta_0+\eta_1 + \ldots + \eta_s = 1.
\end{align*} } }
Below, we formulate computing $\lambda_i$s, $\eta_j$s and $h_1$
and $h_2$ as a semi-definite programming problem.
Let
\[W=\left( \begin{matrix}
1 & x^T & \yy^T & \zz^T\\
x & xx^T & x\yy^T & x\zz^T\\
\yy & \yyx^T & \yy\yy^T & \yy\zz^T\\
\zz & \zzx^T & \zz\yy^T & \zz\zz^T
\end{matrix}
\right )\]
\begin{comment}
\begin{align} \label{pq:def}
f_i = \left<P_i,\left( \begin{matrix}
1 & x^T & \yy^T & \zz^T\\
x & xx^T & x\yy^T & x\zz^T\\
\yy & \yyx^T & \yy\yy^T & \yy\zz^T\\
\zz & \zzx^T & \zz\yy^T & \zz\zz^T
\end{matrix}
\right ) \right > ,~~
g_j = \left<Q_j,\left( \begin{matrix}
1 & x^T & \yy^T & \zz^T\\
x & xx^T & x\yy^T & x\zz^T\\
\yy & \yyx^T & \yy\yy^T & \yy\zz^T\\
\zz & \zzx^T & \zz\yy^T & \zz\zz^T
\end{matrix}
\right ) \right >,
\end{align}
\end{comment}
\begin{align} \label{pq:def}
f_i = \langle P_i, W\rangle ,~~ g_j = \langle Q_j, W\rangle,
\end{align}
where $P_i$ and $Q_j$ are $(1+d+u+v) \times (1+d+u+v)$ matrices,
and
\begin{comment}
\begin{align*}
&h_1 = \left<
M,
\left( \begin{matrix}
1 & x^T & \yy^T & \zz^T\\
x & xx^T & x\yy^T & x\zz^T\\
\yy & \yyx^T & \yy\yy^T & \yy\zz^T\\
\zz & \zzx^T & \zz\yy^T & \zz\zz^T
\end{matrix}
\right )
\right > ,
h_2 = \left<
\hat{M},
\left( \begin{matrix}
1 & x^T & \yy^T & \zz^T\\
x & xx^T & x\yy^T & x\zz^T\\
\yy & \yyx^T & \yy\yy^T & \yy\zz^T\\
\zz & \zzx^T & \zz\yy^T & \zz\zz^T
\end{matrix}
\right )
\right >,
\end{align*}
\end{comment}
\begin{align*}
h_1 = \langle M, W\rangle , ~~ h_2 = \langle \hat{M}, W\rangle,
\end{align*}
where $M=(M_{ij})_{4\times4}, \hat{M}=(\hat{M}_{ij})_{4\times4}$
\begin{comment}
\begin{align*}
&M=\left( \begin{matrix}
M_{11} & M_{12} & M_{13} & M_{14}\\
M_{21} & M_{22} & M_{23} & M_{24}\\
M_{31} & M_{32} & M_{33} & M_{34}\\
M_{41} & M_{42} & M_{43} & M_{44}
\end{matrix} \right),~~~~~~~
\hat{M} = \left( \begin{matrix}
\hat{M}_{11} & \hat{M}_{12} & \hat{M}_{13} & \hat{M}_{14}\\
\hat{M}_{21} & \hat{M}_{22} & \hat{M}_{23} & \hat{M}_{24}\\
\hat{M}_{31} & \hat{M}_{32} & \hat{M}_{33} & \hat{M}_{34}\\
\hat{M}_{41} & \hat{M}_{42} & \hat{M}_{43} & \hat{M}_{44}
\end{matrix} \right)
\end{align*}
\end{comment}
with appropriate dimensions, for example $M_{12} \in \mathbb{R}^{1 \times d}$ and
$\hat{M}_{34} \in \mathbb{R}^{u \times v}$.
Then, with $\mathbf{NSOSC}$, by Theorem~\ref{the:int}, Problem 1 is reduced to
the following $\mathbf{SDP}$ feasibility problem.
\textbf{Find:}
\begin{center}
$\lambda_1,\ldots, \lambda_r,\eta_0,\ldots,\eta_s \in \mathbb{R}$ and real symmetric matrices $M, \hat{M} \in \mathbb{R}^{(1+d+u+v)\times (1+d+u+v)}$
\end{center}
subject to
\begin{eqnarray*}
\left\{ ~ \begin{array}{l}
\sum_{i=1}^{r} \lambda_i P_i +\sum_{j=1}^{s} \eta_j Q_j + \eta_0 E_{1,1} + M+\hat{M} = 0$, $\sum_{j=0}^{s} \eta_j=1,\\[1mm]
M_{41}=(M_{14})^T=0,M_{42}=(M_{24})^T=0,M_{43}=(M_{34})^T=0,M_{44}=0,\\[1mm]
\hat{M}_{31}=(\hat{M}_{13})^T=0,\hat{M}_{32}=(\hat{M}_{23})^T=0,\hat{M}_{33}=0,\hat{M}_{34}=(\hat{M}_{43})^T=0,\\[1mm]
M\succeq 0, \hat{M}\succeq 0,\lambda_i \ge0, \eta_j \ge 0, \mbox{ for }
i=1,\ldots,r,j=0,\ldots,s,
\end{array}
\right.
\end{eqnarray*}
where $E_{1,1}$ is a $(1+d+u+v) \times (1+d+u+v)$ matrix, whose $(1,1)$ entry is $1$ and the others are $0$.
This is a standard $\mathbf{SDP}$ feasibility problem, which can be
solved efficiently by well known $\mathbf{SDP}$ solvers, e.g., CSDP
\cite{CSDP},
SDPT3
\cite{SDPT3}, SeDuMi \cite{SeDuMi}, etc., with
time complexity polynomial in $n=d + u + v$.
\begin{remark} \label{remark:1}
Problem 1 is a typical quantifier elimination (QE) problem, which can be solved symbolically. However, it is very hard to solve large problems by general QE algorithms because of their high complexity. So, reducing Problem 1 to $\mathbf{SDP}$ problem makes it possible to solve many large problems in practice. Nevertheless, one may doubt whether we can use numerical result in verification. We think that verification must be rigorous and numerical results should be verified first. For example, after solving the above $\mathbf{SDP}$ problem numerically, we verify that whether $-(\sum_{i=1}^{r} \lambda_i f_i +\sum_{j=1}^{s} \eta_j g_j + \eta_0)$ is an SOS by the method of Lemma \ref{lem:split}, which is easy to do. If it is, the result is guaranteed and output. If not, the result is unknown (in fact, some other techniques can be employed in this case, which we do not discuss in this paper.). Thus, our algorithm is sound but not complete.
\end{remark}
\subsection{General Case}
The case of
$\textit{Var}(\phi) \subset \textit{Var}(\psi)$ is
not an issue since $\phi$ serves as an interpolant of $\phi$
and $\psi$.
We thus assume
that $\textit{Var}(\phi) \nsubseteq \textit{Var}(\psi)$.
We show below how an interpolant can be generated in the general case.
If $\phi$ and $\psi$ do not satisfy the $\mathbf{NSOSC}$
condition, i.e., an SOS polynomial $h(x, \yy, \zz)$ can be computed from nonstrict
inequalities $f_i$s using nonpositive constant multipliers, then
by the lemma below, we can construct ``simpler'' interpolation
subproblems $\phi', \psi'$
from $\phi$ and $\psi$ by constructing from $h$ an SOS polynomial
$f(x)$ such that $\phi \models f(x) \ge 0$ as well as $\psi
\models - f(x) \ge 0$.
Each $\phi'$ $\psi'$ pair has the following characteristics
because of which the algorithm
is recursively applied to $\phi'$ and $\psi'$.
\begin{enumerate}
\item[(i)] $\phi' \wedge \psi' \models \bot$,
\item[(ii)] $\phi',\psi'$ have the same form as $\phi,\psi$, i.e., $\phi'$ and $\psi'$
are defined by some $f_i' \ge 0$ and $g_j'>0$, where $f_i'$ and $g_j'$ are CQ,
\item[(iii)] $\#(\textit{Var}(\phi') \cup \textit{Var}(\psi')) <
\#(\textit{Var}(\phi) \cup \textit{Var}(\psi))$ to ensure
termination of the recursive algorithm, and
\item[(iv)] an interpolant
$I$ for $\phi$ and $\psi$ can be computed
from an interpolant $I'$ for $\phi'$ and $\psi'$ using $f$.
\end{enumerate}
\oomit{Now, suppose $\textit{Var}(\phi) \nsubseteq \textit{Var}(\psi)$ and the condition $\mathbf{NSOSC}$ does not hold,
i.e., there exist
$\delta_1,\ldots,\delta_r \ge 0$ such that
$(-\sum_{i=1}^r \delta_i f_i)$ is a nonzero SOS polynomial, we construct such
$\phi'$ and $\psi'$ satisfy $(i)-(iv)$ below. }
\begin{lemma} \label{lemma:dec}
If Problem 1 does not satisfy the $\mathbf{NSOSC}$ condition, there exists $f \in \mathbb{R}[x]$,
such that
$\phi \Leftrightarrow \phi_1 \vee \phi_2$ and
$\psi \Leftrightarrow \psi_1 \vee \psi_2$,
where,
\begin{align}
&\phi_1= (f > 0 \wedge \phi) ,~~
\phi_2 = (f = 0 \wedge \phi),\label{phi2}\\
&\psi_1 = (-f > 0 \wedge \psi),~~
\psi_2 = (f = 0 \wedge \psi).\label{psi2}
\end{align}
\end{lemma}
\begin{proof}
Since $\mathbf{NSOSC}$ does not hold, there exist $\delta_1,\ldots, \delta_r \in \mathbb{R}^+$ such that $-\sum_{i=1}^r \delta_i f_i$ is a nonzero SOS. Let $h(x,\yy,\zz)$ denote this quadratic SOS polynomial.
Since $(-\sum_{i=1}^{r_1} \delta_i f_i) \in \mathbb{R}[x,\yy]$
and $(-\sum_{i=r_1+1}^r \delta_i f_i) \in \mathbb{R}[x,\zz]$, the coefficient of any term
$\yy_i \zz_j, 1\le i \le u,1 \le j \le v,$ is 0 after expanding
$h$. By Lemma \ref{h:sep}
there exist two quadratic SOS polynomials $h_1 \in \mathbb{R}[x,\yy]$ and
$h_2 \in \mathbb{R}[x,\zz]$ such that $h=h_1+h_2$
with the following form:
{\small \begin{align*}
\mathrm{(H1)}: & ~ h_1(x,\yy)=a_1 (\yy_1 -l_1(x,\yy_2,\ldots,\yy_u))^2 + \ldots + a_u (\yy_u -l_u(x))^2+\\
&\frac{a_{u+v+1}}{2}(x_1 - l_{u+v+1}(x_2,\ldots,x_d))^2 + \ldots + \frac{a_{u+v+d}}{2} (x_d - l_{u+v+d})^2 +\frac{a_{u+v+d+1}}{2}, \\[3mm]
\mathrm{(H2)}:& ~ h_2(x,\zz)=a_{u+1} (\zz_1 -l_{u+1}(x,\zz_2,\ldots,\zz_v))^2 + \ldots + a_{u+v} (\zz_v -l_{u+v}(x))^2+\\
&\frac{a_{u+v+1}}{2}(x_1 - l_{u+v+1}(x_2,\ldots,x_d))^2 + \ldots + \frac{a_{u+v+d}}{2} (x_d - l_{u+v+d})^2+\frac{a_{u+v+d+1}}{2}.
\end{align*} }
Let
\begin{align} \label{f:form}
f= \sum_{i=1}^{r_1} \delta_i f_i + h_1=-\sum_{i=r_1+1}^r \delta_i f_i-h_2.
\end{align}
Obviously, $f \in \mathbb{R}[x,\yy]$ and $f \in \mathbb{R}[x,\zz]$,
this implies $f \in \mathbb{R}[x]$.
Since $h_1, h_2$ are SOS, it is easy to see that
$\phi \models f(x) \ge 0, ~~ \psi \models -f(x) \ge 0$.
Thus,
$\phi \Leftrightarrow \phi_1 \vee \phi_2$,
$\psi \Leftrightarrow \psi_1 \vee \psi_2$.
\qed
\end{proof}
Using the above lemma, an interpolant $I$ for $\phi$ and $\psi$
can be constructed from an interpolant $I_{2,2}$ for $\phi_2$ and
$\psi_2$.
\begin{theorem} \label{lemma:p22}
Let $\phi$, $\psi$, $\phi_1, \phi_2, \psi_1, \psi_2$ as defined in
Lemma \ref{lemma:dec}, then given an interpolant $I_{2,2}$ for $\phi_2$ and $\psi_2$,
$I:= (f >0 ) \vee (f \ge0 \wedge I_{2,2})$
is an interpolant for $\phi$ and $\psi$.
\end{theorem}
\begin{proof}
It is easy to see that $f > 0$ is an interpolant for both
$(\phi_1, \psi_1)$ and $(\phi_1, \psi_2)$, and
$f \ge 0$ is an interpolant for $(\phi_2, \psi_1)$.
Thus, if $I_{2,2}$ is an interpolant for $(\phi_2,\psi_2)$, then $I$
is an interpolant for $\phi$ and $\psi$. \qed
\end{proof}
An interpolant for $\phi_2$ and $\psi_2$ is constructed
recursively since the new constraint included in $\phi_2$
(similarly, as well
as in $\psi_2$) is: $\sum_{i=1}^{r_1} \delta_i f_i + h_1=0$ with
$h_1$ being an SOS.
Let $\phi'$ and $\psi'$ stand for the formulas constructed
after analyzing $\phi_2$ and $\psi_2$ respectively.
Given that $\delta_i$ as well as $f_i \ge
0$ for each $i$, case analysis is performed on $h_1$ depending upon
whether it has a positive constant
$a_{u+v+d+1} > 0$ or not.
\begin{theorem} \label{the:gcase:1}
Let $\phi'\hat{=} (0>0)$ and $\psi'\hat{=} (0>0)$. In the proof of Lemma \ref{lemma:dec}, if $a_{u+v+d+1} > 0$, then
$\phi'$ and $\psi'$ satisfy $(i)-(iv)$.
\end{theorem}
\begin{proof}
$(i),(ii)$ and $(iii)$ are trivially satisfied.
Since $a_{u+v+d+1} > 0$, it is easy to see that
$h_1 >0$ and $h_2 >0$.
From (\ref{phi2}), (\ref{psi2}) and (\ref{f:form}), we have
$\phi_2 \models h_1 = 0$, and $\psi_2 \models h_2=0$.
Thus $\phi_2 \Leftrightarrow \phi' \Leftrightarrow\bot$ and
$\psi_2 \Leftrightarrow \psi' \Leftrightarrow\bot$.
\qed
\end{proof}
\oomit{For the case $a_{u+v+d+1} > 0$, we construct $\phi'$ and $\psi'$ in Theorem \ref{the:gcase:1},
then we construct $\phi'$ and $\psi'$ on the case $a_{u+v+d+1} = 0$ below.}
In case
$a_{u+v+d+1} = 0$, from the fact that $h_1$ is an SOS and has
the form $\mathrm{(H1)}$, each nonzero square term in $h_1$ is
identically 0. This implies that some of the variables in $x, \yy$ can be
linearly expressed in term of other variables; the same argument applies
to $h_2$ as well. In particular, at least
one variable is eliminated in both $\phi_2$ and $\psi_2$,
reducing the number of variables appearing in $\phi$ and
$\psi$, which ensures the termination of the algorithm. A
detailed analysis is given in following lemmas, where it is shown how
this elimination of variables is performed,
generating $\phi'$ and $\psi'$ on which the algorithm can be
recursively invoked; an a theorem is also proved to ensures this.
\begin{lemma} \label{lemma:elim}
In the proof of Lemma \ref{lemma:dec}, if $a_{u+v+d+1} = 0$, then $x$ can be
represented as $(x^1, x^2)$, $\yy$ as $(\yy^1, \yy^2)$ and $\zz$ as $(\zz^1, \zz^2)$,
such that
\begin{align*}
&\phi_2 \models ( (\yy^1 = \Lambda_1 \left( \begin{matrix} x^2 \\ \yy^2 \end{matrix} \right) + \gamma_1)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) ), \\
&\psi_2 \models ((\zz^1 = \Lambda_2 \left( \begin{matrix} x^2 \\ \zz^2 \end{matrix} \right) + \gamma_2)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) ),
\end{align*}
and $\#(\textit{Var}(x^1)+\textit{Var}(\yy^1)+\textit{Var}(\zz^1)) > 0$,
for matrixes $\Lambda_1,\Lambda_2,\Lambda_3$ and vectors $\gamma_1,\gamma_2,\gamma_3$.
\end{lemma}
\begin{proof}
From (\ref{phi2}), (\ref{psi2}) and (\ref{f:form}) we have
\begin{align} \label{phi-psi:2}
\phi_2 \models h_1 = 0, ~~~~ \psi_2 \models h_2=0.
\end{align}
Since $h_1+h_2 =h$ is a nonzero polynomial, $a_{u+v+d+1} = 0$ ,
then there exist some $a_i \neq 0$, i.e. $a_i > 0$, for $1\le i \le u+v+d$.
Let
\begin{align*}
&N_1:=\{i \mid a_i > 0 \wedge 1 \le i \le u \}, \\
&N_2:=\{i \mid a_{u+i} > 0 \wedge 1 \le i \le v \}, \\
&N_3:=\{i \mid a_{u+v+i} > 0 \wedge 1 \le i \le d \}.
\end{align*}
Thus, $N_1$, $N_2$ and $N_3$ cannot all be empty. In addition, $h_1=0 $ implies that
\begin{align*}
&\yy_i=l_i(x,\yy_{i+1},\ldots,\yy_u), ~~~~for ~ i \in N_1,\\
&x_i=l_{u+v+i}(x_{i+1},\ldots,\zz_d), ~for ~ i \in N_3.
\end{align*}
Also, $h_2=0 $ implies that
\begin{align*}
&\zz_i=l_{u+i}(x,\zz_{i+1},\ldots,\zz_v), ~for ~ i \in N_2,\\
&x_i=l_{u+v+i}(x_{i+1},\ldots,\zz_d), ~for ~ i \in N_3.
\end{align*}
Now, let \oomit{Let divide each of $\yy,\zz,x$ into two parts by $N_1,N_2,N_3$}
\begin{align*}
& \yy^1 = (y_{i_1},\ldots, y_{i_{|N_1|}}),
\yy^2= (y_{j_1}, \ldots, y_{j_{u-|N_1|}}), \\
& \quad \quad \mbox{ where }
\{i_1,\ldots, i_{|N_1|}\} =N_1, \{j_1,\ldots,j_{u-|N_1|}\} = \{1,\ldots, u\} -N_1,\\
& \zz^1 = (z_{i_1},\ldots, z_{i_{|N_2|}}),
\zz^2= (z_{j_1}, \ldots, z_{j_{u-|N_2|}}), \\
& \quad \quad \mbox{ where }
\{i_1,\ldots, i_{|N_2|}\} =N_2, \{j_1,\ldots,j_{v-|N_2|}\} = \{1,\ldots, v\} -N_2,\\
& x^1 = (x_{i_1},\ldots, x_{i_{|N_3|}}),
x^2= (x_{j_1}, \ldots, x_{j_{u-|N_3|}}), \\
& \quad \quad \mbox{ where }
\{i_1,\ldots, i_{|N_3|}\} =N_3, \{j_1,\ldots,j_{d-|N_3|}\} = \{1,\ldots, d\} -N_3.
\end{align*}
Clearly, $\#(\textit{Var}(x^1)+\textit{Var}(\yy^1)+\textit{Var}(\zz^1)) > 0$.
By linear algebra, there exist three matrices $\Lambda_1,\Lambda_2,\Lambda_3$ and three vectors $\gamma_1,\gamma_2,\gamma_3$ s.t.
\begin{align*}
&\yy^1 = \Lambda_1 \left( \begin{matrix} x^2 \\ \yy^2 \end{matrix} \right) + \gamma_1,\\
&\zz^1 = \Lambda_2 \left( \begin{matrix} x^2 \\ \zz^2 \end{matrix} \right) + \gamma_2,\\
&x^1 = \Lambda_3 x^2 + \gamma_3.
\end{align*}
Since
$\phi_2 \models h_1 = 0, ~~~~ \psi_2 \models h_2=0$,
then,
\begin{align*}
&\phi_2 \models ( (\yy^1 = \Lambda_1 \left( \begin{matrix} x^2 \\ \yy^2 \end{matrix} \right) + \gamma_1)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) ), \\
&\psi_2 \models ((\zz^1 = \Lambda_2 \left( \begin{matrix} x^2 \\ \zz^2 \end{matrix} \right) + \gamma_2)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) ).
\end{align*}
\qed
\end{proof}
So, replacing $(x^1,\yy^1)$ in $f_i(x,\yy)$ and $g_j(x,\yy)$ by $\Lambda_3 x^2 + \gamma_3$
$\Lambda_1 \left( \begin{matrix} x^2 \\ \yy^2 \end{matrix} \right) + \gamma_1$ respectively,
results in new polynomials $\hat{f_i}(x^2,\yy^2)$ and $\hat{g_j}(x^2,\yy^2)$, for $i=1,\ldots,r_1$, $j=1,\ldots,s_1$.
Similarly, replacing $(x^1,\zz^1)$ in $f_i(x,\zz)$ and $g_j(x,\zz)$ by $ \Lambda_3 x^2 + \gamma_3$ and
$\Lambda_2 \left( \begin{matrix} x^2 \\ \zz^2 \end{matrix} \right) + \gamma_2$ respectively, derives new polynomials $\hat{f_i}(x^2,\zz^2)$ and $\hat{g_j}(x^2,\zz^2)$, for $i=r_1+1,\ldots,r$, $j=s_1+1,\ldots,s$. Regarding the resulted polynomials above, we have the following property.
\begin{lemma} \label{concave-hold}
Let $\xi \in \mathbb{R}^m$ and $\zeta \in \mathbb{R}^n$ be two vector variables,
$g(\xi,\zeta)= \left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right)^T G \left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right) + a^T \left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right) + \alpha$ be a CQ polynomial on $(\xi,\zeta)$,
i.e. $G \preceq 0$. Replacing $\zeta$ in $g$ by $\Lambda \xi + \gamma$ derives
$\hat{g}(\xi) = g(\xi, \Lambda \xi + \gamma)$, then $\hat{g}(\xi)$ is a CQ
polynomial in $\xi$.
\end{lemma}
\begin{proof}
$G \preceq 0$ iff $- \left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right)^T G \left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right)$ is an SOS. Thus, there exist $l_{i,1} \in \mathbb{R}^m$,
$l_{i,2} \in \mathbb{R}^n$, for
$i=1,\ldots, s$, $s \in \mathbb{N}^{+}$ s.t.
$\left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right)^T G \left( \begin{matrix} &\xi \\
&\zeta \end{matrix} \right) = - \sum_{i=1}^s (l_{i,1}^T \xi + l_{i,2}^T \zeta)^2$.
Hence,
\begin{align*}
\left( \begin{matrix} &\xi \\
&\Lambda \xi + \gamma
\end{matrix} \right)^T G \left( \begin{matrix} &\xi \\
&\Lambda \xi + \gamma
\end{matrix} \right) = - \sum_{i=1}^s (l_{i,1}^T \xi + l_{i,2}^T (\Lambda \xi + \gamma))^2\\
=- \sum_{i=1}^s ((l_{i,1}^T + l_{i,2}^T \Lambda) \xi + l_{i,2}^T \gamma)^2 \\
=- \sum_{i=1}^s ((l_{i,1}^T + l_{i,2}^T \Lambda) \xi)^2 + l(\xi),
\end{align*}
where $l(\xi)$ is a linear function in $\xi$.
Then we have
\begin{align*}
\hat{g}(\xi) = - \sum_{i=1}^s ((l_{i,1}^T + l_{i,2}^T \Lambda) \xi)^2 + l(\xi)+
\va^T \left( \begin{matrix} &\xi \\
&\Lambda \xi + \gamma
\end{matrix} \right) + \alpha.
\end{align*}
Obviously, there exist $\hat{G} \preceq 0$, $\hat{\va}$ and $\hat{\alpha}$ such that
\begin{align*}
\hat{g} = \xi \hat{G} \xi^T + \hat{\va}^T \xi + \hat{\alpha}.
\end{align*}
Therefore, $\hat{g}$ is concave quadratic polynomial in $\xi$. \qed
\end{proof}
\begin{theorem} \label{the:gcase:2}
In the proof of Lemma \ref{lemma:dec}, if $a_{u+v+d+1} = 0$, then
Lemma \ref{lemma:elim} holds. So, let $\hat{f_i}$ and $\hat{g_j}$ as above,
and
\begin{align*}
&\phi' = \bigwedge_{i=1}^{r_1} \hat{f_i} \ge 0 \wedge
\bigwedge_{j=1}^{s_1} \hat{g_j} >0, \\
&\psi' = \bigwedge_{i=r_1+1}^{r} \hat{f_i} \ge 0 \wedge
\bigwedge_{j=s_1+1}^{s} \hat{g_j} >0.
\end{align*}
Then $\phi'$ and $\psi'$ satisfy $(i)-(iv)$.
\end{theorem}
\begin{proof}
From Lemma \ref{lemma:elim}, we have
\begin{align*}
&\phi_2 \models ( (\yy^1 = \Lambda_1 \left( \begin{matrix} x^2 \\ \yy^2 \end{matrix} \right) + \gamma_1)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) ), \\
&\psi_2 \models ((\zz^1 = \Lambda_2 \left( \begin{matrix} x^2 \\ \zz^2 \end{matrix} \right) + \gamma_2)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) ).\\
\end{align*}
Let
\begin{align*}
&\phi_2' := ( (\yy^1 = \Lambda_1 \left( \begin{matrix} x^2 \\ \yy^2 \end{matrix} \right) + \gamma_1)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) \wedge \phi ), \\
&\psi_2' := ((\zz^1 = \Lambda_2 \left( \begin{matrix} x^2 \\ \zz^2 \end{matrix} \right) + \gamma_2)\wedge (x^1 = \Lambda_3 x^2 + \gamma_3) \wedge \psi ).\\
\end{align*}
Then $\phi_2 \models \phi_2'$, $\phi_2 \models \phi_2'$ and $\phi_2'\wedge\psi_2'\models\bot$. Thus any interpolant for $\phi_2'$ and $\psi_2'$ is also an interpolant of $\phi_2$ and $\psi_2$.
By the definition of $\phi'$ and $\psi'$, it follows $\phi' \wedge \psi' \models \bot$ iff $\phi_2^{'}\wedge\psi_2^{'}\models\bot$,
so $\phi' \wedge \psi' \models \bot$, $(i)$ holds.
Moreover, $\phi_2{'}\models \phi'$, $\psi_2{'}\models \psi'$, $\textit{Var}(\phi') \subseteq \textit{Var}(\phi_2{'})$ and $\textit{Var}(\psi') \subseteq \textit{Var}(\psi_2{'})$, then any interpolant for
$\phi'$ and $\psi'$ is also an interpolant for $\phi_2{'}$ and $\psi_2{'}$, then also an
interpolant for $\phi_2$ and $\psi_2$. By Theorem \ref{lemma:p22}, $(iii)$ holds.
Since $\#(\textit{Var}(\phi)+\textit{Var}(\psi)) - \#(\textit{Var}(\phi')+\textit{Var}(\psi'))=\#(x^1,\yy^1,\zz^1) >0$,
then $(vi)$ holds.
For $(ii)$, $\phi',\psi'$ have the same form with $\phi,\psi$, means that
$\hat{f_i}, i=1,\ldots,r$ are CQ and $\hat{g_j}, j=1,\ldots,s$
are CQ. This is satisfied directly by
Lemma \ref{concave-hold}.
\qed
\end{proof}
\begin{comment}
\begin{theorem} \label{the:gcase:2}
In the proof of Lemma \ref{lemma:dec}, if $a_{u+v+d+1} = 0$,
by eliminating (at least one) variables in $\phi$ and
$\psi$ in terms of other variables (Lemma \ref{lemma:elim}), resulting in
$\hat{f_i}$s and $\hat{g_j}$s defined as in Lemma
\ref{concave-hold}, mutually contradictory formulas with fewer variables
\begin{align*}
&\phi' = \bigwedge_{i=1}^{r_1} \hat{f_i} \ge 0 \wedge
\bigwedge_{j=1}^{s_1} \hat{g_j} >0, \\
&\psi' = \bigwedge_{i=r_1+1}^{r} \hat{f_i} \ge 0 \wedge
\bigwedge_{j=s_1+1}^{s} \hat{g_j} >0,
\end{align*}
are generated that satisfy $(i)-(iv)$.
\end{theorem}
\end{comment}
The following simple example illustrates how the above
construction works.
\begin{example} \label{exam1}
Let $f_1 = x_1, f_2 = x_2,f_3= -x_1^2-x_2^2 -2x_2-z^2, g_1= -x_1^2+2 x_1 - x_2^2 + 2 x_2 - y^2$. Two formulas $\phi:=(f_1 \ge 0) \wedge (f_2 \ge0) \wedge (g_1 >0)$,
$\psi := (f_3 \ge 0)$. $\phi \wedge \psi \models \bot$.
The condition $\mathbf{NSOSC}$ does not hold, since
\begin{align*}
-(0 f_1 + 2 f_2 + f_3) = x_1^2 +x_2^2 + z^2 {\rm ~ is ~ a ~ sum ~ of ~ square}.
\end{align*}
Then we have $h=x_1^2 +x_2^2 + z^2$, and
\begin{align}
h_1 = \frac{1}{2}x_1^2+\frac{1}{2}x_2^2,~~ h_2 =\frac{1}{2}x_1^2+ \frac{1}{2}x_2^2 + z^2. \label{h:choose}
\end{align}
Let $f = 0 f_1 + 2 f_2 + h_1 =
\frac{1}{2}x_1^2+\frac{1}{2}x_2^2+2x_2$.
For the recursive call, we have $f = 0$ as well as $x_1 = 0, x_2
= 0$ from $h_1=0$ to construct
$\phi'$ from $\phi$; similarly $\psi'$ is constructing
by setting $x_1=x_2=0,z=0$ in $\psi$ as derived from $h_2 = 0$.
\begin{align*}
\phi' =0 \ge 0 \wedge 0 \ge 0 \wedge -y^2 > 0 = \bot, ~~\psi^{'} = 0 \ge 0 = \top.
\end{align*}
Thus, $I(\phi',\psi'):=(0 > 0)$ is an interpolant for $(\phi',\psi')$.
An interpolant for $\phi$ and $\psi$ is thus
$(f(x) >0 ) \vee (f(x)=0 \wedge I(\phi',\psi'))$, which is
$\frac{1}{2}x_1^2+\frac{1}{2}x_2^2+2x_2 > 0$.
\end{example}
\oomit{ By Theorem \ref{the:gcase:1} and Theorem \ref{the:gcase:2}, when
$\textit{Var}(\phi) \nsubseteq \textit{Var}(\psi)$ and the
condition $\mathbf{NSOSC}$ does not hold, we can solve Problem 1 in a
recursive way. From $(vi)$ we know that this recursion must
terminate at most $d+u+v$ times. If it terminates at
$\phi',\psi'$ with $\mathbf{NSOSC}$, then Problem 1 is solved by Theorem
\ref{the:int}; otherwise, it terminates at $\phi',\psi'$ with
$\textit{Var}(\phi') \subseteq \textit{Var}(\psi')$, then
$\phi'$ itself is an interpolant for $\phi'$ and $\psi'$.
}
\subsection{Algorithms} \label{sec:alg}
Algorithm $\mathbf{IGFCH}$ deals with the case when $\phi$ and $\psi$
satisfy the $\mathbf{NSOSC}$ condition.
\begin{algorithm}[!htb]
\label{alg:int}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\mathbf{IGFCH}$ }}
\Input{Two formulas $\phi$, $\psi$ with $\mathbf{NSOSC}$ and $\phi \wedge \psi \models \bot$, where
$\phi= f_1 \ge 0 \wedge \ldots \wedge f_{r_1} \ge 0 \wedge g_1 >0 \wedge \ldots \wedge g_{s_1} > 0 $,
$\psi= f_{r_1+1} \ge 0 \wedge \ldots \wedge f_{r} \ge 0 \wedge g_{s_1+1} >0 \wedge \ldots \wedge g_{s} > 0 $,
$f_1, \ldots, f_{r}, g_1, \ldots, g_s$ are all concave quadratic polynomials,
$f_1, \ldots, f_{r_1}, g_1, \ldots, g_{s_1} \in \mathbb{R}[x,\yy]$,
$f_{r_1+1}, \ldots, f_{r}, g_{s_1+1}, \ldots, g_{s} \in \mathbb{R}[x,\zz]$
}
\Output{A formula $I$ to be a Craig interpolant for $\phi$ and $\psi$}
\SetAlgoLined
\BlankLine
\textbf{Find} $\lambda_1,\ldots,\lambda_r \ge 0,\eta_0,\eta_1,\ldots,\eta_s \ge 0, h_1 \in \mathbb{R}[x,\yy], h_2 \in \mathbb{R}[x,\zz]$ by SDP s.t.
\begin{align*}
& \sum_{i=1}^{r} \lambda_i g_j+\sum_{j=1}^{s} \eta_j g_j + \eta_0 + h_1+h_2 \mathbf{Eq}uiv 0,\\
& \eta_0+\eta_1 + \ldots + \eta_s = 1,\\
& h_1, h_2 {\rm ~ are ~ SOS~polynomial};
\end{align*}\\
\tcc{This is essentially a $\mathbf{SDP}$ problem, see Section \ref{sec:hold}}
$f:=\sum_{i=1}^{r_1} \lambda_i g_j+\sum_{j=1}^{s_1} \eta_j g_j + \eta_0 + h_1$\;
\textbf{if } $\sum_{j=0}^{s_1} \eta_j > 0$ \textbf{ then } $I:=(f>0)$;
\textbf{else} $I:=(f\ge 0)$\;
\KwRet $I$ \label{subalg:return}
\end{algorithm}
\begin{theorem}[Soundness and Completeness of $\mathbf{IGFCH}$] \label{thm:correctness-1}
$\mathbf{IGFCH}$ computes an interpolant $I$ of mutually contradictory $\phi, \psi$ with CQ
polynomial inequalities satisfying the
$\mathbf{NSOSC}$ condition .
\end{theorem}
\begin{proof}
It is guaranteed by Theorem \ref{the:int}. \qed
\end{proof}
The recursive algorithm $\mathbf{IGFCH}$ is given below. For the base
case when $\phi, \psi$ satisfy the $\mathbf{NSOSC}$ condition, it invokes $\mathbf{IGFCH}$.
\begin{algorithm}[!htb]
\label{alg:int}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\mathbf{IGFQC}$ }\label{prob:in-out}}
\Input{Two formulas $\phi$, $\psi$ with $\phi \wedge \psi \models \bot$, where
$\phi= f_1 \ge 0 \wedge \ldots \wedge f_{r_1} \ge 0 \wedge g_1 >0 \wedge \ldots \wedge g_{s_1} > 0 $,
$\psi= f_{r_1+1} \ge 0 \wedge \ldots \wedge f_{r} \ge 0 \wedge g_{s_1+1} >0 \wedge \ldots \wedge g_{s} > 0 $,
$f_1, \ldots, f_{r}, g_1, \ldots, g_s$ are all CQ polynomials,
$f_1, \ldots, f_{r_1}, g_1, \ldots, g_{s_1} \in \mathbb{R}[x,\yy]$, and
$f_{r_1+1}, \ldots, f_{r}, g_{s_1+1}, \ldots, g_{s} \in \mathbb{R}[x,\zz]$
}
\Output{A formula $I$ to be a Craig interpolant for $\phi$ and $\psi$}
\SetAlgoLined
\BlankLine
\textbf{if} $\textit{Var}(\phi)\subseteq \textit{Var}(\psi)$ \textbf{then} $I:=\phi$; \KwRet $I$\; \label{alg2:1}
\textbf{Find} $\delta_1,\ldots,\delta_r \ge 0, h \in \mathbb{R}[x,\yy,\zz]$ by SDP s.t. $\sum_{i=1}^r \delta_i f_i +h \mathbf{Eq}uiv 0$ and $h$ is
SOS; \label{alg2:2}\\
\tcc{Check the condition $\mathbf{NSOSC}$}
\textbf{if} \emph{no solution} \textbf{then} $I := \mathbf{IGFCH}(\phi, \psi)$; \label{cond:hold}
\KwRet $I$\; \label{alg2:3}
\tcc{ $\mathbf{NSOSC}$ holds}
Construct $h_1 \in \mathbb{R}[x,\yy]$ and $h_2 \in \mathbb{R}[x,\zz]$ with the forms $\mathrm{(H1)}$ and $\mathrm{(H2)}$\;
\label{alg2:4}
$f:=\sum_{i=1}^{r_1} \delta_i f_i +h_1 =-\sum_{i=r_1}^{r} \delta_i f_i -h_2 $\; \label{alg2:5}
Construct
$\phi'$ and $\psi'$ using Theorem \ref{the:gcase:1} and Theorem
\ref{the:gcase:2} by eliminating variables due to
$h_1 = h_2 = 0$\; \label{alg2:6}
$I' = \mathbf{IGFQC}(\phi', \psi')$\; \label{alg2:7}
$I:=(f>0) \vee (f \ge 0 \wedge I')$\; \label{alg2:8}
\KwRet $I$ \label{alg2:9}
\end{algorithm}
\begin{theorem}[Soundness and Completeness of $\mathbf{IGFQC}$] \label{thm:correctness-2}
$\mathbf{IGFQC}$ computes an interpolant $I$ of mutually contradictory $\phi, \psi$ with CQ
polynomial inequalities.
\end{theorem}
\begin{proof}
If $\textit{Var}(\phi) \subseteq \textit{Var}(\psi)$, $\mathbf{IGFQC}$ terminates at step \ref{alg2:1}, and
returns $\phi$ as an interpolant. Otherwise, there are two cases:
(i) If $\mathbf{NSOSC}$ holds, then $\mathbf{IGFQC}$ terminates at step \ref{alg2:3} and
returns an interpolant for $\phi$ and $\psi$ by calling
$\mathbf{IGFCH}$. Its soundness and completeness follows from the
previous theorem.
(ii) $\textit{Var}(\phi) \nsubseteq \textit{Var}(\psi)$ and
$\mathbf{NSOSC}$ does not hold: The proof is by induction on the number
of recursive calls to $\mathbf{IGFQC}$, with the case of 0 recursive
calls being (i) above.
In the induction step, assume that for a $k^{th}$-recursive call to
$\mathbf{IGFQC}$ gives a correct interpolant $I'$ for $\phi'$ and
$\psi'$, where $\phi'$ and $\psi'$ are constructed by Theorem \ref{the:gcase:1} or
Theorem \ref{the:gcase:2}.
By Theorem \ref{the:gcase:2},
the interpolant $I$ constructed from $I'$ is the correct answer
for $\phi$ and $\psi$.
The recursive algorithm terminates in all three cases: (i)
$\textit{Var}(\phi) \subseteq \textit{Var}(\psi)$, (ii) $\mathbf{NSOSC}$
holds, which is achieved at most $u+v+d$ times by Theorem
\ref{the:gcase:2}, and (iii) the number of variables in $\phi',
\psi'$ in the recursive call is smaller than the number of
variables in $\phi, \psi$.
\oomit{Meanwhile, for these two basic cases,
we have already know that this algorithm return the right
answer, by inductive method, the algorithm return the right
answer with input $\phi$ and $\psi$.} \qed
\end{proof}
\subsection{Complexity analysis of $\mathbf{IGFCH}$ and $\mathbf{IGFQC}$}
It is well known that an $\mathbf{SDP}$ problem
can be solved in polynomial time complexity. We analyze the
complexity of the above algorithms assuming that
the complexity of an $\mathbf{SDP}$ problem is of time complexity $g(k)$,
where $k$ is the input size.
\begin{theorem} \label{thm:complexity-1}
The complexity of $\mathbf{IGFCH}$ is $\mathcal{O}(g(r+s+n^2))$, where $r$
is the number of nonstrict inequalities $f_i$s and $s$ is the
number of strict inequalities $g_j$s, and $n$
is the number of variables in $f_i$s and $g_j$s.
\end{theorem}
\begin{proof}
In this algorithm we first need to solve a constraint solving problem in step $1$, see Section
\ref{sec:hold}, it is an $\mathbf{SDP}$ problem with size $\mathcal{O}(r+s+n^2)$, so the complexity
of step $1$ is $\mathcal{O}(g(r+s+n^2))$. Obviously, the complexity of steps
$2-4$ is linear in $(r+s+n^2)$, so the complexity of $\mathbf{IGFCH}$ is $\mathcal{O}(g(r+s+n^2))$. \qed
\end{proof}
\begin{theorem} \label{thm:complexity-2}
The complexity of $\mathbf{IGFQC}$ is $\mathcal{O}(n* g(r+s+n^2) )$,
where $r, s, n$ are as defined in the previous theorem.
\end{theorem}
\begin{proof}
The algorithm $\mathbf{IGFQC}$ is a recursive algorithm, which is called
at most $n$ times, since in every recursive call, at least one
variable gets eliminated. Finally, it terminates at step $1$ or step
$3$ with complexity $\mathcal{O}(g(r+s+n^2))$.
The complexity of each recursive call, i.e., the complexity
for step $2$ and steps $4-9$, can be analyzed as follows:
For step $2$, checking if $\mathbf{NSOSC}$ holds is done by solving the following problem: \\
\textbf{ find: }
$\delta_1,\ldots,\delta_r \ge 0$, and an
SOS polynomial $ h \in \mathbb{R}[x,\yy,\zz]$
s.t.
$\sum_{i=1}^r \delta_i f_i +h \mathbf{Eq}uiv 0$,
\noindent which is equivalent to the following linear matrix inequality ($\mathbf{LMI}$), \\
{\bf find: }
$\delta_1,\ldots,\delta_r \ge 0$, $M \in R^{(n+1 \times (n+1)}$,
s.t.
$M=-\sum_{i=1}^r \delta_i P_i$, $M\succeq 0$,
where $P_i \in R^{(n+1) \times (n+1)}$ is defined as (\ref{pq:def}).
Clearly, this is an $\mathbf{SDP}$ problem with size $\mathcal{O}(r+n^2)$, so the complexity of this step is $\mathcal{O}( g(r+n^2) )$.
For steps $4-9$, by the proof of Lemma \ref{h:sep}, it is easy to see that to represent $h$ in the
form $\mathrm{(H)}$ in Lemma \ref{lem:split}
can be done with complexity
$\mathcal{O}(n^2)$,
$h_1$ and $h_2$ can be computed with complexity $\mathcal{O}(n^2)$. Thus,
the complexity of step $4$ is $\mathcal{O}(n^2)$. Step $5$ is much easy. For step $6$,
using linear algebra operations, it is
easy to see that the complexity is $\mathcal{O}(n^2+r+s)$.
So, the complexity is $\mathcal{O}(n^2+r+s)$ for step $4-9$.
In a word, the overall complexity of $\mathbf{IGFQC}$ is
\begin{eqnarray*}
\mathcal{O}(g(r+s+n^2))+n \mathcal{O}(n^2+r+s)
& = & \mathcal{O}(n * g(r+s+n^2) ).
\end{eqnarray*}
\qed
\end{proof}
\section{Combination: quadratic concave polynomial inequalities
with uninterpreted function symbols (\textit{EUF})}
This section combines the quantifier-free theory of quadratic
concave polynomial inequalities with the theory of equality over
uninterpreted function symbols (\textit{EUF}).
\oomit{Using hierarchical
reasoning framework proposed in \cite{SSLMCS2008} which was
applied in \cite{RS10} to generate interpolants for mutually
contradictory formulas in the combined quantfier-free theory of linear
inequalities over the reals and equality over uninterpreted
symbols, we show below how the algorithm $\mathbf{IGFQC}$ for quadratic concave
polynomial inequalities over the reals can be extended to
generate interpolants for mutually contradictory formulas
consisting of quadratic concave polynomials expressed using terms
built from unintepreted symbols.}
The proposed algorithm for generating interpolants for the combined
theories is presented in Algorithm~\ref{alg:euf}. As the reader would observe,
it is patterned after the algorithm $\text{INTER}_{LI(Q)^\Sigma}$ in Figure 4 in
\cite{RS10} following the hierarchical reasoning and
interpolation generation framework in \cite{SSLMCS2008} with the following key differences\footnote{The
proposed algorithm andd its way of handling of combined theories
do not crucially depend upon using algorithms in \cite{RS10};
however, adopting their approach makes proofs and presentation
easier by focusing totally on the quantifier-free theory of CQ polynomial
inequalities.}:
\begin{enumerate}
\item To generate interpolants for mutually contradictory
conjunctions of CQ polynomial
inequalities, we call $\mathbf{IGFQC}$.
\item We prove below that (i) a nonlinear equality over
polynomials cannnot be generated from CQ
polynomials, and furthermore (ii) in the base case when the $\mathbf{NSOSC}$
condition is satisfied by CQ polynomial
inequalities, linear equalities are deduced only from the linear
inequalities in a problem (i.e., nonlinear inequalities do not play any
role); separating terms for mixed equalities are computed the
same way as in the algorithm SEP in \cite{RS10}, and (iii) as shown in Lemmas \ref{h:sep},
\ref{lem:split} and Theorem \ref{the:gcase:2}, during recursive calls to $\mathbf{IGFQC}$, additional
linear unmixed equalities are deduced which are local to either $\phi$
or $\psi$, we can use these equalities as well as those in
(ii) for the base case to reduce the number of variables
appearing in $\phi$ and $\psi$ thus reducing the complexity of
the algorithm;
{ equalities relating variables of $\phi$ are also
included in the interpolant}.
\end{enumerate}
Other than that, the proposed algorithm reduces to
$\text{INTER}_{LI(Q)^\Sigma}$ if $\phi, \psi$ are purely from $LI(Q)$ and/or
$\textit{EUF}$.
In order to get directly to the key concepts used, we assume the reader's
familiarity with the basic construction of flattening and
purification by introducing fresh variables for the arguments
containing uninterpreted functions.
\oomit{\begin{definition}
Given two formulas $\phi$ and $\psi$ with $\phi \wedge \psi \models \bot$. A formula $I$ is said to be
an interpolant for $\phi$ and $\psi$, if the following three conditions hold, $(i)$ $\phi \models I$, $(ii)$
$I \wedge \psi \models \bot$, and $(iii)$ the variables and function symbols in $I$ is in both $\phi$ and $\psi$,
i.e., $\textit{Var}(I) \subset \textit{Var}(\phi) \cap \textit{Var}(\psi) \wedge FS(I) \subset FS(\phi) \cap FS(\psi)$, where $FS(w)$ means that
the function symbols in $w$.
\end{definition} }
\subsection{Problem Formulation}
Let $\Omega = \Omega_1 \cup \Omega_2 \cup \Omega_3$ be a finite
set of uninterpreted function symbols in $\textit{EUF};$ further, denote
$\Omega_1 \cup \Omega_2$ by $\Omega_{12}$
and $\Omega_1 \cup \Omega_3$ by $\Omega_{13}$.
Let $\mathbb{R}[x,\yy,\zz]^{\Omega}$ be the extension of $\mathbb{R}[x,\yy,\zz]$
in which polynomials can have terms built using function symbols
in $\Omega$ and variables in $x, \yy, \zz$.
\begin{problem} \label{EUF-problem}
Suppose two formulas $\phi$ and $\psi$ with
$\phi \wedge \psi \models \bot$, where
$\phi= f_1 \ge 0 \wedge \ldots \wedge f_{r_1} \ge 0 \wedge g_1 >0
\wedge \ldots \wedge g_{s_1} > 0 $,
$\psi= f_{r_1+1} \ge 0 \wedge \ldots \wedge f_{r} \ge 0 \wedge
g_{s_1+1} >0 \wedge \ldots \wedge g_{s} > 0 $,
where $f_1, \ldots, f_{r}, g_1, \ldots, g_s$ are all CQ polynomial,
$f_1, \ldots, f_{r_1}, g_1, \ldots, g_{s_1} \in
\mathbb{R}[x,\yy]^{\Omega_{12}}$,
$f_{r_1+1}, \ldots, f_{r}, g_{s_1+1}, \ldots, g_{s} \in
\mathbb{R}[x,\zz]^{\Omega_{13}}$,
the goal is to generate an
interpolant $I$ for $\phi$ and $\psi$, expressed using the common
symbols $x, \Omega_1$, i.e., $I$ includes only polynomials in $\mathbb{R}[x]^{\Omega_1}$.
\end{problem}
{\bf Flatten and Purify:} Purify and flatten the formulas $\phi$
and $\psi$ by introducing fresh variables for each term with
uninterpreted symbols as well as for the terms with uninterpreted
symbols. Keep track of new variables introduced exclusively for
$\phi$ and $\psi$ as well as new common variables.
Let $\overline{\phi} \wedge \overline{\psi} \wedge \bigwedge D$
be obtained from $\phi \wedge \psi$ by flattening and purification
where $D$ consists of unit clauses of the form
$\omega(c_1,\ldots,c_n)=c$, where $c_1,\ldots,c_n$ are variables
and $\omega \in \Omega$.
Following \cite{SSLMCS2008,RS10}, using the axiom of
an uninterpreted function symbol, a set $N$ of Horn clauses are generated as follows,
$$
N=\{ \bigwedge_{k=1}^n c_k=b_k \rightarrow c=b \mid \omega(c_1,\ldots,c_n)=c \in D, \omega(b_1,\ldots,b_n)=b \in D \}.
$$
The set $N$ is partitioned into $N_{\phi}, N_{\psi}, N_{\text{mix}}$
with all symbols in $N_{\phi}, N_{\psi}$ appearing in $\overline{\phi}$, $\overline{\psi}$,
respectively, and $N_{\text{mix}}$ consisting of symbols from both $\overline{\phi}, \overline{\psi}$.
It is easy to see that for every Horn clause in $N_{\text{mix}}$, each of
equalities in the hypothesis as well as conclusion is mixed.
\begin{eqnarray} \label{eq:reducedP}
\phi \wedge \psi \models \bot \mbox{ iff } \overline{\phi} \wedge \overline{\psi} \wedge D \models \bot
\mbox{ iff } (\overline{\phi}\wedge N_{\phi}) \wedge (\overline{\psi} \wedge N_{\psi}) \wedge N_{\text{mix}} \models \bot.
\end{eqnarray}
Notice that $ \overline{\phi} \wedge \overline{\psi} \wedge N
\models \bot$ has no uninterpreted function symbols. An interpolant generated
for this problem\footnote{after properly handling
$N_{\text{mix}}$ since Horn clauses have symbols both from
$\overline{\phi}$ and $\overline{\psi}$.} can be used to
generate an interpolant for $\phi, \psi$ after uniformly
replacing all new symbols by their corresponding expressions from $D$.
\subsection{Combination algorithm}
If $N_{\text{mix}}$ is empty, implying there are
no mixed Horn clauses, then the algorithm invokes $\mathbf{IGFQC}$ on a
finite set of subproblems generated from a disjunction of
conjunction of polynomial inequalities obtained after expanding Horn
clauses in $N_{\phi}$ and $N_\psi$ and applying De Morgan's
rules. The resulting interpolant is a disjunction of the
interpolants generated for each subproblem.
The case when $N_{\text{mix}}$ is nonempty is more interesting, but it has the same structure as
the algorithm $\text{INTER}_{LI(Q)^\Sigma}$ in \cite{RS10} except that
instead of $\text{INTER}_{LI(Q)}$, it calls $\mathbf{IGFQC}$.
The following lemma proves that if a conjunction of polynomial
inequalities satisfies the $\mathbf{NSOSC}$ condition and an
equality on variables can be deduced from it, then it suffices to
consider only linear inequalities in the conjunction. This
property enables us to use algorithms used in \cite{RS10} to
generate such equalities as well as separating terms for the
constants appearing in mixed equalities (algorithm SEP in
\cite{RS10}).
\begin{lemma} \label{lemma:qc2lin}
Let $f_i$, $i=1,\ldots,r$ be CQ polynomials, and $\lambda_i \ge 0$, if
$\sum_{i=1}^{r} \lambda_i f_i \mathbf{Eq}uiv 0$,
then for any $1\le i \le r$,
$\lambda_i=0$ or $f_i$ is linear.
\end{lemma}
\begin{proof}
Let $f_i = x^T A_i x + l_i^T x + \gamma_i$, then $A_i \preceq 0$, for $i=1, \ldots, r$.
Since $\sum_{i=1}^{r} \lambda_i f_i = 0$, we have $\sum_{i=1}^{r} \lambda_i A_i = 0$.
Thus for any $1\le i \le r$, $\lambda_i=0$ or $A_i=0$. \qed
\end{proof}
\begin{lemma} \label{lem:linear-part}
Let $\overline{\phi}$ and $\overline{\psi}$ be obtained as above with $\mathbf{NSOSC}$. If $\overline{\phi} \wedge \overline{\psi}$ is satisfiable, $\overline{\phi} \wedge \overline{\psi} \models c_k=b_k$, then
$LP(\overline{\phi}) \wedge LP(\overline{\psi}) \models c_k=b_k$,
where $LP(\overline{\phi})$ ($LP(\overline{\psi})$) is a formula defined by all the linear constraints in
$\overline{\phi}$ ($\overline{\psi}$).
\end{lemma}
\begin{proof}
Since $\overline{\phi} \wedge \overline{\psi} \models c_k=b_k$, then $\overline{\phi} \wedge \overline{\psi} \wedge c_k>b_k \models \bot$. By Theorem \ref{the:int},
there exist $\lambda_i\ge 0$ ($i=1,\cdots,r$), $\eta_j \ge 0$ ($j=0,1,\cdots,s$), $\eta \ge 0$ and two quadratic SOS polynomials $\overline{h}_1 $ and $\overline{h}_2 $ such that
\begin{align}
& \sum_{i=1}^{r} \lambda_i \overline{f}_i +\sum_{j=1}^{s} \eta_j \overline{g}_j+\eta (c_k-b_k) + \eta_0 + \overline{h}_1+\overline{h}_2 \mathbf{Eq}uiv 0, \label{cond:sep:1}\\
& \eta_0+\eta_1 + \ldots + \eta_s +\eta= 1.\label{cond:sep:2}
\end{align}
As $\overline{\phi} \wedge \overline{\psi}$ is satisfiable and $\overline{\phi} \wedge \overline{\psi} \models c_k=b_k$, there exist $x_0,\yy_0,\zz_0,\aa_0,\bb_0,\cc_0$ s.t.
$\overline{\phi}[x/x_0, \yy/\yy_0,\aa/\aa_0,\cc/\cc_0]$, $\overline{\psi}[x/x_0, \zz/\zz_0,\bb/\bb_0,\cc/\cc_0]$,
and $c_k=b_k[\aa/\aa_0,\bb/\bb_0,\cc/\cc_0]$. Thus, it follows that
$\eta_0=\eta_1=\ldots=\eta_s=0$ from (\ref{cond:sep:1}) and $\eta=1$ from (\ref{cond:sep:2}).
Hence, (\ref{cond:sep:1}) is equivalent to
\begin{align} \label{amb}
\sum_{i=1}^{r} \lambda_i \overline{f}_i + (c_k-b_k) + \overline{h}_1+\overline{h}_2 \mathbf{Eq}uiv 0.
\end{align}
Similarly, we can prove that there exist $\lambda_i'\ge 0$ ($i=1,\cdots,r$) and two quadratic SOS polynomials $h_1'$ and $h_2'$ such that
\begin{align} \label{bma}
\sum_{i=1}^{r} \lambda_i' \overline{f}_i + (b_k-c_k) + \overline{h}_1'+\overline{h}_2' \mathbf{Eq}uiv 0.
\end{align}
From (\ref{amb}) and (\ref{bma}), it follows
\begin{align} \label{bmap}
\sum_{i=1}^{r} (\lambda+\lambda_i') \overline{f}_i + \overline{h}_1+\overline{h}_1'+\overline{h}_2+\overline{h}_2' \mathbf{Eq}uiv 0.
\end{align}
In addition, $\mathbf{NSOSC}$ implies $\overline{h}_1\mathbf{Eq}uiv \overline{h}_1' \mathbf{Eq}uiv \overline{h}_2 \mathbf{Eq}uiv \overline{h}_2' \mathbf{Eq}uiv0$. So
\begin{align} \label{amb1}
\sum_{i=1}^{r} \lambda_i \overline{f}_i + (c_k-b_k) \mathbf{Eq}uiv 0,
\end{align}
and
\begin{align} \label{bma1}
\sum_{i=1}^{r} \lambda_i' \overline{f}_i + (b_k-c_k) \mathbf{Eq}uiv 0.
\end{align}
Applying Lemma \ref{lemma:qc2lin} to (\ref{amb1}), we have that
$\lambda_i=0$ or $f_i$ is linear.
So
\begin{align*}
LP(\overline{\phi}) \wedge LP(\overline{\psi}) \models c_k \le b_k.
\end{align*}
Likewise, by applying Lemma \ref{lemma:qc2lin} to (\ref{bma1}), we have
\begin{align*}
LP(\overline{\phi}) \wedge LP(\overline{\psi}) \models c_k \ge b_k. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\qed
\end{align*}
\end{proof}
If $\mathbf{NSOSC}$ is not satisfied, then the recursive
call to $\mathbf{IGFQC}$ can generate linear equalities as stated
in Theorems \ref{the:gcase:1} and \ref{the:gcase:2} which can
make hypotheses in a Horn clause in $N_{\text{mix}}$ true, thus
deducing a mixed equality on symbols .
\begin{algorithm}[!htb]
\label{alg:euf}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\mathbf{IGFQC}eu$} \label{prob:in-out}}
\Input{two formulas $\overline{\phi}$, $\overline{\psi}$, which are
constructed respectively from $\phi$ and $\psi$ by flattening and purification, \\
$N_{\phi}$ : instances of functionality axioms for functions in $D_{\phi}$,\\
$N_{\psi}$ : instances of functionality axioms for functions in $D_{\psi}$,\\
where $\overline{\phi} \wedge \overline{\psi} \wedge N_{\phi} \wedge N_{\psi} \models \bot$,
}
\Output{A formula $I$ to be a Craig interpolant for $\phi$ and $\psi$.}
\SetAlgoLined
\BlankLine
Transform $\overline{\phi}\wedge N_{\phi} $ to a DNF $\vee_i \phi_i$\; \label{alg4:1}
Transform $\overline{\psi}\wedge N_{\psi} $ to a DNF $\vee_j \psi_j$\; \label{alg4:2}
\KwRet $I:= \vee_i \wedge_j \mathbf{IGFQC}(\phi_i, \psi_j)$ \label{alg4:3}
\end{algorithm}
\begin{algorithm}[!htb]
\label{alg:euf}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\mathbf{IGFQC}e$ }\label{prob:in-out}}
\Input{ $\overline{\phi}$ and $\overline{\psi}$: two formulas, which are constructed
respective from $\phi$ and $\psi$ by flattening and purification, \\
$D$ : definitions for fresh variables introduced during flattening and purifying $\phi$ and $\psi$,\\
$N$ : instances of functionality axioms for functions in $D$,\\
where $\phi \wedge \psi \models \bot$, \\
$\overline{\phi}= f_1 \ge 0 \wedge \ldots \wedge f_{r_1} \ge 0 \wedge g_1 >0 \wedge \ldots \wedge g_{s_1} > 0 $, \\
$\overline{\psi}= f_{r_1+1} \ge 0 \wedge \ldots \wedge f_{r} \ge 0 \wedge g_{s_1+1} >0 \wedge \ldots \wedge g_{s} > 0 $,
where \\
$f_1, \ldots, f_{r}, g_1, \ldots, g_s$ are all CQ polynomial,\\
$f_1, \ldots, f_{r_1}, g_1, \ldots, g_{s_1} \in \mathbb{R}[x,\yy]$, and\\
$f_{r_1+1}, \ldots, f_{r}, g_{s_1+1}, \ldots, g_{s} \in \mathbb{R}[x,\zz]$
}
\Output{A formula $I$ to be a Craig interpolant for $\phi$ and $\psi$}
\SetAlgoLined
\BlankLine
\eIf { $\mathbf{NSOSC}$ holds }
{ $L_1:=LP(\overline{{\phi}})$; $L_2:=LP(\overline{{\psi}})$\; \label{alg3:7}
separate $N$ to $N_{\phi}$, $N_{\psi}$ and $N_{mix}$\;
$N_{\phi}, N_{\psi} := \textbf{SEPmix}(L_1, L_2, \emptyset, N_{\phi}, N_{\psi}, N_{mix})$\;
$\overline{I} := \textbf{IGFQCEunmixed}(\overline{\phi}, \overline{\psi}, N_{\phi}, N_{\psi})$\;
}
{ Find $\delta_1,\ldots,\delta_r \ge 0$ and an SOS
polynomial $h$ using SDP
s.t. $\sum_{i=1}^r \delta_i f_i +h \mathbf{Eq}uiv 0$,\; \label{alg3:19}
Construct $h_1 \in \mathbb{R}[x,\yy]$ and $h_2 \in \mathbb{R}[x,\zz]$ with form $(H1)$ and $(H2)$\;
\label{alg3:20}
$f:=\sum_{i=1}^{r_1} \delta_i f_i +h_1 =-\sum_{i=r_1}^{r} \delta_i f_i -h_2 $\; \label{alg3:22}
Construct $\overline{\phi'}$ and $\overline{\psi'}$ by Theorem \ref{the:gcase:1} and Theorem
\ref{the:gcase:2} by eliminating variables due to condition
$h_1 = h_2 = 0$\; \label{alg3:22}
$I' := \mathbf{IGFQC}e(\overline{\phi'}, \overline{\psi'},D,N)$\; \label{alg3:24}
$\bar{I}:=(f>0) \vee (f \ge 0 \wedge I')$\; }
Obtain $I$ from $\overline{I}$\;
\KwRet $I$
\end{algorithm}
\begin{algorithm}[!htb]
\label{alg:euf}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\textbf{SEPmix}$ }\label{prob:in-out}}
\Input{ $L_1,L_2$: two sets of linear inequalities,\\
$W$: a set of equalities,\\
$N_{\phi}, N_{\psi}, N_{mix}$: three sets of instances of functionality axioms.
}
\Output{$N_{\phi}, N_{\psi}$: s.t. $N_{mix}$ is separated into $N_{\phi}$ or $N_{\psi}$.}
\SetAlgoLined
\BlankLine
\eIf {there exists $(\bigwedge_{k=1}^K c_k=b_k \rightarrow c=b) \in N_{mix}$ s.t
$L_1 \wedge L_2 \wedge W \models \bigwedge_{k=1}^K c_k=b_k$}
{
\eIf { $c$ is $\phi$-local and $b$ is $\psi$-local}
{
for each $k \in \{ 1, \ldots, K \}$,
$t_k^{-}, t_k^{+} := \textbf{SEP}(L_1,L_2,c_k,b_k)$\;
$\alpha:=$ function symbol corresponding to $\bigwedge_{k=1}^K c_k=b_k \rightarrow c=b$\;
$t:=$ fresh variable;
$D := D \cup \{ t=f(t_1^{+}, \ldots, t_K^{+}) \}$\;
$C_{\phi}:=\bigwedge_{k=1}^K c_k=t_k^{+} \rightarrow c=t$;
$C_{\psi}:=\bigwedge_{k=1}^K t_k^{+}=b_k \rightarrow t=b$\;
$N_{mix}:=N_{mix}-\{ C \}$; $N_{\phi} := N_{\phi} \cup \{ C_{\phi} \}$\;
$N_{\psi} := N_{\psi} \cup \{ C_{\psi} \}$;
$W:= W \cup \{ c=t,t=d \}$\;
}
{
\eIf {$c$ and $b$ are $\phi$-local}
{
$N_{mix} := N_{mix} -\{ C \}$; $N_{\phi} := N_{\phi} \cup \{ C \}$; $W:=W \cup \{ c = b \}$\;
}
{
$N_{mix} := N_{mix} -\{ C \}$; $N_{\phi} := N_{\phi} \cup \{ C \}$; $W:=W \cup \{ c = b \}$\;
}
}
call $\textbf{SEPmix} (L_1, L_2, W, N_{\phi}, N_{\psi}, N_{mix})$\;
}
{
\KwRet $N_{\phi}$ and $N_{\psi}$\;
}
\end{algorithm}
\begin{algorithm}[!htb]
\label{alg:euf}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\textbf{SEP}$ }\label{prob:in-out}}
\Input{ $L_1,L_2$: two sets of linear inequalities,\\
$c_k, b_k$: local variables from $L_1$ and $L_2$ respectively.
}
\Output{$t^{-}, t^{+}$: expressions over common variables of $L_1$ and $L_2$
s.t $L_1 \models t^{-} \le c_k \le t^{+}$ and $L_2 \models t^{+} \le b_k \le t^{-}$}
\SetAlgoLined
\BlankLine
rewrite $L_1$ and $L_2$ as constraints in matrix form $a - A x \ge 0$ and $b - B x \ge 0$\;
$x_i, x_j$ in $x$ is the variable $c_k$ and $b_k$\;
$e^{+} := \nu^{+} A + \mu^{+} B$; $e^{-} := \nu^{-} A + \mu^{-} B$\;
$\nu^{+},\mu^{+} :=$ solution for $\nu^{+} \ge 0 \wedge \mu^{+} \ge 0 \wedge \nu^{+} a+ \mu^{+} b \le 0 \wedge
e_i^{+}=1 \wedge e_j^{+}=-1 \wedge \bigwedge_{l \neq i,j} e_l^{+}=0$\;
$\nu^{-},\mu^{-} :=$ solution for $\nu^{-} \ge 0 \wedge \mu^{-} \ge 0 \wedge \nu^{-} a+ \mu^{-} b \le 0 \wedge
e_i^{-}=-1 \wedge e_j^{-}=1 \wedge \bigwedge_{l \neq i,j} e_l^{-}=0$\;
$t^{+} := \mu^{+}Bx + x_j - \mu^{+} b$\;
$t^{-} := \nu^{-} Ax + x_i - \nu^{-} a$\;
\KwRet $t^{+}$ and $t^{-}$\;
\end{algorithm}
\begin{theorem} (Soundness and Completeness of $\mathbf{IGFQC}e$)
$\mathbf{IGFQC}e$ computes an interpolant $I$ of mutually contradictory $\phi, \psi$ with CQ
polynomial inequalities and \textit{EUF}.
\end{theorem}
\begin{proof}
Let $\phi$ and $\psi$ are two formulas satisfy the conditions of the input of the
Algorithm $\mathbf{IGFQC}e$, $D$ is the set of definitions of fresh variables introduced during
flattening and purifying $\phi$ and $\psi$, and $N$ is the set of instances of functionality
axioms for functions in $D$.
If the condition $\mathbf{NSOSC}$ is satisfied, then from Lemma \ref{lem:linear-part},
we could deal with $N$ just using the linear constraints in $\phi$ and $\psi$, which
is the same as \cite{RS10}. Since $N$ is easy to be divided into three parts,
$N_{\phi} \wedge N_{\psi} \wedge N_{\text{mix}}$. From the algorithm in \cite{RS10}, $N_{\text{mix}}$
can be divided into two parts $N_{\phi}^{\text{mix}}$ and $N_{\psi}^{\text{mix}}$ and add them to
$N_{\phi}$ and $N_{\psi}$, respectively. Thus, we have
\begin{eqnarray*}
\phi \wedge \psi \models \bot& ~\Leftrightarrow ~ & \overline{\phi} \wedge \overline{\psi} \wedge D \models \bot ~ \Leftrightarrow ~
\overline{\phi} \wedge \overline{\psi} \wedge N_{\phi} \wedge N_{\psi} \wedge N_{\text{mix}} \models \bot \\
& \Leftrightarrow &\overline{\phi} \wedge N_{\phi} \wedge N_{\phi}^{\text{mix}}
\wedge \overline{\psi} \wedge N_{\psi} \wedge N_{\psi}^{\text{mix}} \models \bot.
\end{eqnarray*}
The correctness of step $4$ is guaranteed by Lemma~\ref{lem:linear-part} and Theorem 8 in \cite{RS10}.
After step $4$, $N_{\phi}$ is replaced by $N_{\phi}\wedge N_{\phi}^{\text{mix}}$, and $N_{\psi}$ is
replaced by $N_{\psi} \wedge N_{\psi}^{\text{mix}}$.
An interpolant for
$\overline{\phi} \wedge N_{\phi} \wedge N_{\phi}^{\text{mix}}$
and $\overline{\psi} \wedge N_{\psi} \wedge N_{\psi}^{\text{mix}}$ is generated in step $5$, the correctness of
this step is guaranteed
by Theorem~\ref{thm:correctness-2}.
Otherwise if the condition $\mathbf{NSOSC}$ is not satisfied, we can obtain two polynomials $h_1$ and
$h_2$, and derive two formulas
$\overline{\phi'}$ and $\overline{\psi'}$. By Theorem \ref{lemma:p22}, if there is an
interpolant $I'$ for $\overline{\phi'}$ and $\overline{\psi'}$, then we can get an interpolant $I$
for $\overline{\phi}$ and $\overline{\psi}$ at step $11$.
Similar to the proof of Theorem~\ref{thm:correctness-2},
it is easy to argue that this reduction will terminate at the case when $\mathbf{NSOSC}$ holds in finite steps.
Thus, this completes the proof. \qed
\end{proof}
\begin{example}
Let two formulae $\phi$ and $\psi$ be defined as follows,
\begin{align*}
\phi:=&(f_1=-(y_1-x_1+1)^2-x_1+x_2 \ge 0) \wedge (y_2=\alpha(y_1)+1) \\
&\wedge ( g_1= -x_1^2-x_2^2-y_2^2+1 > 0),
\end{align*}
\begin{align*}
\psi:=&(f_2=-(z_1-x_2+1)^2+x_1-x_2 \ge 0) \wedge (z_2=\alpha(z_1)-1) \\
&\wedge (g_2= -x_1^2-x_2^2-z_2^2+1 > 0),
\end{align*}
where $\alpha$ is an uninterpreted function. Then
\begin{align*}
\overline{\phi}:=&(f_1=-(y_1-x_1+1)^2-x_1+x_2 \ge 0) \wedge (y_2=y+1) \\
&\wedge ( g_1= -x_1^2-x_2^2-y_2^2+1 > 0),\\
\overline{\psi}:=&(f_2=-(z_1-x_2+1)^2+x_1-x_2 \ge 0) \wedge (z_2=z-1) \\
&\wedge (g_2= -x_1^2-x_2^2-z_2^2+1 > 0),\\
D=(&y_1=z_1 \rightarrow y=z).
\end{align*}
The condition NSOSC is not satisfied, since
$-f_1-f_2=(y_1-x_1+1)^2+(z_1-x_2+1)^2$ is a SOS. It is easy to have
$$h_1=(y_1-x_1+1)^2~, ~~h_2=(z_1-x_2+1)^2.$$
Let $f:=f_1+h_1=-f_2-h_2=-x_1+x_2$, then it is easy to see that
$${\phi} \models f \ge0 ~,~~{\psi} \models f \le0.$$
Next we turn to find an interpolant for the following formulae
$$((\phi \wedge f>0) \vee (\phi \wedge f=0)) ~~and ~~ ((\psi \wedge -f>0) \vee (\psi \wedge f=0)).$$
Then
\begin{align}
\label{int:eq:e}
(f>0) \vee (f\ge0 \wedge I_2)
\end{align}
is an interpolant for $\phi$ and $\psi$,
where $I_2$ is an interpolant for $ \phi \wedge f=0$ and $\psi \wedge f=0$.
It is easy to see that
\begin{align*}
\phi \wedge f=0 \models y_1=x_1-1 ~,~~ \psi \wedge f=0 \models z_1=x_2-1.
\end{align*}
Substitute then into $f_1$ in $\overline{\phi}$ and $\overline{\psi}$, we have
\begin{align*}
\overline{\phi'}=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1,\\
\overline{\psi'}=&~~~~x_1-x_2 \ge 0 \wedge z_2=z-1 \wedge g_2>0 \wedge z_1=x_2-1.
\end{align*}
Only using the linear form in $\overline{\phi'}$ and $\overline{\psi'}$ we deduce that $y_1=z_1$ as
\begin{align*}
\overline{\phi'} \models t^{-}=x_1-1 \le y_1 \le t^{+}=x_2-1~~,~~\overline{\psi'} \models x_2-1 \le z_1 \le x_1-1.
\end{align*}
Let $t=\alpha(t)$, then separate $y_1=z_1 \rightarrow y=z$ into two parts,
\begin{align*}
y_1=t^{+} \rightarrow y=t, ~~ t^{+}=z_1 \rightarrow t=z.
\end{align*}
Add them to $\overline{\phi'}$ and $\overline{\psi'}$ respectively, we have
\begin{align*}
\overline{\phi'}_1=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1 \wedge y_1=x_2-1 \rightarrow y=t,\\
\overline{\psi'}_1=&~~~~x_1-x_2 \ge 0 \wedge z_2=z-1 \wedge g_2>0 \wedge z_1=x_2-1 \wedge x_2-1=z_1 \rightarrow t=z.
\end{align*}
Then
\begin{align*}
\overline{\phi'}_1=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1 \wedge \\
& (x_2-1>y_1 \vee y_1>x_2-1 \vee y=t),\\
\overline{\psi'}_1=&~~~~x_1-x_2 \ge 0 \wedge z_2=z-1 \wedge g_2>0 \wedge z_1=x_2-1 \wedge t=z.
\end{align*}
Thus,
\begin{align*}
\overline{\phi'}_1=&\overline{\phi'}_2\vee \overline{\phi'}_3 \vee \overline{\phi'}_4,\\
\overline{\phi'}_2=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1 \wedge x_2-1>y_1,\\
\overline{\phi'}_3=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1 \wedge y_1>x_2-1,\\
\overline{\phi'}_4=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1 \wedge y=t.
\end{align*}
Since
$\overline{\phi'}_3=false$, then
$\overline{\phi'}_1=\overline{\phi'}_2\vee \overline{\phi'}_4$.
Then find interpolant
$$I(\overline{\phi'}_2,\overline{\psi'}_1),~~~~I(\overline{\phi'}_4,\overline{\psi'}_1). $$
$=$ replace by two $\ge$, like,
$y_1=x_1-1$ replace by $y_1\ge x_1-1$ and $x_1-1 \ge y_1$.
Then let $I_2=I(\overline{\phi'}_2,\overline{\psi'}_1) \vee I(\overline{\phi'}_4,\overline{\psi'}_1) $
an interpolant is found from (\ref{int:eq:e}) .
\end{example}
\begin{comment}
\begin{example}
Consider $\phi$ and $\psi$ where $\alpha$ is an uninterpereted
function symbol:
\begin{align*}
\phi:=&(f_1=-(y_1-x_1+1)^2-x_1+x_2 \ge 0) \wedge (y_2=\alpha(y_1)+1) \\
&\wedge ( g_1= -x_1^2-x_2^2-y_2^2+1 > 0),
\end{align*}
\begin{align*}
\psi:=&(f_2=-(z_1-x_2+1)^2+x_1-x_2 \ge 0) \wedge (z_2=\alpha(z_1)-1) \\
&\wedge (g_2= -x_1^2-x_2^2-z_2^2+1 > 0),
\end{align*}
Flattening and purification, with
$D = \{ y = \alpha(y_1), z = \alpha(z_1) \}, ~~ N=(y_1=z_1 \rightarrow y=z)$:
gives
\begin{align*}
\overline{\phi}:=&(f_1=-(y_1-x_1+1)^2-x_1+x_2 \ge 0) \wedge (y_2=y+1) \\
&\wedge ( g_1= -x_1^2-x_2^2-y_2^2+1 > 0),\\
\overline{\psi}:=&(f_2=-(z_1-x_2+1)^2+x_1-x_2 \ge 0) \wedge (z_2=z-1) \\
&\wedge (g_2= -x_1^2-x_2^2-z_2^2+1 > 0),\\
\end{align*}
The condition $\mathbf{NSOSC}$ is not satisfied, since
$-f_1-f_2=(y_1-x_1+1)^2+(z_1-x_2+1)^2$ is an SOS.
We follow the steps given in Subsection \ref{sec:gen}
$h_1=(y_1-x_1+1)^2~, ~~h_2=(z_1-x_2+1)^2.$
This gives $f:=f_1+h_1=-f_2-h_2=-x_1+x_2$;
${\phi} \models f \ge0 ~,~~{\psi} \models f \le0.$
Next we turn to find an interpolant for the following formulas
$$((\phi \wedge f>0) \vee (\phi \wedge f=0)) ~~and ~~ ((\psi \wedge -f>0) \vee (\phi \wedge f=0)).$$
Then $(f>0) \vee (f\ge0 \wedge I_2)$ is an interpolant for $\phi$ and $\psi$, where $I_2$ is an interpolant for $ \phi \wedge f=0$ and $\phi \wedge f=0$.
It is easy to see that
\begin{align*}
\phi \wedge f=0 \models y_1=x_1-1 ~,~~ \phi \wedge f=0 \models z_1=x_2-1.
\end{align*}
Substitute then into $f_1$ in $\overline{\phi}$ and $\overline{\psi}$, we have
\begin{align*}
\overline{\phi'}=&-x_1+x_2 \ge 0 \wedge y_2=y+1 \wedge g_1>0 \wedge y_1=x_1-1,\\
\overline{\psi'}=&~~~~x_1-x_2 \ge 0 \wedge z_2=z-1 \wedge g_2>0 \wedge z_1=x_2-1.
\end{align*}
Only using the linear form in $\overline{\phi'}$ and $\overline{\psi'}$ we deduce that $y_1=z_1$ as
\begin{align*}
\overline{\phi'} \models x_1-1 \le y_1 \le x_2-1~~,~~\overline{\psi'} \models x_2-1 \le z_1 \le x_1-1.
\end{align*}
Let $t=x_1-1$, $\alpha_t=\alpha(t)$, then
\begin{align*}
\overline{\phi'} \models (x_1 \ge x_2 \rightarrow y=\alpha_t)~~,~~\overline{\psi'} \models (x_2 \ge x_1 \rightarrow z=\alpha_t).
\end{align*}
Thus, $\overline{\phi'} \wedge \overline{\phi'} \wedge D \models \perp$ be divide into two parts $\phi$-part and $\psi$-part as follows,
\begin{align*}
(\overline{\phi'} \wedge (x_1 \ge x_2 \rightarrow y=\alpha_t)) \wedge (\overline{\psi'} \wedge (x_2 \ge x_1 \rightarrow z=\alpha_t)) \models \perp.
\end{align*}
Then find an interpolant for $(\overline{\phi'} \wedge (x_1 \ge x_2 \rightarrow y=\alpha_t))$ and $ (\overline{\psi'} \wedge (x_2 \ge x_1 \rightarrow z=\alpha_t))$
without uninterpreted function.
\end{example}
\end{comment}
\section{Proven interpolant}
Since our result is obtained by numerical calculation, it can't guard the solution satisfy the constraints strictly.
Thus, we should verify the solution obtained from a $\mathbf{SDP}$ solver to get a proven interpolant.
In the end of section \ref{sec:sdp}, the remark \ref{remark:1} said one can use Lemma \ref{lem:split} to verify the
result obtained from some $\mathbf{SDP}$ solver. In this section, we illuminate how to verify the result obtained from some
$\mathbf{SDP}$ solver to get a proven interpolant by an example.
\begin{example}
{ \begin{align*}
\phi :&=f_1=4-(x-1)^2-4y^2 \ge 0 \wedge f_2=y- \frac{1}{2} \ge 0, \\
\psi :&=f_3=4-(x+1)^2-4y^2 \ge 0 \wedge f_4=x+2y \ge 0.
\end{align*} }
\end{example}
Constructing SOS constraints as following,
\begin{align*}
&\lambda_1\ge 0, \lambda_2\ge 0, \lambda_3\ge 0, \lambda_4 \ge 0, \\
&-(\lambda_1 f_1+ \lambda_2f_2+\lambda_3 f_3+ \lambda_4f_4+1) \mbox{ is a SOS polynomial}
\end{align*}
Using the $\mathbf{SDP}$ solver \textit{Yalmip} to solve the above constraints for $\lambda_1, \lambda_2, \lambda_3, \lambda_4$,
take two decimal places, we obtain
\begin{align*}
\lambda_1=3.63, \lambda_2=38.39, \lambda_3=0.33, \lambda_4=12.70.
\end{align*}
Then we have,
\begin{align*}
-(\lambda_1 f_1+ \lambda_2f_2+\lambda_3 f_3+ \lambda_4f_4+1)
=3.96x^2+6.10x+15.84y^2-12.99y+6.315.
\end{align*}
Using Lemma \ref{lem:split}, we have
{\small
\begin{align*}
3.96x^2+6.10x+15.84y^2-12.99y+6.315
=3.96(x+\frac{305}{396})^2+15.84(y+\frac{1299}{3168})^2+\frac{825383}{6336},
\end{align*} }
which is a SOS polynomial obviously.
Thus, $I:=\lambda_1f_1+\lambda_2f_2+1>0$, i.e.,
$-3.63X^2-14.52y^2+7.26x+38.39y-7.305>0$, is a proven interpolant for $\phi$ and $\psi$.
\section{Beyond concave quadratic polynomials}
Theoretically speaking, \emph{concave quadratic} is quite restrictive. But in practice, the results obtained above are powerful enough
to scale up the existing verification techniques of programs and hybrid systems, as all well-known abstract domains, e.g. \emph{octagon}, \emph{polyhedra},
\emph{ellipsoid}, etc. are concave quadratic, which will be further demonstrated in the case study below. Nonetheless, we now discuss how to generalize our approach
to more general formulas by allowing polynomial equalities whose polynomials may be neither concave nor quadratic
using Gr\"{o}bner basis.
\oomit{In the above sections, we give an algorithm to find an interpolant when all the constraints are concave quadratic,
which means that if there exists an equation in the constraints it must be linear.
In this section, we allow the polynomial equation join the constraints, i.e., we suppose that all inequation constraints are
the same concave quadratic, but equation constraints are normal polynomial equations without the condition to be linear.
We give a sufficient method to generate an interpolant.}
Let's start the discussion with the following running example.
\begin{example} \label{nonCQ-exam}
Let $G=A \wedge B$, where
\begin{align*}
A:~&x^2+2x+(\alpha(\beta(a))+1)^2 \leq 0 \wedge \beta(a)=2c+z \wedge\\
&2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0,\\
B:~&x^2-2x+(\alpha(\gamma(b))-1)^2 \leq 0 \wedge \gamma(b)=d-z \wedge\\
&d^2+d+y^2+y+z=0 \wedge -d^2+y+2z=0,
\end{align*}
try to find an interpolant for $A$ and $B$.
\end{example}
It is easy to see that there exist some constraints which are not concave quadratic, as some equations are not linear.
Thus, the interpolant generation algorithm above is not applicable directly.
For easing discussion, in what follows, we use $\mathbf{IEq}(S), \mathbf{Eq}(S)$ and $\mathbf{LEq}(S)$ to stand for the sets of polynomials respectively from inequations,
equations and linear equations of $S$, for any polynomial formula $S$. E.g.,
in Example \ref{nonCQ-exam}, we have
\begin{align*}
\mathbf{IEq}(A)&= \{ x^2+2x+(\alpha(\beta(a))+1)^2 \},\\
\mathbf{Eq}(A)&= \{\beta(a)-2c-z, 2c^2+2c+y^2+z, -c^2+y+2z\},\\
\mathbf{LEq}(A)&=\{\beta(a)-2c-z \}.
\end{align*}
\oomit{
\begin{problem} \label{nonCQ-problem}
Let $A(x,\zz)$ and $B(\yy,\zz)$ be
\begin{align}
A\, :\, &f_1(xx,\zz)\ge 0 \wedge \ldots \wedge f_{r_1}(xx,\zz) \ge 0
\wedge g_1(xx,\zz)> 0 \wedge \ldots \wedge g_{s_1}(xx,\zz) > 0 \nonumber\\
\, &\wedge h_1(xx,\zz)= 0 \wedge \ldots \wedge h_{p_1}(xx,\zz) = 0, \\
B \, :\, &f_{r_1+1}(\yy,\zz)\ge 0 \wedge \ldots \wedge f_{r}(\yy,\zz) \ge 0
\wedge g_{s_1+1}(\yy,\zz)> 0 \wedge \ldots \wedge g_{s}(\yy,\zz) > 0 \nonumber\\
\, &\wedge h_{p_1+1}(\yy,\zz)= 0 \wedge \ldots \wedge h_{p}(\yy,\zz) = 0,
\end{align}
where $f_1, \ldots, f_r$ and $g_1, \ldots, g_s$ are concave quadratic polynomials,
$h_1, \ldots, h_t$ are general polynomials, unnecessary to be concave quadratic, and
\begin{align}
A(xx,\zz) \wedge B(\yy,\zz) \models \bot,
\end{align}
try to find an interpolant for $A(xx,\zz)$ and $B(\yy,\zz)$.
\end{problem}
\begin{note}
Let's denote $\mathbf{IEq}(S), \mathbf{Eq}(S)$ and $\mathbf{LEq}(S)$ to be the set of all inequations in $S$,
all equations in $S$ and all linear equations in $S$, where $S$ is a formula. E.g.,
formula $A$ in Example \ref{nonCQ-exam}, we have
\begin{align*}
\mathbf{IEq}(A)&= \{ x^2+2x+(\alpha(\beta(a))+1)^2 \},\\
\mathbf{Eq}(A)&= \{\beta(a)-2c-z, 2c^2+2c+y^2+z=0, -c^2+y+2z\},\\
\mathbf{LEq}(A)&=\{\beta(a)-2c-z \}.
\end{align*}
\end{note} }
In the following, we will use Example \ref{nonCQ-exam} as a running example to explain the basic idea how to
apply Gr\"{o}bner basis method to extend our approach to more general polynomial formulas.
Step $1$: Flatten and purify.
Similar to the concave quadratic case, we purify and
flatten $A$ and
$B$ by introducing fresh variables $a_1,a_2,b_1,b_2$, and obtain
\begin{align*}
A_0:~&x^2+2x+(a_2+1)^2 \leq 0 \wedge a_1=2c+z \wedge \\
& 2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0,\\
D_A:~&a_1=\beta(a) \wedge a_2=\alpha(a_1),\\
B_0:~&x^2-2x+(b_2-1)^2 \leq 0 \wedge b_1=d-z \wedge \\
& d^2+d+y^2+y+2z=0 \wedge -d^2+y+z=0,\\
D_B:~&b_1=\gamma(b) \wedge b_2=\alpha(b_1).
\end{align*}
Step $2$: { Hierarchical reasoning}.
Obviously, $A \wedge B$ is unsatisfiable in
$\PT ( \QQ )^{ \{ \alpha,\beta,\gamma\} }$
if and only if $A_0 \wedge B_0 \wedge N_0$ is unsatisfiable in $\PT ( \QQ )$,
where $N_0$
corresponds to the conjunction of Horn clauses constructed from $D_A \wedge D_B$ using
the axioms of uninterpreted functions (see the following table).
{\small \begin{center}
\begin{tabular}{c|c|c}\hline
D & $G_0$ & $N_0$ \\\hline
$D_A:~a_1=\beta(a) \wedge$ & $A_0:~x^2+2x+(a_2+1)^2 \leq 0 \wedge
a_1=2c+z \wedge$ & \\
\quad ~~~~ $a_2=\alpha(a_1)$ & $2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0$ & $N_0: b_1=a_1 \rightarrow
b_2=a_2$\\
& & \\
$D_B:~b_1=\gamma(b) \wedge$ & $B_0:~x^2-2x+(b_2-1)^2 \leq 0
\wedge b_1=d-z \wedge$ & \\
\quad ~~~~ $b_2=\alpha(b_1)$ & $d^2+d+y^2+y+2z=0 \wedge -d^2+y+z=0$ & \\
\end{tabular}
\end{center} }
To prove $A_0 \wedge B_0 \wedge N_0 \models \bot$,
we compute the Grobner basis of $\mathbb{G}$ of $\mathbf{Eq}(A_0) \cup \mathbf{Eq}(B_0)$ under the order
$c\succ d \succ y\succ z \succ a_1 \succeq b_1$, and have $a_1-b_1 \in \mathbb{G}$. That is,
$A_0 \wedge B_0 \models a_1=b_1$.
Thus, $A_0 \wedge B_0 \wedge N_0$ entails
\begin{align*}
a_2=b_2 \wedge x^2+2x+(a_2+1)^2 \leq 0 \wedge x^2-2x+(b_2-1)^2 \leq 0.
\end{align*}
This implies
$$2x^2+a_2^2+b_2^2+2\leq0,$$
which is obviously unsatisfiable in $\mathbb{Q}$.
Step $2$ gives a proof of $A \wedge B \models \bot$.
In order to find an interpolant for $A$ and $B$, we need to divide $N_0$ into two parts,
$A$-part and $B$-part, i.e., to find a term $t$ only with common symbols, such that
\begin{align*}
A_0 \models a_1=t ~~~B_0 \models b_1=t.
\end{align*}
Then we can choose a new variable $\alpha_t=\alpha(t)$ to be a common variable, since the term $t$ and
the function $\alpha$ both are common. Thus $N_0$ can be divided into two parts as follows,
\begin{align*}
a_2=\alpha_t \wedge b_2=\alpha_t.
\end{align*}
Finally, if we can find an interpolant $I(x,y,z,\alpha_t)$ for
\begin{align*}
(\mathbf{IEq}(A_0) \wedge \mathbf{LEq}(A_0) \wedge a_2=\alpha_t) \wedge (\mathbf{IEq}(A_0) \wedge \mathbf{LEq}(A_0) \wedge b_2=\alpha_t),
\end{align*} using Algorithm $\mathbf{IGFQC}$,
then $I(x,y,z,\alpha(t))$ will be an interpolant for $A \wedge B$.
Step $3$: Dividing $N_0$ into two parts. According to the above analysis, we need to find a witness $t$
such that $A_0 \models a_1=t$, $B_0 \models b_1=t$, where $t$ is an expression over the common symbols of $A$ and $B$.
Fortunately, such $t$ can be computed by Gr\"{o}bner basis method as follows:
First, with the variable order $c \succ a_1 \succ y \succ z$, the Gr\"{o}bner basis $\mathbb{G}_1$ of $\mathbf{Eq}(A_0)$
is computed to be
\begin{align*}
\mathbb{G}_1=&\{ y^4+4y^3+10y^2z+4y^2+20yz+25z^2-4y-8z, \\
&y^2+a_1+2y+4z, y^2+2c+2y+5z \}.
\end{align*}
Thus, we have
\begin{align} \label{eq-a1}
A_0 \models a_1=-y^2-2y-4z.
\end{align}
Simiarly, with the variable order $d \succ b_1 \succ y \succ z$, the Gr\"{o}bner basis $\mathbb{G}_2$ of $\mathbf{Eq}(B_0)$
is computed to be
\begin{align*}
\mathbb{G}_2=&\{ y^4+4y^3+6y^2z+4y^2+12yz+9z^2-y-z, \\
&y^2+b_1+2y+4z, y^2+d+2y+3z \}.
\end{align*}
Thus, we have
\begin{align} \label{eq-b1}
B_0 \models b_1=-y^2-2y-4z.
\end{align}
Whence, $t=-y^2-2y-4z$ is the witness.
Let $\alpha_t=\alpha(-y^2-2y-4z)$, which is an expression constructed from the common symbols of $A$ and $B$.
Next, find an interpolant for following formula
\begin{align*}
(\mathbf{IEq}(A_0) \wedge \mathbf{LEq}(A_0) \wedge a_2=\alpha_t) \wedge (\mathbf{IEq}(B_0) \wedge \mathbf{LEq}(B_0) \wedge b_2=\alpha_t).
\end{align*}
Using $\mathbf{IGFQC}$, we obtain an interpolant for the above formula as
\begin{align*}
I(x,y,z,\alpha_t)=x^2+2x+(\alpha_t+1)\le 0.
\end{align*}
Thus, $x^2+2x+(\alpha(-y^2-2y-4z)+1)\le 0$ is an interpolant for $A \wedge B$.
\begin{problem} \label{nonCQ-problem}
Generally, let $A(x,\zz)$ and $B(\yy,\zz)$ be
\begin{align}
A\, :\, &f_1(xx,\zz)\ge 0 \wedge \ldots \wedge f_{r_1}(xx,\zz) \ge 0
\wedge g_1(xx,\zz)> 0 \wedge \ldots \wedge g_{s_1}(xx,\zz) > 0 \nonumber\\
\, &\wedge h_1(xx,\zz)= 0 \wedge \ldots \wedge h_{p_1}(xx,\zz) = 0, \\
B \, :\, &f_{r_1+1}(\yy,\zz)\ge 0 \wedge \ldots \wedge f_{r}(\yy,\zz) \ge 0
\wedge g_{s_1+1}(\yy,\zz)> 0 \wedge \ldots \wedge g_{s}(\yy,\zz) > 0 \nonumber\\
\, &\wedge h_{p_1+1}(\yy,\zz)= 0 \wedge \ldots \wedge h_{p}(\yy,\zz) = 0,
\end{align}
where $f_1, \ldots, f_r$ and $g_1, \ldots, g_s$ are concave quadratic polynomials,
$h_1, \ldots, h_t$ are general polynomials, unnecessary to be concave quadratic, and
\begin{align}
A(xx,\zz) \wedge B(\yy,\zz) \models \bot,
\end{align}
try to find an interpolant for $A(xx,\zz)$ and $B(\yy,\zz)$.
\end{problem}
According to the above discussion, Problem~\ref{nonCQ-problem} can be solved by Algorithm~\ref{ag:nonCQ} below.
\begin{algorithm}[!htb] \label{ag:nonCQ}
\label{alg:int}
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\caption{ {\tt $\mathbf{IGFQC}$ }\label{prob:in-out}}
\Input{Two formulae $A$, $B$ as Problem \ref{nonCQ-problem} with $A \wedge B \models \bot$}
\Output{An formula $I$ to be a Craig interpolant for $A$ and $B$}
\SetAlgoLined
\BlankLine
\textbf{Flattening, purification and hierarchical reasoning}
obtain $A_0$, $B_0$, $N_A$, $N_B$, $N_{mix}$;\\
$A_0:=A_0 \wedge N_A, B_0:=B_0\wedge N_B$;\\
\While{$(\mathbf{IEq}(A_0) \wedge \mathbf{LEq}(A_0)) \wedge (\mathbf{IEq}(B_0)\wedge \mathbf{LEq}(B_0)) \not\models \bot$}
{
\If{$N_{mix}=\emptyset$}{\textbf{break}}
Choose a formula $a_1=b_1 \rightarrow a_2=b_2 \in N_{mix}$ corresponding to function $\alpha$;\\
$N_{mix}:=N_{mix}\setminus \{ a_1=b_1 \rightarrow a_2=b_2\}$;\\
Computing Grobner basis $\mathbb{G}_1$ for $\mathbf{Eq}(A_0)$ under purely dictionary ordering with
some variable ordering that other local variable $\succ a_1 \succ$ common variable; \\
Computing Grobner basis $\mathbb{G}_2$ for $\mathbf{Eq}(B_0)$ under purely dictionary ordering with
some variable ordering that other local variable $\succ b_1 \succ$ common variable; \\
\If{there exists a expression $t$ with common variable s.t. $a_1 \in \mathbb{G}_1\wedge b_1 \in \mathbb{G}_2$}
{introduce a new variable $\alpha_t=\alpha(t)$ as a common variable;
$A_0:=A_0 \wedge a_2=\alpha_t, B_0:=B_0 \wedge b_2=\alpha_t$}
}
\If{$(\mathbf{IEq}(A_0) \wedge \mathbf{LEq}(A_0)) \wedge (\mathbf{IEq}(B_0)\wedge \mathbf{LEq}(B_0)) \models \bot$}
{
Using $\mathbf{IGFQC}$ to obtain an interpolant $I_0$ for above formula;\\
Obtain an interpolant $I$ for $A \wedge B$ from $I_0$;\\
\KwRet $I$ }
\Else{\KwRet Fail}
\end{algorithm}
\oomit{
Let
\begin{align}
\mathcal{L}:=\{ l_1, \ldots, l_m \}
\end{align}
Step $1$: Choose all the linear polynomial $l_1, \ldots, l_{m_1}$ from $h_1, \ldots, h_{t_1}$ and
choose all the linear polynomial $l_{m_1+1}, \ldots, l_{m}$ from $h_{t_1+1}, \ldots, h_{t}$. Let
\begin{align}
\varphi_1:&f_1(x,\zz)\ge 0 \wedge \ldots \wedge f_{r_1}(x,\zz) \ge 0
\wedge g_1(x,\zz)> 0 \wedge \ldots \wedge g_{s_1}(x,\zz) > 0 \nonumber\\
&\wedge l_1(x,\zz)= 0 \wedge \ldots \wedge l_{m_1}(x,\zz) = 0, \\
\varphi_1:&f_{r_1+1}(\yy,\zz)\ge 0 \wedge \ldots \wedge f_{r}(\yy,\zz) \ge 0
\wedge g_{s_1+1}(\yy,\zz)> 0 \wedge \ldots \wedge g_{s}(\yy,\zz) > 0 \nonumber\\
&\wedge l_{t_1+1}(\yy,\zz)= 0 \wedge \ldots \wedge l_{m}(\yy,\zz) = 0.
\end{align}
Then $\varphi_1(x,\zz)$ and $\psi_1(\yy,\zz)$ are in the concave quadratic case,
if
\begin{align}
\varphi_1(x,\zz)\wedge\psi_1(\yy,\zz) \models \bot,
\end{align}
we can find an interpolant for $\varphi_1(x,\zz)$ and $\psi_1(\yy,\zz)$, which is also an
interpolant for $\varphi(x,\zz)$ and $\psi(\yy,\zz)$, we obtain an interpolant; else jump to step 2.
Step $1'$: Using Grobner basis method (or any other computer algebraic method) to obtain linear equations as much as
possible from $h_1=0, \ldots, h_{t_1}=0$, note as $l_1, \ldots, l_{m_1}$; the same, obtain linear equations as much as
possible from $h_{t_1+1}=0, \ldots, h_{t}=0$, note as $l_{m_1+1}, \ldots, l_{m}$. Let
\begin{align}
\varphi_1:&f_1(x,\zz)\ge 0 \wedge \ldots \wedge f_{r_1}(x,\zz) \ge 0
\wedge g_1(x,\zz)> 0 \wedge \ldots \wedge g_{s_1}(x,\zz) > 0 \nonumber\\
&\wedge l_1(x,\zz)= 0 \wedge \ldots \wedge l_{m_1}(x,\zz) = 0, \\
\varphi_1:&f_{r_1+1}(\yy,\zz)\ge 0 \wedge \ldots \wedge f_{r}(\yy,\zz) \ge 0
\wedge g_{s_1+1}(\yy,\zz)> 0 \wedge \ldots \wedge g_{s}(\yy,\zz) > 0 \nonumber\\
&\wedge l_{t_1+1}(\yy,\zz)= 0 \wedge \ldots \wedge l_{m}(\yy,\zz) = 0.
\end{align}
Then $\varphi_1(x,\zz)$ and $\psi_1(\yy,\zz)$ are in the concave quadratic case,
if
\begin{align}
\varphi_1(x,\zz)\wedge\psi_1(\yy,\zz) \models \bot,
\end{align}
we can find an interpolant for $\varphi_1(x,\zz)$ and $\psi_1(\yy,\zz)$, which is also an
interpolant for $\varphi(x,\zz)$ and $\psi(\yy,\zz)$, we obtain an interpolant; else jump to step 2.
Step $2$: Choose a linear polynomial $L(x,\yy,\zz)$ from the Grobner basis of $h_1, \ldots, h_t$ under some ordering,
which is different from all the element form $\mathcal{L}$. And add $L$ into $\mathcal{L}$.
It is easy to see that we can divide $L(x,\yy,\zz)$ into two part, that
\begin{align}
L(x,\yy,\zz)=L_1(x,\zz)-L_2(\yy,\zz),
\end{align}
where $L_1(x,\zz)$ and $L_2(\yy,\zz)$ both are linear polynomial.
Introduce two new variable $\alpha$ and $\beta$. Compute the Grobner basis $\mathbb{G}_1$ of
$h_1, \ldots, h_{t_1}, \alpha-L_1$ under the ordering $x > \alpha >\zz$; Compute the Grobner basis
$\mathbb{G}_2$ of $h_{t_1+1}, \ldots, h_{t}, \beta-L_2$ under the ordering $x > \alpha >\zz$.
Find a polynomial $\theta(\zz)$ such that
\begin{align}
\alpha-\theta(\zz) \in \mathbb{G}_1 \wedge \beta-\theta(\zz) \in \mathbb{G}_2.
\end{align}
Introduce a new variable $\gamma$($\gamma=\theta(\zz)$), update $\varphi_1(x,\zz)$ and $\psi_1(\yy,\zz)$
as follow
\begin{align}
\varphi_1(x,\zz,\gamma) \leftarrow \varphi \wedge \gamma-L_1(x,\zz)=0,\\
\psi_1(\yy,\zz,\gamma) \leftarrow \psi \wedge \gamma-L_2(\yy,\zz)=0.
\end{align}
It is easy to see that
\begin{align}
\varphi(x,\zz) \models \varphi_1(x,\zz,\gamma), \\
\psi(\yy,\zz) \models \psi_1(\yy,\zz,\gamma).
\end{align}
And $\varphi_1(x,\zz,\gamma)$ and $\psi_1(\yy,\zz,\gamma)$ are in the concave quadratic case,
if
\begin{align}
\varphi_1(x,\zz,\gamma)\wedge\psi_1(\yy,\zz,\gamma) \models \bot,
\end{align}
we can find an interpolant $I(\zz,\gamma)$ for $\varphi_1(x,\zz,\gamma)$ and $\psi_1(\yy,\zz,\gamma)$.
Thus, $I(\zz,\theta(\zz))$ is an
interpolant for $\varphi(x,\zz)$ and $\psi(\yy,\zz)$, we obtain an interpolant; else jump to step 2, repeat.
}
\oomit{
\section{example}
\begin{example} \label{exam}
Let $G=A \wedge B$, where
\begin{align*}
A:~&x^2+2x+(f(g(a))+1)^2 \leq 0 \wedge g(a)=2c+z \wedge\\
&2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0,\\
B:~&x^2-2x+(f(h(b))-1)^2 \leq 0 \wedge h(b)=d-z \wedge\\
&d^2+d+y^2+y+z=0 \wedge -d^2+y+2z=0.
\end{align*}
\end{example}
We show that $A \wedge B$ is unsatisfiable in $\PT ( \QQ )^{ \{ f,g,h\} }$ as
follows:
Step $1$: Flattening and purification.
We purify and
flatten the formulae $A$ and
$B$ by replacing the terms starting with $f$ with new variables. We
obtain the
following purified form:
\begin{align*}
A_0:~&x^2+2x+(a_2+1)^2 \leq 0 \wedge a_1=2c+z \wedge\\
&2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0,\\
D_A:~&a_1=g(a) \wedge a_2=f(a_1),\\
B_0:~&x^2-2x+(b_2-1)^2 \leq 0 \wedge b_1=d-z \wedge\\
&d^2+d+y^2+y+2z=0 \wedge -d^2+y+z=0,\\
D_B:~&b_1=h(b) \wedge b_2=f(b_1).
\end{align*}
Step $2$: { Hierarchical reasoning}.
By Theorem \ref{flat-puri} we have that $A \wedge B$ is unsatisfiable in
$\PT ( \QQ )^{ \{ f,g,h\} }$
if and only if $A_0 \wedge B_0 \wedge N_0$ is unsatisfiable in $\PT ( \QQ )$,
where $N_0$
corresponds to the consequences of the congruence axioms for those
ground terms which
occur in the definitions $D_A \wedge D_B$ for the newly introduced
variables.
\begin{center}
\begin{tabular}{c|c|c}\hline
Def & $G_0$ & $N_0$ \\\hline
$D_A:~a_1=g(a) \wedge a_2=f(a_1)$ & $A_0:~x^2+2x+(a_2+1)^2 \leq 0 \wedge
a_1=2c+z \wedge$ & \\
& $2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0$ & $N_0: b_1=a_1 \rightarrow
b_2=a_2$\\
& & \\
$D_B:~b_1=g(b) \wedge b_2=f(b_1)$ & $B_0:~x^2-2x+(b_2-1)^2 \leq 0
\wedge b_1=d-z \wedge$ & \\
& $d^2+d+y^2+y+2z=0 \wedge -d^2+y+z=0$ & \\
\end{tabular}
\end{center}
To prove that $A_0 \wedge B_0 \wedge N_0$ is unsatisfiable, note that
$A_0 \wedge
B_0 \models a_1=b_1$. Thus, $A_0 \wedge B_0 \wedge N_0$ entails
\begin{align*}
a_2=b_2 \wedge x^2+2x+(a_2+1)^2 \leq 0 \wedge x^2-2x+(b_2-1)^2 \leq 0.
\end{align*}
Plus the second inequation and the third inequation and using the first
equation we have,
$$2x^2+a_2^2+b_2^2+2\leq0,$$
which is inconsistent over $\mathbb{Q}$.
\begin{example}
Consider the clause $a_1=b_1 \rightarrow a_2=b_2$ of $N_0$ in Example
\ref{n-mix}. Since
this clause contains both $A$-local and $B$-local, it should be
divided into $N_{mix}$.
Then we need to separate it into two parts, one is $A$-pure and other
is $B$-pure.
We try to find them as follow:
Firstly, we note that $A_0 \wedge B_0 \models_{\PT ( \QQ )} a_1=b_1$. The
proof is following,
\begin{align*}
A_0:~&x^2+2x+(a_2+1)^2 \leq 0 \wedge a_1=2c+z \wedge\\
&2c^2+2c+y^2+z=0 \wedge -c^2+y+2z=0,\\
B_0:~&x^2-2x+(b_2-1)^2 \leq 0 \wedge b_1=d-z \wedge\\
&d^2+d+y^2+y+2z=0 \wedge -d^2+y+z=0.
\end{align*}
Let $A_{0,i}$, and $B_{o,j}$ be the $i^{th}$ clause in $A_0$ and the
$j^{th}$ clause in
$B_0$ respectively, where $1 \leq i,j \leq 4$.
Then we have:
\begin{align*}
A_{0,2}+A_{0,3}+2A_{0,4}~~ \rightarrow ~~a_1+y^2+2y+4z=0;\\
B_{0,2}+B_{0,3}+~~B_{0,4}~~ \rightarrow ~~b_1+y^2+2y+4z=0.
\end{align*}
Thus,
$$a_1=-y^2-2y-4z=b_1.$$
This complete the proof of $A_0 \wedge B_0 \models_{\PT ( \QQ )} a_1=b_1$.
From the proof, we can see that there exists a term $t=-y^2-2y-4z$
containing only
variables common to $A_0$ and $B_0$ such that $A_0 \models_{\PT ( \QQ )}
a_1=-y^2-2y-4z$ and
$B_0 \models_{\PT ( \QQ )} b_1=-y^2-2y-4z$. From which we can separate the clause
$a_1=b_1 \rightarrow a_2=b_2$ into $A$-part and $B$-part as follow,
\begin{align*}
a_1=-y^2-2y-4z \rightarrow a_2=e_{f(-y^2-2y-4z)};\\
b_1=-y^2-2y-4z \rightarrow b_2=e_{f(-y^2-2y-4z)}.
\end{align*}
\end{example}
Thus we have
\begin{align*}
N_{sep}^A= \{ a_1=-y^2-2y-4z \rightarrow a_2=e_{f(-y^2-2y-4z)} \};\\
A_{sep}^B= \{ b_1=-y^2-2y-4z \rightarrow b_2=e_{f(-y^2-2y-4z)} \}.
\end{align*}
Now, we try to obtain an interpolant for
$$(A_0 \wedge N_{sep}^A) \wedge (B_0 \wedge N_{sep}^B).$$
Note that $(A_0 \wedge N_{sep}^A)$ is logically equivalent to $(A_0
\wedge a_2 = e_{f(-y^2-2y-4z)})$, and $(B_0 \wedge N_{sep}^B)$ is
logically equivalent to
$(B_0 \wedge b_2 = e_{f(-y^2-2y-4z)})$. The conjunction
$(A_0 \wedge a_2 = e_{f(-y^2-2y-4z)}) \wedge (B_0 \wedge b_2 =
e_{f(-y^2-2y-4z)})$ is
unsatisfiable, which is because two circles
$(x+1)^2+(e_{f(-y^2-2y-4z)}+1)^2 \leq 1$ and
$(x-1)^2+(e_{f(-y^2-2y-4z)}-1)^2 \leq 1$ have an empty intersection. Any
curve separate
them could be an interpolant for $(A_0 \wedge a_2 = e_{f(-y^2-2y-4z)})
\wedge (B_0 \wedge b_2 = e_{f(-y^2-2y-4z)})$,
we might choose $I_0: x+e_{f(-y^2-2y-4z)}<0$. It is easy to see
$(A_0 \wedge a_2 = e_{f(-y^2-2y-4z)}) \models_{\PT ( \QQ )} I_0$,
$(B_0 \wedge b_2 = e_{f(-y^2-2y-4z)}) \wedge I_0 \models_{\PT ( \QQ )} \bot$ and
$I_0$ contain only the common variables in $A_0$ and $B_0$.
So, $I_0$ is an interpolant for
$(A_0 \wedge a_2 = e_{f(-y^2-2y-4z)}) \wedge (B_0 \wedge b_2 =
e_{f(-y^2-2y-4z)})$.
Replacing the newly introduced constant $e_{f(-y^2-2y-4z)}$ with the
term it denotes,
let $I : x+f(-y^2-2y-4z)<0$ .
It is easy to see that:
\begin{align*}
(A_0 \wedge D_A) \models_{\PT ( \QQ )^{\Sigma}} I,\\
(B_0 \wedge D_B) \wedge I \models_{\PT ( \QQ )^{\Sigma}} \bot
\end{align*}
Therefore, $I$ is an interpolant for $(A_0 \wedge D_A) \wedge (B_0
\wedge D_B)$, obviously,
is an interpolant for $A \wedge B$.
}
\section{Implementation and experimental results}
We have implemented the presented algorithms in \textit{Mathematica} to synthesize interpolation for concave quadratic polynomial inequalities as well as their combination with \textit{EUF}. To deal with SOS solving and semi-definite programming, the Matlab-based optimization tool \textit{Yalmip} \cite{Yalmip} and the SDP solver \textit{SDPT3} \cite{SDPT3} are invoked. In what follows we demonstrate our approach by some examples, which have been evaluated on a 64-bit Linux computer with a 2.93GHz Intel Core-i7 processor and 4GB of RAM.
\begin{example} \label{exp:4}
Consider the example:
\begin{eqnarray*}
\phi := (f_1 \ge 0) \wedge (f_2 \ge0) \wedge (g_1 >0),\quad
\psi := (f_3 \ge 0). \quad
\phi \wedge \psi \models \bot.
\end{eqnarray*}
where $f_1 = x_1, f_2 = x_2,f_3= -x_1^2-x_2^2 -2x_2-z^2, g_1= -x_1^2+2 x_1 - x_2^2 + 2 x_2 - y^2$.
The interpolant returned after $0.394$ s is
\begin{eqnarray*}
I:= \frac{1}{2}x_1^2+\frac{1}{2}x_2^2+2x_2 > 0
\end{eqnarray*}
\end{example}
\begin{example} \label{exp:5}
Consider the unsatisfiable conjunction $\phi \wedge \psi$:
\begin{eqnarray*}
\phi := f_1 \ge 0 \wedge f_2 \ge 0 \wedge f_3 \ge 0 \wedge g_1>0, \quad
\psi := f_4 \ge 0 \wedge f_5 \ge 0 \wedge f_6 \ge 0 \wedge g_2 >0 .
\end{eqnarray*}
where
$f_1 = -y_1+x_1-2$,
$f_2 = -y_1^2-x_1^2+2x_1 y_1 -2y_1+2x_1$,
$f_3 = -y_2^2-y_1^2-x_2^2 -4y_1+2x_2-4$,
$f_4 = -z_1+2x_2+1$,
$f_5 = -z_1^2-4x_2^2+4x_2 z_1 +3z_1-6x_2-2$,
$f_6 = -z_2^2-x_1^2-x_2^2+2x_1+z_1-2x_2-1$,
$g_1 = 2x_2-x_1-1$,
$g_2 = 2x_1-x_2-1$.
The condition NSOSC does not hold, since
$$-(2f_1 + f_2) = (y_1 -x_1 +2)^2 \textrm{ is a sum of square}.$$
Then we have $h=(y_1 -x_1 +2)^2$, and
\begin{align*}
h_1=h=(y_1 -x_1 +2)^2, \quad h_2=0.
\end{align*}
Let $f=2f_1+f_2+h_1=0$. Then construct $\phi'$ by setting $y_1=x_1-2$ in $\phi$, $\psi'$ is $\psi$. That is
\begin{align*}
\phi':=0\ge0 \wedge 0 \ge 0 \wedge -y_2^2-x_1^2-x_2^2+2x_2 \ge 0 \wedge g_1>0, \quad \psi':=\psi.
\end{align*}
Then the interpolation for $\phi$ and $\psi$ is reduced as
\begin{align*}
I(\phi,\psi)=(f> 0) \vee (f=0 \wedge I(\phi', \psi')) = I(\phi',\psi').
\end{align*}
For $\phi'$ and $\psi'$, the condition NSOSC is still unsatisfied, since $-f_4-f_5 = (z_1-2x_2-1)^2$ is an SOS.
Then we have $h=h_2=(z_1-2x_2-1)^2$, $h_1=0$, and thus $f=0$.
\begin{align*}
\phi''=\phi', \quad \psi''=0\ge0 \wedge 0 \ge 0 \wedge -z_2^2-x_1^2 - x_2^2+2x_1\ge0 \wedge
g_2 >0.
\end{align*}
The interpolation for $\phi'$ and $\psi'$ is further reduced by $ I(\phi',\psi')=I(\phi'',\psi'')$, where
\begin{align*}
\phi'':=(f_1'=-y_2^2-x_1^2-x_2^2+2x_2\ge0) \wedge 2x_2-x_1-1>0,\\
\psi'':=(f_2'=-z_2^2-x_1^2 - x_2^2+2x_1\ge0) \wedge
2x_1-x_2-1 >0.
\end{align*}
Here the condition NSOSC holds for $\phi''$ and $\psi''$, then by SDP we find $\lambda_1=\lambda_2=0.25, \eta_0=0, \eta_1=\eta_2=0.5$ and SOS polynomials $h_1=0.25*((x_1-1)^2+(x2-1)^2+y_2^2)$ and $h_2=0.25*((x_1-1)^2+(x_2-1)^2+z_2^2)$ such that
$\lambda_1 f_1'+\lambda_2 f_2' + \eta_0 + \eta_1 g_1 + \eta_2 g_2 +h_1 +h_2 \mathbf{Eq}uiv 0$ and
$\eta_0+\eta_1+\eta_2=1$.
For $\eta_0+\eta_1=0.5>0$, the interpolant returned after $2.089$ s is $f>0$, i.e. $I:= -x_1 + x_2 > 0$.
\end{example}
\begin{example} \label{exp:6}
Consider the example:
\begin{align*}
\phi:=&(f_1=-(y_1-x_1+1)^2-x_1+x_2 \ge 0) \wedge (y_2=\alpha(y_1)+1) \\
&\wedge ( g_1= -x_1^2-x_2^2-y_2^2+1 > 0), \\
\psi:=&(f_2=-(z_1-x_2+1)^2+x_1-x_2 \ge 0) \wedge (z_2=\alpha(z_1)-1) \\
&\wedge (g_2= -x_1^2-x_2^2-z_2^2+1 > 0).
\end{align*}
where $\alpha$ is an uninterpreted function. It takes $0.369$ s in our approach to reduce the problem to find an interpolant as $I(\overline{\phi'}_2,\overline{\psi'}_1) \vee (\overline{\phi'}_4,\overline{\psi'}_1)$, and another $2.029$ s to give the final interpolant as
\begin{eqnarray*}
I:= (-x_1 + x_2 > 0) \vee (\frac{1}{4}(-4\alpha(x_2-1)-x_1^2-x_2^2)>0)
\end{eqnarray*}
\end{example}
\begin{example} \label{exp:7}
Let two formulae $\phi$ and $\psi$ be defined as
\begin{align*}
\phi:=&(f_1=4-x^2-y^2 \ge 0) \wedge f_2=y \ge 0 \wedge ( g= x+y-1 > 0), \\
\psi:=&(f_4=x \ge 0) \wedge (f_5= 1-x^2- (y+1)^2 \ge 0).
\end{align*}
The interpolant returned after $0.532$ s is $I:= \frac{1}{2}(x^2+y^2+4y)>0$ \footnote{In order to give a more objective comparison of performance with the approach proposed in \cite{DXZ13}, we skip over line 1 in the previous algorithm $\mathbf{IGFQC}$.}.
\oomit{While AiSat gives
\begin{small}
\begin{align*}
I':= &-0.0732 x^4+0.1091 x^3 y^2+199.272 x^3 y+274818 x^3-0.0001 x^2 y^3-0.079 x^2 y^2 \\
&-0.1512 x^2 y-0.9803 x^2+0.1091 x y^4+199.491 x y^3+275217 x y^2+549634 x y \\
&+2.0074 x-0.0001 y^5-0.0056 y^4-0.0038 y^3-0.9805 y^2+2.0326 y-1.5 > 0
\end{align*}
\end{small}}
\end{example}
\begin{example} \label{exp:8}
This is a linear interpolation problem adapted from \cite{RS10}. Consider the unsatisfiable conjunction $\phi \wedge \psi$:
\begin{eqnarray*}
\phi := z-x \ge 0 \wedge x-y \ge 0 \wedge -z > 0, \quad
\psi := x+y \ge 0 \wedge -y \ge 0 .
\end{eqnarray*}
It takes 0.250 s for our approach to give an interpolant as $I:= - 0.8x - 0.2y > 0$.
\end{example}
\begin{example} \label{exp:9}
Consider another linear interpolation problem combined with \textit{EUF}:
\begin{eqnarray*}
\phi := f(x) \ge 0 \wedge x-y \ge 0 \wedge y-x \ge 0, \quad
\psi := -f(y) > 0 .
\end{eqnarray*}
The interpolant returned after 0.236 s is $I:= f(y) \ge 0$.
\end{example}
\begin{example} \label{exp:10}
Consider two formulas $A$ and $B$ with $A \wedge B \models \bot$, where\\
\begin{minipage}{0.6\textwidth}
\begin{align*}
A :=& -{x_1}^2 + 4 x_1 +x_2 - 4 \ge 0 \wedge \\
& -x_1 -x_2 +3 -y^2 > 0,\\
B :=& \mathbf{-3 {x_1}^2 - {x_2}^2 + 1 \ge 0} \wedge x_2 - z^2 \ge 0.
\end{align*}
Note that a concave quadratic polynomial (the bold one) from the \textit{ellipsoid} domain is involved in $B$. It takes 0.388 s using our approach to give an interpolant as $ I:=-3 + 2 x_1 + {x_1}^2 + \frac{1}{2} {x_2}^2 > 0.$ An intuitive description of the interpolant is as the purple curve in the right figure, which separates $A$ and $B$ in the panel of common variables $x_1$ and $x_2$.
\end{minipage}
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=\linewidth]{ellipsoid.png}
\end{minipage}
\end{example}
\begin{example} \label{exp:11}
Consider two formulas $\phi$ and $\psi$ both are defined by an ellipse joint a half-plane:
{\small \begin{align*}
\phi :=4-(x-1)^2-4y^2 \ge 0 \wedge y- \frac{1}{2} \ge 0,~~
\psi :=4-(x+1)^2-4y^2 \ge 0 \wedge x+2y \ge 0.
\end{align*} }
The interpolant returned after 0.248 s is $I:=-3.63x^2-14.52y^2+7.26x+38.39y-7.305 > 0$.
\end{example}
\begin{example} \label{exp:12}
Consider two formulas $\phi$ and $\psi$ both are defined by an octagon joint a half-plane:
{\small \begin{align*}
\phi &:= -3 \le x \le 1 \wedge -2 \le y \le 2 \wedge -4 \le x-y \le 2
\wedge -4 \le x+y \le 2 \wedge x+2y+1\le 0, \\
\psi &:= -1 \le x \le 3 \wedge -2 \le y \le 2 \wedge -2 \le x-y \le 4
\wedge -2 \le x+y \le 4 \wedge 2x-5y+6\le 0.
\end{align*} }
The interpolant returned after 0.225 s is $I:=-13.42x-29.23y-1.7 > 0$.
\end{example}
\begin{example} \label{exp:13}
Consider two formulas $\phi$ and $\psi$ both are defined by an octagon joint a half-plane:
{\small \begin{align*}
\phi &:= 2 \le x \le 7 \wedge 0 \le y \le 3 \wedge 0 \le x-y \le 6
\wedge 3 \le x+y \le 9 \wedge 23-3x-8y\le 0, \\
\psi &:= 0 \le x \le 5 \wedge 2 \le y \le 5 \wedge -4 \le x-y \le 2
\wedge 3 \le x+y \le 9 \wedge y-3x-2\le 0.
\end{align*} }
The interpolant returned after 0.225 s is $I:=12.3x-7.77y+4.12 > 0$.
\end{example}
\vspace*{-2mm}
\begin{table}
\begin{center}
\begin{tabular}{llp{1.9cm}<{\centering}p{1.4cm}<{\centering}p{1.4cm}<{\centering}p{1.4cm}<{\centering}p{1.9cm}<{\centering}}
\toprule
\multirow{2}{*}{\begin{minipage}{1.7cm}
Example
\end{minipage}}
& \multirow{2}{*}{\begin{minipage}{1.5cm}
Type
\end{minipage}}
& \multicolumn{5}{c}{Time (sec)}\\
\cmidrule(lr){3-7}
& & \textsc{CLP-Prover} & \textsc{Foci} & \textsc{CSIsat} & AiSat
& Our Approach\\ \hline
Example \ref{exp:4} & NLA & -- & -- & -- & -- & 0.394 \\
Example \ref{exp:5} & NLA & -- & -- & -- & -- & 2.089 \\
Example \ref{exp:6} & NLA+\textit{EUF} & -- & -- & -- & -- & 2.398 \\
Example \ref{exp:7} & NLA & -- & -- & -- & 0.023 & 0.532 \\
Example \ref{exp:8} & LA & 0.023 & $\times$ & 0.003 & -- & 0.250 \\
Example \ref{exp:9} & LA+\textit{EUF} & 0.025 & 0.006 & 0.007 & -- & 0.236 \\
Example \ref{exp:10} & Ellipsoid & -- & -- & -- & -- & 0.388 \\
Example \ref{exp:11} & Ellipsoid2 & -- & -- & -- & 0.013 & 0.248 \\
Example \ref{exp:12} & Octagon1 & 0.059 & $\times$ & 0.004 & 0.021 & 0.225 \\
Example \ref{exp:13} & Octagon2 & 0.065 & $\times$ & 0.004 & 0.122 & 0.216 \\
\bottomrule
\end{tabular}\\
-- means that the interpolant generation fails, and $\times$ specifies a particularly wrong answer.
\vspace*{0.1in}
\caption{Evaluation results of the presented examples}\label{tab1}
\vspace*{-10mm}
\end{center}
\end{table}
The experimental evaluation on the above examples is illustrated in Table~\ref{tab1}, where we have also compared on the same platform with the performances of AiSat, a tool for nonlinear interpolant generation proposed in \cite{DXZ13}, as well as three publicly available interpolation procedures for linear-arithmetic cases, i.e. Rybalchenko's tool \textsc{CLP-Prover}) in \cite{RS10}, McMillan's procedure \textsc{Foci} in \cite{mcmillan05}, and Beyer's tool \textsc{CSIsat} in \cite{CSIsat}. Table \ref{tab1} shows that our approach can successfully solve all the examples and it is especially the completeness that makes it an extraordinary competitive candidate for synthesizing interpolation. Besides, \textsc{CLP-Prover}, \textsc{Foci}, and \textsc{CSIsat} can handle only linear-arithmetic expressions with an efficient optimization (and thus the performances in linear cases are better than our raw implementation). As for AiSat, a rather limited set of applications is acceptable because of the weakness of tackling local variables, and whether an interpolant can be found or not depends on a pre-specified total degree. In \cite{DXZ13}, not only all the
constraints in formula $\phi$ should be considered but also some of their products, for instance,
$f_1,f_2,f_3 \ge0$ are three constraints in $\phi$, then four constraints $f_1f_2,f_1f_3,f_2f_3,f_1f_2f_3 \ge0$ are added in $\phi$.
Table~\ref{tab1} indicates the efficiency of our tool is lower than any of other tools whenever a considered example is
solvable by both. This is mainly because our tool is implemented in \textit{Mathematica}, and therefore have to invoke some SDP solvers with low efficiency. As a future work, we plan to re-implement the tool using C, thus we can call SDP solver CSDP which is
much more efficient. Once a considered problem is linear, an existing interpolation procedure will be invoked directly, thus,
SDP solver is not needed.
\oomit{
The experimental evaluation on the above motivating examples is illustrated in table \ref{tab1}, where we have also compared on the same platform with the performances of AiSat, a tool for nonlinear interpolant generation proposed in \cite{DXZ13}, as well as three publicly available interpolation procedures for linear-arithmetic cases, i.e. Rybalchenko's tool \textsc{CLP-Prover}) in \cite{RS10}, McMillan's procedure \textsc{Foci} in \cite{mcmillan05}, and Beyer's tool \textsc{CSIsat} in \cite{CSIsat}. Table \ref{tab1} shows that our approach can successfully solve all the examples and it is especially the completeness that makes it an extraordinary competitive candidate for synthesizing interpolation. Besides, \textsc{CLP-Prover}, \textsc{Foci}, and \textsc{CSIsat} can handle only linear-arithmetic expressions with an efficient optimization. As for AiSat, a rather limited set of applications is acceptable because of the weakness of tackling local variables.}
\section{Conclusion}
The paper proposes a polynomial time algorithm for generating
interpolants from mutually contradictory conjunctions of concave
quadratic polynomial inequalities over the reals. Under a
technical condition that if no nonpositive constant combination
of nonstrict inequalities is a sum of squares polynomials, then
such an interpolant can be generated essentially using
the linearization of quadratic polynomials. Otherwise, if this
condition is not satisified, then the algorithm is recursively
called on smaller problems after deducing linear equalities
relating variables. The resulting interpolant is a disjunction of
conjunction of polynomial inequalities.
Using the hierarchical calculus framework proposed in
\cite{SSLMCS2008}, we give an interpolation algorithm for the
combined quantifier-free theory of concave quadratic polynomial
inequalities and equality over uninterpreted function
symbols. The combination algorithm is patterned after a
combination algorithm for the combined theory of linear
inequalities and equality over uninterpreted function symbols.
In addition, we also discuss how to
extend our approach to formulas with polynomial equalities whose polynomials may be neither concave nor quadratic
using Gr\"{o}bner basis.
The proposed approach is applicable to all existing abstract domains like
\emph{octagon}, \emph{polyhedra}, \emph{ellipsoid} and so on, therefore it can be used to improve
the scalability of existing verification techniques for programs and hybrid systems.
An interesting issue raised by the proposed framework for dealing
with nonlinear polynomial inequalities is the extent to which their linearization
with some additional conditions
on the coefficients (such as concavity for quadratic polynomials)
can be exploited. We are also investigating how results reported
for nonlinear polynomial inequalities based on positive
nullstellensatz \cite{Stengle} in \cite{DXZ13} and
the Archimedian condition on variables, implying that every
variable ranged over a bounded interval, can be exploited in the
proposed framework for dealing with polynomial inequalities.
\end{document}
|
\begin{document}
\title{f Fundamental Groups of Commuting Elements in Lie Groups}
\begin{abstract} We compute the fundamental group of the spaces of ordered commuting $n$-tuples
of elements in the Lie groups $SU(2)$, $U(2)$ and $SO(3)$. For $SO(3)$ the mod-2 cohomology of
the components of these spaces is also obtained.
\end{abstract}
\section {\bf Introduction}\label{intro}
In this paper we calculate the fundamental groups of the connected components of the spaces
$$M_n(G):=Hom({\bbb Z}^n,G),\ \mbox{ where $G$ is one of $SO(3),\ SU(2)$ or $U(2)$.}$$
The space $M_n(G)$ is just the space of ordered commuting $n$-tuples of elements from $G,$ topologized as a subset of $G^n.$
The spaces $M_n(SU(2))$ and $M_n(U(2))$ are connected (see~\cite{AC}), but $M_n(SO(3))$ has
many components if $n>1.$ One of the components is the one containing the element $(id,id,\ldots,id);$ see Section~\ref{thespaces}. The other components are all homeomorphic to $V_2(\mathbb R^3)/\mathbb Z_2\oplus\mathbb Z_2,$ where $V_2(\mathbb R^3)$ is the Stiefel manifold of orthonormal $2$-frames in $\mathbb R^3$ and the action of
$\mathbb Z_2\oplus\mathbb Z_2$ on $V_2(\mathbb R^3)$ is given by
$$(\epsilon_1,\epsilon_2)(v_1,v_2)=(\epsilon_1 v_1,\epsilon_2 v_2),\
\mbox{where $\epsilon_j=\pm 1$ and $(v_1,v_2)\in V_2(\mathbb R^3). $}
$$
Let $e_1,e_2,e_3$ be the standard basis of $\mathbb R^3.$ Under the homeomorphism $SO(3)\to V_2(\mathbb R^3)$ given by $A\mapsto (Ae_1,Ae_2)$ the action of
$\mathbb Z_2\oplus\mathbb Z_2$ on $V_2(\mathbb R^3)$ corresponds to the action defined by right multiplication by the elements
of the group generated by the transformations
$$(x_1,x_2,x_3)\mapsto(x_1,-x_2,-x_3),\ (x_1,x_2,x_3)\mapsto(-x_1,x_2,-x_3).$$
The orbit space of this action is homeomorphic to ${\bbb S}^3/Q_8$, where $Q_8$ is
the quaternion group of order eight.
Then $M_n(SO(3))$ will be a disjoint union of many copies of ${\bbb S}^3/Q_8$ and the component containing $(id,\ldots,id).$ For brevity let $\vec{1}$ denote the $n$-tuple $(id,\ldots,id).$
\begin{definition}\label{comps}
{\em Let $M_n^+(SO(3))$ be the component of $\vec{1}$ in $M_n(SO(3))$, and let $M_n^-(SO(3))$ be the complement
$M_n(SO(3))-M_n^+(SO(3))$.}
\end{definition}
Our main result is the following
\begin{theorem}\label{mainth}
For all $n\ge 1$
$$\begin{array}{rcc}
\pi_1(M_n^+(SO(3))) & = & {\bbb Z}_2^n \\
\pi_1(M_n(SU(2))) & =& 0 \\
\pi_1(M_n(U(2))) & = & {\bbb Z}^n \end{array}$$
\end{theorem}
The other components of $M_n(SO(3)),$ $n>1,$ all have fundamental group $Q_8.$
\begin{remark}\label{cover}{\em To prove this theorem we first prove that
$\pi_1(M_n^+(SO(3))) = {\bbb Z}_2^n,$ and then use the following property of spaces of homomorphisms (see~\cite{G}).
Let $\Gamma$ be a discrete group, $p:\tilde{G}\to G$ a covering
of Lie groups, and $C$ a component of the image of the induced map
$p_*:Hom(\Gamma,\tilde{G})\to Hom(\Gamma,G)$. Then $p_*:p_*^{-1}(C)\to C$ is a regular covering with
covering group $Hom(\Gamma,Ker\ p)$.
Applying this to the universal coverings $SU(2)\to SO(3)$ and $SU(2)\times{\bbb R} \to U(2)$ induces coverings
$${\bbb Z}_2^n\to M_n(SU(2))\to M_n^+(SO(3))$$ $${\bbb Z}^n\to M_n(SU(2))\times{\bbb R}^n\to M_n(U(2))$$}
\end{remark}
\begin{remark}{\em The spaces of homomorphisms arise in different contexts (see~\cite{J}).
In physics for instance, the orbit space $Hom({\bbb Z}^n,G)/G$, with $G$ acting by conjugation, is the moduli space of isomorphism
classes of flat connections on principal $G$-bundles over the $n$-dimensional torus. Note that, if $G$ is connected,
the map $\pi_0(Hom({\bbb Z}^n,G))\to\pi_0(Hom({\bbb Z}^n,G)/G)$ is a bijection of sets. The study of these spaces arises
from questions concerning the understanding of the structure of the components of this moduli space and their number.
These questions are part of the study of the quantum field theory of gauge theories over the $n$-dimensional
torus (see~\cite{BFM},\cite{KS}).
}\end{remark}
The organization of this paper is as follows. In Section 2 we study the toplogy of $M_n(SO(3))$ and
compute its number of components. In Section 3 we prove Theorem~\ref{mainth} and apply this result to
mapping spaces. Section 4 treats the cohomology of $M_n^+(SO(3))$. Part of the content of this paper is part
of the Doctoral Dissertation of the first author (\cite{E}).
\section{\bf The Spaces $M_n(SO(3))$}\label{thespaces}
In this section we describe the topolgy of the spaces $M_n(SO(3)),\ n\ge 2.$
If $A_1,A_2$ are commuting elements from $SO(3)$ then there are $2$ possibilities:
either $A_1,A_2$ are rotations about a common axis; or
$A_1,A_2$ are involutions about axes meeting at right angles.
The first possibility covers the case where one of $A_1,A_2$ is the identity since the identity can be considered as a rotation about any axis.
It follows that there are 2 possibilities for an $n$-tuple
$(A_1,\ldots,A_n)\in M_n(SO(3)):$
\begin{enumerate}
\item Either $A_1,\ldots,A_n$ are all rotations about a common axis $L$; or
\item There exists at least one pair $i,j$ such that $A_i,A_j$ are involutions about perpendicular axes. If $v_i,v_j$ are unit vectors representing these axes then all the other $A_k$ must be one of $id,A_i,A_j$ or $A_iA_j=A_jA_i$ (the involution about the cross product $v_i\times v_j).$
\end{enumerate}
It is clear that if $\omega(t)=(A_1(t),\ldots,A_n(t))$ is a path in
$M_n(SO(3))$ then exactly one of the following 2 possibilities occurs:
either the rotations $A_1(t),\ldots,A_n(t)$ have a common axis $L(t)$ for all $t$; or there exists a pair $i,j$ such that $A_i(t),A_j(t)$ are involutions about perpendicular axes for all $t$. In the second case the pair $i,j$ does not depend on $t.$
\begin{proposition}\label{compvec{1}}
$M_n^+(SO(3))$ is the space of $n$-tuples $(A_1,\ldots,A_n)\in SO(3)^n$ such that all the $A_j$ have a common axis of rotation.
\end{proposition}
\begin{proof}
Let $A_1,\ldots,A_n$ have a common axis of rotation $L$. Thus
$A_1,\ldots,A_n$ are rotations about $L$ by some angles $\theta_1,\ldots,\theta_n.$ We can change all angles to $0$ by a path (while maintaining the common axis). Conversely, if $\omega(t)=(A_1(t),\ldots,A_n(t))$ is a path containing $\vec{1}$ then the $A_j(t)$ will have a common axis of rotation for all $t$ (which might change with $t$).
\end{proof}
Thus any component of $M_n^-(SO(3))$ can be represented by an $n$-tuple $(A_1,\ldots,A_n)$ satisfying possibility 2 above.
\begin{corollary} The connected components of $M_2(SO(3))$ are $M_2^{\pm}(SO(3)).$ \end{corollary}
\begin{proof} Let $(A_1,A_2)$ be a pair in $M_2^-(SO(3)).$ Then there are unit vectors $v_1,v_2$ in ${\bbb R}^3$ such that $v_1,v_2$ are
perpendicular and $A_1,A_2$ are involutions about $v_1,v_2$ respectively. The pair $(v_1,v_2)$ is not unique since any one of the
four pairs $(\pm v_1,\pm v_2)$ will determine the same involutions. In fact there is a 1-1 correspondence between pairs $(A_1,A_2)$
in $M_2^-(SO(3))$ and sets $\{(\pm v_1,\pm v_2) \}.$ Thus $M_2^-(SO(3))$ is homeomorphic to the orbit space
$V_2({\bbb R}^3)/{\bbb Z}_2\oplus{\bbb Z}_2.$ Since $V_2({\bbb R}^3)$ is connected so is $M_2^-(SO(3)).$
\end {proof}
Next we determine the number of components of $M_n^-(SO(3))$ for $n>2$. The following example will give an indication of the complexity.
\begin{example}\label{example1}
Let $(A_1,A_2,A_3)$ be an element of $M_3^-(SO(3)).$ Then there exists at least one pair $A_i,A_j$ without a
common axis of rotation. For example suppose $A_1,A_2$ don't have a common axis. Then
$A_1,A_2$ are involutions about perpendicular axes $v_1,v_2$, and there are
$4$ possibilities for $A_3:$
$A_3=id,A_1,A_2\ \mbox{or}\ A_3=A_1A_2.$
We will see that the triples
$$(A_1,A_2,id),(A_1,A_2,A_1),(A_1,A_2,A_2),(A_1,A_2,A_1A_2)$$ belong to different components. Similarly if $A_1,A_3$ or $A_2,A_3$ don't have a common axis of rotation. This leads to 12 components, but some of them are the same component.
An analysis leads to a total of 7 distinct components corresponding to the following 7 triples:
$(A_1,A_2,id)$, $(A_1,A_2,A_1)$, $(A_1,A_2,A_2)$, $(A_1,A_2,A_1A_2)$, $(A_1,id,A_3)$, $(A_1,A_1,A_3)$, $(id,A_2,A_3)$;
where $A_1,A_2$ are distinct involutions in the first 4 cases; $A_1,A_3$ are distinct involutions in the next 2 cases; and $A_2,A_3$ are
distinct involutions in the last case. These components are all homeomorphic to ${\bbb S}^3/ Q_8$.
Thus $M_3(SO(3))$ is homeomorphic to the disjoint union
$$M_3^+(SO(3))\sqcup {\bbb S}^3/Q_8\sqcup\ldots\sqcup {\bbb S}^3/Q_8,$$ where there are 7 copies of ${\bbb S}^3/Q_8$.
\end{example}
The pattern of this example holds for all $n\ge 3.$ A simple analysis shows that
$M_n^-(SO(3))$ consists of $n$-tuples
$\vec{A}=(A_1,\ldots,A_n)\in SO(3)^n$ satisfying the following conditions:
\begin{enumerate}
\item Each $A_i$ is either an involution about some axis $v_i,$ or the
identity.
\item If $A_i,A_j$ are distinct involutions then their axes are at right
angles.
\item There exists at least one pair $A_i,A_j$ of distinct involutions.
\item If $A_i,A_j$ are distinct involutions then every other $A_k$ is one of $id,A_i,A_j$ or $A_iA_j.$
\end{enumerate}
This leads to 5 possibilities for any element
$(B_1,\ldots,B_n)\in M_n^-(SO(3)):$
$$
(B_1,B_2,*,\ldots,*),(B_1,id,*,\ldots,*),(id,B_2,*,\ldots,*),
(B_1,B_1,*,\ldots,*),(id,id,*,\ldots,*),
$$
where $B_1,B_2$ are distinct involutions about perpendicular axes and the asterisks are choices from amongst $id,B_1,B_2,B_3=B_1B_2.$ The choices must satisfy the conditions above.
These 5 cases account for all components of $M_n^-(SO(3)),$ but not all choices lead to distinct components. If
$\omega(t)= (B_1(t),B_2(t),\ldots,B_n(t))$ is a path in
$M_n^-(SO(3))$ then it is easy to verify the following statements:
\begin{enumerate}
\item If some $B_i(0)=id$ then $B_i(t)=id$ for all $t$.
\item If $B_i(0)=B_j(0)$ then $B_i(t)=B_j(t)$ for all $t$.
\item If $B_i(0),B_j(0)$ are distinct involutions then so are $B_i(t),B_j(t)$ for all $t$.
\item If $B_k(0)=B_i(0)B_j(0)$ then $B_k(t)=B_i(t)B_j(t)$ for all $t$.
\end{enumerate}
These 4 statements are used repeatedly in the proof of the next theorem.
\begin{theorem}
The number of components of $M_n^-(SO(3))$ is
$$\left\{\begin{array}{ll}
\displaystyle\frac{1}{6}(4^{n}-3\times 2^{n}+2) & \mbox{if $n$ is even}\\
& \\
\displaystyle \frac{2}{3}(4^{n-1}-1)-2^{n-1}+1 &\mbox{if $n$ is odd}
\end{array}\right.
$$
Moreover, each component is homeomorphic to ${\bbb S}^3/Q_8.$
\end{theorem}
\begin{proof}
Let $x_n$ denote the number of components. The first 3 values of $x_n$ are $x_1=0,$ $ x_2=1$ and $x_3=7,$ in agreement with the statement in the theorem. We consider the above 5 possibilities one by one. First assume $\vec{B}=(B_1,B_2,*,\ldots,*).$ Then different choices of the asterisks lead to different components. Thus the contribution in this case is $4^{n-2}.$ Next assume
$\vec{B}=(B_1,id,*,\ldots,*).$ Then all choices for the asterisks are admissible, except for those choices involving only $id$ and $B_1.$ This leads to
$4^{n-2}-2^{n-2}$ possibilities. However, changing every occurrence of $B_2$ to $B_3,$ and $B_3$ to $B_2,$ keeps us in the same component. Thus the total contribution in this case is $(4^{n-2}-2^{n-2})/2.$ This is the same contribution for cases 3 and 4.
Finally, there are $x_{n-2}$ components associated to $\vec{B}=(id,id,*,\ldots,*).$
This leads to the recurrence relation
$$\displaystyle x_n= 4^{n-2}+\frac{3}{2}(4^{n-2}-2^{n-2})+x_{n-2}$$
Now we solve this recurrence relation for the $x_n.$
Given any element $(B_1,\ldots,B_n)\in M_n^-(SO(n))$ we select a pair of
involutions, say $B_i,B_j,$ with perpendicular axes $v_i,v_j.$ All the other $B_k$ are determined uniquely by $B_i,B_j.$ Thus the element $(v_i,v_j)\in V_2(\mathbb R^3)$ determines $(B_1,\ldots,B_n).$ But all the elements $(\pm v_i,\pm v_j)$ also determine $(B_1,\ldots,B_n).$ Thus the component to which $(B_1,\ldots,B_n)$ belongs is homeomorphic to
$V_2(\mathbb R^3)/\mathbb Z_2\oplus\mathbb Z_2\cong {\bbb S}^3/Q_{8}.$
\end{proof}
\section{\bf Fundamental Group of $M_n(G)$}
In this section we prove Theorem \ref{mainth}, and we start by finding an appropriate description of $M_n^+(SO(3))$.
Let $T^n=({\bbb S}^1)^n$ denote the $n$-torus. Then
\begin{theorem}\label{quotient}
$M_n^+(SO(3))$ is homeomorphic to the quotient space ${\bbb S}^2\times T^n/\sim,$ where $\sim$ is the equivalence relation generated by
$$\displaystyleplaystyle
(v,z_1,\dots,z_n)\sim (-v,\bar{z}_1,\ldots,\bar{z}_n)\ \mbox{ and }\
(v,\vec{1})\sim(v^{\prime},\vec{1})\
\mbox{for all $v,v^{\prime}\in {\bbb S}^2,z_i\in {\bbb S}^1.$}
$$
\end{theorem}
\begin{proof}
If $(A_1,\ldots,A_n)\in M_n^+(SO(3))$ then there exists $v\in {\bbb S}^2$ such that $A_1,\ldots,A_n$ are rotations about $v$. Let $z_j\in {\bbb S}^1$ be the elements corresponding to these rotations. The $(n+1)$-tuple $(v,z_1,\ldots,z_n)$ is not unique. For example, if one of the $A_i$'s is not the identity then $(-v,\bar{z}_1,\ldots,\bar{z}_n)$ determines the same $n$-tuple of rotations. On the other hand, if all the $A_i$'s are the identity then any element $v\in {\bbb S}^2$ is an axis of rotation.
\end{proof}
We will use the notation $[v,z_1,\ldots,z_n]$ to denote the equivalence class of $(v,z_1,\dots,z_n).$
Thus $x_0=[v,\vec{1}]\in {\bbb S}^2\times T^n/\sim$ is a single point, which we choose to be the base point. It corresponds to the $n$-tuple $(id,\dots, id)\in M_n^+(SO(3))$. Then
$M_n^+(SO(3))$ is locally homeomorphic to $\mathbb R^{n+2}$ everywhere except at the point $x_0$ where it is singular.
{\it Proof of Theorem 1.2}: Notice that the result holds for $n=1$ since $Hom({\bbb Z},G)$ is homeomorphic to $G$.
The first step is to compute $\pi_1(M_n^+(SO(3))).$ Let $T^n_0=T^n-\{\vec{1}\}$ and
$M_n^+=M_n^+(SO(3)).$
Removing the singular point $x_0=[v,\vec{1}]$ from $M_n^+$ we have
$M_n^+-\{x_0\}\cong {\bbb S}^2\times T_0^n/\mathbb Z_2,$ see Theorem \ref{quotient}.
If $t$ denotes the generator of $\mathbb Z_2$ then the $\mathbb Z_2$ action
on ${\bbb S}^2\times T_0^n$ is given by
$$t(v,z_1,\ldots,z_n)=(-v,\bar{z}_1,\ldots,\bar{z}_n),\ v\in {\bbb S}^2, z_j\in {\bbb S}^1$$ This action is fixed point free and so there is
a two-fold covering
${\bbb S}^2\times T^n_0\stackrel{p}{\to}M_n^+-\{x_0\}$ and a short exact sequence
$$1\to \pi_1({\bbb S}^2\times T^n_0)\to\pi_1(M_n^+-\{x_0\})\to{\bbb Z}_2\to 1$$
Let ${\bf n}$ denote the north pole of ${\bbb S}^2.$ Then for base points in ${\bbb S}^2\times T^n_0$ and $M_n^+-\{x_0\}$ we take
$({\bf n},-1,\ldots,-1)=({\bf n},-\vec{1})$ and $[{\bf n},-1,\ldots,-1]=[{\bf n},-\vec{1}]$ respectively.
This sequence splits. To see this note that the composite
${\bbb S}^2\to {\bbb S}^2\times T^n_0 \to {\bbb S}^2$ is the identity, where the first map is
$ v\mapsto(v,-\vec{1})$ and the second is just the projection. Both maps are equivariant with respect to the ${\bbb Z}_2$-actions, and therefore $ M_n^+-\{x_0\}$ retracts onto $\mathbb R P^2.$
First we consider the case $n=2.$ Choose $-1$ to be the base point in ${\bbb S}^1.$ The above formula for the action of $\mathbb Z_2$ also defines a ${\bbb Z}_2$ action on ${\bbb S}^2\times({\bbb S}^1\vee {\bbb S}^1).$
This action is fixed point free. The inclusion ${\bbb S}^2\times({\bbb S}^1\vee {\bbb S}^1)\subset {\bbb S}^2\times {\bbb S}^1\times {\bbb S}^1$ is equivariant and there exists a ${\bbb Z}_2$-equivariant strong deformation retract from
${\bbb S}^2\times T^2_0$ onto ${\bbb S}^2\times({\bbb S}^1\vee {\bbb S}^1).$ Let $a_1,a_2$ be the generators
$({\bf n},{\bbb S}^1,-1)$ and $({\bf n},-1,{\bbb S}^1)$ of $\pi_1({\bbb S}^2\times T^2_0)={\bbb Z}*{\bbb Z}.$ See the Figure below.
The involution $t:{\bbb S}^2\times T^2_0\to {\bbb S}^2\times T^2_0$ induces isomorphisms
$$\begin{array}{ccc}
\pi_1({\bbb S}^2\times({\bbb S}^1\vee {\bbb S}^1),\{{\bf n},-1,-1\})&\stackrel{c}{\to}&
\pi_1({\bbb S}^2\times({\bbb S}^1\vee {\bbb S}^1),\{{\bf s},-1,-1\})\\
\pi_1({\bbb S}^2\vee({\bbb S}^1\vee {\bbb S}^1),\{{\bf n},-1,-1\})&\stackrel{c}{\to}&
\pi_1({\bbb S}^2\vee({\bbb S}^1\vee {\bbb S}^1),\{{\bf n},-1,-1\})
\end{array}$$
where ${\bf s}=-{\bf n}$ is the south pole in ${\bbb S}^2.$
We have the following commutative diagram
$$\xymatrix{
{\bbb S}^2\vee_{{\bf n}}({\bbb S}^1\vee {\bbb S}^1)\ar[d]^{i_{{\bf n}}}\ar[rr]^t & & {\bbb S}^2\vee_{{\bf s}}({\bbb S}^1\vee {\bbb S}^1)\ar[d]^{i_{{\bf s}}} \\
{\bbb S}^2\times_{{\bf n}}({\bbb S}^1\vee {\bbb S}^1)\ar[rr]^t\ar[rd]_p & & {\bbb S}^2\times_{{\bf s}}({\bbb S}^1\vee {\bbb S}^1)\ar[ld]^p \\
& M_2^+-\{x_0\} & }$$
where $i_{{\bf n}}$ and $i_{{\bf s}}$ are inclusions. Here the subscripts ${\bf n}$ and ${\bf s}$ refer to the north and south poles respectively, which we take to be base points of ${\bbb S}^2$ in the one point unions. The inclusions $i_{\bf n},i_{\bf s}$ induce isomorphims on $\pi_1$ and therefore
$p_*\pi_1({\bbb S}^2\vee_{\bf n}({\bbb S}^1\vee {\bbb S}^1))=p_*\pi_1({\bbb S}^2\vee_{\bf s}({\bbb S}^1\vee {\bbb S}^1)).$ Thus $t$ sends $a_1$ to the loop based at $s$ but with
the opposite orientation (similarly for $a_2$). See the Figure below.
\begin{center}
\includegraphics [scale=0.5] {figure1.eps}
\end{center}
We now have $\pi_1(M_2^+-\{x_0\})=
<a_1,a_2,t\ |\ t^2=1, a_1^t=a_1^{-1}, a_2^t=a_2^{-1}>$.
For the computation of $\pi_1(M_n^+-\{ x_0\}),\ n\ge 3,$ note that the inclusion $T^n_0 \subset T^n$ induces an isomorphism on $\pi_1.$ Therefore
$\pi_1(T^n_0)=<a_1,\ldots,a_n\ |\ [a_i,a_j]=1\ \forall\ i,j>.$
The various inclusions of $T_0^2$ into $T_0^n$ (corresponding to pairs of generators) show that the action of $t$ on the
generators is still
given by $a_i^t=a_i^{-1}.$
Thus
$$\pi_1(M_n^+-x_0)=<a_1,\ldots,a_n,t\ |\ t^2=1,[a_i,a_j]=1, a_i^t=a_i^{-1}>,\
\mbox{ for $n\geq 3$.}$$
The final step in the calculation of $\pi_1(M_n^+)$ is to use van Kampen's theorem. To do this let
$U\subset {\bbb S}^1$ be a small open connected neighbourhood of $1\in {\bbb S}^1$ which is invariant under conjugation. Here small means
$-1\not\in U$. Then $N_n={\bbb S}^2\times U^n/\sim$ is a contractible neighborhood of $x_0$ in $M_n^+.$
We apply van Kampen's theorem to the situation
$M_n^+=\displaystyle (M_n^+-\{x_0\})\cup N_n.$
The intersection $N_n\cap(M_n^+-\{x_0\})$ is homotopy equivalent to
$({\bbb S}^2\times{\bbb S}^{n-1})/{\bbb Z}_2$ where ${\bbb Z}_2$ acts by multiplication by
$-1$ on both factors. Therefore $\pi_1(N_n\cap(M_n^+-\{x_0\})) \cong{\bbb Z}$ when $n =2$, and ${\bbb Z}_2$ when $n\geq 3$.
Thus we need to understand the homomorphism induced by the inclusion
$N_n\cap(M_n^+-\{ x_0\} )\to M_n^+-\{ x_0\}$.
When $n=2$ the inclusion of $N_2\cap(M_2^+-\{x_0\})$ into $M_2^+-\{x_0\}$ induces the following commutative diagram
$$\xymatrix{
{\bbb Z}\ar[r]\ar[d]^2 & {\bbb Z}*{\bbb Z}\ar[d] \\
\pi_1(N_2\cap(M_2^+-\{x_0\}))\ar[r]\ar[d] & \pi_1(M_2^+-\{x_0\})\ar[d] \\
{\bbb Z}_2\ar[r]^= & {\bbb Z}_2 }$$
where the map on top is the commutator map. So if the generator of $\pi_1(N_2\cap(M_2^+-\{x_0\}))$ is sent to
$w\in\pi_1(M_2^+-\{x_0\})$, then $w^2=[a_1,a_2]$, and the image of $w$ in ${\bbb Z}_2$ is $t$.
Thus we can write $w=a_1^{n_1}a_2^{m_1}\cdots a_1^{n_r}a_2^{m_r}t$ with $n_i,m_i\in{\bbb Z}$.
Then
$$w^2=a_1^{n_1}a_2^{m_1}\cdots a_1^{n_r}a_2^{m_r}a_1^{-n_1}a_2^{-m_1}\cdots a_1^{-n_r}a_2^{-m_r}=
a_1 a_2 a_1^{-1}a_2^{-1}$$
which occurs only if $r=1$ and $n_1=m_1=1$. It follows that $w=a_1a_2 t$.
Thus $$\pi_1(M_2^+)=<a_1,a_2,t\ |\ t^2=1, a_1^t=a_1^{-1}, a_2^t=a_2^{-1}, a_1a_2 t=1>$$ and routine computations
show that this is the Klein four group.
For $n\geq 3$ the inclusion map $N_n\cap(M_n^+-\{x_0\})\to M_n^+-\{x_0\}$ can be understood by looking at the following diagram
$$\xymatrix{
{\bbb S}^2\times {\bbb S}^1\ar[rr]\ar[dd]\ar[rd] & & {\bbb S}^2\times T^2_0\ar[dd]\ar[rd] & \\
& {\bbb S}^2\times S^{n-1}\ar[rr]\ar[dd] & & {\bbb S}^2\times T^n_0\ar[dd] \\
N_2\cap(M_2^+-\{x_0\})\ar[rr]\ar[rd] & & M_2^+-\{x_0\}\ar[rd] & \\
& N_n\cap(M_n^+-\{x_0\})\ar[rr] & & M_n^+-\{x_0\}
}$$
Note that the map $N_2\cap(M_2^+-\{x_0\})\to N_n\cap(M_n^+-\{x_0\})$ induces the canonical projection ${\bbb Z}\to{\bbb Z}_2$. A chase
argument shows that the inclusion $N_n\cap(M_n^+-\{x_0\})\to M_n^+-\{x_0\}$ imposes the relation $a_1a_2t$ as well, and
therefore
$$\pi_1(M_n^+)=<a_1,\ldots,a_n,t\ |\ t^2=1,[a_i,a_j]=1, a_i^t=a_i^{-1}, a_1a_2 t=1>.$$
By performing some routine computations we see that this group is isomorphic to ${\bbb Z}_2^n$.
This completes the proof of Theorem~\ref{mainth} for $SO(3)$. The cases of $SU(2)$ and $U(2)$ follow from Remark~\ref{cover}.
$\Box$
Since the map $\pi_1(\vee_n G)\to\pi_1(G^n)$ is an epimorphism, it follows that the inclusion maps
$$M_n^+(G)\to G^n\hspace{.6cm}if\ \ G=SO(3)$$
$$M_n(G)\to G^n\hspace{.5cm}if\ \ G=SU(2),U(2)$$
are isomorphisms in $\pi_1$ for all $n\geq 1$.
Recall that there is a map $Hom(\Gamma,G)\to Map_*(B\Gamma,BG)$, where $Map_*(B\Gamma,BG)$ is the space of pointed
maps from the classifying space of $\Gamma$ into the classifying space of $G$. Let $Map_*^+(T^n,BG)$ be the component of
the map induced by the trivial representation.
\begin{corollary}\label{5.2} The maps
$$M_n^+(G)\to Map_*^+(T^n,BG)\hspace{.5cm}if\ \ G=SO(3)$$
$$M_n(G)\to Map_*^+(T^n,BG)\hspace{.5cm}if\ \ G=U(2)$$
are injective in $\pi_1$ for all $n\geq 1$.
\end{corollary}
\begin{proof} By induction on $n$, with the case $n=1$ being trivial. Assume $n>1$, and
note that there is a commutative diagram
$$\xymatrix{
M_n^+(SO(3))\ar[r]\ar[d] & Map_*^+(B\pi_1(T^n),BSO(3))\ar[d] \\
Hom(\pi_1(T^{n-1}\vee {\bbb S}^1),SO(3))\ar[r]\ar[d] & Map_*^+(B\pi_1(T^{n-1}\vee {\bbb S}^1),BSO(3))\ar[d] \\
Hom(\pi_1(T^{n-1}),SO(3))\times SO(3)\ar[r] & Map_*^+(B\pi_1(T^{n-1}),BSO(3))\times SO(3) }$$
in which the bottom map is injective in $\pi_1$ by inductive hypothesis, the lower vertical maps are homeomorphisms,
and the upper left vertical map is injective in $\pi_1$. Thus the map on top is also injective as wanted.
The proof for $U(2)$ is the same.
\end{proof}
\begin{remark}{\em We have the following observations.
\begin{enumerate}
\item The two-fold cover ${\bbb Z}_2\to {\bbb S}^3\times {\bbb S}^3\to SO(4)$ allows us to study $Hom({\bbb Z}^n,SO(4))$.
Let $M_n^+(SO(4))$ be the component covered by $Hom({\bbb Z}^n,{\bbb S}^3\times {\bbb S}^3)$. Since $Hom({\bbb Z}^n,{\bbb S}^3\times {\bbb S}^3)$ is
homeomorphic to $Hom({\bbb Z}^n,{\bbb S}^3)\times Hom({\bbb Z}^n,{\bbb S}^3)$, it follows that
$$\pi_1(M_n^+(SO(4)))={\bbb Z}_2^n$$
\item The space $Hom({\bbb Z}^2,SO(4))$ has two components. One is $M_2^+(SO(4)),$ which is covered by $\partial^{-1}_{SU(2)^2}(1,1)$,
and the other is covered by $\partial^{-1}_{SU(2)^2}(-1,-1)$, where $\partial_{SU(2)^2}$ is the commutator map of $SU(2)\times SU(2)$.
Recall $\partial^{-1}_{SU(2)}(-1)$ is homeomorphic to $SO(3)$ (see~\cite{AM}), so $\partial^{-1}_{SU(2)^2}(-1,-1)$ is
homeomorphic to $SO(3)\times SO(3)/{\bbb Z}_2\times{\bbb Z}_2$, where the group ${\bbb Z}_2\times{\bbb Z}_2$ acts by left diagonal multiplication when it is
thought of as the subgroup of $SO(3)$ generated by the transformations $(x_1,x_2,x_3)\mapsto(x_1,-x_2,-x_3)$ and
$(x_1,x_2,x_3)\mapsto(-x_1,x_2,-x_3)$.
\item Corollary~\ref{5.2} holds similarly for $SO(4)$, and trivially for $SU(2)$.
\end{enumerate}
}\end{remark}
\section{\bf Homological Computations}
In this section we compute the ${\bbb Z}_2$-cohomology of $M_n^+(SO(3))$. The ${\bbb Z}_2$-cohomology of the other components
of $M_n(SO(3))$ is well-known since these are all homeomorphic to ${\bbb S}^3/Q_8$.
To perform the computation we will use the description of $M_n^+(SO(3))$ that we
saw in the proof of Theorem~\ref{mainth}. The ingredients we have to consider are the spectral sequence of the fibration
${\bbb S}^2\times T^n_0\to(M_n^+-\{ x_0\})\to {\bbb R} P^\infty$ whose $E_2$ terms is
$${\bbb Z}_2[u]\otimes E(v)\otimes E(x_1,\ldots,x_n)/(x_1\cdots x_n)$$
with $deg(u)=(1,0)$, $deg(v)=(0,2)$ and $deg(x_i)=(0,1)$;
and the spectral sequence of the fibration ${\bbb S}^2\times{\bbb S}^{n-1}\to N_n\cap (M_n^+-\{ x_0\})\to{\bbb R} P^\infty$ whose
$E_2$-term is $${\bbb Z}_2[u]\otimes E(v)\otimes E(w)$$
with $deg(u)=(1,0)$, $deg(v)=(0,2)$ and $deg(w)=(0,n-1)$. Note that in both cases $d_2(v)=u^2$, whereas $d_2(x_i)=0$ since
$H^1(M_n^+-\{ x_0\},{\bbb Z}_2)={\bbb Z}_2^{n+1}$. Therefore the first spectral sequence collapses at the third term.
As $d_n(w)=u^n$ and $d_j(w)=0$ for $j\neq n$, the second spectral sequence collapses at the third term when $n=2$
and at the fourth term when $n\geq 3$.
The last step is to use the Mayer-Vietoris long exact sequence of the pair $(M_n^+-\{ x_0\},N_n)$ which yields the following: for $n=2,3$,
$$H^q(M_2^+(SO(3)),{\bbb Z}_2)=\left\{\begin{array}{ccl}
{\bbb Z}_2 & & q=0\\
{\bbb Z}_2\oplus{\bbb Z}_2 & & q=1\\
{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2 & & q=2\\
{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2 & & q=3\\
{\bbb Z}_2 & & q=4\\
0 & & q\geq 5 \end{array}\right.$$
$$H^q(M_3^+(SO(3)),{\bbb Z}_2)=\left\{\begin{array}{ccl}
{\bbb Z}_2 & & q=0\\
{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2 & & q=1\\
{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2 & & q=2\\
{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2 & & q=3\\
{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2\oplus{\bbb Z}_2 & & q=4\\
{\bbb Z}_2 & & q=5\\
0 & & q\geq 6 \end{array}\right.$$
whereas for $n\geq 4$,
$$H^q(M_n^+(SO(3)),{\bbb Z}_2)=\left\{\begin{array}{ccl}
{\bbb Z}_2 & & q=0\\
& & \\
{\bbb Z}_2^n & & q=1\\
& & \\
{\bbb Z}_2^{{n\choose 1}+{n\choose 2}} & & q=2\\
& & \\
\displaystyle {\bbb Z}_2^{{n\choose q-2}+{n\choose q-1}+{n\choose q}} & & 3\leq q\leq n\\
& & \\
{\bbb Z}_2^{{n\choose n-1}+1} & & q=n+1\\
& & \\
{\bbb Z}_2 & & q=n+2 \\
& & \\
0 & & q\geq n+3 \end{array}\right.$$
So the Euler characteristic of $M_n^+(SO(3))$ is given by
$$\chi(M_n^+(SO(3)))=\left\{
\begin{array}{ccl}
0 & & n=2\ or\ odd \\
& & \\
\displaystyle 2+n(n-1)-{n\choose k-1}-{n\choose k}-{n\choose k+1} & & n=2k,\ \ k\geq 2
\end{array}\right.$$
\
{
\parbox{6cm}{Denis Sjerve\\
{\it Department of Mathematics},\\
University of British Columbia\\
Vancouver, B.C.\\
Canada\\
{\sf [email protected]}}\
{
}\
\parbox{6cm}{Enrique Torres-Giese\\
{\it Department of Mathematics},\\
University of British Columbia\\
Vancouver, B.C.\\
Canada\\
{\sf [email protected]}}}\
\end{document}
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\title{A Note on an Asymptotically Good Tame Tower}
\author{Siman Yang}
\thanks{The author was partially supported by the program for Chang Jiang Scholars and Innovative Research Team in University}
\begin{abstract}
The explicit construction of function fields tower with many rational points relative to the genus in the tower play a key role for the construction of asymptotically good algebraic-geometric codes. In 1997 Garcia, Stichtenoth and Thomas [6] exhibited two recursive asymptotically good Kummer towers over any non-prime field. Wulftange determined the limit of one tower in his PhD thesis [13]. In this paper we determine the limit of another tower [14].
\end{abstract}
\maketitle
\noindent \underline{Keywords:} Function fields tower, rational
places, genus.
\ \\
\section{Introduction}
Let $K={\mathbb F}_q$ be the finite field of cardinality $q$, and let $\mathcal{F}=(F_i)_{i\geq 0}$ be a sequence of
algebraic function fields each defined over $K$. If $F_i\varsubsetneqq F_{i+1}$ and $K$ is the full constant field for all $i\geq 0$, and $g(F_j)>1$ for some $j\geq 0$, we call $\mathcal{F}$ a tower.
Denoted by $g(F)$ the genus of the function field $F/{\mathbb F}_q$ and $N(F)$ the number of ${\mathbb F}_q$-rational places of $F$.
It is well-known that for given genus $g$ and finite field ${\mathbb F}_q$, the number of ${\mathbb F}_q$-rational places of a function field is upper bounded due to the Weil's theorem (cf. [11]). Let $N_q(g):=\max \{N(F)|F\,\,\mbox{is a function field of genus}\, g\,\mbox{over}\,{\mathbb F}_q\}$ and let
\begin{equation*}
A(q)=\displaystyle\limsup_{g\rightarrow \infty}N_q(g)/g,
\end{equation*}
the Drinfeld-Vladut bound [2] provides a general upper bound of $A(q)$
\begin{equation*}
A(q)\leq \sqrt{q}-1.
\end{equation*}
Ihara[7], and Tsfasman, Vladut and Zink [12] independently showed that this bound is met when $q$ is a square by the theory of Shimura modular curves and elliptic modular curves, respectively. For non-square $q$ the exact value of $A(q)$ is unknown. Serre[10] first showed that $A(q)$ is positive for any prime power $q$
$$
A(q)\geq c\cdot \log q
$$
with some constant $c>0$ irrelevant to $q$. It was proved in [6]
that for any tower $\mathcal{F}=(F_i)_{i\geq 0}$ defined over
${\mathbb F}_q$ the sequence $N(F_n)/g(F_n)_{n\geq 0}$ is convergent. We
define the limit of the tower as
$$
\lambda(\mathcal{F})=\displaystyle\lim _{i\rightarrow \infty} N(F_i)/g(F_i).
$$
Clearly, $0\leq \lambda(\mathcal{F})\leq A(q)$. We call a tower $\mathcal{F}$ asymptotically good if
$\lambda(\mathcal{F})>0$.
To be useful towards the aim of yielding asymptotically good codes, a tower must be asymptotically good.
Practical implementation of the codes also requires explicit equations for each extension step in the tower. In 1995, Garcia and Stichtenoth [4] exhibited the first explicit tower of Artin-Schreier extensions over any finite field of square cardinality which met the upper bound of Drinfeld and Vladut.
In 1997 Garcia, Stichtenoth and Thomas [6] exhibited two explicit asymptotically good Kummer towers over any non-prime field which were later generalized by Deolalikar [1].
For other explicit tame towers, readers may look at [3], [5], [9].
The two asymptotically good Kummer towers in [6] are given as below.
Let $q=p^e$ with $e>1$, and let $F_n={\mathbb F}_q(x_0, \cdots, x_n)$ with
\begin{equation}
x_{i+1}^{\frac{q-1}{p-1}}+(x_i+1)^{\frac{q-1}{p-1}}=1 \,\,\,\,(i=0, \cdots, n-1).
\end{equation}
Then $\mathcal{F}=(F_0, F_1, \cdots)$ is an asymptotically good
tower over ${\mathbb F}_q$ with $\lambda (\mathcal{F})\geq 2/(q-2)$.
Let $q$ be a prime power larger than two, and let $F_n={\mathbb F}_q(x_0, \cdots, x_n)$ with
\begin{equation}
x_{i+1}^{q-1}+(x_i+1)^{q-1}=1 \,\,\,\,(i=0, \cdots, n-1).
\end{equation}
Then $\mathcal{F}=(F_0, F_1, \cdots)$ is an asymptotically good tower over ${\mathbb F}_{q^2}$ with $\lambda (\mathcal{F})\geq 2/(q-2)$.
Wulftange showed in [13] that $\lambda (\mathcal{F})=2/(q-2)$ for the first tower, we will show in the next section
that the limit of the second tower is also $2/(q-2)$.
\section{The limit of the tower}
\begin{lemma}
Let $F_1=K(x,y)$ defined by Eq. (2).\\
Over $K(x)$ exactly the zeroes of $x-\alpha$, $\alpha \in {\mathbb F}_q \backslash \{-1\}$ are ramified in $F_1$, each of ramification index $q-1$.\\
Over $K(y)$ exactly the zeroes of $y-\alpha$, $\alpha \in {\mathbb F}_q^*$ are ramified in $F_1$, each of ramification index $q-1$.
\end{lemma}
\begin{proof} By applying the theory of Kummer extension (cf. [11, Chap. III.7.3]).
\end{proof}
\begin{proposition}
Let $P_\alpha \in {\mathcal P} (F_0)$ be a zero of $x_0-\alpha$, $\alpha \in {\mathbb F}_q\backslash \{-1\}$. Then, $P_\alpha$ is totally ramified in $F_{n+1}/F_n$ for any $n\geq 0$.
\end{proposition}
\begin{proof} Let $P\in {\mathcal P} (F_n)$ lying above $P_\alpha$ for some $\alpha \in {\mathbb F}_q\backslash \{-1\}$. From Eq. (2), one can check $x_1(P)=x_2(P)=\cdots=x_n(P)=0$. Thus the ramification index of the extension of the restriction $P$ in $K(x_i, x_{i+1})/K(x_i)$ is $q-1$ for $i=0, 1, \cdots, n$, also the ramification index of the extension of the restriction $P$ in $K(x_i, x_{i+1})/K(x_{i+1})$ is $1$ for $i=0, 1, \cdots, n$. The proof is finished by diagram chasing and repeated application of Abhyankar's lemma.
\end{proof}
Let $Q \in {\mathcal P} (F_n)$ be a place ramified in $F_{n+1}$. Then $P:=Q\cap K(x_n)$ is ramified in $K(x_n, x_{n+1})$ due to Abhyankar's lemma. From Lemma 2.1, $x_n(P)=\alpha$ for some $\alpha \in {\mathbb F}_q\backslash \{-1\}$. If $\alpha \not= 0$, $P$ is ramifed in $K(x_{n-1}, x_n)$ of ramification index $q-1$ due to Lemma 2.1, and due to Abhyankar's lemma, the place in $K(x_{n-1},x_n)$ lying above $P$ is unramified in $K(x_{n-1}, x_n, x_{n+1})$, again by Abhyankar's lemma, $Q$ is unramified in $F_{n+1}$. Thus $Q$ is a zero of $x_n$. This implies $Q$ is a zero of $x_{n-1}-\beta$ for some $\beta \in {\mathbb F}_q\backslash \{-1\}$. From Eq. (2), one has the following possibilities for a place $Q \in {\mathcal P} (F_n)$ ramified in $F_{n+1}$.
(a) The place $Q$ is a common zero of $x_0, x_1, \cdots, x_n$.
(b) There is some $t$, $-1\leq t <n-1$ such that
(b1) $Q$ is a common zero of $x_{t+2}, x_{t+3}, \cdots, x_n$.
(b2) $Q$ is a zero of $x_{t+1}-\alpha $ for some $\alpha \in {\mathbb F}_q^*\backslash \{-1\}$.
(b3) $Q$ is a common zero of $x_0 +1, x_1 +1, \cdots, x_t +1$.
(Note that condition (b2) implies (b1) and (b3)).
\begin{lemma}
Let $-1\leq t <n$ and $Q\in {\mathcal P} (F_n)$ be a place which is a zero of $x_{t+1}-\alpha$ for some $\alpha \in {\mathbb F}_q^*\backslash \{-1\}$. Then one has
(i)\,\, If $n<2t+2$, then $Q$ is unramified in $F_{n+1}$.
(ii) If $n\geq 2t+2$, then $Q$ is ramified in $F_{n+1}$ of ramification index $q-1$.
\end{lemma}
\begin{proof}
The assertion in (i) and (ii) follow by diagram chasing with the help of Lemma 2.1 and repeated applications of Abhyankar's lemma.
\end{proof}
For $0 \leq t <\lfloor n/2 \rfloor $ and $\alpha \in {\mathbb F}_q^*\backslash \{-1\}$, set
$X_{t, \alpha}:=\{ Q\in {\mathcal P} (F_n)| Q \,\,\text{is a zero of}\,\, x_{t+1}-\alpha \}$
and $A_{t, \alpha}:=\displaystyle\sum _{Q\in X_{t, \alpha}}Q$. Denote by $Q_{t+1}$ the restriction of $Q$ to $K(x_{t+1})$, we have $[F_n:K(x_{t+1})]=(q-1)^n$ and $e(Q|Q_{t+1})=(q-1)^{n-t-1}$. Then deg $A_{t, \alpha}=(q-1)^{t+1}$ follows from the fundemental equality $\sum e_i f_i =n$. Combining the above results one obtains
\begin{align}
\text{deg Diff}(F_{n+1}/F_n)&=(q-1)(q-2)+\displaystyle\sum _{\alpha \in {\mathbb F}_q^*\backslash \{-1\}} \displaystyle\sum _{t=0}^{\lfloor n/2 \rfloor -1}(q-2)(q-1)^{t+1}\\
&=(q-2)(q-1)^{\lfloor n/2 \rfloor +1}.
\end{align}
Now we can easily determine the genus of $F_n$ by applying the transitivity of different exponents and Hurwitz genus formula. The result is:
\[g(F_{n+1})=\left\{
\begin{array}{cccccc}
(q-2)(q-1)^{n+1}/2-(q-1)^{n/2+1}+1,\,\, \mbox{if}\,\, n\,\, \mbox{is even}, \\
(q-2)(q-1)^{n+1}/2-q(q-1)^{(n+1)/2}/2+1 ,\,\, \mbox{if}\,\, n\,\, \mbox{is odd}.
\end{array}\right.
\]
Thus $\gamma (\mathcal{F}):=\displaystyle\lim_{n\rightarrow \infty} g(F_n)/[F_n:F_0]=(q-2)/2$.
\begin{remark}
Note that from the proof of [6, Theorem 2.1 and Example 2.4],
$\gamma(\mathcal{F})$ is upper bounded by $(q-2)/2$.
\end{remark}
Next we consider the rational places in each function field $F_n$.
First we consider places over $P_{\infty}$. It is easy to see that
$P_{\infty}$ splits completely in the tower. From Prop. 2.2,
there's a unique ${\mathbb F}_q$-rational place in $F_n$ over $P_{\alpha}$
for any $\alpha \in {\mathbb F}_q\backslash \{-1\}$. Then we consider the
$K$-rational place over $P_{-1}$ in $F_n$. Let $0 \leq t <n$ and
$Q\in {\mathcal P} (F_n)$ be a place which is a zero of $x_{t+1}-\alpha$
for some $\alpha \in {\mathbb F}_q^*\backslash \{-1\}$. We study the
condition for such place $Q$ to be $K$-rational.
\begin{lemma}
Let $Q'$ be a place of $F_2$ and $Q'$ is a zero of $x_1-\beta$ for some $\beta \in {\mathbb F}_q^*\backslash \{-1\}$. Then, if char$(\mathcal{F})\not=2$, $Q'$ is not a ${\mathbb F}_q$-rational place and $Q'$ is a ${\mathbb F}_{q^2}$-rational place if and only if $\beta =-1/2$; if
char$(\mathcal{F})=2$, $Q'$ is not a ${\mathbb F}_{q^2}$-rational place.
\end{lemma}
\begin{proof}
Note that $x_2$ and $x_0+1$ both are $Q'$-prime elements. Eq. (2) implies $(\frac{x_2}{x_0+1})^{q-1}=\frac{x_1}{1+x_1}$, which is equivalent to $\beta /(1+\beta)$. $(\frac{x_2}{x_0+1})^{q-1}(Q')\not=1$ implies $Q'$ is not ${\mathbb F}_q$-rational, and $(\frac{x_2}{x_0+1})^{q^2-1}(Q')=1$ if and only if $\beta =-1/2$ as $\beta \in {\mathbb F}_q^*\backslash \{-1\}$.
\end{proof}
We generalize this result to the following proposition.
\begin{proposition}
Assume char$(\mathcal{F})$ is odd. Fix positive integers $t\leq m$. There are $2^{t-1}(q-1)$ many ${\mathbb F}_{q^2}\backslash {\mathbb F}_q$-rational places $Q$ in ${\mathbb F}_{q^2}(x_{m-t}, x_{m-t+1}, \cdots, x_{m+t})$ which are zeroes of $x_m-\beta$ for some $\beta \in {\mathbb F}_q^*\backslash \{-1\}$ if $q\equiv -1$ (mod $2^t$), with each of them corresponds to a tuple $(\alpha _1, \alpha _2, \cdots, \alpha _t)$ satisfying
\[\left\{
\begin{array}{cccccc}
x_m\equiv -1/2,\\
x_{m+1}/(x_{m-1}+1)&\equiv& \alpha _1,\,\,\mbox{with}\,\, \alpha _1 ^2&=&-1,\\
x_{m+2}/(x_{m-2}+1)&\equiv& \alpha _2,\,\,\mbox{with}\,\,\alpha _2 ^2&=&-1/\alpha_1,\\
\cdots &\equiv&\cdots, \,\, \cdots&=&\cdots,\\
x_{m+t-1}/(x_{m-t+1}+1)&\equiv& \alpha _{t-1},\,\,\mbox{with}\,\,\alpha_{t-1}^2&=&-1/\alpha_{t-2},\\
x_{m+t}/(x_{m-t}+1)&\equiv& \alpha _t,\,\,\mbox{with}\,\,\alpha _t ^{q-1}&=&-\alpha_{t-1}.
\end{array}\right.
\]
\end{proposition}
\begin{proof}
Prove by induction on $t$. For $t=1$, this is the case in Lemma 2.5, here we take $\alpha _0=1$. For $t\geq 1$, it is easily checked $(\frac{x_{m+t+1}}{x_{m-t-1}+1})^{q-1} \equiv \frac{-x_{m+t}}{x_{m-t}+1}$ from definition. Thus, $\alpha _{t+1}\in {\mathbb F}_{q^2}$ if and only if $\alpha_t^{q+1}=1$. By induction hypothesis on $t$, $\alpha _t ^{q-1}=-\alpha _{t-1}$. Therefore $Q$ is a ${\mathbb F}_{q^2}$-rational place implies $\alpha _t^2=-1/\alpha_{t-1}$. Note $\alpha_{t-1}^{2^{t-1}}=-1$. Let $q=2^tk-1$, we have $(-1)^k=1$, thus $k$ is even, i.e., $q\equiv -1$ (mod $2^{t+1}$). This finishes the induction on $t+1$.
\end{proof}
Using this proposition and Lemma 2.3, we yield the following result.
\begin{proposition}
Assume char$(\mathcal{F})$ is odd. Suppose $2^l||(q+1)$. The number of ${\mathbb F}_{q^2}$-rational place in $F_n$ which is a zero of $x_m-\alpha (0<m\leq n)$ for any $\alpha \in {\mathbb F}_q^*\backslash \{-1\}$ is counted as below.
\[\left\{
\begin{array}{cccccc}
2^{m-1}(q-1),\,\, &\mbox{when}&\, 1\leq m \leq n/2\,\, \mbox{and}\,\, m\leq l, \\
0,\,\, &\mbox{when}&\, 1\leq m \leq n/2\,\, \mbox{and}\,\, m>l, \\
2^{n-m-1}(q-1),\,\, &\mbox{when}&\, n>m>n/2\,\, \mbox{and}\,\, n-m\leq l, \\
0,\,\, &\mbox{when}&\, n>m>n/2\,\, \mbox{and}\,\, n-m>l, \\
q-2,\,\, &\mbox{when}&\, m=n.
\end{array}\right.
\]
\end{proposition}
\begin{proof}
Let $0<m<n$ and $a=\min \{m, n-m \}$. If $a=m$ (resp. $a=n-m$), from Lemma 2.5, there exists ${\mathbb F}_{q^2}$-rational place in $F_{2m}$ (resp. $K(x_{2m-n}, \cdots, x_n)$) with $x_m\equiv \alpha$ for some $\alpha \in {\mathbb F}_q^*\backslash \{-1\}$ if and only if $2^a||(q+1)$, the number of such places is $2^{a-1}(q-1)$, and all these places totally ramified in $F_n$ according to Lemma 2.1.
\end{proof}
Hence, the number of ${\mathbb F}_{q^2}$-rational place in $F_n$ lying
above $P_{-1}$ is
\[\left\{
\begin{array}{cccccc}
(q-1)(2^{l+1}-1),\,\, &\mbox{if}& \,\, n>2l, \\
(q-1)(2^{(n+1)/2}-1), \,\, &\mbox{if n is odd}&\mbox{and}\, n\leq 2l, \\
(q-1)(3\times 2^{n/2-1}-1), \,\, &\mbox{if n is even}&\mbox{and}\, n\leq 2l.
\end{array}\right.
\]
\begin{remark}
If char$(\mathcal{F})>2$, among all ${\mathbb F}_{q^2}$-rational place in $F_n$ lying above $P_{-1}$, exactly $q-1$ are ${\mathbb F}_q$-rational, corresponding to $x_n\equiv \alpha$ for some $\alpha \in {\mathbb F}_q^*$, respectively.
If char$(\mathcal{F})=2$, from Lemma 2.5, there are exactly $q-1$ ${\mathbb F}_{q^2}$-rational places in $F_n$ lying above $P_{-1}$, which are all ${\mathbb F}_q$-rational, corresponding to $x_n\equiv \alpha$ for some $\alpha \in {\mathbb F}_q^*$, respectively.
\end{remark}
Next we determine the ${\mathbb F}_{q^2}$-rational place $Q$ in $F_n$ lying
above $P_\alpha$ for some $\alpha \in {\mathbb F}_{q^2}\backslash {\mathbb F}_q$.
Direct calculation gives $x_1(Q)=\alpha _1$ for some $\alpha _1
\not\in {\mathbb F}_q$. Similarly, $x_2(Q)=\alpha -2$, $\cdots$,
$x_n(Q)=\alpha _n$, with $\alpha _i \in \overline{{\mathbb F}}_q \backslash
{\mathbb F}_q$. We observe that $Q$ is ${\mathbb F}_{q^2}$-rational in $F_n$ if and
only if $\alpha, \alpha _1, \cdots, \alpha _n$ are all in
${\mathbb F}_{q^2}$. To verify it, assume $\alpha, \alpha _1, \cdots,
\alpha _n$ are all in ${\mathbb F}_{q^2}$. Then $Q$ is completely splitting
in each extension $F_i/F_{i-1} (i=1, 2, \cdots , n)$, with
$x_i\equiv c\alpha _i$ for some $c\in {\mathbb F}_q ^*$ in each place
respectively.
We have $\alpha _1\in {\mathbb F}_{q^2}$ if and only if $(1+\alpha)^{q-1}+(1+\alpha)^{1-q}=1$. Similarly, $\alpha _i (i=1, 2, \cdots, n-1) \in {\mathbb F}_{q^2}$ if and only if $(1+\alpha _{i-1})^{q-1}+(1+\alpha _{i-1})^{1-q}=1$. Thus, $Q$ is $F_{q^2}$-rational implies $(1+\alpha)^{q-1}, (1+\alpha _1)^{q-1}, \cdots, (1+\alpha _{n-1})^{q-1}$ all are the root of $x^2-x+1=0$.
{\bf Claim.} $(1+\alpha)^{q-1}, (1+\alpha _1)^{q-1}, \cdots, (1+\alpha _{n-1})^{q-1}$ are equal.
{\bf Proof of the claim.} Prove by contradiction. For simplicity assume $(1+\alpha)^{q-1}\not=(1+\alpha _1)^{q-1}$.
Thus $x^2-x+1=(x-(1+\alpha)^{q-1})(x-(1+\alpha _1)^{q-1})$. Comparing the coefficient of $x^1$,
one has $1=(1+\alpha)^{q-1}+(1+\alpha _1)^{q-1}=(2+\alpha _1-\alpha _1^{q-1})/(1+\alpha_1)$. This implies $\alpha _1 \in {\mathbb F}_q$, which is a contradiction.
Let $p=$char$({\mathbb F}_q)$, we consider the following two cases respectively.
{\bf Case 1: $p=3$.}
Since the unique root of $x^2-x+1=0$ is $-1$, $1-\alpha _1^{q-1}=-1$, and $(1+\alpha_1)^{q-1}=-1$. It is easily checked these two equalities lead to a contradiction.
{\bf Case 2: $p\not=3$.}
Thus, $-1$ is not a root of $x^2-x+1=0$, which implies $(1+\alpha)^{q-1}$ and $(1+\alpha)^{1-q}$ are distinct roots of $x^2-x+1=0$. Thus, $(1+\alpha)^{q-1}+(1+\alpha)^{1-q}=1$. By assuming $(1+\alpha)^{q-1}+(1+\alpha)^{1-q}=1$, we have $x_1(Q)=\alpha_1$, with $\alpha_1 ^{q-1}=(1+\alpha)^{1-q}$. Hence, $\alpha _1=c/(1+\alpha)$ for some $c\in {\mathbb F}_q^*$. Since $(1+\alpha)^{q-1}=(1+\alpha _1)^{q-1}$, direct calculation gives $c=\frac{(1+\alpha)^{2q-1}-(1+\alpha)^q}{1-(1+\alpha)^{2q-2}}$. Iterating this procedure, we have a ${\mathbb F}_{q^2}$-rational place $Q$ in $F_n$ lying above $P_\alpha$ for some $\alpha \in {\mathbb F}_{q^2}\backslash {\mathbb F}_q$ is one-one corresponding to a tuple $(c_1, c_2, \cdots, c_n)$ satisfying
\[\left\{
\begin{array}{cccccc}
x_0\equiv \alpha,\,\,&\mbox{with}&\,\, (1+\alpha)^{q-1}+(1+\alpha)^{1-q}=1, \\
x_1\equiv c_1/(1+\alpha):=\alpha_1 \,\, &\mbox{with}& \,\,c_1=\frac{(1+\alpha)^{2q-1}-(1+\alpha)^q}{1-(1+\alpha)^{2q-2}}\in {\mathbb F}_q^*, \\
x_2\equiv c_2/(1+\alpha_1):=\alpha_2 \,\, &\mbox{with}& \,\,c_2=\frac{(1+\alpha_1)^{2q-1}-(1+\alpha_1)^q}{1-(1+\alpha_1)^{2q-2}}\in {\mathbb F}_q^*, \\
\cdots &\mbox{with}& \,\,\cdots \\
x_{n-1}\equiv c_{n-1}/(1+\alpha_{n-2}):=\alpha_{n-1} \,\, &\mbox{with}& \,\,c_{n-1}=\frac{(1+\alpha_{n-2})^{2q-1}-(1+\alpha_{n-2})^q}{1-(1+\alpha_{n-2})^{2q-2}}\in {\mathbb F}_q^*, \\
x_n\equiv c_n/(1+\alpha_{n-1}),\,\, &\mbox{with}& \,\, c_n\in{\mathbb F}_q^*.
\end{array}\right.
\]
Therefore, for any $\alpha \in {\mathbb F}_{q^2}\backslash {\mathbb F}_q$, the number of ${\mathbb F}_{q^2}$-rational places in $F_n$ lying above $P_\alpha$ is zero if char$(\mathcal{F})=3$; and $(q-1)\#\{\alpha \in {\mathbb F}_{q^2}\backslash {\mathbb F}_q: (1+\alpha)^{q-1}+(1+\alpha)^{1-q}=1\}$ if char$(\mathcal{F})\not=3$.
As we have determined all ${\mathbb F}_{q}$-rational places and ${\mathbb F}_{q^2}$-rational places in $F_n$, we are now able to determine the value of $\nu(\mathcal{F})$. If char$(\mathcal{F})\not=2$ and the constant field is ${\mathbb F}_q$, then $\nu(\mathcal{F})=0$, and $\nu(\mathcal{F})=1$ if the constant field is ${\mathbb F}_{q^2}$.
If char$(\mathcal{F})=2$, and the constant field is ${\mathbb F}_q$ ($q>2$), then $\nu(\mathcal{F})=1$.
\begin{remark}
One can check that the function field tower recursively defined by Eq. (2) is isomorphic in some extension field of ${\mathbb F}_{q^2}$, to a tower recursively defined by $y^{q-1}=1-(x+\alpha)^{q-1}$, where $\alpha$ is any nonzero element of ${\mathbb F}_q$.
\end{remark}
From above discussion, Eq.(2) defines an asymptotically bad tower over any prime field (it does not define a tower over ${\mathbb F}_2$). Lenstra showed in [8, Theorem 2] that there does not exist a tower of function fields $\mathcal{F}=(F_0, F_1, \cdots)$ over a prime field which is recursively defined by $y^m=f(x)$, where $f(x)$ is a polynomial $f(x)$, $m$ and $q$ are coprime, such that the infinity place of $F_0$ splits completely in the tower, and the set $V(\mathcal{F})=\{P\in {\mathcal P}(F_0)|P \,\,\mbox{is ramified in}\,\,F_n/F_0\,\, \mbox{for some}\,\, n\geq 1\}$ is finite. A tower recursively defined by Eq. (2) falls in this form with a finite set $V(\mathcal{F})$, but no place of $F_0$ splits completely in the tower. Thus arises a problem: can one find an asymptotically good, recursive tower of the above form, over a prime field, with a finite set $V(\mathcal{F})$ and a finite place splitting completely in the tower?
\vskip .2in
\noindent
Siman Yang\\
Department of Mathematics, East China Normal University,\\
500, Dongchuan Rd., Shanghai, P.R.China 200241. \ \ e-mail: [email protected]
\\ \\
\end{document}
|
\begin{document}
\title[On the smallest trees with the same restricted $U$-polynomial]{On the smallest trees with the same restricted $U$-polynomial and the rooted $U$-polynomial}
\author{Jos\'e Aliste-Prieto \and Anna de Mier \and Jos\'e Zamora}
\address{Jos\'e Aliste-Prieto. Departamento de Matematicas, Universidad Andres Bello, Republica 498, Santiago, Chile}
\email{[email protected]}
\address{Jos\'e Zamora. Departamento de Matem\'aticas, Universidad Andres Bello, Republica 498, Santiago, Chile
}
\email{[email protected]}
\address{Anna de Mier. Departament de Matem\`atiques, Universitat Polit\`ecnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain
}
\email{[email protected]}
\maketitle
\begin{abstract}
In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this construction is a generalization of the $U$-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed from its rooted $U$-polynomial.
\end{abstract}
\tikzstyle{every node}=[circle, draw, fill=black,
inner sep=0pt, minimum width=4pt,font=\small]
\section{Introduction}\label{sec:intro}
The chromatic symmetric function \cite{stanley95symmetric} and the $U$-polynomial \cite{noble99weighted} are powerful graph invariants as they generalize many other invariants like, for instance, the chromatic polynomial, the matching polynomial and the Tutte polynomial. It is well known that the chromatic symmetric function and the $U$-polynomial are equivalent when restricted to trees, and there are examples of non-isomorphic graphs with cycles having the same $U$-polynomial (see \cite{brylawski1981intersection} for examples of graphs with the same polychromate and \cite{sarmiento2000polychromate,merino2009equivalence} for the equivalence between the polychromate and the $U$-polynomial) and also the same is true for the chromatic symmetric function (see \cite{stanley95symmetric}) . However, it is an open question to know whether there exist non-isomorphic trees with the same chromatic symmetric function (or, equivalently, the same $U$-polynomial). The negative answer to the latter question, that is, the assertion that two trees that have the same chromatic symmetric function must be isomorphic, is sometimes referred to in the literature as \emph{Stanley's (tree isomorphism) conjecture}. This conjecture has been so far verified for trees up to 29 vertices~\cite{heil2018algorithm} and also for some classes of trees, most notably caterpillars \cite{aliste2014proper,loebl2019isomorphism} and spiders \cite{martin2008distinguishing}.
A natural simplification for Stanley's conjecture is to define a truncation of the $U$-polynomial, and then search for non-isomorphic trees with the same truncation. A study of these examples could help to better understand the picture for solving Stanley's conjecture.
To be more precise, suppose that $T$ is a tree with $n$ vertices. Recall that a parition $\lambda$ of $n$ is a sequence $\lambda_1,\lambda_2,\ldots,\lambda_l$ where $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_l$.Recall that $U(T)$ can be expanded as
\begin{equation}
\label{eq:Uintro}
U(T)=\sum_{\lambda} c_\lambda \mathbf x_\lambda,
\end{equation}
where the sum is over all partitions $\lambda$ of $n$, $\mathbf x_\lambda=x_{\lambda_1}x_{\lambda_2}\cdots x_{\lambda_l}$ and the $c_\lambda$ are non-negative integer coefficients (for details of this expansion see Section 2).
In a previous work~\cite{Aliste2017PTE}, the authors studied the $U_k$-polynomial defined by restricting {the sum in \eqref{eq:Uintro} to the partitions of length smaller or equal than $k+1$}, and then showed the existence of non-isomorphic trees with the same $U_k$-polynomial for every $k$. This result is based on a remarkable connection between the $U$-polynomial of a special class of trees and the Prouhet-Tarry-Escott problem in number theory. Although the Prouhet-Tarry-Escott problem is known to have solutions for every $k$, in general it is difficult to find explicit solutions, specially if $k$ is large. Hence, it was difficult to use this result to find explicit examples of trees with the same $U_k$-polynomial.
The main result of this paper is to give an explicit and simple construction of non-isomorphic trees with the same $U_k$-polynomial for every $k$. It turns out that for $k=2,3,4$ our examples coincide with the minimal examples already found by Smith, Smith and Tian~\cite{smith2015symmetric}. This leads us to conjecture that for every $k$ our construction yields the smallest non-isomorphic trees with the same $U_k$-polynomial. We also observe that if this conjecture is true, then Stanley's conjecture is also true.
To prove our main result, we first introduce and study a generalization of the $U$-polynomial to rooted graphs, which we call the rooted $U$-polynomial or $U^r$-polynomial. As it is the case for several invariants of rooted graphs, the rooted $U$-polynomial distinguishes rooted trees up to isomorphism. Under the correct interpretation, it can also be seen as a generalization of the pointed chromatic symmetric function introduced in \cite{pawlowski2018chromatic} (See Remark \ref{pawlowski}). The key fact for us is that the rooted $U$-polynomial exhibits simple product formulas when applied to some joinings of rooted graphs. These formulas together with some non-commutativity is what allows our constructions to work.
Very recently, another natural truncation for the $U$-polynomial was considered in \cite{heil2018algorithm}. Here, they restrict the range of the sum in \eqref{eq:Uintro} to partitions whose parts are smaller or equal than $k$. They also verified that trees up to $29$ vertices are distinguished by the truncation with $k=3$ and proposed the conjecture that actually $k=3$ suffices to distinguish all trees.
This paper is organized as follows. In Section \ref{sec:rooted}, we introduce the rooted $U$-polynomial and prove our main product formulas. In Section \ref{sec:dist}, we show that the rooted $U$-polynomial distinguishes rooted trees up to isomorphism. In Section \ref{sec:main}, we recall the definition of the $U_k$-polynomial and prove our main result.
\section{The rooted $U$-polynomial}\label{sec:rooted}
We give the definition of the $U$-polynomial first introduced by
Noble and Welsh~\cite{noble99weighted}.
We consider graphs where we allow loops and parallel edges.
Let $G = (V, E)$ be a graph. Given $A\subseteq E$, the restriction $G|_A$ of $G$ to $A$ is the subgraph of $G$ obtained from $G$ after deleting every edge that is not contained in $A$ (but keeping all the vertices). The \emph{rank} of $A$ is defined as $r(A) = |V| - k(G|_A)$, where $k(G|_A)$ is the number of connected components of $G|_A$.
The \emph{partition induced by $A$}, denoted by $\lambda(A)$, is the partition of $|V|$ whose parts are the sizes of the connected components of $G|_A$.
Let $y$ be an indeterminate and $\mathbf{x} = x_1,x_2,\ldots$ be an infinite set of commuting indeterminates that commute with $y$. Given an integer partition $\lambda=\lambda_1,\lambda_2,\cdots,\lambda_l$, define $\mathbf{x}_\lambda:=x_{\lambda_1}\cdots x_{\lambda_l}$. The \emph{$U$-polynomial} of a graph $G$ is defined as
\begin{equation}
\label{def:W_poly}
U(G;\mathbf x, y)=\sum_{A\subseteq E}\mathbf x_{\lambda(A)}(y-1)^{|A|-r(A)}.
\end{equation}
Now we recall the definition of the $W$-polynomial for weighted graphs, from which the $U$-polynomial is a specialization. A \emph{weighted graph} is a pair $(G,\omega)$ where $G$ is a graph and $\omega:V(G)\rightarrow {\mathbb P}$ is a function. We say that $\omega(v)$ is the weight of the vertex $v$.
Given a weighted graph $(G,\omega)$ and an edge $e$,
the graph $(G-e,\omega)$ is defined by deleting the edge $e$ and leaving $\omega$ unchanged. If $e$ is not a loop, then the graph $(G/e,\omega)$ is defined by first deleting $e$ then by identifying the vertices $u$ and $u'$ incident to $e$ into a new vertex $v$. We set $\omega(v)=\omega(u)+\omega(u')$ and leave all other weights unchanged.
The $W$-polynomial of a weighted graph $(G,\omega)$ is defined by the following properties:
\begin{enumerate}
\item If $e$ is not a loop, then $W(G,\omega)$ satisfies
\[W(G,\omega) = W(G-e,\omega) + W(G/e,\omega);\]
\item If $e$ is a loop, then
\[W(G,\omega) = y W(G-e,\omega);\]
\item If $G$ consists only of isolated vertices $v_1,\ldots,v_n$ with weights $\omega_1,\ldots,\omega_n$, then
\[W(G,\omega) = x_{\omega_1}\cdots x_{\omega_n}.\]
\end{enumerate}
In \cite{noble99weighted}, it is proven that the $W$-polynomial is well-defined and that $U(G)=W(G,1_G)$ where $1_G$ is the weight function assigning weight $1$ to all vertices of $G$. The deletion-contraction formula is very powerful, but in this paper we will only use it in the beginning of the proof of Theorem \ref{theo:YZ} in Section \ref{sec:main}.
A \emph{rooted graph} is a pair $(G,v_0)$, where $G$ is a graph and $v_0$ is a vertex of $G$ that we call the \emph{root} of $G$.
Given $A\subseteq E$, define $\lambda_r(A)$ to be the size of the component of $G|_A$ that contains the root $v_0$, and $\lambda_-(A)$ to be
the partition induced by the sizes of all the other components. The \emph{rooted $U$-polynomial} is
\begin{equation}
\label{def:U_poly_rooted}
U^r({G,v_0};\mathbf x, y, z)=\sum_{A\subseteq E}\mathbf x_{\lambda_{-}(A)}z^{\lambda_{r}(A)}(y-1)^{|A|-r(A)},
\end{equation}
where $z$ is a new indeterminate that commutes with $y$ and $x_1,x_2,\ldots$.
We often write $G$ instead of $(G,v_0)$ when $v_0$ is clear from the context, and so we will write
$U^r(G)$ instead of $U^r(G,v_0)$. Also, if $(G,v_0)$ is a rooted graph, we will write $U(G)$ for the $U$-polynomial of $G$ (seen as an unrooted graph). If we compare $U^r(G)$ with $U(G)$, then we see that for each term of the form $\mathbf{x}_\lambda y^n z^m$ appearing in $U^r(G)$ there is a corresponding term of the form $\mathbf{x}_\lambda y^n x_m$ in $U(G)$. This motivates the following notation and lemma, whose proof follows directly from the latter observation.
\begin{notation}
If $P(\mathbf{x}, y, z)$ is a polynomial, then $(P(\mathbf{x},y,z))^*$ is the polynomial obtained by expanding $P$ as a polynomial in $z$ (with coefficients that are polynomials in $\mathbf{x}$ and $y$) and then substituting $z^n\mapsto x_n$ for every $n\in{\mathbb N}$. For instance,
if $P(\mathbf{x},y,z) = x_1yz-x_2x_3z^3$, then $P(\mathbf{x},y,z)^* = x_1^2y-x_2x_3^2$. Note that in general
$(P(\mathbf{x},y,z)Q(\mathbf{x},y,z))^* \neq P(\mathbf{x},y,z)^*Q(\mathbf{x},y,z)^*$.
\end{notation}
\begin{lemma}
For every graph $G$ we have
\begin{equation}
(U^r(G))^* = U(G).
\end{equation}
\end{lemma}
\begin{remark}
We could also define a rooted version of the $W$-polynomial, but we will not need this degree of generality for the purposes of this article.
\end{remark}
\subsection{Joining of rooted graphs and product formulas}
In this section we show two product formulas for the rooted $U$-polynomial. These will play a central role in the proofs of the results in the following sections.
Let $(G,v)$ and $(H,v')$ be two rooted graphs. Define
$G\odot H$ to be the rooted graph obtained after first taking the disjoint union of $G$ and $H$ and then by identifying $v$ and $v'$. We refer to $G\odot H$ as \emph{the joining} of $G$ and $H$. Note that from the definition it is clear that $G\odot H = H\odot G$.
We also define $G\cdot H$ to be the rooted graph obtained after first taking the disjoint union of $G$ and $H$, then adding an edge between $v$ and $v'$ and finally declaring $v$ as the root of the resulting graph. Since we made a choice for the root, in general $G\cdot H$ and $H\cdot G$ are isomorphic as unrooted graphs, but not as rooted graphs.
\begin{figure}
\caption{Example of two rooted graphs $G$ and $H$ and their different products $G\cdot H$ and $G\odot H$.}
\end{figure}
\begin{lemma}
\label{lemma:joining}
Let $G$ and $H$ be two rooted graphs. We have
\begin{equation}
\label{eq:pseudo}
U^r(G\odot H) = \frac{1}{z}U^r(G)U^r(H).
\end{equation}
\end{lemma}
\begin{proof}
By substituting the definition of $U^r$ to $G$ and $H$ in the r.h.s. of \eqref{eq:pseudo}
\begin{equation}
\label{W_poly_rooted}
\sum_{A_G\subseteq E(G)}\sum_{A_H\subseteq E(H)}\mathbf x_{\lambda_{-}(A_G)\cup\lambda_{-}(A_H)}z^{\lambda_{r}(A_G)+\lambda_{r}(A_H)-1}(y-1)^{|A_G|+|A_H|-r(A_G)-r(A_H)}.
\end{equation}
Given $A_G\subseteq E(G)$ and $A_H\subseteq E(H)$, set $A = A_G\cup A_H$. By the definition of the joining, there is a set $A'\subseteq E(G\odot H)$ corresponding to $A$ such that $\lambda_{-}(A') = \lambda_{-}(A_G)\cup\lambda_{-}(A_H)$ and $\lambda_r(A) = \lambda_r(A_G)+\lambda_r(A_H)-1$. From these equations, one checks that $r(A)=r(A_G)+r(A_H)$. Plugging these relations into \eqref{W_poly_rooted} and then rearranging the sum yields $U^r(G\odot H)$ and the conclusion now follows.
\end{proof}
\begin{lemma}
Let $G$ and $H$ be two rooted graphs. Then we have
\begin{equation}
\label{eq:sep_concat}
U^r(G\cdot H) = U^r({G})(U^r(H) + U(H)).
\end{equation}
\end{lemma}
\begin{proof}
By definition, $E(G\cdot H) = E(G)\cup E(H)\cup\{e\}$, where $e$ is the edge joining the roots of $G$ and $H$. Thus, given $A\subseteq E(G\cdot H)$, we can write it as $A = A_G\cup A_H\cup F$ where $A_G\subseteq E(G)$, $A_H\subseteq E(H)$ and $F$
is either empty or $\{e\}$. Let $\delta_F$ equal to one if $F=\{e\}$ and zero otherwise. The following relations are easy to check:
\begin{eqnarray*}
\lambda_{-}(A)&=&\begin{cases}
\lambda_{-}(A_G)\cup \lambda(A_H), &\text{if $F=\emptyset$,}\\
\lambda_{-}(A_G)\cup \lambda_{-}(A_H),&\text{otherwise};
\end{cases}\\
\lambda_r(A)&=& \lambda_r(A_G)+\lambda_r(A_H)\delta_{F};\\
r(A)&=& r(A_G)+r(A_H)+\delta_{F};\\
|A|&=&|A_G|+|A_H|+\delta_F.
\end{eqnarray*}
Now replacing the expansions of $U^r(G)$, $U^r(H)$ and $U(H)$ into the r.h.s. of
\eqref{eq:sep_concat} yields
\begin{multline}
\sum_{A_G\subseteq E(G),A_H\subseteq E(H)} \mathbf x_{\lambda_{-}(A_G)\cup\lambda_{-}(A_H)} z^{\lambda_{r}(A_G)+\lambda_{r}(A_H)}(y-1)^{|A_G|-r(A_G)+|A_H|-r(A_H)}
\\+
\sum_{A_G\subseteq E(G),A_H\subseteq E(H)} \mathbf x_{\lambda_{-}(A_G)\cup\lambda(A_H)}z^{\lambda_{r}(A_G)}(y-1)^{|A_G|-r(A_G)+|A_H|-r(A_H)}
\end{multline}
Using the previous relations we can simplify the last equation to
\begin{multline}
\sum_{A = A_G\cup A_H\cup\{e\}} \mathbf x_{\lambda_{-}(A)} z^{\lambda_{r}(A)}(y-1)^{|A|-r(A)}
\\+
\sum_{A = A_G\cup A_H} \mathbf x_{\lambda_{-}(A)}z^{\lambda_{r}(A)}(y-1)^{|A|-r(A)},
\end{multline}
where in both sums $A_G$ ranges over all subsets of $E(G)$ and $A_H$ ranges over all subsets of $E(H)$. Finally, we can combine the sums to get $U^r(G\cdot H)$, which finishes the proof.
\end{proof}
\begin{remark}
\label{pawlowski}
It is well-known (see \cite{stanley95symmetric,noble99weighted}) that the chromatic symmetric function of a graph can be recovered from the $U$-polynomial by
\[X(G) = (-1)^{|V(G)|}U(G;x_i=-p_i,y=0).\]
In \cite{pawlowski2018chromatic}, Pawlowski introduced the rooted chromatic symmetric function. It is not difficult to check that
\[X^r(G,v_0) = (-1)^{|V(G)|}\frac{1}{z}U^r(G,v_0;x_i=-p_i,y=0).\]
By performing this substitution on \eqref{eq:pseudo} we obtain Proposition 3.4 in \cite{pawlowski2018chromatic}.
\end{remark}
\section{The rooted $U$-polynomial distinguishes rooted trees}
\label{sec:dist}
In this section we will show that the rooted $U$-polynomial distinguishes rooted trees up to isomorphism. Similar results for other invariants of rooted trees appear in \cite{bollobas2000polychromatic,gordon1989greedoid,hasebe2017order}. The proof given here follows closely the one in \cite{gordon1989greedoid} but one can also adapt the proof of \cite{bollobas2000polychromatic}. Before stating the result we need the two following lemmas.
\begin{lemma}
\label{lem:degree}
Let $(T,v)$ be a rooted tree. Then, the number of vertices of $T$ and the degree of $v$ can be recognized from $U^r(T)$.
\end{lemma}
\begin{proof}
It is easy to see that {$U^r(T)=z^{n}+q(z)$} where $q(z)$ is a polynomial in $z$ of degree less than $n$ with coefficients in ${\mathbb Z}[y,\mathbf x]$ and $n$ is the number of vertices of $T$. Hence, to recognize the number of vertices of $T$, it suffices to take the term of the form $z^j$ with the largest exponent in $U^r(T)$ and this exponent is the number of vertices. To recognize the degree of $v$, observe that a term of $U^r(T)$ has the form $zx_\lambda$ for some $\lambda$ corresponding to $A$ if and only the edges of $A$ are not incident with $v$. In particular, the term of this form with smaller degree correspond to $A=E\setminus I(v)$ where $I(v)$ denotes the set of edges that are incident with $v$ and in fact the term is $zx_{n_1}x_{n_2}\ldots x_{n_d}$ where $n_1,n_2,\ldots,n_d$ are the number of vertices in each connected component of $T-v$. Since each connected component is connected to $v$ by an edge, this means that the degree of $v$ is equal to $d$ and hence it is the degree of this term minus one.
\end{proof}
\begin{lemma}
\label{lem:irreducible}
Let $(T,v)$ be a rooted tree. Then,
$\frac{1}{z}U^r(T,v)$ is irreducible if and only if the degree of $v$ is one.
\end{lemma}
\begin{proof}
Let $n$ denote the number of vertices of $T$. Suppose that the degree of $v$ is one. We will show that $\frac{1}{z}U^r(T,v)$
is irreducible. Denote by $e$ the only edge of $T$ that is incident with $v$. It is easy to check that $\lambda_r(A)\geq 1$ for all $A\subseteq E$ and that, if $A=E-e$, then $\lambda(A)=(n-1,1)$. Consequently,
\[\frac{1}{z}U^r(T,v) = x_{n-1}+
\sum_{A\subseteq E, A\neq E-e}\mathbf x_{\lambda_{-}(A)}z^{\lambda_{r}(A)-1} \]
where the second sum is a polynomial in ${\mathbb Z}[z,x_1,x_2,\ldots,x_{n-2}]$. This implies that $\frac{1}{z}U^r(T,v)$ is a monic polynomial in $x_{n-1}$ of degree one, and hence it is irreducible.
To see the converse, it suffices to observe that if the degree of $v$ is equal to $l>1$ then there are $T_1,T_2,\ldots, T_l$ rooted trees having a root of degree one and
$(T,v) = T_1\odot T_2\odot\cdots\odot T_l$. This implies that
\[\frac{1}{z}U^r(T) = \frac{1}{z}U^r(T_1)\frac{1}{z}U^r(T_2)\ldots\frac{1}{z}U^r(T_l)\]
and hence $\frac{1}{z}U^r(T)$ is not irreducible.
\end{proof}
We say that a rooted tree $(T,v)$ can be reconstructed from its $U^r$-polynomial if we can determine $(T,v)$ up to rooted isomorphism from $U^r(T,v)$. We show the following result.
\begin{theorem}
\label{teo8}
All rooted trees can be reconstructed from its $U^r$-polynomial.
\end{theorem}
\begin{proof}
By Lemma \ref{lem:degree} we can recognize the number of vertices of a rooted tree from its $U^r$-polynomial. Thus, we proceed by induction on the number of vertices. For the base case, there is only one tree with $1$ vertex, hence the assertion is trivially true. Now suppose that all rooted trees with $k-1$ vertices can be reconstructed from their $U^r$-polynomial and let $U^r(T,v)$ be the $U^r$-polynomial of some unknown tree $(T,v)$ with $k$ vertices. Again by Lemma \ref{lem:degree} we can determine the degree $d$ of $v$ from $U^r(T)$. We distinguish two cases:
\begin{itemize}
\item $d=1$: In this case, let $T'=T-v$ rooted the unique vertex of $T$ that is incident to $v$. This means that $T = 1\cdot T'$ where $1$ is the rooted tree with only one vertex. From \eqref{eq:pseudo} it follows
that \[U^r(T) = z(U^r(T') + U(T'))=(\frac{1}{z}U^r(T'))z^2+U(T')z.\]
Since the variable $z$ does not appear in $U(T')$, we can determine $U^r(T')$
from $U^r(T)$ by collecting all the terms in the expansion of $U^r(T)$ that are divisible by $z^2$ and then dividing them by $z$. Since $T'$ has $k-1$ vertices, by the induction hypothesis, we can reconstruct $T'$ and hence the equality $T=1\cdot T'$ allows us to reconstruct $T$.
\item $d>1$: In this case, we know that $\frac{1}{z}U^r(T)$ is not irreducible by Lemma \ref{lem:irreducible} and hence it decomposes as
\[\frac{1}{z}U^r(T) = P_1P_2\cdots P_d,\]
where the $P_i$ are the irreducible factors in ${\mathbb Z}[z,x_1,\ldots]$. On the other hand, as in the proof of Lemma \ref{lem:irreducible}, $T$ can be decomposed into $d$ branches $T_1,T_2,\ldots, T_d$, which are
rooted trees with the root having degree one, $T = T_1\cdot T_2\cdot T_d$ and
\[\frac{1}{z}U^r(T) =\frac{1}{z}U^r(T_1)\frac{1}{z}U^r(T_2)\cdots \frac{1}{z}U^r(T_d).\]
Since ${\mathbb Z}[z,x_1,\ldots]$ is a unique factorization domain, up to reordering factors, we have $U^r(T_i) = zP_i$ for all $i\in\{1,\ldots,d\}$. Since $d>1$ and by the definition of the $T_i$'s they have at least one edge (and hence two vertices), it follows that each $T_i$ has at most $k-1$ vertices. Since we know each of their $U^r$-polynomials, by the hypothesis induction, we can reconstruct each of them, and so we can reconstruct $T$.
\end{itemize}
\end{proof}
\begin{corollary}
The $U^r$-polynomial distinguishes trees up to rooted isomorphism.
\end{corollary}
\begin{figure}
\caption{The reconstructed tree from Example \ref{example}
\end{figure}
\begin{example}
\label{example}
Suppose $U^r(T,v)=x_{1}^{5} z + 3 \, x_{1}^{4} z^{2} + 4 \, x_{1}^{3} z^{3} + 4 \, x_{1}^{2} z^{4} + 3 \, x_{1} z^{5} + z^{6} + 2 \, x_{1}^{3} x_{2} z + 5 \, x_{1}^{2} x_{2} z^{2} + 4 \, x_{1} x_{2} z^{3} + x_{2} z^{4} + x_{1}^{2} x_{3} z + 2 \, x_{1} x_{3} z^{2} + x_{3} z^{3}$. From the term $z^6$, we know that $T$ has $6$ vertices. The terms of the form $z\mathbf{x}_\lambda$ are $x_1^5z+2x_1^3x_2z+x_1^2x_3z$. Thus, the degree of $v$ is $3$. Moreover, if we factorize $\frac{1}{z}U^r(T,v)$ into irreducible factors we obtain
\[\frac{1}{z}U^r(T,v)={\left(x_{1}^{3} + x_{1}^{2} z + x_{1} z^{2} + z^{3} + 2 \, x_{1} x_{2} + x_{2} z + x_{3}\right)} {\left(x_{1} + z\right)}{\left(x_{1} + z\right)}.\]
This means that
\begin{eqnarray}
U_r(T_1,v_1)&=& x_{1}^{3}z + x_{1}^{2} z^2 + x_{1} z^{3} + z^{4} + 2 \, x_{1} x_{2} z + x_{2} z^2 + x_{3}z,\\
U_r(T_2,v_2)&=&x_{1}z + z^2,\\
U_r(T_3,v_3)&=&{x_{1}z + z^2}.
\end{eqnarray}
From the terms $z^4$ and $x_3z$ in $U^r(T_1)$ it is easy to see that $T_1$ has $4$ vertices and $v_1$ has degree 1. Hence, $T_1=1\cdot T_1'$,
where
\[U^r(T_1') = \frac{1}{z}\left(x_{1}^{2} z^2 + x_{2} z^2 + x_{1} z^{3} + z^{4}\right) = x_{1}^{2} z + x_{2} z + x_{1} z^{2} + z^{3}. \]
Similarly $T_1'=1\cdot T_1''$, where
\[U^r(T_1'') = \frac{1}{z}\left( x_{1} z^{2} + z^{3}\right)=x_{1} z + z^{2}.\]
From this, it is not difficult to see that $T_2,T_3$ and $T_1''$ are rooted isomorphic to $1\cdot 1$. Finally, we have
\[T = (1\cdot (1\cdot (1\cdot 1)))\odot (1\cdot 1)\odot (1\cdot 1).\]
\end{example}
\section{The restricted $U$-polynomial}
\label{sec:main}
Let $T$ be a tree with $n$ vertices.
It is well known that in this case $r(A)=|A|$ for every $A\subseteq E(T)$. Hence, $U(T)$ and $U^r(T)$ (if $T$ is rooted) do not depend on $y$. Given an integer $k$, the $U_k$-polynomial of $T$ is defined by
\begin{equation}
U_k(T;\mathbf x)=\sum_{A\subseteq E,|A|\geq n-k}\mathbf x_{\lambda(A)}.
\end{equation}
Observe that since $T$ is a tree, every term in $U_k(T)$ has degree at most $k+1$ and that restricting the terms in the expansion of $U(T)$ to those of degree at most $k+1$ yields $U_k(T)$.
As noted in the introduction, it is proved in \cite{Aliste2017PTE} that for every integer $k$ there are non-isomorphic trees $T$ and $T'$ that have the same $U_k$-polynomial but distinct $U_{k+1}$-polynomial. However, the trees found in \cite{Aliste2017PTE} are not explicit. In this section, with the help of the tools developed in previous sections, we will explicitly construct such trees.
We start by defining two sequences of rooted trees. Let us denote the path on three vertices, rooted at the central vertex, by $A_0$ and the path on three vertices, rooted at one of the leaves, by $B_0$. The trees $A_k$ and $B_k$ for $k\in{\mathbb N}$ are defined inductively as follows:
\begin{equation}
\label{AK}
A_k := A_{k-1}\cdot B_{k-1}\quad\text{and}\quad
B_k := B_{k-1}\cdot A_{k-1}.
\end{equation}
\begin{figure}
\caption{The rooted trees $A_2$ and $B_2$}
\end{figure}
We first observe that $A_0$ and $B_0$ are isomorphic as unrooted trees but not isomorphic as rooted trees, which means that they have different $U^r$. In fact, a direct calculation shows that
\[\Delta_0:=U^r(A_0)-U^r(B_0) = x_1z^2-x_2z.\]
By applying Lemma \ref{lemma:joining} we deduce:
\begin{proposition}
\label{prop}
For all $k\in{\mathbb N}$, the trees $A_k$ and $B_k$ are isomorphic but not rooted-isomorphic. Moreover, we have
\begin{equation}
\label{DeltaK}
U^r(A_k) - U^r(B_k) = \Delta_0P_k,
\end{equation}
where $P_k := U(A_0)U(A_1)\cdots U(A_{k-1})$.
\end{proposition}
\begin{proof}
The proof is done by induction. The basis step is clear from the definition of $\Delta_0$. For the induction step,
we assume that for a given $k$, the graphs $A_{k-1}$ and $B_{k-1}$ are isomorphic and that $U^r(A_{k-1}) - U^r(B_{k-1}) = \Delta_0P_{k-1}$.
From \eqref{AK}, it is easy to see that $A_k$ and $B_k$ are isomorphic as unrooted trees. Also, combining \eqref{AK} with \eqref{eq:sep_concat} we get
\[U^r(A_k) = U^r(A_{k-1})(U^r(B_{k-1})+U(B_{k-1})).\]
Similarly for $B_k$ we get
\[U^r(B_k) = U^r(B_{k-1})(U^r(A_{k-1})+U(A_{k-1})).\]
Subtracting these two equations, using that $U(A_{k-1})=U(B_{k-1})$ and plugging the induction hypothesis yields
\[U^r(A_k) - U^r(B_k) = U(A_{k-1})\big(U^r(A_{k-1})-U^r(B_{k-1})\big) = U(A_{k-1})P_{k-1}\Delta_0 = P_{k}\Delta_0.\]
Hence, by induction, \eqref{DeltaK} holds for every $k$. To finish the proof, notice that since $A_k$ and $B_k$ have distinct $U^r$, they are not rooted-isomorphic by Theorem~\ref{teo8}.
\end{proof}
Observe that all the terms of $P_k$ have degree at least $k$. Now we can state our main result.
\begin{theorem}\label{theo:YZ}
Given $k,l\in{\mathbb N}$, let
\begin{equation}
Y_{k,l}={(A_k\odot A_l)\cdot (B_k\odot B_l)}\quad \text{and}\quad
Z_{k,l} = (A_l \odot B_k)\cdot (B_l\odot A_k).
\end{equation}
Then the graphs $Y_{k,l}$ and $Z_{k,l}$ (seen as unrooted trees) are not isomorphic, have the same $U_{k+l+2}$-polynomial and distinct $U_{k+l+3}$-polynomial.
\end{theorem}
Before giving the proof, we need the following lemma, which is
a corollary of Lemma \ref{lemma:joining} and Proposition \ref{prop}.
\begin{lemma}\label{l:D}
Let $T$ be a rooted tree and $i$ an integer. Then
\begin{equation}
\label{eq:lD}
U(A_i\odot T) - U(B_i\odot T) = P_i\mathcal{D}(T),
\end{equation}
where \begin{equation}
\label{eq:DT}
\mathcal{D}(T) = x_1(z U^r(T))^* - x_2 U(T).
\end{equation} In particular all the terms in $\mathcal{D}(T)$ have degree at least $2$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma:joining}, we have
\[
U^r(A_i \odot T)- U^r(B_i \odot T) = z^{-1}U^r(T) \big(U^r(A_i)-U^r(B_i)\big).
\]
Applying Proposition \ref{prop} to the last term yields
\[U^r(A_i \odot T)- U^r(B_i \odot T) = P_iU^r(T)\frac{\Delta_0}{z}.
\]
The conclusion now follows by taking the specialization $z^n\rightarrow x_n$ in the last equation to obtain (note that $P_i$ does not depend on $z$)
\[U(A_i\odot T) - U(B_i\odot T) = P_i \left[U^r(T)(x_1z-x_2)\right]^* = P_i\mathcal{D}(T).\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{theo:YZ}]
We start by applying the deletion-contraction formula to the edges corresponding to the $\cdot$ operation in the definitions of $Y_{k,l}$ and $Z_{k,l}$; it is easy to see that
\begin{equation} \label{eq:first_difference}
U(Y_{k,l}) - U(Z_{k,l}) = U(A_k \odot A_l) U(B_k\odot B_l) - U(A_l\odot B_k)U(B_l\odot A_k),
\end{equation}
since after contracting the respective edges we get isomorphic weighted trees.
We apply Lemma~\ref{l:D} twice, to $T = A_k$ and $i=l$ first, and then to $T=B_k$ and $i=l$, and replace the terms $U(A_k\odot A_l)$ and $U(A_l\odot B_k)$ in~\eqref{eq:first_difference}. Recalling that $\odot$ is commutative and after some cancellations, we obtain
$$U(Y_{k,l})-U(Z_{k,l}) = P_l\Big( \mathcal{D}(A_k) U(B_k\odot B_l) -\mathcal{D}(B_k) U(B_l\odot A_k) \Big).$$
We use Lemma~\ref{l:D} once more, with $T=B_l$ and $i=k$, to arrive at
\begin{equation}
\label{eq:ykzk:final}
U(Y_{k,l})-U(Z_{k,l}) = P_l \Big(\big(\mathcal{D}(A_k)-\mathcal{D}(B_k)\big) U(B_l\odot A_k) - \mathcal{D}(A_k)\mathcal{D}(B_l)P_k\Big)
\end{equation}
Using \eqref{eq:DT} and Proposition~\ref{prop} we get
\[\mathcal{D}(A_k)-\mathcal{D}(B_k) = x_1P_k(z\Delta_0)^* = x_1(x_1x_3-x_2^2)P_k,\]
and substituting this into \eqref{eq:ykzk:final} yields
\begin{equation}
\label{eq:last}
U(Y_{k,l})-U(Z_{k,l}) = P_lP_k\Big((x_1^2x_3-x_1x_2^2)U(B_l\odot A_k) - \mathcal{D}(A_k)\mathcal{D}(B_l)\Big).
\end{equation}
This implies that all the terms that appear in the difference have degree at least $l+ k + 4$. Hence $Y_{k,l}$ and $Z_{k,l}$ have the same $U_{k+l+2}$-polynomial. To see that they have distinct $U_{k+l+3}$-polynomial, from \eqref{eq:last} we can deduce that the only terms of degree $l+k+4$ come from terms of degree $4$ in the difference
\[
\Big((x_1^2x_3-x_1x_2^2)U(B_l\odot A_k) - \mathcal{D}(A_k)\mathcal{D}(B_l)\Big).\]
An explicit computation of these terms yields
\[ (x_1^2x_3-x_1x_2^2)x_{n(l)+n(k)-1}-(x_1 x_{n(k)+1}-x_2x_{n(k)})(x_1x_{n(l)+1}-x_2x_{n(l)}),\]
where $n(k)$ is the number of vertices of $A_k$ (and also $B_k$). From this last equation, the conclusion follows.
\end{proof}
We may consider the following quantity:
\[\Phi(m) := \min\{l: \exists \text{ non-isomorphic trees $H,G$ of size $l$ s.t. $U_m(H)=U_m(G)$}\}.\]
\begin{proposition}
\label{prop:Phi}
We have
\[\Phi(m)\leq
\begin{cases}
6\cdot 2^{\frac{m}{2}}-2, &\text{if $m$ is even}\\
6\cdot 3\cdot 2^{\lfloor\frac{m}{2}\rfloor-1}-2,& \text{if $m$ is odd.}
\end{cases}\]
In particular $\Phi(m)$ is finite.
\end{proposition}
\begin{proof}
By Theorem \ref{theo:YZ}, we see that $\Phi(m)\leq|Y_{k,l}|$ for all $(k,l)$ such that $k+l+2=m$. It is easy to check that $|A_i|=|B_i|=3\cdot 2^i$ for all $i$. Thus,
\[|Y_{k,l}|= 2(|A_k|+|B_l|-1)=6(2^k+2^l)-2\quad\text{for all $(k,l)$}.\]
If $m=k+l+2$ is fixed, then we see that $|Y_{k,l}|$ is minimized
when $k=l=\frac{m}{2}-1$ if $m$ is even and otherwise is minimized when $k=\lfloor\frac{m}{2}\rfloor$ and $l=\lfloor\frac{m}{2}\rfloor-1$. Replacing the values of $k$ and $l$ yields the desired inequality.
\end{proof}
Observe that when $(k,l) \in \{ (0,0), (1,0), (1,1)\}$ (respectively), the graphs $Y_{k,l}$ and $Z_{k,l}$ are the smallest examples of non-isomorphic trees with the same $U_m$ for $m\in \{2,3,4\}$ (respectively). This fact was verified computationally in \cite{smith2015symmetric}. This leads us to make the following conjecture
\begin{conjecture}\label{c:YZ}
If $m$ is even, then $Y_{m/2-1,m/2-1}$ and $Z_{m/2-1,m/2-1}$ are the smallest non-isomorphic trees with the same $U_m$-polynomial and if $m$ is odd, then
the same is true for $Y_{\lfloor m/2\rfloor,\lfloor m/2\rfloor-1}$ and $Z_{\lfloor m/2\rfloor,\lfloor m/2\rfloor-1}$. In other words,
\[\Phi(m) =
\begin{cases}
6\cdot 2^{\frac{m}{2}}-2, &\text{if $m$ is even}\\
6\cdot 3\cdot 2^{\lfloor\frac{m}{2}\rfloor-1}-2,& \text{if $m$ is odd.}
\end{cases}\]
\end{conjecture}
The following proposition relates $\Phi$ with Stanley's conjecture.
\begin{proposition}
The following assertions are true:
\begin{enumerate}[a)]
\item For every $m$, Stanley's conjecture is true for trees with at most $\Phi(m)-1$ vertices.
\item Stanley's conjecture is true if and only if $\lim_m\Phi(m)=\infty$.
\item Conjecture \ref{c:YZ} implies Stanley's conjecture.
\end{enumerate}
\end{proposition}
\begin{proof}
To show a), observe that the existence of non-isomorphic trees $T$ and $T'$ of size smaller than $\Phi(m)$ with the same $U$-polynomial contradicts the definition of $\Phi(m)$. To see b), if $\lim_m\Phi(m)=\infty$, then by a), then clearly Stanley's conjecture is true for all (finite) trees. For the converse, suppose that $\Phi(m)$ is uniformly bounded by $N$, and let $T_m,T_m'$ be two non-isomorphic trees of size smaller or equal than $N$ with the same $U_m$-polynomial. Since there finitely many pairs of trees of size smaller or equal than $N$, it follows that there exist $T$ and $T'$ two trees such that $T=T_m$ and $T'=T'_m$ for infinitely many $m$. This implies that $U(T)=U(T')$ and this would contradict Stanley's conjecture. This finish the proof of b). Assertion c) follows directly from Conjecture \ref{c:YZ} and b).
\end{proof}
\section*{Acknowledgments}
The first and third author are partially supported by CONICYT FONDECYT Regular 1160975 and Basal PFB-03 CMM Universidad de Chile. The second author is partially supported by the Spanish
Ministerio de Economía y Competitividad project MTM2017-82166-P. A short version of this work appeared in \cite{aliste2018dmd}.
\end{document}
|
\begin{document}
\preprint{HEP/123-qed}
\title[Short Title]{Generating optimal states for a homodyne Bell test}
\author{Sonja Daffer}
\email{[email protected]}
\author{Peter L. Knight}
\affiliation{
Blackett Laboratory,
Imperial College London,
Prince Consort Road,
London SW7 2BW,
United Kingdom
}
\date{\today}
\begin{abstract}
\noindent We present a protocol that produces a
conditionally prepared state that can be used for a Bell test
based on homodyne detection. Based on the results of Munro [PRA
1999], the state is near-optimal for Bell-inequality violations
based on quadrature-phase homodyne measurements that use
correlated photon-number states. The scheme utilizes the Gaussian
entanglement distillation protocol of Eisert \textit{et.al.}
[Annals of Phys. 2004] and uses only beam splitters and
photodetection to conditionally prepare a non-Gaussian state from
a source of two-mode squeezed states with low squeezing parameter,
permitting a loophole-free test of Bell inequalities.
\\
\end{abstract}
\pacs{03.65.Ud, 42.50.Xa, 42.50.Dv, 03.65.Ta}
\maketitle
Bell's theorem is regarded by some as one of the most profound
discoveries of science in the twentieth century. Not only does it
provide a quantifiable measure of correlations stronger than any
allowed classically, which is a key resource in many quantum
information processing applications, it also addresses fundamental
questions in the foundations of quantum mechanics. In 1964, Bell
quantified Bohm's version of the Einstein, Podolsky, and Rosen
(EPR) gedanken experiment, by introducing an inequality that
provides a test of local hidden variable (LHV) models
\cite{bell1964}. A violation of Bell's inequality forces one to
conclude that, contrary to the view held by EPR, quantum mechanics
can not be both local and real. In order to experimentally support
this conclusion in a strict sense, a Bell test that is free from
loopholes is required. Although it is still quite remarkable that
such seemingly metaphysical questions can even be put to the test
in the laboratory, a loophole-free Bell test has yet to be
achieved.
For more than three decades, numerous experiments have confirmed
the predictions of the quantum theory, thereby disproving local
realistic models as providing a correct description of physical
reality\cite{aspect1982}. However, all experiments performed to
date suffer from at least one of the two primary loopholes -- the
detection loophole and the locality loophole. The detection
loophole arises due to low detector efficiencies that may not
permit an adequate sampling of the ensemble space while the
locality loophole suggests that component parts of the
experimental apparatus that are not space-like separated could
influence each other. The majority of Bell tests have used
optical systems to measure correlations, some achieving space-like
separations but still subjected to low efficiency photodetectors
(see, \textit{e.g.}, Ref. \cite{weihs1998}). Correlations in the
properties of entangled ions were shown to violate a Bell
inequality using high efficiency detectors eliminating the
detection loophole; however, the ions were not space-like
separated \cite{rowe2001}. A major challenge that has yet to be
achieved is to experimentally realize a single Bell test that
closes these loopholes.
The ease with which optical setups address the locality loophole
coupled with the currently achievable high efficiencies ($> 0.95$)
of homodyne detectors make Bell tests using quadrature-phase
measurements good candidates for a loophole-free experiment.
Furthermore, continuous quadrature amplitudes are the optical
analog of position and momentum and more closely resemble the
original state considered by EPR. Unlike photon counting
experiments which deal with the microscopic resolution of a small
number of photons, by mixing the signal with a strong field,
homodyne measurements allow one to detect a macroscopic current
\cite{reid1997}.
In this article, we propose a test of Bell inequalities using
homodyne detection on a conditional non-Gaussian ``source" state,
prepared using only passive optics and photon detection. Events
are pre-selected -- using event-ready detection one knows with
certainty that the desired source state has been produced --
requiring no post-processing. Photon detectors are only used in
the pre-selection process and only affect the probability of
successfully creating the source state whereas the actual
correlation measurements are performed using high efficiency
homodyne detectors. The source is a correlated photon-number
state that is near-optimal for Bell tests using homodyne
detection, opening the possibility of a conclusive, loophole-free
test.
We consider a two-mode quantum state of light that can be written
as
\begin{equation} \label{eq:correlated photon state}
| \Psi \rangle = \sum_{n=0}^{\infty} c_n | n,n \rangle,
\end{equation}
which is correlated in photon number $| n,n \rangle = | n
\rangle_A \otimes | n \rangle_B$ for modes $A$ and $B$. For
example, the two-mode squeezed state $| \psi_\lambda \rangle$ has
coefficients given by $c_n=\lambda^n \sqrt{1-\lambda^2} $, where
$\lambda=\tanh(s)$ is determined by the squeezing parameter $s$
\cite{knight1985}. Such states are experimentally easy to
generate; however, because they possess a Gaussian Wigner
distribution in phase space, they are unsuitable for tests of Bell
inequalities using quadrature-phase measurements
as it is a requirement that the Wigner function possesses negative
regions \cite{{bell1964}}. Alternative, theoretically predicted
two-mode quantum superposition states called circle states, also
generated from vacuum fields through nondegenerate parametric
oscillation, having coefficients given by $c_n=r^{2n}/n!
\sqrt{I_0(2r^2)}$, do exhibit a violation for quadrature-phase
measurements with a maximum violation occurring for $r=1.12$
\cite{Gilchrist1998}. Unfortunately, unlike the two-mode squeezed
states, circle states are difficult to realize experimentally. A
recently proposed solution towards an experimentally realizable
state of light that is suitable for a homodyne Bell test is the
photon-subtracted two-mode squeezed state
\cite{Nha2004,GarciaPatron2004}, having coefficients
$c_n=\sqrt{(1-\lambda^2)^3/(1+\lambda^2)}(n+1)\lambda^n$, which
utilizes non-Gaussian operations on a Gaussian state. In this
scheme, a photon is detected from each mode of a two-mode squeezed
state and only the resulting conditional state is used for
correlation measurements in the Bell test. While the two-mode
squeezed state has a positive-everywhere Wigner function, the
conditional state after photon subtraction does not.
To date, all proposed states for a Bell test using
quadrature-phase measurements are not optimal states, meaning that
they do not produce the maximum possible violation of Bell
inequalities. The scheme presented here produces a two-mode
photon-entangled state that is near-optimal, using only beam
splitters and photon detection. The beam splitter may be described
by the unitary operator \cite{wodkiewicz1985}
\begin{equation} \label{eq:BS unitary}
U_{ab}=T^{a^\dag a}e^{-R^* b^\dag a}e^{R b
a^\dag}T^{-b^\dagger b},
\end{equation}
which describes the mixing of two modes $a$ and $b$ at a beam
splitter with transmissivity $T$ and reflectivity $R$. On-off
photon detection is described by the positive operator-valued
measure (POVM) of each detector, given by
\begin{equation}
\Pi_0 = |0 \rangle \langle 0 |, \hspace{.2in} \Pi_1 = I-|0 \rangle \langle 0
|.
\end{equation}
The on-off detectors distinguish between vacuum and the presence
of any number of photons. The procedure is event-ready, a term
introduced by Bell, in the sense that one has a classical signal
indicating whether a measurable system has been produced. The
states demonstrating a violation of local realism presented here
do not rely on the production of exotic states of light; in fact,
only a parametric source generating states with a low squeezing
parameter is required, making the procedure experimentally
feasible with current technology. As depicted by Fig. 1, there
are three parties involved: Alice, Bob, and Sophie. Sophie
prepares the source states that are sent to Alice and Bob, who
perform correlation measurements. We first describe the procedure
Sophie uses to generate the source states, which is shown by the
diagram in Fig. 2, and then discuss the measurements performed by
Alice and Bob.
\begin{figure}\label{fig:f1BellSchematic8}
\end{figure}
In the first step, two-mode squeezed states are mixed pairwise at
unbalanced beam splitters followed by the non-Gaussian operation
associated with the POVM element $\Pi_1$. Specifically, a
non-Gaussian state is generated by
\begin{equation} \label{eq:BellFockStateOperation}
( \Pi_{1,c} \otimes \Pi_{1,d})
(U_{ac} \otimes U_{bd}) |\psi_\lambda \rangle
|\psi_\lambda \rangle,
\end{equation}
where $|\psi_\lambda \rangle$ denotes the two-mode squeezed state
with $c_n=\lambda^n \sqrt{1-\lambda^2}$. For sufficiently small
$\lambda$, the operator $\Pi_1$ describing the presence of photons
at the detector approaches the rank-one projection onto the single
photon number subspace $|1\rangle\langle 1|$, which is still a
non-Gaussian operation. Under this condition, (un-normalized)
states of the form
\begin{equation} \label{eq:BellFockState}
| \psi^{(0)} \rangle = | 0,0 \rangle + \xi | 1,1 \rangle
\end{equation}
can be produced. That is, even though the output state of
(\ref{eq:BellFockStateOperation}) will in general be a mixed
state, when $\lambda \in [0,1)$ is very small, the resulting
states can be made arbitrarily close in trace-norm to an entangled
state with state vector given by Eq. (\ref{eq:BellFockState}),
provided the appropriate choice of beam splitter transmittivity
$|T(\lambda)|=|\xi-\sqrt{\xi^2+8 \lambda^2}|/4 \lambda$ is used
\cite{browne2003}. It should be emphasized that the state $|
\psi^{(0)}\rangle$, having a Bell-state form, can be generated for
arbitrary $\xi$.
It is interesting to note that the state given by Eq.
(\ref{eq:BellFockState}) does not violate a Bell inequality for
quadrature-phase measurements for any $\xi$, even when it has the
form of a maximally entangled Bell state, as was shown in Ref.
\cite{munro1999}, in which a numerical study of the optimal
coefficients for Eq. (\ref{eq:correlated photon state}) was
performed. For certain values of $\xi$, Eq.
(\ref{eq:BellFockState}) describes a state that possesses a Wigner
distribution that has negative regions, showing that negativity of
the Wigner function is a necessary but not sufficient condition
for a violation of Bell inequalities using quadrature-phase
measurements.
The second step is to combine two copies of the state given by Eq.
(\ref{eq:BellFockState}) pairwise and locally at 50:50 beam
splitters described by the unitary operator of Eq. (\ref{eq:BS
unitary}). Detectors that distinguish only between the absence and
presence of photons are placed at the output port of each beam
splitter and when no photons are detected, the state is retained.
The resulting un-normalized state is
\begin{equation}
|\psi^{(i+1)} \rangle = \langle 0,0 | U_{ac} \otimes U_{bd}
| \psi^{(i)} \rangle | \psi^{(i)} \rangle = \sum_{n=0}^\infty
c_n^{(i+1)}
|n,n \rangle,
\end{equation}
where the coefficients are given by \cite{opatrny2000}
\begin{equation}
c_n^{(i+1)}=2^{-n} \sum_{r=0}^n
\left(
\begin{array}{c}
n \\
r
\end{array}
\right)
c_{r}^{(i)} c_{n-r}^{(i)}.
\end{equation}
It is optimal to iterate this procedure three times so that Sophie
prepares the state $|\psi^{(3)} \rangle$. Each iteration leads to
a Gaussification of the initial state
\cite{browne2003,eisert2004}, which builds up correlated photon
number pairs in the sum of Eq. (\ref{eq:correlated photon state}).
Further iterations would Gaussify the state too much and destroy
the nonlocal features for phase space measurements.
The final step is to reduce the vacuum contribution by subtracting
a photon from each mode of the state $|\psi^{(3)} \rangle$,
obtaining a state proportional to $ab |\psi^{(3)} \rangle$. This
is done by mixing each mode with vacuum at two beam splitters with
low reflectivity. A very low reflectivity results in single
photon counts at each detector with a high probability when a
detection event has occurred. Thus, the unitary operation
describing the action of the beam splitter is expanded to second
order in the reflectivity and the state is conditioned on the
result $N=1$ at each detector. The final photon-subtracted state,
given by
\begin{equation}
|\psi^{(3)} \rangle_{PS}= {\mathcal N} \sum_{n=0}^\infty (n+1)c^{(3)}_{n+1}
|n,n \rangle,
\end{equation}
where ${\mathcal N}$ is a normalization factor, is a near-optimal
state for homodyne detection. Figure 3 compares the previously
proposed states -- the circle state and the photon-subtracted
two-mode squeezed state -- with the near-optimal state
$|\psi^{(3)} \rangle_{PS}$, as well as the numerically optimized
state in Ref. \cite{munro1999}. The conditioning procedure alters
the photon number distribution of the input state and behaves
similarly to entanglement distillation.
\begin{figure}\label{fig:f2BellTree}
\end{figure}
Although the procedure used to create the correlated photon source
is probabilistic, with the success probability determined by the
amount of two-mode squeezing and the transmittivity of the
unbalanced beam splitters, it is event-ready -- Sophie has a
record of when the source state was successfully prepared. Low
efficiency photon detectors used in the state preparation only
affect the success probability and do not constitute a detection
loophole. Each mode of the source state $|\psi^{(3)}\rangle_{PS}$
is distributed to a separate location where correlation
measurements using high efficiency homodyne detectors are
performed by the two distant (space-like separated) parties, Alice
and Bob. Alice and Bob each mix their light modes with independent
local oscillators (LO) and randomly measure the relative phase
between the beam and the LO, taking into account the timing
constraint that ensures fair sampling. Alice measures the rotated
quadrature $x_{\theta}^A=x^A \cos \theta + p^A \sin \theta $ and
Bob measures the rotated quadrature $x_{\phi}^B=x^B \cos \phi +
p^B \sin \phi.$ Correlations are considered for two choices of
relative phase: $\theta_1$ or $\theta_2$ for Alice and $\phi_1$ or
$\phi_2$ for Bob. Finally, Alice, Bob and Sophie compare their
experimental results to determine when the source state was
successfully generated and which correlation measurements to use
for the Bell inequalities.
Two types of Bell inequalities will be examined -- the
Clauser-Horne-Shimony-Holt (CHSH) and Clauser-Horne (CH)
inequalities \cite{clauser1969}. To apply these inequalities,
which are for dichotomous variables, the measurement outcomes for
Alice and Bob are discretized by assigning the value $+1$ if $x
\geq 0$ and $-1$ if $x<0$. Let $P^{AB}_{++}(\theta,\phi)$ denote
the joint probability that Alice and Bob realize the value $+1$
upon measuring $\theta$ and $\phi$, respectively and
$P^{A}_{+}(\theta)$ denote the probability that Alice realizes the
value $+1$ regardless of Bob's outcome, with similar notation for
the remaining possible outcomes. From LHV theories, the following
joint probability distribution can be derived:
\begin{equation}
P^{AB}_{ij}(\theta,\phi)=\int \rho(\lambda) p_i^A(\theta,\lambda)
p_j^B(\phi,\lambda) d \lambda
\end{equation}
with $i,j=\pm$, by postulating the existence of hidden variables
$\lambda$ and independence of outcomes for Alice and Bob. Quantum
mechanically, the joint probability distribution is given by the
Born rule $P(x^A_\theta,x^B_\phi)=|\langle x^A_\theta,x^B_\phi
|\psi^{(3)} \rangle_{PS}|^2.$ The probability for Alice and Bob
to both obtain the value $+1$ is
$P^{AB}_{++}(\theta,\phi)=\int_0^\infty \int_0^\infty
P(x^A_\theta,x^B_\phi) d x^A_\theta d x^B_\phi$. The joint
distribution is symmetric and a function of only the sum of the
angles $\chi=\theta+\phi$ permitting the identification
$P^{AB}_{++}(\theta,\phi)=P^{AB}_{++}(\chi)=P^{AB}_{++}(-\chi)$
and $P^{AB}_{++}(\chi)=P^{AB}_{--}(\chi)$. The marginal
distributions $P_+^{A}(\theta)=P_+^{B}(\phi)=1/2$ are independent
of the angle. Given the probability distributions, the predictions
of quantum theory can be tested with those of LHV theory.
First, we consider the Bell inequality of the CHSH type, which
arises from linear combination of correlation functions having the
form
\begin{equation} \label{eq:bellcombo}
\emph{B}= E(\theta_1,\phi_1)+E(\theta_1,\phi_2)+E(\theta_2,\phi_1)-E(\theta_2,\phi_2),
\end{equation}
where $E(\theta_i,\phi_j)$ is the correlation function for Alice
measuring $\theta_i$ and Bob measuring $\phi_j$. These
correlations are in turn determined by
\begin{equation} \label{eq:correlation function}
E(\theta,\phi)=P^{AB}_{++}(\theta,\phi)+P^{AB}_{--}(\theta,\phi)
-P^{AB}_{+-}(\theta,\phi)-P^{AB}_{-+}(\theta,\phi),
\end{equation}
obtained through the many measurements that infer the
distributions $P^{AB}_{ij}(\theta,\phi)$. With the aid of the
symmetry and angle factorization properties, the CHSH inequality
takes the simple form $\emph{B}= 3 E(\chi)-E(3 \chi)$ with LHV
models demanding that $|\emph{B}| \leq 2$. The strongest
violation of the inequality is obtained for the value
$\chi=\pi/4$, thus, a good choice of relative phases for Alice and
Bob's measurements is $\theta_1=0$, $\theta_2=\pi/2,$
$\phi_1=-\pi/4$, and $\phi_2=\pi/4$. Using homodyne detection
with optimal correlated photon number states, the maximum
achievable violation is 2.076 whereas using the source states
presented here, a Bell inequality violation of $\emph{B}=2.071$ is
achievable.
Let us also consider the Clauser-Horne (strong) Bell inequality
formed by the linear combination
\begin{equation} \label{eq:chcombo}
\frac{P^{AB}_{++}(\theta_1,\phi_1)-P^{AB}_{++}(\theta_1,\phi_2)+
P^{AB}_{++}(\theta_2,\phi_1)+P^{AB}_{++}(\theta_2,\phi_2)}
{P_+^{A}(\theta_2)+P_+^{B}(\phi_1)},
\end{equation}
denoted by $\emph{S}$, for which local realism imposes the bound
$|\emph{S}| \leq 1$. Again, using the properties of the
probability distributions, the simplification $\emph{S}= 3
P^{AB}_{++}(\chi)-P^{AB}_{++}(3 \chi)$ is possible. With the
following choice of the phases: $\theta_1=0,$ $\theta_2=\pi/2$,
$\phi_1=-\pi/4$, and $\phi_2=\pi/4$, a violation of
$\emph{S}=1.018$ is attainable given the states in Eq.
(\ref{eq:BellFockState}) with parameter value $\xi=1/\sqrt{2}$,
which is quite close to the maximum value of 1.019 achieved by the
numerical, optimal states in Ref. \cite{munro1999} .
\begin{figure}\label{fig:OptimalStateCompare}
\end{figure}
We have shown how it is possible to prepare a near-optimal
state for a Bell test that uses quadrature-phase homodyne
measurements. Only very low squeezed states, passive optical
elements and photon detectors are required, making the procedure
experimentally feasible at present.
An initial state with a positive-everywhere Wigner function was
succeeded by both non-Gaussian and Gaussian operations to prepare
a state that exhibits a strong violation of both the CHSH and CH
Bell inequalities. Efforts are currently being made towards an
experimental realization of entanglement distillation for Gaussian
states. The procedure presented here offers the opportunity for
another possible experiment, as it utilizes a subset of an
entanglement distillation procedure. Of course, any observed
violation of a Bell inequality is sensitive to inefficiencies in
the experiment that tend to deplete correlations. A full analysis
involving dark counts and detection inefficiencies as addressed in
Ref. \cite{eisert2004} is necessary. Near-optimal states for
homodyne detection may allow a larger window for experimental
imperfections and offer the opportunity for a conclusive,
loophole-free Bell test.
We thank Bill Munro and Stefan Scheel for useful comments and
discussions. This work was supported by the U.S. National Science
Foundation under the program NSF01-154, by the U.K. Engineering
and Physical Sciences Research Council, and the European Union.
\end{document}
|
\begin{document}
\title{\rqmtitle}
\preprint{Version 2}
\author{Ed Seidewitz}
\email{[email protected]}
\affiliation{14000 Gulliver's Trail, Bowie MD 20720 USA}
\date{\today}
\pacs{03.65.Pm, 03.65.Fd, 03.30.+p, 11.10.Ef}
\begin{abstract}
Earlier work presented a spacetime path formalism for
relativistic quantum mechanics arising naturally from the
fundamental principles of the Born probability rule,
superposition, and spacetime translation invariance. The
resulting formalism can be seen as a foundation for a number
of previous parameterized approaches to relativistic quantum
mechanics in the literature. Because time is treated similarly
to the three space coordinates, rather than as an evolution
parameter, such approaches have proved particularly useful in
the study of quantum gravity and cosmology. The present paper
extends the foundational spacetime path formalism to include
massive, nonscalar particles of any (integer or half-integer)
spin. This is done by generalizing the principle of
translational invariance used in the scalar case to the
principle of full Poincar\'e invariance, leading to a
formulation for the nonscalar propagator in terms of a path
integral over the Poincar\'e group. Once the difficulty of the
non-compactness of the component Lorentz group is dealt with,
the subsequent development is remarkably parallel to the
scalar case. This allows the formalism to retain a clear
probabilistic interpretation throughout, with a natural
reduction to non-relativistic quantum mechanics closely
related to the well known generalized Foldy-Wouthuysen
transformation.
\end{abstract}
\maketitle
\section{Introduction} \label{sect:intro}
Reference \onlinecite{seidewitz06a} presented a foundational formalism
for relativistic quantum mechanics based on path integrals over
parametrized paths in spacetime. As discussed there, such an approach
is particularly suited for further study of quantum gravity and
cosmology, and it can be given a natural interpretation in terms of
decoherent histories \cite{seidewitz06b}. However, the formalism as
given in \refcite{seidewitz06a} is limited to scalar particles. The
present paper extends this spacetime path formalism to non-scalar
particles, although the present work is still limited to massive
particles.
There have been several approaches proposed in the literature for
extending the path integral formulation of the relativistic scalar
propagator \cite{feynman50,feynman51,teitelboim82,hartle92} to the
case of non-scalar particles, particularly spin-1/2 (see, for example,
\refcites{bordi80, henneaux82, barut84, mannheim85, forte05}). These
approaches generally proceed by including in the path integral
additional variables to represent higher spin degrees of freedom.
However, there is still a lack of a comprehensive path integral
formalism that treats all spin values in a consistent way, in the
spirit of the classic work of Weinburg \cite{weinberg64a, weinberg64b,
weinberg69} for traditional quantum field theory. Further, most
earlier references assume that the path integral approach is basically
a reformulation of an \emph{a priori} traditional Hamiltonian
formulation of quantum mechanics, rather than being foundational in
its own right.
The approach to be considered here extends the approach from
\refcite{seidewitz06a} to non-scalar particles by expanding the
configuration space of a particle to be the Poincar\'{e} group (also
known as the inhomogeneous Lorentz group). That is, rather than just
considering the position of a particle, the configuration of a
particle will be taken to be both a position \emph{and} a Lorentz
transformation. Choosing various representations of the group of
Lorentz transformations then allows all spins to be handled in a
consistent way.
The idea of using a Lorentz group variable to represent spin degrees
of freedom is not new. For example, Hanson and Regge \cite{hanson74}
describe the physical configuration of a relativistic spherical top as
a Poincar\'e element whose degrees of freedom are then restricted.
Similarly, Hannibal \cite{hannibal97} proposes a full canonical
formalism for classical spinning particles using the Lorentz group for
the spin configuration space, which is then quantized to describe both
spin and isospin. Rivas \cite{rivas89, rivas94, rivas01} has made a
comprehensive study in which an elementary particle is defined as ``a
mechanical system whose kinematical space is a homogeneous space of
the Poincar\'e group''.
Rivas actually proposes quantization using path integrals, but he does
not provide an explicit derivation of the non-scalar propagator by
evaluating such an integral. A primary goal of this paper to provide
such a derivation.
Following a similar approach to \refcite{seidewitz06a}, the form of
the path integral for non-scalar particles will be deduced from the
fundamental principles of the Born probability rule, superposition,
and Poincar\'e invariance. After a brief overview in
\sect{sect:background} of some background for this approach,
\sect{sect:non-scalar:propagator} generalizes the postulates from
\refcite{seidewitz06a} to the non-scalar case, leading to a path
integral over an appropriate Lagrangian function on the Poincar\'e
group variables.
The major difficulty with evaluating this path integral is the
non-compactness of the Lorentz group. Previous work on evaluating
Lorentz group path integrals (going back to \refcite{bohm87}) is based
on the irreducible unitary representations of the group. This is
awkward, since, for a non-compact group, these representations are
continuous \cite{vilenkin68} and the results do not generalize easily
to the covering group $SL(2,\cmplx)$ that includes half-integral
spins.
Instead, we will proceed by considering a Wick rotation to Euclidean
space, which replaces the non-compact Lorentz group $SO(3,1)$ by the
compact group $SO(4)$ of rotations in four dimensions, in which it is
straightforward to evaluate the path integral. It will then be argued
that, even though the $SO(4)$ propagator cannot be assumed the same as
the true Lorentz propagator, the propagators should be the same when
restricted to the common subgroup $SO(3)$ of rotations in three
dimensions. This leads directly to considerations of the spin
representations of $SO(3)$.
Accordingly, \sect{sect:non-scalar:euclidean} develops the Euclidean
$SO(4)$ propagator and \sect{sect:non-scalar:spin} then considers the
reduction to the three-dimensional rotation group and its spin
representations. However, rather than using the usual Wigner approach
of reduction along the momentum vector \cite{wigner39}, we will reduce
along an independent time-like four-vector \cite{piron78, horwitz82}.
This allows for a very parallel development to \refcite{seidewitz06a}
for antiparticles in \sect{sect:non-scalar:antiparticles} and for a
clear probability interpretation in
\sect{sect:non-scalar:probability}.
Interactions of non-scalar particles can be included in the formalism
by a straightforward generalization of the approach given in
\refcite{seidewitz06a}. \Sect{sect:non-scalar:interactions} gives an
overview of this, though full details are not included where they are
substantially the same as the scalar case.
Natural units with $\hbar = 1 = c$ are used throughout the following
and the metric has a signature of $(- + + +)$.
\section{Background}
\label{sect:background}
Path integrals were originally introduced by Feynman \cite{feynman48,
feynman65} to represent the non-relativistic propagation kernel
$\kersym(\threex_{1} - \threex_{0}; t_{1}-t_{0})$. This kernel gives the
transition amplitude for a particle to propagate from the position
$\threex_{0}$ at time $t_{0}$ to the position $\threex_{1}$ at time
$t_{1}$. That is, if $\psi(\threex_{0}; t_{0})$ is the probability
amplitude for the particle to be at position $\threex_{0}$ at time
$t_{0}$, then the amplitude for it to propagate to another position
at a later time is
\begin{equation*}
\psi(\threex; t) = \intthree \xz\,
\kersym(\threex - \threex_{0}; t-t_{0})
\psi(\threex_{0}; t_{0}) \,.
\end{equation*}
A specific \emph{path} of a particle in space is given by a
position function $\threevec{q}(t)$ parametrized by time (or, in
coordinate form, the three functions $q^{i}(t)$ for $i = 1,2,3$).
Now consider all possible paths starting at $\threevec{q}(t_{0}) =
\threex_{0}$ and ending at $\threevec{q}(t_{1}) = \threex_{1}$. The
path integral form for the propagation kernel is then given by
integrating over all these paths as follows:
\begin{equation} \label{eqn:A0a}
\kersym(\threex_{1} - \threex_{0}; t_{1}-t_{0})
= \zeta \intDthree q\,
\delta^{3}(\threevec{q}(t_{1}) - \threex_{1})
\delta^{3}(\threevec{q}(t_{0}) - \threex_{0})
\me^{\mi S[\threevec{q}]} \,,
\end{equation}
where the phase function $S[\threevec{q}]$ is given by the classical
action
\begin{equation*}
S[\threevec{q}] \equiv
\int_{t_{0}}^{t_{1}} \dt\, L(\dot{\threevec{q}}(t)) \,,
\end{equation*}
with $L(\dot{\threevec{q}})$ being the non-relativistic Lagrangian in
terms of the three-velocity $\dot{\threevec{q}} \equiv
\dif\threevec{q} / \dt$.
In \eqn{eqn:A0a}, the notation $\Dthree q$ indicates a path integral
over the three functions $q^{i}(t)$. The Dirac delta functions
constrain the paths integrated over to start and end at the
appropriate positions. Finally, $\zeta$ is a normalization factor,
including any limiting factors required to keep the path integral
finite (which are sometimes incorporated into the integration measure
$\Dthree q$ instead).
As later noted by Feynman himself \cite{feynman51}, it is possible to
generalize the path integral approach to the relativistic case. To do
this, it is necessary to consider paths in \emph{spacetime}, rather
than just space. Such a path is given by a four dimensional position
function $q(\lambda)$, parametrized by an invariant \emph{path
parameter} $\lambda$ (or, in coordinate form, the four functions
$\qmul$, for $\mu = 0,1,2,3$).
The propagation amplitude for a free scalar particle in spacetime is
given by the Feynman propagator
\begin{equation} \label{eqn:A0b}
\prop = -\mi(2\pi)^{-4}\intfour p\,
\frac{\me^{\mi p\cdot(x-\xz)}}
{p^{2}+m^{2}-\mi\epsilon} \,.
\end{equation}
It can be shown (in addition to \refcite{feynman51}, see also, e.g.,
\refcites{seidewitz06a, teitelboim82}) that this propagator can be
expressed in path integral form as
\begin{equation} \label{eqn:A0c}
\prop = \int_{\lambdaz}^{\infty} \dif\lambda\,
\zeta \intDfour q\,
\delta^{4}(q(\lambda) - x) \delta^{4}(q(\lambdaz) - \xz)
\me^{\mi S[q]} \,,
\end{equation}
where
\begin{equation*}
S[q] \equiv
\int_{\lambdaz}^{\lambda} \dif\lambda'\, L(\qdot)(\lambda')) \,,
\end{equation*}
and $L(\qdot)$ is now the relativistic Lagrangian in terms of the the
four-velocity $\qdot \equiv \dif q / \dl$.
Notice that the form of the relativistic expression differs from the
non-relativistic one by having an additional integration over
$\lambda$. This is necessary, since the propagator must, in the end,
depend only on the change in position, independent of $\lambda$.
However, as noted in \refcite{seidewitz06a}, \eqn{eqn:A0c} can be
written as
\begin{equation} \label{eqn:A0d}
\prop = \int_{\lambdaz}^{\infty} \dif\lambda\, \kerneld \,,
\end{equation}
where the \emph{relativistic kernel}
\begin{equation} \label{eqn:A0e}
\kerneld = \zeta \intDfour q\,
\delta^{4}(q(\lambda) - x) \delta^{4}(q(\lambdaz) - \xz)
\me^{\mi S[q]}
\end{equation}
now has a form entirely parallel with the non-relativistic case. The
relativistic kernel can be considered to represent propagation over
paths of the specific length $\lambda - \lambdaz$, while \eqn{eqn:A0d}
then integrates over all possible path lengths.
Given the parallel with the non-relativistic case, define the
\emph{parametrized} probability amplitudes $\psixl$ such that
\begin{equation*}
\psixl = \intfour \xz\, \kerneld \psixlz \,.
\end{equation*}
Parametrized amplitudes were introduced by Stueckelberg
\cite{stueckelberg41, stueckelberg42}, and parametrized approaches to
relativistic quantum mechanics have been developed by a number of
subsequent authors \cite{nambu50, schwinger51, cooke68, horwitz73,
collins78, piron78, fanchi78, fanchi83, fanchi93}. The approach is
developed further in the context of spacetime paths of scalar
particles in \refcite{seidewitz06a}.
In the traditional presentation, however, it is not at all clear
\emph{why} the path integrals of \eqns{eqn:A0a} and \eqref{eqn:A0b}
should reproduce the expected results for non-relativistic and
relativistic propagation. The phase functional $S$ is simply chosen to
have the form of the classical action, such that this works. In
contrast, \refcite{seidewitz06a} makes a more fundamental argument
that the exponential form of \eqn{eqn:A0e} is a consequence of
translation invariance in Minkowski spacetime. This allows for
development of the spacetime path formalism as a foundational
approach, rather than just a re-expression of already known results.
The full invariant group of Minkowski spacetime is not the
translation group, though, but the Poincar\'e group consisting of both
translations \emph{and} Lorentz transformations. This leads one to
consider the implications of applying the argument of
\refcite{seidewitz06a} to the full Poincar\'e group.
Now, while a translation applies to the position of a particle, a
Lorentz transformation applies to its \emph{frame of reference}. Just
as we can consider the position $x$ of a particle to be a translation
by $x$ from some fixed origin $O$, we can consider the frame of
reference of a particle to be given by a Lorentz transformation
$\Lambda$ from a fixed initial frame $I$. The full configuration of a
particle is then given by $(x,\Lambda)$, for a position $x$ and a
Lorentz transformation $\Lambda$---that is, the configuration space of
the particle is also just the Poincar\'e group. The application of an
arbitrary Poincar\'e transformation $(\Delta x,\Lambda')$ to a
particle configuration $(x,\Lambda)$ results in the transformed
configuration $(\Lambda' x + \Delta x, \Lambda' \Lambda)$.
A particle path will now be a path through the Poincar\'e group, not
just through spacetime. Such a path is given by both a position
function $q(\lambda)$ \emph{and} a Lorentz transformation function
$M(\lambda)$ (in coordinate form, a Lorentz transformation is
represented by a matrix, so there are \emph{sixteen} functions
$\hilo{M}{\mu}{\nu}(\lambda)$, for $\mu,\nu = 0,1,2,3,$). The
remainder of this paper will re-develop the spacetime path formalism
introduced in \refcite{seidewitz06a} in terms of this expanded
conception of particle paths. As we will see, this naturally leads to
a model for non-scalar particles.
\section{The Non-scalar Propagator}
\label{sect:non-scalar:propagator}
This section develops the path-integral form of the non-scalar
propagator from the conception of Poincar\'e group particle paths
introduced in the previous section. The argument parallels that of
\refcite{seidewitz06a} for the scalar case, motivating a set of
postulates that lead to the appropriate path integral form.
To begin, let $\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)$
be the transition amplitude for a particle to go from the
configuration $(\xz, \Lambdaz)$ at $\lambdaz$ to the configuration
$(x, \Lambda)$ at $\lambda$. By Poincar\'e invariance, this amplitude
only depends on the relative quantities $x-\xz$ and
$\Lambda\Lambdaz^{-1}$. By parameter shift invariance, it only depends
on $\lambda-\lambdaz$. Similarly to the scalar case (\eqn{eqn:A0d}),
the full propagator is given by integrating over the kernel path
length parameter:
\begin{equation} \label{eqn:A1a}
\propsym(x-\xz,\Lambda\Lambdaz^{-1})
= \int_{0}^{\infty} \dl\,
\kersym(x-\xz,\Lambda\Lambdaz^{-1};\lambda) \,.
\end{equation}
The fundamental postulate of the spacetime path approach is that a
particle's transition amplitude between two points is a superposition
of the transition amplitudes for all possible paths between those
points. Let the functional $\propsym[q, M]$ give the transition
amplitude for a path $q(\lambda), M(\lambda)$. Then the transition
amplitude $\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)$
must be given by a path integral over $\propsym[q, M]$ for all paths
starting at $(\xz, \Lambdaz)$ and ending at $(x, \Lambda)$ with the
parameter interval $[\lambdaz, \lambda]$.
\begin{postulate}
For a free, non-scalar particle, the transition amplitude
$\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)$ is given
by the superposition of path transition amplitudes $\propsym[q,
M]$, for all possible Poincar\'e path functions $q(\lambda),
M(\lambda)$ beginning at $(\xz, \Lambdaz)$ and ending at $(x,
\Lambda)$, parametrized over the interval $[\lambdaz, \lambda]$.
That is,
\begin{multline} \label{eqn:A1}
\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)
= \zeta \intDfour q\, \intDsix M\, \\
\delta^{4}(q(\lambda) - x)
\delta^{6}(M(\lambda)\Lambda^{-1} - I)
\delta^{4}(q(\lambdaz) - \xz)
\delta^{6}(M(\lambdaz)\Lambdaz^{-1} - I)
\propsym[q, M] \,,
\end{multline}
where $\zeta$ is a normalization factor as required to keep the
path integral finite.
\end{postulate}
As previously noted, the notation $\Dfour q$ in \eqn{eqn:A1} indicates
a path integral over the four path functions $\qmul$. Similarly,
$\Dsix M$ indicates a path integral over the Lorentz group functions
$\hilo{M}{\mu}{\nu}(\lambda)$. While a Lorentz transformation matrix
$[\hilo{\Lambda}{\mu}{\nu}]$ has sixteen elements, any such matrix is
constrained by the condition
\begin{equation} \label{eqn:A2}
\hilo{\Lambda}{\alpha}{\mu}\eta_{\alpha\beta}
\hilo{\Lambda}{\beta}{\nu} = \eta_{\mu\nu} \,,
\end{equation}
where $[\eta_{\mu\nu}]=\mathrm{diag}(-1,1,1,1)$ is the flat Minkowski
space metric tensor. This equation is symmetric, so it introduces ten
constraints, leaving only six actual degrees of freedom for a Lorentz
transformation. The Lorentz group is thus six dimensional, as
indicated by the notation $\Dsix$ in the path integral.
To further deduce the form of $\propsym[q, M]$, consider a family of
particle paths $q_{\xz,\Lambdaz}, M_{\xz,\Lambdaz}$, indexed by the
starting configuration $(\xz, \Lambdaz)$, such that
\begin{equation*}
q_{\xz,\Lambdaz}(\lambda)
= \xz + \Lambdaz \tilde{q}(\lambda)
\quad \text{and} \quad
M_{\xz,\Lambdaz}(\lambda) = \Lambdaz \tilde{M}(\lambda) \,,
\end{equation*}
where $\tilde{q}(\lambdaz) = 0$ and $\tilde{M}(\lambdaz) = I$. These
paths are constructed by, in effect, applying the Poincar\'e
transformation given by $(\xz, \Lambdaz)$ to the specific functions
$\tilde{q}$ and $\tilde{M}$ defining the family. (Note how the
ability to do this depends on the particle configuration space being
the same as the Poincar\'e transformation group.)
Consider, though, that the particle propagation embodied in
$\kersym[q,M]$ must be Poincar\'e invariant. That is, $\kersym[q',M']
= \kersym[q,M]$ for any $q',M'$ related to $q,M$ by a fixed Poincar\'e
transformation. Thus, all members of the family $q_{\xz,\Lambdaz},
M_{\xz,\Lambdaz}$, which are all related to $\tilde{q}. \tilde{M}$ by
Poincar\'e transformations, must have the same amplitude
$\kersym[q_{\xz,\Lambdaz}, M_{\xz,\Lambdaz}] = \kersym[\tilde{q},
\tilde{M}]$, depending only on the functions $\tilde{q}$ and
$\tilde{M}$.
Suppose that a probability amplitude $\psi(\xz, \Lambdaz)$ is given
for a particle to be at in an initial configuration $(\xz,\Lambdaz)$
and that the transition amplitude is known to be $\kersym[\tilde{q},
\tilde{M}]$ for specific relative configuration functions $\tilde{q},
\tilde{M}$. Then, the probability amplitude for the particle to
traverse a specific path $(q_{\xz,\Lambdaz}(\lambda),
M_{\xz,\Lambdaz}(\lambda))$ from the family defined by the functions
$\tilde{q}, \tilde{M}$ should be just $\kersym[q_{\xz,\Lambdaz},
M_{\xz,\Lambdaz}] \psi(\xz, \Lambdaz) = \kersym[\tilde{q}, \tilde{M}]
\psi(\xz, \Lambdaz)$.
However, the very meaning of being on a specific path is that the
particle must propagate from the given starting configuration to the
specific ending configuration of the path. Further, since the paths in
the family are parallel in configuration space, the ending
configuration is uniquely determined by the starting configuration.
Therefore, the probability for reaching the ending configuration must
be the same as the probability for having started out at the given
initial configuration $(\xz,\Lambdaz)$. That is,
\begin{equation*}
\sqr{\kersym[\tilde{q}, \tilde{M}]\psi(\xz,\Lambdaz)}
= \sqr{\psi(\xz,\Lambdaz)} \,.
\end{equation*}
But, since $\kersym[\tilde{q}, \tilde{M}]$ is independent of $\xz$ and
$\Lambdaz$, we must have $\sqr{\kersym[q, M]} = 1$ in general.
This argument therefore suggests the following postulate.
\begin{postulate}
For any path $(q(\lambda),M(\lambda))$, the transition amplitude
$\propsym[q,M]$ preserves the probability density for the particle
along the path. That is, it satisfies
\begin{equation} \label{eqn:A3}
\sqr{\propsym[q,M]} = 1 \,.
\end{equation}
\end{postulate}
The requirements of \eqn{eqn:A3} and Poincar\'e invariance mean that
$\propsym[q,M]$ must have the exponential form
\begin{equation} \label{eqn:A4}
\propsym[q,M] = \me^{\mi S[\tilde{q}, \tilde{M}]} \,,
\end{equation}
for some phase functional $S$ of the \emph{relative} path functions
\begin{equation*}
\tilde{q}(\lambda) \equiv M(\lambdaz)^{-1}(q(\lambda)-q(\lambdaz))
\quad \text{and} \quad
\tilde{M}(\lambda) \equiv M(\lambdaz)^{-1}M(\lambda) \,.
\end{equation*}
As discussed in \refcite{seidewitz06a}, we are actually justified in
replacing these relative functions with path derivatives under the
path integral, even though the path functions $q(\lambda)$ and
$M(\lambda)$ may not themselves be differentiable in general. This is
because a path integral is defined as the limit of discretized
approximations in which path derivatives are approximated as mean
values, and the limit is then taken over the path integral as a
whole, not each derivative individually. Thus, even though the
individual path derivative limits may not be defined, the path
integral has a well-defined value so long as the overall path
integral limit is defined.
However, the quantities $\tilde{q}$ and $\tilde{M}$ are expressed in a
frame that varies with the $M(\lambdaz)$ of the specific path under
consideration. We wish instead to construct differentials in the fixed
``laboratory'' frame of the $q(\lambda)$. Transforming $\tilde{q}$ and
$\tilde{M}$ to this frame gives
\begin{equation*}
M(\lambdaz)\tilde{q}(\lambda) = q(\lambda)-q(\lambdaz)
\quad \text{and} \quad
M(\lambdaz)\tilde{M}(\lambda)M(\lambdaz)^{-1}
= M(\lambda)M(\lambdaz)^{-1} \,.
\end{equation*}
Clearly, the corresponding derivative for $q$ is simply
$\qdot(\lambda) \equiv \dif q / \dl$, which is the tangent vector to
the path $q(\lambda)$. The derivative for $M$ needs to be treated a
little more carefully. Since the Lorentz group is a \emph{Lie group}
(that is, a continuous, differentiable group), the tangent to a path
$M(\lambda)$ in the Lorentz group space is given by an element of the
corresponding \emph{Lie algebra} \cite{warner83, frankel97}. For the
Lorentz group, the proper such tangent is given by the matrix
$\Omega(\lambda) = \dot{M}(\lambda)M(\lambda)^{-1}$, where
$\dot{M}(\lambda) \equiv \dif M / \dl$.
Together, the quantities $(\qdot, \Omega)$ form a tangent along the
path in the full Poincar\'e group space. We can then take the
arguments of the phase functional in \eqn{eqn:A4} to be $(\qdot,
\Omega)$. Substituting this into \eqn{eqn:A1} gives
\begin{multline} \label{eqn:A5}
\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)
= \zeta \intDfour q\, \intDsix M\, \\
\delta^{4}(q(\lambda) - x)
\delta^{6}(M(\lambda)\Lambda^{-1} - I)
\delta^{4}(q(\lambdaz) - \xz)
\delta^{6}(M(\lambdaz)\Lambdaz^{-1} - I)
\me^{\mi S[\qdot, \Omega]} \,,
\end{multline}
which reflects the typical form of a Feynman sum over paths.
Now, by dividing a path $(q(\lambda), M(\lambda))$ into two paths at
some arbitrary parameter value $\lambda$ and propagating over each
segment, one can see that
\begin{equation}
S[\qdot, \Omega;\lambda_{1},\lambdaz]
= S[\qdot, \Omega;\lambda_{1},\lambda] +
Ê S[\qdot, \Omega;\lambda,\lambdaz] \,,
\end{equation}
where $S[\qdot, \Omega;\lambda',\lambda]$ denotes the value of
$S[\qdot,\Omega]$ for the path parameter range restricted to
$[\lambda,\lambda']$. Using this property to build the total value of
$S[\qdot,\Omega]$ from infinitesimal increments leads to the following result
(whose full proof is a straightforward generalization of the proof
given in \refcite{seidewitz06a} for the scalar case).
\begin{theorem}[Form of the Phase Functional]
The phase functional $S$ must have the form
\begin{equation*}
S[\qdot,\Omega] = \int^{\lambda_{1}}_{\lambdaz} \dl' \,
L[\qdot,\Omega;\lambda'] \,,
\end{equation*}
where the parametrization domain is $[\lambdaz,\lambda_{1}]$ and
$L[\qdot, \Omega;\lambda]$ depends only on $\qdot$, $\Omega$ and
their higher derivatives evaluated at $\lambda$.
\end{theorem}
Clearly, the functional $L[\qdot,\Omega; \lambda]$ plays the traditional role
of the Lagrangian. The simplest non-trivial form for this functional
would be for it to depend only on $\qdot$ and $\Omega$ and no higher
derivatives. Further, suppose that it separates into uncoupled parts
dependent on $\qdot$ and $\Omega$:
\begin{equation*}
L[\qdot,\Omega; \lambda] =
L_{q}[\qdot;\lambda] + L_{M}[\Omega; \lambda] \,.
\end{equation*}
The path integral of \eqn{eqn:A5} then factors into independent parts
in $q$ and $M$, such that
\begin{equation} \label{eqn:A5a}
\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz) =
\kerneld \kersym(\Lambda\Lambdaz^{-1}; \lambda-\lambdaz) \,.
\end{equation}
If we take $L_{q}$ to have the classical Lagrangian form
\begin{equation*}
L_{q}[\qdot;\lambda] = L_{q}(\qdot(\lambda))
= \frac{1}{4}\qdot(\lambda)^{2} - m^{2} \,,
\end{equation*}
for a particle of mass $m$, then the path integral in $q$ can be
evaluated to give \cite{seidewitz06a,teitelboim82}
\begin{equation} \label{eqn:A5b}
\kerneld = (2\pi)^{-4}\intfour p\, \me^{ip\cdot(x-\xz)}
\me^{-\mi(\lambda-\lambdaz)(p^{2}+m^{2})} \,.
\end{equation}
Similarly, take $L_{M}$ to be a Lorentz-invariant scalar function of
$\Omega(\lambda)$. $\Omega$ is an antisymmetric matrix (this can be
shown by differentiating the constraint \eqn{eqn:A2}), so the scalar
$\tr(\Omega) = \hilo{\Omega}{\mu}{\mu} = 0$. The next simplest choice
is
\begin{equation*}
L_{M}[\Omega;\lambda] = L_{M}(\Omega(\lambda))
= \frac{1}{2}\tr(\Omega(\lambda)\Omega(\lambda)\T)
= \frac{1}{2}\Omega^{\mu\nu}(\lambda)
\Omega_{\mu\nu}(\lambda) \,.
\end{equation*}
\begin{postulate}
For a free non-scalar particle of mass $m$, the Lagrangian
function is given by
\begin{equation*}
L(\qdot,\Omega) = L_{q}(\qdot) + L_{M}(\Omega) \,,
\end{equation*}
where
\begin{equation*}
L_{q}(\qdot) = \frac{1}{4}\qdot^{2} - m^{2}
\end{equation*}
and
\begin{equation*}
L_{M}(\Omega) = \frac{1}{2}\tr(\Omega\Omega\T) \,.
\end{equation*}
\end{postulate}
Evaluating the path integral in $M$ is complicated by the fact that
the Lorentz group is not \emph{compact}, and integration over the
group is not, in general, bounded. The Lorentz group is denoted
$SO(3,1)$ for the three plus and one minus sign of the Minkowski
metric $\eta$ in the defining pseudo-orthogonality condition
\eqn{eqn:A2}. It is the minus sign on the time component of $\eta$
that leads to the characteristic Lorentz boosts of special relativity.
But since such boosts are parametrized by the boost velocity,
integration of this sector of the Lorentz group is unbounded. This is
in contrast to the three dimensional rotation subgroup $SO(3)$ for the
Lorentz, which is parameterized by rotation angles that are bounded.
To avoid this problem, we will \emph{Wick rotate} \cite{wick50} the
time axis in complex space. This replaces the physical $t$ coordinate
with $\mi t$, turning the minus sign in the metric to a plus sign,
resulting in the normal Euclidean metric $\mathrm{diag}(1,1,1,1)$. The
symmetry group of Lorentz transformations in Minkowski space then
corresponds to the symmetry group $SO(4)$ of rotations in
four-dimensional Euclidean space. The group $SO(4)$ \emph{is} compact,
and the path integration over $SO(4)$ can be done \cite{bohm87}.
Rather than dividing into boost and rotational parts, like the Lorentz
group, $SO(4)$ instead divides into two $SO(3)$ subgroups of rotations
in three dimensions. Actually, rather than $SO(3)$ itself, it is more
useful to consider its universal covering group $SU(2)$, the group of
two-dimensional unitary matrices, because $SU(2)$ allows for
representations with half-integral spin \cite{weyl50, weinberg95,
frankel97}. (The covering group $SU(2) \times SU(2)$ for $SO(4)$ in
Euclidean space corresponds to the covering group $SL(2,\cmplx)$ of
two-dimensional complex matrices for the Lorentz group $SO(3,1)$ in
Minkowski space.)
Typically, Wick rotations have been used to simplify the evaluation of
path integrals parametrized in time, like the non-relativistic
integral of \eqn{eqn:A0a}. In this case, replacing $t$ by $\mi t$
results in the exponent in the integrand of the path integral to
become real. Unlike this case, the exponent in the integrand of a
spacetime path integral remains imaginary, since the Wick rotation
does not affect the path parameter $\lambda$. Nevertheless, the path
integral can be evaluated, giving the following result (proved in the
Appendix).
\begin{theorem}[Evaluation of the $SO(4)$ Path Integral]
Consider the path integral
\begin{multline} \label{eqn:A6a}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
= \euc{\eta} \intDsix \ME\,
\delta^{6}(\ME(\lambda)\LambdaE^{-1}-I)
\delta^{6}(\ME(\lambdaz)\LambdaEz^{-1}-I) \\
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \tr(\OmegaE(\lambda')
\OmegaE(\lambda')\T)
\right]
\end{multline}
over the six dimensional group $SO(4) \sim SU(2) \times SU(2)$,
where $\OmegaE(\lambda')$ is the element of the Lie algebra $so(4)$
tangent to the path $\ME(\lambda)$ at $\lambda'$. This path integral
may be evaluated to get
\begin{multline} \label{eqn:A6}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz) \\
= \sum_{\ell_{A},\ell_{B}}
\exp^{-\mi( \Delta m_{\ell_{A}}^{2}
+ \Delta m_{\ell_{B}}^{2})
(\lambda - \lambdaz)}
(2\ell_{A}+1)(2\ell_{B}+1)
\chi^{(\ell_{A}\ell_{B})}(\LambdaE\LambdaEz^{-1}) \,,
\end{multline}
where the summation over $\ell_{A}$ and $\ell_{B}$ is from $0$ to
$\infty$ in steps of $1/2$, $\Delta m_{\ell}^{2} = \ell(\ell+1)$
and $\chi^{(\ell_{A},\ell_{B})}$ is the group character for the
$(\ell_{A},\ell_{B})$ $SU(2) \times SU(2)$ group representation.
\end{theorem}
The result of \eqn{eqn:A6} is in terms of the \emph{representations}
of the covering group $SU(2) \times SU(2)$. A (matrix) representation
$L$ of a group assigns to each group element $g$ a matrix $D^{(L)}(g)$
that respects the group operation, that is, such that
$D^{(L)}(g_{1}g_{2}) = D^{(L)}(g_{1}) D^{(L)}(g_{2})$. The
\emph{character function} $\chi^{(L)}$ for the representation $L$ of a
group is a function from the group to the reals such that
\begin{equation*}
\chi^{(L)}(g) \equiv \tr(D^{(L)}(g)) \,.
\end{equation*}
The group $SU(2)$ has the well known \emph{spin representations},
labeled by spins $\ell = 0, 1/2, 1, 3/2, \ldots$ \cite{weyl50,
weinberg95} (for example, spin 0 is the trivial scalar representation,
spin 1/2 is the spinor representation and spin 1 is the vector
representation). A $(\ell_{A},\ell_{B})$ representation of $SU(2)
\times SU(2)$ then corresponds to a spin-$\ell_{A}$ representation for
the first $SU(2)$ component and a spin-$\ell_{B}$ representation for
the second $SU(2)$ component.
Of course, it is not immediately clear that this result for $SO(4)$
applies directly to $SO(3,1)$. In some cases, it can be shown that the
evolution propagator for a non-compact group is, in fact, the same as
the propagator for a related compact group. Unfortunately, the
relationship between $SO(4)$ and $SO(3,1)$ (in which an odd number,
three, of the six generators of $SO(4)$ are multiplied by $\mi$ to get
the boost generators for $SO(3,1)$) is such that the evolution
propagator of the non-compact group does not coincide with that of the
compact group \cite{krausz00}.
Nevertheless, $SO(4)$ and $SO(3,1)$ both have compact $SO(3)$
subgroups, which are isomorphic. Therefore, the restriction of the
$SO(4)$ propagator to its $SO(3)$ subgroup should correspond to the
restriction of the $SO(3,1)$ propagator to its $SO(3)$ subgroup. This
will prove sufficient for our purposes. In the next section, we will
continue to freely work with the Wick rotated Euclidean space and the
$SO(4)$ propagator as necessary. To show clearly when this is being
done, quantities effected by Wick rotation will be given a subscript
$E$, as in \eqn{eqn:A6}.
\section{The Euclidean Propagator} \label{sect:non-scalar:euclidean}
For a scalar particle, one can define the probability amplitude
$\psixl$ for the particle to be at position $x$ at the point $\lambda$
in its path \cite{seidewitz06a, stueckelberg41, stueckelberg42}. For a
non-scalar particle, this can be extended to a probability amplitude
$\psixLl$ for the particle to be in the Poincar\'{e} configuration
$(x,\Lambda)$, at the point $\lambda$ in its path. The transition
amplitude given in \eqn{eqn:A1} acts as a propagation kernel for
$\psixLl$:
\begin{equation*}
\psixLl = \intfour \xz\, \intsix \Lambdaz\,
\kersym(x-\xz,\Lambda\Lambdaz^{-1};\lambda-\lambdaz)
\psilz{\xz, \Lambdaz} \,.
\end{equation*}
The Euclidean version of this equation has an identical form, but in
terms of Euclidean configuration space quantities:
\begin{equation} \label{eqn:B1}
\psixLEl = \intfour \xEz\, \intsix \LambdaEz\,
\kersym(\xE-\xEz,\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
\psixLElz \,.
\end{equation}
Using \eqn{eqn:A5a}, substitute into \eqn{eqn:B1} the Euclidean scalar
kernel (as in \eqn{eqn:A5b}, but with a leading factor of $\mi$) and
the $SO(4)$ kernel (\eqn{eqn:A6}), giving
\begin{multline} \label{eqn:B1a}
\psixLEl = \sum_{\ell_{A},\ell_{B}}
\intfour \xEz\, \intsix \LambdaEz\, \\
\kersym^{(\ell_{A},\ell_{B})}(\xE-\xEz;\lambda-\lambdaz)
\chi^{(\ell_{A},\ell_{B})}(\LambdaE\LambdaEz^{-1})
\psixLElz \,,
\end{multline}
where
\begin{equation*}
\kersym^{(\ell_{A},\ell_{B})}(\xE-\xEz;\lambda-\lambdaz)
\equiv \mi(2\pi)^{-4}\intfour \pE\, \me^{i\pE\cdot(\xE-\xEz)}
\me^{-\mi(\lambda-\lambdaz)
(\pE^{2}+m^{2}+\Delta m_{A}^{2}+\Delta m_{B}^{2})}\,.
\end{equation*}
Since the group characters provide a complete set of orthogonal
functions \cite{weyl50}, the function $\psi(\xEz,\LambdaEz;\lambdaz)$
can be expanded as
\begin{equation*}
\psixLElz = \sum_{\ell_{A},\ell_{B}}
\chi^{(\ell_{A},\ell_{B})}(\LambdaEz)
\psi^{(\ell_{A},\ell_{B})}(\xEz;\lambdaz) \,.
\end{equation*}
Substituting this into \eqn{eqn:B1a} and using
\begin{equation*}
\chi^{(\ell_{A},\ell_{B})}(\LambdaE) = \intsix \LambdaEz\,
\chi^{(\ell_{A},\ell_{B})}(\LambdaE\LambdaEz^{-1})
\chi^{(\ell_{A},\ell_{B})}(\LambdaEz)
\end{equation*}
(see \refcite{weyl50}) gives
\begin{equation*}
\psixLEl = \sum_{\ell_{A},\ell_{B}}
\chi^{(\ell_{A},\ell_{B})}(\LambdaE)
\psi^{(\ell_{A},\ell_{B})}(\xE;\lambda) \,,
\end{equation*}
where
\begin{equation} \label{eqn:B1b}
\psi^{(\ell_{A},\ell_{B})}(\xE;\lambda)
= \intfour \xEz\,
\kersym^{(\ell_{A},\ell_{B})}(\xE-\xEz;\lambda-\lambdaz)
\psi^{(\ell_{A},\ell_{B})}(\xEz;\lambdaz) \,.
\end{equation}
The general amplitude $\psixLEl$ can thus be expanded into a sum of
terms in the various $SU(2) \times SU(2)$ representations, the
coefficients $\psi^{(\ell_{A},\ell_{B})} (\xE;\lambdaz)$ of which each
evolve separately according to \eqn{eqn:B1b}. As is well known,
reflection symmetry requires that a real particle amplitude must
transform according to a $(\ell,\ell)$ or $(\ell_{A},\ell_{B}) \oplus
(\ell_{B},\ell_{A})$ representation. That is, the amplitude function
$\psixLEl$ must either have the form
\begin{equation*}
\psixLEl = \chi^{(\ell,\ell)}(\LambdaE)
\psi^{(\ell,\ell)} (\xE;\lambda)
\end{equation*}
or
\begin{equation*}
\psixLEl
= \chi^{(\ell_{A},\ell_{B})}(\LambdaE)
\psi^{(\ell_{A},\ell_{B})} (\xE;\lambda)
+ \chi^{(\ell_{B},\ell_{A})}(\LambdaE)
\psi^{(\ell_{B},\ell_{A})} (\xE;\lambda) \,.
\end{equation*}
Assuming one of the above two forms, shift the particle mass to
$m'^{2} = m^{2} + 2\Delta m_{\ell}^{2}$ or $m'^{2} = m^{2} + 2\Delta
m_{\ell_{A}}^{2} + 2\Delta m_{\ell_{B}}^{2}$, so that
\begin{equation*}
\psixLEl = \intfour \xz, \intsix \Lambdaz\,
\chi^{(L)}(\LambdaE\LambdaEz^{-1})
\kersym(\xE-\xEz;\lambda-\lambdaz) \psixLElz \,,
\end{equation*}
where $\kersym$ here is (the Euclidean version of) the scalar
propagator of \eqn{eqn:A5b}, but now for the shifted mass $m'$, and
$(L)$ is either $(\ell,\ell)$ or $(\ell_{A},\ell_{B})$. That is, the
full kernel must have the form
\begin{equation} \label{eqn:B1c}
\kersym^{(L)}(\xE-\xEz,\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
= \chi^{(L)}(\LambdaE\LambdaEz^{-1})
\kersym(\xE-\xEz;\lambda-\lambdaz) \,.
\end{equation}
As is conventional, from now on we will use four-dimensional spinor
indices for the $(1/2,0) \oplus (0,1/2)$ representation and vector
indices (also four dimensional) for the $(1,1)$ representation, rather
than the $SU(2) \times SU(2)$ indices $(\ell_{A},\ell_{B})$ (see, for
example, \refcite{weinberg95}). Let $\DLElpl$ be a matrix
representation of the $SO(4)$ group using such indices. Define
correspondingly indexed amplitude functions by
\begin{equation} \label{eqn:B2}
\lpl{\psi}(\xE;\lambda) \equiv \intsix \LambdaE\, \DLElpl \psixLEl
\end{equation}
(note the \emph{double} indexing of $\psi$ here).
These $\lpl{\psi}$ are the elements of an algebra over the $SO(4)$
group for which, given $\xE$ and $\lambda$, the $\psixLEl$ are the
\emph{components}, ``indexed'' by the group elements $\LambdaE$ (see
Section III.13 of \refcite{weyl50}). The product of two such algebra
elements is (with summation implied over repeated up and down indices)
\begin{equation*}
\begin{split}
\hilo{\psi_{1}}{l'}{\lbar}(\xE; \lambda)
\hilo{\psi_{2}}{\lbar}{l}(\xE; \lambda)
&= \intsix {\LambdaE}_{1}\, \intsix {\LambdaE}_{2}\,
\hilo{\Dmat}{l'}{\lbar}({\LambdaE}_{1})
\hilo{\Dmat}{\lbar}{l}({\LambdaE}_{2})
\psi_{1}(\xE, {\LambdaE}_{1};\lambda)
\psi_{2}(\xE, {\LambdaE}_{2}; \lambda) \\
&= \intsix \LambdaE\, \DLElpl \intsix {\LambdaE}_{1} \,
\psi_{1}(\xE, {\LambdaE}_{1}; \lambda)
\psi_{2}(\xE, {\LambdaE}_{1}^{-1}\LambdaE; \lambda) \\
&= \lpl{(\psi_{1}\psi_{2})}(\xE;\lambda) \,,
\end{split}
\end{equation*}
where the second equality follows after setting ${\LambdaE}_{2} =
{\LambdaE}_{1}^{-1}\LambdaE$ from the invariance of the integration
measure of a Lie group (see, for example, \refcite{warner83}, Section
4.11, and \refcite{weyl50}, Section III.12---this property will be
used regularly in the following), and the product components
$(\psi_{1}\psi_{2})(\xE, \LambdaE; \lambda)$ are defined to be
\begin{equation*}
(\psi_{1}\psi_{2})(\xE, \LambdaE; \lambda)
\equiv \intsix \LambdaE'\, \psi_{1}(\xE, \LambdaE'; \lambda)
\psi_{2}(\xE, \LambdaE^{\prime -1}\LambdaE; \lambda) \,.
\end{equation*}
Now substitute \eqn{eqn:B1} into \eqn{eqn:B2} to get
\begin{multline*}
\lpl{\psi}(\xE;\lambda)
= \intsix \LambdaE\, \intfour \xEz\, \intsix \LambdaEz\, \\
\DLElpl
\kersym(\xE-\xEz,\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
\psixLElz \,.
\end{multline*}
Changing variables $\LambdaE \to \LambdaE'\LambdaEz$ then gives
\begin{equation*}
\begin{split}
\lpl{\psi}(\xE;\lambda)
&= \intfour \xz\,
\left[
\intsix \LambdaE'\,
\hilo{\Dmat}{l'}{\lbar}(\LambdaE')
\kersym(\xE-\xz,\LambdaE';\lambda - \lambdaz)
\right] \\
&\qquad\qquad\qquad\qquad\qquad\qquad
\intsix \LambdaEz\, \hilo{\Dmat}{\lbar}{l}(\LambdaEz)
\psixLElz \\
&= \intfour \xz\,
\hilo{\kersym}{l'}{\lbar}(\xE-\xz;\lambda-\lambdaz)
\hilo{\psi}{\lbar}{l}(\xz;\lambdaz) \,,
\end{split}
\end{equation*}
where the kernel for the algebra elements $\lpl{\psi}(\xE;\lambda)$ is
thus
\begin{equation*}
\kersymlpl(\xE-\xEz;\lambda-\lambdaz)
= \intsix \LambdaE\, \DLElpl
\kersym(\xE-\xEz,\LambdaE;\lambda-\lambdaz) \,.
\end{equation*}
Substituting \eqn{eqn:B1c} into this, and using the definition of the
character for a specific representation, $\chi(\LambdaE) \equiv
\tr(\DLE)$, gives
\begin{equation*}
\kersymlpl(\xE-\xEz;\lambda-\lambdaz)
= \left[
\intsix \LambdaE\, \DLElpl
\hilo{\Dmat}{\lbar}{\lbar}(\LambdaE)
\right]
\kersym(\xE-\xEz;\lambda-\lambdaz) \,.
\end{equation*}
Use the orthogonality property
\begin{equation*}
\intsix \LambdaE\, \DLElpl \lohi{\Dmat}{\lbar'}{\lbar}(\LambdaE)
= \hilo{\delta}{l'}{\lbar'} \lohi{\delta}{l}{\lbar} \,,
\end{equation*}
where the $SO(4)$ integration measure has been normalized so that $\intsix
\LambdaE = 1$ (see \refcite{weyl50}, Section 11), to get
\begin{equation} \label{eqn:B3}
\kersymlpl(\xE-\xEz;\lambda-\lambdaz)
= \lpl{\delta}\kersym(\xE-\xEz;\lambda-\lambdaz) \,.
\end{equation}
The $SO(4)$ group propagator is thus simply $\lpl{\delta}$. As
expected, this does not have the same form as would be expected for
the $SO(3,1)$ Lorentz group propagator. However, as argued at the end
of \sect{sect:non-scalar:propagator}, the propagator restricted to the
compact $SO(3)$ subgroup of $SO(3,1)$ \emph{is} expected to have the
same form as for the $SO(3)$ subgroup of $SO(4)$. So we turn now to
the reduction of $SO(3,1)$ to $SO(3)$.
\section{Spin} \label{sect:non-scalar:spin}
In traditional relativistic quantum mechanics, the Lorentz-group
dependence of non-scalar states is reduced to a rotation representation
that is amenable to interpretation as the intrinsic particle spin.
Since, in the usual approach, physical states are considered to have
on-shell momentum, it is natural to use the 3-momentum as the vector
around which the spin representation is induced, using Wigner's
classic ``little group'' argument \cite{wigner39}.
However, in the spacetime path approach used here, the fundamental
states are not naturally on-shell, rather the on-shell states are
given as the time limits of off-shell states \cite{seidewitz06a}.
Further, there are well-known issues with the localization of on-shell
momentum states \cite{newton49, hegerfeldt74}. Therefore, instead of
assuming on-shell states to start, we will adopt the approach of
\refcite{piron78, horwitz82}, in which the spin representation is
induced about an arbitrary timelike vector. This will allow for a
straightforward generalization of the interpretation obtained in the
spacetime path formalism for the scalar case \cite{seidewitz06a}.
First, define the probability amplitudes $\lpl{\psi}(x;\lambda)$ for a
given Lorentz group representation similarly to the correspondingly
indexed amplitudes for $SO(4)$ representations from
\sect{sect:non-scalar:euclidean}. Corresponding to such amplitudes,
define a \emph{set} of ket vectors $\ketpsi\lol$, with a \emph{single}
Lorentz-group representation index. The $\ketpsi\lol$ define a
\emph{vector bundle} (see, for example, \refcite{frankel97}), of the
same dimension as the Lorentz-group representation, over the
scalar-state Hilbert space.
The basis position states for this vector bundle then have the form
$\ketxll$, such that
\begin{equation*}
\lpl{\psi}(x;\lambda)
= \gmat^{l'\lbar}\,{}_{\lbar}\innerxlpsi\lol \,,
\end{equation*}
with summation assumed over repeated upper and lower indices and
$\gmat$ being the invariant matrix of a given Lorentz group
representation such that
\begin{equation*}
\Dmat\adj \gmat \Dmat = \Dmat \gmat \Dmat\adj = \gmat \,,
\end{equation*}
for any member $\Dmat$ of the representation, where $\Dmat\adj$ is the
Hermitian transpose of the matrix $\Dmat$. For the scalar
representation, $\gmat$ is $1$, for the (Weyl) spinor representation
it is the Dirac matrix $\beta$ and for the vector representation it is
the Minkowski metric $\eta$.
In the following, $\gmat$ will be used (usually implicitly) to
``raise'' and ``lower'' group representation indices. For instance,
\begin{equation*}
\lpbraxl \equiv \gmat^{l'l}\,{}\lol\braxl \,,
\end{equation*}
so that
\begin{equation} \label{eqn:C0}
\lpl{\psi}(x;\lambda) = \hilp\innerxlpsi\lol \,.
\end{equation}
The states $\ketxll$ are then normalized so that
\begin{equation} \label{eqn:C0d}
\hilp\inner{x';\lambda}{x;\lambda}\lol
= \lpl{\delta}\, \delta^{4}(x' - x) \,,
\end{equation}
that is, they are orthogonal at equal $\lambda$.
Consider an arbitrary Lorentz transformation $M$. Since $\psixLl$ is a
scalar, it should transform as $\psipxLl = \psil{M^{-1}x',
M^{-1}\Lambda'}$. In terms of algebra elements,
\begin{equation} \label{eqn:C0e}
\begin{split}
\lpl{\psi'}(x';\lambda)
&= \intsix \Lambda'\, \Dmatlpl(\Lambda')
\psil{M^{-1}x', M^{-1}\Lambda'} \\
&= \intsix \Lambda\,
\hilo{\Dmat}{l'}{\lbar'}(M)
\hilo{\Dmat}{\lbar'}{l}(\Lambda)
\psil{M^{-1}x', \Lambda} \\
&= \hilo{\Dmat}{l'}{\lbar'}(M)
\hilo{\psi}{\lbar'}{l}(M^{-1}x; \lambda) \,.
\end{split}
\end{equation}
Let $\UL$ denote the unitary operator on Hilbert space corresponding
to the Lorentz transformation $\Lambda$. Then, from \eqn{eqn:C0},
\begin{equation*}
\lpl{\psi'}(x';\lambda) = \hilp\inner{x';\lambda}{\psi'}\lol \\
= \hilp\bral{x'} \UL \ketpsi\lol \,.
\end{equation*}
This and \eqn{eqn:C0e} imply that
\begin{equation*}
\UL^{-1} \ketll{x'} = \ketllp{\Lambda^{-1}x'}\,
\lpl{[\DL^{-1}]} \,,
\end{equation*}
or
\begin{equation} \label{eqn:C0f}
\UL \ketxll = \ketllp{\Lambda x}\, \DLlpl \,.
\end{equation}
Thus, the $\ketxll$ are localized position states that transform
according to a representation of the Lorentz group.
Now, for any future-pointing, timelike, unit vector $n$ ($n^{2} = -1$
and $n^{0} > 0$) define the standard Lorentz transformation
\begin{equation*}
L(n) \equiv R(\threen) B(|\threen|) R^{-1}(\threen) \,,
\end{equation*}
where $R(\threevec{n})$ is a rotation that takes the $z$-axis into the
direction of $\threevec{n}$ and $B(|\threevec{n}|)$ is a boost of
velocity $|\threevec{n}|$ in the $z$ direction. Then $n = L(n) e$,
where $e \equiv (1, 0, 0, 0)$.
Define the \emph{Wigner rotation} for $n$ and an arbitrary Lorentz
transformation $\Lambda$ to be
\begin{equation} \label{eqn:C1}
\WLn \equiv L(\Lambda n)^{-1} \Lambda L(n) \,,
\end{equation}
such that $\WLn e = e$. That is, $\WLn$ is a member of the
\emph{little group} of transformations that leave $e$ invariant. Since
$e$ is along the time axis, its little group is simply the rotation
group $SO(3)$ of the three space axes.
Substituting the transformation
\begin{equation*}
\Lambda = L(\Lambda n) \WLn L(n)^{-1} \,,
\end{equation*}
into \eqn{eqn:C0f} gives
\begin{equation*}
\UL \ketxll = \ketl{\Lambda x}\lolp\,
\lpl{\left[
\Dmat \left(
L(\Lambda n) \WLn L(n)^{-1}
\right)
\right]} \,.
\end{equation*}
Defining
\begin{equation} \label{eqn:C2}
\ketlml{x,n} \equiv \ketxllp \lpl{[\Lmat(n)]} \,,
\end{equation}
where $\Lmat(n) \equiv \Dmat(L(n))$, we see that $\ketlml{x,n}$
transforms under $\UL$ as
\begin{equation} \label{eqn:C3}
\UL \ketlml{x,n}
= \ketlmlp{\Lambda x, \Lambda n}\,
\lpl{\left[
\Dmat \left( \WLn \right)
\right]} \,,
\end{equation}
that is, according to the Lorentz representation subgroup given by
$\Dmat(\WLn)$, which is isomorphic to some representation of the
rotation group.
The irreducible representations of the rotation group (or, more
exactly, its covering group $SU(2)$) are just the spin
representations, with members given by matrices $\Dsps$, where the
$\sigma$ are spin indices. Let $\ketpsi\los$ be a member of a Hilbert
space vector bundle indexed by spin indices. Then there is a linear,
surjective mapping from $\ketpsi\lol$ to $\ketpsi\los$ given by
\begin{equation*}
\ketpsi\los = \ketpsi\lol \uls \,,
\end{equation*}
where
\begin{equation} \label{eqn:C4}
(\ulspt)^{*} \uls = \hilo{\delta}{\sigma'}{\sigma} \,.
\end{equation}
The isomorphism between the rotation subgroup of the Lorentz
group and the rotation group then implies that, for any
rotation $W$, for all $\ketpsi\lol$,
\begin{equation*}
\ketpsi\lolp \ulpsp \sps{[D(W)]} = \ketpsi\lolp \lpl{[\Dmat(W)]} \uls
\end{equation*}
(with summation implied over repeated $\sigma$ indices, as well as
$l$ indices) or
\begin{equation} \label{eqn:C5}
\ulpsp \sps{[D(W)]} = \lpl{[\Dmat(W)]} \uls \,,
\end{equation}
where $D(W)$ is the spin representation matrix corresponding to $W$.
Define
\begin{equation} \label{eqn:C6}
\ketls{x,n} \equiv \ketlml{x,n}\, \uls \,.
\end{equation}
Substituting from \eqn{eqn:C2} gives
\begin{equation} \label{eqn:C7}
\ketls{x,n} = \ketxll\, \uls(n) \,.
\end{equation}
where
\begin{equation} \label{eqn:C8}
\uls(n) \equiv \hilo{[\Lmat(n)]}{l}{l'}\,
\ulps \,.
\end{equation}
Then, under a Lorentz transformation $\Lambda$, using \eqns{eqn:C3}
and \eqref{eqn:C5},
\begin{equation*}
\begin{split}
\UL \ketls{x,n}
&= \ketlmlp{\Lambda x, \Lambda n}\, \lpl{[\Dmat(\WLn)]} \uls \\
&= \ketlmlp{\Lambda x, \Lambda n}\, \hilo{u}{l'}{\sigma'}
\sps{[D(\WLn)]} \\
&= \ketlsp{\Lambda x, \Lambda n}\, \sps{[D(\WLn)]} \,,
\end{split}
\end{equation*}
that is, $\ketls{x,n}$ transforms according to the appropriate spin
representation.
Now consider a past-pointing $n$ ($n^{2} = -1$ and $n^{0} < 0$). In
this case, $-n$ is future pointing so that $-n = L(-n)e$, or $n =
L(-n)(-e)$. Taking $L(-n)$ to be the standard Lorentz transformation
for past-pointing $n$, it is thus possible to construct spin states
in terms of the future-pointing $-n$. However, since the spacial
part of $n$ is also reversed in $-n$, it is conventional to consider
the spin sense reversed, too. Therefore, define
\begin{equation} \label{eqn:C9}
\vls(n) \equiv (-1)^{j+\sigma} \hilo{u}{l}{-\sigma}(-n) \,,
\end{equation}
for a spin-$j$ representation, and, for past-pointing $n$, take
\begin{equation*}
\ketls{x,n} = \ketxll\, \vls(n) \,.
\end{equation*}
The matrices $\uls$ and $\vls$ are the same as the spin coefficient
functions in Weinberg's formalism in the context of traditional field
theory \cite{weinberg64a} (see also Chapter 5 of
\refcite{weinberg95}). Note that, from \eqn{eqn:C5}, using
\eqn{eqn:C1},
\begin{equation*}
\ulpsp \sps{[D(\WLn)]}
= \lpl{[\Dmat(\WLn)]} \uls
= \lpl{[\Lmat(\Lambda n)^{-1} \DL \Lmat(n)]} \uls \,,
\end{equation*}
so, using \eqn{eqn:C8},
\begin{equation} \label{eqn:C10}
\ulsp(\Lambda n) \sps{[D(\WLn)]} = \lpl{[\DL]} \uls(n) \,.
\end{equation}
Using this with \eqn{eqn:C9} gives
\begin{equation*}
\lpl{[\DL]} \vls(n)
= (-1)^{\sigma-\sigma'} \vlpsp(\Lambda n)
\hilo{[D(\WLn)]}{-\sigma'}{-\sigma}\,.
\end{equation*}
Since
\begin{equation*}
(-1)^{\sigma - \sigma'} \hilo{D(W)}{-\sigma'}{-\sigma}
= [\sps{D(W)}]^{*}
\end{equation*}
(which can be derived by integrating the infinitesimal case), this
gives,
\begin{equation} \label{eqn:C11}
\vlpsp(\Lambda n) [\sps{D(\WLn)}]^{*}
= \lpl{[\DL]} \vls(n) \,.
\end{equation}
As shown by Weinberg \cite{weinberg64a, weinberg95}, \eqns{eqn:C10}
and \eqref{eqn:C11} can be used to completely determine the $u$ and
$v$ matrices, along with the usual relationship of the Lorentz group
scalar, spinor and vector representations to the rotation group
spin-0, spin-1/2 and spin-1 representations.
Since, from \eqns{eqn:C4} and \eqref{eqn:C8},
\begin{equation*}
\begin{split}
\ulspt(n)^{*}\uls(n) &= [\lohi{\Lmat(n)}{l}{\lbar'}]^{*}
(\lohi{u}{\lbar'}{\sigma'})^{*}
\hilo{[\Lmat(n)]}{l}{\lbar}\,
\hilo{u}{\lbar}{\sigma} \\
&= (\lohi{u}{\lbar'}{\sigma'})^{*}
\hilo{[\Lmat(n)^{-1}]}{\lbar'}{l}
\hilo{[\Lmat(n)]}{l}{\lbar}\,
\hilo{u}{\lbar}{\sigma} \\
&= (\lohi{u}{\lbar}{\sigma'})^{*}
\hilo{u}{\lbar}{\sigma} \\
&= \sps{\delta} \,,
\end{split}
\end{equation*}
\eqns{eqn:C0d} and \eqref{eqn:C7} give
\begin{equation} \label{eqn:C12a}
\hisp\inner{x',n;\lambda}{x,n;\lambda}\los
= \sps{\delta} \delta^{4}(x'-x)
\end{equation}
(and similarly for past-pointing $n$ with $\vls$), so that, for given $n$
and $\lambda$, the $\ketls{x,n}$ form an orthogonal basis. However,
for different $\lambda$, the inner product is
\begin{equation} \label{eqn:C12b}
\hisp\inner{x,n;\lambda}{\xz,n;\lambdaz}\los
= \kernelsps \,,
\end{equation}
where $\kernelsps$ is the kernel for the rotation group. As previously
argued, this should have the same form as the Euclidean kernel of
\eqn{eqn:B3}, restricted to the rotation subgroup of $SO(4)$. That is
\begin{equation}
\kernelsps = \sps{\delta} \kerneld \,.
\end{equation}
As in \eqn{eqn:A1a}, the propagator is given by integrating the
kernel over $\lambda$:
\begin{equation*}
\propsps = \sps{\delta} \prop \,,
\end{equation*}
where (using \eqn{eqn:A5b})
\begin{equation*}
\prop = \int_{\lambdaz}^{\infty} \dl\, \kerneld
= -\mi(2\pi)^{-4}\intfour p\,
\frac{\me^{\mi p\cdot(x-\xz)}}
{p^{2}+m^{2}-\mi\epsilon} \,,
\end{equation*}
the usual Feynman propagator \cite{seidewitz06a}. Defining
\begin{equation*}
\kets{x,n} \equiv \int_{\lambdaz}^{\infty} \dl\, \ketls{x,n}
\end{equation*}
then gives
\begin{equation} \label{eqn:C13}
\hisp\inner{x,n}{\xz,n;\lambdaz}\los = \propsps \,.
\end{equation}
Finally, we can inject the spin-representation basis states
$\ketls{x,n}$ back into the Lorentz group representation by
\begin{equation*}
\ketll{x,n} \equiv \ketls{x,n}\ulst(n)^{*} \,,
\end{equation*}
(and similarly for past-pointing $n$ with $\vlst$). Substituting
\eqn{eqn:C7} into this gives
\begin{equation} \label{eqn:C13a}
\ketll{x,n} = \ketxllp\Plpl(n) \,,
\end{equation}
where
\begin{equation} \label{eqn:C14}
\Plpl(n) \equiv \ulps(n)\ulst(n)^{*} = \vls(n)\vlst(n)^{*}
\end{equation}
(the last equality following from \eqn{eqn:C9}). Using
\eqns{eqn:C12a} and \eqref{eqn:C12b}, the kernel for these states is
\begin{equation*}
\hilp\inner{x,n;\lambda}{\xz,n;\lambdaz}\lol
= \Plpl(n) \kerneld \,.
\end{equation*}
However, using \eqns{eqn:C10} and \eqref{eqn:C11}, it can be shown
that the $\ketll{x,n}$ transform like the $\ketxll$:
\begin{equation*}
\UL \ketll{x,n}
= \ketllp{\Lambda x, \Lambda n}\, \Dmatlpl(\lambda) \,.
\end{equation*}
Taking
\begin{equation*}
\ket{x,n}\lol \equiv \int_{\lambdaz}^{\infty} \dl\, \ketll{x,n}
\end{equation*}
and using \eqn{eqn:C13} gives the propagator
\begin{equation} \label{eqn:C15}
\hilp\inner{x,n}{\xz,n;\lambdaz}\lol = \Plpl(n) \prop \,.
\end{equation}
Now, the $\ketll{x,n}$ do not span the full Lorentz group Hilbert
space vector bundle of the $\ketxll$, but they do span the subspace
corresponding to the rotation subgroup. Therefore, using
\eqn{eqn:C13a} and the idempotency of $\Plpl(n)$ as a projection
matrix,
\begin{equation} \label{eqn:C16}
\begin{split}
\ket{x,n}\lol &= \intfour \xz\,
\hilp\inner{\xz,n;\lambdaz}{x,n}\lol
\ketlz{\xz,n}\lolp \\
&= \intfour \xz\, \Plpl(n)\prop^{*}
\hilo{P}{\lbar'}{l'}(n) \ketxlz_{\lbar'} \\
&= \intfour \xz\, \Plpl(n)\prop^{*} \ketxlzlp \,.
\end{split}
\end{equation}
\section{Particles and Antiparticles}
\label{sect:non-scalar:antiparticles}
Because of \eqn{eqn:C13}, the states $\kets{x,n}$ allow for a
straightforward generalization of the treatment of particles and
antiparticles from \refcite{seidewitz06a} to the non-scalar case. As in
that treatment, consider particles to propagate \emph{from} the past
\emph{to} the future while antiparticles propagate from the
\emph{future} into the \emph{past} \cite{stueckelberg41,
stueckelberg42, feynman49}. Therefore, postulate non-scalar particle
states $\ketans{x}$ and antiparticle states $\ketrns{x}$ as follows.
\begin{postulate}
Normal particle states $\ketans{x}$ are such that
\begin{equation*}
\hisp\inner{\adv{x},n}{\xz,n;\lambdaz}\los
= \thetaax \propsps
= \thetaax \propasps \,,
\end{equation*}
and antiparticle states $\ketrns{x}$ are such that
\begin{equation*}
\hisp\inner{\ret{x},n}{\xz,n;\lambdaz}\los
= \thetarx \propsps
= \thetarx \proprsps \,,
\end{equation*}
where $\theta$ is the Heaviside step function, $\theta(x) = 0$,
for $x < 0$, and $\theta(x) = 1$, for $x > 0$, and
\begin{equation*}
\proparsps = \sps{\delta} (2\pi)^{-3}
\intthree p\, (2\Ep)^{-1}
\me^{\mi[\mp\Ep(x^{0}-\xz^{0})
+\threep\cdot(\threex-\threex_{0})]} \,,
\end{equation*}
with $\Ep \equiv \sqrt{\threep^{2} + m^{2}}$.
\end{postulate}
Note that the vector $n$ used here is timelike but otherwise
arbitrary, with no commitment that it be, e.g., future-pointing for
particles and past-pointing for antiparticles.
This division into particle and antiparticle paths depends, of course,
on the choice of a specific coordinate system in which to define the
time coordinate. However, if we take the time limit of the end point
of the path to infinity for particles and negative infinity for
antiparticles, then the particle/antiparticle distinction will be
coordinate system independent.
In taking this time limit, one cannot expect to hold the 3-position of
the path end point constant. However, for a free particle, it is
reasonable to take the particle \emph{3-momentum} as being fixed.
Therefore, consider the state of a particle or antiparticle with a
3-momentum $\threep$ at a certain time $t$.
\begin{postulate}
The state of a particle ($+$) or antiparticle ($-$) with
3-momentum $\threep$ is given by
\begin{equation*}
\ketarns{t,\threep}
\equiv (2\pi)^{-3/2} \intthree x\,
\me^{\mi(\mp\Ep t + \threep\cdot\threex)}
\ketarns{t,\threex} \,.
\end{equation*}
\end{postulate}
Now, following the derivation in \refcite{seidewitz06a}, but carrying
along the spin indices, gives
\begin{equation} \label{eqn:D1}
\begin{split}
\ketans{t,\threep} &=
(2\Ep)^{-1} \int_{-\infty}^{t} \dt_{0}\,
\ketanlzs{t_{0},\threep} \quad \text{and} \\
\ketrns{t,\threep} &=
(2\Ep)^{-1} \int_{t}^{+\infty} \dt_{0}\,
\ketrnlzs{t_{0},\threep} \,,
\end{split}
\end{equation}
where
\begin{equation} \label{eqn:D1a}
\ketarnlzs{t,\threep}
\equiv (2\pi)^{-3/2} \intthree x\,
\me^{\mi(\mp\Ep t + \threep\cdot\threex)}
\ketlzs{t,\threex,n} \,.
\end{equation}
Since
\begin{equation*}
\hisp\inner{\advret{t',\threepp},n; \lambdaz}
{\advret{t,\threep},n; \lambdaz}\los =
\sps{\delta} \delta(t'-t) \delta^{3}(\threepp - \threep) \,,
\end{equation*}
we have, from \eqn{eqn:D1},
\begin{equation*}
\hisp\inner{\advret{t,\threep},n}
{\advret{t_{0}, \threep_{0}{}},n; \lambdaz}\los =
(2\Ep)^{-1} \sps{\delta} \theta(\pm(t-t_{0}))
\delta^{3}(\threep - \threep_{0}) \,.
\end{equation*}
Defining the time limit particle and antiparticle states
\begin{equation} \label{eqn:D2}
\ketarthreepns \equiv \lim_{t \to \pm\infty}
\ketarns{t,\threep} \,,
\end{equation}
then gives
\begin{equation} \label{eqn:D3}
\hisp\inner{\advret{\threep},n}
{\advret{t_{0}, \threep_{0}{},n}; \lambdaz}\los
= (2\Ep)^{-1} \sps{\delta}
\delta^{3}(\threep - \threep_{0}) \,,
\end{equation}
for \emph{any} value of $t_{0}$.
Further, writing
\begin{equation*}
\ketarnlzs{t_{0}, \threep}
= (2\pi)^{-1/2} \me^{\mp\mi\Ep t_{0}}
\int \dif p^{0}\, \me^{\mi p^{0}t_{0}}
\ketlzs{p,n} \,,
\end{equation*}
where
\begin{equation} \label{eqn:D4}
\ketlzs{p,n} \equiv (2\pi)^{-2} \intfour x\, \me^{\mi p \cdot x}
\ketlzs{x,n}
\end{equation}
is the corresponding 4-momentum state, it is straightforward to see
from \eqn{eqn:D1} that the time limit of \eqn{eqn:D2} is
\begin{equation} \label{eqn:D5}
\ketarthreepns \equiv \lim_{t \to \pm\infty} \ketarns{t,\threep}
= (2\pi)^{1/2} (2\Ep)^{-1} \ketarnlzs{\pm\Ep,\threep} \,.
\end{equation}
Thus, a normal particle ($+$) or antiparticle ($-$) that has
3-momentum $\threep$ as $t \to \pm\infty$ is \emph{on-shell}, with
energy $\pm\Ep$. Such on-shell particles are unambiguously normal
particles or antiparticles.
For the on-shell states $\ketarthreepns$, it now becomes reasonable to
introduce the usual convention of taking the on-shell momentum vector
as the spin vector. That is, set $\npar \equiv (\pm\Ep, \threep)/m$
and define
\begin{equation*}
\varketarthreep\los \equiv \kets{\advret{\threep},\npar}
\end{equation*}
and
\begin{equation*}
\varketar{t,\threep}\los \equiv \kets{t,\advret{\threep},\npar} \,,
\end{equation*}
so that
\begin{equation*}
\varketarthreep\los =
\lim_{t\to\pm\infty} \varketar{t,\threep}\los \,.
\end{equation*}
Further, define the position
states
\begin{equation} \label{eqn:D6}
\begin{split}
\varketax\lol
&\equiv (2\pi)^{-3/2}\intthree p\,
\me^{\mi(\Ep x^{0} - \threep\cdot\threex)}
\varketa{x^{0},\threep}\los \ulst(\npa)^{*}
\text{ and } \\
\varketrx\lol
&\equiv (2\pi)^{-3/2}\intthree p\,
\me^{\mi(-\Ep x^{0} - \threep\cdot\threex)}
\varketr{x^{0},\threep}\los \vlst(\npr)^{*} \,.
\end{split}
\end{equation}
Then, working the previous derivation backwards gives
\begin{equation*}
\hilp(\advret{x}\ketxlzl = \thetaarx \proparlpl \,,
\end{equation*}
where
\begin{equation*}
\proparlpl \equiv
(2\pi)^{-3} \intthree p\, \Plpl(\npar)
(2\Ep)^{-1} \me^{\mi[\pm\Ep (x^{0}-\xz^{0}) -
\threep\cdot(\threex-\threex_{0})]} \,.
\end{equation*}
Now, it is shown in \refcites{weinberg64a, weinberg95} that the
covariant non-scalar propagator
\begin{equation*}
\proplpl = -\mi(2\pi)^{-4} \intfour p\, \Plpl(p/m)
\frac{\me^{\mi p\cdot(x-\xz)}}{p^{2}+m^{2}-\mi\varepsilon} \,,
\end{equation*}
in which $\Plpl(p/m)$ has the polynomial form of $\Plpl(n)$, but $p$
is not constrained to be on-shell, can be decomposed into
\begin{equation*}
\proplpl = \thetaax\propalpl + \thetarx\proprlpl
+ \Qlpl\left(-\mi\pderiv{}{x}\right)
\mi\delta^{4}(x-\xz) \,,
\end{equation*}
where the form of $\Qlpl$ depends on any non-linearity of $\Plpl(p/m)$
in $p^{0}$. Then, defining
\begin{equation*}
\varketx\lol \equiv \intfour \xz\, \proplpl^{*} \ketxlz\lolp \,,
\end{equation*}
$\varketax\lol$ and $\varketrx\lol$ can be considered as a
particle/antiparticle partitioning of $\varketx\lol$, in a similar
way as the partitioning of $\ket{x,n}\los$ into $\keta{x,n}\los$ and
$\ketr{x,n}\los$:
\begin{equation*}
\begin{split}
\thetaarx\hilp(x\ketxlzl &= \thetaarx \proplpl \\
&= \thetaarx \proparlpl \\
&= \hilp(\advret{x}\ketxlzl \,.
\end{split}
\end{equation*}
Because of the delta function, the term in $\Qlpl$ does not
contribute for $x \neq \xz$.
The states $\ket{x,n}\lol$ and $\varketx\lol$ both transform according
to a representation $\Dmatlpl$ of the Lorentz group, but it is
important to distinguish between them. The $\ket{x,n}\lol$ are
projections back into the Lorentz group of the states $\kets{x,n}$
defined on the rotation subgroup, in which that subgroup is obtained
by uniformly reducing the Lorentz group about the axis given by $n$.
The $\varketx\lol$, on the other hand, are constructed by
inverse-transforming from the momentum states
$\varketar{t,\threep}\los$, with each superposed state defined over a
rotation subgroup reduced along a different on-shell momentum vector.
One can further highlight the relationship of the $\varketx\lol$ to
the momentum in the position representation by the formal equation
(using \eqn{eqn:C16})
\begin{equation*}
\varketx\lol = \intfour \xz\,
\Plpl\left(
\mi m^{-1} \pderiv{}{x}
\right)
\prop^{*} \ketxlzlp
= \ket{x, \mi m^{-1} \partial/\partial x}\lol
= \Plpl\left(
\mi m^{-1} \pderiv{}{x}
\right) \ketx\lolp \,.
\end{equation*}
The $\varketx\lol$ correspond to the position states used in
traditional relativistic quantum mechanics, with associated on-shell
momentum states $\varketarthreep$. However, we will see in the next
section that the states $\ket{x,n}\lol$ provide a better basis for
generalizing the scalar probability interpretation discussed in
\refcite{seidewitz06a}.
\section{On-Shell Probability Interpretation}
\label{sect:non-scalar:probability}
Similarly to the scalar case \cite{seidewitz06a}, let $\HilbH^{(j,n)}$
be the Hilbert space of the $\ketnlzs{x}$ for the spin-$j$
representation of the rotation group and a specific timelike vector
$n$, and let $\HilbH^{(j,n)}_{t}$ be the subspaces spanned by the
$\ketnlzs{t,\threex}$, for each $t$, forming a foliation of
$\HilbH^{(j,n)}$. Now, from \eqn{eqn:D1a}, it is clear that the
particle and antiparticle 3-momentum states $\ketarnlzs{t,\threep}$
also span $\HilbH^{(j,n)}_{t}$. Using these momentum bases, states in
$\HilbH^{(j,n)}_{t}$ have the form
\begin{equation*}
\ketarnlzs{t, \psi}
= \intthree p\, \sps{\psi}(\threep) \ketarnlzsp{t, \threep} \,,
\end{equation*}
for matrix functions $\psi$ such that $\tr(\psi\adj\psi)$ is
integrable. Conversely, it follows from \eqn{eqn:D3} that the
probability amplitude $\sps{\psi}(\threep)$ is given by
\begin{equation} \label{eqn:E0}
\sps{\psi}(\threep) = (2\Ep)\hisp\inner{\advret{\threep},n}
{\advret{t,\psi},n; \lambdaz}\los \,.
\end{equation}
Let $\HilbH^{\prime (j,n)}_{t}$ be the space of linear functions dual
to $\HilbH^{(j,n)}_{t}$. Via \eqn{eqn:E0}, the bra states
$\his\braathreep$ can be considered as spanning subspaces
$\advret{\HilbH}^{\prime (j,n)}$ of the $\HilbH^{\prime (j,n)}_{t}$,
with states of the form
\begin{equation*}
\his\bra{\advret{\psi},n}
= \intthree p\, \lohi{\psi}{\sigma'}{\sigma}(\threep)^{*}\;
\hisp\bra{\advret{\threep},n} \,.
\end{equation*}
The inner product
\begin{equation*}
(\psi_{1},\psi_{2})
\equiv \his\inner{\advret{\psi_{1}{}},n}
{\advret{t,\psi_{2}{}},n;\lambdaz}\los
= \int \frac{\dthree p}{2\Ep}
\lohi{\psi_{1}}{\sigma'}{\sigma}(\threep)^{*}
\sps{\psi_{2}}(\threep)
\end{equation*}
gives
\begin{equation*}
(\psi,\psi)
= \int \frac{\dthree p}{2\Ep}
\sum_{\sigma'\sigma} \sqr{\sps{\psi}(\threep)}
\geq 0 \,,
\end{equation*}
so that, with this inner product, the $\HilbH^{(j,n)}_{t}$ actually
are Hilbert spaces in their own right.
Further, \eqn{eqn:D3} is a \emph{bi-orthonormality relation} with the
corresponding resolution of the identity (see \refcite{akhiezer81} and
App.\ A.8.1 of \refcite{muynk02})
\begin{equation*} \label{eqn:E1}
\intthree p\,
(2\Ep) \ketarnlzs{t, \threep}\;\his\bra{\advret{\threep},n}
= 1 \,.
\end{equation*}
The operator $(2\Ep) \ketarnlzs{t, \threep}\;\his\braar{\threep,n}$
represents the quantum proposition that an on-shell, non-scalar
particle or antiparticle has 3-momentum $\threep$.
Like the $\lpl{\psi}$ discussed in \sect{sect:non-scalar:euclidean} for
the Lorentz group, the $\sps{\psi}$ form an algebra over the rotation
group with components $\psi(\threep, B)$, where $\Bsps$ is a member
of the appropriate representation of the rotation group, such that
\begin{equation} \label{eqn:E2}
\sps{\psi}(\threep)
= \intthree B\, \Bsps \psi(\threep, B) \,,
\end{equation}
with the integration taken over the 3-dimensional rotation group.
Unlike the Lorentz group, however, components can also be reconstructed
from the $\sps{\psi}(\threep)$ by
\begin{equation} \label{eqn:E3}
\psi(\threep, B) = \beta^{-1}\hilo{(B^{-1})}{\sigma}{\sigma'}
\sps{\psi}(\threep) \,
\end{equation}
where
\begin{equation*}
\beta \equiv \frac{1}{2j+1} \intthree B \,,
\end{equation*}
for a spin-$j$ representation, is finite because the rotation group is
closed. Plugging \eqn{eqn:E3} into the right side of \eqn{eqn:E2}
and evaluating the integral does, indeed, give $\sps{\psi}(\threep)$,
as required, because of the orthogonality property
\begin{equation*}
\intthree B\, \Bsps \hilo{(B^{-1})}{\sbar}{\sbar'}
= \beta \hilo{\delta}{\sigma'}{\sbar'} \lohi{\delta}{\sigma}{\sbar}
\end{equation*}
(see \refcite{weyl50}, Section 11). We can now adjust the group
volume measure $\dthree B$ so that $\beta = 1$.
The set of all $\psi(\threep, B)$ constructed as in \eqn{eqn:E3} forms
a subalgebra such that each $\psi(\threep, B)$ is uniquely determined
by the corresponding $\sps{\psi}(\threep)$ (see \refcite{weyl50},
pages 167ff). We can then take $\sqr{\psi(\threep, B)} =
\sqr{\hilo{(B^{-1})}{\sigma}{\sigma'}\sps{\psi}(\threep)}$ to be the
probability density for the particle or antiparticle to have
3-momentum $\threep$ and to be rotated as given by $B$ about the axis
given by the spacial part of the unit timelike 4-vector $n$. The
probability density for the particle or antiparticle in 3-momentum
space is
\begin{equation*}
\intthree B\, \sqr{\psi(\threep, B)}
= \lohi{\psi}{\sigma'}{\sigma}(\threep)^{*}
\sps{\psi}(\threep) \,
\end{equation*}
with the normalization
\begin{equation*}
(\psi,\psi)
= \int \frac{\dthree p}{2\Ep}
\lohi{\psi}{\sigma'}{\sigma}(\threep)^{*}
\sps{\psi}(\threep)
= 1 \,.
\end{equation*}
Next, consider that $\ketnlzs{t,\threex}$ is an eigenstate of the
three-position operator $\op{\threevec{X}}$, representing a particle
localized at the three-position $\threex$ at time $t$. From
\eqn{eqn:E0}, and using the inverse Fourier transform of \eqn{eqn:D4}
with \eqn{eqn:D5}, its three momentum wave function is
\begin{equation} \label{eqn:E4}
(2\Ep)\, \hisp\inner{\advret{\threep},n}
{t,\threex;\lambdaz}\los
= (2\pi)^{-3/2} \sps{\delta}
\me^{\mi(\pm\Ep t - \threep\cdot\threex)} \,.
\end{equation}
This is just a plane wave, and it is an eigenfunction of the operator
\begin{equation*}
\me^{\pm\mi\Ep t} \mi \pderiv{}{\threep} \me^{\mp\mi\Ep t} \,,
\end{equation*}
which acts as the identity on the spin indices and is otherwise the
traditional momentum representation $\mi \partial/\partial\threep$ of
the three-position operator $\op{\threevec{X}}$, translated to time
$t$.
This result exactly parallels that of the scalar case
\cite{seidewitz06a}. Note that this is only so because of the use of
the independent vector $n$ for reduction to the rotation group,
rather than the traditional approach of using the three-momentum
vector $\threep$. Indeed, it is not even possible to define a
spin-indexed position eigenstate in the traditional approach,
because, of course, the momentum is not sharply defined for such a
state \cite{piron78, horwitz82}.
On the other hand, consider the three-position states $\varketarx\lol$
introduced at the end of \sect{sect:non-scalar:antiparticles}. Even
though these are Lorentz-indexed, they only span the rotation
subgroup. Therefore, we can form their three-momentum wave functions
in the $\his\varbra{\advret{\threep}}$ bases. Using \eqns{eqn:D6} and
\eqref{eqn:D3},
\begin{equation} \label{eqn:E5}
(2\Ep)\, \his\varinner{\advret{\threep}}{\advret{x}}\lol
= (2\pi)^{-3/2} \ulst(\np)^{*}
\me^{\mi(\pm\Ep t - \threep\cdot\threex)} \,.
\end{equation}
At $t = 0$, up to normalization factors of powers of $(2\Ep)$, this is
just the Newton-Wigner wave function for a localized particle of
non-zero spin \cite{newton49}. It is an eigenfunction of the position
operator represented as
\begin{equation} \label{eqn:E6}
\ulpspt(\np)^{*} \me^{\mi\Ep t} \mi \pderiv{}{\threep}
\me^{-\mi\Ep t} \ulsp(\np)
\end{equation}
for the particle case, with a similar expression using $\vls$ in the
antiparticle case. Other than the time translation, this is
essentially the Newton-Wigner position operator for non-zero spin
\cite{newton49}.
Note that \eqn{eqn:E4} is effectively related to \eqn{eqn:E5} by a
generalized Foldy-Wouthuysen transformation \cite{foldy50, case54}.
However, in the present approach it is \eqn{eqn:E4} that is seen to
be the primary result, with a natural separation of particle and
antiparticle states and a reasonable non-relativistic limit, just as
in the scalar case \cite{seidewitz06a}.
\section{Interactions} \label{sect:non-scalar:interactions}
It is now straightforward to extend the formalism to multiparticle
states and introduce interactions, quite analogously to the scalar
case \cite{seidewitz06a}. In order to allow for multiparticle states
with different types of particles, extend the position state of each
individual particle with a \emph{particle type index} $\nbase$, such
that
\begin{equation*}
\hilp\inner{x',\nbase';\lambda}{x,\nbase;\lambda}\lol
= \delta^{l'}_{l}\delta^{\nbase'}_{\nbase}\delta^{4}(x'-x) \,.
\end{equation*}
Then, construct a basis for the Fock space of multiparticle states as
sym\-me\-trized/anti\-sym\-me\-trized products of $N$ single particle
states:
\begin{multline*}
\ket{\xnliN}\lolN
\equiv (N!)^{-1/2} \sum_{\text{perms }\Perm} \delta_{\Perm}
\ket{\xni{\Perm 1};\lambda_{\Perm 1}}\loli{\Perm 1} \cdots \\
\ket{\xni{\Perm N};\lambda_{\Perm N}}\loli{\Perm N} \,,
\end{multline*}
where the sum is over permutations $\Perm$ of $1,\ldots,N$, and
$\delta_{\Perm}$ is $+1$ for permutations with an even number of
interchanges of fermions and $-1$ for an odd number of interchanges.
Define multiparticle states $\ket{\xniN}\lolN$ as similarly
sym\-me\-trized/anti\-sym\-me\-trized products of $\ketx\lol$ states.
Then,
\begin{equation} \label{eqn:F1}
\hilpN\inner{\xnpiN}{\seqN{\xnlzi}}\lolN
= \sum_{\text{perms }\Perm} \delta_{\Perm}
\prod_{i = 1}^{N}
\proplplij{\Perm i}{i} \,,
\end{equation}
where each propagator is also implicitly a function of the mass of the
appropriate type of particle. Note that the use of the same parameter
value $\lambdaz$ for the starting point of each particle path is
simply a matter of convenience. The intrinsic length of each particle
path is still integrated over \emph{separately} in $\ket{\xniN}\lolN$,
which is important for obtaining the proper particle propagator
factors in \eqn{eqn:F1}. Nevertheless, by using $\lambdaz$ as a common
starting parameter, we can adopt a similar notation simplification as
in \refcite{seidewitz06a}, defining
\begin{equation*}
\ket{\xnlziN}\lolN \equiv \ket{\seqN{\xnlzi}}\lolN \,.
\end{equation*}
It is also convenient to introduce the formalism of creation and
annihilation fields for these multiparticle states. Specifically,
define the creation field $\oppsitl(x,\nbase;\lambda)$ by
\begin{equation*}
\oppsitl(x,\nbase;\lambda)\ket{\xnliN}\lolN
= \ket{x,\nbase,\lambda;\xnliN}_{l,\listN{l}} \,,
\end{equation*}
with the corresponding annihilation field $\oppsil(x,\nbase;\lambda)$
having the commutation relation
\begin{equation*}
[\oppsilp(x',\nbase';\lambda), \oppsitl(x,\nbase;\lambdaz)]_{\mp}
= \delta^{\nbase'}_{\nbase}\propsymlpl(x'-x;\lambda-\lambdaz) \,,
\end{equation*}
where the upper $-$ is for bosons and the lower $+$ is for fermions.
Further define
\begin{equation*}
\oppsil(x,\nbase) \equiv
\int_{\lambdaz}^{\infty} \dl\, \oppsil(x,\nbase;\lambda) \,,
\end{equation*}
so that
\begin{equation*}
[\oppsilp(x',\nbase'), \oppsitl(x,\nbase;\lambdaz)]_{\mp}
= \delta^{\nbase'}_{\nbase}\propsymlpl(x'-x) \,,
\end{equation*}
which is consistent with the multi-particle inner product as given in
\eqn{eqn:F1}. Finally, as in \refcite{seidewitz06a}, define a
\emph{special adjoint} $\oppsi\dadj$ by
\begin{equation} \label{eqn:F2}
\oppsi\dadj\lol(x,\nbase) = \oppsitl(x,\nbase;\lambdaz) \text{ and }
\oppsi\dadj\lol(x,\nbase;\lambdaz) = \oppsitl(x,\nbase) \,,
\end{equation}
which allows the commutation relation to be expressed in the more
symmetric form
\begin{equation*}
[\oppsilp(x',\nbase'), \oppsi\dadj\lol(x,\nbase)]_{\mp}
= \delta^{\nbase'}_{\nbase}\propsymlpl(x'-x) \,.
\end{equation*}
We can now readily generalize the postulated interaction vertex
operator of \refcite{seidewitz06a} to the non-scalar case.
\begin{postulate}
An interaction vertex, possibly occurring at any position in
spacetime, with some number $a$ of incoming particles and some
number $b$ of outgoing particles, is represented by the operator
\begin{equation} \label{eqn:F3}
\opV \equiv g\hilpn{a}{}\loln{b} \intfour x\,
\prod_{i = 1}^{a} \oppsi\dadj_{l'_{i}}(x,\nbase'_{i})
\prod_{j = 1}^{b} \oppsi^{l_{j}}(x,\nbase_{j}) \,,
\end{equation}
where the coefficients $g\hilpn{a}{}\loln{b}$ represent the
relative probability amplitudes of various combinations of indices
in the interaction and $\oppsi\dadj$ is the special adjoint
defined in \eqn{eqn:F2}.
\end{postulate}
Given a vertex operator defined as in \eqn{eqn:F3}, the interacting
transition amplitude, with any number of intermediate interactions,
is then
\begin{multline} \label{eqn:F4}
G(\xnpiNp | \xniN)\hilpn{N'}{}\lolN \\
= \hilpn{N'}\bra{\xnpiN} \opG \ket{\xnlziN}\lolN \,,
\end{multline}
where
\begin{equation*}
\opG \equiv \sum_{m=0}^{\infty} \frac{(-\mi)^{m}}{m!}\opV^{m}
= \me^{-i\opV} \,.
\end{equation*}
Each term in this sum gives the amplitude for $m$ interactions,
represented by $m$ applications of $\opV$. The $(m!)^{-1}$ factor
accounts for all possible permutations of the $m$ identical factors
of $\opV$.
Clearly, we can also construct on-shell multiparticle states
$\ket{\parnpiN}\lospn{N'}$ and $\ket{\tparnlziN}\losN$ from the
on-shell particle and antiparticle states $\ketarthreep\los$ and
$\ketarlz{t,\threep}\los$. Using these with the operator $\opG$:
\begin{multline} \label{eqn:F5}
G(\parnpiN | \parniN)\hispn{N'}{}\losN \\
\equiv \left[ \prod_{i=1}^{N'} 2\E{\threepp_{i}} \right]
\hispn{N'}\bra{\parnpiN} \opG \ket{\tparnlziN}\losN \,,
\end{multline}
results in a sum of Feynman diagrams with the given momenta on
external legs. Note that use of the on-shell states requires
specifically identifying external lines as particles and
antiparticles. For each incoming and outgoing particle, $+$ is chosen
if it is a normal particle and $-$ if it is an antiparticle. (Note
that ``incoming'' and ``outgoing'' here are in terms of the path
evolution parameter $\lambda$, \emph{not} time.)
The inner products of the on-shell states for individual incoming and
outgoing particles with the off-shell states for interaction vertices
give the proper factors for the external lines of a Feynman diagram.
For example, the on-shell state $\keta{\threepp}\los$ is obtained in
the $+\infty$ time limit and thus represents a \emph{final} (i.e.,
outgoing in \emph{time}) particle. If the external line for this
particle starts at an interaction vertex $x$, then the line
contributes a factor
\begin{equation*}
(2\Epp) \hisp\inner{\adv{\threepp}}{x;\lambdaz}\lol
= (2\pi)^{-3/2}
\me^{\mi(+\Epp x^{0} - \threepp \cdot \threex)}
\ulspt(\threepp)^{*} \,.
\end{equation*}
For an incoming particle on an external line ending at an
interaction vertex $x'$, the factor for this line is (assuming
$x^{\prime 0} > t$)
\begin{equation*}
(2\Ep) \hilp\inner{x'}{\adv{t,\threep};\lambdaz}\los
= (2\pi)^{-3/2}
\me^{\mi(-\Ep x^{\prime 0} + \threep \cdot \threexp)}
\ulps(\threep) \,.
\end{equation*}
Note that this expression is independent of $t$, so we can take $t \to
-\infty$ and treat the particle as \emph{initial} (i.e., incoming in
time). The factors for antiparticles are similar, but with the time
sense reversed. Thus, the effect is to remove the propagator factors
from external lines, exactly in the sense of the usual LSZ reduction
\cite{lsz55}.
Now, the formulation of \eqn{eqn:F5} is still not that of the usual
scattering matrix, since the incoming state involves initial particles
but final antiparticles, and vice versa for the outgoing state. To
construct the usual scattering matrix, it is necessary to have
multi-particle states that involve either all initial particles and
antiparticles (that is, they are composed of individual asymptotic
particle states that are all consistently for $t \to -\infty$) or all
final particles and antiparticles (with individual asymptotic states
all for $t \to +\infty$). The result is a formulation in terms of the
more familiar scattering operator $\opS$, which can be expanded in a
Dyson series in terms of a time-dependent version $\opV(t)$ of the
interaction operator. The procedure for doing this is exactly
analogous to the scalar case. For details see \refcite{seidewitz06a}.
\section{Conclusion} \label{sect:conclusion}
The extension made here of the scalar spacetime path approach
\cite{seidewitz06a} begins with the argument in \sect{sect:background}
on the form of the path propagator based on Poincar\'e invariance.
This motivates the use of a path integral over the Poincar\'e group,
with both position and Lorentz group variables, for computation of the
non-scalar propagator. Once the difficulty with the non-compactness of
the Lorentz group is overcome, the development for the non-scalar case
is remarkably parallel to the scalar case.
A natural further generalization of the approach, particularly given
its potential application to quantum gravity and cosmology, would be
to consider paths in curved spacetime. Of course, in this case it is
not in general possible to construct a family of parallel paths over
the entire spacetime, as was done in \sect{sect:non-scalar:propagator}.
Nevertheless, it is still possible to consider infinitesimal
variations along a path corresponding to arbitrary coordinate
transformations. And one can certainly construct a family of
``parallel'' paths at least over any one coordinate patch on the
spacetime manifold. The implications of this for piecing together a
complete path integral will be explored in future work.
Another direction for generalization is to consider massless
particles, leading to a complete spacetime path formulation for
Quantum Electrodynamics. However, as has been shown in previous work
on relativistically parametrized approaches to QED (e.g.,
\refcite{shnerb93}), the resulting gauge symmetries need to be handled
carefully. This will likely be even more so if consideration is
further extended to non-Abelian interactions. Nevertheless, the
spacetime path approach may provide some interesting opportunities for
addressing renormalization issues in these cases \cite{seidewitz06a}.
In any case, the present paper shows that the formalism proposed in
\refcite{seidewitz06a} can naturally include non-scalar particles. This
is, of course, critical if the approach is to be given the
foundational status considered in \refcite{seidewitz06a} and the
cosmological interpretation discussed in \refcite{seidewitz06b}.
\endinput
\appendix*
\section{Evaluation of the $SO(4)$ Path Integral}
\label{app:path}
\begin{theorem*}
Consider the path integral
\begin{multline*}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
= \euc{\zeta} \intDsix \ME\,
\delta^{6}(\ME(\lambda)\LambdaE^{-1}-I)
\delta^{6}(\ME(\lambdaz)\LambdaEz^{-1}-I) \\
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \tr(\OmegaE(\lambda')
\OmegaE(\lambda')\T)
\right]
\end{multline*}
over the six dimensional group $SO(4) \sim SU(2) \times SU(2)$,
where $\OmegaE(\lambda')$ is the element of the Lie algebra $so(4)$
tangent to the path $\ME(\lambda)$ at $\lambda'$. This path integral
may be evaluated to get
\begin{multline} \label{eqn:A1A}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz) \\
= \sum_{\ell_{A},\ell_{B}}
\exp^{-\mi( \Delta m_{\ell_{A}}^{2}
+ \Delta m_{\ell_{B}}^{2})
(\lambda - \lambdaz)}
(2\ell_{A}+1)(2\ell_{B}+1)
\chi^{(\ell_{A}\ell_{B})}(\LambdaE\LambdaEz^{-1}) \,,
\end{multline}
where the summation over $\ell_{A}$ and $\ell_{B}$ is from $0$ to
$\infty$ in steps of $1/2$, $\Delta m_{\ell}^{2} = \ell(\ell+1)$
and $\chi^{(\ell_{A},\ell_{B})}$ is the group character for the
$(\ell_{A},\ell_{B})$ $SU(2) \times SU(2)$ group representation.
\end{theorem*}
\begin{proof}
Parametrize a group element $\ME$ by a six-vector $\theta$ such
that
\begin{equation*}
\ME = \exp(\sum_{i=1}^{6}\theta_{i}J_{i}) \,,
\end{equation*}
where the $J_{i}$ are $so(4)$ generators for $SO(4)$. Then
$\tr(\OmegaE\OmegaE\T) = \dot{\theta}^{2}$, where the dot denotes
differentiation with respect to $\lambda$. Dividing the six
generators $J_{i}$ into two sets of three $SU(2)$ generators, the
six-vector $\theta$ may be divided into two three-vectors
$\theta_{A}$ and $\theta_{B}$, parametrizing the two $SU(2)$
subgroups. The path integral then factors into two path integrals
over $SU(2)$:
\begin{multline*}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz) \\
= \euc{\zeta}^{1/2} \intDthree W_{A}\,
\delta^{3}(W_{A}(\lambda)B_{A}^{-1}-I)
\delta^{6}(W_{A}(\lambdaz)B_{A0}^{-1}-I)
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \dot{\theta_{A}}^{2})
\right] \\
\times \euc{\zeta}^{1/2} \intDthree W_{B}\,
\delta^{3}(W_{B}(\lambda)B_{B}^{-1}-I)
\delta^{6}(W_{B}(\lambdaz)B_{B0}^{-1}-I)
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \dot{\theta_{B}}^{2})
\right] \,,
\end{multline*}
where $\LambdaE = B_{A} \otimes B_{B}$ and $\LambdaEz = B_{A0}
\otimes B_{B0}$.
The $SU(2)$ path integrals may be computed by expanding the
exponential in group characters \cite{kleinert06,bohm87}. The
result is
\begin{multline} \label{eqn:A2A}
\euc{\zeta}^{1/2} \intDthree W\,
\delta^{3}(W(\lambda)B^{-1}-I)
\delta^{6}(W(\lambdaz)B_{0}^{-1}-I)
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \dot{\theta}^{2})
\right] \\
= \sum_{\ell}\exp^{-\mi\Delta m_{\ell}^{2}
(\lambda - \lambdaz)}
(2\ell+1)
\chi^{(\ell)}(B B_{0}^{-1}) \,,
\end{multline}
where $\chi^{(\ell)}$ is the character for the spin-$\ell$
representation of $SU(2)$ and the result includes the correction
for integration ``on'' the group space, as given by Kleinert
\cite{kleinert06}. The full $SO(4)$ path integral is then given by
the product of the two factors of the form \eqn{eqn:A2A}, which is
just \eqn{eqn:A1A}, since \cite{weyl50}
\begin{equation*}
\chi^{(\ell_{A},\ell_{B})}(\LambdaE\LambdaEz^{-1}) =
\chi^{(\ell_{A})}(B_{A}B_{A0}^{-1})
\chi^{(\ell_{B})}(B_{B}B_{B0}^{-1}) \,.
\end{equation*}
\end{proof}
\endinput
\end{document}
|
\begin{document}
\title{On finite groups with few automorphism orbits}
\begin{abstract}
Denote by $\omega(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $\omega(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $\omega(G) = 6$ and show that if $G$ is a nonsolvable finite group with $\omega(G) = 6$, then either $G \simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.
\end{abstract}
\title{On finite groups with few automorphism orbits}
\section{Introduction}
The groups considered in the following are finite. The problem of the classification of the groups with a prescribed number of {\emph conjugacy classes} was suggested in \cite{B}. For more details for this problem we refer the reader to \cite{VS}. In this paper we consider an other related invariant. Denote by $\omega(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. If $\omega(G) = n$, then we say that $G$ has $n$ automorphism orbits. The trivial group is the only group with $\omega(G) = 1$. It is clear that $\omega(G) = 2$ if and only if $G$ is an elementary abelian $p$-group, for some prime number $p$ \cite[3.13]{D}. In \cite{LD}, Laffey and MacHale give the following results:
\begin{itemize}
\item[(i)] Let $G$ be a finite group which is not of prime-power order. If $w(G) = 3$, then $|G| = p^nq$ and $G$ has a normal elementary abelian Sylow $p$-subgroup $P$, for some primes $p$, $q$, and for some integer $n > 1$. Furthermore, $p$ is a primitive root mod $q$.
\item[(ii)] If $w(G)\leqslant 4$ in a group $G$, then either $G$ is solvable or $G$ is isomorphic to $A_5$.
\end{itemize}
Stroppel in \cite[Theorem 4.5]{S} has shown that if $G$ is a nonabelian simple group with $\omega(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5$, $A_6$, $PSL(2,7)$ or $PSL(2,8)$. In the same work he suggested the following problem: \\
{\noindent}{\bf Problem.}{
(Stroppel \cite[Problem 9.9]{S}) Determine the finite nonsolvable groups $G$ in which $\omega(G) \leqslant 6$.} \\
In answer to Stroppel's question, we give a complete classification for the case $\omega(G)\leq 5$ in Theorem A and provide a characterization of $G$ when $\omega(G)= 6$ in Theorem B. Precisely:
\begin{thmA}\label{th.A}
Let $G$ be a nonsolvable group in which $\omega(G) \leqslant 5$. Then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$.
\end{thmA}
Using GAP, we obtained an example of a nonsolvable and non simple group $G$ in which $\omega(G) = 6$ and $|G|=960$. Moreover, there exists a characteristic subgroup $N$ of $G$ such that $G/N \simeq A_5$, where $N$ is an elementary abelian $2$-subgroup. Actually, we will prove that this is the case for any nonsolvable non simple group with $6$ automorfism orbits.
\begin{thmB} \label{th.B}
Let $G$ be a nonsolvable group in which $\omega(G) = 6$. Then one of the following holds:
\end{thmB}
\begin{itemize}
\item[(i)] $G \simeq PSL(3,4)$;
\item[(ii)] There exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.
\end{itemize}
According to Landau's result \cite[Theorem 4.31]{Rose}, for every positive integer $n$, there are only finitely many groups with exactly $n$ conjugacy classes. It is easy to see that no exists similar result for automorphism orbits. Nevertheless, using the classification of finite simple groups, Kohl has been able to prove that for every positive integer $n$ there are only finitely many nonabelian simple groups with exactly $n$ automorphism orbits \cite[Theorem 2.1]{Kohl}. This suggests the following question. \\
{\noindent}{\it Are there only finitely many nonsolvable groups with $6$ automorphism orbits?}
\section{Preliminary results}
A group $G$ is called AT-group if all elements of the same order are conjugate in the automorphism groups. The following result is a straightforward corollary of \cite[Theorem 3.1]{Z}.
\begin{lem} \label{Z.th}
Let $G$ be a nonsolvable AT-group in which $\omega(G) \leqslant 6$. Then $G$ is simple. Moreover, $G$ is isomorphic to one of the groups $A_5$, $A_6$, $PSL(2,7)$, $PSL(2,8)$ or $PSL(3,4)$.
\end{lem}
The spectrum of a group is the set of orders of its elements. Let us denote by $spec(G)$ the spectrum of the group $G$.
\begin{rem} \label{rem.spec} The maximal subgroups of $A_5$, $A_6$, $PSL(2,7)$, $PSL(2,8)$ and $PSL(3,4)$ are well know (see for instance \cite{Atlas}). Then
\begin{itemize}
\item[(i)] $spec(A_5) = \{1,2,3,5\}$;
\item[(ii)] $spec(A_6) = \{1,2,3,4,5\}$;
\item[(iii)] $spec(PSL(2,7)) = \{1,2,3,4,7\}$;
\item[(iv)] $spec(PSL(2,8)) = \{1,2,3,7,9\}$;
\item[(v)] $spec(PSL(3,4)) = \{1,2,3,4,5,7\}$
\end{itemize}
\end{rem}
For a group $G$ we denote by $\pi(G)$ the set of prime divisors of the orders of the elements of $G$.
Recall that a group $G$ is a characteristically simple group if $G$ has no proper nontrivial characteristic subgroups.
\begin{lem} \label{ch.simple}
Let $G$ be a nonabelian group. If $G$ is a characteristically simple group in which $\omega(G) \leqslant 6$, then $G$ is simple.
\end{lem}
\begin{proof}
Suppose that $G$ is not simple. By \cite[Theorem 1.5]{G}, there exist a nonabelian simple subgroup $H$ and an integer $k \geqslant 2$ such that
$$
G = \underbrace{H \times \ldots \times H}_{k \ times}.
$$
By Burnside's Theorem \cite[p. 131]{G}, $\pi(G) =\{p_1,\ldots, p_s\}$, where $s \geqslant 3$. Then, there are elements in $G$ of order $p_ip_j$, where $i,j \in \{1,\ldots,s\}$ and $i \neq j$. Thus, $\omega(G) \geqslant 7$.
\end{proof}
\begin{lem} \label{simple-lem}
Let $G$ be a nonsolvable group and $N$ a characteristic subgroup of $G$. Assume that $|\pi(G)| = 4$ and $N$ is isomorphic to one of the groups $A_5, A_6, PSL(2,7)$ or $PSL(2,8)$. Then $\omega(G) \geqslant 8$.
\end{lem}
\begin{proof}
Let $P$ be a Sylow $p$-subgroup of $G$, where $p \not\in \pi(N)$. Set $M = NP$. Since $p$ and $|Aut(N)|$ have coprime orders, we conclude that $M = N \times P$. Arguing as in the proof of Lemma \ref{ch.simple} we deduce that $\omega(G) \geqslant 8$.
\end{proof}
\begin{rem} \label{prop.spec}
(Stroppel, \cite[Lemma 1.2]{S}) Let $G$ be a nontrivial group and $K$ a characteristic subgroup of $G$. Then
$$
\omega(G) \geqslant \omega(K)+ \omega(G/K) - 1.
$$
\end{rem}
\begin{lem} \label{lem.spec}
Let $G$ be a nonsolvable group in which $|spec(G)|\geqslant 6$. Then either $G$ is simple or $\omega(G) \geqslant 7$.
\end{lem}
\begin{proof}
Assume that $\omega(G)= 6$. Then, $G$ is AT-group. By Lemma \ref{Z.th}, $G$ is simple. Moreover, $G \simeq PSL(3,4)$.
\end{proof}
\begin{prop} \label{A5-prop}
Let $G$ be a group and $N$ a proper characteristic subgroup of $G$. If $N$ is isomorphic to one of the following groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$, then $\omega(G) \geqslant 7$.
\end{prop}
\begin{proof}
By Lemma \ref{simple-lem}, there is no loss of generality in assuming $\pi(G) = \pi(N)$. In particular, there is a subgroup $M$ in $G$ such that $|M| = p|N|$, for some prime $p \in \pi(G)$. Excluding the case $N \simeq PSL(2,8)$ and $p=7$, a GAP computation shows that $|spec(M)| \geqslant 6$. By Lemma \ref{lem.spec}, $\omega(G) \geqslant 7$. Finally, if $|M| = 7|N|$, where $N \simeq PSL(2,8)$, then $M/C_M(N) \lesssim Aut(N)$. Since $\pi(Aut(N)) = \pi(N)$, we have $C_M(N) \neq \{1\}$. Thus, $\omega(G) \geqslant 7$.
\end{proof}
The following result gives us a description of all nonabelian simple groups with at most $5$ automorphism orbits.
\begin{thm} (Stroppel, \cite[Theorem 4.5]{S}) \label{S.th}
Let $G$ be a non-abelian simple group in which $\omega(G) \leqslant 5$. Then $G$ is isomorphic to one of the groups $A_5$, $A_6$, $PSL(2,7)$ or $PSL(2,8)$.
\end{thm}
\section{Proofs of the main results}
\begin{thmA}
Let $G$ be a nonsolvable group in which $\omega(G) \leqslant 5$. Then $G$ is isomorphic to one of the following groups $A_5$, $A_6$, $PSL(2,7)$ or $PSL(2,8)$.
\end{thmA}
\begin{proof}
According to Theorem \ref{S.th}, all simple groups with at most $5$ automorphism orbits are $A_5, A_6, PSL(2,7)$ and $PSL(2,8)$. We need to show that every non simple group $G$ with $\omega(G) \leqslant 5$ is solvable.
Suppose that $G$ is not simple. Note that, if $G$ is caracteristically simple and $\omega(G) \leqslant 5$, then $G$ is simple (Lemma \ref{ch.simple}). Thus, we may assume that $G$ contains a proper nontrivial characteristic subgroup, say $N$. By Remark \ref{prop.spec}, $\omega(N)$ and $\omega(G/N) \leqslant 4$. By \cite[Theorem 3]{LD}, it suffices to prove that $N$ and $G/N$ cannot be isomorphic to $A_5$. If $N \simeq A_5$, then $\omega(G) \geqslant 7$ by Proposition \ref{A5-prop}. Suppose that $G/N \simeq A_5$. Then $N$ is elementary abelian $p$-group, for some prime $p$. For convenience, the next steps of the proof are numbered.
\begin{itemize}
\item[(1)] Assume $p\neq 2$.
\end{itemize}
Since a Sylow 2-subgroup of $G$ is not cyclic, we have an element in $G$ of order $2p$ \cite[p. 225]{G}. Therefore $\omega(G) \geqslant 6$.
\begin{itemize}
\item[(2)] Assume $p=2$.
\end{itemize}
In particular, $|g| \in \{2,4\}$ for any $2$-power element outside of $N$. Note that, if $|g|=4$, then $G$ is AT-group. By Lemma \ref{Z.th}, $G$ is simple, a contradiction. So, we may assume that there exists an involution $g$ outside of $N$. We have, $(gh)^2 \in N$, for any $h \in N$. In particular, $gh = hg$, for any $h \in N$. Therefore $N < C_G(N)$. Since $G/N \simeq A_5$, it follows that $N \subseteq Z(G)$. So $\omega(G) \geqslant 6$. Thus $G$ is solvable, which completes the proof.
\end{proof}
It is convenient to prove first Theorem B under the hypothesis that $|\pi(G)| > 3$ and then extend the result to the general case.
\begin{prop} \label{aux-prop}
Let $G$ be a nonsolvable group in which $\omega(G) = 6$. If $|\pi(G)| > 3$, then $G \simeq PSL(3,4)$.
\end{prop}
\begin{proof}
Assume that $G$ is not characteristically simple. Let $N$ be a proper nontrivial characteristic subgroup of $G$. By Remark \ref{prop.spec}, $N$ and $G/N$ have at most $5$ automorphism orbits. Since $G$ is nonsolvable, we have $N$ or $G/N$ nonsolvable.
Suppose that $N$ is nonsolvable. By Theorem A, $N$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. By Lemma \ref{simple-lem}, $\omega(G) \geqslant 8$. Thus, we may assume that $G/N$ is nonsolvable. By Theorem A, $G/N$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8).$ Let $\pi(G) = \{2,3,p,q\}$ and $\pi(G/N) = \{2,3,p\}$. Since $\omega(N) \leqslant 3$, it follows that there exists a characteristic elementary abelian $q$-subgroup $Q$ in $N$ (\cite[Theorem 2]{LD}). Without loss of generality we can assume that $Q=N$. By Schur-Zassenhaus Theorem \cite[p. 221]{G}, there exists a complement for $N$ in $G$ (that is, there exists a subgroup $K$ such that $G = KN$ and $K \cap N = 1$). In particular, $K \simeq G/N$. Since $|Aut(K)|$ and $|N|$ are coprime numbers, it follows that $G$ is the direct product of $N$ and $K$. Arguing as in the proof of Lemma \ref{ch.simple} we deduce that $\omega(G) \geqslant 8$.
We may assume that $G$ is characteristically simple. By Lemma \ref{ch.simple}, $G$ is simple. Using Kohl's classification \cite{Kohl}, $G \simeq PSL(3,4)$. The result follows.
\end{proof}
\begin{ex} \label{ex.N}
Using GAP we obtained one example of nonsolvable and non simple group $G$ such that $|G| = 960$ and $\omega(G)=6$. Moreover, there exists a normal subgroup $N$ of $G$ such that
$$
G/N \simeq A_5 \ \mbox{e} \ N \simeq C_2 \times C_2 \times C_2 \times C_2.
$$
\end{ex}
\begin{thmB} \label{th.B}
Let $G$ be a nonsolvable group in which $\omega(G) = 6$. Then one of the following holds:
\end{thmB}
\begin{itemize}
\item[(i)] $G \simeq PSL(3,4)$;
\item[(ii)] There exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.
\end{itemize}
\begin{proof}
By Lemma \ref{ch.simple}, if $G$ is characteristically simple, then $G$ is simple. According to Kohl's classification \cite{Kohl}, $G \simeq PSL(3,4)$. In particular, by Proposition \ref{aux-prop}, if $|\pi(G)| \geqslant 4$, then $G \simeq PSL(3,4)$. So, we may assume that $|\pi(G)| = 3$ and $G$ is not characteristically simple. We need to show that for every non simple and nonsolvable group $G$ with $\omega(G) = 6$, there exists a proper characteristic subgroup $N$ such that $G/N \simeq A_5$, where $N$ is an elementary abelian $2$-subgroup.
Let $N$ be a proper characteristic subgroup of $G$. For convenience, the next steps of the proof are numbered.
\begin{itemize}
\item[(1)] Assume that $\omega(N) = 2$.
\end{itemize}
So, $\omega(G/N) = 4$ or $5$ and $N$ is elementary abelian $p$-group, for some prime $p$. According to Theorem A and Example \ref{ex.N}, it is sufficient to consider $G/N$ isomorphic to one of the groups $A_6, PSL(2,7)$ or $PSL(2,8)$. Since the Sylow 2-subgroup of $G/N$ is not cyclic, it follows that the subgroup $N$ is elementary abelian $2$-subgroup \cite[p.\ 225]{G}. Suppose that $G/N \simeq PSL(2,8)$. Arguing as in the proof of Theorem A we deduce that $G$ is $AT$-group, a contradiction. Now, we may assume that $G/N \in \{ A_6, PSL(2,7)\}$. Without loss of generality we can assume that there are elements $a \in G \setminus N$ and $h \in N$ such that $|a| = 2$ and $|ah| = 4$. Then there exist the only one automorphism orbit in which it elements has order $4$, $\{(ah)^{\varphi} \mid \ \varphi \in Aut(G) \}$. On the other hand, $aN$ has order $2$ and $\omega(G) = 6$. Therefore $G/N$ cannot contains elements of order $4$, a contradiction.
\begin{itemize}
\item[(2)] Assume that $\omega(N) = 3$.
\end{itemize}
There exists a characteristic subgroup $Q$ of $N$ and of $G$ (\cite[Theorem 2]{LD}). As $G/N$ and $G/Q$ are simple, we have a contradiction.
\begin{itemize}
\item[(3)] Assume that $\omega(N) = 4$ or $5$.
\end{itemize}
In particular, $\omega(G/N) \leqslant 3$. Arguing as in $(2)$ we deduce that $\omega(G/N) = 2$. By Theorem A, $N$ is
simple. Hence $$N \in \{A_5, A_6, PSL(2,7), PSL(2,8)\}.$$ By Proposition \ref{A5-prop}, $\omega(G) \geqslant 7$.
\end{proof}
\subsection*{Acknowledgment}
The authors wishes to express their thanks to S\'ilvio Sandro for several helpful comments concerning ``GAP''.
\end{document}
|
\begin{document}
\title{Exact order of extreme $L_p$ discrepancy of infinite sequences in arbitrary dimension}
\author{Ralph Kritzinger and Friedrich Pillichshammer\thanks{The first author is supported by the Austrian Science Fund (FWF), Project F5509-N26, which is a part of the Special Research Program ``Quasi-Monte Carlo Methods: Theory and Applications''.}}
\date{}
\maketitle
\begin{abstract}
We study the extreme $L_p$ discrepancy of infinite sequences in the $d$-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star $L_p$ discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension $d$ and any $p>1$ the extreme $L_p$ discrepancy of every infinite sequence in $[0,1)^d$ is at least of order of magnitude $(\log N)^{d/2}$, where $N$ is the number of considered initial terms of the sequence. For $p \in (1,\infty)$ this order of magnitude is best possible.
\end{abstract}
\centerline{\begin{minipage}[hc]{130mm}{
{\em Keywords:} extreme $L_p$-discrepancy, lower bounds, van der Corput sequence\\
{\em MSC 2020:} 11K38, 11K06, 11K31}
\end{minipage}}
\section{Introduction}
Let $\mathcal{P}=\{\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots,\boldsymbol{x}_{N-1}\}$ be an arbitrary $N$-element point set in the $d$-dimensional unit cube $[0,1)^d$. For any measurable subset $B$ of $[0,1]^d$ the {\it counting function} $$A_N(B,\mathcal{P}):=|\{n \in \{0,1,\ldots,N-1\} \ : \ \boldsymbol{x}_n \in B\}|$$ counts the number of elements from $\mathcal{P}$ that belong to the set $B$. The {\it local discrepancy} of $\mathcal{P}$ with respect to a given measurable ``test set'' $B$ is then given by $$\Delta_N(B,\mathcal{P}):=A_N(B,\mathcal{P})-N \lambda (B),$$ where $\lambda$ denotes the Lebesgue measure of $B$. A global discrepancy measure is then obtained by considering a norm of the local discrepancy with respect to a fixed class of test sets.
In the following let $p \in [1,\infty)$.
The classical {\it (star) $L_p$ discrepancy} uses as test sets the class of axis-parallel rectangles contained in the unit cube that are anchored in the origin. The formal definition is
$$ L_{p,N}^{{\rm star}}(\mathcal{P}):=\left(\int_{[0,1]^d}\left|\Delta_N([\boldsymbol{z}ero,\boldsymbol{t}),\mathcal{P})\right|^p\,\mathrm{d} \boldsymbol{t}\right)^{1/p}, $$
where for $\boldsymbol{t}=(t_1,t_2,\ldots,t_d)\in [0,1]^d$ we set $[\boldsymbol{z}ero,\boldsymbol{t})=[0,t_1)\times [0,t_2)\times \ldots \times [0,t_d)$ with area $\lambda([\boldsymbol{z}ero,\boldsymbol{t}))=t_1t_2\cdots t_d$.
The {\it extreme $L_p$ discrepancy} uses as test sets arbitrary axis-parallel rectangles contained in the unit cube. For $\boldsymbol{u}=(u_1,u_2,\ldots,u_d)$ and $\boldsymbol{v}=(v_1,v_2,\ldots,v_d)$ in $[0,1]^d$ and $\boldsymbol{u} \leq \boldsymbol{v}$ let $[\boldsymbol{u},\boldsymbol{v})=[u_1,v_1)\times [u_2,v_2) \times \ldots \times [u_d,v_d)$, where $\boldsymbol{u} \leq \boldsymbol{v}$ means $u_j\leq v_j$ for all $j \in \{1,2,\ldots,d\}$. Obviously, $\lambda([\boldsymbol{u},\boldsymbol{v}))=\prod_{j=1}^d (v_j-u_j)$. The extreme $L_p$ discrepancy of $\mathcal{P}$ is then defined as
$$L_{p,N}^{\mathrm{extr}}(\mathcal{P}):=\left(\int_{[0,1]^d}\int_{[0,1]^d,\, \boldsymbol{u}\leq \boldsymbol{v}}\left|\Delta_N([\boldsymbol{u},\boldsymbol{v}),\mathcal{P})\right|^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p}. $$
Note that the only difference between standard and extreme $L_p$ discrepancy is the use of anchored and arbitrary rectangles in $[0,1]^d$, respectively.
For an infinite sequence $\mathcal{S}_d$ in $[0,1)^d$ the star and the extreme $L_p$ discrepancies $L_{p,N}^{\bullet}(\mathcal{S}_d)$ are defined as $L_{p,N}^{\bullet}(\mathcal{P}_{d,N})$, $N \in \mathbb{N}$, of the point set $\mathcal{P}_{d,N}$ consisting of the initial $N$ terms of $\mathcal{S}_d$, where $\bullet \in \{{\rm star},{\rm extr}\}$.
Of course, with the usual adaptions the above definitions can be extended also to the case $p=\infty$. However, it is well known that in this case the star and extreme $L_{\infty}$ discrepancies are equivalent in the sense that $L_{\infty,N}^{{\rm star}}(\mathcal{P}) \le L_{\infty,N}^{{\rm extr}}(\mathcal{P}) \le 2^d L_{\infty,N}^{{\rm star}}(\mathcal{P})$ for every $N$-element point set $\mathcal{P}$ in $[0,1)^d$. For this reason we restrict the following discussion to the case of finite $p$.
Recently it has been shown that the extreme $L_p$ discrepancy is dominated -- up to a multiplicative factor that depends only on $p$ and $d$ -- by the star $L_p$ discrepancy (see \cite[Corollary~5]{KP21}), i.e., for every $d \in \mathbb{N}$ and $p \in [1,\infty)$ there exists a positive quantity $c_{d,p}$ such that for every $N \in \mathbb{N}$ and every $N$-element point set in $[0,1)^d$ we have
\begin{equation}\label{monLpstex}
L_{p,N}^{{\rm extr}}(\mathcal{P})\le c_{d,p} L_{p,N}^{{\rm star}}(\mathcal{P}).
\end{equation}
For $p=2$ we even have $c_{d,2}=1$ for all $d \in \mathbb{N}$; see \cite[Theorem~5]{HKP20}. A corresponding estimate the other way round is in general not possible (see \cite[Section~3]{HKP20}). So, in general, the star and the extreme $L_p$ discrepancy for $p< \infty$ are not equivalent, which is in contrast to the $L_{\infty}$-case.
\paragraph{Bounds for finite point sets.} For finite point sets the order of magnitude of the considered discrepancies is more or less known. For every $p \in(1,\infty)$ and $d \in \mathbb{N}$ there exists a $c_{p,d}>0$ such that for every finite $N$-element point set $\mathcal{P}$ in $[0,1)^d$ with $N \ge 2$ we have
\begin{equation*}
L_{p,N}^{\bullet}(\mathcal{P}) \ge c_{p,d} (\log N)^{\frac{d-1}{2}}, \quad \mbox{where } \bullet \in \{{\rm star},{\rm extr}\}.
\end{equation*}
For the star $L_p$ discrepancy the result for $p \ge 2$ is a celebrated result by Roth~\cite{Roth} from 1954 that was extended later by Schmidt~\cite{S77} to the case $p \in (1,2)$. For the extreme $L_p$ discrepancy the result for $p \ge 2$ was first given in \cite[Theorem~6]{HKP20} and for $p \in (1,2)$ in \cite{KP21}. Hal\'{a}sz~\cite{H81} for the star discrepancy and the authors \cite{KP21} for the extreme discrepancy proved that the lower bound is even true for $p=1$ and $d=2$, i.e., there exists a positive number $c_{1,2}$ with the following property: for every $N$-element $\mathcal{P}$ in $[0,1)^2$ with $N \ge 2$ we have
\begin{equation}\label{lbdl1D2dipts}
L_{1,N}^{\bullet}(\mathcal{P}) \ge c_{1,2} \sqrt{\log N}, \quad \mbox{where } \bullet \in \{{\rm star},{\rm extr}\}.
\end{equation}
On the other hand, it is known that for every $d,N \in \mathbb{N}$ and every $p \in [1,\infty)$ there exist $N$-element point sets $\mathcal{P}$ in $[0,1)^d$ such that
\begin{equation}\label{uplpps}
L_{p,N}^{{\rm star}}(\mathcal{P}) \lesssim_{d,p} (\log N)^{\frac{d-1}{2}}.
\end{equation}
(For $f,g: D \subseteq \mathbb{N} \rightarrow \mathbb{R}^+$ we write $f(N) \lesssim g(N)$, if there exists a positive number $C$ such that $f(N) \le C g(N)$ for all $N \in D$. Possible implied dependencies of $C$ are indicated as lower indices of the $\lesssim$ symbol.)
Due to \eqref{monLpstex} the upper bound \eqref{uplpps} even applies to the extreme $L_p$ discrepancy. Hence, for $p \in (1,\infty)$ and arbitrary $d \in \mathbb{N}$ (and also for $p=1$ and $d=2$) we have matching lower and upper bounds. The result in \eqref{uplpps} was proved by Davenport~\cite{D56} for $p= 2$, $d= 2$, by Roth~\cite{R80} for $p= 2$ and arbitrary $d$ and finally by Chen~\cite{C80} in the general case. Other proofs were found by Frolov~\cite{Frolov}, Chen~\cite{C83}, Dobrovol'ski{\u\i}~\cite{Do84}, Skriganov~\cite{Skr89, Skr94}, Hickernell and Yue~\cite{HY00}, and Dick and Pillichshammer~\cite{DP05b}. For more details on the early history of the subject see the monograph \cite{BC}. Apart from Davenport, who gave an explicit construction in dimension $d=2$, these results are pure existence results and explicit constructions of point sets were not known until the beginning of this millennium. First explicit constructions of point sets with optimal order of star $L_2$ discrepancy in arbitrary dimensions have been provided in 2002 by Chen and Skriganov \cite{CS02} for $p=2$ and in 2006 by Skriganov \cite{S06} for general $p$. Other explicit constructions are due to Dick and Pillichshammer \cite{DP14a} for $p=2$, and Dick \cite{D14} and Markhasin \cite{M15} for general $p$.
\paragraph{Bounds for infinite sequences.} For the star $L_p$ discrepancy the situation is also more or less clear. Using a method from Pro{\u\i}nov~\cite{pro86} (see also \cite{DP14b}) the results about lower bounds on star $L_p$ discrepancy for finite sequences can be transferred to the following lower bounds for infinite sequences: for every $p \in(1,\infty]$ and every $d \in \mathbb{N}$ there exists a $C_{p,d}>0$ such that for every infinite sequence $\mathcal{S}_d$ in $[0,1)^d$
\begin{equation}\label{lbdlpdiseq}
L_{p,N}^{{\rm star}}(\mathcal{S}_d) \ge C_{p,d} (\log N)^{d/2} \ \ \ \ \mbox{for infinitely many $N \in \mathbb{N}$.}
\end{equation}
For $d=1$ the result holds also for the case $p=1$, i.e., for every $\mathcal{S}$ in $[0,1)$ we have
\begin{equation*}
L_{1,N}^{{\rm star}}(\mathcal{S}) \ge C_{1,1} \sqrt{\log N} \ \ \ \ \mbox{for infinitely many $N \in \mathbb{N}$.}
\end{equation*}
Matching upper bounds on the star $L_p$ discrepancy of infinite sequences have been shown in \cite{DP14a} (for $p=2$) and in \cite{DHMP} (for general $p$). For every $d \in \mathbb{N}$ there exist infinite sequences $\mathcal{S}_d$ in $[0,1)^d$ such that for any $p \in [1,\infty)$ we have $$L_{p,N}^{{\rm star}}(\mathcal{S}_d) \lesssim_{d,p} (\log N)^{d/2} \quad \mbox{ for all $N \in \mathbb{N}$.}$$
So far, the extreme $L_p$ discrepancy of infinite sequences has not yet been studied. Obviously, due to \eqref{monLpstex} the upper bounds on the star $L_p$ discrepancy also apply to the extreme $L_p$ discrepancy. However, a similar reasoning for obtaining a lower bound is not possible. In this paper we show that the lower bound \eqref{lbdlpdiseq} also holds true for the extreme $L_p$ discrepancy. Thereby we prove that for fixed dimension $d$ and for $p\in (1,\infty)$ the minimal extreme $L_p$ discrepancy is, like the star $L_p$ discrepancy, of exact order of magnitude $(\log N)^{d/2}$ when $N$ tends to infinity.
\section{The result}
We extend the lower bound \eqref{lbdlpdiseq} for the star $L_p$ discrepancy of infinite sequences to extreme $L_p$ discrepancy.
\begin{thm}\label{thm2}
For every dimension $d \in \mathbb{N}$ and any $p>1$ there exists a real $\alpha_{d,p} >0$ with the following property: For any infinite sequence $\mathcal{S}_d$ in $[0,1)^d$ we have $$L_{p,N}^{{\rm extr}}(\mathcal{S}_{d}) \ge \alpha_{d,p} (\log N)^{d/2} \ \ \ \ \mbox{ for infinitely many }\ N \in \mathbb{N}.$$ For $d=1$ the results even holds true for the case $p=1$.
\end{thm}
For the proof we require the following lemma. For the usual star discrepancy this lemma goes back to Roth~\cite{Roth}. We require a similar result for the extreme discrepancy.
\begin{lem}\label{le1}
For $d\in \mathbb{N}$ let $\mathcal{S}_d=(\boldsymbol{y}_k)_{k\ge 0}$, where $\boldsymbol{y}_k=(y_{1,k},\ldots,y_{d,k})$ for $k \in \mathbb{N}_0$, be an arbitrary sequence in the $d$-dimensional unit cube with extreme $L_p$ discrepancy $L_{p,N}^{{\rm extr}}(\mathcal{S}_d)$ for $p \in [1,\infty]$. Then for every $N\in \mathbb{N}$ there exists an $n \in \{1,2,\ldots,N\}$ such that $$L_{p,n}^{{\rm extr}}(\mathcal{S}_d) \ge \frac{2^{1/p}}{2}\, L_{p,N}^{{\rm extr}}(\mathcal{P}_{N,d+1})- \frac{1}{2^{d/p}},$$ with the adaption that $2^{1/p}$ and $2^{d/p}$ have to be set 1 if $p =\infty$, where $\mathcal{P}_{N,d+1}$ is the finite point set in $[0,1)^{d+1}$ consisting of the $N$ points $$\boldsymbol{x}_k=(y_{1,k},\ldots,y_{d,k},k/N) \ \ \mbox{ for }\ k \in \{0,1,\ldots ,N-1\}.$$
\end{lem}
\begin{proof}
We present the proof for finite $p$. For $p=\infty$ the proof is similar.
Choose $n \in \{1,2,\ldots,N\}$ such that $$L_{p,n}^{{\rm extr}}(\mathcal{S}_d) =\max_{k=1,2,\ldots,N} L_{p,k}^{{\rm extr}}(\mathcal{S}_d).$$
Consider a sub-interval of the $(d+1)$-dimensional unit cube of the form $E=\prod_{i=1}^{d+1}[u_i,v_i)$ with $\boldsymbol{u}=(u_1,u_2,\ldots,u_{d+1}) \in [0,1)^{d+1}$ and $\boldsymbol{v}=(v_1,v_2,\ldots,v_{d+1}) \in [0,1)^{d+1}$ satisfying $\boldsymbol{u}\leq\boldsymbol{v}$. Put $\overline{m}=\overline{m}(v_{d+1}):=\lceil N v_{d+1}\rceil$ and $\underline{m}=\underline{m}(u_{d+1}):=\lceil N u_{d+1}\rceil$. Then a point $\boldsymbol{x}_k$, $k \in \{0,1,\ldots, N-1\}$, belongs to $E$, if and only if $\boldsymbol{y}_k \in \prod_{i=1}^d[u_i,v_i)$ and $N u_{d+1} \le k < N v_{d+1}$. Denoting $E'=\prod_{i=1}^d[u_i,v_i)$ we have $$A_N(E,\mathcal{P}_{N,d+1})=A_{\overline{m}}(E',\mathcal{S}_d)-A_{\underline{m}}(E',\mathcal{S}_d)$$ and therefore
\begin{align*}
\Delta_N(E,\mathcal{P}_{N,d+1}) = & A_N(E,\mathcal{P}_{N,d+1}) -N \prod_{i=1}^{d+1} (v_i - u_i)\\
= & A_{\overline{m}}(E',\mathcal{S}_d)-A_{\underline{m}}(E',\mathcal{S}_d) - \overline{m} \prod_{i=1}^d (v_i-u_i) + \underline{m} \prod_{i=1}^d (v_i-u_i)\\
& + \overline{m} \prod_{i=1}^d (v_i-u_i) - \underline{m} \prod_{i=1}^d (v_i-u_i) - N \prod_{i=1}^{d+1} (v_i - u_i)\\
= & \Delta_{\overline{m}}(E',\mathcal{S}_d) - \Delta_{\underline{m}}(E',\mathcal{S}_d) + \left(\overline{m}-\underline{m}-N(v_{d+1}-u_{d+1})\right) \prod_{i=1}^d (v_i-u_i).
\end{align*}
We obviously have $|\overline{m}-N v_{d+1}| \le 1$, $|\underline{m}-N u_{d+1}| \le 1$ and $|\prod_{i=1}^d (v_i-u_i)| \le 1$. Hence $$|\Delta_N(E,\mathcal{P}_{N,d+1})| \le |\Delta_{\overline{m}}(E',\mathcal{S}_d)| + |\Delta_{\underline{m}}(E',\mathcal{S}_d)| +2,$$
which yields
\begin{align*}
L_{p,N}^{{\rm extr}}(\mathcal{P}_{N,d+1}) \le & \left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}}\Big| |\Delta_{\overline{m}}(E',\mathcal{S}_d)| + |\Delta_{\underline{m}}(E',\mathcal{S}_d)| +2\Big|^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p}\\
\le & \left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}} |\Delta_{\overline{m}}(E',\mathcal{S}_d)|^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p}\\
& + \left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}} |\Delta_{\underline{m}}(E',\mathcal{S}_d)|^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p}\\
& + \left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}} 2^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p},
\end{align*}
where the last step easily follows from the triangle-inequality for the $L_p$-semi-norm.
For every $u_{d+1},v_{d+1} \in [0,1]$ we have $L_{\overline{m},p}^{{\rm extr}}(\mathcal{S}_d) \le L_{n,p}^{{\rm extr}}(\mathcal{S}_d)$ and $L_{\underline{m},p}^{{\rm extr}}(\mathcal{S}_d) \le L_{n,p}^{{\rm extr}}(\mathcal{S}_d)$, respectively.
Setting $\boldsymbol{u}'=(u_1,\dots,u_d)$ and $\boldsymbol{v}'=(v_1,\dots,v_d)$, we obtain
\begin{eqnarray*}
\lefteqn{\left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}} |\Delta_{\overline{m}}(E',\mathcal{S}_d)|^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p}}\\
& = & \left( \int_0^1 \int_{0,\, u_{d+1} \le v_{d+1}}^1 \int_{[0,1]^d}\int_{[0,1]^d,\, \boldsymbol{u}'\leq \boldsymbol{v}'} |\Delta_{\overline{m}}(E',\mathcal{S}_d)|^p\,\mathrm{d} \boldsymbol{u}'\,\mathrm{d}\boldsymbol{v}' \,\mathrm{d} u_{d+1} \,\mathrm{d} v_{d+1}\right)^{1/p}\\
& = & \left( \int_0^1 \int_{0,\, u_{d+1} \le v_{d+1}}^1 (L_{\overline{m},p}^{{\rm extr}}(\mathcal{S}_d))^p \,\mathrm{d} u_{d+1} \,\mathrm{d} v_{d+1}\right)^{1/p}\\
& \le & \left( \int_0^1 \int_{0,\, u_{d+1} \le v_{d+1}}^1 (L_{n,p}^{{\rm extr}}(\mathcal{S}_d))^p \,\mathrm{d} u_{d+1} \,\mathrm{d} v_{d+1}\right)^{1/p}\\
& = & \frac{1}{2^{1/p}}\, L_{p,n}^{{\rm extr}}(\mathcal{S}_d).
\end{eqnarray*}
Likewise we also have $$\left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}} |\Delta_{\underline{m}}(E',\mathcal{S}_d)|^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p} \le \frac{1}{2^{1/p}}\, L_{p,n}^{{\rm extr}}(\mathcal{S}_d).$$ Also $$\left(\int_{[0,1]^{d+1}}\int_{[0,1]^{d+1},\, \boldsymbol{u}\leq \boldsymbol{v}} 2^p\,\mathrm{d} \boldsymbol{u}\,\mathrm{d}\boldsymbol{v}\right)^{1/p}= \frac{2}{2^{(d+1)/p}}.$$
Therefore we obtain
$$L_{p,N}^{{\rm extr}}(\mathcal{P}_{N,d+1}) \le \frac{2}{2^{1/p}}\, L_{p,n}^{{\rm extr}}(\mathcal{S}_d) + \frac{2}{2^{(d+1)/p}}.$$
From here the result follows immediately.
\end{proof}
Now we can give the proof of Theorem~\ref{thm2}.
\begin{proof}[Proof of Theorem~\ref{thm2}]
We use the notation from Lemma~\ref{le1}. For the extreme $L_p$ discrepancy of the finite point set $\mathcal{P}_{N,d+1}$ in $[0,1)^{d+1}$ we obtain from \cite[Corollary~4]{KP21} (for $d \in \mathbb{N}$ and $p>1$) and \cite[Theorem~7]{KP21} (for $d=1$ and $p=1$) that $$L_{p,N}^{{\rm extr}}(\mathcal{P}_{N,d+1}) \ge c_{d+1,q} (\log N)^{d/2}$$ for some real $c_{d+1,q}>0$ which is independent of $N$. Let $\alpha_{d,p} \in (0,2^{\frac{1}{p}-1}c_{d+1,p})$ and let $N \in \mathbb{N}$ be large enough such that $$\frac{2^{1/p} c_{d+1,p}}{2} \, (\log N)^{d/2} -\frac{1}{2^{d/p}} \ge \alpha_{d,p} (\log N)^{d/2}.$$ According to Lemma~\ref{le1} there exists an $n \in \{1,2,\ldots,N\}$ such that
\begin{eqnarray}\label{eq1}
L_{p,n}^{{\rm extr}}(\mathcal{S}_d) & \ge & \frac{2^{1/p}}{2}\, L_{p,N}^{{\rm extr}}(\mathcal{P}_{N,d+1})-\frac{1}{2^{d/p}}\nonumber\\
& \ge & \frac{2^{1/p} c_{d+1,p}}{2} \, (\log N)^{d/2} -\frac{1}{2^{d/p}}\nonumber\\
& \ge & \alpha_{d,p} (\log n)^{d/2}.
\end{eqnarray}
Thus we have shown that for every large enough $N \in \mathbb{N}$ there exists an $n \in \{1,2,\ldots,N\}$ such that
\begin{equation}\label{eq2}
L_{p,n}^{{\rm extr}}(\mathcal{S}_d) \ge \alpha_{d,p} (\log n)^{d/2}.
\end{equation}
It remains to show that \eqref{eq2} holds for infinitely many $n \in \mathbb{N}$. Assume on the contrary that \eqref{eq2} holds for finitely many $n \in \mathbb{N}$ only and let $m$ be the largest integer with this property. Then choose $N \in \mathbb{N}$ large enough such that $$\frac{2^{1/p} c_{d+1,p}}{2} \, (\log N)^{d/2} -\frac{1}{2^{d/p}} \ge \alpha_{d,p} (\log N)^{d/2} > \max_{k=1,2,\ldots,m} L_{p,k}^{{\rm extr}}(\mathcal{S}_d).$$ For this $N$ we can find an $n \in \{1,2,\ldots,N\}$ for which \eqref{eq1} and \eqref{eq2} hold true. However, \eqref{eq1} implies that $n > m$ which leads to a contradiction since $m$ is the largest integer such that \eqref{eq2} is true. Thus we have shown that \eqref{eq2} holds for infinitely many $n \in \mathbb{N}$ and this completes the proof.
\end{proof}
As already mentioned, there exist explicit constructions of infinite sequences $\mathcal{S}_d$ in $[0,1)^d$ with the property that
\begin{align}\label{ub:extrlpinf}
L_{p,N}^{{\rm extr}}(\mathcal{S}_d) \lesssim_{p,d} (\log N)^{d/2}\ \ \ \ \mbox{ for all $N\ge 2$ and all $p \in [1,\infty)$.}
\end{align}
This result follows from \eqref{monLpstex} together with \cite[Theorem~1.1]{DHMP}. These explicitly constructed sequences are so-called order 2 digital $(t, d)$-sequence over the finite field $\mathbb{F}_2$; see \cite[Section~2.2]{DHMP}. The result \eqref{ub:extrlpinf} implies that the lower bound from Theorem~\ref{thm2} is best possible in the order of magnitude in $N$ for fixed dimension $d$.
\begin{rem}\rm
Although the optimality of the lower bound in Theorem~\ref{thm2} is shown by means of matching upper bounds on the star $L_p$ discrepancy we point out that in general the extreme $L_p$ discrepancy is really lower than the star $L_p$ discrepancy. An easy example is the van der Corput sequence $\mathcal{S}^{{\rm vdC}}$ in dimension $d=1$, whose extreme $L_p$ discrepancy is of the optimal order of magnitude
\begin{equation}\label{optvdclpex}
L_{p,N}^{{\rm extr}}(\mathcal{S}^{{\rm vdC}}) \lesssim_p \sqrt{\log N}\quad \mbox{all $N\geq 2$ and all $p\in [1,\infty)$,}
\end{equation}
but its star $L_p$ discrepancy is only of order of magnitude $\log N$ since
\begin{equation}\label{exordvdc}
\limsup_{N \rightarrow \infty} \frac{L_{p,N}^{{\rm star}}(\mathcal{S}^{{\rm vdC}})}{\log N}=\frac{1}{6 \log 2} \quad \mbox{ for all $p \in [1,\infty)$.}
\end{equation}
For a proof of \eqref{exordvdc} see, e.g., \cite{chafa,proat} for $p=2$ and \cite{pil04} for general $p$. A proof of \eqref{optvdclpex} can be given by means of a Haar series representation of the extreme $L_p$ discrepancy as given in \cite[Proposition~3, Eq.~(9)]{KP21}. One only requires good estimates for all Haar coefficients of the discrepancy function of the first $N$ elements of the van der Corput sequence, but these can be found in~\cite{KP2015}. Employing these estimates yields after some lines of algebra the optimal order result \eqref{optvdclpex}.
\end{rem}
\begin{rem}\rm
The periodic $L_p$ discrepancy is another type of discrepancy that is based on the class of periodic intervals modulo one as test sets; see \cite{HKP20,KP21}. Denote it by $L_{p,N}^{{\rm per}}$. The periodic $L_p$ discrepancy dominates the extreme $L_p$ discrepancy because the range of integration in the definition of the extreme $L_p$ discrepancy is a subset of the range of integration in the definition of the periodic $L_p$ discrepancy, as already noted in \cite[Eq.~(1)]{HKP20} for the special case $p=2$. Furthermore, it is well known that the periodic $L_2$ discrepancy, normalized by the number of elements of the point set, is equivalent to the diaphony, which was introduced by Zinterhof~\cite{zint} and which is yet another quantitative measure for the irregularity of distribution; see \cite[Theorem~1]{Lev} or \cite[p.~390]{HOe}. For $\mathcal{P}=\{\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots,\boldsymbol{x}_{N-1}\}$ in $[0,1)^d$ it is defined as $$F_N(\mathcal{P})=\left(\sum_{\boldsymbol{h} \in \mathbb{Z}^d} \frac{1}{r(\boldsymbol{h})^2} \left| \frac{1}{N} \sum_{k=0}^{N-1} {\rm e}^{2 \pi \mathtt{i} \boldsymbol{h} \cdot \boldsymbol{x}_k}\right|^2\right)^{1/2},$$ where for $\boldsymbol{h} =(h_1,h_2,\ldots,h_d)\in \mathbb{Z}^d$ we set $r(\boldsymbol{h})= \prod_{j=1}^d \max(1,|h_j|)$. Now, for every $p>1$ for every infinite sequence $\mathcal{S}_d$ in $[0,1)^d$ we have for infinitely many $N \in \mathbb{N}$ the lower bound
$$\frac{(\log N)^{d/2}}{N} \lesssim_{p,d} \frac{1}{N}\, L_{p,N}^{{\rm extr}}(\mathcal{S}_d) \le \frac{1}{N}\, L_{p,N}^{{\rm per}}(\mathcal{S}_d).$$ Choosing $p=2$ we obtain $$\frac{(\log N)^{d/2}}{N} \lesssim_{d} \frac{1}{N}\, L_{2,N}^{{\rm per}}(\mathcal{S}_d) \lesssim_d F_N(\mathcal{S}_d)\quad \mbox{ for infinitely many $N \in \mathbb{N}$.}$$ Thus, there exists a positive $C_d$ such that for every sequence $\mathcal{S}_d$ in $[0,1)^d$ we have
\begin{equation}\label{lb:dia}
F_N(\mathcal{S}_d) \ge C_d \, \frac{(\log N)^{d/2}}{N} \quad \mbox{ for infinitely many $N \in \mathbb{N}$.}
\end{equation}
This result was first shown by Pro{\u\i}nov~\cite{pro2000} by means of a different reasoning. The publication \cite{pro2000} is only available in Bulgarian; a survey presenting the relevant result is published by Kirk~\cite{kirk}. At least in dimension $d=1$ the lower bound \eqref{lb:dia} is best possible, since, for example, $F_N(\mathcal{S}^{{\rm vdC}}) \lesssim \sqrt{\log N}/N$ for all $N \in \mathbb{N}$ as shown in \cite{progro} (see also \cite{chafa,F05}). Note that this also yields another proof of \eqref{optvdclpex} for the case $p=2$. A corresponding result for dimensions $d>1$ is yet missing.
\end{rem}
\noindent {\bf Author's address:} Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Austria, 4040 Linz, Altenberger Stra{\ss}e 69. Email: [email protected], [email protected]
\end{document}
|
\begin{document}
\title{Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator
hanks{This work was supported in part by the NSF under Grant DMS-1115118.}
\newcommand{\slugmaster}{
\slugger{siims}{xxxx}{xx}{x}{x--x}}
\begin{abstract}
We propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation.
The diffusion matrix is defined based on the input image. The eigenfunctions and the projection
of the input image in some eigenspace capture key features of the input image.
An important property of the model is that for many input images, the first few eigenfunctions are close to being piecewise
constant, which makes them useful as the basis for a variety of applications such as image segmentation and edge detection.
The eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from
several commonly used discrete spectral clustering models.
The relation provides a better understanding and helps developing more efficient numerical implementation
and rigorous numerical analysis for discrete spectral segmentation methods.
The new continuous model is also different from energy-minimization methods such as geodesic active contour
in that no initial guess is required for in the current model. The multi-scale feature is a natural consequence of
the anisotropic diffusion operator so there is no need to solve the eigenvalue problem at multiple levels.
A numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy
is presented. It is shown that the numerical scheme gives much more accurate results on eigenfunctions than
uniform meshes. Several interesting features of the model are examined in numerical examples and
possible applications are discussed.
\end{abstract}
\noindent{\bf Key Words.}
eigenvalue problem, image segmentation, finite-element schemes, mesh adaptation, anisotropic diffusion.
\noindent{\bf AMS 2010 Mathematics Subject Classification.}
65N25, 68U10, 94A08
\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{IMAGE SEGMENTATION WITH EIGENFUNCTIONS}{IMAGE SEGMENTATION WITH EIGENFUNCTIONS}
\section{Introduction}
We are concerned with image segmentation using
the eigenvalue problem of an anisotropic linear diffusion operator,
\begin{equation}
-\nabla \cdot (\mathbb{D}\nabla u)=\lambda u, \quad \text{ in }\Omega
\label{eq:HW-eigen}
\end{equation}
subject to a homogeneous Dirichlet or Neumann boundary condition, where the diffusion matrix
$\mathbb{D}$ is symmetric and uniformly positive definite on $\Omega$. In this study, we consider
an anisotropic diffusion situation where $\mathbb{D}$ has different eigenvalues and is defined based
on the gray level of the input image.
A method employing an eigenvalue problem to study image segmentation is referred to as a spectral clustering
method in the literature. This type of methods have extracted great interest from researchers
in the past decade; e.g., see \cite{Grady-01,shi_normalized_2000,Wu-Leahy-1993}.
They are typically derived from a minimum-cut criterion on a graph.
One of the most noticeable spectral clustering methods is the normalized cut method proposed
by Shi and Malik \cite{shi_normalized_2000} (also see Section~\ref{SEC:relation-1} below)
which is based on the eigenvalue problem
\begin{equation}
(D-W) {\bf u} = \lambda D {\bf u} ,
\label{eq:Malik-Shi-eigen}
\end{equation}
where ${\bf u}$ is a vector representing the gray level value on the pixels, $W$ is a matrix defining
pairwise similarity between pixels, and $D$ is a diagonal matrix formed with the degree
of pixels (cf. Section 2.2 below). The operator $L=D-W$ corresponds to the graph Laplacian in graph spectral
theory. An eigenvector associated with the second eigenvalue is used
as a continuous approximation to a binary or $k$-way vector that
indicates the partitions of the input image. Shi and Malik suggested that
image segmentation be done on a hierarchical basis where
low level coherence of brightness, texture, and etc. guides a binary (or
$k$-way) segmentation that provides a big picture while high
level knowledge is used to further partition the low-level segments.
While discrete spectral clustering methods give impressive partitioning results in general,
they have several drawbacks. Those methods are typically defined and operated
on a graph or a data set. Their implementation cost depends on the size of the graph or data set.
For a large data set, they can be very expensive to implement.
Moreover, since they are discrete, sometimes their physical and/or geometrical meanings
are not so clear. As we shall see in Section~\ref{SEC:relation-1}, the normalized cut
method of Shi and Malik \cite{shi_normalized_2000} is linked to an anisotropic diffusion
differential operator which from time to time can lead to isotropic diffusion.
The objective of this paper is to investigate the use of the eigenvalue problem
(\ref{eq:HW-eigen}) of an anisotropic diffusion operator for image segmentation.
This anisotropic model can be viewed as a continuous, improved anisotropic generalization
of discrete spectral clustering models such as (\ref{eq:Malik-Shi-eigen}).
The model is also closely related to the Perona-Malik anisotropic filter.
The advantages of using a continuous model for image segmentation
include (i) It has a clear physical interpretation (heat diffusion or
Fick's laws of diffusion in our case);
(ii) Many well developed theories of partial differential equations can be used;
(iii) Standard discretization methods such as finite differences, finite elements,
finite volumes, and spectral methods can be employed;
and (iv) The model does not have be discretized on a mesh associated with the given
data set and indeed, mesh adaptation can be used to improve accuracy and efficiency.
As mentioned early, we shall define the diffusion matrix $\mathbb{D}$ using
the input image and explore properties of the eigenvalue problem.
One interesting property is that for many input images, the first few eigenfunctions of the model
are close to being piecewise constant, which are very useful for image segmentation.
However, this also means that these eigenfunctions change abruptly between
objects and their efficient numerical approximation requires mesh adaptation.
In this work, we shall use an anisotropic mesh adaptation strategy developed
by the authors \cite{Huang-Wang-13} for differential eigenvalue problems.
Another property of (\ref{eq:HW-eigen}) is that
eigenfunctions associated with small eigenvalues possess coarse, global
features of the input image whereas eigenfunctions associated with larger eigenvalues
carry more detailed, localized features.
The decomposition of features agrees with the view of Shi and Malik \cite{shi_normalized_2000}
on the hierarchical structure of image segmentation but in a slightly different sense
since all eigenfunctions come from low level brightness knowledge.
The paper is organized as follows. In Section~\ref{SEC:eigen}, we give a detailed description
of the eigenvalue problem based on an anisotropic diffusion operator and
discuss its relations to some commonly used discrete spectral clustering models
and diffusion filters and some other models in image segmentation.
Section~\ref{SEC:implement} is devoted to
the description of the finite element implementation of the model and
an anisotropic mesh adaptation strategy.
In Section~\ref{SEC:numerics}, we present a number of applications in image segmentation
and edge detection and demonstrate several properties of the model.
Some explanations to the piecewise constant property of eigenfunctions are given in
Section~\ref{SEC:piecewise}.
Concluding remarks are given in Section~\ref{SEC:conclusion}.
\section{Description of the eigenvalue problem}
\label{SEC:eigen}
\subsection{Eigenvalue problem of an anisotropic diffusion operator}
We shall use the eigenvalue problem (\ref{eq:HW-eigen}) subject to a Dirichlet or Neumann boundary condition
for image segmentation.
We are inspired by the physics of anisotropic heat transport
process (e.g., see \cite{Gunter-Yu-Kruger-Lackner-05,Sharma-Hammett-07}),
treating the dynamics of image diffusion as the transport of energy
(pixel values) and viewing the eigenvalue problem as the steady state of the dynamic process.
Denote the principal diffusion direction by
$v$ (a unit direction field) and its perpendicular unit direction by $v^{\perp}$.
Let the conductivity coefficients along these directions be
$\chi_{\parallel}$ and $\chi_{\perp}$. ($v$, $\chi_{\parallel}$, and $\chi_{\perp}$ will be defined below.)
Then the diffusion matrix can be written as
\begin{equation}
\mathbb{D}=\chi_{\parallel}vv^{T}+\chi_{\perp}v^{\perp}(v^{\perp})^{T} .
\label{D-1}
\end{equation}
When $\chi_{\parallel}$ and $\chi_{\perp}$ do not depend on $u$, the diffusion operator
in (\ref{eq:HW-eigen}) is simply a linear symmetric second order elliptic operator.
The anisotropy of the diffusion tensor $\mathbb{D}$ depends on the choice of the conductivity coefficients.
For example, if $\chi_{\parallel}\gg\chi_{\perp}$, the diffusion
is preferred along the direction of $v$. Moreover, if $\chi_{\parallel}=\chi_{\perp}$,
the diffusion is isotropic, having no preferred diffusion direction.
To define $\mathbb{D}$, we assume that an input image is given. Denote its gray level by $u_0$.
In image segmentation, pixels with similar values of gray level will be grouped and
the interfaces between those groups provide object boundaries. Since those interfaces are orthogonal
to $\nabla u_0$, it is natural to choose the principal diffusion direction as $v=\nabla u_{0}/|\nabla u_{0}|$.
With this choice, we can rewrite (\ref{D-1}) into
\begin{equation}
\mathbb{D} = \frac{\chi_{\parallel}}{|\nabla u_{0}|^{2}}
\begin{bmatrix}|\partial_x u_{0}|^{2}+\mu |\partial_y u_{0}|^{2} & (1-\mu)\left|\partial_x u_{0}\partial_y u_{0}\right|\\
(1-\mu)\left|\partial_x u_{0} \partial_y u_{0}\right| & |\partial_y u_{0}|^{2}+\mu |\partial_x u_{0}|^{2} \end{bmatrix}
\label{D-2}
\end{equation}
where $\mu = \chi_{\perp}/\chi_{\parallel}$.
We consider two choices of $\chi_{\parallel}$ and $\mu$. The first one is
\begin{equation}
\chi_{\parallel}=g(|\nabla u_{0}|), \quad \mu = 1,
\label{D-3}
\end{equation}
where $g(x)$ is a conductance function that governs the behavior of diffusion.
This corresponds to linear isotropic diffusion. As in \cite{Perona-Malik-90},
we require $g$ to satisfy $g(0)=1$, $g(x)\ge0$, and $g(x)\to 0$ as $x \to \infty$.
For this choice, both $\chi_{\parallel}$ and $\chi_{\perp}$ becomes very small
across the interfaces of the pixel groups and therefore, almost no diffusion is allowed
along the normal and tangential directions of the interfaces.
The second choice is
\begin{equation}
\chi_{\parallel}=g(|\nabla u_{0}|), \quad \mu=1+|\nabla u_{0}|^{2}.
\label{D-4}
\end{equation}
This choice results in an anisotropic diffusion process. Like the first case, almost no diffusion is allowed
across the interfaces of the pixel groups but, depending on the choice of $g$, some degree of diffusion
is allowed on the tangential direction of the interfaces.
We shall show later that with a properly chosen $g$ the eigenfunctions of (\ref{eq:HW-eigen})
capture certain ``grouping'' features of the input image $u_{0}$
very well. This phenomenon has already been observed and explored
in many applications such as shape analysis \cite{Reuter-09,Reuter-06},
image segmentation and data clustering \cite{Grady-01, shi_normalized_2000, Shi-Malik-2001, Wu-Leahy-1993},
and high dimensional data analysis and machine learning
\cite{Belkin_towards_2005,Nadler_diffusion_2005,Nadler_diffusion_2006,Luxburg-2007}.
In these applications, all eigenvalue problems are formulated on a
discrete graph using the graph spectral theory, which is different from what is
considered here, i.e., eigenvalue problems of differential operators.
The application of the latter to image segmentation is much less known.
We shall discuss the connection of these discrete eigenvalue problems
with continuous ones in the next subsection.
It is noted that the gray level function $u_0$ is defined only at pixels.
Even we can view $u_{0}$ as the ``ground truth'' function (assuming
there is one function whose discrete sample is the input image),
it may not be smooth and the gradient cannot be defined
in the classical sense. Following \cite{Alvarez-Lions-92,Catte-Lions-92},
we may treat $u_{0}$ as a properly regularized approximation of the ``true image''
so that the solution to the eigenvalue problem (\ref{eq:HW-eigen}) exists.
In the following, we simply take $u_{0}$ as the linear interpolation
of the sampled pixel values (essentially an implicit regularization
from the numerical scheme). More sophisticated regularization
methods can also be employed.
We only deal with gray level images in this work. The approach
can be extended to color or texture images when a diffusion matrix
can be defined appropriately based on all channels. In our computation, we use
both Dirichlet and Neumann boundary conditions, with the latter
being more common in image processing.
\subsection{Relation to discrete spectral clustering models}
\label{SEC:relation-1}
The eigenvalue problem (\ref{eq:HW-eigen}) is closely related to a family of discrete spectral
clustering models, with the earliest one being the normalized
cut method proposed by Shi and Malik \cite{shi_normalized_2000}.
To describe it, we define the degree of dissimilarity (called $cut$) between
any two disjoint sets $A,B$ of a weighted undirected graph $G=(V,E)$ (where $V$ and $E$ denote the sets of the nodes
and edges of the graph) as the total weight of
the edges connecting nodes in the two sets, i.e.,
\[
cut(A,B)=\sum_{p\in A,\; q\in B}w(p,q).
\]
Wu and Leahy \cite{Wu-Leahy-1993} proposed to find $k$-subgraphs by
minimizing the maximum cut across the subgroups and use them for
a segmentation of an image. However, this approach
usually favors small sets of isolated nodes in the graph. To address
this problem, Shi and Malik \cite{shi_normalized_2000} used
the normalized cut defined as
\[
Ncut(A,B)=\frac{cut(A,B)}{assoc(A,A\cup B)}+\frac{cut(A,B)}{assoc(B,A\cup B)} ,
\]
where $assoc(A,A\cup B)=\sum_{p\in A,\; q\in A\cup B}w(p,q)$. They sought the minimum
of the functional $Ncut(A,B)$ recursively to obtain a $k$-partition
of the image. The edge weight $w(p,q)$ is chosen as
\[
w(p,q)=\begin{cases}
e^{-|u_{q}-u_{p}|^2/\sigma^2}, & q\in\mathcal{N}_{p},\\
0, & {\rm otherwise,}
\end{cases}
\]
where $\mathcal{N}_{p}$ is a neighborhood of pixel $p$ and $\sigma$ is a positive parameter.
Shi and Malik showed that the above optimization problem is NP-hard
but a binary solution to the normalized cut problem can be
mapped to a binary solution to the algebraic eigenvalue problem (\ref{eq:Malik-Shi-eigen})
with $D$ being a diagonal matrix with diagonal entries $d_{p}=\sum_{q}w(p,q)$
and $W$ being the weight matrix $(w(p,q))_{p,q}^{N\times N}$. Eigenvectors of
this algebraic eigenvalue problem are generally not binary. They are used to approximate
binary solutions of the normalized cut problem through certain partitioning.
To see the connection between the algebraic eigenvalue problem (\ref{eq:Malik-Shi-eigen})
(and therefore, the normalized cut method) with the continuous eigenvalue problem (\ref{eq:HW-eigen}),
we consider an eigenvalue problem in the form of (\ref{eq:HW-eigen}) with the diffusion matrix defined as
\begin{equation}
\mathbb{D} = \begin{bmatrix}e^{-|\partial_{x}u_{0}|^2/\sigma^2} & 0\\
0 & e^{-|\partial_{y}u_{0}|^2/\sigma^2} \end{bmatrix}
\label{D-5}
\end{equation}
A standard central finite difference discretization of this problem on a rectangular mesh gives rise to
\begin{equation}
\frac{(c_{E_{i,j}}+c_{W_{i,j}}+c_{N_{i,j}}+c_{S_{i,j}})u_{i,j}-c_{E_{i,j}}u_{i+1,j}-c_{W_{i,j}}u_{i-1,j}-c_{N_{i,j}}u_{i,j+1}-c_{S_{i,j}}u_{i,j-1}}{h^{2}}=\lambda u_{i,j},
\label{eq:orthotropic-fe-scheme}
\end{equation}
where $h$ is the grid spacing and the coefficients $c_{E_{i,j}},\; c_{W_{i,j}},\; c_{N_{i,j}},\; c_{S_{i,j}}$ are given as
\begin{eqnarray*}
c_{E_{i,j}} = e^{-|u_{i+1,j}-u_{i,j}|^2/\sigma^2},\quad
c_{W_{i,j}} = e^{-|u_{i-1,j}-u_{i,j}|^2/\sigma^2},\\
c_{N_{i,j}} = e^{-|u_{i,j+1}-u_{i,j}|^2/\sigma^2},\quad
c_{S_{i,j}} = e^{-|u_{i,j-1}-u_{i,j}|^2/\sigma^2}.
\end{eqnarray*}
It is easy to see that (\ref{eq:orthotropic-fe-scheme}) is almost the same as (\ref{eq:Malik-Shi-eigen})
with the neighborhood $\mathcal{N}_{i,j}$ of a pixel location $(i,j)$ being chosen to include the four closest
pixel locations $\{(i+1,j),(i-1,j),(i,j+1),(i,j-1)\}$. The difference lies in that (\ref{eq:Malik-Shi-eigen}) has
a weight function on its right-hand side. Moreover, it can be shown that (\ref{eq:orthotropic-fe-scheme})
gives {\it exactly} the algebraic eigenvalue problem for the average cut problem
\[
{\rm min}\frac{cut(A,B)}{|A|}+\frac{cut(A,B)}{|B|} ,
\]
where $|A|$ and $|B|$ denote the total numbers of nodes in $A$ and $B$, respectively.
Notice that this problem is slightly different from the normalized cut problem and its solution is known
as the Fiedler value. Furthermore, if we consider the following generalized eigenvalue problem
(by multiplying the right-hand side of (\ref{eq:HW-eigen}) with a mass-density function),
\begin{equation}
-\nabla \cdot \left(\begin{bmatrix}e^{-|\partial_{x}u_{0}|^2/\sigma^2} & 0\\
0 & e^{-|\partial_{y}u_{0}|^2/\sigma^2} \end{bmatrix}
\nabla u\right)= (e^{-|\partial_x u_{0}|^2/\sigma^2} + e^{-|\partial_y u_{0}|^2/\sigma^2}) \lambda u,
\label{eq:pm-aniso-1}
\end{equation}
we can obtain (\ref{eq:Malik-Shi-eigen}) exactly with a proper central finite difference discretization.
The above analysis shows that either the average cut or normalized cut model can be approximated
by a finite difference discretization of the continuous eigenvalue problem
(\ref{eq:HW-eigen}) with the diffusion matrix (\ref{D-5})
which treats diffusion differently in the $x$ and $y$ directions.
While (\ref{D-5}) is anisotropic in general, it results in isotropic diffusion
near oblique interfaces where $\partial_x u_0 \approx \partial_y u_0$ or
$\partial_x u_0 \approx - \partial_y u_0$. This can be avoided
with the diffusion matrix (\ref{D-2}) which defines diffusion differently along
the normal and tangential directions of group interfaces.
In this sense, our method consisting of (\ref{eq:HW-eigen}) with (\ref{D-2})
can be regarded as an improved version of (\ref{eq:HW-eigen}) with (\ref{D-5}),
and thus, an improved continuous generalization of the normalized cut or the average cut method.
It should be pointed out that there is a fundamental difference between discrete spectral clustering
methods and those based on continuous eigenvalue problems.
The former are defined and operated directly on a graph or data set
and their cost depends very much on the size of the graph or data.
On the other hand, methods based on continuous eigenvalue problems
treat an image as a sampled function and are defined by
a discretization of some differential operators. They have the advantage
that many standard discretization methods such as finite difference, finite
element, finite volume, and spectral methods can be used.
Another advantage is that they do not have to be operated directly
on the graph or the data set. As shown in \cite{Huang-Wang-13},
continuous eigenvalue problems can be solved efficiently on
adaptive, and especially anisotropic adaptive, meshes (also see Section~\ref{SEC:numerics}).
It is worth pointing out that the graph Laplacian
can be connected to a continuous diffusion operator by defining the latter on a
manifold and proving it to be the limit of the discrete Laplacian.
The interested reader is referred to the work of
\cite{Belkin_towards_2005,Nadler_diffusion_2005,Nadler_diffusion_2006,Singer-06,Luxburg-2007}.
\subsection{Relation to diffusion models}
The eigenvalue problem (\ref{eq:HW-eigen}) is related to several diffusion models used in image processing.
They can be cast in the form
\begin{equation}
\frac{\partial u}{\partial t}=\nabla \cdot \left(\mathbb{D}\nabla u\right)
\label{eq:linear-diffusion}
\end{equation}
with various definitions of the diffusion matrix. For example, the Perona-Malik nonlinear filter
\cite{Perona-Malik-90} is in this form with $\mathbb{D} = g(|\nabla u|) I$, where $g$ is the same
function in (\ref{D-3}) and $I$ is the identity matrix. The above equation with $\mathbb{D}$ defined in (\ref{D-2})
with $\mu=1$ and $\chi_{\parallel}=g(|\nabla u_{0}|)$ gives rise to a linear diffusion process
that has similar effects as the affine Gaussian smoothing process \cite{Nitzberg-Shiota-92}.
The diffusion matrix we use in this paper in most cases is in the form (\ref{D-2}) with $\mu$ and $\chi_{\parallel}$
defined in (\ref{D-4}). A similar but not equivalent process
was studied as a structure adaptive filter by Yang et al. \cite{Yang-Burger-96}.
The diffusion matrix (\ref{D-2}) can be made $u$-dependent by choosing $\mu$ and $\chi_{\parallel}$
as functions of $\nabla u$.
Weickert \cite{Weickert-1996} considered a nonlinear anisotropic diffusion model with a diffusion matrix
in a similar form as (\ref{D-2}) but with $\nabla u_0$ being replaced by the gradient of a smoothed
gray level function $u_\sigma$ and with $\chi_{\parallel} = g(|\nabla u_\sigma|)$
and $\mu = 1/g(|\nabla u_\sigma|)$.
Interestingly, Perona and Malik \cite{Perona-Malik-90} considered
\begin{eqnarray}
\frac{\partial u}{\partial t} = \nabla \cdot \left(\begin{bmatrix}g(|\partial_{x}u |) & 0\\
0 & g(|\partial_{y} u|) \end{bmatrix}\nabla u\right)
\label{eq:pm-linear-diffusion}
\end{eqnarray}
as an easy-to-compute variant to the Perona-Malik diffusion model (with $\mathbb{D} = g(|\nabla u|) I$).
Zhang and Hancock in \cite{Zhang-Hancock-08} considered
\begin{eqnarray}
\frac{\partial u}{\partial t} = -\mathcal{L}(u_0) u,
\label{eq:Zhang-Hancock}
\end{eqnarray}
where $\mathcal{L}$ is the graph Laplacian defined on the input image $u_0$
and image pixels are treated as the nodes of a graph. The weight between
two nodes $i,j$ is defined as
\[
w_{i,j}=\begin{cases}
e^{-(u_{0}(i)-u_{0}(j))^{2}/\sigma^{2}}, & \quad \text{ for }\|i-j\|\le r\\
0, & \quad \text{otherwise}
\end{cases}
\]
where $r$ is a prescribed positive integer and $\sigma$ is a positive parameter.
As in Section~\ref{SEC:relation-1}, it can be shown that this model can be regarded
as a discrete form of a linear anisotropic diffusion model.
It has been reported in \cite{Buades-Chien-08, Nitzberg-Shiota-92, Yang-Burger-96,Zhang-Hancock-08}
that the image denoising effect with this type of linear diffusion model is
comparable to or in some cases better than nonlinear evolution models.
\section{Numerical Implementation}
\label{SEC:implement}
The eigenvalue problem (\ref{eq:HW-eigen}) is discretized using the standard
linear finite element method with a triangular mesh for $\Omega$. The finite element
method preserves the symmetry of the underlying continuous problem
and can readily be implemented with (anisotropic) mesh adaptation.
As will be seen in Section~\ref{SEC:numerics}, the eigenfunctions
of (\ref{eq:HW-eigen}) can have very strong anisotropic behavior,
and (anisotropic) mesh adaptation is essential to improving the efficiency
of their numerical approximation.
While both Dirichlet and Neumann boundary conditions are considered in our
computation, to be specific we consider only a Dirichlet boundary condition
in the following. The case with a Neumann boundary condition can be discussed similarly.
We assume that a triangular mesh $\mathcal{T}_{h}$ is given for $\Omega$.
Denote the number of the elements of $\mathcal{T}_h$ by $N$ and
the linear finite element space associated with $\mathcal{T}_h$ by $V^{h}\subset H_{0}^{1}\left(\Omega\right)$.
Then the finite element approximation to the eigenvalue problem (\ref{eq:HW-eigen}) subject
to a Dirichlet boundary condition is to find $0 \not\equiv u^h \in V^{h}$ and $\lambda^h \in \mathbb{R}$ such that
\begin{equation}
\int_{\Omega}(\nabla v^h)^t \mathbb{D}\nabla u^h =\lambda^h \int_{\Omega} u^h v^h,\qquad\forall v^h\in V^{h}.
\label{eq:fem-1}
\end{equation}
This equation can be written into a matrix form as
\[
A {\bf u} = \lambda^h M {\bf u},
\]
where $A$ and $M$ are the stiffness and mass matrices, respectively, and ${\bf u}$ is the vector
formed by the nodal values of the eigenfunction at the interior mesh nodes.
An error bound for the linear finite element approximation of the eigenvalues is given by
a classical result of Raviart and Thomas \cite{Raviart-Thomas-83}. It states that
for any given integer $k$ ($1\le k\le N$),
\[
0\le\frac{\lambda_{j}^{h}-\lambda_{j}}{\lambda_{j}^{h}}\le C(k)\sup_{v\in E_{k},\|v\|=1}\| v-\Pi_{h}v\|_E^{2},
\qquad1\le j\le k
\]
where $\lambda_j$ and $\lambda_j^h$ are the eigenvalues (ordered in an increasing order) of the continuous and
discrete problems, respectively, $E_{k}$ is the linear space spanned by the first $k$ eigenfunctions of the
continuous problem, $\Pi_{h}$ is the projection operator from $L^{2}(\Omega)$ to the finite element space $V^{h}$,
and $\| \cdot \|_E$ is the energy norm, namely,
\[
\| v-\Pi_{h}v\|_E^{2} = \int_\Omega \nabla (v-\Pi_{h}v)^t \mathbb{D} \nabla (v-\Pi_{h}v).
\]
It is easy to show (e.g., see \cite{Huang-Wang-13}) that the project error can be bounded
by the error of the interpolation associated with the underlying finite element space,
with the latter depending directly on the mesh. When the eigenfunctions change abruptly
over the domain and exhibit strong anisotropic behavior, anisotropic mesh adaptation
is necessary to reduce the error or improve the computational efficiency (e.g. see \cite{Boff-10,Huang-Wang-13}).
An anisotropic mesh adaptation method was proposed for eigenvalue problems
by the authors \cite{Huang-Wang-13}, following the so-called $\mathbb{M}$-uniform mesh approach
developed in \cite{Huang-05,Huang-06,Huang-Russell-11} for the numerical solution
of PDEs. Anisotropic mesh adaptation provides one advantage over isotropic one
in that, in addition to the size, the orientation of triangles is also adapted to be aligned with
the geometry of the solution locally. In the context of image processing, this mesh alignment
will help better capture the geometry of edges than with isotropic meshes.
The $\mathbb{M}$-uniform mesh approach of anisotropic mesh adaptation views
and generates anisotropic adaptive meshes as uniform ones in the metric specified
by a metric tensor $\mathbb{M} = \mathbb{M}(x,y)$.
Putting it in a simplified scenario, we may consider a uniform mesh defined on the surface of
the gray level $u$ and obtain an anisotropic adaptive mesh by projecting the uniform mesh
into the physical domain. In the actual computation, instead of using the surface of $u$ we employ
a manifold associated with a metric tensor defined based on the Hessian of the eigenfunctions.
An optimal choice of the metric tensor (corresponding to the energy norm) is given \cite{Huang-Wang-13} as
\[
\mathbb{M}_{K}=\det\left(H_{K}\right)^{-1/4}\max_{(x,y)\in K}\|H_{K}\mathbb{D}(x,y)\|^{1/2}\left(\frac{1}
{|K|}\|H_{K}^{-1}H\|_{L^{2}(K)}^{2}\right)^{1/2}H_{K},\qquad\forall K\in\mathcal{T}_{h}
\]
where $K$ denotes a triangle element of the mesh, $H$ is the intersection of the recovered Hessian
matrices of the computed first $k$ eigenfunctions, and $H_{K}$ is the average of $H$ over $K$.
A least squares fitting method is used for Hessian recovery. That is,
a quadratic polynomial is constructed locally for each node via least squares fitting to neighboring
nodal function values and then an approximate Hessian at the node is obtained by differentiating the
polynomial. The recovered Hessian is regularized with a prescribed small positive constant which is taken
to be $0.01$ in our computation.
An outline of the computational procedure of the anisotropic adaptive mesh finite
element approximation for the eigenvalue problem (\ref{eq:HW-eigen}) is given
in Algorithm~\ref{alg:aniso}. In Step 5, BAMG (Bidimensional Anisotropic Mesh Generator)
developed by Hecht \cite{Hecht-Bamg-98} is used to generate the new mesh
based on the computed metric tensor defined on the current mesh.
The resultant algebraic eigenvalue problems are solved using the Matlab eigenvalue solver
{\tt eigs} for large sparse matrices. Note that the algorithm is iterative. Ten iterations are used in our
computation, which was found to be enough to produce an adaptive mesh
with good quality (see \cite{Huang-05} for mesh quality measures).
\begin{algorithm}[h]
\begin{raggedright}
1. Initialize a background mesh.
\par\end{raggedright}
\begin{raggedright}
2. Compute the stiffness and mass matrices on the mesh.
\par\end{raggedright}
\begin{raggedright}
3. Solve the algebraic eigenvalue problem for the first $k$ eigenpairs.
\par\end{raggedright}
\begin{raggedright}
4. Use the eigenvectors obtained in Step 3 to compute the metric tensor.
\par\end{raggedright}
\begin{raggedright}
5. Use the metric tensor to generate a new mesh (anisotropic, adaptive) and go to Step 2.
\par\end{raggedright}
\caption{Anisotropic adaptive mesh finite element approximation for eigenvalue problems.}
\label{alg:aniso}
\end{algorithm}
\section{Numerical results}
\label{SEC:numerics}
In this section, all input images are of size $256\times256$ and
normalized so that the gray values are between 0 and 1. The domain
of input images is set to be $[0,1]\times[0,1]$. All eigenfunctions
are computed with a homogeneous Neumann boundary condition unless otherwise specified.
When we count the indices of eigenfunctions, we ignore the first trivial
constant eigenfunction and start the indexing from the second one.
\subsection{Properties of eigenfunctions}
\subsubsection{Almost piecewise constant eigenfunctions}
A remarkable feature of the eigenvalue problem (\ref{eq:HW-eigen}) with the diffusion matrix (\ref{D-2})
is that for certain input images, the first few eigenfunctions are close
to being piecewise constant. In Fig.~\ref{fig:synth1}, we display
a synthetic image containing 4 objects and the first 7 eigenfunctions.
The gaps between objects are 4 pixel wide. To make the problem more interesting,
the gray level is made to vary within each object (so the gray value of the
input image is not piecewise-constant).
We use the anisotropic diffusion tensor $\mathbb{D}$ defined in (\ref{D-2}) and (\ref{D-4}) with
\begin{equation}
g(x)=\frac{1}{(1+x^{2})^{\alpha}},
\label{D-6}
\end{equation}
where $\alpha$ is a positive parameter. Through numerical experiment (cf. Section~\ref{SEC:4.1.6}), we observe that
the larger $\alpha$ is, the closer to being piecewise constant the eigenfunctions are.
In the same time, the eigenvalue problem (\ref{eq:HW-eigen}) is also harder to solve numerically since
the eigenfunctions change more abruptly between the objects.
We use $\alpha = 1.5$ in the computation for Fig.~\ref{fig:synth1}.
The computed eigenfunctions are normalized such that
they have the range of $[0,255]$ and can be rendered as gray level images.
The results are obtained with an adaptive mesh of 65902 vertices and re-interpolated to
a $256\times256$ mesh for rendering.
The histograms of the first 3 eigenfunctions together with the plot
of the first 10 eigenvalues are shown in Fig.~\ref{fig:synth7hist}.
It is clear that the first 3 eigenfunctions are almost piecewise constant.
In fact, the fourth, fifth, and sixth are also almost piece constant whereas the seventh
is clearly not. (Their histograms are not shown here to save space but this can be seen
in Fig.~\ref{fig:synth1}.)
Fig.\ref{fig:nzsynth1x} shows the results obtained an image with a mild level of noise.
The computation is done with the same condition as for Fig.~\ref{fig:synth1} except that
the input image is different. We can see that the first few eigenfunctions are also piecewise
constant and thus the phenomenon is relatively robust to noise.
\begin{figure}
\caption{The input synthetic image and the first 7 eigenfunctions (excluding
the trivial constant eigenfunction), from left to right, top to bottom. The results are obtained with
the diffusion matrix defined in (\ref{D-2}
\label{fig:synth1}
\end{figure}
\begin{figure}
\caption{The first 10 eigenvalues and the histograms of the first 3 eigenfunctions in Fig.\ref{fig:synth1}
\label{fig:synth7hist}
\end{figure}
\begin{figure}
\caption{A noisy synthetic image and the first 7 eigenfunctions, left to right,
top to bottom. \label{fig:nzsynth1x}
\label{fig:nzsynth1x}
\end{figure}
\subsubsection{Eigenvalue problem (\ref{eq:HW-eigen}) versus Laplace-Beltrami operator}
Eigenfunctions of the Laplace-Beltrami operator (on surfaces) have been studied for image segmentation
\cite{Shah-00,Sochen-Kimmel-Malladi-98} and shape analysis \cite{Reuter-09,Reuter-06}.
Thus, it is natural to compare the performance of
the Laplace-Beltrami operator and that of the eigenvalue problem (\ref{eq:HW-eigen}).
For this purpose, we choose a surface such that the Laplace-Beltrami operator has the same
diffusion matrix as that defined in (\ref{D-2}), (\ref{D-4}), and (\ref{D-6}) and takes the form as
\begin{equation}
- \nabla \cdot \left(\frac{1}{\sqrt{1+|\nabla u|^{2}}}\begin{bmatrix}1+|\partial_y u_{0}|^{2} & -|\partial_x u_{0}\partial_y u_{0}|\\
-|\partial_x u_{0}\partial_y u_{0}| & 1+|\partial_x u_{0}|^{2}
\end{bmatrix}\nabla u\right) = \lambda \sqrt{1+|\nabla u_0|^{2}}\; u .
\label{LB-1}
\end{equation}
The main difference between this eigenvalue problem with (\ref{eq:HW-eigen}) is that there is a weight function on the right-hand side of (\ref{LB-1}), and
in our model the parameter $\alpha$ in (\ref{D-6}) is typically greater than 1.
The eigenfunctions of the Laplace-Beltrami operator obtained with a clean input image of Fig.~\ref{fig:nzsynth1x}
are shown in Fig.~\ref{fig:LB-eigenfunctions}. From these figures one can see that the eigenfunctions of
the Laplace-Beltrami operator are far less close to being piecewise constant, and thus, less suitable for
image segmentation.
\begin{figure}
\caption{The clean input image of Fig.~\ref{fig:nzsynth1x}
\label{fig:LB-eigenfunctions}
\end{figure}
\subsubsection{Open or closed edges}
We continue to study the piecewise constant property of eigenfunctions of (\ref{eq:HW-eigen}).
Interestingly, this property seems related to whether the edges of the input image form a closed curve.
We examine the two input images in Fig.~\ref{fig:openarc}, one containing
a few open arcs and the other having a closed curve that makes a jump in the gray level.
The first eigenfunction for the open-arc image changes gradually where
that for the second image is close to being piecewise constant.
\begin{figure}
\caption{From left to right, input image with open arcs, the corresponding first eigenfunction,
input image with connected arcs, the corresponding first eigenfunction.}
\label{fig:openarc}
\end{figure}
\subsubsection{Anisotropic mesh adaptation}
For the purpose of image segmentation, we would like the eigenfunctions to be as close to
being piecewise constant as possible. This would mean that they change abruptly in narrow regions
between objects. As a consequence, their numerical approximation can be difficult, and
(anisotropic) mesh adaptation is then necessary in lieu of accuracy and efficiency.
The reader is referred to \cite{Huang-Wang-13} for the detailed studies of convergence and advantages
of using anisotropic mesh adaptation in finite element approximation of anisotropic eigenvalue problems
with anisotropic diffusion operators. Here, we demonstrate the advantage of using an anisotropic adaptive
mesh over a uniform one for the eigenvalue problem (\ref{eq:HW-eigen}) with the diffusion matrix defined
in (\ref{D-2}), (\ref{D-4}), and (\ref{D-6}) and subject to the homogeneous Dirichlet boundary condition.
The input image is taken as the Stanford bunny; see Fig.~\ref{fig:bunny41}. The figure also shows
the eigenfunctions obtained on an adaptive mesh and uniform meshes of several sizes.
It can be seen that the eigenfunctions obtained with the adaptive mesh have very sharp
boundaries, which are comparable to those obtained with a uniform mesh of more than
ten times of vertices.
\begin{figure}
\caption{Top row: the image of the Stanford bunny and the first 3 eigenfunctions
computed with an anisotropic adaptive mesh with 45383 vertices; Bottom row: the
first eigenfunction on a uniform mesh with 93732, 276044, 550394 vertices and on
an adaptive mesh with 45383 vertices, respectively. All eigenfunctions are computed
with the same diffusion matrix defined in (\ref{D-2}
\label{fig:bunny41}
\end{figure}
\subsubsection{Anisotropic and less anisotropic diffusion}
Next, we compare the performance of the diffusion matrix (\ref{D-2}) (with
(\ref{D-4}), (\ref{D-6}), and $\alpha = 1.5$) and that of a less anisotropic diffusion matrix
(cf. (\ref{eq:pm-linear-diffusion}), with (\ref{D-6}) and $\alpha = 1.5$)
\begin{equation}
\mathbb{D} = \begin{bmatrix}g(|\partial_{x}u_0 |) & 0\\ 0 & g(|\partial_{y} u_0|) \end{bmatrix} .
\label{D-7}
\end{equation}
The eigenfunctions of (\ref{eq:HW-eigen}) with those diffusion matrices
with the Stanford bunny as the input image are shown in
Fig.~\ref{fig:ani-vs-iso}. For (\ref{D-7}), we compute the
eigenfunction on both a uniform mesh of size $256\times256$ and
an adaptive mesh of 46974 vertices. The computation with (\ref{D-2})
is done with an adaptive mesh of 45562 vertices.
The most perceptible difference in the results is that the right ear of the bunny
(not as bright as other parts) almost disappears in the first eigenfunction with
the less anisotropic diffusion matrix. This can be recovered if the conductance
is changed from $\alpha = 1.5$ to $\alpha = 1.0$, but in this case,
the eigenfunction becomes farther from being piecewise-constant.
The image associated with the first eigenfunction for (\ref{D-2}) seems sharper
than that with (\ref{D-7}).
\begin{figure}
\caption{The first eigenfunction of (\ref{eq:HW-eigen}
\label{fig:ani-vs-iso}
\end{figure}
\subsubsection{Effects of the conductance $g$}
\label{SEC:4.1.6}
We now examine the effects of the conductance and consider four cases:
$g_1$ ((\ref{D-6}) with $\alpha = 1.0$), $g_2$ ((\ref{D-6}) with $\alpha = 1.5$),
$g_3$ ((\ref{D-6}) with $\alpha = 3.0$), and
\[
g_{4}(x)=\begin{cases}
(1-(x/\sigma)^{2})^{2}/2, & \text{ for }|x|\le\sigma\\
\epsilon, & \text{ for }|x|>\sigma
\end{cases}
\]
where $\sigma$ and $\epsilon$ are positive parameters.
The last function is called Tukey's biweight function and considered
in \cite{Black-Sapiro-01} as a more robust choice of the edge-stopping
function in the Perona-Malik diffusion. We show the results with
(\ref{D-2}) on the Stanford bunny in Fig.~\ref{fig:g-choice}.
We take $\sigma=9$ and $\epsilon=10^{-6}$ for Tukey's biweight function. Increasing
the power $\alpha$ in $g(x)$ defined in (\ref{D-6}) will make
eigenfunctions steeper in the regions between different objects
and thus, closer to being piecewise constant. Tukey's biweight function
gives a sharp result but the body and legs are indistinguishable.
\begin{figure}
\caption{Top row: the graphs of $g_1, g_2, g_3, g_4$. Middle row: the first eigenfunctions on the bunny image for $g_1, g_2, g_3, g_4$, respectively. Bottom row: the histograms of the corresponding first eigenfunctions.}
\label{fig:g-choice}
\end{figure}
\subsection{Applications in Edge Detection and Image Segmentation}
Eigenfunctions can serve as a low level image feature extraction device
to facilitate image segmentation or object edge detection. Generally speaking,
eigenfunctions associated with small eigenvalues contain ``global'' segmentation features
of an image while eigenfunctions associated with larger eigenvalues carry more information
on the detail. Once the eigenfunctions are obtained, one can use
numerous well developed edge detection or data clustering techniques
to extract edge or segmentation information. We point out that spectral
clustering methods also follow this paradigm. In this section,
we focus on the feature extraction step and employ only simple, well known
techniques such as thresholding by hand, $k$-means
clustering, or Canny edge detector in the partitioning step. More
sophisticated schemes can be easily integrated to automatically detect
edges or get the segmentations.
We point out that boundary conditions have an interesting effect on
the eigenfunctions. A homogeneous Dirichlet boundary condition
forces the eigenfunctions to be zero on the boundary and may wipe out
some structures there (and therefore, emphasize objects inside the domain).
It essentially plays the role of defining ``seeds'' that indicates
background pixels on the image border. The idea of using user-defined seeds or
intervene cues has been widely used in graph based image segmentation methods
\cite{Grady-01}, \cite{Rother-Blake-04}, \cite{Shi-Yu-04}, \cite{Malik-Martin-04}.
The PDE eigenvalue problem (\ref{eq:HW-eigen}) can also be solved with more sophisticated
boundary conditions that are defined either on the image border or inside the image.
On the other hand, a homogeneous Neumann
boundary condition tends to keep those structures.
Since mostly we are interested in objects inside the domain, we consider here
a homogeneous Dirichlet boundary condition. The diffusion matrix
defined in (\ref{D-2}), (\ref{D-4}), and (\ref{D-6}) ($\alpha = 1.5$) is used.
In Fig.~\ref{fig:Lenna_ef1}, we show the first eigenfunctions obtained with Dirichlet and Neumann
boundary conditions with Lenna as the input image.
For the edge detection for Lenna, it is natural to extract the ``big picture'' from the first eigenfunction
and get the edge information from it. We show the edges obtained by thresholding a few level lines
in the top row of Fig.~\ref{fig:lenna-contour}. Since any level line
with value $s$ is the boundary of the level set $L_{s}=\{(x,y):I(x,y)\ge s\}$
of an image $I$, and $L_{s}$ is non-increasing with respect to $s$,
the level line is ``shrinking'' from the boundary of a wider shape
to empty as $s$ increases from 0 to 255. Some intermediate steps
give salient boundaries of the interior figure. However, to make the ``shrinking''
automatically stop at the correct edge, other clues potentially from
mid or high level knowledge in addition to the low level brightness
info should be integrated in the edge detection step. We also use
the MATLAB function {\tt imcontour} to get major contours, and apply $k$-means
clustering to the eigenfunctions with $k=2,3,4,5$, shown in the second
row of Fig.~\ref{fig:lenna-contour}.
\begin{figure}
\caption{From left to right, Lenna, first eigenfunctions obtained with Dirichlet and Neumann
boundary conditions, respectively.}
\label{fig:Lenna_ef1}
\end{figure}
\begin{figure}
\caption{Top row: contour drawing by MATLAB (with no level parameters specified),
level line 50, 240, 249; bottom row: segmentation with $k$-means, $k=2,3,4,5$.}
\label{fig:lenna-contour}
\end{figure}
We next compute for an image with more textures from \cite{MartinFTM01} (Fig.~\ref{fig:tiger-gallery}).
This is a more difficult image for
segmentation or edge detection due to many open boundary arcs
and ill-defined boundaries. We display the the first eigenfunction
and the $k$-means clustering results in Fig.~\ref{fig:tiger-gallery}.
The $k$-means clustering does not capture the object as well as in
the previous example. Better separation of the object and the background
can be obtained if additional information is integrated into the clustering strategy.
For instance, the edges detected by the Canny detector (which uses
the gradient magnitude of the image) on the
first eigenfunction clearly give the location of the tiger. Thus,
the use of the gradient map of the first eigenfunction
in the clustering process yields more accurate object boundaries.
For comparison, we also show the edges detected from the input image
with the Canny detector.
\begin{figure}
\caption{Top row: the input image, the edges of the input image with the Canny detector,
Level lines with value 50, 240, 254. bottom row: the first eigenfunction,
the edges of the first eigenfunction with the Canny detector, $k$-means
clustering with $k=2,3,4$.}
\label{fig:tiger-gallery}
\end{figure}
Another way to extract ``simple'' features is to change the conductance
$g$ (e.g., by increasing $\alpha$ in (\ref{D-6})) to make
the eigenfunctions closer to being piecewise constant. This makes eigenfunctions more clustered
but wipes out some detail of the image too. To avoid this difficulty,
we can employ a number of eigenfunctions and use the projection of
the input image into the space spanned by the eigenfunctions to construct
a composite image. A much better result obtained in this way with 64 eigenfunctions
is shown in Fig.~\ref{fig:tigerfin}.
\begin{figure}
\caption{From left to right, the first eigenfunction with $\alpha = 2$ in (\ref{D-6}
\label{fig:tigerfin}
\end{figure}
It should be pointed out that not always the first few eigenfunctions cary most useful information
of the input image. Indeed, Fig.~\ref{fig:sports-gallery} shows that
the first eigenfunction carries very little information. Since the
eigenfunctions form an orthogonal set in $L^{2}$, we can project
the input image onto the computed eigenfunctions.
The coefficients are shown in Fig.~\ref{fig:sports-components}.
We can see that the coefficients for the first two eigenfunctions are very small
compared with those for the several following eigenfunctions.
It is reasonable to use the eigenfunctions with the greatest magnitudes of
the coefficients. These major eigenfunctions will provide most useful information;
see Fig.~\ref{fig:sports-gallery}.
\begin{figure}
\caption{Top row: from left to right, the input image and the first 6 eigenfunctions.
Bottom row: from left to right, the edges on the input image (Canny),
the edges on the 3rd and 4th eigenfunctions (Canny), the $k$-means
clustering results with $k=3$ for the 3rd and the 4th eigenfunctions,
respectively; Level line of value 205 of the 3rd eigenfunction, level
line of value 150 of the 4th eigenfunction.}
\label{fig:sports-gallery}
\end{figure}
\begin{figure}
\caption{The coefficients of the input image projected onto the first 64 eigenfunctions in
Fig.~\ref{fig:sports-gallery}
\label{fig:sports-components}
\end{figure}
\section{The piecewise constant property of eigenfunctions}
\label{SEC:piecewise}
As we have seen in Section~\ref{SEC:numerics},
the eigenfunctions of problem (\ref{eq:HW-eigen})
are localized in sub-regions of the input image and the first few of them are close
to being piecewise constant for most input images except for two types of images.
The first type of images is those containing
regions of which part of their boundaries is not clearly defined (such
as open arcs that are common in natural images). In this case, the first eigenfunction
is no longer piecewise-constant although the function values can still be well clustered.
The other type is input images for which the gray level changes gradually and its gradient
is bounded (i.e., the image contrast is mild).
In this case, the diffusion operator simply behaves like the Laplace operator and has
smooth eigenfunctions. For other types of images, the gray level has an abrupt change
across the edges of objects, which causes the conductance $g(|\nabla u_0|)$
to become nearly zero on the boundaries between the objects. As a consequence,
the first few eigenfunctions are close to being constant within each object. This property
forms the basis for the use of the eigenvalue problem (\ref{eq:HW-eigen}) (and its eigenfunctions)
in image segmentation and edge detection.
In this section, we attempt to explain this property from the physical, mathematical, and graph spectral points of view.
We hope that the analysis, although not rigorous, provides some insight of the phenomenon.
From the physical point of view, when the conductance $g(|\nabla u_0|)$
becomes nearly zero across the boundaries between the objects, the diffusion flux will be nearly zero
and each object can be viewed as a separated region from other objects. As a consequence,
the eigenvalue problem can be viewed as a problem defined on multiple separated subdomains, subject to
homogeneous Neumann boundary conditions (a.k.a. insulated boundary conditions)
on the boundary of the whole image and the internal
boundaries between the objects. Then, it is easy to see that the eigenfunctions corresponding to the eigenvalue 0
include constant and piecewise constant (taking a different constant value on each object) functions.
This may explain why piecewise constant eigenfunctions have been observed for most input images.
On the other hand, for images with mild contrast or open arc object edges, the portion of the domain associated
any object is no longer totally separated from other objects and thus the eigenvalue problem may not have piecewise constant
eigenfunctions.
\begin{figure}
\caption{A piecewise smooth function representing an input image with two objects.}
\label{fig:a-fun}
\end{figure}
Mathematically, we consider a 1D example with an input image with two objects. The gray level of the image is sketched
in Fig.~\ref{fig:a-fun}. The edge is located at the origin, and the
segmentation of the image is a 2-partition of $[-L,0]$ and
$[0,L]$. The 1D version of the eigenvalue problem (\ref{eq:HW-eigen}) reads as
\begin{equation}
- \frac{d}{d x}\left(g(|u_0'(x) |)\frac{d u}{d x} \right) =\lambda u ,
\label{eq:example1}
\end{equation}
subject to the Neumann boundary conditions $u'(-L)=u'(L)=0$. We take the conductance
function as in (\ref{D-6}) with $\alpha = 2$. Although $u_0$ is not differentiable,
we could imagine that $u_0$ were replaced by a smoothed function which has a very large
or an infinite derivative at the origin. Then, (\ref{eq:example1}) is degenerate since $g(|u_0'(x)|$
vanishes at $x=0$. As a consequence, its eigenfunctions can be non-smooth. Generally speaking,
studying the eigenvalue problem of a degenerate elliptic operator is a difficult task,
and this is also beyond the scope of the current work. Instead of performing a rigorous analysis, we
consider a simple approximation to $g(|u_0'(x)|)$,
\[
g_{\epsilon}(x)=\begin{cases}
g(|u_0'(x)|), & \text{ for }-L\le x<-\epsilon{\rm \ or\ }\epsilon<x\le L\\
0, & \text{ for }-\epsilon \le x\le \epsilon
\end{cases}
\]
where $\epsilon$ is a small positive number.
The corresponding approximate eigenvalue problem is
\begin{equation}
- \frac{d}{d x}\left(g_\epsilon(|u_0'(x) |)\frac{d u}{d x} \right) =\lambda u .
\label{eq:exampe1-approx}
\end{equation}
The variational formulation of this eigenvalue problem is given by
\begin{equation}
\min_{u\in H^1(-L,L)}\int_{-L}^{L}g_{\epsilon}(|u_0'(x)|)(u')^{2}, \quad \text{ subject to } \int_{-L}^{L}u^{2}=1 .
\label{eq:ex1-variation-prob}
\end{equation}
Once again, the eigenvalue problem (\ref{eq:exampe1-approx}) and the variational problem (\ref{eq:ex1-variation-prob})
are degenerate and should be allowed to admit non-smooth solutions.
The first eigenvalue of (\ref{eq:exampe1-approx}) is 0, and a trivial eigenfunction associated with this eigenvalue
is a constant function. To get other eigenfunctions associated with 0, we consider
functions that are orthogonal to constant eigenfunctions, i.e., we append to
the optimization problem (\ref{eq:ex1-variation-prob}) with the constraint
\[
\int_{-L}^{L}u=0.
\label{eq:ex1-constraints}
\]
It can be verified that an eigenfunction is
\begin{equation}
u_{\epsilon}(x)=\begin{cases}
-c, & \text{ for }x\in [-L,-\epsilon)\\
\frac{c x}{\epsilon}, & \text{ for } x\in [-\epsilon, \epsilon]\\
c, & \text{ for } x\in (\epsilon, L]
\end{cases}
\label{1D-solution}
\end{equation}
where $c = (2 (L-2\epsilon/3))^{-1/2}$. This function is piecewise constant for most part of the domain
except the small region $[-\epsilon, \epsilon]$. Since the original problem (\ref{eq:example1}) can be viewed
to some extent as the limit of (\ref{eq:exampe1-approx}) as $\epsilon \to 0$, the above analysis may
explain why some of the eigenfunctions of (\ref{eq:example1}) behave like piecewise constant functions.
The piecewise constant property can also be understood in the context of the graph spectral
theory. We first state a result from \cite{Mohar-Alavi-91,Luxburg-2007}.
\begin{prop}[\cite{Mohar-Alavi-91,Luxburg-2007}]\;
Assume that $G$ is a undirected graph with $k$ connected components and
the edge weights between those components are zero. If the nonnegative weights matrix $W$
and the diagonal matrix $D$ are defined as in Section~\ref{SEC:relation-1},
then the multiplicity of the eigenvalue 0 of the matrix $D-W$ equals the number of the connected
components in the graph. Moreover, the eigenspace of
eigenvalue 0 is spanned by the indicator vectors $1_{A_{1}},\cdots,1_{A_{k}}$
of those components ${A_{1}},\cdots, {A_{k}}$.
\label{prop-1}
\end{prop}
This proposition shows that the eigenvalue zero of the algebraic eigenvalue problem (\ref{eq:Malik-Shi-eigen})
could admit multiple eigenvectors as indicators of the components. As shown in Section~\ref{SEC:relation-1},
(\ref{eq:Malik-Shi-eigen}) can be derived from a finite difference discretization of the continuous
eigenvalue problem (\ref{eq:HW-eigen}) (with a proper choice of $\mathbb{D}$). Thus,
the indicators of the components can also be regarded as discrete approximations of some continuous
eigenfunctions. This implies that the latter must behave like piecewise constant functions.
Interestingly, Szlam and Bresson \cite{Szlam-Bresson-10} recently proved
that global binary minimizers exist for a graph based problem called Cheeger Cut where the minimum of
the cut is not necessarily zero.
In the continuous setting, a properly designed conductance $g(|\nabla u_0|)$
can act like cutting the input image $u_0$ into subregions along the boundary
of the subregions and forcing the eigenvalue problem to be solved
on each subregion. In a simplified case, we can have the following continuous
analogue of Proposition~\ref{prop-1}. The proof is straightforward.
\begin{prop}\;
Suppose $\Omega\subset \mathbb{R}^2 $ is a bounded Lipschitz domain, $u_0 \in SBV(\Omega)$
(the collect of special functions of bounded variation) and the discontinuity set $K$ of $u_0$ is a finite
union of $C^{1}$ closed curves, $g(\cdot)$ is a bounded positive continuous function.
We define $g(|\nabla u_0|)=0$
for $(x,y)\in K$, and $g(|\nabla u_0|)$ takes its usual meaning for $(x,y)\in\Omega\backslash K$.
For any function $u\in SBV(\Omega)$, assuming $\Gamma$ is the discontinuity
set of $u$, we define the Rayleigh quotient on $u$ as
\[
R(u)=\frac{\widetilde{\int}_{\Omega}(\nabla u)^{T}g(|\nabla u_0|)\nabla u}{\int_{\Omega}u^{2}} ,
\]
where
\[
\widetilde{\int}_{\Omega}(\nabla u)^{T}g(|\nabla u_0|)\nabla u=\begin{cases}
\int_{\Omega\backslash K}g(\nabla u_0|)|\nabla u|^{2}, & \text{ for }\Gamma\subseteq K\\
\infty, & \text{ for } \Gamma\nsubseteq K .
\end{cases}
\]
Then, the minimum of $R(u)$ is zero and any piecewise constant function
in $SBV(\Omega)$ with discontinuity set in $K$ is a minimizer.
\label{prop-2}
\end{prop}
The eigenvalue problem related to the above variational problem can be formally written as
\begin{equation}
-\nabla \cdot \left(g(|\nabla u_0|)\nabla u\right)=\lambda u.
\label{eq:general-eigen-prob}
\end{equation}
The equation should be properly defined for all $u_0$, $u$ possibly in the space
of $BV$. This is a degenerate elliptic problem which could admit discontinuous
solutions, and it seems to be far from being fully understood.
In the following proposition, we suggest a definition of weak solution
in a simplified case. The property indicates that problem (\ref{eq:general-eigen-prob})
is quite different from a classical elliptic eigenvalue problem if
it has a solution in $BV$ that takes a non-zero constant value on an
open set. The proof is not difficult and thus omitted.
\begin{prop}\;
Suppose $\Omega$ is a bounded Lipschitz domain in $R^{2}$, $u_0\in SBV(\Omega)$
and the discontinuity set $K$ of $u_0$ is a finite union of $C^{1}$
closed curves, $g(\cdot)$ is a bounded positive continuous function.
We define $g(|\nabla u_0|)=0$ for $(x,y)\in K$, and $g(|\nabla u_0|)$ takes its usual
meaning for $(x,y)\in\Omega\backslash K$.
We define $u\in SBV(\Omega)$ to be a weak eigenfunction of (\ref{eq:general-eigen-prob})
satisfying a homogeneous Dirichlet boundary condition if
\[
\int_{\Omega}(\nabla u)^{T}g(|\nabla u_0|)\nabla\phi=\int_{\Omega}\lambda u\phi,
\quad \forall \phi \in C_{0}^{1}(\Omega)
\]
where, assuming that $\Gamma$ is the discontinuity set of $u$, the integral
on the left side is defined by
\[
\int_{\Omega}(\nabla u)^{T}g(|\nabla u_0|)\phi=\begin{cases}
\int_{\Omega\backslash K}(\nabla u)^{T}g(\nabla u_0|)\phi, & \Gamma\subseteq K\\
\infty, & {\rm \Gamma\nsubseteq K.}
\end{cases}
\]
If a weak eigenfunction $u\in SBV(\Omega)$ exists and takes a non-zero constant value
on a ball $B_{\epsilon}(x_{0}, y_0)\subset\Omega$, then the corresponding eigenvalue $\lambda$ is zero.
\label{prop-3}
\end{prop}
If (\ref{eq:general-eigen-prob}) indeed admits non-zero
piecewise-constant eigenfunctions, one can see an interesting connection
between (\ref{eq:general-eigen-prob}) (for simplicity we assume
a homogeneous Dirichlet boundary condition is used) and Grady's Random Walk image
segmentation model \cite{Grady-01} where multiple combinatorial Dirichlet problems are solved
for a $k$-region segmentation with predefined seeds indicating segmentation labels.
Using a similar argument in Section 2.2, one can show that the numerical
implementation of the method is equivalent to solving a set of Laplace problems
which are subject to a Neumann boundary condition on the image border and Dirichlet boundary conditions
on the seeds and are discretized on a uniform mesh for potentials $u^{i}$,
$i=1,\cdots,k$. These boundary problems read as
\begin{eqnarray}
\nabla \cdot \left(\begin{bmatrix}g(|\partial_{x}u_{0}|) & 0\\
0 & g(|\partial_{y}u_{0}|)
\end{bmatrix}\nabla u^{i}\right) & = & 0,\ {\rm in}\ \Omega\backslash S
\label{eq:Grady-model}\\
\frac{\partial u^{i}}{\partial n} & = & 0,\ {\rm on}\ \partial\Omega \nonumber \\
u^{i} & = & 1,\ {\rm in}\ S_{i} \nonumber \\
u^{i} & = & 0,\ {\rm in}\ S\backslash S_{i}
\nonumber
\end{eqnarray}
where $S_{i}$ is the set of seeds for label $i$ and $S$ is the set of all seeds.
This problem with a proper choice of $g$ also gives
a solution that has well clustered function values, a phenomenon called
``histogram concentration'' in \cite{Buades-Chien-08}.
Note that when $\lambda=0$, the following equation, which
is an anisotropic generalization of (\ref{eq:general-eigen-prob}),
$$
-\nabla \cdot
\left(\begin{bmatrix}g(|\partial_{x}u_{0}|) & 0\\
0 & g(|\partial_{y}u_{0}|)
\end{bmatrix}\nabla u^{i}\right) = \lambda u^i,
$$
becomes exactly the Laplace equation in Grady's model which
is the Euler-Lagrange equation of the energy
\begin{equation}
\int_{\Omega}(\nabla u)^{T}\begin{bmatrix}g(|\partial_{x}u_{0}|) & 0\\
0 & g(|\partial_{y}u_{0}|)
\end{bmatrix}\nabla u .
\label{eq:bv-energy}
\end{equation}
While the proper definition of functional (\ref{eq:bv-energy})
for general $u$, $u_{0}$ possibly in $BV$ is missing, we can still
define it for a simpler case as in Proposition 5.2. Then, there is
no unique minimizer of this energy, and, as stated in Proposition
5.2, any minimizer of the above energy in $SBV$ yields the minimum
value 0 in the ideal case (with proper $g$ and $u_{0}$ as in Proposition
5.2). While the eigenvalue method considers the minimizer of the above
energy on the admissible set $\left\{ u:u|_{\partial\Omega}=0,\int u^{2}\ dx=1\right\} $,
the Random Walk method considers the minimizer satisfying boundary
conditions in (\ref{eq:Grady-model}). Both gives piecewise-constant
minimizers that can be used for image segmentation.
\section{Concluding remarks}
\label{SEC:conclusion}
We have introduced an eigenvalue problem of an anisotropic differential operator
as a tool for image segmentation. It is a continuous and anisotropic generalization
of some commonly used, discrete spectral clustering models for image
segmentation. The continuous formulation of the eigenvalue problem allows
for accurate and efficient numerical implementation, which is crucial
in locating the boundaries in an image. An important observation
from numerical experiment is that non-trivial, almost piecewise constant
eigenfunctions associated with very small eigenvalues exist, and
this phenomenon seems to be an inherent property of the model.
These eigenfunctions can be used as the basis for image segmentation
and edge detection. The mathematical theory behind this is still unknown
and will be an interesting topic for future research.
We have implemented our model with a finite element method and shown
that anisotropic mesh adaptation is essential to the accuracy and efficiency
for the numerical solution of the model. Numerical tests on segmentation
of synthetic, natural or texture images based on computed eigenfunctions
have been conducted. It has been shown that the adaptive mesh implementation
of the model can lead to a significant gain in efficiency.
Moreover, numerical results also show that the anisotropic nature of the
model can enhance some nontrivial regions of eigenfunctions which may
not be captured by a less anisotropic or an isotropic model.
\def$'${$'$}
\end{document}
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